Interactive Decision Making for Hierarchical Multiobjective Linear Programming Problems with Random Variable Coefficients

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1 SCIS & ISIS 200, Dec. 8-2, 200, Okayama Convention Center, Okayama, Japan Interactive Decision Making for Hierarchical Multiobjective Linear Programg Problems with Random Variable Coefficients Hitoshi Yano School of Humanities and Social Sciences, Nagoya City University Nagoya Japan Abstract In this paper, we focus on hierarchical multiobjective linear programg problems with random variable coefficients where multiple decision makers in a hierarchical organization have their own multiple objective linear functions together with common linear constraints. In order to deal with multiple objective linear functions and linear constraints involving random variable coefficients, P-Pareto optimality concept is defined in cumulative distribution function space. After each decision maker specifies his/her own decision power and reference probabilistic values, the corresponding candidate of a satisfactory solution is obtained from among P-Pareto optimal solution set on the basis of linear programg. Interactive processes are demonstrated by means of an illustrative numerical example. I. INTRODUCTION In the real-wod decision making situations, it is often required that the goal of the overall system is achieved in the hierarchical structure, where many decision makers who belong to its sections or divisions are in action to seek their own goals independently and are affected each other. The Stackelberg games [], [3] can be regarded as multilevel programg problems with multiple decision makers. Although many kinds of techniques to obtain a Stackelberg solution have been proposed, almost all of such techniques are unfortunately not efficient in computational aspects. In order to circumvent the computation inefficiency to obtain such a Stackelberg solution and the paradox that the lower level decision power often doates the upper level decision power, Lai [5], Shih et al.[] and Lee et al.[8] introduced concepts of memberships of optimalities and degrees of decision powers and proposed fuzzy approaches to multilevel linear programg problems. In their approaches, each decision maker elicits his/her own membership functions for not only the objective functions but also the decision variables. Following the fuzzy decision [9] together with membership functions, the mathematical programg problem of finding the maximum decision is formulated and solved to obtain a candidate of the satisfactory solution. However, in such fuzzy approaches for multilevel linear programg problems, the decision makers are required to elicit each of membership functions for not only the objective functions but also the decision variables, and to update them in each of the iterations. It seems to be very difficult to elicit membership functions for the decision variables. In order to circumvent such difficulties, we proposed a fuzzy approach for hierarchical multiobjective linear programg problems HMOLP)[4], where multiple decision makers in a hierarchical organization have their own multiple objective linear functions and common linear constraints. In order to deal with HMOLP, a solution concept called a generalized Λ-extreme point is introduced, and an interactive algorithm is proposed to obtain the satisfactory solution from among a generalized Λ-extreme point set. On the other hand, in the actual decision making situations, the decision makers often encounter difficulties to deal with vague information or uncertain data. In order cope with such vague or uncertain decision situations, there exist two kinds of approaches called stochastic programg approach [2], [4] and fuzzy mathematical programg one [6], [9]. Fuzzy mathematical programg approaches are naturally extended for multiobjective programg problems [7], [9], [5]. Stochastic programg approaches are also extended for multiobjective programg problems [0], [2]. Especially, Sakawa and the others [0] formulated multiobjective linear programg problems with random variable coefficients using the transformation techniques of the probability maximization model and chance constrained conditions. They introduced M-Pareto optimal solution concept in membership space of cumulative distribution functions which means that each objective function is less than or equal to a certain permissible