Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters

Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters

Fuzzy Sets and Systems 109 (2000) 3–19 www.elsevier.com/locate/fss Interactive fuzzy programming for multi-level linear programming problems with fu...

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Fuzzy Sets and Systems 109 (2000) 3–19

www.elsevier.com/locate/fss

Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters Masatoshi Sakawaa; ∗ , Ichiro Nishizakia , Yoshio Uemurab a Department

of Industrial and Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan b Products Projects Development Department, Juken Sangyou Co., Ltd., 1-1 Mokuzaikouminami, Hatsukaichi, Hiroshima 738-0022, Japan Received July 1997; received in revised form April 1998

Abstract This paper presents interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. In fuzzy programming for multi-level linear programming problems, recently developed by Lai et al., since the fuzzy goals are determined for both an objective function and decision variables at the upper level, undesirable solutions are produced when these fuzzy goals are inconsistent. In order to overcome such problems, after eliminating the fuzzy goals for decision variables, interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters is presented. In our interactive method, after determining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived eciently by updating the satisfactory degrees of decision makers with considerations of overall satisfactory balance among all levels. Illustrative numerical examples for two-level and three-level linear programming problems are provided to c 2000 Elsevier Science B.V. All rights reserved. demonstrate the feasibility of the proposed method. Keywords: Multilevel linear programming problem with fuzzy parameters; Fuzzy programming; Fuzzy goals; Interactive methods

1. Introduction Two-level programming problems, in which a decision maker (DM) at the upper level makes a decision subject to an optimization problem for a DM at the lower level, have two interpretations. They depend on whether there is a cooperative relationship among DMs or not. Consider a decision problem in a decentralized rm as an example of a decision problem with cooperative ∗

Corresponding author. E-mail address: [email protected] (M. Sakawa)

DMs. Top management, an executive board, or headquarters interests itself in overall management policy such as long-term corporate growth or market share. In contrast, operation divisions of the rm are concerned with coordination of daily activities. After headquarters make a decision in accordance with the overall management policy, each division determines a goal to be achieved and tries to attain the goal, fully understanding the decision by the headquarters. As an example of a decision problem without cooperative DMs, consider the Stackelberg duopoly: Firms 1 and 2 supply homogeneous goods to a market. Suppose Firm 1 dominates Firm 2 in the market, and

c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 1 3 0 - 4

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consequently Firm 1 rst determines a level of supply and then Firm 2 decides its level of supply after it realizes Firm 1’s level of supply. There is essentially a cooperative relationship between the DM at the upper level and the DM at the lower level in the former problem while each DM does not have a motivation to cooperate each other in the latter problem. As the former’s mathematical programming problem, we can model such a problem as a singleobjective large-scale mathematical programming problem used the decomposition method or a multiobjective programming problem with objective functions of all levels. The two-level programming formulation is intended to supplement the decomposition approach, not supplant it [4]. However, the formulation is noteworthy because a hierarchical structure of the decision problem is explicitly included in a mathematical model. Studies on the latter have been seen in the literature on game theory. Such a situation is modeled as a Stackelberg game, in which there are two players, and one player determines a strategy and thereafter the other player decides a strategy [9]. Each player completely knows objective functions and constraints of an opponent and himself=herself, and the DM at the upper level (leader) rst speci es a strategy and then the DM at the lower level (follower) speci es a strategy so as to optimize the objective with full knowledge of the decision of the DM at the upper level. According to the rule, the DM at the upper level also speci es the strategy so as to optimize the objective. Then a solution de ned as the above-mentioned procedure is called the Stackelberg strategy (solution). The Stackelberg strategy has been employed as a solution concept when decision problems are modeled as two-level programming problems, whether there is a cooperative relationship between the DMs or not. Even if the objective functions of both DMs and the common constraint functions are linear, it is known that this problem is a non-convex programming problem with a special structure. In general, the Stackelberg solution does not satisfy Pareto optimality because of its non-cooperative nature. Computation methods for the Stackelberg solution are classi ed roughly into three categories: the vertex enumeration approach based on a characteristic that an extreme point of a set of best responses of the DM

at the lower level is also that of a set of the common constraints, the Kuhn–Tucker approach in which the upper level’s problem with constraints including optimality conditions of the lower level’s problem is solved, and the penalty function approach which adds a penalty term to the upper level’s objective function so as to satisfy optimality of the lower level’s problem. The Kth best method proposed by Bialas and Karwan [4] is one of vertex enumeration approaches. The solution search procedure of the method starts from a point which is an optimal solution to the problem of the upper level and checks whether it is also an optimal solution to the problem of the lower level or not. If the rst point is not the Stackelberg solution, the procedure continues to examine the second best solution to the problem of the upper level and so on. The Kuhn–Tucker method is used by Bialas and Karwan [4] in their parametric complementary pivot algorithm. Bard and Falk [3] replaces the complementarity constraint (complementary slackness condition) with a separable representation and applies a general branch and bound algorithm. Bard [1] formulates a two-level programming problem as an equivalent semi-in nite problem and develops his grid search algorithm through a parametric linear  u [10] proposes an algorithm program technique. Unl based on bicriteria programming by using the result of Bard [1]. White and Anandalingam [13] develops an approach to two-level programming using a duality gap-penalty function format. For obtaining the Stackelberg solution to a multilevel linear programming problem, Bard [2] and Wen and Bialas [11] propose algorithms for three-level problems. Bard [2] formulates a normal nonlinear programming problem by using the Kuhn–Tucker conditions for the problems of the third level and the second level, and proposes a cutting plane algorithm employing a vertex search procedure to solve a threelevel linear programming problem. Wen and Bialas [11] develop a hybrid algorithm to solve a three-level linear programming problem. The algorithm adopts the Kth best algorithm to generate the Kth best extreme point and the complementary pivot algorithm to check feasibility. Recently, Lai [5] and Shih, Lai and Lee [8] have proposed a solution concept, which is di erent from the concept of a Stackelberg solution, for problems such that decisions of DMs in all the levels are sequen-

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tial and all of the DMs essentially cooperate with each other. Their method is based on an idea that the DM at the lower level optimizes an objective function, taking a goal or preference of the upper level into consideration. DMs elicit membership functions of fuzzy goals for their objective functions and, especially, the DM at the upper level also speci es those of fuzzy goals for the decision variables of the DM. The DM at the lower level solves a fuzzy programming problem with a constraint on a satisfactory degree of the DM at the upper level. Unfortunately, there is a possibility that their method leads a nal solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and the decision variables. On the other hand, Sakawa et al. formulate mathematical programming problems with fuzzy parameters from the viewpoint of experts’ imprecise or fuzzy understanding of the nature of parameters in a problem-formulation process, and proposed a fuzzy programming for multi-objective programming problems [6, 7]. In this paper, we present interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. In order to overcome the problem in the methods of Lai et al., after eliminating the fuzzy goals for decision variables, our interactive method for multi-level linear programming problems with fuzzy parameters begins. In our interactive method, after determining the fuzzy goals of the DM at all levels, a satisfactory solution is derived eciently by updating the satisfactory degrees of DMs with considerations of overall satisfactory balance among all levels. Illustrative numerical examples for two-level and three-level linear programming problems are provided to demonstrate the feasibility of the proposed method. 2. Interactive fuzzy programming for two-level linear programming problems with fuzzy parameters Let DM1 denote the DM at the upper level and DM2 denote the DM at the lower level. DM1 rst determines a decision and thereafter DM2 specify a decision so as to optimize their objectives with full knowledge of the decision of DM1. According to the rule, it is supposed that DM1 determines the decision so as to optimize an objective function of DM1.

