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A weighted max–min model for fuzzy goal programming
Fuzzy Sets and Systems 142 (2004) 407 – 420 www.elsevier.com/locate/fss
A weighted max–min model for fuzzy goal programming Chang-Chun Lin∗ Department of Information Management, Kung-Shan University of Technology, 949 Da-wan Road, Yung-Kang, Tainan 710, Taiwan, ROC Received 7 February 2002; received in revised form 25 February 2003; accepted 28 February 2003
Abstract After Narasimhan’s pioneering study of applying fuzzy set theory to goal programming in 1980, many achievements in the 1eld have been recorded. Most of them followed the max–min approach. However, when objectives have di5erent levels of importance, only the weighted additive model of Tiwari et al. seems to be applicable. However, the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave. This study proposes a novel weighted max–min model for fuzzy goal programming (FGP) and for fuzzy multiple objective decision-making. The proposed model adapts well to even the most complicated membership functions. Numerical examples demonstrate that the proposed model can be e5ectively incorporated with other approaches to FGP and is superior to the weighted additive approach. c 2003 Elsevier B.V. All rights reserved. Keywords: Fuzzy mathematical programming; Fuzzy goal programming; Weighted max–min model
1. Introduction Since Narasimhan [13] 1rst applied fuzzy set theory to goal programming in 1980, many achievements in the 1eld fuzzy goal programming (FGP) have been recorded. Hannan [5] 1rst considered the use of interpolated membership functions, or piecewise linear membership functions, to represent the preference of the decision maker (DM) for several objective values. Both the methods of Narasimhan [13] and the method of Hannan [5] are applicable only to FGP problems with concave membership functions, as depicted in Fig. 1. Nakamura [12] extended FGP to include quasiconcave membership functions, as shown in Fig. 2. Yang et al. [19], Inuiguchi et al. [6], Li and Yu [10], Wang and Fu [18], and Lin and Chen [11] also proposed methods for FGP with quasiconcave membership functions. ∗
Corresponding author. Fax: +886-6-273-2726. E-mail address: [email protected] (C.-C. Lin).
c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/S0165-0114(03)00092-7
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i ((ax) i )
1 i ( g i3 )
i ( g i2 ) i ( g i1 )
0
gi 0
g i1
gi 2
g i3
gi 4
(ax )i
Fig. 1. Concave piecewise linear membership function.
i ((ax) i )
1 i ( g i3 )
i ( g i2 )
i ( g i1 )
0
gi 0
g i1
gi 2
g i3
gi 4
(ax )i
Fig. 2. Quasiconcave piecewise linear membership function.
Zimmermann [20] 1rst used the max–min operator of Bellman and Zadeh [1] to solve fuzzy multi-objective linear programming problems. The aforementioned methods of FGP also followed the max–min approach, except that of Tiwari et al. [17]. Several approaches have been developed to handle cases in which objectives are not equally important. The 1rst is the fuzzy weights approach of Narasimhan [14], in which membership functions that represent linguistic priorities are de1ned on goal values. Strictly speaking, the fuzzy weights represent only the relative importance of the goal values of a certain objective rather than the relative importance of di5erent objectives. The second is the weighted model considered by Hannan [5] in which objectives are di5erently weighted to represent their relative importance, and the weighted sum of the deviations from the centers of triangular membership functions is minimized. However, this method uses only isosceles triangular membership functions. The third method is the preemptive structure of Tiwari et al. [16]. Like all preemptive structures, the shortcoming of this method is that higher-level objectives must be achieved before lower-level objectives can be considered. Higher-level objectives are thus in1nitely more important than lower-level objectives. Objectives can only be of equal importance or of extremely di5erent importance. The last method is the additive model of Tiwari et al. [17] in which the weighted sum of the achieved levels of objectives is maximized. The augmented max–min approach of Li [9], later improved by Lai and Hwang [8], also uses weighted sum as part of the objective function. However, the Section 3 will show that the summation of quasiconcave functions may no longer be quasiconcave, which fact is important when quasiconcave functions are involved.
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Above discussion suggests that no weighted max–min model considers objectives of di5erent importance levels while adapting to all kinds of membership functions. This study proposes such a weighted max–min model for FGP. The proposed model can be incorporated with any other FGP model using the max–min operator and serves well for FGP problems with quasiconcave membership functions.
2. Weighted max–min model Consider the following FGP problem with m fuzzy goals Find
x ∼
to satisfy (ax)i ¿ gi ;
i = 1; 2; : : : ; m;
(1)
subject to Bx 6 b; x ¿ 0;
where x is an n-vector with components x1 ; x2 ; : : : ; xn and Bx 6 b are system constraints in vector notation. Any FGP problem can be expressed with Model (1) without a loss of generality because ∼ ∼ every 6 constraint can be converted to an equivalent ¿ constraint and every ∼ = constraint can ∼ ∼ be replaced by a 6 constraint and a ¿ constraint. Since all objectives might not be achieved simultaneously under the system constraints, the decision maker may de1ne a lower tolerance limit and a membership function for each objective to determine the achieved level of that objective. A ∼ membership function i ((ax)i ) for the ith fuzzy goal (ax)i ¿ gi can be expressed as if gi 6 (ax)i ; 1 (2) i ((ax)i ) = fi ((ax)i ) if li 6 (ax)i ¡ gi ; 0 if (ax)i 6 li ; ∼
where li is a lower tolerance limit for the fuzzy goal (ax)i ¿ gi . Function fi ((ax)i ) herein can be a linear or piecewise linear function that is concave or quasiconcave. With Zimmermann’s approach, using max–min as the operator, a max–min model for Problem (1) can be stated as follows: Maximize subject to 6 fi ((ax)i ); Bx 6 b;
i = 1; 2; : : : ; m;
(3)
x ¿ 0: Li [9] proposed a two-phase approach in which the 1rst phase is to use Zimmermann’s approach to search for an optimal value of and to 1nd a possible solution x0 . If the possible solution is unique in phase one, x0 will be an optimal solution. Otherwise, the following program is formulated
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to 1nd an eMcient solution: m Maximize i =m i=1
subject to 6 i 6 fi ((ax)i ); Bx 6 b; x ¿ 0;
i = 1; 2; : : : ; m;
(4)
where is the solution of Program (3). Lai and Hwang [8] proposed the following program to replace the two-phase approach of Li: Maximize +
m
wi fi ((ax)i )
i=1
subject to 6 fi ((ax)i ); Bx 6 b; x ¿ 0;
i = 1; 2; : : : ; m;
(5)
where is a suMciently small positive number; wi is the relative weight of the ith objective and m i=1 wi = 1. Recently, Dubois and Fortemps [3] suggested a multi-step procedure for constraint satisfaction problems, which yields all discrimin- and leximin-optimal solutions and can better discriminate among solutions. Both the approaches of Zimmermann and Li do not consider the relative weights of objectives. Although the model of Lai and Hwang considers the relative weights of objectives, the method is no more than an extension of Zimmermann’s approach. Objectives with heavy weights are not emphasized because is only a very small number. One model that takes into account the objectives’ weights is the additive model of Tiwari et al. [17], which is formulated as follows: Maximize
