A new fuzzy MCDM approach based on centroid of fuzzy numbers

Expert Systems with Applications 38 (2011) 5226–5230

Contents lists available at ScienceDirect

Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A new fuzzy MCDM approach based on centroid of fuzzy numbers A. Hadi-Vencheh a,⇑, M.N. Mokhtarian b a b

Department of Mathematics, Khorasgan Branch, Islamic Azad University, Isfahan, Iran Department of Industrial Management, Mobarakeh Branch, Islamic Azad University, Isfahan, Iran

a r t i c l e

i n f o

Keywords: Fuzzy logic Centroid of fuzzy number Group decision making

a b s t r a c t This paper proposes a new fuzzy MCDM (FMCDM) approach based on centroid of fuzzy numbers for ranking of alternatives. The FMCDM approach allows decision makers (DMs) to evaluate alternatives using linguistic terms such as very high, high, slightly high, medium, slightly low, low, very low or none rather than precise numerical values, allows them to express their opinions independently, and also provides an algorithm to aggregate the assessments of alternatives. Three numerical examples are investigated using the FMCDM approach to illustrate its applications. It is shown that the FMCDM approach offers a flexible, practical and effective way of group decision making. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Fuzzy set theory, initially proposed by Zadeh (1965), has been extensively applied to objectively reflect the ambiguities in human judgment and effectively resolve the uncertainties in the available information in an ill-defined multiple criteria decision making environment. A fuzzy MCDM model is used to assess alternatives versus selected criteria through a committee of decision makers, where suitability of alternatives versus criteria, and the importance weights of criteria, can be evaluated in linguistic values represented by fuzzy numbers (Chen & Hwang, 1992; Hadi-Vencheh & Mirjaberi, in press). Numerous approaches have been proposed to solve fuzzy MCDM problems. A review and comparison of many of these methods can be found in (Carlsson & Fuller, 1996; Chen & Hwang, 1992; Ribeiro, 1996; Triantaphyllou & Lin, 1996). Some recent applications can be found in (Al-Najjar & Alsyouf, 2003; Chen, 2001; Chen & Chiou, 1999; Chen, Lin, & Huang, 2006; Chou, 2007; Kahraman, Ruan, & Dogan, 2003; Liang, 1999; Önüt, Efendigil, & Soner Kara, 2010; Sun, Lin, & Tzeng, 2009; Wu, Tzeng, & Chen, 2009). In most fuzzy MCDM problems, the final evaluation values of alternatives are still fuzzy numbers, and these fuzzy numbers need a proper ranking approach to defuzzify them into crisp values for decision making. However, despite the merits, most of the above papers cannot present membership functions for the final fuzzy evaluation values, nor can they clearly develop defuzzification formulae from the membership functions of the final fuzzy evaluation values, limiting the applicability of the fuzzy MCDM methods available.

⇑ Corresponding author. E-mail address: [email protected] (A. Hadi-Vencheh). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.10.036

To resolve the above problems, this work suggests using centroid of fuzzy numbers to solving the fuzzy MCDM model. Herein, the centroid formula for fuzzy numbers from Wang, Yang, Xu, and Chin (2006) is applied for defuzzification due to its simplicity of implementation. In the proposed model,the ratings of alternatives versus criteria, are assessed in linguistic values represented by fuzzy numbers. The centroid of each alternative under each criterion is computed through a suggested method by Wang et al. (2006). Then, a centroid-based distance method is suggested for ranking alternatives. The method utilizes the Euclidean distances from the origin to the centroid point of fuzzy weighted average of each alternative to compare and rank the alternatives. Finally, three numerical examples demonstrate the applicability of the proposed model.

2. Background Fuzzy sets are generalizations of crisp sets and were first introduced by Zadeh as a way of representing imprecise or vagueness in real world. A fuzzy set is a collection of elements in a universe of information where the boundary of the set contained in the universe is ambiguous, vague and otherwise fuzzy. Each fuzzy set is specified by a membership function, which assigns to each element in the universe of discourse a value within the unit interval [0, 1]. The assigned value is called degree (or grade) of membership, which specifies the extent to which a given element belongs to the fuzzy set or is related to a concept. If the assigned value is 0, then the given element does not belong to the set. If the assigned value is 1, then the element totally belongs to the set. If the value lies within the interval (0, 1), then the element only partially belongs to the set. Therefore, any fuzzy set can be uniquely determined by its membership function.

