Contributions to Management Science
Reza Zanjirani Farahani • Masoud Hekmatfar
Editors
Facility Location
Concepts, Models, Algorithms
and Case Studies
Editors
Dr. Reza Zanjirani Farahani
Centre for Maritime Studies
National University of Singapore
Singapore
cmszfr@nus.edu.sg
Masoud Hekmatfar
Amirkabir University of Technology
Department of Industrial Engineering
Iran
hekmatfar@aut.ac.ir
ISSN 1431-1941
ISBN 978-3-7908-2150-5
e-ISBN 978-3-7908-2151-2
DOI 10.1007/978-3-7908-2151-2
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2009922331
c Springer-Verlag Berlin Heidelberg 2009
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To
Professor Zvi Drezner,
to whom we are greatly indebted for his
generous scientific contribution in the area of
Facility Location
Contents
Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
1
1
Distance Functions in Location Problems . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
Marzie Zarinbal
5
2
An Overview of Complexity Theory . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
Milad Avazbeigi
19
3
Single Facility Location Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
Esmaeel Moradi and Morteza Bidkhori
37
4
Multifacility Location Problem . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
Farzaneh Daneshzand and Razieh Shoeleh
69
5
Location Allocation Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .
Zeinab Azarmand and Ensiyeh Neishabouri Jami
93
6
Quadratic Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 111
Masoumeh Bayat and Mahdieh Sedghi
7
Covering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 145
Hamed Fallah, Ali NaimiSadigh, and Marjan Aslanzadeh
8
Median Location Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 177
Masoomeh Jamshidi
9
Center Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 193
Maryam Biazaran and Bahareh SeyediNezhad
10 Hierarchical Location Problem . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 219
Sara Bastani and Narges Kazemzadeh
vii
viii
Contents
11 Hub Location Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 243
Masoud Hekmatfar and Mirsaman Pishvaee
12 Competitive Location Problem.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 271
Mohammad Javad Karimifar, Mohammad Khalighi Sikarudi,
Esmaeel Moradi, and Morteza Bidkhori
13 Warehouse Location Problem .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 295
Zeinab Bagherpoor, Shaghayegh Parhizi, Mahtab Hoseininia,
Nooshin Heidari, and Reza Ghasemi Yaghin
14 Obnoxious Facility Location . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 315
Sara Hosseini and Ameneh Moharerhaye Esfahani
15 Dynamic Facility Location Problem .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 347
Reza Zanjirani Farahani, Maryam Abedian, and Sara Sharahi
16 Multi-Criteria Location Problem .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 373
Masoud Hekmatfar and Maryam SteadieSeifi
17 Location-Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 395
Anahita Hassanzadeh, Leyla Mohseninezhad, Ali Tirdad,
Faraz Dadgostari, and Hossein Zolfagharinia
18 Storage System Layout .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 419
Javad Behnamian and Babak Eghtedari
19 Location-Inventory Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 451
Mohamadreza Kaviani
20 Facility Location in Supply Chain . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 473
Meysam Alizadeh
21 Classification of Location Models and Location Softwares . . . . . . . . . . . . . 505
Sajedeh Tafazzoli and Marzieh Mozafari
22 Demand Point Aggregation Analysis for Location Models . . . . . . . . . . . . . 523
Ali NaimiSadigh and Hamed Fallah
Appendix: Metaheuristic Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 535
Zohre Khoban and Saeed Ghadimi
Index . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 545
Introduction
The mathematical science of facility locating has attracted much attention in discrete and continuous optimization over nearly last four decades. Investigators have
focused on both algorithms and formulations in diverse settings in both the private
sectors (e.g., industrial plants, banks, retail facilities, etc.) and the public sectors
(e.g., hospitals, post stations, etc.).
Facility location problems locate a set of facilities (resources) to minimize the
cost of satisfying some set of demands (of the customers) with respect to some set of
constraints. Facility location decisions are critical elements in strategic planning for
a wide range of private and public firms. The branches of locating facilities are broad
and long-lasting, influencing numerous operational and logistical decisions. High
costs associated with property acquisition and facility construction make facility
location or relocation projects long-term investments. Decision makers must select
sites that will not only perform well according to the current system state, but also
will continue to be profitable for the facility’s lifetime, even as environmental factors
change, populations shift, and market trends evolve. Finding robust facility locations
is thus a difficult task, demanding decision makers to account for uncertain future
events.
Location science is an area of analytical study that can be traced back to Pierre de
Fermat, Evagelistica Torricelli (a student of Galileo), and Battista Cavallieri. Each
one independently proposed (and some say solved) the basic Euclidean spatial median problem early in the seventeenth century.
The study of location theory started formally in 1909 when Alfred Weber considered how to locate a single warehouse in order to minimize the total distance
between the warehouse and several customers. After that, location theory was driven
by a few applications. Location theory gained researchers’ interest again in 1964
with a publication by Hakimi (1964), who wanted to locate switching centers in a
communications network and police stations in a highway system.
The term “location problem” refers to the modeling, formulation, and solution of
a class of problems that can best be described as locating facilities in some given
spaces. Deployment, positioning, and locating are frequently used as synonymous.
There are differences between location and layout problems: the facilities in location
problems are small relative to the space in which they are sited and the interaction
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 0, c Physica-Verlag Heidelberg 2009
1
2
Introduction
among facilities may occur; but in layout problems, the facilities to be located are
large relative to the space in which they are positioned, and the interaction among
facilities is common.
There are four components that describe location problems: customers, who are
assumed to be already located at points or on routes, facilities that will be located,
a space in which customers and facilities are located, and a metric (standard) that
indicates distances or time between customers and facilities.
Facility location models are used in a variety of applications. Some of them include locating warehouses within a supply chain to minimize the average time to
market, locating noxious material to maximize their distances from the public, locating railroad stations to minimize the unpredictability of delivery schedules, locating
automatic teller machines to serve bank customers better, etc. Facility location models can differ in their objective function, the distance metric applied, the number and
size of the facilities to locate, and several other decision indices. Depending on the
specific application, inclusion and consideration of these various indices in the problem formulation will lead to very different location models.
Facility location books are numerous. Francis et al. (1992) introduced some
prevalent models such as single/multi facility location problems, quadratic assignment location problems (QAP) and covering problems. Mirchandani and Francis
(1990) wrote about discrete location theory. The network based location theory
book by Daskin (1995) focused on discrete location problems. Drezner (1995)
represented some models and applications in location environments. Drezner and
Hamacher (2002) published a book about the theory and applications of facility
location. Nickel and Puerto (2005) extended a complete survey in the area of continuous and network based location models especially about median location problems.
In this book, most of the subjects are seen in an equal trend; classical models
such as single facility location problem, multiple facility location problem, median
problem, center problem and covering problem, contemporary models such as hierarchical facility location problem, hub location problem and competitive location
problem and advanced models such as location in supply chain.
The arrangement of the chapters has a reasonable style in which the predecessors
and successors have been regarded from concepts viewpoints; that is, to solve one
of the P-center models, it has to be converted to some covering problems, therefore
the covering chapter is followed by center chapter.
Most chapters have a similar trend to represent their concepts in which application and classification are included in part one, mathematical modeling, solution
technique and some case studies in parts two, three and four, respectively.
Because of the importance of distances in objective functions of location problems, in Chap. 1 different kinds of distances in location problems are discussed.
Chap. 2 introduces complexities employed in location problems.
Chapters 3–9 discuss some prevalent and classic concepts in location theory. Single facility and multiple facility location problems are treated in Chaps. 3 and 4,
respectively, in which traditional concepts of location problems are introduced.
Some prevalent models in both discrete and continuous spaces are introduced in
these chapters.
Introduction
3
Location area can be divided into three parts: location problems, allocation problems and location-allocation problems. We represent location-allocation problems
in Chap. 5. In some cases of location problems, locating needs to consider distances and interactions between facilities, therefore we face quadratic assignment
problems, which are discussed in Chap. 6. In covering problems, customers need
to be with a specific distance through facilities which are servicing; these problems introduced different kinds of covering problems that are discussed in Chap. 7.
Median problems are considered as the main topics in the location allocation problems. These problems try to find the median points among some candidate points
to minimize the sum of costs, and most of their applications are in private areas. In
Chap. 8, we study these kinds of problems as median problems. Public and emergency services need to be located to satisfy all customers, thus center problems have
emerged to minimize the maximum distances between the facilities and the demand
points (customers). In Chap. 9 we introduce these problems and their applications.
After Chap. 9, we will introduce some contemporary concepts in location problems. The first of them is the hierarchical facility location problem, which is
discussed in Chap. 10. This chapter deals with different levels and categories of facilities, which have to be located with some relationships among them. In Chap. 11,
hub location problems have been addressed. In some cases we want to eliminate
some interactions between demand points (customers) and facilities to reduce the
complexity of their networks, therefore we introduce some facilities as hub points
and reduce these relations. This leads to minimizing the total cost of the network.
In Chap. 12, we cover some concepts about competitive areas not monopolized
as competitive location problems. In these areas, facilities that have to be located
need to compete with other facilities to gain a market share. In some areas facilities are warehouses and have to be located to satisfy customer demands, thus in
Chap. 13 we will introduce warehouse location problems in which different kinds
of siting and solution methods are discussed. In some cases, we need to locate some
hazardous facilities that have to be far from public places. Their objectives minimize these kinds of facilities’ exposure and, are introduced in a separate chapter as
an obnoxious facility location problem (Chap. 14). The nature of facility location
problems leads to considering future uncertainty. Thus in real world, we face problems which have no definite planning horizon. In Chap. 15, we treat these problems
as dynamic facility location problems. In many real world cases, we face some incompatible objective functions, therefore a separate chapter is introduced as a multi
criteria location problem, which deals with conflicted objectives and includes most
facility location topics (Chap. 16).
In Chap. 17, we represent location routing problems which not only discuss locating some new facilities in some candidate points but also set routing between these
facilities and demand points (customers). In this, we treat vehicle routing problems
as a subordinate of the location routing problem. Inventory costs have a major effect
on location problems and there is a close relationship between objective functions
of delivery locating points and inventory costs, therefore it is better to consider inventory costs determining minimum costs in dealing with satisfying demand points
(customers). Products need to be put into storage locations before they can be picked
4
Introduction
to fulfill customer orders, therefore the layout of storage will be an important matter
which leads to better efficiency and delivery. This concept (different with warehouse
location problem) which deals with siting of warehouses rather than their layouts
in location areas as storage system layout is represented in Chap. 18. We represent location-inventory problems in a specific chapter (Chap. 19) which explains
inventory concepts and their parameters in the siting of facilities. Nowadays, supply
chains have been expanded in modern environments. In Chap. 20, two separate concepts are combined: supply chain and location. This chapter discusses relationships
between supply chains and siting problems in modern areas named supply chain in
location. The classification of facility location problems together with introducing
some prevalent facility location softwares are covered in Chap. 21. Location problems often interest to find locations of new facilities that provide services of some
kind to existing facilities. Sometimes finding all new facilities is not an economical task and an analysis is needed to aggregate the demand data by representing
a collection of individuals as one demand point. In Chap. 22, this kind of analysis for demand point aggregation is represented. Finally, an appendix is introduced,
and it contains meta-heuristic algorithms employed to determine and solve facility
location models.
We express our appreciation for editorial who managed to edit successfully the
manuscripts that were characterized by a great variety of individual preferences in
style and layout, and to Dr. Werner A. Müller, Springer Executive Vice President
in Business/Economics and Statistics, Dr. Niels Peter Thomas, Springer Editor in
Business/Economics, Alice Blanck, Business/Economics and Statistics Editorial
and also Indhu Arumugam, SPi Technologies India Private Ltd., project manager
for their support.
References
Daskin MS (1995) Network and discrete location: Models, algorithms, and applications. Wiley,
New York
Drezner Z (1995) Facility location: A survey of application and methods. Springer, Berlin
Drezner Z, Hamacher H (2002) Facility location: Applications and theory. Springer, Berlin
Francis RL, McGinnis LF, White JA (1992) Facility layout and location: An analytical approach.
Prentice Hall, Englewood Cliffs, NJ
Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians
of a graph. Oper Res 12:450–459
Mirchandani PB, Francis RL (1990) Discrete location theory. Wiley, New York
Nickel S, Puerto J (2005) Location theory: A unified approach. Springer, Berlin
Chapter 1
Distance Functions in Location Problems
Marzie Zarinbal
Distance is a numerical description of how far apart objects are at any given moment
in time. In physics or everyday discussion, distance may refer to a physical length,
a period of time, or it is estimated based on other criteria.
While making location decisions, the distribution of travel distances among the
service recipients (clients) is an important issue.
Most classical location studies focus on the minimization of the mean (or total)
distance (the median concept) or the minimization of the maximum distance (the
center concept) to the service facilities. (Ogryczak 2000) In these studies, the location modeling is divided into four broad categories:
1. Analytic models. These models are based on a large number of simplifying assumptions such as the fix cost of locating facility. The travel distances follow the
Manhattan metric.
2. Continuous models. These models are the oldest location models, deal with geometrical representations of reality, and are based on the continuity of location
area. The classic model in this area is the Weber problem. Distances in the Weber
problem are often taken to be straight-line or Euclidean distances but almost all
kind of the distance functions can be used here (Jiang and Xu 2006; Hamacher
and Nickel 1998).
In the study of continuous location theory, it is generally assumed that the customers
may be treated as points in space. This assumption is valid when the dimensions of
the customers are small relative to the distances between the new facility and the
customers. However, it is not always the case. Sometimes, we should not ignore
the dimensions of the customers. Some researchers have treated the customers as
demand regions representing the demand over a region.
Jiang and Xu (2006) discussed that some researchers such as Brimberg and
Wesolowsky in 1997, 2000 and 2002 and Nickel et al. in 2003 used the distance
between the facility and the closest point of a demand region; and in the others, the
distance between the facility and a demand region may be calculated as some form
of expected or average travel distance.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 1, c Physica-Verlag Heidelberg 2009
5
6
M. Zarinbal
3. Network models. Network models are composed of links and nodes. Absolute
1-median, un-weighted 2-center and q-criteria L-median on a tree models are
some well-known models in this area. Distances are measured with respect to the
shortest path.
4. Discrete models. In these models, there are a discrete set of candidate locations.
Discrete N -median, un-capacitated facility location, and coverage models are
some well-known models in this area. Like the distances in continuous models,
all kind of the distance functions can be used here but sometimes it could be specified exogenously (Hamacher and Nickel 1998; Fouard and Malandain 2005).
Distances and norms are usually defined on the finite space E n and take real values.
In discrete geometry, however, we sometimes need to have discrete distances defined
on Z n with their values in Z. Since Z n is not a vector space, the notion of distances
and norms had to be extended.
1.1 Distance and Norms Specifications
Assume X D .x1 , y1 / and Y D .x2 , y2 /. Then d.X; Y / is the distance function between points X and Y , and has these characteristics (Fouard and Malandain 2005).
d.X; Y / 0
8X; Y
Possitivity;
d.X; Y / D 0 , X D Y
d.X; Y / D d.Y; X /
8X; Y
8X; Y
d.X; Y / d.X; R/ C d.R; Y /
(1.1)
Definition;
(1.2)
Symmetry;
8X; Y
(1.3)
Triangular Inequality:
(1.4)
1.2 Distances Function
The distances function between points X D .x1 , x2 ; : : : ; xn / and Y D .y1 ,
y2 ; : : : ; yn / is called dk;p .X; Y / the Minkowski distance of order p, which defines as follows:
! p1
n
X
p
dK;p .X; Y / D
:
(1.5)
ki jxi yi j
i D1
IF k1 D k2 D : : : D kn D kp then we have
dK;p .X; Y / D K
n
X
i D1
jxi yi jp
! p1
:
(1.6)
1
Distance Functions in Location Problems
7
Equation (1.6) is called the weighted dk;p -norm. (K is distance function’s weight)
IF k1 D k2 D : : : D kn D 1 then we have
dK;p .X; Y / D
n
X
p
jxi yi j
i D1
! p1
(1.7)
:
The parameters k1 and k2 of the dk;p -norm can be seen as unequal weights or
non symmetric distance irregularities along the axis directions. An empirical work
showed that the accuracy of distance estimations in the dk;p -norm is better than the
weighted dk;p -norm. (Uster and Love 2003)
In the situation of (1.7), we can define some famous distance functions such as:
IF p D 1 the 1-norm, rectilinear, Manhattan or right angle distances can be
obtained: (1.8)
n
X
(1.8)
dK;p .X; Y / D
jxi yi j:
i D1
Rectilinear distances are applicable when travel is allowed only on two perpendicular directions such as North–South and East–West arteries. This distance is also
popular among researchers because the analysis is usually simpler than employing
other metrics (Drezner and Wesolowsky 2001).
The Rectilinear distance is also called Taxicab Norm distances; because it is the
distance a car would drive in a city lay-out in square blocks (if there are no one-way
streets).
IF p D 2 the 2-Norm or Euclidean distances can be obtained by (1.9)
dK;p .X; Y / D
n
X
i D1
jxi yi j2
! 21
(1.9)
:
It is what would be obtained if the distance between two points were measured with
a ruler: the “intuitive” idea of distance.
Air travel or travel over water can be exactly modeled by Euclidean distances
(Drezner and Wesolowsky 2001).
IF p D 1 the Infinity Norm or Chebyshev distance can be obtained (1.10)
d1 .X; Y / D lim p!1
n
X
i D1
jxi yi jp
! p1
D max.jx1 y1 j ; : : : ; jxn yn j/:
(1.10)
d1 and d1 are obviously discrete distances, but not d2 . The parameter d2 is the most
commonly used continuous distance, because of its rotation invariance.
8
M. Zarinbal
1.3 Different Kinds of Distances
There are also other kinds of distances used in real problems. Some of them are as
follows:
1.3.1 Aisle Distance
As mentioned above Rectilinear or Euclidean distance function are the most common methods used in models, however, these distance measures are not realistic for
some applications such as material handling in plants. Figure 1.1 shows aisles in a
plant.
The interdepartmental aisle travel distances can be found by formulating and
finding the shortest path on a network problem and may be specified to provide
the necessary distance between resources. This makes it possible to evaluate the
actual aisle travel distance for each layout that is generated during the search process
(Norman et al. 2001).
For calculating aisle distance, the strategies of handling systems must be considered. “The routing of a picker follows selective one-way traffic in that he traverses
an entire length of the aisle containing the items to be picked and is not allowed
to turn around or reverse but ends up on the opposite side of the aisle after picking
the items. The optimal route in this strategy is to arrange the items within the batch
such that the items found in the aisle nearest to Input/output station are collected
first followed by the next nearest aisle. When the last item is picked, the picker will
return to the I/O station”. Chew and Tang (1999) is an example of these strategies.
1.3.2 Distance Matrix
Yu and Sarker (2003) indicated that Sarker in 1989 and Sarker et al. in 1994 and
1998 developed a number of amoebic properties of a distance matrix for equally
Fig. 1.1 Aisles in plant
layout
1
Distance Functions in Location Problems
9
spaced linear locations to generate different assignments of machines to locations
that minimize the total unidirectional and/or bi-directional flows. The form of a
distance matrix may vary as its corresponding location assignment changes.
8
< X Y if 1 Y < X L
dXY D jX Y j D Y X if 1 X < Y L :
(1.11)
:
0
if 1 Y D X L
Each location distance can be decomposed into two directional distances that are
defined below.
B
.
Backward: d B is a backward distance matrix, with its element dXY
X Y if 1 Y < X L
B
D
:
dXY
0
else
(1.12)
F
(Yu and
Forward: d F is a forward distance matrix, with its element dXY
Sarker 2003)
Y X if 1 X < Y L
F
dXY
D
:
(1.13)
0
else
1.3.3 Minimum Lengths Path
The distance between two points on P is the minimum length of any path between
those points that lies on P . The “facility center”, or “1-center”, of the facility is the
point of P that minimizes the maximum distance to a facility. There are some algorithms to find minimum lengths path (shortest path) such as Dijkstra Algorithm and
the algorithm of Mitchell et al. which is a continuous version of Dijkstra Algorithm
(Aronov et al. 2005).
1.3.4 Block Distance
Dearing et al. (2005) discussed that block distances are a special case of norm distances which were introduced to location models by Witzgall et al. in 1964, and
Ward and Wendell in 1985. Block distances are used to model travel distance in applications where travel directions are restricted to the fundamental directions. Also
it has a wide usage in barriers problems.
They can also be viewed as a generalization of distances in fixed orientations as
introduced in 1987 by Widmayer et al. (Dearing et al. 2005) where it is assumed
that all fundamental directions have unit length, that is
kak k D 1
8k D 1; 2; : : : : ; 2n;
where jjak jj is the Euclidean norm of ak .
(1.14)
10
M. Zarinbal
The block distance between the points, X1 and X2 with respect to a given set of
fundamental directions a1 , a2 ; : : : ; a2n is denoted by dp .X1 ; X2 / and is defined as
dp .X1 ; X2 / D ˛12 C ˇ12 ;
(1.15)
where ˛12 and ˇ12 are nonnegative scalars so that (Dearing et al. 2005)
X2 X1 D ˛12 ak C ˇ12 akC1 8k D 1; 2; : : : : ; 2n:
(1.16)
1.3.5 Gauges Measures
Most of the references in the literature concerning continuous location problems
have considered distances induced by norms. There are also a number of papers
that consider the use of gauges defined by the Minkowski functional of a compact
convex set (not necessarily symmetric) containing the origin in its interior. These
functions have been used in location theory to model situations where the symmetry
property of a norm does not make sense.
There are also general models where the definiteness property of the gauge of
a compact convex set is relaxed. Relaxing definiteness introduces the existence of
zero-distance regions (Fig. 1.2).
Gauges of compact convex sets have a very interesting property: The distance
between two points is the shortest path between them using only fundamental directions of the unit ball.
Let be a closed convex set containing the origin. The function ' defined by
.x/ D inff˛ > 0 W x 2 ˛ g
(1.17)
is called the gauge of . The set will be called the unit ball associated with '. We
define the distance from y to x by '.y x).
If in addition is symmetric with respect to the origin, ' is a norm and the
symmetry property of a norm ('.y x/ D '.x y/) added to '.y x/ properties.
(Hinojosa and Puerto 2003).
Fig. 1.2 Zero-Distance
Region
1
Distance Functions in Location Problems
11
1.3.6 Variance of Distances
The Variance of the Distances seeks locations that equalize distances from the demand points to the facility and thus seeks equitable location for all customers.
If the distance function is defines as Euclidean distance function, the variance of
the distances between the clients .x/ and the facility .y/, ı 2 .x; y/ is
ı 2 .x; y/ D
n
P
i D1
Ni di2 .x; y/
n
P
i D1
Ni
0P
n
B i D1
B
@
Ni di .x; y/
n
P
i D1
Ni
12
C
C ;
A
(1.18)
where “n” is the number of demand points and “Ni ” the number of clients at demand
point i .i D 1; 2; : : : ; n/ (Drezner and Drezner 2007).
1.3.7 Hilbert Curve
Cantor was the first researcher to map the interval [0, 1] into the square [0, 1]2 . Later
the first space-filling curve, the Peano curve, was presented to construct a curve
that passes through every entry of a two dimensional region. Afterwards, several
different space-filling curves were presented and the Hilbert curve is the most well
known (Chung et al. 2007).
Hilbert curve is a continuous curve that passes through each point in space exactly once. It enables one to continuously map an image onto a line and is an
excellent 2D image to line mapping. The position of each pixel on the mapped line
is called the Hilbert order of that pixel (Song and Roussopoulos 2002). Figure 1.3
shows a simple example of Hilbert curve.
Fig. 1.3 The Hilbert Curve
12
M. Zarinbal
1.3.8 Mahalanobis Distance
Mahalanobis distance is introduced by Mahalanobis in 1936 and widely used in
cluster analysis and other classification techniques (De Maesschalck et al. 2000). It
is closely related to Hotelling’s T -Square Distribution used for multivariate statistical testing. Also, Mahalanobis distance and leverage are often used to detect outliers
especially in the development of linear regression models.
Euclidean and Mahalanobis distance can be calculated both in the original variable space and in the principal component space.
The Euclidean is easy to compute and interpret, but this is less the case for
the Mahalanobis. In the original variable space, the Mahalanobis takes into account the correlation in the data, since it is calculated using the inverse of the
variance–covariance matrix of the data set of interest. However, the computation
of the variance–covariance matrix can cause problems.
The (1.19) shows the original Mahalanobis distance xi from the mean of data
or the center of class (dM .xi / and in the case of two variables, x1 and, variance–
covariance matrix is shown in (1.2) (/ is the mean of data or the center of classes)
(De Maesschalck et al., 2000).
dM .xi / D
AD
q
jxi j A1 jxi jT ;
12
12 1 2
12 1 2
22
!
:
(1.19)
(1.20)
1.3.9 Hamming Distance
The Hamming distance is introduced by Richard Hamming in 1950 and used in
telecommunication to count the number of flipped bits in a fixed-length binary word
as an estimate of error, and therefore is sometimes called the signal distance. Hamming weight analysis of bits is used in several disciplines including information
theory, coding theory, and cryptography (Chae and Fromm 2005).
1.3.10 Levenshtein Distance
In 1965, Vladimir Levenshtein introduces the Levenshtein Distance (LD). In information theory and computer science, the Levenshtein distance is a string metric,
which is one way to measure edit distance. The minimum number of operations
needed to transform one string into the other, where an operation is an insertion,
deletion, or substitution of a single character, gives the Levenshtein distance between two strings (Nickel and Puerto 2005). Thus,
1
Distance Functions in Location Problems
13
LD (“IBM”, “IBN”) D 1, since one substitution is needed to transform IBM to
IBN.
LD (“Success”, “Successful”) D 3, since three additions are needed to transform
Success to Successful.
LD is robust to spelling errors and small local differences between the strings
(Chae and Fromm 2005).
1.3.11 Hausdorff Distance
This kind of distance metric is used in continues models and is defines as follows:
If there are two compact sets, A and B, the Hausdorff distance between them is
dH .A; B/ D max.max d2 .x; B/; max d2 .y; A/;
x2A
(1.21)
y2B
where (Nickel and Puerto 2005)
d2 .x; B/ D min d2 .x; y/:
(1.22)
y2B
Table 1.1 shows various kinds of locations problems, the distance Functions used to
solve them and their developers.
Table 1.1 Distance functions used in location problems
Developed year
Problem
Distance
Developer
References
1909
Continues location
problem
Euclidean
distances
Weber
Hamacher and
Nickel (1998)
1937
Multi facility
location problem
Euclidean
distances
Weiszfeld
Munoz-Perez and
SaamenoRodrõguez (1999)
1963
Multifacility
location problem
Rectilinear
distances in a
network of
aisles
Francis
Munoz-Perez and
SaamenoRodrõguez (1999)
1970
Private and public
sector location
models
Lp distance
ReVelle et al.
Munoz-Perez and
SaamenoRodrõguez (1999)
1973
Multifacility
location problem
Euclidean &
rectilinear
distances
(HAP
procedure)
Eyster et al.
Munoz-Perez and
SaamenoRodrõguez (1999)
(continued)
14
M. Zarinbal
Table 1.1 (continued)
Developed year
Problem
Distance
Developer
References
1977
Traveling salesman
location problem
Rectilinear
sistances
Chan, Hearn
Munoz-Perez and
SaamenoRodrõguez (1999)
1978
Fixed charge plant
location problem
(using LP)
Random and
euclidean
distances
Morris
Schilling et al.
(2000)
1980
Unweighted
1-maximin problem
in a bounded &
convex polyhedron
in Rk
Euclidean
distances
Dasarathy, White
Chae and
Fromm (2005)
1980
Weighted 1
maximin problem
Euclidean
distances
Drezner,
Wesolowsky
Chae and
Fromm (2005)
1981
Generalized
versions of
1-maximin models
Euclidean
distances
Hansen et al.
Munoz-Perez and
SaamenoRodrõguez (1999)
1981
Location problem
with barriers for
median problem
Euclidean
distances
Katz, Cooper
Plastria and
Carrizosa (2004)
1982
Traveling salesman
location problem
Rectilinear,
euclidean, and
Lp distance
problems
Drezner,
Wesolowsky
Munoz-Perez and
SaamenoRodrõguez (1999)
1983
Location problem
with barriers for
median problem
Rectilinear
distances
Larson, Sadiq
Plastria and
Carrizosa (2004)
1986
Location of an
undesirable facility
Weighted
inverse square
distance
Melachrinoudis,
Cullinane
Munoz-Perez and
SaamenoRodrõguez (1999)
1986
Location of an
undesirable facility
Euclidean &
rectilinear
distances
Melachrinoudis,
Cullinane
Munoz-Perez and
SaamenoRodrõguez (1999)
1986
Single facility
location problem
Minimizing
the variance of
distances
Maimon
Chung et al. (2007)
1987
The median shortest
path problem
Shortest path
distance
Current et al.
Hamacher and
Nickel (1998)
1989
Assigning machines
to locations
Distance
matrix
Sarker
Yu and
Sarker (2003)
1992
Improved traveling
salesman location
problem
Rectilinear
distances
Tamir
Munoz-Perez and
SaamenoRodrõguez (1999)
(continued)
1
Distance Functions in Location Problems
15
Table 1.1 (continued)
Developed year
Problem
Distance
Developer
References
1994
Weber facility
location in the
presence of forbidden
regions
Lp distance
Aneja, Palar
Hamacher and
Nickel (1998)
1994
Competitive location
model
Euclidean
distances
T. Drezner
Plastria and
Carrizosa (2004)
1995
Undesirable facility
location by
generalized cutting
planes
Euclidean
distances
Carrizosa,
Plastria
Hamacher and
Nickel (1998)
1995
Bi objective min
quantile max covering
problems
Euclidean
distances
Carrizosa,
Plastria
Munoz-Perez and
SaamenoRodrõguez
(1999)
1996
Locating a point in a
network
Shortest path
distance
Drezner,
Wesolowsky
Munoz-Perez and
SaamenoRodrõguez
(1999)
1996
Location problem
with barriers for
median problem
Euclidean
distances
Butt, Cavalier
Plastria and
Carrizosa (2004)
1997
P-Median problem
(new heuristic
approach)
Euclidean
distances
Dai, Cheung
Hamacher and
Nickel (1998)
1998
Locating a new
facility in a
competitive
environment
Euclidean
distances with
correction
Drezner T,
Drezner Z
Drezner and
Drezner (1998)
1999
A P-center grid
positioning
Rectilinear
distances
Rayco et al.
Hamacher and
Nickel (1998)
2000
Designing distribution
systems
Rectilinear
distances
Erlebacher,
Meller
Hamacher and
Nickel (1998)
2000
Location problem
with barriers for
median problem
The K-centrum multi
facility location
problem
Lp distance
Hamacher,
Klamroth
Hamacher and
Nickel (1998)
K largest distances
in a graph
Tamir
Hamacher and
Nickel (1998)
2001
2002
Location problem
with barriers for
center problem
Rectilinear
distances
Dearing et al.
Dearing et al.
(2005)
2008
Quadratic assignment
problem
Number of
variables with
different values in
the population
members (Ga)
Drezner Z
Drezner (2008)
16
M. Zarinbal
1.4 Summary
The distance functions and its definition play an important role in facility location
problems. As it is shown above, we have various kinds of distance function with different definitions. Each of them has its own domain, advantages, and disadvantages.
For defining the distance function, one must consider the semantic of the problem,
the distance characteristic, and its usage domain.
References
Aronov B, VanKreveld M, VanOostrum R, Varadarajan K (2005) Facility location on a polyhedral
surface. Discrete Comput Geom 30:357–372
Chae A, Fromm H (2005) Supply chain management on demand. Springer, Berlin
Chew EP, Tang LC (1999) Travel time analysis for general item location assignment in a rectangular warehouse. Eur J Oper Res 112:582–597
Chung KL, Huang YL, Liu YW (2007) Efficient algorithms for coding Hilbert curve of arbitrarysized image and application to window query. Inf Sci 177:2130–2151
Dearing PM, Klamroth K, Segars R Jr (2005) Planar location problems with block distance and
barriers. Ann Oper Res 136:117–143
De Maesschalck R, Jouan-Rimbaud D, Massart DL (2000) The Mahalanobis distance. Chem Intell
Lab Syst 50:1–18
Drezner Z (2008) Extensive experiments with hybrid genetic algorithms for the solution of the
quadratic assignment problem. Comput Oper Res 35:717–736
Drezner T, Drezner Z (1998) Facility location in anticipation of future competition. Location Sci
6:155–173
Drezner T, Drezner Z (2007) Equity models in planar location. Comput Manage Sci 4:1–16
Drezner Z, Wesolowsky GO (2001) On the collection depots location problem. Eur J Oper Res
130:510–518
Fouard C, Malandain G (2005) 3-D chamfer distances and norms in anisotropic grids. Image Vision
Comput 23:143–158
Hamacher HW, Nickel S (1998) Classification of location models. Location Sci 6:229–242
Hinojosa Y, Puerto J (2003) Single facility location problems with unbounded unit balls. Math
Method Oper Res 58:87–104
Jiang J, Xu Y (2006) MiniSum location problem with farthest Euclidean distances. Math Methodol
Oper Res 64:285–308
Munoz-Perez J, Saameno-Rodroguez JJ (1999) Location of an undesirable facility in a polygonal
region with forbidden zones. Eur J Oper Res 114:372–379
Nickel S, Puerto J (2005) Location theory: A unified approach. Springer-Verlag, Berlin
Norman BA, Arapoglu R, Smith AE (2001) Integrated facilities design using a contour distance
metric. IIE Trans 33:337–344
Ogryczak W (2000) Inequality measures and equitable approaches to location problems. Eur
J Oper Res 122:347–391
Plastria F, Carrizosa E (2004) Optimal location and design of a competitive facility. Math Program
100:247–265
Schilling DA, Rosing KE, ReVelle CS (2000) Network distance characteristics that affect computational effort in p-median location problems. Eur J Oper Res 127:525–536
1
Distance Functions in Location Problems
17
Song Z, Roussopoulos N (2002) Using Hilbert curve in image storing and retrieving. Inf Syst
27:523–536
Uster H, Love RF (2003) Formulation of confidence intervals for estimated actual distances.
Eur J Oper Res 151:586–601
Yu J, Sarker BR (2003) Directional decomposition heuristic for a linear machine-cell location
problem. Eur J Oper Res 149:142–184
Chapter 2
An Overview of Complexity Theory
Milad Avazbeigi
Computational complexity theory (Shortly: Complexity Theory) has been a central
area of theoretical computer science since its early development in the mid-1960s.
Its subsequent rapid development in the next three decades, has not only established
it as a rich, exciting theory, but also has shown strong influence on many other
related areas in computer science, mathematics, and operation research (Du and
Ko 2000). However, the notions of algorithms and complexity are meaningful only
when they are defined in terms of formal computation models (Du and Ko 2000).
Apparently, we need some models to base the foundation of complexity theory
on them. In this chapter, we introduce only three basic models: deterministic turing machine (DTM), non-deterministic turing machine (NTM) and Oracle machine
models. It should be noted there are also some other models (see Du and Ko 2000).
Using such models, allows us to separate the complexity notion from any physical machine. Hence, we can measure the time complexity of algorithms and
hardness of problems independent from a specific machine which runs the algorithm(s). It should be noted that these are just abstract models; means, are defined
mathematically (Sipser 1996).
The structure of this chapter is as follows. We first discuss why we actually need
complexity theory. Then, we introduce three basic models of computation: DTM
and NTM and Oracle model. Then we present a brief introduction about the concept
of big O notation which is widely used in the complexity theory. In Sect. 2.5, the decision problems as a special form of problems are described. Following this section,
the basic concepts of reduction are presented, which help us to make relationships
between different classes of complexity and also provide a rich tool to identify the
unknown complexity class of a new problem. Finally, we introduce the most popular
classes of complexity: P , NP, NP-complete and NP-hard. In each class, also, some
known problems are presented.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 2, c Physica-Verlag Heidelberg 2009
19
20
M. Avazbeigi
2.1 Advantage of Complexity Theory
As quoted in previous section, by using computation models, we try to generalize
our results of algorithms runs to other problem instances, computers and implementations. However, without such computational models, and by just relying on
physical machines, it would be difficult however to base a theory on the detailed
specification of the physical objects and even if we could, the theory might not be
very useful, because we would need to modify it for every different set of hardware
(Martin 1996). In doing so, we attempt to define the execution time as a function
of the size of the problem. Also Time is not measured in second, minutes or any
another similar measures. Roughly speaking, we try to measure it as the number of
steps that has to be taken to resolve an instance of the problem at hand which is
apparently independent from any specific computer or machine.
2.1.1 Computational Complexity
1. Defines clearly what solving a problem “efficiently” means.
2. Categorizes problems into those that can be solved efficiently and those that
cannot.
3. Estimates the amount of time (or memory) needed to solve problems (Daskin
1995).
These are main reasons underlying the use of “complexity theory”. Using complexity theory, we can evaluate an algorithm in front of the problem at hand to understand
whether the existing algorithm can resolve the given problem completely as the size
of the problem grows or not. Also, we can compare algorithms in respect to the time
and resources they need, to resolve a given problem. Recognition of the complexity
class of a problem is another important help of this theory (2). Most of the time,
the recognized complexity class of a problem, determines our future approach we
choose to resolve the problem. For example, if we realize that the problem at hand is
NP-complete (which is described in next sections), we shift our concentration from
exact solutions to approximate and usually so called heuristic and meta-heuristic
approaches.
2.2 Abstract Models of Computation: Abstract Machines
2.2.1 Preliminary Definitions
2.2.1.1 String
The basic data structure in complexity theory is usually considered as “String”. All
other data structure can be encoded and presented by strings. A string is a finite
2
An Overview of Complexity Theory
21
sequence of symbols. For instance, the word “string” is a string of over the symbols
of English letters (Du and Ko 2000).
2.2.1.2 Language
If A is the set of strings that machine M accepts, we say that A is the language of
machine M and write L.M / D A (Sipser 1996); we say that M accepts A or M
recognizes.
A machine may accept several strings, but it always accepts only one language.
For convenience, we often work only on strings of the alphabet f0; 1g (Du and
Ko 2000). To show that this does not impose a serious restriction on the theory, we
note that a simple method can be constructed of encoding strings over any finite
alphabet into the strings over f0; 1g.
2.2.2 Turing Machine Models
The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 (Turing 1936).
2.2.2.1 Deterministic Turing Machine (DTM) (Du and Ko 2000)
DTM consists of two basic units: the control unit and the memory unit. The control unit contains a finite number of states. The memory unit is a tape that extends
infinitely to both ends. The tape is divided into an infinite number of tape squares
(or tape cells). Each tape square stores one of a finite number of tape symbols. The
communication between the control unit and the tape is through a read/write tape
head that scans a tape square at a time. Figure 2.1 shows a simple single-tape DTM.
An important concept about Turing machine is the concept of configuration. A
configuration of a TM is a record of all information of the computation of the machine at a specific moment, which includes the current state, the current symbols in
the tape, and the current position of the tape head.
B
B
a
b
a
a
Finite control
(Control unit)
Fig. 2.1 Single-tape deterministic Turing Machine
B
B
B
Tape
22
M. Avazbeigi
2.2.2.2 Non-Deterministic Turing Machine (NTM) (Du and Ko 2000)
The Turing machine described in the previous section is a deterministic machine,
because for each configuration of a machine there is at most one move to make,
and hence there is at most one next configuration. If we allow more than one moves
for some configurations, and hence those configurations have more than one next
configuration, then the machine is called a nondeterministic Turing machine (NTM).
In complexity theory, we use the concept of Turing machines to model our computations and as described in Sect. 2.1, to make independent the computations from
hardware of computer. To see examples about these models, see example of Du
and Ko (2000) and Sipser (1997). Speaking in an imprecise manner, a computation
changes the configuration of a machine and takes the machine from one configuration to a new configuration. Finally, a finite number of computations take us from an
initial state of machine to target (desired) state of machine which can be considered
as the answer to the problem to be resolved.
2.2.2.3 Oracle Turing Machine (Du and Ko 2000)
A function-oracle DTM is an ordinary DTM equipped with an extra tape, called the
query tape, and two extra states, called the query state and the answer state. The
oracle machine M works as follows: First, on input x and with oracle function f , it
begins the computation at the initial state and behaves exactly like the ordinary TM
when it is not in any of the special states.
The machine is allowed to enter the query state to make queries to the oracle, but
it is not allowed to enter the answer state from any ordinary state. Before it enters the
query state, machine M needs to prepare the query string y by writing the string y
on the query tape and leaving the tape head of the query tape scanning the square to
the right of the rightmost square of y. After the oracle machine M enters the query
state, the computation is taken over by the “oracle” f , which will do the following
for the machine: it reads the string y on query tape; it replaces y by the string f .y/;
and it puts the tape head of the query tape back scanning the leftmost square of
f .y/; it puts the tape head of the query tape back scanning the left most square of
f .y/; and it puts the machine into answer state. Then the machine continues from
the answer state as usual. The actions taken by the oracle count as only one unit
of time.
2.3 Big-O Notation (Wood 1987)
The complexity of computational problems can be discussed by fixing a model of
computation and then considering how much of the machines resources are required
for the solutions. In order to make a meaningful comparison of the inherent complexity of two problems, it is necessary to look at instances over a range of sizes.
2
An Overview of Complexity Theory
23
The most common approach is to compare the growth rates of the two runtimes,
each viewed as a function of the instance size (Martin 1996).
We measure the time and space complexity of a problem or program by total
function from N to N , since time and space are measured in positive integral units
as is the size of input data. In order to compare time or space complexities of problems or programs we are usually interested only in their order, that is, multiplicative
constants and lower-order terms are ignored. The big-O notation is used for this
purpose.
Given the two functions f , g: N ! N , we write f .n/ D O.g.n//, if there are
positive integers c and d such that, for all n d ,
f .n/ cg.n/;
cf .n/ g.n/:
(2.1)
(2.2)
In this case f is said to be big-O of g.
Similarly, we write f .n/ D ˝.g.n//, if there are positive integers c and d such
that, for all n d ,
In this case we say f is big-omega of g.
If f .n/ D O.g.n// and f .n/ D ˝.g.n//, then we write f .n/ D .g.n//, that
is, f is big-theta of g.
Whenever f .n/ D O.g.n//, then g.n/ is an upper bound for f .n/ and whenever
f .n/ D ˝.g.n//, g.n/ is a lower bound for f .n/.
Remember that the big-O notation compares only the rate of growth of functions
rather than their values, so when f .n/ D .g.n//, f .n/ and g.n/ have the same
rates of growth, but can be very different in their values.
2.3.1 Example
Take the polynomials f .x/ D 6x 4 2x 3 C 5; g.x/ D x 4 . We say f .x/ has order
O.g.x// or O.x 4 /. From the definition of order, jf .x/j cjg.x/j for all x > 1,
where c is a constant.
Proof.
j6x 4 2x 3 C 5j 6x 4 C 2x 3 C 5 where x:1;
(2.3)
j6x 4 2x 3 C 5j 6x 4 C 2x 4 C 5x 4 because x 3 < x 4 ; and so on;
j6x 4 2x 3 C 5j 13x 4 :
(2.4)
(2.5)
So we can say:
f .x/ i s O.g.x// as x ! 1:
(2.6)
24
M. Avazbeigi
2.4 Time Complexity
Now, using big-O notation, we can talk about complexity of algorithms in front
of problems. As mentioned before, big-O notation gives us a tool to talk about
complexity of algorithms in respect to steps (approximately) they take to resolve
the problem at hand, so we make our models independent from a specific hardware
configuration or implementation.
Also it is important to say in analysis of algorithms, we are interested in worst
case analysis of algorithms; the longest time they take to resolve a problem.
2.4.1 Constant Time
In computational complexity theory, constant time, or O.1/ time, refers to the computation time of a problem when the time needed to solve that problem does not
depend on the size of the data it is given as input.
For example accessing any single element in an array takes constant time as only
one operation has to be made to locate it.
It can be noted, if the number of elements is known in advance and does not
change, however, such an algorithm can still be said to run in constant time. For
example, think about a problem as finding of an unknown chose square of a chess
board. It is clear that, growth of board size changes the number of steps has to be
taken to find the square. However, for any specific size of board, it is a constant
predefined value. So our algorithm in front of this problem takes constant time.
2.4.2 Linear Time (Sipser 1996)
In computational complexity theory, an algorithm is said to take linear time, or O.n/
time, if the asymptotic upper bound for the time it requires is proportional to the size
of the input, which is usually denoted n. Informally spoken, the running time increases linearly with the size of the input. For example, finding the minimal value
in an unordered array takes O.n/ time because all the items in array have to be
checked.
2.4.3 Polynomial Time (Papadimitriou 1994)
In computational complexity theory, polynomial time refers to the computation time
of a problem where the run time, m.n/, is no greater than a polynomial function of
the problem size, n. Written mathematically using big O notation, this states that
m.n/ D O.nk / where k is some constant that may depend on the problem.
2
An Overview of Complexity Theory
25
Mathematicians sometimes use the notion of “polynomial time on the length
of the input” as a definition of a “fast” or “feasible” computation, as opposed to
“super-polynomial time”, which is anything slower than that. Exponential time is
one example of a super-polynomial time.
2.4.4 Exponential Time (Sipser 1996)
In complexity theory, exponential time is the computation time of a problem where
the time to complete the computation, m.n/, is bounded by an exponential function
of the problem size, n. In other words as the size of the problem increases linearly,
the time to solve the problem increases exponentially.
Written mathematically, there exists k > 1 such that m.n/ D O.k n / and there
exists c such that m.n/ D O.c n /.
2.5 Decision Problems
In computability theory and computational complexity theory, a decision problem is
a question in some formal system with a yes-or-no answer, depending on the values
of some input parameters. For example, the problem “given two numbers x and y,
does x evenly divide y?” is a decision problem. The answer can be either “yes” or
“no”, and depends upon the values of x and y.
A formal definition of decision problem is “A decision problem is any arbitrary
yes-or-no question on an infinite set of inputs”. Because of this, it is traditional to
define the decision problem equivalently as: the set of inputs for which the problem
returns yes (Martin 1996).
For every optimization problem, there is a Decision Problem version. Hence,
we can convert an optimization problem into a decision problem which means a
question with answer “yes” or “no”. Satisfiability problem is a popular and classic
example of decision problems which is described in Sect. 2.7.
2.6 Reduction
A reduction is a way of converting one problem into another problem in such a
way that, if the second problem is solved, it can be used to solve the first problem
(Sipser 1996).
For example, suppose you want to find your way around a new city. You know
this would be easy if you had a map. This demonstrates reducibility. The problem
of finding your way around the city is reducible to the problem of obtaining a map
of the city (Sipser 1996).
Many examples also can be found in mathematics. For example the problem of
solving a system of linear equations reduces to the problem of inverting matrix.
26
M. Avazbeigi
2.6.1 Linear Reduction
Linear reductions are used widely in complexity theory. Linear reduction in literature is defined as follows (Brassard and Bratley 1988):
Let A and B be two solvable problems. A is linearly reducible to B, denoted
A l B, if the existence of an algorithm for B that works in a time in O.t.n//,
for any function t.n/, implies that there exists an algorithm for A that also works
in a time in O.t.n//. When A l B and B l A both hold. A and B are linearly
equivalent, denoted A l B.
2.6.2 Polynomial Reduction
Another important definition is polynomial reduction (Brassard and Bratley 1988):
Let X and Y be two problems. Problem X is polynomially reducible to problem
Y in the sense of Turing, denoted X T Y; if there exists an algorithm for solving X
in a time that would be polynomial if we took no account of the time needed to solve
arbitrary instances of problem Y . In other words, the algorithm for solving problem
X may make whatever use it chooses of an imaginary procedure that can somehow
magically solve problem Y at no cost. When X T Y and Y T X simultaneously,
then X and Y are equivalent in the sense of Turing, denoted Y T X .
2.6.3 Polynomial Reduction: Many-One Polynomially Reducible
We introduced the decision problems as the problems in which we simply look for
answer “yes” or “no”. The restriction to decision problems allows us to introduce a
simplified notion of polynomial reduction:
Let X I and Y J be two decision problems. Problem X is many-one
reducible to problem Y , denoted by X m Y; if there exists a function f W I ! J
computable in polynomial time, known as the reduction function between X and Y ,
such that
When X m Y and Y m X both hold, then X and Y are many-one polynomially
equivalent, denoted X m Y (Brassard and Bratley 1988).
2.7 Examples
In this section some classic problems that we would refer to them in Sect. 2.8, are
presented. Here our aim is just explanation of problems. In Sect. 2.8, we analyze
these problems from the view of complexity theory.
2
An Overview of Complexity Theory
27
2.7.1 Traveling Salesman Optimization Problem
Given: A graph G.N; A/ with node set N and link set A. Associated with each link
.i; j / in A is a nonnegative link length dij .
Find: circuit that visits all nodes and is of minimum total length (Daskin 1995).
As we said in Sect. 2.5, any optimization problem has a decision version. The
corresponding decision problem to Traveling Salesman Problem (TSP) is: “Given a
number n of cities, n 3 integer, a non negative nx n distance matrix of integers
C D Œcij , and a non negative integer L: Is there a closed tour passing from every
city exactly once, with total length L‹”.
This is the general form of TSP. By specifying the actual graph on which the
traveling salesman problem is to be solved, we are specifying an instance of the
problem.
When we speak of the size of an instance of a problem, we are referring to a way
of characterizing how big the problem is (Daskin 1995).
In TSP the number of nodes and the number of links in a problem will constitute
an adequate description of the size of a problem.
2.7.2 Satisfiability Problem
Given: A Boolean expression – a function of true/false variables.
Question: Is there an assignment of truth values (TRUE or FALSE) to the variables such that the expression is TRUE (Daskin 1995).
As the problem shows, satisfiability (SAT) problem is essentially expressed in
the form of a decision problem. We just need to acquire the answer “yes” or “no”.
A Boolean function is a function whose variable values and function value are
all in fTRUE, FALSEg. We often denote TRUE by 1 and FALSE by 0 (Du and
Ko 2000).
It can be shown that, given a general Boolean formula, we can construct an equivalent one in conjunctive normal form (CNF), that is a formula like:
“C1 AND C 2 AND: : : AND C m”
where Ci; i D 1; : : : ; m are clauses consisting of disjunctions of Boolean variables, simple or negated.
(x1 OR x2 OR x3) and (x1 OR :x2) and (x2 OR :x3) and (x3 OR :x1) and
(:x1 OR :x2 OR :x3).
Now, the decision problem is:
Given a Boolean formula in conjunctive normal form (CNF), is it satisfiable?
That is, is there a set of “true-false” values to be assigned to the various variables,
such that the compound proposition is true?
28
M. Avazbeigi
2.7.3 Hamiltonian Cycle Problem
Given: A graph G.N; A/ where N is the set of nodes or vertices and A is the set of
links.
Question: Does the graph contain a cycle that visits every vertex (i.e., a path that
visits each node exactly once except the first node which is also visited at the last
node on the path)? (Daskin 1995).
2.7.4 Clique Problem
Given: An undirected graph G D .N; A/ where N is the set of nodes or vertices
and A is the set of links.
Question: A clique in G is a set of nodes K N such that fu; vg 2 A for every
pair of nodes u, v 2 K. Given a graph G and an integer k, the k-Clique problem
consists of determining whether there exists a clique of k nodes in G (Brassard and
Bratley 1988).
2.8 Complexity Classes
Now, after introduction of computation models (Turing machine models), big-O
notation, different time complexities, decision problems and concepts of reduction,
we are prepared to talk about complexity classes of problems.
First it is necessary to present a formal definition of complexity classes. Also it
is necessary to note that there are some other classes, which are not presented here
because this chapter aims to present an introduction to complexity theory. For more
information about other complexity classes see references at the end of chapter.
In computational complexity theory, a complexity class is a set of problems of
related complexity. A typical complexity class has a definition of the form: the set
of problems that can be solved by abstract machine M using O.f .n// of resource
R (n is the size of the input) (Du and Ko 2000).
2.8.1 Class P
P is a class of languages that are decidable in a polynomial time on a deterministic
single-tape Turing machine (Sipser 1996).
The class P plays a central role in our theory and is important because:
P is invariant for all models of computation that are polynomially equivalent to
the deterministic single-tape Turing machine.
2
An Overview of Complexity Theory
29
P roughly corresponds to the class of problems that are realistically solvable on
a computer.
Item 1 indicates that problems in P class are not affected by the particulars of
the model of computation that we are using.
Item 2 indicates whenever we prove a problem falls in Class P , an algorithm can
be found which can solve the problem in polynomial time, means the run time, m.n/,
is no greater than a polynomial function of the problem size, n. Written mathematically using big O notation, this states that m.n/ D O.nk / where k is some constant
that may depend on the problem.
We describe algorithms with numbered stages. The notion of stage of an algorithm is analogous to the step of a Turing machine, though of course, implementing
one stage of an algorithm on a Turing machine, in general require many Turing
machine steps (Sipser 1996).
To show an algorithm runs in polynomial time, we need to show two things. First,
we have to give a polynomial upper bound (see Sect. 2.3 about big-O notation) of
the stages that the algorithm uses when it runs on an input of length n. Then we
have to examine the individual stages in the description of the algorithm to be sure
each can be implemented in polynomial time on a reasonable deterministic model
(Sipser 1996). In fact, the number of stages and running time of each stage both are
bounded by polynomial functions. Kozen (2006) states that Cobham and Edmonds
are “generally credited with the invention of the notion of polynomial time”.
As quoted in Sect. 2.1, complexity theory helps us to determine whether an
algorithm is efficient or not. Now we can define an efficient algorithm as: An algorithm is efficient (or polynomial-time) if there exists a polynomial p.n/ such that
the algorithm can solve any instance of size n in a time in O.p.n// (Brassard and
Bratley 1988).
2.8.1.1 Example of Problems in P
P is known to contain many natural problems, including the decision versions of
linear programming, calculating the greatest common divisor.
In 2002, it was shown that the problem of determining “if a number is prime” is
in P (Agrawal et al. 2004). It is clear that this is a decision problem requires “yes”
or “no” answer.
Sorting a set of integers also is another example of P class problems. This is
because, an algorithm can be found capable of solving the problem in polynomial
time. A classic known algorithm for sorting is select method.
2.8.2 Class NP
NP is the class of decision problems for which there exists a proof system such that
the proofs are succinct and easy to check.
30
M. Avazbeigi
In fact, in order to prove a problem is in NP, we do not require that there should
exists an efficient way to find a proof of x when x 2 X; only there should exits an efficient way to check the validity of a proposed short proof (Brassard and
Bratley 1988).
Equivalent to the verifier-based definition is the following characterization: NP
is the set of decision problems solvable in polynomial time by a non-deterministic
Turing machine.
The two definitions of NP as the class of problems solvable by a nondeterministic
Turing machine (TM) in polynomial time and the class of problems verifiable by
a deterministic Turing machine in polynomial time are equivalent (The proof is
described by many textbooks, Sipser 1997, Sect. 2.7.3).
If we remember the definition of Class P , we immediately realize that all problems in P are in NP also. This is because we can verify all decision versions of
problems in P in polynomial time.
2.8.2.1 Example of Problems in NP
For the class NP, we simply require that any “yes” answer is “easily” verifiable. In
order to explain the verifier-based definition of NP, let us consider the subset sum
problem:
Assume there is a set of integers. The task of deciding whether a subset with sum
zero exists is called the subset sum problem.
Assume that we are given some integers, such as f 1; 2; 3; 9; 8g, and we
wish to know whether some of these integers sum up to zero. In this example, the
answer is “yes”, since the subset of integers f 1; 2; 3g corresponds to the sum
.1/ C .2/ C 3 D 0. It is clear that evaluation of each possible answer with n
member take O.n/ operation and hence can be verified in polynomial time.
Also the problem of Clique (described in Sect. 2.7) is in NP. The clique problem
is to determine whether a graph contains a clique of a specified size. To prove clique
is in NP, it is enough to generate a verifier which can check the correctness of an
answer in polynomial time.
For example in the undirected graph in Fig. 2.2, we have a 4-clique:
The decision version of the traveling salesman problem is in NP. The problem
is to determine if there is a route visiting all cities with total distance less than k.
Again the proof arises directly from the fact that for any given possible answer, we
can check whether the given circuit visits all nodes and is less than predetermined
constant k or not.
Fig. 2.2 A graph with
4-clique
2
An Overview of Complexity Theory
31
2.8.3 Class NP-Complete
A decision problem X is NP-complete if: X 2 NP; and for every problem Y 2 NP;
Y T X (Brassard and Bratley 1988)
Item 1 indicates, first we need to prove the given problem belongs to class NP.
From definition of class NP, we need to prove that a certificate exists which can be
verified in polynomial time.
Item 2 indicates that all the other problems in NP, polynomially transform to it.
The concepts of reductions presented in Sect. 2.6.
So if the problem X is NP-complete and the problem Z is in NP,
Z is NP-complete if and only if X T Z.
If X m Z then Z is NP-complete (Brassard and Bratley 1988).
This is so important to us, because suppose we have a pool of problems that have
already been shown to be NP-complete. To prove Z is NP-complete, we can choose
an appropriate problem X from the pool and show X is polynomially reducible to
Z (either many-one to in the sense of Turing). Several thousand problems have been
enumerated in this way.
From a historical view, the concept of “NP-complete” was introduced by Stephen
Cook in a paper entitled “The complexity of theorem-proving procedures” on pages
151–158 of the Proceedings of the 3rd Annual ACM Symposium on Theory of
Computing in 1971.
2.8.3.1 Cooks Theorem
In the celebrated Cook–Levin theorem (independently proved by Leonid Levin),
Cook proved that the Boolean satisfiability problem is NP-complete (See Gary and
Johnson 1979 or Papadimitrious and Steiglits 1982 for proof). In 1972, Richard
Karp proved that several other problems were also NP-complete (Karp 1972); thus
there is a class of NP-complete problems (besides the Boolean satisfiability problem). Since Cook’s original results, thousands of other problems have been shown
to be NP-complete by reductions from other problems previously shown to be NPcomplete; many of these problems are collected in Garey and Johnson’s 1979 book
Computers and Intractability: A Guide to NP-completeness.
From reduction concepts, a key characteristic of NP-complete problems is that
if a polynomial time algorithm can be found for any such problem, then it will
also solve all NP-complete problems in polynomial time. If we could find such an
algorithm we would have shown that P D NP.
2.8.3.2 P D NP Problem
An important aspect of the complexity theory is to categorize computational problems and algorithms into complexity classes. The most important open question of
complexity theory is whether the complexity class P is the same as the complexity
32
M. Avazbeigi
Fig. 2.3 Open problem
P D NP
NP
P = NP
P
class NP, or is merely a subset as is generally believed (Fig. 2.3). Shortly after
the question was first posed, it was realized that many important industry problems in the field of operations research are of an NP subclass called NP-complete.
NP-complete problems have the property that solutions to these problems are quick
to check, yet the current methods to find solutions are not “efficiently scalable”.
More importantly, if the NP class is larger than P , then no efficiently scalable solutions exist for these problems.
The openness of the P -NP problem prompts and justifies various research areas
in the computational complexity theory, such as identification of efficiently solvable
special cases of common computational problems, study of the computational complexity of finding approximate or heuristic solutions, as well as research into the
hierarchies of complexity classes.
Nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great unsolved
problems of mathematics.
The point is, because of many known unresolved problems in NP-complete class,
the trend is more toward P ¤ NP.
2.8.3.3 The Importance of NP-completeness Phenomenon
The phenomenon of NP-completeness is important for both theoretical and practical
reasons (Sipser 1996):
On the theoretical side, a researcher trying to show that P is unequal to NP
only needs to look up to an NP-complete problem. If any problem in NP requires
more than polynomial time, an NP-complete one does. Furthermore, a researcher
attempting to prove that P equals NP only needs to find a polynomial time algorithm
for an NP-complete problem to achieve this goal.
On the practical side, the phenomenon of NP-completeness may prevent wasting time searching for nonexistent polynomial time algorithm to solve a particular
problem. Even though we may not have necessary mathematics prove that the
problem is not polynomial time solvable (P D NP problem), we believe that P is
unequal to NP, so proving that a problem is NP-complete is strong evidence of its
nonpolynomiality.
2
An Overview of Complexity Theory
33
2.8.3.4 Example of Problems in NP-complete
Since the introduction of NP-complete class, many problems have been proved to
be in NP-complete class. Here there is an in-complete list of problems (Du and
Ko 2000):
Boolean satisfiability problem (SAT)
Knapsack problem
Hamiltonian cycle problem
Traveling salesman problem
Sub graph isomorphism problem
Subset sum problem
Clique problem
N -puzzle
Vertex cover problem
Independent set problem
Graph coloring problem
Figure 2.4 shows a diagram of some of the problems and the reductions typically
used to prove their NP-completeness. In this diagram, an arrow from one problem to
another indicates the direction of the reduction. Note that this diagram is misleading
Circuit - SAT
SAT
Subset problem
3-CNF SAT
Clique problem
Vertex cover problem
Hamiltonian Cycle
Travelling Salesman
Fig. 2.4 Some NP-complete problems, indicating the reductions typically used to prove their
NP-completeness (see http://en.wikipedia.org/wiki/NP-complete)
34
M. Avazbeigi
as a description of the mathematical relationship between these problems, as there
exists a polynomial-time reduction between any two NP-complete problems; but it
indicates where demonstrating this polynomial-time reduction has been easiest.
The Hamiltonian cycle problem was shown to be NP-complete by Karp (1972).
As the Fig. 2.4 shows, TSP is also a NP-complete problem. To show that the TSP
decision problem is NP-complete, we need to show two things: (a) that the TSPdecision problem is in class NP and (b) that a known NP-complete problem reduces
to the TSP-decision problem (For this problem Hamiltonian cycle problem). To
show (a), we note that, given any cycle, we can compute the cost of the cycle in
polynomial time and therefore determine in polynomial time if the cycle has length
less than or equal to B (in which case it would be a “yes” instance to the TSPdecision problem). Thus the TSP-decision problem is in class NP. To show (b), we
construct a complete graph with the same vertex set as that found in the HCP. For
each link in the new graph, if the corresponding link exits in the instance of the HCP,
let the link length be 1; otherwise let the link length be 2. Clearly, the HCP has a
solution if and only if the TSP on this complete graph has a solution with values
less than or equal to n where n is the number of nodes in the vertex set (this proof is
chose from Daskin 1995).
2.8.4 Class NP-Hard
NP-hard (nondeterministic polynomial-time hard), in computational complexity
theory, is a class of problems informally “at least as hard as the hardest problems
in NP.” A problem H is NP-hard if and only if there is an NP-complete problem L
that is polynomial time Turing reducible to H , i.e. L T H . In other words, L can
be solved in polynomial time by an oracle machine with an oracle for H . Informally
we can think of an algorithm that can call such an oracle machine as subroutine for
solving H , and solves L in polynomial time if the subroutine call takes only one
step to compute (Gary and Johnson 1979). NP-hard problems may be of any type:
decision problems, search problems, optimization problems.
Such problems are ones such that an NP-complete problem polynomially reduces
to the problem in question, but the problem under study is not provable in the class
NP. Formally, the term NP-hard is also used to describe the optimization versions
of the decision problems that are NP-complete (Daskin 1995).
2.8.4.1 Example of Problems in NP-Hard
An example of an NP-hard problem is the decision problem SUBSET-SUM. We
already described this problem. The problem is, given a set of integers, does any
non-empty subset of them add up to zero? That is a yes/no question, and happens
to be NP-complete. Another example of an NP-hard problem is the optimization
2
An Overview of Complexity Theory
35
problem of finding the least-cost route through all nodes of a weighted graph or
traveling salesman problem that we described it in Sect. 2.7.
There are also decision problems that are NP-hard but not NP-complete, for example the halting problem. This is the problem “given a program and its input, will
it run forever?” That’s a yes/no question and hence, a decision problem. It is easy
to prove that the halting problem is NP-hard but not NP-complete. For example the
Boolean satisfiability problem can be reduced to the halting problem by transforming it to the description of a Turing machine that tries all truth value assignments
and when it finds one that satisfies the formula it halts and otherwise it goes into
an infinite loop. It is also easy to see that the halting problem is not in NP since
all problems in NP are decidable in a finite number of operations, while the halting
problem, in general, is not (Garey and Johnson 1979).
2.9 Further Reading
Some classic books on complexity theory and network flows are Garey and Johnson
(1979), Ahuja et al. (1993), Karp (1972), Papadimitriou and Steiglitz (1982), and
Sahni and Horowitz (1978).
References
Agrawal M, Kayal N, Saxena N (2004) PRIMES is in P. Ann Math 160(2):781–793
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications.
Prentice-Hall, Englewood Cliffs, NJ. ISBN: 013617549X
Brassard G, Bratley P (1988) Algorithmics theory and practice. Prentice-Hall, Englewood
Cliffs, NJ
Cook S (1971) The complexity of theorem-proving procedures. Proceedings of the third annual
ACM symposium on theory of Computing, pp 151–158
Daskin MS (1995) Network and discrete location models: Algorithms, and applications. Wiley,
New York
Du D, Ko K (2000) Theory of computational complexity. Wiley, New York
Garey MR, Johnson DS (1979) Computers and intractability: A guide to the theory of NPcompleteness. W.H. Freeman, San Francisco, CA
Karp RM (1972) Reducibility among combinational problems. In: Miller R, Thatcher J (eds) Complexity of computer computations. Plenum Press, New York, pp 86–103
Kozen DC (2006) Theory of computation. Springer-Verlag, Berlin
Martin JC (1996) Introduction to languages and the theory of computation. McGraw-Hill,
New York
Papadimitriou CH (1994) Computational complexity. Addison-Wesley, Reading, MA
Papadimitrious CH, Steiglits K (1982) Combinatorial optimization: Algorithms and complexity.
Prentice-Hall, Englewood Cliffs, NJ
Sahni S, Horowitz E (1978) Combinational problems: Reducibility and approximation. Oper Res
26:718–759
36
M. Avazbeigi
Sipser M (1996) Introduction to the theory of computation. Preliminary Edition. PWS Publishing
Sipser M (1997) Introduction to the theory of computation. PWS Publishing
Turing A (1936) On computable numbers with an application to the entscheidnungs problem. Proc
Lond Math Soc ser 2, 42:230–265
Wood D (1987) Theory of computation. Wiley, New York
Chapter 3
Single Facility Location Problem
Esmaeel Moradi and Morteza Bidkhori
This chapter will focus on the simplest types of location problems, single facility
location problem. These problems occur on a regular basis when working, layout
problems (e.g., we may need to locate a machine in a shop, or items inside a warehouse). Also, on a larger scale, they can occur in, say, choosing the location of a
warehouse to serve customers to whom goods must be delivered.
The models shall be studied as being “quick and dirty.” They are “quick” in the
sense that they can be used quickly and easily, and “dirty” in the send that they
are approximate. The use of these models should be considered particularly when
some location decision must be made quickly and with limited resource available
for decision analysis.
When we wish to locate a single new facility in the plane, we often would like
to minimize an objective function involving Euclidean or rectilinear distances between the new facility and a collection of existing facilities having known planar
locations. The first objective function we consider is that of total travel distance, or
total travel cost.
A number of interesting one-facility location problems exist and are amenable to
the analysis presented in this chapter. Some typical examples of one-facility location
problems are the location of:
1.
2.
3.
4.
5.
New warehouse relative production facilities and customers.
Hospital, fire station or library in a metropolitan area.
New classroom building on a college campus.
New airfield to be used to provide supplies for a number of military bases.
Component in an electrical network.
In practice, many factors have an impact on location decisions. The relative importance of these factors depends on whether the scope of a particular location
problem is international, national, statewide, or communitywide. For example, if we
are trying to determine the location of a manufacturing facility in a foreign country,
factors such as political stability, foreign exchange rates, business climate, duties,
and taxes play a role. If the scope of the location problem is restricted to a few
communities, then factors like community services, property tax incentive, local
business climate, and local government regulations are more important.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 3, c Physica-Verlag Heidelberg 2009
37
38
E. Moradi and M. Bidkhori
It is often extremely difficult to find a single location that meets all these objectives at the desired level. For example, a location in the Midwest may offer a highly
skilled labor pool, but construction and land costs may be too high.
This chapter is organized as follows. In Sect. 3.1, we consider a general problem
formulation with rectilinear or square Euclidean or Euclidian or lp-norm distances,
and Sect. 3.2, we consider solution techniques for discrete and continuous space.
Continuous space is divided to MiniSum and MiniMax problem with various distances. MiniMax problem involving Euclidean distances is called the circle covering
problem, which can be interpreted as the problem of covering all existing facility locations with a circle of minimum radius. MiniMax problems are more specialized
than MiniSum problems and seem to be of interest, principally in cases where a
worst case analysis is quite important. Finally, Sect. 3.3 represents one real world
case studies briefly.
3.1 Problem Formulation
In this section, we represent a general problem formulation that is involving the distance traveled per trip with rectilinear or square Euclidean or Euclidian or lp-norm
distances. In final section we discus about regional facilities.
3.1.1 A General Formulation of the Problem
3.1.1.1 Model Inputs
Model inputs of this model are as follows:
i : the index of existing facilities
n: the number of existing facilities
3.1.1.2 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
X D .x; y/: coordinates of the location of new facility
d.xi ; yi /: the distance between of new facility and existing facility i
3.1.1.3 Parameters
Parameters of this model are as follows:
Pi D .ai ; bi /: coordinates of the location of existing facility i
wi : weights of existing facility i
3
Single Facility Location Problem
39
3.1.1.4 A General Formulation
A general formulation of the problem considered in this chapter may be given as
follows:
m
X
f .X / D
wi d.X; Pi /:
(3.1)
t D1
The one-facility location problem is to determine the location of the new facility,
say X that minimizes f .X /, the annual transportation cost.
3.1.2 Rectilinear Distance with Point Facilities
The rectilinear distance location problem combines the property of being a very appropriate distance measure for a large number of location problems and the property
of being very simple to treat analytically.
Figure 3.1 illustrates that several different paths between x; pi for each the rectilinear distance are the same. The number of such paths is, of course, infinite (Francis
and White 1974).
The rectilinear distance location problem can be stated mathematically as
Min f .x; y/ D
m
X
wi .jx ai j C jy bi j/ :
(3.2)
i D1
From (2) it is seen that the problem can be equivalently stated as
Min f .x; y/ D Min
m
X
wi jx ai j C
i D1
m
X
wi jy bi j:
i D1
Pi
X
Fig. 3.1 Different rectilinear paths between X and Pi
(3.3)
40
E. Moradi and M. Bidkhori
where each quantity on the right-hand side can be treated as separate optimization
problems:
Min f .x; y/ D
Min f .x; y/ D
m
X
i D1
m
X
wi jx ai j;
(3.4)
wi jy bi j :
(3.5)
i D1
3.1.3 Square Euclidean Distance with Point Facilities
In some facility location problems, cost is not a simple linear function of distance.
As an example, the cost associated with the response of a fire truck to a fire is
expected to be nonlinear with distance. Depending on the location problem, f .X /
can take on a number of different formulations. One nonlinear form of f .X / treated
in this chapter is the gravity problem. Suppose that cost is proportional to the square
of the Euclidean distance between X and Pi . Thus, the function becomes
f .X / D
m
X
t D1
wi .x ai /2 C .y bi /2 :
(3.6)
Location problem having the formulation given by (3.6) are referred to as gravity
problems (Francis and White 1974).
3.1.4 Euclidean Distance with Point Facilities
The function of Euclidean distance is
f .X / D
m
X
t D1
0:5
wi .x ai /2 C .y bi /2 :
(3.7)
Euclidian distance applies for some network location problems as well as some
instances involving conveyors and air travel. Some electrical wiring problems and
pipeline design problems are also examples of Euclidean distance problems (Francis
and White 1974).
3.1.5 LP-Norm Distance with Point Facilities
Norms are usually employed as the basis for distance predicting functions in continuous location models. Since norms are convex functions, incorporating a norm in
3
Single Facility Location Problem
41
the objective function of a continuous location problem provides the useful property
of convexity in the optimization model (Uster and Love 2001).
The lp -norm distance between any two points u D .u1 ; u2 / and v D .v1 ; v2 / is
given by
1=p
lp .u; v/ D ju1 v1 jp C ju2 v2 jp
; p 1:
(3.8)
3.1.6 Regional Facilities Problem (Drezner 1986)
We consider the single-facility of the MiniSum type of location facilities on the
plane. Both demand location and the facilities to be located are assumed to have
circular shapes, and demand and service is assumed to have a uniform probability
density inside each shape.
The problem reduces to the question of the effective distance de that should stand
for distance between the demand area and the facility. de is actually the expected
distance between the shapes with uniform probability distribution of demand and
service. In Fig. 3.2, a circle of radius R representing a facility and a circle of radius
r representing a demand area depicted. The distance between the circles center is d .
Let d.x; y/ be the distance between points X and Y (by any metric), F be the facility, and D be the demand. The probability that service and demand are generated
at dF and dD, are dF=s x2F dF, and dD=s x2F dD, respectively. Therefore,
de D
R
R
d.X; Y / dF dD
x2F y2D
R
x2F
R
dF
dD
(3.9)
:
y2D
The MiniSum problem objective function is a sum of terms associated with pairs of
facilities and demand points. de should represent the distance in the term associated
with the facility F and the area demand D. This distance should be multiplied by
the appropriate weight and the sum of all these terms should be minimized (Drezner
1986).
R
r
d
Fig. 3.2 Facility demand area
42
E. Moradi and M. Bidkhori
3.2 Solution Techniques
3.2.1 Techniques for Discrete Space Location Problems
(Heragu 1997)
Our focus is primarily on the single-facility location problem. We provide both discrete space and continuous space models. The single facility for which we seek
a location may be the only one that will serve all the customers, or it may be an
addition to a network of existing facilities that are already serving customers.
3.2.1.1 Qualitative Analysis
The location scoring method is a very popular, subjective decision-making tool that
is relatively easy to use. It consists of these steps:
Step 1. List all the factors that are important-that have an impact on the location
decision.
Step 2. Assign an appropriate weight (typically between 0 and 1) to each factor
based on the relative importance of each.
Step 3. Assign a score (typically between 0 and 100) to each location with respect
to each factor identified in step 1.
Step 4. Compute the weighted score for each factor for each location by multiplying its weight by the corresponding score.
Step 5. Compute the sum of the weighted scores for each location and choose a
location based on these scores.
Although step 5 calls for the location decision to be made solely on the basis
of the weighted scores, those scores were arrived at in a subjective manner, and
hence a final location decision must also take into account objective measures such
as transportation cost, loads, and operating costs (Heragu 1997).
3.2.1.2 Quantitative Analysis
Several quantitative techniques are available to solve the discrete space, single facility location problem. Each is appropriate for a specific set of objectives and
constraints.
For example, the so-called MiniMax location model is appropriate for determining the location of an emergency service facility, where the objective is to minimize
the maximum distance traveled between the facility and any customer.
The reader may be wondering: If the set of plants including their locations is
given, where is the location problem? To answer this question, consider the following problem: We have m plants in a distribution network that serves n customers.
Due to an increase in demand at one or more of these n customers, it has become
3
Single Facility Location Problem
43
necessary to open an additional plant the new plant could be located at p possible
sites. To evaluate which of the p sites will minimize distribution (transportation)
costs, we can set up p transportation models, each with n customers and m C 1
plants, where the .m C 1/ plant corresponds to the new location being evaluated.
Solving the model will tell us not only the distribution of goods from the m C 1
plant (including the new one from the location being evaluated) but also the cost of
the distribution. The location that yields the least overall distribution cost is the one
where the new facility should be located (Heragu 1997).
3.2.1.3 Hybrid Analysis
A disadvantage of the qualitative method is that a location decision is made based
entirely on a subjective evaluation. Although the quantitative method overcomes this
disadvantage, it does not allow us to incorporate unquantifiable factors that have a
major impact on the location decision. For example, the quantitative techniques can
easily consider transportation and operational costs, but intangible factors, such as
the attitude of a community toward businesses, potential labor unrest, and reliability
of auxiliary service providers, though important in choosing a location, are difficult
to capture. We need a method that incorporates subjective as well as quantifiable
cost and other factors.
This model classifies the objective and subjective factors important to the specific
location problem being addressed as:
Critical;
Objective;
Subjective.
The meaning of the latter two factors is obvious, but the meaning of critical
factors needs some discussion. In every location decision, at least one factor usually
determines whether or not a location will be considered for further evaluation. For
example, if water is used extensively in a manufacturing process (e.g., a brewery),
then a site that does not have an adequate water supply now or in the future is
automatically removed from consideration. This is an example of a critical factor.
Some factors can be objective and critical or subjective and critical. For example,
the adequacy of skilled labor may be a critical factor as well as a subjective factor.
After the factors are classified, they are assigned numeric values:
CF ij : if location i satisfies critical factor j , 0 otherwise
OF ij : cost of objective factor j at location i
SF ij : numeric value assigned (on a scale of 0–1) to subjective factor j for location i .
Wj : weight assigned to subjective factor J.0 Wj 1/.
Assume that we have m candidate locations and p critical, q objective, and r
subjective factors. We can determine the overall critical factor measure .CFM j /,
44
E. Moradi and M. Bidkhori
objective factor measure .OFM j / and subjective factor measure .CFM j / for each
location i with these equations:
CFM i D CF i1 CF i 2 : : : CFiP D
P
Y
CF ij ; i D 1; : : : ; m;
(3.10)
j D1
max
OFM I D
max
SFMi D
R
X
"
"
q
P
j D1
q
P
#
OFij
j D1
#
q
P
OF ij
j Di
OFij min
"
q
P
j Di
OFij
# ; i D 1; : : : ; m;
wj SFij ; i D 1; : : : ; m:
(3.11)
(3.12)
j D1
The location measure LM j for each location is then calculated as:
LM i D CFMŒ˛OFM i C .1 ˛/SFM i ;
(3.13)
where is the weight assigned to the objective factor measure? Notice that even if one
critical factor is not satisfied by a location i , then CFM j and hence LM j are equal
to zero. The OFM j values are calculated so that the location with
P
P the maximum
OF ij gets an OFM j value of zero and the one with the smallest OF ij value gets
an OFM j value of one. Equation (3.13) assumes that the objective factors are cost
based. If any of these factors are profit based, then a negative sign has to be placed in
front of each such objective factor and (3.13) can still be used. This works because
maximizing a linear profit function z is the same as minimizing z.
After LM j is determined for each candidate location, the next step is to select
the one with the greatest LM j value. Because the a weight is subjectively assigned
by the user, it may be a good idea for the user to evaluate the LM j values for various appropriate a weights, analyze the trade off between objective and subjective
measures, and choose a location based on this analysis (Heragu 1997).
3.2.2 Techniques for Continues Space Location Problems
3.2.2.1 MiniSum Problems for Rectilinear Distance
Median method with point facilities. As the name implies, the median method finds
the median location (defined later) and assigns the new facility to it. This method
is used for single-facility location problems with rectilinear distance. Consider m
facilities in a distribution network. Due to marketplace reasons (e.g., increased customer demand), it is desired to add another facility to this network. The interaction
between the new facility and existing ones is known. The problem is to locate the
new facility to minimize the total interaction cost between each existing facility and
the new one.
3
Single Facility Location Problem
45
We can rewrite expression (1) as follows:
Min f .X / D
m
X
wi jx i xj C
i D1
m
X
wi jyi yj:
(3.14)
i D1
Because the x and y terms can be separated, we can solve the optimal x and y
coordinates independently. Here is the median method:
Median method’s steps
Step 1. List the existing facilities in no decreasing order of the x coordinates.
Step 2. Find the j th x coordinate in the list (created in step I) at which the
cumulative weight equals or exceeds half the total weight for the first time;
j 1
X
wi <
i D1
m
X
wi
i D1
2
and
j
X
wi
i D1
m
X
wi
i D1
2
:
(3.15)
Step 3. List the existing facilities in no decreasing order of the y coordinate.
Step 4. Find the kth y coordinate in the list (created in step 3) at which the
cumulative weight equals or exceeds half the total weight for the first time:
k1
X
wi <
i D1
m
X
wi
i D1
2
and
k
X
wi
i D1
m
X
wi
i D1
2
:
(3.16)
The optimal location of the new facility is given by the j th x coordinate and the kth
y coordinate identified in steps 2 and 4, respectively.
Programmed mathematical method with point facilities. Although the median
method is the most efficient algorithm for the rectilinear distance, single facility
location problem, we present programmed mathematical method for solving it. It involves transforming the nonlinear, unconstrained model given by (3.14) into an
equivalent linear. Consider the following notation:
D
(
.xi x/
0
if .xi x/ > 0
;
otherwise
(3.17)
xi D
(
.xi x/
0
if .xi x/ 0
:
otherwise
(3.18)
xiC
We can observe that
jxi xj D xiC C xi ;
xi C x D xiC xi :
(3.19)
(3.20)
46
E. Moradi and M. Bidkhori
A similar definition of yiC ; yi yield
jyi yj D yiC C yi ;
yi C y D yiC yi :
(3.21)
(3.22)
Thus the transformed linear model is:
n
X
wi .xiC C xi C yiC C yi /:
(3.23)
xi C x D xiC C xi ; i D 1; : : : ; n;
(3.24)
yi C y D yiC yi ; i D 1; : : : ; n;
xiC ; xi ; yiC ; yi 0; i D 1; : : : n;
(3.25)
x; y unrestricted in sign:
(3.27)
Min
i D1
Subject to
(3.26)
For this model to be equivalent to (3.14), the solution must be such that either xiC
or xi , but not both, is greater than zero. [If both are, then the values of x and x
do not satisfy their definition in (3.17) and (3.18).] Similarly, only one of yiC ; yi
must be greater than zero. Fortunately these conditions are automatically satisfied
in the preceding linear model. This can be easily verified by contradiction. Assume
that in the solution to the transformed model, xiC and yi take on values p and q,
where p; q > 0. We can immediately observe that such a solution cannot be optimal
because one can choose another set of values for, xiC ; xi as follows:
xiC D p minfp; qg; xi D q minfp; qg:
(3.28)
And obtain a feasible solution to the model that yields a lower objective value than
before because the new xiC ; xi values are less than their previously assumed values. More over, at least one of the new values of xiC ; xi is zero according to the
expression (3.28). This means that the original set of values for xiC ; xi could not
have been optimal. Using a similar argument, we can show that either y C or y
will take on a value of zero in the optimal solution.
The model described by expressions (3.19), (3.21), (3.23) and (3.27), can be
simplified by noting that xi C can be substituted as x xi C xiC from equality
(3.20) and the fact that x is unrestricted in sign. Also y may be substituted similarly,
resulting in a model with 2n fewer constraints and variables.
Contour line method for point facilities. Contour lines are important because if
the optimal location determined is infeasible, we can move along the contour line
and choose a feasible point that will have a similar cost. Also, if subjective factors
need to be incorporated, we can use contour lines to move away from the optimal
location determined by the median method to another point that better satisfies the
subjective criteria.
3
Single Facility Location Problem
47
We now provide an algorithm to construct contour lines, describe the steps, and
illustrate with a numeric example. Algorithm for drawing contour lines is as follows:
Step 1. Draw a vertical line through the x coordinate and a horizontal line through
the y coordinate of each facility.
Step 2. Label each vertical line vi ; i D 1; 2; : : : ; p, and horizontal line Hi ; i D
1; 2; : : : ; q, where
Vi D sum of weights of facilities whose x coordinates fall on vertical line i
Hi D sum of weights of facilities whose y coordinates fall on horizontal line j
m
P
wi .
Step 3. Set i D j D 1 and N0 D D0 D
i D1
Step 4. Set Ni D Ni 1 C 2Vi ; and Di D Di 1 C 2Hi . Increment i D i C 1 and
j D j C 1.
If i p or j q, repeat 4. Otherwise, set i D j D 0.
Steps 5. Determine Sij , the slope of the contour lines through the region bounded
by vertical lines i and i C 1 and horizontal lines j and j C 1 using the equation
Si D Ni =Di . Increment i D i C 1 and j D j C 1.
Step 6. If i p or j q, go to step 5. Otherwise, select any point .x; y/ and
draw a contour line with slope Sij in the region Œi; j in which .x; y/ appears so
that the line touches the boundary of this region. From one of the endpoints of
this line, draw another contour line through the adjacent region with the corresponding slope. Repeat this until you get a contour line ending at point .x; y/.
You now have a region bounded by contour lines with .x; y/ on the boundary of
the region.
There are four points about this algorithm. First, the numbers of vertical and
horizontal lines need not be equal. Two facilities may have the same x coordinate
but not the same y coordinate, thereby requiring one horizontal line and two vertical
lines. In fact, this is why the index i of Vi ; ranges from one to p and that of Hi
ranges from one to q.
Second, the Ni and Di computed in steps 3 and 4 correspond to the numerator
and denominator, respectively, of the slope equation of any contour line through the
region bounded by the vertical lines i and i C 1 and the horizontal lines j and j C 1.
To verify this, consider the objective function (14) when the new facility is located
at some point .x; y/ that is, x D x; y D y:
f .X / D
m
X
wi jx i xj C
i D1
m
X
wi jyi yj:
(3.29)
i D1
By noting that the Vi0 s and Hi0 s calculated in step 2 of the algorithm correspond
to the sum of the weights of facilities whose x; y coordinates are equal to the
x; y coordinates, respectively, of the i; j distinct lines and that we have p; q such
coordinates or lines .p m; q m/, we can rewrite (29) as follows:
f .X / D
m
X
i D1
Vi jx i xj C
m
X
i D1
Hi jyi yj:
(3.30)
48
E. Moradi and M. Bidkhori
Suppose that x is between the s and .S Cl/, (distinct) x coordinates or vertical lines
(since we have drawn vertical lines through these coordinates in step 1). Similarly,
let y be between the t and .t C l/ vertical lines, then
f .X / D
S
X
P
X
Vi .x xi / C
i D1
Vi .xi x/ C
i DS C1
t
X
Hi .y yi / C
i D1
q
X
Hi .yi y/:
i Dt C1
(3.31)
Rearranging the variable and constant terms and added and subtracted terms we can
reach this equation (for details see Heragu 1997):
"
f .X / D 2
s
X
i D1
Vi
m
X
#
"
wi x C 2
i D1
t
X
i D1
Hi
m
X
i D1
#
wi y C c:
(3.32)
Equation (3.26) is f .X / D Ns x C Dt y C c, which can be rewritten as
yD
NS
x C .f .X / c/:
Dt
(3.33)
This expression for the total cost function at x; y or, in fact, any other point in the
region Œs; t has the form y D mx C c, where the slope N D Ns =Dt . This is
exactly how the slopes are computed in step 5 of the algorithm.
We have shown that the slope of any point x; y within a region Œs; t bounded by
vertical lines sand s C 1 and horizontal lines t and t C 1 can be easily computed.
Thus the contour line (or is cost line) through x; y in region Œs; t may be readily
drawn. Proceeding from one line in one region to the next line in the adjacent region
until we come back to the starting point .x; y/ then gives us a region of points in
which any point has a total cost less than or equal to that of .x; y/.
Third, the lines V0 ; VP C1 and H0 ; HP C1 are required for defining the “exterior”
regions. Although they are not included in the algorithm steps, the reader must take
care to draw these lines.
Fourth, once we have determined the slopes of all the regions, the user may
choose any point .x; y/ other than a point that minimizes the objective function
and draw a series of contour lines in order to get a region that contains points (i.e.,
facility locations) yielding as good or better objective function values than .x; y/.
Thus step 6 could be repeated for several points to yield several such regions. Beginning with the innermost region, if any point in it is feasible, we use it as the optimal
location. If not, we can go to the next innermost region to identify a feasible point.
We repeat this procedure until we get a feasible point (Heragu 1997).
3.2.2.2 MiniSum Square Euclidean Distance
Programmed mathematical method with point facilities. This problem can be formulated as follows:
3
Single Facility Location Problem
49
Min f .x; y/ D
m
X
t D1
wi .x ai /2 C .y bi /2 :
Any point .x ; y / that minimizes (34) must satisfy the conditions
@f .x ; y / @f .x ; y /
;
D .0; 0/:
@x
@x
(3.34)
(3.35)
Computing the partial derivatives of (3.34) with respect to x and y and then setting
them to zero gives the following unique solution:
x D
m
P
wi ai
i D1
m
P
;
(3.36)
:
(3.37)
wi
i D1
y D
m
P
wi bi
i D1
m
P
wi
i D1
The coordinates x and y of the new facility may thus be interpreted as weighted
averages of the x and y coordinates of the existing facilities, and are, in fact, the
coordinates that minimize (3.34). Conditions (3.35) can be shown to be both necessary and sufficient for a minimum. Thus, the gravity problem has a simple solution.
The solution is sometimes referred to as the cancroids or center of gravity solution.
Contour line method for point facilities. Contour lines for this problem can be
obtained quite easily. We have two cases; in the first case there exists a single facility.
In the second case there is equal item movement between the new facility and each
of the two existing facilities. Consequently, it is easy to imagine that the contour
lines will be concentric circles centered on the optimum location.
Now, what do you think the contour lines will look like when we have any number of existing facilities with unequal item movement? If you suspect the contour
lines will still be concentric circles centered on the optimum location, your intuition
is remarkable, for that is the case. To see why this is true, notice that from (3.34) we
want to determine the set of all points .x; y/ such that
kD
m
X
i D1
wi .x ai /2 C .y bi /2 :
(3.38)
In this section k is a constant value. Consequently, on expanding the squared terms
we find that
k D x2
m
X
i D1
wi 2x
m
X
i D1
wi ai C
m
X
i D1
wi ai2 C y 2
m
X
i D1
wi 2y
m
X
wi bi C wi bi2 :
i D1
(3.39)
50
E. Moradi and M. Bidkhori
If we let
W D
m
X
wi :
(3.40)
i D1
Divide (3.40) bi W , and employ the relations (3.36) and (3.37), we find that
m
m
X
X
k
wi ai2
wi bi2
D x 2 2xx C
C y 2 2yy C
:
w
W
W
i D1
i D1
(3.41)
On adding .X /2 and .y /2 to both sides of (3.41) and simplifying, we obtain the
equation for a circle,
(3.42)
r 2 D .x x /2 C .y y /2 :
Centered on the point .x ; y / with radius
rD
"
#0:5
m
X
wi ai2 C bi2
k
2
2
C .x / C .y /
:
W
W
i D1
(3.43)
This is an interesting and, to us, a nonnutritive result. Based on this result, if you are
unable to locate the new facility at the optimum location .x ; y / and must evaluate
alternative sites, you should always choose the one that has the smallest straight-line
distance to the point .x ; y / (Francis and White 1974).
Solution method for regional facilities. For simplicity of notation, the effective
distance is denoted in the square Euclidean case by De 2 . Polar coordinates are used.
A point inside the facility circle is .x; /, and a point inside the demand circle is
.y; / (where the origin is the center of that circle). Note that the denominator of
(3.9) is the product of the areas of the two circles. It follows from (3.9) that
De2
D
2
R R
0
rR 2 R R
0 0
0
Œ.d Cy cos x cos /2 C .y sin x sin /2 x:dx:d:y:dy:d
R2 r 2
:
(3.44)
Straightforward calculations lead to
De2 D d 2
R2 C r 2
:
2
(3.45)
Formula (3.45) leads to the following simple theorem:
When demand points and/or facilities have a circular area, then the squaredEuclidean MiniSum problem has the same optimal locations of facilities as the
problem defined whit point at the center of the circles.
By (3.45), the objective function consists of two parts. The first (the weighted
sum of d 2 ) is identical to that of the problem defined with points. The second part
[the weighted sum of .R2 C r 2 /=2] is a constant for given weights and radii of
circles. The theorem clearly follows.
3
Single Facility Location Problem
51
Note that even though the optimal locations of the facilities are the same, the
minimal cost increase by the constant value of the second term of the objective
function (Drezner 1986).
3.2.2.3 MiniSum Euclidean Distance
Weisfeld method with point facilities. The approach that immediately comes to mind
in solving the Euclidean distance problem is again to compute the partial derivative
of (3.7) and set them to zero. Assuming .x; y/ ¤ .ai ; bi /; i D 1; 2; : : : ; m, the
partial derivatives are
m
X
@f .x; y/
D
h
@x
i D1
wi .x ai /
.x ai /2 C .y bi /2
m
X
@f .x; y/
D
h
@y
i D1
wi .y bi /
.x ai /2 C .y bi /2
i0:5 ;
(3.46)
i0:5 :
(3.47)
Notice that If, for any i; .x; y/ D .ai ; bi /, then (3.46) and (3.47) are undefined.
Consequently, we see that difficulties arise when the location for the new facility
coincides (mathematically) with the location of some existing facility. If there were
some guarantee that any optimum location of the new facility would never be the
same as the location of an existing facility, then (3.46) and (3.47) would still give
necessary and sufficient conditions for a least cost location of the new facility. Unfortunately, there is no such guarantee available. Consequently, a modification of the
partial derivative approach is required. The modification is based on the two-tupelo
R.x; y/, which is defined as follows, if .x; y/ ¤ .ai ; bi /; i D 1; : : : ; m:
@f .x; y/ @f .x; y/
;
:
(3.48)
R.x; y/ D
@x
@y
And if .x; y/ D .ak ; bk /; k D 1; 2; : : : ; m,
(
.0;
0/; if uk wk
:
R.x; y/ D R.ak ; bk / D
uk wk
sk ; ukuwk tk ; if uk wk
u
k
(3.49)
k
where
sk D
m
X
i D1
¤k
tk D
m
X
i D1
¤k
wi .ak ai /
h
.ak ai /2 C .bk bi /2
h
.ak ai /2 C .bk bi /2
wi .bk bi /
i0:5 :
(3.50)
i0:5 :
(3.51)
52
E. Moradi and M. Bidkhori
The two-tuple R.x; y/ is defined for all points in the plane. A necessary and sufficient condition for .x ; y / is established to be a least-cost new facility location
is that R.x ; y / D .0; 0/, Consequently, the location of some existing facility
.ak ; ak /, will be the optimum location for the new facility if and only if uk < wk
Thus, one should compute the value of uk and compare it with the value of Wk if
it is suspected that the optimum new facility location coincides with the location of
existing facility k.
Although we have available necessary and sufficient conditions for an optimum solution to the Euclidean problem, we still do not have a way of determining
.x ; y /. The two-tuple R.x; y/, referred to subsequently as Kuhn’s modified gradient, can also be manipulated to provide the basis for a computational procedure
for finding the location .x ; y /. Notice that, on setting (3.46) equal to zero, we
obtain the expression
x
m
X
i D1
wi
h
2
.x ai / C .y bi /
2
i0:5
D
i D1
If we let
gi .x; y/ D
m
X
wi ai
i0:5 ;
h
2
.x ai / C .y bi /2
wi
Œ.x ai /2 C .y bi /2 0:5
:
(3.52)
(3.53)
Then (3.53) can be given as
m
P
ai gi .x; y/
i D1
xD
gi .x; y/
:
(3.54)
:
(3.55)
Likewise, from (3.47) we obtain
m
P
bi gi .x; y/
i D1
yD
gi .x; y/
So long as g.x; y/ is defined, we can employ the following iterative procedure:
xk D
m
P
ai gi .x .k1/ ; y .k1/ /
i D1
m
P
;
(3.56)
gi .x .k1/ ; y .k1/ /
i D1
yk D
m
P
bi gi .x .k1/ ; y .k1/ /
i D1
m
P
i D1
:
gi .x .k1/ ; y .k1/ /
(3.57)
3
Single Facility Location Problem
53
The superscripts denote the iteration number. Thus, a starting value .x 0 ; y 0 / is required to determine. The value of .x 1 ; y 1 / is used to determine the value of .x 2 ; y 2 /,
and so forth. The iterative procedure continues until no appreciable improvement occurs in the estimate of the optimum location for the new facility, or until a location
is found that satisfies Kuhn’s modified gradient condition.
HAP method with point facilities. An alternative iterative solution procedure can
be employed to solve the Euclidean problem without employing Kuhn’s modified
gradient procedure. The procedure is almost identical to that given by (3.56) and
(3.57). With the exception that gi .X; y/ is defined as
gi .x; y/ D h
wi
i0:5 I i D 1; 2; : : : ; m;
.x ai /2 C .y bi /2 C 2
(3.58)
where " is an arbitrarily small, it is positive valued constant. Notice that (3.58) is
always defined. Furthermore, as the value of " approaches zero, the new function
approaches the original function. We have found that the use of (3.58) in (3.56) and
(3.57) produce a very efficient solution procedure for the Euclidean problem.
Contour line method for point facilities. Unfortunately, exact methods for constructing contour lines are not available for the Euclidean problem, except for
the simplest cases where there are one or two existing facilities. As illustrated in
Fig. 3.3. The contour lines for case (a) are for a single existing facility and for case
(b) are for two existing facilities, each having equal item movement with the new
facility.
It is relatively simple to obtain approximate contour lines by evaluating the cost
function over, say, a rectangular grid of points covering the ranges of .x; y/ values of
interest. The contour lines can then be drawn by interpolating between grid points.
Alternatively, one can assign a given value k to f .x; y/ in (7), pick a value of x,
and search over y for the two values that yield the value k. The process is continued
for successive values of x until a family of points is obtained for the contour line
having value k (Francis and White 1974).
Solution method for regional facilities. For similarity of notation, the expression
for de in this case is similar to that of (3.44). The change is that the integrand in
(3.44) is put under a square root. This small change turns the four-dimensional integration into a real challenge.
Fig. 3.3 Contour lines for two simple Euclidean location problems (Francis and White 1974)
54
E. Moradi and M. Bidkhori
First calculate an approximation for d 2 for a large d , i.e., d R; R. let I be
the integrand. Then
I 2 D .d C y cos x cos /2 C .y sin x sin /2 :
(3.59)
With appropriate integrand, we have
de2 Š d 2 C
R2 C r 2
.R2 C r 2 /2
C
:
4
64d 2
(3.60)
And if the last term is ignored (since d is large), then (3.60) is similar to (3.45).
Therefore, for a large d ,
p
d 2 C .R2 C r 2 /=4:
de Š
(3.61)
Is a good approximation of de . Note that the approximation (3.60) is not defined for
d D 0 and tat it is far off for a small d , while de turns out to be quite accurate for
small d , also.
The exact calculation of de is long and tedious. You can see the summarized of
exact calculation of de in Drezner (1986).
The goodness of the approximation of de to de was checked in Drezner (1986),
the ratio between the two was calculated. The ratio de =de was found to be between
0.75 (for r D d D 0) and 1 (for large d and any r).
Finally, it was checked the goodness of the approximation:
dA D
r
4
d 2 C .R2 C r 2 /:
9
(3.62)
The number 4/9 was chosen so that dA D de for r D d D 0. It was found that
de dA 1:07de , which is a better approximation than de .
The point facilities location problem in two regions with different norms. Suppose
the plane is divided by a straight line into two regions with different norms. We
find the location of a single new facility such that the sum of the distances from
the existing facilities to this point is minimized. This is a non-convex optimization
problem. We have the optimal solution lies in the rectangular hull of the existing
points.
Suppose the plane, R2 , is divided by a straight line, y D mx, into two region,
1 , with an lP1 , and 2 with an lP2 norm .p1 ; p2 1/. Suppose also that there are
respectively, n1 and n2 points on each side. The problem is to find the location of
a new point such that the sum of the weighted distances from the existing n1 C n2
points to this is minimized. Mathematically the problem can be stated as
p1 W min
x
(
X
pi 21
wi d.x; pi / C
X
pi 22
)
wi d.x; pi / ;
(3.63)
3
Single Facility Location Problem
55
where d.x; pi / is the shortest distance, induced by the norms, between the existing
point pi D .ai ; bi /, and the new point x D .x; y/I wi ’s > 0 are given weights
assigned to the pi ’s. when x and pi are both on the same side of the dividing line,
then the problem reduces to a single norm problem; for the l1 and l2 ; d.x; pi / is the
rectilinear or Euclidean distances between them. The difficulty arises when x and
pi are on different sides of the boundary line.
We consider the special case of l1 and l2 with the boundary line y D mx. Note
that without loss of generality we assume that the line passes through the origin.
There are also make the more realistic assumption that for the points on the 1 side,
the shortest distance may involve passing through the boundary.
Model properties. There are some properties of the problem for the special case of
l1 and l2 norms. We have a fixed point p 2 2 , an x 2 1 , and a fixed (straight) line
segment L of any orientation. Let
d.x; p/ D min fk1 .x; z/ C k2 .z; p/g ;
z2l
(3.64)
where k1 and k2 are arbitrary norms. Then d is a convex function of X .
There is a characterization of the crossing (gate) points on the boundary line and
the shortest path connecting two points on different sides of the line.
Suppose R2 is divided by a straight line, y D mx, into two regions, 1 and 2 ,
whit l1 , and l2 norms, respectively. Assume, without loss of generality, that m > 0.
Then for any pair of given points p1 D .a1 ; b1 / 2 1 and p2 D .a1 ; b1 / 2 2
the shortest path form p1 to p2 passes through the line segment connecting points
.a1 ; m a1 / and .b1 =m; b1 / on the boundary line.
The intersection of the shortest path connecting points p1 2 1 and p2 2 2
whit the boundary line y D mx .m > 0/ is either one of the points (a1 ; ma1 ) and
.b1 =m; b1 /, or the point .x; mx/, with x given above, if a1 x b1 =m.
There is optimal solution to the overall problem in the rectangular hull of the
existing points. The rectangular hull of a set of points is defined as the smallest
rectangle with sides parallel to the (x and y axes) containing the set.
Solution procedure. Big square small square (BSSS) method is a geometrical branch
and bound algorithm.
The procedure start with the rectangular hull, R, of the existing points, it contains
the optimal solution then we partition R into four rectangles by drawing vertical and
horizontal lines through the middle of its sides. At partitioning level k, denote these
rectangles by Rk1 ; Rk2 ; RK3 , and, Rk4 . We take the points at the center of each
rectangle and evaluate the objective at these points. The best solution provides an
upper bound for the overall problem.
The distance between points inside a rectangle is taken to be zero. The distance
between a point pi outside a rectangle R and the point inside that rectangle is taken
to be the distance between pi and the closest point to it on the boundary of R which
could either be a corner point of R, or the projection of pi onto R. note that since R
is closed and convex, such a point indeed exists.
56
E. Moradi and M. Bidkhori
A sub-square for which the lower bound exceeds the value of the best known
solution is fathomed. The process continues until the larger side of the sub-rectangle
is less than a given tolerance, ". The steps of the algorithm are outlined below. The
input to the algorithm is a set of existing points, and the termination tolerance ".
The output is the optimal location of the new facility, Xb , and the optimal objective
value, fb .
Algorithm
1. Find the rectangular hull of existing facilities, R; set L D 0; fb D 1; and let
d D max fx max xmin ; y max ymin g;
2. Set l D lC1, and partition R into four equal sub-rectangles Rl;1 ; Rl;2 ; Rl;3 ; Rl;4 .
Calculate the objective value flr at the midpoint of each sub-rectangle. If minrfflrg
< fb , update fb to this value.
3. For each demand point
P pi find the minimum distance dir from pi to Rl;r ;
r D 1; : : : ; 4; lbl;r D i wi dir. If lbL;r > fb , fathom RL;r and set lbL;r D 1.
4. Set lb0 D minrflbL;r g and r 0 D argminrflbL;r g. If lb0 D 1, go to step(6); else,
if (0.5)L d < " go to step (5); else, set R D RL;r 0 : fathom RL;r 0 and set
RL;r 0 D 1, and go to step (2).
5. If lb 0 < fb , set fb D lb0 , define Xb as the center of sub-rectangle RL;r 0 , and
fathom this rectangle.
6. Set L D L 1; if L D 0 go to step (7); else, if unfathomed sub-rectangles at
level L are found, choose the one with the most favorable lb ; donet it as R, and
return to step (2); else, repeat step (6).
7. Terminate the algorithm with optimal new facility location Xb having the objective function value fb (Zaferanieh et al. 2008).
3.2.2.4 MiniSum LP-Norm Distance (Francis et al. 1992)
Weiszfeld procedures. The Weiszfeld procedure depends upon the convexity of the
Euclidean metric, and thus, utilizes the first order necessary and sufficient conditions. Since it is impossible to express the unknown variables x1 and x2 in closed
form equations, the first order derivatives cannot be solved directly. Instead, an iteration function is obtained by using these derivatives. In order to eliminate the obvious
difficulty caused by the discontinuities in the derivatives, we use an approximation
of the lp -norm in the objective function. We employ the following hyperbolic approximation of the lp -norm.
lNp .u; v/ D
p=2
p=2 1=p
.u1 v1 /2 C 2
; where p 0; 2> 0:
C .u2 v2 /2 C 2
(3.65)
We use the notation sQ .X / to denote the objective functions of the approximated
lp -norm for single facility location problem.
3
Single Facility Location Problem
xtkC1 D
n
P
wj
j D1
n
P
j D1
57
.p=2/1
1p
2
xtk ajt C 2
ajt
lNp x k ; aj
wj
xtk
ajt
2
; t D 1; 2 : (3.66)
.p=2/1
1p
k
N
lp x ; aj
C2
It should be noted that in order to deal with a well-formulated problem, we assume
that all new facilities are chained. New facility i is chained if there exists a positive
w1ij where j is any existing facility or if there exists a positive w1ij where r is any
chained new facility (Uster and Love 2001).
In addition, the convergence of the Weiszfeld algorithm is discussed in Uster and
Love (2000).
Bounding method. The Weiszfeld procedure is basically an iterative steepestdescent algorithm with a predetermined step size. Therefore, to terminate the
iterative procedure, a stopping rule or a bound for the best objective function value is
required. The rectangular bound at iteration is obtained by solving a rectangular distance location problem. The bound problem involves locating the same number of
facilities in the original problem with respect to the existing facility locations with
newly created weights. At iterations, the percent difference between the optimum
objective function value of the rectangular bound problem and the current objective
function value of the original problem is calculated. If this difference is smaller than
a termination value that prespecified by the user, the procedure is terminated (Uster
and Love 2001). A rectangular bound for the iterative can be obtained by using the
Holder inequality given by
N
X
j˛i ˇi j
i D1
N
X
i D1
p
j˛i j
!1=p
N
X
i D1
jˇi j
q
!1=q
(3.67)
;
where ˛ and ˇ are N-dimensional vectors, p > 1 and 1=p C 1=q D 1. Taking
N D 2 for the planar location model and letting
1=2
2
;
x1 aj1 C 2
.p1/=2
2
ˇ1 D x1k aj1 C 2
;
1=2
2
˛2 D x2 aj 2 C 2
;
.p1/=2
2
ˇ2 D x2k aj 2 C 2
:
˛1 D
(3.68)
(3.69)
(3.70)
(3.71)
We obtain
˛1 ˇ1 C ˛2 ˇ2 ..˛1 /p C .˛2 /p /
1=p
..ˇ1 /q C .ˇ2 /q /
1=q
:
(3.72)
58
E. Moradi and M. Bidkhori
Rearranging terms, we have
p=2
p=2 1=q
2
k
2
k
Nlp .x; aj /
C x2 aj 2 C 2
x1 aj1 C 2
˛1 ˇ1 C ˛2 ˇ2 :
(3.73)
Rewriting the second term on the left-hand side, we obtain
p1
lNp .x; aj / lNp x k ; aj
˛1 ˇ1 C ˛2 ˇ2 :
(3.74)
In order to obtain the cost function of the minimum model, we multiply both sides
by wj and sum for j D 1; : : : ; n. Thus, we have
S .X / D
n
X
wi lp .x; aj /
j D1
n
X
wi
j D1
˛1 ˇ1
C
.lp .x k ; ai //p1
n
X
wi
j D1
˛2 ˇ2
:
.lp .x k ; ai //p1
(3.75)
Minimizing both sides of the inequality over x gives
S .x / min
x
8
n
<X
:
wi
j D1
˛1 ˇ1
.lp .x k ; ai //p1
C
n
X
j D1
wi
9
=
˛2 ˇ2
.lp .x k ; ai //p1 ;
:
(3.76)
Without changing the direction of the inequality, the terms ˛1 and ˛2 can be simplified as jx1 ˛j1 j and jx2 ˛j 2 j, respectively. Thus, the bound as a rectangular
distance problem, SQ B K (X), is found as
S B K .x R / D min
n
X
x2R j D1
n
X
ˇ
ˇ
ˇ
ˇ
uj ˇx1R aj1 ˇ ;
uj ˇx1R aj1 ˇ C min
x2R j D1
(3.77)
where
uj D wj
ˇ1
and
.lQp .x k ; aj //p1
vj D wj
ˇ2
.lQp .x k ; aj //p1
;
j D 1; : : : ; n:
(3.78)
For notational convenience, we denote the solution of a rectangular distance location
problem by x R , and thus, the bound at an iteration k is given by SQ B K .x R /. Let
sQj .X /; j D 1; : : : ; n, denote the terms in sQ .X /. Then the first derivatives of sQj .X /
with respect to x1 and x2 are
@SQj .x/
D wj
@x1
x1 aj1
2
C2
.p1/=2
.lQp .x k ; aj //p1
.x1 aj1 /
:
˛1
(3.79)
3
Single Facility Location Problem
59
And
@SQj .x/
D wj
@x2
.p1/=2
2
x2 aj 2 C 2
.x2 aj 2 /
;
˛2
.lQp .x k ; aj //p1
j D 1; : : : ; n:
(3.80)
By letting e ! 0 and using the equality .xt ˛jt / D sign.xt ˛jt /jxt ˛jt j, for
t D 1; 2, we can simplify the last terms and obtain
@SQj .x/
D wj
@x1
x1 aj1
2
C2
2
.p1/=2
C2
.lQp .x k ; aj //p1
sign.x1 aj1 /;
(3.81)
And
@SQj .x/
D wj
@x2
x2 aj 2
.p1/=2
.lQp .x k ; aj //p1
sign.x2 aj 2 /; j D 1; : : : ; n:
Thus, uj and vj can be rewritten as
ˇ
ˇ
ˇ
ˇ
ˇ @SQ .x k / ˇ
ˇ @SQ .x k / ˇ
ˇ
ˇ
ˇ j
ˇ j
uj D ˇ
ˇ and vj D ˇ
ˇ ; j D 1; : : : ; n:
ˇ @x1 ˇ
ˇ @x2 ˇ
(3.82)
(3.83)
3.2.2.5 MiniMax Problems
There is another class of single facility location problems that we should mention,
called MiniMax problems. One of the best known such problems, called the circle
covering problem, involves enclosing m known points in the plane within a circle
of minimum radius. The circle covering problem is equivalent to the problem of
locating a new facility with respect to m existing facilities so as to minimize the
maximum Euclidean distance from the new facility to the existing facilities. The
circle covering problem may be of interest in locating a transmitter of some kind,
or a receiver, so as to “cover” m stations with as strong signal strength as possible.
Also, the problem of stationing a helicopter so as to minimize the maximum time
for it to respond to an emergency at anyone of m sites is closely related to the circle
covering problem. Contrary to what one might think, the circle covering problem
cannot be solved by inspection (or at least no one has yet been able to do so). There
are, however, very efficient and relatively simple algorithms for solving the problem.
If we draw a circle about each of the m points of radius r. The intersection of the
m circles. Consists of all points whose maximum Euclidean distance to the m given
points is r or less. If we imagine reducing r until the intersection of the circles
is a single point, we obtain the solution to the circle covering problem. Such an
approach is now quite feasible when one has a computer terminal with the facility
for displaying circles, provided that m is not too large.
60
E. Moradi and M. Bidkhori
Another problem, which we might term the diamond covering problem, occur
when we replace the Euclidean distances of the circle covering problem by rectilinear distances. Here by “diamond” we mean a square with each edge making an
angle of ˙45ı with an axis; the radius of the diamond is half the length of the line
segment joining opposite vertices. The diamond covering problem is easy to solve
by applying a 45ı rotation we obtain an equivalent square covering problem which
we solve by constructing a smallest enclosing rectangle. If the rectangle is square,
its center is a MiniMax location. Otherwise, we extend the shortest of a pair of edges
to have the same length as the longer pair, and take the center of a square so constructed as a MiniMax location. Of course, we must apply a revel 45ı rotation to
translate our answer back to a solution of the original problem in some cases there
will be more than one smallest enclosing diamond, resulting in alternative optimum
locations. As with the circle covering problem, it is easy construct contour sets. The
set of all points such that the maximum rectilinear distance between the points and
the m existing facilities is at most r consists of t intersection of m diamonds, with
diamond i having a center at point i and a radius of r, for i D 1; : : : ; m. Much the
same comments apply to these contour sets as made above for the circle covering
problem.
Circle covering problem. In this section we consider two approaches for solving the
circle covering problem.The first approach is basically geometrical in nature and is
particularly well suited for planar circle covering problems. The second approach is
more general, in the sense that it can be used not only for planar problems but for
analogous problems in three or more dimensions. For the first approach we present
an algorithm. For the second approach we show how to convert the problem into
an equivalent quadratic programming problem. Most available algorithms for solving quadratic programming can then be applied to solve the equivalent problem.
