ORDERING OBJECTS ON THE BASIS OF POTENTIAL FUZZY RELATION FOR GROUP EXPERT EVALUATION

ORDERING OBJECTS ON THE BASIS OF POTENTIAL FUZZY RELATION FOR GROUP EXPERT EVALUATION B. Kh. Sanzhapov and R. B. Sanzhapov Volgograd State University of Architecture and Civil Engineering, Akademicheskaya st. 1, Volgograd, Russia E-Mail: sbkh@mail.ru ABSTRACT The problem of ordering objects (determination of importance coefficients) indecision support systems is often connected with processing of expert information. The method for solving this problem with the presence of several fuzzy relations, which simulate the preferences of different expert groups, is proposed. Unlike the well-known approaches to the ordering of objects according to the fuzzy estimations of paired comparisons, the proposed method makes it possible to determine the resulting weights (potentials) of objects without the preliminary approximation of expert information.a numerical example is given. Keywords: ordering objects, fuzzy binary relation, system of potentials, transitive matrix. 1. INTRODUCTION In many decision support systems the standard task is to determine importance coefficients of objects - systems, subsystems, and means implementing solutions. Information can be based on the expert estimates, represented in the form of fuzzy relations. As a rule, even in the case of considering the only single fuzzy relation (FR), the constructed relation does not possess the property of the transitivity, necessary for the ordering of objects [1]. Construction of fuzzy potential relation (FPR), approximating a singular fuzzy binary relation, will allow to determine the weights of the objects (so-called, potentials) and on that basis to order set of objects [2]. Nevertheless, in some cases, the evaluation of superiority of one object over another object is not a single number. This can happen when there is no unity in the evaluation of superiority in a group of experts when considering a particular pair of objects, or paired comparison is made by several expert groups. Unlike the existing approaches, we propose a method to construct the resulting PFR closest to expert evaluation when considering initial fuzzy relations on the set of pairs of objects without the use of preliminary aggregation processes, which increases the accuracy of the calculations. The task of ordering a set of objects X (determination of their weights) in the case of a single matrix of expert estimates was successfully solved in papers [2-8]. In practical problems, the evaluation of superiority of the object i over the object j given by matrix = { } whose elements =,, are determined by experts. Here values x i are the weights of the objects. Equations describing the full consistency of expert estimates are presented as [8] = ۰,,. Equalities (2) characterize the transitivity of the matrix R [8]. Because the matrix R is based on expert estimates, equations (2) are often not carried out. The inconsistency of such expert evaluation makes it necessary use the following transitive matrix instead of the matrix R for ordering objects: = ( ),,. This matrix is somewhat similar to the matrix of expert evaluation R. The proximity matrices of paired comparisons are treated differently. For example, the distance, between the matrices R and T is axiomatically defined. A matrix similar to R is selected as T [2,3]. The papers [4-6] consider the functional,, determining the measure of approximation of matrix R by the transitive matrix T. Matrix T is calculated by solving the corresponding optimization problems. In one paper [7] the weights of objects x i are determined based on the calculation of the maximum eigenvalue of matrix R, and further definition of its eigenvector. In contrast to the above mentioned papers, the proposed method determines the value of the importance of objects (their potentials) on the basis of the approximation of initial FR by one PFR. In some cases it can be rather difficult to directly apply the results of papers [2-8] for the search of the approximating PFR in solving the problem of ordering objects at several FR on basis of the exhaustive sorting of all possible options, each of which is described by a singular FR. This is due to difficulties of computational nature. Indeed, even in the case of two independent expert groups, for n objects the required number of computations will be proportional to. 2. THE MATHEMATICAL STATEMENT OF THE PROBLEM For many methods of the ordering objects [2-8], including ours, the initial information consists of a finite set of objects = {,,, }, and a set of object pairs U = { i, j X X, i, j X}. Fuzzy relations, =,, characterizing opinions of different experts, are established on the set of such object pairs. For each such FR the membership function : [, ] is 8544

