Choose Best Criteria for Decision Making Via Fuzzy Topsis Method

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1 Mathematics and Computer Science 2017; 2(6: doi: /j.mcs ISSN: (rint; ISSN: (Online Choose Best Criteria for Decision Making Via uzzy Topsis Method Muhammad Zulqarnain azal Dayan * Department of Mathematics University of Management and Technology Lahore akistan address: fazaldayan1@gmail.com (. Dayan * Corresponding author To cite this article: Muhammad Zulqarnain azal Dayan. Choose Best Criteria for Decision Making Via uzzy Topsis Method. Mathematics and Computer Science. Vol. 2 No pp doi: /j.mcs Received: October ; Accepted: October ; ublished: November Abstract: Multi Criteria Decision Making (MCDM uses different techniques to find a best alternative from multialternative and multi-criteria conditions. TOSIS is an important practical technique for ranking and selection of different alternatives by using distance measures. Classical TOSIS uses crisp techniques for the linguistic assessments but due to imprecise and fuzziness nature of the linguistic assessments we faced some problems to find out the solution of these problems. We proposed a uzzy TOSIS with its example in this work and use this method for decision making. Keywords: uzzy TOSIS uzzy Set Aggregated uzzy Decision Matrix Normalized uzzy Decision Matrix Weighted Normalized uzzy Decision Matrix 1. Introduction Hwang and Yoon invented the Technique for Order reference by Similarity to Ideal Solution (TOSIS in order to solve MCDM problem with many alternatives [1]. rom crisp to fuzzy data Chen & Hwang remodelled TOSIS [2]. urthermore Chen widened the TOSIS for roup Decision Making in fuzzy atmosphere []. Awasthi et al. used fuzzy TOSIS for the evaluation and selection of the best location planning for urban distribution centres []. Chu [5] and Yong [6] applied fuzzy TOSIS for choosing plant location with minimum costs and maximum use of resources. The relative closeness coefficients were obtained as fuzzy numbers and after defuzzification alternatives were ranked [7]. Wang and Elhang found that their method is much closer to the fuzzy weighted approach presented by Dong and Wong [8]. Mahmoodzadeh et al. incorporating fuzzy AH and TOSIS method gave a new procedure for the project selection problem. Improved fuzzy AH was used to compute the weights of each criterion at first and then TOSIS algorithm was engaged for ranking the projects to be selected [9]. Yong Tsao et al. utilized fuzzy TOSIS for supporting contractors to choose suitable project for bidding with the help of MADM. Triangular fuzzy numbers (TNs were assigned to each linguistic variable for alternative ratings and criteria weights [10]. Zadeh proposed Type-2 S to encompass uncertainty about the membership function in fuzzy set theory (ST [11]. Saremi and Montazer to cope with the decision-making problems having data with large uncertainty used a Type-2 fuzzy TOSIS based method. A Study of TOSIS in Classical uzzy Intuitionistic uzzy and Neutrosophic Environments Interval-valued fuzzy (IV set is a special type of Type-2 S. Saremi and Montazer applied IV sets having lower and upper triangular membership functions; they ranked the alternatives [12]. Chen and Tsao broadened IV-TOSIS to solve MADM problem [1]. Zulqarnain. M. and Saeed. M. proposed and proved the credibility of interval valued fuzzy soft matrix (IVSM in decision making. They discussed its different properties [1]. They also studied fuzzy soft matrix (SM and IVSM and redefined the product of IVSM. Later they used IVSM and SM in decision making problem with examples and compare the results. They observed that SM method is more appropriate for decision making [15]. They proposed a new decision making method on IVSM named as interval valued fuzzy soft max-min decision making method with the help of interval valued fuzzy soft max-min decision making function and used this method for decision making [16].

