Finds and Results from the Swedish Cyprus Expedition: A Gender Perspective at the Medelhavsmuseet, 2004
The k-Centrum problem consists in finding a point that minimises the sum of the distances to the ... more The k-Centrum problem consists in finding a point that minimises the sum of the distances to the k farthest points out of a set of given points. It encloses as particular cases to two of the most known problems in Location Analysis: the center, also named as the minimum enclosing circle, and the median. In this paper the k-Centrum criteria is applied to obtaining a straight line-shaped facility. A reduced finite dominant set is determined and an algorithm with lower complexity than the previous one obtained.
This paper considers a bicriteria model to locate a semi-obnoxious facility within a convex polyg... more This paper considers a bicriteria model to locate a semi-obnoxious facility within a convex polygon, while employing Euclidean push and pull covering criteria. The partial covering context is introduced into an ordinary bicriteria location framework. Although both objectives are neither concave nor convex, low complexity polynomial algorithms to find the efficient solutions and the tradeoffs involved are developed with the help of higher-order Voronoi diagrams. Comparing the tradeoff for the full covering with the others enable decision makers to understand what to extent the maximin and minimax criteria are improved at the expense of uncovering. This is illustrated via numerical examples.
A facility has to be located within a given region taking two criteria of equity and efficiency i... more A facility has to be located within a given region taking two criteria of equity and efficiency into account. Equity is sought by minimizing the inequality in the inhabitant-facility distances, as measured by the sum of the absolute differences between all pairs of squared Euclidean distances from inhabitants to the facility. This measure meets the Pigou-Dalton condition of transfers, and can easily be minimized. Efficiency is measured through optimizing the sum of squared inhabitant-facility distances, either to be minimized or maximized for an attracting or repellent facility respectively. Geometric localization results are obtained for the whole set of Pareto optimal solutions for each of the two resulting bicriteria problems within a convex polygonal region. A polynomial procedure is developed to obtain the full bicriteria plot, both trade-off curves and the corresponding efficient sets.
Some time in the early seventeenth century, the following geometrical optimization problem was po... more Some time in the early seventeenth century, the following geometrical optimization problem was posed:
We improve and extend sufficient conditions for an optimal solution to happen at a fixed point in... more We improve and extend sufficient conditions for an optimal solution to happen at a fixed point in a single facility minisum location model with mixed transportation modes recently proposed and studied by Brimberg, Love and Mladenović. In particular, conditions are derived that are valid for general mixed metrics, while for mixed p -norms, possibly with rotated axes, much stronger conditions are obtained. An example demonstrates the superiority of the new conditions.
In this paper we prove that there always exists a firfite set that includes an optimal solution f... more In this paper we prove that there always exists a firfite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an e-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.
ABSTRACT We indicate a number of shortcomings in the second stage in the approach of Bell et al. ... more ABSTRACT We indicate a number of shortcomings in the second stage in the approach of Bell et al. [1] that may lead to unsatisfactory results. Some possible remedies are investigated.
A center hyperplane in the d-dimensional space minimizes the maximum of its distances from a fini... more A center hyperplane in the d-dimensional space minimizes the maximum of its distances from a finite set of points A with respect to possibly different gauges. In this note it is shown that a center hyperplane exists which is at (equal) maximum distance from at least d + 1 points of A. Moreover the projections of the points among these which lie above the center hyperplane cannot be separated by another hyperplane from the projections of those that are below it. When all gauges involved are smooth, all center hyperplanes satisfy these properties. This geometric property allows us to improve and generalize previously existing results, which were only known for the case in which all distances are measured using a common norm. The results also extend to the constrained case where for some points it is prespecified on which side of the hyperplane (above, below or on) they must lie. In this case the number of points lying on the hyperplane plus those at maximum distance is at least d + 1. It follows that solving such global optimization problems reduces to inspecting a finite set of candidate solutions. Extensions of these results to a separation problem are outlined.
In this work, we consider a classification problem where the objects to be classified are bags of... more In this work, we consider a classification problem where the objects to be classified are bags of instances which are vectors measuring d different attributes. The classification rule is defined in terms of a ball, whose center and radius are the parameters to be computed. Given a bag, it is assigned to the positive class if at least one element is strictly included inside the ball, and it is labelled as negative otherwise. We model this question as a margin optimization problem. Several necessary optimality conditions are derived leading to a polynomial algorithm in fixed dimension. A VNS type heuristic is proposed and experimentally tested.
