Solving the Fuzzy BiLevel Linear Programming with Multiple Followers through Structured Element Method

The optimal solution of fuzzy bilevel linear programming withmultiple followers (MFFBLP)model is shown to be equivalent to the optimal solution of the bilevel linear programming with multiple followers by using fuzzy structured element theory. The optimal solution to this model is found out by adopting the Kuhn-Tucker approach. Finally, an illustrative numerical example for this model is also provided to demonstrate the feasibility and efficiency of the proposed method.


Introduction
Bilevel programming introduced by Stackelberg in 1952 [1] has been developed to solve the decentralized planning problem in which decision makers are often arranged within a hierarchical administrative structure.A bilevel programming problem occurs when two decision makers are located at different hierarchical levels.In general, a decision maker at the upper level is termed as the leader and the lower level is termed as the follower [2,3].In the context of bilevel programming, the leader first specifies a strategy; the follower then specifies a strategy so as to optimize the objective with full knowledge of the action of the leader.
Many researches on bilevel programming so far have centered on the linear version of the problem [2][3][4][5][6][7].Also two fundamental issues in theory and practice of both bilevel programming problems are mostly concerned: one is how to model a real world bilevel programming and the other is how to find properties and an optimal solution to the bilevel programming problem.There are many such hierarchical optimization problems in the fields of industry, agriculture, finance transportation, and so on [8][9][10][11].But in many practical hierarchical decision making systems, the coefficients of objective functions and constraints sometimes cannot be described by precise values.Hence, it is necessary for us to formulate the decentralized decision making problem with uncertainty as fuzzy models.
At present, the linear bilevel programming in which the coefficients are characterized by fuzzy numbers is called fuzzy linear bilevel programming [12,13].Sakawa et al. [4,5,[14][15][16][17] formulated cooperative fuzzy bilevel programming problems and proposed an interactive fuzzy programming approach to solve the problems.From this approach, the concept of a bilevel programming was introduced based on fuzzy number -level sets.At the same time, some researches applied fuzzy set technique to deal with bilevel programming problems.Shih and Stanely Lee [18] applied fuzzy set theory to overcome the computational difficulties in solving bilevel problems.Sinha [19] started from the fuzzy mathematical programming approach to obtain the solution of multilevel linear programming problems.Recently, Zhang et al. [13,[20][21][22] studied fuzzy bilevel programming problem, which focuses on the situation where the leader or the follower has multiple objectives with fuzzy parameters and all followers share their decision variables, and provided related algorithms based on the y function in fuzzy set theory.Moreover, they have first solved the fuzzy linear bilevel programming problems with a specialized form of membership functions, triangular form, in the fuzzy parameters [6,12].Nevertheless so far the fuzzy linear bilevel programming problem remains the focus of fuzzy multilevel programming problem.
This paper discusses the fuzzy bilevel linear programming with multiple followers (MFFBLP) model.Based on the homeomorphism properties between the bounded real fuzzy number and the monotone functions on [−1, 1], the comparison of a fuzzy number is changed into a new comparison of monotone function by the definition of fuzzy numbers structured element weighted order.Then the optimal solutions of new derived model is proved equivalent to the optimal solution of the MFFBLP model.The feasibility of the proposed approach is proved by giving a numerical example.
The following of this paper is arranged as follows.In Section 2, some concepts and properties of the fuzzy numbers structured element weighted order are introduced.In Section 3, we study the optimal solution of the MFFBLP model.In Section 4, one numerical example is shown for illustrating the proposed models and approach.Finally, we give some conclusions in Section 5.

Preliminaries
In this section, some necessary backgrounds and notions of fuzzy structured element theory are presented.
Lemma 4 (see [23]).Let  be a fuzzy structured element and () is its membership function; the function () is continuous and monotone on [−1, 1]; then () is a fuzzy number, and the membership function of () is ( −1 ()), where  −1 () is rotational symmetry function for variables  and , if  is a strictly monotone function, then  −1 () is the inverse function of ().
Lemma 6 (see [25]).Let the triangular fuzzy number Ã = (, , ),  is a fuzzy structured element, and its membership function is Then arbitrary bounded triangular fuzzy number can be generated by  and the monotone bounded function is Therefore, we easily get the conclusion Ã = ().
Remark 7. The class of all bounded fuzzy numbers is denoted by Ñ (), and let  be a canonical fuzzy structured element.

The first follower
The second follower The ith follower The sth follower

Fuzzy Bilevel Linear Programming with Multiple Followers
Figure 1 shows the fuzzy bilevel linear programming with multiple followers model (MFFBLP), and this mathematical model is defined as follows: Proof.By Definition 8, we know that comparing the size of fuzzy number Z1 5), then because  1  () and  1  () are monotonous increasing functions; then In the same way, we get Mathematical Problems in Engineering By Lemmas 6 and 9, we obtain that The proof is completed.

Algorithm and Numerical Example
4.1.Algorithm.By using Theorem 10, we give all steps of the approach for the proposed MFFBLP model.
Step Step 2. Computation formula is as follows: Step 3.According to Theorem 10, the MFFBLP model is transformed into the classical bilevel linear programming with multiple followers model whose optimal solution of model ( 6) can be derived from Kuhn-Tucker's approach [7].
Step 4. After the optimal solution of model ( 6) is plugged into model (5), we get the optimal solution of the MFFBLP model.