2 11 Muhammad Zulqarnain and azal Dayan: Choose Best Criteria for Decision Making Via uzzy Topsis Method 2. relimnires In this section we discussed about S and fuzzy TOSIS which is proposed by (Zadeh 1965 with examples and some important propositions Definition 1 [22] If A be a set and U be a universal set. Set A contains different elements of set U then we assign all elements of A any membership value between the interval [0 1] is called S Definition 2 [18] A fuzzy number is a convex and normal fuzzy subset of the universe of discourse X. 2.. Definition [22] If A and B are two S than A is called fuzzy subset of B if µ A (t µ B (t for all t U. It can be represented as A B. 2.. Example 2 Let U = { } be a universal set A and B are any two S such that A is subset of B. Let A = { } B = { } f A = µ A (t = { 1 /0. /0.1. /0.7 5 /0. 6 /0.9 7 /0.2} and f B = µ B (t = { 1 /0. 2 /0.2 /0.2 /0.9 5 /0.6 6 /0.9 7 /0.}. Clearly µ A (t is subset of µ B (t for all t U therefore Definition [22] A S in which µ A (t = 0 for all t U is called empty S Definition [22] A S in which µ A (t = 1 for all t U is called universal S Definition 5 [22] If A and B are two S than they are called equal S if µ A (t = µ B (t for all t U. It can be represented as A = B Definition 6 [22] Let A be a S than its complement is defined as 2.9. Definition 7 [22] f A c (t = 1- f A (t for all t U. If A and B are two S whose membership functions are µ A (t and µ B (t respectively than their union is also a S i.e. C = A U B which is defined as f C (t = max (µ A (t µ B (t for all t U Definition 8 [22] If A and B are two S whose membership functions are µ A (t and µ B (t respectively than their intersection is also a S i.e. C = A B which is defined as f C (t = min (µ A (t µ B (t for all t U.. uzzy Topsis Algorithm In order to deal with data and information containing nonstatistical uncertainties the best tool is the theory of S which was proposed by Zadeh. It is a matter of fact that we are not always in a position to express our viewpoint exactly. Many opinions are only unclear and uncertain. To model such scenarios more accurately in 1965 Zadeh proposed a new theory named as ST [19]. As far as fuzzy TOSIS is concerned Chen extended TOSIS to uzzy-tosis using TNs to replace the numeric linguistic scales for rating the alternatives and to assign the weights to the criteria [20]. One of the major differences between uzzy-tosis and Classical-TOSIS is that the later utilizes precisely known ratings and weights of the criteria [21]. uzzy-tosis is an application of uzzy logic and S. The Algorithm of uzzy TOSIS is given below. Consider a set of m Alternatives A = {.. } a set of n Evaluation Criteria C = {.. } and a set of i Decision Makers DMs = {DM 1 DM 2... DM i }. Step 1: Selection of a fuzzy rating scale for linguistic variables since alternatives as well as criteria both are in the form of linguistic variables. Step 2: Criteria weightage and fuzzy linguistic ratings for the alternatives given by DM s Taking as fuzzy ratings by k th decision maker for the i th alternative w. r. t the j th criterion represented by the following equation. = ( And is the weight allocated by the k th decision maker for the j th criterion represented by the equation given below. = { } Step : Compute the aggregated fuzzy ratings for the alternatives taking as aggregated fuzzy ratings for the i th alternative w. r. t the j th criterion = ( = { } { } = = and w j is the aggregated fuzzy weights for the j th criterion. = ( = { } = =

3 Mathematics and Computer Science 2017; 2(6: { } Step : Construction of ADM and Aggregated uzzy Weight Matrix (AWM. Now the uzzy MCDM problem will be converted to an ADM as given below = # where is aggregated fuzzy rating for the i th alternative w. r. t the j th criterion. Moreover AWM is defined as $% = [... ] transpose where is the aggregated fuzzy weight of j th criterion. Step 5: Normalization of the DM: The NDM is given below '(= [ ] m n = # where = * + / = 1 2 (benefit criteria = * 0 and - + / + = (cost criteria The Linear Scale Transformation (LST is used to normalize the Decision Matrix. The important point of this normalization method is that the normalized TN are within the interval [0 1]. Step 6: WNDM The WNDM 6(= is given below 6( = [7 ] m n = [ (. ] = (. (. (. (. (. (. (. (. (. Step 7: Determination of IS and NIS: Equations are used to find the IS and the NIS are given below. = ( where 7 = ( and = 1 (>?-@A@BC 2 =7 = ( where 7 = ( and = # 125 (DC-@A@BC 2 =7 i = m j = n Step 8: Calculation of distances E and E : The distances E and E of each weighted alternative 7 ; m from IS and NIS are calculated by using the distance formulas. E = E(7 7 where i = m E = E(7 7 where i = m where E is the distance between two fuzzy numbers. Step 9: Determination of Closeness Coefficient CC i : In order to rank the alternatives closeness coefficient is computed by the following equation.? + CC i =? +?+ for all i = m Step 10: Ranking the alternatives: The alternatives are ranked according to the fact that an alternative with CCi closer to 1 indicates that the alternative is close to the IS and distant form NIS. A large value of closeness index indicates a good performance of the alternative [2].. Application of uzzy Topsis.1. roblem Scenario A company Alpha has to choose a best one from two alternatives and. or the selection of best alternative the company hire a team of decision makers which consist on three members D = {DM 1 DM 2 DM }. our evaluation criteria s are mentioned for the selection of alternative which are given below. our Evaluation Criteria (n= represented by 5H2I 2I2 C = = JKI 2I2.2. Solution by uzzy Topsis N L L MN O N O {N Q Step 1: Selection of a uzzy Ratings Scale for Linguistic Variables Rating scale for linguistic variables is given below in table 1. Table 1. uzzy Ratings for Linguistic Variables. Criteria Weights Alternatives TN Very Low (VL Very oor (V (11 Low (L oor ( (15 Medium (M air ( (57 High (H ood ( (579 Very High (VH Very ood (V (799 Step 2: Criteria weightage and fuzzy linguistic ratings for the alternatives given by DM s In order to integrate the opinions of all the DMs each DM allocates some weights to the criteria given in the table 2.

