A MODIFICATION OF FUZZY TOPSIS BASED ON DISTANCE MEASURE. Dept. of Mathematics, Saveetha Engineering College,

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1 International Journal of Pure and pplied Mathematics Volume 116 No , ISSN: (printed version; ISSN: (on-line version url: ijpam.eu MODIFICTION OF FUZZY TOPSIS BSED ON DISTNCE MESURE 1 V.nandan, 2 G.Uthra 1 R & D Centre, Bharathiar University, Coimbatore and Dept. of Mathematics, Saveetha Engineering College, 2 Dept. of Mathematics, Pachaiyappa s College, Chennai, 1 anandanworld@gmail.com, 2 uthragopalsamy@gmail.com bstract: Multi Criteria Decision Making problems consider various criteria to obtain the best alternative from the available alternatives. Sometimes ranking the alternatives becomes tedious due to partial and incomplete information. Several MCDM techniques like HP, TOPSIS, ELECTRE, PROMETHEE deals this concern to help decision making. In this paper, a new distance approach which is alternate to euclidean distance has been proposed to determine the distance between the ideal solutions in TOPSIS. The improvement of the proposed modified TOPSIS method has been validated using a numerical example. The proposed and traditional TOPSIS methods are compared further to justify the efficiency of the method hence it can be applied in various industrial, manufacturing, transport and environmental problems. Keywords: MCDM, TOPSIS, Fuzzy Set Theory, Euclidean Distance 1. Introduction Decision making is the process of obtaining the best alternative among the available and feasible alternatives. MCDM technique has been generally used to deal situations, where there exist many alternatives and criteria with different ratings and weights. Hwang and Yoon introduced the classical TOPSIS [1-5] method using the Euclidean distance between the positive and negative ideal solution to rank the alternatives. The alternative with the farthest distance from the negative ideal solution and shortest distance from the ideal solution is said to be optimal. L..Zadeh [6,7] originally proposed fuzzy sets theory to deal decision making process which has illdefined data and also introduced linguistic variable through which each partial or subjective variable can be associated with a numerical fuzzy value. Using Z-numbers, bdul Malek Yaakob and lexander Gegov [8] developed Z-TOPSIS to provide the ranking concept and validated the method by Spearman s rank correlation. Jyh-Jiuan Lin et al [9] extended TOPSIS under Minkowski s L λ metric with λ=2 (Euclidean distance measure for MCDM. Using entropy weights and possibility concepts, S. S. ppadoo et al [10] introduced (φ, λ mixed solution strategy for group fuzzy TOPSIS. Sung-Hyuk Cha [11] has done a comprehensive survey on distance and similarity measures by adopting the correlation coefficient and hierarchical clustering technique to reveal similarities among numerous distance/similarity measures. Further, the distance measures are classified into 10 different families. Seung- Seok Choi et al [12] classified 76 distance measures through hierarchical clustering and observed that few measures have very close relationships between them. In this paper, a new distance approach has been introduced to modify the traditional TOPSIS method and the comparison indicates that the method is effective under fuzzy environment. 2. Definitions.Trapezoidal Fuzzy Number trapezoidal fuzzy number can be represented as a=a,a,a,a whose membership function is given 0, a, a x a by µ x= 1, a x a, a x a 0, a x B. Operations on Trapezoidal Fuzzy Numbers The operations of any two fuzzy numbers a = a,a,a,a and b =b,b,b,b can be expressed as: 109

