A general fuzzy TOPSIS model in multiple criteria decision making

Int J Adv Manuf Technol (2009) 45:406–420 DOI 10.1007/s00170-009-1971-5 ORIGINAL ARTICLE A general fuzzy TOPSIS model in multiple criteria decision making Iraj Mahdavi & Armaghan Heidarzade & Bahram Sadeghpour-Gildeh & Nezam Mahdavi-Amiri Received: 9 August 2007 / Accepted: 10 February 2009 / Published online: 14 March 2009 # Springer-Verlag London Limited 2009 Abstract Decision making is the process of finding the best option among the feasible alternatives. In classical multiple criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. Owning to vagueness of the decision data, the crisp data are inadequate for real-life situations. Since human judgments including preferences are often vague and cannot be expressed by exact numerical values, the application of fuzzy concepts in decision making is deemed to be relevant. In this paper, we proposed the application of a fuzzy distance formula in order to compute a crisp value for the standard deviation of fuzzy data. Then, we use this crisp value of the standard deviation to normalize the fuzzy data using the distance formula again. In our normalization approach, we have enough flexibility to consider various types of fuzzy numbers (such as triangular, trapezoidal, and interval). Finally, we use the technique for order preference by similarity to I. Mahdavi (*) Department of Industrial Engineering, College of Technology, Mazandaran University of Science & Technology, P.O. Box 734, Babol, Iran e-mail: irajarash@rediffmail.com A. Heidarzade Department of Industrial Engineering, Payame noor University, Sari, Iran B. Sadeghpour-Gildeh Department of Statistics, University of Mazandaran, Babolsar, Iran N. Mahdavi-Amiri Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran an ideal solution to determine the ranking order of the alternatives. A numerical example from the literature is solved to demonstrate this applicability of the proposed model. We also compare our proposed approach with similar methods in the literature using some examples with known results and a number of randomly generated test problems. The results point to the applicability of our method and signify its effectiveness in identifying solutions. Keywords TOPSIS . Fuzzy number . MCDM . Fuzzy distance . Standard deviation . Normalization 1 Introduction Decision making is the process of selecting a possible course of action from a set of alternatives. In almost all problems, the multiplicity of criteria for judging an alternative prevails. That is, for many problems, the decision maker wants to attain more than one goal in selecting the course of action while satisfying the constraints dictated by environment processes and resources [1]. Technique for order preference by similarity to an ideal solution (TOPSIS), known as a classical multiple criteria decision-making (MCDM) method, has been developed by Hwang and Yoon [2] for solving the MCDM problems. The basic principle of the TOPSIS is that the chosen alternative should have the “shortest distance” from the positive ideal solution and the “farthest distance” from the negative ideal solution. The TOPSIS introduces two “reference” points, but it does not consider the relative importance of the distances from these points. As the assessment values have various types of vagueness/imprecision or subjectiveness, one cannot always use the classical decision-making techniques for these decision problems. In the past few Int J Adv Manuf Technol (2009) 45:406–420 years, numerous attempts to handle this vagueness, imprecision, and subjectiveness have been carried out to apply fuzzy set theory to multiple criteria evaluation methods [3– 8 ]. The overall utility of the alternatives with respect to all criteria is often represented by a fuzzy number, which is named the fuzzy utility and is often referred to by fuzzy multicriteria evaluation methods. The ranking of the alternatives is based on the comparison of their corresponding fuzzy utilities [4, 5, 9, 10]. Multicriteria evaluation methods are used widely in fields such as information project selection [10, 11], material selection [12], and many other areas of management decision problems [12–16] and strategy selection problems [13, 17–19]. Tsaur et al. [20] first convert a fuzzy MCDM problem into a crisp one via centroid defuzzification and then solve the non-fuzzy MCDM problem using the TOPSIS. Chen and Tzeng [13] transform a fuzzy MCDM problem into a crisp MCDM problem using fuzzy integral. Instead of using distance, they employ a gray relation grade to define the relative closeness of each alternative. Chu [21, 22] also changes a fuzzy MCDM problem into a crisp one and solves the crisp MCDM problem using the TOPSIS. Differing from the others, he first derives the membership functions of all the weighted ratings in a weighted normalization decision matrix using interval arithmetic of fuzzy numbers and then defuzzifies them into crisp values using the ranking method of mean of removals. Chen [14] extends the TOPSIS to fuzzy group decision-making situations by defining a crisp Euclidean distance between any two fuzzy numbers. Triantaphyllou and Lin [23] develop a fuzzy version of the TOPSIS based on fuzzy arithmetic operations, which leads to a fuzzy relative closeness for each alternative. Hsu and Chen [24] discuss an