level. They proposed an interactive fuzzy satisficing method to derive a satisficing solution from among M-Pareto optimal solution set. In this paper, we focus on hierarchical multiobjective linear programg problems with random variable coefficients, where multiple decision makers in a hierarchical organization have their own multiple objective linear functions and common linear constraints with random variable coefficients. In order to obtain a satisfactory solution, which reflects a hierarchical decision making structure, each decision maker s own preference structure for their multiple objective functions, and the probabilistic uncertainty of random variable coefficients, an interactive algorithm based on linear programg is proposed. In section 2, hierarchical multiobjective linear programg problems with random variable coefficients are formulated and the corresponding solution concept called a P-Pareto optimal solution in cumulative distribution function 032

2 space is introduced. In section 3, an interactive algorithm is proposed to obtain the satisfactory solution from among a P- Pareto optimal solution set. In section 4, interactive processes of the proposed method are demonstrated by means of an illustrative numerical example. II. HIERARCHICAL MULTIOBJECTIVE LINEAR PROGRAMMING PROBLEMS WITH RANDOM VARIABLE COEFFICIENTS We consider the following hierarchical multiobjective linear programg problem with Random Variable Coefficients HMOLP-RV), where each decision maker DM r ) has his/her own multiple objective linear functions together with common linear constraints, and random variable coefficients are involved in each objective function and the right-hand side of constraints. [HMOLP-RV] first level decision maker : DM z x, ω) = z x, ω),, z k x, ω)) p-th level decision maker : DM q z q x, ω) = z q x, ω),, z qkq x, ω)) Ax bω), x 0 In HMOLP-RV, x = x, x 2,, x n ) T is n-dimensional decision column vector, A is m n coefficient matrix, and each objective function of DM r is defined by z x, ω) = c ω)x + α ω), l =,, k r, ) c ω) = c + t ω)c 2, 2) α ω) = α + t ω)α 2, 3) where c ω), l =,, k r are n dimensional random variable row vectors, α ω), l =,, k r are random variables, and t ω) is a random variable with mean t, whose cumulative distribution function T ) is assumed to be strictly monotone increasing and continuous. In the right-hand-side of the constraints of HMOLP-RV, bω) = b ω),, b m ω)) T 4) are random variables independent each other, whose cumulative distribution function F i ), i =,, m are assumed to be strictly monotone increasing and continuous. Similar to the formulations of multilevel linear programg problems proposed by Lee and Shih[8], it is assumed that the upper level decision makers make their decisions with consideration of the overall benefits for the hierarchical organization, although they can take priority for their objective functions over the lower level decision makers. In order to deal with HMOLP-RV, we adopt stochastic linear programg techniques for HMOLP-RV. Here, for notational convenience, the linear constraints Ax bω) are expressed as a i x b i ω), i =,, m. We adopt chance constrained conditions with satisfactory constraint levels β i for the constraint a i x b i ω). Pra i x b i ω)) β i 5) For the objectives in HMOLP-RV, we substitute the imization of z x, ω) for the maximization of the probability that z x, ω) is less than or equal to a certain permissible objective level f specified by the decision maker DM r ) in his/her subjective manner. p x, f ) = Prz x, ω) f ), r =,, q, l =,, k r 6) Then, HMOLP-RV can be transformed as the following probability maximization model called HMOLPf, β) where the parameters β = β i ) are specified by the first level decision maker DM ) and f r = f r,, f rkr ), r =,, q are specified by each decision maker DM r ) in his/her subjective manner. [HMOLPf, β)] first level decision maker : DM max p x, f ) = p x, f ),, p k x, f k )) q-th level decision maker: DM q max p q x, f q ) = p q x, f q ),, p qkq x, f qkq )) Pra i x b i ω)) β i, i =,, m, x 0. By using cumulative distribution function F i ), the constraint Pra i x b i ω)) β i can be expressed as follows. Pra i x b i ω)) β i, F i a i x) β i, a i x F i β i ) where F i ) means a pseudo-inverse function corresponding to F i ). Similay, under the assumption that c 2 x + α2 > 0, r =,, q, l =,, k r, using cumulative distribution function T ), the objective function p x, f ) in HMOLPf, β) is expressed as follows. p x, f ) = Prz x, ω) f ) = Prc ω)x + α ω) f ) = Pr t ω) f c x + α ) ) c 2 x + α2 f c = T x + α ) ) c 2 x + α2 Therefore, HMOLPf, β) can be converted as the following problem called HMOLP2f, β), where cumulative distribution functions F i ), i =,, m, T ), r =,, q, l =,, k r are explicitly involved. [HMOLP2f, β)] first level decision maker: DM f c max T x + α ) ) c 2 x +, α2 ) ) fk c k, T x + αk ) k c 2 k x + αk 2 q-th level decision maker: DM q 033