m
wi fi ((ax)i )
i=1
subject to Bx 6 b; x ¿ 0;
(6)
Dubois and Prade [4] suggested another method that uses maximum and minimum operators and simultaneously considers the weights of objectives. Let be a possibility distribution describing the fuzzy set of relevant goals. i is the grade of relevance of the ith goal and i∗ = 1 for some i∗ , that is, at least one of the goals is totally relevant. Using the concept of possibility and necessity of fuzzy events, one can evaluate the possibility (Fx ) that x attains a relevant goal, and the necessity N (Fx ) that x attains all the relevant goals as follows: (Fx ) = max min(i ; i ((ax)i ))
(7)
N (Fx ) = min max(1 − i ; i ((ax)i )):
(8)
i
and i
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In (7), i is a threshold above which the ith goal cannot enhance the overall evaluation when a disjunction of goals is considered. In (8), the overall evaluation is lower-bounded by 1 − i if the degree of satisfaction of the ith goal deteriorates too much. Hence i express the degree of importance of the ith goal in the evaluation. However, the solution obtained using the criterion of maximizing possibility tends to satisfy the goal with i = 1 and ignore the other goals. On the other hand, the problem becomes a global optimization when the criterion of maximizing necessity is used. Both the criteria require iterative solving procedures, which can be observed in their applications of Dubois et al. [2], Inuiguchi et al. [7] and Sabbadin et al. [15]. In fact, these schemes are prioritized rather than weighted in the sense that the i are satisfaction thresholds rather than weights proper. When the DM provides relative weights for fuzzy goals with corresponding membership functions, the ratio of the achieved levels should be as close to the ratio of the objective weights as possible to reOect their relative importance. The additive model of Tiwari et al. gives objectives of heavy weight higher achieved levels than others. However, the ratio of the achieved levels is not necessarily the same as that of the objective weights. Thus, this study proposed a weighted max–min model for Problem (1) as follows: Maximize subject to wi 6 fi ((ax)i ); Bx 6 b; x ¿ 0:
i = 1; 2; : : : ; m;
This model is equivalent to solving Program that 1 if w i i ((ax)i ) i ((ax)i ) = = fi ((ax)i ) if wi wi 0 if
(9)
(3) with new membership functions, i ((ax)i ), such gi 6 (ax)i ; li 6 (ax)i ¡ gi ;
(10)
(ax)i 6 li :
Notably, the function values of i ((ax)i ) can exceed unity since wi ¡ 1. So does the optimal achieved level ∗ to Model (4). Nevertheless, the actual achieved level for each objective may never exceed unity. Let x∗ be the optimal solution to Model (9). The actual achieved level for each objective can then be obtained using Function (2). The objective of the weighted max–min model is to 1nd an optimal solution within the feasible area such that the ratio of the achieved levels is as close to the ratio of the weights as possible. This can be veri1ed by the following proposition. Proposition 1. The weighted max–min model 3nds an optimal solution within the feasible area such that the ratio of the achieved levels is as close to the ratio of the weights as possible. Proof. In search of optimal solution, linear programming tends to use as many resources available as possible. As a result, the slack=surplus variables would be reduced as small as possible. Let si denote the slack variable such that wi + si = i ((ax)i );
i = 1; 2; : : : ; m:
(11)
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Since membership function i ((ax)i ) is a bounded quasiconcave function, its value cannot be in1nitely large. Thus, si would be minimized as is maximized. The achieved level of Objective i would become as close to wi as possible; that is, the ratio of the achieved levels would be as close to the ratio of the weights as possible. In an ideal case where all the slack variables are zero, the ratio of the achieved levels would be the same as the ratio of the weights. Example 1. The following example problem, considered by Hannan [5], illustrates how the proposed model can be used to solve FGP problems with concave piecewise linear membership functions. The problem is as given below: Find
x ∼
to satisfy z1 = 3x1 + x2 + x3 ¿ 7; ∼
z2 = x1 − x2 + 2x3 ¿ 8; ∼
z3 = x1 + 2x2 ¿ 5; subject to 4x1 + 2x2 + 3x3 6 10;
(12)
x1 + 3x2 + 2x3 6 8; x3 6 5; x1 ; x2 ; x3 ¿ 0: The corresponding membership functions for the three goals are given as 1 if 7 6 z1 ; 0:2(z1 − 6) + 0:8 if 6 6 z1 ¡ 7; 1 (z1 ) = 0:3(z1 − 5) + 0:5 if 5 6 z1 ¡ 6; 0:5(z1 − 4) if 4 6 z1 ¡ 5; 0 if z1 ¡ 4; 1 if 8 6 z2 ; 0:15(z − 4) + 0:4 if 4 6 z ¡ 8; 2 2 2 (z2 ) = 0:2(z − 2) if 2 6 z 2 2 ¡ 4; 0 if z2 ¡ 2 and
1 0:2(z − 4) + 0:8 3 3 (z3 ) = 0:4(z 3 − 2) 0
if if if if
5 6 z3 ; 4 6 z3 ¡ 5; 2 6 z3 ¡ 4; z3 ¡ 2:
(13)
(14)
(15)
Suppose the relative weights of the three objectives, given by the DM, are w1 = 0:4, w2 = 0:35 and w3 = 0:25. Based on the weighted max–min model and the approach of Yang et al. [19] for concave
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piecewise linear membership functions, a linear program for this problem can be formulated as Maximize subject to 0:4 6 0:2(z1 − 6) + 0:8; 0:4 6 0:3(z1 − 5) + 0:5; 0:4 6 0:5(z1 − 4); 0:35 6 0:15(z2 − 4) + 0:4; 0:35 6 0:2(z2 − 2); 0:25 6 0:2(z3 − 4) + 0:8; 0:25 6 0:4(z3 − 2); 4x1 + 2x2 + 3x3 6 10; x1 + 3x2 + 2x3 6 8; x3 6 5; x1 ; x2 ; x3 ¿ 0:
(16)
The optimal solution obtained by LINDO is x1 = 0:60, x2 = 0:95 and x3 = 1:89 with achieved levels 1 = 0:328, 2 = 0:287 and 3 = 0:205. The equalities of the ratios can be easily veri1ed 0:287 0:205 0:328 = = = 0:82; 0:4 0:35 0:25
(17)
which value is also the optimal achieved level to Program (16). Table 1 compares the solutions obtained by di5erent approaches. The solution obtained by Lai and Hwang’s approach is the same as that obtained by Zimmermann’s approach. The fact veri1es the argument that objectives with heavy weights are not emphasized because is only a very small number. Table 1 also indicates that the solution obtained by the additive model is far beyond acceptable. The ratio of the achieved levels is not the same as that of the objective weights. The achieved level of the second objective is lower than that of the third objective even though the weight of the former is heavier than that of the latter. When the objectives are equally weighted, Program (16) degenerates to a weightless max–min model and the optimal solution becomes x1 = 0:50, x2 = 1:08 and x3 = 1:95 with achieved levels, Table 1 Solutions to Example 1 by di5erent approaches Approaches
z1 z2 z3 1 2 3
Proposed model
Zimmermann (weightless)
Lai and Hwang
Additive model
4.656 3.435 2.513 0.328 0.287 0.205
4.526 3.315 2.658 0.263 0.263 0.263
4.526 3.315 2.658 0.263 0.263 0.263
7.000 2.000 3.100 1.000 0.000 0.440
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1 = 2 = 3 = 0:263. Comparing the solutions obtained by using these two di5erent sets of weights reveals that the proposed model manages to 1nd an optimal solution that gives objective 1 a higher achieved level than the others while the achievement of objective 1 is emphasized by giving it a heavy weight than the others. Thus, the weighted max–min model 1nds an optimal solution such that the ratio of the achieved levels is the same as the weights of objectives.