5

Such strategies are called Stackelberg solutions when the two DMs are in a noncooperative situation, and a two-level linear programming problem for obtaining a Stackelberg solution is formulated as: z1 (x1 ; x2 ) = c11 x1 + c12 x2

minimize x1

where x2 solves minimize

z2 (x1 ; x2 ) = c21 x1 + c22 x2

subject to

A1 x1 + A2 x2 6b;

x2

(1)

x1 ¿0; x2 ¿0; where xi , i = 1; 2 is an ni -dimensional decision variable, ci1 , i = 1; 2 is an n1 -dimensional constant row vector, ci2 , i = 1; 2 is an n2 -dimensional constant row vector, b is an m-dimensional constant column vector, and Ai , i = 1; 2 is an m × ni constant matrix. For the sake of simplicity, we use the following notations: x = (x1 ; x2 ),   c c c = 11 12 ; c21 c22 and A = [A1 A2 ]. In the two-level linear programming problem (1), z1 (x1 ; x2 ) and z2 (x1 ; x2 ), respectively, represent objective functions of the upper and the lower levels and x1 and x2 , respectively, represent decision variables of the upper and the lower levels. In contrast with the above formulation, in this paper, we consider two-level linear programming problems in a situation that each of the DMs at both levels takes overall satisfactory balance between both levels into consideration and tries to minimize an objective function of the DM, paying serious attention to preferences of the other DM. We consider hierarchical decision problems in which DMs have their own objective functions but they can coordinate their decisions, that is, they are essentially cooperative. Such a two-level linear programming problem is formulated as DM1 minimizes x

z1 (x1 ; x2 ) = c11 x1 + c12 x2

DM2 minimizes z2 (x1 ; x2 ) = c21 x1 + c22 x2 x

subject to

A1 x1 + A2 x2 6b;

(2)

x1 ¿0; x2 ¿0: In this paper, we will propose an interactive fuzzy programming in order to resolve problem (2).

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However, when formulating a mathematical programming problem which closely describes and represents a real-world decision situation, various factors of the real-world system should be re ected in the description of objective functions and constraints. Naturally, these objective functions and constraints involve many parameters whose possible values may be assigned by experts. In the conventional approaches, such parameters are required to be xed at some values in an experimental and=or subjective manner through the experts’ understanding of the nature of the parameters in the problem-formulation process. It must be observed here that, in most real-world situations, the possible values of these parameters are often only imprecisely or ambiguously known to the experts. With this observation in mind, it would be certainly more appropriate to interpret the experts’ understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy sets of the real line known as fuzzy numbers. The resulting mathematical programming problem involving fuzzy parameters would be viewed as a more realistic version than the conventional one [6, 7]. From this viewpoint, we assume that parameters involved in the objective functions and the constraints of the two-level linear programming problem are characterized by fuzzy numbers. As a result, a problem with fuzzy parameters corresponding to problem (2) is formulated as DM1 minimizes z1 (x1 ; x2 ) = c˜11 x1 + c˜12 x2 x

subject to

˜ A˜1 x1 + A˜2 x2 6b;

˜ A) ˜ = { (c; b; A) | c˜i1; r (ci1; r )¿ ; (c; ˜ b; i = 1; 2; r = 1; : : : ; n1 ; c˜i2; r (ci2; r )¿ ; i = 1; 2; r = 1; : : : ; n2 ; b˜ k (bk )¿ ; k = 1; : : : ; m; a˜1; kj (a1; kj )¿ ; k = 1; : : : ; m; j = 1; : : : ; n1 ; a˜2; kj (a2; kj )¿ ; k = 1; : : : ; m; j = 1; : : : ; n2 }:

DM1 minimizes x

z1 (x) = c11 x1 + c12 x2

x

(3)

x1 ¿0; x2 ¿0; where c˜i1 = (c˜i1; 1 ; : : : ; c˜i1; n1 ), i = 1; 2; ci2 = (c˜i2; 1 ; : : : ; c˜i2; n2 ), i = 1; 2; b˜ = (b˜ 1 ; : : : ; b˜ m )T ; A˜1 = (a˜1; kj ), k = 1; : : : ; m, j = 1; : : : ; n1 , A˜2 = (a˜2; kj ), k = 1; : : : ; m, j = 1; : : : ; n2 are fuzzy parameters. The superscript T means transposition. For the sake of simplicity, we use the following notations:   c˜11 c˜12 c˜ = c˜21 c˜22 and A˜ = [A˜1 A˜2 ]. Assuming that these fuzzy parameters are characterized by fuzzy numbers, let the corresponding mem-

(4)

Now, suppose that DM1 considers that the degree of all of the membership functions of the fuzzy numbers involved in the two-level linear programming problem should be greater than or equal to some value . Then, for such a degree , problem (3) can be interpreted as the following nonfuzzy two-level linear programming problem which depends on a coecient vector ˜ A) ˜ [6, 7]. (c; b; A) ∈ (c; ˜ b;

DM2 minimizes z2 (x) = c21 x1 + c22 x2

x

DM2 minimizes z2 (x1 ; x2 ) = c˜21 x1 + c˜22 x2

bership functions be: (c˜i1; 1 (ci1; 1 ); : : : ; c˜i1; n1 (ci1; n1 )), i = 1; 2; i = 1; 2; (c˜i2; 1 (ci2; 1 ); : : : ; c˜i2; n2 (ci2; n2 )), (b˜ 1 (b1 ); : : : ; b˜ m (bm )), a˜1; kj (a1; kj ), k = 1; : : : ; m, j=1; : : : ; n1 and a˜2; kj (a2; kj ), k=1; : : : ; m, j=1; : : : ; n2 . We introduce the -level set of the fuzzy numbers c, ˜ ˜ A) ˜ in which b˜ and A˜ de ned as the ordinary set (c; ˜ b; the degree of their membership functions exceeds level :

subject to

A1 x1 + A2 x2 6b;

(5)

x1 ¿0; x2 ¿0: Observe that there exists an in nite number of such a problem (5) depending on the coecient vector ˜ A) ˜ and the values of (c; b; A) are ar(c; b; A) ∈ (c; ˜ b; ˜ A) ˜ in the sense that bitrary for any (c; b; A) ∈ (c; ˜ b; the degree of all of the membership functions for the fuzzy numbers in problem (5) exceeds level . However, if possible, it would be desirable for each DM ˜ A) ˜ in problem (5) so as to to choose (c; b; A) ∈ (c; ˜ b; minimize the objective function under the constraints. From such a point of view, for a certain degree , it seems to be quite natural to have understood the two-level linear programming problem with fuzzy parameters as the following nonfuzzy -two-level linear

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

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programming problem [6, 7]: DM1 minimizes z1 (x; c11 ; c12 ) = c11 x1 + c12 x2 x; c; b; A

DM2 minimizes z2 (x; c21 ; c22 ) = c21 x1 + c22 x2 x; c; b; A

subject to

A1 x1 + A2 x2 6b;

(6)

x1 ¿0; x2 ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b;

Fig. 1. Linear membership function.

It is assumed that DM1 chooses a degree of the level. It should be noted that the parameters (c; b; A) are treated as decision variables rather than constants. It is natural that DMs have fuzzy goals for their objective functions when they take fuzziness of human judgments into consideration. For each of the objective functions zi (x; ci1 ; ci2 ), i = 1; 2 of (6), assume that the DMs have fuzzy goals such as “the objective function zi (x; ci1 ; ci2 ) should be substantially less than or equal to some value, say, pi ”. Let = 0, and then the individual minimum zimin = zi (xio ; coi1 ; coi2 ) = min{zi (x; ci1 ; ci2 ) | A1 x1 + A2 x2 6b; ˜ A) ˜ 0} ˜ b; xi ¿0; i = 1; 2; (c; b; A) ∈ (c;

For the sake of simplicity, in this paper, we adopt a linear membership function, which characterizes the fuzzy goal of the DM at each level. The corresponding linear membership function i (zi ) is de ned as i (zi (x; ci1 ; ci2 ))  0; zi (x; ci1 ; ci2 )¿zi0 ;    z (x; c ; c ) − z 0 i i1 i2 i = ; zi1 ¡zi (x; ci1 ; ci2 )6zi0 ; 1 − z0  z  i i  1; zi (x; ci1 ; ci2 )6zi1 ; (9)