Let, first state the problem of interest precisely. We wish to minimize the function
g.x; y/ defined by
g.x; y/ D maxfŒ.x ai /2 C .y bi /2 1=2 W 1 i mg:
(3.84)
Here, as usual, the points .ai ; bi / are m existing facility locations, and .x; y/ is a new
facility to be located in such a way as to minimize g.x; y/. A problem equivalent to
minimizing g.x; y/ is as follows:
Min Z
(3.85)
Subject to
Œ.x ai /2 C .y bi /2 1=2 Z;
1 i m:
(3.86)
The equivalent problem has the following geometrical interpretation. The constraints state that each existing facility location must lie in a circle with center .x; y/
and radius z, so that the geometrical problem is to find a smallest circles that encloses
all the existing facility locations.
3
Single Facility Location Problem
61
We now consider briefly an algebraic approach to the circle covering problem
which is valid in any number of dimensions. With X and Pi denoting the location of
the new facility and existing facility i , respectively, the Euclidean distance between
the two locations is the square root of the following term (where the superscript
denotes the transpose operation):
.x pi /T .x pi /:
(3.87)
Hence an equitant version of the circle covering problem is as follows:
Min u
(3.88)
.x pi /T .x pi / u; i D 1; : : : ; m:
(3.89)
.x pi /T .x pi / D x T x 2piT X C PIT pi :
(3.90)
Subject to
Because
An equivalent way to write the constraints is as follows:
x T x 2piT X C PIT pi u; i D 1; : : : ; m:
(3.91)
But now if we make the following change of variables,
v D x T x u:
(3.92)
We obtain the following equivalent version of the problem:
Min x T x u:
(3.93)
2piT x v piT PI ; i D 1; : : : ; m
(3.94)
Subject to
The latter problem is a quadratic programming problem with a convex objective
function and linear constraints, and thus is solvable by most quadratic programming
algorithms.
MiniMax location problems with rectilinear distances. The problem we consider
now is one of finding a new facility location that will minimize the following
function:
g.x; y/ D maxfWi Œjx ai j C jy bi j C hi W 1 i mg:
(3.95)
As a possible example of the problem, suppose that .x; y/ is the location of a “convenience” center and that “users” of the center are located at the existing’ facility
locations, the points .ai ; bi / through .am ; bm /. User i require a time of hi to prepare
to go to the center and then travels to the center at a time per unit distance of wi , so
62
E. Moradi and M. Bidkhori
that wi Œjx ai j C jy bi j C hi is the total time to prepare to go to the center and
then go there. The center is to be located so that the maximum such time for any
user will be minimized.
An alternative but equivalent formulation of the problem of minimizing g.x; y/
is the following one:
Min z
(3.96)
Subject to
WI Œjx ai j C jy bi j C hi z; i D 1; : : : ; m:
(3.97)
Because all the weights are positive, we can also write the problem as follows:
Min z
(3.98)
Subject to
jx ai j C jy bi j
z hi
; i D 1; : : : ; m
wi
(3.99)
The constraints of the latter formulation state that the existing facility location
.ai ; bi / is to be in a diamond with center .x; y/ and radius .z hi /=wi for
i D 1; : : : ; m. Let us denote the problem with all unit weights and all zero addends by UP, which represents “unweighed problem.” Similarly, we let WPA denote
the weighted problem with some nonzero addends, and we let UPA denote the unweighed problem with some nonzero addends. For UP, the simplest problem, we
conclude that the problem is one of finding a diamond of minimum radius that will
contain all the existing facility locations. We now give an approach to solve the more
general problem UPA. This approach is based on the fact inequality.
jx ai j C jy bi j ri z hi :
(3.100)
Equivalent is to the following four inequalities:
x ai C y bi ri ;
x ai y C bi ri ;
(3.101)
(3.102)
x C ai y C bi ri ;
x C ai C y bi ri :
(3.103)
(3.104)
We can use this fact to transform UPA to a linear program which can be solved as
follows. Compute the following numbers:
c1 D minfai C bi hi W i D 1; : : : ; mg;
c2 D minfai C bi C hi W i D 1; : : : ; mg;
(3.105)
(3.106)
c3 D minfai C bi hi W i D 1; : : : ; mg;
c4 D minfai C bi C hi W i D 1; : : : ; mg;
(3.107)
(3.108)
c5 D minfc2 c1 ; c4 c3 g:
(3.109)
3
Single Facility Location Problem
63
The minimum objective function value is c5 =2, and the MiniMax locations are the
locations on the line segment L joining the following two points:
1
x1 ; y1 D .c1 c3 ; c1 C c3 C c5 /;
2
1
x2 ; y2 D .c2 c4 ; c2 C c4 c5 /:
2
(3.110)
(3.111)
Consider the construction of level lines for the function g.x; y/ of the problem
WPA. To construct level lines we construct level sets; the boundaries of the level
sets are the level lines. Level lines are of interest for exactly the same reasons we
discussed earlier for the MiniSum problems; they allow us to evaluate easily location
other than the optimal locations [those that minimize the function g.x; y/]. Let us
denote a level set of g.x; y/ of boundary value Z by S.z/, so that S.z/ D f.x; y/ W
g.x; y/ zg. We construct S.z/ as follows. Given a value of Z of interest, we first
compute the following numbers:
z hi
W i D 1; : : : ; m ;
c1 .z/ D min ai C bi C
wi
z C hi
W i D 1; : : : ; m ;
c2 .z/ D min ai C bi C
wi
z hi
c3 .z/ D min ai C bi C
W i D 1; : : : ; m ;
wi
z C hi
c4 .z/ D min ai C bi C
W i D 1; : : : ; m :
wi
(3.112)
(3.113)
(3.114)
(3.115)
The level S.z/ is then as follows:
S.z/ D f.x; y/ W c2 .z/ x C y c1 .z/; c4 .z/ x C y c3 .z/g:
(3.116)
If at least one of the inequalities c2 .z/ c1 .z/; c4 .z/ c3 .z/ does not hold, you
have chosen a value of Z that is smaller than the minimum value of g.x; y/ and the
level set will be empty; thus you mast pick another value of Z which is large enough
so that both of inequalities hold. Supposing the level set to be nonempty, you should
be able to see that the level set is a rectangle, with two parallel sides making a C 45ı
angle with the x axis and the other two parallel sides making a 45ı angle with
the x axis. The vertices of the level set, starting at the top corner and proceeding
clockwise, are as follows:
1
.c1 .z/ c3 .z/; c1 .z/ C c3 .z//;
2
1
v2 .z/ D .c1 .z/ c4 .z/; c1 .z/ C c4 .z//;
2
v1 .z/ D
(3.117)
(3.118)
64
E. Moradi and M. Bidkhori
1
.c2 .z/ c4 .z/; c2 .z/ C c4 .z//;
2
1
v4 .z/ D .c2 .z/ c3 .z/; c2 .z/ C c3 .z//:
2
v3 .z/ D
(3.119)
(3.120)
Hence you can plot a level set of value Z by first choosing Z, computing c1 .z/
through c1 .z/, checking to be sure that the inequalities c2 .z/ c1 .z/; c4 .z/ c3 .z/,
are satisfied by the Z you have chosen, computing v1 .z/ through v4 .z/ and plotting the four points, and then constructing lines joining v1 .z/ and v2 .z/; v2 .z/ and
v3 .z/; v3 .z/ and v4 .z/; v4 .z/ and v1 .z/. The rectangle the lines enclose is the level
set of value Z, and its boundary is the level lines of value Z.
MiniMax problems with Tchebychev and rectilinear distances. In Sect. 3.2.2.4 we
saw that the diamond covering problem could be interpreted, given a 45ı rotation,
as a square covering problem, and that this interpretation led to a simple solution
procedure for the diamond covering problem. What we do now is to exploit this
discovery in a systematic way. As a result, we will obtain an efficient means of
solving WAP, the MiniMax weighted rectilinear distance problem with addends.
Consider two points X and Y in the plane. Suppose that X and Y are the endpoints of the hypotenuse H of a right triangle, with the other two sides of the
triangle, denoted by A and B, being parallel to the horizontal and vertical axes, respectively. We have seen earlier that the length of the hypotenuse H is the Euclidean
distance between X and Y , while the sum of the lengths of sides A and B is the rectilinear distance between X and Y . We now introduce a new distance, called the
Tchebychev distance between X and Y , defined to be the maximum of the lengths
of sides A and B. In other words, if A is longer than B, then A is the Tchebychev
distance between X and Y , while if A is not longer than B, then B is the Tchebychev distance between X and Y . We denote the Tchebychev distance between X
and Y by t.X; V /. Thus if X D .X1 ; X2 /, and Y D .Y1 ; Y2 /, then
t.X; Y / D maxfjx1 y1 j ; jx2 y2 jg:
(3.121)
What does a contour set of Tchebychev distance look like? Consider the set of all
points X whose Tchebychev distance from the origin is at most 1, that is, the 1
set of all points X D .X1 ; X2 / satisfying t.x; 0/ 1, or, equivalently, satisfying
max fjX1 jjX2 j 1g. The latter inequality is equivalent to jX1 j 1 and jX2 j 1.
But these last two inequalities are in turn equivalent to
1 x1 1 and 1 x2 1:
(3.122)
Hence the set of all points X whose Tchebychev distance from the origin is at most
1 is a square with its center at the origin and each side of length 2. This square is
the Tchebychev analog of a circle with center at the origin and radius 1 and the
Tchebychev analog (for rectilinear distance) of a diamond with center at the origin
and radius 1. Of course, if we rotate a diamond by 45ı , we obtain a square, a result
that should give you a good clue about the relationship between Tchebychev, and
rectilinear distances.
3
Single Facility Location Problem
65
Let us now explore the relationship between Tchebychev and rectilinear distances. It is convenient to introduce the following linear transformation, which we
denote by Q.X; Y /:
1 1
Q.x; y/ D .x; y/
D .x C y; x C y/:
1 1
(3.123)
You should be able to verify that the inverse transformation, denoted by Q1 .u; v/,
is given by
3
2
1 1
1
7
6
(3.124)
Q.u; v/ D .u; v/ 4 2 2 5 D .u v; u C v/:
1 1
2
2 2
Another aspect is that the rectilinear distance between any two vertices of D, namely
2, is the same as the Tchebychev distance between the corresponding transformed
points, the vertices of the square S . This result is no coincidence. Given any points
X and Y in the plane, let r.x; y/ denote the rectilinear distance between X and Y ,
if we compute points X 0 and Y 0 using the equations x 0 D Q.X / and Y 0 D Q.X /,
it is known that
(3.125)
r.X; Y / D t.X 0 ; Y 0 /:
That is, the rectilinear distance between X and Y is the same as the Tchebychev
distance between the transformed points X 0 and Y 0 . Equivalently, given any points
X 0 and Y 0 in the plane, if we compute points X and Y using the equations.
X D Q1 .X 0 / and Y D Q1 .Y /:
(3.126)
t.X 0 ; Y 0 / D r.X; Y /:
(3.127)
We conclude that
The consequence of the two equations above is that we can transform a planar location problem involving rectilinear distances into an equivalent problem involving
Tchebychev distances, and vice versa. Hence we obtain equivalence between planar
location problems involving Tchebychev and rectilinear distances. This equivalence
is useful since it is often the case that one problem is easier to analyze than the other.
Let us consider converting the general MiniMax problem with rectilinear distance, denoted by WPA, into an equivalent problem with Tchebychev distances.
Recall that WPA is the problem of minimizing the function g.x; y/, where
g.x; y/ D maxfWI Œjx ai j C jy bi j C hi W 1 i mg:
(3.128)
Let .u; v/ be the result of applying the transformation Q to the point(x; y), and let
.˛i ; ˇi / be the result of applying the transformation Q to the point .˛i; ˇi /. We
know that
WI Œjx ai j C jy bi j C hi D wi maxŒju ˛i j ; jv ˇi j C hi D
maxŒwi ju ˛i j C hi ; wi jv ˇi j C hi
(3.129)
66
E. Moradi and M. Bidkhori
We can thus conclude that
g.x; y/ D maxfmaxŒwI Œju ˛i jChi ; wi Cjv ˇi jChi W i D 1; : : : ; mg: (3.130)
Suppose that we know define the function g1 .u/ and g2 .u/ as follows:
g1 .u/ D maxfwI ju ˛i j C hi W i D 1; : : : ; mg;
g2 .u/ D maxfwI jv ˇi j C hi W i D 1; : : : ; mg:
(3.131)
(3.132)
It then follows that
maxfmaxŒwI ju ˛i j C hi ; wi jv ˇi j C hi W i D 1; : : : ; mg D
maxfg1 .u/; g2 .u/g:
(3.133)
The reason for the latter equality is that regardless of the order in which we compute
the maximum of a collection of numbers, we obtain the same result.
If you examine the latter equation, you can see that the term on the left is the
same as the term on the right in our most recent equation for g.x; y/; hence we
obtain a very useful result as follows: Given
.u; v/ D Q.x; y/:
(3.134)
g.x; y/ D maxfg1 .u/; g2 .u/g:
(3.135)
We have
The consequence of our result is that we can minimize g.x; y/ by solving two independent minimization problems as follows:
1. We minimize g1 .u/ and obtain a minimizing point, say u.
2. Next, we minimize g2 .u/ and obtain a minimizing point, say v.
3. We can then apply the inverse transformation to .u ; v / to obtain a point, say
.x ; y /, and conclude that .x ; y / minimizes g.x; y/.
4. Further, the minimum value of g.x; y/ is equal to max fg1 .u /; g2 .u /g (Francis
et al. 1992).
3.3 Case Study (Heragu 1997)
We now present a relocation project undertaken by a small facility. A small manufacturing company currently located in a university “tech park” has witnessed major
growth since introducing an innovative technology into the marketplace. Its owner
now wants to find a new location and build a bigger facility. In January she hired
senior industrial and management engineering (IME) students at the university to
investigate several potential locations and select the one that best suits her needs.
3
Single Facility Location Problem
67
The student group adopted the following five step approach, which is based on the
hybrid analysis discussed earlier.
Step 1. Determination of requirements: The students conducted interviews with
the owner and facility manager to determine these company specific requirements
for the new facility:
– The company will relocate in New York or Vermont.
– At least 15,000 ft of space is required.
– A power source of three phase, 440 W, and 200 A electrical service is mandatory to power the atomizers used in the manufacturing process.
– The current rent is $7.50 per square foot per year; the company wants to pay
between $3.50 and $4.50 per square foot.
– The company wants to move within the next 8 months.
– All suppliers and vendors should be within 100 miles of the facility.
– A lease of 1–2 years is preferred.
– There should be adequate room for expansion.
– An industrial park or shared facility is preferred.
– The facility should be located close to major highways and airports.
– A loading bay is required; easy access to the bay is desired.
– The new facility should be built to suit.
– The facility maintenance costs should be low or the owner of the building
must pay the maintenance and related expenses.
– The general condition of the building should be good.
– The building should not be considered high risk by insurance companies.
– It is desirable to have secretarial services available nearby.
– The local and state taxes must be reasonable.
Step 2. Classification of location factors: Based on the interview with the owner,
the IME students classified the requirements into three categories:
Critical factors
– Minimum space requirement
– Three phase, 440 W , 200 A electrical service
– Support service providers and vendors within 100 miles
Objective factors
– Rent
– Space rented or leased
– Maintenance and insurance costs and taxes
Subjective factors
–
–
–
–
–
Shared facility
Build to suit
Condition of loading bay
Proximity to airport and major highways
Lease length
68
E. Moradi and M. Bidkhori
– Secretarial support
– Condition of building
Step 3. Data collection: This step requires the most time, but it is very important
and should be done carefully. Information on potential sites and locations was
obtained from sources such as these:
–
–
–
–
Chamber of Commerce
Economic Development Council
Real estate brokers
Facility owners
Step 4. Elimination of sites not meeting critical objectives and development of
a rating chart: From the ten sites for which data were collected, four did not
satisfy the first or the second critical requirement. For the remaining six sites,
machine shops and other support service providers as well as vendors were within
100 miles. The IME student group devised a chart showing the weights of the
objective and subjective factor.
Step 5. Site visits and site evaluation: The students visited all six sites. Data and
rated collected for the sites.
After careful evaluation, the six sites were rated on this evaluation; the Cohoes,
New York, site appears to be the best location, with Bennington, Vermont, as a
(close) second best location (Heragu 1997).
References
Drezner Z (1986) Location of regional facilities. Naval Res Logist Quart 33:523–529
Francis Rl, White JA (1974) Facility layout and location: An analytical approach. Prentice Hall,
Englewood Cliffs, NJ
Francis Rl, McGinnis LF, White JA (1992) Facility layout and location: An analytical approach.
Prentice Hall, Englewood Cliffs, NJ
Heragu SS (1997) Facilities design. PWS publishing company, a division of International Thomson
Publishing Inc. Boston
Uster H, Love RF (2000) The convergence of the Weiszfeld algorithm. Management
Science/Information Systems Area
Uster H, Love RF (2001) A generalization of the rectangular bounding method for continuous
location models. Comput Math Appl 44:181–191
Zaferanieh M, Taghizadeh KH, Brimberg J, Wesolowsky GO (2008) A BSSS algorithm for the single facility location problem in two regions with different norms. Eur J Oper Res 190(1):79–89
Chapter 4
Multifacility Location Problem
Farzaneh Daneshzand and Razieh Shoeleh
In the previous chapter, we studied the case of a single new facility to be located
relative to a number of existing facilities. In this chapter we consider the problem
of optimally locating more than one new facilities with respect to locations of a
number of existing facilities (demand points), the locations of which are known.
While the problems are natural extension of those of single facility location, there
are two important conditions:
1. At least two facilities are to be located
2. Each new facility is linked to at least one other new facility
If the first condition contracted, this problem is considered as a single facility location problem (SFLP) and if the second condition contracted, we can consider the
problem as some of independent single facility location problems. Thus the SFLP
can be considered as a spatial case of the multifacility location problem (MFLP).
4.1 Applications and Classifications
As it is expected, applications of MFLPs occur in the same contexts as discussed
in the chapter “Single Facility Location Problem” by Esmaeel Moradi and Morteza
Bidkhori, this volume for SFLP. Ostresh (1977) represented some applications of
the MFLP as follows:
1. A system of warehouses is to be established to serve a set of predetermined
regions.
2. Industrial and commercial establishments tend to be more concentrated than expected on the basis of minimizing transport costs alone.
3. In large organizations (such as the Federal government) face to face communication must take place between adjacent (usually) levels of the hierarchy.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 4, c Physica-Verlag Heidelberg 2009
69
70
F. Daneshzand and R. Shoeleh
A classification of different types of MFLP and their properties is shown in the
following statement:
Area solution: discrete, continual
The space in which facilities are located: planer location, sphere location
Objective function: MiniMax, MiniSum
Type of the distance: rectangular distance, Euclidean distance, squared Euclidean
distance, lp distance
Parameters: stochastic, deterministic
How facilities are assumed: point, region
Researchers have worked on a variety of MFLPs. However no research has been
conducted on many types of MFLPs resulted from multiplying the above items.
4.2 Models
In this section we introduce MFLP models represented and developed as mathematical models.
4.2.1 MiniSum
The MiniSum multifacility location problem consisting of finding locations of new
facilities which will minimize a total cost function consists of a sum of costs directly proportional to the distances between the new facilities, and costs directly
proportional to the distances between new and existing facilities.
The general-model for this problem can be stated as follows:
4.2.1.1 Model Assumptions
The assumptions of this model are as follows:
The area solution is continual
The space in which facilities are located is planer
The objective function is MiniSum
Type of the distance can be either rectangular, Euclidean, squared Euclidean or
lp distance
Parameters are deterministic
Facilities are assumed as points
4
Multifacility Location Problem
71
4.2.1.2 Model Inputs
Model inputs of this model are:
n: Number of new facilities
m: Number of existing facilities
wij : Nonnegative weight between new facility i and existing facility j by a unit
distance
vi k : Nonnegative weight between new facilities i and k by a unit distance
d.Xj ; Pi /: Distance between the location of new facility j and existing facility i
d.Xj ; Xk /: Distance between the location of new facilities j and k
Pj W .aj ; bj / The location coordinates of existing facility j
4.2.1.3 Model Output (Decision Variable)
Model output of this model is:
Xi W .xi ; yi / The location coordinates of new facility i
4.2.1.4 Objective Function
Min
X
vjk :d.Xj ; Xk / C
m
n X
X
wji :d.Xj ; Pi /:
(4.1)
j D1 i D1
1j <kn
Thus each of the m new facilities is to be located with respect to the n existing
facilities and also with respect to the other new facilities. The location of Xj may
depend on the location of some point Xk because of the terms involving vjk . For
convenience in the presentation, we assume all the wji and all the vj k are positive.
Notice that it is the cost proportional to the distance between new facilities which
distinguish the MFLP from SFLP. In fact, when terms vji are zero then (4.1) may be
written as
m
n X
X
Min
wji :d.Xj ; Pi /:
(4.2)
j D1 i D1
And (4.2) just defines a one-facility total cost expression. So that (4.2) is the sum of
n different one-facility cost expression and may be written as
Min
n
X
j D1
Min
m
X
i D1
!
wji :d.Xj ; Pi / :
(4.3)
Since the location of one new facility has no effect upon the cost of locating other
new facilities for the special case where all vij are zero, locations of new facilities
may be found by solving n SFLP independently.
72
F. Daneshzand and R. Shoeleh
4.2.1.5 Rectangular Distance MiniSum Location Problem
In this section, we consider MFLP when we have rectangular distance. The objective of this problem is minimization of the weighted sum of the rectangular distance
between the locations of new facilities and new and existing facilities. For formulation of this problem if in (4.1) we replace the d.Xj ; Pi / and d.Xj ; Xk / by (4.4)
and (4.5) The Rectangular MFLP problem can be formulated as (4.6) this problem
is of added interest since it will be shown that its optimal solution can be used to
compute both a lower and an upper bound for the value of the optimal solution to
the Euclidean problem.
ˇ
ˇ ˇ
ˇ
(4.4)
d.Xj ; Xk / D ˇxj xk ˇ C ˇyj yk ˇ ;
ˇ
ˇ ˇ
ˇ
(4.5)
d.Xj ; Pi / D ˇxj ai ˇ C ˇyj bi ˇ ;
X
ˇ ˇ
ˇ
ˇ
Min
vjk ˇxj xk ˇ C ˇyj yk ˇ
1j <kn
C
n X
m
X
j D1 i D1
ˇ ˇ
ˇ
ˇ
wji ˇxj ai ˇ C ˇyj bi ˇ :
(4.6)
The objective function is converted into two minimum problems:
Min f D Min f1 .x/ C Min f2 .x/:
where:
f1 .X / D
X
1j <kn
f2 .Y / D
X
1j <kn
m
n X
ˇ X
ˇ
ˇ
ˇ
vjk ˇxj xk ˇ C
wji ˇxj ai ˇ;
(4.7)
(4.8)
j D1 i D1
n X
m
ˇ
ˇ X
ˇ
ˇ
vjk ˇyj yk ˇ C
wji ˇyj bi ˇ:
(4.9)
j D1 i D1
Just like single facility location case in the chapter “Single Facility Location Problem” by Esmaeel Moradi and Morteza Bidkhori, this volume, optimum x coordinate
of new facilities can be found independent from optimum y coordinates. As you
see the objective function is nonlinear and we should make it linear. The method
of linearization is very similar to the method we used in single facility location in
the chapter “Single Facility Location Problem” by Esmaeel Moradi and Morteza
Bidkhori, this volume.
Linearization
Given numbers a, b, p and q,
If
a b p C q D 0;
p: q D 0;
p 0; q 0:
4
Multifacility Location Problem
73
Then
ja bj D p C q:
So minimizing the objective function is equivalent to the following problem:
Min
X
vjk .pjk C qjk / C
m
n X
X
wji .rji C sji /;
(4.10)
j D1 i D1
1j <kn
Subject to
xj xk pjk C qjk D 0
xj rji C sji D ai
pjk ; qjk 0
1 j < k n;
i D 1; : : : ; m; j D 1; : : : ; n;
1 j < k n;
(4.11)
(4.12)
(4.13)
rji ; sji 0
i D 1; : : : ; m; j D 1; : : : ; n;
(4.14)
pjk qjk D 0
1 j < k n;
(4.15)
rji :sji D 0
i D 1; : : : ; m; j D 1; : : : ; n;
(4.16)
xj unrestricted;
j D 1; 2; : : : ; n:
(4.17)
As in the single facility location for any basic feasible solution, if pjk is in the basic
feasible solution, qjk will not be and vice versa. Likewise, if rji is in the basic feasible
solution, sji will not be and vice versa. Since variables not in the basic feasible
solution are zero, the multiplicative constraints will be therefore satisfied for every
feasible solution. Minimizing f2 is the same as what was done for f1 .
4.2.1.6 Squared Euclidean Distance MiniSum Location Problem
In this section, we consider MFLP when we have squared Euclidean distance. The
objective of this problem is minimization of the weighted sum of the square of the
Euclidean distance between the locations of new facilities and new and existing
facilities. The problem considered in this section is also referred to as a “quadratic
facility location problem” and the “gravity problem”.
Although there aren’t too many situations where there are physical reasons for
using squared Euclidean distance, there are at least two reasons for the gravity
problem. First, in some cases the solution to the gravity problem can be used
to approximate the solution to the location problem where costs increase linearly
with Euclidean distance. Second, there exist location problems where costs increase
quadratically with the Euclidean distance between facilities.
For formulation of this problem if in (4.1) we replace the d.Xj , Pi / and
d.Xj ; Xk / by (4.18) and (4.19) the squared Euclidean MFLP problem can formulate
as (4.20)
d.Xj ; Xk / D .xj xk /2 C .yj yk /2 ;
(4.18)
74
F. Daneshzand and R. Shoeleh
d.Xj ; Xi / D .xj ai /2 C .yj bi /2 ;
X
Min f D
vjk : .xj xk /2 C .yj yk /2
(4.19)
1j <kn
C
n X
m
X
j D1 i D1
wji : .xj ai /2 C .yj bi /2 :
(4.20)
4.2.1.7 Contour Lines for Squared Euclidean MFLP
In MFLP construction of contour lines is possible expect for certain special case
when n D 2. In this section we want to observe the property of contour lines for
squared Euclidean MFLP
Let f .x; y/ given in (4.20) be written as f .x; y/ D f .x/ C f .y/ where
f .x/ D
X
vjk .xj xk /2 C
X
vjk .yj yk /2 C
wji .xj ai /2 :
(4.21)
wji .yj bi /2 :
(4.22)
j D1 i D1
1j <kn
And
f .y/ D
n X
m
X
n X
m
X
j D1 i D1
1j <kn
If n D 2, contour lines for f .x/ are concentric ellipses centered on (x ; y / and
contour lines for f .y/ are concentric ellipses centered on (x ; y /
Because of the squared Euclidean MFLP is separable and symmetric in x and y,
only f .x/ is considered and contour line is defined as the set of all points (x1 ; x2 /
for which
k D v12 .x1 x2 / C
m
X
w1i .x1 ai /2 C
i D1
m
X
w1i .x2 ai /2 :
(4.23)
i D1
And k is a constant denoting the value of the contour line. By expanding and collecting terms it is seen that (4.23) can be expressed in the form of a general conic
section:
(4.24)
Ax12 C Bx1 x2 C C x22 C Dx1 C Ex2 C F D 0;
where
A D v12 C
m
X
w1i ;
i D1
B D 2v12 ;
m
X
C D v12 C
w2i ;
i D1
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Multifacility Location Problem
75
D D 2
E D 2
m
X
i D1
m
X
w1i ai ;
w2i ai ;
i D1
DD
m
X
w1i ai2 C
i D1
m
X
w2i ai2 k:
i D1
A sufficient condition given by Thomas (1968) for (4.24) to be an ellipse is for the
discriminate, B 2 4AC , to be negative. Direct substitution gives
2
B 4AC D
4v212 4v212 4v12
m
X
i D1
w1i C
m
X
!
w2i 4
i D1
m
X
w1i
i D1
!
m
X
i D1
w2i
!
< 0:
(4.25)
Assuming the problem is well formulated (4.24) is the equation for a rotated ellipse
as noted by the presence of the x1 x2 term. Due to symmetry a similar result is found
for the y variables. Furthermore, by definition of a contour line since k achieves its
smallest value at (x1 ; x2 / the contour line is centered on the optimum location with
respect to the nonrotated axes (Eyster and White 1973).
4.2.1.8 Euclidean Distance MiniSum Location Problem
The Euclidean multifacility location problem often assumes that the transportation
costs from the new facility to a demand point are proportional to the Euclidean
distance between these points, with the factor of proportionality (weight) depending
on the demand point.
Here, this problem is mathematically formulated:
Min
P
1j <kn
C
m
n P
P
j D1 i D1
1=
vjk : .xj xk /2 C .yj yk /2 2
2
wji : .xj ai / C .yj bi /
1
2 =2
(4.26)
:
It is interesting to note that when all the existing facility locations are collinear, that
is, all lie on a single line, the Euclidean model essentially includes the rectilinear
distance model.
4.2.2 MiniMax
Minimization of a sum of weighted distances that was introduced in Sect. 4.1.1 may
not be a proper goal when the facilities to be located must provide services of an
76
F. Daneshzand and R. Shoeleh
urgent nature. Therefore in this section we introduce some multifacility location
problems with MiniMax objective function and by different type of distances.
4.2.2.1 Model Assumptions
Model assumptions of this model are as follows:
The area solution is continual
The space in which facilities are located is planer
The objective function is MiniMax
Type of the distance can be either rectangular, Euclidean, squared Euclidean or
lp distance
Parameters are deterministic
Facilities are assumed as points
4.2.2.2 Model Inputs
We have all the inputs in previous model.
4.2.2.3 Model Outputs (Decision Variables)
The outputs of this model are similar to the previous model.
4.2.2.4 Objective Function
Min f D Maxf vjk :d.Xj ; Xk /; wji :d.Xj ; Pi /g:
(4.27)
4.2.2.5 Rectangular Distance MiniMax Location Problem
The problem considered is that of locating n new facilities among m existing facilities with the objective of minimizing the maximum weighed Rectangular distance
among all facilities.
ˇ
ˇ ˇ
˚ ˇ
Min f D Max wji ˇxj ai ˇ C ˇyj bi ˇ j D 1; : : : ; n; i D 1; : : : ; mI
ˇ ˇ
ˇ
ˇ
(4.28)
vjk ˇxj xk ˇ C ˇyj yk ˇ 1 j < k < n :
4
Multifacility Location Problem
77
4.2.2.6 Euclidean Distance MiniMax Location Problem
The problem considered is that of locating n new facilities among m existing facilities with the objective of minimizing the maximum weighed Euclidean distance
among all facilities.
Elzinga et al. formulated this problem in 1976:
˚
Min f D Max wji Œ.xj ai /2 C .yj bi /2 1=2 j D 1; : : : ; n; i D 1; : : : ; mI
vjk Œ.xj xk /2 C .yj yk /2 1=2 1 j < k < n :
(4.29)
4.2.3 Other Models
4.2.3.1 Rectangular Multi Product Multifacility Location Problem
This model considers multi products in MFLP. Sherali and Shetty (1978) formulated
the rectangular multi product MFLP.
4.2.3.2 Model Assumptions
Model assumptions of this model are as follows:
The area solution is continual
The space in which facilities are located is planer
The objective function is MiniSum
The distances are rectangular
Parameters are deterministic
Facilities are assumed as points
4.2.3.3 Model Inputs and Outputs (Decision Variables)
Model inputs and outputs of this model are as follows:
cijk : The cost per unit of product k to be transported from a new facility i to a
new or existing facility j
uijk : The known amount of product k to be transported from a new facility i to a
new or existing facility j at a cost of cijk per unit of the product, per unit distance.
(xj ; yi /: decision variables for i D 1; 2; : : :, n and are fixed and known for each
facility n C i , i D 1; 2; : : :, m’.
78
F. Daneshzand and R. Shoeleh
4.2.3.4 Objective Function
The objective function of this model and its related constraints are as follows:
0
Min
p
n nCm
X
X
X
kD1 i D1 j D1
ˇ ˇ
ˇ
˚ˇ
cijk uijk ˇxi xj ˇ C ˇyi yj ˇ :
(4.30)
The objective function may be written as:
Min
p
n
X
X
kD1 i D1
wij D
p
X
ˇ ˇ
ˇ
˚ˇ
wij ˇxi xj ˇ C ˇyi yj ˇ ;
.cijk uijk C cjik ujik /
j D n C 1; : : : ; n C m0 :
(4.31)
(4.32)
kD1
4.2.3.5 Multifacility Location Problem on Sphere
As it is known, one of the assumptions when locating facilities is concerned with
the size of the area covering the destinations (or the demand points). If the area
covering the demand points is sufficiently small, then this part of the earth’s surface
can be approximated by a plane. When the destination points are widely separated,
the area covering these points can no longer be approximated by a plane and the
formulations we discussed so far is not suitable.
Problems concerning location of international headquarters, distribution/
marketing centers, detection station placement, and placement of radio transmitters
for long range communication may fall into this category.
The location problem on a sphere is more complicated than its counterpart on the
plane, because unlike on plane the solution space is not convex on sphere.
Dhar and Rao (1982) formulated this problem. Any point on the sphere can be
defined by its latitude =2 ˚ =2 and longitude and dij is the
shortest distance between points i and j .
4.2.3.6 Model Assumptions
Model assumptions of this model are as follows:
The area solution is continual
The space in which facilities are located is on sphere
The objective function is MiniSum
Parameters are deterministic
Facilities are assumed as points
The distance is defined as
dij D Arc cosŒcos ˚i cos ˚j cos.i j / C sin ˚i sin ˚j :
(4.33)
4
Multifacility Location Problem
79
4.2.3.7 Model Inputs
We have all the inputs in MiniSum models.
4.2.3.8 Model Outputs (Decision Variables)
The outputs of this model are similar to the MiniSum models.
4.2.3.9 Objective Function
The objective function of this model and its constraints are as follows:
Min˚j j
X
vjk .djk / C
1j <kn
n X
m
X
wji .dji /:
(4.34)
j D1 i D1
Subject to
=2 ˚j =2;
(4.35)
:
(4.36)
4.2.3.10 Multifacility Location Problem with Rectangular Regions
So far we assumed each existing and new facility as a point. Wesolowsky and Love
applied the concept of Rectangular regions to MFLP in 1971a, b. They stated that in
many different contexts it is proper to treat the destination to be served as a region,
rather than a point such as in the location of a library or emergency services or
other public service facility designed to serve either a neighborhood or a densely
populated urban area.
Even when the users of the new facility are discretely distributed, the number of
users may become so large that it may be infeasible in terms of data collection and
computational efficiency to represent each customer by a point. In this situation it is
necessary to assume regional destinations in order to solve the problem.
4.2.3.11 Model Assumptions
Model assumptions of this model are as follows:
The area solution is continual
The space in which facilities are located is planer
The objective function is MiniSum
The distances are rectangular
80
F. Daneshzand and R. Shoeleh
Parameters are deterministic
Facilities are assumed as regions
4.2.3.12 Model Inputs
We have all
in MiniSum models in addition to the following inputs:
the inputs
Ri D ai1 ; ai2 Rectangular region i , where ai1 < ai2 and bi1 < bi2
Ai : The area of Ri
4.2.3.13 Model Outputs (Decision Variables)
The outputs of this model are similar to MiniSum models.
4.2.3.14 Objective Function
The objective function of this model and its constraints are as follows:
Min
Z Z
m
n X
X
ˇ ˇ
ˇ
ˇ
wij
ˇxj ai ˇ C ˇyj bi ˇ dai dbi
Ai
j D1 i D1
Ri
C
X
i j <kn
ˇ ˇ
ˇ
ˇ
vjk ˇxj xk ˇ C ˇyj yk ˇ :
(4.37)
4.2.3.15 Stochastic Multifacility Location Problem
In many cases the parameters of the model are not deterministic. This model is
represented and solved by Seppalla (1975). In order to define a stochastic decision
model, we should first determine the stochastic parameters, how they are distributed,
whether they are correlated, and which decision criterion will be used in industrial
applications. The set of stochastic elements of planning models are often restricted
to consist only of demands for products, while other parameters or variables, such
as unit costs, capacities and locations, are considered to be fixed.
4.2.3.16 Model Assumptions
Model assumptions of this model are as follows:
The area solution is continual
The space in which facilities are located is planer
The objective function is MiniSum
The distances are Euclidean
4
Multifacility Location Problem
81
Parameters are stochastic
Facilities are assumed as points
The weights vjk , wji for all i; j and k are normally distributed random variables
4.2.3.17 Model Inputs
The inputs of this model are similar to the MiniSum model in addition to the following:
˛: is a predetermined probability
ı: is an assisting
P f.g: is a probability operator
4.2.3.18 Model Outputs (Decision Variables)
The output of this model is similar to the MiniSum model.
4.2.3.19 Objective Function
The objective function of this model and its constraints are as follows:
Min ı:
(4.38)
Subject to
P
8
< X
:
1j <kn
C
1=2
vjk : .xj xk /2 C .yj yk /2
n X
m
X
j D1 i D1
9
=
1=2
˛:
ı
wji : .xj ai /2 C .yj bi /2
;
(4.39)
4.3 Solution Techniques
4.3.1 MiniSum
4.3.1.1 Rectangular Distance MiniSum Location Problem
The rectangular distance multifacility location problem always has a minimum cost
solution where the x coordinate of each new facility is equal to the x coordinate of
some existing facility and the same is true for y coordinate (Francis et al. 1992).
82
F. Daneshzand and R. Shoeleh
If we call p1 the linear programming problem obtained in (4.10) a straight forward linear programming of p1 can be time consuming when a large number of new
and existing facilities are involved.
Depending on the characteristics of a particular multifacility Rectangular location problem, the optimum solution can be sometimes obtained in an iterative mode
by solving some single facility location problems. Also, a dual formulation of the
linear programming problem (p1 ) can provide more efficient solution to the rectangular MFLP (Francis et al. 1992).
Francis (1964) solved a special case of the MFLP, with rectilinear distance when
the weights are equal.
Cabot et al. (1970) decomposed the location problem into two independent subproblems, each of which is equivalent to a linear programming problem which
is essentially the dual of a minimal cost network flow problem. The dual variables in each of the optimum tableaus to the two flow problems give the x and y
coordinates respectively of the optimum locations of the new facilities.
Pritsker and Ghare (1970) suggested a gradient technique for this problem. The
basic contribution of them was a derivation of necessary conditions for an optimal
solution and an algorithm for obtaining optimal solutions to the decomposed
problems.
Rao (1973) considered a direct search approach to the RMFLP in detail, however
he demonstrated that the gradient technique was basically a primal simplex-based
linear programming approach, and in the presence of degeneracy, the optimality
conditions were not sufficient. A necessary condition for optimality to be sufficient in special cases and the main difficulties associated with the direct search
approach were discussed.
Wesolowsky and Love (1971a, b) and Morris (1975) showed that the problem
with linear locational constraints could be solved by linear programming.
A thorough set of necessary and sufficient optimality conditions were developed
by Juel and Love (1976).
Idrissi et al. (1989) developed a dual problem for the constrained multifacility
minisum location problems involving mixed norms. General optimality conditions were obtained providing new algorithms which are decomposition methods
based on the concept of partial inverse of a multifunction.
A nonlinear approximation method was developed by Wesolowsky and Love
(1972), where any number of linear and (or) nonlinear constraints defining a
convex feasible region can be included.
The hyperboloid approximation procedure for solving the perturbed rectilinear
distance MFLP was also proposed by Eyster et al. (1973).
Morris (1975) used the dual problem of Rectangular MFLP and then reduced it to
a problem with substantially fewer variables and constraints. He stated that linear
and pairwise constraints limiting distances between new points and between new
and existing points can be imposed to restrict the location of new points.
Picard and Ratliff (1978) solved the problem via at most (m 1) minimum cut
problems on derived networks containing at most (n 2) vertices. They showed
4
Multifacility Location Problem
83
that the optimum location of new facilities is dependent on the relative orderings
of old facilities but not on the distances between them.
Subsequently, Kolen (1981) exhibited the equivalence of the method of Sherali
and Shetty and Picard and Ratliff. The main difference between these two procedures was principally in the computational implementation. Moreover, this type
of approach is known to be the most effective way of solving the rectilinear distance MFLP.
A modified version of the method of Picard and Ratliff (1978) was proposed by
Cheung (1980).
Dax (1986) presented a new method that, as he stated, handles efficiently the
rectilinear distance Problem MFLP having large clusters, i.e. where several new
facilities are located together at one point. This paper states and proves a new necessary and sufficient optimality condition. This condition transforms the problem
of computing a descent direction into a constrained linear least-squares problem. The latter problem is solved by a relaxation method that takes advantage
of its special structure. The new technique is incorporated into the direct search
method.
As an alternative to linear programming, a simple approach which sometimes
finds optimal locations was presented by (Francis et al. 1992) as coordinate descent. By deleting the term that shows the relationship between new facilities in
objective function, the problem is converted to some single facility location problems to which we can apply median conditions. The first coordinate we choose
is the first variable and the second coordinate is the second variable, and so on. It
is continued until we obtain the same vector by coordinate descent that we have
obtained previously by coordinate descent, at which point we stop.
Allen (1995) developed a dual-based lower bound to the multifacility `p distance
location problem and he stated that the bound is as good or better than other
bounds.
4.3.1.2 Squared Euclidean Distance MiniSum Location Problem
In this problem, it is obvious that the function that is to be minimized is strictly convex, and unlike the Euclidean distance case, it has continuous first partial derivatives
with respect to x and y.
Consequently the optimal solution is unique and the approach to finding optimal
solution is the same for the SFLP; partial derivation of (4.20) with respect to each
variable are computed and set to equal to zero.
The result of the partial derivation computation is two sets of line equation on
involving the x coordinate (and y coordinate) of the new facilities.
To compute the partial derivations, it is convenient to define a new quantity vO i k
where
vij k > j
:
(4.40)
vO ik D
vkj k j
84
F. Daneshzand and R. Shoeleh
Computing the partial derivation of (4.21) with respect to xi and (4.22) with respect
to yi gives, for j D 1; : : :; n,
n
m
X
X
@f
D2
vO ik .xj xk / C 2
wji .xj ai /:
@xj
i D1
(4.41)
kD1
And
n
m
X
X
@f
D2
vO ik .yj yk / C 2
wji .yj bi /:
@yj
i D1
(4.42)
kD1
If we set the (4.41) and (4.42) to zero and solve it, the optimum values of the x and
y coordinates for the new facilities are related by the following expressions:
xj D
n
P
vO ik xk C
kD1
n
P
vO ik C
yj D
n
P
vO ik yk C
kD1
n
P
wji ai
i D1
m
P
:
(4.43)
:
(4.44)
wji
i D1
kD1
And
m
P
vO ik C
m
P
wji bi
i D1
m
P
wji
i D1
kD1
Furthermore, White (1971) established that the optimum solution to the multifacility
problem is given by
x D A1 W a:
(4.45)
and
y D A1 W b;
(4.46)
where x and y are n 1 column vectors giving the optimum coordinate locations
for the new facilities, a and b are m1 column vectors giving the x and y coordinate
locations, respectively, for the existing facilities W is an n m matrix containing
the weights wij , and A is an n n nonsingular matrix given as follows:
2
n
P
vO 1k C
P
6 kD1
i
6
6
6
O
v
21
6
AD6
6
:
6
6
:
6
4
Ovn1
Ov12
w1i
n
P
vO 2k C
P
3
Ov1n
::::
Ov2n
w2i : : :
i
kD1
:
:
Ovn2
:
:
::::
:
:
n
P
kD1
vO nk C
P
i
wni
7
7
7
7
7
7:
7
7
7
7
5
(4.47)
4
Multifacility Location Problem
85
The matrix A is strictly diagonally dominant, allowing (4.45) and (4.46) to be solved
using the method of simultaneous displacements, an iterative solution procedure
based on (4.43) and (4.44) (Eyster and White 1973).
It is interesting to note that the solution of the squared Euclidean distance problem has been used to obtain a good starting solution for the corresponding Euclidean
MFLP (Francis et al. 1992).
4.3.1.3 Euclidean Distance MiniSum Location Problem
The objective function of Euclidean MFLP is convex, since it is the sum of norms
that are convex functions and Francis and Cabot (1972) have proven that a necessary and sufficient condition for the objective function to be strictly convex is that
for each new facility i , the set Si D fj W wij > 0g is nonempty and that the location of the points in Si are non-collinear. As it is known, the optimal solution of
Euclidean MFLP problem exists and lies in the convex hull of the existing facilities,
and therefore, this optimal solution can be expressed as the convex combination of
the existing facilities (Francis et al. 1983).
In the previous treatment of the rectilinear distance problem and the squared
Euclidean distance problem; we found that multifacility problems were not substantially more difficult to solve than the corresponding single facility version.
Such is not the case for Euclidean distance problem. The main difficulty with the
Euclidean SFLP arises because its objective function is not differentiable at the
points a1 ; : : :; an , For Euclidean MFLP the function f is nondifferentiable not only
at a set of isolated points, but also on linear subspaces Xi D Xk .
Eyster et al. (1973) used an extension of the Weiszfeld algorithm. In this procedure, in order to avoid difficulties of the partial derivatives of the distance function
not existing at points Pi and at points where other new facilities have been located
d.Xj ; Xk / D Œ.xj xk /2 C .yj yk /2 C "1=2 ;
(4.48)
2
(4.49)
2
1=2
d.Xj ; Xi / D Œ.xj ai / C .yj bi / C "
:
This procedure is labeled the hyperboloid approximation procedure (HAP) and is
probably the most common procedure for solving the multifacility location problem, using Euclidean distances or even rectilinear distances. Rosen and Xue (1993)
proved the global convergence of HAP when applied to the problems MFLP. Hap
comes from computing partial derivatives of the function f , setting them to zero,
and solving for new facility locations.
To simplify, we define terms ˛t .X1 ; : : :; Xn /; ˇt .X1 ; : : :; Xn /; t .X1 ; : : :; Xn /,
and for t D 1; : : :; n as follows:
˛t .X1 ; : : : ; Xn / D
n
X
j D1
C
m
X
i D1
wt i
vtj
xj
Œ.xt xj /2 C .yt yj /2 C "1=2
ai
Œ.xt ai /2 C .yt bi /2 C "1=2
t D 1; : : : ; n;
(4.50)
86
F. Daneshzand and R. Shoeleh
ˇt .X1 ; : : : ; Xn / D
n
X
vtj
j D1
C
m
X
i D1
wt i
Œ.xt ai
t .X1 ; : : : ; Xn /
D
/2
n
X
j D1
C
m
X
i D1
Œ.xt xj
/2
yj
C .yt yj /2 C "1=2
bi
C .yt bi /2 C "1=2
Œ.xt ai
(4.51)
vij
Œ.xt xj
/2
1
C .yt yj /2 C " =2
wt i
/2
t D 1; : : : ; n;
1
C .yt bi /2 C " =2
t D 1; : : : ; n:
(4.52)
Then it may be shown that the partial derivatives of f with respect to xt and yt is
as follows:
@f
D
@xt
t .X1 ; : : : ; Xn /xt
˛t .X1 ; : : : ; Xn /
t D 1; : : : ; n;
(4.53)
@f
D
@yt
t .X1 ; : : : ; Xn /yt
ˇt .X1 ; : : : ; Xn /
t D 1; : : : ; n:
(4.54)
and
By setting the partial derivatives to zero for x1 and y1 we have:
xt D
1
˛t .X1 ; : : : ; Xn /
t .X1 ; : : : ; Xn /
t D 1; : : : ; n:
(4.55)
yt D
1
ˇt .X1 ; : : : ; Xn /
.X
;
t
1 : : : ; Xn /
t D 1; : : : ; n:
(4.56)
And
The iteration procedure determines x kC1 and y kC1 in terms of x k and y k using the
following:
xtkC1 D
ytkC1 D
1
˛t .X1k ; : : : ; Xnk /
k
k
t .X1 ; : : : ; Xn /
1
k
k
t .X1 ; : : : ; Xn /
ˇt .X1k ; : : : ; Xnk /
t D 1; : : : ; n;
(4.57)
t D 1; : : : ; n:
(4.58)
.0/
.0/
By making an initial choice of new facility locations, say ˛t .X1 ; : : :; Xn /, use the
algorithm to compute new answers for new facility locations. In using HAP to solve
multifacility location problems, it has been observed that the larger the value of ©
the faster the convergences to optimum value of approximating function. However,
the accuracy of approximation decrease with increasing values of " consequently, in
4
Multifacility Location Problem
87
solving location problem using HAP a large value of " is used initially; the solution
obtained used as starting solution, using a smaller value of "; and the process is
continued by successively reducing the value of " until no signification decrease in
the value of either (xi ; xj /.interestingly, HAP can be used to solve the rectangular
location problem and can also be used to handle situation involving a mixture of
recliner and Euclidean distance (Eyster et al. (1973).
4.3.1.4 Some other Kinds of Algorithms in Euclidean MFLP
Eyster et al. (1973) used an extension of the Weiszfeld algorithm.
Calamai and Conn (1980) have proposed a pseudo-gradient technique that classifies the new facilities into distinct categories based on their coincidence with
other facilities in order to derive a descent method for solving Euclidean MFLP.
Chatelon (1978) have also approached Euclidean MFLP by using a general
e-subgradient method in which search directions are generated based on the subdifferential of the objective function over a neighborhood of the current iterate.
Sequential unconstrained minimization techniques used by Love (1969) and the
Weiszfeld fixed-point iterative method as utilized by Rado (1988), are also among
other efforts to solve Euclidean MFLP.
Several second-order methods have also been designed to solve the Euclidean
MFLP. Calamai and Conn (1980) were the first to propose a projected gradientbased algorithm.
Various quadratic convergence approaches have also been developed by Calamai
and Conn (1982, 1987), in which specialized line-searches are used in conjunction with projected second-order techniques.
Rosen and Xue (1992) developed an algorithm which, from any initial point,
generates a sequence of points that converges to the closed convex set of optimal
solutions to the Problem Euclidean MFLP.
For the multifacility location problem with no constraints on the location of the
new facilities, Juel and Love (1980) derived some sufficient conditions for the
coincidence of facilities that are valid in a general symmetric metric.
The results of Juel and Love (1980) were later extended by Lefebvere et al.
(1991) to be applicable to some location problems having certain locational constraints.
Mazzerella and Pesamosca (1996) have used the optimality conditions of
Euclidean MFLP as a tool for obtaining both stopping rules for some computational algorithms such as the projected Newton procedure of Calamai and
Conn (1987), and the analytical solution of many simple problems.
Love (1969) applied convex programming to the problem in three dimensions.
Carrizosa et al. (1993) derived the geometrical characterizations for the set of
efficient, weakly efficient and properly efficient solutions of the Euclidean MFLP
when it includes certain convex locational constraints.
Love and Yoeng (1981), Elzinga and Hearn (1983), Juel (1984), and Love and
Dowling (1989) explored the bounding method that continuously updates a lower
88
F. Daneshzand and R. Shoeleh
bound on the optimal objective function value during each iteration. This method
is based on the idea that the convex hull and the current value of the gradient
determine an upper bound on the objective function’s improvement.
Wendell and Petersen (1984) have derived a lower bound from the dual to
Euclidean MFLP.
Love (1974) developed the dual problem corresponding to a hyperbolic approximation of the constrained multifacility location problem with lp distances.
White (1976) gave a Varignon frame interpretation of the dual problem.
Sinha (1966) have used duality results involving general quadratic forms.
Francis (1972) derived a differentiable, convex quadratically constrained dual
optimization problem, and achieved several useful relationships between the dual
and Euclidean MFLP.
Xue et al. (1996) have suggested the use of polynomial-time interior point algorithm to solve this dual problem based on this idea, they presented a procedure
in which an approximate optimum to Euclidean MFLP can be recovered by solving a sequence of linear equations, each associated with an iterate of the interior
point algorithm used to solve the dual problem.
Love and Kraemer (1973) gave a dual decomposition method for solving the
constrained Euclidean MFLP.
Love (1974) developed the dual problem corresponding to a hyperbolic approximation of the objective function for the constrained MFLP with lp distance.
4.3.2 MiniMax
4.3.2.1 Rectangular Distance MiniMax Location Problem
Some procedures for solving rectangular MiniMax MFLP are shown here:
Wesolowsky (1972) converted the rectangular MiniMax MFLP into a parametric
linear programming problem with 5mn C 5=2n.n 1/ constraints and 2mn C
n.n 1/ C 2n variables in addition to the parameters.
Elzinga and Hearn (1973) recognized some simplifications in linear program presented by Wesolowsky.
Dearing and Francis (1974) showed that the problem can be decomposed into two
sub problems that have identical structures and that may be solved independently
each of which had 2mn C n.n 1/ constraints and n C 1 variables. Each problem
is solved efficiently by converting it into an equivalent network flow problem.
Morris (1973) has introduced this problem with linear constraints which (a) limit
the new facilities location and (b) enforce upper bounds on the distances between
new and existing facilities and between new facilities. He uses dual variables that
provide information about the complete range of new facility locations which
satisfies the MiniMax criterion.
4
Multifacility Location Problem
89
Drezner and Wesolowsky (1978) presented a method involved the numerical integration of ordinary differential equations and was computationally superior to
methods using nonlinear programming.
4.3.2.2 Euclidean Distance MiniMax Location Problem
The application of nonlinear duality theory shows Euclidean minimax MFLP can
always be solved by maximizing a continuously differentiable concave objective
subject to a small number of linear constraints. This leads to a solution procedure
which produces very good numerical results. Love et al. (1973) presented a nonlinear programming method for computing the solution to MFLPs using Euclidean
distances when the MiniMax criterion is to be satisfied.
4.3.3 Solution Techniques for other Models
In this section we introduce some solution techniques for the models shown in
Sect. 1.3
A specialized simplex based-algorithm was derived by Sherali and Shetty (1978)
for solving rectangular multiproduct MFLP.
Dhar and Rao (1982) presented an iterative solution for MFLP on sphere.
The procedure involved the approximation of the domain of objective function
which in the limit approaches to that of the original objective function. Aykin
and Babu (1987) considered Euclidean, squared Euclidean and the great circle
distances. They formulated an algorithm and investigated its convergence properties.
Wesolowsky and Love (1971a, b) considered the problem of MFLP with rectangular regions for the cases n D 1 or 2. They used a simple gradient reduction
technique to solve the single facility problem. As they stated, the procedure becomes very complex when n > 2. Aly and Marucheck (1982) stated that if
there is interfacility interaction among the new facilities, a gradient-free nonlinear search algorithm is utilized. Computational experience suggests that this
algorithm is expedient even in the solution of large problems.
Seppalla (1975) stated that in order to be able to solve the stochastic MFLP, we
must transform it into the deterministic equivalent forms.
4.3.4 Some Heuristic and Metaheuristic Methods
There are a few heuristics methods for solving MFLP:
Vergin and Rogers (1967) introduced a simple heuristic for solving MFLP with
Euclidean distance. This procedure locates each of new facilities in a temporary
90
F. Daneshzand and R. Shoeleh
location at each step and locates the next new facility according to the facilities
located so far. After all n new facilities are located in this manner the process is
repeated and the readjustment process is continued until no further movements
occur during a complete round of adjustment evaluations.
Davoud Pour and Nosraty (2006) solved the MFLP with ant-colony optimization
metaheuristic when the distances are rectangular and Euclidian. This algorithm
produces optimal solutions for problem instances of up to 20 new facilities.
4.4 Case Study
Smallwood (1965) introduced a model for the placement of n detection stations so
as to maximize the probability that at least one of them will detect any enemy event
occurring within the area. This research was done within boundaries of USA and
USSR. The model is based on five assumptions that one of them is the assumption
of plane area that was made in order to allow the use of the relatively convenient
Cartesian coordinate system. Each of these assumptions can be relaxed to reduce
the error of the model.
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Chapter 5
Location Allocation Problem
Zeinab Azarmand and Ensiyeh Neishabouri Jami
Location-allocation (LA) problem is to locate a set of new facilities such that the
transportation cost from facilities to customers is minimized and an optimal number
of facilities have to be placed in an area of interest in order to satisfy the customer
demand.
This problem occurs in many practical settings where facilities provide homogeneous services such as the determination and location of warehouses, distribution
centers, communication centers and production facilities.
Since LA problem was proposed by Cooper (1963) and spread to a weighted
network by Hakimi (1964), network LA problem and many models were presented
by Badri (1999).
For solving these models, numerous algorithms have been designed, involving branch-and-bound algorithms (Kuenne and Soland 1972), simulated annealing
(Murray and Church 1996) and Tabu search (Brimberg and Mladenovic 1996;
Ohlemüller 1997) and P-Median plus Weber (Hansen et al. 1998). Some hybrid
algorithms have been also suggested, such as the one based on simulated annealing
and random descent method (Ernst and Krishnamoorthy 1999) and the one utilizing the Lagrange relaxation method and genetic algorithm (Gong et al. 1997).
Brimberg et al. (2000) improved present algorithms and proposed variable neighborhood search, which is proved to obtain the best results when the number of
facilities to locate is large.
5.1 Classification of Location-Allocation Problem
The basic components of location-allocation problems can be thought to consist
of facilities, locations, and customers. The definitions and properties of these basic components will be discussed in this part along with some different types of
location-allocation models. The presentation provided in this part has been influenced by the paper of Scaparra and Scutellà (2001) which proposes a unified
framework for characterizing the different aspects of location problems.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 5, c Physica-Verlag Heidelberg 2009
93
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Z. Azarmand and E.N. Jami
5.1.1 Classifications of Facilities
The facilities are usually characterized by, among other things, their number, type,
and costs. Other facility-connected properties can include for instance profit, capacity, attraction range (within which the customers are drawn to the facility), and the
type of service provided.
One of the properties characterizing is the number of new facilities, the simplest
case is the single-facility problem in which only one new facility is to be established.
The more general case is called the multi-facility problem in which the aim is to
locate simultaneously more than one facility.
The type of a facility is another important property, in the simplest case, all the
facilities are supposed to be identical with respect to their size and the kind of service
they offer. However, it is often necessary to locate facilities that differ from one
another, for instance hospitals and smaller health care units. Location-allocation
models can also be differentiated into single-service and multi-service types, based
on whether the facilities can provide only one or many services.
It can also be taken into consideration whether the facilities can supply an infinite
demand or whether their production and supply capacity is limited. In this respect,
the problems are often classified into uncapacitated and capacitated.
5.1.2 Classified on the Physical Space or Locations
The set of eligible locations has three possible representations: discrete, continuous,
and network.
Continuous space models are sometimes referred to as site-generation models
since the generation of appropriate sites is left to the model at hand.
Discrete space models are sometimes referred to as site-selection models since
we have a priori knowledge of the site candidates.
The network-based model is the third type of location models that can be distinguished with respect to the locations. Problems defined on networks can be either
continuous or discrete depending on whether the links of the network are considered
as a continuous set of candidate locations or whether only the nodes are eligible for
the placement of new facilities.
5.1.3 Classifications of the Demand
The demands of customer are deterministic or probabilistic:
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5.2 Models
The models of location-allocation problem are divided into two main parts: the general models which are discussed in Sect. 5.2.1 and developed models which are
discussed in Sects. 5.2.2 and 5.2.4.
For modeling these problems we use operation research, to minimize the total
cost. The LA model has some variables as follows:
Facilities’ number
Facilities’ location
Amount of allocation of the facilities to customers
Capacity of each facility
5.2.1 General LA Model (Cooper 1963)
Cooper (1963) introduced and solved a general model of LA with two new facilities
and seven demand’s points for the first time.
5.2.1.1 Model Assumptions
Model assumptions are as follows (The problem is also known as the multisource
Weber problem):
The solution space is continuous
Each customer’s demand can supply by several facilities ignoring the opening
cost of new facility
Facilities are uncapacitated
Parameters are deterministic supplying all the demand
No relationship between new facilities
5.2.1.2 Model Inputs
Model inputs are as follows:
n: number of customer (existing facilities)
r: customer’s demand j D 1; 2 : : : ; n
aj : the coordinates of the customers j D 1; 2 : : : ; n
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5.2.1.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
': Total cost for transportation the goodsnservices between facilities and customer
m: Number of facilities
xi : The coordinates of the new facilities i D 1; 2 : : : ; m
wij : Quantity supplied to customer j by facility i
d.xi ; aj /: The distance between a customer j and new facility i
5.2.1.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min ' D
n
m X
X
wij d.xi ; aj /:
(5.1)
wij D rj ; j D 1; 2; : : : n;
(5.2)
i D1 j D1
Subject to
m
X
i D1
wij 0;
i D 1; 2; : : : ; m
:
j D 1; 2; : : : ; n
(5.3)
Equation (5.1) minimizes the total transportation cost. Equation (5.2) ensures that
all customer demand is satisfied. Because there are no capacity constraints on the
facilities, an optimal solution will have the demand at each customer served by the
facility that is closest to it. Equation (5.3) is standard constraints.
As you see there are some assumptions in the general model, if we change some
of these, then the model will be converted to a developed LA model. We can see the
importance of modeling in such problems.
5.2.2 LA Model Each Customer Covered by Only One Facility
5.2.2.1 Model Assumptions
In this model the second assumption in general model changes so that, each customer can use only one facility. The other assumptions of this model are similar to
the general model.
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5.2.2.2 Model Inputs
The inputs of this model are similar to the general model.
5.2.2.3 Model Outputs (Decision Variables)
We have all the outputs in general model in addition to the following output:
zij D 1 if an customer j is assigned to a new facility i ;
0 otherwise;
And wij D demand of each customer so we don’t have wij in this model
5.2.2.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min D
m X
n
X
Zij rj d.xi ; aj /:
(5.4)
i D1 j D1
Subject to
m
X
Zij D 1; j D 1; 2; : : : n;
(5.5)
i D1
Zij 2 f0; 1g;
i D 1; 2; : : : ; m
:
j D 1; 2; : : : ; n
(5.6)
Equation (5.4) minimizes the total transportation cost. Equation (5.5) ensures that
all customer demand is satisfied. Equation (5.6) is standard constraints
In this form, the LA problem may involve the determination of z, the allocation
matrix.
5.2.3 LA Model with Facility’s Opening Cost
5.2.3.1 Model Assumptions
In this model, the third assumption in general model changes to, spot the facility’s
opening cost. The other assumptions of this model are similar to the general model.
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5.2.3.2 Model Inputs
We have all the inputs in general model in addition to the following input:
f .xi / D facility’s opening cost.
5.2.3.3 Model Outputs (Decision Variables)
The outputs of this model are similar to the general model.
5.2.3.4 Objective Function and its Constraints
The differences between the objective function of this model and general model are
as follows:
Equation (5.1) is eliminated and the following term is added to the objective
function:
m
X
f .xi /:
(5.7)
i D1
The constraints of this model are similar to the general model.
If we assume that the opening cost is independent of facility location and m is
known, the problem reduces to the well-known multi source Weber problem since
the sum of opening costs is now constant and may be deleted from the objective
function.
5.2.4 Capacitated LA Model with Stochastic Demands
(Zhou and Liu 2003)
This model was proposed by Zhou and Liu (2003).
5.2.4.1 Model Assumptions
In this model, the fourth and fifth assumption in general model changes so that,
facilities are capacitated and demand of customer are probabilistic. The other assumptions of this model are similar to the general model.
5.2.4.2 Model Inputs
We have all the inputs in general model in addition to the following input:
j : stochastic demand of customer j
j .!/: a realization of stochastic vector j
si : capacity of facility i
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5.2.4.3 Model Outputs (Decision Variables)
The outputs of this model are similar to the general model. But in a deterministic
LA problem, the allocation w is a decision variable which is fixed all the time.
However, in a stochastic LA problem, the decision w will be made every period
after the demands of customers are realized.
5.2.4.4 Objective Function and its Constraints
The objective of this model is similar to general model:
Min D
m X
n
X
wij d.xi ; aj /:
(5.8)
i D1 j D1
Subject to
m
X
wij D j .!/; j D 1; 2; : : : n;
(5.9)
i D1
m
X
wij S ;
(5.10)
j D1
wij 0;
i D 1; 2; : : : ; m
:
j D 1; 2; : : : ; n
(5.11)
Equations (5.9) and (5.10) ensure that w is feasible. Equations (5.11) are standard
constraints.
If we don’t find the feasible point for w; then demands of some customers are
impossible to be met, and the right-hand side of (5.8) becomes meaningless. As a
penalty, we define:
n
m X
X
Min D
j .!/d.xi ; aj /:
(5.12)
i D1 j D1
5.3 Solution Techniques
The calculation aspects of solving certain classes of LA problems are, exact equations, heuristic methods and meta-heuristic methods that are presented for solving
these problems.
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Z. Azarmand and E.N. Jami
5.3.1 Exact Solutions (Cooper 1963)
In order to find the set of .xi ; yi / that will minimize ' we differentiate (13) with
respect to xi and yi solve the equations resulting from setting the derivatives (into
.xxi ; yxi /) equal to zero. Thus:
D
n
m X
X
(5.13)
Zij Wi dij ;
i D1 j D1
n
X
@
D
Zij W
@Xxi
j D1
n
X
@
D
Zij W
@Yxi
j D1
@dij
@Xxi
D 0I i D 1; : : : ; m;
(5.14)
@dij
@Yxi
D 0I i D 1; ::; m:
(5.15)
Solving (5.14) and (5.15) will yield the 2 m values of .xi ; yi / which will cause ' to
be a minimum, for some particular set of Zij .
It is well to consider the amount of computation that may be involved. For m
sources and n destinations there are S(n,m) possible assignments of n destinations
to m sources is given by:
m
1 X m
.1/K .m k/m :
(5.16)
S.n; m/ D
k
mŠ
kD0
For very large n, these Stirling numbers can be formidably large. Equation (5.14)
and (5.15) would have to be solved S(n, m) times to find which particular allocation of sources to destinations, among this minimum set, is the absolute minimum
we seek.
For small-scale problems this is feasible using a digital computer. However, for
large-scale problems of industrial importance the amount of computation is prohibitive.
Suppose it is desired to minimize Euclidean distance between sources and destinations, under the assumption that cost is proportional to distance. For this case
we have:
m X
n
h
X
2 i1=2
2
Zij Wi Xxi Xaj C Yxi Yaj
:
(5.17)
D
i D1 j D1
Differentiating to find the minimum yields:
9
8̂
>
>
ˆ
n <
=
X
Zij Wi Xxi Xaj
D 0I i D 1; ::; m;
1= >
2
ˆ
j D1 :̂
;
Xxi Xaj C Yxi Yaj 2 >
(5.18)
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101
8̂
ˆ
n
X<
9
>
>
=
Zij Wi Yxi Yaj
D 0I i D 1; : : : ; m:
1=2 >
2
ˆ
>
j D1 :̂
;
Xxi Xaj C Yxi Yaj
(5.19)
Cooper (1963) has found from experience that, of several methods tried, the following “method of iteration” works best.
These equations are solved iteratively. Let the superscript indicate the iteration
parameter. The iteration equations for xi and yi are simply:
XxkC1
D
i
2
Zij Wi Xaj
n 6
X
6 Dijk
6P
4 n Zij Wi
j D1
j D1
YxkC1
D
i
2
Dijk
Zij Wi Yaj
n 6
X
6 Dijk
6 n
4 P Zij Wi
j D1
j D1
Dijk
3
7
7
7;
5
(5.20)
3
7
7
7:
5
(5.21)
A set of convenient starting values that always yields a convergent algorithm is
simply the weighted mean coordinates:
Xx0i D
3
2
n 6
7
X
6 Zij Wi Xaj 7
7I i D 1; : : : ; m;
6 P
5
4 n
j D1
Zij
(5.22)
j D1
Yx0i D
3
2
n 6
7
X
6 Zij Wi Yaj 7
7I i D 1; : : : ; m:
6 n
5
4 P
j D1
Zij
(5.23)
j D1
For large numbers of destinations .>10/ the method is not computationally
attractive.
5.3.2 Heuristic Methods
Many heuristics have been proposed for classic location-allocation, as well as a
few exact algorithms. Heuristics are needed to quickly solve large problems and to
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Z. Azarmand and E.N. Jami
provide good initial solutions for exact algorithms. The first heuristics is the wellknown iterative location-allocation algorithm of Cooper (1964).
Cooper’s heuristic generates p subset of fixed points and then solved each one
using the exact method for solving a single-facility location problem. The fixed
points’ set is divided into p subsets. For each of these p subsets, using an initial facility location, the exact location method is applied to find the optimal single facility
location. Each fixed point is then reallocated to the nearest facility. After all fixed
points have been completely reallocated, the exact location method is applied again
to improve the location of those facilities where the set of customers assigned has
changed. This process, alternating between the location and the allocation phases,
is repeated until no further improvement can be made. The solution found by the alternate algorithm is a local minimum. Eilon et al. (1971) showed that for a problem
with p D 5 and n D 50, using 200 randomly generated starting solution, 61 local
optima were found, and the worst solution deviates from the best one by 40.9%. In
order to have the closest local minimum to the optimal one, the method is repeated
several times using different starting locations at random.
Cooper (1963) proved that the objective function is neither concave nor convex,
and may contain several local minima. Hence, the classic location-allocation falls in
the realm of global optimization problem (Henrik and Robert 1982).
If two points in K dimensional space are given by q D .q1 ; : : : : : ; qk / and s D
.s1 ; : : : : : ; sk / then the lp distance between q and s is given by
lq .q; s/ D kq skp D
"
k
X
jqi si jp
i D1
#1=p
:
(5.24)
The location-allocation problem with lp distances in two dimensions is given by:
Min
n
m X
X
wij xi aj
p
:
(5.25)
i D1 j D1
Subject to
m
X
wij D rj I j D 1; : : : ; m;
(5.26)
i D1
wij D 0I i D 1; : : : ; mI j D 1; : : : ; n:
(5.27)
If we consider the wij to be given constants, then the dual of the objective function
of (5.27) for p > 1 is given by Love as:
Max g.u/ D
n
m X
X
i D1 j D1
aj Uij0 :
(5.28)
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103
Subject to
n
X
Uij0 D 0; i D 1; : : : ; m;
(5.29)
j D1
Uij wij ; i D 1; : : : ; mI j D 1; : : : ; n;
q D p=.p 1/; Uij D .uij1 ; uij2 /:
(5.30)
(5.31)
And a prime denotes transpose. The Uij are the dual variables.
This is a nonlinear programming problem with a linear objective function and
constraints that are either linear or well-behaved in the sense that the feasible constraint set is convex. This latter property is ensured since kU kq is convex in U . we
next state a theorem about nonlinear programming problems.
We may state the location-allocation problem given by (5.27) as a concave minimization problem. The vector w be defined as
w D .0; : : : ; 0; w11 ; w12 ; : : : ; w1m ; w21 ; : : : ; wmn /:
(5.32)
Then, using the dual formulation of the location problem given by (5.28), let
G .w/ D max :g.u/ D
m X
n
X
aj u0ij
(5.33)
i D1 j D1
Subject to
n
X
Uij D 0; i D 1; : : : ; m;
(5.34)
j D1
Uij
q
wij ; i D 1; : : : ; m; j D 1; : : : ; n:
(5.35)
We may write the location-allocation problem as
MinG .w/:
(5.36)
Subject to
m
X
wij D rj I j D 1; : : : ; m;
(5.37)
i D1
wij D 0; i D 1; : : : ; mI j D 1; : : : ; n:
(5.38)
G .w/ is concave and this is a concave minimization problem.
A small number of computational algorithms for concave minimization problems
are available. Each of these was considered but proved to be computationally inefficient for the problem at hand. It was decided, therefore, to test a set of algorithm
104
Z. Azarmand and E.N. Jami
which exploits the properties of the location-allocation problem. It which follows
from the concavity of G .w/ and which was previously given by Cooper, is that
the optimal solution must lie at an extreme point of the constraint set of (5.31). The
formulation given by (5.31) presents the location-allocation problem as a concave
minimization problem subject to linear constraints. An algorithm which proceeds
only to adjacent extreme points is not guaranteed to reach optimality for concave
minimization problem in general. A more elaborate scheme is to test pairs of new
variables for entry into the basis at each iteration in addition to investigating adjacent
extreme points. Procedures are analogous in linear programming to testing all possible pairs of non-basic variables for entry into the solution at each step of the simplex
method in addition to testing all single non-basic variables for entry to the solution.
For minimization programming problems with linear or convex objective functions,
the examination of pairs of variables for entry is not required, since single variable
entry methods converge to optimality. However, since the location-allocation problem may have many local minima, testing pairs of variables for possible entry to
the solution at each iteration enables the algorithm to “step over” a local minimum,
thus greatly increasing the algorithm’s chance of the globally minimum point. As
an example, two procedures are as follows:
Procedure 1
Step 1: Locate the new facilities arbitrarily.
Step 2: Given the locations, find the optimal allocation.
Step 3: Given the allocations, find the optimal locations.
Step 4: If the locations find in step 3 differ from the locations previously tried,
go to step 2; Otherwise, the current local minimum has been found, its cost is
computed, and the perturbation scheme can be started.
Step 5: Set i D 0.
Step 6: Set i D i C 1; if i > m stop.
Step 7: Set j D 0.
Step 8: Set j D j C 1; if j > m go to step 6.
Step 9: If existing facility j is allocated to new facility i in the current local
minimum, go to step 8; otherwise, tentatively allocate existing facility j to new
facility i .
From this tentative solution, a new local minimum is found by alternative locating and allocating.
Step 10: If the cost of the new local minimum exceeds the lowest cost found
so far, go to step 8; otherwise, update the current local minimum and restart the
perturbation scheme.
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105
Procedure 2
Step 1: Locate the new facilities arbitrarily.
Step 2: Given the locations, find the optimal allocations.
Step 3: Given the allocations, find the optimal locations.
Step 4: If the locations find in step 3 differ from the locations previously tried,
go to step 2; Otherwise, the current local minimum has been found, its cost is
computed, and the perturbation scheme can be started.
Step 5: Set i D 0.
Step 6: Set i D i C 1; if i > m stop.
Step 7: Set j D 0.
Step 8: Set j D j C 1; if j > m go to step 6.
Step 9: If existing facility j1 is allocated to new facility i in the current local
minimum, go to step 11.
Step 10: Set j2 D j1 .
Step 11: Set j2 D j2 C 1; if j2 > n go to step 8.
Step 12: If existing facility j2 is allocated to new facility i in the current local
minimum, go to step 11; otherwise, tentatively allocate existing facility j1 and j2
to new facility i .
From this tentative solution, a new local minimum is found by alternative locating and allocating.
Step 13: If the cost of the new local minimum exceeds the lowest cost found
so far, go to step 11; otherwise, update the current local minimum and restart the
perturbation scheme.
In 102 test problems for which the optimal solutions are known, two of algorithms achieved the exact optimum solutions in all cases.
5.3.3 Metaheuristic Methods (Salhi and Gamal 2003)
Location-allocation problem is NP-hard problem, metaheuristic methods have been
shown to the best way to tackle larger NP-hard problems. Meta-heuristic such as
simulated annealing, Tabu search, genetic algorithm (GA), variable neighborhood
search, and ant systems increase the chance of avoiding local optimally.
5.4 Case Study
In this section we will introduce some real-world case studies related to locationallocation problems.
106
Z. Azarmand and E.N. Jami
5.4.1 A Facility Location Allocation Model for Reusing Carpet
Materials (Louwers et al. 1999)
Each year tons of carpets are disposed. In Western Europe alone, 1.6 million tons
were deposed in 1996. At the moment, almost all this “waste” is disposed in landfills, using up a lot of space, whereas most of this “waste” can be reused as a material
or fuel. Therefore, it is not surprising that in more countries, including Germany and
The Netherlands, the legislation concerning the disposal of carpet waste has become
more stringent.
This article focuses on the design of the logistic structure of the above network,
i.e. the physical locations where the different activities should take place, the capacities of the facilities at these locations for storing and preprocessing the carpets that
are disposed, the allocation of disposed carpet waste flows to these locations and
facilities, as well as the transportation modes to be used.
The first activity concerns the collection of carpets that are no longer desired by
their owners. There are many different sources, including households and companies, collecting used pieces of carpet from houses, offices, cars, etc.
The above suggests that identifying, sorting, separating and compressing carpet
waste should take place at the same location. In the rest of this paper the centers
where this takes place are denoted by RPC (regional preprocessing center).
Carpets are disposed in many different ways, varying from rolls up to very small
irregular shaped pieces. The quality of these carpets varies from new, unused to
used-up, and contaminated.
There are different types of carpets, containing different materials like wool and
synthetic fibers that are interesting for different parties. This requires that carpet
waste is identified and sorted.
In order to determine the economically optimal logistic structure of the carpet
waste reuse network, the number and capacities of the RPCs for which the calculations are to be done have been calculated. As mentioned before, at an RPC only
certain preprocessing capacities can be installed. For the two applications of Europe
and the USA, only three possibilities existed denoted by P C1 ; P C2 and P C3 , where
P C1 < P C2 < P C3 .
The optimal locations and capacities of the RPCs are resulted from the calculations for one of the scenarios for Germany and the Benelux. In order to have a
readable figure, the flows to and from the RPCs have not been indicated in Fig. 5.1.
So for this scenario four RPCs should be opened, three of them with a capacity of
150 kt/yr and one with a capacity of 50 kt/yr.
Disposer
Source
RPC
Processor
Fig. 5.1 The main activities in the context of reusing carpet waste
Re-user
5
Location Allocation Problem
107
5.4.2 A New Organ Transplantation Location-Allocation Policy
(Bruni et al. 2006)
In this part they propose a location model for the optimal organization of transplant
system. Instead of simulation approach, which is typical when facing many health
care applications, their approach is distinctively based on a mathematical programming formulation of the relevant problem. The allocation of transplantable organs
across regions with the objective of attaining regional equity in health care is the
aim of this part.
The allocation of scarce donated organs is both an increasingly complex clinical
and social problem, as well as a challenging example of mathematical programming
problem formulation. The process leading to donor identification, consent, organ
procurement and allocation continue to dominate debates and efforts in the field
of transplantation. A huge storage of donors remains while the number of patients
needing organ transplantation increases.
The problem of long waiting times for transplant candidates and the continual
growth in waiting list size underscores a simple reality: supply of organs does not
meet the need. This evidence has produced substantial debate about the mechanisms
for allocation of organs to potential recipients, with issues of fairness, efficiency and
regional versus national interests.
The following discussion focuses on the key issues that are most relevant for the
purposes of their study:
The factors considered in the development of an organ allocation policy
The design of an equitable transplant system
As far as the design of the transplant system is concerned, we have to take into
account the spatial distribution of some organizations (referred as OPO – Organ
Procurement Organization), that play a crucial role in the design of a fair and equitable transplant system.
The transplantation is performed on condition that the recipients reach the transplant hospital in the reasonable time. Therefore, once the organ is allocated, the
access of the transplant service can be measured in terms of distance to the nearest
or time taken to travel to the health facility. The probability of the waiting list in
which the patient is registered, this size is influenced by the location of the OPOs
within the national area.
Our contribution is placed in this respect. In particular, we propose a mixed
integer linear programming model to face the problem of transplant system organization. The proposed model can be used to analyze simultaneously:
The location problem of OPOs and referring donor hospital and transplant centers
The districting problem
The waiting lists balancing problem
These issues have a crucial role for the fairness and the equity of the transplant in
the transplant system.
108
Z. Azarmand and E.N. Jami
The goal of an efficient and fair transplant system is to allocate organs in an
equitable and timely fashion. A fair and efficient transplant system should guarantee
the greatest survival rate, for patients and for organs used.
When donor organs become available after an individual dies, an organ procurement organization (OPO) takes the organs into custody. The OPO then matches
the donor organs with the appropriate transplant patients, by gathering information
about the donor organs.
The elapsed time between removal and implantation is critical for the organ because they cannot be used if they have been waiting more than an upper bound time
known to clinician as “cold-ischemia time”. The time spent in the transplant process
has two components.
For the organ, the first segment of travel time is between the explanation and the
arrival to the transplant center. The second segment of waiting refers to the time
spent by the potential host traveling to the transplant center.
They design and formulate transplant location allocation model (TRALOC
model for short); TRALOC is clearly not able to fully capture the complexity of the
transplant system. Nevertheless, the assumptions they made are quite general and
allow an elegant and easy formulation. They make clear that the objective of their
study is to make the appropriate simplifications that lead to a tractable model.
With the aim to asses and validate TRALOC model. They applied it to the Italy
transplant network. The national area is divided into 20 regions and 105 provinces.
In particular each province represents a demand node and, therefore, the problem is
a 105 node network.
To solve this model different values of p are being tested based on the minimum required amount, where p is the number of OPOs that should be established.
By determining the number of new facilities, the NP-hard problem changes to NPcomplete problem, then Lingo 6 is applied to solve it. Finally decision makers and
policy makers were offered the results in order to choose the optimum form according to their own point of view.
References
Badri MA (1999) Combining the analytic hierarchy process and goal programming for global
facility location-allocation problem. Int J Prod Econ 62:237–248
Brimberg J, Mladenovic N (1996) Solving the continuous Location-Allocation problem with tabu
search. Stud Locational Ann 8:23–32
Brimberg J, Hansen P, Mladenovic N, Taillard ED (2000) Improvements and comparison of heuristics for solving the uncapacitated multi source Weber problem. Oper Res 48:444–460
Bruni M, Conforti D, Sicilia N, Trotta S (2006) A new organ transplantation Location-Allocation
policy: A case study of Italy. Health Care Manage Sci 9:125–142
Cooper L (1963) Location-Allocation problems. Oper Res 11(3):331–343
Cooper L (1964) Heuristic methods for location-allocation problems. SIAM Rev 6(1):37–53
Eilon S, Watson-Gandy CDT, Christofides N (1971) Distribution Management. New York, Hafner
Ernst AT, Krishnamoorthy M (1999) Solution algorithms for the capacitated single allocation hub
location problem. Ann Oper Res 86:141–159
5
Location Allocation Problem
109
Gong D, Gen M, Yamazaki G, Xu W (1997) Hybrid evolutionary method for capacitated locationallocation problem. Comput Ind Eng 33:577–580
Hakimi S (1964) Optimum distribution of switching centers in a communication network and some
related graph theoretic problems. Oper Res 13:462–475
Hansen P, Jaumard B, Taillard E (1998) Heuristic solution of the multi source Weber problem as a
p-median problem. Oper Res Lett 22:55–62
Henrik J, Robert FL (1982) Properties and solution methods for large location-allocation problems.
J Oper Res Soc 33(5):443–452
Kuenne RE, Soland RM (1972) Exact and approximate solutions to the multi source Weber problem. Math Program 3:193–209
Louwers D, Kip BJ, Peters E, Souren F, Flapper SDP (1999) A facility location-allocation model
for reusing carpet materials. Comput Ind Eng 36:855–869
Murray AT, Church RL (1996) Applying simulated annealing to location-planning models.
J Heuristics 2:31–53
Ohlemüller M (1997) Tabu search for large location-allocation problems. J Oper Res Soc
48(7):745–750
Scaparra MP, Scutellà MG (2001) Facilities, locations, customers: Building blocks of location
models: A survey. Technical report TR-01–18, Computer Science Department, University of
Pisa, Italy
Salhi S, Gamal MDH (2003) A Genetic algorithm based approach for the uncapacitated continuous
location–allocation problem. Ann Oper Res 123:203–222
Zhou J, Liu B (2003) New stochastic models for capacitated location-allocation problem. Comput
Ind Eng 45:111–125
Chapter 6
Quadratic Assignment Problem
Masoumeh Bayat and Mahdieh Sedghi
The quadratic assignment problem (QAP) in location Theory is the problem of
locating facilities the cost of placing a facility depends on the distances from other
facilities and also the interaction with other facilities. QAP was introduced by
Koopmans and Beckman in 1957 who were trying to model a facilities location
problem.
It is possible to formulate some classic problems of combinatorial optimization,
such as the traveling salesman, maximum clique and graph partitioning problems as a QAP. The QAP belongs to the class of NP-complete problems and
is considered one of the most difficult combinatorial optimization problems. Exact solution strategies for the QAP have been unsuccessful for large problem
(approximately N 25).
In Fig. 6.1 the QAP search trends and tendencies in about 50 years is shown
(Hahn et al. 2007). Figure 6.1 shows the distribution of QAP publications with respect to the categories: applications, theory and algorithms.
Figure 6.2 distributes the number of articles by 5-year periods since 1957, for
each period, the work is also classified according to the same categories of Fig. 6.1.
Figure 6.2 shows that an explosion of interest in theory and algorithm development
occurred in the period from 1992 to the present.
Figure 6.3 shows a steady increase in interest in the QAP in recent years.
Figure 6.4, which covers the recent years, shows that the interest in algorithms
continues to be very strong, with a cyclical trend in theoretical developments. Applications continue to be of interest, but to a lesser extent. It is noteworthy that there
is recently a growing interest in applications to communication link design and to
the optimization of communications networks.
As a first example for introducing the problem we describe the problem of assigning facilities to locations in an office. Let have four facilities that each facility is
designated to exactly one location and vice-versa. Our problem is to find a minimum
cost allocation of facilities into locations taking the costs as the sum of all possible distance-flow products. For example in Fig. 6.5 an assignment D .3; 2; 4; 1/
is shown:
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 6, c Physica-Verlag Heidelberg 2009
111
112
M. Bayat and M. Sedghi
200
150
100
50
0
Appl
Theory
Alg
Fig. 6.1 Publications classification according to their contents (Hahn et al. 2007)
Appl
Theory
Alg
50
40
30
20
10
0
1
-6
57
6
-6
62
1
-7
66
6
-7
72
-8
77
1
6
-8
1
-9
87
82
6
-9
92
01
20
97
05
20
02
20
Fig. 6.2 Number of publications in 50 years according to their content (Hahn et al. 2007)
30
25
20
15
10
1999
2000
2001
2002
2003
2004
2005
Fig. 6.3 Number of QAP papers published in recent years (Hahn et al. 2007)
Appl
Theory
Alg
15
10
5
0
2000
2001
2002
2003
2004
2005
Fig. 6.4 Recent advances. (Hahn et al. 2007)
For this assignment, the objective function value is the product of matrix D, the
distance matrix describing the distance between two facilities, and matrix F , the
flow matrix corresponding with this assignment. For this example, given D and F
as follows:
6
Quadratic Assignment Problem
113
f12
1
d12
1
2
f14
f13
f24
f32
3
4
d14
2
d13
1
2
3
2
1
4
d24
3
d32
f3
d34
Facilities
Locations
3
4
4
Assignment π
Fig. 6.5 Assignment D (3, 2, 4, 1)
Table 6.1 QAP’s application in different science
Authors
Subject
Koopmans & Beckmann
Steinberg
Proposed the QAP as a mathematical model
Minimize the number of connections in a
backboard wiring with QAP
Applied QAP to economic problems
For assigning a new facility
Scheduling problems
Applied QAP to archeology
Applied QAP in statistical analysis
Used QAP in the analysis of reaction chemistry
In numerical analysis
Error control in communications
The problem of memory layout optimization in
signal processors
Heffley
White & Francis
Geoffrion & Graves
Krarup & Pruzan
Hubert
Forsberg
Brusco & Stahl
Ben-David & Malah
Wess and Zeitlhofer
0
1
0 10 20 30
B10 0 15 18C
C
DDB
@20 15 0 11A ;
30 18 11 0
Publication years
0
02
B2 0
F DB
@3 7
21
1957
1961
1972
1974
1976
1978
1987
1994
2000
2005
2004
1
32
7 1C
C:
0 5A
50
D represents the distance matrix corresponding to assignment .
0
1
20 10 30 0
B15 0 18 10C
C
D D B
@ 0 15 11 20A ;
11 18 0 30
0
023
B2 0 7
F DB
@3 7 0
215
1
2
1C
C:
5A
0
Then, the total cost for this assignment is DF D 2; 117 and our object is minimizing total cost by permutation of matrix D columns. Any assignment will result in
a feasible solution, therefore, there are N Š feasible solutions.
Since its first formulation, the QAP has been applied in many problems in different sciences. In Table 6.1 its application in different sciences and in Table 6.2. its
application in location problems has been indicated.
114
M. Bayat and M. Sedghi
Table 6.2 QAP’s applications in location problem
Authors
Hopkins & Dickney
Pollatschek
Elshafei
Bos
Beenjaafar
Miranda
Subject
Assignment of buildings in a University campus
Design of typewriter keyboards and control panels
In a hospital planning
Forest management
In facilities layout for minimizing work-in-process
(WIP)
Placement of electronic components
Publication years
1972
1976
1977
1993
2002
2005
In Sect. 6.1 we describe QAP’s formulations that are represented in literature. In
Sect. 6.2 we point out some problems that are related to QAP. In Sect. 6.3 QAP’s
lower bounds and solution methods are discussed and in Sect. 6.4 some case studies
are presented.
6.1 Formulations of QAP
In this section we introduce QAP formulations represented in literature.
6.1.1 Integer Programming Formulations (ILP)
6.1.1.1 Model Assumptions
The objective function is MiniSum.
Each facility is designated to exactly one location and vice-versa
The solution space is discrete and finite.
The number of location and facilities is known (exogenous).
All decision variables of the model are binary (0–1) variables.
6.1.1.2 Model Inputs
F D Œfij : flow matrix between facility i and j
D D Œdkp :distances matrix between locations k and p
B D Œbik : allocation costs of facility i to location k
6.1.1.3 Model Outputs (Decision Variables)
xi k D
(
1;
0;
if facility i assigned in location k
otherwise
6
Quadratic Assignment Problem
115
6.1.1.4 Koopman and Beckman (1957) Formulation
This formulation is the first mathematical formulation of QAP. Koopman and
Beckman (1957) represented this formulation. Let N D f1; : : : ; ng, our objective
is finding permutation ˚ of set N to minimize the objective function.
6.1.1.5 Objective Function and its Constraint
Min
n
X
bi'.i / C
i D1
n X
n
X
fij d'.i /'.j /;
(6.1)
j D1 i D1
Subject to
n
X
xij D 1 8j D 1; : : : ; n;
(6.2)
xij D 1 8i D 1; : : : ; n;
(6.3)
xij 2 f0; 1g 8i; j D 1; : : : ; n
i; j; k; p D 1; : : : ; n
(6.4)
i D1
n
X
j D1
Equation (6.1) assures that to each location assigned only one facility and (6.3)
assure that each facility assigned only to one location.
6.1.1.6 Lawler (1963) Formulation
Lawler (1963) represented a general model of QAP that Koopman and Beckman
introduced. He proposed cijkp coefficient that have the special form:
Cijkp D dkp fij :
6.1.1.7 Objective Function and its Constraint
Min
n X
n
X
Cijkp xik xjp ;
(6.5)
j D1 i D1
Subject to
n
X
i D1
xij D 1 8j D 1; : : : ; n;
(6.6)
116
M. Bayat and M. Sedghi
n
X
xij D 1 8i D 1; : : : ; n;
(6.7)
j D1
xij 2 f0; 1g
8i; j D 1; : : : ; n
i; j; k; p D 1; : : : ; n
(6.8)
6.1.1.8 Linearization
For linearization of the model Lawler (1963) proposed to replace the product xij xkp
by n4 new binary variables yijkp :
yijkp D xij xkp ;
and showed that the QAP is equivalent to a linear integer program with O.n4 / variables and constraints.
Theorem 6.1. The QAP is equivalent to the integer linear program of the form:
Min
n X
n
X
Cijkp yijkp ;
(6.9)
j D1 i D1
Subject to
n X
n X
n X
n
X
yijkp D n2 ;
(6.10)
i D1 j D1 kD1 pD1
xij C xkp 2yijkp 0 8i; j; k; p D 1; : : : ; n;
(6.11)
n
X
xij D 1
8j D 1; : : : ; n;
(6.12)
n
X
xij D 1
8i D 1; : : : ; n;
(6.13)
i D1
j D1
xij 2 f0; 1g
8i; j D 1; : : : ; n
i; j; k; p D 1; : : : ; n
(6.14)
6.1.2 Mixed Integer Programming Formulations (MILP)
The linearization of the problem is inconvenient, because of the large additional
amount of variables and constraint. Finding linearization with fewer variable and
constraint is proper.
6
Quadratic Assignment Problem
117
6.1.2.1 Kaufman and Broeckx (1978) MILP Formulation
Kaufman and Broeckx (1978) developed another linearization by mixed integer programming. They used Glover’s linearization technique which is extremely favorable
in this type of problems, where one only has positive cost-coefficients. They have
shown that the MILP formulation and QAP are equivalent. Z rewrite as:
zD
XX
i
k
0
xik @
XX
p
j
1
fij dkp xjp A :
(6.15)
And continuous variable introduce as:
wik D xik
n
n X
X
fij dkp xjp :
(6.16)
kD1 pD1
There are n2 such new continuous variables, and the new objective is to minimize:
zD
XX
i
(6.17)
wik :
k
6.1.2.2 Objective Function and Constraints
Min z D
n X
n
X
wij ;
(6.18)
j D1 i D1
Subject to
n
X
i D1
n
X
xij D 1
8j D 1; : : : ; n;
(6.19)
xij D 1 8i D 1; : : : ; n;
(6.20)
j D1
hij xij C
n X
n
X
cijkp xkp wij hij
8i; j D 1; : : : ; n;
(6.21)
kD1 pD1
xij 2 f0; 1g ; wij 0 8i; j D 1; : : : ; n;
where hij D
n X
n
X
kD1 pD1
Cijkp :
(6.22)
(6.23)
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M. Bayat and M. Sedghi
6.1.2.3 Frieze and Yadegar (1983) MILP Formulation
Frieze and Yadegar (1983) proposed a mixed integer linear programming model to
discuss the relationship between Gilmore-Lawler lower bounds with decomposition
for the quadratic assignment problem and a Lagrangian relaxation of a particular
integer programming formulation. Their model has n4 real variables, n2 Boolean
variables and n4 C 4n3 C n2 C 2n constraints. Frieze and Yadegar have shown that
this MILP model and QAP are equivalent.
6.1.2.4 Objective Function and Constraints
Min
n X
n X
n
n X
X
fij dkp yij kp ;
(6.24)
i D1 j D kD1 pD1
Subject to
n
X
i D1
n
X
kD1
n
X
i D1
n
X
xik D 1 8k D 1; : : : ; n;
(6.25)
xik D 1 8i D 1; : : : ; n;
(6.26)
yijkp D xjp
8i; j; k; p D 1; : : : ; n;
(6.27)
yijlp D xjp
8i; j; k; p D 1; : : : ; n;
(6.28)
yijlp D xjp
8i; j; k; p D 1; : : : ; n;
(6.29)
yijlp D xjp
8i; j; k; p D 1; : : : ; n;
(6.30)
j D1
n
X
kD1
n
X
pD1
yikik D xik
8i; k D 1; : : : ; n;
xi k 2 f0; 1g 8i; k D 1; : : : ; n
0 yij kp 1 8i; j; k; p D 1; : : : ; n;
(6.31)
(6.32)
6.1.3 Formulation by Permutations
Let Sn be the set of all permutations with n elements and 2sn consider fij the
flows between facilities i and j and d.i /.j / the distances between locations .i /
and .j /. Each permutation represents an allocation of facilities to locations.
6
Quadratic Assignment Problem
119
Xij D
(
1
0
if .i / D j
if.i / ¤ j :
6.1.3.1 Objective Function and Constraint
Min
2sn
n X
n
X
fij d'.i /'.j /:
(6.33)
i D1 j D1
Has the same (6.1)–(6.4).
6.1.4 Trace Formulation
Trace formulation is supported by linear algebra and use the trace function to determine QAP lower bound for the cost. This approach allows for the application of
spectral theory, which makes possible the use of semi-definite programming to the
QAP (Hahn et al. 2007).
Trace.A/ D
n
X
aii
i D1
6.1.4.1 Edwards (1980) Formulation
Objective function and constraint
Min tr.F:X:D:X t /;
(6.34)
X 2 ;
(6.35)
x2sn
Subject to
where F and D are n n matrices, tr denotes the trace of a matrix and is the set
of n n permutation matrices.
6.1.5 Graph Formulation
The QAP can be defined in terms of graphs in the following way.
120
M. Bayat and M. Sedghi
6.1.5.1 Models Inputs
G f D .V f ; E f /: an undirected flow graph,
V f D fvi I i D 1; : : : ; ng: vertices that represent facilities,
.vi ; vj / 2 Ef : the edges that represent the existence of a flow between facilities
i and j ,
fij D The cost of edge .vi ; vj /,
G d D .V d ; E d / be a distance graph,
V d D fvi I i D 1; : : : ; ng vertices that represent locations d˚.i /˚.j / .
The edge costs are the distances between the corresponding locations.
6.1.5.2 Objective Function and Constraints
Min
2sn
X
fij d'.i /'.j /
(6.36)
.vi ;vj /2E f
Has the same (6.1)–(6.4).
6.2 QAP Related Problems
6.2.1 The Quadratic Bottleneck Assignment Problem (QBAP)
Steinberg (1961) proposed QBAP for minimizing the maximum-wire-length norms
in backboard wiring problem. The QBAP general program is obtained from the
QAP formulation by changing its minisum objective function with minimax, which
suggests the term bottleneck function.
6.2.1.1 Objective Function
Min Max ffij d.i /.j / I 1 i; j n g:
(6.37)
6.2.2 The Biquadratic Assignment Problem (BiQAP)
The biquadratic assignment problem (BiQAP) is a generalization of the quadratic
assignment problem (QAP). The problem was first introduced and studied by
Burkard et al. (1994). It is a nonlinear integer programming problem where the
6
Quadratic Assignment Problem
121
objective function is a fourth degree multivariable polynomial and the feasible domain is the assignment polytope. BiQAP motivated by a practical problem arising
in VLSI synthesis.
6.2.2.1 Objective Function and its Constraints
Min
n X
n X
n X
n X
n X
n X
n X
n
X
fijkl dmpst xim xjp xks x lt :
(6.38)
i D1 j D1 kD1 lD1 mD1 pD1 sD1 rD1
Has the same (6.1)–(6.4)
6.2.3 The Quadratic Semi-Assignment Problem (QSAP)
This is a special case used to model clustering and partitioning problems by
Hansen and Lih (1992). This problem belongs to the class of the NP-hard problems
some task-assignment problems in distributed systems can be easily formulated as
quadratic semi-assignment problems.
6.2.3.1 Objective Function and Constraints
Min
m X
n
X
Cij xik xjk :
(6.39)
kD1 i;j D1
Subject to
m
P
xik D 1
1 i n;
kD1
xij2f0;1g
(6.40)
1 i; j n:
6.2.4 The Multiobjective QAP (MQAP)
6.2.4.1 Knowles and Corne Formulation
Knowles and Corne (2003) introduced the multiobjective QAP (mQAP), with multiple flow matrices that naturally models any facility layout problem where we are
concerned with the flow of more than one type of item or agent. This problem has
different flow matrices but it always keeps the same distance matrix.
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M. Bayat and M. Sedghi
6.2.4.2 Objective Function and its Constraints
˚
Min C ./ D C 1 ./; : : : ; C m ./ ;
XX
fij d.i /.j / ; 8k D 1; : : : ; m:
where C k ./ D
2sn
(6.41)
(6.42)
Has the same (6.1)–(6.4).
6.2.4.3 Hamacher Formulation
Hamacher et al. (2003) introduced a different formulation of a multiobjective QAP
that considers different matrices of flows and distances for modeling problems found
in facility layout for social institutions.
0
n
P
B i;j;k;lD1
B
B
Min B :::
B n
@ P
i;j;k;lD1
fik djl xij xkl
Q
Q
fik dkl xij xkl
1
C
C
C
C;
C
A
(6.43)
where the problem is to find assignment x as:
xij D
(
1;
0;
if facility i assigned in location j
:
otherwise
Has the same (6.1)–(6.4).
6.2.5 The Quadratic Three-Dimensional Assignment Problem
(Q3AP)
Pierskalla (1967a, b) introduced the quadratic three-dimensional assignment problem (Q3AP) in a technical memorandum. The Q3AP is an extension of the QAP
that can be represented as a three-dimensional assignment problem.
6.2.5.1 Objective Function and Constraints
Where I; J and P are disjoint sets of constraints
6
Quadratic Assignment Problem
123
8̂
9
>
ˆ
>
ˆ
>
ˆ
>
<P
=
n
n
n P
n P
n
n P
n P
n
n
P
P
P
P
Min
bijp xijp C
Cijpk nq xijp xk nq ;
ˆi D1 j D1 pD1
>
i D1 j D1 pD1
ˆ
>
ˆ
>
kD1 nD1 qD1
>
:̂
;
k¤i n¤j q¤p
(6.44)
where x 2 I \ J \ P; x binary
8
9
n
n X
<
=
X
I D x0W
xijp D 1 for i D 1; : : : ; n ;
:
;
(6.45)
j D1 pD1
8
n X
n
<
X
xijp D 1
J D x0W
:
i D1 pD1
P D
8
<
:
x0W
n X
n
X
i D1 j D1
xijp D 1
9
=
for j D 1; : : : ; n ;
;
9
=
for p D 1; : : : ; n :
;
(6.46)
(6.47)
6.2.6 The Generalized Quadratic Assignment Problem (GQAP)
Lee and Ma (2004) indicated that the GQAP had been proposed by them in 1991
which is a generalized problem of the QAP in that there is no restriction that one
location can accommodate only single equipment. This problem arises in many real
world applications such as facility location problem and logistics network design.
6.2.6.1 Model Inputs
M : a set of equipments .D f1; 2; : : : ; mg/,
N : a set of locations .D f1; 2; : : : ; ng/,
si : space requirement of equipment i ,
Sk : total available space at location k,
ci k : the cost of installing equipment i at location k,
qij : the flow volume from equipment i to j in the planning horizon,
dkh : the distance between location k and h .D dhk /,
v: the travel cost per unit distance and per unit flow volume.
6.2.6.2 Model Outputs (Decision Variables)
xi k D
(
1;
0;
if equipment i is assigned to location k;
otherwise:
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M. Bayat and M. Sedghi
6.2.6.3 Assumptions
Just to avoid trivial cases, we will assume:
(a) There is no single location that is big enough to hold all the equipments or:
X
i 2M
si > Sk
8k 2 N:
(b) There is no equipment which is too big to be assigned to any location or:
Sk maxi si ; 8k 2 N
(c) All the locations are distinctive or:
dij ¤ 0; 8j ¤ h 2 N
(d) The distance between locations is symmetric.
6.2.6.4 Objective Function and its Constraints
Min
m X
n
X
cik xik C
j D1 kD1
m X
m X
n X
n
X
qij dkh xik xjh ;
(6.48)
i D1 j D1 kD1 hD1
Subject to
n
X
xik D 1;
8i 2 f1; : : : ; mg ;
(6.49)
m
X
si xik Sk;
8k 2 f1; : : : ; ng ;
(6.50)
kD1
i D1
xik 2 f 0; 1 g ;
8i; k:
(6.51)
6.2.7 Stochastic QAP (SQAP)
SQAP is basically a network design problem, whenever there is a need to assign
activities to the nodes of a network in order to minimize congestion; the SQAP
is an appropriate model of the problem. In SQAP we have distance matrix and
P matrix that is the probability of having interaction between facility i and j
(Smith and Li 2001).
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Quadratic Assignment Problem
125
6.3 Solution Techniques
The methods used in combinatorial optimization problems can be either exact,
heuristic or metaheuristic shown in Fig. 6.6. The most frequently used strategies
are branch-and-bound or dynamic programming general methods and also there are
a number of heuristic techniques using different conceptions. In what follows, we
discuss about computational complexity and both approaches and bring their most
important references.
6.3.1 Computational Complexity
In fact, the QAP belongs to the class of computationally hard problems known as
NP complete. The proof that the QAP is indeed NP-complete was first shown by
Sahni and Gonzales (1976) and they also proved that any routine that finds even
an ©-approximate solution is also NP-complete, thus making the QAP among the
“hardest of the hard” of all combinatorial optimization problems. Belonging to this
class of problems suggests that an algorithm which solves the problem to optimality
in polynomial time is unlikely to exist. Therefore, to practically solve the QAP one
has to apply heuristic algorithms which find very high quality solutions in short
computation time.
6.3.2 Lower Bounds
Lower bounds are useful for measuring heuristic search worst-case solutions quality,
and are an essential component in the construction of Branch and Bound (B&B)
methods for achieving optimal solutions. We are trying to derive bound that is not
being hard to compute, can easily be evaluated for subsets of the problem which
occur after some branching, and finally is tight. There are many different proposals
for deriving bounds. In the following we briefly survey bounds for QAP problem.
100
75
50
25
0
Heuristic
Exact
Fig. 6.6 Publications: solution techniques (Hahn et al. 2007)
Metahurestic
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M. Bayat and M. Sedghi
6.3.2.1 The Gilmore and Lawler Lower Bound (GLB)
The Gilmore–Lawler bound (GLB) which was proposed by Gilmore (1962) and
Lawler (1963) is one of the best-known lower bounds for the small QAP’s problem.
It is easy to compute, but it deteriorates fast by increasing in the size of the problem.
GLB D LAP.F / QAP:
This bounding technique is combinatorial in nature and requires the solution of
a LAP for a Koopman–Beckmann problem that can be computed in O.n3 / time
(Anstreicher 2003). Several authors proposed different approaches to improve the
GLB, Frieze and Yadegar (1983) proposed GLB with decomposition, Assad and Xu
(1985) proposed the AX bound that is obtained iteratively, where n2 C 1 assignment
of size n are solved in each iteration. Hence, the running time to compute is O.k n5 /
where k is the number of iterations.
6.3.2.2 Bounds Based on LP and Dual-LP
Frieze and Yadegar (1983), discuss the relationship between GLB with decomposition and a Lagrangian relaxation of a mixed integer programming formulation
(6.24)–(6.32).
Drezner (1995) proved that the optimal solution for a MILP-formulation is a
lower bound for the corresponding QAP that is at least as good as the classical
Gilmore-Lawler lower bound. Adams and Johnson (1994) develop a dual ascent
procedure for approximating the dual of the MILP relaxation and proved that each
dual solution of the linear programming is also a lower bound for the QAP. These
bounds are quite expensive to compute and can only be applied to problems of
dimension n 30.
6.3.2.3 Eigenvalue Related Bounds (EVB)
Finke et al. (1987) proposed Eigenvalue bounds which are based on the trace formulation of QAP. The basic idea to derive this bound consists in minimizing objective
function of trace formulation over orthogonal rather than just permutation matrices.
Let 1 ; 2 ; : : : ; n be the eigenvalues of the symmetric matrix F and let
1 ; 2 ; : : : ; n be the eigenvalue of the symmetric matrix D. since F and D are
symmetric, the eigenvalues are real and we can assume the ordering
1 2 : : : n and 1 2 : : : n :
For all permutation ˚
n
X
i D1
i i
n X
n
X
i D1 j D1
fij d'.i /'.j /
n
X
i D1
i ni :
(6.52)
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127
Hadley et al. (1992a) consider also the case of non-symmetric QAPs and develop for
them eigenvalue bounds by means of Hermitian matrices. These bounds have good
quality in comparison to Gilmore–Lawler-like bounds; however, they are expensive
in terms of computation time and deteriorate quickly when lower levels of a branch
and bound tree are searched. The simple EVB is too weak to be computationally
useful, but several schemes for improving the bound have been considered. Hadley
et al. (1992b) proposed improved eigenvalue bound which have been derived from
the projected QAP (PD). EVD and PD both require O.n3 / computing time.
6.3.2.4 Bounds Based on Reformulations
Reformulation linearization technique (RLT) is designed to generate a hierarchy
of linear programming (LP) relaxations leading from the ordinary continuous relaxation to the convex hull representation for mixed-integer 0–1 programming
problems (Sherali and Adams 1990). Sherali et al. (2000) proposed reduced reformulation linearization technique to improve RLT. Adams et al. (2007) calculate
bounds using a level-2 reformulation linearization technique (2-RLT) that provides
sharp and tight lower bounds.
6.3.2.5 Variance Reduction Bounds
Initially proposed by Li et al. (1994a), these bounds are based on reduction schemes
and are defined from the variance of F and D matrices. These bounds, when used
in a branch-and-bound algorithm, take less computational time and generally obtain better performance than GLB. They show more efficiency when the flow and
distance matrices have high variances (Hahn et al. 2007).
6.3.2.6 Semi-Definite Programming and Reformulation–Linearization
Bounds
Semi-definite programming (SDP) has proven to be very successful in providing tight relaxations for hard combinatorial problems. SDP relaxations for the
quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP (Zhao
et al. 1998).
Anstreicher (2001) compares SDP relaxations and eigenvalue bounds;
Anstreicher and Brixius (2001) describe a new convex quadratic programming
bound for the QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off
between bound quality and computational effort.
Burer and Vandenbussche (2006) applied Lagrangian relaxation on a lift-andproject QAP relaxation, following the ideas in Lova’sz and Schrijver (1991), thus
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obtaining very tight SDP bounds. A report, by Rendl and Sotirov (2003), discusses
a very good semi-definite programming (SDP) lower bound for the QAP. In 2003,
when the report was written, it reported the tightest lower bounds for a large number
of QAPLIB instances.
Hahn et al. (2007) indicated that according to the research of Povh and Rendl in
2006, the strongest relaxation (R3) from Rendl and Sotirov (2003) and the relaxation
from Burer and Vandenbussche (2006) are actually the same The differing lower
bound values in the two papers are due to the fact that Rendl and Sotirov use the
bundle method, which gives only underestimates of the true bound, while Burer and
Vandenbussche are able to compute this bound more accurately (Hahn et al. 2007).
6.3.3 Exact Algorithms
The different methods used to achieve a global optimum for the QAP include
branch-and-bound, cutting planes or combinations of these methods, like branchand-cut and dynamic programming. These algorithms are time consuming for large
scale problem.
6.3.3.1 Branch-and-Bound
Among all three methods Branch-and-bound are the most known and used algorithms. There are several references concerning QAP branch-and-bound algorithms:
The first two branches and bound algorithms were developed by Gilmore (1962)
and Lawler (1963) which assign single facilities to single locations. The main difference between two algorithms is in computing the lower bounds. Land (1963) and
Gavett and Plyter (1966) proposed a algorithm which assign pairs of facilities to
pairs of locations. Graves and Whinston (1970) and Lawler (1963) developed procedure which assign one unassigned facility to vacant location. Bazaraa and Kirca
(1983) proposed the algorithm eliminates “mirror image” branches, thus reducing
the search space. Mans et al. (1995) applied a parallel depth first search branch
and bound algorithm for the quadratic assignment problem. Hahn and Grant (1998)
presented a new branch-and-bound algorithm for solving the quadratic assignment
problem (QAP). The algorithm is based on a dual procedure (DP) similar to the
Hungarian method for solving the linear assignment problem. Hahn et al. (1998)
applied a branch-and-bound algorithm for the quadratic assignment problem based
on the Hungarian method.
6.3.3.2 Cutting Plane
Cutting plane methods introduced by Bazaraa and Sherali (1980), initially, did not
present satisfactory results. The employed technique is not widely used so far, but
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129
good quality solutions for QAP cases are being presented. The slow convergence
of this method makes it proper only for small instances. There are some references
presented as follows:
Burkard and Bonniger (1983) which improved a cutting plane method to
solve the QAP, Kaufman and Broeckx (1978), Bazaraa and Sherali (1982),
Miranda et al. (2005).
6.3.3.3 Branch-and-Cut
The branch-and-cut technique, a variation proposed by Padberg and Rinaldi (1991),
appears to be an alternative cutting strategy that exploits the polytope defined by
the feasible solutions of the problem. Its main advantage over cutting planes is that
the cuts are associated with the polypore’s facets. Cuts associated with facets are
more effective than the ones produced by cutting planes, so the convergence to an
optimal solution is accelerated. The dearth of knowledge about the QAP polytope is
the reason why polyhedral cutting planes are not widely used for this problem, and
also Padberg and Rinaldi (1991), Kaibel (1998), Jünger and Kaibel (2000, 2001a, b),
Blanchard et al. (2003).
6.3.3.4 Dynamic Programming
Dynamic programming is a technique used for QAP special cases where the flow
matrix is the adjacency matrix of a tree.
Christofides and Benavent (1989) studied this case using a MILP approach to the
relaxed problem. It was then solved with a dynamic programming algorithm, taking
advantage of the polynomial complexity of the instances.
6.3.4 Heuristic Algorithms
Heuristic algorithms do not guarantee optimality of the best solution obtained and is
a procedure dedicated to search good quality solutions. So, all approximate methods
are included in this category.
Heuristic procedures include the following categories: constructive, limited enumeration, improvement methods and simulated approach.
6.3.4.1 Constructive Method
Gilmore (1962) introduced a constructive method that completes a permutation
(i.e., feasible solution) at each iteration of the algorithm. There are some references
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in this field: Armour and Buffa (1963), Buffa et al. (1964), Tansel and Bilen (1998),
Burkard et al. (1991), Arkin et al. (2001), Gutin and Yeo (2002), Yu and Sarker
(2003).