aggregation of fuzzy opinions under group decision making. Li [25] proposes a simple and efficient fuzzy model to deal with multi-judges/ MCDM problems in a fuzzy environment. Li [26] proposes several linear programming models and methods for multiattribute decision making under “intuitionistic fuzziness” where the concept of intuitionistic fuzzy sets is a generalization of the concept of fuzzy sets. Liang [27] incorporates fuzzy set theory and the basic concepts of positive ideal and negative ideal points and extends MCDM to a fuzzy environment. Ölçer and Odabaşi [28] propose a new fuzzy multi-attribute decision-making method, which is suitable for multiple attributive group decision-making problems in a fuzzy environment, and the method can deal with the problems of ranking and selection. Olson and Wu [29] present a simulation of fuzzy multi-attribute models based on the concept of gray relations, reflecting either interval input or commonly used trapezoidal input. This model is a simulated fuzzy MCDM model that can be applied to multiattribute decision-making problems effectively. Yeh et al. [15] propose a fuzzy MCDM method based on the concepts 407 of positive ideal and negative ideal points to evaluate performance of the bus companies. Despite the applicability of these methods to many decision-making problems, typical fuzzy multicriteria analyses require the comparison of fuzzy numbers. However, the comparison process can be quite complicated and may produce unreliable results [15, 28, 30, 31]. Jahanshahloo et al. [32, 33] extend the TOPSIS to decision-making problems with interval data and fuzzy data. Here, we first convert the decision matrix into a fuzzy decision matrix and construct a weighted fuzzy decision matrix once the decision makers’ fuzzy ratings have been pooled. The new process of normalization by use of fuzzy distance value and normal fuzzy deviation approach are applied for normalization and detection of the crisp value. According to the concept of TOPSIS, we define the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS). Finally, a closeness coefficient is applied to calculate the ranking order of all alternatives. The higher value of the closeness coefficient indicates that an alternative is closer to FPIS and farther from FNIS simultaneously. The rest of the paper is organized as follows: Section 2 describes the basic definitions and notations concerning fuzzy numbers and linguistic variables. In Section 3, we propose an approach for normalization of the fuzzy data using a fuzzy distance formula. We then present an algorithm to extend TOPSIS to deal with fuzzy data. In Section 4, the proposed algorithm is illustrated with seven examples. Finally, the conclusions are pointed out in Section 5. 2 Fuzzy numbers and linguistic variables The representation of multiplication operation on two or more fuzzy numbers is a useful tools for decision makers in the fuzzy multiple criteria decision-making environment for ranking all the candidate alternatives in selecting an optimal one. In this section, some basic definitions of fuzzy sets, fuzzy numbers, and linguistic variables are reviewed from Buckley [3], Kaufmann and Gupta [34], and Zadeh [6–8]. The basic definitions, commonly being used in fuzzy set theory, are given in Appendix. Next, the definitions of linguistic variable, Dpq distance and standard deviation of a fuzzy numbers being used here are given. Definition 1 A linguistic variable is a variable whose value is given in linguistic terms [9, 23]. Linguistic terms have been found intuitively easy to use in expressing the subjectiveness and/or qualitative imprecision of a decision maker’s assessments [15, 23, 35]. 408 Int J Adv Manuf Technol (2009) 45:406–420 Definition 2 A fuzzy MCDM problem with m alternatives and n criteria can be concisely expressed in a fuzzy decision matrix as: A1 A2 ~ D ¼ A3 .. . Am 2 ~C1 x 11 6 ~x21 6 6 ~x31 6 6 .. 4 . ~x m1 C2 ~x 12 ~x 22 ~x32 .. . ~x m2 C3 . . . Cn 3 ~x . . . ~x1n 13 ~x . . . ~x2n 7 23 7 ~x33 . . . ~x3n 7 7; .. 7 .. . . . 5 . . ~x . . . ~xmn m3 ð1Þ ~ ~ ;w ~ ; . . . ;w ~ Š; W ¼½w 1 2 n ~j , j=1,2,…,n, are where ~xij , i=1,2,…,m, j=1,2,…,n, and w ~ linguistic fuzzy numbers. Note that wj represents the weight of the jth criterion, Cj, and ~xij is the performance rating of the ith alternative, Ai, with respect to the jth criterion, Cj, evaluated by k evaluators. This study applies the method of average value ~xij for k evaluators concerning the same evaluation criteria, that is,  1  ð2Þ x~ij ¼  ~x1ij  ~x2ij 


 ~xijk ; k ~ b~ Þ ¼ Dp;q ð a; 8h > < ð1 > : ð1 where ~x pij is the rating of alternative Ai with respect to criterion Cj evaluated by the pth evaluator. The weighted fuzzy decision matrix is: 2~ v11 6 v~21 6 ~ 6~ V ¼ 6 v31 6 .. 4 . v~m1 2~ w1 ~ 6w 6 1 6 6 ¼6 ~ 6w 6 1 6 4 ~1 w 0<a1 0<a1 The analytical properties of Dp,q depend on the first parameter p, while the second parameter q of Dp,q characterizes the subjective  weight  attributed to the end points of the support, i.e., aþ ; a a a of the fuzzy numbers. If there is no reason for distinguishing any side of the fuzzy numbers, Dp;12 is recommended. Having q close to 1 results in considering the right side of the support of the fuzzy numbers more favorably. .. . .. . v~13 v~23 v~33 .. . v~m3








.. . ~ ~2 x11 w ~ ~2 x21 w ~ x12 ~ x22 ~ xi1 ~2 w ~ ~2 xm1 w v~1n v~2n v~3n .. .