In this work, a semi-obnoxious facility must be located in the euclidean plane to give service to... more In this work, a semi-obnoxious facility must be located in the euclidean plane to give service to a group of customers. Simultaneously, a set of populated areas, with shapes approximated via polygons, must be protected from the negative effects derived from that facility. The problem is formulated as a margin maximization model, following a strategy successfully used in Support Vector Machines. Necessary optimality conditions are studied and a finite dominating set of solutions is obtained, leading to a polynomial algorithm.
One recently proposed criterion to separate two data sets in Classification is to use a hyperplan... more One recently proposed criterion to separate two data sets in Classification is to use a hyperplane that minimizes the sum of distances to it from all the misclassified data points, where misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study an extension of this problem: we seek the hyperplane minimizing the sum of concave nondecreasing functions of the distances of misclassified points to it. It is shown that an optimal hyperplane exists containing at least d affinely independent points. This extends the result known for the minimization of the sum of distances, and enables to use combinatorial localsearch heuristics for this problem. As a corollary, the same result is obtained for the approximation problem in which a hyperplane minimizing the sum of concave nondecreasing functions of the distances from a set of data points is sought.
ABSTRACT A single facility has to be located in the plane in competition with fixed existing faci... more ABSTRACT A single facility has to be located in the plane in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, which fully patronise the facility to which it is most attracted. Attraction by a facility is expressed by some general attractiveness of the facility divided by a power of its Euclidean distance to demand. For existing facilities attractiveness is fixed, while the costs connected with the new facility are an increasing function of its attractiveness. Each demand point attracted by the new facility generates a given amount of income. The aim is to find that location for the new facility which maximizes the resulting profits. It is shown that this problem is well-posed under the additional assumption that consumers are novelty oriented, i.e. attraction ties are resolved in favor of the new facility. The problem then reduces to a parametric maxcovering problem with inflated Euclidean distances, which is solvable in polynomial time.
ABSTRACT The well-known majority theorem for Fermat-Weber location problems states that when all ... more ABSTRACT The well-known majority theorem for Fermat-Weber location problems states that when all distances are measured by a fixed pseudometric, then any destination with weight at least half of the total weight of all destination is an optimal site. We study the implications of such majority when both attracting (positive weight) and repelling (negative weight) destinations are present. When no constraints are present, and when majority holds at an attracting destination, the classical majority theorem is still valid, while when there is a repelling strict majority in an unbounded space, the objective is unbounded below. We then consider the constrained case where the location is restricted to lie within a given compact region. When majority is at an attracting destination then an optimal solution exists which is “first-reachable” from this destination, a generalization of visibility to general pseudometric spaces. When majority is at a repelling destination an optimal solution exists which is “last reachable” from this destination.
Page 1. Support Vector Regression for imprecise data ∗ Emilio Carrizosa, José Gordillo Universida... more Page 1. Support Vector Regression for imprecise data ∗ Emilio Carrizosa, José Gordillo Universidad de Sevilla (Spain) {ecarrizosa,jgordillo}@us.es Frank Plastria Vrije Universiteit Brussel (Belgium) Frank.Plastria@vub.ac.be 30th October 2007 Abstract ...
Huff location problems have been extensively analyzed within the field of competitive continuous ... more Huff location problems have been extensively analyzed within the field of competitive continuous location. In this work, two Huff location models on networks are addressed, by considering that users go directly to the facility or they visit the facility in their way to a destination. Since the problems are multimodal, a branch and bound algorithm is proposed, in which two different bounding strategies, based on Interval Analysis and DC optimization, are used and compared. Computational results are given for the two bounding procedures, showing that problems of rather realistic size can be solved in reasonable time.
ABSTRACT Facility location models in the literature usually consider the facility to be either pu... more ABSTRACT Facility location models in the literature usually consider the facility to be either purely attractive, and then seek the facility locations minimizing the overall transportation costs, or purely undesirable, and then seek the sites minimizing some social cost such as the environmental impact caused. When facilities, although necessary for the community, also have some negative impact on the population or the environment – pollution, noise, risk of accidents, etc. – these two contradicting aspects should be taken into account simultaneously. This leads to models which are more realistic, but are usually also much less tractable from a computational viewpoint. In this note we present a critical overview of the mathematical methods commonly used in this emerging field.
