An iterative goal programming approach for solving fuzzy multiobjective integer linear programming problems
Introduction
In literature there are several approaches available to deal with multiobjective integer linear programming problems. Goal programming is one powerful approach that has been proposed for the modeling, analysis and solution of multiobjective optimization problems. Previous results on the use of a wide variety of goal programming models to solve these problems are reported in [2], [6], [7], [8], [13], [14], [15], [18].
Effective integer goal programming methods for all-integer, mixed-integer and zero–one multiobjective problems have been introduced by Lee and Morris [10]. These methods were based on the cutting-plane algorithm, branch-and-bound method, and implicit enumeration technique.
In Ignizio [6], a hybrid approach combining generalized goal programming and generalized networks has been presented for the modeling of certain large-scale multiobjective integer programming problems.
Hughes and Saad [5] discussed stability of efficient solutions in the decision space for parametric multiobjective integer linear programming problems via goal programming tools. The approach presented in [5] depends on Balinski algorithm [1] together with the iterative approach suggested by Dauer and Krueger [3] with the help of Gomory cuts [9], [16]. Later on, Osman et al. [11] investigated stability of efficient solutions for zero–one multiobjective linear programming problems using goal programming.
More recently, Saad and Sharif [12] presented an iterative goal programming approach for solving multiobjective integer linear programming problems. The definition of, and an algorithm to determine the stability set of the first kind for these problems have been characterized. The present paper is an extension of the study introduced in [12] to cover the problem investigated previously, but with fuzzy parameters in the right-hand side of the constraints. To our knowledge, this problem has not yet studied before in the integer case and under fuzzy environment.
The roots of the present paper lie in the following sections: Section 2 presents the formulation of multiobjective integer linear programming problem involving fuzzy parameters in the right-hand side of the constraints with the associated fuzzy integer linear goal programming model. Section 3 contains some basic definitions in the fuzzy theory together with the statement of the nonfuzzy version of the formulated goal programming model. In Section 4, a solution algorithm is suggested and described in sequential steps to solve such programs. Section 5 provides a numerical example to illustrate the theory and the solution algorithm. Finally, Section 6 contains the conclusions.
Section snippets
Problem formulation
In this section, we begin by introducing the following multiobjective integer linear programming problem with fuzzy parameters in the right-hand side of the constraints:whereand Z:Rn → Rk, Z(x) = (z1(x), z2(x), …, zk(x)) is a vector-valued criterion with zi(x), i = 1, 2, …, k, are real-valued linear objective functions, is a vector of fuzzy parameters and Rn is the set of all ordered n-tuples of real
Fuzzy concepts
The fuzzy theory has been advanced by L.A. Zadeh at the University of California in 1965. This theory proposes a mathematical technique for dealing with imprecise concepts and problems that have many possible solutions. The concept of fuzzy mathematical programming on a general level was first proposed by Tanaka et al. (1974) in the framework of the fuzzy decision of Zadeh and Bellman [17].
For the development that follows, we shall introduce some definitions concerning trapeziodal fuzzy numbers
Solution algorithm
In this section we develop a solution algorithm to solve the integer linear goal program (4.a), (4.b), (4.c), (4.d). The outline of this algorithm is as follows:
- Step 0.
Set α = α* = 0.
- Step 1.
Determine the points (a1, a2, a3, a4) for each fuzzy parameter , in program (2.a), (2.b), (2.c) with the corresponding membership function for the vector of fuzzy parameters .
- Step 2.
Convert program (2.a), (2.b), (2.c) into the nonfuzzy integer linear goal program
An illustrative example
Consider the following integer linear goal program involving fuzzy parameters in the right-hand side of the constraints:
Assume that the membership function corresponding to the fuzzy parameters is in the form:where corresponds to each . In addition, we assume also that the fuzzy
Conclusions
In this paper, an iterative goal programming approach has been proposed for solving multiobjective integer linear programming problems. These problems involve fuzzy parameters in the right-hand side of the constraints. The algorithm presented in this paper has the advantage of dealing with many integer linear goal programs by varying the parameters in the right-hand side of the constraints. Certainly, there are many other points for future research in the area of fuzzy integer linear goal
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