5227

A. Hadi-Vencheh, M.N. Mokhtarian / Expert Systems with Applications 38 (2011) 5226–5230

e of the universe Let X be the universe of discourse. A fuzzy set A of discourse X is said to be convex if and only if for all x1 and x2 in X there always exists:

leA ðkx1 þ ð1  kÞx2 Þ P MinðleA ðx1 Þ; leA ðx2 ÞÞ; e and where le is the membership function of the fuzzy set A A e of the universe of discourse X is said to be k 2 [0, 1]. A fuzzy set A normal if there exists a xi 2 X satisfying leðxi Þ ¼ 1: Fuzzy numbers A are special cases of fuzzy sets that are both convex and normal. A fuzzy number is a convex fuzzy set, characterized by a given interval of real numbers, each with a grade of membership between 0 and 1. Its membership function is piecewise continuous and satisfies the following conditions: 1. 2. 3.

leA ðxÞ ¼ 0 outside some interval [a, d]; leA ðxÞ is non-decreasing (monotonic increasing) on [a, b] and

non-increasing (monotonic decreasing) on [c, d]; leðxÞ ¼ 1 for each x 2 [b, c], A

where a 6 b 6 c 6 d are real numbers in the real line R: The most commonly used fuzzy numbers are triangular and trapezoidal fuzzy numbers, whose membership functions are respectively defined as:

leA ðxÞ ¼ 1

8 xa > < ba

dx > db

:

0

8 xa > ba > > <1 leA ðxÞ ¼ dx > 2 > > : dc 0

~ 1; . . . ; w ~n where ~ x1 ; . . . ; ~xn are n fuzzy numbers to be weighted and w are fuzzy weights. Let z1 = (a, b) and z2 = (c, d) be two complex numbers. Then basic arithmetic operations on these complex numbers are defined as:

Addition : z1 þ z2 ¼ ða þ c; b þ dÞ; Multiplication : z1 :z2 ¼ ðac  bd; ad þ bcÞ; ! ac þ bd bc  ad Division : z1  z2 ¼ ; : c2 þ d2 c2 þ d2

3. The proposed FMCDM approach In fuzzy MCDM problems, criteria/attribute values and the relative weights are usually characterized by fuzzy numbers. Suppose a fuzzy MCDM problem has n alternatives, A1, . . . , An, and m decision criteria/attributes, C1, . . . , Cm. Each alternative is evaluated e ¼ ð~ with respect to the m criteria/attributes. Let X xij Þnm be a fuzzy f ~ 1; . . . ; w ~ m Þ be fuzzy weights. Then the decision matrix and W ¼ ðw fuzzy weighted average for each alternative can be rewritten as:

~ ~ ~ ~ ~ ~ ~hi ¼ w1 xi1 þ w2 xi2 þ . . . þ wm xim ~2 þ ... þ w ~m ~1 þ w w

ð3:1Þ

a 6 x 6 b; Fuzzy arithmetic operations are found not suitable for computing ~ hi because the weight variables appear in both denominator and numerator simultaneously. Here for simplicity we focus on the cen j ¼ ðcj ; dj Þ be the centroid of ~xij and troid of ~ hi . Let xij ¼ ðaij ; bij Þ and w e j , respectively. Besides, let  w hi ¼ ðgi ; ki Þ be the centroid of ~ hi and ~ j the above equation can using the centroid of each ~xij and each w be written as below:

b 6 x 6 d; Otherwise; a 6 x 6 b; b 6 x 6 c; c 6 x 6 d;

Pm h

Otherwise:

For brevity, triangular and trapezoidal fuzzy numbers are often denoted as (a, b, d) and (a, b, c, d). It is obvious that triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers with b = c. e ¼ ða1 ; a2 ; a3 Þ and B e ¼ ðb1 ; b2 ; b3 Þ be two positive triangular Let A fuzzy numbers. Then basic fuzzy arithmetic operations on these fuzzy numbers are defined as Kaufmann and Gupta (1991).

hi ¼ ðgi ; ki Þ ¼ Pm

P m



a c  bij dj ;