3. Models with quasiconcave membership functions Only the weighted additive model of Tiwari et al. [17] seems to be applicable in the case where the DM feels that the objectives are of di5erent levels of importance, while giving membership functions that include some quasiconcave ones. However, the additive model encounters a serious problem when quasiconcave membership functions are involved in FGP. Fig. 3 shows that the summation of two quasiconcave functions is not necessarily a quasiconcave function. This can cause the solution of FGP to stall at a local optimum. Moreover, further investigation is required to determine whether the methods for FGP problems with quasiconcave membership functions, for example, the methods of Inuiguchi et al. [6] or the method of Yang et al. [19], can be integrated with the additive model. This section describes how the weighted max–min model can be used to solve FGP problems with quasiconcave membership functions and objective weights. The FGP problem used in the subsequent examples is the numerical example from Inuiguchi et al. [6]. The problem is 1rst treated with the method of Yang et al. [19] in Example 2, and then with the method of Inuiguchi et al. in Example 3. Example 2. Consider the problem below: Find
x
∼
to satisfy z1 = −x1 + 2x2 ¿ 12; ∼
z2 = 2x1 + x2 ¿ 21; subject to −x1 + 3x2 6 21; x1 + 3x2 6 27; 4x1 + 3x2 6 45; 3x1 + x2 6 30; x1 ; x2 ¿ 0:
(18)
The corresponding membership functions for the two goals are 1 0:08(z1 − 2) + 0:2 1 (z1 ) = 0:04(z1 + 3) 0
if if if if
12 6 z1 ; 2 6 z1 ¡ 12; − 3 6 z1 ¡ 2; z1 ¡ −3
(19)
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415
1
1 + 2
2
i
1
0
Fig. 3. Summation of two quasiconcave functions.
and
1 0:1(z − 17) + 0:6 2 2 (z2 ) = 0:06(z2 − 7) 0
if if if if
21 6 z2 ; 17 6 z2 ¡ 21; 7 6 z2 ¡ 17; z2 ¡ 7;
(20)
Suppose the DM gives weights, w1 = 0:6 and w2 = 0:4. Incorporating the method of Yang et al. [19] for quasiconcave membership functions with the weighted max–min model gives the following linear program. Maximize subject to 0:6 6 0:08(z1 − 2) + 0:2 + M1 ; 0:6 6 0:04(z1 + 3) + M (1 − 1 ); 0:4 6 0:1(z2 − 17) + 0:6 + M2 ; 0:4 6 0:06(z2 − 7) + M (1 − 2 ); −x1 + 3x2 6 21; x1 + 3x2 6 27; 4x1 + 3x2 6 45; 3x1 + x2 6 30; x1 ; x2 ¿ 0; 1 ; 2 = 0; 1;
(21)
where M is a large positive number. The optimal solution is x1 = 4:59 and x2 = 7:47, and the achieved levels are 1 = 0:868 and 2 = 0:579. Since the approaches of Zimmermann, and Lai and Hwang are unsuitable for weighted problems, Table 2 only compares the solution obtained by the proposed model and that by the additive model. Again, by the proposed model, the ratio of the achieved levels (0:868=0:579 = 1:5) equals the ratio of the weights (0:6=0:4 = 1:5). However, the ratio of the achieved levels (1=0:48 = 2:08) exceeds the ratio of the weights 1.5. Thus, the solution obtained by the proposed model is better than that by the additive model. Using the approach of Yang et al., the linear program for solving Problem (18) can be formulated straightforwardly. This is not the case when the method of Inuiguchi et al. [6] is used. Though
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C.-C. Lin / Fuzzy Sets and Systems 142 (2004) 407 – 420 Table 2 Solutions to Example 2 by di5erent approaches Approaches
z1 z2 1 2
Proposed model
Additive model
10.350 16.650 0.868 0.579
12.000 15.000 1.000 0.480
the method of Yang et al. is easily applied, it requires zero-one variables to formulate the linear program, causing computational ineMciency in obtaining the solution. The method of Inuiguchi et al. translates a set of quasiconcave membership functions into a set of equivalent concave membership functions. The method uses no zero-one variables and guarantees that the optimal solution, obtained using the new concave membership functions, is exactly the same as that obtained using the original quasiconcave membership functions. As stated above, the weighted max–min model is equivalent to the unweighted max–min model with new membership functions that is obtained by dividing the original membership function by their corresponding weights. Thus, the membership functions must be divided by their corresponding weights to become new membership functions before the method of Inuiguchi et al. can be applied to the weighted max–min model. Membership function i ((ax)i ) is divided by wi to become i ((ax)i ), which is then translated to concave function, i ((ax)i ) by the process of Inuiguchi et al. to be used for solving the problem. The following example illustrates how the method of Inuiguchi et al. can be integrated with the weighted max–min model. Example 3. Consider the same problem as in Example 2. When the weights are equal, the problem remains an ordinary FGP problem. Translating membership functions (19) and (20) using the translation process of Inuiguchi et al. (refer to [6] for detailed algorithm), yields the following equivalent concave membership functions:
1 (z1 ) =
and
2 (z2 ) =
1 3
(z 65 1 1 (z 13 1
− 7) + + 3)
10 13
0
1 3 z2 − 10 13 + 52 3 (z − 7) 26 2 0
if if if if
5 13
12 6 z1 ; 7 6 z1 ¡ 12; − 3 6 z1 ¡ 7; z1 ¡ −3
(22)
if 21 6 z2 ; if 10 13 6 z2 ¡ 21; 1 if 7 6 z2 ¡ 10 ; 3 if z2 ¡ 7:
(23)
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417