(7)

and the individual maximum zimax = max{zi (x; ci1 ; ci2 ) | A1 x1 + A2 x2 6b; ˜ A) ˜ 0 } (8) ˜ b; xi ¿0; i = 1; 2; (c; b; A) ∈ (c; of the objective function are referred to when each DM elicits membership function prescribing the fuzzy goal for the objective function zi (x; ci1 ; ci2 ), i = 1; 2. The DM determines the membership function i (zi (x; ci1 ; ci2 )), which is strictly monotone decreasing for zi (x; ci1 ; ci2 ), consulting the variation ratio of degree of satisfaction in the interval between the individual minimum (7) and the individual maximum (8). The domain of the membership function is the interval [zimin ; zimax ], i = 1; 2, and the DM speci es the value zi0 of the objective function for which the degree of satisfaction is 0 and the value zi1 of the objective function for which the degree of satisfaction is 1. For the value undesired (larger) than zi0 , it is de ned that i (zi (x; ci1 ; ci2 )) = 0, and for the value desired (smaller) than zi1 , it is de ned that i (zi (x; ci1 ; ci2 )) = 1.

where zi0 and zi1 denote the values of the objective function zi (x) such that the degree of membership function is 0 and 1, respectively, and it is assumed that the DM subjectively assesses zi0 and zi1 . Suppose that applying the way suggested by Zimmermann [14] and setting = 0, the DM speci es zi0 and zi1 in the following. That is, using the individual minimum (7) together with zim = zi (x jo ; coj1 ; coj2 );  1 if i = 2; i = 1; 2; j = 2 if i = 1;

(10)

the DM determines the linear membership function as in (9) by choosing zi1 = zimin , zi0 = zim , i = 1; 2. After eliciting the membership functions, DM1 subjectively speci es a minimal satisfactory level ˆ ∈ [0; 1] for the membership function 1 (z1 (x; c11 ; c12 )), and then DM2 maximizes the membership function 2 (z2 (x; c21 ; c22 )) subject to a condition that DM1’s membership function 1 (z1 (x; c11 ; c12 )) is larger than or equal to ˆ together with the given

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M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

constraints, that is, DM2 solves the following problem:

For problem (13), consider the auxiliary problem: maximize  x; ; c; b; A

maximize 2 (z2 (x; c21 ; c22 ))

subject to

x; c; b; A

subject to

1 (z1 (x; c11 ; c12 ))¿;

A1 x1 + A2 x2 6b; ˆ 1 (z1 (x; c11 ; c12 ))¿;

2 (z2 (x; c21 ; c22 ))¿;

(11)

x1 ¿0; x2 ¿0;

0661;

˜ A) ˜ : (c; b; A) ∈ (c; ˜ b;

x1 ¿0; x2 ¿0;

 = min(1 (z1 (x; c11 ; c12 )); 2 (z2 (x; c21 ; c22 )))

(12)

and the following problem is substituted for problem (11): maximize



subject to

A1 x1 + A2 x2 6b; ˆ 1 (z1 (x; c11 ; c12 ))¿¿; 2 (z2 (x; c21 ; c22 ))¿; 0661; x1 ¿0; x2 ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b;

(14)

˜ A) ˜ : (c; b; A) ∈ (c; ˜ b;

Constraints on fuzzy goals for decision variables are eliminated in our formulation (11) while they are involved in the formulations by Lai et al. [5, 8]. If an optimal solution to problem (11) exists, it follows that DM1 obtains a satisfactory solution having a satisfactory degree larger than or equal to the minimal satisfactory level ˆ speci ed by DM1’s own self. However, the larger the minimal satisfactory level is assessed, the smaller DM2’s satisfactory degree becomes. Consequently, a relative di erence between the satisfactory degrees of DM1 and DM2 becomes larger and it is feared that overall satisfactory balance between both levels cannot be maintained. To take account of the overall satisfactory balance between both levels, DM1 needs to compromise with DM2 on DM1’s minimal satisfactory level. To do so, a satisfactory degree of both DMs is de ned as

x; ; c; b; A

A1 x1 + A2 x2 6b;

(13)

By solving problem (14), we obtain a solution maximizing a smaller satisfactory degree between those of both DMs. Unfortunately, problem (14) is not a linear programming problem even if all the membership functions i (zi (x; ci1 ; ci2 )), i = 1; 2 are linear. To solve problem (14) by using the linear programming technique, we introduce the set-valued functions: Si (ci1 ; ci2 ) = {(x; ) | i (zi (x; ci1 ; ci2 ))¿}; i = 1; 2; Tj (bj ; A1; j ; A2; j ) = {x | A1; j x1 + A2; j x2 6bj };

(15)

j = 1; : : : ; m; where Ai; j is a row vector corresponding to the jth row of the m × ni matrix Ai . Then it can be easily veri ed that the following relations hold for Si (ci1 ; ci2 ) and Tj (bj ; A1; j ; A2; j ) when x¿0 [7, 6]. Proposition 1. (1) If c1i1 6ci12 ; then Si (c1i1 ; ·) ⊇ Si (ci12 ; ·); and if c1i2 6ci22 ; then Si (·; c1i2 ) ⊇ Si (·; ci22 ). (2) If b1j 6b2j ; then Tj (b1j ; ·; ·) ⊆ Tj (b2j ; ·; ·). (3) If A11; j 6A21; j ; then Tj (·; A11; j ; ·) ⊇ Tj (·; A21; j ; ·); and if A12; j 6A22; j ; then Tj (·; ·; A12; j ) ⊇ Tj (·; ·; A22; j ). From the properties of the -level set for the vectors of fuzzy numbers c˜11 , c˜12 , c˜21 , c˜22 , b˜ and the matrices of fuzzy numbers A˜1 , A˜2 , it should be noted that the feasible regions for c11 , c12 , c21 , c22 , bj , A1j and A2j can be denoted, respectively, by the closed intervals [c L11 ; c R11 ], [c L12 ; c R12 ], [c L21 ; c R21 ], [c L22 ; c R22 ], [b Lj ; b Rj ], [AL1; j ; AR1; j ] and [AL2; j ; AR2; j ]. Therefore, through the use of Proposition 1, we can obtain an optimal solution to problem (14) by solving

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

the following linear programming problem: maximize  x; 

subject to

AL1 x1 + AL2 x2 6bR ; 1 (z1 (x; cL11 ; cL12 ))¿; 2 (z2 (x; cL21 ; cL22 ))¿;

(16)

0661; x1 ¿0; x2 ¿0: If the optimal solution (x∗ ; ∗ ) to problem (14) obtained by solving problem (16) satis es the condition ˆ it follows that the solution that 1 (z1 (x∗ ; c L11 ; c L12 ))¿, is satisfying for DM1. However, the solution (x∗ ; ∗ ) does not always maintain overall satisfactory balance of both levels. Then the ratio of satisfactory degree between both levels =

2 (z2 (x∗ ; cL21 ; cL22 )) ; 1 (z1 (x∗ ; cL11 ; cL12 ))

(17)

which is de ned by Lai [5], is useful. Let L and U denote the lower bound and the upper bound of  speci ed by DM1, respectively. If ¿U , i.e., 2 (z2 (x∗ ; c L21 ; c L22 ))¿U 1 (z1 (x∗ ; c L11 ; c L12 )), then DM1 updates the minimal satisfactory level ˆ by ˆ Receiving the updated level ˆ0 , DM2 increasing . solves problem (11) with ˆ0 , and then DM1 obtains a larger satisfactory degree and DM2 accepts a smaller satisfactory degree. Conversely, if ¡L , i.e., 2 (z2 (x∗ ; c L21 ; c L22 ))¡L 1 (z1 (x∗ ; c L11 ; c L12 )), then DM1 updates the minimal satisfactory level ˆ by deˆ and DM1 accepts a smaller satisfactory creasing , degree and DM2 obtains a larger satisfactory degree. At an iteration ‘, let 1 (z1‘ ), 2 (z2‘ ) and ‘ denote DM1’s and DM2’s satisfactory degrees and a satisfactory degree of both levels, respectively, and let ‘ = 2 (z2‘ )=1 (z1‘ ) denote a ratio of satisfactory degrees of the upper and the lower levels. Let a corresponding solution be x ‘ . When DM2 proposes the solution to DM1 and the following two conditions are satis ed, DM1 concludes the solution as a satisfactory solution and the iterated interactive process terminates.