3 max ) fq c qx + αq) T q c 2 q x +, α2 q, T qkq fqkq c qk q x + α qk q ) c 2 qk q x + α 2 qk q ) ) x Xβ) = {x 0 a i x F i β i ), i =,, m} In order to deal with HMOLP2f, β), we introduce Pareto optimal solution concept in cumulative distribution function space, called P-Pareto optimal solution. Definition. x Xβ) is said to be a P-Pareto optimal solution to HMOLP2f, β), if and only if there is no x Xβ) such that p x, f ) p x, f ), r =,, q, l =,, k r. In the following, we assume that q decision makers in the hierarchical decision to situation reach an agreement that they choose their satisfactory solution to HMOLP2f, β) from among P-Pareto optimal solution set. Then, for generating a candidate of the satisfactory solution from among P-Pareto optimal solution set, each decision maker DM r ) is asked to specify his/her reference probabilistic values p, l =,, k r [9] which are reference levels of achievement of cumulative distribution function T ). Once the reference probabilistic values are specified, the corresponding P-Pareto optimal solution, which is, in a sense, close to their requirement, is obtained by solving the following imax problem. [MINMAXf, β, p)] x n+ x Xβ),x n+ E f c p T x + α ) ) c 2 x + x n+, α2 r =,, q, l =,, k r Theorem. ) If x, x n+) is a unique optimal solution to MINMAXf, β, p), then x Xβ) is a P-Pareto optimal solution. 2) If x Xβ) is a P-Pareto optimal solution, then x, x n+) is an optimal solution to MINMAXf, β, p) for some reference probabilistic values p, where p = p x, f ) + x n+, r =,, q, l =,, k r. It should be noted here that, in general, P-Pareto optimal solution obtained by solving MINMAXf, β, p) does not reflect the hierarchical structure between q decision makers where the upper level decision maker can take priority for his/her cumulative distribution functions over the lower level decision makers. In order to cope with such a hierarchical preference structure between q decision makers in MINMAXf, β, p), we introduce the decision power w r, r =,, q [5] for the inequality constraints p T ) x n+, l =,, k r in MINMAXf, β, p), where the r-th level decision maker DM r ) can specify the decision power w r+ in his/her subjective manner and the last decision maker DM q ) has no decision power. In order to reflect the hierarchical preference structure between multiple decision makers, the decision powers w = w, w 2,, w q ) T have to satisfy the following inequality condition. w = w 2 w q w q > 0 7) Then, the corresponding modified MINMAXf, β, p) is reformulated as follows. [MINMAX2f, β, p, w)] x Xβ),x n+ E x n+ f c p T x + α ) ) c 2 x + x n+ /w r, α2 r =,, q, l =,, k r Since each cumulative distribution function T ) is strictly monotone increasing and continuous, MINMAX2f, β, p) can be equivalently transformed as follows. [MINMAX3f, β, p, w)] x Xβ),x n+ E x n+ f c x + α ) T p x n+ /w r ) c 2 x + α 2 ) r =,, q, l =,, k r 8) where T ) is a pseudo-inverse function corresponding to T ). It should be noted here that the constraints 8) of MINMAX3f, β, p, w) for the fixed value of x n+, can be reduced to linear inequalities. Therefore, the optimal value x n+ to MINMAX3f, β, p, w) can be obtained as the maximum value of x n+ so that there exists an admissible set satisfying the linear constraints of MINMAX3f, β, p, w) for some fixed value of x n+. From the property of T ), the inequality conditions 0 < p x n+ /w r <, r =,, q, l =,, k r must be satisfied. As a result, x n+ satisfies the following inequalities. x n+ [ max w r p ) + δ, r=,,q,l=,,k r w r p δ] r=,,q,l=,,k r where δ > 0 is sufficiently small and positive. Therefore, we can obtain the imum value of x n+ to MINMAX4f, β, p, w) by combined use of the bisection method and phase one of linear programg technique. After calculating x n+ on the basis of linear programg, one of the corresponding optimal solutions x can be obtained by solving the following linear fractional programg problem. [LFPf, β, p, w, x n+)] ) x Xβ) f c x + α) c 2 x + α2 034

