Fractional programming approach to fuzzy weighted average

Fuzzy Sets and Systems 120 (2001) 435–444

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Fractional programming approach to fuzzy weighted average Chiang Kao a ; ∗ , Shiang-Tai Liu b a

Department of Industrial Management, National Cheng Kung University, Tainan 70101, Taiwan, ROC of Industrial Management, Van Nung Institute of Technology, Chung-Li 320, Taiwan, ROC

b Department

Received 3 March 1998; received in revised form 18 March 1999; accepted 2 July 1999

Abstract This paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average. Based on the -cut representation of fuzzy sets and the extension principle, a pair of fractional programs is formulated to 2nd the -cut of fuzzy weighted average. Owing to the special structure of the fractional programs, in most cases, the optimal solution can be found analytically. Consequently, the exact form of the membership function can be derived by taking the inverse function of the -cut. For other cases, a discrete but exact solution to fuzzy weighted average is c 2001 Elsevier Science B.V. All rights provided via an e6cient solution method. Examples are given for illustration.
reserved. Keywords: Fuzzy weighted average; Fractional programming; Parametric programming; Decision analysis

1. Introduction Selecting an appropriate alternative is to undertake an important task in decision making. When the alternatives are evaluated in terms of a single criterion, the decision is relatively simple, in that a score is assigned to each alternative and the one with the best score is selected. In reality, the selection problem is more complex because usually multiple criteria are involved. To acquire a perception of the overall attractiveness of alternatives, one method frequently used is to calculate the weighted average score for each alternative:  n n

y= w i xi wi ; (1) i=1

i=1

where wi is a weighting coe6cient representing the relative importance of the ith criterion and xi is a score representing the relative merit of alternative x in the ith criterion. The alternative with the best weighted average score is then selected. ∗

Corresponding author. Tel.: + 886-6-275-3396; fax: + 886-6-236-2162. E-mail address: [email protected] (C. Kao).

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 9 9 ) 0 0 1 3 7 - 2

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C. Kao, S.-T. Liu / Fuzzy Sets and Systems 120 (2001) 435–444

n In (1), if we treat xi as the bene2t acquired under state i and wi = i=1 wi as the probability that state i will occur, then y is the expected return of alternative x, and the weighted average problem falls into the category of decision making under risk in decision analysis [9]. Model (1) has wide applications in engineering [5], science and management [9,14]. Zeleny [14] classi2es the problems for decision making into four basic groups according to, whether the criterion considered is single or multiple and whether the alternatives are well de2ned or poorly de2ned. Clearly, the most di6cult group is the one with multiple criteria and fuzzily de2ned causal relationships. In this case, the weighting coe6cients wi and the scores of merit xi in (1) become fuzzy numbers, and the weighted average y becomes a fuzzy number as well. Dong and Wong [7] have developed a computational algorithm based on the -cut representation of fuzzy sets and interval analysis to provide a discrete but exact solution to fuzzy weighted average. Liou and Wang [11] modi2ed Dong–Wong’s method to facilitate computational e6ciency. There are also other more e6cient methods [8,10] proposed for fuzzy weighted average. In this paper we follow the idea of Dong and Wong [7] to use -cut representation to calculate fuzzy weighted averages. For each membership value , a pair of fractional programming problems is formulated to 2nd the -cut of the fuzzy weighted average. As  varies, the family of fractional programs is modeled by using parametric programming technique. For most problems, the exact membership function of the fuzzy weighted average can be derived. For others, e6cient numerical solution method still exists to provide good approximation for the membership function. In the following sections, 2rstly, the basic concept of fuzzy weighted average is introduced. Secondly, the fractional programming approach for constructing the membership function for fuzzy weighted average is developed. Finally, the examples discussed in Dong and Wong [7] are solved analytically by using the proposed approach. 2. Fuzzy weighted average Consider the general fuzzy weighted average with n criteria. The relative importance of each criterion, W˜i , and the score of merit of alternative x in criterion i, X˜i , are approximately known and are represented by the following convex fuzzy sets: W˜i = {(wi ; W˜i (wi )) | wi ∈ Wi }; X˜i = {(xi ; X˜i (xi )) | xi ∈ Xi };

i = 1; : : : ; n;