Pm

j¼1 dj

j¼1 ½ ij j

P m

c

a

j¼1 ½ ij dj

Pm

j¼1 dj



þ bij cj 



ðl; mÞ 2 P 2 ; m þ j¼1 cj j¼1 dj

¼ Pm

ð3:2Þ

where

For trapezoidal fuzzy number (a, b, c, d), the centroid formulae turns out to be

l¼ m¼

ð2:1Þ

m m m m X X X X ½aij cj  bij dj  cj þ ½aij dj þ bij cj  dj ; j¼1

j¼1

j¼1

m X

m X

m X

m X

j¼1

j¼1

j¼1

j¼1

½aij dj þ bij cj 

cj 

j¼1

½aij cj  bij dj 

dj :

Now Compute overall score for alternative Ai by the following equation

Especially when b = c, the above formulas become

Si ¼ ð2:2Þ

The fuzzy weighted average of fuzzy numbers is referred to as fuzzy weighted average, which is defined as:

~ ~ ~ ~ ~ ~ ~h ¼ w1 x1 þ w2 x2 þ    þ wn xn ; ~ ~ ~ w1 þ w2 þ    þ wn

Pm

c

j¼1 j ;

j¼1 j ;

eB e  ða1 b1 ; a2 b2 ; a3 b3 Þ; Multiplication : A   eB e  a1 ; a2 ; a3 : Division : A b1 b2 b3

e ¼ aþbþc; xð AÞ 3 1 e ð AÞ ¼ : y 3

a c  bij dj ; aij dj þ bij cj Þ P m

eB e ¼ ða1  b3 ; a2  b2 ; a3  b1 Þ; Subtraction : A

  e ¼ 1 a þ b þ c þ d  cd  ab ; xð AÞ 3 cþdab   cb e ¼1 1þ ð AÞ : y 3 cþdab

i ðaij ; bij Þ:ðcj ; dj Þ Pm j¼1 ðcj ; dj Þ

j¼1 ð ij j

¼

¼

eþB e ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 Þ; Addition : A

j¼1

ð2:3Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2i þ k2i :

ð3:3Þ

As a summary, the fuzzy MCDM method based on centroid of fuzzy numbers can be summed up as follows: ~ij and w ~ j by Eq. (2.1) or (2.2).  Determine the centroid of each x  Calculate the centroid of fuzzy weighted average of each alternative by Eq. (3.2).  Generate overall score of alternative Ai by Eq. (3.3).

5228

A. Hadi-Vencheh, M.N. Mokhtarian / Expert Systems with Applications 38 (2011) 5226–5230

 Rank and prioritize alternatives according to their overall scores: big score means high priority.

Table 1 Linguistic variables for the relative importance weights of five criteria. Linguistic variable

Fuzzy number

4. Numerical examples

Very low (VL) Low (L) Medium low (ML) Medium (M) Medium high (MH) High (H) Very high (VH)

(0, 0, 0.1) (0, 0.1, 0.3) (0.1, 0.3, 0.5) (0.3, 0.5, 0.7) (0.5, 0.7, 0.9) (0.7, 0.9, 1.0) (0.9, 1.0, 1.0)

In this section, we examine three numerical examples using the proposed FMCDM method. These examples are taken from Chen (2000), Chen and Hwang (1992), Hwang and Yoon (1981), Triantaphyllou and Lin (1996) for the purpose of comparison. Example 1. Reconsider the example investigated by Chen (2000), in which a software company desires to hire a system analysis engineer among from three candidates, A1, A2, and A3, who are evaluated by a committee of three decision makers (DMs) against five benefit criteria, i.e. emotional steadiness (C1), oral communication skills (C2), personality (C3), past experience (C4) and selfconfidence (C5). The relative importance weights of the five criteria are described using linguistic variables such as Low, Medium, High etc., which are defined in Table 1. The ratings (i.e. criteria values) are also characterized by linguistic variables such as poor, fair, good, and the like, which are defined in Table 2. The three DMs express their opinions on the importance weights of the five criteria and the ratings of each candidate with respect to the five criteria independently. Tables 3 and 4 show the original assessment information provided by the three DMs, where aggregated fuzzy numbers are obtained by averaging the fuzzy ~ j ¼ ðw ~ 1j þ w ~ 2j þ w ~ 3j Þ=3, where opinions of the three DMs. That is w k ~ j wj is the relative importance weight given by the kth DM and w ~ j. is centroid of w The results are presented in Table 5 which give the ranking of A2 A3 A1, where the symbol means is superior or preferred to. The relative closenesses of the three candidates obtained by Chen (2000) are 0.62 for A1, 0.77 for A2, and 0.71 for A3, which are significantly lower than the above values. It is easy to see that Chen’s approach leads to the same ranking as ours.

