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MICROSOFT EXCEL ADD-IN FOR SOLVING MULTICRITERIA DECISION PROBLEMS IN FUZZY ENVIRONMENT Radomír Perzina, Jaroslav Ramík Silesian University in Opava School of Business Administration in Karviná e-mail: (perzina, ramik)@opf.slu.cz Keywords analytic hierarchy process, analytic network process, multi-criteria decision making, pair-wise comparisons, feedback, fuzzy Extended abstract The main goal of every economic agent is to make a good decision, especially in economic environment with many investment alternatives and evaluation criteria. In this paper a new decision model based on analytic network process (ANP) is proposed. Comparing to classical approach it is modified for solving the decision making problem with fuzzy pair-wise comparisons and a feedback between the criteria. The evaluation of the weights of criteria, the variants as well as the feedback between the criteria is based on the data given in pair-wise comparison matrices with fuzzy elements. Extended arithmetic operations with fuzzy numbers are proposed as well as ordering fuzzy relations to compare fuzzy outcomes. An illustrating numerical example is presented to clarify the methodology. The solution is compared with the same problem solved by the classical analytic hierarchy process (AHP), i.e. with non-fuzzy evaluations in the pair-wise comparisons and without the feedback. All calculations are performed in Microsoft Excel add-in software named FVK that was developed for solving the proposed model. Comparing to other software products for solving multicriteria problems, FVK is free, able to work with fuzzy data, allows easy manipulation with data and utilizes capabilities of widespread spreadsheet Microsoft Excel. 1. Introduction When applying Analytic Hierarchy Process (AHP) in decision making one usually meets two difficulties: when evaluating pair-wise comparisons on the nine point scale we do not incorporate uncertainty or when decision criteria are not independent as they should be. In this paper these difficulties are solved by a proposal of the new method which incorporates uncertainty using pair-wise comparisons by triangular fuzzy numbers, and takes into account interdependences between criteria. The first difficulty is solved by fuzzy evaluations: instead of saying e.g. “with respect to criterion C element A is 2 times more preferable to element B” we say “element A is possibly 2 times more preferable to element B”, where “possibly 2” is expressed by a triangular fuzzy number. In some real decision situations, dependency of the decision criteria occur quite frequently, e.g. the criterion price is naturally influenced by the quality criterion. Here, the dependency is modeled by a feedback matrix, which expresses the grades of influence of the individual criteria on the other criteria. The interface between hierarchies, multiple objectives and fuzzy sets have been investigated by the author of AHP T.L. Saaty [6]. Later on, Laarhoven and Pedrycz [9] extended AHP to fuzzy pairwise comparisons. Saaty extended AHP to a more general process with feedback called Analytic Network Process (ANP) [7], [8]. In this paper we extend the approaches from [1], [2], [8] to the case of feedbacks between the decision criteria as it was specified in [5], moreover we also supply an illustrating realistic example to demonstrate the proposed method, documented by the outputs from Microsoft Excel add-in FVK that was developed for solving the proposed model. 2. Multi-criteria decisions In Analytic hierarchy process (AHP) we consider a three-level hierarchical decision system: On the first level we consider a decision goal G, on the second level, we have n independent evaluation n criteria: C1, C2,...,Cn, such that wCi 1 , where w(Ci) > 0, i = 1,2,...,n, w(Ci) is a positive real number – i 1 weight, or, relative importance of criterion Ci subject to the goal G. On the third level m variants m (alternatives) of the decision outcomes V1, V2,...,Vm are considered such that again wV , C 1 , r 1 r i where w(Vr,Ci) is a non-negative real number - an evaluation (weight) of Vr subject to the criterion Ci, i = 1,2,...,n. This system is characterized by the supermatrix W, see [8]: 0 W = W21 0 0 0 W32 0 0 , I (1) where W21 is the n1 matrix (weighing vector of the