v~mn .. . .. . ~ xi2 ~ xm2 p < 1; p ¼ 1: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ffi u 2 u X 4 X 1 ~ b~ Þ ¼ u t 4 ðbi ai Þðbiþ1 aiþ1 Þ5: ð bi ai Þ 2 þ D2; 12 ð a; 6 i¼1 i2f1;3g ð6Þ 7 7 7 7 7 5 ~j ... w ~j ... w .. . ~j ... w .. . ~j ... w .. . .. . ~ x1j ~ x2j ~ xij ~ xmj ~n ... w ~n ... w .. . ~n ... w .. . ~n ... w 3 ~ x1n ~ x2j 7 7 .. 7 7 . 7 ~ xin 7 7 7 .. 5 . ~ xmn ð3Þ ð4Þ Due to the use of standard deviation of fuzzy numbers in the normalization process, p=2 is more useful in calculating standard deviation, and hence, we assume that p=2. Since the significance of the end points of the support of the fuzzy numbers is assumed to be the same, then we consider q ¼ 12. For triangular fuzzy numbers ã=(a1,a2,a3) and e b ¼ ðb1 ; b2 ; b3 Þ, the above distance with p=2 and q ¼ 12 is then calculated as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u 2 u X 3 X 1 u ~ b~ Þ ¼ t 4 D2; 12 ð a; ðbi ai Þðbiþ1 aiþ1 Þ5: ð bi ai Þ 2 þ ð b2 a2 Þ 2 þ 6 i¼1 i2f1;2g ~ And if a~ ¼ ða1 ; a2 ; a3 ; a4 Þ and b ¼ ðb1 ; b2 ; b3 ; b4 Þare trapezoidal fuzzy numbers, the distance is calculated as: 3 Definition 3 The Dp,q distance, indexed by parameters 1<p ~ <∞ and 0<q<1, between two fuzzy numbers ã and b is a nonnegative function given by [36]: i1 p  R  R1  a  da þ q 1 aþ bþ p da p ; b a a a a 0 0      bþ qÞ sup aa ba  þ q inf aþ a a ; qÞ v~12 v~22 v~32 .. . v~m2 ð5Þ Definition 4 The standard deviation of a fuzzy number is obtained as [36]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 2  ~ DS ¼ D2; 1 vij ; vj Þ; j 2 n ð7Þ Int J Adv Manuf Technol (2009) 45:406–420 409 where v~ij , i=1,2,…,m, j=1,2,…,n are fuzzy numbers and v~j , j=1,2,…,n are the mean values of fuzzy numbers calculated by:  n 1 ~ vj ¼   v~ij : ð8Þ n i¼1 Step 3-4: Obtain the absolute normalized fuzzy value of each element by Eq. 10. Step 3-5: Identify the sign of normalized fuzzy values by steps 3-6, 3-7, and 3-8. Step 3-6: Compute γmax as the maximal support ~ of components of V . Identify the fuzzy number g~max [for example, if compo~ nent of V are triangular or trapezoidal fuzzy numbers, then the fuzzy value of g~max becomes (γmax, γmax, γmax) or (γmax, γmax, γmax, γmax), respectively]. Definition 5 The normalized weighted fuzzy decision matrix, denoted by R, is: R ¼ rij mn : ð9Þ If v~ij , i=1,2,…,m, j=1,2,…n, are the fuzzy numbers in the weighted fuzzy decision matrix, then the normalization process can be performed by:   Dp;q v~ij ; v~j rij ¼ ; ð10Þ DS j where v~j is the mean of the  jth column in the weighted fuzzy decision matrix, Dp;q v~ij ; v~j and DSj are defined as in Eqs. 4 and 7, respectively.  For  triangular and trapezoidal fuzzy numbers, the Dp;q v~ij ; v~j and DSj are defined as in Eqs. 5 and 6, respectively. Note that the values of rij in the normalized weighted fuzzy decision matrix are crisp numbers. 3 The proposed algorithm Based on the discussions of Section 2, we now present an algorithm for solving the MCDM problem using the fuzzy TOPSIS with capability of general fuzzy numbers. 3.1 Algorithm GFTOPSIS-MCDM: general fuzzy TOPSIS in MCDM Step 1: Choose the linguistic ratings or fuzzy values ( x~ij , i= 1,2,…,m, j=1,2,…,n), for alternatives with respect to criteria and choose the appropriate ~ j , j=1,2,…n) as weights of linguistic variables ( w the criteria. ~ Step 2: Construct the weighted fuzzy decision matrix V ¼ v~ij mn by Eq. 3. Step 3: Determine the normalized weighted fuzzy decision matrix R ¼ rij mn using the following steps. Step 3-1: Calculate the mean value of each ~ column of matrix V by Eq. 8. Step 3-2: Obtain the distance of each element of ~ matrix V from mean value of the corresponding column by Eq. 4. Step 3-3: Identify the standard deviation of each ~ column of the matrix V by Eq. 7. ~ Calculate the distance between each component of V and   g~max : Di;j ¼ D2;12 v~ij ; g~max , i=1,2,…,m and j=1,2,…,n. Calculate the distance between the mean value in each  column and g~max : Dj ¼ D2;12 v~j ; g~max , j=1,2,…,n Step 3-7: For j=1,2,...n do For i=1,2,...m do if Dij < Dj then let Dij =−Dij. Step 3-8: Select the minimal value in matrix D and add its absolute value to all components of D. The result of this step is the normalized