Fuzzy trees in decision support systems

European Journal of Operational Research 174 (2006) 293–310 www.elsevier.com/locate/ejor Decision Support Fuzzy trees in decision support systems Tomaž Savšek b a,* , Marjan Vezjak b, Nikola Pavešić b a Ministry of Defence, Kardeljeva pl. 25, 1000 Ljubljana, Slovenia Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia Received 7 May 2001; accepted 25 February 2005 Available online 6 June 2005 Abstract This paper is based on the following assumption: that there exists a fuzzy tree structure and a distance between fuzzy trees which provides the basis for fuzzy decision-making. The paper provides the following: (1) a new definition of the fuzzy relational tree structure, (2) the development of a new comparative method for fuzzy trees and its experimental testing and evaluation, (3) a new descriptive method of military structures in a fuzzy tree format and the development of a fuzzy decision support system.  2005 Elsevier B.V. All rights reserved. Keywords: Fuzzy sets; Graph theory; Decision support systems 1. Introduction 1.1. Decision-making in complex and ill-defined systems Complex systems and ill-defined information are one of the most influential factors in the decision-making process. When making a decision, one has to face the uncertainty of future events and the uncertainties which accompany the transmission, transfer and reception of information. Classical approaches to system modelling can hardly cope with complex systems and systems involving uncertainty for it is extremely difficult to find a global function or analytical structure for a non-linear system. Fuzzy set theory provides a mathematical frame for system modelling when knowledge and information about a system are incomplete and experience-based rather than systematic. Such problems are often encountered when applied to control, pattern recognition or MCDM (Andriole, 1990; Carlsson, 1994). * Corresponding author. Tel.: +386 1 471 1999; fax: +386 1 471 2356. E-mail address: tomaz.savsek@mors.si (T. Savšek). 0377-2217/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.068 294 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 Therefore, it is obvious that in real life many processes are closer to fuzzy processes than mathematically precisely defined procedures. One often gets the impression that, in real life situations, problems are most efficiently solved using the human factor because machines (computers) have proved too complicated and impractical. As a consequence, there is an increasing need to improve ‘‘machine’’ capacities so that we can solve fuzzy decision-making problems. These can be defined as making decisions in a fuzzy environment. Many methods that attempt to answer the problem of decision-making have been developed. Most of them have so far been based mainly on probability calculus, statistics, mathematical linear and non-linear programming and similar classical mathematical tools. However, the increasing complexity of systems on the one hand, and a lack of information on the other hand, demand new approaches and methods for the decision-making process. Modern approaches to decision-making are based on structural descriptions of objects and the use of expert systems. One of them, fuzzy logic, has the advantage of a better overview of ill-defined systems compared to classical data-probability approaches. In accordance with such principles, we suggest a new method of solving fuzzy decision-making problems. Data presentation in the shape of trees enables us to save structural information in the form of structures with hierarchical relations between basic elements. Such models have proved very efficient as support for decisions. Due to their mathematical characteristics they can be easily manipulated using efficient algorithms. In this paper we present a decision support system that is based on the comparison of fuzzy trees. As an example of a decision-making process we introduce a fuzzy system that supports decision-making for the commander of a combat unit. 1.2. Fuzzy relational structures Graph theory plays an important role in the modelling of complex structures. There are three definitions of graph fuzzification: fuzzification of nodes, fuzzification of relations and simultaneous fuzzification of nodes and relations. Fuzzy trees, which are a special case of fuzzy graphs, are helpful for representing soft or ill-defined structures, especially in decision analysis and in decision support systems. When the knowledge about the system is incomplete, or when the system is complex and data is lacking, or when we deal with information that is either not correct or even intentionally false, the principles of fuzzy set theory can be applied. The theory of fuzzy graphs and trees has important links to the theory of fuzzy classification and decision analysis. The distance between two trees is defined in terms of the least number of substitutions, insertions, and deletions of nodes required to obtain one from the other. In this paper we propose the language transformation of fuzzy trees, namely, substitution, deletion and insertion of fuzzy nodes. With the increasing popularity of using hierarchical data structure and tree construction schemes, we have to face the problem of comparing two tree-like data structures. The problem may arise in several ways: from comparing the tree structure of an input with that of the templates for classification and decision support, or from sorting the data according to the similarity between tree structures for fast retrieval. These situations initiated the study of distance measures for trees. 