A Fuzzy Interactive Approach for Solving Bilevel Programming 1

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1 A Fuzzy Interactive Approach for Solving Bilevel Programming 1 He Ju Lin 2, Wan Zhong Ping 3 School of Mathematics and Statistics Wuhan University, Wuhan Wang Guang Min and Lv Yi Bing Institute of System Engineering, Wuahn University, Wuhan Abstract In this paper, we focus bilevel programming problem with common decision variable in the upper level and lower level programming. After introducing fuzzy goals for objective functions, consulting the ratios of satisfaction between the leader at the upper level and the follower at the lower level, interactive fuzzy decision making method to derive a satisfactory solution for decision makers is presented. A numerical example illustrates the proposed method. Key Words: bilevel programming, interactive fuzzy method, decision making, satisfactory solution. Subject Classification (GB/O22.2,O159): 90C30,90C70 0 Introduction Decision-Making(DM) in most real life problems, such as engineering design and management,economic policy,traffic problem etc.,can be made in hierarchical order system.the bilevel programming,which is a nested optimization problem with two levels in a hierarchy, namely the upper and lower level decisionmakers, is a practical and useful tool for solving DM problems. Therefore bilevel programming has been developed and researched by many authors such as [1, 2, 3, 4, 5, 6] etc.the bilevel programming problems are generally difficult to solve due to the complexity of the problem even for the simplest linear problems. In a bilevel programming, upper decision maker(leader) optimizes his objective function independently and is affected by the reaction of the lower level maker(follower) who makes his decision after the former.their objective functions are generally conflict each other. However, most problems encountered in practice fall into the situation in which they depend on the degree of interactive or cooperation between the two levels,but the information between them is incomplete and vague. Simultaneously,we notice that although the lower level maker cannot control the decision of the upper level maker,final decision of the lower level does eventually influence the upper level and overall results. Fuzzy programming method has been applied to bilevel programming problems [7, 8, 9], and it has given us a strong impression.lai [7] introduced a new concept the satisfactory strategy for multilevel programming problems on the basis of the fuzzy set theory and proposed a new algorithm to obtain the satisfactory solution of the leaders and followers. Sakawa et al [9, 10, 11, 12] proposed a interactive 1 Foundation item: Research partly supported by the National Key Basic Research Special Fund (2003CB415200) and the National Science Foundation( ). 2 Biography: He Julin, Male,Associate professor,research direction:optimization algorithms and applications. 3 Corresponding author. zpwan@public.wh.hb.cn 1

2 fuzzy programming for various bilevel programming problems. To solve the bilevel programming problem with common decision variable in the upper level and lower level programming, we present also an interactive fuzzy programming decision making strategy, which is differ from fuzzy programming method mentioned the above in this paper. It is mainly emphasize the fact that we consider not only the leaders is the dominant action but also treat the satisfactory degree of the followers impartially at decision making process. This paper is organized as follows: Bilevel programming problem and formulation are introduced in Section 2.In Section 3, an interactive fuzzy decision making algorithm is given. A numerical example is reported in Section 4 and the conclusion is given in Section 5. 1 Problem Formulation In this paper, we deal with the following the bilevel programming problems with common decision vector in the upper level and lower level programming: (BP) min x, y F 1(x, y, z) (1) s.t. g(x, y) 0 (2) where y, z solves min y, z F 2(x, y, z) (3) s.t. h(x, y, z) 0. (4) where F 1 (x, y, z) is the objective function of the leader and F 2 (x, y, z) is the objective function of the follower according to the relate space, g : R n1 n2 R p, h : R n2 n3 R q. All decision variables are column vectors. For notational convenience,let (x, y, z) denote (x T, y T, z T ) T, the others is similar. The bilevel programming problem(1)-(4) is presented in [5], by considering a bilevel programming model of optimal bidding strategies, between the power Seller and Buyer for contract arrangements of the long-term contracts and the spot markets transactions under uncertain electricity spot market, in which Seller(or Buyer) at the upper level and Buyer(or Seller) at the lower level make decisions. It is natural that the leader and follower have fuzzy goals for their objective functions when they take fuzziness of human judgments into consideration [7, 8, 11]. For each of the objective functions of the BP problem, assume that the leader and follower have fuzzy goals using fuzzy theory [13]. Through the interaction with decision makers, these fuzzy goals can be quantified by eliciting the corresponding membership functions, which are denoted by µ 1 (F 1 (x, y, z)) for the leader and µ 2 (F 2 (x, y, z)) for the follower respectively. Let S = {(x, y, z) : g(x, y) 0, h(x, y, z) 0}, which denotes the constrained region of the BP problem. Definition 1. (x, y, z ) is said to be a complete optimal solution, if and only if there exists (x, y, z ) S such that F 1 (x, y, z ) F 1 (x, y, z) and F 2 (x, y, z ) F 2 (x, y, z) for all (x, y, z) S. However, in general, such a complete optimal solution that simultaneously minimizes both the leader s and follower s objective functions does not always exist. Instead of a complete optimal solution, 2