(2a)

i = 1; : : : ; n;

(2b)

where Wi and Xi are the crisp universal sets of the relative importance and the relative merit, and W˜i and X˜i are, respectively, the membership functions of the fuzzy numbers W˜i and X˜i . Similar to the weighted average de2ned in (1) for crisp values, fuzzy weighted average can be de2ned as  n n

Y˜ = W˜i X˜i W˜i : (3) i=1

i=1

Since W˜i and X˜i are fuzzy numbers, the weighted average Y˜ is also fuzzy. According to the extension principle [12,13,15], the fuzzy weighted average Y˜ has the following membership function:   n   n 

 Y˜i (y) = sup min W˜i (wi ); X˜i (xi ); i = 1; : : : ; n y = (4) wi x i wi :  x;w i=1

i=1

C. Kao, S.-T. Liu / Fuzzy Sets and Systems 120 (2001) 435–444

437

In a straightforward manner, the above equation can be formulated as the following nonlinear programming problem: Y˜(y) = max s:t:

z z6W˜i (wi ); i = 1; : : : ; n; z6X˜i (xi ); i = 1; : : : ; n; y=

n


 w i xi

i=1

n


wi ;

i=1

wi ∈ Wi ; xi ∈ Xi ; i = 1; : : : ; n:

(5)

Given a value y, the membership value of Y˜ is solved from Model (5). Since this model is nonlinear and, more importantly, the functions W˜i and X˜i usually are not diKerentiable, it is very di6cult to solve [1]. Instead of viewing the problem from the domain of Y˜, Dong and Wong [7] tackle the problem from the -cut of Y˜. Denote the -cuts of W˜i and X˜i as (Wi ) = {wi ∈ Wi | W˜i (wi )¿};

(6a)

(Xi ) = {xi ∈ Xi | X˜i (xi )¿}:

(6b)

The -cuts are crisp intervals which can be expressed in another form:   (Wi ) = min{wi ∈ Wi | W˜i (wi )¿}; max{wi ∈ Wi | W˜i (wi )¿} wi

wi

= [(Wi )L ; (Wi )U  ];

(7a)

  (Xi ) = min{xi ∈ Xi | X˜i (xi )¿}; max{xi ∈ Xi | X˜i (xi )¿} xi

= [(Xi )L ; (Xi )U  ]:

xi

(7b)

The idea of Dong and Wong [7] in deriving the membership function of Y˜ is to 2nd the -cut of Y˜. For each membership value , use (7) to 2nd the end points of (Wi ) and (Xi ) . With n criteria, there are 22n distinct permutations of the 2n-ary array (w1 ; : : : ; wn ; x1 ; : : : ; xn ), where wi and xi can be one of the two end points L U 2n (Wi )L and (Wi )U permutations of (w1 ; : : : ; wn ; x1 ; : : : ; xn ) can  , and (Xi ) and (Xi ) , respectively. Each of the 2 be substituted into (1) to calculate a weighted average yk ; k = 1; : : : ; 22n . The interval [min{yk ; k = 1; : : : ; 22n }; max{yk ; k = 1; : : : ; 22n }] is then the -cut of Y˜. This process can be repeated for any membership value . The number of  values used determines the closeness of the discretized membership function to the true membership function. The method of Dong and Wong [7] provides a discrete but exact solution to fuzzy weighted averages. Liou and Wang [11] modify Dong–Wong’s method to reduce the number of permutations to [n(n + 1) + 2]. The method of Guh et al. [8] is more e6cient which only requires 2(n − 1) permutations provided the scores of merit of every criterion are sequenced in decreasing or increasing order. Lee and Park [10] propose another