2. Fundamentals of fuzzy set theory 2.1. Some conceptual foundations In the last few years, fuzzy set theory has become of practical value in fields which were not long ago considered to be the domain of classical methods. As a consequence, fuzzy sets are being increasingly applied to decision-making, management, planning, anticipation and optimisation. Information itself, which T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 295 is gaining in importance in all of the fields mentioned above, is a good with market value. Information is always associated with a certain degree of fuzziness and uncertainty, for it can easily be transformed into incomplete, inexact or tardy information or disinformation. 2.2. Approximation of human reasoning Human reasoning is a much more complex mental process than classical logical reasoning. People are capable of comprehending information despite the fact that it frequently contains incomplete definitions, unclear conceptions or inexact data and facts. Given this assumption, the reasoning process cannot be explained simply in terms of probability models (Carlsson, 1994). Classical logical reasoning leaves no room for doubt as only two alternatives are available: yes/no, right/ wrong, black/white, true/false. Each alternative can be ascribed probability, but this still leaves us with two alternatives. Real life, however, indicates that decisions do not involve only two possibilities. We sometimes use the word ‘‘maybe’’ because this is not a black and white world. It is also interwoven with grey nuances. This leads us to the conclusion that by using fuzzy logic and fuzzy set theory one can draw conclusions which are a good approximation of human reasoning in decision support systems. 2.3. Definition of a fuzzy set Let X be a classical set of objects, called the universe or universal set, whose generic elements are denoted x. Membership in a classical subset A of X is often defined as the characteristic function lA, which transforms elements from X to {0, 1} such that (Dubois and Prade, 1980):  1; iff x 2 A; lA ðxÞ ¼ 0; iff x 62 A. If the set {0, 1} is expanded to the interval of real numbers [0, 1], then A is called a fuzzy set. lA stands for the degree of membership of the element x to a fuzzy set A. The closer the value of lA to 1, the more x belongs to A. In other words, A is a subset of X without a sharp boundary. Fuzzy set A is completely characterized by the pair: A ¼ fx; lA ðxÞ; x 2 X g. In 1972 Zadeh proposed a different notation of a fuzzy set (Zadeh, 1972). When X is a finite set {x1, x2, . . . , xn}, a fuzzy set A on X is expressed as n X lA ðxi Þ=xi . A ¼ lA ðx1 Þ=x1 þ    þ lA ðxn Þ=xn ¼ i¼1 When X is not finite, a fuzzy set A is expressed as Z A¼ lA ðxÞ=x. X 2.4. Membership function lA(x) The main difference between classical and fuzzy sets is the classification of an element into set membership. According to classical or non-fuzzy classification, the degree of membership of an element to a set equals zero (0) if the element does not belong to a set and equals one (1) if the element belongs to a set. In the case of fuzzy sets, element membership becomes a real value, lying in a closed interval between 0 296 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 and 1. Non-fuzzy sets thus describe a membership set with the elements 0 and 1, whereas fuzzy sets describe a membership interval [0, 1]. A membership function lA(x) represents the degree of membership of the element x in a fuzzy set A. Degrees of membership represent the arrangement of objects in the elementary set X. The value of the degree of membership lA(x) can also be interpreted as: • degree of compatibility, i.e. set contents predicate the relation between set A and element x, • degree of possibility that element x corresponds to the value of the parameter which is fuzzily defined by A. 3. Relational structures In order to find a general structural mathematical model, researchers from different science branches have studied relational structures. They have tried to determine rules and definitions that would describe the elementary elements in a set as well as mutual relations within the set. Developments in theoretical and mathematical methods led to an increasing number of applied solutions in different branches of science, such as pattern recognition (Miclet, 1986; Niemann, 1990; Vezjak, 1991), system theory, computer science (Alavi et al., 1985) and operational research (Ball et al., 1995; Gass and Harris, 1996). The advantages and applicability of relational structures are shown primarily in the processing of complex systems and data. Descriptions using numerical methods are no longer sufficient and