Formulating and solving the problem with Functions (22) and (23) gives the optimal solution, x1 = 5:6 and x2 = 7:13, with achieved levels of 1 = 2 = 0:733. Suppose, however, that the DM gives weights, w1 = 0:6 and w2 = 0:4 as in Example 2. Firstly, membership Functions (19) and (20) are divided by w1 and w2 to become 5 if 12 6 z1 ; 3 2 (z − 2) + 1 if 2 6 z ¡ 12; 1 1 3 (24) 1 (z1 ) = 15 1 (z + 3) if − 3 6 z1 ¡ 2; 15 1 0 if z1 ¡ −3 and 2 (z2 ) =
5 2 1
(z − 17) 4 2 3 (z − 7) 20 2
+
3 2
0
if if if if
21 6 z2 ; 17 6 z2 ¡ 21; 7 6 z2 ¡ 17; z2 ¡ 7:
(25)
Notably, the membership degree of both of these membership functions may exceed unity. The following process illustrates how membership Functions (24) and (25) can be translated into equivalent concave membership functions. According to the process of Inuiguchi et al., the vertices vj = (v1j ; v2j ) at which the two membership functions have same degrees are
83 43 53 73 ; v3 = and v5 = v1 = (−3; 7); v2 = 2; ; 17 ; v4 = 12; ; 21 : 9 4 3 4 (26) By
= pj min pj+1
vkj+2 − vkj+1 vkj+1 − vkj
k=1;2
and letting p1 = 1,
(27)
83 83 7 43 −2 (2 − (−3)); 17 − −7 = ; 4 9 9 4
43 3 7 43 53 83 p3 = × min = 12 − −2 ; − 17 17 − 4 4 4 3 9 20 p2 = 1 × min
and 3 × min p4 = 20
73 43 53 3 53 − 12 12 − ; 21 − − 17 = : 4 4 3 3 4
Then, pj pj = pj
(28)
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C.-C. Lin / Fuzzy Sets and Systems 142 (2004) 407 – 420
yields,
7 35 3 3 = ; + + 4 20 4 53
7 3 3 3 3 + + = p2 = 20 4 20 4 53 7 p1 = 4
and 3 p3 = 4
7 3 3 + + 4 20 4
=
15 : 53
The values of the two new concave membership functions at every vertex can be obtained as follows: 1 (−3) = 2 (7) = 0;
20 20 83 =0+ = ; 1 (2) = 2 9 73 73
43 20 35 55 = 2 (17) = 0 + 1 + = ; 4 73 73 73
20 35 3 58 53 =0+ + + = 1 (12) = 2 3 73 73 73 73 and
1
73 4
= 2 (21) = 0 +
3 15 20 35 + + + = 1: 73 73 73 73
Finally, the new concave membership functions are obtained 73 1 if 6 z1 ; 4
12 z − 43 + 55 if 43 6 z ¡ 73 ; 1 1 4 73 4 4 1 (z1 ) = 365 4 43 if − 3 6 z1 ¡ ; (z1 + 3) 73 4 0 if z1 ¡ −3 and
1 if 21 6 z2 ;
9 20 83 83 z2 − + if 6 z2 ¡ 21; 9 73 9 2 (z2 ) = 146 83 9 (z − 7) if 7 6 z2 ¡ ; 2 9 73 0 if z2 ¡ 7:
(29)
(30)
Formulating and solving the problem as the unweighted max–min model in Example 1 gives the optimal solution, x1 = 4:59 and x2 = 7:47. The achieved levels are 1 = 0:868 and 2 = 0:579,
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419
as obtained using the method of Yang et al. in Example 2. If the translation process is performed before the original quasiconcave membership functions are divided by their corresponding weights, then the resulting concave membership functions are membership Functions (22) and (23). Solving a weighted max–min model with these concave membership functions gives the optimal solution, x1 = 3:69 and x2 = 7:77. The achieved levels are 1 = 0:988 and 2 = 0:489, of which the ratio is 0:988=0:489 = 2:02 rather than 0:6=0:4 = 1:5. These results indicate that the dividing must take place before the translation process when the method of Inuiguchi et al. is used. 4. Conclusions This study proposes a simple but e5ective weighted max–min model for FGP and for fuzzy multiple objective decision-making. The proposed model outperforms the widespread weighted sum approach in at least two ways. Firstly, the summation of quasiconcave membership functions may not remain quasiconcave. Secondly, the proposed model gives solutions in which the ratio of the achieved levels of the goals or objectives is close to or even the same as the ratio of the objective weights. The simplicity of the proposed model is such that it can be combined e5ectively with other approaches for FGP. Examples illustrate how the proposed model can be combined with the methods of Inuiguchi et al. [6] and of Yang et al. [19] to solve a FGP problem with quasiconcave membership functions. Notably, the original membership functions must be divided by their corresponding weights before the translation process of obtaining concave membership functions takes place whenever the method of Inuiguchi et al. is used. Reversing this order would mislead the solution such that a worse solution might be obtained. For other methods that also use certain techniques to translate the membership functions from quasiconcave to concave, the membership functions should 1rstly be divided by their weights. Acknowledgements The author would like to thank the editors-in-chief and the anonymous referees for their valuable comments on the earlier versions of this paper. References [1] R. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Sci. 17B (1970) 141–164. [2] D. Dubois, H. Fargier, H. Prade, Possibility theory in constraint satisfaction problems: handling priority, preference and uncertainty, Appl. Intell. 6 (1996) 287–309. [3] D. Dubois, F. Fortemps, Computing improved optimal solutions to max–min Oexible constraint satisfaction problems, European J. Oper. Res. 118 (1999) 95–126. [4] D. Dubois, H. Prade, Weighted minimum and maximum operations, Inform. Sci. 39 (1986) 205–210. [5] E.L. Hannan, Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235–248. [6] M. Inuiguchi, H. Ichihashi, Y. Kume, A solution algorithm for fuzzy linear programming with piecewise membership functions, Fuzzy Sets and Systems 34 (1990) 15–31.