weighted fuzzy decision matrix. Note that the values of the normalized weighted fuzzy decision matrix are crisp numbers. Step 4: TOPSIS: Determine the positive ideal solution (PIS) and negative ideal solution (NIS) by: PIS ¼ Max rij ; j 2 J [ Min rij ; j 2 J 0 NIS ¼ Min rij ; j 2 J [ Max rij ; j 2 J 0 i i i i n o ¼ r1* ; r2* ; :::; rn* ;   ¼ r1 ; r2 ; :::; rn ; where J is associated with benefit criteria, and J′ is associated with the cost criteria. Step 5: Calculate the separation measures, using the ndimensional Euclidean distance. The separation of each alternative from the positive ideal solution is given by: ( )1 2 2 n  P dþ ¼ r r* ; 1  i  m: i j¼1 ij j Similarly, the separation from the negative ideal solution is given by: ( )1 2 2 n  P di ¼ ; 1  i  m: rij rj j¼1 Step 6: Calculate the relative closeness to the ideal solution. The relative closeness of alternative Ai with respect to PIS is defined by: CCi* ¼ di di þdiþ ; 1  i  m: 410 Int J Adv Manuf Technol (2009) 45:406–420 Table 2 Importance weights of the criteria from three decision makers Criteria C1 C2 C3 C4 C5 Fig. 1 Hierarchical structure of decision problem Since di  0 and diþ  0, then clearly, CCi* 2 ½0; 1Š. Step 7: Rank the preference order. For ranking alternatives using this index, we can rank alternatives in decreasing order. 4 Numerical illustrations Here, first we work out a numerical example, taken from [37], to illustrate the GFTOPSIS-MCDM algorithm for solving the decision-making problems with trapezoidal fuzzy data and then compare the performance of our method with similar methods in the literature. 4.1 Numerical example A high-technology manufacturing company desires to select a suitable material supplier to purchase the key components of new products [37]. After preliminary screening, five candidates (A1, A2, A3, A4, A5) remain for further evaluation. A 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), Decision makers D1 D2 D3 H VH VH H H H VH VH H H H VH H H H 3. technological capability (C3), 4. conformance quality (C4), and 5. conflict resolution (C5). The hierarchical structure of this decision problem is shown in Fig. 1. We apply our fuzzy TOPSIS algorithm (algorithm GFTOPSIS-MCDM) to solve this problem. We now give a summary of the computational procedure. Three decision makers use the linguistic weighting variables shown in Table 1 to assess the importance of the criteria. The importance weights of the criteria determined by these three decision makers are shown in Table 2. Decision makers use the linguistic rating variables shown in Table 3 to evaluate the ratings of candidates with respect to each criterion. The ratings of the five candidates by the decision makers under the various criteria are shown in Table 4. The linguistic evaluations shown in Tables 2 and 4 are converted into trapezoidal fuzzy numbers to construct the fuzzy decision matrix and determine the fuzzy weight of each criterion, as given in Table 5. The weighted fuzzy decision matrix and the mean value of fuzzy numbers in each column are constructed as in Table 6. The distance value of each element and the standard ~ deviation of each column of matrix V with the corresponding mean value have been shown in Table 7. The absolute standard normal value of each element of ~ matrix V is given in Table 8. ~ The maximum support of the fuzzy numbers in matrix V is γmax =10 and the fuzzy number of γmax is (10, 10, 10, Table 1 Linguistic variables for the importance weight of each criterion Table 3 Linguistic variables for the ratings Very low (VL) Low (L) Medium low (ML) Medium (M) Medium high (MH) High (H) Very high (VH) Very poor (VP) Poor (P) Medium poor (MP) Fair (F) Medium good (MG) Good (G) Very good (VG) (0; 0; 0.1; 0.2) (0.1; 0.2; 0.2; 0.3) (0.2; 0.3; 0.4; 0.5) (0.4; 0.5; 0.5; 0.6) (0.5; 0.6; 0.7; 0.8) (0.7; 0.8; 0.8; 0.9) (0.8; 0.9; 1.0; 1.0) (0; (1; (2; (4; (5; (7; (8; 0; 2; 3; 5; 6; 8; 9; 1; 2) 2; 3) 4; 5) 5; 6) 7; 8) 8; 9) 10; 10) Int J Adv Manuf Technol (2009) 45:406–420 411 Table 4 Rating of the five candidates by decision makers under various criteria Criteria C1 C2 C3 C4 C5 Suppliers Decision makers D1 D2 D3 A1 A2 A3 A4 A5 A1 A2 A3 A4 A5 MG G VG G MG MG VG VG G