4 f c x + α ) T p x n+/w r ) c 2 x + α 2 ) r =,, q, l =,, k r, r, l), ) Using the Charnes-Cooper transformation [3], s = c 2 x +, y = s x, s > 0 9) α2 LFPf, β, p, w, x n+) can be transformed to the linear programg problem. [LPf, β, p, w, x n+)] c y + α f ) s y 0,s>δ T p x n+/w r ) c 2 y + α 2 s) + c y + α f ) s 0, r =,, q, l =,, k r, r, l), ) Ay F i β i ) s 0 where δ > 0 is sufficiently small and positive. Strictly speaking, in order to guarantee that the optimal solution y, s ) to LPf, β, p, w, x n+) is unique, Pareto optimality test [0] should be done. III. AN INTERACTIVE ALGORITHM After obtaining a P-Pareto optimal solution x by solving MINMAX3f, β, p, w) on the basis of linear programg, each decision maker DM r ) must either be satisfied with the current values of his/her cumulative distribution functions p x, f ), l =,, k r, or update his/her decision power w r+ and/or his/her reference probabilistic values p r = p r,, p rkr ). In order to help each decision maker update his/her reference probabilistic values, trade-off information is very useful. Such trade-off information is obtainable since it is related to the simplex multipliers of LPf, β, p, w, x n+) [0]. Theorem 2. [0] Let y, s x ) = c 2 x + α 2, c 2 x + α 2 ) be a unique and nondegenerate optimal solution of LPf, β, p, w, x n+), and let the constraints with the reference probabilistic values p r = p r,, p rkr ) be active. Then, the following relation holds. p ) ) x) p x) = π c 2 x + α 2 p x=x c 2 x + α 2 x ) p x ) 0) where π > 0 is the corresponding simplex multipliers for the constraint of LPf, β, p, w, x n+), and p x ) means a differential coefficient of T ) at x, i.e., f c dt x + α ) ) p x c 2 ) = x + α2 f c d x + α ) ) c 2 x + α2 x=x Now, we can construct the interactive algorithm to derive the satisfactory solution of multiple decision makers DM r, r =,, q) in a hierarchical organization from among P-Pareto optimal solution set for HMOLP-RV. Step The first level decision maker DM ) sets the satisfactory constraint levels 0 β i, i =,, m for the constraint a i x b i ω) in his/her subjective manner. Step 2: Under the chance constrained conditions Pra i x b i ω)) β i, i =,, m, calculate the individual imum and maximum of Ec ω)x + α ω)) = c + t c 2 )x + α + t α 2 ), r =,, q, l =,, k r. Considering such values, each decision maker DM r ) sets the initial permissible objective levels f, l =,, k r for the objective function z x, ω) in HMOLP-RV. Step 3: For the given parameters f, β), formulate HMOLPf, β) where the cumulative distribution functions T ), r =,, q, l =,, k r are adopted as the objective function p x, f ). In HMOLPf, β), initial decision powers are set as w r =, r =,, q, and initial reference probabilistic values are set as p =, r =,, q, l =,, k r. Step 4: For the given parameters f, β, p, w), solve MINMAX3f, β, p, w) using the bisection method and the phase one of the two-phase simplex method to obtain the optimal value x n+. If the optimal value x n+ 0, then go to Step 5. Otherwise, after updating the reference probabilistic values as p p x n+/w r, r =,, q, l =,, k r, solve MINMAX3f, β, p, w) in order to guarantee the optimal value x n+ 0, and go to Step 5. Step 5: For the optimal value x n+ 0 of MINMAX3f, β, p, w), solve LPf, β, p, w, x n+) and obtain the corresponding optimal solution y, s ). y, s ) is equivalently transformed to the optimal solution x to MINMAX3f, β, p, w), and compute the trade-off rates between p x, f ), r =,, q, l =,, k r. Step 6: If each decision maker DM r ) is satisfied with the current values of his/her cumulative distribution functions p x, f ), l =,, k r, then go to Step 7. Otherwise, let the s-th level decision maker DM s ) be the uppermost of the decision makers who are not satisfied with the current values of his/her cumulative distribution functions p sl x, f sl ), l =,, k s. Considering the current values and trade-off rates, DM s updates his/her decision power w s+ and/or his/her reference probabilistic values p sl, l =,, k s according to the following two rules, and return to Step 4. Rule : When the decision maker DM s ) updates his/her decision power w s+, w s+ w s must be satisfied in order to guarantee the inequality conditions 7). After updating w s+, if there exists some index t > s + such that w s+ < w t, then the corresponding decision power w t must be replaced as w t w s+. Rule 2: When the decision maker DM s ) updates his/her reference probabilistic values p s, i =,, k s, the reference probabilistic values of the other decision makers DM r, r =,, q, r s) must be set as the current vales of the cumulative distribution functions p x, f ), l =,, k r, i.e. p p x, f ), r =,, q, r s, l =,, k r. Step 7: For the permissible objective levels f, l =,, k r, each decision maker DM r ) is satisfied with the 035