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[7] M. Inuiguchi, H. Ichihashi, H. Tanaka, Possibilistic linear programming with measurable multiattribute value functions, ORSA J. Comput. 1 (1989) 146–158. [8] Y.J. Lai, C.L. Hwang, Possibilistic linear programming for managing interest rate risk, Fuzzy Sets and Systems 49 (1992) 121–133. [9] R.J. Li, Multiple objective decision making in a fuzzy environment, Ph.D. Dissertation (1990), Department of Industrial Engineering, Kansas State University, Manhattan, KS. [10] H.L. Li, C.S. Yu, A fuzzy multiobjective program with quasiconcave membership functions and fuzzy coeMcients, Fuzzy Sets and Systems 109 (2000) 59–81. [11] C.C. Lin, A.P. Chen, A generalization of Yang et al.’s method for fuzzy programming with piecewise linear membership functions, Fuzzy Sets and Systems 132 (2002) 347–352. [12] K. Nakamura, Some extensions of fuzzy linear programming, Fuzzy Sets and Systems 14 (1984) 211–229. [13] R. Narasimhan, Goal programming in a fuzzy environment, Decision Sci. 11 (1980) 325–338. [14] R. Narasimhan, On Fuzzy goal programming—some comments, Decision Sci. 12 (1981) 532–538. [15] R. Sabbadin, D. Dubois, P. Grenier, H. Prade, A fuzzy constraint satisfaction problem in the wine industry, J. Intell. Fuzzy Systems 6 (1998) 367–374. [16] R.N. Tiwari, S. Dharmar, J.R. Rao, Priority structure in fuzzy goal programming, Fuzzy Sets and Systems 19 (1986) 251–259. [17] R.N. Tiwari, S. Dharmar, J.R. Rao, Fuzzy goal programming—an additive model, Fuzzy Sets and Systems 24 (1987) 27–34. [18] H.F. Wang, C.C. Fu, A generalization of fuzzy programming with preemptive structure, Comput. Oper. Res. 24 (1997) 819–828. [19] T. Yang, J.P. Ignizio, H.J. Kim, Fuzzy programming with nonlinear membership functions: piecewise linear approximation, Fuzzy Sets and Systems 41 (1991) 39–53. [20] H.-J. Zimmermann, Fuzzy programming and linear programming with multiple objective functions, Fuzzy Sets and Systems 1 (1978) 45–55.
A weighted max–min model for fuzzy goal programming Chang-Chun Lin∗ Department of Information Management, Kung-Shan University of Technology, 949 Da-wan Road, Yung-Kang, Tainan 710, Taiwan, ROC Received 7 February 2002; received in revised form 25 February 2003; accepted 28 February 2003
Abstract After Narasimhan’s pioneering study of applying fuzzy set theory to goal programming in 1980, many achievements in the 1eld have been recorded. Most of them followed the max–min approach. However, when objectives have di5erent levels of importance, only the weighted additive model of Tiwari et al. seems to be applicable. However, the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave. This study proposes a novel weighted max–min model for fuzzy goal programming (FGP) and for fuzzy multiple objective decision-making. The proposed model adapts well to even the most complicated membership functions. Numerical examples demonstrate that the proposed model can be e5ectively incorporated with other approaches to FGP and is superior to the weighted additive approach. c 2003 Elsevier B.V. All rights reserved. Keywords: Fuzzy mathematical programming; Fuzzy goal programming; Weighted max–min model
1. Introduction Since Narasimhan [13] 1rst applied fuzzy set theory to goal programming in 1980, many achievements in the 1eld fuzzy goal programming (FGP) have been recorded. Hannan [5] 1rst considered the use of interpolated membership functions, or piecewise linear membership functions, to represent the preference of the decision maker (DM) for several objective values. Both the methods of Narasimhan [13] and the method of Hannan [5] are applicable only to FGP problems with concave membership functions, as depicted in Fig. 1. Nakamura [12] extended FGP to include quasiconcave membership functions, as shown in Fig. 2. Yang et al. [19], Inuiguchi et al. [6], Li and Yu [10], Wang and Fu [18], and Lin and Chen [11] also proposed methods for FGP with quasiconcave membership functions. ∗
Corresponding author. Fax: +886-6-273-2726. E-mail address: [email protected] (C.-C. Lin).
c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/S0165-0114(03)00092-7
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i ((ax) i )
1 i ( g i3 )
i ( g i2 ) i ( g i1 )
0
gi 0
g i1
gi 2
g i3
gi 4
(ax )i
Fig. 1. Concave piecewise linear membership function.
i ((ax) i )
1 i ( g i3 )
i ( g i2 )
i ( g i1 )
0
gi 0
g i1
gi 2
g i3
gi 4
(ax )i
Fig. 2. Quasiconcave piecewise linear membership function.
Zimmermann [20] 1rst used the max–min operator of Bellman and Zadeh [1] to solve fuzzy multi-objective linear programming problems. The aforementioned methods of FGP also followed the max–min approach, except that of Tiwari et al. [17]. Several approaches have been developed to handle cases in which objectives are not equally important. The 1rst is the fuzzy weights approach of Narasimhan [14], in which membership functions that represent linguistic priorities are de1ned on goal values. Strictly speaking, the fuzzy weights represent only the relative importance of the goal values of a certain objective rather than the relative importance of di5erent objectives. The second is the weighted model considered by Hannan [5] in which objectives are di5erently weighted to represent their relative importance, and the weighted sum of the deviations from the centers of triangular membership functions is minimized. However, this method uses only isosceles triangular membership functions. The third method is the preemptive structure of Tiwari et al. [16]. Like all preemptive structures, the shortcoming of this method is that higher-level objectives must be achieved before lower-level objectives can be considered. Higher-level objectives are thus in1nitely more important than lower-level objectives. Objectives can only be of equal importance or of extremely di5erent importance. The last method is the additive model of Tiwari et al. [17] in which the weighted sum of the achieved levels of objectives is maximized. The augmented max–min approach of Li [9], later improved by Lai and Hwang [8], also uses weighted sum as part of the objective function. However, the Section 3 will show that the summation of quasiconcave functions may no longer be quasiconcave, which fact is important when quasiconcave functions are involved.
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Above discussion suggests that no weighted max–min model considers objectives of di5erent importance levels while adapting to all kinds of membership functions. This study proposes such a weighted max–min model for FGP. The proposed model can be incorporated with any other FGP model using the max–min operator and serves well for FGP problems with quasiconcave membership functions.