Finds and Results from the Swedish Cyprus Expedition: A Gender Perspective at the Medelhavsmuseet, 2004
The k-Centrum problem consists in finding a point that minimises the sum of the distances to the ... more The k-Centrum problem consists in finding a point that minimises the sum of the distances to the k farthest points out of a set of given points. It encloses as particular cases to two of the most known problems in Location Analysis: the center, also named as the minimum enclosing circle, and the median. In this paper the k-Centrum criteria is applied to obtaining a straight line-shaped facility. A reduced finite dominant set is determined and an algorithm with lower complexity than the previous one obtained.
This paper considers a bicriteria model to locate a semi-obnoxious facility within a convex polyg... more This paper considers a bicriteria model to locate a semi-obnoxious facility within a convex polygon, while employing Euclidean push and pull covering criteria. The partial covering context is introduced into an ordinary bicriteria location framework. Although both objectives are neither concave nor convex, low complexity polynomial algorithms to find the efficient solutions and the tradeoffs involved are developed with the help of higher-order Voronoi diagrams. Comparing the tradeoff for the full covering with the others enable decision makers to understand what to extent the maximin and minimax criteria are improved at the expense of uncovering. This is illustrated via numerical examples.
A facility has to be located within a given region taking two criteria of equity and efficiency i... more A facility has to be located within a given region taking two criteria of equity and efficiency into account. Equity is sought by minimizing the inequality in the inhabitant-facility distances, as measured by the sum of the absolute differences between all pairs of squared Euclidean distances from inhabitants to the facility. This measure meets the Pigou-Dalton condition of transfers, and can easily be minimized. Efficiency is measured through optimizing the sum of squared inhabitant-facility distances, either to be minimized or maximized for an attracting or repellent facility respectively. Geometric localization results are obtained for the whole set of Pareto optimal solutions for each of the two resulting bicriteria problems within a convex polygonal region. A polynomial procedure is developed to obtain the full bicriteria plot, both trade-off curves and the corresponding efficient sets.
Some time in the early seventeenth century, the following geometrical optimization problem was po... more Some time in the early seventeenth century, the following geometrical optimization problem was posed:
We improve and extend sufficient conditions for an optimal solution to happen at a fixed point in... more We improve and extend sufficient conditions for an optimal solution to happen at a fixed point in a single facility minisum location model with mixed transportation modes recently proposed and studied by Brimberg, Love and Mladenović. In particular, conditions are derived that are valid for general mixed metrics, while for mixed p -norms, possibly with rotated axes, much stronger conditions are obtained. An example demonstrates the superiority of the new conditions.
In this paper we prove that there always exists a firfite set that includes an optimal solution f... more In this paper we prove that there always exists a firfite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an e-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.
ABSTRACT We indicate a number of shortcomings in the second stage in the approach of Bell et al. ... more ABSTRACT We indicate a number of shortcomings in the second stage in the approach of Bell et al. [1] that may lead to unsatisfactory results. Some possible remedies are investigated.
A center hyperplane in the d-dimensional space minimizes the maximum of its distances from a fini... more A center hyperplane in the d-dimensional space minimizes the maximum of its distances from a finite set of points A with respect to possibly different gauges. In this note it is shown that a center hyperplane exists which is at (equal) maximum distance from at least d + 1 points of A. Moreover the projections of the points among these which lie above the center hyperplane cannot be separated by another hyperplane from the projections of those that are below it. When all gauges involved are smooth, all center hyperplanes satisfy these properties. This geometric property allows us to improve and generalize previously existing results, which were only known for the case in which all distances are measured using a common norm. The results also extend to the constrained case where for some points it is prespecified on which side of the hyperplane (above, below or on) they must lie. In this case the number of points lying on the hyperplane plus those at maximum distance is at least d + 1. It follows that solving such global optimization problems reduces to inspecting a finite set of candidate solutions. Extensions of these results to a separation problem are outlined.
In this work, we consider a classification problem where the objects to be classified are bags of... more In this work, we consider a classification problem where the objects to be classified are bags of instances which are vectors measuring d different attributes. The classification rule is defined in terms of a ball, whose center and radius are the parameters to be computed. Given a bag, it is assigned to the positive class if at least one element is strictly included inside the ball, and it is labelled as negative otherwise. We model this question as a margin optimization problem. Several necessary optimality conditions are derived leading to a polynomial algorithm in fixed dimension. A VNS type heuristic is proposed and experimentally tested.
