ASIAN JOURNAL OF MANAGEMENT RESEARCH Online Open Access publishing platform for Management Research

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1 ASIAN JOURNAL OF MANAGEMENT RESEARCH Online Open Access publishing platform for Management Research Copyright 2010 All rights reserved Integrated Publishing association Review Article ISSN The key of managerial problems in fuzzy world: Technique of order preference by similarity to ideal solution School of Computer Applications, Sanghvi Institute of Management and Science, Indore, MP, India ABSTRACT Managerial Problems include both qualitative and quantitative attributes which are often assessed using imprecise data and human udgments. The paper presents the solution of multi attribute problems by using the technique of order preference by similarity to ideal solution. The fuzzy evaluation values are given by triangular fuzzy numbers. A new distance is defined using which the distance of each alternative from the positive and negative ideal solutions are calculated. Finally a closeness coefficient is defined to determine the ranking order of the alternative. Keywords: Managerial Problems, Triangular Fuzzy Numbers, TOPSIS Method, Fuzzy TOPSIS, Multi criteria Decision Problems. 1. Introduction Proper decision in management planning, controlling and motivation processes is an important factor in any organization. Decisions whether to accept or reect a new ob, to buy or not to buy shares of a particular company etc. are taken after analyzing in each situation. Generally, in real world situations because of incomplete or non obtainable information the data are usually fuzzy, so we extend technique of order preference by similarity to ideal solution for fuzzy data. The concept of fuzzy logic or fuzzy measures has practical applications in industry and business or in decision making problems. 2. Fuzzy world Whenever it is not sure that the particular element belongs to the particular set; it may or may not belong to the set, then the fuzzy concept is introduced. Thus a fuzzy set is completely characterized by a membership function. This fuzzy set theory was formalized by Professor Lofti Zadeh at the University of California in 1965 (Bellman & Zadeh, 1970). If U is a collection of obects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs: A = {( x,µ ( x) ) x U} A where µ A ( x) is called membership function and µ A : U [0,1]. Thus, by using the concept of membership function we can define fuzzy set, where each member is written in the form of ordered pairs, where it is a mapping from universal set to closed set ASIAN JOURNAL OF MANAGEMENT RESEARCH 149

2 [0, 1] (Yager, 1981, Shih, Shyur and Lee, 2007). The central notion of fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value in the range [0, 1], with 0.0 representing absolute falseness and 1.0 representing absolute truth (Chen & Tan, 1994). Thus, in short, Fuzzy means a form of knowledge representation suitable for notions that cannot be defined precisely (Fodor and Roubens, 1994). This concept has wide applications in all sciences. In this paper its one of the applications in management sciences is taken. 3. Technique for order preference by similarity to the ideal solution (TOPSIS) TOPSIS means the technique for order preference by similarity to the ideal solution. This is one of the multi criteria decision techniques that provide the basis for decisions which promote sustainable development of our economy (Hwang and Lin, 1987 and Saaty, 1980). Decisions made at different decision levels differ in complexity or level of uncertainty (Bui, 1987). The aim of decision support is to assist decision maker to make decisions that are consistent with their values, goals and performances. TOPSIS method determines the ranking of the critical success factors (Vincke, 1992). This Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was first developed by Hwang and Yoon (Hwang & Yoon, 1981), based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest from the negative-ideal solution (NIS) for solving a multiple criteria decision making problem (Delgado, 1983). In short, the ideal solution is composed of all best values attainable of criteria, whereas the negative ideal solution is made up of all worst values attainable of criteria. In this method two artificial alternatives are hypothesized: 1. Ideal alternative: the one which has the best level for all attributes considered. 2. Negative ideal alternative: the one which has the worst attribute values. TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal alternative (Lai, Liu & Hwang, 1994) Fuzzy TOPSIS Method Consider m alternatives 1, 2, 3,..m and n attributes 1, 2, 3,.n. Score of each alternative with respect to each attribute is taken (Li and Lee, 1990). It is noted that all the scores are fuzzy numbers (Chen, 2000, Carlsson, 1988). Now consider following steps: Step 1: Construct normalized decision matrix: X = (x i ) m n matrix. Here m indicates alternatives or options and n indicates attributes or criteria, (Lai and Hwang, 1994). Normalize scores or data as follows: r i = x i / (Σx 2 i) for i = 1, 2, 3, m; = 1, 2, 3,.n. ASIAN JOURNAL OF MANAGEMENT RESEARCH 150

3 Step 2: Construct the weighted normalized decision matrix. Assume a set of weights for each attribute w for = 1, 2, 3,..n such that each w (0, 1) and n w = 1 say each w is a normalized fuzzy number (Solymosi and Dombi, 1986). =1 or simply we can Multiply each column of the normalized decision matrix by its associated weight. An element of the new matrix is: v i = w r i Step 3: Determine the ideal and negative ideal solutions. Ideal solution: A* = { v * 1,, v * n }, where v * ={ max (v i ) if J ; min (v i ) if J' } Negative ideal solution: A' = { v 1 ',, v n ' }, where v' = { min (v i ) if J ; max (v i ) if J' } Here J be the set of benefit attributes or criteria (more is better) and J' be the set of negative attributes or criteria (less is better) (Nurmi, 1981). Step 4: Calculate the separation measures for each alternative. The separation from the ideal alternative is: S i * = [ Σ (v * v i ) 2 ] ½ i = 1,, m. Similarly, the separation from the negative ideal alternative is: S' i = [ Σ (v ' v i ) 2 ] ½ i = 1,, m. Step 5: Calculate the relative closeness to the ideal solution C i * C i * = S' i / (S i * +S' i ), 0 < Ci* < 1 Select the alternative with C i * closest to Application Consider four alternatives 1, 2, 3 and 4 and four attributes A, B, C and D. W = set of weights of attributes. W = {0.1, 0.4, 0.3, 0.2} Table 1: Preliminary Entries Weight Attributes A B C D ASIAN JOURNAL OF MANAGEMENT RESEARCH 151