9

Termination conditions of the interactive process for two-level linear programming problems C1. DM1’s satisfactory degree is larger than or equal to the minimal satisfactory level ˆ speci ed by ˆ DM1, i.e., 1 (z1‘ )¿. ‘ C2. The ratio  of satisfactory degrees is in the closed interval between its lower and its upper bounds speci ed by DM1. The condition C1 means DM1’s required condition for solutions proposed by DM2. The condition C2 is provided in order to keep overall satisfactory balance between both levels. Unless the conditions are satis ed simultaneously, ˆ DM1 needs to update the minimal satisfactory level . Procedure for updating the minimal satisfactory level ˆ P1. If the condition C1 is not satis ed, then DM1 ˆ decreases the minimal satisfactory level . ‘ P2. If the ratio  exceeds its upper bound U , ˆ then DM1 increases the minimal satisfactory level . ‘ Conversely, if the ratio  is below its lower bound L , then DM1 decreases the minimal satisfactory ˆ level . Let ˆ0 denote the updated minimal satisfactory level. DM2 solves the following maximization problem with the updated minimal satisfactory level ˆ0 . maximize

2 (z2 (x; c21 ; c22 ))

subject to

A1 x1 + A2 x2 6b;

x; c; b; A

1 (z1 (x; c11 ; c12 ))¿ˆ0 ;

(18)

x1 ¿0; x2 ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b; We can obtain an optimal solution to problem (18) by solving the following problem through a transformation similar to the transformation from problem (14) to problem (16). maximize

L L 2 (z2 (x; c21 ; c22 ))

subject to

AL1 x1 + AL2 x2 6bR ;

x

L L ; c12 ))¿ˆ0 ; 1 (z1 (x; c11

x1 ¿0; x2 ¿0:

(19)

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M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

x1 ¿0; : : : ; xt ¿0;

The above-mentioned algorithm is summarized as follows: Algorithm of the interactive fuzzy programming for solving two-level linear programming problems with fuzzy parameters Step 1: Set ‘ = 1. DM1 determines a degree of the ˜ A) ˜ , elicits the membership function -level set (c; ˜ b; 1 (z1 ) of the fuzzy goal for the objective function of DM1, and speci es the minimal satisfactory level ˆ and the lower and the upper bounds of the ratio of satisfactory degrees . Step 2: DM2 elicits the membership function 2 (z2 ) of the fuzzy goal for the objective function of DM2. Step 3: DM2 solves the auxiliary problem (14) through problem (16), and then proposes a solution x‘ to problem (14), (z1‘ ; z2‘ ), ‘ , 1 (z1‘ ), 2 (z2‘ ) and ‘ to DM1. Step 4: If the solution proposed by DM2 to DM1 satis es the termination conditions, DM1 concludes the solution as a satisfactory solution and the algorithm stops. Otherwise ‘ = ‘ + 1. Step 5: DM1 updates the minimal satisfactory level ˆ in accordance with the procedure of updating minimal satisfactory level. Step 6: DM2 solves problem (18) through problem (19) and proposes an obtained solution to DM1. Return to Step 4. 3. Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters

˜ A) ˜ ; (c; b; A) ∈ (c; ˜ b;

(20)

where xj , j = 1; : : : ; t is an nj -dimensional decision variable, cij , i = 1; : : : ; t, j = 1; : : : ; t is an nj dimensional constant row vector, b is an mdimensional constant column vector, and Aj , j = 1; : : : ; t is an m × nj constant matrix. Let x = (x1 ; : : : ; xt ),   c= 

 · · · c1t  .. ; . 

c11

···

ct1   c˜ = 

A = [A1 · · · At ];

ctt

 · · · c˜1t  ..  . · · · c˜tt

c˜11 c˜t1

and A˜ = [ A˜1 · · · A˜t ]. It is assumed that a DM at each level elicits a linear membership function of a fuzzy goal i (zi (x; ci1 ; : : : ; cit )), i = 1; : : : ; t, and that all the DMs except for DMt at the lowest level determine minimal satisfactory levels ˆi ∈ [0; 1], i = 1; : : : ; t − 1. In a multi-level case, a problem corresponding to problem (11) for a two-level problem can be extended as maximize x; c; b; A

t (zt (x; ct1 ; : : : ; ctt ))

We extend the interactive fuzzy programming for two-level linear programming problems with fuzzy parameters to that for multi-level problems, i.e., t-level linear programming problems with fuzzy parameters. In a way similar to a two-level problem (3), we can formulate an -t-level linear programming problem as

subject to

DM1 minimizes z1 (x) = c11 x1 + · · · + c1t xt

De ning a satisfactory degree of DMs at all the levels as

x; c; b; A

.. . DMt minimizes

zt (x) = ct1 x1 + · · · + ctt xt

subject to

A1 x1 + · · · + At xt 6b;

x; c; b; A

A1 x1 + · · · + At xt 6b; i (zi (x; ci1 ; : : : ; cit ))¿ˆi ; i = 1; : : : ; t − 1;

(21)

x1 ¿0; : : : ; xt ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b;

 = min(1 (z1 (x; c11 ; : : : ; c1t )); : : : ; t (zt (x; ct1 ; : : : ; ctt )));

(22)

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

we substitute the following problem for problem (21). maximize x; ; c; b; A



subject to A1 x1 + · · · + At xt 6b; i (zi (x; ci1 ; : : : ; cit ))¿ˆi ¿; i = 1; : : : ; t − 1; t (zt (x; ct1 ; : : : ; ctt ))¿;

(23)

0661; x1 ¿0; : : : ; xt ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b; To solve problem (23), we formulate the auxiliary problem: maximize x; ; c; b; A



subject to A1 x1 + · · · + At xt 6b; i (zi (x; ci1 ; : : : ; cit ))¿; i = 1; : : : ; t;

(24)

0661; x1 ¿0; : : : ; xt ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b; An optimal solution to problem (24) can be obtained by solving the problem: maximize x; 



subject to A1L x1 + · · · + AtL xt 6bR ; L ; : : : ; citL ))¿; i = 1; : : : ; t; i (zi (x; ci1

(25)

0661; x1 ¿0; : : : ; xt ¿0: A ratio of satisfactory degrees is also extended as i =

L L i+1 (zi+1 (x∗ ; c i+1;1 ; : : : ; c i+1; t )) ; L L i (zi (x∗ ; c i1 ; : : : ; c it ))

i = 1; : : : ; t − 1:

(26)

11

At an iteration ‘, let i (zi‘ ), i = 1; : : : ; t and ‘ denote a satisfactory degrees of the DM (DMi) at the ith level, i = 1; : : : ; t, and a satisfactory degree of all ‘ )=i (zi‘ ) the levels, respectively, and let ‘i = i+1 (zi+1 denote a ratio of satisfactory degrees of the ith and the (i + 1)th levels. Let a corresponding solution be x‘ . For all i = 1; : : : ; t − 1, DM(i + 1) proposes the solution to DMi. Then it follows that the DMs obtain the satisfactory solution and the iterated interactive process terminates if the following two conditions are satis ed. Termination conditions of the interactive process for multi-level linear programming problems C1. For all i = 1; : : : ; t−1, DMi’s satisfactory degree is larger than or equal to the minimal satisfactory level ˆi speci ed by DMi, i.e., i (zi‘ )¿ˆi , i = 1; : : : ; t − 1. C2. For all i = 1; : : : ; t−1, the ratio ‘i of satisfactory degrees is in the closed interval between its lower and its upper bounds speci ed by DMi. Suppose that the DMs from the (q + 1)th level to the (t − 1)th level, i.e., DM(q + 1), DM(q + 2); : : : ; and DM(t −1), satisfy the proposed solution but DMq does not satisfy it. Then DMq, DM(q + 1); : : : ; and DM(t − 1) need to update their minimal satisfactory levels ˆi , i = q; q + 1; : : : ; t − 1. Procedure for updating the minimal satisfactory level ˆ P1. DMi who does not satisfy the condition C1, i (zi‘ ) ˆi , decreases the minimal satisfactory level ˆi . P2. If the ratio ‘i exceeds its upper bound, then DMi increases the minimal satisfactory level ˆi . Conversely, if the ratio ‘i is below its lower bound, then DMi decreases the minimal satisfactory level ˆi . P3. If the ratio ˆi+1 = ˆi of the minimal satisfactory levels also is not in the valid interval of ‘i , then the minimal satisfactory level ˆi is updated in a way similar to the updating step (P2). Let ˆ0i denote the updated minimal satisfactory level. DMt solves the following maximization problem with the updated minimal satisfactory level ˆ0i :