6.3.4.2 Enumerative Methods
Enumeration can guarantee that the obtained solution is optimum only if they can go
to the end of the enumerative process. Nissen and Paul (1995) applied the threshold
accepting technique to the QAP. Burkard and Bonniger (1983), West (1983).
6.3.4.3 Improvement Methods
Improvement methods correspond to local search algorithms. Most of the QAP
heuristics are in this category. White (1993) proposed a new approach, where the
actual data are relaxed by embedding them in a data space that satisfies an extension of the metric triangle property. Arora et al. (2002) proposed a randomized
procedure for rounding fractional perfect assignments to integral assignments and
a also Heider (1973), Mirchandani and Obata (1979), Bruijs (1984), Pardalos et al.
(1993), Burkard and Cela (1995), Li and Smith (1995), Anderson (1996), Talbi et al.
(1998a), Deineko and Woeginger (2000), Misevicius (2000a), Mills et al. (2003).
6.3.5 Metaheuristic Algorithms
They are characterized by the definition of a priori strategies adapted to the problem
structure. Several of these techniques are based on some form of simulation of a
natural process studied within another field of knowledge (metaphors) so they have
two important categories “metaheuristics based on natural process metaphors” and
“metaheuristics based directly on theoretical and experimental considerations”.
6.3.5.1 Metaheuristics Based on Natural Process Metaphors
This kind of algorithm; categorizes to simulated annealing, genetic algorithm,
scatter search algorithm, ant colony optimization (ACO) and neural networks and
Markov chains.
6.3.5.2 Simulated Annealing Algorithm
Burkard and Rendl (1984) proposed one of the first applications of simulated annealing to the QAP and Wilhelm and Ward (1987) presented the new equilibrium
components for it. Yip and Pao (1994) applied the new technique, called guided
evolutionary simulated annealing (GESA) to QAP, there is also Bos (1993a, b).
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A metaheuristic closely related to SA, was also applied to QAP by Nissen and
Paul (1995). Abreu et al. (1999) applied the technique by trying to reduce the number of inversions associated to the problem solution, together with the cost reduction.
Misevicius (2000b, 2003) propose a modified simulated annealing algorithm for the
QAP-M-SA-QAP and tested the algorithm on a number of instances from the library
of the QAP instances (QAPLIB) and also Bos (1993a, b), Burkard and Cela (1995),
Peng et al. (1996), Mavridou and Pardalos (1997), Chiang and Chiang (1998), Tian
et al. (1996 and 1999), Siu and Chang (2002), Baykasoglu (2004).
6.3.5.3 Genetic Algorithm
The GA can be good in order to find a “quite” good but non-optimal solution in a
fast way. However since we are not sure of the convergence of the algorithm toward
a defined value, this algorithm is not usable.
Drezner (2005a) presented a two-phase genetic algorithm and also TavakkoliMoghaddain and Shayan (1998), Gong et al. (1999), Drezner and Marcoulides
(2003), El-Baz (2004), Wang and Okazaki (2005).
6.3.5.4 Scatter Search Algorithm
Glover (1977) introduced in a heuristic study of integer linear programming problems. Application of scatter search to the QAP can be found in Cung et al. (1997).
6.3.5.5 Ant Colony Optimization (ACO)
Maniezzo and Colorni (1999) first applied the ACO to the QAP. Stützle and Dorigo
(1999) proposed a simple “2-opt” local search with ACO for QAP. Gambardella
et al. (1999) show ant colony as a competitive meta heuristic, mainly for instances
that have few good solutions close to each other. Middendorf et al. (2002) proposed a
new ACO algorithm called s-m-p-Ant and experimented it on instances of the QAP.
Solimanpur et al. (2004)solved the inter-cell layout problem and flow between the
cells which is modeled as a quadratic assignment problem (QAP), by ACO, there
are some more references Colorni et al. (1996), Stützle and Holgez (2000), Talbi
et al. (2001), Ying and Liao (2004), Acan (2005).
6.3.5.6 Neural Networks and Markov Chains
This algorithm is structurally different from metaheuristics, they are also based on
a nature metaphor and they have been applied to the QAP by: Ishii and Sato (1998)
applied constrained neural approaches to QAP.
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M. Bayat and M. Sedghi
Tsuchiya et al. (1996) applied proposed algorithm to QAPLIB.
Bousonocalzon and Manning (1995) proposed the Hopfield neural network as a
method for solving the QAP. Uwate et al. (2004) performance of chaos and burst
noises injected on the Hopfield neural network for quadratic assignment problems,
and there are some more references listed below: Obuchi et al. (1996), Rossin et al.
(1999), Nishiyama et al. (2001), Hasegawa et al. (2002).
6.3.5.7 Metaheuristics Based Directly on Theoretical and Experimental
Considerations
This kind of algorithm; categorizes to Tabu search, GRASP, VNS and hybrid algorithms.
6.3.5.8 Tabu Search
Tabu search is a local search algorithm that was introduced by Glover (1989a, b) to
find good quality solutions for integer programming problems.
Despite the inconvenience of depending on the size of the tabu list and the
way this list is managed, the performances of those algorithms show them as
being very efficient strategies for the QAP, as analyzed by Taillard (1991) and
Battiti and Tecchiolli (1994). In Skorin-Kapov (1990) a TS application is used to
solve QAPs. The method, called Tabu-navigation, uses swap moves (i.e., the exchange in the location of two objects) to search the solution space. Taillard (1991)
applied robust tabu search for the quadratic assignment problem and also Taillard
(1991) developed a TS procedure with less complexity for QAP with incorporates
a quick update for the moves in the candidate list at every iteration, this procedure
allows the complete evaluation of the swap neighborhood.
Chakrapani and Skorin-Kapov (1993) applied a dynamically changing tabu
list sizes, aspiration criterion and long term memory, tabu search for QAP,
Skorin-Kapov (1994) also implemented dynamic tabu list sizes for their algorithm.
Drezner (2005b) extended concentric tabu for the quadratic assignment problem.
6.3.5.9 GRASP
Li et al. (1994b) presented a GRASP for QAPLIB. Mavridou et al. (1998) proposed GRASP for the biquadratic assignment problem. Pitsoulis (2001) presented
GRASPs for solving the following NP-hard nonlinear QAP and also biquadratic
assignment problem (BiQAP). Fleurent and Glover (1999) indicate that memory
based medications improve the solution quality of GRASP for solving the QAP.
Rangel et al. (2000) introduced new GRASP uses a criterion to accept or reject
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Quadratic Assignment Problem
133
a given initial solution, thus trying to avoid potentially fruitless searches. Oliveira
et al. (2004) built a GRASP using the path-relinking strategy, which looks for improvements along the paths joining pairs of good solutions.
6.3.5.10 VNS
This method was introduced by Mladenovic and Hansen (1997). In Gambardella
et al. (1999), three VNS strategies are proposed for the QAP. One of them is a
search over variable neighborhood, according to the basic paradigm. The other two
are hybrids in combination with some of the previously described methods.
6.3.5.11 Hybrid Algorithms
Usually, there are combinations of two or more metaheuristic algorithm. Most of
the time hybrid algorithm produces better solution for the same problem than single
algorithm. Some references are listed below:
Fleurent and Ferland (1994) applied a new hybrid procedure that combines
genetic operators to existing heuristic, local search and tabu search to solve the
quadratic assignment problem (QAP). Li et al. (1994a, b) proposed genetic algorithm for the QAP which incorporates the construction phase of the GRASP to
generate the initial population. Battiti and Tecchiolli (1994), Bland and Dawson
(1994), Chiang and Chiang (1998) use tabu search with simulated annealing.
Bölte and Thonemann (1996) presented a combination of simulated annealing
and genetic algorithm. Dorigo et al. (1997) presented HAS-QAP, an hybrid ant
colony system coupled with a local search, applied to the quadratic assignment problem. HAS-QAP is compared with some of the best heuristics available for the QAP:
two taboo search versions, that is, robust and reactive taboo search, a hybrid genetic
algorithm, and simulated annealing. Ahuja et al. (2000), Drezner (2003) introduced
a genetic algorithm incorporates many greedy principles in its design and, hence
refer to it as a greedy genetic algorithm. Youssef et al. (2003) use tabu search, simulated annealing and fuzzy logic together. Balakrishnan et al. (2003) used GATS,
a hybrid algorithm that considers a possible planning horizon, which combines genetic with tabu search and is designed to obtain all global optima.
Misevicius (2003) propose a modified simulated annealing algorithm for the
QAP-M-SA-QAP combined with a tabu search. Misevicius (2004) introduced new
results for the quadratic assignment problem used an improved hybrid genetic procedure. Dunker et al. (2004) combined dynamic programming with evolutionary
computation for solving a dynamic facility layout problem.Tseng and Liang propose (2005) a hybrid metaheuristic called ANGEL to solve QAP. ANGEL combines
the ant colony optimization (ACO), the genetic algorithm (GA) and a local search
method (LS).
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After introducing metaheuristic methods, we bring a chart to shows the distribution of references by metaheuristic resolution methods.
6.3.6 Comparing QAP Algorithms
From the computational results which were gathered out of many papers, it is clear
that hybrid algorithms almost perform better than single heuristic or metaheuristic
algorithms. And from the frequency of resolution methods (Fig. 6.7) we understand
that, although heuristic and metaheuristic algorithm do not guarantee the optimal solution, because of their lower computational time than exact algorithms, they apply
further. There are some results with their reference that bring below:
Battiti and Tecchiolli (1994) compared simulated annealing (SA) and Tabu
search (TS) on the quadratic assignment problem. A recent work on the same
benchmark suite argued that SA could achieve a reasonable solution quality with
fewer function evaluations than TS.
Orhan and Cigdem (2001) applied a fuzzy tabu on various sized QAPs. The
developed algorithm is different from the tabu search approaches. The obtained
results show that fuzzy tabu search algorithms is more dominant than other algorithms in terms of the quality of the solutions found and the number of the points
searched in solution space (and tabu search appears to be the most effective local search approach to the QAP and also for many QAP instances of QAPLIB
(Bullnheimer 1998).
Ahuja et al. (2000) test greedy genetic algorithm on all the benchmark instances
of QAPLIB, a well-known library of QAP instances. Out of the 132 total instances
in QAPLIB of varied sizes, the greedy genetic algorithm obtained the best known
solution for 103 instances, and for the remaining instances (except one) found solutions within 1% of the best known solutions.
Misevicius (2003) proposed a modified simulated annealing algorithm for the
QAP-M-SA-QAP combined with a tabu search We tested our algorithm on a number
of instances from the library of the QAP instances – QAPLIB. The results obtained
from the experiments show that the proposed algorithm appears to be superior to
earlier versions of the simulated annealing for the QAP. The power of M-SA-QAP
is also corroborated by the fact that the new best known solution was found for the
one of the largest QAP instances.
40
30
20
10
0
SS
VNS
GRASP
GA
NNO
TS
Fig. 6.7 Publications: metaheuristics used to the QAP (Hahn 2007)
AC
SA
HA
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Quadratic Assignment Problem
135
Tseng and Liang (2005) applied hybrid metaheuristic called ANGEL combines
the ant colony optimization (ACO), the genetic algorithm (GA) and a local search
method (LS) to QAP. Over a hundred instances of QAP benchmarks were tested
and the results show that ANGEL is able to obtain the optimal solution with a high
success rate of 90%.
Gambardella et al. (1999), in this paper presents HAS-QAP, an hybrid ant colony
system coupled with a local search, applied to the quadratic assignment problem.
HAS-QAP is compared with some of the best heuristics available for the QAP:
two tabu search versions, that is, robust and reactive tabu search, an hybrid genetic
algorithm, and simulated annealing. Experimental results show that HAS-QAP and
the hybrid genetic algorithm are the best performing on real world, irregular and
structured problems.
Merz and Freisleben (1999) investigated Memetic algorithm (MA) on a set of
problem instances containing between 25 and 100 facilities/locations. The results
indicate that the proposed MA is able to produce high quality solutions quickly.
A comparison of the MA with some of the currently best alternative approachesreactive tabu search, robust tabu search and the fast ant colony system-demonstrates
that the MA outperforms its competitors on all studied problem instances of practical interest.
Tsutsui (2007) applied the cunning ant system (CAS) and ACO to QAP, the result
showed promising performance of CAS on QAP.
6.4 Case Study
In this section we introduce some case study in real world concerning with QAP.
6.4.1 Hospital Layout as a Quadratic Assignment Problem
(Elshafei 1977)
The problem of locating hospital departments so as to minimize the total distance
traveled by patients can be formulated as a quadratic assignment problem. The hospital concerned (the Ahmed Maher Hospital) is located in a rather densely populated
part of Cairo. It is composed of six major departments: out-patient, in-patient, dental
research, accident and emergency, physiotherapy and housekeeping and maintenance, each department occupying a separate building. The out-patient department
is more overcrowded with the average daily number of patients now exceeding 700
people and patients having to move among the 17 clinics in the department. The locations of the clinics relative to each other has been criticized for causing too much
traveling for patients and for causing bottlenecks and serious delays. It was therefore
decided to improvement in the layout of the department to reduce the total distance
traveled by patients and hence in the frequency of bottlenecks and congestions.
136
M. Bayat and M. Sedghi
The cost of the original layout is 13,973,298 the cost of the best solution obtained
is 11,281,887, so we obtain approximately the total 19.2% decreasing in distance.
The full solution required 136 seconds CPU time on an IBM 360140, the initial
solution being obtained after 44 s CPU time so the remaining 92 s being taken by
part B. The initial solution was 16.4% better than the original layout, a further 2.8%
improvement being obtained from part B.
6.4.2 Backboard Wiring Problem (Steinberg 1961)
In 1961, Leon Steinberg’s paper described a backboard wiring problem (SWP)
that concerns the placement of computer components so as to minimize the total
amount of wiring required to connect them. In the particular instance considered by
Steinberg 34 components with a total of 2,625 interconnections are to be placed on a
backboard with 36 open positions. To formulate the wiring problem mathematically
it is convenient to add two Dummy components with no connections to any others
so that the numbers of components and locations are both n D 36 so for showing
the answer he considered a 6 6 rectangle that each pieces shows the location of
the components. After solving the problem with branch- and bound algorithm the
result shows that PB and the related QPB perform very poorly and the performance
of GLB is reasonable, and although the dual LP and polyhedral bounds are better
the computational cost of these bounds is many orders of magnitude higher than that
of GLB. The computation to obtain TDB is also much greater than that required for
PB or GLB.
6.4.3 Minimizing WIP Inventories (Benjaafar 2002)
Benjaaafar (2002) use the model to introduce a formulation of the facility layout
design problem where the objective is to minimize work-in-process (WIP) they
show that layouts obtained using a WIP-based formulation can be very different
from those obtained using the conventional quadratic assignment problem (QAP)
formulation. For example, a QAP-optimal layout can be WIP-infeasible. In general, they show that WIP is not monotonic in material-handling travel distances so
it is possible to reduce overall distances between departments but increase WIP,
Furthermore, they find that the relative desirability of a layout can be affected by
changes in material-handling capacity even when travel distances remain the same.
After solving this problem with branch- and bound algorithm layout that minimize
pf (probability of full handling), does not necessarily minimize pe (probability of
empty handling) and we can have a layout with similar pf but different pe so QAP
model does not guarantee to be feasible.
6
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137
6.4.4 Zoning in Forest (Bos 1993)
Bos (1993), proposed Zoning the Waterbloem a national forest in southern of
Netherlands, where less than 10% covered by forest, and formulate as QAP so it
can take apart of land quality, management goals and interaction between environment of the forest for assigning a particular grid to particular use. Waterbloem is
presented by 84 grids and its environment is 80 grids and four utility identify:
1.
2.
3.
4.
Timber production
Nature conservation
Recreation
Timber production & dispersed recreation
Other area is assign as 5 – residential area, 6 – forest, 7 – agriculture or 8 – road.
QAP model solved using simulated annealing method with different weighted
the table of interaction between uses and the results of problem are available in the
referenced article.
6.4.5 Computer Motherboard Design Problem (Miranda 2005)
The model have N electronic components and N location and the goal is minimize
the distance between components so to control the temperature put all heat resource
component together and call it hot – spot and determine the optimal placement.
Miranda for solving the electronic motherboard design problem described it as a
quadratic assignment problem with additional linear costs and solves it by Bender
decomposition heuristic algorithm. At the end they find that QAP model design
produce lower temperature (there are some picture in the reference article).
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Chapter 7
Covering Problem
Hamed Fallah, Ali NaimiSadigh, and Marjan Aslanzadeh
In many covering problems, services that customers receive by facilities depend on
the distance between the customer and facilities. In a covering problem the customer
can receive service by each facility if the distance between the customer and facility
is equal or less than a predefined number. This critical value is called coverage
distance or coverage radius and shown by Dc.
Church and ReVelle (1974) modeled the maximization covering problem. Covering problems are divided into two branches; tree networks and general networks,
according to their graph. In addition, these problems are divided into two problems:
Total covering and partial covering problems, based on covering all or some demand
points.
The total covering problem is modeled by Toregas (1971). Up to the present
time many developments have occurred about total covering and partial covering
problems in solution technique and assumptions.
Covering problem has many applications such as: designing of switching circuits,
data retrieving, assembly line balancing, air line staff scheduling, locating defend
networks (at war), distributing products, warehouse locating, location emergency
service facility (Francis et al. 1992).
Let us first introduce the concept of covering a demand point with an example.
Consider the tree network Fig. 7.1:
Demand points in this problem are A, B, C and D. The distance between each
two demand points, is shown on their connecting arc. Consider that we want the
demand of point A to be covered. Coverage distance is supposed to be 5. Thus in
order to cover point A, we should locate at least one facility on the network (feasible
space of problem), in a way that it’s distance from point A is equal or less than 5.
Therefore, the demand point A will be covered if at least one facility is located on
one of thick lines, in Fig. 7.2:
Consider three candidate locating sites I, II and III (that III is conformed on
demand point D). Among these three places only II can cover point A; it means that
one facility is located in place II, the demand point A will be covered.
Note that the coverage distance (Dc) can be a kind of time or cost. For example,
if the walking time from a residential region to a store is equal or less than 5 min the
demand of that region will be covered.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 7, c Physica-Verlag Heidelberg 2009
145
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H. Fallah et al.
Fig. 7.1 Covering a demand
point
15
A
C
8
10
B
D
Fig. 7.2 The space that can
cover A
I
5
A
C
5
5
II
B
D III
7.1 Problem Formulation
The covering problem is one of well known problems of binary programming. Feasible solution space of this problem is a network (graph).
In all problems of this chapter, the capacity of facilities is considered to be unlimited. In addition, facilities are desirable; therefore, the nearness of them to the
demand points is interesting.
We want to cover all demand points with minimum possible cost. The general
parameters of the problem are as follow:
i D 1; 2; : : : ; m
Index of demand points
M
The number of demand point
j D 1; 2; : : : ; n
Index of candidate locating points
n
The number of candidate locating point
fj
The cost of locating a facility on candidate locating point j
aij : is 1 if candidate locating point j can cover the demand point i , otherwise is 0.
For perception of the concept of aij consider Fig. 7.3:
The demand points are A and B and the distance between them is 15 km. Assume
than coverage distance is Dc D 5 km, thus the reduced feasible solution space is
containing of points A, B and two intersection points shown by Fig. 7.4.
As an instance since candidate locating point 2 cannot cover the demand of B,
aB2 D 0 and since candidate locating point 1 can cover the demand of A, aA1 D 1;
thus:
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Fig. 7.3 An example for
perception of the concept of
aij
A
B
i=A
i=B
Fig. 7.4 The reduced
solution space
j=1
8̂
aA1 D 1
ˆ
ˆ
<
aA2 D 1
ˆ
ˆaB1 D 0
:̂
aB2 D 0
Decision variable of model:
j=2
j=3
j=4
A
B
i=A
i=B
8̂
aA3 D 0
ˆ
ˆ
<
aA4 D 0
ˆ
ˆaB3 D 1
:̂
aB4 D 1
Xj : is 1 if a facility be located on place j , otherwise is 0
Thus the total covering problem model is as follows:
Min
n
X
fj Xj ;
(7.1)
j D1
Subject to
n
X
aij Xj 1I 8i;
(7.2)
j D1
Xj D 0; 1I 8j:
(7.3)
Equation (7.1) minimized total locating costs and (7.2) ensures that all demand
points will be covered. Note that the (7.2) illustrates the number of located facilities that can cover demand of i . In other words (7.2) explains that for each demand
point i . We should locate at least one facility in one of places that can cover that
demand point.
7.2 Total Covering Problem
The total covering problem is divided into two branches: Tree network and general
network (cyclic network), there is a simple solution for tree networks because of no
cycle and it will be described in Sect. 7.2.1.
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Fig. 7.5 Example of general
network
A
B
C
DC
DC
A
A
B
B
DC
DC
DC
Reduced solution
space
DC
C
C
Fig. 7.6 Reducing solution space from continuous space to set points (Daskin 1995)
The general networks (Cyclic networks), are networks that there are more than
one rout between some or all of their nodes.
These networks have a closed loop as their name implies. Consider network
below:
In Fig. 7.5 the demand points are A, B, C, the coverage distance is Dc and the
feasible solution space of this problem is the set of points of network.
It seems very difficult to determine the location of facilities and to enter them
to the mathematical model as quantities. It is proved that instead of all points of
network (Daskin 1995) we can consider a reduced solution space including demand
points and intersection points. Intersection points are points that their distance from
at least one of demand points is exactly equal to coverage distance. As we can see
in Fig. 7.6 the solution space is reduced from a continuous space to a set of nine
candidate locating points.
Thus all of covering problems have finite candidate locating points (that are always demand points) or we can reduce their space to discreet solution space, so in
this chapter we consider the solution space to be discreet.
If the cost of locating is equal in different places, we have
fj D 1I
j D 1; : : : ; n;
(7.4)
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149
So the objective function of the model will be the minimization of number of required facilities covering all demand points.
X
Xj :
(7.5)
Min
j
The problem above is known as classical total covering problem.
The covering problem often has alternative optimal solutions, so we can add more
objectives to the total covering problem.
7.2.1 Maximizing the Number of Points Covered More than Once
Consider a state that we want to locate the facilities such that the number of demand
points covered two times, is maximized, in addition to cover all demand points with
minimum number of facilities, so that, if a facility is not able to service, another
facility works as a supplementary.
We define:
M D the number of demand points.
Si : is 1 if demand points i is covered at least two times, otherwise is 0
The model of this problem is as follows:
X
X
Si ;
Xj
Min.M C 1/
(7.6)
i
j
Subject to
X
aij Xj Si 1;
(7.7)
j
Xj D ı; 1I
8j ;
(7.8)
Sj D ı; 1I
8i :
(7.9)
P
Equation (7.7) ensures that Si is 1 if aij Xj is greater than or equal 2. Moreover,
since the coefficient Si in the objective function is 1, the variable Si will be 1
to satisfy the optimal solution. The weight .M C 1/ in the objective function ensures that the number of required facilities is not more than primary model of total
covering problem.
7.2.2 Multiple Total Covering Problems (Mirchandani et al. 1990)
Classical total covering problem is indeed a special kind of multiple total covering
problem. In this problem we want to cover each demand point bi times, based on
it’s importance (that i D 1; : : : ; m and b D 1; 2; : : :). In addition at each candidate
locating maximum Uj facilities can be located. .Uj D 1; 2; : : : I j D 1; : : : ; n/:
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The model of this problem is as follows:
Min
n
X
fj Xj ;
(7.10)
i D 1; : : : ; m;
(7.11)
j D1
Subject to
n
X
aij Xj bi ;
j D1
ı Xj uj I
Xj integerI j D 1; : : : ; n:
(7.12)
It is obvious that this problem has feasible solution if:
n
X
aij uj bi I
i D 1; : : : ; n:
(7.13)
j D1
In this model for some point it isn’t sufficient to facilities be available and the demand value or importance of demand of each point specifies the number of needed
facilities (Mirchandani and Francis 1990).
7.2.3 Total Covering Problem with the Preference of Selecting
Location of Existing Facilities (Daskin 1995)
We want to select solutions including points of existing facilities among optimal
solutions of total covering problems.
So we can define:
Je : Set of locations of existing facilities
Jn : Set of new candidate locating points
": an infinitesimal
The model of the problems is as follows:
X
X
Min
Xj C .1 C "/
Xj ;
j 2Je
(7.14)
j 2Jn
Subject to
X
aij Xj C
X
aij Xj 1I
j 2Je
j 2Jn
Xj D ı; 1I
j 2 fJe [ Jn g :
8i ;
(7.15)
(7.16)
In order to number of facilities not be greater than the optimal solution of initial
total covering model (classical model), it is necessary to have " < 1= jJn j. The large
value for coefficient Xj jj 2 Jn in objective function causes the model to tend more
to selecting locations having available facilities (Daskin 1995).
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151
7.2.4 Total Edge Covering Problem (Daskin 1995)
All problems studied in this chapter yet, discussed about covering or not covering
demand points (nodes of network).
There are problems in practice that we should cover edges of graph.
For example in locating police stations.
Edge covering problem similar to covering nodes of graph, can be discussed with
different objectives and constraints. In this chapter we consider total edge covering
problem with objective minimizing cost of locating.
As before we can reduce feasible solution space to set of demand points and
intersection points. To simplify the problem we assume that each edge should be
totally covered only by one node and we can not cover a part of an edge by a node
and cover other part of it by another node.
But to have feasible solution for the problem, we should have:
(Length of longest edge) – (coverage distance) (coverage distance).
To prove this claim, consider Fig. 7.7.
If we can cover the longest edge of above network by one facility, we can certainly cover all shorter edges.
In Fig. 7.7, AB is the longest edge. If we can not cover an edge by any of candidate locating points on it, certainly we can not cover it totally by out of that edge.
Thus to have feasible solution X1 (or X2 ) should cover total edge AB, therefore the
distance between X1 and B (or between X1 and A) should be less than or equal to
coverage distance.
To solve the problem of lack of any feasible solution on edges violating the above
condition, we consider one (or more) artificial edge(s) and thus decompose each
edge to two or more edges in a way that above condition be satisfied. These artificial
nodes will be added to the set of candidate locating points.
A
Dc
X2
X1
Dc
Dc
B
Dc
Dc
C
Fig. 7.7 A network for proving relation (Daskin 1995)
Dc
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k
k
i
j
i
j
Fig. 7.8 Example (Daskin 1995)
Parameters of problem:
G .V; E/
The graph of problem
Set of nodes
Set of edges
V
E
i D 1; : : : ; m
ij 2 E.i < j /
dij
K D 1; : : : ; n
Fk
Dc
indices of nodes of network
indices of edges of network
Distance of nodes i and j
indices of candidate locating points
Cost of locating at candidate locating point k
Coverage distance
aij k : is 1 if candidate locating point k can cover edge k completely, otherwise is 0.
Parameter aijk nodes more explain. Consider Fig. 7.8, in part a candidate locating
point k is on the edge ij. Here the point k can cover all of edge ij if it can cover both
i and j .
It means:
aij k D ai k :aj k
But in part (b) point k can cover all of edge ij if:
˚
Max fı; DC dki g C Max ı; DC dkj dij :
(7.17)
The statement above .Dc dki / shows a part of edge ij that can be covered by node
k by node i and .Dc dkj / shows a part of edge ij that can be covered by node k
by node j . Obviously, whole of edge ij will be covered if sum of these two parts is
greater than or equal to the length of edge ij. Decision variable of problem:
Xk : is 1 if a facility is located of candidate point k, otherwise is 0
The model of this problem is as follows:
Min
n
X
kD1
fk Xk ;
(7.18)
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153
Subject to
n
X
aijk Xj 1I 8ij 2 EI i < j ;
(7.19)
kD1
Xj D 0; 1:
(7.20)
As we can see, the model is similar to the total covering model and the only difference is in computing coefficients of coverage matrix .aij k / (Daskin 1995).
7.2.5 Notes on Total Covering Problems
7.2.5.1 Total Covering Problem with Fuzzy Matrix of Coverage Coefficients
(Chiang et al. 2005)
There are problems in practice that we can not exactly talk about covering a demand
point by a candidate locating location.
In other words, the possibility of covering demand point i by facility j , that is
shown by j .i / has a value which belongs to interval [0,1].
The objective is that with minimum cost of locating facilities the possibility of
covering each demand point is greater than or equal to a predefined value ˛.
Thus the primary model of the problem is as follows:
Min
n
X
(7.21)
Cj Xj ;
j D1
Subject to
1
n
Y
.1 j .i / Xj / ˛I
i D 1; : : : ; m;
(7.22)
j D1
Xj 2 f0; 1g :
(7.23)
That Cj is the cost of locating a facility at point j W.j D 1; 2 : : : n/.
Chiang et al. (2005) proved that we can transform the (7.21) of above model to
the following linear model:
Subject to
n
X
j D1
ln.1 j .i // Xj ln.1 ˛/I
i D 1; : : : ; m;
(7.24)
Note that in this model we don’t consider ˛ D 1 and ı < ˛ < 1 (Chiang et al. 2005).
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7.2.5.2 Symmetrical Total Covering Problem (The Lottery Problem)
(Jans and Degraeve 2008)
In a lottery game, n numbers are randomly drawn from a set of m numbers. On
a lottery ticket, we fill out n number hoping that they will match the n numbers
selected. We want to know the minimum number of tickets we have to fill out in
order to ensure that there is at least one ticket which has p or more matching numbers. We call this a .m; n; p/-lottery problem, with m n p. Of course, we also
want to know which specific numbers we have to fill out. We will show how this
problem can be formulated as an integer linear programming (ILP) problem, more
specifically a set-covering problem.
Define S as the set of all possible tickets. This same set S also defines all possible
draws, as a selection of n numbers out of m defines both a possible draw and a
possible ticket. For the IP formulation, we define a variable for each possible ticket:
xj D 1 if ticket j is filled out, 0 otherwise, 8j 2 S . Further, define Si as the set of
all tickets which have at least p out of n numbers in common with draw i . Due to the
equivalence between a ticket and a draw, Si is also the set of all draws which have at
least p out of n numbers in common with ticket i . Finally, we define the parameters
of the coefficient matrix as follows: aij D 1 if ticket j 2 Si ; D 0 otherwise.
The objective is to minimize the number of tickets that we have to fill out (7.24).
There is only one set of constraints imposing that for each possible outcome, there
must be at least one ticket filled out which has at least p numbers in common with
that outcome (7.25). The variables for the tickets must be binary (7.26). The resulting formulation is a set covering problem:
Min
X
xj ;
(7.25)
j 2S
Subject to
X
aij xj 1 8i 2 S ;
(7.26)
j 2S
xj 2 f0; 1 g :
(7.27)
We characterize the cardinality of the sets for a general (m, n, p)-lottery. The total
number of variables equals the number of combinations of n elements out of m (Jans
and Degraeve 2008):
jS j D m
n :
7.2.5.3 Stochastic Total Covering Problem for Destructive
and Constructive Cases (Hwang 2004)
In all works done yet there was the assumption that the value of inventory always
remains constant.
Special cases of stochastic total covering location are studied for destructive and
constructive instances. This problem can be formulated using stochastic total covering that is solvable with binary programming.
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Covering Problem
155
Objective: minimizing the number of storing facilities in a set of discreet points.
Constraint: probability of covering each point should not be less than a critical
value.
Presumptions:
Passing time causes improvement or decay in inventory
Distances can be of any type
Decision variable: Xj
Xj : is 1 if a facility is located at point j, otherwise is 0
Parameter:
Fij : Sum of transportation costs in each period and costs of constructive or destructive
Ai : Service level needed for demand point i
ri : Critical value of probability of covering demand point i
The small model is as follows (Hwang 2004):
Min
n
P
xj ;
(7.28)
j D1
Subject to
n
P
aij xj 1 8i D 1; : : : ; m;
j D1
xj D f0; 1g 8j D 1; : : : ; n;
(
1 if p.Fij Ai / ri
aij D
:
Otherwise
0
(7.29)
(7.30)
(7.31)
7.3 Partial Covering Problem
Total covering problem can not cover all location problems in real world, because
in many problems budget constraints and other constraints do not let us cover all
points. For example, consider that we need four facilities to cover all demand points
but budget constraint does not let us locate more than three facilities, thus covering
problem doesn’t have feasible solution.
On the other hand, the total covering problem considers all demand points similarly regardless of their demand. For example in a total covering problem, satisfying
a demand point that has 10 units demand, has the same importance with satisfying
a demand point that has 1,000 units demand.
To confront such problems, another covering problem is proposed that called
partial covering problem. In this part we will introduce some samples of it.
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7.3.1 Minimizing Costs Arisen from Not Covering Demand Points
(Mirchandani and Francis 1990)
In this problem, maximum number of facilities is limited and equal to P . In other
words the problem is exogenous and Pi is penalty cost of not covering demand
point i . In addition consider the number of demand points is m and the number of
candidate locating points is n.
Decision variables of problem are:
Xj : is 1 if a facility is located at j , otherwise is 0
Zi : is 1 if the demand point i is not satisfied, otherwise is 0
The model of this problem is as follows:
Min
m
X
(7.32)
pi Zi ;
i D1
Subject to
n
X
j D1
aij Xj C Zi 1 I i D 1; : : : ; m;
n
X
(7.33)
Xj P ;
(7.34)
Xj 2 f0; 1gI j D 1; : : : ; n;
(7.35)
j D1
Zi 2 f0; 1g
I
i D 1; : : : ; m:
(7.36)
Assumption above the objective of this model is to minimize total penalty costs
of
P points that their demand is not satisfied. Equation (7.32) ensures that Zi is 1 if
aij Xj is zero; it means that their demand is not satisfied. Equation (7.33) ensures
that the number of facilities can not be more than P .
If costs of locating facilities at different points are different, it means we have
Fj D the cost of locating a facility at candidate point j .
We can replace constraint of the number of facilities represented by (7.33) with
capital constraint as follows:
n
X
Fj Xj C :
(7.37)
j D1
C is minimum capital to locate facilities (Mirchandani and Francis 1990).
7.3.2 Minimizing Costs of Locating Facilities and Costs Arisen
from Not Covering Demand Points
In this condition, there is no limitation to number of facilities (or maximum budget
of locating facilities), in other words the problem is endogenous.
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157
The objective is to minimize sum of locating costs and penalty costs arisen from
not covering demand points.
Model of this problem is as follows:
Min
n
X
j D1
Fj Xj C
m
X
pi Zi ;
(7.38)
i D1
Subject to
n
X
j D1
aij Xj C Zi 1 I i D 1; : : : ; m;
(7.39)
Xj 2 f0; 1g I j D 1; : : : ; n;
Zi 2 f0; 1g I j D 1; : : : ; m:
(7.40)
(7.41)
Fj is cost of locating a facility at point j and Pi is penalty
Pof not covering demand
aij Xj is zero, it means
point i in (7.37). Equation (7.38) ensures that Zi is 1 if
that demand point i is not covered.
7.3.3 Maximum Covering Location Problems (Berman
et al. 2003)
One of most famous partial covering problem is maximum covering location problem.
This problem proposed by Church and ReVelle (1974), they limited candidate
locating points to the nodes in network (Berman et al. 2003).
The objective of this problem is to maximize total satisfied demands. In this
problem, maximum number of facilities is limited (the problem is exogenous).
The model is as follows:
X
hi Zi ;
(7.42)
Max
i
Subject to
Zi
X
aij Xj I 8i ;
(7.43)
j
X
Xj P ;
(7.44)
Xj D 0; 1I 8j;
(7.45)
Zi D 0; 1I
(7.46)
j
8i:
Equation (7.42) states that we want to minimize total demands of satisfied demand
points. hi is demand point i and Zi is one if demand of point i is covered.
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This model similar to the classic covering problem, we can reduce the size of
problem using (column deletion rule) but we can not use any of (row deletion rules)
(see Sect. 7.2.1).
7.3.4 Expected Maximum Covering Problem (Daskin 1995)
So far, we assumed that in order to cover each demand point, it is enough that each
facility be located in a distance lower than or equal to coverage distance from that
demand point. This assumption is valid if the capacity of facilities is assumed to be
unlimited. Now consider a simple network in Fig. 7.9. Assume that Dc D 8.
The amount of demands and distance of demand points are shown in the picture.
The reduced feasible solution space of this problem is including of points A and B
and two intersection points shown by Fig. 7.10.
For example if we locate a facility at point j D 2 and assuming that the capacity
is unlimited, total demand of network (maximum demand) will be covered.
Now assume that the facility can service only one of demand points each time.
In this condition if demand points A and B need this facility simultaneously the
facility can service one of demand points in practice and the other point will not be
covered.
Thus if the capacity of the facilities is limited, they must be available in addition to cover demand points. But since availability or unavailability of facilities is
not precisely predictable each time, the objective of this problem is to maximize
expected coverage by available facilities.
qD
Sum of hours that facilities worked in a period
:
Number of facilities used
Assume that the probability of being unavailable for each facility at any moment is
similar and equal to q. the amount of q can be estimated using past data. A way to
estimate amount of q is as follows:
Assume that the number of facilities that can cover point i is ni . Thus probability
of covering demand point i is equal to probability that at least one of facilities be
available
A
B
14
10
15
Fig. 7.9 The network with Dc D 8 (Daskin 1995)
j=1
Fig. 7.10 The reduced
network (Daskin 1995)
A
j=2
j=3
j=4
B
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159
Then we have:
1 q ni :
(7.47)
Thus expected value of covered demand is:
m
X
i D1
hi .1 q ni /:
(7.48)
Above statement is not linear, but using (7.48) and property of objective function, it
can be easily transformed to linear form.
1 q ni D .1 q/.1 C q C q 2 C : : : : C q ni 1 /:
(7.49)
Zi k : is 1 If the demand point i is covered at least k times, otherwise is 0
P D Maximum number of facilities
Xj D number of located facilities at candidate locating point j
Note that Xj is not necessarily zero or one. Indeed the more number of facilities
at a candidate locating point, the more probability of availability of at least one
facility for covered demand points by that point. Now we place instead of 1 q n i
.1 q/.Zi1 C qZi 2 C : : : C q p1 Zip /:
(7.50)
And we have the constraint below:
p
X
kD1
Zi k D
X
(7.51)
aij Xj :
j
It means the number of Zi k that is one, is exactly equal to the number of facilities
that can cover demand point i .
Consider graph in Fig. 7.11:
Assume that the coverage distance is 8 and two facilities are located at A and B.
Thus right hand side of equality (7.52) for demand point A is 1 and we have:
ZA;1 C ZA;2 D 1:
(7.52)
Now we should prove that optimal solution is necessarily ZA;1 D 1; ZA;2 D 0 and
Solution ZA;1 D 0 and ZA;2 D 1 is not candidate of optimality. Since the objective
function is max and coefficient Zi k decreases with increasing k, the model prefers to
A
B
14
10
Fig. 7.11 Graph of covered demand (Daskin 1995)
15
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H. Fallah et al.
Zi k with smaller k be one and thus for example in this problem, solution ZA;1 D 0
and ZA:2 D 1 is not candidate of optimality.
The model of expected maximum covering problem is as follows (Daskin 1995):
Max
X
hi
i
P
X
kD1
.1 q/.q k1 Zi k /;
(7.53)
Subject to
P
X
Zix D
P
X
Xj P ;
kD1
j D1
Xj 0
n
X
j D1
and
aij Xj I 8i
(7.54)
(7.55)
integerI 8j;
Zi;k D 0; 1I 8i; k:
(7.56)
(7.57)
7.3.5 Maximum Covering Problem Considering Non-Ascending
Coverage Function (Berman et al. 2003)
One of presumptions of covering problem is that demand points in a critical distance
are covered and they will not be covered out of this distance.
For example in fire station location problem assume that the coverage distance is
3 miles. Thus if the distance of a residential region from the fire station is 2.9 miles,
in classic view of covering problem any amount of demand can not be covered by
that station. This point of view seems unreal in practice. Berman et al. (2003) developed a kind of covering problem that two coverage bounds replaced with coverage
distance.
As we can see in Fig. 7.12 if the distance between demand point and candidate
locating point as less than lower bound of coverage .l/ this point can be covered
totally. If their distance is more than upper bound .u/, the demand point will not
be covered and if their distance is between two critical bounds a proportion of its
demand will be covered .f /.
Berman et al. (2003) also proved that if the function f is convex the solution
space of problem will be reduced to a set of demand points and intersection points.
But here intersection points are points that their distances from at least one of demand points are equal to coverage lower bound or coverage upper bound.
We assume that function f is a linear function and define:
f .d.i; j // D
u d.i; j /
I
ul
i D 1; : : : ; mI
j D 1; : : : ; n:
(7.58)
Covering Problem
161
A proportion of demand of i that can be covered by the facility located at j
7
f
u
I
The distance between demand point and candidate locating point
Fig. 7.12 Relationship of Dc with demand will be covered (Berman et al. 2003)
Thus the proportion of demand of point i that can be covered by candidate locating
point j is equal to:
8̂
I
if d.i; j / l
<1
/
C.i; j / D ud.i;j
I if l < d.i; j / u :
ul
:̂ı
I
if d.i; j / > l
C.i; j /s are parameters of problem and should be determined before solving it.
Assume that hi is the demand of point i and P is maximum number of facilities.
The model of this problem is as follows:
Max
m
X
i D1
˚
hj M ax C.i; j /Xj :
j
(7.59)
Subject to
n
X
j D1
Xj P ;
Xj D 0; 1I
(7.60)
j D 1; : : : ; n:
(7.61)
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Equation (7.59) for each demand point i , choose a point among candidate locating
points j , nearer to demand point i and have more C.i; j /.
This value is indeed maximum coverage that demand point i receives from different facilities. As we can see the objective function of problem states sum of covered
demands.
Equation (7.60) limits maximum number of facilities to P .
Note that the above model can be transformed easily to linear model below
(Berman et al. 2003):
X
hi Yi ;
(7.62)
Max
i
Subject to
n
X
j D1
Xj P
Yi D
n
X
j D1
n
X
(7.63)
C.i; j /Zij ;
(7.64)
j D1
Zij D 1;
Zij Xj ;
Yi 0
Xj D 0; 1
Zij D 0; 1
(7.65)
i D 1; 2; : : : ; m;
j D 1; 2; : : : ; n;
i D 1; 2; : : : ; m & j D 1; 2; : : : ; n:
(7.66)
(7.67)
(7.68)
(7.69)
7.4 The Bi-Objective Covering Tour Problem (Jozefowieza
et al. 2007)
The aim is to investigate the solution of a multi-objective routing problem, namely
the bi-objective covering tour problem (BOCTP), by means of a cooperative strategy
involving a multi-objective evolutionary algorithm and a single-objective branchand-cut algorithm.
The BOCTP aims to determine a minimal length tour for a subset of nodes
while also minimizing the greatest distance between the nodes of another set and
the nearest visited node. The BOCTP can be formally described as follows: let
G D .V [ W; E/ be an undirected graph, where V [ W is the vertex set, and
E D f.vi ; vj /jvi ; vj 2 V [ W; i < j g is the edge set. Vertex v1 is a depot is
the set of vertices that can be visited V is the set of vertices that must be visited .v 2 T /, and W is the set of vertices that must be covered. A distance matrix
C D .cij /, satisfying triangle inequality, is defined for E. The BOCTP consists of
define a tour for a subset of V , which contains all the vertices from T , while at the
7
Covering Problem
163
same time optimizing the following two objectives: .a/ the minimization of the tour
length and .b/ the minimization of the cover. The cover of a solution is defined as
the greatest distance between a node w 2 W , and the nearest visited node v 2 V .
One generic application of the CTP involves designing a tour in a network whose
vertices represent.
Points that can be visited, and from which the places that are not on the tour can
be easily reached.
For instance, Toregas et al. (1971) have used the CTP to model the determination
of a tour for a mobile.
Medical facility in an area of Ghana, where every village cannot be reached
(Jozefowieza et al. 2007).
7.5 A Fuzzy Multi Objective Covering Based Vehicle Location
Model for Emergency Services (Araz et al. 2007)
Timeliness is one of the most important objectives that reflect the quality of emergency services such as ambulance and firefighting systems. To provide timeliness,
system administrators may increase the number of service vehicles available. Unfortunately, increasing the number of vehicles is generally impossible due to capital
constraints. In such a case, the efficient deployment of emergency service vehicles
becomes a crucial issue. The objectives considered in the model are maximization of the population covered by one vehicle, maximization of the population with
backup coverage and increasing the service level by minimizing the total travel distance from locations at a distance bigger than a pre specified distance standard for
all zones.
The model addresses the issue of determining the best base locations for a limited number of vehicles so that the service level objectives are optimized. Three of
the important surrogates that reflect the quality of emergency service systems are
considered as objectives in the model:
Maximization of the population covered by one vehicle
Maximization of the population with backup coverage and
Minimization of the total travel distance from locations at a distance bigger than
a pre specified distance standard for all zones.
The proposed model allows the incorporation of decision maker’s imprecise aspiration levels for the goals by means of FGP approach.
Timeliness can be measured in many ways such as:
Minimization of the total or average time to serve all emergency calls;
Minimization of the maximum travel time to any single call;
Maximization of area coverage (ensures that as many zones in the area as possible
is covered within S minutes of travel)
Maximization of call coverage (ensures that as many calls in the area as possible
is covered within minutes of travel).
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A new model, formed by considering maximal backup coverage model of Hogan
and ReVelle (1986) and capacitated maximal covering model of Pirkul and Schilling
(1991) as the base, is introduced in this section. The proposed model formulation is
stated to allocate a fixed number of emergency service vehicles to previously defined
locations so that three important service level objectives can be achieved.
(
(
1 If dij S
1
eij D
aij D
0 If dij > S
0
(
1 If demand node i is covered once
Yi D
:
0
Otherwise
If dij >S
If dij S
where Ui is the fraction of population in zone i covered by more than one vehicle,
hi is the population in zone i; Pij is the fraction of population in zone i served by a
facility/facilities located in zone j; dij is the travel distance or time from j to i; S
is the distance or time standard, C is the number of vehicles to be located, and kj is
the workload capacity for a vehicle located in zone j . The mathematical statement
of the proposed model is as follows:
Max Z1 D
Max Z2 D
Min Z3 D
X
X
hi Yi ;
(7.70)
hi Ui ;
(7.71)
i
XX
i
eij hi dij Pij ;
(7.72)
j
Subject to
P
j
aij P Y U 0 8i 2 I;
ij
i
i
Ui Yi 0 8i 2 I ;
P
Pij 1 8i 2 I ;
j
P
(7.73)
(7.74)
(7.75)
hi Pij Kj Xj 8j 2 J ;
(7.76)
Xj C;
(7.77)
Ui 1 8i 2 I ;
(7.78)
Xj 1 & integer 8j 2 J ;
(7.79)
Yj D Œ0; 1 8j 2 J ;
(7.80)
Pij 1 8.i; j /:
(7.81)
i
P
j
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165
In this model, first and second objectives maximize the population covered by at
least one vehicle and the population with backup coverage, respectively. Third objective provides the maximization of the service level by minimizing the total travel
distance from locations at a distance bigger than S for all zones. Populations of
the zones are included as weights in the third objective. The first two constraints;
determine which zones receive backup coverage. The first constraint set determine
the number of facilities within the coverage distance of a zone. The second constraint ensures that backup coverage can only be provided if first coverage already
exists. Equation (7.75) provides that the entire population at each zone will be assigned to some facility. Equation (7.76) ensures that the total population assigned to
a facility does not exceed the capacity of that facility. This constraint also ensures
that population will only be assigned to facility sites which have facilities actually
located at them. Equation (7.77) limits the total number of facilities to be located.
In (7.78), Ui , is considered as a continuous variable to allow fractional backup coverage for a zone. We restrict the value of Ui in the range of [0, 1] so that as much
zone as possible can take the advantage of having backup coverage. As can be seen,
locating more than one vehicle in a zone is allowed with (7.79). In contrast to the
basic location covering models it is no longer assumed that the entire population in a
zone will be served by its nearest facility. In this formulation it is also assumed that
the entire population in a zone does not have to be assigned to the same vehicle and
that some portion of the population in a zone may be covered and the rest of it not
covered. It is provided by using the continuous form of the assignment variable, pij .
7.6 Solving Methods
7.6.1 Exact Methods
7.6.1.1 Solving Total Covering Problem in Tree Networks (Francis
et al. 1992)
In total covering problem we want to cover total demand by minimum number of
facilities. We assume that each demand point has its own coverage distance.
As we mentioned before a demand point is covered if at least one facility is
located at its coverage distance. Thus in this problem it should be located at least
one facility at coverage distance of each demand point.
In the following we explain the optimal solution algorithm of this problem by an
example.
Consider tree network shown by Fig. 7.13 (Francis et al. 1992), we want to cover
all demand points. The distances between demand points are shown on the edges of
network.
We suppose that the wavy lines emanating from vertices 1 through 5 are strings
whose lengths are the coverage distances. For example the coverage distance of
node 1, is 10.
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H. Fallah et al.
1
10
14
4
14
5
15
10
5
20
2
6
12
18
3
3
Fig. 7.13 The covering of tree network (Francis et al. 1992)
10
1
4
14
10
5
2
20
12
18
3
6
3
5
15
Fig. 7.14 Coverage distance of node 4 (Francis et al. 1992)
1 Suppose that we can locate a facility at any points on the network. The algorithm
begins from an end node of network and continues its string. As we can see in
Fig. 7.14 this string passing node 6 continues 4 units (the weighted line).
Now we eliminate node 4 and consider the remained part of string as a new string
and connect it to node 6 shown by Fig. 7.15.
We do the same for node 5 shown by Fig. 7.16:
Now node 6 acts as an end node, we consider shortest string connected to this
node. Since this string doesn’t reach to the adjacent node (node 2) we locate the first
facility at the end of this string and then eliminate node 6 shown by Fig. 7.17.
7
Covering Problem
167
10
1
4
14
10
4
5
20
2
6
12
18
3
5
3
15
Fig. 7.15 Eliminating node 4 (Francis et al. 1992)
1
10
4
14
10
4
5
2
20
12
18
3
3
6
3
5
Fig. 7.16 Eliminating node 5 (Francis et al. 1992)
With locating the first facility at X1 node 4, 5 and 6 are covered. Now we begin
from one of end nodes again. For example we continue the string connected to node
1 on the arc between 2 and 3. Since the end of this string does not reach to node 2,
we locate the second facility at the end of this string and eliminate node 1 shown by
Fig. 7.18.
168
H. Fallah et al.
10
1
4
14
10
5
20
2
4
6
X1
12
18
3
3
5
Fig. 7.17 Cating the first facility (Francis et al. 1992)
1
10
4
10
10
X2
4
5
2
20
4
6
X1
12
18
3
3
5
Fig. 7.18 Locating the second facility (Francis et al. 1992)
Since the end of string connected to node 2, reaches X 2, thus the second facility
can cover demand of point 2 and we can eliminate this node too shown by Fig. 7.19.
Now we continue the string connected to node 3 toward node 2. Since the end of
this string does not reach to node 2 we can locate the third facility at the end of this
7
Covering Problem
169
1
4
10
10
X2
4
5
2
20
4
6
X1
12
18
3
3
5
Fig. 7.19 Eliminating node 2 (Francis et al. 1992)
string and then all demand points of network are covered by minimum number of
facilities (3 facilities).
The steps of algorithm are as follows:
A: Continue strings connected to the end points toward the common node connected to them.
B: Locate a facility at the end of strings that is not reached to the adjacent node
and consider the remained part of it as a new string and eliminate the end nodes.
C: Continue the shortest string connected to the common node toward its adjacent
node if this string passed the adjacent node, consider the remained part of it
as a new string, otherwise locate a facility at the end of it and eliminate that
common node.
D: Do these steps until all demand points be covered.
2 Consider that we can locate only at nodes of network. With a simple revision in
algorithm, we can reach to optimal solution of this problem. For this aim, for
nodes that the end of their string does not reach to adjacent node we locate a
facility at the end of string, and for nodes that their string passes the adjacent
node locate a facility at the adjacent node.
7.6.1.2 Reduction Rules in Classic Total Covering Problem (Daskin 1995)
When all of the fixed costs are equal (i.e., fj D 1), we can often reduce the size of
the problem using a variety of reduction rules.
170
H. Fallah et al.
Fig. 7.20 An example
(Daskin 1995)
15
A
B
9
16
10
7
11
E
11
D
C
12
8
13
17
F
Consider network shown by Fig. 7.20 (Daskin 1995), the coverage distance is
Dc D 11:
We begin with a column reduction rule. Consider two columns j and k. If
aij ai k for all demand nodes i and aij < ai k for at least one demand node i ,
then location k covers all demands covered by location j . We say that location k
dominates location j . In this case, column j may be eliminated, since, if we were
to locate at node j , we could always do at least as well by locating at node k. In
addition, we can then set Xj D 0. For example, in the problem above, candidate
site D dominates nodes A and B (since a facility located at D will cover nodes A,
B, D, and E, while a facility located at A will only cover nodes A, B, and D, and a
facility at B will only cover nodes A, B, and D). Thus, we can set XA D XB D 0.
Similarly, candidate site C dominates site F (since a facility located at C covers
demand nodes C, E, and F , while a facility at -F covers only nodes C and F ). We
therefore set XF D 0. (Note that we can also eliminate all but one of a set of equivalent sites, where sites j and k are said to be equivalent if aij D ai k for all demand
nodes i .) After the column reductions described above, the problem becomes
Min XC C XD C XE ;
(7.82)
Subject to
XD 1;
(7.83)
XD 1;
XD C XE 1;
(7.84)
(7.85)
XD C XE 1;
XC C XD C XE 1;
(7.86)
(7.87)
XC 1;
(7.88)
XC ; XD ; XE D 0; 1:
(7.89)
We now consider row reduction P
techniques which allow us to eliminate rows from
the problem. Consider row i . If aij D 1, then there is only one facility site that
j
7
Covering Problem
171
can cover node i . In that case, we find the location j such that aij D 1 and
set Xj D 1. We can then eliminate any row in which Xj appears, since, with
Xj D 1, those constraints will be satisfied (i.e., those nodes will be covered by the
facility at location j ). For example, in the problem above, there is only one nonzero
coefficient in the first constraint. Therefore, we know that XD must equal 1. We set
XD D 1 and then eliminate the constraint corresponding to rows A, B, D, and E,
since these demand nodes will all be covered by the facility at location D. Similarly,
the constraint for node F has only one nonzero coefficient. Thus, we can set Xc D 1
and remove the rows corresponding to nodes C and F . At this point, there are no
remaining rows, and the problem becomes the trivial problem of minimizing XE
subject to the integrality constraint that XE equals either 0 or 1. Clearly, we set
XE D 0. At this point, we know the optimal value of all of the decision variables
and the problem is solved.
Note that despite the fact that this is an NP-complete problem whose optimal solution is technically difficult to obtain, we have been able to solve the
problem without resorting to any formal optimization technique (such as linear
programming).
The row and column reduction rules outlined above may be used iteratively until
neither rule allows us to eliminate a column or a row. Often, application of these
rules will allow us to solve the problem completely. This is not always the case,
however, as the following example shows. This example also allows us to introduce
an additional row reduction rule. In this example, we consider the network shown
in Fig. 7.20, but now use a coverage distance of 18 (i.e., Dc D 18).
After using the column reduction rule outlined above, we can eliminate the
columns corresponding to candidate sites A and F (and set XA D XF D 0).
Since no row has only a single element, we cannot use, the row reduction rule
outlined above to eliminate any rows. However, we can use a second row reduction
rule. Consider two rows m and n. If amj anj for all candidate sites j and amj <
anj for at least one candidate site j . then we can eliminate row n. This is so because
the requirement that demand node m be covered will guarantee that node n is also
covered. (Any facility site that covers demand node m also covers demand node n).
This rule allows us to eliminate the rows corresponding to nodes B and E, since
any facility that covers node A (i.e., a facility located at node B, C, or D) will also
cover nodes B and E. (As before, we can eliminate all but one of a set of equivalent
demand nodes, where demand nodes m and n are said to be equivalent if amj D anj
for all candidate sites j .).
Unfortunately, we cannot reduce the size of this problem any further. Repeated
application of the column reduction rule and the two row reduction rules to this
problem will not eliminate any more rows or columns. Thus, we must find some
other way to solve this problem. One way of doing so is to ignore the integrality
constraint and replace it by a nonnegative constraint.
If we solve the resulting linear programming problem, we find XB D Xc D
XD D XE D 1=3. The objective function is 4=3. Clearly, this is not an all-integer
solution; it does not solve the original set covering problem.
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H. Fallah et al.
7.6.1.3 The Branch and Bound Method (Daskin 1995)
To ensure that we obtain an all-integer solution, additional techniques will generally
be required. One approach is to use branch and bound.
After reducing the problem size using above rules, we can solve the model using optimal solving methods such as branch and bound or Balas algorithm (Daskin
1995).
7.6.2 Heuristic Methods
7.6.2.1 The Greedy Adding Algorithm (Daskin 1995)
This algorithm and its variants may be used to solve (at least approximately) a large
number of other location problems. The algorithm is known as a greedy algorithm
since it does what is best at each step of the algorithm without looking ahead to see
how the current decisions will impact on later decisions and alternatives.
If we were to locate only one facility (i.e., P D 1), we could solve the problem
optimally by simply evaluating
how many demands each candidate site covers (canP
a
h
demands)
and selecting the site that covers the most
didate site j covers
i ij i
demands (Daskin 1995).
7.6.2.2 Lagrangian Relaxation (Daskin 1995)
Lagrangian relaxation is an approach to solving difficult problems (such as integer
programming problems). The approach outlined below is cast in terms of solving a
maximization problem. The approach involves the following general steps:
1. Relax one or more constraints by multiplying the constraint(s) by Lagrange multiplier(s) and bringing the constraint(s) into the objective function.
2. Solve the resulting relaxed problem to find the optimal values of the original
decision variables (in the relaxed problem)
3. (Optional) Use the resulting decision variables from the solution to the relaxed
problem found in step 2 to find a feasible solution to the original problem.
4. Use the solution obtained in step 2 to compute an upper bound on the best value
of the objective function.
5. Examine the solution obtained in step 2 and determine which of the relaxed constraints are violated. Use some method to modify the Lagrange multipliers in
such a way that the violated constraints are less likely to be violated on the subsequent iteration. After new Lagrange multipliers have been identified, return to
step 2 (Daskin 1995).
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Covering Problem
173
7.6.3 Metaheuristic Methods
7.6.3.1 Grasp Algorithm (Bautista and Pereira 2006)
The GRASP metaheuristic is a random iterative optimization procedure. This
metaheuristic has been used to solve diverse problems of optimization, including scheduling, route design, logic, location, graphs, assignment, manufacturing,
transport, and telecommunications problems, among others.
Each iteration in the metaheuristic is made up of two phases: a constructive and
a local search phase. During the constructive phase, the algorithm uses a randomized greedy heuristic to obtain an initial solution to the problem. This is based on
modified greedy procedures, where the greedy rule is substituted by a random selection among a limited list of candidates showing the best values for the greedy
selection rule.
On the other hand, the local search phase permits exploration of the generated solution neighborhood in an attempt to find higher-quality solutions. After
local search, the best solution found during this phase is compared to the best-known
solution, and substitutes it if the objective value is better than the previously known.
Once a stopping criterion is met, the best solution obtained during the procedure is
returned.
In order to solve an optimization problem by means of a GRASP procedure, it is
necessary to define at least the following elements integrated in the heuristics:
The randomized constructive procedure used during this procedure,
The neighborhood of the solution and the procedure to investigate it,
The stopping criterion usually associated to a maximum number of iterations.
One of the major advantages of the GRASP metaheuristic is how easy this general
scheme may be adapted to the solution of particular problems. GRASP requires few
parameters, basically the stopping criterion, associated to the maximum number
of iterations, and a rule to construct the restricted candidate list (RCL) during the
constructive phase (Bautista and Pereira 2006).
7.7 Case Study
7.7.1 Combination of MCDM and Covering Techniques
(Farahani and Asgari 2007)
In this paper, locating some warehouses as distribution centers (DCs) in a real-world
military logistics system will be Investigated. There are two objectives: finding the
least number of DCs and locating them in the best possible locations. The first objective implies the minimum cost of locating the facilities and the latter expresses the
quality of the DCs locations, which is evaluated by studying the value of appropriate
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H. Fallah et al.
attributes affecting the quality of a location. Quality of a location depends on a
number of attributes; so the value of each location is determined by using multi
attribute decision making models, by considering the feasible alternatives, the related attributes and their weights according to decision maker’s (DM) point of view.
Then, regarding the obtained values and the minimum number of DCs, the two objective functions are formed. Constraints imposed on these two objectives cover all
centers, which must be supported by the DCs. Using multiple objective decision
making techniques, the locations of DCs are determined. In the final phase, we use
a simple set partitioning model to assign each supported center to only one of the
located DCs.
The supply chain is composed of three types of facilities:
Origins;
Supportive centers;
Supported centers.
The objective functions and the constraints of the problem are as follows:
Objective function 1: Maximizing the utility of the selected locations. Utility of
a potential point depends on 23 attributes that will be explained later.
Objective function 2: Minimizing the number of supportive centers.
Constraint 1: All of the supported centers must be covered (coverage criteria will
be presented later) by supportive centers.
Constraint 2: Each of the supported centers must be supported by one and only
one of the located supportive centers.
Phase 1. Determining all attributes that influence the utility of a location.
Phase 2. In this phase, we use all possible alternatives as initial inputs of our
process.
Phase 3. Using MADM techniques to assess the quality of the locations Normally.
Phase 4. Using a multiple objective set covering model to find the best locations.
Covering problems hold a central place in location theory. In these problems, we
are given a set of demand points and a set of potential sites for locating facilities.
A demand point is said to be covered by a facility if it lies within a pre-specified
distance of that facility.
Set covering model is as follows (Francis et al. 1992; Daskin 1995; Mirchandani
and Francis 1990):
Min Z D
m
X
Xi ;
(7.90)
i D1
Subject to
m
X
ai k Xi 1I k D 1; 2; : : : ; n;
(7.91)
i D1
Xi 2 f0; 1g I i D 1; 2; : : : ; m;
(7.92)
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Covering Problem
175
xi is a binary variable that is equal to one if the feasible alternative i (a potential location for supportive centers) is suitable for locating supportive centers; otherwise it
is equal to 0. Equation (7.90) ensures that the minimum number of supportive centers are located. Equation (7.90) ensures that all of the supported centers are covered.
In this expression A D Œaki is called covering matrix; aki is equal to 1 if a potential supportive center located in location i.xi D 1/ is able to cover the supported
center located in location k. Generating a covering matrix A D Œaki is based on a
factor called critical distance. Critical distance is the maximum time or distance that
a supportive center can serve. Covering models focus on the worst-case behavior of
the system (Daskin 1995). In our problem, we have used critical distances equal to
8, 12 or 24 h. Here, there are a number of 233 supported centers. We have designed
a modified set covering model with two objective functions as follows (Model 0):
Min Z1 D
33
X
Max Z2 D
33
X
xi ;
(7.93)
i D1
ci xi ;
(7.94)
i D1
Subject to
33
X
aki xi 1I k D 1; : : : ; 233
(7.95)
i D1
xi 2 f0; 1g i D 1; : : : ; 33;
(7.96)
where ci has been already defined as output of Phase 3. Equation (7.93) is similar
to (7.90). Equation (7.94) maximizes the quality of the selected facilities. Equation (7.95) is similar to (7.90). In this problem, it can be considered three scenarios
as follows:
Scenario 0 (Model 0): The problem is solved for the first time and no supportive
center already exists. In this case, the model is the same as Model 0.
Scenario 1 (Model 1): There already exist some active supportive centers (existing facilities) and all of these supportive centers must carry on operating. In
this case, we treat the location of these existing facilities as new locations. But in
Phase 4 we use Model 1 instead of
Model 0 in which we have added the following constraints:
xi D 1˚
8i 2 set of existing facilities g
Scenario 2 (Model 2): There already exist some active supportive centers (existing facilities) and we try to keep them active if possible. In this case, we consider
the location of these existing facilities as new locations. However, in Phase 4 we
use Model 2 instead of Model 0 in which we have used the following objective
function instead of (7.93):
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H. Fallah et al.
Where n1 shows the number of existing supportive centers and .n n1 / shows
the number of new supportive centers. " is a parameter that tries to increase the coefficient of binary variables of the new facilities; this forces the objective function to
use existing facilities as much as possible. Our computational results and sensitivity
analysis show that " D 2=n is an appropriate value in this case.
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Mirchandani PB, Francis RL (1990) Discrete location theory. Wiley, New York
Pirkul H, Schilling DA (1991) The maximal covering location problem with capacities on total
workload. Manage Sci 37:233–248.
Toregas C, Swain R, ReVelle C, Bergman L (1971) The location of emergency service facilities.
Oper Res 19:1363–1373
Chapter 8
Median Location Problem
Masoomeh Jamshidi
The median problem is considered as the main problems identified with the locationallocation problems (see Chap. 5). These problems are intended to find the median
points among the candidate points, so that the sum of costs can be minimized
through this target function. These kinds of problems include the establishment of
the public services including schools, hospitals, firefighting, Ambulance, technical
audit stations of cars, and etc. The target function in the median problems is of the
minisum kind. In fact in these problems we try to quantify the sum of distances
(costs).
We call the first algorithm to be considered the Chinese algorithm. Apparently,
the problem that motivated the development of the Chinese algorithm was the
location of a threshing floor, used to separate the wheat from the chaff after a wheat
harvest. In this case the tree network represents a road network, with the vertices
being the location of wheat fields. The weight for each vertex represents the amount
of wheat to be transported to and from the field and the threshing floor. Locating the
threshing floor at the 1-median causes the total cost of transporting the wheat to and
from the threshing floor to be minimized (Francis et al. 1992).
In median problems, there is the location finding in the kind of network or graph.
The discussion of median problems is focused on graph theory and cannot be beyond
the network or graph.
Fermat (seventeenth century) proposed one problem in order to minimize the
sum of distances. In this problem, one rectangular was considered (three points on
the plane) and the aim was to find a point among three points, so that the sum
of distances among the chosen point could be minimized through three angles of
the rectangular. The problem was developed by Alfred Weber and again was introduced in twentieth century. In this problem, Weber called them as the clients
demand through applying the weight on the angles points of rectangular, and called
the median point as the servicing point. He defined the servicing point among three
pointof clients demand, so that the sum of distances can be minimized. This problem
is called as the first problem of the location-Allocation (see Chap. 5). Later the
problem proposed by Weber was introduced for multiple servicing states (multiple
facilities) and developed more than three points. The problem introduced by Weber
was connection problem. N can be achieved through choosing the median points
on apexes points of the graph or the nodes of network. This problem was quite
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 8, c Physica-Verlag Heidelberg 2009
177
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M. Jamshidi
similar to the candidate of the correlated problem proposed by Weber. Again the
problem was developed by Hakimi (1964), and through application of the weight
on the graph, he began to find the P point on it in order to minimize the sum of the
weighted distance from the points.
8.1 Classification
8.1.1 1-Median
The median problems are intended to find the location of one facility on the network,
so that the total cost can be minimum.
8.1.2 P-Median
The P-median problems it is a disconnected problem and we can choose the candidate points through it.
8.1.3 An Example
Now, with an example, we want showing that how we can change continues problem to discrete problem and find feasible solution. Assume in the city we want
to find some location for change store. This problem is a continuous problem. For
changing this problem into a discrete problem, we can divide the city into smaller regions. These regions can be municipal regions or any other region. In every region,
we determine the demand for chain stores. If we divide the city into five regions,
the continuous map of city with its demands will be changed into a problem with
five demand points. Determining the center of every region and putting whole of
the demand on that point, the problem changes into network. For determining the
center of every region we can use the weight of every region, for example the center
of region can be in the east or west of region of course the location of facility is only
on nodes. Consequently, we face with discrete network. Figure 8.1 is the schematic
representation of this network.
In this network, the nodes are considered among two points of supply. The supply
can be in one of the following forms:
Static demand. That is the demand defined in a fixed or definite form
Probable demand. The demand which involves a variable with a gravity function
Dynamic demand. The demand of network (defined in a function of time.
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179
Fig. 8.1 Distance–cost relationship in three conditions
Since in modeling design, the demand of network in real problems depends on the
population, we can argue that the dynamic demand is more correct and true than any
other kind of demands. These suggestions can lead us to the proper answers.
8.2 Mathematical Models
8.2.1 Classic Model
The median problems are intended to find the location of P facility on the network,
so that the total cost can be at minisum. The cost means the cost of providing services from node i to the nearest node established there. This cost depends on factors
like distance between the node of i and servicing node and volume of demands of
in node. The Classic Median Problem has some assumption as follows:
Linear relationship between cost and distance
Good facilitated
Infinite time horizon
Infinite facility capacity
Don’t have an initial setup cost
Exogenous problem
Same facilities
Stationary facilities
Constant node’s demand
Discrete problem
In this problem;
Xij : is equal to 1 if the demand of node i is covered by the facility that has been
setup at node j , otherwise is 0
Yj : is equal to 1 if a facility is setup at node j , otherwise is 0
dij : The distance between the node of i demand and candidate node to establish
facility j (dij is zero if i D j ).
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M. Jamshidi
P : The number of facilities to be established
hi : Demand of i node
n: Number of nodes
This model was proposed by ReVelle and Swain (1970);
Min
XX
i
hi dij Xij
j
i; j D 1; 2; :::; n:
(8.1)
Subject to
X
j
X
j
Xij D 1
8i;
Yj D P ;
Xij Yj
8i; j;
Xij ; Yj 2 f0; 1g 8i; j:
(8.2)
(8.3)
(8.4)
(8.5)
In this problem, the aim is to minimize the total cost needed to satisfy the needs
of nodes (to minimize sum of demand-distance). Equation (8.2) states that all
demands should and each node is serviced by just one facility (demand limitation).
Equation (8.3) states that there is an endogenous problem and proves the exact P
establishment. Equations (8.4) states that open facilities can meet the demands. Thus
the demand of i node can be provided with facility established in j node .Xij D 1/
through one facility in j node .Yj D 1/.
8.2.2 Capacitated Plant Location Problem Model (CPLPM)
In this part, we introduce the capacitated plant location problem (CPLP) with multiple facilities in the same site (CPLPM), a special case of the classical CPLP where
several facilities can be opened in the same site. Applications of the CPLPM arise in
a number of contexts, such as the location of polling stations. Although the CPLPM
can be modeled and solved as a standard CPLP, this approach usually performs
very poorly. CPLP is a classical combinatorial optimization problem (Bramel et al.
1998). This location problem is of utmost importance for many public and private
organizations and its aim is determining a set of capacitated facilities (warehouses,
plants, polling stations, etc.) in such a way that the sum of facility construction costs
and transportation costs is minimized. Unlike other problem we allow multiple facilities in the same site. The CPLP is strongly NP-hard (Mirchandani and Francis
1990) and has been extensively studied in clustering and location theory. As a result,
an overabundance of solution approaches has been proposed in the past decades.
Exact algorithms have been developed, among others, by Christo Edes and Beasley
(1983), Leung and Magnanti (1989), whereas heuristics have been investigated by
8
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181
Van Roy (1985), Beasley (1988), ChristoEdes and Beasley (1983), GeoIrion and
McBride (1978), Guinard and Kim (1987), Jacobsen (1983), Khumawala (1974).
A systematic comparison of heuristics and relaxations for the capacitated plant location problem is provided by Cornuejols et al. (1991) Based on both a theoretical
analysis and extensive computational results; they suggest the use of a Lagrangian
heuristic to solve large instances of the CPLP.
In this problem;
U : The set of potential facilities,
V : The set of customers,
dj : The demand of customer j .j 2 V /, where dj > 0,
qi : The capacity of facility i .i 2 U /, where qi > 0,
cij : The cost of supplying all the demand of customer j .j 2 V / from facility
i .i 2 U /,
fi : The fixed cost associated with opening facility i .i 2 U /, where fi > 0,
p: The desired number of open facilities (also referred to as medians),
yi : A binary decision variable, which takes the value 1, if facility i .i 2 U / is
open, 0 otherwise,
xij : A continuous decision variable, corresponding to the fraction of the demand
of customer j .j 2 V / supplied from facility i .i 2 U /.
Then CPLP can be formulated as a mixed integer linear programming problem as
follows:
XX
X
.CPLP/ Min
fi yi :
(8.6)
cij xi C
i 2U
i 2U j 2V
Subject to
X
i 2U
X
j 2V
X
i 2U
xij D 1;
i 2 V;
dj xij qi yi;
i 2 U;
yi D p;
xij 0;
yi 2 f0; 1g;
(8.7)
(8.8)
(8.9)
i 2 U; j 2 V;
(8.10)
i 2 U:
(8.11)
The objective function (8.6) expresses the minimization of the total costs.
Equations (8.7) ensure that the demand of each customer is satisfied. Equations (8.8)
establish the connection between .xij / and .yi / variables. They state that no customer can be supplied from a closed facility and the total demand supplied from each
open facility does not exceed the capacity of the facility. Equation (8.9) establishes
that the number of open facilities is p. Equations (8.10) provide lower bounds on the
.xij / variables. It is worth noting that (8.7)–(8.10) imply xij 1 .i 2 U I j 2 V /.
Finally, (8.11) are the integrality constraints.
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M. Jamshidi
In the CPLPM, U represents a set of areas where facilities can be located. Such
set is partitioned into n nonempty U1 ‘ : : : ’Un , each corresponding to a site where
one or more areas are available. Consequently fi D f 0 i , if i I i 2 UK .k 2
f1; : : : ; ng/, and Cij D C 0 ij if i 2 UK .k 2 f1; : : : ; ng/ and j; j 0 2 V .
8.2.3 Capacitated P-median Problem (Lorenaa 2004)
If no fixed costs are associated to the potential facilities, then the CPLP is called the
capacitated p-median problem (CPMP).
Where N D f1; : : : ; ng is the index set of entities to allocate and also of possible
medians with xij D 1 if entity i is allocated to median j , and xij D 0, otherwise;
xjj D 1 if median j is selected and xjj D 0, otherwise.
The CPMP model can be formulated in two ways. The first is the following binary
integer-programming problem (P ):
Min
XX
cij xij :
(8.12)
i 2N j 2N
Subject to
X
i 2N
X
j 2N
X
i 2N
xij D 1;
i 2 N;
xjj D P;
(8.14)
di xjj qi xjj;
xij 2 f0; 1g;
(8.13)
j 2 N;
i 2 N; j 2 N:
(8.15)
(8.16)
Equations (8.13)–(8.14) impose that each entity is allocated to only one median.
Equations (8.15) imposes that a total median capacity must be respected, and (8.16)
provide the integer conditions.
The CPMP problem can also be modeled as the following set partitioning problem with a cardinality constraint (SPP).This is the formulation found in Minoux. The
same formulation can be obtained from the problem P by applying the Dantzing–
Wolfe decomposition.
Where S D fS1 ; S2 ; : : :; Sm g, is a set of subsets of N ;
A D Œai k nm , is a matrix with
ai k D
(
1
0
.SPP/ Min
if i 2 SK 0
otherwise:
m
X
kD1
ck xk :
(8.17)
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183
Subject to
m
X
Ak xk D 1;
(8.18)
xk D P ;
(8.19)
xij 2 f0; 1g:
(8.20)
kD1
m
X
kD1
Equations (8.12)–(8.13) are conserved and respectively updated to (18) and (20),
according the Dantzig–Wolfe decomposition process.
If S is the set of all subsets of N , the formulation can give an optimal solution
to the CPMP. However, the number of subsets may be huge, and a partial set of
columns can be considered instead.
The SPP defined above is also known as the restricted master problem in the
column generation context.
8.3 Solution Techniques
Some kinds of solution for solving the p-median problems are as follows:
Exact methods
Heuristic algorithm
Metaheuristic algorithm
The complete accounting, heuristic and met heuristic algorithm are one of the first
technique that are used them for solving the median problem.
Teitz and Bart (1968) proposed two innovated algorithm to solve the P -median
problems through studying the complete number algorithm of Hakimi. This algorithm is based on choosing a primary series of nodes and then exchanging its
members with other nodes of the network in order to improve the target function.
Jarvinen et al. (1972) proposed a branch and bound algorithm to solve P -median
problem. Meanwhile, Egbo, Samelson and colleagues proposed a branch and bound
algorithm used to determine its limit through method of logerangean reparation
(Narula et al. 1968). Neebe (1978) defined the transportation P -median problem in
which the number of supply points was limited to P . Also he proposed a branch and
bound algorithm for this problem in which the logerangean reparation method was
used to determine its limit. Kariv and Hakimi (1979) proved to find the P -median in
a network was to find NP-Hard, and to determine 1-median for a tree was possible
in O.n2 / stage. Galvao (1980) determined the dual form of P -median problem, and
obtained a low limit for the problem using the solution by an innovative method.
This limit has been used in a branch and bound algorithm and applied to solve the
problems with 30 nodes and desirable P numbers.
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M. Jamshidi
8.3.1 Lemma
The Weber’s problem was developed by Hakimi (1964), and through application of
the weight on the graph, he began to find the P point on it in order to minimize
the sum of the weighted distance from the points. To solve the problem, there was
no limitation far the places of the points, and they could be placed in any points of
the graph. Though the points were not located exactly on the apexes in the resulted
solutions, Hakimi (1964) showed that there could be always a series apexes P to
minimize the target functions. Thus, through the constraint of finding the solution
just on the apexes, again Hakimi (1964) considered this disconnected problem as
the candidate of Weber’s problem. In the median problems, the solution consisting
of the apexed P is called the P -median. Also, these problems are to be studied and
defined just over the graph or networks. Hakimi (1964) generalized the concept of
1-median to multi-median. He used the concept to determine how to distribute the
switching centers in the Tele-communication network. Later the P -median problems were considered as the inseparable part of the location theory and were called
as a main problem.
8.3.2 Solving 1-Median Problem Algorithm on Tree
(Goldman 1971)
Put for all nodes hO i D hi
P
Choice one node beside the i if it is hO i 2hi , put the facility in the node and go
to step 3.
Otherwise add Wi to wk . k is the only node that it has intersection with inside
i node.
Calculate the objective function.
In the algorithm, time of solving the algorithm increases by developing the number of nodes linearly, because any node will consider its application just one time
O.n2 p 2 /. Kariv and Hakimi (1979) submitted one algorithm for finding the place
of facility P on a tree networks with (n)nodes.
According to previous, without considering the type of network that can be a tree
network or general, if at least half of the application of all network appear on a node,
we can get at least one best solution by putting facility on that node. In tree network,
we will lead to getting place of best facility through using the node on top of the
tree and mixing those with other connected nodes. This subject is not possible in
general networks. For better understanding of this subject consider Fig. 8.2. These
different conditions that all tree nodes can be placed on best settlement are changed
according to the length of relating vectors.
8
Median Location Problem
185
Fig. 8.2 Instance of general
network
hA = 3
B
A
hB = 4
γ
β
C
hC = 5
8.3.3 Exact Methods
In solving median problem, mathematic models are used by applying the integer
programs. For solving this model of linear integer program, some ways are submitted that will solve based on relieving model of original model and by the dual theory
and relieving Lagrangian method.
8.3.3.1 Complete Accounting (Teitz and Bart 1968)
First, for solving the median problem, we should consider a condition that a facility
may be put on a network. In this method, by counting all conditions we will simply
get the best result simply. Especially, we know this characteristic of median problems that one best result is obtained by putting the facilities on nodes. In order to
calculate the amount of objective function, we can use this condition:
Zj D
X
hi dij :
(8.21)
j
When we put the facility on j node, we can calculate the amount of Zj for all
nodes and choose the least one for the best result. In this phrase, just one facility has
located from p-facility that should be located and is left p-1 facilities.
The number of forward solution is calculated with this equation:
NŠ
N
:
D
P
P Š.N P /Š
(8.22)
8.3.4 Heuristic Algorithms
Just as mentioned, solving the median problem in time dimension imitate just one
polynomial, but it had been shown by Kariv and Hakimi (1979) that if this subject
be in a general network, it is a NP-complete. So some struggles have drawn for
finding heuristic solutions for solving these problems. The important facts about
these facility algorithms are that how much these are good or possible answers.
186
M. Jamshidi
Although these algorithms result in real world problems very well, they don not
guarantee achieving the optimum or approximating it. In solving some problems,
these solutions may be optimum or near the best. In other cases, these solution may
be very far. The first Heuristic algorithm for solving the median problems:
Greedy-adding algorithm
Alternate algorithm
Vertex substitutions algorithm
According to these tree algorithms, innumerable algorithms and new techniques are
made for solving the median problems that are based on those three algorithms.
From these three basic algorithms, the vertex substitution algorithm is more general
in comparison to the other two algorithms and up to the present time, it was one
of the mostly used algorithms for solving median problems. Another heuristic algorithm is the branch and bound algorithm. (Heuristic branch and bound algorithm
is not the same as one of the linear-program problems techniques named branch
and bound.)
According to a classification, algorithms are classified into two groups of reply
maker algorithms and recovering reply algorithms.
Myopic algorithm is one of the reply maker algorithms. This algorithm tries to
get one primary and possible solution for median problems. The algorithm is similar
to Greedy adding algorithm that is applied for maximum covering problems.
Both exchange and neighborhood search algorithms are kinds of recovering reply
algorithm. These algorithms are not able to get the primary reply for problems, but
they use them to improve reply that how got from maker algorithms. Two algorithms
are similar to both Greedy adding and maximum heuristic algorithms for maximum
covering and substitution problems. When Lagrangian discounting method is used
besides one or more heuristic algorithm that will be introduced this section, they
will get the best or near the best reply.
8.3.5 Metaheuristic Algorithms
Meta heuristic algorithms for solving the median problem:
Genetic algorithm
Concentration algorithm
Neural network
Tabu search algorithm
Simulated annealing
In recent decades metaheuristic algorithms are submitted for solving these kinds of
problems. Among these algorithms we can mention the genetic algorithm, neighborhood search algorithm, tabu search algorithm, concentration algorithm, simulated
annealing algorithm and neural network. Also there are some other algorithms for
obtaining one reply with acceptable quality.
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Metaheuristic methods are usually used for too many nodes (100, 200, 300, 400,
500, 600, 700, 800 and 900).
8.4 Comparison of Methods
In this section, all achieved results that have been concluded from a 12-nodes
network are mentioned. Here there is no solution way and just we consider their
comparisons results.
The solutions that have been achieved by using the myopic algorithms, in the
most cases, appears almost in 30% error of amount of Lagrange objective function.
In each stage, all solution that has been achieved from exchange algorithms
method is always for the each number of facilities better than solution by using the neighborhood search algorithms.
It is better in some situations that neighborhood search algorithms applied after
setting each facility at network in each stage, and in some other situations, it is
better to apply the neighborhood search algorithms after setting all facilities.
We should consider that when the number of settled facilities will be average
figure, the differences between all achieved results differ from various algorithms
together and with Lagrange algorithm method.
When the number of the settled facilities is few, it seams that all algorithms work
better. For instance, when the number of the settled facilities is more, all results
that have been achieved from algorithms are so good and algorithms work so
well. But when the number of the settled facilities is neither few nor more, the
problem is so difficult.
Exchange algorithm is the best method for this network, considering that exchanging appears after putting all facilities by using the myopic algorithm.
8.5 Studying Statically the Methods for Solutions of Median
Problem (Reese 2005)
The aim here is to study the methods used to solve the median problems and the
years of proposing them which were classified for 1963–2005 periods. The following factors were selected among other sources:
Those focused directly on median problems,
Those involved minisum target function,
Those which define median problems on the graph or network,
Sum of answers limited to searching on nodes,
Those that did not have primary cost of establishment,
Those belonging to median problems with limited capacity or unlimited facilities.
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M. Jamshidi
Also we avoided the resources with one of the following conditions:
They had minimax target function,
Study establishment of median problems in a connected space,
There was a probable state demand or cost,
There was a multi-purpose target function,
Considering dislocation of facilities in a time horizon.
The following classification obtained considering the proposed solutions.
8.5.1 Classification of Solving Methods by Period
The number of presented paper in this area before 1970 were 7, between 1970 and
1974 were 9, between 1975 and 1979 were 12, between 1980 and 1984 were 9,
between 1985 and 1989 were 6, between 1990 and 1994 were 11, between 1995 and
1999 were 24 and between 2000 and 2005 the number of paper have the remarkable
growth so that it reaches 42 paper.
It should also be noted that the methods obtained and used in the last 10 years
are so higher than the methods used in 1963–1994.
8.5.2 Classification of Different Solving Methods
According to the researches between 1965 and 2005 the LP Relaxation was used
more than other methods, after that respectively Vertex substitution, approximation
algorithm, genetic algorithm (GA), IP formulation were used, graph theory and surrogate relaxation are in the same level, and other methods were assigned less than
five cases.
8.6 Case Study
In this section we will introduce some real-world case studies related to p-median
problem:
8.6.1 Post Center Locations (Alba and Dominquez 2006)
In Australia in order to determine ten post centers among 200 centers, 10-median
problem formulated and finally ten cities including Sidney, Melborn and Adlid, etc.
were selected as post centers.
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189
8.6.2 Entrance Exam Facilities (Correa et al. 2004)
PARANA State University (UFPR) in Curitiba in Brazil used the median P model
to determine the location of facilities concerning M.A entrance exam in 2001. The
aim was to appropriate 19,710 candidate students to the facilities located nearer to
their homes as possible.
It was determined that 26 facilities needed to meet the demands of 19,710
students among 43 candidate facilities. Finally the number of candidate solutions
should be 421 billion.
43
D 421; 171; 648; 758:
26
The problem was solved by a genetic algorithm.
8.6.3 Polling Station Location (Ghiani et al. 2002)
Polling station location problem in Italy: Number of polling stations was determined according to the number of resident voters in five Italian municipalities (data:
November 2000).
This study of the CPLPM was motivated by a polling station location problem
in an Italian municipality. In Italy, the following binding obligations must be taken
into account:
The number of polling stations is fixed for each municipality and calculated according to the number of resident voters the number of voters assigned to each
polling station may not exceed given lower and upper bounds;
The suitability of the potential sites is established by specific safety measures on
the accessibility and typology of the buildings (for example, in some countries,
only public buildings, such as schools, are eligible).
8.6.3.1 Formulating the Problem as a CPLPM
U : The set of areas where polling stations can be located.
P : Number of polling stations to be activated.
V : The set of streets having at least one resident voter
dj : The number of voters in street j .j 2 V /, where dj > 0.
qi : The upper limit on the total number of voters assigned to a polling station
located in area i .i 2 U /I qi may be the same for all areas, i.e.
qi D q D
X dj
I i 2 U:
p
j 2V
(8.23)
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M. Jamshidi
cij : The transportation cost incurred if all voters resident in street j .j 2 V / are
assigned to a polling station in area i .i 2 U /I cij can be chosen, for simplicity,
equal to dj Sij j , where Sij is the walking distance between area i and the “centre of gravity” of street j; fi D f , for all areas. Consequently, the contribution of
the fixed costs to the objective function (6) can be excluded.
8.6.3.2 Computational Results
As far as the experimental phase is concerned, we have solved first the problem
of the optimal location of polling stations in the municipality of Castrovillari. Town
located in Southern Italy with a population, dated back to November 2000, of 15,709
voters, located in 351 different streets spread in a wide area. The solution actually
adopted by the local government is based on the assignment of all voters resident
in the same street to the same polling station, chosen among the nearest ones. As a
result, the walking distance covered by all voters is about 10,731 km, and the number
of voters for each polling station was reported. It was observed that, with respect to
the ideal capacity, the fourth polling station has 18% of voters less, whereas the 18th
polling station has 21% of voters more. The feasible solution found by the heuristic
is well balanced and has a cost 37.6% less than that of the solution currently adopted
by the municipality of Castrovillari. Computational results show that the average
deviation of the heuristic solution over the lower bound is less than 2%.
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Leung JM, Magnanti TL (1989) Valid inequalities and facets of the capacitated plan location problem Math Program 44:271–291
Lorenaa L, Senneb E (2004) A column generation approach to capacitated p-median problems
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Mirchandani PB, Francis RL (1990) Discrete location theory. Wiley, New York
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Reese J (2005) Methods for solving the p-median problem: An annotated bibliography. Technical
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ReVelle C, Swain R (1970) Central facilities location. Geograph Anal 2:30–42
Teitz MB, Bart P (1968) Heuristic methods for estimating the generalized vertex median of a
weighted graph. Eur J Oper Res 16(5):955–961
Van Roy TJ (1985) A cross decomposition algorithm for capacitated facility location. Eur J Oper
Res 34:145–63
Chapter 9
Center Problem
Maryam Biazaran and Bahareh SeyediNezhad
In the covering problems, the attempt is to determine the location of the minimum
number of facilities necessary to cover all demand nodes. In this type of problems,
the coverage distance is an exogenous data. But sometimes the number of facilities
needed to cover all demand nodes with a predefined coverage distance may be quite
large. In order to overcome this, the maximum covering location problem has been
discussed. In this model, the objective is to maximize the number of covered demand
nodes with a fixed number of facilities. In other words, we relaxed the total coverage
requirement (Daskin 1995).
A different strategy is discussed in this chapter. Now, instead of asking the model
to minimize the number of facilities with a given coverage distance, we will ask the
model to minimize the coverage distance with a given number of facilities, while
maintaining the coverage of all demand nodes. This model is introduced under
the title of p-center problem which is in fact a minimax problem. In this model, the
objective is to find locations of p facilities so that all demands are covered and
the maximum distance between a demand node and the nearest facility (coverage
distance) is minimized. It can be said that we have relaxed the coverage distance
(Daskin 1995).
In the p-center model, each demand point has a weight. These weights may have
different interpretations such as time per unit distance, cost per unit distance or
loss per unit distance (Daskin 1995). So the problem would be seeking a center to
minimize a maximum time, cost or loss. In other words, the concern is about the
worst case and we want to make it as good as possible (Francis et al. 1992).
For example, assume that we need to establish a number of fire stations in a
town. If the time to reach the scene from facility j to demand node i is dij , then this
amount must not be greater than minutes. Therefore, we are looking to provide dij <
for each node i and a closest facility j . If we are to cover all the points, we need
certain number of facilities. To evaluate the number of facilities needed, one needs
to solve a set covering problem. Now assume that there is a budget constraint and
we can not establish enough fire stations, then there are two ways of approaching the
problem. One is to cover maximum points; in this case you are facing a maximum
covering problem. The second approach is to cover all of demand nodes, but through
increasing the radius of coverage distance. Evaluation of the minimum increase in
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 9, c Physica-Verlag Heidelberg 2009
193
194
Fig. 9.1 Example network
illustrating suboptimality of
nodal locations (Daskin 1995)
M. Biazaran and B. SeyediNezhad
A
8
B
the coverage distance with respect to the given number of facilities can be done with
a p-center model.
One should recognize the difference between problems in which new facilities
can be established anywhere on the network (on the nodes and on the links of the
network) and problems in which the new facilities can only be established on the
nodes of the network. These problems are called absolute center problem and vertex
center problem respectively. It can easily be shown that the solution to the absolute center problem may be better than the solution to the vertex one. Consider the
sample network in Fig. 9.1. If one new facility .p D 1/ can be located only on the
nodes, each node can be the optimum answer and the maximum distance between
the facility and the other node is equal to 8. But if we can locate the new facility
anywhere on the network, the optimum place would be the midway between A and
B and the coverage distance can be reduced to 4 (Daskin 1995).
This chapter is organized as follows. In Sect. 9.1, we will discuss applications
and classifications of p-center problems. In Sect. 9.2 mathematical models are presented. Some of the exact solution algorithms for center problems are discussed
in Sect. 9.3 and a number of approximate solution approaches are introduced in
Sect. 9.4. Finally in Sect. 9.5 a short summary of a real-world case study is given.
9.1 Applications and Classifications
Some of the potential applications of the p-center model would be in:
Quick services (hospital emergency services, fire stations, police stations, : : :)
Computer network services (location of the data files)
Distribution (warehouses, garages, : : :)
Military purpose
Government and general (parks, hotels, : : :)
Location-allocation for post boxes and bus stops
Since we have a minimax objective function for the p-center model, it seems that
it would be most applicable to emergency cases. Thus, Pacheco and Casado (2005)
applied the p-center model to find the best locations in which to place special health
resources.
The literature on p-center problem has grown rapidly after Hakimi (1964)
published a paper on the absolute center and median problems. The p-center
problem was defined and formulated by Hakimi (1964, 1965). Many papers have
been published since 1960s. The emphasis of papers was on suggesting solution
procedures.
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The following is a classification of different types of p-center models and their
properties:
Solution area: plane, network
In case of network: tree network, general graph
Facility capacity: limited, unlimited
Number of centers: one center, more than one center
Demand type: only on nodes, on links and nodes (continuous)
Weights of demand points: equal (unweighted), positive (frienly facilities), negative (obnoxious facilities (see Chap. 14)), mixed (pos/neg)
Possible facility locations: anywhere on the network (absolute), only on nodes
(vertex)
The following are a number of generalized p-center problems:
9.1.1 K-Network P-Center Problem
The problem of locating p centers on k underlying networks corresponding to k
periods (Hochbaum and Pathria 1998). There is one network for each period of
time. It is required to locate permanent facilities and the objective is to minimize
the maximum distance over all k periods.
9.1.2 P-Facility -Centdian Problem
The problem consists of finding the p points that minimize a convex combination
of the p-center and p-median objective function (Pérez-Brito et al. 1998). For example, locating a local branch bank may call for the minimization of the average
distance traveled by all prospective customers without being located too far away
from any customer.
9.1.3 K -Centrum Multi-Facility Problem
This problem generalizes and unifies the p-center problem and p-median problem
(Tamir 2001). The objective of this unifying model is to minimize the sum of the
k largest service distances. The p-center and the p-median problems correspond to
the cases where k D 1 and n, respectively.
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9.1.4 P-Center Problem with Pos/Neg Weights
Burkard and Dollani (2003) was the first to introduce this model. The model is
a p-center problem in which nodes can have positive or negative weights, which
indicate friendly and obnoxious facilities (see Chap. 14).
9.1.5 Anti P-Center Problem
This problem was introduced for the first time by Klein and Kincaid (1994). Instead
of minimizing the maximum weighted distance between a demand node and its nearest facility, in anti-p-center, the objective is to maximize the minimum weighted
distance between demand nodes and the nearest facility. In other words, we are locating p obnoxious facilities (see Chap. 14). All weights are negative in this model.
9.1.6 Continuous P-Center Problem
In the case where each point in the network is a demand point, as opposed only to
vertices, the problem will be referred to as the continuous p-center problem (Tansel
et al. 1983). Tamir (1987) discussed this problem in his paper.
9.1.7 Asymmetric P-Center Problem
The problem is said to be asymmetric when for each pair of nodes i and j in the
network, we have dij ¤ dji . Panigrahy and Vishwanathan (1998) were the first to
introduce this type of p-center problem.
9.2 Models
In this section, we introduce mathematical models of p-center.
9.2.1 Vertex P-Center Problem
The vertex p-center problem was formulated by Hakimi (1965) and we present it as
follows:
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9.2.1.1 Model Assumptions
The assumptions of this model are as follows:
The facilities can only be located on the nodes of the network (vertex)
The capacities of the facilities are unlimited
There are p facilities to be located
Demand points are on the nodes of the network
Demand nodes are unweighted
9.2.1.2 Model Inputs
The inputs of this model are as follows:
dij : length of the shortest path between demand node i to candidate facility site j
p: number of facilities to locate
9.2.1.3 Model Outputs (Decision Variables)
The outputs of this model are as follows:
Xj D 1 if a facility is located at candidate site j and 0 otherwise
Yij D 1 if demand node i is assigned to facility at candidate node j and 0
otherwise
z D maximum distance between a demand node and the nearest facility
9.2.1.4 Objective Function and its Constraints
Min z:
(9.1)
Subject to
X
Yij D 1 8i;
(9.2)
X
Xj D p;
(9.3)
Yij Xj 8i; j;
X
dij Yij 8i;
z
(9.4)
(9.5)
Xj 2 f0; 1g 8j;
(9.6)
Y 2 f0; 1g 8i; j;
(9.7)
j
j
j
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M. Biazaran and B. SeyediNezhad
Equation (9.1) in conjunction with (9.5) minimizes the maximum distance between
a demand node and its nearest facility. Equations (9.2) state that all of a demand at
node i must be assigned to a facility at some node j for all nodes i . Equation (9.3)
guarantees that p facilities are located. Equations (9.4) ensure that assignments can
only be made to open facilities. Equations (9.6)–(9.7) are the integrality constraints.
9.2.2 Vertex P-Center Problem with Demand-Weighted Distance
In some cases, we want to consider the demand-weighted distance (Daskin 1995).
9.2.2.1 Model Assumptions
Model assumptions are similar to the ones of vertex p-center model except that
demand nodes are weighted.
9.2.2.2 Model Inputs
We have all inputs in previous model in addition to the following inputs:
hi : demand at node i
9.2.2.3 Model Outputs (Decision Variables)
The outputs of this model are similar to the previous model.
9.2.2.4 Objective Function and its Constraints
The objective function and constraints are similar to the previous model except that
(9.5) must be replaced by:
X
dij Yij 8i:
(9.8)
z hi
j
9.2.3 Capacitated Vertex P-Center Problem
Ozsoy and Pinar (2006) represented the capacitated vertex p-center model. We
introduce it as follows:
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9.2.3.1 Model Assumptions
Model assumptions are similar to the vertex p-center model except that the
capacities of the facilities are limited.
9.2.3.2 Model Inputs
We have all inputs in previous model in addition to the following inputs:
Qj : service capacity of facility site j
9.2.3.3 Model Outputs (Decision Variables)
Model outputs are similar to the previous model.
9.2.3.4 Objective Function and its Constraints
The objective function and constraints are similar to the vertex p-center model in
addition to the following constraints:
X
i
hi Yij Qj 8j:
(9.9)
With (9.9), capacity restrictions of the facilities are incorporated into the vertex
p-center model.
9.3 Exact Solution Approaches
Kariv and Hakimi (1979) showed that the p-center problem on a general graph is
NP-hard. However, when the network is a tree, the optimal solution can be found in
polynomial time. Hence, the first part of this section is dedicated to center problems
on tree networks and in the second part, center problems on general graphs are
discussed.
9.3.1 Center Problems on a Tree Network
9.3.1.1 Vertex 1-Center on a Tree Network (Daskin 1995)
A vertex center in a tree network is a node that has the minimum distance to its
farthest node. Hakimi (1964) presented an algorithm to find this node. First we need
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M. Biazaran and B. SeyediNezhad
Fig. 9.2 An example of a tree
network
3
5
4
5
1
3
2
2
3
4
6
to compute the square matrix D D .dij / of order n:
dij D
(
d.vi ; vj /;
for i; j D 1; 2; : : : ; n and i ¤ j
:
d.vi ; vj / D 0; for i D j; i D 1; 2; : : : ; n
(9.10)
Let dim be the maximum entry in the i th column of D. vc is a vertex center if:
dcm D min.d1m ; d2m ; : : : ; dnm /:
(9.11)
For example, consider the tree network in Fig. 9.2. It is easily determined that v1 is
the center of this tree with d1 m D 7.
2
3
0 3 5 2 7 6
6 3 0 8 5 4 3 7
6
7
6 5 8 0 7 12 11 7
6
7;
D.dij / D 6
7
6 2 5 7 0 9 8 7
4 7 4 12 9 0 7 5
6 3 11 8 7 0
dcm D min.d1m ; d2m ; : : : ; dnm / D 7:
If we have a tree with weighted demand nodes, the procedure is similar to above
except that each row of D.dij / must be multiplied by its respective node’s demand
weight hi .
9.3.1.2 Absolute 1-Center on an Unweighted Tree Network (Daskin 1995)
For the unweighted case, Handler (1973) presented an especially elegant algorithm.
Handler’s method finds any longest path in the tree and locates the absolute center
at the midpoint of the path. The steps are as follows:
1. Pick any vertex on the tree and find the farthest vertex from it. Call this vertex vs .
2. Find a vertex that is farthest from vs and call it vt .
3. This path is the longest path and its midpoint is the unique absolute center of
the tree.
We can illustrate this algorithm using Fig. 9.3.
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201
Fig. 9.3 An example of a tree
network
A
8
9
B
10
D
5
F
6
C
5
E
G
17
A
8
9
0
9
B
10
10
D
F
5
6
5
C
E
14
G
6
15
Fig. 9.4 Calculated distances from node D
0
A
8
8
17
9
B
27
10
D
F
5
6
5
C
13
E
G
23
32
Fig. 9.5 Continued form Fig. 9.4
We begin by picking a vertex, for example D. We then find the distance between
node D and all vertices on the tree. Figure 9.4 shows the result of this calculation.
Node A is the farthest node from D, so we let node A be vs . Figure 9.5 shows
the distances from node A.
Since G is the farthest node from A, we call it vt . the absolute 1-center is at the
point midway between A and G on the unique path from node A to node G, or 1
unit from node D on the edge between D and B, as shown in Fig. 9.6. The objective
function equals 16.
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M. Biazaran and B. SeyediNezhad
A
8
1
9
B
5
10
D
6
C
F
5
E
G
Fig. 9.6 Location of the absolute 1-center in the network of Fig. 9.3
9.3.1.3 Absolute 2-Center on an Unweighted Tree Network (Daskin 1995)
To solve the absolute 2-center on an unweighted tree, we can modify the algorithm
used to find the absolute 1-center on an unweighted tree. Steps of this algorithm are
as follows:
1. Use the algorithm for the absolute 1-center and find the absolute 1-center.
2. If the absolute 1-center is on a node, delete one of the arcs incident on the center
which is on the path from vs to vt , then delete the arc between vs and vt . Now,
we have two disconnected subtrees.
3. Use the absolute 1-center algorithm to find the absolute 1-center of each subtree.
This would be a solution to the absolute 2-center problem.
To illustrate the algorithm, consider again the tree shown in Fig. 9.3. As the Fig. 9.6
indicates, the absolute 1-center lies on the link BD. Removing this link results in the
two trees shown in Fig. 9.7. After applying the absolute 1-center algorithm to each
subtree, we obtain locations of x1 and x2 as shown. The objective function value is
10.5 (maximum between two objective function values).
9.3.1.4 Absolute 1-Center on a Weighted Tree Network (Daskin 1995)
In this section, we are seeking an absolute center on a weighted tree or a tree in
which the weights associated with each of the nodes are not equal. Consider the
simple network in Fig. 9.8. In this case, we want to minimize the maximum demandweighted distance between a demand node and its nearest facility. In this simple
case, to locate the facility, a point at X units from node A is selected such that or
X D 4. Now the demand-weighted distance between each node and the facility is
24. This is also shown in Fig. 9.9. It is obvious that the solution involves locating
closer to the larger demand node.
For a simple tree with only two nodes, computing the absolute 1-center is easy,
but when there is a more complicated tree, it becomes more difficult.
9
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203
A
8
1.5
4.5
B
10
D
F
5
6
5
C
E
G
Fig. 9.7 Absolute 2-center on a tree network
Fig. 9.8 Simple network for
the weighted problem on tree
hA=6
hB=3
12
A
Fig. 9.9 Maximum demandweighted distance D 24
hA=6
B
hB=3
12
A
B
X=4
However we can generalize the approach as follows. Consider two nodes i and j .
In the case that the center is located on the path between them, we must solve the
following equations for the location X :
hi d.i; x/ D hj d.j; x/;
(9.12)
d.i; X / C d.X; j / D d.i; j /:
(9.13)
From the (9.12)–(9.13) we obtain (9.14): (After solving for d.i; X /, we have (9.15)
or (9.16))
hi d.i; X / D hj Œd.j; X / d.i; X /;
(9.14)
d.i; X / D
hj d.i; j /
;
hi C hj
(9.15)
d.X; j / D
hi d.i; j /
:
hi C hj
(9.16)
So, if we were to locate a facility at this point, both nodes i and j would be
hi hj d.i; j /
;
hi C hj
(9.17)
demand-weighted distance units from that center. This center is located d.i; X /
units from node i and on the path between node i and node j . we can compute
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M. Biazaran and B. SeyediNezhad
this value for every pair of nodes in the tree and select the pair that has the largest
value to obtain the optimal solution. The approach is as follows:
Compute ˇij for every pair of nodes i and j :
ˇij D
hi hj d.i; j /
:
.hi C hj /
(9.18)
Find: ˇst D maxij .ˇij /. Corresponding nodes are s and t.
Locate at a point Œht =.hs C ht /d.s; t/ from node s on the unique path from s
to t.
Since n is the number of nodes in the tree, this approach involves computing O.n2 /
terms ˇij . The approach can be simplified by computing only a portion of the bound
matrix B D .ˇij /. The algorithm is as follows:
1. Compute one row of the ˇij elements in the matrix.
2. Find the maximum element in that row (if the maximum element is in a column
that was already computed, stop)
3. Compute the elements ˇij in the column in which the maximum ˇij element
occured in step 2.
4. Find the maximum element in the column that was just computed (if the maximum element is in a row that was already computed, stop).
5. Compute the elements ˇij in the row in which the maximum ˇij occurred in step
4. Go to step 2.
For more explanation of this method consider the network of Fig. 9.10.
In the first step, elements in the row corresponding to node A are computed. The
partial matrix in Table 9.1 shows the results.
As shown in Table 9.1, the element in the column corresponding to node G is the
largest one, so in the 4th step, we will compute elements in corresponding column.
The results are shown in Table 9.2.
9
A
8
10
8
9
B
C
Fig. 9.10 A weighted tree
D
F
5
6
5
6
4
10
E
G
5
12
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205
Table 9.1 A partially computed matrix of ˇij terms
ˇij
A
B
C
D
E
F
G
A
0
37.89
46.8
72
73.92
74.76
164.57
Table 9.2 A partially computed matrix of ˇij terms
ˇij
A
B
C
D
E
F
A
B
C
D
E
F
G
0
37.89
46.8
72
73.92
74.76
Fig. 9.11 Location of the
weighted absolute 1-center for
the weighted tree of Fig. 9.10
G
164:57
130:90
116
72
74:11
15
0
9
A
8
8
10
9
B
4
1
D
F
8.71
5
5
6
C
G
E
6
5
12
The largest element corresponds to the row associated with node A, so we stop.
The optimal location is at the point 13.71 units away from node G on the path from
node G to node A or 8.71 units from node F to node D on the link from node F to
node D shown by Fig. 9.11.
hA
d.A; G/ D 13:71:
hG C hA
(9.19)
Note that we only had to compute 11 of the ˇij elements instead of 21 elements.
(The actual number of elements that must be computed is n.n1/=2 or 21 in
this case).
9.3.1.5 Absolute P-Center on a Weighted Tree Network (Francis et al. 1992)
In this section, the p-center problem on a tree will be solved by solving a sequence
of covering problems.
We are given a set Y D fy1 ; : : : ; yp g of center locations, and each center in some
edge. For each demand node vi ; i D 1; : : : ; n, and any Y , we have the closest center
assumption as follows:
D.Y; vi / D minfd.y1 ; vi /; : : : ; d.yp ; vi /g:
(9.20)
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M. Biazaran and B. SeyediNezhad
It means that each demand node vi receives service from a closest center (facility).
Now the objective function of interest would be as follows:
G.Y / D maxfh1 D.Y; v1 /; : : : ; hn D.Y; vn /g:
(9.21)
We seek an absolute p-center Y which minimizes G.Y /. Note that if p n, the
problem has a trivial solution (one center at each vertex).
To solve this problem, we shall use the fact that there is some 1-center problem
whose minimal value is equal to the minimal value of G.Y /. Thus, the search for the
minimal objective function value can be limited to the positive entries in the bound
matrix B D .ˇij /.
Minimal Objective Function Value Property
The minimal objective function value for the p-center problem, say zp , is
some entry in the bound matrix BD.ˇij /.
Suppose that we have a p-center problem with n nodes and Y D fy1 ; : : : ; yp g in an
absolute p-center. We can write the p-center problem as follows:
Min G.Y /:
(9.22)
G.Y / D maxfh1 D.Y; v1 /; : : : ; hn D.Y; vn /g;
(9.23)
jY j D p:
(9.24)
Subject to
Equivalently we can
Min z:
(9.25)
Subject to
D.Y; vi /
z
i D 1; : : : ; n;
hi
jY j D p:
(9.26)
(9.27)
Consider zp as the minimal objective function value, so zp is the smallest value of z
such that there exist a p-center Y for which Y and z satisfy (9.26)–(9.27). Consider
R to be the set of entries in the B matrix. According to minimal objective function
value property, we can say that zp will be one of the numbers in the set R. So we
conclude that zp is the smallest number z in R for which there exist a Y that Y and
z satisfy (9.26)–(9.27). We let C.z/ denote the following covering problem:
C.z/ W Min jY j :
(9.28)
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Center Problem
207
Subject to
D.Y; vi /
z
i D 1; 2; : : : ; n;
hi
(9.29)
C.z/ is the equivalent covering problem and in this problem, z would be the parameter of the covering problem. So each time we change z, we will have a different
covering problem.
If zp is the minimal objective function value for p-center problem and q.z/ is
the minimal objective function of the covering problem, we conclude that zp is the
minimal element r in R for which q.r/ p.
As you see, the p-center problem can be solved by solving a sequence of covering
problems.
We use discrete values in R as possible values of zp and there is as many as
n.n 1/=2 elements in B. So a good procedure is needed for searching among
these elements to find zp . A search procedure that is often used is called bisection
search. The algorithm is as follows (Francis et al. 1992):
1. Construct the set R consisting of all distinct positive values ˇij .
2. Repeat step 3 until R consists of a single element.
3. Find a median entry in R, say r. Solve the covering problem C.r/ and compute
q.r/. If q.r/ p, delete all elements greater than r form R, otherwise, delete all
elements not greater than r (including r).
4. Let zp be the single element in R. Solve C.zp / and obtain its optimal solution
Y . If jY j D p, take Y as an optimal p-center and stop. If jY j D p1 < p,
choose any convenient p-p1 centers append to Y to obtain an optimal p-center
and stop.
Using bisection search is a good idea, because it reduces the number of elements in
R by half at each iteration. That is, even if R initially consists of 1,000,000 elements,
bisection search would have to solve at most 21 covering problems!
9.3.2 Center Problems on a General Graph
In this section, center problems on a general graphs are discussed. A general graph
is a network N D .V; A/ which has at least one cycle. A path in a network is called
a cycle if the initial and final vertices in the path are identical (Francis et al. 1992).
9.3.2.1 Vertex 1-Center on a General Graph (Francis et al. 1992)
To find a vertex center, we need to compute the Matrix D D .dij /. The entry in row
i and column j in this matrix is d.vi ; vj / that implicates the shortest path between
the node i and the node j in the graph (One can use Floyd’s algorithm to compute
these distances).
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M. Biazaran and B. SeyediNezhad
Fig. 9.12 Example of a
general graph
1
5
4
6
2
6
3
3
4
Table 9.3 Example of a
vertex 1-center on a general
graph
D0
vi
2
0
10
8
14
Weight of node vj
3
4
15
16
0
24
18
0
18
12
g.vi /
2
14
12
6
0
16
24
18
18
In the next step, in order to obtain the new matrix D 0 , every entry in each column
must be multiplied by hj . For each row of the new matrix, the maximum entry g.vi /
is determined and placed in the right margin. Any row with the smallest value in the
right margin identifies a vertex center.
Consider the general graph of Fig. 9.12 with weights of 2, 3, 4, and 2 for vertices
1–4, respectively. Computation results are shown in Table 9.3. With g.v1 / D 16, we
can see that v1 is a vertex 1-center on this general graph.
9.3.2.2 Vertex P-Center Problem on a General Graph (with Integer
Distances) (Daskin 1995)
A solution approach to solve the unweighted vertex p-center on a general graph is
outlined in this section. We assume that all link distances are integer values. (Since
all rational values can be converted to integer values by multiplying them by a sufficiently large number, this is not a restrictive assumption). In this approach, we
search over the range of coverage distances to find the smallest one that allows all
nodes to be covered by a vertex center.
The search procedure is called binary search. In this procedure, we define initial
lower and upper bounds for the objective function value of the p-center problem.
Using the average of the lower and upper bounds, we solve the set covering problem.
If the number of facilities needed to cover all nodes is less than or equal to p, we
will find out that the objective function of the p-center problem cannot be larger
than this coverage distance. So we replace the upper bound to the current coverage
distance. If the number of facilities needed to cover all nodes is greater than p, then
similarly we will find out that the objective function value of the p-center problem
must be larger than the current value, because we have to cover all nodes with a
smaller number of centers and hence a larger coverage distance. In this case we
would replace the lower bound with the current coverage distance plus 1.
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Let q.r/ denote the optimal value of the set covering problem when the coverage
distance is r. also let RL and RH as lower and upper bounds on the p-center objective function value. The initial values of lower and upper bounds are defined as
follows:
RL D 0;
R
H
(9.30)
D .n 1/ maxij .dij /;
(9.31)
where n is the number of nodes in the graph and dij is the length of link .i; j /. The
binary search algorithm would be as follows:
1. Set RL and RH . Using equations in (9.30)–(9.31). Note that by setting these
values in this manner, we make sure that every possible value for the coverage
distance is considered.