2. Weighted max–min model Consider the following FGP problem with m fuzzy goals Find
x ∼
to satisfy (ax)i ¿ gi ;
i = 1; 2; : : : ; m;
(1)
subject to Bx 6 b; x ¿ 0;
where x is an n-vector with components x1 ; x2 ; : : : ; xn and Bx 6 b are system constraints in vector notation. Any FGP problem can be expressed with Model (1) without a loss of generality because ∼ ∼ every 6 constraint can be converted to an equivalent ¿ constraint and every ∼ = constraint can ∼ ∼ be replaced by a 6 constraint and a ¿ constraint. Since all objectives might not be achieved simultaneously under the system constraints, the decision maker may de1ne a lower tolerance limit and a membership function for each objective to determine the achieved level of that objective. A ∼ membership function i ((ax)i ) for the ith fuzzy goal (ax)i ¿ gi can be expressed as if gi 6 (ax)i ; 1 (2) i ((ax)i ) = fi ((ax)i ) if li 6 (ax)i ¡ gi ; 0 if (ax)i 6 li ; ∼
where li is a lower tolerance limit for the fuzzy goal (ax)i ¿ gi . Function fi ((ax)i ) herein can be a linear or piecewise linear function that is concave or quasiconcave. With Zimmermann’s approach, using max–min as the operator, a max–min model for Problem (1) can be stated as follows: Maximize subject to 6 fi ((ax)i ); Bx 6 b;
i = 1; 2; : : : ; m;
(3)
x ¿ 0: Li [9] proposed a two-phase approach in which the 1rst phase is to use Zimmermann’s approach to search for an optimal value of and to 1nd a possible solution x0 . If the possible solution is unique in phase one, x0 will be an optimal solution. Otherwise, the following program is formulated
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to 1nd an eMcient solution: m Maximize i =m i=1
subject to 6 i 6 fi ((ax)i ); Bx 6 b; x ¿ 0;
i = 1; 2; : : : ; m;
(4)
where is the solution of Program (3). Lai and Hwang [8] proposed the following program to replace the two-phase approach of Li: Maximize +
m
wi fi ((ax)i )
i=1
subject to 6 fi ((ax)i ); Bx 6 b; x ¿ 0;
i = 1; 2; : : : ; m;
(5)
where is a suMciently small positive number; wi is the relative weight of the ith objective and m i=1 wi = 1. Recently, Dubois and Fortemps [3] suggested a multi-step procedure for constraint satisfaction problems, which yields all discrimin- and leximin-optimal solutions and can better discriminate among solutions. Both the approaches of Zimmermann and Li do not consider the relative weights of objectives. Although the model of Lai and Hwang considers the relative weights of objectives, the method is no more than an extension of Zimmermann’s approach. Objectives with heavy weights are not emphasized because is only a very small number. One model that takes into account the objectives’ weights is the additive model of Tiwari et al. [17], which is formulated as follows: Maximize
m
wi fi ((ax)i )
i=1
subject to Bx 6 b; x ¿ 0;
(6)
Dubois and Prade [4] suggested another method that uses maximum and minimum operators and simultaneously considers the weights of objectives. Let be a possibility distribution describing the fuzzy set of relevant goals. i is the grade of relevance of the ith goal and i∗ = 1 for some i∗ , that is, at least one of the goals is totally relevant. Using the concept of possibility and necessity of fuzzy events, one can evaluate the possibility (Fx ) that x attains a relevant goal, and the necessity N (Fx ) that x attains all the relevant goals as follows: (Fx ) = max min(i ; i ((ax)i ))
(7)
N (Fx ) = min max(1 − i ; i ((ax)i )):
(8)
i
and i
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411
In (7), i is a threshold above which the ith goal cannot enhance the overall evaluation when a disjunction of goals is considered. In (8), the overall evaluation is lower-bounded by 1 − i if the degree of satisfaction of the ith goal deteriorates too much. Hence i express the degree of importance of the ith goal in the evaluation. However, the solution obtained using the criterion of maximizing possibility tends to satisfy the goal with i = 1 and ignore the other goals. On the other hand, the problem becomes a global optimization when the criterion of maximizing necessity is used. Both the criteria require iterative solving procedures, which can be observed in their applications of Dubois et al. [2], Inuiguchi et al. [7] and Sabbadin et al. [15]. In fact, these schemes are prioritized rather than weighted in the sense that the i are satisfaction thresholds rather than weights proper. When the DM provides relative weights for fuzzy goals with corresponding membership functions, the ratio of the achieved levels should be as close to the ratio of the objective weights as possible to reOect their relative importance. The additive model of Tiwari et al. gives objectives of heavy weight higher achieved levels than others. However, the ratio of the achieved levels is not necessarily the same as that of the objective weights. Thus, this study proposed a weighted max–min model for Problem (1) as follows: Maximize subject to wi 6 fi ((ax)i ); Bx 6 b; x ¿ 0:
i = 1; 2; : : : ; m;
This model is equivalent to solving Program that 1 if w i i ((ax)i ) i ((ax)i ) = = fi ((ax)i ) if wi wi 0 if
(9)
(3) with new membership functions, i ((ax)i ), such gi 6 (ax)i ; li 6 (ax)i ¡ gi ;
(10)
(ax)i 6 li :
Notably, the function values of i ((ax)i ) can exceed unity since wi ¡ 1. So does the optimal achieved level ∗ to Model (4). Nevertheless, the actual achieved level for each objective may never exceed unity. Let x∗ be the optimal solution to Model (9). The actual achieved level for each objective can then be obtained using Function (2). The objective of the weighted max–min model is to 1nd an optimal solution within the feasible area such that the ratio of the achieved levels is as close to the ratio of the weights as possible. This can be veri1ed by the following proposition. Proposition 1. The weighted max–min model 3nds an optimal solution within the feasible area such that the ratio of the achieved levels is as close to the ratio of the weights as possible. Proof. In search of optimal solution, linear programming tends to use as many resources available as possible. As a result, the slack=surplus variables would be reduced as small as possible. Let si denote the slack variable such that wi + si = i ((ax)i );
i = 1; 2; : : : ; m:
(11)
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Since membership function i ((ax)i ) is a bounded quasiconcave function, its value cannot be in1nitely large. Thus, si would be minimized as is maximized. The achieved level of Objective i would become as close to wi as possible; that is, the ratio of the achieved levels would be as close to the ratio of the weights as possible. In an ideal case where all the slack variables are zero, the ratio of the achieved levels would be the same as the ratio of the weights. Example 1. The following example problem, considered by Hannan [5], illustrates how the proposed model can be used to solve FGP problems with concave piecewise linear membership functions. The problem is as given below: Find