criteria), i.e. w(C1 ) W21 , w(Cn ) (2) and W32 is the mn matrix: w(C1 ,V1 ) w(Cn ,V1 ) . W32 w(C1 ,Vm ) w(Cn ,Vm ) (3) The columns of this matrix are evaluations of variants by the criteria, I is the unit matrix. W is a column-stochastic matrix, i.e. the sums of columns are equal to one. Then the limit matrix W (see [3]) exists and we can calculate the resulting priority vector of weights of the variants Z which is given by formula (4). The variants can be ordered according to these priorities. Z = W32W21 (4) In real decision systems with 3 levels there exist typical interdependences among individual elements of the decision hierarchy e.g. criteria or variants. Consider now the dependences among the criteria. This system is then given by the supermatrix W: 0 W = W21 0 0 W22 W32 0 0 , I (5) where the interdependences of the criteria are characterized by nn matrix W22: w(C1 , C1 ) w(Cn , C1 ) . W22 w(C1 , Cn ) w(Cn , Cn ) In general, matrix (5) is not column-stochastic, hence the limiting matrix does not exist. Stochasticity of this matrix can be saved by additional normalization. Then there exists a limiting matrix W and the vector of weights Z can be calculated by formula (6), see [5]. Z W32 I W22 W21 1 (6) As the matrix W22 is close to the zero matrix, as the dependences among the criteria are usually weak, it can be approximately substituted by the first 4 terms of Taylor’s expansion and we get: 3 Z W32 (I W22 W222 W22 )W21 . (7) 3. Fuzzy numbers and fuzzy matrices In practice it is sometimes more convenient for the decision maker to express his/her evaluation in words of natural language saying e.g. “possibly 3”, “approximately 4” or “about 5”. Similarly, he/she could use the evaluations as “A is possibly weak preferable to B”, etc. It is advantageous to express these evaluations by fuzzy sets of the real numbers, e.g. triangular fuzzy numbers. A triangular fuzzy number a is defined by a triple of real numbers, i.e. a = (aL; aM; aU), where aL is the Lower number, aM is the Middle number and aU is the Upper number, aL ≤ aM ≤ aU . If aL = aM = aU, then a is the crisp number (non-fuzzy number). In order to distinguish fuzzy and non-fuzzy numbers we shall denote the ~ (a L ; a M ; aU ) . It is known that the arithmetic fuzzy numbers, vectors and matrices by the tilde, e.g. a operations +,-, * and / can be extended to fuzzy numbers by the Extension principle, see e.g. [2]. If all elements of an mn matrix A are triangular fuzzy numbers then we call A the triangular fuzzy ~ matrix and this matrix is composed of the triples of real numbers. Particularly, if A is a pair-wise comparison matrix, we assume that it is reciprocal and there are units on the diagonal. 4. Decision making process The proposed decision support method of finding the best variant, or ordering of all the variants can be described by an algorithm that we describe in this section: 4.1. Calculate the triangular fuzzy weights We have to calculate the triangular fuzzy weights as evaluations of the relative importance of the criteria, evaluations of the feedback of the criteria and evaluations of the variants according to the ~ ,w ~ ,..., w ~ , individual criteria. We assume that there exists a fuzzy vectors of triangular fuzzy weights w 1 2 n L M U ~ w (w ; w ; w ) , i = 1,2,...,n, which can be calculated by the following formula (see [5]): i i i i ~ (wL ; wM ; wU ) , k = 1,2,...,n, w k k k k where (8) 1/ n n S akj j 1 , S {L, M ,U} . wkS 1/ n n n M aij i 1 j 1 (9) In [2], the method of calculating triangular fuzzy weights by (9) from the triangular fuzzy pair-wise comparison matrix is called the logarithmic least squares method. This method can be used both for calculating the triangular fuzzy weights as relative importance of the individual criteria and also for eliciting relative triangular fuzzy values of the criteria for the individual variants out of the pair-wise comparison matrices and also for calculating feedback impacts of some criteria on the other criteria. 4.2. Calculate the aggregating triangular fuzzy evaluations of the variants Now we calculate the synthesis - the aggregated triangular fuzzy values of the individual variants by formula (6), eventually, the approximate formula (7), applied for triangular fuzzy matrices: ~ ~ ~ 1 ~ Z W32 I ~ W22 W21 , ~ ~ ~ ~ ~3 ~ Z W32 (I ~ W22 ~ W222 ~ W22 )W21 . (6*) (7*) Here, for addition, subtraction and multiplication of triangular fuzzy numbers we use the fuzzy arithmetic operations defined in [5]. 