2. Set r as shown in (9.32). Where bxc denotes the largest integer less than or equal
to x.
H
R C RL
:
(9.32)
rD
2
3. Solve a set covering problem with a coverage distance of r and let the solution
be q.r/.
4. If q.r/ p, reset RH to r; else reset RL to r C 1.
5. If RH ¤ RL , go to step 2; otherwise, stop, r is the optimal value for the objective
function of p-center problem and optimal locations for the p-center problem,
are the locations corresponding to the set covering solution for this coverage
distance.
To illustrate this algorithm, consider the graph of Fig. 9.13. All distances are integer in this graph. Let p D 2. The maximum link distance is 6 and there are five
nodes. Therefore, we initially set RH D 24 and RL D 0. Table 9.4 summarizes
the iterations of the algorithm for this problem. The optimal value for the objective
function is r D 4. From the covering problem, when coverage distance is 4, optimal
locations of two centers are nodes B and C or nodes D and C .
The discussed algorithm can be extended to solve the weighted vertex p-center
problem. All steps are similar in this case except that the initial upper bound RH ,
must be replaced by
RH D .n1/Œmaxij .dij /Œmaxij .hi /:
(9.33)
A
6
5
B
Fig. 9.13 Example of a
general network with integer
distances
4
C
5
4
D
3
E
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M. Biazaran and B. SeyediNezhad
Table 9.4 Summary of Iterations of the vertex 2-center algorithm
Iteration
1
2
3
4
5
6
RL
RH
r
q(r)
0
0
0
4
4
4
24
12
6
6
5
4
12
6
3
5
4
Stop!
1
1
4
1
2
Note that in solving the set covering problem, demand node i is covered by
candidate site j only if hi dij r.
9.3.2.3 Absolute P-Center on a General Graph (Francis et al. 1992)
In this section we will introduce the concept of distance function. Consider the simple graph of Fig. 9.13. If we are to locate a center in a point X on the link between
node B and node C and x units from node B, the distance between this point and
node A would be:
d.X; A/ D minfd.B; X / C d.B; A/; d.X; C / C d.C; A/g:
(9.34)
Let d.B; A/ D dBA is the shortest path between node B and Node A, and d.C; A/ D
dCA . Since we have d.B; X / D x and d.X; C / D lBC x, we can write the
(9.33) again:
(9.35)
d.X; A/ D minfx C dBA ; .lBC x/ C dCA g:
If we substitute the values, we will have:
d.X; A/ D minfx C 6; x C 9g:
(9.36)
Figure 9.14 illustrates d.X; A/, the minimum distance between node A and point X .
so the shortest path between node A and point X is a piecewise function, that depends on the distance of point X from node B, say x.
Now if we consider all other nodes with point X , we will have the graph of
Fig. 9.15.
If we let g.X / denote the maximum of weighted distances involving X , then
we have
(9.37)
g.X / D maxfhA d.X; A/; : : : ; hE d.X; E/g:
With all weights equal to one, the thick piecewise line in Fig. 9.15, illustrates g.X /.
If the absolute center of the graph is to be located on the link BC, the optimal value
of objective function would be the minimum of g.X /. In this case, the minimum is
on node C .
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211
Fig. 9.14 Graph of d(X, A),
for the network of Fig. 9.13
9
6
4
Fig. 9.15 Graphs of distances
on the link BC
0
5
B
C
x
8
6
4
0
x1
B
8̂
<hE d.X; E/ 0 x 1
g.X / D hA d.X; A/ 1 x 2:5 :
:̂
hD d.X; D/ 2:5 x 5
x2
5
x
C
(9.38)
We call x1 and x2 the intersection points. Intersection point is defined by exactly
two distinct weighted distance functions being equal. At this point, one of the two
functions increases locally as x moves away from the point in one direction, while
the other increases as x moves away from the point in the other direction. For example, consider the intersection point x1 in the graph of Fig. 9.15. At this point, we
have hD d.X; D/ D hA d.X; A/, while hA d.X; A/ increases as x decreases locally
from 2.5, hD d.X; D/ increases as x increases locally from 2.5.
As it is indicated, it is also possible to have local minima at the endpoints of
the arc. For example, point C is not defined by an intersection point, but it is the
minimum of g.X /.
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M. Biazaran and B. SeyediNezhad
Now after this introduction, we conclude that to find the absolute center in a
network, we have to find all intersection points on every arc and choose the one
with minimum distance to all other nodes and also evaluate g.vi / at every vertex
and finally choose the best point among all intersection points and vertices.
Consider the p-center problem, in which X D fx1 ; : : : ; xp g is a collection of
centers. We let D.X; vi / be the distance between vertex vi and a closest center in X ,
so the objective would be to find an absolute p-center X that minimizes:
G.X / D maxfh1 D.X; v1 /; : : : ; hn D.X; vn /g:
(9.39)
The VIP property for the p-center problem is the following:
Vertex and Intersection Point (VIP) Property
There exists at least one absolute p-center for which each center is at a vertex or
an intersection point. Thus it is sufficient to consider only vertex and intersection
point locations in order to find an absolute p-center.
This property reduces the p-center problem with infinite number of points to be
considered to a problem with a finite number of points to be considered.
Although now we have a finite set of potential center locations, it would be possibly
very large. To improve our enumeration approach, we will avoid enumeration of
intersection points on some arcs. If we have a current best trial solution, X 0 , with
function value g.X 0 /. If we know a number bpq so that g.X / bpq for all X in
arc Œvp ; vq , and also bpq > g.X 0 /, no location on the Œvp ; vq can be an absolute
center.
Arc Exclusion-Bounding Property
bipq D min fhi d.vp ; vi /; hi d.vq ; vi /g
bpq D max fbipq : i D 1; : : : ; ng
If bpq >g.X 0 /, no point in arc Œvp ; vq is an absolute 1-center.
9.3.2.4 Formulating the P-Center Problem (Francis et al. 1992)
Now that we have limited the solution of absolute p-center to a finite set of potential
center locations using VIP and Arc exclusion-bounding properties, we can formulate the p-center problem. In this case, there is binary decision for every potential
location to include it in the p-center or not. Define Q D fq1 ; q2 ; : : : ; qm / to be the
set of all potential center location. First n members of Q correspond to vertices of
the graph.
We associate each intersection point qk with a bound value bk , that is simply
the weighted distance from on of the two vertices used to define the intersection
point. For example, consider the intersection point x1 in Sect. 3.2.3. x1 is defined by
vertices A and E, so the bound value for x1 is 7.
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213
Decision variables:
xj D 1 if a center is located at qj and 0 otherwise
Now the p-center problem can be formulated as follows:
Min zp :
(9.40)
zp minfhi dij jj 2 .X /gi D 1; : : : ; n;
(9.41)
Subject to
X
xj p;
(9.42)
j
xj 2 f0; 1gj D 1; : : : ; m:
(9.43)
The set .X / in (9.41) is the set of potential center locations that are used in the
solution. This is a mathematical statement of the p-center problem, but there is no
direct solution procedure for it. But the indirect approach is used to solve this problem and consists of solving a sequence of related covering problems. In Sect. 3.2.5,
we will discuss the relationship between p-center problem and covering problem.
9.3.2.5 Relationship Between P-Center Problem and Covering Problem
(Francis et al. 1992)
We used covering problem to solve center problems earlier in this chapter. Now in
this section, we will discuss the relationship between them in general.
Suppose that .z ; X / is the optimal solution to p-center problem. Covering
problem is as follows:
Decision variables:
aij D 1 if hi dij z and 0 otherwise
CP .z/ W Minimize p D
X
xj
(9.44)
aij xj 1 i D 1; : : : ; n;
(9.45)
j
Subject to
X
j
xj 2 f0; 1gj D 1; : : : ; m:
(9.46)
Now suppose that z1 and z are specific values for parameter z, and the optimal
solution to CP.z1 / is .p 1 ; X 1 /. The relationship between two problems is as follows,
offered without proof:
1. If .z ; X / is a solution to p-center problem, then .p; X / solves CP.z /
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M. Biazaran and B. SeyediNezhad
2. If .z ; X / is a solution to p-center problem and .p 1 ; X 1 / solves CP.z1 /, then:
(a) If p 1 > p, then z1 < z .
(b) If p 1 p, then z1 z .
To explain these claims, part 1 states that if we knew the z , we could use CP.z / to
determine X . Part 2 states that if we can guess a value z1 for z , and solve CP.z1 /,
in case (a), we obtain a lower bound on z and in case (b), we obtain a feasible
solution or an upper bound on z . Earlier in this chapter we had a similar discussion
on this, for vertex p-center on a general graph with integer distances.
9.4 Approximate Solution Approaches
Kariv and Hakimi (1979) showed that the p-center problem on a general graph is
NP-hard. In this section, some of the approximate solution approaches suggested
for solving p-center problem are discussed.
Many heuristic and metaheuristic solution approaches were designed for the
p-center problem in recent years; here we will mention some of them.
Pallottino et al. (2002) represented a local search heuristic for the capacitated
vertex p-center.
Bespamyatnikh et al. (2002) represented a polynomial time algorithm for
p-center problem on circular arc graphs.
Burkard and Dollani (2003) represented a number of algorithms with low computational complextity for p-center problem with pos/neg weights on trees and
general graphs.
Caruso et al. (2003) represented four algorithms that solve p-center problems.
Two exact algorithms and two heuristics. All these methods are based on solving
a sequence of set covering problems. Two heuristics can solve problems up to
900 nodes in a few seconds.
Hochbaum and Pathria (1997) represented approximation algorithms for two
closely related problems p-SetSupplier and p-SetCenter.
Khuller et al. (2000) represented polynomial time approximation algorithm to
solve the fault tolerant p-center problem in which every vertex must have at least
˛ centers close to it.
Khuller and Sussmann (2000) represented an approximation algorithm for the
capacitated p-center problem.
Panigrahy and Vishwanathan (1998) represented an approximation algorithm for
the asymmetric p-center. The authors repeatedly employed a greedy set covering
algorithm.
Pelegrin and Canovas (1998) presented new versions of seed point algorithms.
In these algorithms, the first step is to generate p points for the p-center problem and then generating a partition of the demand points. They proposed a new
assignment rule based on partitions.
9
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215
Andersson et al. (1998) represented an approximation algorithm based on demand point aggregation method.
Dyer and Frieze (1985) presented a greedy algorithm in which the first center is
chosen at random. This variant of greedy is called Greedy Plus (GrP).
Drezner (1984) represented two heuristic algorithms for the p-center problem.
These heuristics resemble algorithms for other location-allocation problems.
Hochbaum and Shmoys (1985) represented a 2-approximation heuristic for the
p-center problem.
Hassin et al. (2003) introduced a local search strategy in which solutions are
compared lexicographically rather than by their worst coordinate. They applied
this approach to p-center problem.
Mladenovic et al. (2003) suggested metaheuristic approaches for the first time
as a means for solving p-center problems. They represented a basic Variable
Neighborhood search and two Tabu search metaheuristics and a multi-start local
search for the p-center problem.
Bozkaya and Tansel (1998) proposed a spanning tree approach for the absolute
p-center problem on cyclic networks (general graphs). The approach was motivated by the fact that the problem is NP-hard on general networks but solvable in
polynomial time on trees.
Pacheco and Casado (2005) proposed a metaheuristic procedure based on the
scatter search approach for solving p-center problem. The procedure incorporates procedures based on different strategies such as local search, GRASP and
path relinking. The aim was solving problems with real data.
Cheng et al. (2007) modeled the network of p-center problem as an interval
graph whose edges all have unit lengths and provide a polynomial time algorithm to solve the problem under the assumption that the endpoints of the
intervals are sorted.
9.5 Case Study
In this section we introduce a real-world case study related to p-center problems.
9.5.1 A Health Resource Case (Pacheco and Casado 2005)
In Pacheco and Casado (2005) a real health resource allocation problem was surveyed in Burgos, a rural area in the north of Spain. As mentioned in their paper,
the authorities have limited budget, hence less than ten facilities considered to be
located. With dispersed population in this area, it was essential to design a method
to provide good solutions with a low number of facilities.
Real data in this case refer to the estimation of diabetes cases in that area. Four
hundred and fifty two locations with at least one known case of diabetes were taken
into account within the area and 152 of these locations were considered suitable for
the project.
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M. Biazaran and B. SeyediNezhad
They dealt with two problems, first one was minimizing the maximum distance
between users and facilities (p-center) and the second one was minimizing the population further away from its closest available facility than a previously fixed time
(maximum covering).
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Chapter 10
Hierarchical Location Problem
Sara Bastani and Narges Kazemzadeh
In the basic models presented in previous chapters, it is assumed that there was only
one type of facility being located. As the systems that provide services/products
usually consist of two or more levels of facilities, we discuss hierarchical facility
systems in this chapter.
Many facility systems are hierarchical in nature. These facilities are usually hierarchical in terms of the types of services they provide. For example, the health care
delivery system consists of local clinics, hospitals and medical centers (Fig. 10.1). In
this system a local clinic provides basic health care and diagnostic services. A hospital provides out-patient surgery in addition to those provided by a local clinic; and
a medical center provides out-patient surgery and a full range of in-patient services.
As another example of a hierarchical system, consider a solid waste disposal system.
The solid waste is collected from the source of solid waste and shipped to transfer
stations or landfill stations by trucks. Other examples of a hierarchical system are:
education system, postal system, banking system and production–distribution system (Narula 1986; Daskin 1995).
In the examples above, facilities cannot be located independently at each level.
Thus, there is a need to consider them as a hierarchical system.
In this chapter, location-routing problem, which is confused with the hierarchical
location problem, is not discussed. In the location-routing problem, the locations
of the primary facilities and the demand points are fixed and given; the objective is to locate intermediate facilities and design tours originating at the primary
facilities to serve the secondary facilities and tours emanating from secondary facilities to serve the demand points. The location-routing problem is presented in
Chap. 18.
This chapter is organized as follows. In Sect. 10.1, we give a definition about hierarchical location problem, its applications and classifications. In Sects. 10.2–10.5
mathematical models for the problem are given. We present solution methods in
Sect. 10.3. We conclude the chapter with some case studies in Sect. 10.4.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 10, c Physica-Verlag Heidelberg 2009
219
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S. Bastani and N. Kazemzadeh
Customer 3
Hospital
Customer 2
Medical center
Customer 1
Local clinic
Level 3 service: a full
range of in-patient services
Level 2 service: out-patient
surgery
Level 1 service: basic health
care and diagnostic services
Fig. 10.1 An example of a health care delivery system
10.1 Applications and Classifications
Many applications of hierarchical location problem are:
Health care systems are one of the most studied real-life applications of the
hierarchical facility location problem. Such systems can consist of local clinics, hospitals and medical centers. Some works in this area are Calvo and
Marks (1973), Dökmeci (1977), Okabe et al. (1997), Rahman and Smith (1999),
Marianov and Taborga (2001), Galvão et al. (2002, 2006) and Smith et al. (2009).
Researches for considering location problems in health systems, including hierarchical problems is reviewed by Rahman and Smith (2000).
Solid waste disposal system consists of transfer stations and landfill stations.
A work for this application is Barros et al. (1998).
Production–distribution system consists of factories, warehouses and retail outlets. Some works in this area are Scott (1971), Ro and Tcha (1984), Tcha and
Lee (1984), Van Roy (1989) and Eben-Chaime et al. (2002).
Education system consists of kindergartens, guidance schools and high schools.
Moore and ReVelle (1982) and Teixeira and Antunes (2008) represented some
models in this area. Teixeira and Antunes (2008) considered two levels for school
system: public and private schools.
Emergency medical service (EMS) systems consist of basic life support and advance life support. Some works in this area are Charnes and Storbeck (1980),
Mandell (1998) and Şahin et al. (2007).
Some other applications have been presented for hierarchical facility locations. For
example Banerji and Fisher (1974) presented an application of successively inclusive hierarchical systems for area planning. Chan et al. (2008) is another application
presented for hierarchical models. They applied a hierarchical maximal-coverage
location–allocation for the case of search and rescue.
A hierarchical system of facilities consists of m levels in which level 1 is the
lowest level of service or facility and level m is the highest level. Level 0 presents
demand nodes.
Narula (1986) considers two types of facility hierarchies and three types of service in relation to a hierarchical system. Daskin (1995) refers to this as two ways in
which services are offered and three types in terms of the regions to which services
are provided.
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Hierarchical Location Problem
221
In a successively inclusive facility hierarchy, a facility at level k .k D 1; : : : ; m/
provides services 1 through k.
– A successively inclusive facility hierarchy is said to have a locally inclusive
service hierarchy if a level k facility .k D 1; : : : ; m/ at location j offers services of level 1; : : : ; k only to demands at location j and only level k services
to demands at location i ¤ j .
– A successively inclusive facility hierarchy is said to have a successively inclusive service hierarchy, Daskin (1995) refers to this as globally inclusive
service hierarchy, when a level k facility .k D 1; : : : ; m/ at location j provides level 1; : : : ; k services to all demand nodes.
In a successively exclusive facility hierarchy, a facility level k .k D 1; : : : ; m/
provides only services of level k.
– A successively exclusive facility hierarchy is said to have a successively exclusive service hierarchy if a level k facility at location j provides type k
services to all demand nodes.
Narula (1984) considers two types of flow discipline. The flow is integrated when
it can be from any lower-level facility to any higher-level facility. The flow is discriminating if it is from any lower-level facility k to the next higher-level facility
k C 1.
Şahin and Süral (2007) classify the hierarchical location problem with respect to
flow pattern, service variety, spatial configuration, objective, horizontal interactions
and capacity limits on facilities.
Flow pattern refers to demand flow through levels of a hierarchical system. In a
single-flow pattern, demand begins at level 0, passes through all levels, and finally
ends at the highest level; or vice versa. In a multi-flow pattern, demand begins and
ends at any level k .k D 1; : : : ; m/. Flow pattern is similar to the flow discipline
described in Narula (1984). We can consider flow pattern in a referral or nonreferral system. In a referral system, lower-level facility refers a proportion of
customers to the higher-level facility.
Service variety specifies the service availability at the levels of hierarchy. Nested
and non-nested features, respectively resembles Narula’s successively inclusive
and exclusive systems.
Spatial configuration refers to the coherency of a hierarchical system. A coherent
hierarchical system is one in which all customers of the same low level facility
are also customers of the same high level facility.
For locating facilities in hierarchical location problem models, three types of
objectives are considered: median, fixed charge, and covering objectives.
Teixeira and Antunes (2008) consider three types of spatial pattern according to
which demands are assigned to facilities: closest assignment, single assignment and
path assignment. In closest assignment, a demand node is assigned to the closest
facility located. In single assignment, a demand node is assigned only to one facility located. In path assignment, when a given facility is located at location j , all
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S. Bastani and N. Kazemzadeh
demands near the path travelled by the customers to reach the facility are also assigned to it. In following Sects. 10.2–10.5, we introduce categories of hierarchical
location models presented and developed as mathematical models.
10.2 Flow-Based Hierarchical Location Problem
Aardal et al. (1999) considers flow-based formulation with fixed-charge objective in
which allocations is represented by a variable that is more common to network flow
models and establishes flow from one level of the hierarchy to another. Şahin and
Süral (2007) present this model with median objective.
10.2.1 Flow-Based Formulation for Single-Flow Systems (Şahin
and Süral 2007)
Şahin and Süral (2007) consider a two-level single-flow system, in which flow begins from demand node, passes a level 1 facility and ends at a level 2 facility.
10.2.1.1 Model Assumptions
Model assumptions of this model are as follows:
The objective function is MiniSum.
The solution space is discrete and finite.
The formulation represents a two-level, single-flow, non-nested, non-coherent,
capacitated system.
The number of facilities at each level is known (exogenous model).
10.2.1.2 Model Inputs
Model inputs of this model are as follows:
cjj 0 : unit cost of flow between facilities level 2 at candidate location j 0 and level
1 at candidate location j
cij : unit cost of flow between level 1 facility at candidate location j and demand
node at i
di : the amount of demand at node i
Mj 0 : the capacity of level 2 facility at candidate location j 0
Mj : the capacity of level 1 facility at candidate location j
p 0 : the number of level 2 facilities to be located
p: the number of level 1 facilities to be located
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10.2.1.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
xj 0 D 1 if level 2 facility located at j 0 , zero otherwise
yj D 1 if level 1 facility located at j , zero otherwise
vjj 0 : amount of flow between level 2 facility at location j 0 and level 1 facility at
location j
uij : amount of flow between level 1 facility at location j and demand at location i
10.2.1.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
XX
XX
(10.1)
vjj 0 cjj 0 :
uij cij C
Min
i
j
j
j0
Subject to
X
uij D di 8 i;
X
vjj 0 D
X
uij Mj yj 8 j;
(10.4)
X
vjj 0 Mi xi 8 j 0 ;
(10.5)
X
yj D p;
(10.6)
X
xj 0 D p 0 ;
(10.7)
uij 0 8 i; j;
(10.8)
vjj 0 0 8 j; j 0 ;
(10.9)
j
j0
i
j0
j
j0
X
i
uij 8 j;
(10.2)
(10.3)
yj D 0; 1 8 j;
(10.10)
xj 0 D 0; 1 8 j 0 :
(10.11)
Equation (10.1) minimizes the total demand-weighted distance from demand nodes
to level 1 facilities and from level 1 facilities to level 2 facilities. Equations (10.2)
ensure that the total demand of a node is completely satisfied by level 1
facilities. Equations (10.3) ensure that the demand referred to a level 2 facility by a level 1 facility is equivalent to the total demand of that level 1 facility.
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Equations (10.4)–(10.5) are for facility capacities. Equations (10.6)–(10.7) locate
the required numbers of level 1 and 2 facilities, respectively. Equations (10.8)–
(10.9) are non-negativity; (10.10)–(10.11) are integrality constraints.
10.2.2 Flow-Based Formulation for Multi-Flow Systems (Şahin
and Süral 2007)
Şahin and Süral (2007) consider a two-level multi-flow, nested system, in which
demand at location i is provided either directly or via a level 1 facility for its level 2
demands and can be served by a level 2 facility for its both level 1 and 2 demands.
Since facilities are capacitated, a portion of level 1 demand is provided by a level 1
facility and the remaining portion by a level 2 facility. The same model can be seen
in Narula and Ogbu (1985).
10.2.2.1 Model Assumptions
The assumptions of this model are similar to previous model except that there is the
following difference:
The formulation represents a two-level, multi-flow, nested, non-coherent, capacitated system.
10.2.2.2 Model Inputs
Model inputs are similar to previous model in addition to the following parameter:
cij 0 : unit cost of flow between level 2 facility at candidate location j 0 and demand
node at i
10.2.2.3 Model Outputs (Decision Variables)
Model outputs are similar to previous model in addition to the following decision
variable:
wij 0 : amount of flow between level 2 facility at location j 0 and demand at location i
10.2.2.4 Objective Function and its Constraints
The objective function is similar to previous model in addition to the following term:
XX
wij 0 cij 0 :
(10.12)
i
j0
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The differences between the constraints of this model and previous model are as
follows:
Equations (10.2) and (10.5) are omitted.
The following constraints are added to the formulation:
X
j
X
j0
uij C
X
vjj 0 C
j0
wij 0 D di 8 i;
X
i
wij 0 Mi xi 8 j 0 ;
wij 0 0 8 i; j 0 :
(10.13)
(10.14)
(10.15)
Equation (10.12) in the objective function minimizes the total demand-weighted
distance from demand nodes to level 2 facilities. Equations (10.13) replace
(10.2) for which the first-level demand can be assigned to either level facility.
Equations (10.14) replace (10.5) for which a level 2 facility capacity can be utilized
either by the second-level demand at a node or the referred demand from level 1.
Equations (10.15) are non-negativity.
10.3 Median-Based Hierarchical Location Problem
(Daskin 1995)
Mirchandani (1987) presents a generalized median-based hierarchical location
model which allows various allocation schemes. Daskin (1995) represents medianbased formulations in which demand at location i is assigned to a set of facilities
one from each level.
10.3.1 Median-Based Formulation for Globally Inclusive Service
Hierarchies
A model for a successively inclusive facility hierarchy operating under a globally
inclusive service hierarchy is formulated by Daskin (1995).
10.3.1.1 Model Assumptions
Model assumptions of this model are as follows:
The objective function is MiniSum.
The solution space is discrete and finite.
The formulation represents a successively inclusive facility hierarchy operating
under a globally inclusive service hierarchy.
The number of facilities at each level is known (exogenous model).
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10.3.1.2 Model Inputs
Model inputs of this model are as follows:
hi k : the amount of demand for type k services at node i
dij : the distance between node i and candidate location j
pk : the number of type k facilities to be located
10.3.1.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj k D 1 if a facility of type k is located at candidate location j , zero otherwise
Yij k D 1 if demand at node i for type k services is assigned to a facility at
candidate location j , zero otherwise
10.3.1.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
XXX
i
j
hi k dij Yij k :
(10.16)
k
Subject to
X
Yij k D 1 8 i; k;
(10.17)
X
Xj k D pk 8 k;
(10.18)
j
j
Yij k
m
X
hDk
Xj h 8 i; j; k;
(10.19)
Xj k D 0; 1 8 j; k;
(10.20)
Yij k D 0; 1 8 i; j; k:
(10.21)
Equations (10.16) minimizes the demand-weighted total distance. Equations (10.17)
stipulates that all demand levels at all locations must be assigned to some facility. Equation (10.18) limits the total number of level k facilities located to pk .
Equation (10.19) ensure that demands at node i for level k services are assigned
to the facility at location j when there is a level k or higher level facility located
at location j . They are the linkage constrains. Equations (10.20)–(10.21) are the
integrality constraints.
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10.3.2 Median-Based Formulation for Locally Inclusive Service
Hierarchies
A model for a successively inclusive facility hierarchy operating under a locally
inclusive service hierarchy is formulated by Daskin (1995).
10.3.2.1 Model Assumptions
The assumptions of this model are similar to previous model except that there is the
following difference:
The formulation represents a successively inclusive facility hierarchy operating
under a locally inclusive service hierarchy.
10.3.2.2 Model Inputs
Model inputs are similar to previous model.
10.3.2.3 Model Outputs (Decision Variables)
Model outputs are similar to previous model.
10.3.2.4 Objective Function and its Constraints
The objective function is similar to previous model. The differences between the
constraints of this model and previous model are as follows:
Equations (10.19) are omitted
The following constraints are added to the formulation:
Yij k Xj k 8 j; kI 8 i ¤ j;
Yjj k
m
X
hDk
Xj h 8 j; k:
(10.22)
(10.23)
Equations (10.22) ensure that demands at node i for level k service are assigned
to the facility at location j when there is a level k facility located at location j .
Equations (10.23) state that demands for level k service are assigned to the level
k or higher level facility when the demand node and candidate facility location are
the same.
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10.3.3 Median-Based Formulation for Successively Exclusive
Service Hierarchies
A model for a successively exclusive facility hierarchy operating under a successively exclusive service hierarchy is formulated by Daskin (1995).
10.3.3.1 Model Assumptions
The assumptions of this model are similar to globally inclusive service hierarchy
model except that there is the following difference:
The formulation represents a successively exclusive facility hierarchy operating
under a successively exclusive service hierarchy.
10.3.3.2 Model Inputs
Model inputs are similar to globally inclusive service hierarchy model.
10.3.3.3 Model Outputs (Decision Variables)
Model outputs are similar to globally inclusive service hierarchy model.
10.3.3.4 Objective Function and Its Constraints
The objective function is similar to previous model. The differences between the
constraints of this model and previous model are as follows:
Equations (10.19) are omitted
The following constraints are added to the formulation:
Yij k Xj k 8 i; j; k:
(10.24)
Equations (10.24) ensure that demands at node i for level k service are provided by
the level k facility at location j .
10.4 Coverage-Based Hierarchical Location Problem
(Daskin 1995)
There are basically three different approaches to formulate covering problems. The
set covering location problem seeks to minimize the number of open facilities (or
the cost of opening facilities) while ensuring that all demand points are within the
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coverage area of some open facility. The p-center problem locates a given number
.p/ of facilities while minimizing the maximum distance between demands and
facilities. The most studied covering problem in hierarchical location literature is the
maximum covering problem that seeks to maximize the amount of demand covered.
A hierarchical location problem with a coverage objective is formulated by
Daskin (1995). This formulation considers maximal covering with two approaches
for defining covered demands.
10.4.1 Hierarchical Maximal Covering Location Problem
This model for hierarchical maximal covering location problem is presented by
Daskin (1995).
10.4.1.1 Model Assumption
The objective is to maximize the total number of demands that are covered while
the number of facilities at each level is known.
The formulation represents a nested system.
10.4.1.2 Model Inputs
Model inputs are similar to median-based hierarchical model in addition to the following parameter:
kq
aij D 1 if a level q facility located at candidate location j can provide service k
to demands at node i , zero otherwise.
10.4.1.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj q D 1 if we locate a level q facility at candidate location j , zero otherwise.
Zi k D 1 if demands for service k at node i are covered, zero otherwise.
10.4.1.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Max
XX
i
k
hi k Zi k :
(10.25)
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Subject to
X
j
Xj q D Pq 8q;
(10.26)
kq
(10.27)
Zi k
XX
j
q
aij Xj q 8 i; k;
0 Zi k 1 8 i; k;
(10.28)
Xj q D 0; 1 8j; q:
(10.29)
Equation (10.25) maximizes the total number of demands of all levels that are covered. Equations (10.26) stipulate that exactly Pq level q facilities are to be located.
Equations (10.27) stipulate that demand for service k at node i cannot be counted as
being covered unless we locate at least one facility at one or more of the candidate
locations which are able to provide service k to demand node i . Equations (10.28)–
(10.29) are the nonnegative and standard integrity constraints, respectively. Note
that the coverage variables .Zi k / explicitly do not need to accept only integer values.
10.4.2 Hierarchical Maximal Covering Location Problem
with Covering all Kinds of Demands
In some situations, demands at each demand node can only be counted as being
covered if all of the services that are demanded at the node i are covered. Moore
and ReVelle (1982) introduced a maximal covering model with this assumption.
Daskin (1995) also represented the model as follows.
10.4.2.1 Model Assumption
The model assumptions are similar to previous model, except that in this model,
demands at each demand node can only be counted as being covered if all kind of
the services that are demanded at the node i are covered.
10.4.2.2 Model Inputs
Model inputs are similar to median-based hierarchical model in addition to the following parameters:
hi : The demand at node i . This is probably a function of the service specific
demands .hi k / at node i such as:˙k hi k or ˙k ˛k hi k or maxk fhi k g where k is a
weight associated with demands for service level k.
ıi k D 1 if hi k > 0, zero otherwise
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10.4.2.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Wi D 1 if demands at node i are covered, zero otherwise
10.4.2.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Max
X
hi Wi :
(10.30)
i
Subject to
X
Xj q D Pq 8q;
X X kq
aij Xj q 8i; k;
Zi k
(10.31)
Wi Zi k C .1 ıi k / 8i; k;
(10.33)
0 Zi k 1 8i; k;
(10.34)
Xj q D 0; 1 8j; k;
(10.35)
0 Wi 1 8i:
(10.36)
j
(10.32)
q
Equation (10.30) maximizes the number of covered demands.
Equations (10.31)–(10.32) are identical to (10.26)–(10.27) respectively. Equations (10.33) stipulates that demands at node i cannot be covered unless all services
that are required at demand node i are covered. If there exists demand for service
level k at node i .ıi k D1/, then (10.33) stipulates that Wi Zi k , meaning that
node i cannot be covered unless demands for service level k at node i are covered.
However, if there is no demand for level k services at node i , then node i can be
covered even if Zi k D 0. Finally, (10.34)–(10.36) are the nonnegative and integrity
constraints on the problem’s decision variables.
10.4.2.5 Extensions and Comments on the Model
A complication associated with hierarchical covering location problem is that the
critical distance for lower-level services should be significantly less than that for the
higher-level services. However, in nested systems, the critical distance definition
changes since one can cover demand for lower-level services with a higher-level
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facility as long as the multi-flow setting exists. To consider critical distance, it can
be assumed that demand i is covered if:
Lower-level facility is within a distance dl of node i and higher-level facility is
within a distance dh of node i or
Higher-level facility is within a distance d of node i , where dl < d < dh .
This definition is presented by Moore and ReVelle (1982) and then used by Mandell
(1998), Espejo et al. (2003) and Jayaraman et al. (2003).
Mandell(1998) presented aprobabilisticversion of thismodel.Capacity of facilities
constraintswereaddedtohierarchicalmaximalcoveringlocationproblembyJayaraman
et al. (2003). Marianov and Serra (2001) proposed a hierarchical covering model
within a queuing approach that also guarantees a service level. This approach is also
mentioned in Boffey et al. (2007). Serra et al. (1992) and Marianov and Taborga
(2001) considered this model in condition of competition with existing facilities.
10.5 Median-Based Hierarchical Relocation Problem (Teixeira
and Antunes 2008)
Teixeira and Antunes (2008) presented a model for relocating hierarchical facilities.
The model considers openings and closures.
10.5.1 Median-Based Hierarchical Relocation Problem
with Closest Assignment
The model assumes that customers attend the closest facility offering a particular
level of service (Teixeira and Antunes 2008).
10.5.1.1 Model Assumptions
The objective function is MiniSum.
The solution space is discrete and finite.
The formulation represents a nested and capacitated system.
The type of spatial pattern is single and closest assignment.
The number of facilities at each level is known (exogenous model).
10.5.1.2 Model Inputs
Model inputs are similar to globally inclusive service hierarchy model in addition
to these new parameters:
Jk 0 : the set of locations with existing type k facilities
Bi k : the maximum capacities of level k facility at candidate location j
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bj k : the minimum capacities of level k facility at candidate location j
qk : the number of type k facilities to be closed
Dk : the maximum user-to-facility distance for demand level k
10.5.1.3 Model Outputs (Decision Variables)
Model inputs are similar to globally inclusive service hierarchy model in addition
to one new variable:
Zj k t : the capacity occupied with demand level k of a level t facility located at
candidate location j
10.5.1.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
XXX
i
j
hi k dij Yij k :
(10.37)
k
Subject to
X
j
Yij k D 1 8 i; k;
Yij k
m
X
hDk
m
X
m
X
hDk
Xj h 8 i; j; k;
Zj k t
X
i
ui k Yij k 8 j; k;
(10.38)
(10.39)
(10.40)
Zj k t bj k Xj k 8 j; k;
(10.41)
Zj k t Bj k Xj k 8 j; k;
(10.42)
hDk
m
X
hDk
X
Yihk Xj k 8 i; j; kjt k;
(10.43)
dih dij
X
X j k pk
X
ˇ ˇ
Xj k ˇJk0 ˇ qk 8 k;
8 k;
(10.44)
j
j
(10.45)
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Yij k D 0 8 i; j; kjdij Dk ;
(10.46)
Xj k D 0; 1 8 j; k;
(10.47)
Yij k D 0; 1 8 i; j; k;
(10.48)
Yj k t 0 8 j; k; t:
(10.49)
Equations (10.38) ensure that all demands of all levels from all nodes are satisfied.
Equations (10.39) impose that a given level of demand can only be satisfied by a
facility of equal or higher level. Equations (10.40) define capacity variables Zj k t
by stating that the demand of each level assigned to a node has to be served by
some facility of equal or higher level located there. Equations (10.41)–(10.42) impose maximum and minimum limits on capacity, according to facility type. Closest
assignment Equations (10.43) are written separately per demand level and state that
each demand level must be assigned to the closest facility of equal or higher level.
Equations (10.44)–(10.45) limit the number of new facilities to be opened and existing facilities to be closed. Equations (10.46) limit the maximum travel distance
between demand nodes and facilities. Finally, Equations (10.47)–(10.49) define decision variables and enforce single assignment.
10.5.2 Median-Based Hierarchical Relocation Problem with Path
Assignment
The model assumes that when a given facility is located at location j , all demands
near the path travelled by the customers to reach the facility are also assigned to it
(Teixeira and Antunes 2008).
10.5.2.1 Model Assumptions
The assumptions of this model are similar to previous model except that there is the
following difference:
The type of spatial pattern is single and path assignment.
10.5.2.2 Model Inputs
Model inputs are similar to previous model in addition to one new parameter:
jPij j: the cardinality of the subset of demand nodes that are near the shortest path
from v to j
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10.5.2.3 Model Outputs (Decision Variables)
Model outputs are similar to previous model.
10.5.2.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Equations (10.43) are omitted
The following constraints are added to the formulation:
X
h
ˇ ˇ
Yhj k ˇPij ˇ Yij k 8 i; j; k :
(10.50)
In (10.50) the assignments of different levels are independent, while in (10.43) the
location of higher-level facilities influences lower-level assignments.
10.6 Solving Algorithms for Hierarchical Location Problem
Many algorithms are represented to solve different types of hierarchical facility location problems. Similar to single-level location problems, Lagrangian relaxations,
specialized branch and bound procedures and conventional heuristic methods are
developed for hierarchical location problems. Also a group of studies developing
combinatorial approximation algorithms have appeared in recent years. We review the related literatures of solving algorithms for hierarchical facility location
problems.
Scott (1971) presented a heuristic algorithm for the hierarchical location–
allocation problem. The algorithm involves a steady movement from a high
value of the objective function, to progressively lower values, and this is accomplished through a series of iterative stages each of which seeks out a locally
optimal solution.
Dökmeci (1977) used a heuristic method. In this method at first an ideal theoretic solution is obtained from the heuristic procedure. Then by repeating the
foregoing heuristic and adjustment procedures for each level, the environmentally adjusted optimal locations of the facilities are obtained.
Moore and ReVelle (1982) used relaxed linear programming, supplemented by
branch and bound where necessary, to solve the integer programming problem which is extension of the 2-level hierarchical maximal covering location
problem.
Considering the hierarchical maximum covering location problem presented
by Moore and ReVelle (1982), Espejo et al. (2003) defined a combined
Lagrangian–surrogate (L–S) relaxation which reduces to a 0–1 knapsack
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problem. A Lagrangian and a surrogate relaxation of the problem, which are
particular cases of (L–S) relaxation, are also discussed. Then they developed a
subgradient-based heuristic using the proposed relaxations to solve this problem.
Tcha and Lee (1984) represented a branch and bound algorithm for the multilevel uncapacitated facility location problem. In this method, the dual based
scheme of “dual ascent procedure” is applied. Also the convergence rate to an
optimal solution for this branch and bound algorithm has been further accelerated by employing two other devices, node simplification procedure and primal
descent procedure.
For the two-level uncapacitated facility location problem with some side constraints. Ro and Tcha (1984) proposed a branch and bound solution procedure,
employing a set of new devices for lower bounds and simplifications which are
obtained by exploiting the submodularity of the objective function and the special
structure of the side-constraints.
Narula and Ogbu (1985) developed the Lagrangian relaxation of the uncapacitated 2-hierarchical location–allocation problem and proposed a subgradient
optimization procedure to solve it.
Hodgson (1986) solved the nested hierarchical location–allocation problem with
allocations based on facility size by employing a heuristic method which allows
all levels to be located simultaneously. He reformulated the model based upon
the Reilly law of retail gravitation, because of the difficulties encountered with
the traditional inverse square Reilly law, using a negative exponential allocation
rule which has been a popular way of overcoming the inability to model trips
over short distances.
Van Roy (1989) used standard features and techniques for solving linear
programming (LP) and mixed integer programming (MIP) problems with an
improved formulation based on pre- and post-processing of the problem and
dynamic cut generation procedures for speed up the solution of MIP problems.
Okabe et al. (1997) used the optimization procedure with two steps for a system
of successively inclusive hierarchical facilities. The first step optimizes a system
of exclusive hierarchical facilities by an analytical method. Using this optimal solution, the second step optimizes a system of successively inclusive hierarchical
facilities by a computational search method.
Barros et al. (1998) represented a heuristic method based on linear relaxation.
Aardal (1998) presented some classes of inequalities to strengthen the linear relaxation for obtaining a good lower bound on the optimal solution in branch
and bound.
Aardal et al. (1999) developed a randomized algorithm for the hierarchical facility location problem. The algorithm is a randomized rounding procedure that
use an optimal solution of a linear programming relaxation and its dual to make
a random choice of facilities to be opened.
Goncharov and Kochetov (2000) presented a probabilistic Tabu search algorithm
for multi stage uncapacitated facility location problem.
Marianov and Taborga (2001) proposed a heuristic algorithm to solve
their model.
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A simple dual ascent method for the hierarchical facility location problem is
presented by Bumb and Kern (2001). The algorithm is deterministic and based
on the primal-dual technique.
Bumb (2001) presented a 0.47-approximation algorithm for the maximization
version of the two level uncapacitated facility location problem. For the analysis
of this algorithm the assumption that there are only two levels of facilities is
essential. The generalization of the algorithm to the case when the facilities are
located on k levels, with k > 2, makes the analysis of the algorithm much more
difficult.
For the nested hierarchical queuing covering location problem, Marianov and
Serra (2001) offered a bi-level heuristic with two phases. A construction phase
using a greedy adding procedure with random substitution (GRASP) and an
improvement phase by using a vertex-substitution heuristic and a tabu search
procedure.
Eben-Chaime et al. (2002) proposed a heuristic solution schemes and lower
bounds on objective values for capacitated location–allocation problem on a line.
Galvão et al. (2002) represented Lagrangian-based decomposition of the hierarchical location problem and proposed three heuristic methods to solve this model:
a 3 p-median heuristic which consists of successively locating each level services
and then applying a vertex substitution algorithm, a Lagrangian heuristic which
uses a subgradient optimization algorithm, and a modified Lagrangian heuristic.
Galvão et al. (2006) used a Lagrangian heuristic for capacitated version of their
previous model presented in Galvão et al. (2002).
A Lagrangian relaxation methodology coupled with a heuristic was employed by
Jayaraman et al. (2003) for hierarchical maximal covering location problem.
Ageev et al. (2004) presented improved combinatorial approximation algorithms
for the k-level facility location problem. In this research they applied path reduction and greedy for an approximation algorithm. Also they represented a
recursive path reduction Algorithm for approximate this problem.
A quasi-greedy algorithm for approximating the classical uncapacitated twolevel facility location problem is presented by Zhang (2006).
A hybrid multistart heuristic for the uncapacitated facility location problem
was presented by Resende and Werneck (2006). Their method has two phases.
The first is a multistart routine with intensification using a process called pathrelinking. The second phase is post-optimization.
Ignacio et al. (2008) presented a two-level hierarchical model for the location
of concentrators and routers in computers networks. They used Lagrangian relaxation to provide lower bounds for the problem. A tabu search meta-heuristic
is then developed to provide approximate solutions. A feasible solution of good
quality is obtained by the TS algorithm.
Chan et al. (2008) presented a nonlinear model for generalized search-and-rescue
problem. This problem is modeled using a multiobjective linear integer program
(MOLIP), which is an approximation of a highly nonlinear integer program.
As a solution algorithm, the MOLIP is converted to a two-stage network-flow
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formulation that reduces the number of explicitly enumerated integer variables.
Non-inferior solutions of the MOLIP are evaluated by a value function, which
identifies solutions that are similar to the more accurate nonlinear model.
10.7 Case Study
In this section we will introduce some real-world case studies related to hierarchical
location problems.
10.7.1 A Hierarchical Model for the Location of Maternity
Facilities in the Municipality of Rio de Janeiro
(Galvão et al. 2002 and 2006)
Galvão et al. (2002) represented a 3-level hierarchical model that addresses the location of levels of facilities associated with maternal and perinatal care in Rio de
Janeiro including basic units, maternity homes and neonatal clinics.
Since the overall aim is to develop a location model that aids health care authorities to reduce the perinatal mortality rate in the municipality, good access to
facilities is needed. Therefore the model seeks to optimize the surrogate objective
of total distance travelled by mothers-to-be. By the description of the different types
of facility, they developed a successively inclusive model. Some other assumptions
are considered for this model.
The model is solved for real 1995 data of the municipality of Rio de Janeiro.
Also results are given for available problems, for networks ranging from 10 to 400
vertices.
Galvão et al. (2006) completed their model by adding some form of capacity
constraints, which are needed especially in the higher, resource intensive level of
the hierarchy. This model is solved exactly for a small problem and approximately
for the 152-vertex Rio de Janeiro data.
10.7.2 Locational Analysis for Regionalization of Turkish Red
Crescent Blood Services (Şahin et al. 2007)
Şahin et al. (2007) represented the location problem of the blood services of Turkish
Red Crescent (TRC) at a regional level to increase the countrywide service level of
the blood services, which is an integral part of the national health care system. They
developed location–allocation models to solve the problems of regionalization based
on a hierarchical structure.
10
Hierarchical Location Problem
239
The problem was considered as a 2-level hierarchical system in which the regional blood centers (RBCs) are the upper-level facilities whereas the blood centers,
blood stations, and mobile units are the lower-level facilities.
The entire problem was decomposed into three stages. In Stage 1 they formulated
a 2-level hierarchical pq-median problem in which the locations of the regional
centers, the allocation of blood centers to the regional centers together with their
service-referral levels, and the allocation of demand points to the blood centers
were determined. A hierarchical set covering problem was formulated in Stage 2
to increase the service level. This second sub-problem finds the minimum number
of blood stations such that every demand point is covered by at least one facility within a given maximal service distance or time. To help increase the service
level even further, an integer programming model was formulated as the third subproblem in Stage 3 to determine the fleet size for the mobile units in each region.
10.7.3 School Network Planning in Coimbra, Portugal (Teixeira
and Antunes 2008)
Teixeira and Antunes (2008) presented a study on the redeployment of Coimbra’s
primary school network in Portugal. Three objectives were pursued by the education authorities. Three scenarios for the redeployment of the school network were
considered. First, they solved median-based hierarchical relocation problem with
closest assignment constraints. For this model, no feasible solutions could be found
for any one of the three scenarios. The solutions were obtained for the three scenarios using median-based hierarchical relocation problem with path assignment
constraints.
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Chapter 11
Hub Location Problem
Masoud Hekmatfar and Mirsaman Pishvaee
One of the novel topics in location problems is the hub location problem. There
are plenty of applications for the hub location problem; therefore, this section is
dedicated to introducing this problem to readers. The preface is composed of three
parts: apprehensions, definitions, and classifications of the hub location problem.
In this chapter we discuss services such as, movement of people, commodities
and information which occurs between an origin-destination pair of nodes (see A–B
in Fig. 11.1 as an origin-destination pair). Each origin-destination pair needs a service different from other pairs. Thus, the commodities carried from i to j are not
interchangeable with the commodities carried from j to i .
If we have N nodes and if each node can be either an origin or a destination,
we’ll have N.N 1/ origin-destination pairs of nodes in a network which form a
fully connected network (a network in which all nodes are connected together).
Notice that i j pair is different from j i pair. A sample network with six nodes is
presented in Fig. 11.1 (Daskin 1995).
Assuming that we have different traffic services in this network and that each
vehicle can service five origin-destination pairs every day, with 18 vehicles, we will
be able to service ten nodes every day.
If we set one of the nodes as a hub1 node and connect it to all of the other nodes,
which are introduced as spoke, we will have 2.N 1/ connections to service all
origin-destination node pairs. This network is presented in Fig. 11.2 (Daskin 1995).
In this network, if there are different traffic services and if each vehicle can service
five origin-destination pairs every day, with 18 vehicles, we will be able to service
46 nodes every day.
Thus, with fixed traffic resources, we can service more cities with a hub network
than with a completely connected network2.
1
Hub means the ball in the center of a wheel and Campbell (1994) defined it as “the facilities that
are servicing many origin-destination pairs as transformation and tradeoff nodes, and are used in
traffic systems and telecommunications.”
2
This argument ignores vehicles’ capacity. It should be clear that the volume of goods or the
number of people transported on each link in the hub-and-spoke network will be considerably
greater than the number transported on each link of the fully connected network. This also ignores
differences in the distances between nodes as it assumes that the number of origin-destination pair
trips a vehicle can service is independent from the distances between the nodes.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 11, c Physica-Verlag Heidelberg 2009
243
244
M. Hekmatfar and M. Pishvaee
Fig. 11.1 A fully connected
network with 6 nodes and
30 origin-destination pairs
(Daskin 1995)
A
B
F
C
E
D
Fig. 11.2 A hub and spoke
network with 6 nodes and
30 origin-destination pairs
(Daskin 1995)
A
B
C
D
E
F
The main disadvantage of such system (regarding all nodes except the hub node)
is that more than one trip is required to travel between each origin-destination pair
since we have to pass the hub node to be able to travel from one non-hub node to
another.
In multi hub networks, we assume that the hub nodes are completely connected
to one another and that each non-hub node is connected to exactly (at least) one
hub node such as Fig. 11.3 (Daskin 1995). In Fig. 11.3, there are 15 cities and three
hubs. The number of passengers or commodities carried from one hub to another is
greater than the number of passengers or commodities moved from each non-hub
node to that hub; for example if the traffic between each origin-destination pair is
ten units, there will be 140 traffic units between each non-hub city and the hub it is
connected to, but there will be 250 traffic units between two hubs.
It is not reasonable to establish direct paths between every pair of nodes; for
example assume that we establish direct paths between each pair of nodes in a connective network such as, a route network or a computer network (Camargo et al.
2008). Thus, hub nodes are used to resolve this problem. One of the advantages of
using hubs is that we gain economic profits by establishing more qualitative paths
between the hubs (Camargo et al. 2008). In road networks, when there are a large
number of relationships between two nodes, it is economic to establish a highway
with several lines between those two nodes, and obtain faster movement of vehicles,
and less waste in fuel, and, thus, savings in time and cost.
In computer networks, fiber-optic cables are only used to connect hubs and it is
not economic to use them to join any two nodes.
11
Hub Location Problem
245
Fig. 11.3 Example of a threehub network (Daskin 1995)
D
E
C
A
G
B
F
H
I
J
K
L
O
N
M
This chapter is organized as follows. In Sect. 11.1, some applications of hub location problem and its classifications are given, and Sect. 11.2 presents some models
developed for the problem. Some relevant algorithms to solve hub models are given
in Sect. 11.3. Finally, Sect. 11.4 represents some real world case studies with a short
summary.
11.1 Applications and Classifications
Many applications of hub problem are given in the following:
Airlines and air ports: Some works in this area are Toh et al. (1985), Shaw (1993),
Aykin (1995), Jaillet et al. (1996), Bania et al. (1998), Sasaki et al. (1999),
Martin and Roman (2003) and Adler and Hashai (2005). In Adler and Hashai
(2005), airlines and transportations based on open sky policy are surveyed in
the Middle East. It is interesting that based on their research, the four cities selected as optimized hub airports are, in order of utility, Cairo, Tehran, Istanbul,
and Riyadh. Dubai which now works as an important center for airport transportation, is not an optimal hub airport. This means that, politic and economic
problems have dominated over optimized location.
Transportation and handling problems: Some works in this area are Don et al.
(1995), Lumsdenk et al. (1999), Aversa et al. (2005), Baird et al. (2006), Cunha
and Silva (2007), Yaman et al. (2007) and Eiselt (2007). In Eiselt (2007), finding the optimized land dump locations, with respect to garbage transportation
stations, is discussed.
Post delivery services and fast delivery packing companies: Kuby et al. (1993),
Krishnamoorthy et al. (1994), Ernst and Krishnamoorthy (1996) and Ebery et al.
(2000) represent models in this area.
246
M. Hekmatfar and M. Pishvaee
Telecommunication systems and massage delivery networks: Some works in this
area are Lee et al. (1996) and Klincewicz (1998).
Emergency services: Hakimi (1964) and Berman et al. (2007) represented models
in emergency services. Berman et al. (2007) discussed the location of a hub for
rescue helicopters.
Chain stores in supply chain (like Wall-Mart): A work in this area is Campbell
et al. (2002).
Productive companies in basis transportation correctly: Every assembly plant
probably wants to find an optimized solution for its material storage and handling
such that each production facility can receive its required materials effectively.
Some practical examples of hub problems are as follows:
A classical example is Hungarian railway system with Budapest as its hub.
Two huge American airlines based in Atlanta and Chicago, use hub networks.
Dubai airport is a hub for many flights in the Middle East.
The hub location problem has a short history. The first paper on hub location problem was published by Toh et al. (1985). This paper was on the application of hub
location problem in airlines and airports. Although Hakimi (1964) first published a
paper on hub location, since next paper on this topic was not published until two
decades later, it is assumed that the hub location problem has been first discussed
in 1980s.
O’Kelly (1987) developed hub location models. He had an important role in developing the first modeling of hub problems (O’Kelly 1987, 1992). Later, Campbell
(1994) played a major role in completing hub modeling. His papers are among the
most important papers on types of hub modeling (Campbell 1994, 1996). Also,
some authors played important roles in improving this topic (Aykin 1994, 1995;
Klincewicz 1991, 1992).
Many papers have been published on hub location problem so far (since1980s).
Daskin (1995) has written a book on “Network and Discrete location” and only a
brief section of this book is dedicated to hub location. Figure 11.4 shows that the
number of papers has increased in recent years. The number of published papers
on hubs shows a significant increase in 1996, which is presumably the time when
the modeling of hub problems has reached maturity. The emphasis of papers was on
modeling in the early years, on optimizing and completing the models in the following years, and, finally, on solution methods in recent years. Thus, hub location problem is a rather new topic and a good area for research and development activities.
A classification of different types of hub problems and their properties is represented as follows:
Area solution: discrete, continual
Objective function: MiniMax, MiniSum
Determination of the number of hubs: exogenous, endogenous
Number of hubs: one hub, more than one hub
Hub capacity: unlimited (uncapacitated), limited
11
Hub Location Problem
247
12
11
10
9
8
7
6
5
4
3
2
1
0
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
20
06
Number
Papers
Year
Focus on primary
Modeling
Focus on model
optimization
Focus on large scale
optimization and solution
algorithms
Fig. 11.4 Revolution of Hub location problem papers3
Cost of hub location: no cost, fixed cost, variable cost
Node connections to hub: to one hub, to more than one hub
The cost of connection to a hub: no cost, fixed cost, variable cost
Researchers have worked on a variety of hub modeling problems. However, no research has been conducted on many types of hub modeling problems resulted from
multiplying the above items. On the other hand, some of the new models do not have
any applications in the real world and, therefore, are not really worth modeling.
Since most of the applications of hub problems in real world are discrete, the
models developed so far are mostly discrete models.
11.2 Models
In this section we introduce hub models represented and developed as mathematical
models.
11.2.1 Single Hub Location Problem (O’Kelly 1987)
O’Kelly (1987) was the first to introduce the single hub model which has only one
hub node.
3
This information is based on Springer-Verlag and Elsevier (Science Direct) websites.
248
M. Hekmatfar and M. Pishvaee
11.2.1.1 Model Assumptions
Model assumptions of this model are as follows:
The objective function is MiniSum.
The solution space is discrete and finite.
There is only one hub node.
All of the nodes are connected to the hub node and each non-hub node is connected to every other non-hub node via the hub node.
The number of hub nodes is known (exogenous model).
The installation cost of the hub node is not considered.
The capacity of the hub node is unlimited (uncapacitated model).
All decision variables of the model are binary (0–1) variables.
11.2.1.2 Model Inputs
Model inputs of this model are as follows:
hij : demand or flow between origin i and destination j
Cij : unit cost of local (non-hub to hub) movement between nodes i and j
11.2.1.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj D a hub is located at node j
Yij D node i is connected to a hub located at node j
11.2.1.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
XXX
i
j
k
hi k Cij C Cj k yij ykj :
(11.1)
Subject to
X
j
Xj D 1;
(11.2)
Yij Xj 0 8i;j ;
(11.3)
Xj D 0; 1 8j ;
(11.4)
Yij D 0; 1 8i;j :
(11.5)
11
Hub Location Problem
249
Equation (11.1) minimizes the total cost associated with the transport through the
hub. Equation (11.2) ensures that we locate only one hub. Equation (11.3) ensures
that demand node i cannot be connected to a hub at j unless we locate the hub at j .
Equations (11.4) and (11.5) are standard integrity constraints.
11.2.1.5 Linearizing the Objective Function
The objective function is quadratic since it involves the product of decision variables. However since there is only one hub, if node i is assigned to a hub at node j ,
then all nodes k.k ¤ i / must be assigned to the hub at node j . Thus we can rewrite
(11.1) as follows:
XXX
i
j
k
XX
Cij yij
hi k Cij C Cj k yij ykj D
i
XX
j
Cj i yij
i
X
k
hki
!
D
j
XX
i
j
X
k
!
hi k C
(11.6)
Cij yij .Oi C Di /;
where Oi : the total outflow of node i I Di : the total inflow of node i .
Having transformed (11.1)–(11.6), we can find the optimal 1-hub location with
minimum objective function value by total enumeration in O.N 2 / time.
11.2.2 P-Hub Location Problem (O’Kelly 1987)
This model is referred to as single allocation P-hub location problem, since each
non-hub node is assigned to one hub node (O’Kelly 1987).
11.2.2.1 Model Assumptions
Model assumptions are as follows:
The objective function is MiniSum.
The solution space is discrete and finite.
All of the hub nodes are connected to one another and each non-hub node is
connected to (exactly, at least) a hub.
The number of hub nodes is known (exogenous).
To travel between two non-hub nodes, one or two hubs have to be passed, i.e.
two non hub nodes are never connected directly.
The installation cost of the hub nodes is not considered.
The capacities of the hub nodes are unlimited (uncapacitated model).
All decision variables of the model are binary variables (0–1).
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M. Hekmatfar and M. Pishvaee
11.2.2.2 Model Inputs
We have all the inputs in previous model in addition to the following inputs:
˛ D discount factor for line-haul movement between hubs .0 ˛ < 1/.
As transportation cost between two hubs is less than transportation cost between
a hub node and a non-hub node, we multiply ˛ by Cij in calculating the movement
cost between two hubs.
P D the number of hubs to locate.
11.2.2.3 Model Outputs (Decision Variables)
The outputs of this model are similar to the previous model.
11.2.2.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
XX
i
C˛
k
0
Ci k Yi k @
X
XXXX
i
j
k
j
1
hij A C
XX
k
i
hij Ckm yi k yj m :
Cki Yi k
X
J
hj i
!
(11.7)
m
Subject to
X
yij D 1 8i;
(11.8)
X
xj D P ;
(11.9)
j
j
yij xj 0 8i; j;
(11.10)
xj D 0; 1 8j;
(11.11)
yij D 0; 1 8i; j:
(11.12)
Equation (11.7) minimizes the total cost associated with the P hubs locations and
the assignment of nodes to the hubs. The first term is the cost of connecting all trips
originating at node i to the hub k to which node i is attached. The second term
is the cost of connecting all trips destined for node i to hub k, to which node i is
11
Hub Location Problem
251
attached (sum on all i, k). The third term is a hub-to-hub connection cost (sum of
two hubs). Equation (11.8) ensures that each node i is assigned to exactly one hub.
Equation (11.9) ensures that the number of hub nodes is P . Equation (11.10) states
that demand node i cannot be connected to a hub at j unless we locate the hub at j .
Equations (11.11) and (11.12) are standard integrity constraints.
11.2.3 Multiple Allocation P-Hub Location Model (P-Hub
Median Location Model) (Campbell 1991)
The objective function of the previous model was nonlinear; so Campbell (1991)
represented the following model with a linear objective function. He formulated this
model like P-median problem and called it P-hub median location model. We discuss P-hub median problems in which each non-hub node can be connected to more
than one hub, and therefore, are called multiple allocation P-hub location problem.
Notice that this property is not fixed, so we also discuss models with one allocation
between each hub node and non-hub node (discussed in Sect. 11.2.5).
11.2.3.1 Model Assumptions
Model assumptions are similar to the ones of P-hub problem model except that the
Zijkm variables is relaxed .Zijkm 0/ and each non-hub node can be connected to
more than one hub node.
11.2.3.2 Model Inputs
Model inputs are similar to P-hub problem model except that the Cij variables are
defined as follows:
Cijkm D unit cost of travel between origin i and destination j when going via
hubs at nodes k and m
(11.13)
Cijkm D Ci k C ˛Ckm C Cmj :
11.2.3.3 Model Outputs (Decision Variables)
The model outputs of this model are as follows:
Xj D a hub is located at node j
Zijkm D flow from origin i to destination j uses hubs at candidates sites k and m.
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M. Hekmatfar and M. Pishvaee
11.2.3.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
XXXX
i
j
k
Cijkm hij Zijkm :
(11.14)
m
Subject to
X
k
xk D P ;
XX
k
m
Zijkm D 1 8i; j;
(11.15)
(11.16)
Zijkm xm 8i; j; k; m;
(11.17)
Zijkm xk 8i; j; k; m;
(11.18)
Zijkm 0 8i; j; k; m;
(11.19)
xk D 0; 1 8k:
(11.20)
Equation (11.14) minimizes the demand-weighted total travel cost. Equation (11.15)
stipulates that exactly P hubs should be located. Equations (11.16) ensure that
each origin-destination pair (i, j) must be assigned to exactly one hub pair. Equations (11.17) and (11.18) stipulate that flow from origin i to destination j cannot be
assigned to a hub at location k or m unless a hub is located at these candidate nodes
(when we travel from one node to another node via one hub node, m and k are
coincided with each other). Equations (11.19) are standard integrality constraints.
Equations (11.20) are relaxed decision variable constraints.
One of the key difficulties associated with this hub location model should also be
evident from this formulation. The number of assignment variables .Zijkm / can be
extremely large. In fact, if every origin or destination node is a candidate hub, there
will be O.N 4 / of such variables. For a relatively small problem with 32 origins
and destinations, we would have over one million such decision variables. In short,
the size of these problems grows very quickly with the number of nodes in the
problem unless some a priori means of eliminating candidate hub locations is used.
The use of quadratic formulation (11.7) reduces the number of decision variables
dramatically but does not make solving the problem any easier.
11.2.4 P-Hub Median Location Problem with Fixed Costs
(O’ Kelly 1992)
P-hub median location model with respect to fixed cost of hub locations is represented by O’ Kelly (1992).
11
Hub Location Problem
253
11.2.4.1 Model Assumptions
The model assumptions are similar to the ones of P-hub problem model except that
there are two differences:
The number of hub nodes is not known beforehand (endogenous).
A fixed cost of hub location is incorporated into the model.
11.2.4.2 Model Inputs
Model inputs are similar to the inputs of P-hub problem model in addition to two
new variables:
B D a weight on the capital or fixed costs to allow exploration of the tradeoff
between capital costs and transport (or operating) costs
fk D the fixed cost of hub location in candidate node k
11.2.4.3 Model Outputs (Decision Variables)
The outputs are similar to P-hub problem model.
11.2.4.4 Objective Function and its Constraints
The differences between the objective function of this model and P-hub problem
model are as follows:
Equations (11.9) are eliminated
The following term is added to the objective function:
B
X
fx xk :
(11.21)
k
11.2.5 Single Spoke Assignment P-Hub Median Location
Problem (Single Allocation P-Hub Location Problem)
(Daskin 1995)
Equation (11.14)–(11.20), P-hub median location model, allow each of the spoke
nodes to be assigned to multiple hubs. Thus, for example, the assignment shown in
Fig. 11.5 (Daskin 1995) is entirely possible as a result of solving this model. In this
figure two origin-destination flows are shown: origin i to destination j1 and origin i
to destination j2 . Origin i is assigned to two different hubs: k1 ; k2 .
254
M. Hekmatfar and M. Pishvaee
Fig. 11.5 Schematic representation of multiple spoke
assignment (Daskin 1995)
i
k2
k1
Hubs
Fig. 11.6 Schematic representation of single spoke
assignment (Daskin 1995)
m
m
j1
j2
i
i
k1
k2
k1
k2
m
m
m
m
j1
j2
j1
j2
In many cases, it is desirable (for operational reasons) to have each of the spoke
nodes assigned to a single hub. This might result in one of the assignments shown
in Fig. 11.6 (Daskin 1995).
The single spoke assignment P-hub median location model is formulated as
Daskin (1995)’s model.
11.2.5.1 Model Assumptions
The assumptions of this model are similar to median P-hub model except that there
are the two following differences:
Each non-hub node is assigned to only one hub.
All of the variables are binary variables (0–1).
11.2.5.2 Model Inputs
Model inputs are similar to median P-hub model.
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11.2.5.3 Model Outputs (Decision Variables)
Model outputs are similar to median P-hub model in addition to the following decision variables:
Yi k D non-hub node i is assigned to a hub node k
11.2.5.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
XXXX
i
j
k
Cijkm hij Zijkm :
(11.22)
m
Subject to
X
k
Xk D P;
XX
k
m
Zijkm D 1 8 i; j;
Yi k Xk 8 i; k;
X
Yi k D 1 8i;
(11.23)
(11.24)
(11.25)
(11.26)
k
Yi k C Yj m 2Zijkm 0 8i; j; k; m;
(11.27)
Xk D 0; 1 8k;
(11.28)
Yi k D 0; 1 8i; k;
(11.29)
Zijkm
(11.30)
D 0; 1 8i; j; k; m:
Equation (11.22)–(11.24) are identical to those of (11.14)–(11.16). Equations (11.25) ensure that spoke i cannot be assigned to a hub at location k unless
we locate a hub at location k. Equations (11.26) are key constraints that stipulate
that each spoke node i is assigned to exactly one hub. Equations (11.27) state that
the flow from origin i to destination j cannot be routed through hubs at nodes k
and m, unless spoke node i is assigned to a hub at k and spoke node m is assigned
to a hub at j . Equations (11.28)–(11.30) are standard integrity constraints.
11.2.6 The Extension Model of Fixed Cost for Connecting
a Spoke to a Hub (Campbell 1994)
Instead of forcing each spoke node to be assigned to a single hub, we may want to
stipulate a fixed cost to any spoke/hub connection. Alternatively, we could incorporate a fixed cost for connecting a spoke to a hub. Campbell (1994) shows how these
extensions can be formulated.
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M. Hekmatfar and M. Pishvaee
11.2.6.1 Model Assumptions
The assumptions of this model are similar to median P-hub model except that there
is a fixed cost for connecting a spoke to a hub.
11.2.6.2 Model Inputs
Model inputs are similar to median P-hub model in addition to the following
parameter:
gi k : the fixed cost of connecting spoke node i to a hub at candidate location k.
11.2.6.3 Model Outputs (Decision Variables)
Model outputs are similar to single allocation median P-hub model.
11.2.6.4 Objective Function and its Constraints
The objective function is similar to median P-hub model in addition to the
following term:
XX
gi k Yi k :
(11.31)
i
k
11.2.7 Minimum Value Flow on any Spoke/Hub Connection
Problem (Campbell 1994)
Instead of forcing each spoke node to be assigned to a single hub, we may want
to stipulate that the flow along any spoke/hub connection exceed some minimum
value. Campbell (1994) represented this model, which is similar to single allocation
P-hub location problem.
11.2.7.1 Model Assumptions
The assumptions of this model are similar to median P-hub model except that there
is a minimum flow for each spoke/hub connection.
11.2.7.2 Model Inputs
Model inputs are similar to median P-hub model in addition to the following
parameter:
li k : the minimum flow between spoke i and hub k.
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257
11.2.7.3 Model Outputs (Decision Variables)
Model outputs are similar to single allocation median P-hub model.
11.2.7.4 Objective Function and its Constraints
The objective function is similar to median P-hub model except that the following
constraints are added to median P-hub model’s constraints:
Yi k C Yj m 2Zijkm 0 8i; j; k; m;
XX
XX
hij Zijkm C
hpi ZPski Li k Yi k 8i; k:
m
j
P
(11.32)
(11.33)
s
Equations (11.32) were explained before. The first term of constraints (11.33) are
the flow from spoke node i to hub k and then from there to any hub/destination pair.
The second term is the flow from any origin/hub pair to hub k and then from there
to destination i . The sum of these two flows is the total flow between two nodes i, k.
11.2.8 Capacity Limitation of Hub Location Problem
(Campbell 1994)
Capacity limitation of hub node means that the total flows, incoming or outgoing,
must be less than or equal to a fixed value and is called capacitated hub location
problem. This model is represented by Campbell (1994).
11.2.8.1 Model Assumptions
The assumptions of this model are similar to median P-hub model except that the
capacities of the hub nodes are limited.
11.2.8.2 Model Inputs
Model inputs are similar to median P-hub model in addition to the following
parameter:
k D the capacity of a hub at candidate node k.
11.2.8.3 Model Outputs (Decision Variables)
Model outputs are similar to median P-hub model.
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M. Hekmatfar and M. Pishvaee
11.2.8.4 Objective Function and its Constraints
The objective function is similar to median P-hub model except that the following
constraints are added to median P-hub model’s constraints:
XX X
XXX
hij Zijkm C
hij Zijsk k Xk 8k:
(11.34)
m
i
s
j
i
j
The left side of the above inequality shows the total incoming and outgoing flows
of node k.
11.2.9 P-Hub Center Location Problem (Campbell 1994)
The center location problem is an important problem for its applications such as
emergency facility location or the worst situation scenario. The P-hub center location problem is similar to P-center location problem. If an origin-destination pair
in a hub location problem is introduced as a demand node in P-center problem, the
meaning of a hub center problem is a set of hubs that minimizes the maximum cost
of each origin/destination pair. This problem is used for decomposable goods or
sensitive goods in a hub system. This model is represented by Campbell (1994).
11.2.9.1 Model Assumptions
The assumptions of this model are similar to median P-hub model except that Xk
variables are relaxed and the objective function is MiniMax.
11.2.9.2 Model Inputs
Model inputs are similar to median P-hub model.
11.2.9.3 Model Outputs (Decision Variables)
Model outputs are similar to median P-hub model.
11.2.9.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
o
n
Min Max Cijkm hij Zijkm :
(11.35)
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259
Subject to
X
k
Xk D P ;
XX
k
m
Zijkm D 1 8 i; j;
(11.36)
(11.37)
Zijkm Xk 8i; j; k; m;
(11.38)
Zijkm X 8i; j; k; m;
(11.39)
0 Xk 1 8k;
(11.40)
0 Zijkm 1 8i; j; k; m:
(11.41)
Equation (11.35) minimizes the maximum cost of transportation between each origin/
destination pair. Equations (11.36)–(11.41) are similar to Median P-hub location
problem.
Campbell et al. (2007) represented a new optimization of P-hub center problems
(P-HC) and analyzed the complexity of the model and then represented algorithms
to solve them.
11.2.10 Hub Covering Location Problem (Campbell 1994)
This model can be used to solve P-hub center model. If an origin-destination pair
in a hub location problem is introduced as a demand node in hub covering location
problem, the meaning of a hub covering problem is:
The origin/destination pair (i, j) is covered by (m, k) pair of hub nodes, if the
cost of i node to j node via m, k hubs is less than or equal to a certain fixed value
(number).
(11.42)
Cijkm ij :
Campbell (1994) represented this model and we share this model to two models:
hub set covering location model and hub maximal Covering location model.
11.2.10.1 Hub Set Covering Location Problem
Hub set covering location model is a special case of hub covering location model.
We introduce this model as follows:
Model assumptions. The assumptions of this model are similar to median P-hub
model except that the number of hubs is not known (endogenous) before solving
and a fixed cost of hub location is incorporated in the model.
Model inputs. The model inputs of this model are as follows:
Fk D the fixed hub location cost in candidate node k
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M. Hekmatfar and M. Pishvaee
Cijkm D transportation cost from origin i to destination j via hubs at candidate
nodes k and m
ij D maximum cost for going from origin i to destination j (distance covering)
Vijkm D node hubs m, k cover the origin-destination i, j
Model outputs (decision variables): Model outputs are similar to median
P-hub model.
Objective function and its constraints. The objective function of this model and
its related constraints are as follows:
X
FK XK :
(11.43)
Min
k
Subject to
Zijkm Xk 8i; j; k; m;
(11.44)
Zijkm Xm 8i; j; k; m;
XX
Vijkm Zijkm 1 8i; j;
(11.45)
0 Xk 1 8k;
(11.47)
0 Zijkm 1 8i; j; k; m:
(11.48)
k
(11.46)
m
Equation (11.43) minimizes the total hub location costs. Equations (11.46) ensure
that all of origin-destination pairs are covered at least once. The other equations are
similar to those of the previous models.
11.2.10.2 Hub Maximal Covering Location Problem
Hub maximal covering location model is a special case of hub covering location
model. We introduce it as follows:
Model assumptions. The assumptions of this model are similar to median P-hub
model except that the number of hubs is known (exogenous) before solving the
model and that the fixed cost of hub location is not considered in the model.
Model inputs. The model inputs are as follows:
hij D the demand flow from origin i to destination j
Model outputs (decision variables). Model outputs are similar to median
P-hub model.
Objective function and its constraints. The objective function of this model and
its related constraints are as follows:
XXX
(11.49)
hij Zijkm Vijkm :
Max
i
j
m
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261
Subject to
X
k
Xk D P ;
X X
k
m
Zijkm D 1 8 i; j;
(11.50)
(11.51)
Zijkm Xk 8i; j; k; m;
(11.52)
Zijkm Xm 8i; j; k; m;
(11.53)
0 Xk 1 8k;
(11.54)
0 Zijkm 1 8i; j; k; m:
(11.55)
Equation (11.49) maximizes the covered demand value. The other equations of this
model are similar to P-hub median model introduced before.
11.3 Solution Techniques
Many algorithms are represented to solve difference types of hub location problems.
We review the related literatures and introduce some of the proposed associated
algorithms.
11.3.1 Various Kinds of Algorithms
To solve small hub problems, integer programming optimization methods are used.
However, for larger problems, heuristic methods or Meta heuristic methods are utilized.
In the past, few solving methods were proposed for hub location problems in
which the number of hubs is a decision variable and the fixed cost of establishing a
hub is considered. Nevertheless, with the growth of meta heuristic methods, number
of methods to solve such problems has been increased.
To solve the two-hub case, a procedure is given by Ostresh (1975) to solve the
two-center location–allocation problem (location–allocation problem with no interaction between the sources where each destination is assigned to the closest
source).
O’Kelly (1987) formulated the P-hub median problem (P-HLMP) as a quadratic
integer programming problem. He showed that the problem is NP-hard, and proposed two enumeration-based heuristics to solve it.
Exchange clustering methods are presented by Klincewicz (1991).
Greedy search methods, Tabu search and Grab are given by Klincewicz (1992).
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M. Hekmatfar and M. Pishvaee
Tabu search methods are given by Skorin-Kapov and Skorin-Kapov (1994).
Genetic algorithm methods are given by Abdinnour-Helm and Venkataramanan
(1998).
A mixed method, Tabu search and genetic algorithm by Abdinnour-Helm (1998);
A mixed method, Simulated Annealing and Tabu search given by Chen (2007);
Genetic algorithm method given by Topcouglu et al. (2005);
Solving uncapacitated single allocation hub location problem with installed fixed
cost simulated annealing is represented by Aykin (1995).
Greedy Interchange, Alflo, Maxflo given by Campbell (1996);
Shortest Path method given by Sohn (1998);
Greedy Algorithm for one stop state (traveling form an origin to a destination
through only one hub node) given by Sasaki et al. (1999), are discussed in this
section. Two networks are studied with 32 nodes and 2 or 5 hubs, and 50 nodes
and 2 or 4 hubs, respectively. Comparisons between the greedy and branch and
bound algorithms have shown that the greedy algorithm needs less machine time.
Ebery et al. (2000) represented a heuristic algorithm with shortest path method
to solve capacitated multiple allocation hub location problem (CHLP-M), and
then used the gained upper bound in branch and bound procedure in basic linear
programming.
Ernst and Krishnamoorthy (1999) represented methods to solve capacitated single allocation hub location problem (CHLP-S).
Aykin (1994) represented procedures based on Lagrangian relaxation to solve
capacitated hub location problems (CHLP).
Camargo et al. (2008) represented a bender decomposition method to solve uncapacitated multiple allocation hub location problem (UHLP-M).
Canovas et al. (2007) represented a dual ascent method, and a heuristic method,
to solve uncapacitated multiple allocation hub location problem (UHLP-M).
Rodriguez et al. (2007) represented a solution method based on simulated annealing to solve capacitated hub location problem (CHLP). In this method each
hub is assumed to be an M/M/1 queuing system and has solved for a network
with 52 nodes.
Martine Labbe et al. (2005) represented a solution method based on branch
and cut algorithm to solve uncapacitated single allocation hub location problem
(UHLP-S). In this method, the network connecting the hub nodes is called backbone Network and the connected network of the terminal nodes is called access
network.
Marianov et al. (2003) represented Tabu search algorithm in airlines location in
which each hub node is assumed to be an M/D/C queue. In this method, only a
part of feasible solution is surveyed and the best neighborhood is selected as a
new hub node with respect to continuous iterations on neighborhood nodes of
former hub, even the target function gains a worse solution. This method works
best for networks of up to 50 nodes with 4 or 5 hubs. However, the efficiency
of method decreases as the number of nodes increases and reaching a feasible
solution is guaranteed.
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263
Wagner (2007) represented a clustering method using Tabu search and genetic
algorithm. This method was used to solve traveling salesman problem (TSP).
Pamuk et al. (2001) represented a single relocating heuristic by Tabu search to
solve P-hub center problems (P-HLCP). Two single-allocation schemes were
used in the evaluation phase of the algorithm. A greedy local search was employed to improve the resulting allocations.
Cunha and Silva (2007) represented an efficient hybrid genetic algorithm (GA)
approach for the hub and spoke location problem for the less-than-truckload
(LTL) services in Brazil. This problem can be seen as a modified version of the
uncapacitated hub location problem with single allocation (UHLP-S), in which
the discount factor on the hub to hub links is not constant but may vary according
to the total amount of cargo between hub terminals.
Thomadsen and Larsen (2007) represented branch-and-price algorithm or IP column generation (the combination of column generation and branch-and-bound
algorithm) to solve a two-layered network (a hierarchical network) consisting of
clusters of nodes, each defining an access network and a backbone network. The
two layers in the network are: the backbone network and the access networks.
The backbone network connects disjoint clusters of nodes, each including an access network. The node connecting an access network to the backbone network
is called a hub.
Costa et al. (2008) represented a bi-criteria integer linear programming to solve
the capacitated single allocation hub location problems (CHLP-S).
Bollapragada et al. (2005) represented a new network-planning model and an
effective greedy solution heuristic to solve a model that is most closely related to
the capacitated hub maximum-covering location problem with multi allocations
(CHMCLP-M).
The quality of the heuristic algorithm is evaluated by comparing its coverage with
the optimal (for small problems) or with an upper bound obtained by solving a
linear-programming relaxation.
Rodriguez-Martin et al. (2008) represented a mixed integer linear programming
formulation and described two branch-and-cut algorithms based on decomposition techniques to solve a capacitated multi allocation hub location problem
(CHLP-M)
Yaman (2008) represented a heuristic algorithm based on Lagrangian relaxation
and local search to solve P-hub location median single allocation problems (PHLMP-S).
Wagner (2008) represented LP-relaxation to solve uncapacitated multi allocation
P-hub location median problems (P-HLMP-M).
Kratica et al. (2007) represented two genetic algorithms for solving the uncapacitated single allocation P-hub location median problem (P-HLMP-S).
A wide array of solution methods are represented in Table 11.1. It’s shown that
methods lose their efficiency, need much time to be solved when the number of
nodes is increased over 50.
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Table 11.1 Solution methods for hub location problem
Reference
Problem
Solving method
O’Kelly (1986)
Ostresh (1975)
1-HLP-S
2-HLP
O’Kelly (1987)
P-HLMP
Klincewicz (1991)
Klincewicz (1992)
P-HLMP
P-HLMP
Skorin-Kapov and
Skorin-Kapov (1994)
Skorin-Kapov et al. (1996)
P-HLMP-S
Nearest distance
Location–allocation
(shortest route)
Nearest distance-quadratic
integer
Clustering
Tabu search and greedy
random algorithm
Tabu search
Campbell (1991)
Campbell (1994)
O’ Kelly (1992)
Abdinnour-Helm and
Venkataramanan (1998)
Abdinnour-Helm (1998)
P-HLMP-S &
P-HLMP-M
P-HLMP-M
P-HLMP-M
UHLP-S
UHLP-S
UHLP-S
Chen (2007)
UHLP-S
Topcouglu et al. (2005)
Aykin (1995)
UHLP-S
UHLP-S
Campbell (1996)
Sohn and Park (1998)
Sasaki et al. (1999)
P-HLMP-M
P-HLMP-M
P-HLMP-M
Ebery et al. (2000)
Ebery (2001)
CHLP-M
P-HLMP-S
Aykin (1994)
P-HLMP-S
Camargo et al. (2008)
Canovas et al. (2007)
UHLP-M
UHLP-M
Rodriguez et al. (2007)
Labbe et al. (2005)
Sung and Jin (2001)
CHLP-M
UHLP-S
P-HLMP-M
Wagner (2007)
UHLP-M
Marin (2005)
Marianov and Serra (2003)
Klincewicz (1996)
CHLP-M
UHLP-M
UHLP-M
Efficient number
of nodes (number
of hubs)
Linear programming
Integer programming
Integer programming
Heuristic algorithm
Genetic algorithm
Genetic algorithm and tabu
search (GATS)
Tabu search and simulated
annealing
Genetic algorithm
Simulated annealing
(greedy interchange) B & B
Greedy- interchange
Nearest distance
B&B algorithm and greedy
algorithm
Nearest distance – B & B
Mixed integer linear
programming
B & B (lagrangian
relaxation)
Benders decomposition
Means of a dual ascent
technique- B & B
Simulated annealing
Branch & cut (B & C)
Clustering (dual-based
approach)
Clustering (genetic
algorithm & Tabu search)
Integer linear programming
Tabu search
Dual ascent and dual
adjustment techniques
within a B & B scheme
200
200 (5)
25 (3–4)
50 (2–4)
200
50 (2–3)
200
120
52
40
50 (4–5)
(continued)
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265
Table 11.1 (continued)
Reference
Problem
Solving method
Mayer and Wagner (2002)
Yaman et al. (2007)
UHLP-M
CHLP-S
Cunha and Silva (2007)
Pamuk and Sepil (2001)
UHLP-S
UHLP-M
Thomadsen and Larsen
(2007)
UHLP-M
Costa et al. (2008)
CHLP-S
Bollapragada et al. (2005)
CHMCLP-M
(maximum
covering)
CHLP-M
Dual ascent approach
Tabu search (with greedy
algorithm) – B & C
Genetic algorithm
Tabu search (with greedy
local algorithm)
Branch and price
(combination of column
generation and B & B)
Bi-criteria integer linear
programming
Greedy algorithm
Rodriguez-Martin and
Salazar-Gonzalez (2008)
Yaman (2008)
P-HLMP-S
Pirkul and Schilling (1998)
Wagner (2008)
Kratica et al. (2007)
P-HLP-S
P-HLMP-M
P-HLMP-S
Mixed integer linear
programming with B & C
based on decomposition
method
Lagrangian relaxation and
local search
Lagrangian relaxation
LP relaxation
Genetic algorithm
Efficient number
of nodes (number
of hubs)
49
25
25 (5)
25
40
25 (10)
81 (25)
25
50 (5)
200 (20)
11.3.2 Some Relevant Algorithms
O’Kelly (1987) represented two heuristic methods to solve uncapacitated single allocation hub location problem.
In the first heuristic method each demand node is allocated to the nearest hub
node. It is known that when ˛ D 0, the third part of the nonlinear (11.7) will be zero
and the problem is reduced to a P-Median Problem. This method works well when
˛ < 0:5.
In the second heuristic method all ways to allocate non-hub nodes to the nearest
or second nearest hub nodes are analyzed. If we have N nodes to select P hubs, we
will have N!/P!.N! ways. For each one of the N!/P!.N! ways, all of the 2N –P ways
to allocate non-hub nodes to the nearest and second nearest hub nodes are studied.
By increasing the number of nodes solution time is increased.
The second heuristic method results in better solution compared to the first
heuristic method. Also, the second heuristic gives a tighter upper bound on the objective function than the first one.
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11.3.2.1 Greedy Heuristic Algorithm
Greedy algorithm was first presented by Sasaki et al. (1999) to solve the 1-stop uncapacitated multiple allocation hub location problem. One stop means that to travel
between each origin-destination pair, we have to pass only one hub node. The application of this method is for small networks such as Japan inner airlines network.
11.3.2.2 Genetic Algorithm
Topcuoglu et al. (2005) represented this algorithm to solve uncapacitated single
allocation hub location problem with fixed cost of location.
A genetic algorithm (GA) is a search algorithm for finding the near-optimal solutions in large spaces, which is inspired from population genetics. The general idea
was introduced by Holland (1975). Genetic algorithms have been applied to a large
set of problems in various fields in the literature.
Two example sources for detailed information on GA are the books written by
Goldberg (1989) and Mitchell (1998).
11.3.2.3 Benders Decomposition Method
Camargo et al. (2008) represented this method based on an old algorithm in 1962
which proposed a partitioning method for solving mixed linear and nonlinear integer programming problems. Camargo et al. (2008) defined a relaxation algorithm
for solving a problem through partitioning it into two simpler problems: an integer
problem, known as MP, and a linear problem, known as SP.
The MP is a relaxed version of the original problem with a set of integer variables
and its associated constraints, while SP is the original problem with the values of
the integer variables temporarily fixed by the MP.
The algorithm solves each one of the two simpler problems iteratively, one at a
time. At each iteration, a new constraint, known as Benders cut, is added to the MP.
This new constraint is originated by the dual problem of the SP. The algorithm goes
on until the objective function value of the optimal solution to the MP is equal to
that of the SP, when it stops obtaining the optimal solution of the original mixed
integer problem.
11.4 Case Study
In this section we will introduce some real-world case studies related to hub location
problems.
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267
11.4.1 The Policy of Open Skies in the Middle East
(Adler and Hashai 2005)
In Adler and Hashai (2005), airlines and transportations based on open sky policy
were surveyed in the Middle East.
Conclusions drawn from this investigation may enable both researchers and policy makers to develop a greater understanding of the social welfare impacts of
deregulation in the regional air-transport industry and the economic benefits to individual air carriers, countries and passengers alike, once peace restraints in the
region. This analysis had led to the conclusion that the increase in both leisure and
business air traffic due to the reduction in violence in the Middle East may lead to
an increase of 51% in inter-country passenger flow under the assumption of deregulation of the regional air-transport industry.
It is interesting that based on their research the four cities selected as optimized
hub airports are, in order of utility, Cairo, Tehran, Istanbul, and Riyadh. Dubai which
now works as an important center for airport transportation, is not an optimal hub
airport. This means that, politic and economic problems have dominated over optimized location.
11.4.2 A Hub Port in the East Coast of South America (Aversa
et al. 2005)
Aversa et al. (2005) formulated a mixed planning model for selecting a hub port,
from eleven ports servicing to transportation demands, in the east coast of South
America.
It was recommended that minimisation of transport costs, is often given unwarranted significance, usually by “awkward” river ports, such as Antwerp and
Hamburg, while the importance of port costs, in these ports, is at the same time
purposely downplayed.
In this research many ports were researched such as, ports in Brazil, Argentina
and Uruguay. Their model consists of 3,883 variables and 4,225 constraints. After
solving the model, port Santos, Brazil was chosen as the hub port.
11.4.3 A Hub Model in Brunswick, Canada (Eiselt 2007)
Eiselt (2007) represented an application of location models to the siting of landfills.
The landfill and transfer station location problem was formulated similar to a hub
location problem. The problem included a parameter that measures the discount
factor of the transportation between transfer stations and landfills in comparison to
the unit transportation cost between customers and transfer stations.
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M. Hekmatfar and M. Pishvaee
The Province of New Brunswick was used to compare optimized facility locations with facility locations that had been chosen by the planners. In order to make
such comparison, the major towns and villages were chosen on the basis of the statistical data available from the latest census. The result of the calculations was that,
the optimized solutions were between 10 and 40% less costly as compared to the
observed solutions. Still, the optimized locations of the landfills were quite close
to those found in the optimization runs. The only exception was a case in which
the planners deviated from their plan due to public opposition. Their first choice of
location also appeared in some of the optimized runs. Some additional optimization
was also performed on a smaller subset of the 93 points which were the basis of the
optimization reported above.
11.4.4 A Hub/Spoke Network in Brazil (Cunha and Silva 2007)
Cunha and Silva (2007) represented the problem of configuring hub-and-spoke networks for trucking companies that operate less-than-truckload services in Brazil.
The problem consists of determining the number of consolidation terminals (also
known as hubs), their locations and the assignment of the spokes to the hubs, aiming to minimize the total cost, which is composed of fixed and variable costs.
References
Abdinnour-Helm S (1998) A hybrid heuristic for the uncapacitated hub location problem. Eur J
Oper Res 106:489–99
Abdinnour-Helm S, Venkataramanan MA (1998) Solution approaches to hub location problems.
Ann Oper Res 78:31–50
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Chapter 12
Competitive Location Problem
Mohammad Javad Karimifar, Mohammad Khalighi Sikarudi,
Esmaeel Moradi, and Morteza Bidkhori
A large part of location theory in operational research has been built around the
(mostly implicit) modeling assumption of a spatial monopoly: the facility to be located offers a unique product or service and is the single player in the part of the
market that is considered. Most situations in practice do not fit such models and the
need arises to incorporate competition with other players. This has long been understood by economists who have studied competition, including its spatial aspects, for
some 70 years.
A location model is said to be about competitive facilities when it explicitly incorporates the fact that other facilities are already (or will be) present in the market
and that the new facilities will have to compete with them for its (their) market
share. The apparent simplicity of this statement hides several implicit and explicit
notions which have to be made more precise before a clear and well-defined model
arises.
This chapter is organized as follows. In Sect. 12.1, we consider literatures, definitions and classifications of competitive location problem. Section 12.2 presents
some models with continuous and discrete space developed for the problem, and
also some relevant algorithms to solve these models are given in this section. Finally,
Sect. 12.3 represents some real world case studies with a short summary.
12.1 Applications and Classifications
Hotelling’s work (Hotelling 1929) on two terms competing in a linear market (with
consumers distributed uniformly along the line) set the foundations of what is today
the burgeoning field of competitive location.
During the late thirties and early forties, several papers using the same spatial representation as Hotelling but modifying some of the economic assumptions appeared
in the economic literature (Hoover 1936; Lerner and Singer 1937; Smithies 1941).
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 12, c Physica-Verlag Heidelberg 2009
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There followed several decades of stagnation in the contribution of new insights
in the field of competitive location in linear markets have been past.
Since the late seventies however, a myriad of different models have appeared
in the literature of spatial economics and industrial organization (Gabszewicz and
Thisse 1991).
Parallel to the development of this body of literature, a new field on location modeling was growing in the late sixties and seventies at a fast rate, namely facility location analysis. This field of research, coming basically from the fields of operations
research, regional science and geography, dealt with the problem of locating new
facilities in a spatial market in order to optimize one or several geographical and/or
economic criteria. These criteria included overall distance minimization and transport and manufacturing cost minimization. The literature in facility location analysis
is extensive: good sources of references can be found in Chhajed et al. (1993). Although most of these models used more realistic spatial representations than the line,
such as networks and planes, most of them dealt exclusively with non-competitive
situations, and little attention was paid to the characterization of market equilibrium.
From the late seventies, considerations on the interaction between competing
facilities in discrete space have been developed following several different approaches. An extensive bibliographic survey with over 100 citations on competitive
location can be found in Eiselt et al. (1993).
One of the first questions that is addressed by several authors is whether or not
a set of locations in the vertices of a network exist that will ensure a Nash equilibrium, that is, a position where neither firms have incentives to move. Wendell and
McKelvey (1981) considered the location of two competitive firms with one server
each and tried to and a situation where a firm would capture at least 50% of the
market regardless of the location of its competitor. Results showed that there was
not a general strategy for the firm that would ensure this capture if locations were
restricted to vertices of the network. Hakimi (1986) also analyzed extensively the
problem of competitive location on vertices and proved that, under certain mathematical conditions such as concave transportation costs functions, that there exists a
set of optimal locations on the vertices of the network.
A similar problem was studied by Lederer and Thisse (1990). Their problem not
only looked at the specification of a site but also at the setting of a delivered price.
They formulated the problem as a two stage game, where in the first stage both
firms choose locations and in the second stage them simultaneously set delivery
price schedules, and the result is that there is sub-game perfect Nash equilibrium. As
Hakimi (1992) did, they also proved that if firm’s transport costs are strictly concave,
then the set of locally choices of the firm is reduced to the vertices of the network.
As a consequence, the location problem can be reduced to a 2-median problem if
social costs are minimized. A similar result was obtained by Hakimi et al. (1992).
The problem of two firms competing in a spatial market has also been studied
in the case where the market is represented by a tree. Eiselt et al. (1993) proved
that in such a case there is not a sub-game perfect Nash equilibrium if both prices
and locations are to be determined. Eiselt et al. (1993) extended the problem to the
location of three facilities in a tree. They found that the existence of equilibrium
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depended on the distribution of the demand in the geographical space in question.
In both models, firms were allowed to locate on the edges of the network.
The game-theoretical models presented so far restrict themselves to the location
of firms with one facility each that compete against each other.
Tobin and Friesz (1986) examined the case of a profit-maximizing firm that entered a market with several plants. They considered price and production effects on
the market, since the increase in the overall production level from the opening of
new plants in a spatial market stimulates reactions in the competitors. These reactions might a fact not only production levels, but also prices and locations.
Tobin and Friesz developed two models: (1) spatial price equilibrium model
which determines equilibrium in prices and production levels for a given number
of firms and (2) a Cournot–Nash oligopolistic model in which a few profit maximizing firms compete in spatially separated markets. They used both models to analyze
the case of an entering firm that is going to open several new plants in spatially separated markets, and knows that its policy will have impact on market prices. Since
profits depend on location and price-levels and these depend on the reaction of the
competitors, it is not possible to use a standard plant location model. To tackle the
problem, they used sensitivity analysis on variation inequalities to relate changes in
production to changes in price to obtain optimal locations. The model was solved
using a heuristic procedure where in the first step a spatial competitive equilibrium
model was obtained and, in the second step, a sensitivity analysis of profit to production changes was performed to select locations and production levels likely to
maximize total profits.
This model was generalized by Friesz et al. (1989) to allow the entering firm to
determine not only production levels and the sites of its plants, Due to the mathematical complexity of these models, Miller et al. (1992) developed several heuristic
methods to tackle the problem using the approach of variation inequalities.
Price-location modeling has been studied in a non-competitive model by Hanjoul
et al. (1990). They develop three incapacitated plant location models where different
alternative spatial price policies are considered.
Another body of literature on competitive location deals with the sitting of retail
convenience stores. This type of store is characterized by (1) a limited and very
similar product offering across outlets, (2) similar store image across firms, and (3)
similar prices. Therefore, location is a major determinant of success.
Ghosh and Craig (1984) considered the location of several retail facilities by two
servers. The problem is to locate retail facilities in a competitive market knowing
that a competing firm will also enter this market. They used a MiniMax approach,
where the entering firm maximizes its profit given the best location of the competitor. The firm’s objective is to maximize the net present value of its investment over
a long-term planning horizon. The model did not allow location at the same site for
both firms and therefore did not examine the issue of ties. Ghosh and Craig used a
heuristic algorithm to obtain solutions. The algorithm is as follows: for each possible
set of locations of firm, the best sitting strategy is found for firm B. The final result
is the set of locations where Firm A’s objective maximum given the best reactive
location strategy of its competitor. A Teitz and Bart hill climbing heuristic was used
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to determine the sites for both firms. The model is adapted to examine other strategies such as preemption, i.e., the identification of locations that are robust against
competitive action. In a similar model, Dobson and Karmarkar (1987) introduced
the notion of stability in the location of retail outlets by two profit maximizing firms
in a competitive market.
Most competitive decision location models in discrete space assume that consumers patronize the closest shop. Karkazis (1989) considered two criteria that
customers may use to decide which shop to patronize: a level criterion based on
the preferences of a customer on the size of the facility and a distance criterion
based on closeness to the store. He developed a model that would determine the location and number of servers to enter the market when there are other firms already
operating in the market by maximizing the profit subject to a budget constraint.
Another model that examines competition among retail stores in a spatial market
was developed by ReVelle (1986). The maximum capture problem (MAXCAP) has
formed the foundation of a series of models. These models The MAXCAP model,
based on the classical maximal location covering problem of Church and ReVelle
(1974), consists of the location of servers by an entering firm so as to maximize its
market share capture in a market in which competitor servers are already in position. Eiselt and Laporte (1989a, b) modified the MAXCAP formulation to include
attraction parameters based on gravity models and Voronoi diagrams. ReVelle and
Serra (1991) extended the formulation to allow relocation of existing servers as well
as the location of new servers.
The MAXCAP model has also been adapted to consider facilities that are hierarchical in nature and where there is competition at each level of the hierarchy (Serra
et al. 1992).
Regarding Medianoid of Network, as what Hakami says that only about the equal
situation of MAXCAP, as discussed by ReVelle, is different.
Those individuals who have in the recent years also very much studied in this
connection, have been Drezner, Eiselt, Laporte, Choedhury, Koval, Wesolowsky,
Benati, Berman, Freisz, Tobin and Miller.
From the year 1997 onwards, the number of articles increased very much and the
peak period in these years, was the year 2000.
The recent years’ have focused towards the expansion of applications and assumptions and to cite an example, in an article written by Aboolian et al. (2006)
three objects have been followed. The first one being, is the number of facilities that
must be located and the second one is the location of such facilities and the last one
which quietly differs from the other articles, is the facility terms, from the point of
size, type and volume of services given, etc.
12.1.1 Game Theories (Winston and Wayne 1995)
Game theories or competitive strategies, are mathematical theories, in which competitive opportunities have been noticed. This viewpoint is advantageous, when two
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or more individuals or organizations with specific objects, who have tried to take
decisions and the decision of anyone of them, effect others’ decisions. (This theory
does not show the playing of implementation of the game but in fact, shows the
description of methods and principles of selection of the games).
This theory has been mentioned by Mr. Phone Newman and whose theory, taking
into consideration the base of MiniMax, each competitor, acts in such a manner in
order to reduce his maximum losses into minimum.
Regarding this theory, many examples can be given, such as playing of two players in a chess game, competition between members of parliament and competition
of trading companies for safeguarding their shares in the market. The peculiarities
of competitive games are:
The number of participants or competitors is limited.
The participant has a limited list of those activities that can be implemented and
it’s possible that such a list may not be the same for all.
Each participant is aware of the possible selections made by others but does not
know which one would be selected by them.
A game can only be considered as over, when each participant would have implemented any one of his selected games. (on assumption that all the games would
be played at the same time).
Any actions taken by anyone of the individual participants, has an advantage for
him and which could be either positive or negative.
The advantage that each participant would have does not depend only upon his
own actions but also depends on the actions of others.
In this theory, it’s necessary that explanation for some of the parameters are studied,
some of which are as under:
Game. Actions taken between two or more participants and which has either
profit or loss for him.
Player. Each participant or competitor is called a player in the game.
Strategy. Rules have been set earlier already, based on which, the player while
playing, takes decision about his own actions from the list of his own (actions).
Pure strategy. Rule for taking decision for permanent selection, is a special
action.
Impure strategy. Is a decision that is taken from the pure strategies, with fixed
probabilities.
Zero total game. Game of N player, in which the total advantages are zero.
Optimal strategy. Is a strategy in which the value of Minimax is equal to
MaxiMin.
As already shown in the last explanation, in this theory, we can arrive at an optimal
result, when the value of MiniMax (that is, maximum loss is turned into minimum)
equal to value of MaxiMin (that is, minimum advantage is turned into maximum).
Accordingly, in view of the initial explanations and definitions regarding game
theories, we shall continue in subsequent sections, by continuing with facility
locations.
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Some of studies of game theory deals with situations where there are only two
decision makers (or players), but there are the situations with n (where n > 2) players. For more details about game theory see Winston and Wayne (1995).
12.1.2 Static Competition
The simplest competitive models arise when competition is assumed to be already
present in the market. The whereabouts and characteristics of this competition are
known in advance and assumed to be fixed. Such models correspond to a short term
view: they are based on the assumption that the time and/or effort/cost needed for
the competition to react is sufficiently long to harvest the main benefits of the new
facility. These models also form the basis on which more complex models may be
built. These kind of static situations are discussed in more detail in the remainder of
this part.
12.1.3 Competition with Foresight
The situation becomes quite different when a virgin market is entered in the knowledge that other competing actors will enter it soon afterwards.
It will, then, be necessary to make decisions with foresight about this competition, which itself will enter a market where competition is already present. The
ensuing Stackelberg type models, where each evaluation of the main objective
involves the solution of the competitor’s nontrivial optimization model, quickly become extremely complex. We enter here the realm of sequential models, which were
recently extensively surveyed in (Eiselt and Laporte 1996a, b).
12.1.4 Dynamic Models and Competitive Equilibrium
Existing competition will most probably alter its strategy when it loses part of or
even all of its market shares to a newcomer, implying that the competitive environment changes. This leads to dynamic models which aim at describing the
action/reaction cycles of the competing actors.
One of the traditional questions in this respect is the possible (in) existence of
equilibrium situations, dear to economists, to which such a system might evolve. It
is in fact this point of view that forms the root of competitive location theory thanks
to the seminal paper of Hotelling (1929). This large area of research has been most
often surveyed, see e.g. (Eiselt and Laporte 1989a, b, 1996a, b) and will not be
discussed here.
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12.1.5 Point vs. Regional Demand
Where does demand originate? Is it discrete, i.e., concentrated in a finite set of points
or rather continuously dispersed over a region? In both cases a precise description
is needed of its spatial distribution.
In case of point demand its volume, be it expressed in terms of quantity, frequency and/or currency (then sometimes called “buying power” at each demand
point should be given. For regional demand this is described by a continuous spatial
distribution, often assumed to be uniform.
It may be argued that in principle individual customers form a discrete set, so
should be described by a point distribution. However, there are usually too many
individuals involved and their location in space is not fixed in time, hence a continuous distribution might also be necessary and/or adequate. Observe, however, that
regional demand typically has meaning only in a continuous or network environment (see below).
Next, it must be determined whether demand is elastic or inelastic with respect
to quality. In other words, does the volume of demand, depend on the (conditions
of) supply or may it be considered as fixed.
This will largely be determined by the product type. It is customary to consider
demand for essential goods, such as bread, to be inelastic (within time-periods during which populations may be considered as constant), as opposed to inessential
goods, e.g., luxury, for which demand may be highly supply/price sensitive.
Demand may also vary independently of the supply, due to the inevitable uncertainties in the market’s description, or due to inherent randomness, e.g., weather
effects. Such situations then are patronizing by stochastic demand. For models of
this type we refer to (Drezner and Drezner 1996; Peeters 1997).
12.1.6 Patronizing Behavior
In order to be able to determine the market share of a facility it is necessary to
describe in a precise manner which part of the demand will be captured by each of
the competing facilities. This involves the way customers behave when making the
choice which of several facilities to patronize.
It is generally considered that each customer feels some attraction towards each
of the competing facilities. It is the way these attraction forces determine the actual
patronizing choice which leads to two quite different types of customer behavior
models.
This choice is deterministic when the full demand of each customer is served by
the facility to which it is mostly attracted. This conceptually simple patronizing rule
is the most common one in the literature. Basically it assumes that as long as the
supply side remains unchanged customers will always patronize one and the same
facility.
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The deterministic rule stated above does not clarify what happens in the case of
ties, in other words when a customer feels equally and maximally attracted towards
several facilities, including a new one. Several tie resolution rules may be considered: either the demand is (equally?) split over all tied facilities, or it goes fully
towards the new facility, or may stick to the competing facilities.
A deterministic choice rule does not allow for the “changing mood” of customers.
The choice is probabilistic when each customer splits its volume of demand over
the different facilities, with probabilities determined some way by the attraction felt
towards each facility. At present this seems to be the only alternative proposed to
the deterministic “all or nothing” rule.
12.1.7 Attraction Function
The attraction function describes how a customer’s attraction (also often called utility, particularly in economics) towards a facility is obtained. In location theory it is
always assumed that some notion of distance between customer and facility plays a
crucial role in this attraction.
Typically, attraction will decrease with distance and the attraction function describes in what precise way. In case all competing facilities, existing and new, are
uniform, i.e., apart from their site they are further indistinguishable in the sense that
they offer exactly comparable, and thus substitutable, products and services at the
same prices, the distance will be the sole determinant in the attraction.
It may be observed that in this case it is usually considered that only deterministic behavior applies. In many cases, however, facilities are multiform, i.e., they
do differ in other aspects than the mere site where they are located, and customers
will take these differences into account in the way they feel attracted to them. These
differences should then be incorporated into the attraction function as additional parameters on top of the distance. There are basically two standard ways in which this
may be done.
In Economics it is often considered that the product’s observed price primarily
determines the customer’s choice. A notion of attraction in such a setting is therefore
fully price based. In the traditional mill pricing system the price actually paid for
the product is given at the facility, but the customer should travel to the facility and
its travel cost, determined by the distance of travel, should be included in the full
observed price. Typically this leads to additive attraction functions.
Apart from distance, one may consider the price as just one of several ingredients in the overall attraction process, in which other features like floor area, number
of cashing counters, product mix, publicity, etc. also play an important role. All
these properties (excluding distance) may be summarized into a single measure for
which the unattractive word “attractiveness” has been used, and which we prefer to
simply call the quality. In this context one usually finds proposalsfor multiplicative
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attraction functions. Note that this type of attraction function has recently been criticized since it might imply that a choice of facility may change during the trip
towards the facility.
12.1.8 Decision Space
In competitive location models one finds back the three traditional spatial settings
of location theory: discrete space, a network, and continuous space. Observe that a
complete description of space should also include the distance measure used.
In discrete space there is only a (relatively small) finite list of candidate sites and
the market is always assumed to consist of point demand. Distances may then in
principle be obtained in a very precise way including all possible special features
necessary for an adequate description of reality.
Indeed the full set of demand (candidate) facility site pairs is fully known and
finite. The central difficulty in practice resides in the amount of data to be collected.
In network environment both demand and facilities may lie anywhere along the
edges of the network. The nodes of the network are just the points where edges
meet. Since travel is assumed to be restricted to the network, distance is typically
calculated as shortest path distance. Location of new facilities might be further restricted to only part of the network, as defined by a set of edge segments.
It should be observed that the term network location is often used in a more
restrictive sense, where only nodes of the network are candidate sites. In our terminology such problems are considered as discrete problems, and the network setting
is only used in order to obtain a standard way to calculate distances by shortest paths.
On the other hand, theoretical developments in network setting most often lead
to the identification (and efficient construction) of finite sets of points which are
guaranteed to contain at least one or all optimal sites for the location problem at
hand. Such localization theorems then allow reducing the original network location
problem to a discrete one by restricting the search for optimal sites to this “finite
dominating set”, thereby generating the situation described in previous paragraph.
Similar discrimination results often appear in continuous setting too.
Continuous space refers to a location space determined by a coordinate system
in which in principle any real coordinate values are admissible.
One-dimensional space is equivalent to the simplest possible network: a linear
segment (or line), a very popular setting for competitive equilibrium studies in virtue
of its simplicity. Other settings are of course the geographical space fucoids as a twodimensional plane or possibly a sphere. Applications in e.g., product positioning
may involve even higher dimensional settings.
In continuous space it is necessary to specify the type of distance which is used.
Classically this will often be the patronizing distance, but many other distance measures may be considered, like LP .1 p C1/ of which the rectangular (or
Manhattan) distance .p D 1/ and Euclidean distance .p D 2/ are special instances,
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or block norms, obtained by shortest paths using only a finite set of travel directions
and velocities and even their asymmetric cousins block gauges which usually lead
to easily manageable linearity properties.
12.2 Models
In this part we introduce four models which have different kinds of solution space;
the first of them is in continuous space and other three subsequent models solve in
discrete space.
12.2.1 Gravity Problem
Before starting of the models we can illustrate the continuous space before and after
of competition. In Fig. 12.1 there are just one player and it has all the market but in
Fig. 12.2 there are two players and the market is shared between them.
Fig. 12.1 A continuous space
with one player (no competition)
Fig. 12.2 A continuous space
with two player and sharing
the market (with competition)
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12.2.1.1 Model Assumptions
Model assumptions of this model are as follows:
Existing competition is known and fixed.
Customers’ patronize the most attracting facility with their full demand, i.e., “the
winner gets it all” principle.
The probability that a customer patronizes a facility is proportional to its attractiveness and inversely proportional to a power of the distance to it.
When p new facilities are opened in an area, the total market share T attracted
by these facilities and by those already part of the franchise.
This model finds the best location for new facilities whose individual measures
of attractiveness are known.
12.2.1.2 Model Inputs (Indexes and Sets)
Model inputs of this model are as follows:
i: be the sub index of demand point
j: be the sub index of existing facilities
m: be the set of new facilities
12.2.1.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
.xm ; ym /: the location of new facility m, for m D 1; : : : ; p
12.2.1.4 Parameters
Parameters of this model are as follows:
n: the number of demand points (each demand point represents a small area
around it)
.ai ; bi /: the location of demand point i D 1; : : : ; n
Bi : the available buying power at demand point i for i D 1; : : : ; n
K: the number of existing competing facilities
C: the number of existing facilities which are part of one’s own chain
P: the number of
pnew facility to be locating in the area
.xm ym / D .xm ai /2 C .ym bi /2 : the distance between demand point
i and new facility m. i D 1; : : : ; nI m D 1; : : : ; p
Ej : the measure of attractiveness of existing facility j for j D 1; : : : ; k
Am : the measure of attractiveness of existing facility m for m D 1; : : : ; p
: the power to which the distance is raised
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12.2.1.5 Objective Function and its Constraints
The objective function of this model as follows:
T D
n
X
Bi
i D1
p
P
mD1
p
P
mD1
di
Am
.xm ; ym /
C
di
Am
.xm ; ym /
C
c
P
j D1
k
P
j D1
Ej
dij
(12.1)
:
Ej
dij
The objective is to find the best location in the plane that maximizes the total market
share captured T using (12.1).
A sequence of algebraic manipulations (Drezner 1995) leads to a somewhat simpler minimization problem, one of minimizing the total buying power not attracted
by the chain.
k
P
Ej
n
d
X
j DcC1 ij
:
(12.2)
Bi p
F D
k
P
P
Ej
Am
i D1
C
.x
/
mD1
di
m;
ym
j D1
dij
The total sums for existing facilities are constants. Let
Gi D Bi
k
k
X
X
Ej
Ej
;
H
D
:
i
d
d
j DcC1 ij
j D1 ij
(12.3)
These definitions lead to:
Min W
8̂
ˆ
<
ˆ
:̂
F D
n
X
i D1
Gi
Hi C
p
P
mD1
di
Am
.xm ; ym /
9
>
>
=
:
(12.4)
>
>
;
12.2.1.6 Solution Gravity Problem
Algorithm 1
Randomly generate p sites for the p new facilities.
Perform Weisfeld iteration for each of the facilities while holding the other rooted
in their places (Drezner 1995).
After all p facilities were relocated one by one in step 2, calculate the changes
in their locations. If the location changes are less than a perspective ", terminate the
algorithm.
The following are two practical approaches to solving (12.4) (Drezner 1995):
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The AMPL approach
Instead of pursuing a specialized algorithm for the solution of these problems,
standard non-linear programming codes available for the solution of non-linear programming problems like (Drezner 1995) can be used. The student version of AMPL
(Drezner 1995) was used to solve this problem. Since these standard programs assume that the objective function is convex, one still needs to resolve the problem repeatedly with various starting points and select the best solution for implementation.
Also, the no convexity of the objective function may cause difficulties in the solution
procedure itself. The AMPL modeling program is quite easy to write, is very short
and compact, and is easy to follow. The full program is given in (Drezner 1995).
The excel approach
Spreadsheet software now incorporates optimization capabilities. The “solver”
option in Excel can be used to solve. In Excel 4.0, this option is found under “formula”. A spread sheet is built to calculate the objective function, and the solver
provides the optimal solution with the calculated market share at the point. Since
the problem is not convex, the procedure has to be repeated using many randomly
generated starting location and the best solution is selected for implementation.
12.2.2 The Maximum Capture Problem Model (MAXCAP)
(Serra and ReVelle 1995)
12.2.2.1 Model Assumptions
Model assumptions of this model are as follows:
Existing competition is known and fixed.
Customers’ patronize the most attracting facility with their full demand, i.e., “the
winner gets it all” principle.
This model leads to combinatorial optimization models similar to covering, as
exhaustively reviewed in.
12.2.2.2 Model Inputs (Indexes and Sets)
Model inputs of this model are as follows:
i; I : be the sub index and set of demand points
pi : set of sites s which i would patronize if a new facility would be opened there
Ti : set of sites for i tied with the currently patronized competitor’s facility
12.2.2.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
yi : the fully captured for i
zi : the tied with a competitor for i
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12.2.2.4 Parameters
Parameters of this model are as follows:
wi : the demand of customer for i 2 I
xs : the facility be opened at s
12.2.2.5 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Max
X
i 2I
wi yi C
X wi
i 2I
2
zi :
(12.5)
Subject to
X
xs .i 2 I / ;
(12.6)
X
xs .i 2 I / ;
(12.7)
Yi C Zi 1 .i 2 I /;
X
xs P ;
(12.8)
Yi
Zi
s2pi
s2Ti
(12.9)
s2S
Yi ; Zi 2 f0; 1g xs 2 f0; 1g :
(12.10)
12.2.2.6 Solution Maximum Capture Problem
Discrete models without split demand lead to the maximal covering type location
model introduced in (Church and ReVelle 1974). This is a simplified version of
the maximal capture problem given above, in which all tie sets ti are empty, and
therefore without the auxiliary tie variables zi . Typically, these models will have
many optimal solutions, a feature which is troublesome for standard exact search
branch and bound methods, but makes heuristic approaches particularly appealing.