Table 2 Linguistic variables for the ratings. Linguistic variable

Fuzzy number

Very poor (VP) Poor (P) Medium poor (MP) Fair (F) Medium good (MG) Good (G) Very good (VG)

(0, 0, 1) (0, 1, 3) (1, 3, 5) (3, 5, 7) (5, 7, 9) (7, 9, 10) (9, 10, 10)

Table 3 The relative importance weights of the five criteria by three DMs. Criterion

DM1

DM2

DM3

~j w

j w

C1 C2 C3 C4 C5

H VH VH VH M

VH VH H VH MH

MH VH H VH MH

(0.7, 0.87, 0.97) (0.9, 1.0, 1.0) (0.77, 0.93, 1.0) (0.9, 1.0, 1.0) (0.43, 0.63, 0.83)

(0.846, 0.333) (0.966, 0.333) (0.900, 0.333) (0.966, 0.333) (0.630, 0.333)

Example 2. Reconsider the example investigated by Triantaphyllou and Lin (1996), in which three alternatives A1 A3 are evaluated against four benefit criteria C1 C4. The fuzzy weights and the fuzzy decision matrix are duplicated in Table 6. The results are recorded in Table 7. The Si values for the three alternatives are 0.850 for A1, 0.835 for A2, and 1.184 for A3, which lead to the ranking of A3 A1 A2. The fuzzy relative closenesses of the three alternatives obtained by Triantaphyllou and Lin (1996), who use fuzzy arithmetic operations, are (0.04, 0.42, 5.83) for A1, (0.01, 0.21, 3.99) for A2, and (0.06, 0.79, 10.42) for A3. It is clear that the same ranking is generated for the example.

Table 4 Ratings of three candidates with respect to the five criteria by the three DMs Criteria

Candidates

C1

DMs

A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3

C2

C3

C4

C5

DM1

DM2

DM3

MG G VG G VG MG F VG G VG VG G F VG G

G G G MG VG G G VG MG G VG VG F MG G

MG MG F F VG VG G G VG VG VG MG F G MG

Table 6 Fuzzy weights and fuzzy decision matrix for Example 2. Cj

C1

C2

C3

C4

ej w wj

(0.13, 0.20, 0.31)

(0.08, 0.15, 0.25)

(0.29, 0.40, 0.56)

(0.17, 0.25, 0.38)

(0.213, 0.333)

(0.160, 0.333)

(0.416, 0.333)

(0.266, 0.333)

A1 A2 A3

(0.08, 0.25, 0.94) (0.23, 1.00, 3.10) (0.15, 0.40, 1.48)

(0.25, 0.93, 2.96) (0.13, 0.60, 2.24) (0.13, 0.20, 0.88)

(0.34, 0.70, 1.71) (0.03, 0.05, 0.09) (0.62, 1.48, 3.41)

(0.12, 0.24, 0.92) (0.12, 0.40, 1.48) (0.24, 1.00, 3.03)

Table 5 Numerical results for Example 1. Ai

 xi1

 xi2

 xi3

 xi4

 xi5

 hi

Si

Rank

A1 A2 A3

(7.556, 0.333) (8.110, 0.333) (7.776, 0.333)

(6.890, 0.333) (9.666, 0.333) (8.446, 0.333)

(7.446, 0.333) (9.333, 0.333) (8.446, 0.333)

(9.333, 0.333) (9.666, 0.333) (8.446, 0.333)

(5.000, 0.333) (8.446, 0.333) (8.110, 0.333)

(7.387, 0.278) (9.304, 0.110) (8.263, 0.326)

7.392 9.109 8.269

3 1 2

5229

A. Hadi-Vencheh, M.N. Mokhtarian / Expert Systems with Applications 38 (2011) 5226–5230 Table 7 Numerical results for Example 2. Ai

 xi1

 xi2

 xi3

 xi4

 hi

Si

Rank

A1 A2 A3

(0.423, 0.333) (1.443, 0.333) (0.676, 0.333)

(1.380, 0.333) (0.990, 0.333) (0.403, 0.333)