4 116 Muhammad Zulqarnain and azal Dayan: Choose Best Criteria for Decision Making Via uzzy Topsis Method Table 2. Criteria Weightage by the DMs = { }. DM 1 DM 2 DM H (5 7 9 M ( 5 7 M ( 5 7 C 1 w = (w w w VH (7 9 9 H (5 7 9 H (5 7 9 C 2 w = (w w w VH (7 9 9 H (5 7 9 H (5 7 9 C w = (w w w M ( 5 7 L (1 5 L (1 5 C w = (w w w Table shows the aggregated fuzzy weights = ( for each criteria (j=1 2. Table. The aggregated fuzzy weights = ( for each criteria (j=1 2 are calculated. = ( = ( Where = 125 }= min{5 }= = ( = ( = = [ ] = = ( = ( = 1 } = max{9 7 7} = 9 = ( = ( Similarly and are calculated $% = [ ] Transpose = [( ( ( ( ] Transpose The alternative ratings according to the DM s are listed in the below table. Table. Alternative Ratings = (. Cri. C 1 C 2 C C Alts. Decision Makers DM 1 DM 2 DM x = (a b c = (57 x = (a b c = (57 x = (a b c = (57 x = (a b c = (579 x = (a b c = (579 x = (a b c = (57 V V V x = (a b c = (799 x = (a b c = (799 x = (a b c = (799 V x = (a b c = (579 x = (a b c = (57 x = (a b c = (579 x = (a b c = (15 x = (a b c = (57 x = (a b c = (15 x = (a b c = (15 x = (a b c = (15 x = (a b c = (15 x = (a b c = (57 x = (a b c = (57 x = (a b c = (15 x = (a b c = (15 x = (a b c = (15 x = (a b c = (57 Step : Compute the aggregated fuzzy ratings for the alternatives Aggregated uzzy Ratings for alternatives are calculated by using given equations = (a 11 b 11 c 11 a 11 = 125 } = min { } = = = [5+5+5] = 5 Therefore c 11 = 1 } = max {7 7 7} = 7 = ( Other values are calculated in a similar manner and are given in the below table 5. Step-: Construction of ADM % = [ ] Table 5. Construction of ADM. C 1 C 2 C C =( =( =( =( =( =( =( =( Step 5: Normalization of the ADM The values for the NADM are calculated by using the equations which are given for benefit criteria and cost criteria.

5 Mathematics and Computer Science 2017; 2(6: = X L - LL = X Z [ = X L - OL L / LL = X \ ]..000} = = X LO - O L Y where C 1 is the cost criteria LL Y where = = min { } = L L Y where C / OL 1 is the cost criteria OL Y where = = min {.000 / LO - O - LO - O Y where C 2 is the benefit criteria = X Z \ \ Y where \ \ \ = 1 2 = max { } = = X OO / OO - OO - O - O - O Y where C 2 is the benefit criteria = X [ Z.]]Z \ \ 9.000} = = X L - / L - \ Y where \ = 1 2 = max { L - Y where C is the benefit criteria = X.]]Z Z Z 5.000} = = X O - / O - Z Y where Z = 1 2 = max { O - Y where C is the benefit criteria = X [ Y where Z Z Z = 1 2 = max { } = = X LQ / LQ - LQ - Q - Q - Q Y where C is the benefit criteria = X [ Y where Z Z Z = 1 2 = max { } = Table 6. NADM '% = [ ]. = X OQ / OQ - OQ - Q - Q - Q Y where C is the benefit criteria = X.]]Z Z Y where Z Z Z = 1 2 = max { } = Therefore Normalized Aggregated uzzy Decision Matrix (NADM is tabulated in the given table 6. C 1 C 2 C C = ( = ( = ( = ( = ( = ( = ( = ( Step 6: WNDM To get WNDM the values from above table are multiplied with AWM which are calculated in table 7. Table 7. WNDM 6% = [7 ]. C 1 C 2 C C 7 = ( = ( = ( = ( = ( = ( = ( = ( Step 7: Determination of IS and NIS Table 8 gives the calculated values of IS and NIS respectively. Table 8. Calculated values of IS and NIS. A * 7 = (999 7 = (999 7 = (999 7 = (777 7 = (111 7 = ( = ( = ( Step 8: Calculation of distances E and E To find the distances E and E it is necessary to find out the distance of each weighted normalized alternative from the IS and NIS respectively. d (7 7 j=12 d (7 7 = d(( (9 9 9 for j=1 = 5.50 (( ( ( d (7 7 = d(( (9 9 9 for j=2 (( ( ( d (7 7 = d(( (9 9 9 for j= = (( ( ( d (7 7 = d(( (7 7 7 for j= =.809 (( ( ( The remaining values d (7 7 d (7 7 d (7 7 for j= 1 2 are left for the sake of brevity. In table 9 the distances d( and d( from IS and NIS for the Alternatives are calculated. =.09