A recent proposal (Benati and Laporte 1995) for solving large-scale models of this
type consists of a Tabu search method.
Since in network and continuous setting ties appear whenever a site is chosen
which is exactly at break-even distance from the demand point, the deterministic allocation rule inevitably leads to discontinuity of the capture function at such points,
and previous rule of even split enhances this. Without split one at least retains semicontinuity. Two extreme situations without split demand may now be considered.
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First there is conservatism, stating that in case of tie with a new facility; customers will go on patronizing the existing facility they patronized before. This is the
rule followed in (Hakimi 1990) introducing the NP-hard .r; xp / medianoid problem: find r sites on a network maximizing the total weight of demand vertices lying
closer (in terms of shortest paths) to one of them than to any of the competitors xp .
Second, the opposite assumption of novelty orientation, i.e., in case of tie the new
facility gets all demand (Hanjoul and Thill 1987), gives rise to closed captured markets, a feature of particular importance in the next section. With this tie-resolution
rule there corresponds to each demand point a (closed) subset of the location space
of all sites for the new facility(ies) that would capture that demand. In other words,
we have a collection of subsets, each related to a given demand volume. For a particular site the total captured demand is then obtained by summation of the volumes
of all sets it lies in. Maximizing the captured demand now means finding (a point in)
a no void intersection of such sets corresponding to the highest total volume. In the
Euclidean plane these sets are circular balls and it is easy to show that optimal solutions will always exist among all intersection points of two of the boundary circles,
an easily enumerated finite set of candidates. Dresdner (Drezner 1981) shows how
to do this efficiently for a single facility, while (Mehrez and Stulman 1982) suggest
a dynamic programming approach for the multi facility version.
These types of model have also been advocated in marketing studies for brand
positioning and even in political science, see (Schmalensee and Thisse 1988). In
the first case, product characteristics are taken as coordinates in a feature space,
consumer groups having a common ideal brand-description represent demand, and
attraction towards a brand is expressed by way of the (Euclidean) distance between
the points representing the ideal and the real brand. This leads to ball intersection
problems in higher dimensions, which turn out to be NP-hard when dimension is
part of the input, but polynomial in fixed dimension. Several solution approaches
have been developed recently, all relying on geometric construction and evaluation
of a finite dominating set.
12.2.3 The Maximum Capture Problem with Price Model
(PMAXCAP) (Serra and ReVelle 1998)
We will introduce the max capture model with price which is a kind of location and
design problem that contains the price of the product in its output.
12.2.3.1 Model Assumptions
Model assumptions of this model are as follows:
Existing competition is known and fixed.
Customers’ patronize the most attracting facility with their full demand, i.e., “the
winner gets it all” principle.
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The space is discrete and is defined by a connected graph. On each vertex of
the graph there is a demand whit a distinct size, the set of local markets that are
located on the vertices of the graph.
The consumer’s decision on patronizing a store is based on transportation costs
and price. Consumers always go the outlet with the lowest total price, regardless
of its ownership.
There is an existing firm (from now on, Firm B) operating with q outlets. A new
firm (Firm A) wishes to enter and establish p servers. (The product sold in this
industry is homogeneous).
Suppose the demand is elastic then the demand of local market i is a function
of the market’s characteristics and the price customer faces and is denoted as
Di .…i /.Thus it could be said that Di .p Ctdib / is the demand for the local market
i; shopping at its closest outlet. Therefore, the demand function for each local
market i for Firm A is defined as follows: if biA is the closest Firm A’s server to
i and biB the closest Firm B’s server, then:
DiA . A ; B / D
(
Di . A / if p A C tdi b A < p B C tdi b B
i
i
0 if p A C tdi b A p B C tdi b B
i
:
(12.11)
i
Equation (12.11) is, Firm A will capture the demand of local market i if the total
price …A
i (mill price plus transport costs) faced by consumers is lower than the
total price …Bi that defined the in the same way for B. Assume that production
entails for each outlet a fixed set-up cost fj , and a constant marginal cost v, per
unit. Thus the total cost for each outlet is given by fj C vs . s is the number of
products in this facility. S is the number of products in this facility. If j A is the
set of actual outlet locations .J A J /, then profits for firm A can be written as
follows:
X
X
YA
fj :
(12.12)
D .p A v/
DiA . A ; B /
j 2J A
Equation (12.12) for firm A is determining j A and p A to maximize profits. This
profit also depends on other model parameters such as the location and prices of
the competitor firm and the demand function of consumers among others.
12.2.3.2 Model Inputs (Indexes and Sets)
Model inputs of this model are as follows:
i,I: be the sub index and the set of local markets that are located on the vertices
of the graph
j, J: be the sub index and set of potential locations for Firm A’s outlets.
JB : be the set of actual locations of the outlets of Firm B.
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12.2.3.3 Model Outputs (Decision Variable)
Model outputs of this model are as follows:
P A : as the mill price to its customers irrespective of this location. (Continuous
decision Variable)
yi A D 1, if firm A captures demand node i ; 0, if not
xi A D 1, if firm A locates a server at node j ; 0, if not
12.2.3.4 Parameters
Parameters of this model are as follows:
biA : be the closest firm A’s server to i
biB : be the closest firm B’s server to j
p B : be the mill price to its customers irrespective of this location
dij : be the network (shortest) distance between local markets i and an outlet in j .
nA D number of firm an outlet servers
12.2.3.5 Objective Function and its Constraints
If the demand function (12) of the local markets is totally inelastic with respect to
prices, it can be written as Di D ai , where ai is the total demands that market i will
purchase.
The objective function of this model and its related constraints are as follows:
Max
Y
D .P A v/.
X
i 2I
ai yj /
X
fj xj :
(12.13)
j 2J
Subject to
yiA
n
X
j D1
X
j 2NI .biB /
XjA 8i 2 I;
(12.14)
xjA D nA ;
(12.15)
yiA ; xjA D .0; 1/ 8i 2 I; 8j 2 J:
(12.16)
Where additional notation is:
o
n
Ni .biB / D 8j 2 J; p A C tdij < p B C tdi b B :
i
(12.17)
PMAXCAP is the problem which maximizes profit capture as opposed to population
captured.
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Equations (12.14) depends on the set Ni .biB /, which is known as capture aria of
the demand i . Each one of the demand nodes i has an associated set Ni .biB / which
contains all the potential nodes at which Firm A can locate a server and capture the
demand of local market i . Therefore, if one of the variables xiA belonging to the
corresponding constraint is equal to 1 (a facility is located within the capture area
of node i ), then capture variable yiA is allowed to be 1, which indicates that node i
has been captured by Firm A. Finally, (12.15) sets the number of servers that Firm
A is going to locate. The objective function defines the total profits that Firm A can
achieve. For each local market If yiA D 1, then .p A v/ai is added to the revenues.
Fixed costs are multiplied by xj , so if an outlet established; its associated fixed cost
is subtracted from the objective.
If we want to consider the number of A’s established servers as inborn, then its
enough to eliminate (12.15).
The basic difference between the MAXCAP problem and the PMAXCAP problem formulation are in three points: on the sets Ni in restriction, in the objective
function, and in the equality case. Indeed the above formula does not pay any incidence to equality model. Of course it did not make any mistake because both value
and place are variable and the possibility of the equality is very low.
The Ni .biB / set contains all candidate nodes where if the server is established
there the demand of node i will be captured. While in the MAXCAP problem, these
sets were known preferment for each node, in the PMAXCAP model while in the
MAXCAP problem these sets were known preferment for each node, in the PMAXCAP model these are variable because the price is variable therefore (12.14) cannot
be written extensively. In this formulation that was given for PMAXCAP, the objective function is nonlinear.
12.2.3.6 Solution Maximum Capture Problem with Prices
If the demands function is elastic, then the PMAXCAP problem has to be reformulated using a P-median like approach which is called PMAXMED (Serra and
ReVelle 1998) for solving the PMAXMED model in both cases that said before,
a barred searching based method was given because using the branch and bound
methods is difficult (Serra and ReVelle 1998).
12.2.4 Flow Capturing Location Allocation Problem Model
(FCLAP)
12.2.4.1 Model Assumptions
Model assumptions of this model are as follows:
Nowadays, many customers do their shopping in their daily travels between home
and work or vice versa, instead of set a special travel for buying a good. Thus, in
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such an area, the demand in the network is like a flow instead of a node. the goal
is to establish the servers in the places as they can capture the maximum demand.
In the articles (Berman et al. 1992; Fouska 1988) the writers had studied this
subject. Which Hodgson called “flow capturing location allocation problem”
(FCLAP) (Hodgson 1990). It is concluded in all papers that the optimal set of
locations must be on the nodes.
In most of the studies cited thus far, customers can visit only service facilities located at points on paths of the network. That is, no deviation from the pre-planned
path is allowed. But new articles have overruled this assumption (Berman et al.
1995). Thus customer can pervert his from his predetermined way. The extra
distance traveled in this case is called the “deviation distance”.
D.p; j / the deviation distance from path p to node j , d.i; j / being the shortest
distance from node i to node j in the network; and s; e represent the starting and
ending nodes for path p.
D.p; x/ D d.s; x/ C d.x; e/ d.s; e/:
(12.18)
12.2.4.2 Model Inputs (Indexes and Sets)
Model inputs of this model are as follows:
K: set for existing competing facilities.
M : number of facilities to be located
G.N; A/: N , set of nodes with cardinality n; A, the set of arcs
P : set of nonzero flow paths through network nodes and arcs
p: a path in P
i : be the sub index that are not located on the vertices of the graph.
j : be the sub index that are available
t: be the sub index that are new
12.2.4.3 Model Outputs (Decision Variable)
Model outputs of this model are as follows:
Xpt i D 1, if customers on path p visit facility t located on node i ; 0, if not
Yt i D 1, if facility t is located on node i ; 0, if not.
12.2.4.4 Parameters
Parameters of this model are as follows:
fp : number of customers in path P
M : number of facilities to be located
.xm ; ym /: location of new facility m, for m D 1; 2; : : : ; M
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M.J. Karimifar et al.
dij : distance between demand point i and existing facility j , i D 1; : : : ; nI
j D 1; : : : ; k
dij .xm ; ym /: euclidean distance between demand point i and new facility i D
1; : : : ; nI m D 1; : : : ; M
Ej : measure of attractiveness of existing facility j I j D 1; : : : ; k
Am : measure of attractiveness of new facility m; m D 1; : : : ; M
: power to which distance is raised
T : total market share attracted by facilities of one chain
Np : set of nodes capable of capturing flow on path P
12.2.4.5 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Max
X
fp
P 2P
M
P
P
t D1 i 2N K
M
P
P
t D1 i 2N K
Ai
x
DD .p;i / pt i
Ai
x
DD .p;i / pt i
:
C
P
j 2K
(12.19)
Ej
DD .p;i /
Subject to
M
X
t D1
xpt i 1 8p 2 P; 8i 2 N K;
X
i 2N K
xpt i D 1 8p 2 P; t D 1; : : : ; M;
yt i xpt i 0 8i 2 N K; t D 1; : : : ; M;
M
X
X
yt i D M;
(12.20)
(12.21)
(12.22)
(12.23)
t D1 i 2N K
xpt i ; yt i 2 f0; 1g ;
(12.24)
8p 2 P ; 8i 2 N K; t D 1; : : : ; M:
(12.25)
Equation (12.19) represents the objective to be maximized: flow captured by these
new facilities.
Equations (12.20) requires that for each node, at most one facility can be located
there. Equations (12.21) indicates that each new facility can be located on only one
node. Equations (12.22) guarantees that the flow on path p can be captured only if
the corresponding facility is located. Equation (12.23) assures that M facilities are
to be located.
The binary requirements for all decision variables are expressed in (12.24).
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12.2.4.6 Solution Flow Capturing Location Allocation Problem
Upon model is nonlinear which might make problems in the calculation phase. Wu
and Lin (2003) had changed this model to liner, based on the method presented by
Wu (1997), but the result will not always redound to simplicity. Also those man had
presented an inventive method for solving the model base on a greedily procedure
which is too similar to what Berman et al. (Berman et al. 1995; Berman and Krass
1998) had developed for FCLAP problem.
12.3 Case Study
In this section we will introduce some real-world case studies related to Competitive
location problems.
12.3.1 A Case in Toronto (Aboolian et al. 2006)
A company would like to enter the metropolitan Toronto market by locating one
or several supermarket-type retail facilities there. There are two choices for facility
designs: a regular facility, with approximately 15,000 square feet of retail space, and
a large facility with approximately 45,000 square feet of retail space. These values
were chosen based on data from the Food Marketing Institute which lists 15,000
square feet as the size of a typical supermarket and 40,000–50,000 square feet as
the size of a superstore. The budget that the company has allocated for the Toronto
expansion is sufficient to locate one large facility or up to three regular facilities
anywhere in the city.
In this case, the following competitive situation is assumed: a competitor providing similar service has already entered the Toronto market, using the same facility
design alternatives. For the competitor, the same two choices regarding the number
and sizes of the facilities are considered: one large or three small. For each combination of values of problem parameters described below, optimal locations for the
competitor’s facilities is selected. The company now has to decide on the location
and sizes of its own facilities given the existing locations of competitive facilities.
The network that is used corresponded to the geography of metropolitan Toronto.
The lease costs for retail real estate vary significantly throughout the city, the cost
data per square foot is used from the April 2004 Commercial Reality Watch report
produced by the Toronto Real Estate Board. Since each node of network represents
an area that can contain multiple facilities, own and competitive facilities are allowed to be co-located within the same nodes.
The problem described above is analyzed under three values for demand elasticity parameter, which represent respectively very high elasticity, moderate elasticity,
and low elasticity; three values for the size sensitivity parameter, which represent,
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M.J. Karimifar et al.
respectively, very high sensitivity, moderate sensitivity, and low sensitivity; and two
values for the distance sensitivity, which represent respectively, high and moderate
values. Aboolian et al. (2006) wanted to see the extent to which the optimal location and sizing decisions depend on the problem parameters and on the sizes and
locations of pre-existing competitive facilities.
The following conclusions can be made based on the results of the case study in
Toronto:
It is always optimal to locate three regular facilities when either the size sensitivity is small, or size sensitivity is moderate and distance sensitivity is high.
When distance and size sensitivities are both moderate, three regular facilities
are also preferred, except when demand is very elastic, in which case one large
facility performs better.
When size sensitivity is high, one large facility is generally optimal, except when
distance sensitivity is high and demand is not very elastic.
The optimal sizes of own facilities appear to depend only on problem parameters
the sizes/number of competitive facilities do not seem to affect own design decisions. However the locations of own facilities are certainly influenced by both,
the number and locations of competitive facilities, as well as by the problem
parameters.
When demand is very elastic, co-locations are very common i.e., one or more
of own facilities are located in the same FSAs as the competitive facilities, we
can loosely interpret these as shopping mall solutions. On the other hand, when
demand elasticity is low, co-location never happens owns facilities tend to be
located away from the competitive ones.
12.3.2 A Case in Yuanlin Taiwan (Wu and Lin 2003)
The decision objective for service providers is to find the optimal locations for such
service facilities to Maximize the number of customers “captured”. This sort of
problem is usually called a “flow capturing location allocation problem” (FCLAP),
that in the earlier seasons of this book is discussed. In this article a mathematical model and heuristic solving method is developed based on the greedy solving
method, and this solving method in the instance study of real networks in Yuanlin,
Changhua which was accomplished in Taiwan is surveyed.
Yuanlin is a township in Changhua County, Central Taiwan with an approximate
area of 33 W 72 km2 and a 1998 population of 126,395.
A newly established retail chain is planning to enter the local convenience-store
market and locate a number of new stores in Yuanlin. For this analysis, The Household Registration Office of Yuanlin provided data such as geographical zoning, and
the population of each zone from a 1998 update.
In this analysis, there is a network with 39 nodes and 1482 paths and several
assumptions made in this analysis are described as follows: These criteria consist of
product mix (c1), price (c2), interior layout and design (c3), personnel (c4), overall
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cleanliness (c5), floor space (c6), traffic congestion in the surrounding area (c7),
and frequency of promotional sales (c8). Information regarding the location and
attractiveness score is gathered for each existing store.
The greedy heuristic and the enumeration method developed are used for solving
this authentic network. The results enumeration approaches are nodes 19, 27, 3 and
the results greedy approach summarized are nodes 8, 27, 19. The deviation from the
optimal of the greedy heuristic is 5.73%, with 7.141 s taken, which is more than 340
times faster than the 2,484.658 s required by the enumeration method are.
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Chapter 13
Warehouse Location Problem
Zeinab Bagherpoor, Shaghayegh Parhizi, Mahtab Hoseininia,
Nooshin Heidari, and Reza Ghasemi Yaghin
Finding the optimal location for industrial facilities has always been of a crucial
importance and priority from geographers’ and economists’ view point. Although
economics play the main role in presenting location theories by resorting to its theories, geographers, with their focus on locative and spatial changes leading to natural
phenomenon, have also had a share in complimenting these location theories and
finding optimal points to locate industrial activities location. From this point of
view, materials, market and other manufacturing factors have not been concentrated
in one point, and their spatial separation necessitates traveling distances which, in
turn, brings about costs. That is why optimal industrial location is also a part of
geographical studies.
By location theories, we mean “a series of principals with resorting to which,
the optimal location of industrial activities is determined (point conjoining on the
most benefit)”. The history of location theories goes back to nineteenth century in
Germany. The first industrial location theory was presented by Shuffle in 1878 A.D.
He was successful in applying the gravity model in the way industries are located:
Mij D Pi Pj Dij 2 ;
(13.1)
where P&D represent population and distances, respectively. In this way, industries
are located adjacent to huge cities as a response to demand and market factors.
The most dominating principle in finding the optimal location is based on the
basis of the least cost. Lanhart is another scientist in this field who represented his
theory in 1882. According to his reasoning, the position of industries depends on the
significant factor of transportation. In other words, the ideal point for locating industry is a point where the total transportation cost, including material, final products
and fuel, are at minimum.
At the mercy of myriad business, logistics and government initiatives, including
just in time production, quick response, research on quality, enhanced customer
satisfaction, operators’ safety, and environmental protection, warehouse operations
have been and are continuously revolutionized (Tompkins et al. 1996). Warehouses
can play each of the following roles in a distribution network:
They can be used as a balance and storage point because of the difference
between the time scheduling and manufacturing demand. For this reason, the
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 13, c Physica-Verlag Heidelberg 2009
295
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warehouses are usually located at the manufacturing point. Complete pallets flow
in and out. Warehouses whose role is limited to this can have monthly and seasonal demands until progressing to the next step in distribution.
They can be utilized for storing and commixing outputs coming from different
manufacturing points in one company or more companies, before being send to
the customers. This part usually responses to the weekly or monthly orders.
They can be dispersed to reach the shortest transportation distances to be able to
quickly respond to the customer’s demand. Items are often chosen individually
and similar items can be sent daily.
If possible, two or more of these roles must be consolidated in the same warehouses.
Today’s changes have necessitated this consolidation to make inputs accessible and
to decrease transportation costs. Small items with great value and unpredictable
demands, in particular, are often distributed from a single source through a global
network (Tompkins et al. 1996).
13.1 Classifications
A classification of different types of warehouse location problems is represented as
follows:
Area solution: discrete, continual
Warehouse installation cost: Fixed installation cost, Without fixed installation
cost
Determination of the number of warehouses: exogenous, endogenous
Determination of the number of products: Single product, Multiple product
Determination of number of periods: singe period and multi period.
Warehouse capacity: unlimited capacity, Limited capacity
Constraint on relationship between warehouses and customer: In some models,
customer’s demand could be served by any number of warehouses but in others
there are constraints on relationship between customers and warehouses and each
warehouse can serve only a limited number of customers.
The purpose of these problems is to minimize the total cost of locating warehouses
in candidate points and allocating customers to the warehouses in each period. These
problems also make clear assumptions about costs. The curve in Fig. 13.1 indicates
the required cost to build facilities as a function of output quantity and the curve in
Fig. 13.2 indicates the incurred transportation costs as a function of shipment size
(in fact, transportation costs increase with the size of shipment). It may be difficult
to estimate the cost items. We may also encounter questions such as “How long
does a period last?”, “How many periods should be considered?”, and “How much
are the administering costs of the warehouse?”. Such questions can be examined in
the special problem.
13
Warehouse Location Problem
Fig. 13.1 Warehouse cost
model
297
Cost
p1
1
Quantity
Fig. 13.2 Transportation cost
model
Cost
tij
1
Quantity
13.2 Models
In this section, we introduce warehouse location models represented and developed
as mathematical models.
13.2.1 Warehouse Location Problem without Fixed
Installation Costs (William et al. 1958)
In this section, fixed costs of warehouse installation are not considered in the warehouse location problem and, therefore, there are changes in the problem variables
which convert it to a transportation problem (William et al. 1958).
13.2.1.1 Model Assumptions
Model assumptions are as follows:
The location problem is viewed as a discrete problem.
The capacities of facilities are assumed to be equal and infinite.
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Transportation costs are an increasing function of shipment sizes.
The number of warehouses should not be more than the number of candidate
points.
Each customer’s demand should be estimated and customers should be allocated
to the warehouses.
13.2.1.2 Model Inputs
i : factory .i D 1; 2; : : : ; m/
j : warehouse .j D 1; 2; : : : ; n/
k: retailer .k D 1; 2; : : : ; q
Cij k : cost of the shipment (from factory i to retailer k through warehouse j )
including the relevant inventory cost
Qi : quantity shipped from factory i
Rj : capacity of warehouse j
Sk : quantity required at destination k
13.2.1.3 Model Outputs (Decision Variables)
Xij k : quantity shipped from factory i via warehouse j to retailer k
Aij k : amount of inventory remained in warehouse j from the flow Xij k
13.2.1.4 Objective Function and its Constraints
The problem is to minimize the total delivery costs, i.e., to minimize the following:
X
Cij k Xij k ;
(13.2)
Xij k D Qi i D 1; 2; : : : ; m;
(13.3)
Aij k Xij k Rj j D 1; 2; : : : ; n;
(13.4)
Xij k D Sk k D 1; 2; : : : ; q:
(13.5)
Min Z D
i;j;k
Subject to
X
jk
X
i;k
X
i;j
Equation (13.3): all goods must be shipped out of the factory
Equation (13.4): no warehouse capacity can be exceeded
Equation (13.5): all customers’ demands must be met
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299
The resemblance to the standard transportation problem is obvious and there are
only three differences:
(a) The possible nonlinearity of (13.2)
(b) The presence of the warehouse-capacity (13.4)
(c) The need for a three subscript notation for the variables Xij k (three dimensional X variables) resulting from the necessity of routing each flow through a
warehouse.
Of course, a nonlinear objective function is not necessarily ruled out for the transportation problem, and can be ignored. The warehouse capacity limitations in the
problem, can be also be ignored because a firm never ends up renting more than
a small fraction of the public warehouse space available at any location.
Elimination of difference (c), i.e., the three-subscript notation, becomes a trivial
matter because of the following rule: An optimal (least cost) solution will involve
shipment of all goods that go from factory i to destination k via that (those)
warehouse (s) j for which Ci j k D Min jC 0i j k
In the other ways, it will always pay to make any shipment via the warehouse
that offers the lowest delivery cost. The solution of the program is now simple.
For each factory-destination combination, i k , select a value j for which
(13.5) is satisfied. This can be done by simple inspection of the Cij k data. We
can now revise our notation by letting Xi j k D Xi0 k ; Ci j k D Ci0 k
(since in an optimal solution all other Xi j k ’s will be equal zero). Substituting this
notation in (13.2), (13.4), and (13.5) will obviously leave us with a standard transportation problem whose optimum solution can be found by the standard methods.
13.2.2 Warehouse Location Problem with Fixed Cost
of Establishment (Akinc and Khumawala 1977)
All assumptions in this problem are similar to the previous case; the only difference
is the fixed cost of location.
13.2.2.1 Model Inputs
k: customer .k D 1; 2; : : : ; q/
j : candidate points .j D 1; 2; : : : ; n/
fj : cost of locating a warehouse in point j
rk : customer’s demand
vkj : the cost of shipment to customer k from point j for each unit
ckj D vkj C rk ;
where usually q > n.
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13.2.2.2 Model Outputs (Decision Variables)
xkj : a fraction of rk which is transported from j to customer k
yi : Will be equal one if the candidate point j is selected, otherwise it will be
equal to zero.
13.2.2.3 Objective Function and its Constraints
The warehouse location model is formulated as a mix integer programming problem, as follows:
Min Z D
n
X
j D1
fj yj C
q
n X
X
ckj xkj ;
(13.6)
j D1 kD1
Subject to
n
X
j D1
xkj D 1I
k D 1; : : : ; q ;
(13.7)
xkj C yj 0;
(13.8)
0 xkj 1;
(13.9)
yj 2 f0; 1g :
(13.10)
Notice that rk xkj is the amount transported from point j to customer k (xkj is a
fraction of rk which is transported from point j to customer k).
Equation (13.7) insure that a customer’s demand is satisfied and that the maximum quantity of xkj is equal to 1. Equation (13.8) insure that a warehouse in point
j serves customer k if a warehouse has been located in that point.
This formulation shows that WLP is, in fact, a special type of n-Median in which
the number of facilities is ignored.
13.2.2.4 The Other Way of Formulation
In this section, another WLP formulation is presented which is discussed in the last
section. Here, by adding up of (13.8) over k, the mentioned WLP is converted to the
following problem:
Min z0 D
n
X
fj yj C
j D1
Subject to
n
X
j D1
xkj D 1;
q
n X
X
ckj xkj
.WLP0 /;
(13.11)
j D1 kD1
k D 1; : : : ; q;
(13.12)
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Warehouse Location Problem
q
X
kD1
301
xkj qyj 0;
0 xkj 1;
j D 1; : : : ; n;
(13.13)
k D 1; : : : ; qI j D 1; : : : ; n;
(13.14)
yj 2 f0; 1g :
(13.15)
Both problems have the same optimal solution. However, they have different relaxations. In fact, WLP’ can be solved through scanning, since the following
equation holds:
q
X
xkj D qy j :
(13.16)
kD1
Reducing yj results in the objective function value to reduce as well. Therefore,
the constraints of yj can be solved and replaced in the objective function:
q
n X
X
fj
Min
.
C ckj /xkj ;
q
j D1
(13.17)
kD1
Subject to
n
X
j D1
xkj D 1;
0 xkj 1;
k D 1; : : : ; q;
(13.18)
k D 1; ::; qI j D 1; ::; n:
(13.19)
By this WLP compact method, the warehouse with the least fj =q C ckj is chosen.
The relaxation of WLP’ presents a weak low bound for the WLP and it is not clear
how it can be used to solve WLP.
Therefore, mathematical models with less number of constraints are not necessarily better. For warehouse location models, the extended model of xkj C yj 0
is preferred to the following compact model:
q
X
xkj qyj 0:
(13.20)
kD1
Even if it increases the size of the model (WLP is called strong LP, and WLP’ is
called weak LP) (Akinc and Khumawala 1977).
13.2.3 Capacitated Warehouse Location Problem with
Constraints in Customers Being Serviced (Nagy 2004)
In the pervious models, a customer’s demand could be served by any number of
warehouses, but in this model there is a constraint on the warehouses, and, therefore,
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each warehouse can serve only a limited number of customers and each customer
must be supplied by exactly one open warehouse. The objective is to determine which
warehouses to open, and the allocation of the customers to the opened warehouses,
such that the sum of the maintenance and supply costs is minimized (Nagy 2004).
13.2.3.1 Model Inputs
k: is the number of customer
j : is the number of warehouses
ckj : is a table containing the cost associated to the supply
cf : is the fixed cost
Capacityj : the max number of stores that it can supply
13.2.3.2 Model Outputs (Decision Variables)
Oj : is a vector of Booleans indicating what warehouses have been opened.
Skj : is a matrix of Booleans indicating if customer k is supplied by warehouse j
13.2.3.3 Objective Function and its Constraints
X
X
Skj ckj C cf
Oj ;
Min
(13.21)
j
k;j
Subject to
X
j
X
k
X
k
Skj D 1;
8k;
Skj Capacityj ;
Skj Oj ;
(13.22)
8j;
8k; j :
(13.23)
(13.24)
Equation (13.22) insure that a customer must be supplied by exactly one warehouse.
Equation (13.23) insure that each warehouse has a fixed capacity. Equation (13.24)
insure that a customer can be supplied only by an open warehouse (Nagy 2004).
13.2.4 Single Stage Capacitated Warehouse Location Model
(Sharma and Berry 2007)
In this section, new formulations and relaxations of the single stage capacitated
warehouse location problem (SSCWLP) are described.
13
Warehouse Location Problem
303
13.2.4.1 Model Inputs
i : plant
j : warehouse
k: market
Dk : demand
P for the commodity at market k
dk : Dk = Dk demand at market k as a fraction of total market demand
Si : supply
P available at plant i
si : Si = Dk supply available at plant i as a fraction of the total market demand
fj : fixed cost of locating a warehouse at j
Cij k : cost of transporting a quantity of goods from i to j to market k
CAPj : capacityPof warehouse j
capj : CAPj = Dk capacity of the warehouse at location j as a fraction of the
total market demand
13.2.4.2 Model Outputs (Decision Variables)
Xij k : quantity
Pof commodity transported from plant i to warehouse j to market k
xij k : Xij k = Dk quantity transported as a fraction of total market demand
yj : will be equal 1 if warehouse is located at location j; 0 otherwise
13.2.4.3 Objective Function and its Constraints
XXX
X
Min Z D
fj yj ;
cij k xij k C
i
j
(13.25)
j
k
Subject to
XXX
i
j
XX
j
k
XX
i
xij k D 1;
(13.26)
xij k si ;
8i;
(13.27)
xij k dk ;
8k;
(13.28)
j
XX
i
k
xij k cap j ;
8j;
(13.29)
k
xij k 0;
8i; j; k:
(13.30)
Equation (13.26) ensures that the flow through the entire network is equal to the total
demand of all markets. Equation (13.27) ensure that the outflow from a supply point
is less than its supply. Equation (13.28) ensure that the inflow at a market point meets
the demand of that point. Equation (13.29) are non-negativity constraints (Sharma
and Berry 2007).
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Z. Bagherpoor et al.
Now, we list the constraints that link real variables and the 0–1 integer (location)
variables as follows:
XX
xij k cap j yj ; 8j;
(13.31)
i
k
XX
i
X
i
X
k
k
(13.33)
xij k si yj ;
8i; j;
(13.34)
k
X
i
X
xij k C M 1 yj 0;
xij k C M yj 0;
8j;
xij k M yj 0;
8j;
8j ;
(13.35)
k
XX
i
(13.32)
8j;
XX
i
8j;
xij k dk yj ;
XX
i
xij k yj ;
k
xij k M 1 yj dk ;
xij k C M yj 0;
8j; k;
xij k M yj 0;
8j; k;
8j; k;
(13.36)
i
X
i
X
k
X
xij k M 1 yj si ;
xij k C M yj 0;
8j; i;
xij k M yj 0;
8j; i;
8j; i;
(13.37)
k
X
k
yj D Œ0; 1 ;
yj 0;
8j;
8j:
(13.38)
(13.39)
It should be noted that the (13.32) are the “weak relaxation” constraints and (13.33)
form the “strong relaxation” constraints. Since (13.34) are similar to (13.33), they
are also referred to as “strong relaxation” constraints. Equation (13.31) are as “capacity” constraints. Thus, given one kind of linking constraints (that link 0–1 integer
variables and real variables), we get six formulations for SSCWLP (Sharma and
Berry 2007).
1. GG Weak
min (13.25)
s.t. (13.26)–(13.30), (13.32) and (13.39).
13
Warehouse Location Problem
305
2. GG Strong 1
min (13.25)
s.t. (13.26)–(13.30), (13.33) and (13.38).
3. GG Strong 2
min (13.25)
s.t. (13.26)–(13.30), (13.35) and (13.36).
4. GG BigM 1
min (13.25)
s.t. (13.27)–(13.31), (13.35)–(13.36) and (13.38).
5. GG BigM 2
min (13.25)
s.t. (13.26)–(13.30), (13.36)–(13.37), and (13.38).
6. GG BigM 3
min (13.25)
s.t. (13.26)–(13.30), (13.37), (13.33) and (13.38).
Considering two types of linking constraints (capacity constraints along with
weak, strong (two types) and BigM (three types)) in one formulation, we get
six other formulations for SSCWLP.
7. GG Capacity Weak
min (13.25)
s.t. (13.26)–(13.28), (13.30), (13.31), (13.32) and (13.38).
8. GG Capacity Strong 1
min (13.25)
s.t. (13.26)–(13.28), (13.30), (13.31), (13.33) and (13.38).
9. GG Capacity Strong 2
min (13.25)
s.t. (13.26)–(13.28), (13.30), (13.31), (13.34) and (13.38).
10. GG Capacity BigM 1
min (13.25)
s.t. (13.26)–(13.28), (13.30), (13.31), (13.35) and (13.38).
11. GG Capacity BigM 2
min (13.25)
s.t. (13.26)–(13.28), (13.30), (13.31), (13.36) and (13.38).
12. GG Capacity BigM 3
min (13.25)
s.t. (13.26)–(13.28), (13.30), (13.31), (13.37) and (13.38).
13.2.5 Redesigning a Warehouse Network (Melachrinoudis
and Min 2007)
To take advantage of the economies of scale, a growing number of firms have begun
to explore the possibility of integrating supply chain activities. The advent of such
a possibility would necessitate the redesign of a warehouse network. The consolidation or redesign of warehouses can help a firm save transportation, inventory, and
warehousing costs due to economies of scale (Melachrinoudis and Min 2007).
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In this section, a mixed-integer programming model is developed to solve the
warehouse redesign problem.
13.2.5.1 Problem Definition
Typically, a warehouse redesign problem involves the consolidation of regional
warehouses into a fewer number of master stocking points and the subsequent
phase-out of redundant or underutilized warehouses without deteriorating customer
service (see Table 13.1) (Melachrinoudis and Min 2007).
Table 13.1 Differences in strategic network planning among warehouse redesign alternatives
Key checkpoints
Key factors
Retention of existing
warehouses
Closure of existing
warehouses
Establishment of
new warehouses
Which existing
Warehouses are
still viable for
sustaining
customer services?
How to maintain
the best balance
between customer
service and
logistics costs?
How to determine
the level of
redundancy among
existing
warehouses?
Which existing
warehouses are
considered
redundant with
nearby
warehouses?
When to phase-out
redundant
warehouses
without
disruptions?
Closure cost
Relocation/moving
cost
Warehouse
utilization rate
Severance pay for
laid-off warehouse
employees
How to identify
potential sites for
new warehouse
locations?
Customer service
Maintenance cost
Main advantages
Presence of near
customer locations
Stability
Cost saving
potential
Flexibility
Major shortcomings
Inflexibility
Depreciation
Downsizing
Service disruptions
Obsolescence
Low employee
morale
What is the level
of strategic risks
involved in new
start-up
investment?
Are there any
changes in
locations of
customer bases?
Setup cost
Start-up risk
Labor availability
Regional tax
incentives
Local regulation
Proximity to major
customer bases
Capacity
expansion
Overlap risk
High cost of
investment
Learning curve
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Warehouse Location Problem
307
13.2.5.2 Model Assumptions
1. The warehouses are owned by the company (private).
2. When a warehouse is consolidated into another warehouse, its whole capacity is
relocated.
3. The restructuring plan covers a planning horizon within which no substantial
changes are likely in the customer demands and in the transportation infrastructure.
4. Although the company distributes its products in various quantities, customer
orders are aggregated into a single product.
13.2.5.3 Model Inputs
i : index for manufacturing plants i ,
k: index for customers; k 2 K,
j : index for existing warehouses and new candidate sites for relocation and consolidation; j 2 A,
sets: A D E [ N I .j; l / 2 .E A/
E: set of existing warehouses
N : set of new candidate sites for relocation and consolidation.
Vij : Unit production cost (including storage cost) at manufacturing plant i plus
unit transshipment cost between manufacturing plant i and warehouse j
Sj k : Unit warehousing cost at warehouse j and unit transportation cost between
warehouse j and customer k
rlj : Cost of moving and relocating unit capacity l to consolidated site j .l¤j /
cj : Throughput capacity of existing warehouse j
qi : Production capacity of manufacturing plant i
dk : Demand of customer k
fcj : Cost per unit capacity of warehouse j
fmj : Fixed cost of maintaining warehouse j , excluding capacity cost
fsj : Cost savings resulting from the closure of existing warehouse j
tj k : Truck delivery time (in hours) from warehouse j to customer k
: Maximum of customer access time (in hours) from serving warehouses
C.j / D fkjtj k g;
D.k/ D fj jtj k τg:
13.2.5.4 Model Outputs (Decision Variables)
Xj k : Volume of products shipped from warehouse j to customer k
Yij : Volume of products supplied by plant i to warehouse j
(13.40)
(13.41)
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zj l will be equal to 1 if capacity of warehouse j , is relocated to site l, or if
existing warehouse j remains open and otherwise it will be equal to 0
wj will be equal to 1 if a new warehouse established at site j and otherwise it
will be equal to 0
13.2.5.5 Objective Function and its Constraints
P cP
P P
rlj zlj C
sj k xj k C
cl zlj
fj
i 2I j 2A
j
2A
j 2A k2C.j /
j
2A
l2E
l2E
!
#
"
P
P
P m
P m
P
m
s
zlj C fl
zlj ;
fl 1
C
fj zjj C
fj wj
Min
P P
j 2E
vij yij C
P
P
j 2N
j 2A
l2E
fj 2E;j ¤lg
(13.42)
Subject to
X
j 2A
X
i 2I
yij qi
yij D
X
k2C.j /
X
l2E
X
l2E
X
(13.43)
xj k
(13.44)
X
k2C.j /
xj k
X
cl xlj
l2E
8j 2 A;
8j 2 A;
(13.45)
x j k D dk
8k 2 K;
(13.46)
zlj jEj zjj
8j 2 E;
(13.47)
zlj jEj wj
8j 2 N;
(13.48)
j 2D.k/
X
8i 2 I;
zlj 1
8l 2 E;
(13.49)
xj k 0 8j 2 A; k 2 K;
yij 0 8i 2 I; j 2 A;
zlj ; wj 2 f0; 1g 8l 2 E; j 2 A:
(13.50)
j 2A
(13.51)
(13.52)
Equation (13.42) minimizes total supply chain costs comprised of production, transportation, warehousing, and relocation costs, while maximizing the cost savings resulting from the closure or consolidation of redundant warehouses. Equation (13.43)
assure that the total volume of products shipped to warehouses do not exceed the
capacity of the manufacturing plant supplying those products. Equation (13.44) insure that the total volume of products supplied by the manufacturing plant to each
warehouse matches the total volume of products shipped from that warehouse to
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its customers. In other words, inbound shipping volume for each warehouse must
be equal to its outbound shipping volume. Equation (13.45) insure that the total
volume of products shipped to customers can not exceed the throughput capacity (after consolidation) of the warehouse serving them. Equation (13.46) ensure
that the customer demand is satisfied. Equation (13.47) state that the current resources (i.e., capacity) of an existing warehouse can not be consolidated into another
existing warehouse, unless such a consolidated warehouse remains open. Equation (13.48) state that the capacity of an existing warehouse cannot be relocated to
a new site, unless a warehouse is established at the new site. Equation (13.49) consider various options for an existing warehouse j . These options include: keeping
the warehouse open .zjj D 1/, consolidating its capacity into another existing warehouse j 2 E; j ¤ l.zlj D 1/, relocating
its capacity
to a new site i 2 N.zlj D 1/,
or closing the existing warehouse l zlj D 0 8i 2 A . Equations (13.50)–(13.51) assure the non negativity of decision variables xj k ; yij . Equation (13.52) state that zi l
and wj are zero-one variables (Melachrinoudis and Min 2007).
13.3 Solution Methods
To solve small WLPs, integer programming optimization methods are used. However, for larger problems, heuristic methods or Meta heuristic methods are utilized.
13.3.1 Exact Solution Methods
The UWLP has attracted considerable attention in mathematical programming.
Khumawala (1972) developed a branch and bound algorithm for uncapacitated
warehouse location problems. Akinc and Khumawala (1977) developed a branch
and bound algorithm for capacitated warehouse location problems. Also based
on dual and primal–dual approaches branch and bound algorithms were developed (Erlenkotter 1978; Korkel 1989). Dual-based and primal–dual algorithms are
very effective for the UWLP on the OR Library benchmarks. However, they experience significant difficulties and exhibit exponential behavior on the M instances
(Kratica et al. 2001). These instances stimulate real situations, have a large number of suboptimal solutions, and exhibit a strong tension between transportation
and fixed costs, which makes it difficult to eliminate many warehouses early in the
search (Michel and Hentenryck 2004).
Lagrangian relaxation is a well known technique for calculating lower bounds
in branch and bound algorithms. In order to solve CWLP, Geoffrion and Graves
(1974), Geoffrion and Nauss (1977), and Christofides and Beasley (1983) have used
a Lagrangian relaxation of the demand constraints.
Baker (1986) proposed another Lagrangian relaxation for the demand constraints
to obtain lower band. In order to converge to optimal set of multiplier, he proposed
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a heuristic algorithm. The computational result shows that the algorithm based on
partial dual approach has performed well and generally produced results which are
as good as, or better than those of the previously accepted based on a Lagrangian
relaxation of demand constraints.
Beasley (1988) presented a lower bound for the capacitated warehouse location
problem based upon Lagrangian relaxation of a mixed-integer formulation of the
problem. Feasible solution exclusion constraints are used together with problem reduction tests derived from both the original problem and the Lagrangian relaxation.
By incorporating the lower bound and the reduction tests into a tree search procedure solving problems involving up to 500 potential warehouse locations and 1,000
customers is possible.
Subgradient optimization is a popular method of finding a good set of multipliers for use in Lagrangian relaxation. Baker and Sheasby (1999) developed a
model for obtaining faster convergence. The method was developed within a study
of the generalised assignment problem (GAP) and its application to vehicle routing.
The method was applied to benchmark capacitated warehouse location problems
(CWLPs).
Lee (1993) presented an algorithm for solving multi products capacitated warehouse location problem based on cross-decomposition, to reduce the computational
difficulty by incorporating Benders decomposition and Lagrangian relaxation. The
algorithm solves problems of practical sizes in acceptable times.
Sweeny and Tatham (1976) presented a mixed integer programming formulation for the single period warehouse location model with a dynamic programming
procedure for finding the optimal sequence of configurations over multiple periods.
Kelly and Marucheck (1984) proposed an algorithm for dynamic WLP. First, the
model is simplified and then a partial optimal solution is obtained through iterative
examinations by both upper and lower bounds on savings realized if a site is opened
in a given time period. A complete optimal solution is obtained by solving the reduced model with Benders’ decomposition procedure. The optimal solution is then
tested to determine which time periods contain tentative decisions that may be affected by post-horizon data. The relationship between the lower (or upper) bounds
utilized in the model simplification time period is analyzed.
Dupont (2008) presented a branch and bound algorithm for a facility location
problem with concave site dependent costs.
Sharma et al. (2007) presented new formulations and relaxations for single stage
capacitated warehouse location problem. In this paper different formulations and
relaxations were compared with each other.
13.3.2 Heuristic and Metaheuristic Methods
Because of the nature of many warehouse location problems, the exact methods can
not be used to solve them, so heuristic & Meta heuristic methods can be used.
Genetic algorithms have been shown to be very successful on the UWLP. In a
series of papers spanning over several, Kratica et al. (2001) have shown that genetic
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algorithms find optimal solutions on the OR Library and the M instances (whenever
the optimal solutions are known) with high frequencies and very good efficiency.
Their final algorithm uses clever implementation techniques such as caching and
bit vectors to avoid recomputing the objective function which is quite costly on
large-scale problems. Also, the speed-up of the genetic algorithm over mathematical programming approaches increases exponentially with problem size on the M
instances.
Various heuristic search algorithms have also been proposed but have been less
successful. Alves and Almedia (1992) presented simulated annealing algorithms
which produce high-quality solutions but are quite expensive in computation times
(Michel and Hentenryck 2004).
Al-Sultan and Al-Fawzan (1999) presented a tabu-search algorithm. The algorithm generates 5n neighbors at each iteration and moves to the best neighbor which
is not tabu and improves the current value of the objective function. Each of these iterations takes significant computing time, which limits the applicability of the algorithm. Michel and Hentenryck (2004) presented another tabu search algorithm. The
algorithm uses a linear neighborhood and essentially takes O.m log n/ time per iteration. It finds optimal solutions on the OR Library and the M instances (whenever
the optimal solution is known) with high frequencies. It also outperforms the state of
the-art genetic algorithm of Kratica et al. (2001), both in efficiency and robustness.
Kuehn and Hamburger (1963) and Whitaker (1985) presented Greedy- Bump
and Shift (Interchange) heuristics algorithms. In the Greedy process, warehouses
are located at the most economical positions, one at a time, until no additional warehouses can be added without increasing the total cost. In the Bump process, those
warehouses that became uneconomical as a result of the placement of subsequent
warehouses are eliminated. In the Shift process, a warehouse is shifted to another
potential location in the same territory if the relocation causes a reduction of the
total cost.
Heras and Larrosa (2006) analyzed the effect of heuristic orders at three levels of
increasing overhead: (a) compute the order prior to search and keep it fixed during
the whole solving process (this is called a static order), (b) compute the order at
every search node using current sub problem information (this is called a dynamic
order) and iii) compute a sequence of different orders at every search node and
sequentially enforce the local consistency for each one (this is called dynamic reordering).They performed experiments in three different problems: Max-SAT, MaxCSP and warehouse location problems. They did not find an alternative better than
the rest for all the instances.
Lee (1996) proposed an optimal solution algorithm based on the cross decomposition method for multi type capacitated distribution center location problem. The
cross decomposition method can be applied to solve this problem because both the
primal and dual sub problems are relatively easy to solve and both sub problems
quickly converge. Tight lower and upper bounds can be obtained in just a few iterations of the proposed algorithm. This algorithm can be used as a heuristic which
produces not just a feasible solution but also a confidence interval to measure the
quality of the solution. The algorithm was implemented in FORTRAN.
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13.4 Case Study
In this section, we describe some applications of the warehouse location problem in
real world.
13.4.1 Redesigning a Warehouse Network
(Melachrinoudis and Min 2007)
The model which was described in the last section is a case study of a firm which
plans to redesign its warehouse network and reduce its total logistics costs. Beta
has its main manufacturing plant in Terre Haute, Indiana, and currently operates
21 warehouses to serve a total of 281 customers scattered around the United States
and Canada. The warehouse consolidation problem (WCP) facing Beta differs from
the classical warehouse location problem in that the former is primarily concerned
with determining which existing warehouses to keep open, which new warehouses
to establish, and which warehouses to phase-out among the existing locations,
whereas the latter is primarily concerned with selecting the optimal site among
the alternatives of new locations. In this problem, 25 warehouse locations, including 4 candidate sites were existed. The target problem was solved and analyzed by
LINGO optimization software.
Out of the 281 customers, only four have non-unique warehouse assignments and
each one of those four customers receives partial demand allocations from exactly
two warehouses.
13.4.2 Warehouse Location Problems for Air Freight Forwarders
(Wan et al. 1998)
The move of Hong Kong International Airport from the city centre to a suburban
area in July 1998 provided sufficient capacity to meet the increasing demand of
passenger and air-cargo flows in Hong Kong in the foreseeable future. However, the
move has had adverse side effects such as causing the readjustment of many existing
systems and creating many imminent strategic problems. One of such problems is
the warehouse location of freight forwarders: They have to decide whether they
should locate their warehouses in the new airport, in current locations, or in new
locations somewhere in the city.
In this case, air freight forwarders had then been facing the decision on whether
they should (1) move their warehouses to the new airport, (2) stay in the current
locations, (3) move the warehouses to new locations in the city, (4) keep the current
warehouses and add an additional one in Chek Lap Kok (CLK), and (5) move the
current warehouses to new locations in city and add one in CLK.
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With the special structure of a toll bridge for the (CLK) airport, this problem
is in exactly the same form as the one-median location problem on a tree network
by using a decomposition approach. The problem can be solved by the Chinese
algorithm or its modified version, the majority algorithm.
References
Akinc U, Khumawala BM (1977) An efficient branch and bound algorithm for the capacitated
warehouse location problem. Manage Sci 23(6):585–594
Alves ML, Almedia MT (1992) Simulated annealing algorithm for simple plant location problems.
Revista Investigacao Operationanl 12
Al-Sultan KS, Al-Fawzan MA (1999) A tabu search approach to the uncapacitated facility location
problem Ann Oper Res 86:91–103
Baker BM (1986) A partial dual algorithm for the capacitated warehouse location problem. Eur J
Oper Res 23:48–56
Baker BM, Sheasby J (1999) Accelerating the convergence of subgradient optimization. Eur J Oper
Res 117:136–144
Beasley JE (1988) An algorithm for solving large capacitated warehouse location problems. Eur J
Oper Res 33:314–323
Christofides N, Beasley JE (1983) Extensions to a Lagrangian relaxation approach for the capacitated warehouse location problem. Eur J Oper Res 12:19–28
Dupont L (2008) Branch and bound algorithm for a facility location problem with concave site
dependent costs. International Journal of Production Economics 112(1):245–254
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26(6):992–1009
Geoffrion AM, Graves GV (1974) Multicommodity distribution system design by benders decomposition. Manage Sci 20(5):882–844
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WCSP. J Heuristics 12:287–306
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problem. J Oper Manage 4(3):279–294
Khumawala BM (1972) An efficient branch and bound algorithm for the warehouse location problem. Manage Sci 18(12):585–594
Korkel M (1989) On the exact solution of large-scale simple plant location problems. Eur J Oper
Res 39:157–173
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genetic algorithm. Oper Res 35:127–142
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9:643–666
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Nuass RM (1978) An improved algorithm for the capacitated facility location problem. J Oper Res
Soc 29(12):1195–1201
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Chapter 14
Obnoxious Facility Location
Sara Hosseini and Ameneh Moharerhaye Esfahani
In general, facilities are divided in two groups, the first one are desirable to the
nearby inhabitants which try to have them as close as possible such as hospitals,
fire stations, shopping stores and educational centers. The second group turns out
to be undesirable for the surrounding population, which avoids them and tries to
stay away from them such as garbage dump sites, chemical plants, nuclear reactors,
military installations, prisons and polluting plants. In this sense, Daskin (1995) discussed that Erkut and Neuman in 1989 distinguished between Noxious (hazardous
to health) and Obnoxious (nuisance to lifestyle) facilities, although both can be simply regarded as Undesirable. Moreover, in the last decade, a new nomenclature has
been developed to define these oppositions: NIMBY (not in my back yard), NIMNBY
(not in my neighbor’s back yard), and NIABY (not in anyone’s back yard).
We survey two types here:
Dispersion problems: in which there are only facilities to be located in such
a way as to affect each other the least possible. Usually, Cases such as sales
representative of some organizations are addressed in this type. The mother organization intends to minimize the comparison among its sales representative
and cover more areas via maximizing distance between them. Note that in some
cases, investors interestedly try to minimize the distance between their sales representative, for instance you can imagine two restaurants with same ownership,
although they are placed in one area for making more customers, not only they
don’t lose their customers, but also make more profits.
Undesirable facilities problems: Because of sanitation, security or people’s
welfare, some facilities are not desirable and we try to far them away from
demand centers. Despite these undesirable facilities being necessary, in general,
to the community, for instance garbage dump sites, the location of such facilities might cause a certain disagreement in the population. High opening cost
of these facilities beside high total transport cost of undesirable materials make
decision makers to solve a bi-objective problem (to minimize transport cost and
to maximize distance from demand centers). Furthermore, to maintain security
and because of transportation of hazardous materials generated at these centers,
transport routing should be pointed out. About 5% of GNP of more developing
countries is consumed to campaign with growth of air pollution, noises, accidents
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 14, c Physica-Verlag Heidelberg 2009
315
316
S. Hosseini and A.M. Esfahani
Location of Dispposal center?
Routing of
HazMat residue?
Location of Treatment
centre?
Designing of Emergency Response team?
Routing of
HazMat
waste?
Chemical Material factory
HazMat wastes
Fig. 14.1 The example of Obnoxious problem
and traffic problems. Nevertheless, the advantages of a convenient transportation
system are much more than these costs, so, existence of such system appears
essential. An example of obnoxious problem is shown in Fig. 14.1.
14.1 Applications and Classifications
14.1.1 Applications
Some Applications of these problems can be as follows:
Locating chemical plants
Locating nuclear reactor plants
Locating pollution plants
Locating garbage dump sites
Routing and location – routing of these applications are considered in recent
years.
14.1.2 Revolution of Undesirable Facility Problem
Obnoxious facilities location problem was first introduced at 1975 by Goldman
and Dearing (cited in (Rakas et al. 2004) and first solution was presented at 1978
by Church and Garfinkel in O.mnlogn/ time. McGinnis and White (1978) first
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introduced a bicriteria problem, using the minisum–minimax criteria. Drezner and
Wesolowsky (1983) first introduced the rectilinear maximin problem for locating
an obnoxious facility. Ting devised a linear time algorithm for the 1-maxisum
problem on tree networks at 1984. Mehrez et al. (1986) suggested an improvement of previous algorithm, based on bounds, which reduced the size of model.
Batta and Chiu (1988) present a hazardous material transportation model on a network in which sum of distances of container within a threshold limit from demand
centers, is minimized. Ratick and White (1988) develop a multiobjective model,
minimizing costs and people’s repulsion simultaneously. In this case Wayman and
Kuby (1994) present a model with three objectives: Minimization of risk and
cost and maximization of equity. Gopalan et al. (1990) model the equity of the
HazMat transport risk and present a heuristic solution. At 1995, Melachrinoudis
and Smith extended an algorithm with “m” edges and “n” nodes as obnoxious
facilities, assuming Euclidean spatial distances. But that was not the end of researches and different types of these problems with various assumptions were
reviewed at those years. Marianov and ReVelle (1998) model a linear vehicle routing problem, minimizing cost and risk simultaneously. Erkut and Verter (1998)
model transport risk completely and review various methods. Rogers (1998) propose a model to minimize the global repulsion of the inhabitants of the region.
Results show that insecurity of region is the most important reason for negative reaction of people. Giannikos (1998) use the goal-programming technique
for obnoxious facility location-routing problem. In 1999, Melachrinoudis developed a bicriteria location model for locating a new semi-obnoxious1 facility in
an existing layout which is solved by an adaptation of the Fourier–Motzkin elimination method. Zagrafos and samara presented a methodological framework for
developing a decision support system (DSS) for hazardous materials emergency
response operations (Erkut and Alp 2007). Cases such as Routing emergency response teams and evacuation of region from people and cars were considered in
this paper. Fernandez et al. (2000) address a continuous location within a given
geographical region of an undesirable facility considering environmental aspects.
Akgün et al. (2000) review cases which decision maker seeks for various transport routes. Zhang et al. (2000) devise a routing hazardous materials algorithm,
using GIS2 technique and a Gaussian Plume model for their dispersion model.
One of its applications is to transport urban hazardous wastes. Katz et al. (2002)
propose a maximin model for multi-facility location. Here the distance between
each facility and demand centers should be considered beside distances among
facilities.
Cappanera et al. (2004) addressed the problem of simultaneously locating obnoxious facilities and routing obnoxious materials between a set of built-up areas
and the facilities. Rakas et al. (2004) propose a multiobjective model with fuzzy
data for the location of undesirable facilities. Castillo (2004) reviewed the issues
related to hazmat transportation risks and discussed the modeling tool available for
1
2
These facilities service to demand points; nevertheless, they are undesirable for them.
Geographic information system
318
S. Hosseini and A.M. Esfahani
routing problem of hazardous materials. Also, a route optimization model is developed as a decision support tool. Rodriguez et al. (2006) considered Euclidean travel
distances for obnoxious facilities location problem. Colebrook et al. (2005) have
developed a new bound and a new O(mn) algorithm to solve the network maxian
problem. Erkut and Ingolfsson (2005) formulated the tree design problem as an integer programming problem aiming at trade off between cost and risk. Dı́az-Báňez
et al. (2005) computed shortest paths for transportation of hazardous material in
continuous spaces. They proposed an approximate algorithm based on the bisection
method and reduced the optimization problem to a decision problem, where one
needs to compute the shortest path such that the minimum distance to the demand
points is not smaller than a certain amount r. Pisinger (2006) present an exact algorithm for finding a number of fast upper bounds for P -dispersion problems in which
these bounds can be derived in O.n/ time. Bell (2006) considered mixed rout strategies for the risk-averse shipment of hazardous materials. He believes, for repeated
shipments on the arc with unknown incident probabilities, the safest strategy is to
use a mix of routes.
Berman et al. (2007) extended an optimum model for designing emergency response networks for hazardous materials transportation that would be applicable for
large-scale problems.
Alumur and Kara (2007) develop a comprehensive multi objective model for the
hazardous waste location-routing problem that covers various problems of this field.
For exact solving in a case study, GIS technique and CPLEX software were used.
Akgün et al. (2007) consider effects of weather systems on HazMat transportation.
They introduced weather system effects as circle areas with different probabilities
on accident rate, speed, delays and others. They use time dependent shortest path
problem in form of an exact model and four heuristic models. They concluded in
such weather systems accident rates can be double. Carotenuto et al. (2007) model
a hazmat shipment routing and scheduling problem. They develop a tabu search
algorithm to assign a minimum and equitable risk route to each hazmat shipment
and schedule these shipments on the assigned routes.
14.1.3 Classification of Undesirable Facility Problems
Area solution: Discrete, Continual
Number of Objective function: Multiobjective, Single objective
Category: Location, Routing, Location-Routing
Determine the number of nodes: Exogenous, Endogenous
Facility capacity: Unlimited, Limited
Technology assignment: non-assignment, assignment
Number of type of hazardous waste: one type, more than one type
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14.2 Models
14.2.1 Dispersion Problem (Daskin 1995)
In this problem, suppose that there are some candidate centers for installation facilities. The purpose is to find P points for locating in order that maximize the
minimum distance between located facilities. In other word, the decision maker
knows the number of locatable facilities and candidate centers then, he tries to select set of points that maximize the minimum distance between them.
14.2.1.1 Model Assumptions
The problem is investigated on the discrete network and nodes are representative
of candidate points.
This network is general or it’s possible to have loops.
The facilities often are general. In other word, investor (government) focuses on
keeping away facilities from population centers. In these cases, environmental
costs and some other ones are more important than installation costs.
The numbers of facilities are predetermined.
All of facilities are obnoxious and should be kept away from population centers.
All of facilities are similar and all of services are equal too.
The installation cost is not considered.
The facilities replacement is not considered.
All of parameters are deterministic.
The problem is formulated in static form. It means that, the problem’s inputs are
not dependent to time.
14.2.1.2 Model Inputs
dij : it is distance between i and j
M : it is large elected digit and it is often larger than the maximum distance
between candidate points
P : the number of facility to locate
D: the minimum distance between facilities
14.2.1.3 Model Outputs (Decision variables)
Xj : 1 if facility is installed in node j ; 0 otherwise
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S. Hosseini and A.M. Esfahani
14.2.1.4 Objective Function and its Constraints
Max D
Subject to
X
j
Xj D P
(14.1)
(14.2)
D dij C.M dij /:.1 Xi /C.M dij /:.1 Xj / 8j I 8i j: (14.3)
Xj 2 f0; 1g
8j
(14.4)
Equation (14.1) maximizes the minimum distance between facilities. Equation
(14.2) represents that number of located facilities, should be P .
Equation (14.2) determines the minimum distance along distance between each
pair of facilities and finally the third constraint shows that Xj is 0, 1 variable.
To achieve better perception of (14.2), consider the following explanations:
If Xi D Xj D 1 (install facility in node i; j ), so D dij and it means that
the minimum distance between each pair of facilities can’t be more than from
distance between facilities, certainly it is obvious.
If X i D 0; Xj D 1 or Xi D 1; Xj D 0 means that, only one facility is located
in node i; j .
So, (14.2) will be D M . It’s clear because M is more than all of couple
distances.
Now, consider the case that Xi D 0; Xj D 0. In this case, (14.2) will be M
M axi; j fdij g and, it’s reasonable too.
Researchers didn’t focus much on relationship between dispersion models and
other models. As one example in this case, we can refer to Kuby (1987). He has
illustrated the relationship between P -Dispersion and (P -1) Center.
14.2.2 Undesirable Facility Location Problem (Daskin 1995)
Locating some facilities such as fire stations and shopping centers, are considered
in recent decades. For example in locating a shopping center, we know that it should
be near and available to population centers. But it’s so important that, we consider in
locating factors such as decreasing noise, heavy traffic in that center, related crimes
and people’s tranquility of mind. These factors make problem more complex. The
investigated model in this section is P -Median transformed model and for the reason that facilities are undesirable, the objective function is maximization. In Median
problems because of desirability of facilities such as hospitals and fire stations, the
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objective function is minimization. But in Maxisum models, the subject is locating
facilities such as incinerators that, decision maker attend to far it from demand centers. Of course, it is obvious that we can’t eliminate transportation cost.
14.2.2.1 Model Inputs
P : the number of facilities should be located
14.2.2.2 Model Outputs (Decision Variables)
hi : demand of node i
Distance between demand nodes i and candidate place j dij :
Xj : 1 if facility is located at node j ; 0 otherwise
Yij : 1 if demand of node i was provided by located facility at node j ; 0 otherwise
14.2.2.3 Objective Function and its Constraints
Max
XX
i
X
j
X
j
hi dij yij ;
(14.5)
j
Yij ij D 1 8i;
(14.6)
Xj ij D P;
(14.7)
Yij Xj 8i; j;
Xj D 0; 1 8j;
Yij D 0; 1 8i; j:
(14.8)
(14.9)
(14.10)
Equation (14.5) minimizes the weighted sum of demands and distances.
The (14.6) represent that demands of all nodes should responded by just one
facility. The (14.7) considers the number of all facilities equal P . Equation (14.8)
ensures that if any facility is located at node j , demand of this node should responded by itself. Equations (14.9) and (14.10) are integer limitations.
Example 14.1 was illustrated that the above model in reality doesn’t express nature of mentioned problem.
14.2.2.4 Example 14.1
A,B,C,D are demand centers, digits in boxes are demand amount of each place and
on the line numbers represent distance between places that are connected with line
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S. Hosseini and A.M. Esfahani
Fig. 14.2 Example 14.1
A
3
3
B
4
C
4
2
5
D
5
Table 14.1 Nearest facility to each center
Demand of node i m D 1 M D 2 m D 3 m D 4
A
A
B
C
D
B
B
A
C
D
C
C
B
D
A
D
D
C
B
A
together (Fig. 14.2). After solving the problem with above way, the followed yield
was resulted:
XA D XD D 1
XB D XC D 0
YAD D YBD D YCA D YDA D 1:
Objective function value is 146 that seem so appropriate. In this example, demand
of center A was responded by center D. Of course, it is not reasonable that demand
of center A was responded by center D when there is a facility at node A. but the
above model results center A was responded by center D. This solution obviously
has bad effect on C, D nodes.
Therefore, the model should have some corrections or reformations to meet reality. After adding some constraints into the previous model, the corrected model will
be as followed:
XŒmi C
m
X
Yi Œki 0
8i I m D 1; 2; : : : ; N 1:
(14.11)
kD1
[m]i represents the mth candidate place from demand node i (see Table 14.1). To
illustrate complete formulation of model, additional constraint was explained below:
XB YBB ;
(14.12)
XA YBB C YBA ;
(14.13)
XC YBB C YBA C YBC ;
(14.14)
XD YBB C YBA C YBC C YBD :
(14.15)
Equation (14.12) ensures that, if a facility is located at node B, demand of node
B was provided by node B.
Equation (14.13) says if a facility is located at node A, demand of node B can be
provided by node A or B. Of course, if a facility is located at node B, demand of
14
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323
Table 14.2 Compared Maxisum and median models
P
Model MaxiSum
Center
A
A,B
A,B,C
A,B,C,D
1
2
3
4
Objective function value
86
53
25
0
Model median
Center
B
B,D
A,B,D
A,B,C,D
Objective function value
62
17
8
0
node B was provided by node B and (14.13) will be satisfied. Other constraints are
justified with this way.
The complete formulation of Example 14.1 and final results were brought.
Incidentally, comparison between Median and Maxisum models was brought in
Table 14.2.
M ax 9 YAB C 21 YAC C 36 YAD C 12 YBA C 16 YBC C 36 YBD C 14 YCA
C8 YCB C 10 YCD C 60 YDA C 45 YDB C 25 YDC :
Subject to
XA C XB C XC C XD D P
YAA C YAB C YAC C YAD D 1 YCA C YCB C YC C C YCD D 1
Y BA C YBB C YBC C YBD D 1 Y DA C YDB C YDC C YDD D 1
2
3
YAA XA YBA XA YCA XA YDA XA
6 YAB XB YBB XB YCB XB YDB XB 7
6
7
4 YAC XC YBC XC YC C XC YDC XC 5
YAD XD YBD XD YCD XD YDD XD
2
3 2
3
X A C YAA 0
X B C YBB 0
4 XB C YAA C YAB 0
5 I 4 XA C YBB C YBA 0
5I
XC C YAA C YAB C YAC 0
XC C YBB C YBA C YBC 0
2
3 2
3
X C C YC C 0
X D C YDD 0
4 XB C YC C C YCB 0
5 I 4 XC C YDD C YDC 0
5
XD C YC C C YCB C YCD 0
XB C YDD C YDC C YDB 0
XA ; XB ; XC ; XD D 0; 1
All Yij 0
14.2.3 Hazardous Materials Routing Problem
Hazardous materials that are briefly called HazMats, consist of explosives, flammable substances, oxidizing substances, poisonous substances, radioactive materials,
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S. Hosseini and A.M. Esfahani
infected materials, corrosives and hazardous wastes (Erkut and Verter 1995). Basically these materials are used in many facilities and processes in the industry such as
petroleum refineries, chemical plants and nuclear reactors or hospital’s wastes. Due
to the nature of hazmats, many risks related to production, storage and transportation of these materials could arise. In the case of transporting hazmats, the selection
of routes considering economic and risk issues is a very important problem and
needs to an integrated transport and risk management. Therefore, the term “optimal
routes” means the route that best fulfils the objectives by the stakeholders associated
with transporting and distributing of HazMats within the study area.
It’s not possible to completely avoid risks, nevertheless risk management activities such as mitigation and prevention would be useful. These activities are reviewed
as two groups: proactives and reactives. Some cases such as driver training, restriction of driving hours and container specifications are considered in proactives and
some cases such as firefighting, emergency response and evacuation of region from
inhabitants and cars are of reactives.
To understand its importance, we mention some examples of transporting
HazMats:
Every year, approximately, 4 billions shipments of these materials are transported, in other word, 20% of traveling trucks (one in every five) in USA carry
HazMats.
The difference between the transportation of HazMats and any other ones is, in
this case any accident may cause serious damages for people, animals and plants.
In 1976, as the explosion of a small chemical plant in small city of Seveso and
released Dioxin, 190 persons damaged seriously and thousands of birds were
killed.
Afghanistan, 1982, the explosion of a truck, carrying gasoline, in a tunnel killed
2,700 persons.
India, 1984, the Methil-Iso-Cianate leakage in Bhopal area, killed 400 persons.
Annually, about 2.5 million tons of garbage are generated in Tehran that management cost of these huge volumes is 600 billion rials. It is evident that applying
a quick and efficient transportation system is one of the most important ways for
reduction of such costs. From other hand, management of hospital-garbage is one
of the most important problems that exist in our country.
Basically, routing of HazMats is done by governments, also because of high sensitivity of society toward these accidents, fixed costs of opening facilities are not be
considered in most of these models, in contrast, risk and equity assessment methods are always be considered. Routing of HazMat is as important as locating them,
because these two issues are relevant to each other strictly. In recent decades, researchers had more studies in this matter.
One of the recent research studies was published by Akgün et al. (2007) investigated HazMat routing problem with weather systems considerations. Weather
system effects on accident probability, speed, time duration, risk and cost related
to HazMat transportations. For example it can lead to change routs, reduce speed,
delaying shipment, increase parking times and other issues. Due to the dynamic
14
Obnoxious Facility Location
325
nature of weather system and hence the time varying nature of the weights on links
in the transportation network, time-dependent shortest paths problems (TDSPP)
should be considered. TDSPP was first introduced by Cook and Halsey with using
finite departure times from every node to the destination. TDSPP approach was
developed continuously after that. For example, Ziliakopoulos and Mahmassani
introduced an algorithm that calculates the TDSP’s from all nodes to the destination for every time step over a given time horizon. Cia et al., Erkut and Alp and
Nozik researches were focused on TDSPP. Akgün et al. (2007) focused on the effect of weather systems on hazmat routing. They characterized the time dependent
attributes of a link due to movement of the weather system. They aimed to find
a least risk path for hazmat transportation under variable weather system. They
applied one exact model for routing, Time-dependent shortest Path problem, and
four heuristic model such as k-shortest path heuristic (KPATH), dissimilar path
heuristic (DISSIM), Iterative weather system heuristic (IWS) and finally Myopic
shortest path heuristic (MYOPIC). They compared exact model with heuristics and
heuristics with each others too. They consider a circle area for weather system that
can move by time and weights that in some methods were constant and in some
others were inconstant.
14.2.3.1 Risk Evaluation for Hazardous Waste Transportation
Risk management activities could result in an accurate assessment of the risks and
therefore it would be possible to create strategies for reduction of risk level to its
lowest level. We have expanded models with different conditions in subject of risk
evaluation. Erkut and Verter (1998) suggested much evaluation risk models. The
simplest model in this category yields the product of the accident consequences, the
probability of a hazardous waste accident, activity volume.
Risk D accident consequences probability of a hazardous
waste accident activity volume:
Generally, the accident consequences are measured by population exposure and
the accident probability depends on material type and rout nature. In practice, the
rout is divided to equal segments (k/m or mile) then; the consequences probability
is calculated for all segments by population exposure in accident place.
In this equation assumed that, if an accident happens at center A, people are influenced in circle with radius of R. The assumed radius is considered 0–7 km and the
consequences accident is measured through the Worst-case. It means that without
considering geographical, climate and topography conditions, all exposure people
in assumed circle will perish. Other developed way for calculating the consequences
accident, presented by Raj and Glikman who discussed about effects of wind and air
instability in the consequences accident in 1991 (see Fig. 14.3). It made the model
so complex, but so similar to reality.
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S. Hosseini and A.M. Esfahani
A
B
Fig. 14.3 Accident worst-case
The accident probability of hazardous materials usually ranges from 0.1 to 0.8
in each million mile. Of course it depends on the nature of rout, culture problems,
transportation time (day, night) and security cases. Undoubtedly, to estimate realistic
probability we need accurate information. After recognizing accident probability
and consequences for each segment (with assumption probability is equal in all
segments of rout and, stopping the travel if any accident happened) the formulation
of rout risk will be:
PR D PC C .1 P/ PC C .1 P/2 PC C : : : C .1 P/n1 PC:
(14.16)
P is accident probability and C is accident consequences. According to past
paragraph, the accident probability is nearly 1 million in each mile, therefore we
can eliminate terms with Pn for n > 1 and simplify the above formulation:
PR D
n1
X
.Pj Cj /:
(14.17)
J D1
Erkut and Verter (1998) confirmed that effect of the assumption “negligibility the
product of more than two probabilities” is insignificant especially for long routs.
ReVelle et al. (1991) believed that in transportation of hazardous material, population was exposed in dangerous consequences. So, in the cases such as atomic
waste transportation, the influenced population should be minimized.
Saccomanno and Chan (1985) pretended that in the reason of the accident consequences are so violent, the problem should be modeled through minimizing the
accident probability. It means that, even one accident is not forgivable. Consequently, the accident probability should be nearly zero.
Some researchers claimed that people count an accident with low probability
and high consequences more unreasonable than an accident with high probability
and low consequences. So, in risk evaluation it is bettor to apply P Cq model for
q > 1 because this model increases the value of accident consequence and results
solution with less consequences. There was a difficulty even after Abkowitz’s model
and this was an assumption that the rout was exposed by accident, was usable after accident happened. But it is clear, after accruing accident the rout should be
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Obnoxious Facility Location
327
Table 14.3 Different types of risk evaluation models (Erkut and Ingolfsson 2005)
Model
Traditional risk
Formulation
P
TR .P/ D
Pi Ci
i2p
P
PE .P/ D
Ti
i2p
P
IP .P/ D
Pi
i2p
P
Pi Ci
PR .P/ D
Population exposure
Incident probability
Perceived risk
CR .P/ D
Conditional risk
i2p
P
MV .P/ D
Mean-variance
DU .P/ D
Disutility
Pi Ci
i 2P
P
i 2P
P
i2P
P
Pi
.Pi Ci C kPi Ci2 /
Pi .exp.kCi / 1/
i2P
CR .P/ D maxi2P Ci
Minimax
obstructed temporarily or permanent and occasionally the optimum rout should be
designed again. The rout obstruction cost and also redesigning new rout are such
subjects that lead to brilliancy for importance of correct routing for hazardous material transportation.
Sivakumar et al. (1993, 1995) expanded the conditional risk model that, the consequences minimized in first accident. The final discussion in modeling of hazardous
material transportation risk is looping proposition. It was discussed in Erkut and
Verter (1998).
Table 14.3 reviewed modeling method for hazardous material transportation risk.
14.2.3.2 Risk Equality
Certainly in spite of all discussed conditions, we can not find a rout or place without
any risk or any effect on the environment. Consequently, the decision maker should
divide the risk among segments and reach to minimum variance. This subject is
named “risk equality” in hazardous waste literature and can formulate in different
ways. The general type of it was followed here:
Risk Equity D
X
y
a y.N y/P .y/I a < 0:
(14.18)
In (14.18), N represents population of a center, y determines the number of
accident casualties and constant a is less than zero. It is simply provable that, minimum equality will be obtained, when the number of accident casualties is half of
population. With moving away from each side, the equality increases and in ultimate it reaches to zero value and it is best state. For example in an accident when
328
S. Hosseini and A.M. Esfahani
doesn’t take place any casualties or maybe when all population of center perishes,
the maximum equality is obtained. The fuzzy risk equality modeling seems remarkable direction for future researches.
14.2.3.3 Designing Optimum Routes for Material Transportation
Nowadays with improvement in technology, the hazardous transportation problem is as significant for governments as they establish inflexible regulations for
it. These regulations include prohibiting hazardous material transportation in centers, different times, and beneficiaries’ commitment for at most transportation safety.
Simultaneously the governments design optimum routs and monitor material transportation.
We can classify the problems in designing optimum transportation routs that increases problem. It seems that the best possible solution may be the transportation
at farther distances from the demand centers; however, its extra costs are imposed
to consumers. In the other word, the transport organizations try to reduce the
transport costs, however, the governments emphasize to the reduction of transport risks.
On the other hand, the governments tend to design routs with lowest risk, but they
can’t force transport organizations to drive in these routs, furthermore, designing
a unique path for HazMat transport is not applicable.
Even if that would be possible to design optimum routs for each OD (origindestination) pair, there’s not any assurance for optimization of set of these routs.
Solving a real problem requires so many nodes and edges which is not simply
obtainable with present solution techniques.
Erkut and Alp (2007) proposed using an optimum tree network for simplifying
of solution. In the better word; it’s not permitted to pass through any replaced path.
Their considered network is similar to the “optimum communication spanning tree
network” (OCST), where the objective is to minimize the weighted sum of the length
of the paths between each pair of nodes on a tree.
Let N be the set of nodes on a city road network and E be the set of undirected
edges .i; j / that connect these nodes. Let i < j for the edges in set E. let C be the
set of pairs of nodes .u; v/ such that u < v, where there exists a positive shipment
between nodes u (origin) and v (destination). Moreover, let take:
14.2.3.4 Model Inputs
N : number of nodes in N
A: f.i; j /; .j; i / W .i; j / 2 Eg
Suv : number of shipments between nodes u and v where .u; v/ 2 C
rij : risk per shipment on arc .i; j / 2 A
lij : length of arc .i; j / 2 A
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Obnoxious Facility Location
329
14.2.3.5 Model Outputs (Decision variables)
(
1 if a facility is located at j;
Xj D
0 otherwise:
(
1 if region i uses facility j;
Zijuv D
:
0 otherwise
14.2.3.6 Objective Function and its Constraints
Bi LevelPW
Min
Xij 2f0;1g .i;j /2A
X
.i;k/2A
zuv
ik
P
.u;v/2C
X
uv
Zki
.k;i /2A
(14.19)
suv rij Zijuv ;
8̂
<C1
D 1
:̂
0
i Du
i D v 8i 2 N; .u; v/ 2 C ;
o:w:
Zijuv Xij 8.i; j / 2 AI .u; v/ 2 C;
Zijuv 2 f0; 1g:
(14.20)
(14.21)
(14.22)
Here, we negligee from constraints discussion. The problem can be converted to
a single level optimization problem by writing out the KKT 3 optimally conditions
using additional variables and a large number R.
14.2.3.7 Objective Function and its Constraints
X
Min
Xij 2f0;1g
X
.i;k/2A
3
.i;j /2A
zuv
ik
X
X
suv rij Zijuv ;
(14.23)
.u;v/2C
uv
Zki
.k;i /2A
8̂
<C1
D 1
:̂
0
i Du
i D v 8 i 2 N; .u; v/ 2 C ;
o:w:
(14.24)
Zijuv Xij
8.i; j / 2 AI .u; v/ 2 C;
(14.25)
vuv
ij
Zijuv /
8.i; j / 2 AI .u; v/ 2 C;
(14.26)
Zijuv //
(14.27)
R.1
uv
ij
R.1 .Xij
vuv
ij
uv
ij
0
0
wuv
i
8.i; j / 2 AI .u; v/ 2 C;
free
Karush-Kuhn-Tucker optimality conditions
Z uv
ij
2 f0; 1g Xij 2 f0; 1g:
(14.28)
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S. Hosseini and A.M. Esfahani
Solving this mixed integer programming problem with n nodes, k OD pairs, and
m arcs contains nk C 4mk constraints, m.k C 1/ binary variables, and 2k.m C n/
non-negative continuous variables. For example, with 100 nodes, 30 OD pairs, and
200 arcs, the mathematical problem has 27,000 constraints, 6,200 binary variables,
and 18,000 continuous variables. However the problem size may make it difficult to
solve realistic versions of the problem. We now provide the formulation for OCST.
This formulation is later extended to minimum risk hazmat tree (MRHT).
14.2.3.8 Objective Function and its Constraints
OCST W
X
Min
(14.29)
X
suv rij Zijuv :
.i;j /2A .u;v/2C
Subject to
X
.i;j /2A
zuv
ij
X
.j;i /2A
Zjuvi D
(
1 8i 2 N; .u; v/ 2 C; i D u:
0 8i 2 N; .u; v/ 2 C; i ¤ u; v:
Zijuv C Zjuvi Xij 8.i; j / 2 E; .u; v/ 2 C;
X
Xij D n 1;
(14.30)
(14.31)
(14.32)
.i;j /2E
Zijuv 2 f0; 1g 8i 2 N; .u; v/ 2 C:
(14.33)
Xij 2 f0; 1g 8.i; j / 2 E:
(14.34)
Equation (14.30) provides the flow conservation constraints for each shipment.
Equation (14.31) guarantees that if there is a shipment on an arc, and then it must
be part of the hazmat tree. Equation (14.32) guarantees the construction of a tree
network. Equations (14.33) and (14.34) declare the decision variables as binary.
The nodes are classified in two sets: mandatory and non-mandatory nodes. Origins and destinations are mandatory nodes, and the rest are non-mandatory. Let S
and T W S [ T D N; S \ T D be the set of mandatory and non-mandatory nodes,
respectively.
14.2.3.9 Objective Function and its Constraints
MRHT W Min
X
X
.i;j /2A .u;v/2C
suv rij Zijuv :
(14.35)
14
Obnoxious Facility Location
331
Table 14.4 A comparison of the problem size of the bi-level model and MHRT as a function
Constraints
nk C 4mk
nk C mk=2 C s
Bi-level
MRHT
Binary var.
m.k C 1/
m.k C 1/
Continuous var.
2k.m C n/
0
Subject to
X
zuv
ij
.i;j /2A
Zijuv C
X
X
.j;i /2A
Zjuvi
D
Zjuvi Xij C Xj i
(
1 8i 2 N; .u; v/ 2 C; i D u:
0 8i 2 N; .u; v/ 2 C; i ¤ u; v:
8.i; j / 2 E; .u; v/ 2 C;
(14.36)
(14.37)
Xij D0 f or i D 1;
(14.38)
Xij D1 f or i ¤ 1 and i 2 S;
(14.39)
Xij 1
f or i 2 T;
(14.40)
2 f0; 1g 8.i; j / 2 A; .u; v/ 2 C;
Xij 2 f0; 1g 8.i; j / 2 A:
(14.41)
(14.42)
j 2N
X
j 2N
X
j 2N
Zijuv
For a design problem with n nodes (of which s are mandatory), m arcs and k OD
pairs, Table 14.4 compares the problem sizes for the bi-level model and MRHT as a
function of problem parameters.
For example with 100 nodes (of which 20 arc mandatory), 30 OD pairs, and 200
arcs, MRHT will have 6,200 variables and 6,020 constraints.
14.2.4 Obnoxious Facilities Location-Routing Problem
Obnoxious facilities location problem is usually considered along with routing of
undesirable materials. Most of these obnoxious facilities, such as garbage dump
citing and chemical plants are producer or consumer of Undesirable materials, so
we should consider routing problem with location of these facilities simultaneously.
Cappanera et al. (2004) defined a discrete combined location-routing model, at
2004s, which we refer to as obnoxious facilities location and routing model (OFLR).
OFLR is a NP-Hard problem, for which a Lagrangean heuristic approach is presented. The Lagrangean relaxation proposed allows decomposing of OFLR into a
location sub problem and a routing sub problem. An effective branch and bound
algorithm is then presented, which aims at reducing the gap between the above
mentioned lower and upper bounds. This is accomplished by using a bundle method
to solve at each node the Lagrangean dual. Before 1970s scientists have mainly
focused their attention on classical location problem i.e. service facilities location
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S. Hosseini and A.M. Esfahani
such as hospitals, supermarkets, post-offices and warehouses in which minimizing
the distance between facilities and demand centers is one of objective functions.
Gradually, some problems such as reactor dump citing and chemical plants locating
with maximizing objectives were surveyed. However costs of material transportation to these facilities grow up by increasing distances. On the other hand, recently
increasing attention of environmental-friend groups and repulsion of people forces
governments to consider routing problems beside location of undesirable facilities
and that imposes high costs to governments. In the last ten years for instance, the
Italian electric society has attended to locate at least four electric power supplier
networks and failed in all the cases due to the public opposition. Here, the obnoxious facility location and routing (OFLR) problem is formulated as a capacitated
minimum cost network flow model.
14.2.4.1 Model Assumptions
The model is single commodity, i.e. a single obnoxious material is considered.
The affected sites are represented as point in the plane.
In this network there are some nodes that they don’t use hazardous material and
they don’t produce hazardous material too, but they influenced by hazardous material transportation consequences.
For each site, location and routing exposure thresholds are given.
14.2.4.2 Model Inputs
The model is considered as G D .V; A/ graph, A is set of arcs and the set of vertices
V is the union of the following three sets:
R D f1 : : : mg: the set of affected sites
N D f1 : : : ng: the set of candidate location to establish the new facilities
T D f1 : : : tg: the set of transshipment nodes
Let us define:
ıi .ıi < D 0/: the demand of vertex i . Each node i 2 R can be a hazardous
material source.
˛ij k : the exposure caused by a unitary flow along the arc .i; j / to affected site K
k : the threshold of affected site K relative to the exposure induced by the establishment of obnoxious materials
aij : the exposure caused by the opening of a facility in location j to affected site i
ti : the threshold of affected site I relative to the exposure induced by the establishment of obnoxious facilities
Uj : the capacity of located facility in site j
Cj : the opening cost of facility located in site jth
ij : the transporting cost of a unitary flow along the arc .i; j /
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Obnoxious Facility Location
333
14.2.4.3 Model Outputs (Decision Variables)
Yj : 1 if a new facility is established at node j ; 0 otherwise
Xij : the amount of transported flow along the arc .i; j /
An artificial node P was added to set V , each site j 2 N is connected
P to this
ıi and
node via artificial arc .j; p/ of zero cost. The demand of this node is
i 2<
represents demand of each influenced site i 2 R is completely disposed of.
V 0 and A0 are the extended set of nodes and arcs respectively,
i:e: V 0 D V
Subject to
[
0
fP g; A D A
X
OFLR W MINIMIZE
X
0
j W.j;i /2A
X
˛ijk Xij
.i;j /2A
X
j 2N
Xj i
Xij 0
X
0
j W.i;j /2A
k
aij Yj ti
j 2N
[
Cj Yj C
Xij D ıi
8k 2 R;
8i 2 R;
f.j; p/ W j 2 N g:
X
ij Xij ;
(14.43)
(14.44)
.i;j /2A
8i 2 V 0 ;
(14.45)
(14.46)
(14.47)
8.i; j / 2 A;
(14.48)
Yj 2 f0; 1g 8j 2 N:
(14.49)
The OFLR model is like above model. The objective and constraints are explained below:
Equation (14.44) minimizes sum of establishment and transportation costs.
Equation (14.45) represents that demand of each site is disposed of entirely.
Equation (14.46) considers that the sum of yield affected of hazardous material
transportation among network arcs, must not exceed the threshold k .
Equation (14.47) represents that the sum of yield affected of establishing facilities for each site, must not be over the fixed threshold.
Equation (14.48) ensures that if any facility is not located at node j , this node
just can be applied as transshipment node, i.e. we can’t vacate undesirable flow at
node j and if a facility is located at this node, vacated amount shouldn’t be more
than capacity. Finally the (14.49) is the common constraint of location problems.
The above model is compared with the models by zagrafos and samara (cited
in (Erkut and Alp 2007)) and by List and Mirchandani (cited in (Erkut and Alp
2007)). A goal programming model was presented by zagrafos and samara which
simultaneously minimizes Routing risk, Location risk and routing cost. The main
differences between OFLR and the model by Zografos and Samara are following:
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S. Hosseini and A.M. Esfahani
In goal programming model each affected site suffers from the nearest open
facility only, whilst in OFLR, each affected site is exposed to a set of open obnoxious facilities, according to the value of the aij ’ s.
In goal programming model the total risk given by location and routing activities
is minimized, i.e. risks are summed up over all of the affected sites differently
from OFLR where, for each affected site, given exposure thresholds must not be
exceeded
In goal programming model the number of facilities to open is fixed while in
OFLR is variable
In goal programming model there are capacity constraints on the arcs of network
versus uncapacitated arcs in OFLR.
A multi-objective model is proposed by List and Mirchandani (cited in (Erkut
and Alp 2007)) which risk, cost and risk equality are considered jointly. In this
problem a set of non-overlapping zones are considered as the region of concern.
For each zone, the risk from routing of obnoxious material along nearby arcs and
the establishment of nearby facilities is taken into account and the total risk is
minimized, which is given by the sum of the zonal risks. Equality is regarded
by minimizing the maximal zonal risk. The main issues that characterize the
model proposed by List and Mirchandani (cited in (Erkut and Alp 2007)) are the
following:
In the multi-objective model against OFLR, different types of obnoxious wastes
and materials are considered
Impact from both routing and location are assumed to be additive while in OFLR
they are considered separately
The multi-objective model is uncapacitated.
Alumur and Kara (2007) presented a comprehensive multi-objective model for
the hazardous waste location-routing problem.
Hazardous waste management includes the collection, transportation, treatment
and disposal of hazardous wastes. The aim of proposed model is to answer the following cases: 1 – locating treatment center and assigning proportional technologies,
2 – locating disposal centers, 3 – routing different types of hazardous wastes to compatible treatment technologies and 4 – routing waste residues to disposal centers.
The exclusive limitation of the model that was not incorporated into previous
models is the treatment technology compatibility limitation with type of hazardous
waste should be treated.
The other specialties of model are: 1 – considering different types of hazardous
waste, 2 – grouping them for transportation and treatment and 3 – possibility to
recycle them in treatment and disposal centers.
14.2.4.4 Mathematical Model
The Schematic of Mathematical model is shown in Fig. 14.4.
14
Obnoxious Facility Location
335
b w,q yw,q,i
a w,i gw,i
gw,i
Generation
node
Xw,i,j
yw,q,i
Zi,j
di
Treatment
center
Disposal
center
fq,i ∈ {0,1}
dzi ∈ {0,1}
Fig. 14.4 Schematic modeling
14.2.4.5 Model Inputs
N D .V; A/ transportation network
G D f1 : : : gg generation nodes
T D f1 : : : tg potential treatment nodes
D D f1 : : : d g potential disposal nodes
Tr D f1 : : : t rg transshipment nodes
W D f1 : : : wg hazardous waste types
Q D f1 : : : qg treatment technologies
ci;j : cost of transporting one unit of hazardous waste on link (i,j)
czi;j : cost of transporting one unit of waste residue on link (i,j)
f cq;i : fixed annual cost of opening a treatment technology q 2 Q at treatment
node i 2 T
f di : fixed annual cost of opening a disposal facility at disposal node i 2 D
POPwij : number of people in the bandwidth for hazardous waste type w 2 W
along link .i; j / 2 A
gw;i : amount of hazardous waste type w 2 W generated at generation node i 2 G
˛w;i : recycle percent of hazardous waste type w 2 W generated at generation
node i 2 G
ˇw;q : recycle percent of hazardous waste type w 2 W treated with technology
q2Q
rw;q : percent mass reduction of hazardous waste type w 2 W treated with technology q 2 Q
tq;i : capacity of treatment technology q 2 Q at treatment node i 2 T
t m q;i : minimum amount of hazardous waste for treatment technology q 2 Q at
treatment node i 2 T
dci : disposal capacity of disposal site i 2 D
comw;q : 1 if waste type w 2 W is compatible with technology q 2 Q; 0 otherwise
14.2.4.6 Model Outputs (Decision Variables)
xw;i;j : amount of hazardous waste type w transported through link (i,j)
zi;j : amount of waste residue transported through link (i,j)
336
S. Hosseini and A.M. Esfahani
yw;q;i : amount of hazardous waste type w to be treated at treatment node i with
technology q
di : amount of waste residue to be disposed of at disposal node i
fq;i :1 if treatment technology q is established at treatment node i ; 0 otherwise
d zi : 1 if disposal site is located at disposal node i ; 0 otherwise
14.2.4.7 Objective Function and its Constraints
X X
Min
.i;j /2A w
ci;j xw;i;j C
X
.i;j /
czi;j zi;j C
C
Min
XX
X
XX
i
f cq;i fq;i
q
(14.50)
f di d zi ;
i
(14.51)
POPw;i;j xw;i;j ;
.i;j / w
Subject to
.1 ˛w;i /gw;i D
X
j W.i;j /2A
xw;i;j
C
XX
q
X
w
X
q
X
xw;i;j
j W.j;i /2A
yw;q;i ; w 2 W; i 2 V ;
yw;q;i .1 rw;q /.1 ˇw;q / di D
X
j W.i;j /2A
zi;j
(14.52)
X
j W.j;i /2A
zj;i ; i 2 V ;
(14.53)
yw;q;i tq;i fq;i ; q 2 Q; i 2 T ;
(14.54)
di dci d zi ; i 2 D;
X
m
fq;i ; q 2 Q; i 2 T ;
yw;q;i tq;i
(14.55)
w
(14.56)
w
yw;q;i tq;i comw;q ; w 2 W; q 2 Q; i 2 T ;
XX
yw;q;i D 0; i 2 .V T /;
(14.58)
di D 0; i 2 .V T /;
(14.59)
xw;i;j ; zw;i;j 0; w 2 w; .i; j / 2 A;
(14.60)
yw;q;i 0; w 2 W; q 2 Q; i 2 T ;
(14.61)
q
(14.57)
w
di 0;
i 2 D;
(14.62)
fq:i 2 f0; 1g ;
q 2 Q; i 2 T ;
(14.63)
d zi 2 f0; 1g ;
i 2 D:
(14.64)
14
Obnoxious Facility Location
337
Equation (14.50) is the cost objective minimizes the total cost of transporting
hazardous wastes and residues and the fixed annual cost of opening a treatment
technology and a disposal facility.
Equation (14.51) is the risk objective that minimizes the transportation risk.
Equation (14.52) is the flow balance constraint for hazardous waste and ensures
that all generated non-recycled hazardous waste is transported and treated at a treatment facility.
Equation (14.53) is the flow balance constraint for waste residues.
Equations (14.54) and (14.55) are the capacity constraints for treatment and disposal centers.
Equation (14.56) is the minimum amount of requirement constraint for opening
a treatment center.
Equation (14.57) is the compatibility constraint, which ensures that a hazardous
waste type is treated only with a compatible treatment technology.
Equations (14.58) and (14.59) ensure that treatment and disposal centers should
be located in candidate centers.
The model’s advantages are:
It has capability to apply in large scale problems. In spite of the prior models
were responsible only to a network with 10–15 nodes and 3–4 candidate centers,
this model can cover a network with 90 nodes and 15–20 candidate places in
reasonable time.
Considering the residue waste issue.
Proposing several method for treatment (with compatibility constraint)
The above model was solved with CPLEX software. We can focus on heuristic
techniques for future direction.
This model can be applied in real problems. For example, a age scale implementation of the model in the Central Anatolian region of Turkey is presented in
Sect. 14.4.1.
14.2.5 Multiobjective Obnoxious Facilities Location Problem
(Rakas et al. 2004)
In the real world, usually we should consider multiple objectives to ensure the adaptation of model with real conditions.
In this way Rakas et al. (2004) proposed a multiobjective model as followed:
14.2.5.1 Model Inputs
i D 1; 2; : : : ; : representatives of demand points
j D 1; 2; : : : ; N : representatives of supply points
Pi : population of region i
Cp: the amount of produced waste by everyone
338
S. Hosseini and A.M. Esfahani
di D Cp Pi : the amount of produced waste by each region
Cj Min : minimum capacity of each candidate point
Cj : the maximum capacity of every point
Lij : the length between two points i and j
Cij D Cw Lij : the transport cost for wastes between i and j
fj (k): the cost of opening facility in j with size k
Vi : the error cost of experiment in the region i
Oi : repulsion reason in region i
Sj : capacity of each candidate point
14.2.5.2 Model Outputs
Xj D
(
Yij D
(
1
0
if a facility is located at j;
otherwise
1
0
if region i uses facility j;
otherwise
14.2.5.3 Objective Function and its Constraints
C : the total cost
TC: the transport cost
IIC: primary investment cost
C D T C C IIC;
TC D di Cij Yij ;
IIC D fj .Sj / Xj ;
1
0
N
N
M
X
X
X
@ .di Cij Yij /A C
Min Z 1 D
.fj .Sj / Xj /;
i D1
Min Z 2 D
j D1
M X
N
X
(14.65)
j D1
.Oi Yij /:
(14.66)
i D1 j D1
Subject to
N
X
j D1
Yij D 1
Yij Xj
8i;
(14.67)
8i;8j;
(14.68)
14
Obnoxious Facility Location
339
M
X
.d i Yij / Sj
i D1
CjMin
Sj CjMax
8j ;
(14.69)
8j:
(14.70)
Equation (14.65) minimizes sum of opening and transportation cost and (14.66)
minimizes the people opposition. Equation (14.67) declares each region is covered
by only one facility. Equation (14.68) shows that region i can use facility j only if
one facility is located in j . Equation (14.69) is capacity constraint of each candidate
point and (14.70) proposes the capacity of every point between two limits.
For solving this continuous model, two models are solved separately and Z1 Opt ;
Opt
Z2
are obtained. Then using weighting method, two objective functions are integrated and following model is solved:
Min Z3 D Weight 1
Z1
Opt
Z1
C Weight 2
Z2
Opt
Z2
:
(14.71)
Because of uncertainty in data estimation, required data are solved as fuzzy data
that are described in (Rakas et al. 2004) in detail.
14.3 Solutions and Techniques
In this section we review some researches in obnoxious facility location field focusing on solutions and techniques.
Melachrinoudis (1999) developed a bicriteria location model for locating a new
semi-obnoxious facility in an existing layout which is solved by an adaptation of
the Fourier–Motzkin elimination method
Zhang et al. (2000) devise a routing hazardous materials algorithm, using GIS
technique and a Gaussian Plume model for their dispersion model
Akgün et al. (2000) to find dissimilar paths for transporting HazMat, solved the
capacitated flow problem:
– Iterative penalty method(IPM)
– Minimax method
– Gateway shortest path (GSP) method
By increasing the size of problem, calculations volume rises up strongly. They
used GIS method as a facilator to test different network roads. Finally they proposed a P -Dispersion problem and solved it with constructing a candidate set m
of K-shortest paths.
Kara et al. (2003) present two path- selection algorithms for hazmat transport
problems with selection of a minimum risk path. One of the proposed procedures
is a modified version of shortest path algorithm and the other is an adaptation of
a link-labeling algorithm developed for urban transportation.
340
S. Hosseini and A.M. Esfahani
Cappanera et al. (2004) proposed a Lagrangean heuristic approach allows to decompose obnoxious facility location routing (OFLR) into a location subproblem
and routing subproblem. Then, an effective branch and bound algorithm was offered for reduction of the gap between lower and upper bounds.
Colebrook et al. (2005) presented a new bound for the undesirable 1-median
problem (Maxian) on networks. they proposed an undesirable center and median
models and implemented Multicriteria-anti-cent-dian problem (MACDP). They
created a heuristic algorithm for MACDP.
Dı́az-Báňez et al. (2005) for solving the obnoxious short path problem in continuous spaces presented an approximate algorithm based on the bisection method.
They transformed optimization problem into decision problem and could provide
efficient algorithm. They proposed:
– A general method of voroni diagram with daguare geometry
– Shortest path avoiding obstacles
Erkut et al. (2005) formulated the tree design problem as an integer programming problem aiming at trade off between cost and risk. They developed a simple
heuristic to expand the solution of the tree design problem by adding node segments.
Erkut and Alp (2007) formulate the tree design problem for hazardous materials
routes in and through a major population center as an integer programming model
with an objective of minimizing the total transport risk. They propose the use of
a new model for finding a minimum risk hazmat tree called Optimum Communication Steiner Tree. Then a construction heuristic that adds paths incrementally
and permits local authorities to balance between risk and cost was developed.
Zhang et al. (2005) proposed a new model that is a variant of the vehicle routing problem with time windows (VRPTW) by adding load upper bounds on the
road segments. The objective is to find a schedule to guarantee the safety of all
vehicles. They propose a sophisticated Tabu search heuristic with novel neighborhood operators such as dynamic penalty mechanism to obtain good solutions.
Zhang et al. (2005) used a heuristic algorithm based on greedy to solve a multiobjective approach (analysing cost minimization, potential risk minimization
and risk equity maximization) to assist decision makers in analysing combined
location/routing decisions involving hazmats.
Dadkar et al. (2008) Identified geographically diverse routes for the transportation of hazardous materials by applying k-shortest path algorithm with stochastic
objective and varying over time. They also implemented Mixed Integer Programming.
Erkut and Gzara (2008) develop and test a heuristic for a bilevel network design
problem for hazmat transportation that finds stable solutions. This heuristic exploits the network flow structure at both levels to overwhelm the difficulty and
instability of the bilevel integer programming model.
Pisinger (2006) present an exact algorithm for finding a number of fast upper
bounds for P-dispersion problems in which these bounds can be derived in O.n/
time.
14
Obnoxious Facility Location
341
Bell (2006) considered mixed rout strategies for the Risk-Averse shipment of
hazardous materials. A minimax problem was formulated to determine the safest
set of routs and safest share of traffic between these routs. A simple heuristic
based on a shortest path algorithm and the method of successive averages is proposed. Connections to game theory can be seen in the nature of the solution.
Yapicioglu et al. (2007) offer a new model for the semi obnoxious facility location problem composed of a weighted minisum function for transportation cost
and a distance- based piecewise function for the obnoxious effects of the facility.
They devise a single objective particle swarm optimizer (PSO) and a bi-objective
PSO to solve this problem.
Carotenuto et al. (2007) For finding minimum and equitable risk routes for
HazMat shipment present a mathematical model and solved with Lagrangian relaxation and two heuristic algorithms based on the Yen’s k-shortest path:
– Greedy algorithm (GD)
– Randomized Greedy algorithm (RGD)
Carotenuto et al. (2007) implemented a Tabu Search algorithm for scheduling and
routing of HazMat transportation. They focus on scheduling to decrease traffic
volume of HazMat transportation as well as on a network road.
14.4 Case Study
14.4.1 Obnoxious Facility Location and Routing in Anatolian
Region of Turkey (Alumur and Kara 2007)
In Central Anatolian region of Turkey, there are 184 administrative districts. The
population of districts ranges from 3,700 to 77,000 people. It assumed that the districts with a population more than 25,000 produce hazardous waste.
The data on the amount of hazardous waste produced by each districts in Turkey
are not available presently. So, assumed that the amount of hazardous waste generated by each district is proportional to the population times the industrial activity
level of the district. The GIS system is applied in collecting geographical information. In this region three types of hazardous waste are generated and two treatment
methods are determined. The first type waste can be incinerated, the second type is
suitable for chemical treatment, and the third type can be treated by both incineration and chemical process. The proposed model assumed that treatment and disposal
centers can be located at the same nodes. The candidate point’s selection is so important because of cost considerations. So that, the model was considered in two
conditions: A: 15 candidate places, B: 20 candidate places. After solving the problem with CPLEX resulted that final answers are not mainly different. Therefore, in
the reason of economic considerations the 15 candidate place state was selected.
342
S. Hosseini and A.M. Esfahani
14.4.2 Designing Emergency Response Network for Hazardous
Materials Transportation (Berman et al. 2007)
In addition to government’s policies for decreasing HazMats accidents, detrimental
effects of these accidents are such that we should consider various alternatives for
locating, routing and improving of emergency teams performance to reduce these
consequences. Critical factor is quick arrival of emergency teams at the accident
site. Arrival time, itself, is related to selected paths and location of teams, therefore
such designs are essential to consider.
In this way, Berman et al. (2007), proposed a maximal arc-covering model for
optimal design of such networks. Their model improves solution capability relative
to previous models by adding some binary variables. Moreover, they developed a
greedy heuristic to solve their model efficiently.The advantages of new formulation are:
Reducing the number of variables,
Reducing time of solution,
Compatibility with existing software and large scale problems,
Proposed model is explicit and easy to understand,
The new model constitutes a basis for an efficient heuristic.
Proposed model was efficiently applied for emergency response infrastructure
for gasoline incidents on the highway networks of Quebec and Ontario in Canada.
They derived the model parameters from a GIS-based representation of the two
provinces. This application shows the possibility of a significant improvement in response capability via relocation of the existing stations. Also, a noticeable decrease
in the threshold time of coverage for gasoline incidents is considered.
14.4.3 Locating Waste Pipelines to Minimize their Impact
on Marine Environment (Ceceres et al. 2007)
Ceceres et al. (2007), suggested a methodology for locating waste pipelines to
minimize their impact on marine environment. A waste pipeline, as an undesirable facility, is to be located in a coastal region. The coastal fringe is the territory
where marine, air and terrestrial environments interrelate. Here very diverse and
fragile ecosystems coexist, although subjected in many cases to increasing degradation due to industrial and urban developments disrespectful to the environment.
Pasidonia oceanica (Linnaeus) Delile is a plant with leaves, flowers and fruit,
similar to those plants which live in forests and gardens, but which lives in the sea
between the surface and a depth of 50 m.
It is endemic to the Mediterranean Sea and, by providing the principal source of
oxygenation of the Mediterranean Sea; it is its most important ecosystem. Due to
its ecological role, this sea-grass is a protected species in Spain and in France. Here
14
Obnoxious Facility Location
343
Fig. 14.5 The area Voronoi
diagram
4
5
6
7
3
LAND
8
11
2
1
10
9
a methodology is described to obtain an efficient set of points where the extreme
of a marine pipeline should be located. Two criteria are taken into account, the
Euclidean distance from a given set of protected areas and a utility function related
to the pipeline length, both to be maximized.
The region under study has been assumed to be a rectangle which includes zones
of biological interest, geometrically modeled as rectangles and circles. They used
Voronoi diagram shown in Fig. 14.5.
The length of the pipe has been associated to a utility function described by a
generalized gamma function. Since intensity of pollution has been assumed to be
inversely proportional to the Euclidean distance, the location problem was formulated as a bi-criterion problem in which both the minimum distance to protected
zones and the utility function had to be maximized.
An efficient set of solutions has been identified along edges of the zone Voronoi
diagram by means of an approach based upon the NISE4 method.
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Chapter 15
Dynamic Facility Location Problem
Reza Zanjirani Farahani, Maryam Abedian, and Sara Sharahi
Facility location is a strategic management decision. This decision is usually made,
however, with respect to the current parameters (like weights) which represent population, infrastructure, service requirements and others (Drezner 1995; Francis and
Lowe 1992; Mirchandani and Francis 1990). Much of the research published on
location theory is drawn from the models such as single/multi facility location, covering, P -median, P -center problems, and their applications and extensions. Solving
many of these problems can be extremely difficult. Thus, it is not surprising that so
much work has focused on statistic and deterministic problem formulations. While
such formulations are reasonable research topics, they do not capture many of the
characteristics of real-world location problems.
The strategic nature of facility location problems requires that any reasonable
model consider some aspects of future uncertainty. Since the investment required
by location or relocation facilities is usually large, facilities are expected to remain
operable for an extended time period (Owen and Daskin 1998). Thus, the problem
of facility location truly involves an extended planning horizon. Decision makers
must not only select robust locations which will effectively serve changing demand
over time, but must also consider the timing of facility expansions and relocations
in the long term (Daskin et al. 1992).
It is generally true of facilities that they are expected to serve over a long period
of time. During this time, many of the “constants” of the problem, such as demands
and distribution costs, are likely to change. It is also generally true that relocation
cannot be accomplished without cost (Wesolowsky 1973). Thus, decision makers
must select sites that will not simply perform well according to the current system
state, but that will continue to be profitable for the facility’s lifetime (Owen and
Daskin 1998). Considering the fact that changes in static problem parameters can
be forecasted, optimization requires the balance between the benefits of planned
location changes and their costs (Wesolowsky 1973).
There are two different types of dynamic facility location Problems: Location
and Location–Relocation. These two types are different in the following ways:
In a time-dependent location problem, the decision maker selects a site which is
profitable for a defined time horizon.
In location–relocation problem, the decision maker selects a primary location,
relocation or development times and the facility’s location after relocation.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 15, c Physica-Verlag Heidelberg 2009
347
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In the remainder of this chapter, first, some classifications of time-dependant facility
location problems will be presented. Then, some of the related mathematical models will be reviewed. Next, some lemmas and solution techniques will briefly be
presented. And finally, some real-world case studies will be introduced.
15.1 Classifications
Facility location problems can be divided into static and dynamic problems, These
can also be further classified on the basis of different criteria. This section provides
a brief description of different classifications of dynamic facility problem.
Cause of change. The first and the most important way for the classification of
dynamic facility location problem is based on the cause of change and uncertainty. According to this criterion, dynamic facility location problems are divided
into two categories (Owen and Daskin 1998):
– Uncertainty related to planning for future conditions, and
– Uncertainty due to the limited knowledge of model input parameters.
Note that in the first category, although there are changes in the conditions, they
are deterministic and time-dependent. In other words, the pattern of changes is distinctive and deterministic. In the second category, although the pattern of changes
may be distinctive, it is not deterministic and time dependent, rather, it is stochastic
(Rosental et al. 1978).
Denoting the weight of facility parameter by “Wi ”, deterministic and stochastic
changes can be classified as follows:
– Deterministic changes can be further classified based on whether or not:
1. “Wi ” changes in each period of time horizon but it is fixed and distinctive in
every period.
2. The weight parameter, “Wi ”, is a function of time, say wi .t/.
Stochastic changes can be classified into three types where:
1. “Wi ” has a probabilistic distribution with fixed parameters.
2. “Wi ” has a probabilistic distribution with variable parameters where the parameters are functions of time.
3. There is no information about the alteration of the parameters and the weight of
the facility.
Dependency variable: Time-dependent weights, Distance-dependent weights.
This chapter is mainly dedicated to time-dependent weights. However, some
works such as the one by Huang et al. (1990) consider the weights as a function
of distance between the new facility and the existing facilities; this application
can be found in marketing area.
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Dynamic Facility Location Problem
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The number of relocations: Single relocation, Multiple relocation. In the first
category, there is only one relocation (Emamizadeh and Z.-Farahani 1997a, b)
whereas in the second category the location of the new facility is allowed to
change several times during the time horizon, i.e. n changes are allowed during
the time span.
Classification by the number of relocating facilities (Owen and Daskin 1998;
Chand 1988): Single facility, Multiple facility. In the first category, only one
facility can be relocated whereas in the second, more than one facility can be
considered for relocating (Scott 1971).
Classification by the Relocation Time: Discrete, Continuous. In the first category,
relocation can take place only at pre-determined points (Wesolowsky 1973) of
time whereas in the second category, relocation can take virtually any time in the
defined time horizon (Drezner and Wesolowsky 1991).
Time horizon: Finite, Infinite. Daskin et al. (1992) acknowledge that the difficulty
in solving dynamic facility location problems arises from the uncertainty surrounding future conditions. Even establishing an appropriate time horizon length
is a non-trivial problem which is ignored in most formulations. They argue that
the best way to mange uncertainty is postponing decision making as much as
possible, collecting information and improving forecasts as time advances. Since
the first period decisions are the ones to be implemented immediately, the authors claim that the goal of dynamic location planning should not be to determine
locations and/or relocations for the entire horizon, but to find an optimal or nearoptimal first period solution for the problem over an infinite horizon.
15.2 Mathematical Formulations
15.2.1 Static Model (Wesolowsky 1973)
Perhaps the best known of static location models is the generalized Weber problem.
The problem requires the location of a facility among m destinations. Costs are
assumed to be proportional to distances. Location is achieved by solving:
Min
m
X
wi :di .x; y/
(15.1)
i D1
where:
di .x; y/: is the distance between the facility to be located at .x; y/ and destination i located at .ai ; bi /:
wi : is a constant transforming distances into costs.
To simplify notation in the succeeding section, model (15.1) is set in common
notation.
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Min
Xm
i D1
(15.2)
fi .x; y/:
The static model given in (15.2) is now extended to a model with a planning horizon of r time periods. This means that the demands, costs and destination locations
are forecasted for rtime periods in advance, and the optimal planned location of the
facility in each period must be found. Consider the following problem:
Min
mk
r P
P
kD1 i D1
zk D 0
fki .xk ; yk / C
if
r
P
Ck :zk ;
kD2
(15.3)
dk;k1 D 0:
where:
mk : is the number of destinations in period k.
fki .xk ; yk /: is the present value of the cost of shipping from the facility at
.xk ; yk / in period k to destination i:
Ck : is the cost of moving at the beginning of period k.
dk1; k : the distance the facility is moved at the beginning of period k.
The above model makes the following assumptions:
Each term fkd .xk ; yk / is adjusted to represent its present value at time 0. It can
also be adjusted by a factor reflecting confidence in the forecast.
Each “moving” cost Ck is independent from the distance the facility is to be
moved and is also independent from the number of periods it will remain at the
new location (Wesolowsky 1973).
15.2.2 Dynamic P-Median Model (Owen and Daskin 1998)
We will examine the P -median problem under the scenario planning approach.
15.2.2.1 Model Inputs
k: index of possible scenarios
hi k : demand at node i under scenario k
dij k : distance from node i to facility site j under scenario k
vO k : optimal P -median solution value for scenario k
qk : scenario probability for scenario k.
15.2.2.2 Decision Variables
Yij k D 1 if demand node i is assigned to facility j under scenario k, otherwise is 0
Xj D 1 if facility site j locates at potential, otherwise is 0
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The regret associated with scenario k is, thus, given by:
Rk D vk vO k ;
where vk the value of the demand is weighted total distance (i.e., the P -median
objective value) under the compromise locations
X X
vk D
hi k :dij k :Yij k :
i
j
15.2.2.3 Objective Function and its Constraints
The expected regret problem can thus be formulated as follows:
Minimise
X
(15.4)
qk Rk ;
k
X
Xj D p;
X
Yij k D 1
j
j
(15.5)
8i; k;
(15.6)
Yij k Xj 0
8i; j; k;
XX
Rk
.hi k d ij k Yij k VOk / D 0 8k;
(15.8)
Xj D f0; 1g
(15.9)
i
(15.7)
j
Yij k D f0; 1g
8j;
8i; j; k:
(15.10)
The objective function minimizes the expected regret, with regret defined in
(15.8). The remaining constraints are the scenario planning equivalents to the standard P-median constraints.