there is a need for a structural approach. In this approach, systems or data are divided into basic components and connections between these basic components are described. Relational structures are made up of these components and connections between these components provide their structure. With regard to the types of components used and ways of connecting them, there are three basic types of structural record: (a) A string or chain of signs is the simplest form of arrangement in which a pattern is composed of chaining components. A typical representative of the chain is FreemanÕs code (Duda and Hart, 1972; Niemann, 1990). (b) A tree is a more complex arrangement in which a pattern is composed by nesting one part of the pattern into another (Miclet, 1986). (c) A graph is the most complex arrangement where no constraints whatsoever are applied to the relations between elementary parts of the pattern (Miclet, 1986). 3.1. Comparison of tree structures The increasing popularity of the use of hierarchically organized structures and tree-form schemata leads to the problem of data comparison when it takes a tree structure form. This problem can be tackled from various points of view ranging from the comparison of tree structures for classification and organizational requirements to the problem of arranging organized data in order to meet the need for speedier searching and reviewing. If there is a need to use tree structures, as already described in the previous section, a metric which allows the comparison of two trees should be defined (Fu, 1974). There are two basic approaches to comparing tree structures: (a) The first approach is based on the use of the decomposition of complex structures into simple ones. In tree structures, trees are divided into simpler subtrees applied to an adequate metric (Boorman and Oliver, 1973). T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 297 (b) The second approach is based on the idea of so-called language transformations. Comparison of tree structures is based on the distance between the trees or an overall estimation of the transformation of one tree into another (Wagner and Fisher, 1974). In the past, the second approach has proved to be more appropriate, especially in the areas of pattern recognition and support to decision-making (Batagelj, 1989). 3.2. Levenshtein’s metric LevenshteinÕs metric is based on a syntax analysis, which involves the correction of mistakes. This syntax analysis is derived from the conclusion that all possible mistakes in a certain structured object are simulated with three elementary transformations. These transformations are: deletion, insertion and substitution of elements. The distance between two structured objects X and Y is thus defined as the smallest number of elementary transformations needed to convert object X into object Y. LevenshteinÕs distance d is defined as: dðx; yÞ ¼ minfðI i þ Di þ S i Þg; i i ¼ 1; 2; . . . ; N ; where Ii is the number of inserted elements, Di is the number of deleted elements, Si is the number of substituted elements and N is the number of possible conversions of object X into object Y. Through the introduction of special weights for each individual transformation, the so-called weighted LevenshteinÕs distance can be obtained. LevenshteinÕs distance can be obtained by adjusting cost to individual base transformations, which also imply procedure optimisation. The overall cost of converting one object into another is the sum of the individual transformation costs. The calculation of this sum and distance is based on dynamic programming, i.e. the successive minimization of partial sums. The basic methods for calculating LevenshteinÕs distance are as follows: • • • • • Fisher–WagnerÕs algorithm (Wagner and Fisher, 1974), MooreÕs algorithm (Moore, 1979), Dynamic Warping (Rabiner, 1978), SelkowÕs algorithm (Selkow, 1977), LuÕs algorithm (Lu, 1979). 3.3. Lu’s algorithm Lu (Shin-Yee Lu) calculated the weighted distance between two trees by using the idea of language transformations. Lu defined a transformation between trees such that, for any given trees, it is possible to derive one from the other by repeatedly applying transformations. Assume that each transformation used has some cost associated with it. The distance between two trees is defined to be the minimum-cost sequence required to derive one from the other (Lu, 1979). Definition 1 (Transformation). Let T1 and T2 be two trees. Define a transformation Q : T1 ! T2. Q consists of the following three basic types of operations: • deletion, D(b) ! k, p; • insertion, I(k) ! a, q; • substitution, S(b) ! a, r; where b 2 T1, a 2 T2, k is a null symbol, and p, q, and r are costs associated with D, I and S, respectively. 