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method. In terms of complexity, the computation eKort of their method is O(n log n), while for the methods of Dong and Wong [7], Liou and Wang [11], and Guh et al. [8], they are O(2n ), O(n2 ), and O(n), respectively. If the computational eKort of sorting of the method of Guh et al. [8] is taken into account, then the complexity becomes O(n log n), which is the same as that of Lee and Park [10]. In this paper we approach the problem via mathematical programming technique. A pair of fractional programs is developed to 2nd the -cut of Y˜ based on the extension principle. The following section explains this idea. 3. Fractional programming approach From the extension principle stated in (4), Y˜(y) is the minimum of W˜i (wi ), X˜i (xi ); i = 1; : : : ; n. To tackle from the membership value, one needs W˜i (w in)¿ and X˜i (xi )¿; i = 1; : : : ; n and at least one W˜i (wi ) n or X˜i (xi ) equal to  such that y = i=1 wi xi = i=1 wi to satisfy Y˜(y) = . Suppose all -cuts form a L U L U L nested structure with respect to , that is, [(Wi )L1 , (Wi )U 1 ] ⊆ [(Wi )2 ; (Wi )2 ] and [(Xi )1 ; (Xi )1 ] ⊆ [(Xi )2 ; U (Xi )2 ] for 0 ¡ 2 61 61. Moreover, W˜i (wi )¿ and W˜i (wi ) = , and X˜i (xi )¿ and XM i (xi ) = , respectively, have the same domain. Thus, based on (4), the lower and upper bounds of the -cut of Y˜ can be solved as  n n

(Y )L = min y = w i xi wi i=1

i=1

s:t: (Wi )L 6wi 6(Wi )U ;

i = 1; : : : ; n;

(Xi )L 6xi 6(Xi )U ; (Y )U 

= max

y=

n
i=1

s:t:

w i xi

i = 1; : : : ; n;

 n


(8a)

wi

i=1

(Wi )L 6wi 6(Wi )U ; (Xi )L 6xi 6(Xi )U ;

i = 1; : : : ; n; i = 1; : : : ; n

(8b)

which is a pair of nonlinear fractional programs. In this model, the upper and lower bounds change as a function of the parameter , it falls into the category of parametric programming. Since the constraints in (8) are simple independent bounds, the denominators of the objective functions do not involve xi , and the weight wi is nonnegative, it is obvious that the minimum of y must occur at the smallest value of xi : (Xi )L ; and the maximum must occur at the largest value of xi : (Xi )U  . Hence, Model (8) is simpli2ed to the following linear fractional programs:  n n

(Y )L = min y = wi (Xi )L wi i=1

s:t: (Wi )L 6wi 6(Wi )U ;

i=1

i = 1; : : : ; n;

(9a)

C. Kao, S.-T. Liu / Fuzzy Sets and Systems 120 (2001) 435–444

(Y )U 

= max

n


y=

wi (Xi )U 

 n


i=1

s:t:

wi

i=1

(Wi )L 6wi 6(Wi )U ;

i = 1; : : : ; n:

Following the variable substitution of Charnes and Cooper [4], by letting t = 1= (9) can be transformed to the following linear programs: (Y )L = min

n


y=

439

(9b) n

i=1

wi and vi = twi , Model

vi (Xi )L

i=1

s:t: t(Wi )L 6vi 6t(Wi )U ; n


i = 1; : : : ; n;

vi = 1;

i=1

t¿0; (Y )U  = max

y=

(10a) n


vi (Xi )U 

i=1

s:t:

t(Wi )L 6vi 6t(Wi )U ; n


i = 1; : : : ; n;

vi = 1;

i=1

t¿0:

(10b)

n Each linear program has 2n bounds and only one real constraint: i=1 vi = 1. If the simplex method for bounded variables [2] is applied, in that the bound constraints are handled implicitly in a fashion similar to the conventional simplex method for handling the nonnegativity constraints x¿0; then only one constraint remains. Since the optimal solution of a linear program always occur at an extreme point, the only constraint n v = 1 implies that at most, n iterations are required. The implication is that at most, n permutations of i=1 i (w1 ; : : : ; wn ; x1 ; : : : ; xn ) are needed to 2nd either (Y )L or (Y )U  . From the computational experience on a large number of linear programs solved over several years, it is found that the average number of iterations rarely exceeds 3m [6], where m is the number of constraints. In our case, there is only one constraint, therefore, the average number of permutations of either (10a) or (10b) is expected to be 3. The -cut of Y˜ is the crisp interval [(Y )L ; (Y )U  ] solved from (10). By enumerating diKerent  values, the membership function Y˜ can be constructed. As an illustration, consider an example of three criteria: Y˜ = (W˜1 X˜1 + W˜2 X˜2 + W˜3 X˜3 )=(W˜1 + W˜2 + W˜3 ): The three fuzzy weights have triangular membership functions expressed as: W˜1 = [0; 0:3; 0:9]; W˜2 = [0; 0:7; 1]; and W˜3 = [0:6; 0:8; 1]; and the three fuzzy scores of merit of an alternative have trapezoidal membership functions X˜ 1 = [−2; 1; 2; 3]; X˜ 2 = [1; 2; 4; 5], and X˜ 3 = [2; 3; 6; 7]. The -cuts of the corresponding membership

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C. Kao, S.-T. Liu / Fuzzy Sets and Systems 120 (2001) 435–444

functions are (W1 ) = [(W1 )L ; (W1 )U  ] = [0:3; 0:9 − 0:6]; (W2 ) = [(W2 )L ; (W2 )U  ] = [0:4 + 0:3; 1 − 0:3]; (W3 ) = [(W3 )L ; (W3 )U  ] = [0:6 + 0:2; 1 − 0:2];

(11)

(X1 ) = [(X1 )L ; (X1 )U  ] = [−2 + 3; 3 − ]; (X2 ) = [(X2 )L ; (X2 )U  ] = [1 + ; 3 − 2]; (X3 ) = [(X3 )L ; (X3 )U  ] = [2 + ; 7 − ]: The linear programs for solving (Y )L and (Y )U  , according to (10), are (Y )L = min

y = (−2 + 3)v1 + (1 + )v2 + (2 + )v3

s:t: (0:3)t6v1 6(0:9 − 0:6)t; (0:4 + 0:3)t6v2 6(1 − 0:3)t; (0:6 + 0:2)t6v3 6(1 − 0:2)t; v1 + v2 + v3 = 1; t¿0; (Y )U 

= max s:t:

y = (3 − )v1 + (3 − 2)v2 + (7 − )v3 (0:3)t6v1 6(0:9 − 0:6)t; (0:4 + 0:3)t6v2 6(1 − 0:3)t; (0:6 + 0:2)t6v3 6(1 − 0:2)t; v1 + v2 + v3 = 1; t¿0:

Table 1 lists the results for  = 0; 0:1; : : : ; 1:0. 4. Analytical solution In the linear fractional program of Model (9), the objective function is the ratio of two linear functions which has been proved to be pseudolinear, that is, both pseudoconvex and pseudoconcave [3]. Therefore, it has a minimum at an extreme point of the feasible region and also has a maximum at an extreme point of the feasible region [3]. The bound constraints of Model (9) are independent and constitute a rectangular parallelepiped, Table 1 The -cuts of the fuzzy weighted average at 11  values 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(Y )L (Y )U 

−0:11 6.43

0.18 6.22

0.46 6.02

0.73 5.82

0.98 5.63

1.23 5.44

1.46 5.25

1.68 5.07

1.89 4.90

2.09 4.73

2.28 4.56

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441

i.e., rectangle in high dimensions. Let ∇yL = [∇y1L ; : : : ; ∇ynL ] and ∇yU = [∇y1U ; : : : ; ∇ynU ] denote the gradients of the objective functions of (9a) and (9b), respectively. We have