4 Step 1: Calculate (Σx 2 i) 1/2 for each column and divide each column by that to get r i = x i / (Σx 2 i) Table 2: Calculate r i = x i / (Σx 2 i) Weight Attributes A B C D Step 2: Multiply each column by w to get v i = w r i Table 3: Calculate v i = w r i Attributes A B C D Step 3 (i): Determine ideal solution A*. Choose minimum value in column D and maximum value in A, B and C. Thus A* = {0.059, 0.244, 0.162, 0.080}. Table 4: Determination of Ideal Solution Attributes A B C D Step 3 (ii): Determine negative ideal solution A'. Choose maximum value in column D and ASIAN JOURNAL OF MANAGEMENT RESEARCH 152

5 minimum value in A, B and C. Thus A' = {0.040, 0.164, 0.144, 0.118}. Table 5: Determination of Negative Ideal Solution Attributes A B C D Step 4 (i): Determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} S i * = [ Σ (v * v i ) 2 ] ½ for each row. Alternatives Table 6: Determination of Ideal Solution S i * = [ Σ (v * v i ) 2 ] ½ Step 4 (ii): Determine separation from negative ideal solution A' = {0.040, 0.164, 0.144, 0.118}, S' i = [ Σ (v ' v i ) 2 ] ½ Table 7: Separation from Negative Ideal Solution Alternatives S' i = [ Σ (v ' v i ) 2 ] ½ Step 5: Calculate the relative closeness to the ideal solution Ci* = S' i / (S i * +S' i ) Table 8: Relative Closeness to Ideal Solution Alternatives S' i /(S * i +S' i ) Ci* / / / / ASIAN JOURNAL OF MANAGEMENT RESEARCH 153

6 According to the value of Ci* = S' i / (S i * + S' i ) the best alternative is 1 as 0.74 is nearest to Conclusion Decision making problems involve vagueness or imprecision which are due to, lack of information regarding the environment. Fuzzy set theory is one of the most effective tools to handle imprecision in decision making problems. In the present work TOPSIS method has been extended for fuzzy values. In this approach we considered normalized distance measure to calculate the distance of alternative from positive ideal solution, its distance from negative ideal solution is also considered. That is to say, the less the distance of the alternative under evaluation from the positive ideal solution and the more its distance from the negative ideal solution, the better its ranking. 6. References 1. C. Carlsson, (1988), approximate Reasoning through fuzzy MCDM-models, Operational Research, 87, North-Holland, pp C.L. Hwang and K. Yoon, (1981), Multiple Attribute Decision Making: Methods and Applications. Springer, Berlin, Heidelberg, New York. 3. C.L. Hwang and M.J. Lin, (1987), group decision-making under multiple criteria, Springer Verlag, New-York. 4. C.T. Chen, (2000), extension of the TOPSIS for group decision making under fuzzy environment. Fuzzy sets and systems, 114, pp H. Nurmi, (1981), approaches to collective decision making with fuzzy preference relations, Fuzzy sets and Systems, 6, pp H.S. Shih, H.J. Shyur, E.S. Lee, (2007), an Extension of TOPSIS for Group Decision Making, Mathematical Computational Models, 45, pp 801, J.C. Fodor and M. Roubens, (1994), Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer, Dordrecht. 8. M. Delgado, (1983), a resolution method for Multi-obective Problems, European Journal of Operational Research, 13, pp P. Vincke, (1992), Multicriteria Decision Aid, John Wiley and & Sons, Chichester. 10. R. E. Bellman and L. A. Zadeh, (1970), decision-making in a fuzzy environment, Management Sciences, 17, 1970, pp R.J. Li and E.S. Lee, (1990), Fuzzy Approaches to Multi-criteria De Novo Programs, Journal of Mathematical Analysis and Applications, 153, pp ASIAN JOURNAL OF MANAGEMENT RESEARCH 154

7 12. R.R. Yager, (1981), a new methodology for ordinal multiple aspect decisions based upon fuzzy sets, Decision Science, 12, pp S.M. Chen and J. M. Tan, (1994), handling multi-criteria fuzzy decision making problems based on vague set theory, Fuzzy Sets and Systems, 67, pp T.L. Saaty, (1980), the analytical Hierarchy Process, McGraw Hill, New York. 15. T. Solymosi, J. Dombi, (1986), a method for determining the weights of criteria: the centralized weights, European Journal of Operational Research, 26, pp T.X. Bui, (1987), a group decision support system for cooperative multiple criteria group decision making, Springer Verlag, New-York. 17. Y.J. Lai and C.L. Hwang, (1994), Fuzzy Multiple Obective Decision Making: Methods and Applications, Lecture Notes in Economics and Mathematical Systems, 404, Springer Verlag, New York. 18. Y.J. Lai, T.Y. Liu and C.L Hwang, (1994), TOPSIS for MODM, European Journal of Operational Research, 76(3), pp ASIAN JOURNAL OF MANAGEMENT RESEARCH 155