4.3. Find the best variant The simplest method for ranking a set of triangular fuzzy numbers in (7*) is the center of gravity method. This method is based on computing the x-th coordinates of the center of gravity of each triangle given by the corresponding membership functions of ~z i , i = 1,2,...,n. Evidently, it holds xig ziL ziM ziU . 3 (10) By (10) the variants can be ordered from the best to the worst. There exist more sophisticated methods for ranking fuzzy numbers, see e.g. [4], for a comprehensive review of comparison methods see [2]. In the SW tool FVK described in the next section, we apply two more methods which are based on -cuts of the fuzzy variants being ranked. Particularly, let ~ z be a fuzzy alternative, i.e. fuzzy number, (0,1] be a preselected aspiration level. Then the -cut of ~ z , [~ z ], is a set of all elements x with the membership value at least , i.e. ~ [ z ]= {x| ~z ( x) }. Let ~z and ~z be two fuzzy variants, (0,1]. We say that ~z is R-dominated by ~z at the level if 1 2 1 2 sup[ ~z1 ] ≤ sup[ ~z 2 ]. Alternatively, we also say that ~z 2 R-dominates ~z1 at the level . If ~ z ( z L , z M , z U ) is a triangular fuzzy number, then sup[ ~z ] = zU ( zU z M ) , sup[ ~z ] = zU ( zU z M ), 1 1 1 2 1 2 2 2 as can be easily verified. We say that ~z1 is L-dominated by ~z 2 at the level if inf[ ~z1 ] ≤ inf[ ~z 2 ]. Alternatively, we also say that ~z 2 L-dominates ~z1 at the level . Again, if ~ z ( z L , z M , zU ) , then inf[ ~z1 ] = z1L ( z1M z1L ) , inf[ ~z 2 ] = z2L ( z2M z2L ) . Applying R and/or L domination we can easily rank all the variants (at the level ). 5. Case Study Here we analyze a decision making situation buying an “optimal” refrigerator with 3 decision criteria and 3 variants. The goal of this realistic decision situation is to find the best variant from 3 pre-selected ones according to 3 criteria: e.g. price, design and efficiency. First, we apply the proposed fuzzy ANP algorithm, then classical AHP and finally classical ANP. All calculations are performed in Microsoft Excel add-in software named FVK that was developed for solving the proposed model. 5.1. Importance of the criteria First we express the importance of the criteria that is given by the pair-wise comparison matrix C: C= Crit 1 Crit 2 Crit 3 Crit 1 1 0,2 0,5 1 0,25 0,5 1 22 0,5 1 Crit 2 44 1 1 55 1 2 11 1 1/4 1/4 4 Crit 3 22 1/2 1/2 1 22 11 1 1 By formula (9) we calculate the corresponding triangular fuzzy weights, i.e. the relative fuzzy importance of the individual criteria that are given in matrix W21: W21 = 0,360 0,105 0,227 0,571 0,143 0,286 0,616 - C1 (Price) 0,227 - C2 (Design) 0,454 - C3 (Efficiency) 5.2. Evaluation of variants Next step is to make fuzzy evaluations of the variants according to the individual criteria that are given by the following 3 pair-wise comparison matrices A1, A2, A3: A1 = Var 1 Var 2 Var 3 A2 = Var 1 Var 2 Var 3 A3 = Var 1 Var 2 Var 3 Var 1 1 1 2 1 2 3 1 1/3 1/3 3 3 Var 2 1/2 1/2 1 1 Var 1 1 2 0,25 1 0,167 0,2 1 3 0,5 Var 1 1 0,2 0,333 1 1/4 1/4 4 0,5 11 1 2 Var 2 1/3 1/3 1 1 1/3 1/3 1 1/2 1/2 2 Var 3 1/3 1/3 1/2 1/2 1 1/2 1/2 1 2 1 22 1 1/4 1/4 4 22 1/2 1/2 1 44 11 1 1 Var 3 55 1 1 1 Var 3 Var 2 1 44 0,25 0,333 1/2 1/2 11 66 1 2 33 1 1/3 1/3 3 33 1/2 1/2 1 55 11 1 1 The corresponding fuzzy matrix W32 of fuzzy weights is calculated by (9) as W32 = 0,143 0,236 0,374 0,163 0,297 0,540 0,236 0,428 0,540 0,263 0,263 0,209 0,289 0,379 0,331 0,417 0,526 0,417 0,602 0,100 0,154 0,648 0,122 0,230 0,817 - Variant 1 0,166 - Variant 2 0,263 - Variant 3 5.3. Feedback between criteria In order to evaluate fuzzy feedback between the criteria we apply again pair-wise comparison method, then we obtain the following 3 pair-wise comparison matrices B1, B2, B3: B1 = Crit 2 Crit 3 B2 = Crit 1 Crit 3 B3 = Crit 1 Crit 2 Crit 2 1 0,333 1 0,5 Crit 1 1 3 Crit 1 1 0,333 1 2 1 0,2 Crit 3 22 1 11 1 1 33 1 Crit 3 1/3 1/3 1 1/5 1/5 5 1 1 1/2 1/2 1 1 Crit 2 1 22 0,5 33 55 1 1 1 By using (9), we obtain the fuzzy feedback matrix W22: W22 = 0,000 0,471 0,272 0,000 0,667 0,333 0,000 0,816 0,471 0,194 0,000 0,612 0,250 0,000 0,750 0,306 0,000 0,968 0,612 0,194 0,000 0,750 0,250 0,000 0,968 - C1 (Price) 0,306 - C2 (Design) 0,000 - C3 (Efficiency) 5.4. Total evaluation of the variants Finally we calculate the synthesis – the aggregated triangular fuzzy values of the individual variants. For this purpose we use the approximate formula (7*), by which we get the matrix of fuzzy weights Z for variants. The situation is graphically depicted in Figure 1. Z= Var 1 Var 2 Var 3 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,000 L 0,182 0,112 0,149 M 0,346 0,264 0,390 U 0,870 0,649 0,733 1 2 3 0,200 0,400 0,600 0,800 1,000 Figure 1: Total evaluation of fuzzy variants In the last step we rank the evaluations of the above fuzzy variants resulting in the best decision. Here we use ranking methods as described in section 4.3., i.e. Center of gravity, L domination and R domination. For the last two methods level α = 0.7 was used. The results are in the following table. Rank Center of gravity L dominantion 0,466 1 2 0,342 3 3 0,424 2 1 Variant xgi Var 1 Var 2 Var 3 R dominantion 1 3 2 5.5. Classical AHP Now, we solve the same problem applying classical AHP, i.e. we use non-fuzzy evaluations in the pair-wise comparisons and do not consider the feedback, so the feedback matrix is a zero matrix, i.e. W22 = 0. Then we get the matrix of crisp weights Z for variants and their rank. The situation is graphically depicted in Figure 2. Z= Var 1 Var 2 Var 3 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,000 L 0,320 0,259 0,421 M 0,320 0,259 0,421 U 0,320 0,259 0,421 Rank 2 3 1 1 2 3 0,100 0,200 0,300 0,400 0,500 Figure 2: Total evaluation of variants – AHP 5.6. Classical ANP Finally, we solve the situation again with the same crisp evaluations, however, with a non-zero feedback between the criteria expressed by crisp values (i.e. the classical ANP). Then we get the matrix of crisp weights Z for variants and their rank. The situation is graphically depicted in Figure 3. Z= Var 1 Var 2 Var 3 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,000 L 0,346 0,264 0,390 M 0,346 0,264 0,390 U 0,346 0,264 0,390 Rank 2 3 1 1 2 3 0,100 0,200 0,300 0,400 0,500 Figure 3: Total evaluation of variants – ANP 6. Conclusion In this paper we have proposed decision model based on ANP for solving the decision making problem with fuzzy pair-wise comparisons and a feedback between the criteria. The evaluation of the weights of criteria, the variants as well as the feedback between the criteria is based on the data given in pair-wise comparison matrices. An illustrating case study has been presented to clarify the methodology. Based on the case study we can conclude that fuzzy evaluation of pair-wise comparisons may be more comfortable and appropriate for decision making. Occurrence of dependences among criteria is more realistic. Dependences among criteria influence the final rank of variants and presence of fuzziness in evaluations change the final rank of variants. References [1] Buckley, J.J., Fuzzy hierarchical analysis. Fuzzy Sets and Systems 17, 1985, 1, p. 233-247, ISSN 0165-0114. [2] Chen, S.J., Hwang, C.L. and Hwang, F.P., Fuzzy multiple attribute decision making. Lecture Notes in Economics and Math. Syst., Vol. 375, Springer-Verlag, Berlin – Heidelberg 1992, ISBN 3-54054998-6. [3] Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, 1990, ISBN 0521305861. [4] Ramik, J., Duality in fuzzy linear programming with possibility and necessity relations. Fuzzy Sets and Systems 157, 2006, 1, p. 1283-1302, ISSN 0165-0114. [5] Ramik, J., Perzina, R., Fuzzy ANP – a New Method and Case Study. In Proceedings of the 24th International Conference Mathematical Methods in Economics 2006, University of Western Bohemia, 2006, ISBN 80-7043-480-5. [6] Saaty, T.L., Exploring the interface between hierarchies, multiple objectives and fuzzy sets. Fuzzy Sets and Systems 1, 1978, p. 57-68, ISSN 0165-0114. [7] Saaty, T.L., Multicriteria decision making - the Analytical Hierarchy Process. Vol. I., RWS Publications, Pittsburgh, 1991, ISBN. [8] Saaty, T.L., Decision Making with Dependence and Feedback – The Analytic Network Process. RWS Publications, Pittsburgh, 2001, ISBN 0-9620317-9-8. [9] Van Laarhoven, P.J.M. and Pedrycz, W., A fuzzy extension of Saaty's priority theory. Fuzzy Sets and Systems 11, 1983, 4, p. 229-241, ISSN 0165-0114. This research was partly supported by the grant project of GACR No. 402060431