2 International Journal of Pure and pplied Mathematics (i ddition: a+b =a +b,a +b,a +b,a +b, 0,b 0 (ii Multiplication: a b =a b,a b,a b,a b, a 0,b 0 (iii Subtraction: a b =a b,a b,a b,a b, 0,b 0 (iv Division: a/b =a /b,a /b,a /b,a /b, a 0,b 0 C. Euclidean Distance Let a=a,a,a and b =b,b,b be any two fuzzy numbers, then the distance between them is given by da,b=% 1 6 a b +2a b +2a b +a b 2 Normalization is done using n -3 =, i = 1,..,m; j 7 a = 1,..,n. (1 3 Weighted normalized decision matrix are obtained by v -3 =w 3 n -3, (2 i = 1,.,m; j = 1,.,n. where w 3 is weight of the j th ; criterion and 3/ w 3 = 1. a 4 The positive ideal and negative ideal solution are determined using = = >v =,..,v = ; = Bmax - v -3, j=1,..,nd, = >v,..,v ; = Bmin - v -3, j=1,..,nd, 5 Find the separation measure using the proposed distance measure d(a, b = PIS: d - = = NIS: d - = G F 45 G =F F 45 G H F 45 G =F H F H 45 G I F 45 G =F I F I 45 6 Calculate the relative closeness to ideal solution by D. Minkowski Family I K C - = 4 K H 4 = K I, i = 1,.,m (5 4 Let a=a,a,a,a and b =b,b,b,b be any 7 Rank preference are given according to ascending order of two trapezoidal fuzzy numbers, then the distance between the TOPSIS grade C - a and b is given by: 8 4. Numerical Example 45,i=1,,m (3,i=1,,m (4 Minkowski d(a,b=*+ a - b -. -/ City Block d(a,b=*+ a - b - Euclidean Chebyshev Proposed Distance Measure -/ d(a,b=*+ a - b - -/ d(a, b = max a - b - - -/a - b - d(a,b= -/a - +b - a - b - manufacturing company desires to select a suitable material supplier to purchase the key components of new products. fter preliminary screening, five candidates (1, 2, 3, 4 and 5 remain for further evaluation. committee of three decision makers, D1, D2 and D3 has been formed to select the most suitable supplier. Five benefit criteria are considered: 1 Profitability of supplier (C1, 2 Relationship closeness (C2, 3 Technological capability (C3, 4 Conformance quality (C4, 5 Conflict resolution (C5. The linguistic rating variables to the suppliers of 3 decision makers with respect to each criterion are evaluated and shown as fuzzy decision matrix in Table 1. Several modifications were made in TOPSIS, using the minkowski distance family [City Block, p=1; Euclidean, p=2 and Chebyshev, p= ]. This work proposes a new distance which is alternate to the above mentioned family of distances. 3. The Proposed Method 1 Frame a fuzzy decision matrix with trapezoidal fuzzy numbers. 110

3 International Journal of Pure and pplied Mathematics Table 1. Fuzzy Decision Matrix C1 C2 C3 C4 C5 1 (5,6,7,8 (5,7,8,10 (7,8,8,9 (7,8,8,9 (7,8,8,9 2 (7,8,8,9 (8,9,10,10 (8,9,10,10 (7,8.7,9.3,10 (8,9,10,10 3 (7,8.7,9.3,10 (7, 8.3, 8.7, 10 (7,8.7,9.3,10 (8,9,10,10 (7,8.3,8.7,10 4 (7,8,8,9 (5,7.3,7.7,9 (5,6.7,7.3,9 (7,8,8,9 (7,8.3,8.7,10 5 (5,6,7,8 (5,7.3,7.7,9 (5,6,7,8 (5,6.7,7.3,9 (5,6,7,8 Weight (0.7,0.8,0.8,0.9 (0.8,0.9,1.0,1.0 (0.7,0.87,0.93,1.0 (0.7,0.8,0.8,0.9 (0.7,0.8,0.8,0.9 Using Eqn (1 and (2, The weighted normalized matrix are obtained and shown in Table ( Table 2. Weighted Normalized Matrix C1 C2 C3 C4 C5 (0.35,0.48,0.56,0.72 (0.49,0.70,0.74,0.90 (0.49,0.64,0.64,0.81 (0.49,0.64,0.64,0.81 (0.40,0.63,0.80,1 (0.49,0.64,0.64,0.81 (0.49,0.70,0.74,0.90 (0.56,0.72,0.80,0.90 (0.64,0.81,1,1 (0.56,0.78,0.93,1 (0.49,0.70,0.74,0.90 (0.56,0.72,0.80,0.90 (0.49,0.66,0.70,0.90 (0.56,0.75,0.87,1 (0.49,0.76,0.86,1 (0.49,0.64,0.64,0.81 (0.40,0.66,0.77,0.90 (0.35,0.58,0.68,0.90 (0.49,0.64,0.64,0.81 (0.49,0.66,0.70,0.90 (0.35,0.48,0.56,0.72 (0.40,0.66,0.77,0.90 (0.35,0.52,0.65,0.80 (0.35,0.54,0.58,0.81 (0.35,0.48,0.56, The PIS ( + and NIS ( - are the best and worst values of each criterion, shown in Table 3 Table 3. Positive and Negative Ideal Solution C1 C2 C3 C4 C5 (0.90,0.90,0.90,0.90 (0.90,0.90,0.90,0.90 (0.90,0.90,0.90,0.90 (1,1,1,1 (1,1,1,1 - (0.35,0.35,0.35,0.35 (0.40,0.40,0.40,0.40 (0.35,0.35,0.35,0.35 (0.35,0.35,0.35,0.35 (0.35,0.35,0.35,0.35 The relative closeness coefficient grade and ranking of alternatives are calculated using the proposed distance measure (3, (4 & (5 and shown in Table 4. Table 5 shows the closeness coefficient grade and ranking of alternatives using traditional fuzzy TOPSIS method Table 4. Modified TOPSIS Grade & Rank Table 5. Traditional TOPSIS = d - d - Grade Rank Grade Rank