MG MG G VG G MG MG VG G G G MG G G G MG VG VG G MG G A1 A2 A3 A4 A5 A1 A2 A3 A4 A5 A1 A2 A3 A4 A5 G VG VG MG MG G G VG G MG G VG G G MG G VG VG MG MG G VG VG G MG G VG VG G MG G VG G G MG G VG VG G G G VG G VG MG ~ 10). We calculated distance value of each element V and corresponding mean value in each column with the fuzzy number of γmax to achieve the sign of each element. The result is shown in Table 9. The minimum negative value of Table 9 is −3.9554 and its absolute value is added to all the elements of the above matrix. The result of this step is the normalized weighted fuzzy decision matrix and is given in Table 10. The positive ideal (PIS) and negative ideal (NIS) solutions in the normalized weighted fuzzy decision matrix are determined: PIS ¼½6:9597; 7:5316; 6:8237; 6:8985; 6:5919Š NIS ¼½1:4678; 2:0570; 0:9571; 0:5468; 0Š: The calculated distance of each Ai, i=1,2,…,5 from PIS and NIS with respect to each criterion, respectively, are shown in Tables 11 and 12. The calculated diþ and di of five possible suppliers Ai, i= 1,2,…,5 and the closeness coefficient of each supplier are shown in Table 13. According to the closeness coefficient, the ranking order of all the alternatives can be determined. In this case, the best selection is candidate A2. The five alternatives are ordered as A2, A3, A4, A1, and A5. 4.2 Comparison results Here, we compare our proposed GFTOPSIS-MCDM algorithm with similar methods in two stages. At the first stage, six numerical examples with three alternatives and two criteria are used based on known results of Kuo et al. [38]. Here, we compare our algorithm with the three methods given by Chen and Hwang [5], Li [25], and Chen [14], respectively. In the first example, the preference orders of the three alternatives with respect to the first criterion are equal, and the ranking order of the three candidates with respect to the second criterion is A3, A2, and A1. Consequently, the total ranking of the alternatives is A3, A2, and A1. The results of the algorithms are shown in Table 14. We see that our proposed method and the method of Li [25] both find the correct solution, but the ones by Chen and Hwang [5] and Chen [14] fail to do so. The second example shows that preference of the alternatives is A2, A3, and A1. The results, shown in Table 15, indicate that the correct solution is found only by our method, and the other three methods obtain an incorrect result. The ranking order of the third example, shown in Table 16, is A3, A2, Table 5 Fuzzy decision matrix and fuzzy weights of five candidates A1 A2 A3 A4 A5 Weight C1 C2 C3 C4 C5 (5,6,7,8) (7,8,8,9) (7,8.7,9.3,10) (7,8,8,9) (5,6,7,8) (0.7,0.8,0.8,0.9) (5,7,8,10) (8,9,10,10) (7,8.3,8.7,10) (5,7.3,7.7,9) (5,7.3,7.7,9) (0.8,0.9,1.0,1.0) (7,8,8,9) (8,9,10,10) (7,8.7,9.3,10) (5,6.7,7.3,9) (5,6,7,8) (0.7,0.87,0.93,1.0) (7,8,8,9) (7,8.7,9.3,10) (8,9,10,10) (7,8,8,9) (5,6.7,7.3,9) (0.7,0.8,0.8,0.9) (7,8,8,9) (8,9,10,10) (7,8.3,8.7,10) (7,8.3,8.7,10) (5,6,7,8) (0.7,0.8,0.8,0.9) 412 Int J Adv Manuf Technol (2009) 45:406–420 Table 6 The weighted fuzzy decision matrix A1 A2 A3 A4 A5 Mean value C1 C2 C3 C4 C5 (3.5,4.8,5.6,7.2) (4.9,6.4,6.4,8.1) (4.9,6.96,7.44,9) (4.9,6.4,6.4,8.1) (3.5,4.8,5.6,7.2) (4.2,5.872,6.368, 7.92) (4,6.3,8,10) (6.4,8.1,10,10) (5.6,7.47,8.7,10) (4,6.57,7.7,9) (4,6.57,7.7,9) (4.8,7.002,8.42, 9.6) (4.9,6.96,7.44,9) (5.6,7.83,9.3,10) (4.9,7.569,8.649,10) (3.5,5.829,6.789,9) (3.5,5.22,6.51,8) (4.48,6.6816,7.7376, 9.2) (4.9,6.4,6.4,8.1) (4.9,6.96,7.44,9) (5.6,7.2,8,9) (4.9,6.4,6.4,8.1) (3.5,5.36,5.84,8.1) (4.76,6.464,6.816, 8.46) (4.9,6.4,6.4,8.1) (5.6,7.2,8,9) (4.9,6.64,6.96,9) (4.9,6.64,6.96,9) (3.5,4.8,5.6,7.2) (4.76,6.336,6.784, 8.46)   Table 7 D2;12 v~ij ;v~j C1 C2 C3 C4 C5 A1 A2 A3 A4 A5 DSj 0.8217 0.4430 0.9924 0.4430 0.8217 0.3303 0.5572 1.2118 0.5138 0.6433 0.6433 0.3388 0.3052 1.1684 0.7677 0.7802 1.2213 0.4073 0.2790 0.4749 0.8365 0.2790 0.9688 0.2842 0.2732 0.8760 0.3088 0.3088 1.3143 0.3323 Table 10 Normalized weighted fuzzy decision matrix C1 C2 C3 C4 C5 2.4876 1.3412 3.0044 1.3412 2.4876 1.6444 3.5762 1.5163 1.8983 1.8983 0.7493 2.8684 1.8848 1.9155 2.9983 0.9818 1.6708 2.9432 0.9818 3.4086 0.8223 2.6365 0.9294 0.9294 3.9554 