5 current values of his/her cumulative distribution functions p x, f ), l =,, k r. In order to understand the relationships between the cumulative distribution function values p x, f ), r =,, q, l =,, k r and the permissible objective levels f, each graph of p x, f ) with respect to f is shown for each decision maker DM r ). Considering such information, the first level decision maker DM ) decides whether the interactive processes should be terated, or continued go back to Step 2). It should be noted here that, when the decision maker D s ) updates his/her reference probabilistic values p sl, l =,, k s according to Rule 2 at Step 6, any improvement of one cumulative distribution function can be achieved only at the expense of at least one of the other functions for the fixed decision powers w r, r =,, q. Similay, when the decision maker DM s ) updates his/her decision power w s+ according to Rule at Step 6, the cumulative distribution functions p x, f ), r =,, s, l =,, k r will be improved by the less value for w s+ at the expense of the other cumulative distribution functions p x, f ), r = s +,, q, l =,, k r for the fixed reference probabilistic values p, r =,, q, l =,, k s. IV. A NUMERICAL EXAMPLE In order to demonstrate the proposed method and the interactive processes, we consider the following hierarchical twoobjective linear programg problem with random variable coefficients under two hypothetical decision makers DM and DM 2 ). [HMOLP-RV] first level decision maker : DM c + t ω)c 2 )x + α + t ω)α 2 ) c 2 + t 2 ω)c 2 2)x + α 2 + t 2 ω)α 2 2) second level decision maker : DM 2 c 2 + t 2 ω)c 2 2)x + α 2 + t 2 ω)α 2 2) c 22 + t 22 ω)c 2 22)x + α 22 + t 22 ω)α 2 22) a i x b i ω), i =,, 7, x 0 In the above HMOLP-RV, x = x, x 2,, x 0 ) T is the decision column vector, a i, i =,, 7, c, c2, r =, 2, l =, 2 are the constant coefficient row vectors which are shown in Table, and α = 8, α 2 = 5, α2 = 27, α2 2 = 6, α2 = 0, α2 2 = 4, α22 = 27, α22 2 = 6. t ω), r =, 2, l =, 2 and b i ω), i =,, 7 are Gaussian random variables defined as follows: t ω) N4, 2 2 ), t 2 ω) N3, 3 2 ), t 2 ω) N3, 2 2 ), t 22 ω) N3, 3 2 ), b ω) N64, 30 2 ), b 2 ω) N 90, 20 2 ), b 3 ω) N 84, 5 2 ), b 4 ω) N99, 22 2 ), b 5 ω) N 50, 7 2 ), b 6 ω) N54, 35 2 ), b 7 ω) N42, 42 2 ). According to the proposed interactive algorithm, at Step, the first level decision maker DM ) specifies the satisfactory constraint levels for the constraints as β = β,, β 7 ) = 0.85, 0.95, 0.8, 0.9, 0.85, 0.8, 0.9). Table. Constant coefficients x x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 0 c c c c c c c c a a a a a a a At Step 2, the individual imum and maximum of Ec ω)x + α ω)), r =, 2, l =, 2 are calculated under the chance constrained conditions Xβ) = Pra i x b i ω)) β i, i =,, m. Considering such values, the decision makers DM r, r =, 2) sets the initial permissible objective levels as f = 250, f 2 = 450, f 2 = 950, f 22 = 200. At Step 3, formulate HMOLPf, β) where the cumulative distribution functions p x, f ), r =, 2, l =, 2 are defined. For the cumulative distribution functions p x, f ), r =, 2, l =, 2, the initial decision powers and the initial reference probabilistic values are set as w r =, p =, r =, 2, l =, 2. At Step 4, solve MINMAX3f, β, p, w) on the basis of the bisection method and the phase one of the two-phase simplex method, and obtain the optimal value x = > 0. At Step 5, for the optimal value x = > 0 of MINMAX3f, β, p, w), solve LPf, β, p, w, x n+) and obtain the corresponding optimal solution: p x, f ) = , p 2 x, f ) = , p 2 x, f 2 ) = , p 22 x, f 2 ) = At Step 6, for the optimal solution x, the first level decision maker DM ) updates his/her decision power as w 2 = 0.8 in order to improve p x, f ), p 2 x, f ) at the expense of p 2 x, f 2 ), p 22 x, f 2 ), and go back to Step 4. At Step 4, solve MINMAX3f, β, p, w), and obtain the optimal value x = > 0. At Step 5, for the optimal value x = > 0 of MINMAX4f, β, p, w), solve LPf, β, p, w, x n+) and obtain the corresponding optimal solution: p x, f ) = , p 2 x, f ) = , p 2 x, f 2 ) = , p 22 x, f 2 ) = At Step 6, for the optimal solution x, the first level decision maker DM ) is satisfied with the current value of the cumulative distribution functions p l x, f l ), l =, 2, and the second level decision maker DM 2 ) is not satisfied with the current value of the cumulative distribution functions p 2l x, f 2l ), l =, 2. The second level decision maker DM 2 ) updates his/her reference probabilistic values as p 2 = 0.6, p 22 = 0.59 in order to improve p 2 x, f 2 ) at the 036