4 International Journal of Pure and pplied Mathematics Fig.1 Proposed Distance Measure Fig.2 Comparison of Relative Closeness d i + d i - Proposed Traditional 5. Results and Discussion Table 4 shows the TOPSIS grade and the rank preferences are given in ascending order whereas the traditional TOPSIS rank takes descending order shown in Table 5. The ranking of proposed and traditional TOPSIS are almost same except the first two positions. The ranking order will vary depending upon the relative closeness coefficient. Fig.1 shows the distance between the positive and negative solution of proposed method. It is observed that there exists a significant difference between the distances of positive and negative ideal solution which indicates that the method assures ideal ranking. Fig.2 shows the comparison of ranking with the traditional method. 6. Conclusion In this paper, 1 the fuzzy TOPSIS method has been modified by a new distance measure, 2 the proposed method is used to rank and identify the best alternative from the given alternatives, 3 the method is suitable for problems involving different criterion weights and conflicting criteria, 4 the method is good enough under fuzzy environment when the data are of qualitative in nature or ill defined, 5 the proposed method can be used in all industrial, manufacturing, transport and environmental problems where there exists n number of alternatives with multiple criteria, 6 the comparison of the proposed method with the traditional TOPSIS method justify the method is useful and effective tool for MCDM problems. Reference [1] C.L. Hwang, K. Yoon, Multiple ttribute Decision Making Methods and pplications, Springer, Berlin Heidelberg, [2] S.J. Chen, C.L. Hwang, Fuzzy Multiple ttribute Decision Making: Methods and pplications, Springer, Berlin, [3] C.T. Chen, Extensions to the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems 114 ( [4] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Izadikhah, Extension of the TOPSIS method for decisionmaking problems with fuzzy data, pplied Mathematics and Computation 181 ( [5] Jiang Jiang, Yu-wang Chen, Ying-wu Chen, Kewei Yang, TOPSIS with fuzzy belief structure for group belief multiple criteria decision making, Expert Systems with pplications 38 ( [6] L.. Zadeh, Fuzzy sets, Information and Control 8 ( [7] L.. Zadeh., The Concept of a Linguistic Variable and its pplication to pproximate Reasoning-I: Computer Sciences Division, Department of Electrical Engineering and Computer Sciences, and the Electronics Research Laboratory, University of California, Berkely, California [8] bdul Malek Yaakob and lexander Gegov, Interactive TOPSIS Based Group Decision Making Methodology Using Z-Numbers, International Journal of Computational Intelligence Systems, Volume 9,Issue 2, (2016, [9] Jyh-Jiuan Lin, Miao-Chen Chiang & Ching-Hui Chang, comparison of usual indices and extended TOPSIS methods in mutual funds performance 112

5 International Journal of Pure and pplied Mathematics evaluation, Journal of Statistics and Management Systems, Volume 10, Issue 6, (2007, [10] S. S. ppadoo, S. K. Bhatt & C. R. Bector, mixed solution strategy for group multi-attribute TOPSIS model with application to supplier selection problem, Journal of Information and Optimization Sciences, Volume 33, Issue 1, (2012, [11] Sung-Hyuk Cha, Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions, International Journal of Mathematical Models and Methods In pplied Sciences, Issue 4, Volume 1, 2007 [12] Seung-Seok Choi, Sung-Hyuk Cha, Charles C. Tappert, Survey of Binary Similarity and Distance Measures, Systemics, Cybernetics and Informatics, VOLUME 8 - NUMBER 1,

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