C3 C4 C5 1.4678 5.2966 6.9597 5.2966 1.4678 2.3110 7.5316 5.4716 2.0570 2.0570 4.7047 6.8237 5.8401 2.0399 0.9571 2.9736 5.6262 6.8985 2.9736 0.5468 3.1330 6.5919 4.8848 4.8848 0 A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 30.1610 2.7650 0 2.7650 30.1610 27.2547 0 4.2436 29.9712 29.9712 4.4931 0 0.9675 22.8847 34.4170 15.4048 1.6187 0 15.4048 40.3441 11.9640 0 2.9142 2.9142 43.4531 Table 12 The distance of each Ai (i=1,2,…,5) from NIS Table 9 The sign of each element A1 A2 A3 A4 A5 C2 Table 11 The distance of each Ai (i=1,2,…,5) from PIS Table 8 The absolute normalized value A1 A2 A3 A4 A5 A1 A2 A3 A4 A5 C1 C1 C2 C3 C4 C5 −2.4876 1.3412 3.0044 1.3412 −2.4876 −1.6444 3.5762 1.5163 −1.8983 −1.8983 0.7493 2.8684 1.8848 −1.9155 −2.9983 −0.9818 1.6708 2.9432 −0.9818 −3.4086 −0.8223 2.6365 0.9294 0.9294 −3.9554 A1 A2 A3 A4 A5 C1 C2 C3 C4 C5 0 14.6597 30.1610 14.6597 0 0.0645 29.9712 11.6595 0 0 14.0393 34.4170 23.8437 1.1725 0 5.8894 25.8003 40.3441 5.8894 0 9.8157 43.4531 23.8613 23.8613 0 0.00 0.999998 1.00 3 1 1 0.37069 0.42631 0.42631 3 2 1 0.01228 0.99102 0.99103 3 1 1 0.70909 0.84151 0.84151 3 2 1 (0.15, 0.45, 0.90) (0.499, 0.799, 0.999) (0.5, 0.8, 1) 1 1 1 (0.673, 0.878, 1) (0.673, 0.878, 1) (0.673, 0.878, 1) Ranking Index value Ranking Index value Ranking Index value Ranking [9] (0.489, 0.5, 0.511) Ranking [9] (0.375, 0.511, 0.668) Weight A1 A2 A3 Proposed model Chen [14] and A1. The results show that only the method of Li [25] finds the incorrect solution. The fourth example indicates that the preferences of the alternatives are A1, A2, and A3. The results, shown in Table 17, demonstrate that only the solution of Li [25] is incorrect. In Table 18, we see that the superiority of the alternatives is A3, A2, and A1. Our method and the approach of Li [2] find the correct result, while the methods of Chen and Hwang [5] and Chen [14] fail to do so. The final example shows that preference of the alternatives is A3, A1, and A2. The results, shown in Table 19, indicate that the incorrect solution is found only by Li [25], and the other three methods obtain the correct result. The above results, although based on a small number of examples, showed that our proposed method found the solution in all cases, while the methods of Chen and Hwang [5], Li [25], and Chen [14] failed to find the solution in at least two cases of the first six problems. For a more extensive comparison, in the second stage, we generated random problems of different sizes as proposed in [23] (examples 6-1 and 6-2). The generated problems are used to test the algorithms and compare the results using the so-called contradiction rates as defined in [23]. Three kinds of contradiction rates were recorded for each case by running each case with 500 random replications. The first contradiction rate is named as R1 and defined as the rate at which the fuzzy WSM [23] and another fuzzy method disagree in the indication of the best alternative. The second one is named as R2 and defined as the rate at which the fuzzy WSM [23] and another fuzzy method disagree on the entire ranking of the alternatives, and the last one is named as R3 and defined as the rate at which a method changes the indication of the best alternative when a non-optimal alternative is replaced by a worse alternative. A MATLAB 7.0 computer program is written to generate random data and to solve fuzzy TOPSIS problems with all possible combinations of 3, 5, 7,…, 21 alternatives and 3, 5, 7,…, 21 criteria. As in [23], we first generate an m (alternatives) by n (criteria) matrix with uniformly generated numbers between 1 and 9, using the MATLAB function, random. We then round the components of the matrix, Li [25] 4 1 2 3 5 Chen and Hwang [5] 0.2503 0.9713 0.9411 0.3814 0 C2 29.8088 148.3014 129.8695 45.5828 0 C1 89.2776 4.3846 8.1253 73.9409 178.3464 Rank Table 14 Comparison results CCi* A1 di A2 A3 A1 A2 A3 A4 A5 diþ Index value Table 13 Values of diþ , di and CCi* 3 2 1 413 Ranking Int J Adv Manuf Technol (2009) 45:406–420 414 A1 A2 A3 Table 15 Comparison results C1 Weight A1 A2 A3 C2 Chen and Hwang [5] Li [25] Chen [14] Proposed model (0.375, 0.511, 0.668) Ranking [9] (0.489, 0.5, 0.511) Ranking [9] Index value Ranking Index value Ranking Index value Ranking Index value Ranking (0.45, 0.65, 0.85) (0.55, 0.9, 0.95) (0.6, 0.8, 1) 3 1 2 (0.15, 0.45, 0.90) (0.45, 0.9, 0.95) (0.5, 0.8, 1) 3 1 2 0.60898 0.81875 0.81910 3 2 1 0.00857 0.98953 0.99129 3 2 1 0.32596 0.41663 0.41670 3 2 1 0 1 0.9457 3 1 2 A1 A2 A3 C2 C1 Weight A1 A2 A3 Chen and Hwang [5] Li [25] Chen [14] Proposed model (0.375, 0.511, 0.668) Ranking [9] (0.489, 0.5, 0.511) Ranking [9] Index value Ranking Index value Ranking Index value Ranking Index value Ranking (0.6, 0.75, 1) (0.6, 0.9499, 1) (0.6, 0.98, 1) 3 2 1 (0.7, 0.78, 1) (0.7, 0.9799, 1) (0.7,0.98, 1) 3 2 1 0.83985 0.90722 0.90725 3 2 1 0 0.99979 0.99979 3 1 1 0.42554 0.46764 0.46780 3 2 1 0 0.98857 1 3 2 1 Int J Adv Manuf Technol (2009) 45:406–420 Table 16 Comparison results A2 A3 Table 17 Comparison results C1 Weight C2 Chen and Hwang [5] Li [25] Chen [14] Proposed model (0.375, 0.511, 0.668) Ranking [9] (0.489, 0.5, 0.511) Ranking [9] Index value Ranking Index value Ranking Index value Ranking Index value Ranking (9, 10, 10) (7, 9, 10) (0, 0, 1) 1 2 3 (9, 10, 10) (7, 9, 10) (0, 0, 1) 1 2 3 0.9932 0.8932 0.06875 1 2 3 1 1 0 1 1 2 0.4872 0.4470 0.2899 1 2 3 1 0.977425 0 1 2 3 A1 A2 A3 Int J Adv Manuf Technol (2009) 45:406–420 A1 A1 A2 A3 Table 18 Comparison results C2 C1 Weight A1 A2 A3 (0.375, 0.511, 0.668) Ranking [9] (0.673, 0.878, 1) (0.673, 0.878, 1) (0.673, 0.878, 1) 1 1 1 (0.489, 0.5, 0.511) (0.15, 0.45, 0.90) (0.5001, 0.8, 0.9999) (0.5, 0.8, 1) Chen and Hwang [5] Li [25] Chen [14] Proposed model Ranking [9] Index value Ranking Index value Ranking Index value Ranking 3 1 2 0.70909 0.84151 0.84151 3 1 1 0.01228 0.99104 0.99103 3 1 2 0.37069 0.42631 0.42631 3 1 1 Index value 0 1 0.9999999982 Ranking 3 1 2 415 2 3 1 0.0016 0 1 2 3 1 0.42023 0.41946 0.46407 2 2 1 0 0 1 Index value Ranking Index value Ranking 2 3 1 0.83334 0.83171 0.92356 2 3 1 (0.7, 0.77, 0.79) (0.7, 0.75, 0.8) (0.85, 0.95, 1) 1 1 1 (0.673, 0.878, 1) (0.673, 0.878, 1) (0.673, 0.878, 1) A1 A2 A3 Ranking [9] (0.489, 0.5, 0.511) Fig. 2 Contradiction rate R1 when the number of alternatives is equal to 3 using the MATLAB function, round. Finally, each component aij of the rounded matrix is turned into a triangular fuzzy number (āij, aij, ãij) where, aij ¼ (0.375, 0.511, 0.668) Weight C1 Table 19 Comparison results C2 Ranking [9] Index value Ranking Index value Proposed model Chen and Hwang [5] Li [25] Chen [14] Ranking Int J Adv Manuf Technol (2009) 45:406–420 A2 A3 A1 416 0:5 if aij ¼ 1; : aij 1 if aij > 1; and a~ij ¼ 9 aij þ 1 if aij ¼ 9; : if aij < 9; as suggested by the Saaty scale [39]. Using the above procedure, we generated 100(=10×10) different examples. We executed the methods on these examples using a laptop, 2.00-GHz speed with 1.00 GB of RAM. For the results obtained, the fuzzy WPM of Triantaphyllou and Lin [23], the fuzzy TOPSIS method of Triantaphyllou and Lin [23], the approach proposed by Chen [14], Fig. 3 Contradiction rate R1 when the number of alternatives is equal to 21 Int J Adv Manuf Technol (2009) 45:406–420 417 μ a~ ( x) 1 0 x a2 a1 a3 Fig. 7 A triangular fuzzy number ã Fig. 4 Contradiction rate R2 when the number of alternatives is equal to 3 μa~ ( x) 1 0 a1 a2 a4 a3 Fig. 8 A trapezoidal fuzzy number ã Fig. 5 Contradiction rate R3 when the number of alternatives is equal to 3 μA~ ( x) 1 α 0 Fig. 6 Contradiction rate R3 when the number of alternatives is equal to 21 Aα− Fig. 9 An example of an a-cut Aα+ x 418 Int J Adv Manuf Technol (2009) 45:406–420 and our methods are marked by TF-WPM, TF-TOPSIS, CF-TOPSIS, and GFTOPSIS-MCDM, respectively. All contradiction rates (R1, R2, and R3) are shown in Figs. 2, 3, 4, 5, and 6 with the corresponding numbers of criteria and alternatives. In Figs. 2, 3, 4, 5, and 6, we observe that the contradiction rate of our method has less error in comparison with the other three methods. Moreover, our method is shown to be robust in the sense that the contradiction rate is not strongly affected by the increase in the number of criteria. function value μÃ(x) is termed as the grade of membership of x inà [35]. 