6 118 Muhammad Zulqarnain and azal Dayan: Choose Best Criteria for Decision Making Via uzzy Topsis Method Table 9. Distances d( and d( from IS and NIS for the Alternatives. C 1 C 2 C C IS d (7 7 = 5.50 d (7 7 =.09 d (7 7 = d (7 7 =.809 IS d (7 7 = 5.88 d (7 7 =.86 d (7 7 = d (7 7 =.926 NIS d (7 7 =.82 d(7 7 =.61 d(7 7 = 5.19 d(7 7 =.15 NIS d (7 7 =.72 d(7 7 =.195 d(7 7 =.617 d(7 7 =.089 The distance E of each weighted alternative from IS is computed using E = E(7 7 : i = 1 2 Now E for the alternative form IS A * is calculated by using this equation E = E(7 7 = E(7 7 + E(7 7 + E(7 7 + E(7 7 utting the values in above equation from above table. E = E(7 7 + E(7 7 + E(7 7 + E(7 7 = = 18.9 Similarly E E and E are calculated but for the sake of brevity only their respective values are given in the table 10. Table 10. The distance of each weighted alternative. j k j l j k j l Step 9: Determination of Closeness Coefficient CC i inally the closeness coefficient CCi of each alternative (i =1 2 is calculated by using the formula CCi = CC 1 = CC 2 =? +? +?+ as follows m.z m.zm.\ = 0.97 ].] ].]n.]z = 0.5 Step 10: Ranking the alternatives Hence the ranking order for the alternatives is i.e. the best choice considering the given criteria. 5. Conclusion Classical TOSIS uses crisp techniques for the linguistic assessments but due to imprecise and fuzziness nature of the linguistic assessments we faced some problems to find out the solution of these problems we proposed the uzzy TOSIS with its example. It is evident that fuzzy TOSIS has the ability to deal situations where ambiguity occurs due to the presence of linguistic variables whereas Classical TOSIS is lacking this property. Acknowledgements The authors are very grateful to the editor and reviewers for their comments and suggestions which are helpful in improving the paper. References [1] Hwang C Yoon K Multiple Attribute Decision Making: Methods and Applications A State of the Art Survey 1 (1981. [2] Chen S. J. J Hwang C. L uzzy Multiple Attribute Decision Making Methods 75 (1992. [] Chen C. T. Extensions of the TOSIS for group decisionmaking under fuzzy environment uzzy Sets Syst 11 (1: 1 9 (2000. [] Awasthi A Chauhan S. S oyal S. K A multi-criteria decision making approach for location planning for urban distribution centers under uncertainty Math. Comput. Model 5 (1 2: (2011. [5] Chu T. C. Selecting plant location via a fuzzy TOSIS approach Int. J. Adv. Manuf. Techno 20 (11: (2002. [6] Yong D. lant location selection based on fuzzy TOSIS Int. J. Adv. Manuf. Technol 28 (7 8: 89 8 (2006. [7] Wang Y. M Elhag T. M. S uzzy TOSIS method based on alpha level sets with an application to bridge risk assessment Expert Syst. Appl. 1 (2: (2006. [8] Dong W. M Wong. S uzzy Weighted Averages and Implementation of the Extension rinciple 21: (1987. [9] Mahmoodzadeh S. Shahrabi J. ariazar M. and Zaeri M. S. roject selection by using fuzzy AH and TOSIS technique Int. J. Soc. Manag. Econ. Bus. Eng 1 (: 2 29 (2007. [10] Yong tao T. Shen L. Langston C. and Liu Li. Construction project selection using fuzzy TOSIS approach J. Model. Manag 5 (: (2010. [11] Zadeh L. A. The concept of a linguistic variable and its application to approximate reasoning-i Inf. Sci. (Ny 8 (: (1975. [12] Saremi H. Q and Montazer. A An application of type-2 fuzzy notions in website structures selection Utilizing extended TOSIS method WSEAS Trans. Comput 7 (1: 8 15 (2008.

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