12

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

maximize x; ; c; b; A



subject to A1 x1 + · · · + At xt 6b; i (zi (x; ci1 ; : : : ; cit ))¿; i = 1; : : : ; q − 1; i (zi (x; ci1 ; : : : ; cit ))¿ˆ0i ; i = q; : : : ; t − 1;

(27)

t (zt (x; ct1 ; : : : ; ctt ))¿; 0661; x1 ¿0; : : : ; xt ¿0; ˜ A) ˜ : (c; b; A) ∈ (c; ˜ b; We can obtain an optimal solution to problem (27) by solving the following problem through a transformation similar to the transformation from problem (24) and (25). maximize x; 



subject to A1L x 1 + · · · + A Lt xt 6bR ; L ; : : : ; citL ))¿; i = 1; : : : ; q − 1; i (zi (x; ci1

0 L ; : : : ; citL ))¿ˆi ; i = q; : : : ; t − 1; i (zi (x; ci1

Step 4: If the solution proposed to DMs at all the levels except for DMt satis es the termination conditions, they conclude the solution as a satisfactory one and the algorithm stops. Otherwise ‘ = ‘ + 1. Step 5: If the DMs from the (q + 1)th level to the (t − 1)th level, i.e., DM(q + 1), DM(q + 2); : : : ; and DM(t−1), satisfy the proposed solution but DMq does not satisfy it, DMi, i = q; : : : ; t − 1 update the minimal satisfactory levels ˆi , i = q; : : : ; t−1 in accordance with the procedure of updating minimal satisfactory level. Step 6: DMt solves problem (27) through problem (28). For all i = t; t−1; : : : ; 2, DMi proposes an optimal solution to problem (27) to DM(i − 1) successively. Return to Step 4. If there does not exist any feasible solution to problem (27), go to Step 7. Step 7: For all i = q; : : : ; t − 1, DMi update the minimal satisfactory levels ˆi by decreasing the values ˆi , and return to Step 6.

4. Numerical examples (28)

L ; : : : ; cttL ))¿; t (zt (x; ct1

0661; x1 ¿0; : : : ; xt ¿0: The above-mentioned algorithm is summarized as follows:

In this section, we provide illustrative numerical examples for two-level and three-level linear programming problems to demonstrate the feasibility of the proposed method. Example 1. Consider the following two-level linear programming problem with fuzzy parameters: DM1 minimizes x

c˜11 x1 + c˜12 x2

DM2 minimizes c˜21 x1 + c˜22 x2 Algorithm of the interactive fuzzy programming for solving multi-level linear programming problems with fuzzy parameters Step 1: Set ‘ = 1. DM1 determines a degree of ˜ A) ˜ . For all i = 1; : : : ; t − 1, DMi the -level set (c; ˜ b; elicits the membership function i (zi ) of the fuzzy goal for the objective function of DMi, and speci es the minimal satisfactory level ˆi and the lower and the upper bounds of the ratio of satisfactory degrees i . Step 2: DMt elicits the membership function t (zt ) of the fuzzy goal for the objective function of DMt. Step 3: DMt solves the auxiliary problem (24) through problem (25). For all i = t; t − 1; : : : ; 2, DMi proposes a solution x‘ to problem (24), (z1‘ ; : : : ; zt‘ ), ‘ , i (zi‘ ), and ‘i to DM(i − 1) successively.

x

subject to

˜ A˜1 x1 + A˜2 x2 6b;

(29)

x1 ¿0; x2 ¿0; where x1 = (x1 ; : : : ; x10 )T , x2 = (x11 ; : : : ; x20 )T ; each entry of 25 × 10 coecient matrices A1 and A2 is a random value in the interval [−50; 50]; each entry of the right-hand side constant column vector b is a sum of entries of the corresponding row vector of A1 and A2 multiplied by 0:6; there are four fuzzy parameters in each objective function and 30 fuzzy parameters in the constraints. Coecients including fuzzy parameters are shown in Tables 1 and 2 where c˜ij; k , a˜1; ij , a˜2; ij and b˜ i denote fuzzy parameters, which are 7% of all the coecients.

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

13

Table 1 Values of coecients for the two-level problem c11 c12 c21 c22

c˜11; 1 −34 −23 34

−15 c˜12; 1 −22 21

−15 −19 c˜21; 3 c˜22; 3

−39 −23 −38 23

−44 −15 −29 15

−12 −42 −34 42

−27 −29 c˜21; 7 29

−22 −12 −1 c˜22; 8

−42 c˜12; 9 −13 10

c˜11; 10 −40 −1 40

A1

a˜1; 11 44 −20 6 −38 −38 42 30 −13 1 31 32 9 1 12 −1 −26 11 44 −47 −36 11 3 −41 −44

50 16 −15 a˜1; 42 −21 −48 −21 18 −32 −24 1 −32 13 29 −32 20 22 −46 −39 −15 13 −17 −31 −29 22

29 −27 21 −25 27 −37 −13 22 −25 43 −33 10 15 29 35 −13 18 8 −31 14 −5 −45 39 7 −24

16 −20 19 −23 −15 a˜1; 64 32 3 −9 31 −33 40 −38 5 −35 −48 7 −16 −33 7 42 −1 −46 28 25

−21 1 13 9 42 −9 28 −3 0 a˜1; 10 5 47 50 44 −38 29 a˜1; 16 5 −24 −42 −18 −15 24 2 −2 −21 −1

9 47 −35 36 2 42 13 −2 −32 −7 5 −35 5 24 −46 a˜1; 16 6 40 −23 −14 −3 −48 −21 25 14 15

9 −21 28 −10 −17 a˜1; 67 −35 −10 −26 a˜1; 10 7 −37 28 7 −39 5 5 30 −17 27 34 −18 21 −39 −2 26

−47 31 −38 −4 −7 −13 49 37 −48 −4 −15 −4 34 −49 −21 10 −18 −20 16 a˜1; 20 8 −16 −35 −26 40 −5

−18 8 −28 a˜1; 49 −37 −5 11 32 −39 −8 −8 33 29 −47 −29 47 31 −24 −23 47 −48 41 −47 −44 −28

a˜1; 1 10 −30 42 −41 20 41 −21 −15 −27 16 40 −24 17 8 44 18 −37 −36 −41 a˜1; 20 10 44 16 12 45 −8 b

A2

−1 −24 −6 43 46 20 −18 26 8 −31 −13 −46 2 23 −31 41 45 −16 8 a˜2; 20 1 −45 −38 −17 −42 13

35 −39 −7 25 31 a˜2; 62 −11 −28 15 35 −49 34 −7 4 −30 a˜2; 16 2 −15 48 −12 49 −17 39 −28 38 28

a˜2; 13 −25 −45 a˜2; 43 37 −25 −17 49 −5 −3 −1 47 34 −1 −18 −49 14 48 50 8 −2 −11 −21 −49 1

−42 −18 45 a˜2; 44 −36 45 −29 3 43 a˜2; 10 4 −13 39 −17 42 34 a˜2; 16 4 −11 14 26 −24 45 −5 −32 43 −7

−13 49 −46 32 −30 50 49 −41 43 −18 48 −19 −34 −25 −9 −23 −39 −42 −48 −13 −33 −28 −25 49 −34

21 −12 8 49 −47 a˜2; 66 5 −14 −10 a˜2; 10 6 44 48 31 −15 −4 36 −13 −25 4 34 26 36 34 25 30

17 −47 −34 41 18 39 −28 −15 −15 −17 −45 10 −30 −22 −31 46 47 8 41 47 14 −20 8 −4 −48

a˜2; 18 45 −32 −39 26 −24 −32 −31 15 14 −6 1 11 −30 −43 −1 36 −3 −33 −49 −44 −30 −19 −21 −7