x ∼
to satisfy z1 = 3x1 + x2 + x3 ¿ 7; ∼
z2 = x1 − x2 + 2x3 ¿ 8; ∼
z3 = x1 + 2x2 ¿ 5; subject to 4x1 + 2x2 + 3x3 6 10;
(12)
x1 + 3x2 + 2x3 6 8; x3 6 5; x1 ; x2 ; x3 ¿ 0: The corresponding membership functions for the three goals are given as 1 if 7 6 z1 ; 0:2(z1 − 6) + 0:8 if 6 6 z1 ¡ 7; 1 (z1 ) = 0:3(z1 − 5) + 0:5 if 5 6 z1 ¡ 6; 0:5(z1 − 4) if 4 6 z1 ¡ 5; 0 if z1 ¡ 4; 1 if 8 6 z2 ; 0:15(z − 4) + 0:4 if 4 6 z ¡ 8; 2 2 2 (z2 ) = 0:2(z − 2) if 2 6 z 2 2 ¡ 4; 0 if z2 ¡ 2 and
1 0:2(z − 4) + 0:8 3 3 (z3 ) = 0:4(z 3 − 2) 0
if if if if
5 6 z3 ; 4 6 z3 ¡ 5; 2 6 z3 ¡ 4; z3 ¡ 2:
(13)
(14)
(15)
Suppose the relative weights of the three objectives, given by the DM, are w1 = 0:4, w2 = 0:35 and w3 = 0:25. Based on the weighted max–min model and the approach of Yang et al. [19] for concave
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413
piecewise linear membership functions, a linear program for this problem can be formulated as Maximize subject to 0:4 6 0:2(z1 − 6) + 0:8; 0:4 6 0:3(z1 − 5) + 0:5; 0:4 6 0:5(z1 − 4); 0:35 6 0:15(z2 − 4) + 0:4; 0:35 6 0:2(z2 − 2); 0:25 6 0:2(z3 − 4) + 0:8; 0:25 6 0:4(z3 − 2); 4x1 + 2x2 + 3x3 6 10; x1 + 3x2 + 2x3 6 8; x3 6 5; x1 ; x2 ; x3 ¿ 0:
(16)
The optimal solution obtained by LINDO is x1 = 0:60, x2 = 0:95 and x3 = 1:89 with achieved levels 1 = 0:328, 2 = 0:287 and 3 = 0:205. The equalities of the ratios can be easily veri1ed 0:287 0:205 0:328 = = = 0:82; 0:4 0:35 0:25
(17)
which value is also the optimal achieved level to Program (16). Table 1 compares the solutions obtained by di5erent approaches. The solution obtained by Lai and Hwang’s approach is the same as that obtained by Zimmermann’s approach. The fact veri1es the argument that objectives with heavy weights are not emphasized because is only a very small number. Table 1 also indicates that the solution obtained by the additive model is far beyond acceptable. The ratio of the achieved levels is not the same as that of the objective weights. The achieved level of the second objective is lower than that of the third objective even though the weight of the former is heavier than that of the latter. When the objectives are equally weighted, Program (16) degenerates to a weightless max–min model and the optimal solution becomes x1 = 0:50, x2 = 1:08 and x3 = 1:95 with achieved levels, Table 1 Solutions to Example 1 by di5erent approaches Approaches
z1 z2 z3 1 2 3
Proposed model
Zimmermann (weightless)
Lai and Hwang
Additive model
4.656 3.435 2.513 0.328 0.287 0.205
4.526 3.315 2.658 0.263 0.263 0.263
4.526 3.315 2.658 0.263 0.263 0.263
7.000 2.000 3.100 1.000 0.000 0.440
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1 = 2 = 3 = 0:263. Comparing the solutions obtained by using these two di5erent sets of weights reveals that the proposed model manages to 1nd an optimal solution that gives objective 1 a higher achieved level than the others while the achievement of objective 1 is emphasized by giving it a heavy weight than the others. Thus, the weighted max–min model 1nds an optimal solution such that the ratio of the achieved levels is the same as the weights of objectives.
3. Models with quasiconcave membership functions Only the weighted additive model of Tiwari et al. [17] seems to be applicable in the case where the DM feels that the objectives are of di5erent levels of importance, while giving membership functions that include some quasiconcave ones. However, the additive model encounters a serious problem when quasiconcave membership functions are involved in FGP. Fig. 3 shows that the summation of two quasiconcave functions is not necessarily a quasiconcave function. This can cause the solution of FGP to stall at a local optimum. Moreover, further investigation is required to determine whether the methods for FGP problems with quasiconcave membership functions, for example, the methods of Inuiguchi et al. [6] or the method of Yang et al. [19], can be integrated with the additive model. This section describes how the weighted max–min model can be used to solve FGP problems with quasiconcave membership functions and objective weights. The FGP problem used in the subsequent examples is the numerical example from Inuiguchi et al. [6]. The problem is 1rst treated with the method of Yang et al. [19] in Example 2, and then with the method of Inuiguchi et al. in Example 3. Example 2. Consider the problem below: Find
x
∼
to satisfy z1 = −x1 + 2x2 ¿ 12; ∼
z2 = 2x1 + x2 ¿ 21; subject to −x1 + 3x2 6 21; x1 + 3x2 6 27; 4x1 + 3x2 6 45; 3x1 + x2 6 30; x1 ; x2 ¿ 0:
(18)
The corresponding membership functions for the two goals are 1 0:08(z1 − 2) + 0:2 1 (z1 ) = 0:04(z1 + 3) 0
if if if if
12 6 z1 ; 2 6 z1 ¡ 12; − 3 6 z1 ¡ 2; z1 ¡ −3
(19)
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415
1
1 + 2
2
i
1
0
Fig. 3. Summation of two quasiconcave functions.
and
1 0:1(z − 17) + 0:6 2 2 (z2 ) = 0:06(z2 − 7) 0
if if if if
21 6 z2 ; 17 6 z2 ¡ 21; 7 6 z2 ¡ 17; z2 ¡ 7;
(20)
Suppose the DM gives weights, w1 = 0:6 and w2 = 0:4. Incorporating the method of Yang et al. [19] for quasiconcave membership functions with the weighted max–min model gives the following linear program. Maximize subject to 0:6 6 0:08(z1 − 2) + 0:2 + M1 ; 0:6 6 0:04(z1 + 3) + M (1 − 1 ); 0:4 6 0:1(z2 − 17) + 0:6 + M2 ; 0:4 6 0:06(z2 − 7) + M (1 − 2 ); −x1 + 3x2 6 21; x1 + 3x2 6 27; 4x1 + 3x2 6 45; 3x1 + x2 6 30; x1 ; x2 ¿ 0; 1 ; 2 = 0; 1;
(21)
where M is a large positive number. The optimal solution is x1 = 4:59 and x2 = 7:47, and the achieved levels are 1 = 0:868 and 2 = 0:579. Since the approaches of Zimmermann, and Lai and Hwang are unsuitable for weighted problems, Table 2 only compares the solution obtained by the proposed model and that by the additive model. Again, by the proposed model, the ratio of the achieved levels (0:868=0:579 = 1:5) equals the ratio of the weights (0:6=0:4 = 1:5). However, the ratio of the achieved levels (1=0:48 = 2:08) exceeds the ratio of the weights 1.5. Thus, the solution obtained by the proposed model is better than that by the additive model. Using the approach of Yang et al., the linear program for solving Problem (18) can be formulated straightforwardly. This is not the case when the method of Inuiguchi et al. [6] is used. Though
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C.-C. Lin / Fuzzy Sets and Systems 142 (2004) 407 – 420 Table 2 Solutions to Example 2 by di5erent approaches Approaches
z1 z2 1 2
Proposed model
Additive model
10.350 16.650 0.868 0.579
12.000 15.000 1.000 0.480