Note that the locations are common to all scenarios and must be determined
before knowing which scenario is realized. The demand assignments, however,
are scenario-specific. In essence, they are the result of optimizing the assignments
conditional to the chosen sites after we know which scenario is realized. This formulation requires the decision maker to input probability values qk for each scenario,
values which must be typically estimated. To avoid making such estimates, we can
instead minimize the maximum regret across all scenarios. This objective is more
conservative, and is formulated with the same constraints as above, but with the
following objective function:
Min Max Rk :
k
(15.11)
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15.2.3 Multiperiod Model (Wesolowsky and Truscott 1975)
A dynamic or multiperiod location–allocation formulation is the problem of locating
G facilities among M possible sites to serve N demand points.
The goal is to devise a plan of optimal locations and relocations in response
to predicted changes in the demand volume originating at demand points over a
planning horizon of K periods.
15.2.3.1 Model Inputs
Aj i k : the present value of the cost of assigning node i to nodej in period k:
=
cj k : the present value of the cost of removing a facility from site j in period k:
==
cj k : the present value of the cost of establishing a facility at site j in period k:
mk : the maximum number of facility location changes allowed in period k:
15.2.3.2 Decision Variables
1 if node i is assigned to node j in period k
xj i k D
:
0 Otherwise
1 if a facility is removed from site j in period k
yj0 k D
:
0 Otherwise
1 if a facility is established at site j in period k
yj00k D
:
0 Otherwise
15.2.3.3 Objective Function and its Constraints
The model is:
Min
M
K X
N X
X
kD1 i D1 j D1
Aj i k xj i k C
K X
M
X
kD2 j D1
.cj0 k yj0 k C cj00k yj00k /;
(15.12)
Subject to
M
X
j D1
N
X
i D1
xj i k D 1 8i; k;
(15.13)
xj i k N xjj k
(15.14)
8j; k;
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Dynamic Facility Location Problem
M
X
j D1
M
X
j D1
353
xj i k D G
8k;
(15.15)
yj0 k mk
8k 2;
(15.16)
xjj k xjj k1 C yj0 k yj00k D 0 8j; k 2;
(15.17)
xj i k 0 8i ¤ j yj0 k ; y 00j k 0; 8j; k xjj k 2 f0; 1g; 8j; k:
(15.18)
The objective function is to minimize the costs of distribution from the facilities
to the demand centers. Based on (15.13), each node i is assigned to exactly one
node j . In (15.14) node i can be assigned to node j only if node j is self assigned.
Equation(15.15) guarantee that G self-assignments are made among the M nodes.
Equation (15.16) limit the number of sites vacated in each of periods 2 through
K. Since constant number of facilities, G, is required in all periods by (117), placing an upper bound on the number of facility removals in a period is equivalent to
limiting the number of facility location changes in the period. Equation (15.18) in
conjunction with the second term of (15.12) ensure that the appropriate relocation
costs are charged. The required minimization of costs forces the following binary
values of yj0 k and yj00k for each possible combination of values for xjj k and xjj;k1
in (15.17).
15.2.4 Probabilistic Model (Rosental et al. 1978)
The specific model concerns making dynamic relocation decisions for a new facility
(server) that must interact with existing facilities (customers) whose relocations are
stochastic processes. The two distinguishing features of the problems considered
here are (1) probabilistic location of existing facilities; and (2) dynamic relocation
of new and existing facilities. These two features are treated together.
Costs are location-dependent and are incurred in two ways: (1) when the server
makes choice relocations; and (2) when the server interacts with customers.
We allow both new facility and existing facility to change locations, but the distinction between a new facility and an existing facility is that the location of a new
facility is under a decision maker’s control. The model is relocation policy that minimizes the expected discounted sum of costs.
Xt : server location at time t, decision variable,
At : customer location at time t, stochastic,
N : f1,..., ng: known set of possible locations for both,
F : known server relocation cost matrix, n n,
G: known service cost matrix, n n,
P : known Markov transition matrix for customer location, n n,
B: known discount factor.
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A discrete-time process evolves as follows:
The decision-maker observes (Xt 1 ; At 1 ) and chooses Xt ,
The relocation cost f .xt 1 ; xt / is incurred,
The chance probabilistic At , is realized and
The service cost g.Xt ; At / is incurred.
Then the process repeats. The problem is to find a policy for choosing server
locations, so as to minimize the expected present worth of all costs, i.e.,
Minimize E
" a
X
t D1
#
fF .Xt 1 ; Xt / C G.Xt ; At /gB t 1 :
(15.19)
15.3 Solution Techniques
Some of the common methods used in dynamic Facility location are as follows:
Exact Techniques as in:
– Mixed integer programming. Some works in this area are Wesolowsky and
Truscott (1975), Hormozi and Khumawala (1997) and Z.-Farahani et al.
(2009).
– Complete enumeration. Wesolowsky (1973) represented a method based on
complete enumeration.
– Nonduplicating enumeration. This method was used by Wesolowsky (1973).
– Dynamic programming (DP). Some examples in this area are Hormozi and
Khumawala (1997), Chand (1988), Canel et al. (2001) and Romauch and
Hartl (2005).
– B&B. This method was used by Canel et al. (2001) and Wesolowsky and
Truscott (1975).
– Graphs and networks. An example of this method is Andretta and Mason
(1994).
– Primal-dual. Dias et al. (2004, 2007) used this method to solve their problems.
Heuristic technique. This method was used by Romauch and Hartl (2005).
Meta heuristic techniques as in:
– Simulated annealing (SA). Antunes and Peeters (2000) used SA to solve their
problem.
– Lagrangean relaxation (LR). This method was used by Snyder et al. (2007).
– Tabu search (TS). Rajagopalan et al. (2008) used TS for their problem.
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Dynamic Facility Location Problem
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15.3.1 Fundamental Lemmas
In this section, we will first propound some lemmas regarding the location and
relocation of new facilities.
Lemma 15.1. When there are no relocations and the facility is located only at the
start time, such that the total location costs are minimized, we use the following
weights, and by solving a simple single facility location problem, we can find the best
location. The resulting point is optimal location (Drezner and Wesolowsky 1991).
wi .t/: weight associated with demand point i .
m: Number of current facilities.
wi D
ZT
(15.20)
wi .t/dt:
0
Lemma 15.2. When there are no relocation costs (Drezner and Wesolowsky 1991)
FkC1 Fk ;
(15.21)
where Fk is the cost of optimal ktime break solution. Based on this Lemma, the cost
of optimal k C 1 time break solution can only be lower than the cost of optimal k
time breaks solution. Thus, we should use all opportunities for relocating the facility to the best locations. The best way to manage uncertainty is postponing decision
making as much as possible, collecting information and improving forecasts as time
advances. Since the first period decisions are the ones to be implemented immediately, the goal of dynamic location planning should not be to determine locations
and/or relocations for the entire horizon, but to find an optimal or near-optimal first
period solution for the problem over an infinite horizon (Z.-Farahani et al. 2009).
Lemma 15.3. The objective cost function is additive.
Consider Fig. 15.1 bj1 ; bj and bj C1 are some points in the time horizon. It is
obvious that the location of the new facility can be determined independently during
[.bj1 ; bj ) and (bj ; bj C1 /], given that the relocation takes place at bj .
(x j,y j)
bj–1
(x j+1,y j+1)
bj
Fig. 15.1 New facility locations and relocation time
bj+1
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R.Z. Farahani et al.
The objective function is:
F D
Zbj X
m
d.X
bj 1 i D1
j 1;j
; pi /wi .t/dt C
bZj C1
bj
m
X
d.X j;j C1; pi /wi .t/dt
i D1
D g1 .x j 1 ; y j 1 / C g2 .x j ; y j /;
(15.22)
where Xj k is the optimal facility location during the [bj ,bk /. This shows the additivity of the objective function. Therefore, given bj , the optimal location of the
new facility before and after the relocation time can be determined independently
(Z.-Farahani et al. 2009).
Therefore, if there are predetermined time breaks, we can find the best location
at that time by solving a simple single facility location problem.
Lemma 15.4. For optimal facility location during the [bj , bj C1 ] use Lemma 1:
j
wD
i
Z
bj C1
(15.23)
wi .t/:dt :
bj
Lemma 15.5. The cost of every relocation is at least equal to cost of no relocation.
If F 1.B/ is the cost of optimum location with one relocation at time B; 0 <
B < T , and F .0/ is the cost of optimum location with no relocation, then:
F1 .B/ F .0/:
(15.24)
Based on lemma 5 F1 .B/ F .B D 0/ and what we have: F1 .B/ F
.B D T /.
15.3.2 Single Relocation at Discrete Time
If we consider Fig. 15.2 without t1 and tn , there will be n2
Q candidates for relocation.
We must select one point for relocation, such that the total location and relocation
costs is minimized.
Suppose that tj is selected, according to Fig. 15.3 we must locate facility at this
time and calculate cost function.
(x1, y1)
t1
(x2, y2)
t2
Fig. 15.2 Relocation at discrete time
(x3, y3)
t3
(xn–1, yn–1)
tn–1
tn=T
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Dynamic Facility Location Problem
357
(x1j,y1j)
t1= 0
(x jn,y jn)
tj
tn=T
Fig. 15.3 Relocation at tj
According to Lemma 4, we solve the optimal location problem for intervals
Œt1 ; tj ; Œtj ; tn and .x 1j ; y 1j /; .x j n ; y j n / are reached. Location and relocation
costs are:
Fj D
Ztj X
m
0
i D1
1j
wi .t/:d.x ; pj /dt C
ZT X
m
tj
wi .t/:d.x j n ; pj /dt:
(15.25)
i D1
If this procedure is repeated for tj .j D 2; : : ::; n1/,
Q
there will be n2
Q cost functions as Fj .j D 2; : : :; .n1//.
Q
The minimum Fj is the optimal location and relocation cost, tj is the optimal relocation time, .x 1j ; y 1j / is the optimal location of facility during Œ0; tj and
.x j n ; y j n / is the optimal location of facility during Œtj ; T .
15.3.3 Multiple Relocations at Discrete Times
Without Relocation Costs (Z.-Farahani et al. 2009)
This dynamic problem requires the location of a specified number of facilities
among a predetermined set of potential sites and the allocation of demand centers
to these facilities.
A more interesting problem arises when the location of the new facility is allowed to change several times during the time horizon i.e. n changes are allowed
during the time span Œ0; T . The variables to be determined are the time breaks
B D .b1 ; : : :; b n / at which the changes take place and the associated optimal solution. Defining b0 D 0 and bnC1 D T then, we will have n time breaks. Of course,
T can be infinite.
Given the specific weight functions in time, it could be more economical to
relocate the new facility some time in the future, so that the total location and relocation costs is minimized. It is assumed that the relocation can take place only at
pre-determined points in time. The total location cost, therefore, is the sum of the
location costs before and after the relocation. The total cost depends on the optimal
relocation time and the facility’s optimal locations before and after the relocation.
Refering to Fig. 15.4; Cj k is the cost of locating the facility during period
Œbj ; bk /. In the simplest case (without relocation cost and with finite time horizon
Œ0; T ), the goal is to find the shortest path from b0 to bnC1 .
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R.Z. Farahani et al.
C0n+1
C0n
b0=0
C01
b1
.
. C
. 1j
.
.
.
bj
.
.
. Cjk .
.
.
.
.
.
.
.
bk
.
.
. Ckn.
.
.
bn
Cn,n+1
T
C1n
Fig. 15.4 New facility locations and relocations time
Lemma 3 helps us to find the best relocation times and new locations. However,
using Lemma 3 in different situations related to time horizon and relocation cost,
may be different. The time horizon can be either finite or infinite and relocation
costs can exist or not; in the existence of relocation cost, various alternatives can
occur. These situations will be investigated in following sections. Therefore when
there are no relocation costs, we apply Lemma 2.
15.3.4 Multiple Relocations at Discrete Times
with Relocation Costs
When more than one break is permissible, we have a multiple relocation problem.
Drezner and Wesolowsky (1991) investigate multiple time break and linear change
of weights over time. They consider location on multiple time breaks and devise two
algorithm for minimax location problem. They solve the problem optimally for any
number of time breaks and any distance metric.
First, it can be shown that for a vector Z D .z2 ; z3 ; : : :; zr / problem (3) can be
decomposed into static location problems, where:
zj D
1 Relocation at the beginning of period j
:
0 Otherwise
If n0 be the number of 1’s in the vector Z, the number of different locations for
the facility over the planning period, will be n D nK C 1.
If Sj is the number of periods in the j th sequence that begins with either the
first period or with a period p where zp D 1, and ends either with the last period or
before a period i where zl D 1. the number of periods in sequence j will be iQ p C 1.
The location of the facility during sequence j can be found by solving the static
problem by a method appropriate to the model (Wesolowsky 1973):
Min
mk
l1 X
X
kDp i D1
fki .xk ; yk /:
(15.26)
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Dynamic Facility Location Problem
359
Hence, for any vector Z, the solution of n static problems will specify the successive optimal locations. The optimal solution to the original problem (3) can now
be found by evaluating the cost of the location patterns for every possible Z vector
(Wesolowsky 1973).
15.3.5 Complete Enumeration
The optimal solution to the problem (15.3) can now be found by evaluating the cost
of the location patterns for every possible Z vector. There are 2r1 such vectors
and each vector requires the solution of .Qr 1/=2 C 1 static problems on the average.
Hence, the total number of static problems that must be solved is 2rQ1 Œ1 C .Qr 1/=2.
r1
P
i D0
.i C 1/
r 1
i
D
r1
P
i D0
i:
r1
P r 1
r 1
C
i
i
i D0
D .r 1/:2r2 C 2r1 D 2r1 :
r1
2
:
(15.27)
C1
This total enumeration may be feasible for a small r but it is not necessary, since
there exists a much more efficient way of obtaining a solution (Wesolowsky 1973).
15.3.6 Non-Duplicating Enumeration
Reflection on the above described enumeration procedure yields the following
observation. Once a move is specified by the Z vector, subsequent succeeding locations are independent of the previous one. This leads to the conclusion that many of
the static problems solved in the process of the total enumeration of the Z vectors
are identical a better method of enumeration, one that avoids the recalculation of
static problems.
In this method, one begins the evaluation at the rth period. If there is a move at
the beginning of the rth period, Cj .1/, the sum of the cost of the static problem and
the moving cost Cr , can be calculated. Similarly, in the rt1th period, Cs .1; 1/ can be
found by adding the sum of the costs of the static problem in the rt1th period and
the change cost Crt1 to Cs .1/.
Also Cs .1; 0/ can be found by adding the cost of the static problem in the last
two periods to the change cost Cr1 .
One can thus evaluate the cost of the 2r1 vectors Z without duplicating the
calculation of any static problems. It turns out that this enumeration method calls
for the solution of 2rQ1 static problems (Wesolowsky 1973).
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R.Z. Farahani et al.
15.3.7 Incomplete DP
Noted that there is no need to evaluate all of the branches. At each stage, as one
proceeds from the last period to the first, only those branches where c; is not greater
than the stage minimum need be kept. The saving in the number of static problems
that needs to be evaluated is considerable. If one chooses only one of the branches
in the case where a tie exists at any stage, the total number of static problems to be
evaluated is r.r C 1/=2 (Wesolowsky 1973).
1 C 2 C 3 C r D r.r C 1/=2:
(15.28)
15.3.8 An Especial BIP
There are some points for relocation and we must select some of them, so that the
total cost is minimized. For example, we calculate the cost of the policy that there
are two relocations at time 3 and 4.
f13 D
f34 D
f46 D
m
P
i D1
m
P
i D1
m
P
i D1
wi .t/:d.x 13 ; pi / t 2 Œt1 ; t3 /
wi .t/:d.x 34 ; pi / t 2 Œt3 ; t4 / :
(15.29)
wi .t/:d.x 46 ; pi / t 2 Œt4 ; t6 /
The total cost of this policy is:
F D
Z
t3
t1
f13 dt C
Z
t4
t3
f34 dt C
Z
t6
f46 dt:
(15.30)
t4
This policy can be shown with binary variable: Z13 D Z34 D Z46 D 1. We propose a binary integer programming (BIP) model composed of six steps (Z.-Farahani
et al. 2009).
Step 1 There are n2
Q time candidates for relocation. Adding t1 D 0 and tn D T
as relocation points yields the total of n points. The interval [0,T ] is divided into n1
Q
subintervals. There are m demand points. Calculate:
j
wD
i
tZ
j C1
tj
wi .t/dt
j D 1; :::; n 1
i D 1; :::; m
(15.31)
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Dynamic Facility Location Problem
361
jk
Step 2 Define w W
i
jk
wD
i
Ztk
wi .t/dt
k D j C 1; :::; n
j D 1; 2; :::; n 1
i D 1; 2; :::; m
j < k:
tj
(15.32)
jk
For every demand point i; Calculate the value of w for all values of j and k. This
i
generates the integrated weight associated with the i th demand point for the location
of the facility during the time interval Œtj ; tk /:
Step 3 For each interval [bj , bk /, find the optimal location of the facility (xj k ,
jk
yj k /. By the value of w and the coordinates of the existing facilities (ai ; bi /, the
i
optimal solution for facility location is used.
Step 4 If the new facility has the same location during the time interval [tj , tk /,
calculate Cj k , the location cost, using the following relationship:
Cj k D
m
X
jk
jk
(15.33)
wi : d.X:Pi /;
i D1
where d.X j k ; pi / is the distance between the optimal location of the new facility
and demand point i for Œtj ; tk / and Xj k is calculated in step 3.
Step 5 Using the cost coefficients in step 4, the following model is used to find
the optimal relocation times:
m: number of existing facilities
j
j
wi : the weight associated with demand point i in period j
wi .t/: the weight associated with demand point i at time t
jk
w : the integrated weight associated with the i th demand point during the time
i
interval of [tj , tk /:
.x j k ; y j k /: the optimal location of the new facility during the time interval of
Œtj ; tk /:
Sj : is the relocation cost at time j .
d.X j k ; pi /: the distance between the new facility and i th demand point during
the time interval of [tj , tk /:
Zj k : binary variable equal to 1 if the current relocation takes place at tj and the
next one at tk , and is 0 otherwise.
Min F D
n
X
kD2
n1 X
n
X
j D1 kDj C1
Z1k D 1;
Cj k :Zj k C
n1 X
n
X
Sj Zj k ;
(15.34)
j D2 kDj C1
(15.35)
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R.Z. Farahani et al.
k1
X
j D1
n1
X
j D1
Zj k D
n
X
Zkl
lDkC1
Zj n D 1
8k D 2; : : : ; n 1;
8j; k;
(15.36)
(15.37)
Zj k D 0; 1 8j; k:
(15.38)
This is a binary integer programming (BIP) model with n constraints and n.nC1/=2
variables.
Equation (15.35) ensures that the relocation decision starts at time t1 D 0.
Equation (15.36) enforce the consideration of the next relocation time exactly
after the last relocation time.
Equation (15.37) guarantee that the decision making will continue to the end of
planning horizon, T . The BIP model can be solved using optimization software such
as LINGO and CPLEX.
The solution to this BIP model yields the times for relocation. The facility location for each time interval is found in step 3. The total cost of this policy is F .
Step 6 After solving the model, the following solution is obtained:
Z1b D Zbc D : : : D Zst D Zt n D 1 and all the other variable are equal to 0.
So t1 ; tb ; : : :; tc ; ts ; tt are relocation times and .x t n ; y t n /,(x st ; y st /,. . . ,.x bc ; y bc /,
.x 1b ; y 1b / are the locations of the new facilities after relocation. The total cost of
this policy is F .
In this model there are no constraints on the number of relocations. The maximum number of possible relocations is:
N D
n1 X
n
X
Zj k :
(15.39)
j D1 kDj C1
If we wish to limit this number to L, the following constraint is added to model.
n1 X
n
X
j D1 kDj C1
Zj k L:
(15.40)
15.3.8.1 Location Dependent Relocation Cost
Another situation closer to real world applications is the case in which relocation
cost depends on the location of the facility; i.e. the facility relocation cost at the
time tj is Sj .X D x; y/ instead of Sj , where X D .x; y/ is the location of facility.
For instance, one major part of the cost for locating facility is the price of land. Price
of land for locating a facility will differ from one location to another.
In this model, we solve the problem similar to the procedure described in the
pervious section. The only difference is that the cost of the best location during
[tj ,tk / will be the minimum of the function:
15
Dynamic Facility Location Problem
Cj k D Sj .X / C
363
m
X
i D1
jk
wi d.X j k ; pi /:
(15.41)
This means that the second term of the objective function is not independent of
the first term.
If Sj .X D x; y/ is differentiable, the minimum of Cj k could be calculated directly. When squared Euclidean distances are used, Cj k is differentiable and there
is no need to use Hyper approximation procedure (HAP) for the calculation of the
optimal Cj k (Z.-Farahani et al. 2009).
15.3.8.2 Relocation Cost Depend on the Location of Facility
Before and After Relocation
Sometimes, relocation cost at the time tj depends on the location of facility before
and after tj . In practical cases, there are situations in which the cost of relocating to a
closer location is lower than relocating to a farther location. For example, in mobile
facilities, the relocation cost will be a function of distance between the location of
facility before and after relocation time. In this case the objective function is:
Cj k D u.tj / d.X hj ; X j k / C
m
X
i D1
jk
wi d.X j k ; pi /:
(15.42)
where th is the last relocation point before tj , d.X hj ; X j k / is the distance between the location of facility before tj .X hj / and after tj .X hk /. In addition, u.tj / is
the unit distance cost for relocating a facility from its location to another new location, at the time tj . In this case, Lemma 3 is not satisfied and the objective function
is not additive.
For a finite time horizon, we propose to perform complete enumeration and find
all of feasible paths from t0 to tnC1 . When n is large we must resort to heuristic
algorithms. For each feasible solution, we should calculate the objective function
and select the minimum. The number of these feasible paths can be calculated with
respect to the number of relocations (0, 1, 2) as follows:
1C
n
n
n
C
C ::: C
D 2n :
1
2
n
(15.43)
For solving the objective function of each feasible path, note that if both d.X j k ; pi /
and d.X hj ; X j k / are rectilinear distance, we can solve an equivalent LP substituting
the absolute function with new variables and constraints. If both d.X j k ; pi / and
d.X hj ; X j k / are squared Euclidean distance, the objective function is differentiable
and the problem can easily be solved. If the type of distances related to d.X j k ; pi /
and d.X hj ; X j k / is different or both are Euclidean, we can use the HAP (Francis
and Lowe 1992).
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15.3.9 Relocation at Continues Times
As it was noted, relocation can take place only at pre-determined points in time or,
at any point in the time horizon. In this section, first, we will propound a lemma
and then we will study the facility location–relocation problem with linear time
dependent weights.
Roll Theorem (Leithold 1976). If f .x/ is continuous in [0,T ] and differentiable
in (0,T ) and if f .a/ D f .b/, then there is a c such that a < c < b and f .c/ D 0.
Suppose that there are no candidate times for relocation, and relocation can take
place at any point in the time horizon [0,T ]. Therefore, time break (B/ enters the
cost objective function. The cost objective function will be:
F .B; x1 ; y1 ; x2 ; y2 / D
ZB X
m
0
i D1
wi .t/:d.X1 ; pi /:dt C
ZT X
m
B
wi .t/:d.X2 ; pi /:dt;
i D1
(15.44)
where 0 < B < T .
Given a distinct time break .B/, other variables are calculated, and for every B
there is only one point (x1 ; y1 / and (x2 ; y2 /. In other words, (x1 ; y1 / and (x2 ; y2 /
functions of B).
x1 D f1 .B/ y1 D f2 .B/ x2 D f3 .B/ y2 D f4 .B/:
(15.45)
According to (15.44) and (15.45), the cost function is:
F .B; x1 ; y1 ; x2 ; y2 / D F .B; f1 .B/; f2 .B/; f3 .B/; f4 .B//:
(15.46)
Relation (15.46) shows that the cost function is function of time break and we
can use F as F .B/ according to (15.44):
F .B D 0/ D F .B D T /:
(15.47)
The approximate rectilinear distance is (Francis and Lowe 1992):
jx ai j C jy bi j D
p
p
.x ai /2 C " C .y bi /2 C ":
(15.48)
Using this approximation, the cost function of (15.47) is continuous in [0, T ] and
derivable in (0, T ).
According to Roll Theorem, there is at least one B, such that 0 < B < T and is
an extreme point for cost function.
It is important to note that the weights, wi .t/; cannot be negative in the [0; T ].
For the linear weights the following holds:
ui 0;
vi
ui
:
T
(15.49)
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Dynamic Facility Location Problem
365
For single relocation, linear weight function is put in the cost function (15.44):
ZB X
m
i D1
0
ZB X
m
i D1
0
wi .t/:d.X1 ; pi /:dt C
ZT X
m
B
i D1
d.X1 ; pi /:.ui C vi :t/:dt C
wi .t/:d.X2 ; pi /:dt D
ZT X
m
i D1
B
d.X2 ; pi /:.ui C vi :t/:dt: (15.50)
Integration and sigma are replaced:
B
m Z
X
T
d.X1 ; pi /:.ui C vi :t/:dt C
i D1 :
m
X
m Z
X
i D1 B
m
X
d.X2 ; pi /:.ui C vi :t/:dt D
ˇ
ˇ
ˇ
ˇ
1
1
2 ˇB
2 ˇT
d.X2 ; pi /:.ui :t C vi :t / ˇ D
d.X1 ; pi /:.ui :t C vi :t / ˇ C
2
2
0
B
i D1
i D1
m
X
1
d.X2 ; pi /:.ui :.T B/
d.X1 ; pi /:.ui :B C vi :B 2 / C
2
i D1
i D1
m
X
1
C vi :.T 2 B 2 //:
2
(15.51)
For approximate rectilinear function, (15.51) will be differentiable and concave
and, therefore, we will have:
* m
+
@ X
1
@F
2
D
d.X1 ; pi /:.ui :B C vi :B /
@B
@B i D1
2
+
* m
1
@ X
d.X2 ; pi /:.ui :.T B/ C vi :.T 2 B 2 // : (15.52)
C
@B i D1
2
Integration and sigma are replaced:
X
m
m
X
1
@
@
@F
d.X1 ; pi /:.ui :B C vi :B 2 / C
D
hd.X2 ; pi /:.ui :.T B/i
@B
@B
2
@B
i D1
i D1
m
X
1
C vi :.T 2 B 2 // D
d.X1 ; pi /:.ui C vi :B/
2
i D1
C
D
m
X
i D1
m
X
i D1
d.X2 ; pi /:.ui vi :B/
d.X1 ; pi /:.ui C vi :B/
m
X
i D1
(15.53)
d.X2 ; pi /:.ui C vi :B/:
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In the optimal point (15.53) must be equal to zero:
m
X
i D1
hd.X1 ; pi /:.ui C vi :B/ d.X2 ; pi /:.ui C vi :B/i
D0 )
D0)
m
X
i D1
m
X
i D1
m
X
i D1
h.ui Cvi :B/.d.X1 ; pi / d.X2 ; pi //i
ui :.d.X1 ; pi / d.X2 ; pi //
vi B:.d.X1 ; pi / d.X2 ; pi // D 0:
(15.54)
Therefore, B will be equal to:
B D
m
P
i D1
m
P
i D1
ui :.d.X1 ; pi / d.X2 ; pi //
(15.55)
:
vi B:.d.X1 ; pi / d.X2 ; pi //
Note that to make sure that B is at minimum, the second derivation must be
positive.
@2 F
>0
@B 2
m
@ X
@B
i D1
@2 F
@
D
2
@B
@B
@F
@B
d.X1 ; pi /:.ui C vi :B/
m
D
m
X
i D1
!
d.X2 ; pi /:.ui C vi :B/ )
X
@2 F
D
vi :.d.X1 ; pi / d.X2 ; pi // > 0:
2
@B
i D1
(15.56)
According to lemmas 3 and 4, if the relocation times are determined then:
w.1/
i D
ZB
0
w.2/
i D
ZT
B
ˇ
1
1 2 ˇˇ B
.ui C vi t/:dt Dui t C vi t ˇ D ui B C vi B 2
2
2
0
ˇ
ˇT
1
1
.ui C vi t/:dt Dui t C vi t 2 ˇˇ D ui .T B/ C vi .T 2 B 2 /:
2
2
B
.1/
(15.57)
Therefore, using wi we can solve facility location problem and obtain the location of the facility before relocation, (x1 ; y1 /, then using w.2/ i , the location of
facility after relocation .x1 ; y1 / is reached.
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Dynamic Facility Location Problem
367
Obviously, doing these calculations with hand, particularly for large problems,
is very difficult. However, nowadays, with the technological advances, doing this
calculation with computer is not so difficult and time consuming. Thus, in this subsection, we present an algorithm that obtains optimal point in this problem.
15.3.10 Iterative Algorithm for Obtaining Optimal Solution
.1/
.2/
In this algorithm, first an optional B in interval [0, T ] is selected, and wi and wi
are calculated using (15.57). Then, .x1 ; y1 / and .x2 ; y2 / are calculated through solv.2/
.1/
ing the two obtained location problems based on wi and wi . Next, by substituting
these four variables in (15.55), a new B value is obtained. All the mentioned steps
are repeated using this new B. It can be shown by Roll theorem that these iterations
converge to a B point.
Not that (15.56) must be satisfied in order to have the F function minimized
for the given B value, otherwise, we cannot claim that the obtained B minimizes
F . Even if (15.56) is satisfied, there is no proof that the obtained point is global
minimum. To solve this problem, repeat the algorithm with different Bs as a starting
point. Therefore, if a local minimum exists, we can obtain global minimum.
As it was explained before, the steps of the global minimum algorithm for the
obtained break time and optimal location are as fallows:
Step 1 Generate a reasonable number for search steps. The smaller this value, the
more accurate and the slower the algorithm, and vice versa. Set count D 0; k D 1,
F D 1. Enter coordinates of the existing facilities, linear functions parameters
(ui ; vi ), and permissible errors.
Step 2 Set count D count C step if count T then go to step 10 else go to next
step.
Step 3 Set B k1 D count.
Step 4 As described before and by considering B k1 , calculate values of
k1
.x1 ; y1k1 / and .x2k1 ; y2k1 /.
Step 5 calculate the result of (15.55) as follows:
m
P
i D1
Bk D
m
P
i D1
ui :.d.X1k1 ; pi / d.X2k1 ; pi //
:
(15.58)
vi B:.d.X1k1 ; pi / d.X2k1 ; pi //
ˇ
ˇ
Step 6 If ˇB k B k1 ˇ > ı then set k D k C 1 and go to step 4 else go to the
next step.
Step 7 calculate the result of (15.56) as follows:
m
X
vi :.d.X1k1 ; pi / d.X2k1 ; pi //:
(15.59)
i D1
If this value is negative then set k D 1 and go to step 2 else go to the next step.
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Step 8 calculate the cost using the following equation:
F D
m
P
i D1
C
.d.X1k1 ; pi / d.X2k1 ; pi //:.ui :B k C 21 vi :B k2 /
m
P
i D1
(15.60)
d.X2k1 ; pi /:.ui :T C 21 vi :T 2 //:
Step 9 If F < F then set F D F; B D B k ; .x1 ; y1 / D .x1k1 ; y1k1 /;
y2 / D .x2k1 ; y2k1 /; k D 1 and go to step 2 else without changes go to step 2
Step 10 B ; .x1 ; y1 /; .x2 ; y2 / and F are optimal solutions.
Note that there is also another approach for solving single relocation and multi
relocation time-independent problems that interested readers are referred to Owen
and Daskin (1998) for detailed information.
.x2 ;
15.3.11 Static Stochastic Techniques
The dynamic models described in the previous sections attempt to locate facilities assuming that input parameters are known values or vary deterministically
over time.
In stochastic location problems, any number of system parameters might be taken
as uncertain. Stochastic location problems can be broken down into probabilistic
approach and the scenario planning approach (Owen and Daskin (1998).
15.3.11.1 Probabilistic Approach
Sometimes, we examine models which capture the stochastic aspects of facility location through explicit consideration of the probability distributions associated with
modeled random quantities.
These distributions can be incorporated into standard mathematical programs, or
in a queuing framework (Owen and Daskin 1998).
15.3.11.2 Scenario Planning Approach
Scenario planning is a method in which decision makers capture uncertainty by
specifying a number of possible future states. The objective is to find solutions
which perform well under all scenarios. In some applications, scenario planning replaces forecasting as a way to evaluate trends and potential changes in the business
environment. Firms can thus develop strategic responses to a range of environmental
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Dynamic Facility Location Problem
369
changes, more adequately preparing themselves for the uncertain future. Under such
circumstances, scenarios are qualitative descriptions of plausible future states, derived from the present state with consideration of potential major industry events. In
other applications, scenario planning is used as a tool for formulating and solving
specific operational problems (Owen and Daskin 1998).
The goal of scenario planning is to specify a set of scenarios, which represent
the possible realizations of unknown problem parameters and to consider the range
of scenarios in determining a compromise (robust) location solution (Owen and
Daskin 1998). There are at least three approaches to incorporating scenario planning
into location modeling:
Optimizing the expected performance over all scenarios,
Optimizing the worst-case performance, and
Minimizing the expected or worst-case regret across all scenarios. (Owen and
Daskin 1998).
Scenario planning approach, though invaluable in dealing with conditions where
there is uncertainty about input parameter information, has two disadvantages:
Defining scenario and assigning probabilities to them is difficult
Because of the high calculation efforts required, the number of evaluated scenarios are kept low and therefore, the future condition is restricted (might not be
realistic enough) (Snyder 2006).
The regret associated with a scenario is calculated by comparing the performance
of the optimal locations for the scenario (had planners known for certain that the
scenario would be realized) with the performance of the compromise locations when
the scenario is realized. (Owen and Daskin 1998)
In other words, by extent of loss, also referred to as “opportunity loss” it is meant
what the decision maker pays for performing under the conditions of not knowing
about the occurrence of a scenario as compared with his/her performance in the
compromise locations when the scenario has already realized (Snyder 2006).
Thus, using a regret-based objective allows us to evaluate robust solution alternatives with respect to the optimal solution obtained under data certainty.
One should use this measure in objectives which:
Require assessment of scenario realization probabilities, and
Assume that all scenarios are of equal probability.
15.4 Case Study
In this section, we will introduce some case studies related to Dynamic Facility
location problems.
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15.4.1 A Dynamic Model for School Network Planning (Antunes
and Peeters 2000)
This case study describes a dynamic multi-period optimization model that has been
used in Portugal to formulate planning proposals for the evolution of several school
networks.The solution to this problem gives information about where and when new
schools should be built, what their sizes should be, which schools should be kept
open and which ones should be closed. Additionally, for the schools which remain
open, it identifies which schools should maintain the same size and which should be
resized, becoming larger or smaller.
This model considers the following features:
Over the planning horizon, each site will not change its state more than once;
Both setup and operation costs have a component dependent on capacity and a
component dependent on attendance; and
Facility size is limited to pre-defined standards, expressed in terms of an architectural module, for instance, a given number of classrooms.
The objective function of this model minimizes the total discounted (socioeconomic) costs of a set of facilities. Facility costs are divided into three parts:
fixed costs; capacity-variable costs proportional to the number of modules; and
attendance-variable costs proportional to the number of users, a significant part of
which will normally consist of transport costs.
The approaches that had been used to solve this model are as following:
Branch and bound (B&B), but this method is unsuccessful to solve a few
10 10 3 .centers sites periods/ problems.
Dual-based and LR exact methods. However, this kind of method would hardly
be successful in the current case.
The “myopic” and “panoramic” approaches (heuristics). These methods in the
presence of severe capacity inadequacy and important demand decreases the
situation encountered in many Portuguese regions when the expansion of elementary education was decided didn’t have successful function.
SA: For 50 problems in 17, SA was unable to find the optimum B&B solution.
Moreover, in all these 17 problems, it came quite close to it, as the difference
was always smaller than 1%. In the remaining 33 problems, SA behaved at least
as efficiently as B&B.
15.4.2 A Multiperiod Set Covering for Dynamic Redeployment
of Ambulances (Rajagopalan et al. 2008)
The dynamic available coverage location model was formulated to determine the
minimum number of ambulances and their locations for each time cluster in which
significant changes in demand pattern take place while meeting coverage requirements with a predetermined reliability.
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Dynamic Facility Location Problem
371
A straightforward search algorithm was developed to determine the minimum
number of servers to provide the coverage and availability requirements. The algorithm starts with an estimated fleet size for the first time interval, which is called
a reactive Tabu search (RTS) algorithm. A look-ahead procedure (LAP) within the
RTS was used to compute server specific busy probabilities and the resulting coverage. Results showed that the RTS with LAP takes them very close to number of
servers required to satisfy the coverage constraint generally 2–3 times faster than
the algorithm without LAP.
15.4.3 A Multi-period Model for Combat Logistics (Gue 2003)
In the past, logistics support for amphibious warfare has depended on a large, landbased infrastructure, with trucks accomplishing most of the distribution. New war
fighting concepts for the US Marine Corps emphasize small, highly-mobile forces
supported instead from the sea. The goal of logistics planners is to support these
forces with as little inventory on land as possible. But the best distribution system
depends on a number of operational levers.
For better conception, this study has considered a sea base containing combat
and support units. Each combat unit is required to reach a particular set of objectives
on land; support units are positioned to provide supplies as needed. Combat units
consume food, water, ammunition, and fuel during each time period. Quantities may
vary depending on the intensity of conflict or other concerns. Supply units are free to
deploy, move, and to build up and deplete inventories as necessary to meet demand.
A fleet of vehicles is available to transport combat units to objectives or intermediate
points, to move entire supply units, or to transport supplies between units. Naturally,
vehicle types are constrained to transport only between feasible origin–destination
pairs. The problem is to determine the locations of supply units for each time period
and the shipments of each commodity between units, such that there is as little landbased inventory as possible.
In this case a multi-period, facility location and multi-commodity flow model is
formulated as a mixed integer program. The battle space is modeled as a network
of two types of nodes, combat and supply nodes. In model, it is assumed that the
combat nodes are given in a battle plan and that supply units may not occupy them.
And also that intelligence could provide a set of candidate locations for supply units.
The objective is to minimize the total inventory of land-based support units, in keeping with the primary purpose of sea-based logistics. Decisions in the model are, for
each time period, the locations of support units, inventories held by the units, and
the amounts shipped between units.
Two limitations of this model are:
Transportation capacity is modeled in units of lb mile, the model could propose
a solution that is impossible to implement in practice.
A solution could also require more transporters than are available.
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References
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Chapter 16
Multi-Criteria Location Problem
Masoud Hekmatfar and Maryam SteadieSeifi
Decision-making is the process of selecting a possible subset of decisions from
all the available alternatives (feasible space). We introduced many decision-making
models including one objective function, so far. Almost there are many criteria for
judging the optimality of decision. In this situation, we will be faced with the multicriteria decision-making (MCDM). There are many methods to solve the MCDM
problems, but all of them are common in some factors that are as follows (Hwang
and Masud 1979; Hwang and Yoon 1981):
Multiple objectives/attributes
Conflict among criteria
Incomparable units
Design or selection: solutions to these problems consist of two classes, one of
them designs the best alternative and the second one selects the best solution
among the specified finite alternatives
The MCDM is classified into two categories:
The multi-objective decision-making (MODM), and
The multi-attribute decision-making (MADM)
First, briefly, we define some prevalent important terms used in this area (Hwang
and Masud 1979; Hwang and Yoon 1981):
Criteria: Criteria are standards of judgment or rules to test acceptability
Goal: Goals are a priori levels of aspiration that are determined by decision makers (DMs) with explicit term
Objective: Objectives are the directions that increase the inspiration of DMs
(Maybe it seems that the two recent terms have a common meaning, while objectives give the desired direction, goals give a desired level to achieve.)
Optimal solution: Optimal solution is one which results in maximum value of
each of the objective functions. Because of the conflicting objectives, this solution will not usually exist
Efficient solution (Nondominated solution): xQ is an efficient solution if there exists no other feasible solution that will yield an improvement in one objective
without causing a reduction in at least one other objective
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 16, c Physica-Verlag Heidelberg 2009
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Preferred solution: A preferred solution is selected by DM from efficient
solutions
Satisfing solution: A Satisfing solution Satisfies a minimum acceptable level for
all attributes of the problem
The MCDM comprises some general techniques that can be used in many types of
application including facilities location.
The remainder of this chapter will first present some classifications, objective
functions and mathematical models of MCLPs. Next, solution techniques will be
briefly reviewed. Finally, in Sect. 16.3 some real-world case studies will be introduced.
16.1 Applications and Classifications
Multi-criteria analysis of location problems has received considerable attention
within the scope of continuous and network models in recent years. There are several problems that are accepted as classical ones:
The point-objective problem (Wendell and Hurter 1973; Hansen et al. 1980;
Pelegrin and Fernandez 1988; Carrizosa et al. 1993)
The continuous multi-criteria min–sum facility location problem (FLP)
(Hamacher et al. 1996; Puerto and Fernandez 1999)
The network multi-criteria median location problem (Hamacher and Nickel 1998;
Wendell et al. 1977)
So far, multi-criteria analysis of discrete location models has attracted less attention.
Several authors have dealt with problems and applications of multi-criteria decision
analysis in this field. For instance,
Ross and Soland (1980) worked on multi-activity multi-facility problems and
proposed an interactive solution method to compute non-dominated solutions to
compare and choose from
Lee et al. (1981) studied an application of integer goal programming for facility
location with multiple competing objectives
Solanki (1991) applies an approximation scheme to generate a set of nondominated solutions to a bi-objective location problem
Ogryczak (1999) looks for symmetrically efficient location patterns in a multicriteria discrete location problem
In general, none of the above papers focuses on the complete determination of the
whole set of non-dominated solutions. The only exception is the paper by Ross
and Soland (1980) that gives a theoretical characterization but does not exploit its
algorithmic possibilities.
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16.2 Models
Although location problems have a long history, we can assume that the modern
location has been started from Weber research on one vehicle location problem with
the objective function based on minimizing the total rectilinear distances. This trend
was continued up to 1960s, minimizing the total distances (min–sum).
In 1960s, Hakimi (1964) represented a new kind of problems, minimizing the
maximal distance (min–max). In this decade, many operation researchers worked
on location problems (Eiselt and Laporte 1995).
In the mid 1970s, it seems that the topic of obnoxious location problem was
represented by some researchers such as Goldman and also Church and Granfinkel
for the first time (Eiselt and Laporte 1995). Maybe we can assume that after coming
up of this topic in multi-objective problems, the MCDM techniques were started to
be used to solve the MCDM problems.
Since the aim of this section is FLPs from objective functions point of view, in
the following sections we try to represent the main classification of objective functions, conflicting of contradictory objectives and at last some case study of location
problems.
We concentrate to display these problems and their classifications based on their
objective functions.
16.2.1 Private and Public Facilities
Problems that are naturally multi-objectives are generally discussed in public facilities, semi-noxious facilities and desirable facilities. Cohon (1978) represented
MCLPs in a separate section. He divided FLPs into two main classes:
Facility location in private sections
Facility location in public sector
In the first group, problems have almost only one objective and the main purpose
is almost minimizing the cost of construction, production and transportation, or the
purpose is maximizing the profit earned from constructing the facility. This kind of
problems is located in class of classic location problems.
In the second group, problems that are efficient in governments’ final decisionmaking (directly or indirectly), are divided into three classes:
Facilities with ordinary services
Facilities with emergency services
Facilities with undesirable effect
In ordinary services like libraries, hygienic clinics etc., the purpose usually consists
of the maximizing of service to more people.
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In emergency services like fire stations the purpose is usually consisted of maximizing the minimal covering.
In undesirable facilities, there is an undesirable facility like a place for repulsing
refuses, wastes, etc., and the purpose is consisted of minimizing the effect of this
area on the residents near it.
It seems that the Cohon’s (1978) classification has some problems and it can be
possible to represent better classification. We represent some of them in later parts.
In public sectors, the main problem is the lack of proper criteria and their different natures. For example in a fire station location problem, there are two conflicting
objectives, one of them is consisted of minimizing the cost of facility and the other
one is consisted of preserving the human life. so how the human life can be evaluated?
In Cohon (1978), two location problems are represented, emergency services
location problem (construction of a fire station in Baltimore) and power generation
location problem in a power station. Readers can refer to Cohon (1978) for more
details.
16.2.2 Balancing Objective Functions
A group of objective functions are discussed in public facility and services and the
purpose is consisted of maximizing the equity and balanced services among users.
These problems are introduced as fairness functions, balancing functions and equity
functions.
Displaying the value of equity is varied in different problems, but it can be possible to generate a new class of objective functions based on a development from
min–sum functions and a statistic view.
In prevalent min–sum functions, it is easy to show that minimizing the total
distances of demand points from facility is statistically equal to minimizing the average of distances with assuming one facility and locating in one dimension. (see
Eq. 16.1).
n
P
.xi L/
n
X
i D1
! n d
(16.1)
.xi L/ D n
min W
n
i D1
where L is the situation of facility.
We can assume some other functions like the total squared distance of demands
from facility or cubical distance of demands.
From statistics point of view, the total distances (first torque) are related to average distances, total squared distances (second torque) are related to variance of
distances, and cubical distances (third torque) are related to scale of symmetry for
point distribution around facility location place. In other words, by minimizing the
second torque we can obtain a set of solutions in which the distance oscillation of
demand points from the facility location are minimized.
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With minimizing the total cubical distances, the set of solutions will finally have
less asymmetry. It means that there is not anyone who lives very close or very far
from the facility.
Balancing objective functions can reach a better solution in dealing with public
FLPs and if we see these three functions simultaneously, it will be necessary to use
techniques of multi-objective programming (Eiselt and Laporte 1995).
16.2.3 Pull, Push and Pull–Push Objectives
The various types of facilities from customer point of view were represented by
Eiselt and Laporte (1995) as follows:
Desirable
Undesirable
Partly desirable
Partly undesirable
Indifferent
According to an acceptable classification which was represented by Krarup
et al. (2002), Curtin and Church (2003), and Eiselt and Laporte (1995), objective functions are divided into two main classes: Pull and Push. The combination
of these two classes is introduced as pull–push term and is a major topic in multiobjective real problems.
Another kind of classifications is related to service distribution and establishment
of equity which is represented as balancing objective by Eiselt and Laporte (1995).
Pull Objectives The classic location problems, desirable facilities and generally
older location problems have pull objective functions. For example, it is tried to minimize the distance between production plants and consuming markets which yielded
minimizing the cost of transportation, maximizing the availability of customers and
increasing part of the market.
The classic pull-objective functions are generally divided into four classes:
P-median (min–sum), P-center (min–max), UFLP1 (min–sum) and QAP (min–
sum).
Push Objectives Most of the undesirable location problems such as obnoxious
and noxious and also dispersion models have push objective functions. It means that
in these problems the purpose is consisted of decreasing the noxious effect of facility
on demand points. Since in most cases the noxious effect has an inverse relation with
the distance, the purpose of these problems is consisted of maximizing the distance
between facility and demand points. A complete survey of these problems has been
represented by Curtin and Church (2003).
Erkut and Neuman (1990) made a comparison of dispersion models and divided
them into four main classes: P-dispersion (max–min–min), P-defense (max–sum–
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min), Anti-hub (max–min–sum) and Maxi-sum dispersion/P-maxian (max–sum–
sum). It can be seen that they employed a three-syllable naming convention for
objective functions with maximizing aim to distinguish between different types of
dispersion. Using this convention, the first syllabus for each model was “Max” denoting that all models attempt to maximize the amount of dispersion among selected
facility sites. Both the second and third syllables were either “sum” or “min”. In the
second syllable, a “sum” operator indicates a concern for overall system performance while a “min” operator indicates a concern for worst-case performance (here
least distance). The third syllable is again either “sum” or “min”, and it refers to
which facility interactions comprise the minimum distance to be considered by the
formulation. When the “min” operator is employed, the objective function is only
forced by the minimum distance between any two facilities in a given solution. The
“sum” operator indicates that the distances from each facility to all other facilities
will force the objective function.
Push–Pull Objectives As it was indicated in the previous sections, in most of the
real problems, simultaneously there are some conflicted objectives. It means that objective functions are combined with desirable and undesirable effects. It is noticed
that these problems deal with interference of government like municipal, ecological organization, etc. as an example of these conflicted objectives, assume that the
purpose is consisted of both increasing the distance between production plants and
residential places and also increasing the cost of transportation, decreasing the attraction of professions, etc.
There is a clear example about these types of problems. In establishing a superstore or a sport club, the purpose of investors is decreasing the distance between
their buildings and residential places to attract more customers, but residents prefer
to live not too far and not too close to these facilities. Such problems are introduced
with terms like “close but not too close” and “never in my back yard”. These classes
of problems had not been surveyed until 1970s.
A complete review about discrete location problems with push–pull objectives is
represented by Krarup et al. (2002).
As it has been mentioned in previous sections, there is a classification of classical location problems with pull-objectives and desirable facilities: P -median, UFLP,
QAP with min–sum objective function and P -center with min–max objective function. In QAP demand points and facilities are the same things and the demand is
equal to material flow or information between facilities.
Also there is a classification of classical location problems with push-objectives:
The first group is consisted of undesirable facilities which have noxious effect on
health like chemical wastes and obnoxious effect on environment like the noise of
sports clubs. The second group is consisted of dispersion models (Erkut and Neuman 1990) which there are classically undesirable effects between their facilities.
Krarup et al. (2002) represented that there is a famous problem, apparently first
formulated in the early 1600s by Fermat, together with kind of a companion problem
termed complementary problem (ComP). For a given triangle, Fermat asks for a
fourth point such that the sum of its Euclidean distances, each weighted by C1, to
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the three given points is minimized. ComP differs from Fermat in that the weight
associated with one of these points is 1 instead of C1.
It can be possible to divide push–pull problems into two major groups:
Problems that only inter-facility interactions are considered, some being desirable, and some undesirable, this group is called a semi-dispersion model. There
are some cases in this group such as: QAP with undesirable effect, quadratic
knapsack problem that a number of facilities may be established at some predefined locations and then undesirable effect will be occurred, P -defense-sum
problem and P -dispersion (Krarup et al. 2002)
Problems that involve a set of demand points which pull the facilities towards
them, whereas the undesirable part will be concerned with another set of subjects
which push the facilities away. These problems are modeled as bi-objective with
the attractions and the repulsions as two quite opposing objectives, each of which
might still be multi-objective (Krarup et al. 2002)
The strategies that are used to solve these bi-objective problems are summarized
as follows:
Treat the bi-objective problem as really bi-objective and derive (approximations
to) the Pareto-set
Fix a bound on the obnoxious effects as a (set of) constraint(s) and optimize
the desirable objective. This leads to standard pull objectives with additional
restrictions
Fix a bound for the desirable objective part and optimize the undesirable objective. This leads to standard push objectives with an additional constraint
Combine the push and pull objectives into a single objective
16.2.4 Mathematical Models
In this section, some multi-objective problems with their mathematic models which
Krarup et al. (2002) called them general push–pull will be represented. First, there
are some remarks, which are represented as follows:
The type of modeling has a major effect on final solution. It means that the style
of exhibition in an undesirable effect (as you can see in following sections) will
be affected on type of the objective function (the problem will be single objective
or multiple objective)
The type of objective functions is usually very different between desirable effect and undesirable effect. For example in a network problem, assume that the
purpose is consisted of establishing a plant which pollutes the air. In one part of
objective function with desirable effect, we can use shortest route method on the
network to represent objective function with minimizing the transportation costs.
In another part of objective function with undesirable effect (e.g. air pollution),
we should use Euclidean distance to represent objective function, therefore the
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problem will be NP-hard and we can not use the network properties to find the
optimal solution
Objectives with distance measuring unit are ascending functions in desirable effect and are descending functions in the undesirable effect with a negative (too
negative) slope in minimal covering. Since one of the purposes of multi-objective
functions is consisted of conflicting between objectives, it is necessary to use
techniques of multiple objective functions to solve location problems (e.g. push–
pull objectives)
16.2.4.1 Push–Pull Models
These models consider FLPs in which a set of individuals are specified, which are
negatively (potentially) affected by the facilities. All the models considered are NPhard to solve and thus the solution techniques must rely on some kind of enumerative
methods. Heuristic approaches may also be applied for the solution of large-size
instances (Krarup et al. 2002).
UFLPs with Additive Noxious Effects: The UFLP has some assumptions as
follows:
Finite set I of demand points (users)
Finite set J of candidate points for demands (users)
Each demand point (user) uses a certain facility
Facilities are uncapacitated
The number of facilities are exogenous
Costs include variable cost of customer services and fixed cost fj , the cost of
establishing the facility in j th point
In this problem, the set k is consisted of a set of points such as candidate points and
demand points. akj express the noxious effect of facility j on point k.
The initial model is formulated as follows:
1 0
1
0
X
XX
XX
cij yij C
fj xj A C @
min Z W @
(16.2)
akj xj A
i 2I j 2J
k2K j 2J
Subject to
X
j 2I
yij D 1; i 2 I
yij xj ;
i 2 I;
(16.3)
j 2 J;
xj 2 f0; 1g; yij 2 f0; 1g;
i 2I
(16.4)
j 2 J:
(16.5)
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If we change the model like fQj D fj C
P
akj , then the model will be converted
k2K
to a UFLP standard model facility with single objective. We can see the importance
of modeling in such problems.
UFLPs with Minimal Covering as Noxious Effects: In several situations, however, it is not reasonable to assume that the obnoxious effects are additive. For
example, the facilities poison the subsoil water, every affected subject in some predefined radius will have to connect to the public water supply instead of using its
own pump.
If pollution occurs in a certain radius, after that the other facilities do not have
a noxious effect in that area. In other words, pollution occurs by establishing one
facility and increasing in the number of facilities has not any effect on that.
Therefore, the cost of undesirable effect is constant and does not depend on the
number of facilities located close to the subject. Since this is an obnoxious effect
of minimal covering type, we obtain a different model which may be formulated as
follows:
1
0
X
X
XX
fj xj A C
cij yij C
ak zk
(16.6)
min Z @
i 2J j 2J
j 2J
k2K
Subject to
X
j 2I
yij D 1;
yij xij
xj zk ;
i 2I
(16.7)
i 2 I; j 2 J;
j 2 Ck ; k 2 K;
xj 2 f0; 1g;
yij ; zk 2 f0; 1g;
(16.8)
(16.9)
i 2 I;
j 2 J;
k 2 K:
(16.10)
where the variable zk here is used to indicate whether subject k is affected by any
facility or not.
If a facility is established around Ck from point k, it will be resulted in zk D 1
and ak 0.
This problem will be solved by a bi-objective problem with minsum–minsum
objective function. The represented method to solve this problem is approximation
of Pareto-set and finally comparing and selecting by the DM (Krarup et al. 2002).
Another more stringent push objective is the worst-case like maxmin type, where
one wants to minimize the maximal effect any subject feels from any of the facilities.
When this objective is considered next to the minsum pull objective of the UFLP,
another bi-objective model arises.
As before, we use akj to express the noxious effect of a facility at site j on
subject k. Introducing aQ j D maxk2K akj and an auxiliary variable z for the second
objective, we arrive at the linear, bi-objective MIP formulation:
min
X X
cij yij C
X
fj xj
(16.11)
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M. Hekmatfar and M. SteadieSeifi
min z
Subject to
X
yij D 1;
yij xj ;
xj 2 f0; 1g;
(16.12)
i 2I
i 2 I;
(16.13)
j 2 J;
yij ; zk 2 f0; 1g;
(16.14)
i 2 I;
j 2 J;
k 2 K:
(16.15)
Since undesirable facility problems have multi-objective functions in real world,
there are many case studies in the new topic of location science. Therefore, we will
introduce some literature review of multi-objectives briefly.
Rakas et al. (2004) represented a multi-objective model for undesirable facilities
based on fuzzy programming.
Erkut and Neuman (1989) displayed the necessity of using multi-objectives in
undesirable FLPs. They added that the weakness of single objective problems has
been emphasized in location science like repulsion place of urban wastes.
ReVelle (2000) has emphasized the importance of equity functions in undesirable FLPs. When wastes are generated by residents of an area, it is not fair to
consider just negative effects of wastes on this area; therefore we should use the
multi-objective models.
Ratick and White (1988) developed a three-objective for undesirable FLPs.
Erkut and Tansel (1992) expanded the Ratick and White (1988) models and
solved the problem under condition of facilities with variable capacity by enumerating the possible solutions for possible capacities. This method is useful only for
small size problems.
Wayman and Kuby (1994) represented a three-objective model for undesirable
FLPs. They assumed the following objectives:
Minimizing the fixed cost and transportation cost, minimizing the risk and minimizing the inequity.
Rahman and Kuby (1995) developed a multi-objective model for locating a facility to transport wastes. They represented a bi-objective model with minimizing
the installation cost and minimizing the public opposition as their objectives. The
public opposition is a descending function from distance to residential places.
Giannikos (1998) represented a goal programming model for locating and carrying the dangerous wastes by transportation networks. In this model, it is not only
assumed to minimize the operation costs and the total available risks, but also it is
assumed to consider the distribution of risk and the desirability among populated
centers.
Berman et al. (2000) developed a multi-objective location routing problem (LRP)
on the network.
Berman and Drezner (2000) represented a bi-objective to find the best point on
the network to establish a dangerous waste repulsion center.
It was a brief review of undesirable FLPs and it needs more research to find
further subjects in this area.
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16.2.4.2 The MCLP on the Network
Hamacher et al. (1996, 1998, 2002) surveyed MCLP on the network in different situations such as models with desirable effect and semi-noxious effect. Their models
are usually consisted of single objective function but multiple weighted vectors (the
first group of MCLPs). The results of their researches are consisted of representing
a set of solution algorithms with polynomials under states of set-pareto and lexicography solution. Most of the proofs are based on a trick series and graph theory.
A review of solved problems in this area, their solution codes, complexity of
these problems can be found in their papers. Readers can refer to Hamacher et al.
(1996, 1998, 2002) to find more related subjects. In the followings, MCLP on the
network based on properties and final solution can be seen in brief:
On the network, graph without any direction, distance node to node, function
min–sum, lexicography solution (final solution on the node)
On the network, graph without any direction, distance node to allocation point,
function min–sum, lexicography solution (final solution on the node)
Graph with direction, distance node to node and node to allocation point, function
min–sum, lexicography solution (final solution on the node)
Graph without any direction and with direction, distance node to node, pareto
solution (final solution on the node)
Graph without any direction, distance node to allocation point, pareto solution
(final solution on the node and vectors)
Graph with direction, distance node to allocation point, pareto solution (final
solution on the node) and
Graph without any direction and with direction, function min–sum with respect
to desirable and undesirable effects (final solution on the node)
There are some complicated algorithms with anti-center objective function that
are represented by Hamacher et al. (1996, 1998, 2002), too
16.3 Solution Techniques
16.3.1 The MCDM Techniques
Naturally, these techniques include both the MADM and the MODM techniques.
16.3.1.1 The MADM Techniques
The distinguishing feature of the MADM problems is that there are usually a limited number of predetermined alternatives. These alternatives satisfy each objective
in a specified level and the DM selects the best solution (or solutions) among
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all alternatives according to the priority of each objective and the interaction between them. There are many techniques, which are used to tackle the MADM
problems. The most popular ones are as follows: dominant, maximin, maximax,
conjunctive method, disjunctive method, lexicographic method, elimination by aspects, permutation method, linear assignment method, simple additive weighting
(SAW), hierarchical additive weighting, ELECTRE, TOPSIS, hierarchical tradeoffs, LINMAP, interactive SAW method and MDS with ideal point (Hwang and
Yoon 1981).
16.3.1.2 The MODM Techniques
In the MODM problems the purpose is to design the best alternative by considering
the various interactions within the design constraints which best satisfy the DM by
way of attaining some acceptable levels of a set of some objectives. The MODM
problems have various components, but the common characteristics of them are as
follows:
A set of quantifiable and quality objectives
A set of well defined constraints
A process of obtaining some tradeoff information
There are many techniques, which are used to tackle the MODM problems. The
most popular ones are as follows: global criterion method, utility function, metric
L-P methods, bounded objective method, lexicographic method, goal programming,
goal attainment method, method of Geoffrion, interactive GP, surrogate worth tradeoff, method of satisfactory goals, method of Zionts-Wallenius, STEM and related
method, SEMOPS and SIGMOP method, method of displaced ideal, GPSTEM
method, method of Steuer, parametric method, C-constraint method and adaptive
search method (Hwang and Lin 1987; Hwang and Masud 1979; Szidarovszky
et al. 1986; Ulungu and Teghem 1994; Zionts 1979).
However, in solving the MODM problems, without respect to the used technique,
investigating the following general steps is needed:
Conflicting Objectives: It is in the nature of the MODM problems to have conflicting objectives
Efficient Solution: An ideal solution to a MODM problem is one that results in
the optimum value of each of the objective functions simultaneously. An efficient
solution (Ulungu and Teghem 1994) (also known as non-inferior solution or nondominated solution) is one in which no one objective function can be improved
without a simultaneous detriment to the other objectives
A preferred Solution: A preferred solution (also known as the best solution) is
an efficient solution, which is chosen by the DM as the final decision. We have
used some of simple MODM methods and some sensibility analysis to choose a
preferred solution
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16.3.2 Metaheuristics for the MODM
Various approaches have been utilized to solve multi objective optimization problems. In general, these approaches can be divided into three categories. The approaches in first category, which is named classical, such as weighted sum approach
attempt to convert the multi objective problem into a single objective problem
and optimize new single objective problem (Hajela and Lin 1992; Murata and
Ishibuchi 1995). Optimizing this single objective problem yields a single solution;
but the DMs need diverse options in the real condition. In addition, some classical
approaches require knowing the optimal solution of each objective; acquiring this
information about problem is expensive and time consuming. Their dependency to
chosen weights, special in the case of non deterministic situation, is another fault
of these approaches. Therefore, determination of these weights is really difficult.
The second category of approaches includes those act based on Pareto optimal. In
this case, a set of solutions are acquired when the problem is solved. These approaches, often, utilize evolutionary algorithms to solve multi objective problem
which are more complicated than those that can be solved by deterministic optimization approaches like linear programming. So, these approaches, due to utilizing
evolutionary algorithms, require little information about the problem in addition
to yield a set of solutions. Some of these approaches are MOGA2 (Fonseca and
Fleming 1993), NSGA3 (Srinivas and Deb 1994) and NSGA II4 (Deb et al. 2002).
Nevertheless, there are some approaches which neither convert the multi objective
into single objective nor act based on Pareto optimal. These approaches are settled
in the third category. Some of these approaches are: Vector Evaluated Genetic Algorithm (Schaffer 1985), Lexicographic Ordering, Weight Min–Max Method and
Distance Method.
16.3.3 Multi-Objective Combinatorial Optimization
Nowadays, multi-objective combinatorial optimization (MOCO) (see Ehrgott and
Gandibleux 2000; Ulungu and Teghem 1994) provides an adequate framework
to tackle various types of discrete multi-criteria problems. Within this emergent
research area several methods are known to handle different problems such as
dynamic programming enumeration (Villarreal and Karwan (1981), for a methodological description; Klamroth and Wiecek (2000), for a recent application to knapsack problems) and implicit enumeration (Zionts and Wallenius 1980; Zionts 1979;
Klein and Hannan 1982; Rasmussen 1986; Ramesh et al. 1986). Another approach
based on labeling algorithms can be seen in Captivo et al. (2000). It is worth
noting that most of MOCO problems are NP-hard and intractable (Ehrgott and
2
Multi Objective Genetic Algorithm
Non-dominate Sorting Genetic Algorithm
4
Fast Non-dominate Sorting Genetic Algorithm
3
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M. Hekmatfar and M. SteadieSeifi
Gandibleux 2000, for further details). Even in most of the cases where the singleobjective problem is polynomially solvable, the multi-objective version becomes
NP-hard. This is the case of spanning tree problems and min-cost flow problems,
among others. In the case of uncapacitated plant location problem, the singleobjective version is already NP-hard (Krarup and Pruzan 1983). This ensures that
the multi-objective formulation is not solvable in polynomial time. In this context,
when time and efficiency become real issues, different alternatives can be used to approximate the Pareto optimal set. One of them is the use of general-purpose MOCO
heuristics (Gandibleux et al. 2000). Another possibility is designing “ad hoc” methods based on one of the following strategies:
Computing the supported non-dominated solutions
Performing a partial enumeration of the solutions space
Obviously, the second strategy does not guarantee the non-dominated character of
all the generated solutions since we only consider the solutions obtained during the
partial search. Nevertheless the reduction in computation time can be remarkable.
16.4 Case Study
In this section we will introduce some real-world case studies related to MCLP.
16.4.1 LRP (Lin and kwok 2006)
In this paper an integrated logistic system are surveyed, where decisions on location of depot, vehicle routing and assignment of routes to vehicles are considered
simultaneously. Total cost and workload balance are common criteria influencing
decision-making. Literature on LRPs addressed the location and vehicle routing decisions with a common assumption of assigning one route to one vehicle. However,
the cost of acquiring vehicles (and crew) is often more significant than the routing
cost. This notion of assigning several routes to a vehicle during the routing procedure is explored in their integrated model. Meta heuristics of tabu search and
simulated annealing are applied on real data and simulated data, to compare their
performances under two versions: simultaneous or sequential routes assignment to
vehicles. Results show that the simultaneous versions have advantage over the sequential versions in problems where routes are capacity-constrained.
Logistics operations often involve sending out staff from their offices or depots
to customers in various areas to perform on-site tasks. There are several decisions
to be considered as follows:
Selecting the depot locations
Scheduling routes from selected depots to customers
Assigning routes to vehicles/crew
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These decisions are often inter-dependent and each change has influenced other
decisions.
The total cost is an obvious and primary concern of management. This includes
the fixed cost of selected depots, traveling cost of the routes and cost of vehicles and
crew.
This paper examined the integrated model on the total cost and workload balance
criteria that are important to both management and operational staff.
LRP consists of two sub problems:
FLP
Vehicle Routing Problem
Both of them are shown to be NP-hard.
In the MCDM problems, after establishing the location of depots and delivery
routes for vehicles, demands of customers will be satisfied and management will
reach its purposes.
Readers can refer to Lin and Kwok (2006) to find further subjects about structure
of used algorithms.
16.4.2 Facility Layout (Chiang et al. 2006)
Most of researches have been managed on the facility layout problem, much of it
based on the QAP formulation. The QAP objective is to minimize the transportation
cost which expressed as the product of the quantity of workflow and the distance
traveled.
In addition of the common purpose to minimize the transportation costs, the focus
of this research is also consisted of minimizing the intersections between transportation routes. This problem is used in production lines of plants and traffic problems,
because of the increasing number of intersections, it will derive more queues and
more delays in transportation.
To illustrate the effect of conducting a layout analysis that fails to account for
workflow interference, they consider an eight-department example with the workflow matrix. The solution to this problem using traditional layout analyses (QAP is
solved using Euclidean distances) is provided. This layout arrangement minimizes
the total distance traveled.
Chiang et al. (2006) modeled workflow interference in facility layout design as a
quadratic assignment problem, and developed a branch-and-bound procedure and a
tabu search heuristic to solve the problem.
In this paper, to solve the problem, first the problem is separated into two certain
problems which their objective functions are consisted of minimizing the product of
traveled distance and the number of transportations and minimizing the number of
intersections among routes respectively.
The first problem can be expressed as a QAP and the second problem will be
converted to QAP after adding some constraints to the original model.
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M. Hekmatfar and M. SteadieSeifi
16.4.3 Fire Station Locations
Location of fire stations is an important factor in its fire protection capability. Yang
et al. (2007) aim to determine the optimal location of fire station facilities. The
proposed method is the combination of a fuzzy multi-objective programming and
a genetic algorithm. The fuzzy multiple objectives are converted to a single unified
min–max goal, which makes it easy to apply a genetic algorithm for the problem
solving. Compared with the existing methods of fire station location this approach
has three distinguishing features:
Considering fuzzy nature of a DM in the location optimization model
Fully considering the demands for the facilities from the areas with various fire
risk categories
Being more understandable and practical to DM
The case study was based on the data collected from the Derbyshire fire and rescue
service and tried to illustrate the application of the method for the optimization of
fire station locations.
Determination of where to locate fire stations and how many fire stations to have
in a given area is the most important decision faced by any Chief Fire Officer. The
optimum solution is deduced to minimize the sum of losses from fire and the cost
of providing the service. The fire station location problems have multi-objective
functions which are NP-hard.
Multiple objectives often conflict with each other and require multi-objective approaches. Tzeng and Chen (1999) considered three objectives in the optimal location
of airport fire stations:
Minimizing the total setup cost of fire stations and total loss cost of an incident
Minimizing the longest distance from the fire stations to any incident point
Minimizing the longest distance from any fire station to the high-risk area
Badri et al. (1998) considered 11 objectives in a general MODM for locating fire
stations.
Minimize fixed cost
Minimize total annual operating cost
Maximize service of those areas that have most probability based on number of
forecasted accidents
Minimize average distance travelled from station to accident sites
Minimize maximum distance travelled from station to accident sites
Minimize average time travelled from station to accident sites
Minimize maximum time travelled from station to accident sites
Attain the number of fire stations required
Minimize service overlaps of fire stations
Attain the favored condition of areas
Minimize locating where water availability could be a problem
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The main concept of multi-objective problems based on fuzzy approach is consisted
of finding the highest level of availability among conflicting constraints. Two common goals usually used in many emergency-location (ambulance, fire station etc.)
related studies:
To minimize the fixed cost and the total loss cost of incidents
To minimize the distance from the fire station to any incident site
Tzeng and Chen (1999) represented an approach for the first objective, to obtain the
optimal number of fire stations, and they use a fuzzy multi-objective model for
the second objective to address the recommendation made by the Home Office on
the time limits for attendance at incident sites.
16.4.4 The 2-Facility Centdian Network Problem (Perez-Brito
et al. 1998)
The P-facility centdian network problem consists of finding the P points that minimize a convex combination of the P -center and P -median objective functions. The
vertices and local centers constitute a dominating set for the 1-facility centdian.
In this paper, the problem of selecting several points of a network is considered in
order to minimize a function that is distance dependent with respect to given points
of the network. The median and the center problems are two well-known problems
with numerous possible applications.
In many real world problems, the objective to be optimized in a model is a combination of (possibly conflicting) goals. For example, when locating a fire station,
there may be two objectives as follows:
To minimize the distance between stations and farthest potential customer
To minimize the distance between stations and majority of customers
The first function is a center problem and the second one is a median problem. It is
reasonable, because to locate the fire station centers, the purpose consisted of both
minimizing the distance between center and the farthest potential incident point
(center problem) and minimizing the distance between center and the most densely
populated point (median problem).
Halpern (1978) represented this multi-objective approach for locating a facility on a network (who coined the term centdian for this problem). Hakimi (1964)
demonstrated that the set of vertices (nodes) is a finite dominating set for the
P -median problem. Moreno (1985) proved that the set of vertices and local centers (points, in the internal of the edges, which have equal distance and balanced
with respect to two vertices) is a finite dominating set for the P -center problem.
Halpern (1978) proved that the set of vertices and local centers of the network is a
finite dominating set for the single facility centdian problem. Hooker et al. (1991)
considered a theoretical result that extended the finite dominating sets of the single facility problems to the corresponding P-facility problems, and applied it to the
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M. Hekmatfar and M. SteadieSeifi
P-facility centdian problem. It is not reasonable; therefore a counterexample has
been given by Perez-Brito et al. (1998). In this paper finally an optimal solution
consisting of vertices and local centers is presented as a set of final solution for
solving the 2-centdian on the network. It also presented a solution procedure for a
network that improves the complexity of the exhaustive search in the dominating
set. It also provided a very efficient algorithm that solves the 2-centdian on a tree
network with complexity O.n2 /.
16.4.5 Military Logistics (Z-Farahani and Asgari 2007)
The case of this research relates to the MCDM (including the MADM and the
MODM) and set covering problem. In this paper, it is represented to locate some
warehouses as distribution centers in a real-world military logistics system. This
study looks for finding the least number of distribution centers and locating them in
the best possible locations. The objectives of the problem are as follows:
Maximizing the utility of the selected locations. Utility of a potential point depends on 23 attributes
Minimizing the number of supportive centers
The first objective deduced the minimum cost of locating the facilities and the latter expresses the quality of the distribution centers locations, which is evaluated by
studying the value of appropriate attributes affecting the quality of a location. Quality of a warehouse location depends on a number of attributes; so the value of each
location is determined by using the MADM methods (hereby, TOPSIS). Then, regarding the obtained values and the minimum number of distribution centers, the
two objective functions are formed. Constraints imposed on these two objectives
cover all centers, which must be supported by the distribution centers. Using the
MODM techniques (hereby, utility function), the locations of distribution centers
are determined. In the final phase, a simple set partitioning model is represented to
assign each supported center to only one of the located distribution centers.
16.4.6 A Paper Recycling System (Pati et al. 2008)
Since reducing and reutilizing waste being defined as one of the requirements for
substantial development, the reverse logistic concept of a supply chain has been
used to optimize the recycling system of the paper industry of India.
There are five levels of facilities in this reverse distribution network: customer,
dealer, godown owner, supplier and the manufacturer. Also, there are two varieties
of wastepaper. Based on the requirements of the problem, the goals/objectives are:
1. Minimizing the reverse logistics cost (minimizing the positive deviation from the
planned budget)
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2. Minimizing the quantity on non-relevant wastepaper in the reverse distribution
network (minimizing the positive deviation from the maximum limit of nonrelevant wastepaper target)
3. Maximizing the wastepaper collection at source (minimizing the negative deviation from the minimum desired waste collection)
A mixed integer goal programming (MIGP) model has been proposed for this FLP
in which these objectives are lexicographically minimized and the Pareto inefficient
or dominated solutions are produced, based on six priority structures CWN,CNW,
NCW, NWC, WCN and WNC where C, N and W represent each of above objectives
respectively.
Afterwards, the Pareto optimality of the solution is detected by constructing a
new achievement function from non-weighted deviational variables and is compared
with the initial optimal solution. The new achievement objective is to maximize the
sum of deviational variables which are not present in the achievement function of
the MIGP model.
In conclusion, for each priority, the network sub-entities to be closed or opened
are determined along with the associated route and quantity of flow for different
varieties of papers in this multi-item, multi-echelon and multi-facility network.
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Chapter 17
Location-Routing Problem
Anahita Hassanzadeh, Leyla Mohseninezhad, Ali Tirdad,
Faraz Dadgostari, and Hossein Zolfagharinia
The aim of this chapter is to survey the state of the art in location-routing. The
location-routing problem (LRP) is a research area within location analysis, with the
distinguishing property of paying special attention to underlying issues of vehicle
routing.
Since the Vehicle-Routing Problem (VRP), which is a complex problem itself,
is known as a basic component of LRPs, the main concepts of VRP are introduced
in the first section. In the second section, the definition, applications and classifications of LRP are discussed. The next section is dedicated to the LRP models and
introduces some basic mathematical models of this field. Solution techniques are
presented in the fourth section. Finally, three case studies regarding applications of
LRP in real world are briefly reviewed.
17.1 An Introduction to VRP
Since LRPs are extensions of classical VRPs, here we provide a brief introduction
to VRP including its definition, major applications and classification.
17.1.1 Definition of VRP
The VRP is one of the most studied among the combinatorial optimization problems, due both to its practical relevance and to its considerable difficulty. The VRP
is concerned with the determination of the optimal routes used by a fleet of vehicles,
based on one or more depots, to serve a set of customers.
Typical applications of this problem are, for instance, solid waste collection,
street cleaning, school bus routing, dial-a-ride systems, transportation of handicapped persons, routing of salespeople, and of maintenance units.
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 17, c Physica-Verlag Heidelberg 2009
395
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A. Hassanzadeh et al.
The distribution of goods, in a given time period, concern the service of a set of
customers by a set of vehicles, which are located in one or more depots, are operated
by a set of crews (drivers), and perform their movements by using an appropriate
road network. In particular, the solution of a VRP calls for the determination of a
set of routes, each performed by a single vehicle that starts and ends at its own depot, such that all the requirements of the customers are fulfilled, all the operational
constraints are satisfied, and the global transportation cost is minimized. Here, we
describe the typical characteristics of the routing and scheduling problems by considering their main components (road networks, customers, depots, vehicles, and
drivers), the different operational constraints that can be imposed on the construction of routes, and the possible objectives to be achieved in the optimization process.
The road networks, used for the transportation of goods, is generally described
through a graph, whose arcs represent the road sections and whose vertices correspond to the road junctions and to the depot and customer locations. The arcs (and
consequently the corresponding graphs) can be directed or undirected, depending
on whether they can be traversed in only one direction (for instance, because of the
presence of one-way streets, typical of urban or motorway networks) or in both directions, respectively. Each arc is associated with a cost, which generally represents
its length, and a travel time, which is possibly dependent on the vehicle type or on
the period during which the arc is traversed.
Typical characteristics of customers are:
Vertex of the road graph in which the customer is located
Amount of goods (demand), possibly of different types, which must be delivered
or collected at the customer
Periods of the day (time windows) during which the customer can be served (for
instance, because of specific periods during which the customer is open or the
location can be reached, due to traffic limitations)
Time required to deliver or collect the goods at the customer location (unloading
or loading times, respectively), possibly dependent on the vehicle type
Subset of the available vehicles that can be used to serve the customer (for
instance, because of possible access limitations or loading and unloading requirements)
Sometimes, it is not possible to fully satisfy the demand of each customer. In these
cases, the amounts to be delivered or collected can be reduced, or a subset of
customers can be left unserved. To deal with these situations, different priorities
associated with the partial or total lack of service, can be assigned to the customers.
The routes performed to serve customers start and end at one or more depots,
and are located at the vertices of the road graph. Each depot is characterized by
the number and types of vehicles associated with it and by the global amount of
goods it can deal with. In some real-world applications, the customers are a priori
partitioned among the depots, and the vehicles have to return to their home depot
at the end of each route. In these cases, the overall VRP can be decomposed into
several independent problems, each associated with a different depot.
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Transportation of goods is performed by using a fleet of vehicles whose composition and size can be fixed or can be defined according to the requirements of the
customers. Typical characteristics of the vehicles are:
Home depot of the vehicle, and the possibility to end service at a depot other than
the home one
Capacity of the vehicle, expressed as the maximum weight, or volume, or number
of pallets, the vehicle can load
Possible subdivision of the vehicle into compartments; each characterized by its
capacity and by the types of goods that can be carried
Devices available for the loading and unloading operations;
Subset of arcs of the road graph which can be traversed by the vehicle
Costs associated with utilization of the vehicle (per distance unit, per time unit,
per route, etc.)
Drivers operating the vehicles must satisfy several constraints laid down by union
contracts and company regulations (for instance, working periods during the day,
number and duration of breaks during service, maximum duration of driving periods, overtime). In the following, the constraints imposed on drivers are imbedded in
those associated with the corresponding vehicles.
The routes must satisfy several operational constraints, which depend on the
nature of the transported goods, on the quality of the service level, and on the characteristics of the customers and the vehicles. Some typical constraints are the
following: along each route, the current load of the associated vehicle can not exceed
the vehicle capacity; the customers served in a route can require only the delivery
or the collection of goods; or both possibilities can exist; and customers can be
served only within their time windows and the working periods of the drivers associated with the vehicles visiting them. Precedence constraints can be imposed on
the order in which the customers served in a route are visited. One type of precedence constraint requires that a given customer be served in the same route serving
a given set of other customers and that the customer must be visited before (or after)
the customers belonging to the associated subset. This is the case, for instance, of
the so-called pickup and delivery problems, wherein the routes can perform both the
collection and the delivery of routes, and the goods collected from the pickup
customers must be carried to the corresponding delivery customers by the same
vehicle.
Another type of the precedence constraints impose that if customers by different
types are served in the same route, the order in which customers are visited is fixed.
This situation arises, for instance, for the so-called VRP with back hauls, wherein
again, the routes can perform both the collection and the delivery of goods, but
constraints associated with the loading and unloading operations, and the difficulty
in rearranging the load of the vehicle along the route, mean that all deliveries must
be performed before the collections (Toth and Vigo 2002).
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17.1.2 The Traveling Salesman Problem
There exist a few exact methods formed to solve LRPs, and most of them are represented in recent years. These algorithms are usually on the based on the linear
integer models in the VRPs, which include the locating dimension, too. Hence, it
is useful to have an overview of an exact method that is considered in the field of
VRPs.
The first integer linear programming formulation for the Traveling Salesman
Problem (TSP) belongs to Dantzig and Fulkerson (1954).
Define:
N D f1; 2; :::; ng a set of nodes .points customers or cities/
C D .cij /; .i; j 2 I [ J / W a distance matrix .undefined if i; j 2 I /
xij : a 0–1 variable equal to 1 if and only if edge .i; j / is chosen in the optimal
solution. xij only need to be defined if i j . Here and elsewhere, xij must be
interpreted as xij whenever i j .
The problem is then to
X
cij xij
(17.1)
Minimize
i;j 2J
Subject to
X
i k
xi k C
X
i;j 2S
X
k j
xij
xkj D 2 .k 2 N /
1
.S / 1
2
.S N I 2 jS j n 2/
xij D 0; 1 .i 2 I /
(17.2)
(17.3)
(17.4)
In this formulation, (17.2) are degree constrains: they specify the degree of each
node. Equation (17.3) are subtour elimination constrains: they prevent the formation of subtours over proper subsets of N. Equation (17.4) are integrality constrains
(Laporte et al. 1988).
17.1.3 A Classification of Capacitated VRP
Here we concentrate on the basic version of the VRP, the Capacitated VRP (CVRP).
In the CVRP, all the customers correspond to deliveries and the demand are deterministic, known in advance, and may not be split. The vehicles are identical and
based on a single central depot, and only the capacity restrictions for the vehicles
are imposed. The objective is to minimize the total cost to serve all the customers
(Toth and Vigo 2002).
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Route Length
CVRP
DCVRP
Time
Windows
Backhauling
VRPB
Mixed Service
VRPTW
VRPBTW
VRPPD
VRPPDTW
Fig. 17.1 The basic problems of the VRP class and their interconnections (Toth and Vigo 2002)
Figure 17.1 shows a classification corresponding to the basic problems of the
VRPs and their interconnections. In this figure, CVRP stands for Capacitated VRP,
DCVRP for Capacitated and Distance Constrained VRP, VRPB for VRP with Backhauls, VRPTW for VRP with Time Windows, VRPPD for VRP with Pickup and
Delivery, VRPBTW for VRP with Backhauls and Time Windows, VRPPDTW for
VRP with Pickup and Delivery and Time Windows.
17.2 LRP
The basic concepts of LRPs were introduced by Boventer (1961), Maranzana
(1965), Webb (1968), Lawrence and Pengilly (1969), Higgins (1972) and
Christofides and Eilon (1969). These primary studies did not consider the complexity of LRP as a combined problem. Introduction and extensions of LRP as a
combined problem commenced at the late 1970s and early 1980s. These studies
include articles of Or and Pierskalla (1979), Jacobsen and Madsen (1978) and
Laporte and Norbert (1981). Such popularity of LRP studies almost parallels the
advent of an integrated logistics concept and the growth of international trade which
necessitated distribution efficiency ( Min et al. 1998).
According to the articles which have been published from 1972 to 1996, the
potential future research areas are:
Stochastic LRPs
Time windows LRPs
Dynamic LRPs
LRPs with multiple objectives
We may distinguish between two types of dynamic problems, in one, the depots
are located sequentially. In the other, the depots are located at the beginning of the
planning horizon and vehicle routes vary with the variations in customer demand.
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The former case is more applicable if demand is increasing and the latter if demand
is fluctuating (Nagy and Salhi 2007).
In order to understand the relation between LRP and the classical location problems, it seems necessary to define its place among location problem. Therefore, we
firstly concentrate on the types of trips in which the main difference between the
LRP and the classical location-allocation problem (see Chap. 5) lies. Usually, it is
assumed that users or customers are directly connected to facilities. These types of
trips between customers and facilities are called out-and-back, direct, or return trips.
But there exist many cases in which the trip commences from a facility and covers
many customers. In addition to identifying the number and location of the facilities,
the followings must be determined:
The allocation of customers to the facilities
The allocation of customers to the routes
The order of visiting the customers in a route
These two kinds of trips are presented in Fig. 17.2 (Daskin 1995).
From the customer servicing point of view, location problems can be divided to
two categories:
The customers are being serviced in their own locations
The customers take trips to facilities to get serviced
The common examples of the second category are schools and hospitals.
In general, two cases occur in the first category. The server must return to the
facility after serving the customer, like fire engines (direct trips) or the server can
visit many customers in a tour, like repairers and postmen (tour trips). Figure 17.3
demonstrates this division.
Therefore, if there exist direct trips, the problem is a location-allocation problem
(see Chap. 5), and if there exist tour trips, the problem is a LRP. Hence, a LRP
contains both location and tours ( Min et al. 1998; Lin and Kwok 2006).
a
Facility
Direct Trips
b
Fig. 17.2 Two types of trips
between customers and facilities (Daskin 1995) (a) direct
trips (b) tour trips
Facility
Tour Trips
17
Location-Routing Problem
401
Location
Problems
Selecting the location of
new facilities
Customers are being
serviced in their own
locations
Customers visit the
facility
Direct Trips
(ambulances,
fire engines)
Tours
(postmen, repairers)
Fig. 17.3 Different types of servicing the customers
LRPs are clearly related to both classical location problem and VRP. In fact, both
of the latter problems can be viewed as special cases of the LRP. If we require all
customers to be directly linked to a depot, the LRP becomes a standard location
problem. If, on the other hand, we fix the depot locations, the LRP reduces to a
VRP (Nagy and Salhi 2007).
After defining the place of the problem, the LRP can be defined as follows: A
feasible set of potential facility sites and locations and expected demands of each
customer which are given. Each customer is to be assigned to a facility which will
supply its demand. The shipments of customer demand are carried out by vehicles
which are dispatched from the facilities, and operate on routes that include multiple
customers. The location of distribution facilities and the distribution of products
from these facilities to customers are two key components of a distribution system.
In various different settings, these two components are interdependent; therefore
it is necessary to consider the facility location and the distribution decisions simultaneously (Tuzun and Burke 1999).
Figure 17.4 is a simple representation of LRP sub problems and their
interdepenence.
17.2.1 Applications of LRP
Although most of the applications of location-routing focus on distribution of consumer goods or parcels, there are also some applications in health, military and
communications. Some of these applications are in:
Food and drink distribution
Waste collection
402
A. Hassanzadeh et al.
Fig. 17.4 The relationship
between LRP components
(Min et al. 1998)
Location
Allocation
Routing
Table 17.1 A summary of LRP applications
Author
Application area
Country/region
Watson et al. (1973)
Bednar and Strohmeier (1979)
Or and Pierskalla (1979)
Jacobsen and Madsen (1980)
Nambiar et al. (1989)
Perl and Daskin (1985)
Labbe and Laporte (1986)
Nambiar et al. (1989)
Semet and Taillard (1993)
Kulcar (1996)
Murty and Djang (1999)
Bruns et al. (2000)
Chan et al. (2001)
Lin et al. (2002)
Lee et al. (2003)
Wasner and Zäpfel (2004)
Billionnet et al. (2005)
Gunnarsson et al. (2006)
Food and drink distribution
Consumer goods distribution
Blood bank location
Newspaper distribution
Rubber plant location
Goods distribution
Postbox location
Rubber plant location
Grocery distribution
Waste collection
Military equipment location
Parcel delivery
Medical evacuation
Bill delivery
Optical network design
Parcel delivery
Telecom network design
Shipping industry
United Kingdom
Australia
United States
Denmark
Malaysia
United States
Belgium
Malaysia
Switzerland
Belgium
United States
Switzerland
United States
Hong Kong
Korea
Australia
France
Europe
Blood bank location
Newspaper distribution
Table 17.1 summarizes the main characteristics of papers describing practical
applications (Nagy and Salhi 2007).
17.2.2 Classifications of LRP
Various classification schemes are available in the literature to categorize LRPs
(Min et al. 1998).
A classification of LRP with regard to its problem perspective is presented in
Table 17.2.
17
Location-Routing Problem
403
Table 17.2 Classification of LRP with regard to its problem perspective (Min et al. 1998)
I.
Hierarchical level
A. Single stage
B. Two stages
II.
Nature of demand
A. Deterministic
B. Stochastic
Number of facilities
A. Single facility
B. Multiple facilities
III.
IV.
Size of vehicle fleets
A. Single vehicle
B. Multiple vehicles
V.
Vehicle capacity
A. Uncapacitated
B. Capacitated
VI.
Facility capacity
A. Uncapacitated
B. Capacitated
VII.
Facility layer
A. Primary
B. Secondary
VIII.
Planning horizon
A. Single period
B. Multiple periods
IX.
Time restriction
A. Unspecified time with no deadline
B. Soft time windows with loose deadlines
C. Hard time windows with strict deadlines
X.
Objective function
A. Single objective
B. Multiple objectives
XI.
Types of model data
A. Hypothetical
B. Real world
These classification schemes do not require a specific description. The only vague
ones may be:
Hierarchical level, which can be in one or two stages. One stage means one type
of facility and two stages mean two types. (For example one manufacturing plant
and one distribution center)
Facility layer, which will be described in Sect. 17.2.2.1
Objective function, which is usually a single objective of cost reduction, but several other objectives may be considered (Averbakh and Berman 1995a, b; Jamil
et al. 1994)
404
A. Hassanzadeh et al.
17.2.2.1 Classification of LRP Regarding Layer Diagram
If we consider users or customers on the one hand, and facilities to be located on
the other hand, we can theoretically arrive at the definition of four different types
of problems depending on whether users or facilities are assumed to belong to discrete or continues sets (Laporte 1988). We also temporarily assume that users and
facilities belong to disjoint sets.
Several distribution centers can be represented by a layer diagram such as the one
depicted by Fig. 17.5. In this example, there are three layers which have been identified as primary facilities, secondary facilities, and users. Frequently, the primary
facilities will represent factories, secondary facilities will correspond to depots or
warehouses, and users will be customers. Primary facilities and users are usually
situated at known and fixed locations. On the other hand, the location of secondary
facilities will frequently not be determined a priori: the number or locations of
these facilities, together with the associated distribution routes constitute decision
variables.
Several problems can be defined according to the distribution mode Mt used by
vehicles based on a facility located at layer t. Usually, these vehicles will make trips
to layer t C 1. We consider two distribution modes:
Mt D R: all trips from layer t must be return trips. (i.e. trips to and from a single
user or facility);
Mt D T : trips from layer t may be tours. (i.e. round-trips through several users
or facilities)
The distribution modes used for the whole system will be represented by the
expression =M1 =M2 = : : : =M1 , where is the number of layers. Thus, in a threelayer system, we will have the four following possibilities:
3=R=R: mostly, Occurs when shipments of a generally bulky material (e.g. lumber, cement) have to be made in full loads between successive layers, as is depicted
in Fig. 17.5.
3=R=T : Here, large shipments arriving at the secondary facilities are broken
up and dispatched in smaller loads to customers, as is depicted in Fig. 17.6. This
situation is encountered in food industry, for example.
Secondary
Facilities
Customers
Primary
Facilities
P1
Fig. 17.5 3=R=R layer
diagram (Laporte 1988)
P2
17
Location-Routing Problem
405
Fig. 17.6 3=R=T layer
diagram (Laporte 1988)
Secondary
Facilities
Customers
Primary
Facilities
P1
P2
Fig. 17.7 3=T =R layer
diagram (Laporte 1988)
Secondary
Facilities
Customers
Primary
Facilities
P1
P2
Fig. 17.8 3=T =T layer
diagram (Laporte 1988)
Secondary
Facilities
Customers
Primary
Facilities
P1
P2
3=T =R: As in Fig. 17.7, in this situation, trips are often made from the users who
bring goods to the secondary facilities. These goods are then collected in round-trips
and brought to primary facilities.
3=T =T : This case is frequently encountered in the newspaper industry. Here, the
primary facilities are printing plants; newspapers are dispatched daily to secondary
facilities (transfer points) and then again to retail outlets. This situation is presented
in Fig. 17.8.
In LRPs,
– Location decisions must be made for at least one layer (otherwise, the problem
reduces to a pure routing problem)
– Tours .Mt D T / must be allowed to at least once (otherwise the problem becomes a pure location problem)
Therefore, problems of the form =R=R=: : :=R will not be covered in this study.
406
A. Hassanzadeh et al.
Table 17.3 LRP applications (Laporte 1988)
Type
2/T
2/T
3/R/T
Application
Optimal
location of
blood banks
Aircraft
operating
locations
Distribution
of consumer
goods
3/T/R
Rubber
collection
3/T/T
Newspaper
delivery
Layers (with number of sites)
1
2
a
Blood banks Hospitals
(3)
(17)
3
Or and
Pierskalla
(1979)
McLain
et al. (1984)
Military
airportsa (5)
Military
bases (84)
Factories
Depotsa
Rubber
processing
factoriesa (8)
Printing
plants (21)
Collection Small holders
stations (50) (3750)
4/R/R/T Distribution Factories
of consumer
goods
a
indicates locational decision
References
4
Customers
Transfer
points (37)
Retailers
(4500)
Ware
housesa
Depotsa
WatsonGandy and
Dohrn
(1973)
Nambiar
et al. (1989)
Customers
Jacobsen and
Madsen
(1980)
Mercer
et al. (1978)
Case studies describing various LRP implementations have been reported by a
number of authors. These cover several fields of government and economic activities. Some of the most interesting cases are summarized in Table 17.3.
17.3 Models
17.3.1 Classifications
Min et al. (1998) represent the Laporte’s classification of exact methods as follows:
Direct tree search
Dynamic programming
Integer programming
Nonlinear programming
Since the majority of solution approaches consist of integer programming, here we
concentrate on them and examine them further.
Using the classification of Magnanti (1981), Laporte (1988) puts the integer programming algorithms into the following four categories:
Set partitioning algorithms
Commodity flow algorithms
Vehicle flow algorithms
17
Location-Routing Problem
407
Here in this chapter, the described integer models are related to the third class; therefore, a brief description of it is provided here.
Vehicle flow models deal with the optimal circulation of vehicles and users in the
system and do not include costs and constraints directly related to the actual flow of
goods. The TSP and the p-median problem fall into this category.
It is often convenient to classify these models according to the number of indices
of the flow vehicles. Common cases are:
Two-index vehicle flow formulations
Three-index vehicle flow formulations
There are certain advantages and disadvantages in using two-index or three-index
variables in VRP or LRP formulations. Two-index formulations are concise and involve a relatively small number of variables (particularly in the symmetrical case).
However, they can not take into account different vehicle costs and characteristics and can therefore only be applied under the assumption that the vehicle fleet
is homogenous. Successful implementations of such formulations are reported by
Laporte et al. (1985), for example. Three-index formulations are more versatile but,
at the same time, more costly. Classical examples of such formulations are the vehicle flow models proposed by Golden et al. (1977) However, Laporte (1988) states
that three-index formulations have never led to the successful implementation of an
exact algorithm for the VRP.
17.3.2 Mathematical Models
The exact methods generally act on the basis of mathematical programming. They
often include relaxations and redefining constrains as follows:
Subtour elimination: all tours must include only one distribution center
Chain barring: the distribution centers should not be connected through the routes
Integrality: the variables should be integers (or binary)
Exact methods provide significant insights into problems, but due to the complexity of location-routing they can only tackle relatively small instances. General
location-routing instances with up to 40 potential depot locations or 80 customers
have been solved to optimality (Laporte et al. 1988; Laporte and Norbert 1981).
17.3.2.1 The Classical 3/R/T LRP Model
The classical 3=R=T model presented by Laporte 1988) consists of simultaneously
Selecting facility sites at the second layer
Determining the composition of the vehicle fleet based at each facility
Constructing optimal delivery routes from the supply sources to the facilities
located at the second layer and from these facilities to users
408
A. Hassanzadeh et al.
The solution must be such that
The total system cost is minimized
All user requirements are satisfied without exceeding vehicle capacities
The number of vehicles used does not exceed a given bound
Route length are within a given maximum distance
Route durations do not go beyond a present time
Vehicle routes pass through only one facility
Facility throughput capacities are respected
In order to formally express these constrains and objective, the following notation
is introduced. Define sets as following.
L: the set of all supply sources (first layer)
I : the set of all potential facility sites (second layer)
J : the set of all users (third layer)
K: the set of all vehicle routes
Parameters are defined as below:
m
N D jKj: the maximum allowed number of vehicle routes
C D .cij /; .i; j 2 I [ J /: a distance matrix (undefined if i; j 2 I )
T D .tijk /; .i; j 2 I [ J I k 2 K/: a three-dimensional travel time array for all
routes and for all pairs of facilities and users
gi : the fixed cost of establishing facility i 2 I
vi : the variable cost per throughput unit at facility i 2 I
Vi : the maximum throughput at facility i 2 I
sli : the unit transportation cost from l 2 L to i 2 I
dj : the requirement of user j 2 J
Tkj : the time required by the vehicle used on route k to unload at user j 2 J
Dk : the capacity of the vehicle used on route k
Ek : the maximum allowable length (in distance units) of route k
Tk : the maximum allowed duration of route k
pk : the cost per distance unit of delivery vehicle on route k
qij : a fixed cost incurred for delivering from facility i 2 I to user j 2 J
Variables are as following:
Xij k : is equal to 1 if i immediately precedes j on route k, otherwise is 0
Yj : is equal to 1 if a facility is located at site I , otherwise is 0
Zij : is equal to 1 if user j is served from facility i , otherwise is 0
wli : quantity of goods shipped from supply source l 2 L to facility i 2 I .
Then the problem formulation is as follows:
Min
X
i 2I
C
gi yi C
X X
XX
l2L i 2I
X
k2K i 2.I [J / j 2.I [J /
sli wli C
pk cij xijk
XX
i 2I j 2J
.vi dj C qij /zij
(17.5)
17
Location-Routing Problem
409
Subject to
X
X
xi j k D 1 .j 2 J /
(17.6)
X
X
dj xi j k Dk .k 2 K/
(17.7)
k2 K i 2.I [J /
j 2 J i 2.I [J /
X
X
i 2.I [J / j 2.I [J /
X
X
j 2J i 2.I [J /
X X
ci j xij k Ek
k j xi j k C
X
k2 K i 2 S j 2 .I [ J /S
XX
i2 I j 2 J
X
j 2.I [J /
xj i k
X
wl i
X
wl i Vi yi
l2 L
zi j C
i 2.I [J / j 2.I [J /
t i j k x i j k Tk
.k 2 K/
xi j k 1 .2 jSj jI [ J jI S jI [ J jI S \ J ¤ O/
u2 I [J
yi D 0; 1
zi j D 0; 1
X
j 2 .I [J /
.k 2 K; i 2 I [ J /
.i 2 I /
.i 2 I /
.xiuk C xujk / 1
.i 2 I I j 2 J; k 2 K/
.i 2 I /
.i 2 I I j 2 J /
.l 2 LI i 2 I /
(17.9)
(17.10)
(17.11)
xi j k D 0
dj zij D 0
j2J
X
xijk D 0; 1
wli 0
X
X
(17.8)
xi j k 1 .k 2 K/
X
l2 L
.k 2 K/
(17.12)
(17.13)
(17.14)
.i 2 I I j 2 J; k 2 K/
(17.15)
(17.16)
(17.17)
(17.18)
(17.19)
In this formulation, the objective function is the sum of facility fixed costs, first
level delivery costs, variable warehousing costs and delivery costs, respectively.
Equations (17.6) ensure that every user belongs to one and only one route. Equations
(17.7)–(17.9) guarantee that vehicle capacities, maximum route length and maximum route durations, respectively, are respected. Equation (17.10) are connectivity
constrains: they ensure that every user is on a route connected to the set of facilities. Equation (17.11) are flow conservation equations: any point of (I [ J ) must
be entered and left by the same vehicle. Equation (17.12) stipulated that a vehicle can depart only from a facility. This prevents cases where a vehicle leaves the
same facility for two different users and also passes through more than one facility.
Equation (17.13) ensure that the flow entering a facility is equal to the flow exiting
that facility. Equation (17.14) limit the flow through a facility to the capacity of that
facility. Equation (17.15) specify that zij must be equal to 1 if facility i and user j
belong to the same route k. Finally, (17.16)–(17.19) impose bounds and integrality
conditions on the variables.
410
A. Hassanzadeh et al.
17.3.2.2 The TSP with Single Depot
We described the TSP model in part 1, as a basic model in VRP. Here, we extend
this model in order to solve the LRP.
In this very simple form of LRP, we consider a set N of users and a subset I N
of potential sites for a single facility, in order to minimize the total routing cost for
a fleet of exactly m vehicles based at that facility (Laporte 1988).
The first step is to define a binary variable yi equal to 1 if and only if the facility
is located at node i . The remaining notation is as in TSP. The problem is then to
Min
X
(17.20)
cij xij
i;j 2 J
Subject to
X
i 2I
yi D 1 .i 2 I /
(17.21)
X
xi k C
X
xkj D 2 C 2.m 1/yi
X
xi k C
X
xkj D 2
i k
i k
X
i;j 2S
kj
kj
.k 2 I /
.k 2 N I /
xij 21 .S / 1 .S N I 2 jS j n 2/
yi D 0; 1 .i 2 I /
(17.22)
(17.23)
(17.24)
(17.25)
Most constrains of this model are self-explanatory. Equation (17.24) are derived
as follows: any feasible solution must correspond to a connected graph, therefore
the following constrains must hold:
X
i 2 S; j 2 SN
or
i 2 SN ; j 2 S
Xij 2
.S N I 2 jS j n 2/
(17.26)
In Laporte and Norbert (1981), this model is solved by means of a constraint relaxation method inspired by the branch and bound algorithm for the TSP. The model
is solved through initially relaxing the subtour elimination constrains and the integrality conditions on the variables (but by retaining their upper bound). Integrality is
gradually regained by branch and bound; at an integer solution, a check for violated
subtour elimination constrains is made, the constraint corresponding to the subtour
involving the least nimber of nodes is introduced and the problem is reoptimized.
17
Location-Routing Problem
411
17.3.2.3 The TSP with Multi Depot
All the assumptions and objectives of this category are just like the single-depot
problem, the only difference is locating at most p facilities, instead of one. In addition, the triangle ineguality is assumed to be satisfied in I , i.e.
cki C cij ckj
.i 2 I /
(17.27)
Under this condition, it can be shown that there will be only one vehicle for each
opened facility in the optimal solution. (i.e. the m-TSP solution is then dominated
by a TSP tour). As in previous model, we define gi as the fixed cost of opening a
facility at site i 2 I . Then the problem can be formulated as following, with m D 1:
number of vehicles per new facility, and p: maximum number of facilities.
Min
X
cij xij C
i;j" N
X
gi yi
(17.28)
i 2I
Subject to
X
i" I
X
i k
X
yi p.i 2 I /
xik C
i;j" S
X
kj
xk j D 2
xij 12 .S / 1
yi D 0; 1 .i 2 I /
(17.29)
.k 2 N /
.S N I 2 jS j n 2/
(17.30)
(17.31)
(17.32)
Equation (17.29) has been changed as compared with the previous model, ensuring that the number of new facilities should not exceed a maximum of p ones.
Equation (17.22) used in the single-depot model is not applicable here because with
respect to the triangle inequality m will be equal to 1, and (17.23)–(17.24) are here
appeared in form of (17.30).
Problems were solved by adapting a former Reverse Algorithm for the TSP that
checks for violated subtour elimination constrains even at fractional solutions by
Laporte and Norbert (1981). The solution procedure may be summerized in three
steps:
Step 1. Solve the multi-depot TSP by relaxing (17.31) and the integrality conditions on the variables
Step 2. Identify violated subtour elimination constrains (even if the solution at
hand is not necessarily integer) and introduce one such constrain for each illegal
component remains
Step 3. Reach integrality by gradually introducing Gomory cuts. Stop if a feasible
solution has been reached. Otherwise, proceed to step 2
412
A. Hassanzadeh et al.
17.4 Solution Techniques
Some solution methods for LRPs were mentioned during the previous chapters.
Since the exact models are frequently so complicated with numerous constrains and
variables, they are often solved by applying approximate methods. So the mathematical models are applicable for special cases. Some examples of the exact methods
and the solution techniques of them are presented in Table 17.4. Table 17.4 shows
some examples of these methods.
17.4.1 Heuristic Methods
Karp (1972) showed that location is NP Hard and also vehicle routing decisions is
NP-hard, thus the LRPs also belongs to the class of NP-hard problems. Due to its
complexity, exact solution approaches to the LRP have been very limited and so it
is more common to use the heuristic methods.
This algorithms often divide the problem into its components and then solve
it. The components are: facility location; allocation of users to facilities; and vehicle routing. These three sub-problems are closely interrelated and cannot be
optimized seperately without running the risk of arriving at a suboptimal solution.
Laporte (1988) classifies these algorithms in two categories:
Location–Allocation–Routing Algorithms: facilities are located, users are then
allocated to the facilities and routes are finally defined. Watson-Gandy and
Dohrn (1973) execute these three steps sequentially while other authors combine
the location and allocation steps (see, for example, Bedner and Strohmeier (1979)
Or and Pierskalla (1979))
Table 17.4 Solution methods for LRP exact models (Nagy and Salhi 2007)
Problem
type
Solution
method
Paper
General deterministic
LRP
Cutting
planes
Laporte
et al. (1983)
Branchand-bound
Numerical
optimisation
Branchand-cut
Graph theoretical
Branchand-cut
Laporte
et al. (1988)
Drezner (1982)
Round-trip
location
Eulerian location
Minmax TS
location
Plant cycle
location
Ghiani and
Laporte (1999)
Averbakh and
Berman (2002)
Labbé
et al. (2004)
Facilities
Customers
40
40
3
80
1
10,000
50
1
30
200
Not given
120
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Location-Routing Problem
413
Allocation–Routing–Location: the allocation and routing steps are often simultaneous: sets of routes are first constructed, assuming all facilities “open”; locations
are then selected by dropping various facilities from the system and updating
the location and routing decisions. The SAV-DROP heuristic of Jacobsen and
Madsen (1980) and the algorithm of Srikar and Srivastava (1984) constitute examples of these algorithms
Although this classification is applicable for many heuristic algorithms, not all the
heuristic algorithms can be categorized into them. For example, the algorithm proposed by Nambiar et al. (1989). did not contain the allocation phase.
Also Nagy and Salhi (2007), have divided the heuristic methods as below:
1. Clustering-based method
2. Iterative method
3. Hierarchical method
17.4.2 Metaheuristic Methods
One recent development in the solution of combinatorial problems is the introduction of metaheuristics such as tabu search, genetic algorithms, simulated annealing,
and neural networks (Tuzun and Burke 1999).
All of these metaheuristics aim to search the solution space more electively than
conventional approaches using deferent strategies. They show great promise in solution of difficult combinatorial problems such as the LRP. Among these, TS explores
the solution space by moving from a solution to its best neighbor, even if this results
in a deterioration of the objective function value. This strategy allows the search to
move out of the local optima and explore other regions of the solution space (see
Glover 1995; Glover et al. 1993, for overviews of TS).
17.5 Case Study
17.5.1 Bill Delivery Services (Lin et al. 2002)
In this research, a local case of integrated facility location and distribution planning problem is described. Distribution of bills to customers is a major issue to
a telecommunication service company in Hong Kong that serves a large customer
base of business and residential customers. The customer region is divided to 31
housing states and four potential depots. This problem is constrained by operating
constraints such as:
Daily working hours
Capacity of a vehicle
The capacity of a potential depot
414
A. Hassanzadeh et al.
There is no time window constraint as customers need not be present during the
delivery. If multiple routes could be assigned to a vehicle without exceeding the
staff working hours, further cost reduction could be achieved.
Considering the above interdependent decisions, the basic questions to address
are: given a set of demand nodes (target housing estates) and a set of potential facilities (delivery depot sites), where should the facilities be established? How should
routes be formed from a facility and what is the routing sequence to the demand
nodes such that all demand nodes are served? What is the smaller number of vehicles (vehicle fleet size) to be used for a given set of established routes? The primary
objective of the above location, routing and loading problems is to minimize the
sum of facility costs, staff costs, vehicle rental and operating costs.
For solving the problem, based on five different subproblems, the solutions and
CPU time of five algorithms TA, SA, (TA, SA), (SA, TA), and branch and bound
are calculated and finally the (TA, SA) algorithm is applied to the actual problem.
The solution demonstrates that with two depots and 11 routes, the whole customers
can be serviced while the costs are minimized.
17.5.2 Contaminated Waste Disposal (Caballero et al. 2007)
This work presents a model to find the best location for up to two incineration plants
shared between several preestablished locations in Andalusia (Spain) that will be
used to dispose solid animal waste and simultaneously find the best routes to transport the waste from each slaughterhouse to the plants opened. Thus, we are dealing
with a location (deciding which plants should be opened) routing (designing the
routes to transport the waste from the slaughterhouses) problem.
When evaluating potential locations for the new plants, different factors should
be considered; therefore, a multi-objective problem does exist. Five different objectives are formulated while taking into account two major aspects:
Economic objectives (start-up, maintenance, and transport costs)
Social objectives (social rejection by towns on the truck routes, maximum risk as
an equity criterion, and the negative implications for towns close to the plant)
In the application, we face a problem with six possible locations (where at most
we can choose two of them), 93 clients to serve, five objectives to optimize, and
constraints on truck capacity and on the duration of the route.
To solve this problem, the researchers have adapted a metaheuristic for
multi-objective combinatorial optimization problems based on tabu search, the
multi-objective metaheuristic using an adaptative memory procedure method. This
adaptation consists of some specific neighborhood definitions inspired by the movements used for a similar problem in the literature, a single-objective LRP with
capacitated vehicles. The solution presents a set of efficient solutions considering
the five objectives to the decision maker.
17
Location-Routing Problem
415
17.5.3 Logistics System (Lin and Kwok 2006)
This paper addresses an integrated logistic system where decisions on location
of depot, vehicle routing and assignment of routes to vehicles are considered simultaneously. Total cost and workload balance are common criteria influencing
decision-making. Metaheuristics of tabu search and simulated annealing on real data
and simulated data are applied, to compare their performances under two versions:
simultaneous or sequential routes assignment to vehicles. A new statistical procedure is proposed to compare two algorithms on the strength of their multi-objective
solutions.
This research has the following contributions:
A multi-objective LRP integrated with the routes assignment decisions is considered, and real instances from a local delivery service are used
Certain methods to estimate unavailable data are proposed: the use of GIS to
estimate the travel times and a regression method to estimate the on-site service
(delivery) time
A new statistical procedure based on the hypothesis testing of difference between
two population proportions (Z-test) is proposed, in order to compare the relative
non-dominance of multi-objective solutions between tabu search and simulated
annealing
The effect of allowing multiple use of vehicles in the routes formation stage with
the sequential approach of routing before assignment is examined
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Chapter 18
Storage System Layout
Javad Behnamian and Babak Eghtedari
A warehouse consists of a number of parallel aisles. The items are stored on both
sides of the aisles. Order pickers are assumed to be able to traverse the aisles in
both directions and to change direction within the aisles. Their major roles include: buffering the material flow along the supply chain to accommodate variability
caused by factors such as product seasonality and/or batching in production and
transportation; consolidation of products from various suppliers for combined delivery to customers; and value-added-processing such as kitting, pricing, labeling,
and product customization.
Usually the items in a warehouse exhibit varying characteristics with respect
to dimensions, weight, demand, and other properties. It is natural to apply certain
storage and retrieval strategies depending upon the product families or individual
products within families. Products need to be put into storage locations before they
can be picked to fulfill customer orders. A storage policy is considered optimal if
it minimizes the average time required to store and retrieve a unit load while satisfying the various constraints placed upon the system. A storage assignment policy
is a set of rules which determines where the unit loads of different products will
be located in a warehouse. With regard to storing unit loads, two major classes of
storage policies can be distinguished (Koster et al. 2007; Goetschalckx and Ratliff
1990; Van den Berg et al. 1999) in other word, The storage location assignment
problem (SLAP) is to assign incoming products to storage locations in storage
departments/zones in order to reduce material handling cost and improve space
utilization. Different warehouse departments might use different SLAP policies depending on the department-specific SKU profiles and storage technology (Gu 2005).
A basic rule in assigning products to storage locations is storing “better” products in the “better” locations in the order picking system. A “the most desirable
locations” is a location, which provides faster and more ergonomic access to the
product stored.
The definition of “the most desirable locations” depends on the system as well as
the travel pattern. For example, if traversal routing policy is used for traveling in a
conventional multi-parallel-aisle system, the desirability of locations are measured
in terms of aisles where the most desirable locations are in the aisle that is closest
to the I/O point (This leads to the so-called organ pipe storage location assignment)
(Gu et al. 2007).