18 −43 33 3 22 33 15 −17 46 37 28 41 45 32 −49 17 −9 39 −44 50 −17 −45 −3 23 25

41 17 33 23 36 19 27 29 4 42 −5 −12 −43 −43 25 −12 26 −42 −49 a˜2; 20 10 −16 −45 −22 27 −13

b˜1 −28 −38 b˜4 35 b˜6 27 43 −64 b˜10 −8 144 76 −67 −116 b˜16 74 −105 −101 b˜20 −82 −104 −142 51 −20

14

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19 Table 2 Coecients represented by fuzzy numbers for the two-level problem c˜11; 1 c˜12; 1 c˜21; 3 c˜22; 3

−50 −25 −37 14

−45 −20 −32 19

−40 −15 −27 24

c˜11; 10 c˜21; 9 c˜21; 7 c˜22; 8

−16 −14 −38 7

−11 −9 −33 12

−6 −4 −28 17

a˜1; 11 a˜2; 13 a˜1; 42 a˜2; 43 a˜1; 64 a˜2; 62 a˜1; 10 5 a˜2; 10 4 a˜1; 16 5 a˜2; 16 2 a˜1; 20 8 a˜2; 20 1

21 7 −38 43 8 −44 −23 27 20 −45 −46 32

26 12 −33 48 13 −39 −18 32 25 −40 −41 37

31 17 −28 53 18 −34 −13 37 30 −35 −36 42

a˜1; 1 10 a˜2; 18 a˜1; 49 a˜2; 44 a˜1; 67 a˜2; 66 a˜1; 10 7 a˜2; 10 6 a˜1; 16 6 a˜2; 16 4 a˜1; 20 10 a˜2; 20 10

14 −27 0 6 −6 −7 −34 −11 20 −30 10 −42

19 −22 5 11 −1 −2 −29 −6 25 −25 15 −37

24 −17 10 16 4 3 −24 −1 30 −20 20 −32

78 30 40

83 35 45

88 40 50

87 50 52

92 55 57

97 60 62

b˜1 b˜6 b˜16

Suppose that DM1 determines a degree of the level set as = 0:8, the initial minimal satisfactory level as  = 1:0, and the lower and the upper bounds of  as [0:6; 1:0]. The membership functions (9) of the fuzzy goals are assessed by using values (7) and (10). The individual minima and the corresponding solutions are shown in Table 3. Then problem (16) for this numerical example can be formulated as maximize x; 



(z1 (x) + 384:32)=(−783:98 + 384:32)¿;

that DM1 changes the minimal satisfactory level from ˆ = 1:0 to ˆ0 = 0:75. Then a problem corresponding to problem (19) is formulated as maximize x

L L 2 (z2 (x; c21 ; c22 ))

subject to A1L x1 + A2L x2 6bR ;

(31)

(z1 (x) + 384:32)=(−783:98 + 384:32)¿0:75; x1 ¿0; x2 ¿0:

subject to A1L x1 + A2L x2 6bR ;

b˜4 b˜10 b˜20

(30)

(z2 (x) − 84:39)=(−127:09 − 84:39)¿; 0661; x1 ¿0; x2 ¿0: Data of the rst iteration including an optimal solution to problem (30) are shown in Table 4. The condition C1 of termination of the interactive process is not satis ed because the satisfactory degree 1 (z11 ) = 0:703874 of DM1 does not exceed the minimal satisfactory level ˆ = 1:0. Consequently, suppose

Data of the second iteration including an optimal solution to problem (31) are shown in Table 5. At the second iteration, the satisfactory degree 1 (z12 ) = 0:75 of DM1 becomes equal to the minimal satisfactory level ˆ0 = 0:75 and the ratio 2 = 0:793534 of satisfactory degrees is in the valid interval [0:6; 1:0] of the ratio. Therefore, this solution satis es the conditions of termination of the interactive process and becomes a satisfactory solution for both DMs. Example 2. As an example for a three-level linear programming problem, consider the following problem with fuzzy parameters:

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

15

Table 3 The individual minima and the corresponding solutions for the two-level problem z1min

−783.988457

x1

3.358022 0 4.636045 0

x2 z2min

−127.097148

x1

0.581670 0 0.887641 0

x2

1.406026 3.026433 2.793502 0.034993

0.070977 1.462620 0.195681 2.309067

1.554751 0.855733 2.386315 0

0.647816 2.894935 2.002535 0

0.881096 1.652822 0.532639 0.116233

1.430671 0.958683 0.620470 1.213629

2.019822 0.929059 0 0

0.720285 1.487727 1.249805 0

DM1 minimizes c˜11 x1 + c˜12 x2 + c˜13 x3

Table 4 The rst iteration for the two-level problem

DM2 minimizes c˜21 x1 + c˜22 x2 + c˜23 x3

1

0.703874

DM3 minimizes c˜31 x1 + c˜32 x2 + c˜33 x3

x11

1.140632 0 3.110285 0

x x

(32)

x

subject to

x21

˜ A˜1 x1 + A˜2 x2 + A˜3 x3 6b; x1 ¿0; x2 ¿0; x3 ¿0; T

z11 T

where x1 = (x1 ; : : : ; x5 ) , x2 = (x6 ; : : : ; x10 ) , x3 = (x11 ; : : : ; x15 )T ; each entry of ve-dimensional row constant vectors cij , i; j = 1; 2; 3 and each entry of 16 × 5 coecient matrices A1 , A2 and A3 are random values in the interval [−50; 50]; each entry of the right-hand side constant column vector b is a sum of entries of the corresponding row vector of A1 , A2 and A3 multiplied by 0:6; a little less than 30% of all of the coecients are fuzzy parameters. Coecients including fuzzy parameters are shown in Tables 6 and 7 where c˜ij; r , a˜k; ij and b˜i denote fuzzy parameters. Suppose that DM1 determines a degree of the level set as = 0:8, and DM1 and DM2 determine the initial minimal satisfactory levels as 1 = 2 = 1:0, and the lower and the upper bounds of 1 and 2 as [0:6; 1:0]. The membership functions (9) of the fuzzy goals are assessed by using values (7) and (10). The individual minima and the corresponding solutions are shown in Table 8. Then problem (25) for this numerical example can be formulated as

2.715410 2.144598 1.314700 0.304960

1.382066 0.929700 0.335627 1.006295

1.794216 0.834103 1.944610 0

−665:637844

1 (z11 )

0.703874

z21

−64:467789

2 (z21 )

0.703874

1

1.000000

2.333759 1.441670 2.865077 0

Table 5 A satisfactory solution for the two-level problem x12

1.471500 0 3.338873 0

x22 z12

z22 2

−684:072563 −41:473153 0.793534

maximize x; 

2.524859 2.221156 1.555675 0.254776

1.166568 0.997998 0.310529 1.234265

1.784212 0.877159 2.022162 0

1 (z12 ) 2 (z22 )

0.750000 0.595151

2.064206 1.695350 2.712697 0



subject to A1L x1 + A2L x2 + A3L x3 6bR ; (z1 (x) − 95:44)=(−340:88 − 95:44)¿; (z2 (x) − 184:11)=(−257:12 − 184:11)¿; (z3 (x) − 209:33)=(−144:86 − 209:33)¿; 0661; x1 ¿0; x2 ¿0; x3 ¿0:

(33)

16

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

Table 6 Values of coecients for the three-level problem c11 c˜11; 1 c21 −10 c31 49 A1

−1 47 −19 −26 a˜1; 51 26 37 45 43 47 −24 3 −15 a˜1; 14 1 −36 a˜1; 16 1

−20 −44 50

c˜11; 3 −9 20

c˜11; 4 −4 c˜31; 4

−23 29 16 a˜1; 42 1 10 −20 36 11 −40 6 9 20 22 17 28

27 a˜1; 23 46 0 −5 a˜1; 63 21 a˜1; 83 −1 a˜1; 10 3 −25 a˜1; 12 3 −1 −26 42 a˜1; 16 3

a˜1; 14 a˜1; 24 50 −11 −20 29 39 −35 28 −1 1 a˜1; 12 3 −31 −4 −11 a˜1; 16 4

−38 c˜21; 5 38

c12 c22 c32

4 −4 48

c˜21; 2 c˜22; 2 24

c˜21; 3 23 c˜22; 3 −23 34 11

25 −25 46

35 47

A2

8 −9 49 40 a˜2; 51 −10 −23 a˜2; 81 a˜2; 91 −40 7 a˜2; 12 1 16 a˜2; 14 1 a˜2; 15 1 27