298 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 a a T1 = a d a a d D(b) b a I(d) d a d d d a S(d,c) a a a c d I(c) a a = T2 c d c Fig. 1. Transformation of one tree into the other. Definition 2 (Distance). The distance between two trees T1 and T2, denoted d(T1, T2) is the minimum cost necessary to derive T1 from T2 using a series of transformations Q, that satisfies the following criteria: (a) the predecessor-descendant relation does not change, (b) nodes in T2 do not split or merge, (c) the sequence of post-fix ordering does not change after applying the series of transformations. Assume that the costs associated with D and I—transformations are 1, and the cost associated with S— transformation is 1 if the transformation causes a substitution, and 0 otherwise. Then the distance becomes the minimum number of deletions, insertions, and substitutions necessary to obtain one from the other. Example 1. Fig. 1 represents trees T1 and T2. T1 is transformed into T2 by using four basic operations. If costs p, r and q of basic operations are p = r = q = 1, then the distance between trees is d(T1, T2) = 4. 4. Fuzzy graph and fuzzy tree 4.1. Fuzzy graph The definition of a classic graph G is based upon a collection of elements known as nodes X = {x}, and upon the family U = {u} of the Cartesian product elements X · X, which are known as connections, margins or arcs. A connections set can be alternatively defined as a binary relation l : X · X ! {0, 1} where l(x, y) = 1 means that there is a direct connection, or arc, leading from node x to node y, whereas l(x, y) = 0 denotes the lack of such a connection. Numerous applications have shown that the separation into the complete presence or the complete absence of connection between two elements is not necessarily always the most appropriate (Yeh and Bang, 1975). This has led to much research. In clustering analysis, similarity measures have been used for the grouping of elements (nodes) based upon the similarity level for each pair of elements. These measures are usually normalized or based on fuzzy relation, so the similarity level l(x, y), of element x to element y, is such that 0 6 l(x, y) 6 1. In clustering analysis, the common characteristic of similarity measures is that l is a symmetric, namely l(x, y) = l(y, x) (Halpern, 1975). Whenever the relations between the elements or nodes of a graph do not exist or are not completely absent, one may describe them as fuzzy relations. Using fuzzy set theory, fuzzy graphs can be defined. They are characterized by the fact that the similarity level between any arbitrary pair of nodes can assume any value existing within a closed unit interval [0, 1]. In literature there are different definitions of fuzzy graphs G*. Some authors (Yeh and Bang, 1975; Halpern, 1975) defined a fuzzy graph as a pair G*(X, U*) where X is a set of non-fuzzy nodes and U* is a set of T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 299 fuzzy relations. Such a graph can be named a fuzzy relational graph. Rosenfeld (Rosenfeld, 1975) defined a fuzzy graph as a pair G*(X*, U*) where X* is a set of fuzzy nodes and U* is a set of fuzzy relations. Such a graph can be named a fully fuzzy graph. A fuzzy graph G* can also be defined as a pair G*(X*, U) where X* is a set of fuzzy nodes and U is a set of non-fuzzy relations. We can use such a graph whenever the relations between elements are precisely defined, but the elements are defined as fuzzy nodes (Savšek, 1996). This type of fuzzy graph is also used in our proposed fuzzy system. Definition 3 (Fuzzy node graph). A fuzzy node graph G* is defined as a pair (X*, U), where X* is a fuzzily defined node set and U a relation set on the Cartesian product X* · X*, such that: X  ¼ fx g; x ¼ fal ; lðal Þg; U ¼ flðx ; y  Þg; 8al 2 L; l ¼ 1; . . . ; R; where lðal Þ : L ! ½0; 1 8al 2 L; lðx ; y  Þ : X   X  ! f0; 1g 8ðx ; y  Þ 2 X   X  ;  1; if x and y  are connected;   lðx ; y Þ ¼ 0; otherwise. Each node is defined as a fuzzy set over a set of labels of nodes L with R elements. Relations are defined as crisp (non-fuzzy) relations between nodes. Example 2. Fig. 2 represents a graph G* with fuzzy nodes and classical (non-fuzzy) relation G*(X*, U), where X  ¼ fx1 ; x2 ; x3 ; x4 g and U ¼ fðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx3 ; x1 Þ; ðx2 ; x4 Þ; ðx4 ; x3 Þg. Nodes of a graph are described by fuzzy sets with elements of a set of labels of nodes L = {a, b, c}, such that x1 ¼ fða; lx1 ðaÞÞ; ðb; lx1 ðbÞÞ; ðc; lx1 ðcÞÞg; x2 ¼ fða; lx2 ðaÞÞ; ðb; lx2 ðbÞÞ; ðc; lx2 ðcÞÞg; x3 ¼ fða; lx3 ðaÞÞ; ðb; lx3 ðbÞÞ; ðc; lx3 ðcÞÞg; x4 ¼ fða; lx4 ðaÞÞ; ðb; lx4 ðbÞÞ; ðc; lx4 ðcÞÞg. 4.2. Fuzzy tree A graph without cycles is called an acyclical graph or forest. A linked forest on which all nodes are linked is called a tree. A fuzzy graph can be defined as a forest if a graph, which contains all its arcs, takes the x*2 x*1 x*4 x*3 Fig. 2. Fuzzy node graph. 300 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 shape of a forest. An undirected fuzzy graph can be defined as a tree if a graph, which contains all its arcs, takes the shape of a linked forest. 5. Comparison of fuzzy trees In previous sections, the fundamentals of the fuzzy set theory, classical graphs and relational trees were introduced. Fuzzy graphs and fuzzy trees were defined and some comparative methods for classical relational trees listed. This knowledge is needed in order to represent the proposed system for the comparison of relational trees in a fuzzy environment. A system was developed which generates the distance between relational trees. Fig. 3 shows a model of a fuzzy system containing: 1. 