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 18, c Physica-Verlag Heidelberg 2009
419
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J. Behnamian and B. Eghtedari
A measure of “goodness” of an item could have been the frequency with which
it is requested. If an item is requested frequently, it is logical to keep that item
in an easily accessible location. But if the item is too heavy, it may be too much
time consuming to replenish that item to that favored location. Another measure of
“goodness” for an item is occupying smaller space. On the other hand if an item is
requested very infrequently, it is not necessary at all to assign it to a favored position,
just because it occupies little space. If that practice were followed, the “best” locations could be filled with lots of small products that are not really requested much.
Another basic rule in assigning products to storage locations is taking the
dimensions into consideration. Cube matching of the items with the storage locations is essential to eliminate space inefficiencies. Shelf dimensions should be
spacious enough to allow easy picking, but tight enough to avoid unused space.
Here is a bad usage of shelf space vs. good usage.
An effective storage location assignment policy may reduce the mean travel times
for storage/retrieval and order-picking. Also, by distributing the activities evenly
over the warehouse subsystems, congestion may be reduced and activities may be
balanced better among subsystems, thus increasing the throughput capacity (Van
den Berg et al. 1999).
18.1 Assumptions and Classifications
The storage system considered here contains multiple products with stores and
retrieves performed in a single-command mode1 (i.e., there is only one store or
one retrieve on each round trip). Most of the concepts extend in a straight forward
manner to the dual-command mode (i.e., a store and a retrieve on each round trip).
Although there is the potential for more efficient storage using a dual-command
mode, the single-command mode continues to be widely used. This is primarily
because of the additional coordination required to unload one truck and simultaneously load another.
We assume that all products in the system are stored and moved in unit loads. We
also assume that each unit load requires the same space and that the expected cost of
storing or retrieving a unit load from a storage location is independent of which unit
is stored in that location. This assumption allows computation of the expected travel
cost for each location before any storage assignment has been made (Goetschalckx
and Ratliff 1990).
1
In a single-command cycle either a storage or a retrieval is performed between two consecutive
visits of the input and output station. In a dual-command cycle the S=R machine consecutively
performs storage, travels empty to a retrieval location and performs retrieval. The empty travel
between the storage and retrieval location is referred to as interleaving travel (Van den Berg
et al. 1999).
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The physical location where arriving items will be stored Subject to performance
criteria and constraints such as:
1.
2.
3.
4.
Storage capacity and efficiency
Picker capacity and efficiency based on the picker cycle time
Response time
The compatibility between products and storage locations and the compatibility
between products
5. Item retrieval policy such as FIFO (first-in, first-out), LIFO (last-in, first-out),
BFIFO (batch first-in, first-out). When using the BFIFO policy, items that arrived
in the same replenishment batch are considered to be equivalent.
In typical warehouse operations, the physical storage infrastructure and its characteristics are known when planning the storage location assignment. The availability
of storage locations is always known in automated warehouses and often known in
mechanized warehouses (Gu 2005).
Trade-offs inevitably occur between throughput and storage space in designing
storage systems. The term throughput is used as a measure of the number of storages
and retrievals performed per time period. It can be expressed directly as a rate (e.g.,
320 storages per 8-h day). Alternatively it can be given inversely in terms of the
time required to perform storage (e.g., 1.5 min per storage). Space is a measure of
the static nature of storage. Throughput however is a measure of the activity or the
dynamic nature of storage; it represents the flow occurring in storage.
The size of the storage system depends on a number of parameters and variables.
For example, the size of the storage system is influenced by storage, throughput, and
cost parameters. The decision variables that influence the size of storage include the
storage methods and the storage layout.
The material characteristics and inventory profile establish the storage and
throughput parameters. Included in the former are the characteristics that influence
the way material is stored, handled, and controlled. The material characteristics of
interest include size, weight, shape, value, shelf life, stack ability, toxicity, flammability, explosiveness; and environmental requirements, among others. The inventory
profile includes both the amount of each product stored over time and the input/output functions that generate the activity requirements for storing and retrieving
material.
The input/output functions will depend on; the mission of the storage system. As
an example, consider a distribution center for finished goods produced at various
company-owned manufacturing plants. If a push system is used, the production
plants push inventory to the distribution center and the input function will be determined by the production schedules at the plants. The output or demand function
might be represented by a forecasting equation developed for the marketing department. The cumulative differences in the input and output functions will determine
the storage requirements in the distribution center over time.
Among the cost parameters that influence the size of the storage system are the
costs of providing storage vs. the costs of not providing storage. The former includes
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J. Behnamian and B. Eghtedari
the costs of providing storage internally vs. leasing space or contracting with a public warehouse to provide the storage space. The costs of providing storage space
include the costs associated with space, personnel, and equipment resources.
The cost of not providing storage reflects the impact of a space shortage and
includes the cost of lost business, goodwill costs, and the cost to the total business
due to inadequate space.
The storage method used includes the specification of the unit load and/or container to be stored, handled, and controlled, as well as the storage/retrieval device,
storage equipment, and other material handling equipment. A number of alternative
methods exist for storing and retrieving material, including manually storing items
on shelving, storing unit loads in pallet rack with lift trucks, storing unit loads in
pallet rack with automated storage/retrieval machines, and manually storing small
parts in carousel conveyors. Material can be moved to/from storage manually or
mechanically via conveyors and industrial trucks, or automatically via automatic
guided vehicles and automated monorails, among others.
The storage system layout includes the height, length, and width of storage, the
location of the individual items in storage, and the location and configuration of any
required support functions. Both the storage capacity and the throughput capacity
of the storage system will be influenced by the layout used.
In this chapter we consider the layout of the storage system. However, to accomplish our objective, it will be necessary to determine the size of the storage
requirement. Storage size depends on the number of storage locations required; in
turn, the number of storage locations depends on the storage location policy used.
The storage location assignment problem (SLAP) is formally defined as follows:
Information on the storage area, including its physical configuration and storage
layout.
Information on the storage locations, including their availability, physical dimensions, and location.
Information on the set of items to be stored, including their physical dimensions,
demand, quantity, arrival and departure times.
The storage assignment problem can be divided into three classes depending on the
amount of information known about the arrival and departure of the products stored
in the warehouse:
1. Product information,
2. Item information
3. No information.
Different operational policies exist for each of these classes, and their implementation and performance have been discussed extensively in the literature. Most of
the research has focused on unit-load warehouses. Of course, these SLAP policies
can be applied to non unit-load warehouses as well, but it is usually much more
difficult to provide analytical results because of the complexity of computing the
associated material handling times and cost involved in a non unit-load warehouse
(for example when batching and routing are used) (Gu 2005).
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Storage System Layout
423
18.2 Storage Location Assignment Problem Based on Product
Information
Most of the times, only product information is known about the items to be stored,
and items are instances of products. Products may be classified into product classes.
The assignment problem now assigns an individual item to a product class based
on its product characteristics, and assigns a product class to storage locations. The
location of an item in its class is most often done using some simple rule, such
as nearest location, or randomly. If the number of classes is equal to the number
of products, then this policy is called dedicated storage. If the number of classes
is equal to one, this policy is denoted as random storage. In real-life warehousing
operations, a small number of classes ranging from 3 to 5 are used. This policy is
called class-based storage (Gu 2005).
18.2.1 Dedicated Storage Location Policy
In this method each product is stored at a fixed location, which is called dedicated storage (Koster et al. 2007). The dedicated storage location policy is shown
Fig. 18.1.
Dedicated storage is used when an SKU is assigned to a specific storage location
or a set of locations. The term fixed slot is used to describe the dedicated storage.
Two methods of dedicated storage are commonly used:
Store items in parts number sequence
Dedicate a location for an SKU based on its activity and inventory level.
The latter method is preferred when there are significant differences in either the
activity level or the inventory level for SKUs. Dedicated storage has low space utilization, but the warehouse is easier to manage since it has a permanent assignment
of products to locations.
The class-based storage policy and the dedicated storage policy attempt to reduce
the mean travel times for storage/retrieval by storing products with high demand at
C
C
B
A
Fig. 18.1 Dedicated storage
location policy
I/O
424
J. Behnamian and B. Eghtedari
locations that are easily accessible (Van den Berg et al. 1999). This method requires
more storage space than class-based storage since sufficient storage locations have
to be reserved for the maximum inventory of each product, and therefore increases
warehouse space cost and material handling cost. On the other hand, dedicated storage has the advantage that the controlling of the warehouse is very simple, since
items of a product will always be stored in the same locations and sufficient space
is always available for all of the items in replenishment batches. The simplicity
advantage is decreasing in importance because the introduction of information technologies such as WMS, bar coding, and radio frequency tags provides a real-time
accurate inventory map of the warehouse. The advantages of robustness and simplicity of dedicated storage must be traded off against the increased required storage
space and material handling cost (Gu 2005).
A disadvantage of dedicated storage is that a location is reserved even for products that are out of stock. Moreover, for every product sufficient space has to be
reserved such that the maximum inventory level can be stored. Thus, the space utilization is lowest among all storage policies. An advantage is that order pickers
become familiar with product locations. In retail warehouses, often the product-tolocation assignment matches the layout of the stores. This can save work in the
stores, because the products are logically grouped. Finally, dedicated storage can be
helpful if products have different weights. Heavy products have to be on the bottom
of the pallet and light products on top. By storing products in order of weight and
routing the order pickers accordingly, a good stacking sequence is obtained without
additional effort. Dedicated storage can be applied in pick areas, with a bulk area for
replenishment that may have, for example, random storage. In this way, the advantages of dedicated storage still hold, but the disadvantages are only minor because
dedicated storage is applied only to a small area (Koster et al. 2007).
Kallina and Lynn (1976) discussed the implementation of the COI rule in practice. The COI rule is easy to implement and has the intuitive appeal of locating
compact, fast-moving items in readily accessible locations. Furthermore, the COI
rule is proved to be optimal for dedicated storage when the following assumptions
are satisfied:
1. The objective is to minimize the long-term average order picking cost.
2. The travel cost depends only on locations. Examples that do not satisfy this
assumption include the case when the travel cost is item dependent or when
there are multiple I/O points, and products have different probability of moving from/to the I/O points, i.e., it does not satisfy the factoring assumption as
defined in Mallette and Francis (1972).
3. When dual or multi-command order picking is used, there is no dependence between the picked items in the same picking tour.
4. Certain routing policies are assumed for multi-command order picking, e.g.,
Jarvis and McDowell (1991) assume using the traversal routing policy for the
conventional multi-aisle order picking system.
5. There are no compatibility constraints that limit the storage location assignment,
e.g., certain items must and/or cannot be put together (Gu 2005).
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18.2.1.1 Space Requirements
With dedicated storage, products are assigned to specific locations. Also, one and
only one product is assigned to a storage location. Hence the number of storage locations assigned to a product must be capable of satisfying the maximum storage
requirement for the product. With multiproduct storage, the storage space required
equals the sum of the maximum storage requirements for each of the products
(Francis et al. 1992).
18.2.1.2 Sizing on the Basis of Service Levels
One approach that can be used to size storage under dedicated storage conditions
is a service-level approach. Specifically, when demand for storage is a random
variable, storage capacity can be determined on the basis of the probability of a
shortage of space. With dedicated storage, Qj storage sots are assigned t product
i for i D 1; : : : ; n. Therefore, the probability of there bring a sufficient number of
storage positions for product i is simply the probability of storage demand being
less than or equal to Qj . Thus the probability is given by the cumulative distribution
function Fj (Qj ).
If the storage demands for the various products are statistically independent, the
probability of there being one or more shortages 0 storage space is given by
Pr .1 or more shortages/ D 1 Pr .no shortages/ :
(18.1)
Since the terms on the right-hand side of (18.2) are the cumulative probabilities,
(18.2) can be expressed as
Pr .no shortages/ D
Pr .no shortages/ D
n
Y
pr ;
(18.2)
Fj .Qj /:
(18.3)
j D1
n
Y
j D1
Therefore, on substituting (18.3) in (18.1), we obtain
Pr .1 or more shortages/ D 1
n
Y
Fj .Qj /:
(18.4)
j D1
18.2.1.3 Sizing on the Basis of Costs
The previous analysis of space requirements for dedicated storage was based entirely on, service-level considerations. Under deterministic conditions, the size
426
J. Behnamian and B. Eghtedari
of the storage system was determined to be equal to the sum of the maximum
requirements for each product. When random conditions exist, two approaches were
considered in order to minimize the amount of space required to ensure that the
probability of a shortage is no greater than a prespecified quantity and, given the
storage capacity, allocate the space among the products so that the probability of no
shortage is maximized.
Alternatively, storage size can be determined using cost models. Such models
might reflect the costs of owning and operating space vs. contracting space or incurring a space shortage. To motivate the consideration of cost of models in sizing
storage, consider a situation in which the cost to provide Qj storage slots for product
j is equal to the sum of a fixed cost of building Qj slots, a variable cost of storing
product j each time period, and a variable cost that occurs when the requirement
for space exceeds Qj . One formulation of such a situation, under deterministic conditions, follows:
Min TC .Q1 : : : Qn / D
C
T
P
t D1
n
P
j D1
C0 Qj C
(18.5)
C1;t Œmin .dt;j ; Qj / C C2;t Œmax .dt;j ; Qj ; 0/g:
Qj D owned storage capacity for product j
T D length of the planning horizon in time periods,
dt;j D storage space required for product j during period t
TC.Q1 : : : Qn / D total cost over the planning horizon as a function of the set of
storage capacities.
C0 D discounted present worth cost per unit storage capacity owned during the
planning horizon of T time periods.
C1;t D discounted present worth cost per unit stored in, owned space during time
period t.
C2;t D discounted present worth cost per unit stored in leased space or per unit
of space shortage during time period t.
where
min .dt;j ; Qj / D dt;j if dt;j < Qj ;
min .dt;j ; Qj / D Qj if dt;j Qj ;
max .dt;j Qj ; 0/ D 0 if dt;j Qj < 0;
max .dt;j Qj ; 0/ D dt;j Qj if dt;j Qj 0:
In (18.5) the discounted present worth cost of building the space for product j
is C0 Qj . The operating cost each time period is based on the amount of product j in storage, either the storage requirement .dt;j / or the storage capacity .Qj /,
whichever i the smallest. If the storage requirement is greater than the storage capacity, a space shortage occurs. Under such conditions, we assume that the excess
18
Storage System Layout
427
requirement .dt;j Qj / is met via leased storage at an incremental cost of C2;t per,
unit stored in leased space during period t.
Due to the separable nature of (18.5), the optimum storage capacity can be
determined independently for each product. The total cost function given by (18.5)
can be shown to be piecewise linear and convex. Consequently, a simple solution
procedure can be used to determine the optimum capacity. Before stating the procedure, let C 0 D C0 =C2 C1 the optimum capacity can be obtained as follows:
1. Sequence the demands for space in decreasing order.
2. Sum the demand frequencies over the sequence.
3. When the partial sum is first equal to or greater than C, stop; the optimum capacity equals that demand level.
18.2.1.4 Assigning Products to Storage/Retrieval Locations
With dedicated storage, products are assigned to storage/retrieval locations in an
attempt to minimize the time required to perform the storage and retrieval operations. Of course, for dedicated storage to be feasible, you must have a sufficient
number of storage slots to dedicate slots to products. In such a situation, the assignment problem becomes a matter of assigning products to slots according to an
appropriate criterion. In our case the criterion will be to minimize some function 0
the distance traveled to store and retrieve the assigned products. To formulate the
dedicated storage assignment problem, let
s D number of storage slots or locations
n D number of products to be stored
m D number of input/output (I/O) points
Sj D storage requirement for product j , expressed in number of storage slots;
Tj D throughput requirement or activity level for product j , expressed by the
number of storage/retrievals performed per unit time i
Pi;j D percent of the storage/retrieval trips for product j that are from/to input/out put (I/O) point ti;k D time required to travel between I/O point i and
storage/retrieval location k
Xj;k D 1, if product j is assigned to storage/retrieval location k D 0, otherwise
I (x) D expected time required to satisfy the throughput requirement for the
system the formulation of the dedicated storage assignment problem is
Min
m X
n X
s
X
Tj
.pi;j ti;k xj;k /:
Sj
i D1 j D1
(18.6)
kD1
Subject to
n
X
j D1
xj;k D 1
k D 1; : : : ; s;
(18.7)
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J. Behnamian and B. Eghtedari
s
X
j D1
xj;k D Sj
j D 1; : : : ; n;
xj;k D 0; 1
8j; k:
(18.8)
(18.9)
Equation (18.6) gives, the expected time required to perform the required storages and retrievals during a time period. In particular, if product j is assigned to
storage/retrieval location k .Xj;k D 1/, then it takes ti;k time units to travel from
input point i to storage location k and it takes ti;k time units to travel from retrieval
location k to output point i . Since the total number of storage/retrieval locations
for product j equals Sj , the probability of the storage/retrieval trip being from/to
storage/retrieval location k is 1=Sj for those locations assigned to product j the total number of storage/retrieval trips performed per time period for. product j equals
1j; however, only pi;j percent of the total trips for product j are performed from/to
I/O point i , Hence the expected time required to travel between storage/retrieval
location k and I/O point i for product j is given by the product of Tj =Sj and
Pi;j :ti;k :xj;k Summing over all I/O points, products, and storage locations yields
I.x/. Equation (18.7) ensures that only one product is assigned to storage/retrieval
location k, (18.8) ensures that the number of storage/retrieval locations assigned to
product j equals Sj .
Again, our formulation of the storage/retrieval location assignment problem assumes that each of the Sj loads of product j is equally likely to be retrieved and
that each of the Sj storage locations assigned to product j is equally likely to be
selected for storage. If a first-in, first-out retrieval policy is used and storage is always performed at the location that has been empty the longest period of time, our
assumptions will be valid.
On examining (18.6), notice that it can be written equivalently as
F .x/ D
s
n
m
X
X
Tj X
xj;k
.pi;j ; ti;k /:
Sj
i D1
j D1
(18.10)
kD1
The term in parentheses represents the average amount of time required for product
j to travel between storage/retrieval location k and the m I/O points. Letting
t j;k D
m
X
pi;j ti;k :
(18.11)
i D1
The objective function can be written as
F .x/ D
s
n X
X
cj;k xj;k ;
(18.12)
j D1 kD1
ı
where Cj;k D .Tj Sj / t j;k . Thus the dedicated storage assignment problem can be
formulated as a transportation problem (Francis et al.1992).
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Storage System Layout
429
18.2.2 Cube-Per-Order Index (COI)
Cube-per-order index (COI) rule is one of the earliest dedicated storage algorithms
(Lai et al. 2002, Zapfel et al. 2006). The cube per order index (COI) is perhaps the
most common storage dispatching rule. It is defined as the ratio of the number of
storage addresses allocated to an item, to the number of transactions per period. It is
applied by routing incoming items with the lowest COI values to the most accessible
storage addresses of a facility (Malmborg et al. 2000).
The algorithm consists of locating the items with the lowest COI closest to the
dock, and assigning items to locations progressively farther away from the dock
by increasing COI. Harmatuck showed that COI yields an optimal solution when
the system is: of a single-command (one clamp truck trip fulfils one task, stock or
retrieve); has a single I/O, with no compatibility constraints; and the traveling of
different items are independent. Malmborg and Krishnakumar (2000) have shown
that under the Euclidean distance, COI produces the shortest traveling cycle time
for a multiple-command system (Lai et al. 2002).
18.2.3 Class-Based Storage Location Policy
The class-based storage location policy distributes the products, based on their demand rates, among a number of classes and reserves a region within the storage area
for each class. Accordingly, an incoming load is stored at an arbitrary open location
within its class (Van den Berg et al. 1999). The concept of class-based storage combines some of the methods mentioned so far. In class based storage if the number
of classes is equal to the number of products, then this policy is called dedicated
storage (Gu et al. 2007). The class-based storage location policy is shown Fig. 18.2.
In inventory control, a classical way for dividing items into classes based on
popularity is Pareto’s method. The idea is to group products into classes in such a
way that the fastest moving class contains only about 15% of the products stored but
contributes to about 85% of the turnover each class is then assigned to a dedicated
area of the warehouse. Storage within an area is random. Classes are determined by
some measure of demand frequency of the products, such as COI or pick volume.
BC
A
Fig. 18.2 Class-based storage location policy
I/O
430
J. Behnamian and B. Eghtedari
Fast moving items are generally called A-items. The next fastest moving category
of products is called B-items, and so on. Often the number of classes is restricted to
three, although in some cases more classes can give additional gains with respect to
travel times (Koster et al. 2007).
This storage policy in the literature seems to be a compromise between the dedicated and the randomized storage policies. It divides products into classes based on
their turnover ratio. The class with the highest ratio is located closest to the I/O. The
implementation of class based storage (i.e., the number of classes, the assignment of
products to classes, and the storage locations for each class) has significant impact
on the required storage space and the material handling cost in a warehouse (Gu
et al. 2007).The implementation of class-based storage (i.e., the number of classes,
the assignment of products to classes, and the storage locations for each class) has
significant impact on the required storage pace and the material handling cost in a
warehouse. Research on this problem has been largely focused on AS/RS, especially
single-command AS/RS.
18.2.3.1 Criteria for Assigning a Product (Class) to Storage Locations
Different criteria can be used to assign a product (class) to storage locations. The
three most frequently used criteria are (Gu 2005):
1. Popularity (defined as the number of storage/retrieval operations per unit time
period). For the popularity policy, product classes are ranked by decreasing popularity and the classes with the highest popularity are assigned the most desirable
locations.
2. Maximum inventory (defined as the maximum warehouse space allocated to a
product class). For the maximum inventory policy, product classes are ranked by
increasing maximum inventory and the classes with the lowest maximum inventory are assigned the most desirable locations.
3. Cube-per-order index (COI, which is defined as the ratio of the retrieval operations per unit time/ maximum allocated storage space to the number of storage).
The COI policy takes into consideration both a SKU’s popularity and its storage
space requirement. Product classes are ranked by increasing COI value and the
classes with the high COI are stored in the most desirable locations.
The implementation of the above policies depends on the types of warehouse systems and, therefore there may be have different variations in that, for example:
1. If storage space is measured in units (e.g., shelves and bays), each unit can be
treated as an individual product by appropriately apportioning demand. This is
most commonly used in unit load warehouses and sometimes in less-than-unitload warehouses. Since each unit load occupies the same amount of storage
space, the popularity policy based on the apportioned popularity is essentially
the same as the COI policy. However, it is different from the popularity policy
without apportioning. For example, suppose that product A has three unit loads
and a popularity of three picks per day, and product B has one unit load and a
popularity of two picks a day. The popularity policy without apportioning will
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Storage System Layout
431
rank product A ahead of product B. On the other hand, if product A is treated
as three products (denoted as A1, A2, and A3), each of them will have an apportioned popularity of 1 pick per day. So the popularity policy based on the
apportioned popularity will now rank product B ahead of product A1, A2, and
A3, which can be easily verified to be equivalent to the COI policy.
2. The definition of “the most desirable locations” depends on the system as well as
the travel pattern. For example, if traversal routing policy is used for traveling in
a conventional multi-parallel-aisle system, the desirability of locations are measured in terms of aisles where the most desirable locations are in the aisle that
is closest to the I/O point. This leads to the so-called organ pipe storage location
assignment.
3. The above three policies are simple and flexible enough to be implemented in
different warehouse systems. Among them, the COI policy has been the most
comprehensively studied one. The COI policy was first described without a proof
of its optimality.
18.2.3.2 Assigning a Product (Class) to Storage Locations Based
on Popularity
Model parameters:
Qk D Number of storage locations requested at any single operational period for
the storage of SKU i
S D Quantity of storage at any single operational period for the storage of SKU i
R D Quantity of retrieval at any single operational period for the storage of
SKU i
Storage policy: calculate S=R ratio
If S=R < 1: locate nearer to receiving
If S=R > 1: locate nearer to shipping
If S=R D 1: does not matter
18.2.4 Class-Based Dedicated Storage Location Policy (COI)
As a compromise between dedicated storage and randomized storage, class-based,
dedicated storage is frequently used. With class-based dedicated storage, products
are divided into three, four, or five classes based on their throughput .T /-to-storage
(S ) ratios. The relatively few fast movers are categorized as class 1 products, next
are class 2 products, and then class 3 products, and so on. Dedicated storage is used
for the classes and randomized storage is used within a class.
It should be noted that the entire discussion of dedicated storage given in
Sect. 18.2.1 applies to class-based dedicated storage, if, instead of dealing with;
products, one deals with classes of products. For this reason, our class-based dedicated storage treatment focuses on the formation of classes of products.
432
J. Behnamian and B. Eghtedari
18.2.4.1 Model for Class-Based Storage Location
Van den Berg (1999) presents a polynomial time dynamic programming algorithm
that partitions products and locations into classes such that the mean single command cycle time is minimized. The algorithm works under any demand curve, any
travel time metric, any warehouse layout and any positions of the input station and
output station. We use the following notation:
Qi D independent random variables representing the number of unit-loads present of product i at an arbitrary epoch,
Pk D set of products in class k D 1; 2; : : : ; K. Due to the demand and supply
processes the inventory level fluctuates. We estimate the storage space requirement such that the storage space in every class suffices for at least a fraction
0 < ˛ < 1 of the time. In other words, the probability of a stock overflow is less
than 1˛-. Let Qk be a random variable representing the inventory level of class
k at an arbitrary epoch, i.e.
Qk D
X
(18.13)
Qi :
i 2Pk
Now, we want to find the smallest size Sku for the class-region of class k such that
P .Qk S k / ˛:
(18.14)
Let tj i n denote the travel time between the input station and location j and let tj out
denote the travel time between the output station and location j .
Every stored unit-load is retrieved some time later, so that over a long time period
half of the single command cycles are storages and half are retrievals.
Accordingly, the mean single command cycle time to location j 2 L equals:
1 in
.2t C 2tjout / D .tji n C tjout /:
2 j
(18.15)
The single command cycle time, E.S C /, is defined as
E.SC/ D
k
X
kD1
P
E.Di /
i 2Pk
P
i 2P
E.Di /
:
X .tji n C tjout /
jLk j
;
(18.16)
where Lk denotes the set of storage locations of class k. The First factor represents
the probability that a request concerns class k. The second factor represents the
mean travel time to a location in class k.
In order to minimize the expected single command cycle time, we assign the
products i that constitute the largest demand per reserved space and the locations
j with the smallest .tj i n C tj out / to the First class and we assign the products i
that constitute the next largest demand per reserved space and the locations j with
18
Storage System Layout
433
the next smallest .tj i n C tj out / to the second class, and so on. Accordingly, the
locations are ranked according to non-decreasing .tj i n C tj out / and the products are
ranked according to nonincreasing demand per reserved space. We define gk .p; l/
as the contribution of classes 1; 2; : : : ; k to (18.16) when products 1; 2; : : : ; p and
storage locations 1; 2; : : : ; l are distributed among these classes such that gk .p; l/
is minimal. Then gk .p; l/ satisfies
n
o
j C1;l
gk .p; l/ D min 1i p;1j l hi C1;p C gk1 .i; j / ;
(18.17)
where hi C1; p j C1;l denotes the contribution to (18.16) if the products i C1; : : : :; p
and the locations j C 1; : : : ; l form one class k. Recalling that the number of locations required in each class is determined by (18.15) the values gk .p; l/ are found
by iteratively solving the dynamic programming (18.17) Each gk .p; l/ corresponds
to an optimal solution of the sub problem with k classes and the First p products and
the First l storage locations when ranked as indicated before. We may use the algorithm to determine the optimal class-partition for 1; 2; : : : ; k classes. Subsequently,
the number of classes among 1; 2; : : : ; k may be selected that constitutes an acceptable mean travel time and space requirement (Van den Berg et al. 1999).
Research on this problem has been largely focused on AS/RS, especially single
command AS/RS. Hausman et al. (1976) show that for single-command AS/RS with
the Chebyshev metric, the ideal shape of storage regions is L-shaped. For such systems, the problem reduces to determining the number and boundaries of the classes.
Explicit analytical solutions for the class boundaries can be derived for the case
with 2 or 3 classes, as shown by Hausman et al. (1976), Kouvelis and Papanicolaou
(1995), and Eynan and Rosenblatt (1994). For the general n-class case, Rosenblatt
and Eynan (1989) and Eynan and Rosenblatt (1994) suggest a one-dimensional
search procedure to find the optimal boundaries. The implementation of class-based
storage in multi-command AS/RS is discussed in Guenov and Raeside (1992).
18.2.5 Full Turn-over Based Storage
For the turnover policy, products are ranked by the ratio of their demand rate divided
by their maximum inventory. Products with the highest turnover are stored in the
most desirable locations. The turnover policy is the most comprehensively studied
one in the literature.
This policy distributes products over the storage area according to their turnover.
The products with the highest sales rates are located at the easiest accessible locations, usually near the depot. Slow moving products are located somewhere towards
the back of the warehouse. An early storage policy of this type is the cube-perorder index (COI) rule, A practical implementation of full turnover policies would
be easiest if combined with dedicated storage (Koster et al. 2007). It is applicable
when space requirements for individual items are identical and, like the COI, has
the objective of maximizing throughput capacity (Malmborg et al. 2000). Gu (2005)
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J. Behnamian and B. Eghtedari
showed that the turnover based policy for dedicated storage was first described by
Heskett in 1963 and 1964 as the cube-per-order index (COI) rule without a proof of
its optimality.
Gu (2005) discussed that some researchers compared randomized storage, dedicated storage, and class-based storage in single-command and dual-command
AS/RSs using both analytical models and simulations. The results show that the
turnover-based policy for class-based storage with relatively few classes could
achieve good performance in terms of both material-handling cost and storage
capacity.
The turnover or COI policy has been shown to be optimal for the case with
restrictive assumptions such as single command, dedicated storage, and productindependent travel costs. However, simulation typically has been used to show that
the turnover policy nearly always performs the best in more general cases (Gu 2005).
The main disadvantage is that demand rates vary constantly and the product assortment changes frequently. Each change would require a new ordering of products
in the warehouse resulting in a large amount of reshuffling of stock. A solution
might be to carry out the restocking once per period. The loss of flexibility and
consequently the loss of efficiency might be substantial when using full-turnover
storage. Based on simulation experimental results, Petersen et al. (2004) show that
with regards to the travel distance in a manual order-picking system, full turnover
storage outperforms class-based storage (Koster et al. 2007). Meanwhile, a class
based storage policy allocates zones to specific product groups, often based upon
their turnover rate (Gu et al. 2007).
18.2.5.1 Solution Algorithm for Full Turn-over Based Storage
1. Rank all the available storage locations in increasing distance from the I/O point,
dj .
2. Rank all SKU’s in decreasing “turns”, TH i =Ni
3. Move down the two lists, assigning to the next most highly ranked SKU i , the
next Ni locations.
Model inputs
dj D Storage locations distance from the I/O point
Ni D Number of storage locations
TH i D Number of units handled per unit of time
Note
In case that the material transfer is performed through a forklift truck (or a similar
type of material handling equipment), a proper distance metric is the, so-called,
rectilinear or Manhattan metric (or L1 norm):
dj D jxj xI=O j C jyj yI=O j:
(18.18)
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Storage System Layout
435
For an AS/RS type of storage mode, where the S/R unit can move simultaneously
in both axes, with uniform speed, the most appropriate distance metric is the, socalled Tchebychev metric (or L1 norm):
˚
dj D max jxj xI=O j; jyj yI=O j :
(18.19)
Decision variables: Xij D 1 if location j is allocated to SKU i ; 0 otherwise.
18.2.5.2 Problem Representation
The schematic of problem presentation for full turn-over based storage is shown in
Fig. 18.3.
Formulation
Min
X X THi
i
j
Ni
: dj xij :
(18.20)
Subject to
X
xij D Ni
X
xij D 1
j
i
8i ;
(18.21)
8j ;
(18.22)
xij 2 f0; 1g;xij 0
8i; j:
SKU
Fig. 18.3 Problem
presentation for full turn-over
based storage
(18.23)
Location
N
1
1
Ni
i
j
N
s
L
THi
dj
Cij =
Ni
436
J. Behnamian and B. Eghtedari
Remarks
The previous problem representation corresponds P
to a balanced transportation
problem: Implicitly it has been assumed that: L D Ni
i
P
For the problem to be feasible, in general, it must hold that: L Ni
i
18.3 Storage Location Assignment Problem
Based on Item Information (SLAP/II)
In the SLAP/II problem, it is assumed that complete information is known about the
arrival and departure time of the individual items. It is very unlikely that information
on individual items will be available in typical warehousing operations, but it may
be available in the case of short term planning of container ports or airport gates
(Gu 2005).
1. Assignment problem and vector assignment problem
2. Shared storage policies for balanced input and output (Goetschalckx and Ratliff
1990)
3. Duration-of-stay storage policy
4. Shared storage policies for unbalanced input and output (Goetschalckx and
Ratliff 1990)
Static shared storage policies
Adaptive shared storage policies
18.3.1 Assignment Problem and Vector Assignment Problem
The resulting problem is a specially structured assignment problem (AP), where
items are assigned to storage locations. The special structure derives from the property that two items can occupy the same storage location, provided that they do not
occupy it at the same time. This problem has been called the vector assignment
problem (VAP), since the occupation is no longer expressed as a single binary status variable but as a vector over the different time periods (Goetschalckx 1998).
The optimal solution of this problem for typical warehousing operations is computationally impractical because of the very large problem instances. The problem is
of interest to academic research in warehouse operations because it provides a cost
lower bound or performance upper bound. An example of a heuristic SLAP/II policy
is the duration-of-stay (DOS) policy of Goetschalckx and Ratliff (1990).
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Storage System Layout
437
18.3.2 Shared Storage Policies
Shared storage policies allow for more flexible use of space than that allowed by
dedicated storage policies. This provides the potential both to reduce the maximumeffective storage area and to better utilize the more desirable storage locations. Both
of these factors reduce average travel time.
The biggest disadvantage of shared storage is the increased data and computational requirements to keep track of where each retrieved load is located and to
determine where each received load should be stored. Using dedicated storage systems. A location will always be used to store the same product and the picker can
“learn” where each product is stored (Goetschalckx and Ratliff 1990).
Dedicated storage policies require the warehouse to be large enough to store the
maximum inventories of all products simultaneously. If shared storage is allowed
and if products are not replenished simultaneously, then total storage requirements
vary over time depending on how the material input and output are distributed over
time. The required rack size is equal to the maximum number of locations during
the planning horizon used by the storage policy. The ideal situation occurs when the
input and output flows are balanced. That is, arrival of each unit of new stock occurs
just at the time when an old unit is removed from the system. For this ideal case,
the warehouse need only be large enough to hold the sum of the average product
inventories.
Turnover rate is a product characteristic since all unit loads of the same product
have the same turnover rate. The time in storage or “duration-of-stay” for different
units of the same product is a unit characteristic since it may differ among different
unit loads of the same product. To illustrate this difference, assume that a replenishment batch of units of a product has just arrived and there is no inventory of the
product already in storage. The first unit retrieved will stay the length of one demand
interarrival time. The last unit retrieved will stay, on the average, the length of the
demand interarrival time multiplied by the batch size of the product. It is shown in
Sect. 3.2.1 that a duration-of-stay-based policy is optimal under an assumption of
perfectly balanced inputs and outputs.
18.3.2.1 Optimal Shared Storage Policy for a Perfectly Balanced System
A system is balanced if for every period t, the number of arriving units is equal to
the number of departing units. In a balanced system there is never an empty location
at the end of any period and there is always an open location available for an arriving
unit. In this case the warehouse is of minimal size and is equal to the sun1 of average
inventories of all products.
A system is perfectly, balanced if for any period t, the number of departing units
that have duration of stay of p is equal to the number of arriving units that have a
DOS of p for all p. The number of units arriving in period t with a DOS of y will
be denoted by np .t/. The system is then perfectly balanced if for all p and t the
following relationship holds: np .t/ D np .t C p/.
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J. Behnamian and B. Eghtedari
For a balanced system the aggregate input flow is equal to the aggregate output
flow. For a perfectly balanced system the input flow is equal to the output flow
for each class of units that have the same DOS. Hence perfectly balanced implies
balanced, but not the reverse.
For a perfectly balanced system, let zp be the total number of units arriving during
any p consecutive time periods that have a DOS equal top. Then zp equals to the
number’s of slots in the warehouse required for storing units which have DOS equal
to p, since the units arriving at time period pC1 will exactly replace the units arrived
at time period one. Note, from the definition of the perfectly balanced system, that
zp is constant over time. For a perfectly balanced system the expression for zp is
Zp D
p
X
np .i /:
(18.24)
i D1
The following theorem provides a procedure for determining an optimal shared storage policy for a perfectly balanced system
Theorem 18.1. A shared storage policy which minimizes both travel time and
required storage space for a perfectly balanced system is to allocate for each
p D 1; 2; : : : the zp , unallocated storage locations having the smallest travel time
to unit loads with DOS equal to p.
18.3.3 Duration-of-Stay Storage Policy
In an attempt to reduce the storage space requirement for dedicated storage, some
warehouse managers use a variation of dedicated storage in which the assignment
of products to spaces is managed carefully. In particular, over time, different products use the same storage slot, albeit only one product occupies the slot when it is
occupied. The location policy used is here referred to as shared storage.
To motivate our consideration of shared storage, consider the arrival of 100 pallet
loads of a particular fast mover product to be stored in pallet rack. Pallet loads will
be retrieved and shipped at a rate of five pallet loads per day over a 20-day period.
With randomized storage, 100 empty storage slots are randomly selected for the
product; no recognition is given to the fact that the product is a fast mover.
With dedicated storage, on the other hand, at least 100 empty slots must be
available among the premium locations assigned to the fast-mover product. If randomized storage is used, each time a pallet load is removed from storage, the slot
is available for use by the next product requiring storage. However, with dedicated
storage, each removal of a pallet from storage creates an empty slot that will not
possibly be filled until, at the earliest, the arrival of the next shipment of the same
product. Shared storage recognizes that while the product might be considered to
be a fast mover, each pallet load stays in storage different lengths of time. Depending on the amount of the product in inventory at the time the shipment arrives, it is
possible that five pallet loads will be in storage for only 1 day, whereas five other
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Storage System Layout
439
pallet loads within the same shipment will be in storage for 20 days. From the perspective of the storage positions in the warehouse, five pallet loads appear to be
super fast movers; the remaining pallet loads are viewed as being less fast, perhaps
even medium or slow movers. Shared storage recognizes and takes advantage of the
inherent differences in lengths of time that individual pallet loads remain in storage
(Francis et al. 1992).
An example of a heuristic SLAP/II policy is the duration-of-stay policy of
Goetschalckx and Ratliff (1990). In DOS-based storage policies, the expected DOS
of the i th unit of a SKU with replenishment lot size Q is i= for i D 1; 2 : : : Q,
where is the demand rate of that SKU.
DOS D i=:
(18.25)
Then the items of all the different products having the shortest DOS are assigned to
the closest locations. Hence, the items of a single replenishment batch of a single
product are not stored together in the warehouse. Under some unrealistic assumptions on the scheduling and size of product replenishments, it can be shown that
the DOS storage policy is optimal for both materials handling effort and required
storage capacity. In practice, DOS-based policies are difficult to implement since
it requires the tracking and management of each stored unit in the warehouse. Also
the performance of DOS-based policies depends greatly on factors such as the skewness of demands, balance of input and output flows, inventory control policies, and
the detailed implementations. Kulturel et al. (1999) compared class-based storage
and DOS-based storage using simulation and showed that the former consistently
outperforms the latter in practical settings.
Goetschalckx and Ratliff (1990) and Thonemann and Brandeau (1998) theoretically showed that DOS-based storage policies are the most promising policy in
terms of minimizing traveling costs. In practice, DOS-based policies are difficult to
implement since they require the tracking and management of each stored unit in the
warehouse. Also, the performance of DOS-based policies depends greatly on factors such as the skewness of demands, balance of input and output flows, inventory
control policies, and the detailed implementations (Gu 2005).
18.3.3.1 Space Requirements
The storage space requirements for shared storage range from that required for randomized storage to that required for dedicated storage, depending on the amount
of information available regarding the inventory levels over time for each product.
As noted above, the distinction between shared storage and randomized storage is
that the former involves total specificity regarding the storage locations for products,
whereas with the latter, the locations depend solely on the mergence of empty slots
within the warehouse. Shared storage and dedicated storage differ due to the distinction made by the former regarding the time that each load of a product spends,
formally the shared storage policy. Relatively little experience has been gained in
applying the policy to large-scale problems optimally; furthermore, as will be seen,
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J. Behnamian and B. Eghtedari
it is not immediately obvious how one might go about making optimum location
decisions for large-scale applications. We present the policy because (a) multiple
products sharing storage space is quite common in some industries, and (b) its features are sufficiently attractive to warrant our further attention. Since the material
might be of less interest to those interested in tried and true storage policies, this
section may be omitted without jeopardizing an understanding of the remaining
material in the chapter (Francis et al.1992).
In storage; dedicated storage assigns the total replenishment lot of a product to
a number of storage positions based on the average time spent in storage for the
replenishment lot.
A situation that naturally suggests the use of shared storage is a production line
that is used to produce multiple products. Since products are produced sequentially rather than simultaneously, inventory replenishments are distributed over time.
A beverage bottling line is an example of the type of situation we have in mind; other
examples include production lines for paint, bleach, and industrial chemicals.
In each case cited, the same production equipment is used to produce different
package sizes of the same product, as well as different products. A new setup or
changeover is required between the production of different sizes or products Hence
it is not possible for the inventory levels of the various sizes and products produced
on the same production line to be increasing at the same time. While one product
is being produced, the inventory levels of the other products are decreasing. Not all
products can be at their maximum inventory levels simultaneously. Hence the use i
of dedicated storage would result in some empty storage slots, existing at all times.
18.3.4 Shared Storage Policies for Unbalanced Input and Output
A perfectly balanced system is a much idealized situation which is unlikely to occur
in real storage systems. However, the analysis of the perfectly balanced systems
provides:
1. A bound on how much improvement can be expected from the use of shared
storage policies, and
2. Insights which allow development of attractive heuristics for more realistic situations.
Shared storage policies for systems which are not perfectly balanced need to be distinguished in terms of when information is available to make the storage decisions.
We will define static storage policies as those where all the information affecting the
storage and retrieval decision is available at the beginning of the planning period.
In this case, all assignments of items to locations can be made prior to the beginning of the planning horizon. We define adaptive policies as those using information
which becomes available during the planning period to influence the storage and retrieval decisions. Based on the knowledge gained from the perfectly balanced case,
we develop heuristics for both static and adaptive policies and then compare them
(Goetschalckx and Ratliff 1990).
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18.3.5 Static Shared Storage Policies
First consider a static system where we know the arrival and departure time of each
item during the planning horizon. From the results we conclude that for all shared
storage systems that satisfy the travel independence condition, any storage policy
that relies only on the ranking of the travel times to the locations (and not on their
actual value) is optimal if and only if it simultaneously maximizes the number of
items stored in the first, first two, : : : , first N locations. This is a very restrictive
condition and thus finding optimal policies based only on the ranking of the storage
locations will most likely be restricted to special cases. Goetschalckx (1992) shows
that solving the optimal storage policy will require a prohibitive computational effort
in a warehousing setting. However, if we know the exact arrival and departure time
of each unit during the planning horizon (i.e., the system is static), then an efficient
heuristic (GREEDY) can be formulated.
GREEDY is based on the fact that we can efficiently maximize the number of
units stored in the first location. Given the assignment of units to the first location,
we can maximize the number of units stored in the second location. In general, given
the assignment of units to the first locations we can maximize the number of units
stored in location n C 1. To accomplish this we first order all the units, which will be
stored during the planning horizon, by nondecreasing departure time. If there are any
ties, then they are broken by ordering the units by nondecreasing arrival time. This
tie-breaker fills up early locations as much as possible, thereby increasing flexibility
for later decisions. The storage locations are ordered by nondecreasing travel time.
We then assign the units in order to the open location with the smallest travel time.
The computational effort required by GREEDY is small, since it requires only
a sort of the locations and a sort of the items. The effort is of the order of
O.K log .K// C O.N log .N // where K is the number of units stored during the
planning horizon and N is the number of slots in the warehouse (Goetschalckx and
Ratliff 1990).
18.3.6 Adaptive Shared Storage Policies
Most real life storage systems are adaptive systems, because complete and perfect
information is not available at the beginning of the planning horizon. Most of the
time, the only information available is with regard to average material flow and, in
some cases, what is arriving during the next period. The policy developed in this
section involves establishment of classes within the warehouse based on average
arrivals and departures of items with different durations of stay. This can be viewed
as a variation of the Hausman et al. (1976) class based storage concept, but with the
classes based on unit duration of stay rather than product turnover rate.
Let np .t/ be the number of units with DOS equal to p which arrives in period t.
Let nN p be the average number of items arriving in a period with DOS equal to p. For
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J. Behnamian and B. Eghtedari
a perfectly balanced system nN p is equal to zp =p. Hence, for those items with DOS
equal top, we define a zone of size p nN p which is exactly the right size each period.
For systems which are not perfectly balanced, some periods have np .t/ different
from the number of open locations in the zone of size p nN p . Hence, a zone of size
pi & is too large for some periods and too small for other periods. One approach
to handling this variation is to establish the zones in exactly the same fashion as for
the perfectly balanced case. That is, let zp D p nN p be the zone size for items with
DOS equal to p. Then for p D 1; 2 : : : reserve the z, remaining locations with the
smallest travel time for items with DOS equal to y.
There are two difficulties with this approach which must be overcome. The first
is that z, may not be an integer number of slots since p nN p is not necessarily integer.
This can be handled by rounding z, to the nearest integer. When z, is small the
relative impact of this rounding is greater than when z, is larger. For example if
z1 D 1:5 the zone size could double, based on which way z1 value is rounded. An
alternative is to aggregate items into a class until the z, for the aggregate class can be
rounded without large relative Impact. It is shown that for that case the increase in
average distance traveled which results from combining two adjacent DOS classes
is n2p C =2 where c is the slot width. This result indicates that the impact of combining classes is strongly affected by the number of items arriving in a class, but not
by the DOS itself. Hence, we would be more inclined to combine DOS classes with
small np .
The second difficulty occurs when an item arrives with DOS equal to p and the
zone allocated for items with DOS equal to p is full. It seems logical to assign the
item either to the nearest zone higher than p with an open location or the nearest
zone lower than p with an open location. To better understand the impact of putting
an item in a zone different from its own, consider the following situation. Suppose
that one slot in a zone corresponding to DOS equal to p is always used for items
with DOS equal to q and one slot in a zone corresponding to DOS equal to q is
always used for items with DOS equal to p. Further, assume that both slots remain
full. Let tp and tq , be the average travel time to zones p and q respectively.
If the slots had been used for the correct items, then the expected travel time
would have been tp =p C tp =q. By interchanging the slot usage, the expected travel
time becomes tp =p C tp =q which is an increase of
ı
.q p/.tq tp / pq:
(18.26)
From this we see that the impact of interchange becomes less a p and q get larger
(i.e., the penalty for placing an item in the wrong zone decreases with increasing
duration of stay).
For the computation experiments discussed in the next section, the zones for
the dynamic shared storage policy were determined by the following ADAPTIVE
algorithm with cumulative rounding. The average time between two demands for an
item of product i is denoted by diti .
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18.3.6.1 Adaptive Algorithm
1. For each duration of stay p do
np D 0:
(18.27)
For each item k D 1 : : : qi in replenishment batch of produce i do
P D k:d in :
np D np C 1=.qi :diti /:
(18.28)
(18.29)
2. for each duration of stay p do
zp D np :p:
(18.30)
Cumulative remainder D 0, prev bound D 0, zone D 0.
3. for each duration of stay p do
Cumulative remainder D cumulative remainder Czp
If cumulative remainder > 1 then,
Zone D zone C 1,
Zone size (zone) D round to integer (cumulative remainder),
Lower dos (zone) D prev bound C1,
Upper dos (zone) D p,
Cumulative remainder D cumulative remainder – zone size (zone),
Prev bound D p.
In step 1 of the algorithm the average number of items arriving for each duration
of stay is computed, based on the fact that during qi diti , time periods one item belonging to product i with duration of stay p will arrive in the system. In step 2
the required zone size zp is computed for each duration of stay and the bookkeeping variables for step three are initialized. In step 3 of the algorithm, for each dos
the required zone size is added to the cumulative remainder, until the cumulative
remainder is larger than one location.
Then, the new zone is set equal to the cumulative remainder size rounded to
the nearest integer and the DOS bounds for this new zone are set. The size of this
new zone is then subtracted from the cumulative remainder. This is consistent with
the properties that zones with large zn ’s and corresponding large zn ’s should not
be combined and zones with small zn ’s can be combined without incoming a large
penalty. It also solves the problem on how to round the fractional zone sizes to
integer numbers.
Once the zones are determined, ADAPTIVE executes in the same way as the
optimal policy. At the time of its arrival, the DOS of every unit is estimated and
that unit is stored in its corresponding zone. If that zone is full, the unit is stored in
the open location with the smallest value for expression (Goetschalckx and Ratliff
1990).
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J. Behnamian and B. Eghtedari
18.4 Storage Location Assignment Problem
Based on No Information (SAP/NI)
If no information is available on the characteristics of the arriving items, only very
simple storage policies can be constructed. In this case the most frequently used
policies are
1.
2.
3.
4.
Closest-open-location (COL),
Farthest-open-location (FOL),
Random (RAN)
Longest-open-location (LOL).
The first two policies pick an open location based on its distance to the receiving
dock; the last policy picks the location that has been vacant for the longest time. It
is not known if there is any significant performance difference between them.
18.4.1 Randomized Storage Location Policy
Randomized storage, also referred to as floating slot storage, allows the storage
location for a particular product to change or float over time. In practice, randomized
storage is defined as follows (Fig. 18.4).
When a load arrives for storage it is placed in the closest open feasible location;
retrievals occur on a first-in, first-out basis. If there s more than one input point, the
storage location selected is the one closest to the input point through which the unit
load enters the storage facility. In modeling randomized storage, it is commonly assumed that each empty storage slot is equally likely to be chosen for storage when
a storage operation is performed; similarly, it is assumed that each unit of a particular product is equally likely to be retrieved when multiple storage locations exist
for a product and a retrieval operation is performed. When the warehouse is relatively full, there are no significant differences in the travel distances obtained via
the “equal likelihood” assumptions and those resulting from the closest open slot
practice. However, for a “sparse warehouse” there can be significant differences in
the travel distances obtained.
ABC
Fig. 18.4 Randomized
storage location policy
I/O
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The random storage policy will only work in a computer-controlled environment
(Malmborg et al. 2000). If the order pickers can themselves choose the location
for storage we would probably get a system known as closest open location storage. The first empty location that is encountered by the employee will be used to
store the products. This, typically, leads to a warehouse where racks are full around
the depot and gradually more empty towards the back (if there is excess capacity).
Hausman et al. (1976) argue that closest open location storage and random storage
have a similar performance if products are moved by full pallets only (Koster et al.
2007). Closest open location refers to the placement of stored items in the closest
available address to minimize expected cycle times in randomized storage systems
(Malmborg et al. 2000).
18.4.1.1 Space Requirements
With randomized storage, products can be stored in any available storage slot Hence,
the storage space requirements will equal the maximum of the aggregate storage.
requirements for the products.
We have considered two approaches in sizing a storage facility determining the
minimum size that satisfies; service-level objective (expressed as a probability of a
space shortage), and determining the size on the basis of trade-offs in the costs of
providing space vs. having insufficient space. In this section, both approaches are
considered in sizing a storage facility for randomized storage.
Due to the dynamic conditions that typically exist in the replenishment of products it is very difficult to exactly predict the storage requirement for randomized
storage. For this reason, storage capacity levels are sometimes established by treating inventory levels of the products as random variables (Francis et al. 1992).
18.4.1.2 Sizing Randomized Storage Based on “Service Level” Requirements
Q D max number of storage locations requested at any single operational period
(a random variable)
Pk D Pr .Q D k/; k D 0; 1; 2 : : : .Probability mass function for Q/; (18.31)
X
pj .Cumulative distribution function for Q/: (18.32)
F .k/ D Pr .Qk / D
j D0:::k
Problem formulation. Find the smallest number of locations N , which will satisfy a
requested service level s for storage availability, i.e.,
min N:
(18.33)
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J. Behnamian and B. Eghtedari
Subject to
F .N / s;
(18.34)
N 0:
(18.35)
Solution
N D min fk W
X
pj sg:
(18.36)
j D0;:::;k
18.5 Comparing Storage Policies
The class-based storage location policy and the dedicated storage location policy
attempt to reduce the mean travel times for storage/retrieval by storing products
with high demand at locations that are easily accessible (Van den Berg et al.
1999).
Dedicated storage requires more space than randomized storage. If an out-ofstock condition exists for a given SKU, the empty slot continues to remain
“active” with dedicated storage; whereas, it would not be the case with randomized storage. If multiple slots are assigned for a given SKU, as the inventory level
decreases, the number of empty slots will increase (Koster et al. 2007).
Comparing dedicated storage with random storage, the former has the advantage
of locating fast moving and compact SKUs close to the I/O points, and therefore is beneficial for efficient material handling. However, it also requires more
storage space since sufficient storage locations must be reserved for the maximum inventory of each product. In class-based storage, additional decisions are
to determine the number of classes and to assign products to classes (Gu et al.
2007).
Dedicated storage policies require that a particular storage location be reserved
for units of a single product during the entire planning horizon. Shared storage
policies allow the successive storage of units of different products in the same
location (Goetschalckx and Ratliff 1990).
18.6 Family Grouping
All storage assignment policies discussed so far have not entailed possible relations
between products. For example, customers may tend to order a certain product together with another product. In this case, it may be interesting to locate these two
products close to each other. An example of this is called family-grouping, where
similar products are located in the same region of the storage area. Clearly, grouping of products can be combined with some of the previously mentioned storage
policies. For example, it is possible to use class-based storage and simultaneously
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group related items. However, the decision in which class to locate the products has
to depend on a combination of the properties of all products in the group. Roll and
Rosenblatt (1983) compare the space requirements for the random and grouped storage for a port warehouse and show that the grouped storage assignment increases
space requirements compared to random storage assignment. Rosenblatt and Roll
(1984) set up a model for warehousing costs, taking the effect of space requirements into account. To apply family grouping, the statistical correlation between
items (e.g. frequency at which they appear together in an order, should be known
or at least be predictable. In the literature, two types of family grouping are mentioned. The first method is called the complementary-based method, which contains
two major phases. In the first phase, it clusters the items into groups based on a
measure of strength of joint demand (“complementary”).
In the second phase, it locates the items within one cluster as close to each other
as possible (Wascher 2004). Rosenwein (1994) shows that the clustering problem
can be formulated as a p-median problem. For finding the position of clusters, Liu
(1999) suggests that the item type with the largest demand should be assigned to the
location closest to the depot (volume-based strategy), while Lee (1992) proposes
to take into account also the space requirement (COI-based strategy). The second
type of family-grouping method is called the contact based method. This method is
similar to the complementary method, except it uses contact frequencies to cluster
items into groups. For a given (optimal) routing solution, a contact frequency between item type i and item type j is defined as the number of times that an order
picker picks either item type I directly after item type j , or item type j directly
after item type i . However, the routing decision is dependent on the location of the
item types, which demonstrates the strong interrelationship between item location
and routing. Due to the fact that finding a joint optimal solution for both problems is
not a realistic approach, at least not for problem instances of the size encountered in
practice, contact-based solution methods alternate between the two problem types
(Koster et al. 2007). The contact-based method is considered, for example, in Van
Oudheusden et al. (1988).
18.7 Continuous Warehouse Layout
In this section we assume that the set of storage locations can be represented adequately as “continuous” planar region, a set of positive area, rather than a set of
discrete locations. The reasons for studying continuous warehouse layout are threefold. First, results from continuous formulations can provide insights concerning the
underlying discrete problem. Second, many storage problems involve such a large
number of storage locations that a continuous representation is quite appropriate.
Third, the continuous problem may be easier to solve than the corresponding discrete problem.
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J. Behnamian and B. Eghtedari
18.7.1 Storage Region Configuration
Perhaps after some reflection it should be; apparent that the shape of the continuous
storage region for a product can be defined by a contour line. In this chapter apply
these concepts in defining the boundaries of storage regions within a warehouse. In
this chapter we let the existing facilities be the source (input) and destination output)
points for travel to/from storage; the percentage of travel between storage and the
I/O points represent the weights between existing and new facilities By definition,
a contour line encloses all points having an expected distance raveled less than or
equal to the value of the contour line; we call the set of such points a level set or
contour set. In this chapter we will be interested in constructing level sets of a given
area in defining the storage regions to be used in a warehouse.
A storage region defined by a level set of area A will have an expected distance
less than or equal to the expected distance for any other set of equal area. Based on
the use of contour lines to define storage regions, the problem of determining storage
configurations for continuous warehouse layouts reduces to a geometry problem.
18.8 Dynamic Storage Location Assignment Problems
(Gu 2005)
The version of the SLAP problem studied in the literature is most often static, i.e.,
it assumes that the incoming and outgoing material flow patterns are stationary over
the planning horizon. In reality, the material flow changes dynamically due to factors
such as seasonality and the life cycles of products. Therefore, the storage location
assignment should be adjusted to reflect changing material flow requirements. One
possibility is to relocate those items whose expected retrieval rate has increased (decreased) closer to (farther from) the I/O point. Such relocations are only beneficial
when the expected saving in order picking outweighs the corresponding relocation
cost. Therefore, decisions must be made carefully concerning which set of items
to be relocated, where to relocate them, and how to schedule the relocations. Another type of relocation might take place as a result of the uncertainty in incoming
shipments. For example, Roll and Rosenblatt (1987) describes the situation when
the storage area is divided into separate zones and any incoming shipment must be
stored within a single zone. It might happen that none of the zones has sufficient
space to accommodate an incoming shipment. In such cases, it is advisable to free
some space in a certain zone to accommodate the incoming shipment by shifting
some stored products in that zone to other zones (Gu 2005).
18.9 Case Study
This section lists some real industrial case studies, which not only provide applications of the various design and operation methods in practical contexts, but more
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Table 18.1 A summary of the literature on SLAP case studies
Citation
Zeng et al. (2002)
Problems studied
Storage location assignment;
warehouse dimensioning;
storage and order picking
policy
Type of warehouse
A distribution center
Van Oudheusden
et al. (1988)
Storage location assignment;
batching; routing
A man-on-board AS/RS
in an integrated steel mill
Kallina and Lynn
(1976)
Storage location assignment
using the COI rule
A distribution center
importantly also identify possible future research challenges from the industrial
point of view. Table 18.1 lists these case studies with the problems and the types of
warehouse they investigated. The detailed results and discussions are too cumbersome to be presented here. Interested readers should refer to the original papers (Gu
2005).
References
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Chapter 19
Location-Inventory Problem
Mohamadreza Kaviani
Traditionally, logistics analysts have divided decision levels into strategic, tactical
and operational (Miranda and Garrido 2006). There are also three important decisions within a supply chain: facilities location decisions; inventory management
decisions; and distribution decisions (Shen and Qi 2007). For example, in a distribution network, we could mention location of Distribution centers (DCs) as a strategic
decision, distribution decisions as a tactical decision and inventory service level as a
tactical or operational decision. Often, for modeling purposes, these levels are considered separately. And this may conduce to make non-optimal decisions, since in
reality there is interaction between the different levels (Miranda and Garrido 2006).
For example, most well-studied location models do not consider inventory costs, and
shipment costs are estimated by direct shipping. Although one may argue that tactical inventory replenishment decisions and shipment schemes are not at the strategic
level, and we should not consider them in the strategic planning phase, however,
failure to take the related inventory and shipment costs into consideration when deciding the locations of facilities can lead to sub-optimality, since strategic location
decisions have a big impact on inventory and shipment costs (Shen and Qi 2007). On
the other hand firms would like to consider cost and service levels simultaneously. It
is good to have many DCs, since this reduces the cost of transporting product to customers (/retailers) and will provide better service. Also, it is good to have few DCs,
since this reduces the cost of holding inventory via pooling effects, and reduces the
fixed costs associated with operating DCs via economies of scale (Erlebacher and
Meller 2000).
Nozick and Turnquist (1998) discussed that in 1997, Thomas provided the data
that show total logistics costs (inventory plus transportation) in the U.S. declined
from about 13% of Gross Domestic Product (GDP) in 1985, to about 10.5% in 1993,
largely as a result of increasing emphasis on controlling inventories throughout the
supply chain. “However, since 1993, logistics costs (as a percentage of GDP) have
been essentially constant. The “easy” cost reductions have been made, and further
improvements will require more effective tools” (Nozick and Turnquist 1998).
Nozick and Turnquist (2001) also discussed that in 1997, Thomas provided inventory and freight transportation costs for the US economy in 1985, and through
the 1990s. “Freight transportation costs have been nearly constant (at about 6%
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 19, c Physica-Verlag Heidelberg 2009
451
452
M. Kaviani
of Gross National Product) over the entire period of the data. Inventory costs fell
significantly between 1985 and 1992, as companies implemented just-in-time systems and the economy grew steadily. However, since 1992 there has been almost no
change in inventory costs, and total physical distribution costs seem to have reached
a plateau” (Nozick and Turnquist 2001).
On the basis of the above, managing inventory has become a major challenge
for firms as they simultaneously try to reduce costs and improve customer service
in today’s increasingly competitive business environment (Daskin et al. 2002). The
goal of cost reduction is to provide motivation for centralization of inventories. On
the other hand, the goal of customer responsiveness is to provide motivation for having goods as near to the final consumer as possible. Thus, there is a basic conflict
between these objectives, and locating DCs is a critical decision in finding an effective balance between them (Nozick and Turnquist 2001). Many companies face the
strategic decision of deciding on the number of DCs, their location, and which customers they serve. One objective for a company facing this decision is to maintain
acceptable service while minimizing the fixed costs of operating the DCs, inventory holding costs at the DCs, and transportation costs between plants and DCs, and
DCs and customers (Erlebacher and Meller 2000). Inventory theory literature tends
to focus on finding optimal inventory replenishment strategies at the DCs and the
retailer outlets. This work usually assumes that the number and locations of the DCs
are given. On the other hand, location theory tends to focus on developing models
for determining the number of DCs and their locations, as well as the DC-retailer
assignments. This work usually includes fixed facility setup costs and transportation
costs, but the operational inventory and shortage costs are typically ignored (Daskin
et al. 2002).
The location-inventory problem is presented bellow. A set of suppliers and customers are spatially dispersed in a geographical region. Typically, the customers
face stochastic demand for an array of products offered by the suppliers (Miranda
and Garrido 2008). We consider a three-tiered system consisting of one or more
suppliers, DCs and customers shown in Fig. 19.1. We assume that the locations of
the suppliers and the customers are known and that the suppliers have infinite capacity at least from the perspective of the system being modeled. The problem is
to determine the optimal number of DCs, their locations, the customers assigned to
each DC, and the optimal ordering policy at the DCs (Daskin et al. 2002).
For example, consider a blood bank that supplied roughly 30 hospitals in the
greater Chicago area. Our focus was on the production and distribution of platelets,
the most expensive and most perishable of all blood products. If a unit of platelets
is not used within 5 days of the time it is produced from whole blood, it must be
destroyed. The demand for platelets is highly variable as they are needed in only
a limited number of medical contexts. The hospitals supplied by the blood bank
collectively owned the blood bank and set prices. As a result they could return a
unit of platelets up to the time it outdated and not be charged for it. Thus, there
was little incentive to manage inventories in an efficient manner. Many of the larger
hospitals ordered almost twice the number of platelet units that they used each year
resulting in the need to destroy thousands of units of this expensive blood product.
19
Location-Inventory Problem
453
Customers
DCs
Supplier
Fig. 19.1 A three-tiered system
Other hospitals ordered almost all of their needed platelets on an emergency basis.
The blood bank often had to ship the units to these hospitals using a taxi or express courier at significant expense to the system. Clearly an improved system was
needed (Daskin et al. 2002). Or General Motors is instituting a “customer express
delivery” system in an attempt to provide better availability of a wide range of vehicles to customers, while reducing the amount of inventory held on dealer lots. With
the customer express delivery program, new vehicles would be held at DCs and be
made available to the dealers on order, using a 24-h delivery standard (Nozick and
Turnquist 2001).
This chapter is organized as follows. In Sect. 19.1, literature and classification of
location-inventory problem are given, and Sect. 19.2 presents some models developed for the problem. Some relevant approaches to solve Location Inventory models
are given in Sect. 19.3. Finally, Sect. 19.4 represents a short summary of a real world
case study.
19.1 Applications and Classifications
The literature presents different approaches to model and solve the problem. The
inventory literature tends to ignore the strategic location decision and its associated
costs, whereas the location literature tends to ignore the operational inventory and
shortage costs, as well as the demand uncertainty and the effects that reorder policies
have on these and shipping costs (Shen et al. 2003).
In the literature, many papers have been seen that study the integration and
coordination of any two of the three important decisions: location routing models,
inventory-routing models, and location-inventory models as below:
Nozick and Turnquist (1998) demonstrate a way of approximating the costs of
safety stock for a set of products as a linear function of the number of DCs. This
allows the costs of safety stock to be included directly in a fixed-charge form of
the facility location model
454
M. Kaviani
Erlebacher and Meller (2000) formulate a highly nonlinear integer locationinventory model. They attack the problem by using a continuous approximation
as well as a number of constructions and bounding heuristics
Daskin et al. (2002) and Shen et al. (2003) studied the joint location-inventory
model in which location, shipment and nonlinear safety stock inventory costs are
included in the same model. They developed an integrated approach to determine
the number of DCs to establish, the location of the DCs, and the magnitude of
inventory to maintain at each center
Shen et al. (2003) derived a cross-level model to analyze decisions about inventory control and facility location, specially suited to urban settings, where
the storage space is scarce and the vehicles’ capacity is usually restricted. Both
conditions, on the one hand make the problem difficult to solve optimally, on
the other hand make it more realistic and useful in practice. Shen et al. (2003)
reformulate the problem as a Set Covering Problem, which is then solved
through a hybrid heuristic mixing columns generation and branch and bound
methods. While Daskin et al. (2002) apply Lagrangian relaxation to solve this
problem
Miranda and Garrido (2006) present a simultaneous nonlinear-mixed integer
model of inventory control and facility location decisions, which considers two
novel capacity constraints. The first constraint states a maximum lot size for the
incoming orders to each warehouse, and the second constraint is a stochastic
bound to inventory capacity. This model is NP-hard and presents nonlinear terms
in the objective function and a nonlinear constraint
Snyder et al. (2007) present a stochastic version of the location model with risk
pooling that optimizes location, inventory, and allocation decisions under random
parameters. The goal of their model is to find solutions that minimize the total
cost (including location, transportation, and inventory costs) of the system across
all scenarios. The location model explicitly handles the economies of scale and
risk-pooling effects that result from consolidating inventory sites. They present
a Lagrangian-relaxation–based exact algorithm for the model. The Lagrangian
sub problem is a non-linear integer program, but it can be solved by a low-order
polynomial algorithm
Shen and Qi (2007) formulated a nonlinear integer programming model to minimize the total cost that includes location costs and inventory costs at the DCs, and
distribution costs in the supply chain, for which they propose a Lagrangian relaxation based solution algorithm. By exploring the structure of the problem, they
find a low-order polynomial algorithm for the nonlinear integer programming
problem that must be solved in solving the Lagrangian relaxation sub-problems
Miranda and Garrido (2008) developed an efficient heuristic to solve a joint
location–distribution–inventory model for a three layered supply chain. The
solution approach is based on Lagrangian relaxation, improved with validity
constraints derived from the finite set of all possible combinations of mean demand and variance. The optimal solution’s lower bound is found through the
optimal solution of the dual problem
19
Location-Inventory Problem
455
A classification of different types of location inventory problems and their properties
is represented as follows:
Area solution: Discrete, Continual
Objective function: MiniMax, MiniSum
Variables: integer, mixed integer
Determine the number of DCs: Exogenous, Endogenous
DC capacity: Unlimited, Limited
Cost of DC location: No cost, fixed cost, Variable cost
Solution methods: Exact, Heuristic, Meta heuristic
19.2 Models
In this section, we introduce location-inventory models represented and developed
as mathematical models.
19.2.1 Model of Shen et al. (2003)
The first model that is present here is easy to understand. Shen et al. (2003) assume
that location costs are incurred when DCs are established. Line-haul transportation costs are incurred for shipments from the single supplier to the DCs. Local
transportation costs are incurred in moving the goods from the DC to the retailers.
Inventory costs are incurred at each DC and consist of the carrying cost for the average inventory used over a period of time, as well as safety stock inventory that
is carried to protect against stock outs that might result from the uncertain retailer
demand. Inventory costs are also incurred at each retailer.
19.2.1.1 Model Inputs
Model inputs of this model are as follows:
i : Mean (yearly) demand at retailer i , for each i 2 I
i2 : Variance of (daily) demand at retailer i , for each i 2 I
fi : fixed (annual) cost of locating a regional DC (RDC) at retailer j , for each
j 2I
vj .x/ : cost to ship x units from the main supplier (the plant) to a RDC located
at retailer j , for each j 2 I
dij : cost per unit to ship from retailer j to retailer i , for each i 2 I and j 2 I
˛ : Desired percentage of retailer’s orders satisfied
ˇ : Weight factor associated with the transportation cost
456
M. Kaviani
: Weight factor associated with the inventory cost
z˛ : Standard normal deviate such that P .z z˛ / D ˛
h : Inventory holding cost per unit of product per year
wj .x/ : Total annual cost of working inventory held at DC j if the expected daily
demand at j is x for each j 2 I
Fj : fixed cost of placing an order at DC j , for each j 2 I
L : lead time in days
19.2.1.2 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj D 1, if retailer j is selected as a DC location, and 0 otherwise for each j 2 I
Yij D 1, if retailer i is served by a DC based at retailer j , and 0 otherwise for
each i 2 I and each j 2 I
19.2.1.3 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
X
j
(
fj Xj C ˇ
X
i
i dij Yij C wj
X
i Yij
i
Subject to
X
j
Yij D 1
Yij Xj
!
Chz˛
sX
i
i2 Yij
9
=
;
(19.1)
8i
(19.2)
8i; j
(19.3)
Yij 2 f0; 1g 8i; j
(19.4)
Xj 2 f0; 1g 8j
(19.5)
Equation (19.1) minimizes the weighted sum of the following four costs: the
fixed cost of locating facilities, the shipping cost from the DCs to the non-DC retailers, the expected working inventory cost, and the safety stock costs. Equation (19.2)
ensures that each retailer is assigned to exactly one DC. Equation (19.3) states that
retailers can only be assigned to candidate sites that are selected as DCs. Equations
(19.4) and (19.5) are standard integrity constraints.
19
Location-Inventory Problem
457
19.2.2 Model of Nozick and Turnquist (1998)
Assume that there are n retail outlets assigned to the DC, each with a Poisson demand process whose mean rate is i where 1 i n. Each time an item is
demanded at a retail outlet, a replacement is ordered immediately from the DC.
Therefore, the demand at the DC is Poisson with a mean rate:
X
i
(19.6)
ƒD
i
Assume that the DC has a total of s items either in inventory or on order, and it
orders from the plant each time an item is sent to a retail outlet. We will define
and 2 to be the mean and variance of the delivery time of a product from the plant
to the DC once an order is placed. Then the performance measures for the single
DC and assigned retail outlets are the same as those for an M/G/s queue with an
arrival rate of ƒ, a mean and variance for the service time distribution of and 2 ,
respectively, and s servers.
The stock out rate is the percentage of demand that can not be satisfied from
on-hand inventory. In the queuing system representation, this is analogous to the
probability that when a customer enters the system, a server is not available (i.e.
wait time, denoted by W, is greater than zero). The probability of waiting (stock out),
P .W 0/ in an M/G/s queue can be approximated quite accurately by P .W 0/
in an M/M/s queue.
They use two equations to approximate r D P .W 0/ for two conditions to find
the minimum inventory necessary for a given stock out rate.
If there is a constant total expected demand across all retail outlets, the total
safety stock required is related to the number of DCs used. Consider a product with
an expected demand of 800 units per year. Furthermore, assume that this expected
demand is divided equally among N DCs. Based on these assumptions; they provided a histogram for this case that shows the safety stock necessary for various
numbers of DCs, given a stock out rate of 5%, and the predicted safety stock based
on a linear regression model:
Safety stock D a C bN D 22 C 1:8N
where N D number of DCs.
19.2.2.1 Model Inputs
Model inputs of this model are as follows:
fj : fixed cost of locating a facility at candidate site j
hi : demand at location i
dij : distance from demand location i to candidate site j
˛ : cost per unit distance per unit demand
458
M. Kaviani
19.2.2.2 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj D 1 if a facility is to be located at candidate site j , and 0 otherwise
Yij D fraction of demand at location i which is served by a facility at j
19.2.2.3 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
X
j
fj Xj C ˛
XX
i
hi dij Yij
(19.7)
j
Subject to
X
j
Yij D 1
Yij Xj
8i
(19.8)
8i; j
(19.9)
Xj D 0; 1 8j
Yij 0 8i; j
(19.10)
(19.11)
Equation (19.7) minimizes the sum of the following two costs: the fixed cost of
locating facilities and the shipping cost from the DCs to the demand locations assigned to the DC. Equation (19.8) ensures that each retailer is assigned to exactly
one DC. Equation (19.9) states that retailers can only be assigned to candidate sites
that are selected as DCs. Equations (19.10) and (19.11) are standard integrity constraints.
19.2.3 Model of Erlebacher and Meller (2000)
19.2.3.1 Model Assumptions
Model assumptions of this model are as follows:
The problem is represented by a unit-square grid structure with C columns and
R rows
Uniform customer demand across any grid
Distances are measured rectilinearly between plants and DC locations, and DC
locations and continuously-represented customer locations
Each DC operates under a continuous-review inventory system
The location and capacity of each plant is known and fixed
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Location-Inventory Problem
459
19.2.3.2 Model Inputs
Model inputs of this model are as follows:
i : Index for DCs; i D 1; : : :; N
j : Index for customer grids; j D 1; : : :; M
p: Index for plants; p D 1; : : :; P
.cj ; rj /: Location of customer grid j (right-hand side of grid, x D cj ; bottom of
grid, y D rj /
M
P
dj
dj : Average demand for customer j ; D D total average demand D
j D1
.ap ; bp /: Location of plant p
vp : Volume capacity at plant p
F : Annual cost of operating a DC
s: Unit DC-to-customer transportation (shipping) cost
l: Unit plant-to-DC transportation (logistic) cost
A; h; z; : Order cost, holding cost, safety-stock parameter, and standard deviation of total demand during lead-time, respectively
19.2.3.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Zi D 1 if DCi opened, and 0 otherwise
P
N D number of DCs to open D iZi
.xi ; yi / D location of DCi
wij D 1 if DCi serves customer grid j , and 0 otherwise
tip D average distance from DCi to customer grid j
upi D demand shipped from plant p to DCi
qpi D distance from plant p to DCi
19.2.3.4 Objective Function and Its Constraints
The objective function of this model and its related constraints are as follows:
v
u
p
p
r
M
X
X
uX
z
2Ah C p h t
dj wij C l
Zi upi qpi
Min F
Zi C
Zi
D
j D1
pD1 i D1
i D1
i D1
r
X
Cs
r
X
r X
M
X
i D1 j D1
y
Zi tijx C tij dj wij
(19.12)
where:
ˇ
ˇ
ˇ ˇ
qpi D ˇxi ap ˇ C ˇyi bp ˇ
(19.13)
460
M. Kaviani
tijx D
y
1
Œx .cj 1/2 C 21 .cj
2 i
ˇ
ˇ
ˇ
1 ˇ
xi .cj 1/ˇ C 12 ˇcj
2
tij D
Subject to
xi /2 I if cj 1 xi cj
ˇ
xi ˇ I otherwi se
1
Œy .rj 1/2 C 12 .rj
2 i
ˇ
ˇ
ˇ
1ˇ
yi .rj 1/ˇ C 21 ˇrj
2
M
X
i D1
yi /2 I if rj 1 yi rj
ˇ
yi ˇ I otherwi se
wij D 1
M
X
wij Zi
P
X
upi D
j D1
pD1
r
X
j D1
M
X
8j
dj wij
upi vp
8p
(19.15)
(19.16)
8i
j D1
(19.14)
(19.17)
8i
(19.18)
(19.19)
Zi 2 f0; 1g 8i
(19.20)
wij 2 f0; 1g 8i; j
(19.21)
upi 0 8p; i
(19.22)
Equation (19.12) minimizes the sum of the following four costs: the fixed cost
of locating facilities, total DC inventory costs, and total transportation costs from
plants to DCs, and from DCs to customers. Equations (19.13)–(19.15) compute
the rectilinear distances from the plants to the DCs, and the DC locations to the
continuously-represented customer locations. Equation (19.16) ensures that each
retailer is assigned to exactly one DC. Equation (19.17) states that retailers can only
be assigned to candidate sites that are selected as DCs. Equation (19.18) ensures that
each DC is fully supplied. Equation (19.19) checks each plant’s capacity. Equations
(19.20)–(19.22) are standard integrity constraints.
19.2.4 Model of Daskin et al. (2002)
The formulation that they obtain is a mixed integer non-linear programming problem which can be viewed as an extension of the traditional uncapacitated fixed
charge facility location problem. In addition to the standard facility location and
19
Location-Inventory Problem
461
local distribution costs, the model includes cost components representing working
and safety stock inventories at the DCs as well as transport costs from the supplier(s)
to the DCs. These inventory and supplier-to-DC transport costs introduce significant
nonlinearities into the model.
19.2.4.1 Model Assumptions
The formulation allows for multiple production plants but assumes that the uncapacitated DCs receive the product from the (uncapacitated) plant with the
smallest total shipping cost to the DC
Plant to DC lead time is the same for all plant/DC combinations
DC using a .Q; r/ inventory model
– Axsater (1996) notes that it is common to approximate the .Q; r/ model using
two steps with the order quantity determined by an EOQ model in which
the mean demand is used to represent the stochastic demand process and the
reorder point is determined in a second step
We know which customers are to be assigned to a specific DC
With these assumptions, the model is mathematically equivalent to assuming that
there is only a single plant.
19.2.4.2 Model Inputs
Model inputs of this model are as follows:
I : set of retailers indexed by i
J : set of candidate DC sites indexed by j
fj : fixed cost of locating a DC at candidate site j , for each i 2 I and j 2 J
dij : cost per unit to ship between retailer i and candidate DC site j , for each
i 2 I and j 2 J
: Days per year (used to convert daily demand and variance values to annual
values)
i : Daily mean demand at each retailer
i2 : Daily variance demand at each retailer
S : Set of customers assigned to the DC
L: Lead time in days for deliveries from the supplier to the DC
Z˛ : Is a standard Normal deviate .P .Z Z˛ / D a/
D: Expected annual demand (D D ˙i i /
h: holding cost per item per year
F : fixed cost of placing an order from the DC to the supplier
n: (unknown) number of orders per year
v.x/: cost of shipping an order of size x from the supplier
462
M. Kaviani
ˇ and are weights that we assign to transportation and inventory costs respectively so that we can later test the effects of varying the importance of these costs
relative to the fixed facility costs
g: Fixed cost of a shipment from the plant to the DC
a: Volume dependent cost of a shipment from the plant to the DC
19.2.4.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj D 1 if a facility is to be located at candidate site j , and 0 otherwise
Yij D 1 if demands at retailer i are assigned to a DC at candidate site j , and 0
otherwise
The annual cost of ordering inventory from the supplier at the DC is:
D
F n C ˇv
n
nC
hD
2n
(19.23)
Taking the derivative of this expression with respect to n, the number of orders
per year, and setting the derivative to zero, we obtain:
F C ˇv
nD
D
n
ˇnv0
D
n
hD
D
2 D0
n2
2n
p
hD=.2.F C ˇg//
(19.24)
(19.25)
Substituting this into the cost function we obtain an annual working inventory
cost of
p
2hD.F C ˇg/ C ˇaD
(19.26)
19.2.4.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
X
j 2J
0
0
fj Xj C @ˇ
XX
j 2J i 2I
1
dij i Yij A
1
Xs
X
X X
C@
2h.Fj C ˇgj /
i Yij C
aj
i Yij A
j 2J
ChZ˛
i 2I
X sX
j 2J
i 2I
Li2 Yij
j 2J
i 2I
(19.27)
19
Location-Inventory Problem
463
Subject to
X
8i
(19.28)
8i; j
(19.29)
Xj 2 f0; 1g 8j
(19.30)
Yij 2 f0; 1g 8i; j
(19.31)
j 2J
Yij D 1
Yij Xj
Equation (19.27) minimizes the sum of the following four costs: the fixed cost of
locating facilities, local delivery cost, total working inventory cost, and the safety
stock inventory cost. Equation (19.28) ensures that each retailer is assigned to exactly one DC. Equation (19.29) states that retailers can only be assigned to candidate
sites that are selected as DCs. Equations (19.30) and (19.31) are standard integrity
constraints.
19.2.5 Model of Shen and Qi (2007)
They propose a supply chain design model, which considers the impacts of the
strategic facility location decisions on the tactical inventory and shipment decisions
in a three-tiered supply chain. They assume that each customer has uncertain demand that follows a certain probability distribution. They assume that the customers
are uniformly scattered in a connected area. Several DCs will be opened and each
DC is served directly by the supplier and distributes products to customers. Each
customer will order at the beginning of the period, the DC combines the orders
from different customers and order from the supplier. The number and locations of
DCs are not given a priori. They assume that at every DC, there is a delivery truck
with fixed capacity. A key problem is that the demand that is seen by each DC is a
function of the demands at the customers assigned to that DC, which is a function
of the assignment of customers to the DC. They assume the transportation costs and
the inventory costs exhibit economies of scale under which the average unit cost
decreases as the total volume of activity increases. This realistic assumption will
result in several nonlinear terms in the formulation.
19.2.5.1 Model Assumptions
The customers are uniformly scattered in a connected region, S
The area of S is A
The customer demands are independent and follow Normal distributions
464
M. Kaviani
19.2.5.2 Model Inputs
Model inputs of this model are as follows:
I : Set of customers
J : Set of candidate DC locations
i : Mean (yearly) demand at customer i , for each i 2 I
i2 : Variance of (daily) demand at retailer i , for each i 2 I
fj : Fixed (annual) cost of locating a DC at j , for each j 2 J
˛ : Desired percentage of customer’s orders satisfied (fill rate)
ˇ : Weight factor associated with the shipment cost
: Weight factor associated with the inventory cost
Z˛ : Standard normal deviate such that P .Z Z˛ / D ˛
h : Inventory holding cost per unit of product per year
Fj : Fixed administrative and handling cost of placing an order at DC j , for each
j 2J
L : DC order lead time in days
gj : Fixed shipment cost per shipment from the plant to DC j
aN j : Cost per unit of a shipment from the plant to candidate site j
: Denote the number of visits in a year
m : Total number of customers served by a specific DC j
qN : Vehicle capacity
dij : Distance between customer i and DC j
Tj : Length of the optimal traveling salesman tour that visits DC j and the
customers it serves
19.2.5.3 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xj : 1, if j is selected as a DC location, and 0, otherwise
Yij : 1, if customer i is serviced by a DC based at j , and 0, otherwise
The annual cost of ordering inventory from the supplier at the DC is provided
exactly similar to Daskin et al. (2002) as:
q
2hDj .Fj C ˇgj / C ˇ aN j D
(19.32)
And for approximate Routing cost they assume each DC sends a truck to visit
its customers at a fixed frequency, say every day or every week and estimate the
routing distance from DC j to the customers it serves by the following formulation
(see Chap. 17: LRP):
,
#
"
P
X
P q
d
Y
2
i ij ij
bOij Yij
Vj
qN C 1 1=qN ˆ Yij NA D
i
i
i
(19.33)
19
Location-Inventory Problem
465
19.2.5.4 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
X
(
fj Xj C ˇ aN j
CKj
X
i
sX
i
i Yij C RCj
i Yij C q
sX
i
Subject to
X
j
Yij D 1
Yij Xj
X
i
bOij Yij
i2 Yij
9
=
!
(19.34)
;
8i
(19.35)
8i; j
(19.36)
Yij 2 f0; 1g 8i; j
(19.37)
Xj 2 f0; 1g
(19.38)
8j
where:
q
2.Fj C ˇgj /h
p
q D hZ˛ L
Kj D
Equation (19.34) minimizes the sum of the following five costs: the fixed cost
of locating facilities, the annual shipment cost from the supplier to the DCs, The
annual shipment cost from the DCs to the customers, working inventory cost, the
safety stock inventory cost. Equation (19.35) ensures that each retailer is assigned
to exactly one DC. Equation (19.36) states that retailers can only be assigned to
candidate sites that are selected as DCs. Equations (19.37) and (19.38) are standard
integrity constraints.
The constraints of the model are identical to those of the uncapacitated facility
location (UFL) problem, thus the problem we are studying is more difficult than the
standard UFL problem, which is already a notorious NP-hard problem.
19.2.6 Model of Miranda and Garrido (2008)
The problem is that of locating a set of facilities with constrained storage and ordering capacities, to serve a set of known and fixed spatially distributed clients with
stochastic demands, which must be satisfied to a given level of service.
466
M. Kaviani
19.2.6.1 Model Inputs
Model inputs of this model are as follows:
M : Number of clients to be served
N : Number of potential DCs
i : Index for DC i
j : Index for customer j
dj : Mean daily demand for each customer j
vj : Variance of the daily demand for each customer j
Qmax : Order size capacity
Icap : DC inventory capacity
F Ci : Daily fixed installation and operating costs for DC i
T Cij : Total daily transportation cost from DC i to client j
UOCi : Unit cost per order to be transported from a plant to DC i
OCi : Fixed ordering costs for DC i
H Ci : Daily holding cost per unit of product at DC i
1 ˛ : Level of service linked to the safety stock level at each DC
1 ˇ : Level of service linked to the inventory capacity constraint at each DC
Z : Value of the cumulative standard normal distribution up to a probability of
LTi : Deterministic lead time when ordering from DC i
19.2.6.2 Model Outputs (Decision Variables)
Model outputs of this model are as follows:
Xi D 1 if a DC is installed at site i , and 0 otherwise
Yij D 1 if DC i serves client j , and 0 otherwise
Di D Mean daily demand to be assigned to DC i
Vi D Variance of the daily demand assigned to DC i
Qi D Order size at warehouse i
19.2.6.3 Objective Function and its Constraints
The objective function of this model and its related constraints are as follows:
Min
X
i
F Ci Xi C
C
X
i
XX
i
j
H Ci Z1˛
.UOCi dj C T Cij /Yij C
p
LTi Vi
X
Qi
Di
C H Ci
OCi
Qi
2
i
(19.39)
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Location-Inventory Problem
467
Subject to
X
Yij D 1 8j
(19.40)
Yij Xj 8i; j
X
Yij dj 8i
Di D
(19.41)
i
(19.42)
j
Vi D
X
Yij vj
j
0 Qi Qmax
8i
8i
p
Qi C .Z1˛ C Z1ˇ / LTi Vi Icap Xi
Xj 2 f0; 1g 8j
(19.43)
(19.44)
8i
Yij 2 f0; 1g 8i; j
(19.45)
(19.46)
(19.47)
Equation (19.39) represents the total daily system costs and minimizes the sum
of the following four costs: the fixed and operating cost of locating facilities, daily
transport cost between each DC and its assigned customers, plus the transportation
and ordering costs between the plant and DCs, total ordering and working inventory
cost, and the safety stock inventory cost. Equation (19.40) ensures that each retailer
is assigned to exactly one DC. Equation (19.41) states that retailers can only be
assigned to candidate sites that are selected as DCs. Equations (19.42) and (19.43)
link the mean and the variance of each DC demand, to the mean and the variance of
the demand of each customer assigned to it. Equation (19.44) sets the accepted range
for the lot size. Equation (19.45) is the stochastic inventory capacity restriction.
Equations (19.46) and (19.47) are standard integrity constraints.
19.3 Solution Approaches
Among algorithms that are represented to solve different types of location-inventory
problems, we introduce some of the proposed associated approaches:
19.3.1 Solution Approach of Erlebacher and Meller (2000)
Since the problem they are solving is NP-hard, finding an optimal solution could
require enumerating overall values of N and all customer-to-DC allocations. This
enumeration could be time consuming for realistic-sized problems. To that end, they
first develop an analytical model based on some simplifying assumptions to determine a good starting point for a search on N . Then present bounds of N to limit the
search. And finally, they consider heuristics for allocating customers to DCs.
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M. Kaviani
19.3.1.1 Solution Approach
Determining the optimal number of DCs
– Stylized model for the optimal number of DCs
– Examples with stylized model
– Considering plant locations in the stylized model
Using bounds to determine a range on the optimum number of DCs
Location-allocation heuristics
– Location problems
– The allocation heuristics
– Considering plant locations in the heuristic
19.3.2 Solution Approach of Daskin et al. (2002)
To solve this problem, they use Lagrangian relaxation embedded in branch and
bound.
19.3.2.1 Solution Approach
Using a Lagrangian relaxation:
– Finding a lower bound for number of DCs
– Finding an upper bound for number of DCs
Retailer reassignments (try to improve the bounds)
DC exchange algorithm improvements (try to improve the bounds)
– For each DC in the current solution, find the best substitute DC that is not in the
current solution and retailers are assigned in a greedy manner to the DC
Variable fixing technique
Branch and bound
19.3.3 Solution Approach of Shen and Qi (2007)
Shen and Qi (2007) propose a Lagrangian relaxation based solution algorithm embedded in branch and bound to solve the problem, similar to Daskin et al. (2002). By
exploring the structure of the problem, they find a low-order polynomial algorithm
for the nonlinear integer programming problem that must be solved in solving the
Lagrangian relaxation sub-problems.
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19.3.3.1 Solution Approach
Finding a lower bound
Finding an upper bound
Variable fixing
19.3.4 Solution Approach of Miranda and Garrido (2006)
Their model is NP-hard and presents nonlinear terms in the objective function
and a nonlinear constraint. A heuristic solution approach is introduced, based on
Lagrangian relaxation and the sub gradient method.
19.3.5 Solution Approach of Miranda and Garrido (2008)
Their mathematical model is highly complex due to the incorporation of non-linear
terms into the constraints and objective function, making the standard FLP methods
difficult to apply. For this reason, a new solution method is developed based on
Lagrangian relaxation in conjunction with a complex heuristic (which, due to its
complexity is used only in a fraction of the iterations). The heuristic is based on a
greedy procedure and local improvement methods of the k-opt type. Finally, they
propose a set of inequalities that help to tighten the dual bounds as a part of the
relaxation process. These inequalities arise from the feasible boundaries that emerge
from all the possible combinations of means and variances of different clusters of
demand points.
19.3.5.1 Solution Approach
Relaxation and decomposition
Characterization of the feasible space
An exact procedure
Lagrangian heuristic
19.4 Case Study
In this section, we will introduce a real-world case study related to locationinventory problem:
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M. Kaviani
19.4.1 Distribution System for Finished Automobiles in US
(Nozick and Turnquist 1998)
Consider a distribution system for finished automobiles. The current distribution
system for finished automobiles relies on large networks of independent dealers to
provide the inventory of vehicles from which customers choose. Dealers order new
vehicles from the manufacturer in the models and colors. These orders typically
require 6–12 weeks to fill and make up the dealer stock. But most customers are
unwilling to wait about 6–12 weeks for a vehicle, so most vehicles are sold out of
the finished vehicle inventory held on dealer lots. The system of relying on dealers
to maintain separate pools of inventory is costly because dealers are forced to maintain a large inventory of cars. As an illustration of the magnitude of inventory costs
in the auto industry, consider that in a typical year about 15 million new cars and
light trucks are sold in the U.S., and the average value of those vehicles, is about
$18,500. If the average dealer holds 60 days’ inventory, total on-hand inventory is
approximately 2.5 million vehicles, representing an investment of over $46 billion.
If inventory carrying costs are about 22% of inventory value, then dealer stock generates annual inventory holding costs on the order of $10 billion.
One alternative method of managing this inventory is through a series of RDCs
which would hold most of the inventory and dealers served by a given RDC and
would have access to its entire inventory.
The questions of how many RDCs are necessary, and where should they be located, are critical to the success of the venture.
This example will focus on identifying how many RDCs are necessary and where
they should be located in order to minimize transportation, fixed facility, and inventory costs for new vehicles. In this case study the manufacturer serves demands
across the continental U.S., organized into 698 demand areas. For the purposes of
this analysis, the demand within each of the demand areas is assumed to occur at
the centroid of the demand area.
The fixed-charge model yields a recommendation of 23 RDCs, also determines
the assignment of demand areas to RDCs, and gives an indication (using line thickness) of the volume of demand allocated to each RDC.
They also illustrate the sensitivity of this solution to changes in the per-RDC
fixed-charge coefficient. It is important to notice that within a range of ˙10%, the
number of RDC’s recommended does not change.
References
Axsater S (1996) Using the deterministic EOQ formula in stochastic inventory control. Manage
Sci 42:830–834
Daskin MS, Coullard CR, Shen ZM (2002) An inventory-location model: formulation, solution
algorithm and computational results. Ann Oper Res 110:83–106
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Location-Inventory Problem
471
Erlebacher SJ, Meller RD (2000) The interaction of location and inventory in designing distribution
systems. IIE Trans 32:155–166
Miranda PA, Garrido RA (2006) A simultaneous inventory control and facility location model with
stochastic capacity constraints. Netw Spat Econ 6:39–53
Miranda PA, Garrido RA (2008) Valid inequalities for Lagrangian relaxation in an inventory location problem with stochastic capacity. Transport Res E 44:47–65
Nozick LK, Turnquist MA (1998) Integrating inventory impacts into a fixed-charge model for
locating distribution centers. Transport Res E-Log 34(3):173–186
Nozick LK, Turnquist MA (2001) Inventory, transportation, service quality and the location of
distribution centers of distribution centers. Eur J Oper Res 129:362–371
Shen ZM, Qi L (2007) Incorporating inventory and routing costs in strategic location models. Eur
J Oper Res 179:372–389
Shen ZM, Coullard C, Daskin MS (2003) A joint location-inventory model. Transport Sci
37(1):40–55
Snyder LV, Daskin MS, Teo CP (2007) The stochastic location model with risk pooling. Eur J Oper
Res 179:1221–1238
Chapter 20
Facility Location in Supply Chain
Meysam Alizadeh
A supply chain (SC) is the network of facilities and activities that performs the
function of product development, procurement of material from vendors, the movement of materials between facilities, the manufacturing of products, the distribution
of finished goods to customers, and after-market support for sustainment (Mabert
and Venkataramanan 1998). The supply chain not only includes the manufacturer
and suppliers, but also transporters, warehouses, retailers, and customer themselves.
Within each organization, the supply chain include, but not limited to, new product development, marketing, operations, distribution, finance, and customer service
(Chopra 2003).
A supply chain is dynamic and involves the constant flow of information, product, and funds between different stages. A typical supply chain may involve a variety
of stages. These supply chain stages include:
Customers
Retailers
Wholesalers/distributors
Manufacturers
Component/raw material suppliers
SC systems can be studied and analyzed from several viewpoints. Yet there are
three major perspectives of SC systems: (a) “material flow”, (b) “information flow”,
and (c) “buyer-seller relations” (Fazel Zarandi et al. 2002). The objective of every
supply chain is to maximize the overall value generated. Supply chain management
involves the management of flows between and among stages in a supply chain to
maximize totally supply chain profitability (Chopra 2003).
20.1 Design Phases in Supply Chain
Successful supply chain management requires many decisions relating to the flow
of information, product, and funds. These decisions fall into three categories or
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 20, c Physica-Verlag Heidelberg 2009
473
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phases, depending on the frequency of each decision and the time frame over which
a decision phase has an impact:
1. Supply chain strategy or design;
2. Supply chain planning or tactical level; and
3. Supply chain operations.
In the supply chain design phase, strategic decisions, such as facility location and
technology selection decisions play major roles. While the supply chain configuration is determined, the focus shifts to decisions at the tactical and operational levels,
such as inventory control and distribution decisions. The five major supply chain
drivers include:
Production: what, how, and when to produce;
Inventory: how much to make and how much to store;
Location: where best to do what activities;
Transportation: how and when to move product;
Information: the basis for making these decisions.
Facility location may be the most critical and most difficult of the decisions needed
to realize an efficiency supply chain. Transportation, inventory, and information
sharing decisions can be readily re-optimized in response to changes in the underlying conditions of the supply chain; but facility location decisions are often fixed
and difficult to change even in the intermediate term.
Therefore, the facility location decision is a strategic decision in supply chain
management and play a crucial role in the logistics activities involved in SCM.
20.2 Network Design in Supply Chain
20.2.1 The Role of Network Design in the Supply Chain
Supply chain network design decisions include the location of manufacturing, storage, or transportation-related facilities and the allocation of capacity and roles to
each facility. Supply chain network design decisions are classified as follows:
1. Facility role: what role should each facility play? What processes are performed
at each facility?
2. Facility location: where should facilities be located?
3. Capacity allocation: how much capacity should be allocated to each facility?
4. Market and supply allocation: what market should each facility serve?
Which supply sources should feed each facility?
All network design decisions affect each other and must be made taking this fact
into consideration.
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Facility location decisions have a long-term impact on a supply chain’s performance because it is very expensive to shut down a facility or move it to a different
location. A good location decisions can help a supply chain to be responsive while
keeping its costs low.
20.2.2 Factors Influencing Network Design Decisions
There are seven factors that influence network design decisions in supply chains:
Strategic factors,
Technological factors,
Macroeconomic factors,
Political factors,
Infrastructure factors,
Competitive factors, and
Operational factors.
In the next section we discuss models for making facility location and capacity
allocation decisions during Phase II and IV.
20.3 Classical Models
Managers use network design models in two different situations. First, these models
are used to decide on locations where facilities will be established and the capacity
to be assigned to each facility. Second, these models are used to assign current
demand to the available facilities and identify lanes along which product will be
transported. The following information must be available before the design decision
can be made:
Location of supply sources and markets,
Location of potential facility sites,
Demand forecast by market,
Facility, labor, and material costs by site,
Transportation costs between each pair of sites,
Inventory costs by site as well as a function of quantity,
Sale price of product in different regions,
Taxes and tariffs as product is moved between locations,
Desired response time and other service factors.
Given this information, either gravity or network optimization models may be used
to design the network.
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In this section, we review several traditional facility location models,beginning
with the classical fixed charge location model. We then show how the model can be
extended to incorporate additional facets of the supply chain design problem.
20.3.1 Fixed Charge Facility Location Problem (Daskin
et al. 2005)
This problem is a classical location problem and forms the basis of many location
models that have been used in supply chain design. The problem can be stated simply as follows. We are given a set of customer locations with known demands and a
set of candidate facility locations. If we select to locate a facility at a candidate site,
we incur a known fixed location cost. There is a known unit shipment cost between
each candidate site and each customer location. The problem is to find the locations
of the facilities and the shipment pattern between the facilities and the customers to
minimize the combined facility location and shipment costs subject to a requirement
that all customer demands be met (Daskin et al. 2005).
Specifically, we introduce the following notation:
20.3.1.1 Model Inputs
I : Set of customer locations, indexed by i
J : Set of candidate facility locations, indexed by j
hi : demand at customer location i 2 I
fi : fixed cost of locating a facility at candidate site j 2 J
cij : unit cost of shipment between candidate facility site j 2 J and customer
location i 2 I
20.3.1.2 Model Outputs (Decision Variables)
Xj D
1;
0
if we locate at candidate site j 2 J
if not
;
Yij : Fraction of the demand at customer location i 2 I that is served by a facility at
site j 2 J .
20.3.1.3 Objective Function and its Constraints
The problem is then formulated as the following integer program (Balinski 1965):
Min †j 2J fj Xj C †j 2J †i 2I hi cij Yij :
(20.1)
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Subject to
†j 2J Yij D 1 i 2 I:
Yij Xj 0 i 2 I I j 2 J;
Xj 2 f0; 1g j 2 J;
Yij 0 i 2 I I j 2 J:
(20.2)
(20.3)
(20.4)
(20.5)
The objective function (20.1) minimizes the total cost (fixed facility location C shipment) of setting up and operating the network. Equation (20.2) stipulates that each
demand node is fully assigned. Equation (20.3) states that a demand node cannot
be assigned to a facility unless we open that facility. Equation (20.4) enforces that
each candidate site is either open or closed and (20.5) are a simple non-negativity
equations.
20.3.2 Uncapacitated Facility Location Model with Single
Sourcing
The formulation given in previous section assumes that facilities have unlimited
capacity; the problem is sometimes referred to as the uncapacitated fixed charge
location problem. It is well known that at least one solution to this problem involves
assigning all of the demand at each customer location i 2 I fully to the nearest
open facility site j 2 J (Daskin et al. 2005). in other words, Yij 2 f0; 1g. Many
firms insist on or strongly prefer such single sourcing solutions as they make the
management of the supply chain considerably simpler.
20.3.3 Capacitated Facility Location Model
One natural extension of the problem is to consider capacitated facilities. If we
let bj be the maximum demand that can be assigned to a facility at candidate site
j 2 J (20.1)–(20.5) can be extended to incorporate facility capacities by including
the following additional equation (Daskin et al. 2005):
†i 2I hi Yij bj Xj 0 j 2 J:
(20.6)
Equation (20.6) limits the total assigned demand at facility j 2 J to a maximum
of bj . From the perspective of the integer programming problem, this constraints
obviates the need for (20.3) since any solution that satisfies (20.5) and (20.6) will
also satisfy (20.3).
For fixed values of the facility location variables, Xj , the optimal values of the
assignment variables can be found by solving a traditional transportation problem.
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M. Alizadeh
The embedded transportation problem is most easily recognized if we replace hi
Yij by Zij , the quantity shipped from distribution center j to customer i . The
transportation problem for fixed facility locations is then (Daskin et al. 2005):
Min †i 2I †j 2J cij Zij :
(20.7)
Subject to
X
j 2J
X
Zij D hi
i 2I
i 2 I;
Zij bj Xj
(20.8)
j 2 J;
Zij 0 i 2 I I j 2 J;
(20.9)
(20.10)
where we denote the fixed (known) values of location variables by XO .
The solution to the transportation problem (20.7)–(20.10) may involve fractional
assignments of customers to facilities. This means that the addition of (20.6) will not
automatically satisfy the single sourcing condition. To restore the single sourcing
condition, we can replace the fractional definition of the assignment variables by a
binary one: 8̂
<1; if demands at customer site i are served by facility at
Yij D
candidate site j
:̂
0;
if not
The problem becomes considerably more difficult to solve since there are now far
more integer variables.
Daskin and Jones (1993) observed that, in many practical contexts, the number of customers is significantly greater than the number of distribution centers
that will be sited. As such, each customer represents a small fraction of the total capacity of the distribution center to which it is assigned. Also, if the single
sourcing requirement is relaxed, the number of multiply sourced customers is less
than or equal to the number of distribution centers minus one. Thus, relatively few
customers will be multiply-sourced in most contexts. They further noted that warehouse capacities, when measured in terms of annual throughput as is commonly
done, are rarely known with great precision, as they depend on many factors, including the number of inventory turns as the warehouse. They therefore proposed a
procedure for addressing the single sourcing problem that involves (20.1) ignoring
the single sourcing constraint and solving the transportation problem, (20.2) using
duality to find alternate optima to the transportation problem that require fewer customers to be multiply sourced, and (20.3) allowing small violations of the capacity
constraints to identify solutions that satisfy the single sourcing requirement.
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20.3.4 Locating Plants and Distribution Centers with Multiple
Commodity
In a classic paper, Geoffrion and Graves (1974) extend the traditional fixed charge
facility location problem to include shipments from plants to distribution centers
and multiple commodities. They introduce the following additional notation:
20.3.4.1 Model Inputs
K: Set of plant locations, indexed by k
L: Set of commodities, indexed by l
Dli : demand for commodity l 2 L at customer i 2 I
Slk : supply of commodity l 2 L at plant k 2 K
Vj ; Vj : minimum and maximum annual throughput allowed at distribution center j 2 J
Vj : variable unit cost of throughput at candidate site j 2 J
clkji : unit cost of producing and shipping commodity l 2 L between plant k 2 K,
candidate facility site j 2 J and customer location i 2 I
20.3.4.2 Model Outputs (Decision Variables)
8̂
<1; if demands at customer site i are served by facility at
Yij D
candidate site j
:̂
0;
if not
Zlkji D quantity of commodity l 2 L shipped between plant k 2 K, candidate
facility site j 2 J and customer location i 2 I
20.3.4.3 Objective Function and its Constraints
The problem is then formulated as the following (Geoffrion and Graves 1974):
Min †j 2J fj Xj C †j 2J vj .†i 2l †l2L Dli Yij / C †l2L †k2K †j 2I †i 2I Clkji Zlkji :
(20.11)
Subject to
†j 2J †i 2I Zlkji Slk k 2 KI 1 2 L;
X
Zlkji D Dli Yij 1 2 LI j 2 J I i 2 J ;
k2K
(20.12)
(20.13)
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M. Alizadeh
X
j 2J
Yij D 1
Vj X j
X
i 2 I;
Dli Yij Vj Xj
Xj 2 f0; 1g j 2 J;
(20.14)
j 2 J;
Yij 2 f0; 1g i 2 I I j 2 J;
Zlkj i 0 i 2 I I j 2 J I k 2 KI 1 2 L:
(20.15)
(20.16)
(20.17)
(20.18)
The objective function (20.11) minimizes the total fixed and variable costs of the
supply chain network. Equation (20.12) states that the total amount of commodity l 2 L shipped from plant k 2 K cannot exceed the capacity of the plant to
produce that commodity. Equation (20.13) says that the amount of commodity
l 2 L shipped to customer i 2 I via distribution center (DC) j 2 J must equal to
amount of that commodity produce at all plants that is destined for that customer
and shipped via that DC. Equation (20.14) is the now-familiar single-sourcing constraint. Equation (20.15) imposes lower and upper on the throughput processed at
each distribution center that is used.
20.4 Integrated Decision Making Models
As Shen (2007) discussed in his paper, in the literature, we have seen many papers that study the integration and coordination of any two of the three important
supply chain decisions. He review these papers based on the following three categories: (1) location/routing (LR) models; (2) inventory/routing (IR) models; and (3)
location/inventory (LI) models.
20.4.1 Integrated Location-Routing Models (LR)
An important limitation of the fixed charge location model, and even a multiechelon, multi-commodity extension of Geoffrion and Graves, is the assumption
that full truckload quantities are shipped from a distribution center to a customer
(Daskin et al. 2005). In many contexts, shipments are made in less-than-truckload
(LTL) quantities from a facility to customers along a multiple-stop route. In the case
of full truckload quantities, the cost of delivery is independent of the other deliveries made, whereas in the case of LTL quantities, the cost of delivery depends on the
other customers on the route and the sequence in which customers are visited. During the past three decades, a sizeable body of literature has developed on integrated
location/routing models.
Integrated location/routing problems combine three components of supply chain
design: facility location, customer allocation to facilities and vehicle routing.
Laporte (1998) reviews early work on location routing problems; he summarizes
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481
the different types of formulations, solution algorithms and computational results
of work published prior to 1988. More recently, Min et al. (1998) develop a hierarchical taxonomy and classification scheme that they is used to review the existing
location routing literature. They categorize papers in terms of problem characteristics and solution methodology. One means of classification is the number of
layers of facilities. Typically, three-layer problems include flows from plants to
distribution centers to customers, while two-layer problems focus on flows from
distribution centers to customers.
Like three-layer formulation of Perl, two-layer location routing formulations
(e.g., Laporte et al. 1998) usually are based on integer linear programming formulations for the vehicle routing problem (VRP). Flow formulations of the VRP are
often classified according to the number of indices of the flow variable. The size and
structure of these formulations make them difficult to solve using standard integer
programming or network optimization techniques. Motivated by the successful implementation of exact algorithms for set-partitioning-based routing models, Berger
(1997) formulates a two-layer location/routing problem, she formulates the routes
in terms of paths, where a delivery vehicle may not be required to return to the distribution center after the final delivery is made. In the latter case, the time to return
from the last customer to the distribution center is much less important than the time
from the facility to the last customer. Thee is a recent study from Chan et al. (2001),
in which they apply the three-dimensional space-filling curve. More recent literature
review can be found in Nagy and Salhi (2007).
20.4.2 Integrated Inventory-Routing Models (IR)
There are four characteristics associated with IR models: demands; fleet size, i.e.,
the number of available vehicles, which is limited or unlimited; length of the planning horizon; and number of demand points visited on a vehicle trip.
Kleywegt et al. (2002) provide an excellent survey and classification of IR models using the above four categories plus a fifth category on the research contribution
of the models. More recent literature review can be found in Kleywegt et al. (2002)
and Adelman (2003).
20.4.3 Integrated Location-Inventory Models (LI)
The fixed charge location problem ignores the inventory impacts of facility location
decisions; it deals only with the tradeoff between facility costs, which increase with
the number of facilities, and the average travel cost, which decreases approximately
as the square root of the number of facilities located (call it N ). Baumol and Wolfe
(1958) recognized the contribution of inventory to distribution costs. Eppen (1979)
argued that safety stock costs also increase as the square root of N .
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M. Alizadeh
While the contribution of inventory to distribution costs has been recognized,
many papers study the location, inventory, and distribution coordination issues, but
most of these papers either ignore the inventory costs, or approximate the nonlinear
costs with linear functions. Recently there are a few papers that consider nonlinear
cost terms in their models. Erlebacher and Meller (2000) formulate a non-linear integer location/inventory model. Shen et al. (2003), Daskin et al. (2001) propose the
joint location/inventory in which location, shipment and nonlinear safety stock inventory costs are included in the same model. They develop an integrated approach
to determine the number of DCs to establish, the location of the DCs, and the magnitude of inventory to maintain at each center.
All of the models that will be reviewed in the remainder of this chapter are built
on the basic model described in the next section.
20.5 Basic Model Formulation
Shen (2007) considers the design of a three-tiered supply chain system consisting
of one or more suppliers, DCs, and retailers. Each retailer has uncertain demand.
The problem is to determine how many DCs to locate, where to locate them, which
retailers to assign to each DC, how often to reorder at the DC and what level of
safety stock to maintain, so as to minimize total cost, while ensuring a specified
level of service.
20.5.1 Model Inputs
I : set of retailers
J : set of candidate DC location
i : mean (daily) demand at retailer i , for each i 2 I
i 2 : variance of (daily) demand at retailer i , for each i 2 I
fj : fixed (annual) cost of locating a DC at j , for each j 2 J
dij : cost of shipping a unit from DCj to retailer i , for each i 2 I and j 2 J
˛: desired percentage of retailer orders satisfied (fill rate)
ˇ: weight factor associated with the shipment cost
: weight factor associated with the inventory cost
h: inventory holding cost per unit of product per year
Fj : fixed administrative and handling cost of placing an order at DC j , for each
j 2J
L: DC order lead time in days
gi : fixed shipment cost per shipment from the supplier to DCj
aN j : cost per unit of a shipment from the supplier to DCj
: a constant used to convert daily demand into annual demand
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20.5.2 Model Outputs (Decision Variables)
Xj D 1, if retailer j is selected as a DC location, and 0 otherwise, for each
j 2J
Yij D 1, if retailer i is served by a DC based at location j , and 0 otherwise, for
each i 2 I and j 2 J
20.5.3 Objective Function and its Constraints
Min †j 2J ffj Xj Œ†i 2a .ˇi dij C ˇaj j /χYij g C
q
q
q
2h.Fj C ˇgj / †i Yij C hzα †Li2 Yij :
(20.19)
Subject to
†j 2J Yij D 1 8i 2 I;
(20.20)
Yij Xi 0 8i;
(20.21)
Yij 2 f0; 1g 8i 2 I; j 2 J;
(20.22)
Xj 2 f0; 1g j 2 J:
(20.23)
The first two terms of the objective function are structurally identical to those of
the uncapacitated facility location (UFL) model, which include the fixed cost of
locating facilities, and the delivery costs from the DCs to the retailers as well as the
marginal cost of shipping a unit from a supplier to a DC. The last two terms are
related to inventory costs. Equation (20.20) requires that each retailer is assigned
to exactly one DC. Equation (20.21) states that retailers can only be assigned to
candidate site that are selected as DCs. Equations (20.22) and (20.22) are standard
integrality constraints.
Daskin et al. (2001) and Shen et al. (2003) have discovered some interesting
properties of this model.
20.6 Model with Routing Cost Estimation
In this section, we consider a more realistic modeling of the shipment costs from
DCs to retailers. We consider a three-tiered supply chain system consisting of one
or more suppliers, DCs, and retailers. We assume that each retailer has uncertain demand that follows a certain probability distribution. The locations of the
suppliers are known, but the exact locations of the retailers are not known. We
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M. Alizadeh
assume the retailers are uniformly scattered in a connected area. We assume that
the transportation cost exhibits economies of scale under which the average unit
cost decrease as the travel distance increases.