−4 −18 a˜2; 32 −11 a˜2; 52 −27 −36 24 46 a˜2; 10 2 a˜2; 11 2 −13 a˜2; 13 1 32 −21 −17

−45 19 −15 45 a˜2; 53 25 a˜2; 73 16 a˜2; 93 19 −34 −2 −25 −16 −6 15

−19 A3 41 3 a˜2; 45 9 −11 26 −10 a˜2; 95 16 a˜2; 11 5 28 42 50 −10 35

a˜1; 35 a˜1; 45 −40 −23 34 −5 41 10 23 11 −31 a˜1; 14 5 −13 −36

−35 a˜2; 24 −7 22 −1 13 −30 a˜2; 84 −19 a˜2; 10 4 34 26 36 15 −45 −21

c13 −14 c23 c˜23; 1 c33 c˜33; 1

c˜13; 2 c˜23; 2 c˜33; 2

−33 28 −28

−36 3 −3

c˜13; 5 c˜23; 5 c˜33; 5

b

a˜3; 11 −7 −43 −40 a˜3; 51 20 −49 17 a˜3; 91 13 a˜3; 11 1 a˜3; 12 1 15 8 28 −14

−19 21 3 a˜3; 42 −5 10 a˜3; 72 43 −33 a˜3; 10 2 a˜3; 11 2 a˜3; 12 2 3 −12 −48 a˜3; 16 2

a˜3; 13 2 a˜3; 33 a˜3; 43 −22 −16 32 37 15 19 −30 −29 −39 −41 −25 −17

−29 a˜3; 24 a˜3; 34 −44 −43 a˜3; 64 49 7 a˜3; 94 −5 −19 11 6 46 −12 −11

0 a˜3; 25 −33 −29 48 −20 16 −40 42 a˜2; 10 5 a˜3; 11 5 20 19 9 −7 −1

−97 b˜ 1 b˜ 2 −1 b˜ 3 −3 53 b˜ 4 b˜ 5 22 b˜ 6 −4 −11 b˜ 7 b˜ 8 −43

Table 7 Coecients represented by fuzzy parameters for the three-level problem c˜11; 1 c˜12; 3 c˜22; 2 c˜23; 5 c˜33; 5

−28 44 −55 14 −14

−24 37 −46 12 −12

−20 30 −37 10 −10

c˜11; 3 c˜13; 2 c˜22; 3 c˜31; 4

−52 −10 −44 30

−44 −9 −37 25

−36 −8 −30 20

c˜11; 4 c˜13; 5 c˜23; 1 c˜33; 1

−58 −21 51 −51

−49 −18 43 −43

−40 −15 35 −35

c˜12; 2 c˜21; 5 c˜23; 2 c˜33; 2

55 −33 40 −40

46 −28 34 −34

37 −23 28 −28

a˜1; 14 a˜1; 42 a˜1; 63 a˜1; 14 5 a˜2; 24 a˜2; 52 a˜2; 84 a˜2; 10 2 a˜2; 12 1 a˜3; 11 a˜3; 33 a˜3; 51 a˜3; 94 a˜3; 11 2 a˜3; 16 2

−6 −37 12 28 −22 35 40 −46 4 −48 −30 −21 32 40 −18

−5 −31 14 34 −19 43 50 −39 5 −40 −25 −18 40 50 −15

−4 −25 16 40 −16 51 60 −32 6 −32 −20 −15 48 60 −12

a˜1; 23 a˜1; 45 a˜1; 10 3 a˜1; 16 1 a˜2; 32 a˜2; 53 a˜2; 91 a˜2; 10 4 a˜2; 13 2 a˜3; 13 a˜3; 34 a˜3; 64 a˜3; 10 2 a˜3; 11 5

−32 −16 −49 −55 24 −49 28 −7 −42 −15 9 −55 36 39

−27 −14 −14 −46 30 −14 35 −6 −35 −13 11 −46 45 48

−22 −12 −33 −37 36 −33 42 −5 −28 −11 13 −37 54 57

a˜1; 24 a˜1; 51 a˜1; 12 3 a˜1; 16 3 a˜2; 45 a˜2; 73 a˜2; 93 a˜2; 11 2 a˜2; 14 1 a˜3; 24 a˜3; 42 a˜3; 72 a˜3; 10 5 a˜3; 12 1

32 6 −39 21 32 −31 −25 40 21 −22 20 16 33 −6

39 7 −33 26 39 −26 −21 50 26 −19 24 19 41 −5

46 8 −27 31 46 −21 −17 60 31 −16 28 22 49 −4

a˜1; 35 a˜1; 55 a˜1; 14 1 a˜1; 16 4 a˜2; 51 a˜2; 81 a˜2; 95 a˜2; 11 5 a˜2; 15 1 a˜3; 25 a˜3; 43 a˜3; 91 a˜3; 11 1 a˜3; 12 2

12 −16 30 −30 4 36 −9 −32 −6 −27 28 −42 16 −57

14 −14 37 −25 5 45 −8 −27 −5 −22 34 −35 20 −48

16 −12 44 −20 6 54 −7 −22 −4 −17 40 −28 24 −39

59 88

73 110

87 132

38 38

47 47

56 56

b˜ 3 b˜ 7

−58 86

−49 107

−40 128

b˜ 4 b˜ 8

132 −109

164 −91

196 −73

b˜ 1 b˜ 5

b˜ 2 b˜ 6

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19 Table 8 The individual minima and the corresponding solutions for the three-level problem z1min x1 x2 x3

−340.881010 0 0.014790 0 3.349043 0 0.478848 0.898495 0 1.251230 0 4.363841 0 2.452510 2.835250 0

z2min −257.126188 x1 x2 x3

0 1.175118 0 0.758357 0.533208 0 2.420444 2.843138 0.517513 0 0.756503 0 0.417428 0.890328 0.141371

z3min −144.869060 x1 x2 x3

0 0 0 2.555717 0 0 1.301624 0 1.051190 0 3.967885 0 2.446645 2.882009 0

17

Table 10 The second iteration for the three-level problem 2

0.474601

x12 x22 x32

0

0

1.379780

0.652184

0.102005

0 1.205264

0.972268 0.487693

1.908900 0.434839

1.353654 2.732312

0 1.534944

1 (z12 )

0.474601

3 (z32 )

0.474601

z12

−111:638195

z22 z32

−58:567806

21 22

1.158869

41:228475

2 (z22 )

0.550000

0.862911

maximize x; 



subject to A1L x1 + A2L x2 + A3L x3 6bR ; (z1 (x) − 95:44)=(−340:88 − 95:44)¿;

Table 9 The rst iteration for the three-level problem

(z2 (x) − 184:11)=(−257:13 − 184:11)¿0:55;

1

0.512274

x11 x21 x31

0 0

0 0.994996

1.328993 1.822949

0.891419 1.380408

0 0

1.444770

0.457168

0.554134

2.808694

1.487567

1 (z11 )

0.512274

z11 z21

−128:075606

11

1.000000 1.000000

z31

12

−41:921351 27:884712

2 (z21 ) 3 (z31 )

0.512274 0.512274

Data of the rst iteration including an optimal solution to problem (33) are shown in Table 9. The condition C1 of termination of the interactive process is not satis ed because the satisfactory degree 2 (z21 ) = 0:512274 of DM2 does not exceed the minimal satisfactory level ˆ2 = 1:0. Consequently, suppose that DM2 changes the minimal satisfactory level from ˆ2 = 1:0 to ˆ02 = 0:55. Furthermore, the ratio of minimal satisfactory levels 2 =1 = 0:55 is not in the valid interval [0:6; 1:0] of 1 . Suppose that DM1 also changes the minimal satisfactory level from ˆ1 = 1:0 to ˆ01 = 0:8. Then a problem corresponding to problem (28) is formulated as

(z3 (x) − 209:33)=(−144:87 − 209:33)¿; 0661; x1 ¿0; x2 ¿0; x3 ¿0:

(34)