(0.916, 0.333) (0.056, 0.333) (1.836, 0.333)

(0.426, 0.333) (0.666, 0.333) (1.423, 0.333)

(0.777, 0.344) (0.728, 0.409) (1.160, 0.237)

0.850 0.835 1.184

2 3 1

Table 8 Decision information given by the DM for Example 3. Air-fighters

C1

C2

C3

C4

C5

C6

A1 A2 A3 A4

2.0 2.5 1.8 2.2

1500 2700 2000 1800

20000 18000 21000 20000

5.5 6.5 4.5 5.0

M L H M

VH M H M

Example 3. An extended air-fighter selection problem (Chen & Hwang, 1992; Hwang & Yoon, 1981) is investigated in this example. Suppose one country D plans to buy air-fighters from another country H. The Defense Department of the country H would provide the country D with characteristic data for four candidate air-fighters A1, A2, A3 and A4 The decision maker takes into consideration the following six criteria in evaluating the airfighters, including maximum speed (C1), cruise radius (C2), maximum loading (C3), price (C4), reliability (C5) and maintenance (C6). C5 and C6 are qualitative criteria and their ratings are expressed using linguistic variables. The data and ratings of all air-fighters on every criterion are given by the decision maker as in Tables 8 and 9. The corresponding relations between linguistic variables and positive trapezium fuzzy numbers are given in Table 10. Table 9 The relations between linguistic variables and positive trapezium fuzzy numbers for Example 3. Linguistic variable

Fuzzy number

Very high (VH) High (H) Medium (M) Low (L) Very low (VL)

(0.7, 0.8, 0.9, 1.0) (0.5, 0.6, 0.7, 0.8) (0.3, 0.4, 0.5, 0.6) (0.1, 0.2, 0.3, 0.4) (0.0, 0.1, 0.2, 0.3)

Table 10 Normalized fuzzy decision matrix for Example 3. Air-fighters

C1

C2

C3

C4

C5

C6

A1 A2 A3 A4

0.80 1.00 0.72 0.86

0.55 1.00 0.74 0.67

0.95 0.86 1.00 0.95

0.82 0.69 1.00 0.90

M L H M

VH M H M

f¼ Following Chen and Hwang (1992) we assume that W ½ð0:600; 0:675; 0:675; 0:750Þ; ð0:400; 0:500; 0:500; 0:600Þ; ð0:400; 0:500; 0:500; 0:600Þ; ð0:400; 0:500; 0:500; 0:600Þ; ð0:750; 0:825; 0:825; 0:900Þ; ð0:900; 0:950; 0:950; 1:000Þ, hence we have W ¼ ½ð0:675; 0:333Þ; ð0:500; 0:333Þ; ð0:500; 0:333Þ; ð0:500; 0:333Þ; ð0:825; 0:333Þ; ð0:950; 0:333Þ. For the sake of comparison with Chen and Hwang result, normalization is performed in this study. Table 10 shows the normalized fuzzy decision matrix. Note that the normalization is carried out only on C1, C2, C3 and C4 using rij ¼ xij , for benefit criteria C1, C2 and C3 and rij ¼ x j =xij , for cost criterion C4. Using the proposed approach Table 11 shows the obtained results for this example. So the ranking order of four air-fighters is generated as follows:

A3 A1 A4 A2 : Obviously, the best selection is the air-fighter A3. Our ranking is the same as obtained by Chen and Hwang (1992), Hwang and Yoon (1981).

5. Conclusion MCDM has found wide applications in the solution of real world decision making problems. Most MCDM problems include both quantitative and qualitative criteria which are often assessed using imprecise data and human judgments. Fuzzy set theory is well suited to dealing with such decision problems. In this paper, a new fuzzy MCDM method is proposed to solve multi-criteria decision making problems in fuzzy environments. Linguistic variables as well as crisp numerical values are used to assess qualitative and quantitative criteria. In particular, fuzzy numbers are used in this paper to assess alternatives with respect to qualitative criteria. Based on such an argument, we have proposed a fuzzy MCDM method based on centroid of fuzzy numbers. The proposed fuzzy MCDM method uses arithmetic operations on complex numbers. The technique can be used to generate consistent and reliable ranking order of alternatives in question. The developed method is illustrated using three example. It is expected to be applicable to decision problems in many areas, especially in situations where multiple decision makers are involved and the criteria are fuzzy.