3 a new solution concept, called Pareto optimality, is introduced in the BP. Definition 2. (x, y, z ) is said to be a Pareto optimal solution, if and only if there does not exist (x, y, z) S such that F 1 (x, y, z) F 1 (x, y, z ), F 2 (x, y, z) F 2 (x, y, z ) and F 1 (x, y, z) F 1 (x, y, z ) or F 2 (x, y, z) F 2 (x, y, z ). According to the definition of [14], we can assume that S is nonempty and compact and that a Pareto optimal solution for the BP problem exists throughout this paper. The following linear function is an example of a membership function µ j (F j ) j = 1, 2 which characterizes the fuzzy goal of the leader and follower respectively and to facilitate computation for obtaining solutions, we assume the membership functions to be linear(for j = 1, 2): 0 if F j > Fj U F µ j (F j ) = U j Fj Fj U F if F L j L j < F j Fj U (5) 1 if F j Fj L, where F U j and F L j denote the upper bound and lower bound of the objective function F j (x, y, z) such that the degrees of membership function are 0 and 1, respectively. Without loss of generally we can assume that F U j and F L j are optimal solution of the following programming problem,respectively: F U j = max x,y,z F j(x, y, z) s.t. g(x, y) 0 (6) h(x, y, z) 0, and F L j = min x,y,z F j(x, y, z) s.t. g(x, y) 0 (7) h(x, y, z) 0. 2 Interactive Fuzzy Decision Making Method Here, we instead of the Pareto optimal solution concept in the Definition 2.1, we introduce the of M-Pareto optimal solution, where M refers to membership. Definition 3. (x, y, z ) f(α 1 ) = df {(x, y, z) S : α 1 µ 1 (F 1 (x, y, z))} is said to be a M- Pareto optimal solution to BP problem, if and only if there does not exist (x, y, z) f(α 1 ) such that µ 2 (F 2 (x, y, z)) µ 2 (F 2 (x, y, z )). After the leader specifies the minimal acceptable level α 1 for his membership function, a candidate the satisfying solution of the follower can be generated by solving the following programming: (P) min t s.t. α 1 µ 1 (F 1 (x, y, z)) α 2 µ 2 (F 2 (x, y, z)) t (8) (x, y, z) S, t 1. where α 2 is the minimal acceptable reference level specified by the follower for his membership function. 3

4 The problem (P) can be written as an equivalent programming problem by the definition of the membership function (5): min t s.t. F 1 (x, y, z) F1 U α 1 (F1 U F1 L ) F 2 (x, y, z) F2 U + t(f2 U F2 L ) α 2 (F2 U F2 L ) (9) (x, y, z) S, t 1. Remark: A solution to the problem (P) is always in existence by assumption that S is nonempty and compact set for some α i (i = 1, 2). In addition, if t < 0, then the acceptable level of the follower will be reduced because of exceeding his the maximal acceptable level, otherwise, we should increase the minimal acceptable level α 2. Simultaneously,we need to take account of the overall satisfactory balance between both levels, the leader needs to compromise with the follower on leader s own minimal satisfactory level. It is no difficult to prove the following theorem: Theorem. Suppose that (x, y, z, t ) is an optimal solution of the problem (P), and t = 0, then (x, y, z ) is an optimal M-Pareto solution to the problem (BP). Since the decision makers subjectively specify membership function and the minimal acceptable level,the balance among the both degree of satisfaction between the leader and the follower should be considered. Therefore we must consider that the overall satisfactory balance between both decision makers. We now give the following definition of the autonomy level (the balance satisfactory degree) [7] : δ = µ 2(F 2 (x, y, z)) µ 1 (F 1 (x, y, z)). (10) which means the balance among the both degree of satisfaction between the leader and the follower,the leader decides whether it is necessary to change the minimal acceptable level. When δ 1 > ε (where ε is given tolerance in advance), we then must update the minimal acceptable level α i (i = 1, 2). If δ > 1, then increase α 1 to reinforce the leader s decision making power, and it is possible that reduce α 2 ; otherwise increase α 2 to reinforce the follower s decision making power. On the other hand, the follower must either be satisfied with the current M-Pareto optimal solution, or act on this solution by updating his the minimal acceptable reference value based on the solution of programming problem(p). Simultaneously, the follower expects that a maximal acceptable level is obtained for his membership function. Consequently, we will continuously update the minimal acceptable level of the leader and follower according as δ and the optimal solution to problem (P) in the solving process. We are now ready to construct an interactive fuzzy decision making method for deriving the satisfactory solution for the leader and the follower from the problem (BP), which is summarized as follows: A Fuzzy Interactive Approach for Bilevel Programming 4