All of the models we reviewed so far assume that the locations of the retailers
are given. However, during the supply chain design phase, usually not much information is given about retailer locations. Thus, they should not be fixed in the supply
chain design stage. Continuous approximation models, which use continuous functions to represent distributions of retailer location and demand, have been developed
to provide insights into complicated mathematical programming models. It is also
widely recognized that continuous models should supplement mathematical programming models, not replace them (Geoffrion 1976; Hall 1986). In this section,
we plan to use continuous models to approximate the routing cost in our integrated
supply chain model.
We study an integrated stochastic supply chain design model in which we relax
two major assumptions in the models from Sect. 20.4.3. Instead of assuming that the
decision maker knows exactly where the retailers are located, we assume that the
retailers are scattered in a connected region according to a certain distribution. Furthermore, we model the shipment from a DC to its retailers using a vehicle routing
model instead of the linear direct shipping model. We also assume that there is a
dedicated truck in each DC that delivers to the retailers every period using a certain
route. It is reasonable to assume that, under some conditions (e.g., the driver does
not have to work overtime), the transportation cost related to this route is concave
in the distance travelled.
Let Nı be the vehicle capacity, dij be the distance between retailer i and DCj . The
integrated supply chain model can then be formulated as:
r
q
X
σ2i Yij :
(20.24)
Subject to (20.20)–(20.23)
p
where gO ij D 2 i dij =Nı C .1 1=Nı/˚ A=N 0.
20.7 Model with Capacitated DCs
Ozsen et al. (2003) introduce a capacitated version of the model in Sect. 20.4.3.
They assume that the DCs have capacity restrictions. The capacity constraints are
defined based on how the inventory is managed. Thus, the relationship between the
capacity of a DC and the inventory levels are embedded in the model.
Assume that the inventory is managed by a (r; Q) model with a type-I service
level constraint at each DC. Define the following notation for each j 2 J :
Cj : the capacity of DCj
rj : the reorder point at DCj
Qj : the reorder quantity at DCj
20
Facility Location in Supply Chain
485
It is easy to see that the inventory at DCj reaches its maximum when there is no
demand during the lead time. Thus, the maximal accumulation at DCj equals Qj C
rj , and rj D safety stockC E[demand during lead time]. Let Dj be the expected
annual demand of retailers served by DCj We can write the capacity constraint for
DCj as:
Qj C rj Cj :
Adding this constraint to (20.20)–(20.23) in Sect. 20.4.3, we obtain a new model
with nonlinear terms in both the objective function and the constraints.
20.8 Model with Multiple Levels of Capacity (Amiri 2006)
This section addresses the distribution network design problem in a supply chain
system that involves locating production plants and distribution warehouses, and
determining the best strategy for distributing the product from the plants to the warehouses and from the warehouses to the customers. The goal is to select the optimum
numbers, locations and capacities of plants and warehouses to open so that all customer demand is satisfied at minimum total costs of the distribution network. Unlike
most of past models, this model allows for multiple levels of capacities available
to the warehouses and plants. Amiri (2006) develop a mixed integer programming
model and define the following notations:
20.8.1 Model Inputs
N : index set of customers zones,
M : index set of potential warehouse sites,
L: index set of potential plant sites,
R: index set of capacity levels available to the potential warehouses,
H : index set of capacity levels available to the potential plants,
Cij : cost of supplying one unit of demand to customer zone i from warehouse at
site j ,
Cj k cost of supplying one unit of demand to warehouse at site j from plant at
site k,
Fjr : fixed cost per unit of time for opening and operating warehouse with capacity
level r at site j ,
Gkh : fixed cost per unit of time for opening and operating plant with capacity level
h at site k,
ai : demand per unit of time of customer zone I ,
bjr : capacity with level r for the potential warehouse at site j ,
ekh : capacity with level h for the potential plant at site k.
486
M. Alizadeh
20.8.2 Model Outputs (Decision Variables)
Uij : fraction (regarding ai ) of demand of customer zone i delivered from warehouse at site j ,
Yjrk : fraction (regarding bjr ) of shipment from plant at site k to warehouse at site
j withcapacity level r,
1; if a warehouse with capacity level r is located at site j
Xjr D
0;
otherwise
1; if a plant with capacity level h is located at site k
Vkh D
0;
otherwise
20.8.3 Objective Function and its Constraints
In terms of the above notation, Amiri (2006) formulated the problem as follows:
Min †i 2N †j 2M Cij ai Xij C †r2R †j 2M †k2L Cjk bjr Yjkr
XX
XX
C
Fjr Ujr C
Gkh Vkh :
j 2M r2R
(20.25)
k2L h2H
Subject to
†j 2M Xij D 1 8i 2 N;
X
X
b r U r 8j 2 M;
ai Xij
r2R j j
i 2N
X
U r 1 8j 2 M;
r2R j
X
X
X
b r Y r 8j 2 M;
ai Xij
k2L
r2R j j k
i 2N
X
X
X
e h V h 8k 2 L;
br Y r
h2H k k
j 2M
r2R j jk
X
vhk 1 8k 2 L;
h2H
Xij 0
8i 2 N and j 2 M;
urj 2 .0; 1/ 8j 2 M and r 2 R;
Yjrk 0 8k 2 L; j 2 M and r 2
Vkh 2 .0; 1/ 8k 2 L and h 2 H:
R;
(20.26)
(20.27)
(20.28)
(20.29)
(20.30)
(20.31)
(20.32)
(20.33)
(20.34)
(20.35)
20.9 Model with Service Considerations
When designing supply chains, firms are often faced with the competing objectives
of improving customer service and reducing cost. We extend the basic model in
Sect. 20.4.3 to include a customer service element and develop practical methods for
20
Facility Location in Supply Chain
487
quick and meaningful evaluation of cost/service tradeoffs. Service is measured by
the fraction of all demands that are located within an exogenously specified distance
of the assigned DC.
Ross and Soland (1980), in one of the earliest papers on multi-objective location
problems, argue that practical problems involving the location of public facilities
should be modeled as multi-objective problems. Cost and service are the typical
objectives, although there exist several distinct objectives in each of those two
categories: fixed investment cost, fixed operating cost, variable operating cost, total operating cost, and total discounted cost are all reasonable cost objectives to
consider; and both demand served and response time (or distance traveled) are
appropriate objectives for service measurement. They treat such multi-objective
problems in the framework of a model for selecting a subset of M sites where
public facilities are established to serve client groups located at N distinct points.
For a review of other multi-objective location models, we refer the reader to Shen
(2005).
Model (20.20)–(20.23) in Sect. 20.4.3 captures important facility location, transportation and inventory costs. Some retailers may be served very well, in the sense
that they are located very close to the DCs to which they are assigned, while other
retailers may be served very poorly by this criterion. The maximal covering location
problem (Church and ReVelle 1974) maximizes the number of customers that can be
covered by a fixed number of facilities. Customer i is covered if node i is assigned
to a facility that is within dc of node i , where dc is the coverage distance. Instead
of maximizing covered demand volume, for fixed total demand, we can minimize
the uncovered demand volume. In a manner similar to Daskin (1995), we can then
formulate a model that simultaneously minimizes the fixed costs of the facilities and
a weighted sum of the uncovered demand volume as follows:
Min
X
j 2J
fj Xj C
X
j 2J
X
i 2I
dOij Yij :
(20.36)
Subject to (20.20)–(20.23)
Where W is the weight on the uncovered demand volume and Uij D i , if dij >
dc , and 0, if not.
This model is structurally identical to basic model. The only difference is that
we penalize all assignments of demand nodes to DCs that are more than dc away
from the DC. By varying the weight W on uncovered demand volume, we can trace
out an approximation to the set of non-inferior solutions to the tradeoff between
location-inventory costs and customer responsiveness. Altiparmak et al. (2006)
presented mixed-integer non-linear programming model for multi-objective optimization of SCN. They considered three objectives: (1) minimization of total cost
comprised of fixed costs of plants and distribution centers (DCs), inbound and outbound distribution costs, (2) maximization of customer services that can be rendered
to customers in terms of acceptable delivery time (coverage), and (3) maximization
of capacity utilization balance for DCs (i.e., equity on utilization ratios). To deal
with multi-objective and enable the decision maker to evaluate a greater number
of alternative solutions, two different weight approaches were implemented in the
proposed GA.
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M. Alizadeh
20.10 Profit Maximizing Model with Demand Choice Flexibility
Minimizing total cost has been the primitive objective in most of the supply chain
network design models. These models typically require that every customer’s demand has to be satisfied. However, for a profit-maximizing business, it may not
always be optimal to satisfy all potential demands, especially if the additional cost
is higher than the additional revenue associated with servicing these customers.
Furthermore, if the company is facing competition, sometimes it might be more
profitable to lose some potential customers to competitors since the cost of maintaining these customers can be prohibitively high. Most supply chain related costs,
such as location, transportation, and inventory related costs of an item depend on
total demand volume, and no clear method exists for determining a customer’s profitability a priori, based solely on the characteristics of this customer. Shen addresses
this problem by proposing a profit-maximizing supply chain design model in which
a company can choose whether to satisfy a customer’s demand.
The literature on integrating cost-minimizing supply chain network design models with profit-maximizing microeconomic theory is limited. Hansen et al. (1987)
and Hansen et al. (1995) provide surveys on profit-maximizing location models.
Zhang (2001) considers a profit-maximizing location model, where a firm needs
to decide where to locate a single DC to serve the customers and to set the price
for its product, so as to maximize its total profit. He shows that under certain assumptions on the complexity counts, a special case of this problem can be solved in
polynomial time. However, he does not consider any cost related terms in his model,
not even the DC location costs.
Another location model is proposed by Meyerson (2001), where he considers a
profit earning facility location problem for a given set of demand points, and the decision maker needs to open some facilities such that every demand may be satisfied
from a local facility and the total profit is maximized. Instead of incurring a cost,
opening a facility can gain a certain profit which is a function of the amount of demand the facility satisfies. Since this profit-earning problem is NP-hard, Meyerson
develops an approximation algorithm that is based on linear program rounding.
Demand choice flexibility in inventory models is proposed by Geunes et al.
(2004). Their models address economic ordering decisions when a producer can
choose whether to satisfy multiple markets. They assume the per unit revenue from
serving a specific customaries exogenously given, not a decision variable. Furthermore, their models are pure inventory models; supply chain design related decisions,
such as location decisions, are absent from the models.
Shen (2006) considers not only location decisions and customer-DC assignment
decisions, but also production and inventory related decisions within a supply chain
network. It is assumed that after production, the product is shipped to several DCs,
and then from DCs to each customer the company has decided to serve. Contrary
to Meyerson’s model, assumes opening a facility incurs a certain cost, however, by
serving the customers assigned to this facility, a certain amount of profit will be
gained, which is also a function of the total demand served by the facility.
20
Facility Location in Supply Chain
489
The profit-maximizing supply chain network design problem can be described as
following: Given a production site o, a set I of customers and a set J of candidate
DC locations, the decision maker wants to determine (a) the number and locations of
the DCs, (b) the assignments of customers to DCs, (c) the inventory replenishment
strategy at each DC, and (d) the sale price of the product at each region, so that
the total profit is maximized. The profit is equal to the revenue minus the total cost,
which includes the DC location cost, inventory holding cost at DCs, and delivery
cost to DCs.
To model this problem, we define the following additional notation:
fj .x/ (annual) cost of locating and operating a DC at j , for each j 2 J . it is a
concave function of the annual demand x flowing through DCj
K: fixed cost of delivering product from the production site to a DC
doj : distance from the production site o to DCj , j 2 J
dij : distance from the DCj to customer i , for i 2 I; j 2 J
c.doj / delivery cost per unit from the production site o to DCj it has the following structure:
8̂
c1 ; if 0 < doj < d1
ˆ
ˆ
<c ; d < d < d
2
1
oj
2
C.doj / D
:
ˆ: : : ;
ˆ
:̂
dt 1 < doj < dt
ct ;
It is typically assumed that c1 < c2 < : : : < ct
pj : sale price per unit at the region served by DC j
ui : customer i ’s reserve price g.dij / delivery cost per unit charged to customer
if delivery is made from DCj . it has the following structure:
8̂
v1 ; if 0 < dij < d1
ˆ
ˆ
<v ; d < d < d
2
1
ij
2
g.dij / D
:
ˆ
ˆ: : : ;
:̂
vt ;
dt 1 < dij < dt
it is typically assumed that v1 v2 : : : vt
Shen (2006) assumes the company will lose a customer if the price a customer
has to pay is higher than the customer’s reserve price, i.e., if ui < pj C g.dij /. He
defines
(
pj C g.dij / c.doj /; pj C g.dij / ui
:
Ri .pi C dij / D
0;
pj C g.dij / > ui
Shen (2006) formulates the decision problem as a set-covering model, and present a
branch-and-price algorithm to solve this model. To solve the resulting pricing problem, it suffices to find, for every DCj 2 J , a maximum-reduced-cost set Rj I ,
490
M. Alizadeh
that uses j as the designated DC. Thus, the pricing problem reduces to finding Rj
for each j 2 J . To find Rj , the following integer programming problem must be
solved:
r
r
X
X
X
X
ai Yij 2kh
Min
i Yi hZ˛
i Yi :
i2 Yi fj
i 2S
i 2S
i 2S
i 2S
(20.37)
Subject to
Yi 2 f0; 1g;
8i 2 S; S :
(20.38)
20.11 Model with Multiple Commodities
Nowadays, retailers carry thousands of different products and the amount of data
involved in a supply chain design model can be overwhelming. It is not necessary,
and may not be possible, to account for all distinct products in the strategic supply
chain design phase. Aggregated information at the product category level should be
used instead (Simchi-Levi et al. 2000).
Shen (2007) states that Warszawski and Peer (1973) and Warszawski (1973)
are among the first to study the multi-commodity location problem. These models
consider fixed location costs and linear transportation costs, and assume that each
warehouse can be assigned at most one commodity.
Geoffrion and Graves (1974) consider the capacitated version of the multicommodity location problem in which they impose capacity constraints on the
suppliers and the DCs. They also assume that each customer must be served with all
the products it requires from a single DC or directly from a supplier.
Shen (2005) multi-commodity supply chain design model can be stated as follows: We are given a set of alternative facility locations, a set of retailers, a set of
different products, and a certain activity whose cost can be modeled using a concave
function. The objective is to design a supply chain system that can serve outside demand at minimum cost.
Shen (2007) use the following additional notation:
L: set of commodities;
It : set of retailers that have demand for commodity l 2 L
i l : mean annual demand from retailer i for commodity l.
Decision variables
Xj D 1, if j is selected as facility location, and 0 otherwise, for each j 2 J
Yij l D 1, if the demand for commodity l of retailer i is served by j , and 0
otherwise, for each i 2 I; j 2 J; l 2 L
20
Facility Location in Supply Chain
491
With this notation, Shen (2005) formulates the problem as follows:
Min
X
j 2J
n
X
fj Xj C
l2L
hX
i 2l
.dijl il /Yijl C Gjl
X
i 2l
il Yijl
io
:
(20.39)
Subject to
X
j 2J
Yijl D 1; 8i 2 I; l 2 L;
Yijl Xj 0 8i 2 I; j 2 J; l 2 L;
Yijl 2 f0; 1g:8i 2 I; j 2 J; l 2 L;
Xj 2 f0; 1g:8j 2 J:
(20.40)
(20.41)
(20.42)
(20.43)
20.12 Model with Unreliable Supply
A supply chain network is a complex system in which there are many uncertainties,
such as demands from the customers and delivery reliability. Because most supply
chain design decisions (e.g., facility location) are irreversible, it is critical to take
these uncertainties into account.
As Shen (2007) discussed, Qi and Shen consider the multi-period problem that
in each period, multiple retailers order a specific product from a supplier, and the
supplier ships the product to some intermediate facilities selected from a set of candidate locations. However, the amount of final product delivered on time to a retailer
may not be exactly the amount this retailer requests from the supplier, because of the
quality issues resulting from different production/assembly capabilities in different
facilities, mistakes made during the assembly/packaging operations, the weather or
other factors that may impact the on-time delivery from facilities to retailers. Thus,
the decision maker needs to take this unreliability issue into consideration when
designing the supply chain.
As Snyder and Daskin (2007) surveyed, a more formal review of this body of literature is presented by Snyder et al. (2006). Owen and Daskin (1998), Daskin et al.
(2005), and Snyder (2006) all provide reviews of stochastic location models (generally considering uncertainty in demand, rather than disruptions to facilities). See
Birge and Louveaux (1997) or Higle (2005) for an introduction to general stochastic programming techniques. Snyder and Daskin (2005) introduce several models,
based on classical facility location problems, in which facilities may fail with a
given probability. They minimize a weighted sum of two objectives, one of which
is a classical objective (ignoring disruptions) and the other is the expected cost after accounting for disruptions. Customers are assigned to several facilities, one of
which is the “primary” facility that serves it under normal circumstances, one of
which serves it if the primary facility fails, and so on.
492
M. Alizadeh
As Snyder and Daskin (2007) survived in their paper, Church and Scaparra
(2005) and Scaparra and Church (2005, 2006) consider the fortification, rather than
design, of facilities – that is, the network is assumed to exist and the firm has
resources to prevent disruptions at some of them, thus partially fortifying the network. Their model finds the best facilities to fortify assuming that an interdictor
will attempt to cause worst-case losses for the firm by disrupting a fixed number of
the un-fortified facilities. Similarly, Daskin et al. (2005) allow the firm to choose
whether each facility opened is reliable or unreliable; reliable facilities come at a
higher cost.
20.12.1 Model Inputs
Based on basic model that discussed in Sect. 20.4.3, Qi and Shen define the following additional notation for this problem:
c: purchasing price from the supplier per unit of product
: number of periods per year
Rj : the reliability coefficient associated with facility j , j 2 J , which is a random
variable between 0 and 1. Let j D E.Rj / and j2 D Var (Rj )
pi : retail price at retailer i per unit of product, i 2 I
i : penalty cost of lost good will at retailer i per unit of product, i 2 I
vi : salvage value at retailer i per unit of product, i 2 I
20.12.2 Model Outputs (Decision Variables)
1; facility j 2 J is open
,
0; otherwise
Qij : order quantity at facility j 2 J from retailer i 2 I in each period.
Xj D
20.12.3 Objective Function and its Constraints
Qi and Shen assume that the per-unit purchase and transportation costs are based
on the quantity ordered, not the quantity actually received by the retailers. Based on
this assumption, they formulate the following integrated model:
X
Min
X
C
j 2J
i 2I
fj Xj C c
X
Ti .Q/ D .Q/
i 2I
X
Qij C aj
j 2J
fj Xj :
X
i 2I
Qij C KJ
r
X
i 2I
Qij
(20.44)
20
Facility Location in Supply Chain
493
Subject to
1 e ˇQij Xj
i 2 I; j 2 J;
Qij 0
i 2 I; j 2 J;
Xj 2 f0; 1g
j 2 J:
(20.45)
(20.46)
(20.47)
The objective function maximizes the expected annual profit of the entire system
including all facilities and retailers. The first term represents the facility location
cost for opening facilities and the second term is the annual purchasing cost from
the supplier. The third and fourth term represent the working inventory cost and
the safety stock cost associated with each facility. The last term is profit earned at
retailers. Equation (20.45) requires that retailers can only order from open facilities.
An exponential function is used to formulate this restriction because of the quick
convergence property of the exponential function. The positive constant ˇ is used to
expedite the convergence.
20.13 Model with Facility Failures (Snyder 2003)
Once a set of facilities has been built, one or more of them may from time to time
become unavailable, for example, due to inclement weather, labor actions, natural
disasters, or changes in ownership. These facility “failures” may result in excessive
transportation costs as customers previously served by these facilities must now be
served by more distant ones. In this section, we discuss models for choosing facility locations to minimize fixed and transportation costs while also hedging against
failures within the system. The goal is to choose facility locations that are both inexpensive and reliable.
The models discussed in this section are based on the fixed charge location problem; they address the tradeoff between operating cost (fixed location costs and
day-to-day transportation cost, the classical fixed charge problem objective) and
failure cost (the transportation cost that results after a facility has failed). The first
model considers the maximum failure cost that can occur when a single facility fails,
while the second model considers the expected failure cost given a fixed probability
of failure.
In addition to the notation defined earlier, let
8̂
<1; if facility j 2 J serves as the primary facility and facility
Yij k D
k 2 J serves as the secondary facility for customer i 2 I
:̂
0;
if not
And let V be a desired upper bound on the failure cost that may result if a facility
fails.
494
M. Alizadeh
20.13.1 Objective Function and its Constraints
Snyder (2003) formulates the maximum-failure-cost reliability problem as follows:
Min
Subject to
X
j 2J
X
k2J
X
k2J
X
j 2J
fj Xj C
Yijk D 1
Yijk Xj
X
i 2I
X
j 2J
Yijj D 0
Xj 2 f0; 1g
k2J
hi cij Yijk :
8i 2 I ;
8i 2 I; 8j 2 J;
k2J
(20.48)
(20.49)
Yijk Xk 8i 2 I; 8j 2 J; 8k 2 J;
X X
X X X
hi cij Yijk C
i 2I
j 2J
X
i 2I
(20.50)
(20.51)
k2J
hi cij Yijk V 8j 2 J ;
(20.52)
8i 2 I; 8j 2 J;
(20.53)
8j 2 J;
(20.54)
Yijk f0; 1g 8i 2 I; 8j 2 J; 8k 2 J:
(20.55)
The objective function sums the fixed cost and transportation cost to customers from
their primary facilities. Equation (20.49) requires each customer to be assigned to
one primary and one backup facility. Equation (20.50) and (20.51) prevent a customer from being assigned to a primary or a backup facility, respectively, that has
not been opened. Equation (20.52) is the reliability constraint and requires the failure cost for facility j to be no greater than V . The first summation computes the
cost of serving each customer from its primary facility if its primary facility is not
j , while the second summation computes the cost of serving customers assigned to
j as their primary facility from their backup facilities. Equation (20.53) requires a
customer’s primary facility to be different from its backup facility, and (20.54) and
(20.55) are standard integrality and non-negativity constraints. This model can be
solved for small instances using an off-the-shelf IP solver, but larger instances must
be solved heuristically.
The expected-failure-cost reliability model (Snyder and Daskin 2003) assumes
that multiple facilities may fail simultaneously, each with a given probability q of
failing. In this case, a single backup facility is insufficient, since a customer’s primary and backup facilities may both fail. Therefore, we define:
Yijr D
(
1; if facility j 2 J serves as the level r facility for customer i 2 I
:
0;
if not
A “level-r” assignment is one for which there are r closer facilities that are open.
If r D 0, this is a primary assignment; otherwise it is a backup assignment. The
20
Facility Location in Supply Chain
495
objective is to minimize a weighted sum of the operating cost (the fixed charge
location problem objective) and the expected failure cost, given by
j1
X X jJX
i 2I j 2J rD0
hi cij q r .1 q/Yijr :
Each customer i is served by its level-r facility (call it j ) if the r closer facilities
have failed (this occurs with probability qr) and if j itself has not failed (this occurs
with probability 1 q). The full model is omitted here.
20.14 Planning Under Uncertainty (Snyder et al. 2007)
Long-term strategic decisions like those involving facility locations are always made
in an uncertain environment. However, classical facility location models like the
fixed charge location problem treat data as though they were known and deterministic, even though ignoring data uncertainty can result in highly sub-optimal solutions.
Unfortunately, as Vidal and Goetschalckx (2000) discussed in their paper, critical
parameters such as customer demands, prices, and resource capacities are quite uncertain. Moreover, the arrival of regional economic alliances, for instance the Asian
Pacific Economic Alliance and the European Union, have prompted many corporations to move more and more towards global supply chains, and therefore to become
exposed to risk factors such as exchange rates, reliability of transportation channels,
and transfer prices.
Daskin et al. (2003) said that most approaches to decision making under uncertainty fall into one of two categories: stochastic programming or robust optimization. In stochastic programming, the uncertain parameters are described by discrete
scenarios, each with a given probability of occurrence; the objective is to minimize
the expected cost. In robust optimization, parameters may be described either by
discrete scenarios or by continuous ranges; no probability information is known,
however, and the objective is typically to minimize the worst-case cost or regret.
According Daskin et al. (2003), Sheppard (1974) was one of the first authors
to propose a stochastic approach to facility location. He suggests selecting facility
locations to minimize the expected cost, though he does not discuss the issue at
length. Weaver and Church (1983) and Mirchandani et al. (1985) present a multiscenario version of the P-median problem. Their model can be translated into the
context of the fixed charge location problem as follows. Let S be a set of scenarios.
Each scenario s 2 S has probability of occurring and specifies a realization of random demands and travel costs. Location decisions must be made now, before it is
known which scenario will occur. However, customers may be assigned to facilities
after the scenario is known. Other stochastic facility location models include those
of Louveaux (1986), Franga and Luna (1982), Berman and LeBlanc (1984), Carson
and Batta (1990), and Jornsten and Bjorndal (1994).
496
M. Alizadeh
The earlier models assume the decision maker knows the demand parameters.
That is, the mean i and the standard deviation i of retailer i ’s demand are
known parameters. Snyder et al. (2007) present a stochastic version of the model
in Sect. 20.4.3 that explicitly handles parameter uncertainty by allowing parameters
to be described by discrete scenarios, each with a specified probability of occurrence. The goal is to choose DC locations, assign retailers to DCs, and set inventory
levels at DCs to minimize the total system wide cost. To model this problem, Snyder
et al. (2007) define the following additional notation:
20.14.1 Model Inputs
S : set of scenarios, indexed by s,
i s : mean daily demand at retailer i in scenario s, for i 2 I; s 2 S ,
dijs : per-unit cost to ship from DCj to retailer i in scenario s, for i 2 I; j 2 J ,
qs : probability that scenario s occurs, for s 2 S .
20.14.2 Model Outputs (Decision Variables)
xj D 1, if j is selected as a facility location, and 0 otherwise, for each j 2 J ,
yijs D 1, if retailer i 2 I is served by DCj j 2 J in scenario s 2 S , and 0
otherwise.
20.14.3 Objective Function and its Constraints
X
s2S
X
j 2J
n
X
fj Xj C
ˇi s .dijs C aj /Yijs
i 2I
r
r
X
X
is Yijs C q
Lj i2 Yijs : (20.56)
CKj
i 2I
i 2I
Subject to
P
j 2J
Yijs D 1;
8i 2 I; s 2 S;
Yijs Xj 0 8i 2 I; j 2 J; s 2 S ;
Yijs 2 f0; 1g 8i 2 I; j 2 J; s 2 S ;
Xi 2 f0; 1g 8j 2 J ;
where q D hz˛ .
(20.57)
(20.58)
(20.59)
(20.60)
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Facility Location in Supply Chain
497
20.15 Solution Techniques
A number of solution approaches have been proposed for the supply chain design
models. We review the related literatures.
To solve the uncapacitated fixed charge location model, Maranzana (1964) proposed a neighborhood search improvement algorithm for the closely related
p-median problem that exploits the ease in finding optimal solutions to 1-median
problem: it partitions the customers by facility and then finds the optimal location within each partition. If any facility changes, the algorithm repartitions the
customers and continues until no improvement in the solution can be found.
Teitz and Bart (1968) proposed an exchange or “swap” algorithm for the Pmedian problem that can also be extended to the fixed charge facility location
problem.
Hensen and Mladenovic (1997) proposed a variable neighborhood search algorithm for the P-median problem that can also be used for the fixed charge location
problem.
Heuristics that be applied to the P-median problem will not perform well for
fixed charge facility location problem if the starting number of facilities is suboptimal. Thus, we should apply more sophisticated heuristics to the problem.
Al-Sultan and Al-Fawzan (1999) applied tabu search to the uncapacitated fixed
charge location problem.
Daskin (1995) review the use of Lagrangian relaxation algorithms in solving the
uncapacitated fixed location problem.
When embedded in branch and bound, Lagrangian relaxation can be used to solve
the fixed charge location problem optimally (Geoffrion 1974).
Daskin et al. apply Benders decomposition to locating plants and distribution
centers simultaneously with multiple commodities after noting that, if the location and assignment variables are fixed, the remaining problem breaks down into
jLj transportation problems, one for each commodity.
Shen et al. (2003) outline a column generation approach to solve the basic model
that presented in Sect. 20.4.3. Daskin et al. (2001) proposes a Lagrangian relaxation approach for the same problem.
To solve the model with routing cost estimation, Shen and Qi (2004) develop
a Lagrangian relaxation based solution algorithm. By exploiting the structure of
the problem, they find an O.jI j2 log.jI j// time algorithm for the similar problem
that must be solved in the Lagrangian relaxation subproblems.
Ozsen et al. (2003) apply a Lagrangian relaxation solution algorithm to solve
the capacitated DCs problems that presented in Sect. 20.4.4. The Lagrangian
subproblem is also a non-linear integer program, and they propose an efficient
algorithm for the continuous relaxation of this problem.
Amiri (2006) studied the designing a distribution network in a supply chain system with allow for multiple levels of capacities available to the warehouses and
plants. He develop a mix integer programming model and present a Lagrangian
based solution procedure for the problem.
498
M. Alizadeh
Two solution approaches was presented by Shen and Daskin (2005) for the model
with service consideration, one based on the weighting method and the other
based on genetic algorithms. The genetic algorithm performs very well compared
to the weighting method, and it is the only feasible approach for large-sized problem instances, since the weighting method requires excessive computational time
in such cases.
A branch-and-price algorithm, a variant of branch-and-bound in which the nodes
are processed by solving linear-programming relaxation via column-generation,
was presented by Shen (2006) to solve the profit maximizing model with demand
choice flexibility.
Shen (2005) proposes Lagrangian relaxation embedded in a branch and bound
algorithm to solve the model with multiple commodities.
Yeh (2005) present a hybrid heuristic algorithm for the multistage supply chain
network problem (MSCN). The proposed algorithm employs a simple greedy
method and a hybrid local search method combining the XP, RP and IP to solve
the MSCN. From the computation experiments, both the speed in finding the
solution and the quality of the obtained solutions were good enough to solve the
larger MSCN problem.
Yeh (2006) has also proposed a memetic algorithm (MA) which is a combination
of GA, greedy heuristic, and local search methods for the same problem. The
author has extensively investigated the performance of the MA on the randomly
generated problems.
Romeijn et al. (2007) proposed a framework for the two-echelon supply chain
design problem that incorporates location decisions as well as location-specific
transportation cost and two-echelon inventory costs. Their approach is to formulate the problem as a set-covering model. They propose to solve this problem
using column generation to deal with the fact that the number of variables
(columns) in this formulation is exponentially large.
Altiparmak et al. (2006) propose a new solution procedure based on genetic algorithms to find the set of Pareto-optimal solutions for multi-objective SCN design
problem. To deal with multi-objective and enable the decision maker for evaluating a greater number of alternative solutions, two different weight approaches
are implemented in the proposed solution procedure.
Altiparmak et al. (2007) presents a solution procedure based on steady-state
genetic algorithms (ssGA) with a new encoding structure for the design of a
single-source, multi-product, multi-stage SCN. The effectiveness of the ssGA
has been investigated by comparing its results with those obtained by CPLEX,
Lagrangian heuristic, hybrid GA and simulated annealing on a set of SCN design
problems with different sizes.
Hinojosa et al. (2008) formulating the problem that deal with a facility location
problem where they build new facilities or close down already existing facilities at two different distribution levels over a given time horizon. In addition,
they allow carrying over stock in warehouses between consecutive periods. They
propose a Lagrangian approach which relaxes the constraints connecting the distribution levels. A procedure is developed to solve the resulting, independent
20
Facility Location in Supply Chain
499
subproblems and, based on this solution, to construct a feasible solution for the
original problem.
Jayaraman and Pirkul (2001) have developed a heuristic approach based on
Lagrangean relaxation for the single-source, multi-product, multi-stage SCN design problem. Another heuristic approach based on Lagrangean relaxation and
simulated annealing has been developed by Syam (2002) for a multi-source,
multi-product, multi-location framework.
Jang et al. (2002) have presented a combined model of network design and production/distribution planning for a SCN. While they have used a Lagrangian
heuristic for the design of SCN, a genetic algorithm (GA) has been proposed
for integrated production and distribution planning problem.
Syarif et al. (2002) have developed a spanning tree-based GA approach for the
multi-source, single-product, multi-stage SCN design problem based on Prüfer
numbers.
Jayaraman and Ross (2003) have also proposed a heuristic approach based on
simulated annealing for the designing of distribution network and management
in supply chain environment.
Shen (2007) states that Qi and Shen propose an algorithm based on the bisection
search and the outer approximate algorithm. They show that their algorithm is
more efficient that the outer approximation algorithm when being applied to the
model with unreliable supply.
To solve the model with parameter uncertainty, Snyder et al. (2007) present
a Lagrangian-relaxation based solution algorithm. They show the Lagrangian
subproblem is non-linear integer program, but it can be solved by a low-order
polynomial algorithm. They present qualitative and quantitative computational
results on problems with up to 150 nodes and nine scenarios, and describe both
algorithm performance and solution behavior as key parameters change.
Solution methods discussed above are outlined in Table 20.1. This table shows that
the use of heuristics and meta-heuristics based solution algorithms increase in recently papers.
20.16 Case Study
In this section we will introduce two real-world case studies related to supply chain
design problems.
20.16.1 An Industrial Case in Supply Chain Design
and Multilevel Planning in US (Sousa et al. 2008)
Souse et al. (2007) address a case study, inspired by a real agrochemicals supply
chain, with two main objectives, structured in two stages. In the first stage they
500
M. Alizadeh
Table 20.1 Solution methods for supply chain design problem
References
Maranzana (1964)
Geoffrion (1974)
Problem
Uncapacitated fixed charge
location model
Fixed charge facility location
problem
Fixed charge facility location
problem
Uncapacitated fixed charge
location problem
Uncapacitated fixed charge
location problem
Fixed charge location
Shen et al. (2003)
Basic model
Shen and Qi (2004)
Model with routing cost
estimation
Capacitated Dcs problems
Teitz and Bart (1968)
Hensen and Mladenovic (1997)
Al-Sultan and Al-Fawzan (1999)
Galvo (1993) and Daskin (1995)
Ozsen et al. (2003)
Amiri (2006)
Shen and Daskin (2005)
Shen and Daskin (2005)
Shen (2006)
Shen (2005)
Yeh (2005)
Yeh (2006)
Romeijn et al. (2007)
Altiparmak et al. (2006)
Designing a distribution
network in a supply chain
system with allow for multiple
levels of capacities
Model with service
consideration
Model with service
consideration
Profit maximizing model with
demand choice flexibility
Model with multiple
commodities
Multistage supply chain
network problem (MSCN)
Multistage supply chain
network problem (MSCN)
Two-Echelon supply chain
design problem
Multi-objective SCN design
problem
Altiparmak et al. (2007)
Design of a single-source,
multi-product, multi-stage SCN
Hinojosa et al. (2008)
Dynamic supply chain design
with inventory
Jayaraman and Pirkul (2001)
Single-source, multi-product,
multi-stage SCN design
problem
Design of SCN
Jang et al. (2002)
Solution method
Neighborhood search
improvement algorithm
Exchange or “swap”
algorithm
Variable neighborhood
search algorithm
Tabu search
Lagrangian relaxation
algorithms
Lagrangian relaxation
algorithms
Column generation
approach
Lagrangian relaxation
based solution algorithm
Lagrangian relaxation
solution algorithm
Lagrangian based
solution procedure
Weighting method
Genetic algorithms
Branch-and-price
algorithm
Lagrangian relaxation
embedded in a branch
and bound algorithm
Hybrid heuristic
algorithm
Memetic algorithm (MA)
Column generation
New solution procedure
based on genetic
algorithms
Solution procedure based
on steady-state genetic
algorithms (Ssga)
Lagrangian approach
which relaxes the
constraints connecting the
distribution levels
Heuristic approach based
on Lagrangean relaxation
Lagrangian heuristic
(continued)
20
Facility Location in Supply Chain
501
Table 20.1 (continued)
References
Syarif et al. (2002)
Jayaraman and Ross (2003)
Shen (2007)
Snyder et al. (2007)
Problem
Multi-source, single-product,
multi-stage SCN design
problem
Designing of distribution
network and management in
supply chain environment
Model with unreliable supply
Model with parameter
uncertainty
Solution method
Spanning tree-based GA
approach
Heuristic approach based
on simulated annealing
Algorithm based on the
bisection search and the
outer approximate
algorithm
Lagrangian-relaxation
based solution algorithm
redesign the global supply chain network and optimize the production and distribution plan considering a time horizon of 1 year, providing a decision support tool
for long term investments and strategies. The output decisions from the first stage,
mainly the supply chain configuration and allocation decisions, are the input parameters for the second stage where a short term operational model is used to test
the accuracy of the derived design and plan. The outputs of this stage are detailed
production and distribution plans and an assessment of the customer service level.
20.16.2 Multi-Objective Optimization of Supply Chain Networks
in Turkey (Altiparmak et al. 2006)
The problem considered in this paper has been from a company which is one of the
producers of plastic products in Turkey. The company is planning to produce plastic
profile which is used in buildings, pipelines and consumer materials. The company
wishes to design of SCN for the product, i.e., select the suppliers, determine the
subsets of plants and DCs to be opened and design the distribution network strategy
that will satisfy all capacities and demand requirement for the product imposed by
customers. The problem is a single-product, multi-stage SCN design problem. Considering company managers’ objectives, they formulated the SCN design problem
as a multi-objective mixed-integer non-linear programming model.
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Chapter 21
Classification of Location Models and Location
Softwares
Sajedeh Tafazzoli and Marzieh Mozafari
According to the importance and advantages of classification, the first section of
this chapter is dedicated to some presented classifications of location models, which
help in having more disciplined understanding of location models. In the second
section, some location softwares will be introduced briefly.
21.1 Classification of Location Models
Nowadays, with the increasing development of science in all branches, need for
a systematic arrangement or proposing a classification scheme for easy access to
scientific researches seems necessary. Location science is a branch of optimization
science, which formally introduce by Alfred Weber in 1909. It has been growing so
rapidly for years that now without a systematic classification of models, continuing
the procedure of researches would be so difficult. Therefore, several efforts in classifying location models have been made that, some of them will be mentioned in
this section.
21.1.1 Taxonomy vs. Classification Scheme
Classification scheme and taxonomy are two terms often used interchangeably.
Though there may be subtle differences in different examples. These two terms are
considered as follows:
Taxonomy is a classification of similar objects or concepts into a group, based
on their separating characteristics. For example in location models based on “topological structure” criteria, location problems can be classified into three categories,
planar (continuous), discrete and network. Taxonomy could be presented by proposing few criteria and classifying the concepts based on them.
Classification scheme is a more general term, an arrangement or division of objects into groups based on characteristics common between the objects. The aim of
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 21, c Physica-Verlag Heidelberg 2009
505
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S. Tafazzoli and M. Mozafari
a classification scheme which makes it different from taxonomy is to encompass the
whole subject. But in taxonomy, it is not necessary to cover the whole subject. Regarding, the comprehensiveness of a classification scheme, it is used for encoding
data which could have many benefits. Some of the advantages of using a classification scheme are mentioned below:
Lets a user to find an object in a large collection quickly
Makes detecting duplicate objects easier
Present a meaning for an object, which may not be conveyed by its name or
definition
Makes concise problem statement, as opposed to verbal ambiguous descriptions
Makes data encoding and information retrieval in bibliographical information
system and software libraries simple
Facilitate referencing in literature
Provides a scheme of defined models and help in detecting models which could
be worked on by researchers
Assists in assigning a predefined model to a real problem
21.1.2 Taxonomy
21.1.2.1 Taxonomy of Francis and White (1974)
One of the first taxonomies presented for layout and location problems, is the classification of Francis and White. In this classification six major elements considered:
(1) New facility characteristics, (2) existing facility location, (3) new and existing
facility interaction, (4) solution space characteristics (5) distance measure, and (6)
objective (Fig. 21.1).
21.1.2.2 Taxonomy of Tansel et al. (1983)
Tansel et al. (1983) present a tree structure as a conceptual framework for their
survey of network location models. In this taxonomy, models are classified into
point-location or path-location models, which are shown in Fig. 21.2.
21.1.2.3 Taxonomy of Brandeau and Chiu (1989)
Brandeau and Chiu (1989) present a taxonomy of location models based on three
criteria, objective, decision variable(s) and system parameters in a table format
(Table 21.1). This taxonomy was made according to more than 50 different problem types.
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Classification of Location Models and Location Softwares
507
a
New Facilities Characteristics
Single
Multiple
Point
Area
Layout Problem
Other
Parameter
Decision Variable
Independent
b
Dependent
c
Solution Space
Existing Facility Locations
Static
Dynamic
Deterministic
Probabilistic
Point
Single Dimension
Multidimensional
Discrete
Continuous
Constrained
Unconstrained
Area
Decision Variables
Parameter
d
e
Objective
Qualitative
Minimize Total
Cost
Distance Measure
Quantitative
Rectilinear
Euclidean
Other
Minimize Maximum
Cost
Other
Fig. 21.1 Classification of location models (a) new facilities characteristicsm (b) existing facility
locations (c) solution space (d) objective (e) distance measure, (f) new/existing facility interaction
(Francis and White 1974)
508
S. Tafazzoli and M. Mozafari
f
New/Existing Facility Interaction
Qualitative
Quantitative
Location Dependent
Location Independent
Static
Dynamic
Deterministic
Probabilistic
Parameter
Decision Variable
Fig. 21.1 (continued)
Network Location Problems
Point-Location
Single-Objective
Path-Location
Multi-Objective
MiniSum
Cent-Dian
P-Median
Convex
Objectives
Problem with
mutual communication
Biobjective
Covering
MiniMax
Others
Distance
constraints
Convexity
for Trees
Multifacility
Vector
Minimization
P-Center
Problem with mutual
communication
Fig. 21.2 Tree structure for network location problems (Tansel et al. 1983)
Core
(MiniSum)
Path Center
(Mini-Max)
Spine
(Branch
Weight)
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509
Table 21.1 Taxonomy developed by Brandeau and Chiu (1989)
I. Objective
Optimizing:
Minimize average travel time/average cost
Maximize net income
Minimize average response time
Minimize maximum travel time/cost
Maximize minimum travel time/cost
Maximize average travel time/cost
Minimize server cost subject to a minimum service constraint
Optimize a distance-dependent utility function
Other
Non-optimizing
Type of location dependence of objective function:
Server-demand point distances
Weighted vs. unweighted
Some vs. all demand points
Routed vs. closest
Inter-server distances
Absolute server locationa
Server-distribution facility distances
Distribution facility-demand point distances
Other
II. Decision variables
Server/facility location
Service area/dispatch priorities
Number of servers and/or service facilities
Server volume/capacity
Type of goods produced by each server (in a multi-commodity
situation)
Routing/flows of server or goods to demand points
Queue capacity
Other
III. System parameters
Topological structure:
Link vs. tree vs. network vs. plane vs. n-dimensional spacea
Directed vs. undirected
Travel metric:
Network-constrained vs. rectilinear vs. Euclidean vs. block norm vs.
round norm vs. L, vs. other
Travel time/cost:
Deterministic vs. probabilistic
Constrained vs. unconstrained
Volume-dependent vs. nonvolume-dependent
Demand:
Continuous vs. discrete
Deterministic vs. probabilistic
Cost-Independent vs. cost-dependent
Time-invariant vs. time-varying
Number of servers
Number of service facilities
Number of commodities
Server location:
Constrained vs. unconstrained
Finite vs. infinite number of potential locations
(continued)
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S. Tafazzoli and M. Mozafari
Table 21.1 (continued)
Fixed vs. dependent on system status
Zero vs. nonzero relocation cost
Deterministic vs. probabilistic location
Zero vs. nonzero fixed cost
Server capacity:
Capacitated vs. uncapacitated
Reliable vs. unreliable
Service area and dispatch priorities:
Cooperating vs. noncooperating servers
Closest distance vs. nonclosest-distance service Area
Service discipline:
FCFS vs. priority classes vs. spatially-oriented rule vs. other
Queue capacity
a
For certain product design and product positioning problems.
Table 21.2 Taxonomy proposed by (Daskin 1995)
Criteria
Topological structure
Network type
Distance measure
Number of facilities to locate
Time dependency
Certainty
Product diversity
Public/private sector
Number of objectives
Demand elasticity
Capacity of facilities
Demand allocation type
Hierarchical structure
Desirability of facility
Classification
Planar
Discrete
Network
Tree problems
General graph
Manhattan
Euclidean
Lp
Single facility
Multi-facility
Static
Dynamic
Deterministic
Probabilistic
Single product
Multi product
Public
Private
Single objective Multi objective
Elastic
Inelastic
Capacitated
Uncapacitated
Nearest facility
General allocation
demand
Single-level
Hierarchical
Desirable
Undesirable
21.1.2.4 Taxonomy of Daskin (1995)
Daskin (1995) proposed a classification based on 14 criteria: (1) Topological structure, (2) network type, (3) distance measure, (4) number of facilities to locate, (5)
time dependency, (6) certainty, (7) commodity (product) diversity, (8) public/private
sector, (9) number of objectives, (10) demand elasticity, (11) capacity of facilities,
(12) demand allocation type, (13) hierarchical structure, and (14) desirability of facility. This classification is summarized in Table 21.2.
21.1.2.5 Taxonomy of Nagy and Salhi (2007)
Nagy and Salhi (2007) used eight aspects of problem structure to classify
location-routing problems, (a) Hierarchical structure, (b) Type of input data
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(deterministic/stochastic), (c) Planning period (single/multi-period), (d) Solution
method (exact/heuristic), (e) Objective function, (f) Solution space, (g) Number of
depots (single/multiple), (h) Number and types of vehicles (homogeneous/ heterogeneous), and (i) Route structure.
21.1.2.6 Taxonomy of ReVelle et al. (2008)
ReVelle et al. (2008) provided a brief taxonomy of broad field of facility location
modeling in a bibliography in two branches of discrete location science (1) median
and plant location models and (2) center and covering models. In this taxonomy,
location models are classified into four categories:
1. Analytic models are based on large simplifying assumptions (fixed cost of locating a facility dependent of where it will locate, demand uniformly distributed,
etc). Despite valuable insight which this class of models provides, they cannot be
used for real decision-making purposes
2. Continuous models assume that facilities can be located anywhere in service area,
while demands are often taken as being at discrete locations
3. Network models assume that topological structure of the location model is a
network composed of lines and nodes. Much of the literature in this area is concerned with finding special structures that can be exploited to derive low-order
polynomial time algorithms
4. Discrete models assume that the set of demands and candidate location for facilities are discrete. These problems often formulate in integer or mix-integer
programming that most of them are NP-hard on general network
In addition of these classifications, location models can also be distinguished based
on other attributes (Klose and Drexl 2004; Jia et al. 2007)
21.1.3 Classification Schemes
In several branches of optimization, classification schemes have been proposed and
used by authors successfully (schemes in scheduling (Graham et al. 1979) and queuing theory (Kendall 1951)). Classification schemes are also proposed in location
science that few of them will be mentioned here.
21.1.3.1 Classification Scheme of Handler and Mirchandani (1979)
The first classification scheme was proposed by Handler and Mirchandani (1979).
They suggested a 4-position scheme which is applicable to network location models
with objective functions of the center type.
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Pos1/Pos2/Pos3/Pos4
Position 1: information about the new facilities
Position 2: information about existing facilities
Position 3: number of new facilities
Position 4: network type
21.1.3.2 Classification Scheme of Eiselt et al. (1993)
Eiselt et al. (1993) suggested a 5-position scheme for classifying competitive location models (those based on a game-theory approach).
Pos1/Pos2/Pos3/Pos4/Pos5
Position 1: information about the decision space (linear segment, line, circle,
bounded subset of m-dimensional real space, m-dimensional real space, network,
tree)
Position 2: number of players (specified number, any arbitrary fixed number,
markets with free entry)
Position 3: a description of the pricing policy (mill pricing, uniform delivered
pricing, perfect spatial discriminatory pricing)
Position 4: rules of the game under consideration (Cournot-Nash equilibrium,
subgame perfect Nash equilibrium, Stackelberg equilibrium)
Position 5: behavior of the customers (minimization of distance, minimization of
a deterministic or random utility)
21.1.3.3 Classification Scheme of Carrizosa et al. (1995)
Carrizosa et al. (1995) presented a 6-position scheme for classifying planar models
where both demand rates and service times are given by a probability distribution.
21.1.3.4 Classification Scheme of Hamacher and Nickel (1998)
Hamacher and Nickel (1998) proposed a 5-position classification scheme for location models that covers not only classes of specific location models, as in the
references above, but also covers all of them in a single scheme. Furthermore, this
scheme can describe models which are not of the classical type and problems which
have not been solved yet. The classification scheme has been in use since 1992 and
has been proven to be helpful in research, software development, classroom teaching, and for overview articles. The software library, Library of location algorithms
(LoLA) has been developed (Hamacher et al. 1996) based on this classification
scheme. The classification scheme consists of the following five positions:
Pos1/Pos2/Pos3/Pos4/Pos5,
where position 1 contains information about the number and type of new facilities.
Position 2 indicates type of the location model with respect to the decision space.
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Classification of Location Models and Location Softwares
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Position 3 describes the particulars of the specific location model, such as information about the feasible solutions, capacity restrictions, etc. In position 4, relation
between new and existing facilities is defined by a distance function or by assigned
costs. In position 5 a description of the objective function is given.
A list of symbols is available for each position that covers a large variety of
location models. In Table 21.3, some examples of these symbols, according to the
three main areas of location theory (i.e. continuous location, network location and
discrete location) is given.
If there are not any special assumptions in a position, this is indicated by . For
example, a in position 5 means that any objective function are looked or the in
position 3 means that standard assumptions for the model described in the remaining
four positions hold. It is assumed by default that weights are non-negative and the
objective function is to be minimized. If these assumptions do not hold this has to
be stated respectively in positions 3 and 5.
In the following, some examples from the literature are given that indicate the
ability of the 5-position classification scheme to describe various kinds of location models. Of course the 5-position classification scheme cannot represent the
complete contents of a paper, but it can reflect the major particulars of the models
investigated.
P
Brandeau (1992): 1=P =queue= lP = prob
Erkut and Tansel (1992): 1=T =wm W f .:/=d.V; G/=†
Eiselt and Laporte (1993): 3=T =:=d.V; G/=˙comp
Mirchandani et al (1996): #=D=F D li ne; cap=:=†cov
Burkard et al. (2000): P =G=wm >< 0=d.V; G/=†
Klamroth (2004):1=P =BC = l2;BC =f convex
Gomes et al. (2007): N=P =alloc= l2 =†
Colebrook and Sicilia (2007):1=G=:=d.V; G/=Q CDpar
21.2 Facility Location Softwares
In previous chapters, many models and algorithms have been discussed for solving
different facility location problems. Of course, in large real world problems, it is impossible to implement these algorithms manually. In this point computer programs
or special software come into play.
Some of the major facility location software is introduced in this section. Although they cannot cover all of the facility location models, they are still under
development.
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Table 21.3 Some examples of the classification scheme symbols (Hamacher and Nickel 1998)
Position 1
Location model
Continuous
Network
Discrete
Position 2
Continuous
Network
Position 3
Discrete
Continuous
Symbol
.empty st ri ng/
L
P
A
.empty st ri ng/
P
T
G
n 2 f1; : : : :; ng
#
Rd
P
H
G
Gd
T
D
F
R
B
wm D 1
al loc
Network
Position 4
Discrete
cap
bdg
queue
Continuous
lP
Network
k:k
d.V; V /
pol
d.V; G/
d.V; T /
Discrete
Position 5
Continuous
†
max
CD
R
Description
n points are to be located
n lines are to be located
n paths are to be located
n general areas are to be located
n points are to be located
n paths are to be located
n trees are to be located
n subgroups are to be located
Number of new facilities
Number of new facilities is not
known in advance
D-dimensional space
Euclidean plane i.e., P D R2
General Hilbert space
General undirected graph
General directed graph
A tree
Discrete
Feasible region
Forbidden region
Barrier
Unweighted problem
Allocation of existing facilities to
new facilities
The same possibilities as in the
continuous case
Capacity restrictions
Budget restrictions
Service of new facilities is modeled
using queuing theory
Distance is defined by an lp - norm
General gauge
Polyhedral gauge
General norm
New and existing facilities must be
nodes of the graph
Existing facilities are at nodes and
few facilities can be any points on
the graph.
Analogous to d(V, G), where G is a
tree
Any restrictions and particulars of
given cost Cij can be specified
The classical Weber or sum objective function
The maximum objective function
Cent-Dian objective function
Continuous demand satisfying the
distribution specified in position 3
(continued)
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Table 21.3 (continued)
Location model
Symbol
†prob
Q †par
Network
Discrete
†comp
QAP
†cov C †uncov
†hub
Description
Sum objective with some probabilistic influences, like, for example, different scenarios or weights
which are random variables
Q-criteria Weber model, where we
are looking for Pareto locations
Any of the objective functions
listed for the continuous case which
are meaningful in the network environment
Competitive location model
Quadratic assignment objective
function
Covering objective function
Hub location objective function
21.2.1 LoLA
LoLA is a collection of efficient algorithms for solving planar, network and discrete facility location problems (Hamacher et al. 1996). LoLA uses the classification
scheme of Hamacher and Nickel (1998) for accessing to the implemented algorithms. A part of the algorithms is known from the literature whereas others are the
results of current research. LoLA can solve a number of different location models
including Median, Center, Q-median and Q-center that Q is the number of facilities
in a multi-facility problem or the number of objective functions in a multi-objective
problem. The algorithms available in LoLA can be found in Hamacher et al. (1996).
LoLA has been designed to address several different user groups. It fulfills the
needs of students and teachers as well as researchers and practitioners.
LoLA provides a graphical user interface that allows its simple application in
industrial projects as well as for demonstrations in high school and university teaching. In addition, a Text-based user interface is available to call algorithms of LoLA
from other applications. To solve individual facility location problems, a programming interface allows the direct incorporation of specific algorithms of the program
library into the implementation of extended routines (Callable Library).
21.2.1.1 LoLA and Interface with Geographical Information System
Geographical information systems (GIS) are designed for visualizing real world data
of countries, states, cities, etc. with map data and additional information that allow
decision makers to find solutions to problems such as locating facilities, routing,
etc. The software library LoLA has the ability to be linked with GIS to easily get
real world data and solve real world problems. An example of a GIS is ArcView
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GIS using script language called Avenue. This language has been implemented to
combine ArcView GIS with LoLA (Bender et al. 2002).
Via LoLA ArcView R Link it is possible to read data out of the ArcView GIS
database and convert them into a LoLA input file, then call LoLA Libraries to solve
the model and finally monitor the results and get the solution back to ArcView GIS.
21.2.1.2 LoLA Example
Here a simple example (from software package) is used to illustrate the ideas involved and the basic data required in LoLA.
Consider uncapacitated facility location (UFL) problem, by choosing the characteristics of 5-position classification scheme from software menus, LoLA exhibit
the classification by symbols, #=D= = =† (Pos1: Number of facilities to locate
determine by model, Pos2: Discrete solution space, Pos3: No special conditions in
model, Pos4: Metric define by cost matrix as input, Pos5: Sum (median) objective
function).
Input format of this discrete location model contains information about:
Demand point (x; y; b; n) where x and y are coordinates, b is the customer demand and n is the facility name, for example: 54.00 15.00 5,000 [Palermo]
Cost matrix (n n matrix defines travel or transport cost or time from facility i
to j ).
This model is solved for a UFL problem to identify the cities in Italy from which to
service demand of customer sites in a total of 39 cities (supply point must be placed
on demand points).
The optimal result consists of:
The number and location of supply points (opt1 is Lucca and opt2 is Viterbo,
Fig. 21.3)
objective function value
and allocation of demand points to each supply point (Fig. 21.3)
21.2.2 SITATION
SITATION (Daskin 2002) is a facility location software that accompanies Daskin’s
text (Network and discrete location: models, algorithms, and applications). The SITATION software now solves a number of different discrete and network facility
location problems including:
P-median
P-center
Set covering
Maximal covering
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Sondrio
Trento
Padova
Trieste
Novara Milano Verona Venezia
Torino
Pavia
Ferrara
Piacenza
Genova
Bologna
Parma
Savona
Pesaro
Spezia
Lucca–Opt1 Fano
Pisa Firenza
Ancona
Siena Perugia
Como
Grosseto
Viterbo–Opt2
Avezzann
Poma
Sassari
Napoli
Foggia
Bari
Salerno
Cagliari
Palermo
Catania
Siracusa
Fig. 21.3 LoLA result
Partial set covering
Partial P-center
Uncapacitated fixed charge
1-Source capacitated fix charge problems
SCD integrated inventory location model
Inventory location
Profit maximization
SITATION allows the user to choose from a variety of heuristics and optimizationbased approaches for each of the different models. Some examples of SITATION available algorithms according to different classes of problems are given in
Table 21.4.
In addition, SITATION has the ability to solve multi objective facility location models such as the covering-median tradeoff problem by using the weighting
method.
Menu-OKF, Net Spec and Modify Distances are additional softwares which accompany SITATION and make it more efficient. The program Menu-OKF solves
network problems using the Out-of-kilter flow algorithm. The program Modify
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Table 21.4 Some examples of SITATION available algorithms
Problem to solve
P-median
P-center
Set covering
Maximum covering
Partial set covering
Partial P-center
Uncapacitated fixed charge
Integrated Inv/Loc model (SCD)
Available algorithms
Myopic
Variable neighborhood
Genetic algorithm
Exchange
Lagrangian relaxation
Lagrangian relaxation
Lagrangian relaxation
Greedy adding
Greedy adding and substitution
Lagrangian relaxation
Lagrangian relaxation
Lagrangian relaxation
Add
Drop
Exchange
Variable neighborhood
Lagrangian relaxation
Lagrangian relaxation
Distances modifies distance files used by SITATION. The program Net Spec allows
user to specify the inputs for network problems to be solved in SITATION.
Other SITATION capabilities are as follows:
Entirely menu-driven and relatively easy to understand
Allows the user to add or delete facilities from a solution and also exchange
facilities locations manually, thereby allowing the user to gain confidence in the
solutions determined by the program or to test his or her own solutions
Includes branch and bound capabilities that allow the user to obtain very tight
(usually provably optimal) solutions
Includes good reporting and mapping capabilities for illustrating the solutions
Able to solve problems up to 300 nodes scale (latest version, 5.7.0.26)
21.2.2.1 SITATION Example
Here, one test problem in the software package is used as an example. A P-median
problem where the demand nodes are defined by the data contained in the 150-city
data set found in SITATION. The inter-nodal distances are computed by software,
using great circle distances. The coverage distance and cost per mile are defined 300
and 1 respectively (Daskin 2002).
The model could be run under different number of facilities (different scenarios). Fig. 21.4 illustrates map result of running model for locating 15 facilities with
Lagrangian relaxation.
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Classification of Location Models and Location Softwares
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WA
Chicago Cleveland
Denver
CO
Kansas City
OH
New York
MO
Richmond
CA
VA
Nashville-Davidson (remaTN)
Los Angeles
CA
Mesa
AZ
Dallas
TX
HoustonNew Orleans
LA
Orlando
FL
Fig. 21.4 SITATION result
21.2.3 S-Distance
S-distance is a standalone Spatial Decision Support System, mainly focused on
location-allocation analysis (Sirigos and Photis 2005).
S-distance is able to solve quite large classical discrete and network locationallocation problems, including:
P-median
P-center
Maximal covering
Multi-objective
Current version of the software (version 0.7) offers a number of heuristic and
optimization-based algorithms such as:
Greedy and randomized algorithms
Local search heuristics
Meta-heuristics
Lagrange relaxation
While still being in an early stage, S-distance software is functional and has been
tested on many classical Operation Research instances, as well as on several realworld problems.
Other S-distance capabilities are as follows:
Able to read data from various file formats, such as OR-library files and databases
with network/point topology in dbf format
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Provides all Pairs Shortest Paths calculation using Dijkstra’s (2-ary heap) or
Floyd’s algorithms, for network problems
Incorporate an interactive solution process within a simplified GIS framework
Allows the user straight-forward creation and evaluation of different solutions
both numerically and graphically
21.2.4 Other Facility Location Softwares
There is some other facility location software specialized on solving a specific problem and limited more in their functionalities than the three software introduced
above. Some examples are given in the following:
RLP is a program package for solving restricted 1-facility location problems in a
user friendly environment (Nickel and Hamacher 1992)
Optimal locating air polluting facilities is a general modeling system to evaluate
and optimize the location of an air polluting facility (Fliege 2001)
Jure Mihelic’s K-center algorithms is a program for solving k-Center location
problems (Mihelic 2004)
Minimum enclosing circle applet is a program package for solving the Minimal
Enclosing Circle problem (Eliosoff and Unger 1998). It is useful for planning the
location of a shared facility
Excel template for facility location includes model for center-of-gravity method
for locating distribution centers
GAMBINI is a small GIS-utility which calculates draws and exports multiplicative weighted Voronoi diagrams (Tiefelsdorf and Boots 1997). A point location
data structure can be built on top of the Voronoi diagram in order to find the
object that is nearest to a given point
Mathematical programming softwares such as CPLEX, LINGO, LINDO and
GAMS which are useful when having mathematical models for facility location
problems
References
Bender T, Hennes H, Kalcsics J, Melo MT, Nickel S (2002) Location software and interface with
GIS and supply chain management. In: Drezner Z, Hammacher H (eds) Facility location: applications and theory. Berlin, Springer
Brandeau ML (1992) Characterization of the stochastic median queue trajectory in a plane with
generalized distances. Oper Res 40(2):331–341
Brandeau ML, Chiu SS (1989) An overview of representative problems in location research. Manage Scie 35:645–674
Burkard RE, Cela E, Dollani H (2000) 2-Medians in trees with pos/neg weights. Discrete Appl
Math 105:51–71
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Carrizosa EJ, Conde E, Munoz M, Puerto J (1995) The generalized weber problem with expected
distances. RAIRO 29:35–57
Colebrook M, Sicilia J (2007) A polynomial algorithm for the multicriteria cent-dian location
problem. Eur J Oper Res 179:1008–1024
Daskin MS (1995) Network and discrete location: models, algorithms, and applications. Wiley
Interscience, NY
Daskin MS (2002) SITATION-facility location software. Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL. http://users.iems.
northwestern.edu/msdaskin/
Eiselt HA, Laporte G (1993) The existence of equilibria in the 3-facility Hotelling model in a tree.
Transport Sci 27(1):39–43
Eiselt HA, Laporte G, Thisse JF (1993) Competitive location models: a framework and bibliography. Transport Sci 27:44–54
Eliosoff J, Unger R (1998). MEC – minimum enclosing circle applet. http://www.cs.mcgill.ca/
cs507/projects/1998/jacob/welcome.html
Erkut E, Tansel BC (1992) On parametric medians of trees. Transport Sci 26(2):149–156
Fliege J (2001) OLAF – a general modeling system to evaluate and optimize the location of an air
polluting facility. OR Spektrum 23:117–136
Francis RL, White JA (1974) Facility layout and location: an analytical approach. Prentice-Hall,
Englewood Cliffs
Gomes H, Ribeiro AB, Lobo V (2007) Location model for CCA-treated wood waste remediation
units using GIS and clustering methods. Environ Model Softw 22:1788–1795
Graham RE, Lawler EL, Lenstra JK, Rinnoy Kan AHG (1979) Optimization and approximation in
deterministic sequencing and scheduling: a survey. Ann Discrete Math 4:287–326
Hamacher HW, Nickel S (1998) Classification of location models. Location Sci. 6:229–242
Hamacher HW, Klamroth K, Nickel S, Schoebel A (1996) Library of location algorithms. University of Kaiserslautern. http://www.mathematik.uni-kl.de/lola/
Handler GY, Mirchandani PB (1979) Location on networks theory and algorithms. MIT Press,
Cambridge
Jia H, Ordonez F, Dessouky M (2007) A modeling framework for facility location of medical
services for large-scale emergencies. IIE Trans 39:41–55
Kendall D (1951) Some problems in the theory of queues. J R Stat Soc 13:151–153
Klamroth K (2004) Algebraic properties of location problems with one circular barrier. Eur J Oper
Res 154:20–35
Klose A, Drexl A (2004) Facility location models for distribution system design. Eur J Oper Res
162:4–29
Mihelic J (2004) Jure Mihelic k-center algorithms. Department of Computer and Information Science, University of Ljubljana
Mirchandani P, Kohli R, Tamir A (1996) Capacitated location problems on a line. Transport Sci
30(1):75–80
Nagy G, Salhi S (2007) Location-routing: issues, models and methods. Eur J Oper Res
177:649–672
Nickel S, Hamacher HW (1992) RLP: a program package for solving restricted 1-facility location
problems in a user friendly environment. Eur J Oper Res 62:116–117
ReVelle CS, Eiselt HA, Daskin MS (2008) A bibliography for some fundamental problem categories in discrete location science. Eur J Oper Res 184:817–848
Sirigos S, Photis YN (2005) S-distance software. Department of Planning and Regional Development (DPRD), University of Thessaly, Greece
Tansel BC, Francis RL, Lowe TJ (1983) Location on networks: a survey. Part I: the p-center and
p-median problems. Manage Sci 29:482–497
Tiefelsdorf M, Boots B (1997) GAMBINI multiplicative weighted voronoi diagrams. http://www.
wlu.ca/wwwgeog/special/download/gambini.htm
Chapter 22
Demand Point Aggregation Analysis
for Location Models
Ali NaimiSadigh and Hamed Fallah
When selecting locations for facilities, such as hospitals, fire stations schools, warehouses and retail outlets, one has to take into account the demand for the service
provided by the facility. In many instances, such facilities serve a large number of individuals, and it may not be realistic to model each individual as a separate demand
point (DP). To reduce the problem size to a manageable one, an analyst is usually forced to aggregate the demand (population) data by representing a collection
of individuals as one DP. While this is a practical solution, it perturbs the original
problem and may introduce errors to subsequent analysis (Erkut and Bozkaya 1999).
Many location problems can be formulated as minimizing some location objective function subject to upper bounds on other location constraint functions. When
such functions are sub additive and no decreasing in the distances, worst-case DP
aggregation error bounds are known. It is shown how to solve a relaxation and a
restriction of the aggregated problem in such a way as to obtain lower and upper
bounds on the optimal value of the original problem (Francis et al. 2002).
Location problems often involve finding locations of new facilities that provide
services of some kind to existing facilities, also called DPs. When such problems
take place in urban contexts, each private residence can be a DP. Thus there can
be too many DPs to be modeled individually, and aggregation of the DPs becomes
necessary; indeed, sometimes only aggregated data is available. This aggregation
creates a more tractable model, but also introduces model error.
It is naturally of interest to examine how much error is introduced, and the effect
of the level of aggregation upon the error. Essentially the modeler is faced with a
tradeoff: less model accuracy for more model tractability, or vice versa.
Hillsman and Rhoda (1978) were perhaps the first to study errors associated with
DP aggregation for the K-median problem. Their classification of aggregation errors
was further studied by Current and Schilling (1987), Erkut and Bozkaya (1999) and
Zhao and Batta (1999). Plastria (2001), in an effort to reduce aggregation error,
further refined the Hillsman–Rhoda error classification, and made a strong case for
using the centroid of each aggregation group as the representative point for the group
in the reduced K-median problem. Current and Schilling (1987) considered errors
due to aggregation for the (discrete) covering problem.
Methods for reducing DP aggregation error have been proposed by Current
and Schilling (1987), Bowerman et al. (1999), and Hodgson and Neuman (1993).
R.Z. Farahani and M. Hekmatfar (eds.), Facility Location: Concepts, Models,
Algorithms and Case Studies, Contributions to Management Science,
DOI 10.1007/978-3-7908-2151-2 22, c Physica-Verlag Heidelberg 2009
523
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A. NaimiSadigh and H. Fallah
General background discussions on DP aggregation and resulting errors can be
found in Zhao and Batta (1999, 2000) and Plastria (2001).
Zhao and Batta (2000) point out that there is another kind of aggregation error;
it is perhaps less obvious but often occurs. Due to budgetary constraints, only a
subset of the set of all possible feasible solutions may be considered. For example,
potential location sites of interest may be enumerated in a list. Sites might be added
to the list with more resources to consider sites of interest, additional. This approach
is usually viewed as solving a restriction of the actual problem, but it can also be
viewed as a “solution space” aggregation, since a large collection of potential sites
is, in effect, aggregated into a smaller collection. Zhao and Batta (1999, 2000) also
give a general background discussion of DP aggregation, as do Francis et al. (2002).
22.1 Applications
22.1.1 P-Median Problem
A network location model is defined with respect to a connected graph and a set
of weighted DPs on the graph; generally, p new facilities are to be established to
service these DPs. The problem of optimally locating the p new facilities so that the
sum of the weighted network distances between the DPs and their respective closest
new facilities is smallest is called the p-median problem (Andersson et al. 1998).
The p-median model is arguably the most popular model in the facility location literature dealing with multiple facilities. Given n DPs in some space (such as
Euclidean plane or road network), the goal of the model is to locate p service facilities, and allocate the n DPs to these service facilities such as to minimize the total
distance to be traveled for service. It is possible to solve medium-sized instances
(n D 200) of this problem optimally. Furthermore, efficient heuristics are available
to solve larger instances of the problem (n D 1; 000) to near optimality.
Unfortunately, for most realistic instances of the facility location problem, the
number of DPs is much larger. For example, if we wish to locate facilities (such
as retail outlets, post offices, or bank machines) in a city, then each resident is a
potential DP. Given that the population of many large cities exceeds one million, the
need for some aggregation becomes apparent (Erkut and Bozkaya 1999).
22.1.2 P-Center Problem
A model frequently proposed for locating emergency service facilities is the
p-center model. Its solution recommends locations for such facilities to minimize the maximum time (or distance) in order to respond to an emergency at some
DP. In an urban context, there could be millions of possible emergency DPs. Thus,
it is common to aggregate DPs.
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Demand Point Aggregation Analysis for Location Models
525
Location models are generally NP-hard. Furthermore, real-world location problems are often large in scale, and are not solvable to optimality within reasonable
time and effort. Thus, in practice, approximate representations of problems are created by means of DP aggregation (Rayco et al. 1999).
22.1.3 Covering Problem
Covering models measure effectiveness through assessment of whether demand can
receive service from located facilities (Church and ReVelle 1974). The key concept
in these models is the acceptable proximity or coverage. Usually a maximum value,
known as the service standard, is preset with respect to either distance or travel
time, though the latter can be converted to a distance measure. Demand is said to be
suitably served if it can be reached by any facility within this maximum coverage
standard. For example, fire department response time should be less than 6 min to
an accident or structure fire after a call for service has been received (Tong and
Murray 2006).
22.2 Aggregation Errors
22.2.1 Spatial Aggregation Demand
Spatial aggregation of demand units is a common technique used in the solution
of LA models. As mentioned previously, many location problems, including the
p-median problem, are NP-hard. Spatial aggregation of the demand units results
in a reduction of the size of these problems to make them computationally more
tractable.
The aggregation process reduces the number of distinct demand units so that
the m unaggregated demand units indexed by U are aggregated to q < m aggregated demand units indexed by the set A D f1; : : :; qg with the demand of each
unaggregated demand unit assigned to its respective aggregated demand unit. More
precisely, if the set Uj identifies the unaggregated demand units that are associated
with aggregated demand unit jA then the total demand of this aggregated demand
unit is given by:
!D
X
wi
j 2A
(22.1)
X
wi D.X; ui /
(22.2)
i 2Uj
fU .X / D
i 2U
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A. NaimiSadigh and H. Fallah
In this way, the function f given in (22.2) can be approximated by:
X
!j D.X; aj /
fA .X / D
(22.3)
j 2A
Where aj is the j th aggregated demand unit and D.X; aj / is a measure of the
distance between aj and the element of X that serves the corresponding demand.
Different approximations results by varying the way in which aj and D(X, aj ),
jA, are defined. In any case, solving the p-median problem defined by (22.3) is
computationally more attractive than solving the one defined by (22.2) simply because q < m. In what follows, we will use the terms basic spatial unit (BSU) and
aggregated spatial unit (ASU) to distinguish between unaggregated and aggregated
demand units, respectively. This process of demand aggregation can induce errors
in the evaluation of the cost of a solution to the p-median problem. This error arises
since, in general, the cost of a given solution is different when evaluated using the
ASUs and the BSUs. ˇThe
ˇ total error caused by the aggregation process for a given
solution XN S with ˇXN ˇ D p is given by:
N fU .XN /
fA .X/
(22.4)
Hillsman and Rhoda (1978) categorized this error into three different sources
with two different causes: error due to incorrect distance estimation (Source
A and B) and error caused by misallocating BSUs (Source C). Source A errors
arise since the demand weighted distance from an ASU to a candidate facility site
is not equal to the demand weighted distances between the corresponding BSUs
and the same site. Source B errors are a special case of source A errors and occur
when a potential
ˇ ˇ facility site is at the same position as an ASU. For a given solution
XN S with ˇXN ˇ D p the total source A and B aggregation error is given by:
N
fA .XN / fA;U .X/
Where for X S ,
fA;U .X / D
X X
wj d.X.ai /; uj //
(22.5)
i 2A j 2Uj
Source B errors result in the cost of a solution, underestimating its actual cost,
while source A errors can result in either underestimation or overestimation. Source
C errors occur because all demand in an ASU is allocated to a single facility even
though some of the corresponding
ˇ ˇ BSUs may be closer to a different facility. For a
given solution XN S with ˇXN ˇ D p the total source C error can be calculated as:
N
fA;U .XN / fA .X/
Source C errors result in the cost of the solution to the aggregated problem overestimating the actual cost of the solution.
22
Demand Point Aggregation Analysis for Location Models
527
Hillsman and Rhoda (1978) examined aggregation error for uniformly distributed
demand in three different configurations and calculated aggregation errors of up to
8%. On different data sets, Current and Schilling (1987) discovered aggregation
errors of over 20%.
The discussion so far has concentrated on the total error in the objective function
caused by aggregation. However, Goodchild (1979) noted that “aggregation tends
to produce much more dramatic effects on facility locations than on the value of
the objective function” and Bach (1981) further pointed out that ‘the level of aggregation exerts a strong influence on the optimal location patterns as well as on
the values of the location criteria. Casillas (1987) clarified this notion by distinguishing two classes of errors that aggregation causes in the solution of p-median
problems.
The first is cost estimate error defined as the difference between the estimated
cost of the solution using the aggregated data and the true objective cost calculated
using the unaggregated data. The other is optimality error which is the difference
between the actual costs of the solution found using the aggregated data and the cost
of the solution found using the unaggregated data. If we define:
XA D argX min fA .X /
as the optimal solution to the p-median problem using the ASUs and
XU D argX min fU .X /
As the corresponding optimal solution found using the BSUs, then the corresponding relative cost estimate error is:
ECE D ŒfA .XA / fU .XA /=fU .XA /
(22.6)
While the corresponding relative optimality error is:
EO D ŒfU .XA / fU .XU /=fU .XU /
(22.7)
Note that the cost estimate error ECE D fA .XA / fU .XA / is identical to the
total aggregation error defined in (22.4) when X D XA (Bowerman et al. 1999).
22.2.2 Methods for Reducing Aggregation Errors
22.2.2.1 P-Median Models
The effects of aggregation on the solution of facility location models have been
recognized by many authors. For example, Rushton (1989) stated that the “optimum locations must be sensitive, to an important degree, at some level of spatial
528
A. NaimiSadigh and H. Fallah
aggregation of data. We do not know how to identify this level in advance for
any given application”. Goodchild (1979) also identified the importance of aggregation but further noted that the “effects of aggregation are unique to particular
solutions, and therefore no general rules of aggregation can be found”. In a recent study, Fotheringham et al. (1995) found that solutions to the p-median problem
were highly sensitive to both level of aggregation and the definitions of the ASUs
and that these “results therefore question the reliability of any locational recommendations from a location-allocation model when aggregate demand data are used”.
However, in a similar study, Murray and Gottsegen (1997) found that a “certain
amount of spatial stability [was] in fact present” in the locational configuration of
the p-median solutions and that the error percentage remained relatively low.
Francis and Lowe (1992) also recognized that “too much aggregation can destroy
the accuracy of a location model”. Furthermore, they validated Goodchild’s (1979)
claim of there being “no general rules of aggregation” by establishing that the process of determining an aggregation scheme with minimum worst-case error is an
NP-hard problem. Nevertheless, Francis and Lowe (1992) proposed a method for
determining an aggregation scheme that bounds the level of aggregation error. Their
method involves finding a feasible solution to an associated covering problem. These
results were further developed as a row–column aggregation scheme for p-median
models using rectilinear distances (1996).
Both the effects of aggregation error and the difficulty in finding general aggregation schemes have caused researchers to develop methods for reducing these
errors. For example, Mirchandani and Reilly (1986) proposed a method to reduce
source A and B errors for zonal or polygon-based problems. The travel distance
was decomposed into two components: from a facility to a zone, and the inter-zonal
travel distance. These two components were then combined to obtain an improved
estimate of the distance from a facility site to a demand zone.
Current and Schilling (1987) introduced a weighting scheme to eliminate source
A and source B errors. Their method is based on preventing the loss of locational
information during the aggregation process. In their method, the demand weighted
distance between ASU aj and a potential facility site sk 2 S is replaced by the demand weighted distance of the BSUs associated with the ASU and the site so that:
!j d.sk ; aj / D
X
wi d.sk ; ui /
(22.8)
i 2Uj
where !j is defined as in (22.2). Current and Schilling (1987) tested their method on
several different data sets, and found that on these data sets their method reduced the
magnitude of both the cost estimate error and the optimality error. For these sample problems, their method reduced the average cost estimate error from between
11:8% and 16:4% to between 1.7% and 2.0% and the average optimality error
from between 1.3% and 1.7% to between 0.4% and 1.0%. However, their method
does not consider source C errors because it adjusts the distances before the problem is solved. Since source C errors depend on the facility locations selected in the
solution, these errors cannot be reduced by their method.
22
Demand Point Aggregation Analysis for Location Models
529
Hodgson and Neuman (1993) proposed a method for reducing source C errors in
zonal problems, that is, in problems where the demand is distributed over polygons
referred to as demand polygons. Their method involves finding Thiessen or Voronoi
polygons with the selected set of facilities as control points. Thiessen polygons have
the property that every point in a polygon is closer to that polygon’s control point
than it is to any other control point (1992). These polygons are then topologically
overlaid on the set of demand polygons. Source C error arises wherever any demand
polygon spans more than one Thiessen polygon. The area of each demand polygon
is partitioned into zero or more source C error polygons and a complementary
polygon where demand is allocated to the correct facility. The Thiessen demand
disaggregation method computes the centroid of the error and complementary polygons and assigns them to the nearest facility (Bowerman et al. 1999).
22.2.2.2 The Demand Partitioning Method for Reducing Aggregation Error
None of the currently available methods reported in the literature for reducing aggregation errors consider all three sources of error. In this section, an iterative
method for reducing aggregation errors in p-median problems that addresses all
three sources of errors by combining Current and Schilling’s (1987) method for
handling source A and B errors and Hodgson and Neuman’s (1993) idea of partitioning the ASUs (or in their case, demand zones) to eliminate source C error is
introduced.
The demand partitioning heuristic is an iterative procedure for reducing aggregation error. This procedure is based on applying Current and Schilling’s (1987)
weighting method and solving the aggregated p-median problem. The ASUs are
then partitioned according to the selected facility sites to form a new set of ASUs
so that source C error in the original aggregated problem is eliminated. These new
ASUs are then used to define a new aggregated p-median problem and the process
is repeated until the problem is solved without cost estimate error.
Recall that the weighting method for eliminating source A and B aggregation
errors is based on calculating the demand weighted distances between an ASU and
a facility as the demand weighted sum of the distances between the BSUs aggregated
to that ASU and that facility.
However, this procedure does nothing to eliminate source C errors, which can be
significant for some problems.
The source C errors are reduced through a partitioning procedure. In a manner similar to the Hodgson and Neuman’s (1993) Thiessen demand disaggregation
procedure, the total level of source C errors can be calculated by assigning each
BSU to the nearest facility in the current p-median solution. This calculation gives
the true cost of the solution and eliminates all cost estimate errors for the original solution. This assignment of BSUs to facilities was then used to partition each existing
ASU into one or more new ASUs.
However, the solution found with the original ASUs may not be optimal for the
problem defined with the new set of ASUs. The weighting and partitioning steps can
530
A. NaimiSadigh and H. Fallah
be repeated on this new set of ASUs. This suggests an iterative demand partitioning
procedure for eliminating all aggregation errors from the solution of an aggregated
problem.
Given U , the set of BSUs; A, the set of ASUs; and Uj ¤ ¿; j 2 A the indices
of BSUs aggregated to AS U j , the demand partitioning method can be described as
follows:
.1/
Initialize: Set the iteration count l D 1. Set A.1/ D A and Uj D Uj ; j 2 A
Demand weighting: For each sk S calculate
.1/
d.sk ; aj / D
X
wi d.sk ; ui /=
.1/
X
wi
for all jA.1/
.1/
i 2Uj
i 2Uj
Using these distances eliminates source A and source B errors.
Heuristic p-median solution: For A D A.l/ find
Defining via the current set of ASUs and the distances calculated in step 2. The
corresponding value of the objective function is given by:
fA .X / D
.l/
X
j 2A
!j d.X .aj /; aj / D
XX
wi d.X .aj /; ui /
j 2A i 2Uj
.l/
.l/
where !j D !j ; Uj D Uj ; and aj D aj for all jA, are defined in terms
of the current ASUs. This calculation contains only source C errors and therefore
overestimates fU .X /.
Set X.l/
D X .
. This step is ignored for the first
D X.l1/
Terminate: Stop if l > 1 and X.l/
iteration and otherwise terminates the procedure if the current solution is identical
to the solution from the previous iteration. We know that there is no source C error
in the value of fA.l/ .X.l1/
then there is no aggregation
D X.l1/
/ therefore if X.l/
error in the solution X.l/
.
.l/
Repartition: For each jA.l/ partition Uj , a current ASU, into one or more new
ASUs as follows:
.l/
Assign each BSU in Uj to the facility in the current p-median solution that is
closest to it. The BSUs in each nonempty assignment comprise a new and unique
ASU. Define A.lC1/ to be the index set of the new ASUs.
This step repartitions the BSUs into a new set of ASUs indexed by A.lC1/ so as
to eliminate source C error. All the BSUs in a new ASU were, in the previously
iteration, aggregated to the same ASU and are allocated to the same facility in the
current p-median solution. Conversely, any BSUs that were in the same ASU in
the previous iteration and that are in different ASUs after repartitioning must be
assigned to different facilities
in
p-median solution.
ˇ
ˇ theˇ current
ˇ
Terminate: Stop if ˇA.lC1/ ˇ D ˇA.l/ ˇ. If the number of new ASUs after repartitioning is the same as before repartitioning then the procedure terminates. Since all
BSUs in each new ASU were members of a common ASU in the previous iteration,
having the same number of ASUs implies that there were no incorrectly allocated
BSUs.
22
Demand Point Aggregation Analysis for Location Models
531
Demand weighting: For each sk S calculate the distances between potential
facility sites sk and the new ASUs using:
.lC1/
d.sk ; aj
/D
X
wi d.sk ; ui /=
i 2Uj.lC1/
X
wi
for all j"A.lC1/
i 2Uj.lC1/
Since all BSUs in a new ASU are closest to the same facility in the current
p-median solution we know that:
fA .X / D
X
j 2A
!j D.X ; aj / D
X
i 2U
wi D.X ; ui / D fU .X /
.lC1/
.lC1/
where A D A.lC1/ , X.l/
D X , and !j D !j
and aj D aj
for all jA.
Since fA .X / D fU .X / the solution X.l/
D X has no aggregation error for the
ASU partitions indexed by A.lC1/ .
Iterate: Increment l. Go to step 3.
This iterative procedure will always terminate since the maximum possible number of ASUs is equal to the number of BSUs and the procedure terminates if the
number of ASUs do not increase over iteration (step 6). Thus the maximum possible number of iterations is mq where q is the number of initial ASUs. In practice,
far less iteration is commonly required.
After the procedure terminates at iteration l , the final solution X.l / D X is
the solution of the p-median problem defined by A D A.l / and S with no cost
estimate errors, i.e., fA .X / D fU .X /. This procedure electively reduces the cost
estimate error to zero for the solution of a particular aggregated p-median problem.
There is, of course, no guarantee that the solution to this problem is the same as the
solution to the unaggregated p-median problem.
Computational effort in both distances weighting (steps 2 and 7) and partitioning
(step 5)
Increases only linearly with the number of BSUs for the following reasons:
(a) Demand weighting: Current and Schilling (1987) weighting method requires
the calculation of the demand weighted distance between each ASU and each
potential facility site. Assuming that the distances between the BSUs and the
facility sites are pre-calculated, these distances are calculated as the demand
weighted sum of the BSUs aggregated to an ASU and since the ASUs partition
the BSUs, it takes O(mn)operations, where n is the number of potential facility
sites, to calculate these distances.
(b) Partitioning: Since each BSU must be assigned to the nearest of p facility
sites this assignment requires O(mp) operations. Identifying the new nonempty
ASUs then requires O.m/ plus O.q .l/ p/ operations, where q .l/ is the number of ASUs in iteration l. Since m q .l/ this partitioning requires O.mp/
operations.
532
A. NaimiSadigh and H. Fallah
Thus both procedures scale well to problems with a large number of BSUs. Consequently, the overall computational effort is dominated by whatever heuristic is
used for solving the p-median problem.
An assumption for this algorithm is that there is an initial partitioning of the
BSUs into ASUs.
Often, when using real-world data sets, there is a natural partitioning of the BSUs
through the use of existing political boundaries such as districts, counties, or areas.
This is the approach that is used for the sample data as outlined in the next section.
However, other methods have been suggested to partition the BSUs. Casillas (1987)
randomly selected BSUs as the reference centers for the ASUs.
Each BSU was then aggregated to its nearest reference center. Current and
Schilling (1987) used both visual inspection and solving a p-median problem to
select the reference centers. Finally, as mentioned above, Francis and Lowe (1992)
proposed finding a feasible solution to a covering problem as a method to determine
the reference centers (Bowerman et al. 1999).
22.3 Computational Approach
In this first phase–strategy, our “row–column” DP aggregation procedures involve
partitioning the DPs according to a grid imposed over the demand region (Francis
et al. 1996; Rayco et al. 1997). The widths of rows and columns in each grid are
adjusted by these algorithms to reflect the DP structure. This initial partitioning
provides a coarse aggregation structure. In our earlier work, we used a rectilinear
l-median or l-center as the aggregate DP for each grid cell. There is no guarantee
that this aggregate DP will be on the network. Further, our earlier work was based
only on rectilinear distances, not network distances. Therefore, in this paper, to find
the aggregate DP in each cell, we carry out a second “fine-tuning” step – tactics –
which involves the optimal solution of network location problems on the graph sub
networks induced by the grid partitioning. The location found in each sub network,
an l-median or l-center, determines the aggregate DP that replaces all DPs in the
sub network. Once we place the aggregate DPs on the network, we go through a
Monte Carlo sampling phase, to compute network distances and estimate various
error values of interest.
Consider the “strategy” step for the p-median model. The grid, with c columns
and r rows, is obtained for the p-median problem by solving a c-median problem on
the x-axis projection of the DPs, and an r-median problem on the y-axis projection
of the DPs; see Francis et al. (1996). Next consider the “tactics” step. Each grid cell
induces a sub network whose largest component’s optimal l-median is designated
as an aggregate DP. Finally, as an optional step, each DP can be “assigned” an
aggregate point closest to it; this mapping is given by the Voronoi partition (Hakimi
et al. 1992) of the entire network with respect to the DP set.
The aggregate DP set for the p-center model is found in a similar manner, with
certain modifications. The grid is constructed by solving a pair of one-dimensional
22
Demand Point Aggregation Analysis for Location Models
533
c- or r-center problems defined with respect to the projections onto the coordinate
axes of the images, under a transformation, of the DPs; see Rayco et al. (1997). This
transformation results in a 45ı rotation of the demand data space. The aggregate
DPs for each grid cell, furthermore, are given by the optimal network l-centers of
the largest cell-induced subnetwork components.
The projected median problems on the line are solvable in O..c C r/e/ time using the Hassin and Tamir (1991) algorithm. The one-dimensional center problems
are solvable in O.e log e/ time using a modification of the algorithm by Meggido
et al. (1981) for the p-center problem in trees. Let n D jV j and m D jEj be the number of vertices and edges, respectively, of the network. The splitting of the graph into
subnetworks requires O.e log cr/ effort, and the optimal solution of the l-median
or l-center problems on the subnetworks appears, for typical street networks, to
have an average-case complexity of O.m2 =cr/ (Andersson and Normark 1995). Finally, the Voronoi partition of a graph may be found, using the algorithm by Hakimi
et al. (1992), in O.e C n log m/ time. We use the Voronoi partition to compute
nearest network distances between DPs (aggregated or not) and new facility locations when computing f .X / and f ‘.X /
The aggregate set so obtained may define the relevant approximating problem.
Alternatively, it may serve to initialize an iterative location-allocation procedure,
which is the adaptation to networks of the well-known planar location approach
of Cooper (1963); see also Maranzana (1964). This procedure, however, is computationally intensive, as each iteration requires the solution of network location
problems. The location-allocation procedure locates each aggregate DP (Anderson
et al. 1998).
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Appendix: Metaheuristic Methods
Zohre Khoban and Saeed Ghadimi
Combinatorial optimization (CO) is the process of finding the optimal solution for
problems with a region of feasible solutions. Applications of CO are numerous
in fields of industry and economy, such as routing, scheduling, packing, location,
transportation, telecommunications, financial services, etc. While much progress
has been made in finding exact solutions to some combinatorial optimization problem (COP) such as dynamic programming, many hard COP are still to be solved
exactly and require heuristic methods. Moreover, reaching “optimal solutions” is in
many cases meaningless, so another aim of heuristic methods is quickly producing
good-quality solutions, without necessarily providing any guarantee for reaching the
optimal solution (Resende and De Souse 2004). Metaheuristics are high level procedures that coordinate simple heuristics such as local search, to find solutions that
are of better quality in comparison to those found by the simple heuristics alone.
One of the most important reason of using metaheuristics is to escape from local
optimum that heuristics most of the times get trapped in. One of the main goals in
any metaheuristic method is to create a balance between intensified and diversified
search. Some well-known metaheuristics are: greedy randomized adaptive search
procedure (Glover et al. 2003), scatter search (Glover 1999), variable neighbourhood search (Glover et al. 2003), harmony search (Geem et al. 2001), particle swarm
optimization (Eberhart and Shi 2001), simulated annealing, genetic algorithms, tabu
Search, ant colony optimization and also their hybrids. Hybrids are metaheuristics
with one or more additional heuristic modules or combination of two metaheuristics
for example TS and GA (Glover et al. 1995) which are more efficient in comparison.
In the following sections we describe some of the metaheuristic methods in detail.
Genetic Algorithm
Genetic algorithm (GA) which was first suggested by John Holland (1975) is an
evolutionary strategy. The basic mechanism in GA is that of Darwinian evolution:
good traits survive and mix to form new while the bad traits are eliminated from the
population.
535
536
Z. Khoban and S. Ghadimi
GA is a population based algorithm which starts with an initial population as the
first generation which most of the time selected randomly. Each individual solution
in the population that is called chromosome is a set of linked features and each
feature in one chromosome is named a gene. If one chromosome results in a better
result of fitness function which is considered for each problem, it is fit to survive
and a new generation is created in every iteration by updating the population, using
some genetic operations which make some changes in the survivors.
Fitness function: The fitness of each chromosome is evaluated by a function and
a value is assigned to each chromosome. The fitness function is the only component
of the algorithm that is generally problem specific. Operators have important roles
in GA. The more important ones are:
Selection: The members of the new population are selected based on their fitness.
There are many ways of selection varying in complexity. Low selectivity accepts a
large number of solutions, high selectivity allows a few or one to be dominated,
however a balance needs to be reached in order to prevent the solution from becoming trapped in a local optimum. One simple selection method is roulette wheel
selection in which more fit members of the population have a higher chance of being
selected.
Crossover: This is the most important reproduction genetic operator. The child
of two selected chromosomes in last step gets its features from a random selection
of its parents features. Again there are many kinds of crossover, such as one-point
crossover which first selects a location to cut the two parents. A new chromosome
is then created by copying the first part of the first chromosome, into a bit string,
followed by a copy of the last part of the second chromosome into the same bit
string. The remaining, un-copied parts of the chromosomes are also copied into a
second new chromosome.
Mutation: This would involve making a random change to one of the genes on
the string of one chromosome, done randomly by a given probability.
The result of these operations (selection, crossover and mutation) is a new generation of population, the same size we start with. There are various types of
chromosome’s strings such as bit strings, real numbers, permutations of element,
lists of rules, and etc.
The Simple Genetic Algorithm
1.
2.
3.
4.
5.
Initialize an initial population mostly in a random way.
Select individuals for reproduction.
Generate a new generation by the genetic operations.
Insert children into population and update the population.
If the stopping criteria are satisfied, stop the algorithm, otherwise, return to
step 2
Appendix: Metaheuristic Methods
537
Some of GA’s applications are parametrization (Al-Duwaish 1999), job-shop
scheduling (Cheng et al. 1999), vehicle routing (Tong et al. 2004), time series prediction (Hansen et al. 1999), and chance constrained problem (Poojari and Varghese
2008).
Tabu Search
Tabu search (TS) was first produced by Fred Glover (1986), the basic ideas have
also been sketched by Hansen (1986) and additional efforts of formalization are
reported by Werra and Hertz (1989). Many good examples and applications of TS
with a collection of references could be found in a book by Glover and Languna
(1993). The convergence of TS was proved by Hanafi (2000). TS is a neighborhood
search with an iterative procedure. In each iteration k, from a current solution i goes
to a next solution j which is searched among the solutions in the neighborhood set
N.i; k/ of current solution i . in addition to N.i; k/, there are at least three more
components in a simple TS as below:
1. Evaluation function f(i): Evaluation function which is the problem specification,
evaluates the value for each solution.
2. Tabu list: TS has two kinds of memories, short term and long term, the short
term prevents to selecting identical solutions in limited number of consecutive
iterations by saving last moves in a selection named tabu list T .i / and forbids
moves that might lead to recently visited solutions.
3. Long term memory: The value f .i / of the best solution i * visited so far is
saved in the long term memory and updated when a better solution is found, so
it prevents to getting trapped in local optimum by jumping to the best current
solution when its necessary.
Two Important Concepts
Intensification: Sometimes the search process propend to intensify the search in
some region of S, because iteration by iteration the solutions become better in this
area, so for giving a high priority to the solutions near the current one, it is possible to introduce an additional term in the evaluation function which will penalize
solutions far from this region.
Diversification: Sometimes it is fruitful to diversify the solutions by leaving the
current region and exploring another area of S, because continuing the search in this
region is not very satisfactory for evaluation function; so for giving a high priority to
the solutions far from the current one, it is possible to introduce an additional term
in the evaluation function that will penalize solutions which are close to the present
one (Hertz et al. 1995).
538
Z. Khoban and S. Ghadimi
By using intensification and diversification the evaluation function changes as
below:
f ‘ D f C Penalty of Intensification + Penalty of Diversification.
Aspiration criteria A(i): Aspiration criteria is an additional precaution which is
taken to avoid missing good solutions. Sometimes a solution in spite of belonging
to the tabu list, is better than any solution so far seen; so we would allow the tabu
classification of this solution to be overridden and consider the solution admissible
to be visited.
A Simple Tabu Search Algorithm
1.
2.
3.
4.
5.
6.
7.
Choose an initial solution i in S . Set i D i and k D 0.
Set k D k C 1 and generate N.i; k/ by considering T .i / and A.i /.
Choose the best j in N.i; k/ as a new solution.
Set i D j
If f .i / < f .i / then set i D i .
Update tabu list and aspiration criteria.
If a stopping condition is met, then stop. Else go to Step 2.
By using some immediate stopping conditions of TS have been introduced by
Hertz et al. (1995). Some of TS’s applications are graph coloring (Hertz and Werra
1987), maximum independent set (Friden et al.1990), course scheduling (Hertz
1992), multicommodity location/allocation (Crainic et al. 1993), job shop scheduling (Ponnambalam et al. 2000), quadratic assignment problem (Misevicius 2005).
Ant Colony Optimization
Ant colony optimization (ACO) is a paradigm for designing metaheuristic algorithms for combinatorial optimization problems. The first algorithm which can be
classified within this framework was presented in 1991 (Dorigo et al. 1991) and,
since then, many diverse variants of the basic principle have been reported in the
literature. The inspiring source of ACO is the pheromone trail laying and following
behavior of real ants which use pheromones as a communication medium. Analogous to the biological example, ACO is based on the indirect communication of a
colony of simple agents, called (artificial) ants, mediated by (artificial) pheromone
trails. The pheromone trails in ACO serve as a distributed numerical information
which the ants use to probabilistically construct solutions to the problem being
solved and which the ants adapt during the algorithm’s execution to reflect their
search experience. The essential trait of ACO algorithms is the combination of a
priori information about the structure of a promising solution with a posteriori information about the structure of previously obtained good solutions.
Appendix: Metaheuristic Methods
539
The ACO metaheuristic: After initialization, the metaheuristic iterates over three
phases: at each iteration, a number of solutions are constructed by the ants; these
solutions are then improved through a local search (this step is optional), and finally
the pheromone is updated. The following is a more detailed description of the three
phases (Dorigo et al. 2006).
Construct ant solutions: A set of some artificial ants constructs solutions from
elements of a finite set of available solution components. A solution construction
starts from an empty partial solution. At each construction step, the partial solution
is extended by adding a feasible solution component from the set of its neighborhood.
The choice of a solution component from neighborhood set is guided by a
stochastic mechanism, which is biased by the pheromone associated with each of
its elements. The rule for the stochastic choice of solution components vary across
different ACO algorithms.
Apply local search: Once solutions have been constructed, and before updating
the pheromone, it is common to improve the solutions obtained by the ants through
a local search. This phase, which is highly problem-specific, is optional although it
is usually included in state-of-the-art ACO algorithms.
Update pheromones: The aim of the pheromone update is to increase the
pheromone values associated with good or promising solutions, and to decrease
those that are associated with bad ones. Usually, this is achieved (a) by decreasing
all the pheromone values through pheromone evaporation, and (b) by increasing the
pheromone levels associated with a chosen set of good solutions.
Several ACO algorithms have been proposed in the literature. Some of them are
the original ant system (Dorigo et al. 1991), MAX-MIN ant system (St¨utzle and
Hoos 2000) and ant colony system (Dorigo and Gambardella 1997). Some of ACO’s
applications are traveling salesman problem (Stützle and Hoos 1997), Quadratic assignment (Maniezzo et al. 1994) Scheduling problems (Merkle et al. 2002), vehicle
routing (Gambardella et al. 1999) and graph coloring (Costa and Hertz 1997).
Simulated Annealing
Simulated annealing (SA) is a random-search technique which exploits an analogy
between the way in which a metal cools and freezes into a minimum energy crystalline structure (the annealing process) and the search for a minimum in a more
general system; it forms the basis of an optimization technique for combinatorial
and other problems. SA was developed in 1983 to deal with highly nonlinear problems. SA approaches the global maximization problem similar to using a bouncing
ball that can bounce over mountains from valley to valley. It begins at a high “temperature” which enables the ball to make very high bounces, which enables it to
bounce over any mountain to access any valley, given enough bounces. As the temperature declines the ball cannot bounce so high, and it can also settle to become
trapped in relatively small ranges of valleys. A generating distribution generates
540
Z. Khoban and S. Ghadimi
possible valleys or states to be explored. An acceptance distribution is also defined,
which depends on the difference between the function value of the present generated
valley to be explored and the last saved lowest valley. The acceptance distribution
decides probabilistically whether to stay in a new lower valley or to bounce out of
it. All the generating and acceptance distributions depend on the temperature. It has
been proved that by carefully controlling the rate of cooling of the temperature, SA
can find the global optimum. However, this requires infinite time. Fast annealing
and very fast simulated reannealing (VFSR) or adaptive simulated annealing (ASA)
are each in turn exponentially faster and overcome this problem.
As pointed above (Wang and Zheng 2001), SA is able to converge to the optimum
value, but it could be very expensive to get a desired solution in terms of computational time. For that reason, SA has been improved through hybridizing it with other
methods such as the genetic algorithms (Huang et al. 2001) or by parallelising the
algorithm (Bevilacqua 2002). The evolutionary simulated annealing algorithm was
developed to contribute to the progress in this direction. It offers an evolutionary
process in which a shorter SA algorithm is substituted for the genetic operators of
crossover and mutation to evolve a population of solutions. The SA algorithm is
so compact that one can easily use it in any evolutionary process, where SA can
manipulate the solutions selected from the population. This makes ESA easily implementable in various environments and together with different methods.
Neural Networks
Neural nets have gone through two major development periods – the early 1960s
and the mid 1980s. They were a key development in the field of machine learning.
Artificial neural networks were inspired by biological findings relating to the behavior of the brain as a network of units called neurons. The human brain is estimated
to have around ten billion neurons each connected on average to 10,000 other neurons. Each neuron receives signals through synapses that control the effects of the
signal on the neuron. These synaptic connections are believed to play a key role in
the behavior of the brain. Neural networks have emerged as a field of study within
artificial intelligence (AI) and engineering via the collaborative efforts of engineers,
physicists, mathematicians, computer scientists, and neuroscientists. Although the
strands of research are many, there is a basic underlying focus on pattern recognition
and pattern generation (Widrow et al. 1988), embedded within an overall focus on
network architectures. Many neural network methods can be viewed as generalizations of classical pattern-oriented techniques in statistics and the engineering areas
of signal processing, system identification, optimization, and control theory. There
are also ties to parallel processing, VLSI design, and numerical analysis. A neural
network is first and foremost a graph, with patterns represented in terms of numerical values attached to the nodes of the graph and transformations between patterns
achieved via simple message-passing algorithms. Certain nodes in the graph are
generally distinguished as being input nodes or output nodes, and the graph as a
Appendix: Metaheuristic Methods
541
whole can be viewed as a representation of a multivariate function linking inputs
to outputs. Numerical values (weights) are attached to the links of the graph, parameterizing the input/output function and allowing it to be adjusted via a learning
algorithm. One of the most applied learning algorithms is the backpropagation algorithm. The backpropagation algorithm cycles through two distinct passes, a forward
pass followed by a backward pass through the layers of the network. The algorithm
alternates between these passes several times as it scans the training data. Typically,
the training data has to be scanned several times before the networks “learns” to
make good classifications (Krose and Smagt 1996).
Forward pass: Computation of the outputs of all neurons in the network. The
algorithm starts with the first hidden layer using as input values the independent
variables of a case (often called an exemplar in the machine learning community)
from the training data set. The neuron outputs are computed for all neurons in the
first hidden layer by performing the relevant sum and activation function evaluations. These outputs are the inputs for neurons in the second hidden layer. Again the
relevant sum and activation function calculations are performed to compute the outputs of second layer neurons. This continues layer by layer until we reach the output
layer and compute the outputs for this layer. These output values constitute the neural net’s guess at the value of the dependent variable.
Backward pass: Propagation of error and adjustment of weights.
This phase begins with the computation of error at each neuron in the output
layer. These errors are used to adjust the weights of the connections between the
last-but-one layer of the network and the output layer. The process is repeated for
the connections between nodes in the last hidden layer and the last-but-one hidden
layer. The backward propagation of weight adjustments along these lines continues
until we reach the input layer. At this time we have a new set of weights on which
we can make a new forward pass when presented with a training data observation.
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Index
.Q; r/ inventory model, 461
aggregation, 215
Aisle Distance, 8
algorithm
Weiszfeld, 85
Allocation, 249
Multiple, 251
Single, 249
Ant Colony Optimization, 538
pheromone, 538
Arc Exclusion-Bounding Property, 212
back haul, 397
Big-O notation, 22
binary search, 208, 209
bisection search, 207
Block Distance, 9
bound matrix, 204, 206
Centrum, 195
Class-based dedicated storage location policy,
431
class-based storage location policy, 429, 446
Classification, 505
Classification scheme, 505
Classified on facilities, 94
Classified on the demand, 94
Classified on the physical space, or locations,
94
Combinatorial optimization, 535
Competitive Location Problem, 271
Attraction Function, 278
Competition with Foresight, 276
Decision Space, 279
Flow Capturing Location Allocation
Problem, 288
Game Theories, 274
Gravity Problem, 280
Maximum Capture Problem, 283
Patronizing Behavior, 277
Point vs. Regional Demand, 277
Static Competition, 276
The Maximum Capture Problem with
Price, 285
Complexity Classes, 28
Class NP , 29
Class NP -Hard, 34
Class P , 28
NP -complete, 31
Complexity Theory, 19
Computation models, 19
Deterministic Turing Machine (DTM), 19
non-deterministic turing machine (NTM),
19
Oracle machine models, 19
Computational complexity theory, 19
conflicting objectives, 384
Continuous space models, 94
Contour lines, 74
Cooks theorem, 31
P D NP Problem, 31
Satisfiability problem, 25
Covering models, 525
covering problems, 145
coverage distance, 145
Partial Covering, 145
Total Covering, 145
Cube-per-order index, 429
customer has to be covered by only one
facility, 96
customers, 93
Decision Problems, 25
Dedicated Storage Location Policy, 423, 446
demand partitioning, 529
545
546
demand point, 523
aggregation, 523
error, 523
Dependency Variable, 348
Distance-dependent weight, 348
Time-dependent weight, 348
depot, 396
Designing optimum routes, 328
MRHT, 330
OCST, 328
Deterministic change, 348
Discrete space models, 94
distance function, 6, 210
Distance Matrix, 8
distribution cost, 452
distribution network, 451
customer, 452
Distribution Centers, 451
supplier, 452
Duration-of-Stay storage policy, 438
dynamic facility location problem, 347
Dynamic P -median Model, 350
Location, 347
Location–Relocation, 347
multiperiod location–allocation, 352
Probabilistic Model, 353
static location model, 349
Edge covering, 151
ellipses, 74
endogenous, 253, 259
Exogenous, 179, 193, 248, 249, 260
facilities, 93
facility hierarchy, 221
successively exclusive, 221
successively inclusive, 221
Facility location problem, 348
dynamic, 348
static, 348
facility’s opening cost, 97
Family grouping, 446
Fixed Charge Facility Location
Problem, 476
flow discipline, 221
discriminating, 221
integrated, 221
flow pattern, 221
single-flow, 221
Floyd’s algorithm, 207
Full Turn-over based storage, 433, 434
Index
Gauges Measures, 10
General LA model, 95
general networks, 148
Genetic algorithm, 266, 535
Crossover, 536
Fitness Function, 536
Mutation, 536
selection for reproduction, 536
Grasp Algorithm, 173
gravity problem, 73
greedy, 215
Plus (GrP), 215
Hamming Distance, 12
HAP, 85
Hausdorff Distance, 13
hierarchical, 219
Hilbert Curve, 11
hub, 243, 244
capacitated, 257
Center, 258
Covering, 259
discount factor, 250, 263
Maximal Covering, 260
Median, 251–253
Set Covering, 259
single, 247
hyperboloid approximation
procedure, 85
Integrated Decision Making Models, 480
Intersection point, 211
Inventory cost, 298, 452
lead time, 456
Order cost, 459
safety stock, 453
Lagrangian relaxation, 172, 454
less-than-truckload (LTL), 480
Levenshtein Distance, 12
Location-allocation, 93, 261
locations, 93
logistics cost, 451
Lower Bound, 125
Bounds based on reformulations, 127
Eigenvalue Related Bounds, 126
Gilmore and Lawler lower bound, 126
Semi-definite programming, 127
Variance reduction bounds, 127
LRP, 399, 401
Allocation–Routing–Location, 413
layer diagram, 404
Index
Location–Allocation–Routing, 412
primary facilities, 404
secondary facilities, 404
Mahalanobis Distance, 12
maximum covering, 193
maximum covering location problem, 157
maximum covering problem, 216
MCLP, 383
median problem, 177
CPMP, 182
MFLP, 69
Minimal objective function value
property, 206
Minimum Lengths Path, 9
minisum, 177
Model with Facility Failures, 493
Model with Multiple Commodities, 490
Model with Unreliable Supply, 491
multi-criteria decision-making, 373
Multi-Objective Combinatorial Optimization,
385
Multifacility location
Euclidean Distance, 75
MiniMax, 75
on sphere, 78
Rectangular Distance, 72
Rectangular Multi Product, 77
Squared Euclidean, 73
Stochastic, 80
with Rectangular Regions, 79
Multiple facility
number of relocating facilities, 349
Network
general graph, 195, 207
tree, 195, 199
network, 243
Access, 262, 263
Backbone, 262, 263
completely connected, 243
connective, 244
fully connected, 243
hierarchical, 263
multi hub, 244
node, 243
Neural Networks, 540
backpropagation, 541
hidden layer, 541
non-hub node, 244
number of relocating facilities, 349
Single facility, 349
547
number of relocations, 349
Multiple relocation, 349
Single relocation, 349
Obnoxious, 315
Dispersion problems, 315
multiobjective model, 337
OFLR, 331
Undesirable facilities problems, 315
operational decision, 451
origin-destination pair, 243
p-center problem, 193, 524
absolute, 194
Anti, 196
Asymmetric, 196, 214
capacitated, 198, 214
Continuous, 196
on circular arc graphs, 214
vertex, 194
with pos/neg weights, 196, 214
P -Median, 320
Maxisum, 321
p-median problem, 524
pickup and delivery, 397
Point Single Facilities, 39
Euclidean distance, 40
lp-norm distance, 41
rectilinear distance, 39
square of the Euclidean distance, 40
Profit Maximizing Model, 488
Pull Objectives, 377
Push Objectives, 377
Push-Pull, 378
Models, 380
Objectives, 378
QAP
Generalized Quadratic Assignment
Problem, 123
Multiobjective QAP, 121
Quadratic semi-assignment problem, 121
Quadratic Three-Dimensional Assignment
Problem, 122
Stochastic QAP, 124
The biquadratic assignment problem, 120
The quadratic bottleneck assignment, 120
Quadratic Assignment Problem, 111
quadratic facility location problem, 73
queue, 457
548
Randomized storage location policy, 444
Reduction, 25
Linear reduction, 26
Polynomial reduction, 26
Polynomial reduction: many-one
polynomially reducible, 26
Regional Facilities, 41
Relocation Time, 349
Continuous, 349
Discrete, 349
retailer, 473
retrieval
decision, 440
operation, 430
policy, 421
rate, 448
retrieval location, 428
risks, 324
risk evaluation, 325
robust optimization, 495
route, 396
service hierarchy, 221
globally inclusive, 221
locally inclusive, 221
successively exclusive, 221
successively inclusive, 221
service variety, 221
Nested, 221
non-nested, 221
SFLP, 69
Shared storage policies, 437, 440
Adaptive Shared Storage Policies, 441
Static Shared Storage Policies, 441
Simulated Annealing, 539
acceptance distribution, 540
temperature, 539
single facility location problem, 69
Single Sourcing, 477
Software, 513
S-Distance, 519
LoLA, 515
SITATION, 516
spanning tree, 215
Spatial aggregation, 525
spatial configuration, 221
coherent, 221
spatial pattern, 221
closest assignment, 221
path assignment, 221
single assignment, 221
spoke, 243
Squared Euclidean
Index
Contour lines, 74
Stochastic change, 348
stochastic demands, 98
stochastic programming, 495
storage space, 425, 426
requirement, 439
storage system layout, 422
Storage Location Assignment Problem,
419, 422, 423, 436, 444, 448
strategic decision, 451
supplier, 473
supply allocation, 474
supply chain, 473
supply chain configuration, 474
Supply chain management, 473
Supply chain network design, 474
Supply chain operation, 474
Supply chain planning, 474
Supply chain strategy, 474
Tabu search, 537
Aspiration criteria A(i), 538
Diversification, 537
Intensification, 537
Long term memory, 537
Tabu list T .i /, 537
tactical decision, 451
Taxonomy, 505
The network-based model, 94
three-tiered supply chain, 482
throughput, 421
requirement, 427
Time Complexity, 24
Constant Time, 24
Exponential Time, 25
Linear Time, 24
Polynomial Time, 24
Time horizon, 349
Finite, 349
Infinite, 349
transportation problem, 297
tree networks, 165
trip, 400
direct trip, 400
tour trip, 400
TSP, 263
Uncapacitated, 246
uncertainty, 348
future condition, 348
limited knowledge, 348
Index
Variance of Distances, 11
Vertex and Intersection Point (VIP) Property,
212
VRP, 395, 398
CVRP, 398
DCVRP, 399
VRPB, 399
VRPBTW, 399
VRPPD, 399
VRPPDTW, 399
VRPTW, 399
549
warehouse, 295
shipment size, 296
warehouse layout, 447
warehouse location problem
fixed installation cost, 296
Limited capacity, 296
Multi period, 296
multiple product, 296
Singe Period, 296
single product, 296
unlimited capacity, 296
without fixed installation cost, 296