In this work, a semi-obnoxious facility must be located in the euclidean plane to give service to... more In this work, a semi-obnoxious facility must be located in the euclidean plane to give service to a group of customers. Simultaneously, a set of populated areas, with shapes approximated via polygons, must be protected from the negative effects derived from that facility. The problem is formulated as a margin maximization model, following a strategy successfully used in Support Vector Machines. Necessary optimality conditions are studied and a finite dominating set of solutions is obtained, leading to a polynomial algorithm.
One recently proposed criterion to separate two data sets in Classification is to use a hyperplan... more One recently proposed criterion to separate two data sets in Classification is to use a hyperplane that minimizes the sum of distances to it from all the misclassified data points, where misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study an extension of this problem: we seek the hyperplane minimizing the sum of concave nondecreasing functions of the distances of misclassified points to it. It is shown that an optimal hyperplane exists containing at least d affinely independent points. This extends the result known for the minimization of the sum of distances, and enables to use combinatorial localsearch heuristics for this problem. As a corollary, the same result is obtained for the approximation problem in which a hyperplane minimizing the sum of concave nondecreasing functions of the distances from a set of data points is sought.
ABSTRACT A single facility has to be located in the plane in competition with fixed existing faci... more ABSTRACT A single facility has to be located in the plane in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, which fully patronise the facility to which it is most attracted. Attraction by a facility is expressed by some general attractiveness of the facility divided by a power of its Euclidean distance to demand. For existing facilities attractiveness is fixed, while the costs connected with the new facility are an increasing function of its attractiveness. Each demand point attracted by the new facility generates a given amount of income. The aim is to find that location for the new facility which maximizes the resulting profits. It is shown that this problem is well-posed under the additional assumption that consumers are novelty oriented, i.e. attraction ties are resolved in favor of the new facility. The problem then reduces to a parametric maxcovering problem with inflated Euclidean distances, which is solvable in polynomial time.
ABSTRACT The well-known majority theorem for Fermat-Weber location problems states that when all ... more ABSTRACT The well-known majority theorem for Fermat-Weber location problems states that when all distances are measured by a fixed pseudometric, then any destination with weight at least half of the total weight of all destination is an optimal site. We study the implications of such majority when both attracting (positive weight) and repelling (negative weight) destinations are present. When no constraints are present, and when majority holds at an attracting destination, the classical majority theorem is still valid, while when there is a repelling strict majority in an unbounded space, the objective is unbounded below. We then consider the constrained case where the location is restricted to lie within a given compact region. When majority is at an attracting destination then an optimal solution exists which is “first-reachable” from this destination, a generalization of visibility to general pseudometric spaces. When majority is at a repelling destination an optimal solution exists which is “last reachable” from this destination.
Page 1. Support Vector Regression for imprecise data ∗ Emilio Carrizosa, José Gordillo Universida... more Page 1. Support Vector Regression for imprecise data ∗ Emilio Carrizosa, José Gordillo Universidad de Sevilla (Spain) {ecarrizosa,jgordillo}@us.es Frank Plastria Vrije Universiteit Brussel (Belgium) Frank.Plastria@vub.ac.be 30th October 2007 Abstract ...
Huff location problems have been extensively analyzed within the field of competitive continuous ... more Huff location problems have been extensively analyzed within the field of competitive continuous location. In this work, two Huff location models on networks are addressed, by considering that users go directly to the facility or they visit the facility in their way to a destination. Since the problems are multimodal, a branch and bound algorithm is proposed, in which two different bounding strategies, based on Interval Analysis and DC optimization, are used and compared. Computational results are given for the two bounding procedures, showing that problems of rather realistic size can be solved in reasonable time.
ABSTRACT Facility location models in the literature usually consider the facility to be either pu... more ABSTRACT Facility location models in the literature usually consider the facility to be either purely attractive, and then seek the facility locations minimizing the overall transportation costs, or purely undesirable, and then seek the sites minimizing some social cost such as the environmental impact caused. When facilities, although necessary for the community, also have some negative impact on the population or the environment – pollution, noise, risk of accidents, etc. – these two contradicting aspects should be taken into account simultaneously. This leads to models which are more realistic, but are usually also much less tractable from a computational viewpoint. In this note we present a critical overview of the mathematical methods commonly used in this emerging field.
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