2. 3. 4. Fuzzification of trees, FLU algorithm, Fuzzification of distances, Rules. The input to the system represents two fuzzy relational trees. A fuzzification pre-processor transforms non-fuzzy trees in fuzzy node trees. The output of the system is non-fuzzy value representing the distance between two trees. If a fuzzy output value is required, a fuzzification post-processor is used. 5.1. Fuzzification of trees Fuzzification of input variables is often subjectively coloured, i.e. rules obtained through experience and knowledge play an important role. Fuzzification of sharp input values is done with the help of membership functions. Fuzzification of trees is a process by which sharp input (classical) relational tree structure T is transformed into a fuzzy tree T* : T ! T*. The process of fuzzification is done in three steps: 1. Data input: Both non-fuzzy trees are inserted, T1(X1, U1) and T2(X2, U2). 2. Fuzzification of nodes: Each node is replaced with a fuzzy node, so that a set of nodes X is replaced by a fuzzy set of nodes X* : X ! X*. An appropriate membership function lX  is used in this process. Fuzzification is done for both trees T1 in T2. The result of the fuzzification is fuzzy node trees T 1 ðX 1 ; U 1 Þ and T 2 ðX 2 ; U 2 Þ. Fuzzy system Rules "Fuzzy" output "Non-fuzzy" input T1,T2 Fuzzification pre-processor Fuzzification post-processor d* (T1,T2 ) "Non-fuzzy" output FLU Fig. 3. System of fuzzy tree comparison. d (T1,T2 ) T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 T1 = T1 (x 1,u1 ) T2 = T2 (x 2 ,u 2) Fuzzification of nodes Fuzzification of nodes 301 T2 (x 2*,u2 ) T1 (x1* ,u1) Structural comparison T1*= T1 (x1* ,u1*) T2 *= T2 (x 2*,u*2 ) Fig. 4. Fuzzification and structural comparison of trees. 3. Structural comparison of trees: Input trees are compared with each other regarding structure. If a node is lacking in a tree with respect to the other tree, then a so called fuzzy empty set marked as $ is added to the set: U* : U ! U*. The result of the structural comparison is two trees with the same structure: T 1 ðX 1 ; U 1 Þ and T 2 ðX 2 ; U 2 Þ, where U 1 ¼ U 2 . Fig. 4 illustrates the fuzzification pre-processor of the fuzzy system with the fuzzification and structural comparison. 5.2. FLU algorithm The core of the method of calculating distances between fuzzy trees was named after LuÕs basic procedure for calculating distances between (classical) trees: Fuzzy LuÕs algorithm or FLU. FLU generates distances between two fuzzy tree structures. Section 3.3 describes the method and principle of her classical procedure. The main difference in idea and quality between the classical and suggested fuzzy procedures is in understanding and handling the costs of basic transformations. Transformation of a node xk of fuzzy tree T 1 into a node yk of another fuzzy tree T 2 is described by the membership functions. Transformation is done in two steps: 1. Comparison of a node xk with all possible labels of nodes and calculation of appropriate membership functions l(xk, al) and 2. Comparison of node yk with all possible labels of nodes and calculation of appropriate membership functions l(yk, al), where • L* = {L, $} = {a1, . . . ,al, . . . ,aR, $} is a set of fuzzy defined labels of nodes including an additional fuzzy empty element i.e. fuzzy empty node $; • xk, a fuzzy node of a fuzzy tree T 1 , where "x 2 L*; • yk, a fuzzy node of a fuzzy tree T 2 , where "y 2 L* and • k = 1,2, . . . ,K, where K is the number of nodes of T 1 and T 2 . 302 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 Membership functions of node transformation are written in a form of a membership matrix of relations: 1 0 lx1 =a1 . . . lx1 =al . . . lx1 =aR lx1 =a$ B . .. C .. .. .. B .. . C . . q . C B C B C B l =a    l =a    l =a l =a lðx; aÞ ¼ B xk 1 l R $ C; xk xk xk B . .. C .. .. .. C B . @ . . A . . q . lxK =a1  lxK =al  lxK =aR lxK =a$ where an element lxk =al represents a value of a membership function l(xk, al) of a node xk at the label al. Labels of fuzzy nodes are defined only on a set of fuzzy labels of nodes L*, therefore we define a set of elementary membership functions which can be represented as a membership matrix of elementary relations: 1 0 la1 =a1 . . . la1 =al . . . la1 =aR la1 =a$ B . .. C .. .. .. C B . B . . C . q . . C B B l =a    l =a    l =a lal =a$ C l R C B al 1 al al lL ðx; aÞ ¼ B C. B .. .. C .. .. .. B . . C . . q . C B C B @ laR =a1    laR =al    laR =aR laR =a$ A l$ =a1    l$ =al    l$ =aR l$ =a$ The classical procedure deals with constant costs of basic operations (normally p = q = r = 1). In FLU algorithms the cost of an individual operation is computed using the membership functions of nodes. The following definitions apply for the costs of basic operations: Definition 4 (Deletion cost of a node). Deletion cost pk of a node xk is defined as a scalar power, that is the cardinality of fuzzy defined labels of nodes set L* over node xk: pk ¼ jxk j ¼ Rþ1 X lðxk ; al Þ; l¼1 where l(x, a) is a membership matrix of relations of T 1 . Essentially, this is the cost of deleting, removing or destroying a