5 Step 1. The leader elicits the membership function µ 1 (F 1 (x, y, z) of the fuzzy goal of the leader and the follower elicits the membership functions µ 2 (F 2 (x, y, z) of the follower. Then give tolerance ɛ. Step 2. After the leader sets the initial minimal acceptable level α 1 for his membership function µ 1 (F 1 (x, y, z)), the follower sets the initial reference membership value α 2. Step 3. Solve the problem (P). If there does not exist solution to the problem (P), the leader adjusts the minimal acceptable level α 1 by reducing α 1, until a solution (ˆx, ŷ, ẑ, ˆt) is obtained for the problem (P). Step 4. If ˆt = 0, then goto Step 6. Otherwise, goto the next step. Step 5. If the follower is satisfied with the current solution (ˆx, ŷ, ẑ), then goto Step 6. Otherwise, if t < 0, increase α 2. If t > 0, then ask the leader to update the current minimal acceptable level α 1 and return to Step 3. Step 6. Calculate the balance degree δ. If δ 1 ɛ, then stop. We already obtain a M-Pareto optimal solution (x, y, z ) = (ˆx, ŷ, ẑ), which is the satisfying solution for the leader and the follower. Otherwise, update the minimal acceptable level of the leader or the follower s reference membership value α 2. If δ > 1, then increase α 1 ; otherwise, increase α 2, return to Step 3. 3 Numerical Example To demonstrate the feasibility and efficiency of the proposed interactive fuzzy decision making method, consider the following bilevel linear programming problem involving with 6 nonnegative variables and 9 linear constraints: min F 1 (x, y, z) = 18x 1 10x 2 11y y 2 23z 1 40z 2 min F 2 (x, y, z) = 35x 1 + 9x 2 20y y 2 10z 1 7z 2 s.t. 47x 1 14x 2 y 1 + 4y 2 + z 1 49z x 1 + 2x y 1 35y z z x 1 18x y y z 1 11z x 1 19x 2 y 1 2y 2 49z 1 11z x 1 8x 2 + 2y y z 1 25z x 1 + 3x 2 28y y 2 36z 1 3z x x 2 44y y z 1 2z x 1 13x y y 2 2z 1 + 7z x x y 1 29y 2 49z 1 38z 2 38 x i 0, y i 0, z i 0, i = 1, 2. 5

6 The algorithm takes the following iterative procedure: Iteration 1. Step 1. The membership functions (5) of the fuzzy goal are assessed by using values (6) and (7). The optimal values and the corresponding optimal solutions are shown in table 1. And set the tolerance ɛ = 0.1. Table 1. The initial computation results F1 L (x, y, z) F1 U (x, y, z) F2 L (x, y, z) F2 U (x, y, z) Step 2. The leader sets the initial minimal acceptable level α 1 = 1.0, and the follower sets the initial reference membership value α 2 = 0.4. Step 3. The problem (P) for the numerical example (11) can be formulated as (P) min t (11) s.t F α F 2 α t (x, y, z) S, t 1. where S denotes the feasible region of the problem. There does not exists feasible solution for the problem (11) with α 1 = 1.0, α 2 = 0.4. So the leader adjusts the minimal acceptable level α 1 by reducing α 1 = 1.0 to α 1 = 0.9. The problem (11) is solved with α 1 = 0.9, α 2 = 0.4 and the result is listed in table 2. ˆt Table 2. Results of the iteration (ˆx, ŷ, ẑ) F µ(f 1 ) F µ(f 2 ) Step 4. ˆt = > 0. Step 5. µ(f 2 ) = < 0.5, the follower does not satisfy the solution. so the leader reduces α 1. And the follower satisfy the solution until α 1 = 0.7. Goto step 3. Iteration 2. Step 3. The problem (11) is solved with α 1 = 0.7, α 2 = 0.4 and the result is listed in table 3. 6