defined, where, is interpreted as a measure of superiority of the object i over the object j for the expert with the number k. Further in the article we will use the concept of potential fuzzy relation (PFR)[2]. Definition 1 [2].FRR is a PFR, if there exists a function,, and that = (, ),, =,,,,. Elements of which satisfy the relations, =, +,,,,.,, =,,. The value is called the potential of object, the set of their values = { } is called the system of potentials of PFR. The relationships (3) determine the system of potentials ambiguously, with accuracy up to an arbitrary constant. That is, by adding to all the potentials the same random number; we can obtain a new system of potentials of the same PFR. The value of this constant may be determined so that the potentials will be in the interval [2], Let us consider a matrix =, with elements, =,, for which hold the properties, =,. If FR R is a PFR, then relations hold [2]:, =,,,, =, +,,,,. The matrix W R is the logarithmic analogue of the matrix R. The properties (5), (6) and (1), (2) correspond to each other. In particular, the choice of this matrix W R is due to the possibility of considering absolute superiority of the object i over the object j as, =, =. It should be noted that for any system of potentials, which satisfies restrictions (4), there exists a single reversible PFRR, on the basis of which it is possible to calculate the superiority measure of the object i over the object j. I.e., it is possible to determine the values of membership function, = [ + ]/,,. As mentioned above, in solving practical problems the original information can be given on the basis of expert estimates. Based on them FR =,, =, in general is not PFR. Then it is necessary to clarify the formulation of the problem of the approximation of initial FR by closest PFR. As discussed above, the system of potentials is determined up to an arbitrary constant. Therefore, further down as a source of initial information we will consider a set of matrices B 1,B 2,...,B K, which elements are defined as, =,,,,, =,. Let us choose as PFR 7) Let {T} denote the set of PFR with elements satisfying the equalities (8) and (9). In this case, it is necessary to determine = { },,,,. For each fixed pair of objects (i,j) closest to, value from the set {,,,,,, } is determined. The maximum value for all pairs (i, j) of considered elements will determine the deviation the matrix T from the matrices B 1,B 2,...,B K. The calculation of the minimum value of this deviation for all matrices {T} will define the closest to B 1,B 2,...,B K transitive matrices. It should be noted that in considering only one group of experts, that is, when K=1, the set of matrices will consist of a single element. 3. THE METHOD OF CALCULATING THE SYSTEM OF POTENTIALS For each value : and for each expert k, let us determine the permissible domains for the elements of the transitive matrix = (, ),,.These domains will look as follows, = [max,,, min,, + ],,, =,. Taking into account the relationships for all matrices B 1,B 2,...,B K such as, =,,,, =,, let us rewrite (11) in the format convenient for further calculations:, = [max,,, min,, ],,, =,. Formula (12) allows considering only the lower boundaries of permissible intervals constituting the permissible domains. Then the permissible domain for a possible change of the matrix elements, of PFR will be represented as the union of sets (12):, =,,,. =, In fact, for a fixed value of λ, a graph =,, is considered, where X is the set of vertices, = is the set of arcs, C = { c,, = max, b i, j 8545

λ, i, j X, k =,,, K}is the set of weights of arcs. The weight of each arc of this graph can have K different values. In other words, the graph D can be represented as the union of graphs =,, in which each arc has a single weight. The number = is the number of such graphs. Thus, =, =,,,. We shall consider the graph =,,. Let us denote as a weight of the path = {,,,,,, } which sequentially passes through the vertices,,,. The weight of the path is the sum of weights of arcs that pass through these vertices. As shown in previous paper [2], the condition of existence of transitive matrix, each element of which belongs to the corresponding interval of the permissible weights of the arcs, is the condition for all paths of this graph D l. The existence of transitive matrix for the graph D means the existence of at least one graph D l for which condition for the existence of such a matrix is satisfied. Then the solution of the problem (8) - (10) consists of the search for zero cycle in this graph [2]. In some cases, because of the difficulty of computational nature, it is difficult to examine all the graphs, =,,,. The following is an effective approach to solving this problem (8) - (10). Let us rewrite the task (10) in the following format:,, min, with the performance restrictions (8), (9), (12), (13). Taking into account obtained from (8) relationships for the matrix =,,, =,,,,, and the relationships (9) it is easy to prove that the matrix is uniquely determined by any of its column (or row). Therefore, to construct a permissible solution at a fixed value, which will not be specified further, and satisfying (8), (9), (12), (13), (16), it is sufficient to calculate one column of the matrix T. Further, as such a column, the column number n is selected. Let us consider the square submatrix =,, of the matrix T. All its elements can also be determined by any column (or row). As such column let us select the column with number m: =,,,,, (T is a sign of the transposition). It is to be noted, the diagonals of matrices and consist of zeros. Definition 2. The column with the number m: =,,,,,, which belongs to the submatrix =,, of the matrix T, is called permissible if at fixed value and X=(1,2,...,m) conditions (12),(13),(16) are satisfied. Proposition 1. Elements, +, =, belonging to the permissible column +, are determined by the following conditions:, +, +,, +, +, +. The proof of Proposition 1 follows from the sequential application of the equalities (9). The proposed calculation scheme is to continuously increase the value of λ from zero up until the permissible solution (the permissible column l n ) is achieved. Certainly the value λ is changed in a discrete manner in numerical realization of the algorithm. Continuous change in the value λ is chosen for a clearer demonstration of the algorithm. The calculated λ=λ 0 will be the solution of the problem (8), (9), (12) - (16). The search algorithm for the permissible matrices set { =,,, }, based on the paper [9], is proposed for the fixed value of variable λ. 3.1 Algorithm 1. For the value,, let us construct the column =,,. By its construction this column will be permissible. 2. At this stage, the permissible columns l m, m=3,4,...,n are calculated. Let us fix the arbitrary element,,. From the set of permissible columns {l m-1 } we choose the ones for elements of which the following conditions are satisfied, =, +,,,. =, Thus obtained column =,,,,, on the basis of Proposition 1is permissible. 3. If { = }, then the solution of problem (8), (9), (12), (13) does not exist, and the value of λ should be increased. 4. This stage considers the procedure of a large agreement of the obtained set of matrices with the expert evaluations. Suppose that in the previous stage the value λ=λ 0 was obtained. From the acquired set of matrices defined by {l n }, one with elements satisfying the following conditions are distinguished. Value, of one of the solution matrices belongs to the lower boundary or to the upper boundary of the respective interval (12), and it is unique on this interval. Further, elements,,, of matrix T are determined; those elements are simultaneously upper or lower boundaries of the corresponding intervals from the permissible domains (12). These elements, according to (9), determine the value of,. Let us designate this set of the calculated pairs of elements as U 0. If the number of experts is K=1, then the number of calculated above elements, will be equal to one. For the case of K> 1, the 8546