certain node. Definition 5 (Insertion cost of a node). Insertion cost qk of a node xk is defined as the scalar power or the cardinality of fuzzy defined labels of nodes set L* over the node yk: qk ¼ jy k j ¼ Rþ1 X lðy k ; al Þ; l¼1 where l(y, a) is a membership matrix of relations T 2 . Essentially, these are the costs for forming, inserting or building a certain node. Definition 6 (Substitution cost of a node). The substitution cost rk of the node xk by the node yk is defined as a scalar power, that is the cardinality of the difference between the set L* over the node xk and the set L* over the node yk: T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 rk ¼ jxk  y k j ¼ Rþ1 X 303 jlðxk ; al Þ  lðy k ; al Þj; l¼1 where l(x, a) and l(y, a) are membership matrices of relations of T 1 and T 2 , respectively. The costs of substitution are essentially the price of substituting or replacing one node with another node. Since the fuzzification phase fuzzy empty nodes were introduced, so only the substitution of nodes is used as a basic operation in the FLU algorithm. Deletion and insertion of nodes are only special forms of node substitution. In the case of node deletion, a node is replaced by a fuzzy empty node whereas in the case of a node insertion, a fuzzy empty node is replaced by an appropriate non-empty node. 5.3. Fuzzification of distances The result of the FLU algorithm is a non-fuzzy distance between two trees. As previously described, this distance is computed in terms of the minimum cost of transformation of one tree into another. FLU defines the costs of basic substitutions as a scalar power. The final cost, representing the distance between two trees, is also a scalar power or a sharp (non-fuzzy) value. If a fuzzy output is required, an additional fuzzification post-processor can be used. According to the rules, the fuzzification post-processor transforms the non-fuzzy value into an output fuzzy set. 5.4. Rules Rules play a triple role in the system of fuzzy tree comparison: (a) they define the method of tree fuzzification, (b) they define fuzzy relations with node substitution in the FLU algorithm, (c) they define the method of fuzzification of a distance. Rules are normally formed by experts in a certain field and depend on a set problem. Rule formation can be roughly divided into two methods: • Manual method: on the basis of know-how, interviews and experience, researchers form rules which are manually inserted into a fuzzy system. • Automatic method: with the help of automatic learning systems and accumulation of know-how based on expert bases and systems, rules can be formed which are automatically transferred into a fuzzy system. Rules are expressed in the form of membership functions and are written in the form of a matrix lL ðx; aÞ. 6. Decision-making in combat situations 6.1. Combat decision support A military organization is a typical hierarchical structure. It is relatively complex and involves many attributes. In addition, data on units are often incomplete and force information is often either incorrect or even false. A great military theorist, Prussian general Carl von Clausewitz (1780–1831), wrote (On War): ‘‘War is the province of uncertainty: three-quarters of those things upon which action in war must be calculated are hidden more or less in the clouds of great uncertainty.’’ 304 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 This environment does not in practice allow for the use of the classical logic method. Therefore the application and comparison of fuzzy trees based on the distance between them will be illustrated by using the example of a military structure. The following will have to be taken into consideration: (a) a military organization is a relatively complex and non-homogeneous organization, (b) a certain degree of uncertainty often has to be accounted for, especially when the description of opposing forces is also involved. In such cases, the description of a military organization takes the form of a fuzzy relational structure, that is a fuzzy tree T*(X, U). Each node of a tree x 2 X represents the status of a certain subunit. The status of a subunit can be described by using numerical values (manpower, weapons, ammunition, food, water, . . .). In peacetime it is useful and necessary to use precise values. Such procedures are, however, meaningless in actual war, given all the sources of uncertainty and decision-making complexity listed previously. Therefore, the status of momentarily usable and available systems can be characterized by the following linguistic units (weapons, ammunition, men and morale of the unit): • • • • • e (excellent): the number of active systems is about 100% (Be = 1.0), g (good): the number of active systems is about 90% (Bg = 0.9), a (acceptable): the number of active systems is about 70% (Ba = 0.7), p (poor): the number of active systems is about 50% (Bp = 0.5), $ (missing): the number of active systems is about 0% (B$ = 0.0). Values of the possible status of a unit represent a set of fuzzy labels of nodes L* = {e, g, a, b, $}. A fuzzy tree T* can be formed on this basis, defined over a fuzzy set X with a