2  n n

L L L wi [(Xk ) − (Xi ) ] wi ; k = 1; : : : ; n; (12a) ∇yk = i=1

∇ykU =

n


i=1

 U wi [(Xk )U  − (Xi ) ]

i=1

n


2 wi

;

k = 1; : : : ; n:

(12b)

i=1

To minimize y; wi should be set to its lower bound (Wi )L , when ∇yiL is positive and should be set to its L upper bound (Wi )U  , when ∇yi is negative. To maximize y, on the contrary, wi should be set to its upper U U bound (Wi ) , when ∇yi is positive and set to its lower bound (Wi )L , when ∇yiU is negative. Consequently, it is very straightforward in determining the optimal extreme points. Both (Y )L and (Y )U  are functions of . If they are invertible with respect to , then a left shape function −1 can be obtained, from which the membership L(y) = [(Y )L ]−1 and a right shape function R(y) = [(Y )U ] function Y˜ is constructed as  L(y); (Y )L=0 6y6(Y )L=1 ;    Y˜(y) = 1; (13) (Y )L=1 6y6(Y )U =1 ;    U U R(y); (Y )=1 6y6(Y )=0 : To illustrate this analytical solution method, we consider a two-term weighted average and a three-term weighted average discussed in Dong and Wong [7]. 4.1. Two-term example Consider the fuzzy weighted average: Y˜ = (W˜1 X˜ 1 + W˜2 X˜ 2 )=(W˜1 + W˜2 ); where W˜1 and W˜2 are the same as that de2ned in (11), and X˜ 1 = [0; 1; 2] and X˜ 2 = [2; 3; 4] have the -cuts of their membership functions of (X1 ) = [(X1 )L ; (X1 )U  ] = [; 2 − ]; (X2 ) = [(X2 )L ; (X2 )U  ] = [2 + ; 4 − ]: The corresponding linear fractional programs for solving (Y )L and (Y )U  , according to Model (9), are (Y )L = min s:t:

y = [()w1 + (2 + )w2 ]=(w1 + w2 ) 0:36w1 60:9 − 0:6; 0:4 + 0:36w2 61 − 0:3;

(Y )U 

= max s:t:

y = [(2 − )w1 + (4 − )w2 ]=(w1 + w2 ) 0:36w1 60:9 − 0:6; 0:4 + 0:36w2 61 − 0:3:

The gradients ∇yL and ∇yU , according to (12), are ∇yL = [−2w2 =(w1 + w2 )2 ; 2w1 =(w1 + w2 )2 ]; ∇yU = [−2w2 =(w1 + w2 )2 ; 2w1 =(w1 + w2 )2 ]:

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C. Kao, S.-T. Liu / Fuzzy Sets and Systems 120 (2001) 435–444

Since w1 ; w2 ¿0 and (w1 + w2 )2 ¿0, the 2rst component of both ∇yL and ∇yU is negative and the second component of both ∇yL and ∇yU is positive. Therefore, to 2nd the minimum of y; w1 is set to its upper bound (0:9 − 0:6) and w2 is set to its lower bound (0:4 + 0:3). Likewise, to 2nd the maximum of y; w1 is set to its lower bound 0:3 and w2 is set to its upper bound (1 − 0:3). Thus, (Y )L = [(0:9 − 0:6) + (2 + )(0:4 + 0:3)]=[(0:9 − 0:6) + (0:4 + 0:3)] = (0:8 + 1:9 − 0:32 )=(1:3 − 0:3); (Y )U  = [(2 − )(0:3) + (4 − )(1 − 0:3)]=[0:3 + (1 − 0:3)] = 4 − 1:6: Note that (Y )=0 = [0:8=1:3; 4]; (Y )=0:5 = [1:675=1:15; 3:2]; and (Y )=1 = [2:4; 2:4] are the same as that reported in Dong and Wong [7]. L L U U Both (Y )L and (Y )U  are invertible. Since (Y )=0 = 0:8=1:3; (Y )=1 = 2:4; (Y )=1 = 2:4 and (Y )=0 = 4; according to (13), we have   [(Y )L ]−1 = [(1:9 + 0:3y) − 0:09y2 − 0:42y + 4:57]=0:6; 0:8=1:36y62:4; Y˜(y) = −1 = (4 − y)=1:6; 2:46y64: [(Y )U ] 4.2. Three-term example Consider another case of the three criteria: Y˜ = (W˜1 X˜1 + W˜2 X˜2 + W˜3 X˜3 )=(W˜1 + W˜2 + W˜3 ), where W˜1 ; W˜2 , and W˜3 are the same as that de2ned in (11), and X˜1 and X˜2 are de2ned as in the two-term example. Suppose X˜3 = [4; 5; 6] with the -cut of its membership function of (X3 ) = [(X3 )L ; (X3 )U  ] = [4 + ; 6 − ]: The corresponding linear fractional programs for solving (Y )L and (Y )U  , according to Model (9), are (Y )L = min s:t:

y = [()w1 + (2 + )w2 + (4 + )w3 ]=(w1 + w2 + w3 ) 0:36w1 60:9 − 0:6; 0:4 + 0:36w2 61 − 0:3; 0:6 + 0:26w3 61 − 0:2;

(Y )U  = max s:t:

y = [(2 − )w1 + (4 − )w2 + (6 − )w3 ]=(w1 + w2 + w3 ) 0:36w1 60:9 − 0:6; 0:4 + 0:36w2 61 − 0:3; 0:6 + 0:26w3 61 − 0:2:

The gradients ∇yL and ∇yU , according to (12), are ∇yL = [(−2w2 − 4w3 ); (2w1 − 2w3 ); (4w1 + 2w2 )]=(w1 + w2 + w3 )2 ; ∇yU = [(−2w2 − 4w3 ); (2w1 − 2w3 ); (4w1 + 2w2 )]=(w1 + w2 + w3 )2 : The 2rst component of both ∇yL and ∇yU is always negative and the third component is always positive since w1 ; w2 ; w3 ¿0 and (w1 + w2 + w3 )2 ¿0. Therefore, to minimize y; w1 should be set to its upper bound

C. Kao, S.-T. Liu / Fuzzy Sets and Systems 120 (2001) 435–444

443

(0:9 − 0:6) and w3 should be set to its lower bound (0:6 + 0:2). To maximize y, on the contrary, w1 should be set to its lower bound (0:3) and w3 should be set to its upper bound (1 − 0:2). The second component 2(w1 − w3 )=(w1 + w2 + w3 )2 can be either positive or negative, depending on the values of w1 and w3 . In the minimization case, w1 has already been set to (0:9 − 0:6) and w3 has been set to (0:6 + 0:2). Consequently, (w1 − w3 ) = (0:3 − 0:8), which is positive for 066 38 and negative for 38 ¡61. Therefore, w2 should be set to its lower bound (0:4 + 0:3) for 066 38 and to its upper bound (1 − 0:3) for 38 ¡61. Its geometric meaning is that when  increases from below 38 to above 38 , the optimal extreme L L U U L point changes from [(W1 )U  ; (W2 ) ; (W3 ) ] to [(W1 ) ; (W2 ) ; (W3 ) ]. In the maximization case, w1 has already been set to 0:3 and w3 has been set to (1 − 0:2). Consequently, (w1 − w3 ) = (−1 + 0:5), which is always negative for 0661. Therefore, w2 should be set to its lower bound (0:4 + 0:3). The extreme point [(W1 )L ; (W2 )L ; (W3 )U  ] is the optimal extreme point for 0661. Substituting the optimal extreme points to (Y )L and (Y )U  derives  (Y )L

=

(3:2 + 3:3 − 0:12 )=(1:9 − 0:1); 066 83 ; (4:4 + 2:7 − 0:72 )=(2:5 − 0:7);