5 Conclusions Definition A3 We designed TOPSIS for fuzzy data and developed an algorithm to determine the most preferred choice among all possible alternatives. We converted the decision matrix into a fuzzy decision matrix and constructed a weighted fuzzy decision matrix once the decision makers’ fuzzy ratings have been pooled. The fuzzy distance value and normal fuzzy deviation approach were applied for normalization and determination of the crisp values. Following the approach of TOPSIS, we calculated the distance of each alternative from PIS and NIS, respectively. Finally, a closeness coefficient for each alternative was defined to determine the ranking order of all alternatives. The higher value of closeness coefficient would indicate that an alternative is closer to PIS and farther from NIS simultaneously. We compared our proposed approach with similar methods in the literature using some examples with known results and a number of randomly generated test problems. The results pointed to the applicability of our method and signified its effectiveness in identifying solutions. The proposed method presented here has applications in various decision-making problems such as selection of a suitable material supplier, information project selection, location selection problem, and many other management decision and strategic selection problems. A triangular fuzzy number ã can be defined by a triplet (a1, a2, a3). Its conceptual schema and mathematical form are shown by Eq. 11: 8 0; x < a1 > > < x a1 ; a < x  a2 1 ð11Þ : m a~ ð xÞ ¼ aa23 ax1 ; a < x  a3 > 2 > : a3 a2 0; a3 < x: Definition A2 A fuzzy number à is a fuzzy convex subset of the real line satisfying the following conditions: (a) μÃ(x) is piecewise continuous; (b) μÃ(x) is normalized, that is, there exists m 2 R with μÃ(m)=1, where m is called the mean value of à [3, 35]. A triangular fuzzy number ã in the universe of discourse X that conforms to this definition is shown in Fig. 7. Definition A4 A trapezoidal fuzzy number ã can be defined by a quadruplet (a1, a2, a3, a4). Its conceptual schema and mathematical form are shown by Eq. 12: 8 0; x  a1 > > > > < ax2 aa11 ; a1 < x  a2 m a~ ð xÞ ¼ 1; a2 < x < a3 : > > > aa33 ax4 ; a3 < x  a4 > : 0; a4 > x: ð12Þ A trapezoidal fuzzy number ã in the universe of discourse X that conforms to this definition is shown in Fig. 8. Acknowledgments The first and second authors thank Mazandaran University of Science and Technology, the third author thanks the University of Mazandaran, and the last author thanks Research Council of Sharif University for supporting this work. Definition A5 Appendix The α-cut, Ãα, and strong α-cut, Ãα+, of the fuzzy set à in the universe of discourse X are defined by:    ~ ð13Þ Aa ¼ xm A~ð xÞ  a; x 2 X ; where a 2 ½0; 1Š Definition A1 A fuzzy set à in a universe of discourse X is characterized by a membership function μÃ(x) which associates with each element x in X a real number in the interval [0,1]. The    ~ Aaþ ¼ xmA~ð xÞ > a; x 2 X ; where a 2 ½0; 1Š: ð14Þ The lower and upper points of any α-cut, Ãα, are represented by inf Ãα and sup Ãα, respectively, and we Int J Adv Manuf Technol (2009) 45:406–420 suppose that both are finite. For convenience, we denote inf Ãα by Aa and sup Ãα by Aþ a (see Fig. 9). Definition A6 ~ Assuming that both ã and b are fuzzy numbers and l 2 R, the notions of fuzzy sum, ⊕, fuzzy product by a real number, •, and fuzzy product, ⊗, are defined as follows [34]:     2 mða ~ b~ ð zÞ ¼ sup min m a~ ð xÞ; m b~ð yÞ : ðx; yÞ 2 R and x þ y ¼ z Þ   l 6¼ 0 a~ lz ; : mðl a~Þ ð zÞ ¼ If0g ð zÞ; l ¼ 0 (where I{0}(Z) is the indicator function of ordinary set {0}).   mða~ b~ Þ ðzÞ ¼ sup minð a~ð xÞ; b~ð yÞÞ : ðx; yÞ 2 R2 and x  y ¼ z : ~ Let ã and b be two positive fuzzy numbers, for all α∈[0,1]. The basic operations on positive fuzzy numbers with α-cut operator are as follows: ~ þ ð a~  b Þa ¼ aa þ ba ; aþ a þ ba ~ þ ~ ð a b Þ a ¼ aa  ba ; aa  bþ a and if l 2 Rnf0g, we have: ðl  a~Þa ¼ laa , that is, ðl  a~Þa ¼ laa ; laþ a ; ðl  a~Þa ¼ laþ ; la a a ; if l > 0 if l < 0: References 1. Zeleny M (1982) Multiple criteria decision making. McGraw-Hill, New York 2. Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer, Berlin 3. 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