fuzzy labels of nodes set L* and characterized by a membership function l(x, a). Sinusoid membership functions have proved to be the most suitable for this purpose (Savšek and Vezjak, 1996; Savšek, 1996): lðxk ; al Þ ¼ 1 1 cos pðxk  al Þ þ 2 2 for k ¼ 1; . . . ; K and l ¼ 1; . . . ; R þ 1; ð1Þ where K is a number of nodes in a fuzzy tree. Membership functions defined by (1) and shown at Fig. 5 can also be presented by the matrix of elementary relations: 0 1 1.00=e 0.98=g 0.79=a 0.50=b 0.00=$ B g B 0.98=e 1.00=g B lL ðx; aÞ ¼ a B B 0.79=e 0.90=g B b @ 0.50=e 0.65=g 0.90=a 1.00=a 0.65=b 0.90=b 0.90=a 1.00=b 0.21=a 0.50=b C 0.02=$ C C 0.21=$ C C. C 0.50=$ A 1.00=$ e $ 0.00=e 0.02=g 6.2. Combat decision aid system based on fuzzy tree comparison An important achievement in the development of the automation of command and control functions is the implementation of a system which collects and merges information from diverse sources into an appropriate representation of the tactical situation on the battlefield. Such a system is often also called a battle management system or a combat decision aid system, requiring the following two functions (Waltz and Buede, 1987): T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 305 1 0.8 EXCELLENT 0.6 GOOD ACCEPTABLE 0.4 POOR MISSING 0.2 0 0 B$ 20 40 Bp 60 Ba 80 Bg 100 Be Fig. 5. Diagram of membership functions. Fig. 6. Task group.1 (a) data fusion: collects information from a variety of sources to develop a possible overview of the military situation on the battlefield, (b) decision support: carries out the creation of possible actions and a quantitative evaluation of options on the basis of which a commander selects an appropriate option. One of the most important elements in the decision-making process is the estimation of the situation on the battlefield. For the purposes of our paper we have estimated unit capacity. We simulated an armed struggle, in which a task group was involved (Fig. 6). The task group is represented by means of a fuzzy tree T*. Leaves of a tree (nodes without successors in the tree) represent basic units (e.g. squads and platoons). The rest of the nodes, which are not leaves of the tree, represent superior units (e.g. company, battalion) composed of basic subunits. A leaf of the tree is represented as a pair Ni(Ci, KLi), where • Ci is the manning, described by a fuzzy set C* = {e, g, a, b, $} and an appropriate membership function lCi :   lei lgi lai lpi l$i lCi ¼ . e g a p $ 1 Task group is presented by NATO standard tactical symbols. 306 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 • KLi is the LanchesterÕs efficiency coefficient of unit I (Amacher and Mandallaz, 1986; Lanchester, 1914; Savšek et al., 1994; Savšek and Vezjak, 1995; Stahel, 1975; Willard, 1990). A node, which is not a leaf of the tree, is represented as a pair Ni(Ci, KLi), where • Ci is the manning, defined by a membership function lCi : lC i ¼ S 1 X K Lj lCj K Li j¼1 and S is a number of successors in the tree (subordinate units), KLj LanchesterÕs efficiency coefficient of a subordinate unit of the j and lCj a membership function of the jth successor: lC j ¼   lej lgj laj lpj l$j . e g a p $ • KLi is the LanchesterÕs efficiency coefficient of a combined unit i: K Li ¼ S X K Lj . j¼1 Fig. 7 shows a fuzzy tree that represents a task group. Tree leaves are marked by fuzzy values C* = {e, g, a, p, $} and LanchesterÕs coefficients. Nodes or non-tree leaves are marked by x. In accordance with the FLU algorithm, the cost of an individual basic operation is calculated by using membership functions: pk ¼ jxk j ¼ Rþ1 X lC ðxk ; al ÞBk ; Rþ1 X lC ðy k ; al ÞBk l¼1 qk ¼ jy k j ¼ and l¼1 rk ¼ jxk  y k j ¼ Rþ1 X jlC ðxk ; al ÞBk  lC ðy k ; al ÞBk j; l¼1 where R is the number of labels of nodes (R = 4), Bk is the value of an individual node and l membership functions of relations between transformed nodes. x-# x-# x-# x-# x-# x-# x-# x-# e - 69.74 e - 129.55 e - 129.55 e - 34.87 e - 7.70 e - 3.74 e - 3.74 e - 22.05 e - 518.20 e - 518.20 e - 222.68 e - 24.50 e - 372.38 e - 153.74 e - 3.87 e - 518.20 e - 518.20 e - 222.68 e - 24.50 e - 372.38 e - 153.74 e - 34.87 e - 518.20 e - 518.20 e - 222.68 e - 24.50 e - 273.74 e - 581.00 Fig. 7. Fuzzy tree representation of the task group. e - 3.74 e - 3.74 307 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 6.3. Combat simulation During combat simulation we tracked the number of destroyed systems and the level of manning through the following phases of the combat: 1. 2. 3. 