2 (Y )U  = (7:6 − 0:8 − 0:4 )=(1:4 + 0:4);

3 8 ¡61;

0661:

7:6 5:575 7:1 6:4 6:4 The -cuts at three  values: (Y )=0 = [ 3:2 1:9 ; 1:4 ]; (Y )=0:5 = [ 2:15 ; 1:6 ], and (Y )=1 = [ 1:8 ; 1:8 ] are the same as that reported in Dong and Wong [7]. 3:2 6:4 6:4 7:6 L L U U Both (Y )L and (Y )U  are invertible. Since (Y )=0 = 1:9 ; (Y )=1 = 1:8 ; (Y )=1 = 1:8 ; (Y )=0 = 1:4 , and (Y )L=3=8 = 2:375, according to (13), we have

  L −1  ] = [(3:3 + 0:1y) − 0:01y2 − 0:1y + 12:17]=0:2; 3:2=1:96y62:375; [(Y )  63=8    Y˜(y) = [(Y )L¿3=8 ]−1 = [(2:7 + 0:7y) − 0:49y2 − 3:22y + 19:61]=1:4; 2:3756y66:4=1:8;      −1 [(Y )U = [− (0:8 + 0:4y) + 0:16y2 − 1:6y + 12:8]=0:8; 6:4=1:86y67:6=1:4: ] An exact membership function is derived.

5. Conclusion Weighted average is widely used in engineering, science, and management. When it is extended to fuzzy environment, heavy computation burden is involved. The 2rst study in this area by Dong and Wong [7] requires 22n permutations for problems of n criteria. A later study by Liou and Wang [11] improves the e6ciency to less than [n(n + 1) + 2] permutations. Recently, Guh et al. [8] and Lee and Park [10] proposed two diKerent methods, requiring the same complexity of O(n log n). In this paper we develop a pair of fractional programs to calculate the end points of the -cut of the fuzzy weighted average. By transforming to a pair of linear programs, it is found that the number of permutations needed is less than 2n. On average, it is expected to be less than 6, independent of n. One important feature of the proposed approach which is not shared by other methods is that the family of fractional programs for diKerent  values can be modeled as a parametric program with parameter . For most problems, the exact membership function can be derived by taking the inverse function of the -cut of the fuzzy weighted average. Hence, more information is provided for decision making.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

S. Baas, H. Kwakernaak, Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica 13 (1977) 47– 48. M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear Programming and Network Flows, 2nd ed., Wiley, New York, 1990. M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd ed., Wiley, New York, 1993. A. Charnes, W.W. Cooper, Programming with linear fractional functionals, Naval Res. Logist. Quart. 9 (1962) 181–186. D.I. Cleland, D.F. Kocaoglu, Engineering Management, McGraw-Hill, New York, 1981. G.B. Dantzig, Linear Programming and Extensions, Princeton Univ. Press, Princeton, NJ, 1963. W.M. Dong, F.S. Wong, Fuzzy weighted averages and implementation of the extension principle, Fuzzy Sets and Systems 21 (1987) 183 –199. Y.Y. Guh, C.C. Hong, K.M. Wang, E.S. Lee, Fuzzy weighted average: a max-min paired elimination method, Comput. Math. Appl. 32 (1996) 115 –123. C.A. Holloway, Decision Making Under Uncertainty: Models and Choices, Prentice-Hall, Englewood CliKs, NJ, 1979. D.H. Lee, D. Park, An e6cient algorithm for fuzzy weighted average, Fuzzy Sets and Systems 87 (1997) 39 – 45. T.S. Liou, M.J. Wang, Fuzzy weighted average: an improved algorithm, Fuzzy Sets and Systems 49 (1992) 307– 315. R.R. Yager, A characterization of the extension principle, Fuzzy Sets and Systems 18 (1986) 205 – 217. L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3 – 28. M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982. H.J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd ed., Kluwer-NijhoK, Boston, 1991.