Table 11 Numerical results for Example 3. Ai

 xi1

 xi2

 xi3

 xi4

 xi5

 xi6

 hi

Si

Rank

A1 A2 A3 A4

(0.800, 0.333) (1.000, 0.333) (0.720, 0.333) (0.880, 0.333)

(0.550, 0.333) (1.000, 0.333) (0.740, 0.333) (0.670, 0.333)

(0.950, 0.333) (0.860, 0.333) (1.000, 0.333) (0.650, 0.333)

(0.820, 0.333) (0.690, 0.333) (1.000, 0.333) (0.900, 0.333)

(0.450, 0.416) (0.250, 0.416) (0.650, 0.416) (0.450, 0.416)

(0.850, 0.416) (9.304, 0.416) (8.263, 0.416) (9.304, 0.416)

(0.734, 0.372) (0.670, 0.390) (0.772, 0.381) (0.684, 0.387)

0.823 0.774 0.861 0.786

2 4 1 3

5230

A. Hadi-Vencheh, M.N. Mokhtarian / Expert Systems with Applications 38 (2011) 5226–5230

References Al-Najjar, B., & Alsyouf, I. (2003). Selecting the most maintenance approach using fuzzy multiple criteria decision making. International Journal of Production Economics, 84, 85–100. Carlsson, C., & Fuller, R. (1996). Fuzzy multiple criteria decision making: Recent developments. Fuzzy Sets and Systems, 78, 139–153. Chen, C. T. (2000). Extension of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets and Systems, 114, 1–9. Chen, C. T. (2001). A fuzzy approach to select the location of the distribution center. Fuzzy Sets and Systems, 118, 65–73. Chen, L. H., & Chiou, T. W. (1999). A fuzzy credit-rating approach for commercial loans: A Taiwan case. Omega: International Journal of Management Science, 27, 407–419. Chen, S. J., & Hwang, C. L. (1992). Fuzzy Multiple Attribute Decision Making. Berlin: Springer-Verlag. Chen, C. T., Lin, C. T., & Huang, S. F. (2006). A fuzzy approach for supplier evaluation and selection in supplier chain management. International Journal of Production Economics, 102, 289–301. Chou, C. C. (2007). A fuzzy MCDM method for solving marine transshipment container port selection problems. Applied Mathematics and Computation, 186, 435–444. Hadi-Vencheh, A., & Mirjaberi, M. The seclusion factor method for solving fuzzy multiple criteria decision-making problems, IEEE Transaction on Fuzzy Systems, in press.

Hwang, C. L., & Yoon, K. (1981). Multiple Attributes Decision Making Methods and Applications. Berlin: Springer-Verlag. Kahraman, C., Ruan, D., & Dogan, I. (2003). Fuzzy group decision-making for facility location selection. Information Sciences, 157, 135–153. Kaufmann, A., & Gupta, M. M. (1991). Introduction to fuzzy arithmetic: Theory and applications. New York: Van Nostrand Reinhold. Liang, G. S. (1999). Fuzzy MCDM based on ideal and anti-ideal concepts. European Journal of Operational Research, 112, 682–691. Önüt, S., Efendigil, T., & Soner Kara, S. (2010). A combined fuzzy MCDM approach for selecting shopping center site: An example from Istanbul, Turkey. Expert Systems with Applications, 37, 1973–1980. Ribeiro, R. A. (1996). Fuzzy multiple attribute decision making: A review and new preference elicitation techniques. Fuzzy Sets and Systems, 78, 155–181. Sun, C., Lin, G. T. R., & Tzeng, G. (2009). The evaluation of cluster policy by fuzzy MCDM: Empirical evidence from HsinChu Science Park. Expert Systems with Applications, 36, 11895–11906. Triantaphyllou, E., & Lin, C. T. (1996). Development and evaluation of five fuzzy multi-attribute decision-making methods. International Journal of Approximate Reasoning, 14, 281–310. Wang, Y., Yang, J., Xu, D., & Chin, K. (2006). On the centroids of fuzzy numbers. Fuzzy Sets and Systems, 157, 919–926. Wu, H., Tzeng, G., & Chen, Y. (2009). A fuzzy MCDM approach for evaluating banking performance based on balanced scorecard. Expert Systems with Applications, 36, 10135–10147. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.