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Solving the Fuzzy BiLevel Linear Programming with Multiple Followers through Structured Element Method
The optimal solution of fuzzy bilevel linear programming with multiple 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.
Bilevel programming introduced by Stackelberg in 1952  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 [2–7]. 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 [8–11]. 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, 14–17] 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 λ-level sets. At the same time, some researches applied fuzzy set technique to deal with bilevel programming problems. Shih and Stanely Lee  applied fuzzy set theory to overcome the computational difficulties in solving bilevel problems. Sinha  started from the fuzzy mathematical programming approach to obtain the solution of multilevel linear programming problems. Recently, Zhang et al. [13, 20–22] 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 , 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.
In this section, some necessary backgrounds and notions of fuzzy structured element theory are presented.
Definition 1 (see ). Let be a fuzzy set on and the membership function of . Then, is called a fuzzy structured element, if (i) ; (ii) is a function of monotonous increasing and right continuous on and monotone decreasing and left continuous on (0,1]; (iii) .
Definition 2 (see ). is called a canonical fuzzy structured element, if (i) ; (ii) is continuous and strictly monotone increasing on and strictly monotone decreasing and continuous on .
Definition 3 (see ). is called a symmetrical fuzzy structured element, if .
Lemma 4 (see ). Let be a fuzzy structured element and is its membership function; the function is continuous and monotone on ; then is a fuzzy number, and the membership function of is , where is rotational symmetry function for variables and , if is a strictly monotone function, then is the inverse function of .
Lemma 5 (see ). For a given canonical fuzzy structured element and any finite fuzzy number , there always exists a monotone bounded function on , having the form .
Lemma 6 (see ). Let the triangular fuzzy number , 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 .
Remark 7. The class of all bounded fuzzy numbers is denoted by , and let be a canonical fuzzy structured element.
Definition 8 (see ). Suppose . Its structured element representation is , , respectively, where is given a canonical fuzzy structured element and its membership function is , and are the same sequence monotonic functions on , respectively. By the following formula to determine the binary relation “” is a total order on and then called fuzzy numbers structured element weighted order.
Lemma 9 (see ). Let be a symmetrical fuzzy structured element, and are the same sequence monotonic functions on , the fuzzy number , and , then
when , , and ; when , , and .
The proof of above Lemmas can be found in reference [23–25].
3. 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: where is the leader’s objective function and is the th follower’s objective function; , ; , , , , , , .
Theorem 10. Suppose that , , , , , , , , , and , if is a canonical fuzzy structured element, , , , , , , , , and are monotonous increasing functions, then the model (5) is equivalent to the following model:
Proof. By Definition 8, we know that comparing the size of fuzzy number by in model (5), then
because and are monotonous increasing functions; then
In the same way, we get
By Lemmas 6 and 9, we obtain that
The proof is completed.
4. Algorithm and Numerical Example
By using Theorem 10, we give all steps of the approach for the proposed MFFBLP model.
Step 1. If the fuzzy number is triangular fuzzy number, according to Lemma 6 and expression, we have and then get , , , , , , and .
Step 2. Computation formula is as follows: , , , , , , and , and plug in model (5).
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 .
4.2. Numerical Example
Consider the following MFFBLP problem with , and . Consider where the triangular fuzzy numbers , , , , , , , , , , , , , , and .
Step 1. By Lemma 6, we have
Step 2. Compute , , , , , , , , , , , , , , , , and .
Step 3. By Theorem 10, the original problem is equivalent to the following bilevel linear programming with multiple followers’ problem:
Step 4. We use the Kuhn-Tucker approach to get an optimal solution; the solution of the problem is as follows:
The example illustrates how to solve a MFFBLP problem and obtains an optimal solution by using the proposed approach.
A real world bilevel decision problem may be modeled to have fuzzy coefficients. In this paper, we investigated the MFFBLP model and solved this complex problem by using the fuzzy structured element method. Further study includes the development of models and methods for fuzzy multilevel programming. We will also explore effective applications of the proposed techniques.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work was supported by the Scientific Research Fund of Hunan Provincial Education Department of China (Grant nos. 14C0350 and 13A004), the NSF of Hunan Province of China (Grant nos. 13JJ3109 and 14JJ3127), and the Humanities and Social Science Foundation of the Ministry of Education of China (Grant no. 12YJA630114).
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