Abstract

We address a fully fuzzy bilevel linear programming problem in which all the coefficients and variables of both objective functions and constraints are expressed as fuzzy numbers. This paper is to develop a new method to deal with the fully fuzzy bilevel linear programming problem by applying interval programming method. To this end, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of the problem into a finite number of -level sets. By using -level sets of fuzzy numbers, the fully fuzzy bilevel linear programming problem is transformed into an interval bilevel linear programming problem for each -level set. The main idea to solve the obtained interval bilevel linear programming problem is to convert the problem into two deterministic subproblems which correspond to the lower and upper bounds of the upper level objective function. Based on the th-best algorithm, the two subproblems can be solved sequentially. Based on a series of -level sets, we introduce a linear piecewise trapezoidal fuzzy number to approximate the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem. Finally, a numerical example is provided to demonstrate the feasibility of the proposed approach.

1. Introduction

The bilevel programming problem is a nested optimization problem including two optimization problems in which the feasible region of the upper level problem is determined implicitly by the solution set of the lower level problem. This kind of problem is nonconvex and very hard to solve due to its structure. In the past few decades, the bilevel programming problem has been researched from both the theoretical and computational points of view [1–3] and has been applied so extensively in resource allocation, finance budget, price control, transaction network, and so forth [4].

In conventional bilevel programming models, their parameters are assumed to be well defined and precise. However, in many real-world applications, particularly in some areas linked to human resource planning or actual decision making process, we often need to make a decision on the basis of uncertain data or information. Therefore, inexact optimization methods are desired for supporting environment under uncertainty. The fuzzy set theory provides powerful tools for dealing with imprecise or vague information. In recent years, the fuzzy bilevel programming problem has also become a rapidly progressing research area and has received much attention of some researchers. Sakawa et al. [5] first studied the fuzzy bilevel programming problem and proposed a fuzzy programming method to deal with it on the basis of the definition of optimal solution for bilevel programming proposed by Bard [6]. Zhang and Lu [7] developed an extended Kuhn-Tucher approach to deal with the fuzzy bilevel linear programming problem based on the new definition of optimal solution. The fuzzy Kuhn-Tucker approach, the fuzzy th-best approach, and the fuzzy branch-and-bound approach were designed to handle the fuzzy bilevel programming problem by applying fuzzy set techniques [8]. Dempe and Starostina [9] formulated the fuzzy bilevel programming problem and described one possible approach for formulating a crisp optimization problem being attached to it. In all of the above-mentioned works, those cases of the fuzzy bilevel programming problem have been studied in which all or some coefficients involved in the objective functions and the constraints are assumed to be fuzzy but all variables of the problem are deterministic.

As a matter of fact, the fully fuzzy linear programming in which all the coefficients as well as the variables are represented by fuzzy numbers is an attractive topic for researchers. Buckley and Feuring [10] considered this kind of problem and employed an evolutionary algorithm to deal with it. A new method was proposed to solve a fully fuzzy linear programming problem by applying the concept of comparison of fuzzy numbers [11]. Hosseinzadeh Lotfi et al. [12] transformed the fully fuzzy linear programming problem into two corresponding linear programming problems based on the concept of the symmetric triangular fuzzy number and developed a lexicography method to solve such a problem. Kumar et al. [13] proposed a new method to solve the fuzzy optimal solutions of the fully fuzzy linear programming problems with equality constraints. Subsequently, Najafi and Edalatpanah [14] pointed out that Kumar et al. model [13] was not correct and provided a revised version. Recently, Fan et al. [15] investigated the feasibility of fuzzy solutions of the generalized fuzzy linear programming problem and developed a stepwise interactive algorithm to solve the problem. In all these methods, the fully fuzzy linear programming problem is firstly converted into a crisp linear programming problem and then the obtained crisp linear programming problem is solved to find the fuzzy optimal solutions of the problem. In addition, it should be noted that all these works are considered in the case of one single level fully fuzzy linear programming.