the method of Yang et al. is easily applied, it requires zero-one variables to formulate the linear program, causing computational ineMciency in obtaining the solution. The method of Inuiguchi et al. translates a set of quasiconcave membership functions into a set of equivalent concave membership functions. The method uses no zero-one variables and guarantees that the optimal solution, obtained using the new concave membership functions, is exactly the same as that obtained using the original quasiconcave membership functions. As stated above, the weighted max–min model is equivalent to the unweighted max–min model with new membership functions that is obtained by dividing the original membership function by their corresponding weights. Thus, the membership functions must be divided by their corresponding weights to become new membership functions before the method of Inuiguchi et al. can be applied to the weighted max–min model. Membership function i ((ax)i ) is divided by wi to become i ((ax)i ), which is then translated to concave function, i ((ax)i ) by the process of Inuiguchi et al. to be used for solving the problem. The following example illustrates how the method of Inuiguchi et al. can be integrated with the weighted max–min model. Example 3. Consider the same problem as in Example 2. When the weights are equal, the problem remains an ordinary FGP problem. Translating membership functions (19) and (20) using the translation process of Inuiguchi et al. (refer to [6] for detailed algorithm), yields the following equivalent concave membership functions:
1 (z1 ) =
and
2 (z2 ) =
1 3
(z 65 1 1 (z 13 1
− 7) + + 3)
10 13
0
1 3 z2 − 10 13 + 52 3 (z − 7) 26 2 0
if if if if
5 13
12 6 z1 ; 7 6 z1 ¡ 12; − 3 6 z1 ¡ 7; z1 ¡ −3
(22)
if 21 6 z2 ; if 10 13 6 z2 ¡ 21; 1 if 7 6 z2 ¡ 10 ; 3 if z2 ¡ 7:
(23)
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417
Formulating and solving the problem with Functions (22) and (23) gives the optimal solution, x1 = 5:6 and x2 = 7:13, with achieved levels of 1 = 2 = 0:733. Suppose, however, that the DM gives weights, w1 = 0:6 and w2 = 0:4 as in Example 2. Firstly, membership Functions (19) and (20) are divided by w1 and w2 to become 5 if 12 6 z1 ; 3 2 (z − 2) + 1 if 2 6 z ¡ 12; 1 1 3 (24) 1 (z1 ) = 15 1 (z + 3) if − 3 6 z1 ¡ 2; 15 1 0 if z1 ¡ −3 and 2 (z2 ) =
5 2 1
(z − 17) 4 2 3 (z − 7) 20 2
+
3 2
0
if if if if
21 6 z2 ; 17 6 z2 ¡ 21; 7 6 z2 ¡ 17; z2 ¡ 7:
(25)
Notably, the membership degree of both of these membership functions may exceed unity. The following process illustrates how membership Functions (24) and (25) can be translated into equivalent concave membership functions. According to the process of Inuiguchi et al., the vertices vj = (v1j ; v2j ) at which the two membership functions have same degrees are
83 43 53 73 ; v3 = and v5 = v1 = (−3; 7); v2 = 2; ; 17 ; v4 = 12; ; 21 : 9 4 3 4 (26) By
= pj min pj+1
vkj+2 − vkj+1 vkj+1 − vkj
k=1;2
and letting p1 = 1,
(27)
83 83 7 43 −2 (2 − (−3)); 17 − −7 = ; 4 9 9 4
43 3 7 43 53 83 p3 = × min = 12 − −2 ; − 17 17 − 4 4 4 3 9 20 p2 = 1 × min
and 3 × min p4 = 20
73 43 53 3 53 − 12 12 − ; 21 − − 17 = : 4 4 3 3 4
Then, pj pj = pj
(28)
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C.-C. Lin / Fuzzy Sets and Systems 142 (2004) 407 – 420
yields,
7 35 3 3 = ; + + 4 20 4 53
7 3 3 3 3 + + = p2 = 20 4 20 4 53 7 p1 = 4
and 3 p3 = 4
7 3 3 + + 4 20 4
=
15 : 53
The values of the two new concave membership functions at every vertex can be obtained as follows: 1 (−3) = 2 (7) = 0;
20 20 83 =0+ = ; 1 (2) = 2 9 73 73
43 20 35 55 = 2 (17) = 0 + 1 + = ; 4 73 73 73
20 35 3 58 53 =0+ + + = 1 (12) = 2 3 73 73 73 73 and
1
73 4
= 2 (21) = 0 +
3 15 20 35 + + + = 1: 73 73 73 73
Finally, the new concave membership functions are obtained 73 1 if 6 z1 ; 4
12 z − 43 + 55 if 43 6 z ¡ 73 ; 1 1 4 73 4 4 1 (z1 ) = 365 4 43 if − 3 6 z1 ¡ ; (z1 + 3) 73 4 0 if z1 ¡ −3 and
1 if 21 6 z2 ;
9 20 83 83 z2 − + if 6 z2 ¡ 21; 9 73 9 2 (z2 ) = 146 83 9 (z − 7) if 7 6 z2 ¡ ; 2 9 73 0 if z2 ¡ 7:
(29)
(30)
Formulating and solving the problem as the unweighted max–min model in Example 1 gives the optimal solution, x1 = 4:59 and x2 = 7:47. The achieved levels are 1 = 0:868 and 2 = 0:579,
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419
as obtained using the method of Yang et al. in Example 2. If the translation process is performed before the original quasiconcave membership functions are divided by their corresponding weights, then the resulting concave membership functions are membership Functions (22) and (23). Solving a weighted max–min model with these concave membership functions gives the optimal solution, x1 = 3:69 and x2 = 7:77. The achieved levels are 1 = 0:988 and 2 = 0:489, of which the ratio is 0:988=0:489 = 2:02 rather than 0:6=0:4 = 1:5. These results indicate that the dividing must take place before the translation process when the method of Inuiguchi et al. is used. 4. Conclusions This study proposes a simple but e5ective weighted max–min model for FGP and for fuzzy multiple objective decision-making. The proposed model outperforms the widespread weighted sum approach in at least two ways. Firstly, the summation of quasiconcave membership functions may not remain quasiconcave. Secondly, the proposed model gives solutions in which the ratio of the achieved levels of the goals or objectives is close to or even the same as the ratio of the objective weights. The simplicity of the proposed model is such that it can be combined e5ectively with other approaches for FGP. Examples illustrate how the proposed model can be combined with the methods of Inuiguchi et al. [6] and of Yang et al. [19] to solve a FGP problem with quasiconcave membership functions. Notably, the original membership functions must be divided by their corresponding weights before the translation process of obtaining concave membership functions takes place whenever the method of Inuiguchi et al. is used. Reversing this order would mislead the solution such that a worse solution might be obtained. For other methods that also use certain techniques to translate the membership functions from quasiconcave to concave, the membership functions should 1rstly be divided by their weights. Acknowledgements The author would like to thank the editors-in-chief and the anonymous referees for their valuable comments on the earlier versions of this paper. References [1] R. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Sci. 17B (1970) 141–164. [2] D. Dubois, H. Fargier, H. Prade, Possibility theory in constraint satisfaction problems: handling priority, preference and uncertainty, Appl. Intell. 6 (1996) 287–309. [3] D. Dubois, F. Fortemps, Computing improved optimal solutions to max–min Oexible constraint satisfaction problems, European J. Oper. Res. 118 (1999) 95–126. [4] D. Dubois, H. Prade, Weighted minimum and maximum operations, Inform. Sci. 39 (1986) 205–210. [5] E.L. Hannan, Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235–248. [6] M. Inuiguchi, H. Ichihashi, Y. Kume, A solution algorithm for fuzzy linear programming with piecewise membership functions, Fuzzy Sets and Systems 34 (1990) 15–31.
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