number of such elements, can be represented by a nonunique number. Their combinatorial combinations will divide the initial set of expert estimates into a number of subsets. In all the above defined subsets elements,,,,, are the only numbers on the respective intervals. All such subsets are investigated separately. Elements calculated previously will considered further as constants. They of course can be considered as the lower and the upper boundaries of the respective intervals Taking into account relationships, =,,, let us determine the pairs of objects set Y as = {,, <,, }. For the large agreement of the remained variables the problem is solved again with the initial value of λ=0. Subsequently we shall examine the set of the elements \. Obtained in this case value λ (1) <λ 0. The process of calculations continues until all elements of each subset thus constructed are determined. Thus, the following proposition was proved. Proposition 2. Each matrix calculated according to the algorithm determines PFR. Calculated matrices will make it possible to order the objects set. Each matrix will determine the system of potentials of which differ from each other by some constant. It should be noted, if the expert evaluations are conducted by one group (K=1), the computed matrix T will be unique. 3.2 Numerical example Tables 1 and 2 show the expert evaluations made by the two groups of experts, that is, K = 2. Table-1. Matrix =,. 0,5 0,13 0,25 0,4 0,15 0.8 0,5 0,62 0,5 1 0,5 0,45 0,5 0,85 0,9 0,55 0,2 0,4 0,5 0,85 0,1 0,45 0,55 0,35 0,5 Table-2. Matrix =,. 0,5 0 0,4 0,65 0,65 1 0,5 0,82 0,4 0,65 0,9 0,65 0,5 0,45 0,65 1 0,1 0,3 0,5 0,75 0,4 0,1 0,8 0,4 0,5 Let us calculate matrices and using formula (7). Solution of the problem of approximation of these matrices is the value λ 0 =0,0667. At this value λ 0 there were obtained the values of elements, =,, the upper boundary of the corresponding permissible interval [,;,] λ,, and, =,, the upper boundary of the corresponding permissible interval [,;,] λ,. Equality. =, +, will allow to determine, =, [,;,] λ,,which is the lower boundary of the corresponding permissible interval. The determined elements are assumed constants. Accordingly, the set of element pairs = {,,,,, } were determined. Subsequently we will define elements of the set \. Let us set the initial value λ=0. The approximation problem λ (1) =0,052 at which values, =, [,;,], and, =, [,;,], are determined is solved according to the equation, +, =, =,. Values, =, and, =, are believed to be constants. In this case, the calculated set is = {,,, }. Next, it is necessary to determine the elements of the set \ \.We set the initial value λ = 0. The value λ (2) =0,0415 will be the solution of the approximation problem. From the equation, +, =, =, values, =, [,;,], and, =, [,;,], are determined. The set = {,,, } is determined in this stage. Subsequently we will examine the set \ \ \. The obtained value λ (3) =0,024 allows to determine the value, =, [,;,], from the equation, +, =,. Thus, the obtained set is represented by = {, }. It is further necessary to investigate the set \ \ \ \ with the initial value λ=0. The value λ (4) =0,015 will be the solution of the approximation problem. In this case, the element value, =, [,;,], is calculated from the equation, +, =,. At this stage, the set = {, } has been determined. The last stage is calculated value of λ (5) =0,007, in which the equation, +, =, calculates the value of, =, [,;,],. Thus, based on the calculated values of matrix elements =,, we define the object potentials. To demonstrate, let us assume the maximum value of the potential to be equal to 1. Then, the calculated system of potentials takes the form of = { =,; = ; =,; =,; =,}.The defined system of potentials will make it possible to uniquely order the set of objects as { }. 4. CONCLUSIONS The proposed approach can be used at the stage of the preliminary analysis of the various systems projects, such as technical, social and other. At this stage, expert information typically does not have the property of potentiality. Therefore, when determining the priority of objects, it is desirable to take into account the opinions of all the experts, which can be contradictory. REFERENCES 8547

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