In this paper, we address a fully fuzzy bilevel linear programming problem in which all the coefficients and variables of both objective functions and the constraints are expressed as fuzzy numbers. To our knowledge, until now there are few studies on this type of problem. More recently, Safaei and Saraj [16] discussed the fully fuzzy bilevel linear programming problem by decomposing this problem into three crisp linear programming problems and then dealing with these three deterministic problems to obtain the fuzzy optimal solutions of the problem. However, this work on fuzzy solutions in a fully fuzzy bilevel linear problem is only suitable for triangular fuzzy numbers, which significantly restricts the application scope of the models.

The interval mathematical programming method is an effective approach in tackling uncertainties. This approach does not require distributional information for input parameters and enables a relatively low computational requirement without complicated intermediate models. Over the past few decades, the interval mathematical programming has been heavily studied by many scholars and successfully applied to a variety of practical problems [17–22].

The purpose of this paper is to develop a new method to deal with the fully fuzzy bilevel linear programming problem by applying interval programming method. According to the decomposition principle, the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem can be expressed as the families of its -level sets. Based on this fact, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of the problem into a finite number of -level sets. The fully fuzzy bilevel linear programming problem is transformed into an interval bilevel linear programming problem for any -level set. The obtained interval bilevel linear programming problem is converted into two deterministic subproblems which correspond to the lower and upper bounds of its upper level objective function value. Then the two subproblems can be solved sequentially based on the th-best algorithm. Finally, the linear piecewise trapezoidal fuzzy number is introduced to approximate the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem.

This paper is organized as follows. In Section 2 some basic definitions and results related to interval numbers and fuzzy numbers are reviewed. In Section 3 a new method is proposed to deal with the fully fuzzy bilevel programming problem by applying interval programming method. A numerical example is given to illustrate the proposed method and the obtained results are discussed in Section 4. Section 5 contains the concluding remarks.

2. Preliminaries

In this section, some basic notations and preliminary results of interval numbers and fuzzy numbers are presented.

2.1. Interval Numbers

Let denote the set of all real numbers.

An ordered pair in a bracket defines an interval as where and are the lower and upper bounds of , respectively. When , becomes a deterministic number.

For , , and , we have(i);(ii), , , ;(iii).

Thus

Definition 1 (see [23]). For , the following relationships hold:(i), if and only if and ;(ii), if and only if and .

Definition 2 (see [23]). For , can be defined as follows:

Definition 3 (see [23]). For , its absolute value can be given as follows: Hence,

2.2. Fuzzy Numbers

Fuzzy numbers are one way to describe the vagueness and lack of precision of data. We give some basic concepts of fuzzy numbers as follows.

Definition 4 (see [24]). A fuzzy number is defined as a fuzzy set on , whose membership function satisfies the following conditions: (i)there exists such that ;(ii) is upper semicontinuous;(iii) is convex;(iv)the support of , , and its closure is compact.

The -level set of a fuzzy number is defined by an ordinary set for . It is obvious that the -level set of a fuzzy number is a closed interval, denoted as , for any , .

Definition 5 (see [25]). A fuzzy number is nonnegative if and only if its membership function satisfies for .

3. Fully Fuzzy Bilevel Linear Programming Problem

Consider the following fully fuzzy bilevel linear programming problem in which all the coefficients as well as the variables are represented by fuzzy numbers:where is an -dimensional fuzzy decision vector of the upper level and is an -dimensional fuzzy decision vector of the lower level, respectively; , , are -dimensional fuzzy vectors and is an -dimensional fuzzy vector; is an -dimensional fuzzy vector and are fuzzy matrices in which , for all , , are fuzzy numbers.

Because of the existence of fuzzy coefficients and fuzzy decision variables in the objective functions and the constraints, problem (6) is not well defined and has no clear mathematical meaning. Thus deterministic bilevel optimization techniques cannot be directly applied to solve this kind of problem. Even for one single level, it has been pointed out by Buckley and Feuring [10] that searching for the optimal solutions of a fully fuzzy linear programming problem is a very difficult task. In this section, a potential method is developed to handle the fully fuzzy bilevel linear programming problem.