7 ˆt Table 3. Results of the iteration 2 (ˆx, ŷ, ẑ) F µ(f 1 ) F µ(f 2 ) Step 4. ˆt = 0, goto step 6. Step 6. Calculate δ = 0.4/0.7 = And δ 1 = > ɛ = 0.1. Then update the leader s minimal acceptable level α 1 or the follower s reference membership value α 2. Because δ < 1, increase α 2 = 0.4 to α 2 = 0.5, go to step 3. Iteration 3. Step 3. The problem (11) is solved with α 1 = 0.7, α 2 = 0.5 and the result is listed in table 4. ˆt Table 4. Results of the iteration (ˆx, ŷ, ẑ) F µ(f 1 ) F µ(f 2 ) Step 4. ˆt = > 0, then go to step 5. Step 5. µ(f 2 ) = < 0.5, the follower does not satisfy the solution. So the leader adjusts the minimal acceptable level α 1 by reducing α 1 = 0.7 to α 1 = 0.6, go to step 3. Iteration 4. Step 3. The problem (11) is solved with α 1 = 0.6, α 2 = 0.5 and the result is listed in table 5. ˆt Table 5. Results of the iteration 4 (ˆx, ŷ, ẑ) F µ(f 1 ) F µ(f 2 ) Step 4. ˆt = 0; goto step 6. Step 6. δ = 0.5/0.6 = δ 1 < 1 = > 0.1, and δ < 1, so increase α 2 = 0.5 to α 2 = 0.6, goto step 3. Iteration 5. Step 3. he problem (11) is solved with α 1 = 0.6, α 2 = 0.6 and the result is listed in table 6. ˆt Table 6. Results of the iteration 5 (ˆx, ŷ, ẑ) E F µ(f 1 ) F µ(f 2 ) Step 4. ˆt = 0, then goto step 6. Step 6. δ = 0.6/0.6 = 1 and δ 1 = 0 < 0.1, then stop. Thus, M-Pareto optimal solution (x, y, z ) = ( , , , , , ). And all satisfied degree are µ(f 1 ) = µ(f 2 ) =

8 From the computation results, it can be seen that the proposed algorithm is feasible and efficient. 4 Conclusions In this paper, we have constructed the interactive fuzzy decision making method for bilevel programming problem with common decision variable in the upper level and lower level problem. In our interactive method, by updating the minimal acceptable level of the leader at the upper level with considerations given to overall satisfactory balance between two levels, an overall satisfactory solution has been efficiently derived through our method. An illustrative numerical example has been provided to demonstrate the feasibility and efficiency of the proposed method. References [1] J.F.Bard. Practical Bilevel Optimization: Algorithms and Applications, Kluwer Academic Publishers,Dordrecht,1998. [2] S.Dempe. Foundations of bilevel programming, Kluwer Academic Publishers,2002. [3] G.Liu, J.Han, J.Zhang. Exact penalty functions for convex bilevel programming problems, Journal of Optimization Theory and Applications, (3): [4] A.Migadalas, P.M.Paradalos, P.Värbraud. Multilevel optimization algorithms and applications,kluwer Academic Publishers,1998. [5] Z.Wan et al. Bilevel Programming Model of Optimal Bidding Strategies under the Uncertain Electricity Markets(in Chinese), Automation of Electric Power System 2004,28(19): [6] X.Wang and C.Feng. Optimal Theory in the Bilevel Hierarchy(in Chinese), Bejing, Scinece Press,1995. [7] Y.-J. Lai. Hierarchical optimization: A satisfactory solution, Fuzzy Sets and Systems, 1996, 77: [8] E.S. Lee. Fuzzy multiple level programming, Applied Mathematics and Computation, 2001, 120: [9] M.Sakawa and I.Nishizaki. Interactive fuzzy programming for decentralized two-level linear programming problems, Fuzzy Sets and Systems, 2002, 125: [10] H.-S.Shih and E.S.Lee. Compensatory fuzzy multiple level decision making,fuzzy Sets and Systems, 2000,114: [11],M.Sakawa, I.Nishizaki and Y.Uemura. Interactive fuzzy programming for multilevel linear programming problems,computers Math.Applic.,1998, 31(2):

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