Data of the second iteration including an optimal solution to problem (34) are shown in Table 10. The condition C1 of termination of the interactive process is not satis ed because the satisfactory degree 1 (z12 ) = 0:474601 of DM1 does not exceed the minimal satisfactory level ˆ1 = 0:8. Consequently, suppose that DM1 changes the minimal satisfactory level from ˆ1 = 0:8 to ˆ01 = 0:6. Then a problem corresponding to problem (28) is formulated as maximize x; 



subject to A1L x1 + A2L x2 + A3L x3 6bR ; (z1 (x) − 95:44)=(−340:88 − 95:44)¿0:60; (z2 (x) − 184:11)=(−257:12 − 184:11)¿0:55; (z3 (x) − 209:33)=(−144:86 − 209:33)¿; 0661; x1 ¿0; x2 ¿0; x3 ¿0:

(35)

18

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19

Table 11 A satisfactory solution for the three-level problem 3

0.413870

x13 x23 x33

0

0

1.576829

1.896943

0

0 1.916609

1.106749 0.539322

2.398105 0.397718

1.501166 2.920367

0 1.481124

1 (z13 ) 2 (z23 )

0.600000 0.550000

z13 z23 z33

31 32

−166:352522 −58:567806 62:739388

3 (z33 )

0.413870

0.916667 0.752491

Data of the second iteration including an optimal solution to problem (35) are shown in Table 11. At the third iteration, the satisfactory degree 1 (z13 ) = 0:6 of DM1 becomes equal to the minimal satisfactory level ˆ01 = 0:6 and the satisfactory degree 2 (z23 ) = 0:55 of DM2 becomes equal to the minimal satisfactory level ˆ02 = 0:55. The ratios 31 = 0:916667 and 32 = 0:752491 of satisfactory degrees are in the valid interval [0:6; 1:0] of the ratios 1 and 2 . Therefore, this solution satis es the conditions of termination of the interactive process and then becomes a satisfactory solution for all the DMs.

cision variables of DM1 are introduced. Using the rst numerical example examined at Section 4, we will show an example in which their methods produce an undesirable solution. Suppose that DM1 de nes membership functions of fuzzy goals for decision variables in the following by using a solution xo yielding the individual minimum z1min and a solution xw yielding the individual maximum z1max . For all i = 1; : : : ; 10,  0; xi ¿xio + Ái ;      −xi + (xio + Ái )   ; xio 6xi 6xio + Ái ;  Á i c (xi ) = 1i xi − (xio − Ái )    ; xio − Ái 6xi 6xio ;   Á  i   0; xi ¡xio − Ái ; where Ái = |xio − xiw |. Aggregating the membership functions of fuzzy goals for the objective function and the decision variables, Lai et al. de ned a satisfactory degree of DM1 as c (x1 ); : : : ; 1c 10 (x10 )): 1 = min(1 (z1 (x; c11 ; c12 )); 11

Therefore, problem (30) becomes maximize x; 

5. Conclusions In this paper, we have proposed interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. In our interactive method, after determining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived eciently by updating the satisfactory degrees of DMs at the upper levels with considerations of overall satisfactory balance among all levels. Illustrative numerical examples for two-level and three-level linear programming problems have been provided to demonstrate the feasibility of the proposed method. Appendix A. The problem arising when fuzzy goals for decision variables are introduced In the methods by Lai [5], and Shih et al. [8], fuzzy goals not only for objective functions but also for de-



subject to A1L x1 + A2L x2 6bR ; (z1 (x) + 384:32)=(−783:98 + 384:32)¿; c 1i (xi )¿; i = 1; : : : ; 10;

(z2 (x) − 84:39)=(−127:09 − 84:29)¿; 0661; x1 ¿0; x2 ¿0 and an optimal solution is shown in Table 12. As seen in Table 12, the satisfactory degree of DMs at both levels is 1 = 0:0 because several membership function values of fuzzy goals for decision variables become zero. The DMs cannot help updating their membership functions of fuzzy goals to improve their satisfactory degrees. It seems that a possibility that the overall satisfactory degree becomes zero gets larger if the number of fuzzy goals for decision variables increases.

M. Sakawa et al. / Fuzzy Sets and Systems 109 (2000) 3–19 Table 12 A solution to the problem with fuzzy goals for decision variables 1

0.000000

x11

1.148982 0.000000 1.287428 0.118403

x21 c (x ) 1i i

1.700706 1.567263 0.932108 0.552626

0.383300 0.951902 0.394978 0.264450

0.754668 0.567960 1.130434 0.531488

0.991874 0.889271 1.487149 0.132675

0.342160 0.000000 0.383683 0.485395 0.468897 0.000000 0.000000 0.327894 0.654584 0.000000

z11 z21

−408:924218 −12:713343

1

7:460065

1 (z11 ) 2 (z21 )

0:061550 0:459167

Furthermore, we can point out another disadvantage from the following example. For the sake of simplicity, suppose that there is only one fuzzy goal for a decision variable. Thus, we have c (x1 )): 1 = min(1 (z1 (x; c11 ; c12 )); 11

At the ‘th iteration, suppose that 1 = 0:7; 2 = 0:62;

1 (z1 ) = 0:88;

c 11 (x1 ) = 0:7;

2 (z2 ) = 0:62

and this solution does not satisfy the condition of the bounds of ‘ . At the (‘ + 1)th iteration, suppose that 1 = 0:8; 2 = 0:58;

1 (z1 ) = 0:82;

c 11 (x1 ) = 0:8;

2 (z2 ) = 0:58

and the interactive process terminates as this solution satis es all the conditions. It is seen that DM1 increases the satisfactory degree while DM2 decreases that because the pair (1 ; 2 ) changes from (0:7; 0:62) at the ‘th iteration to (0:8; 0:58) at the (‘ + 1)th iteration. In contrast, for the pair (1 (z1 ); 2 (z2 )) of satisfactory degrees for the objective functions, both DM1 and DM2 decrease their satisfactory degrees because the pair (1 (z1 ); 2 (z2 )) changes from (0:88; 0:62) at the

19

‘th iteration to (0:82; 0:58) at the (‘ + 1)th iteration. It should be noted that when fuzzy goals for decision variables are introduced, the iterative procedure may terminate, having an undesirable (dominated) solution from the viewpoint of the satisfaction of the fuzzy goals for the objective functions. References [1] J.F. Bard, An ecient point algorithm for a linear two-stage optimization problem, Oper. Res. 38 (1983) 556–560. [2] J.F. Bard, An investigation of the linear three-level programming problem, IEEE Trans. Systems Man Cybernet. SMC-14 (1983) 711–717. [3] J.F. Bard, J.E. Falk, An explicit solution to the multi-level programming problem, Comput. Oper. Res. 9 (1982) 77–100. [4] W.F. Bialas, M.H. Karwan, Two-level linear programming, Management Sci. 30 (1984) 1004–1020. [5] Y.J. Lai, Hierarchical optimization: a satisfactory solution, Fuzzy Sets and Systems 77 (1996) 321–335. [6] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993. [7] M. Sakawa, H. Yano, An interactive fuzzy satis cing method for generalized multiobjective linear programming problems with fuzzy parameters, Fuzzy Sets and Systems 35 (1990) 125–142. [8] H.S. Shih, Y.J. Lai, E.S. Lee, Fuzzy approach for multilevel programming problems, Comput. Oper. Res. 23 (1996) 73–91. [9] M. Simaarn, J.B. Cruz, On the Stackelberg strategy in nonzero-sum games, J. Optim. Theory Appl. 11 (1973) 533–555.  u, A linear bilevel programming algorithm based [10] G. Unl on bicriteria programming, Comput. Oper. Res. 14 (1987) 173–179. [11] U.-P. Wen, W.F. Bialas, The hybrid algorithm for solving the three-level linear programming problem, Comput. Oper. Res. 13 (1986) 367–377. [12] U.-P. Wen, S.-T. Hsu, Ecient solutions for the linear bilevel programming problem, European J. Oper. Res. 62 (1991) 354–362. [13] D.J. White, G. Anandalingam, A penalty function approach for solving bi-level linear programs, J. Global Optim. 3 (1993) 397–419. [14] H.-J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978) 45–55.