3.1. Modeling Formulation

Observing that all the coefficients and decision variables of the upper level objective function of problem (6) are fuzzy numbers, the upper level objective function is also a fuzzy number. It is well known that fuzzy numbers can be defined by the families of their -level sets according to the decomposition principle [26]. Then the optimal value of the upper level objective function of problem (6) can be expressed as the families of its -level sets.

Based on this fact, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of problem (6) into a finite number of -level sets. For any , the -level sets of , , , , , , and , , , can be denoted as , , , , and .

Considering that the -level sets of fuzzy numbers are actually closed intervals, the operations on fuzzy numbers are indeed performed by the interval arithmetic operations for -level sets. Then the fully fuzzy bilevel linear programming problem (6) can be transformed into the following interval bilevel linear programming problem for any -level set:

Remark 6. For problem (6), the upper level objective function can be rewritten as . Then we have for any -level set. Obviously, the lower level objective function and the constraints of problem (7) can be obtained by similar algebraic operation.

Under different -level sets, a series of interval bilevel linear programming problems are generated. Here we are interested in computing the lower and upper bounds of the upper level objective functions of these interval bilevel linear programming problems. Based on these results, the optimal value of the upper level objective function of problem (6) can be decomposed as

Hence, we will discuss how to determine the lower and upper bounds of the upper level objective function of the interval bilevel programming problem (7) in the following section.

Since all the coefficients and the decision variables of problem (7) are interval numbers, the optimal upper level objective function values and the feasible regions of the problem are directly affected by different specific values of the coefficients and the decision variables chosen from their ranges. To deal with problem (7), we analyze model characteristics and the interrelationships among the coefficients and the decision variables in both objective functions and the constraints.

Clearly, the lower level problem of problem (7) is a conventional interval linear programming. Huang et al. [17] proposed the two-step method to solve this kind of problem. The basic idea of this method is to first transform the interval linear programming problem into two deterministic subproblems which correspond to the lower and upper bounds of the objective function value and then solve the two subproblems sequentially to obtain the solutions of the problem. However, this method may result in constraint violation in its solution space [22]. To avoid this, Fan and Huang [22] developed a robust two-step method for solving the interval linear programming problem. In this way, we discuss the lower and upper bounds of the objective function of the lower level problem and then give the optimal bounds of the upper level objective function of the interval bilevel linear programming problem (7).

For simplicity, problem (7) can be rewritten as where , , , , , and .

We call the following problem a characteristic version of problem (7):where , , , , , and .

For any , , , and , the following sets are set up by Then we have .

Here we assume that and are nonempty polyhedrons.

Theorem 7. For any , if is an optimal solution and is corresponding objective function value of problem (10), then one has , where and are the lower and upper bounds of the upper level objective function of problem (7).

Proof. For fixed , denote the largest and smallest possible feasible regions of the lower level problem by It is true that .
Denote the two solution sets of the lower level problem by where .
The inducible regions corresponding to the smallest and largest possible feasible regions are
Therefore, the lower and upper bounds of the upper level objective function of problem (7) are such that , and the proof is completed.
Theorem 7 gives the optimal bounds of the interval bilevel programming problem (7).

Definition 8. A general interval linear programming model can be expressed as follows: where , , , , and denotes a set of interval numbers. According to Theorems and in [22], the model has the optimal objective function value and the optimal solutions as follows: , , , for all .

Next we discuss how to obtain solutions for the upper and lower bounds of the upper level objective function values of problem (7).

For given , the lower level problem of problem (7) can be written as

For the coefficient of problem (6), let the former elements be positive and the latter elements negative. According to Lemma 2 [15], for any , the lower and upper bounds of the objective function of problem (17) can be expressed as

For simplicity, denote the term in the constraints of problem (17) by .

According to the robust two-step method [22], a submodel corresponding to is firstly formulated and solved, and then the second submodel corresponding to can be formulated based on the solution of the first one. In detail, the submodel corresponding to , which provides the first step of the solution process, can be formulated as

From submodel (19), solutions of , , and , , can be obtained. Denote the solution sets of model (19) by .

Based on the solutions of submodel (19), the submodel corresponding to , which provides the second step of the solution process, can be formulated as follows:where and , where and .