4. delay operations (t1), battles on the firing line, combined with airborne attack countermeasures (t2), defence operations (t3), counter-attack (t4). Fig. 8 shows fuzzy trees that represent the task group at the end of a certain phase. There are also initial and reference situations. The initial situation shows the units before the beginning of combat, whereas the reference situation represents units which were involved in combat and are no longer capable of fighting. The reference situation is defined on the basis of the experiences of the unit commander and his staff. It can be also defined on the basis of the results from combat simulation systems (Knoll, 1998). Table 1 shows the result of the comparison between fuzzy trees on the basis of the FLU algorithm. The distances between trees represent changes in manning over time considering the initial situation. By monitoring these distances, a commander can easily follow the situation on the battlefield and make quick and efficient decisions. Simulations from the previous section resulted in distances between fuzzy trees. This form of the result is relatively useless; therefore, distances need to be normalized in compliance with a reference distance. The initial situation, representing an excellently manned unit, is ascribed a real value Be = 1.00 while the reference situation, representing a poorly manned unit, is ascribed a real value Bp = 0.50. Normalized distances dn are obtained by the following transformation: e e e e e e e e e e e e e e e e e e e e e e e e e e e e Initial situation e e e e e e e e e e e e e e e e e e e e e Phase 1 e e e e e e e e e e e e e e e e e g e e e e e e e e e e e e e e e e g g e e a a g e e e e e e e e e e e e e e e e e e e Phase 2 e e e e e e e e e e e e e e e e e a g g e e e e e e e e e e e e e e e Phase 3 e e e e e e e e e e e e e e e e e e e e e e e e g g g e p a p e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e p p p e p p p e e e e e e e e e e e e e e e e Reference situation Phase 4 Fig. 8. Dynamics of the combat illustrated by fuzzy trees. Table 1 FLU tree-to-tree distances Situation Initial, d0 Phase, 1 d1 Phase 2, d2 Phase 3, d3 Phase 4, d4 Reference, dr Initial d0 0.00 0.63 3.64 8.40 11.85 22.55 308 T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 Table 2 Normalized distances between fuzzy trees Situation Initial, d n0 Phase 1, d n1 Phase 2, d n2 Phase 3, d n3 Phase 4, d n4 Reference, d nr Initial 1.00 0.98 0.92 0.81 0.74 0.50 d n ¼ Be  Be  Bp d; dr 0 6 d 6 dr; where dr is the distance between trees representing the initial and reference situations. Table 2 shows normalized distances between fuzzy trees. 6.4. Fuzzification of distances In compliance with membership functions, expressed in (1), a fuzzification of normalized distances is possible using the following membership functions: lðd fi Þ ¼ lðd ni ; Bj Þ ¼ 1 1 cos pðd ni  Bj Þ þ ; 2 2 where d ni is a normalized distance and Bj (j = ÔeÕ, ÔgÕ, ÔaÕ, ÔpÕ, Ô$Õ) is the element of fuzzily expressed values. Fuzzy distances df, describing the situation in the battlefield at a given moment, can be expressed in the following way:  Initial situation ðt0 Þ:  Phase 1 ðt1 Þ:  Phase 2 ðt2 Þ:  Phase 3 ðt3 Þ:  Phase 4 ðt4 Þ:  Reference situation ðtr Þ: 1.000 0.976 0.794 0.500 0.000 þ þ þ þ ; e g a g $ 0.999 0.986 0.819 0.531 0.001 þ þ þ þ ; d f1 ¼ e g a g $ 0.984 0.999 0.885 0.624 0.016 þ þ þ þ ; d f2 ¼ e g a g $ 0.914 0.980 0.970 0.781 0.086 d f3 ¼ þ þ þ þ ; e g a g $ 0.842 0.938 0.996 0.864 0.158 þ þ þ þ ; d f4 ¼ e g a g $ 0.500 0.655 0.905 1.000 0.500 þ þ þ þ . d fr ¼ e g a p $ d f0 ¼ Assessment of the condition of a unit at a given moment ti, can be based on the following criterion of the fuzzy decision-making process: max lðd fi Þ ¼ max lðd ni ; Bj Þ. j The results of the assessment of the unitÕs capacity at a certain situation are represented in Table 3. Table 3 Assessment based on distances between fuzzy trees Dynamics of the combat Initial, t0 Phase 1, t1 Phase 2, t2 Phase 3, t3 Phase 4, t4 Reference situations of a unit Situation of a unit Comment e Excellent e Excellent g Good g Good a Acceptable p Poor T. Savšek et al. / European Journal of Operational Research 174 (2006) 293–310 309 7. Conclusion In the past, the main tendency in both science and engineering was the reduction of complex real-world systems into precise mathematical models. Thus operational research began to be applied to real-world decision-making problems. Unfortunately, real-world situations are often not so deterministic. When knowledge about a system is incomplete, or when a system is complex and the data within it are episodic rather than systematic, the principles of fuzzy set theory can be applied. However, graph theory continues to play an important role in the modelling of structures, especially in system analysis and operational research. The theory of fuzzy graphs, therefore, is an important link to the theories of fuzzy classification and decision analysis. Fuzzy graphs are also helpful in the representation of soft or ill-defined structures, for instance in military systems. With the increasing popularity of the use of hierarchical data structures and tree construction schemes, a special class of graphs, the problem of comparing two tree-like data structures arises. We have defined a weighted distance between two fuzzy trees by using the language transformation model. The distance between two fuzzy trees is calculated in terms of the minimum cost of deleting and inserting nodes and substituting labels of nodes. The cost of these basic operations is then calculated on the basis of fuzzy set theory. This makes it possible to handle not only problems in the deterministic environment but also problems that appear in the fuzzy environment. 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