6 expense of p 22 x, f 22 ). According to Rule 2 at Step 4, DM s reference probabilistic values are fixed as the current values of the cumulative distribution functions, i.e., p = p 2 = Similar interactive processes Step 4,5,6) are continued, and the corresponding optimal solution is obtained as follows: p x, f ) = , p 2 x, f ) = , p 2 x, f 2 ) = , p 22 x, f 2 ) = Both hypothetical decision makers DM and DM 2 ) are satisfied with current values for the permissible objective levels f, f 2, f 2, f 22 ) = 250, 450, 950, 450). At Step 7, in order to understand the relationships between the cumulative distribution function values p x, f ), r =, 2, l =, 2 and the permissible objective levels f, each graph of p x, f ) with respect to f is depicted. Considering such information, the first decision maker DM ) decides to terate the interactive processes. [0] M.Sakawa, K.Kato and H.Katagiri, An Interactive Fuzzy Satisficing Method for Multiobjective Linear Programg Problems with Random Variable Coefficients Through a Probability Maximization Model, Fuzzy Sets and Systems, pp , V46, [] H.Shih, Y.-J. Lai and E.S.Lee, Fuzzy approach for multi-level programg problems, Computers and Operations Research, V23, pp. 73-9, 996. [2] I.M.Stancu-Minasian, Stochastic Programg with Multiple Objective Functions, D. Reidel Publishing Company, 984. [3] U.-P.Wen and S.-T.Hsu, Linear bi-level programg problems - a review, Journal of Operational Research Society, V42, pp , 99. [4] H.Yano, A fuzzy approach for hierarchical multiobjective linear programg problems, Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 200, IMECS 200, Hong Kong, pp , 200. [5] H.-J.Zimmermann, Fuzzy programg and linear programg with several objective functions, Fuzzy Sets and Systems, pp.45-55, V, 978. V. CONCLUSIONS In this paper, hierarchical multiobjective linear programg problems with random variable coefficients HMOLP- RV) have been formulated, where multiple decision makers in a hierarchical organization have their own multiple objective linear functions and common linear constraints with random variable coefficients. In order to deal with HMOLP-RV, a P- Pareto optimal solution concept is introduced in cumulative distribution function space for random variable coefficients. A linear programg based interactive algorithm is proposed to obtain a satisfactory solution from among P-Pareto optimal solution set, which reflects a hierarchical decision making structure, each decision maker s own preference structure for their multiple objective functions, and the probabilistic uncertainty of random variable coefficients. In the proposed algorithm, there are many parameters to be controlled by each decision maker DM r ), such as decision powers w r, reference probabilistic values p, l =,, k r, and permissible objective levels f, l =,, k r. Therefore, it will be necessary to find a good way to reduce the mental stress of the decision makers in interactive processes. Simultaneously, applications of the proposed method will require further investigation. REFERENCES [] G.Anandalingam, A mathematical programg model of decentralized multi-level systems, Journal of Operational Research Society, V39, pp , 988. [2] A.Charnes and W.W.Cooper, Chance Constrained Programg, Management Science, pp.73-79, Vol.6, 959. [3] A.Charnes and W.W.Cooper, Programg with Linear Fractional Functions, Naval Research Logistic Quartey, pp.8-86, Vol.9, 962. [4] G.B.Danzig, Linear Programg under Uncertainty, Management Science, pp , Vol., 955. [5] Y.-J.Lai, Hierarchical optimization : a satisfactory solution, Fuzzy Sets and Systems, V77, pp , 996. [6] Y.-J.Lai and C.L.Hwang, Fuzzy Mathematical Programg, Springer, 992. [7] Y.-J.Lai and C.L.Hwang, Fuzzy Multiple Objective Decision Making, Springer, 994. [8] E.S.Lee and H.Shih, Fuzzy and Multi-level Decision Making, Springer, 200. [9] M.Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press,