Yan-Kuen Wu

MINIMIZING A LINEAR OBJECTIVE FUNCTION WITH FUZZY RELATION EQUATION CONSTRAINTS

An minimization problem with a linear objective function subject to fuzzy relation equations using max-product composition has been considered by Loetamonphong and Fang. They first reduced the problem by exploring the special structure of the problem and then proposed a branch-and-bound method to solve this 0-1 integer programming problem. In this paper, we provide a necessary condition for an optimal solution of the minimization problems in terms of one maximum solution derived from the fuzzy relation equations. This necessary condition enables us to derive efficient procedures for solving such optimization problems. Numerical examples are provided to illustrate our procedures.

Minimizing a linear function under a fuzzy max–min relational equation constraint

In this paper we investigate the problem of minimizing a linear objective function subject to a fuzzy relational equation constraint. A necessary condition for optimal solution is proposed. Based on this necessary condition, we propose three rules to simplify the work of computing an optimal solution. Numerical examples are provided to illustrate the procedure. Experimental results are reported showing that our new procedure systematically outperforms our previous work.

An accelerated approach for solving fuzzy relation equations with a linear objective function

In literature, the optimization model with a linear objective function subject to fuzzy relation equations has been converted into a 0-1 integer programming problem by Fang and Li (1999). They proposed a jump-tracking branch-and-bound method to solve this 0-1 integer programming problem. In this paper, we propose an upper bound for the optimal objective value. Based on this upper bound and rearranging the structure of the problem, we present a backward jump-tracking branch-and-bound scheme for solving this optimization problem. A numerical example is provided to illustrate our scheme. Furthermore, testing examples show that the performance of our scheme is superior to the procedure in the paper by Fang and Li. Several testing examples show that our initial upper bound is sharp.

Weighted coefficients in two-phase approach for solving the multiple objective programming problems

Abstract Two-phase approach with equal weighted coefficients has been proposed to yield an efficient solution for multiple objective programming problems. In this note, we will show that the two-phase approach, as long as the weighted coefficients are positive, not necessarily equal, will generate an efficient solution. A counterexample is given to the case that some weighted coefficients are zero.

A Note on Fuzzy Relation Programming Problems with Max-Strict- t -Norm Composition

The fuzzy relation programming problem is a minimization problem with a linear objective function subject to fuzzy relation equations using certain algebraic compositions. Previously, Guu and Wu considered a fuzzy relation programming problem with max-product composition and provided a necessary condition for an optimal solution in terms of the maximum solution derived from the fuzzy relation equations. To be more precise, for an optimal solution, each of its components is either 0 or the corresponding components value of the maximum solution. In this paper, we extend this useful property for fuzzy relation programming problem with max-strict-t-norm composition and present it as a supplemental note of our previous work.

An Efficient Procedure for Solving a Fuzzy Relational Equation With Max–Archimedean t-Norm Composition

In the literature, a necessary condition for minimal solutions of a fuzzy relational equation with max-product composition shows that each of its components is either zero or the corresponding components value of the greatest solution. In this paper, we first extend this necessary condition to the situation with max-Archimedean triangular-norm (t-norm) composition. Based on this necessary condition, we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently. Furthermore, rather than work with the actual equations, we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions. Numerical examples with specific cases of the max-Archimedean t-norm composition are provided to illustrate the procedure.

A COMPROMISE MODEL FOR SOLVING FUZZY MULTIPLE OBJECTIVE LINEAR PROGRAMMING PROBLEMS

ABSTRACT Two-phase approach had been proposed to generate an efficient solution for the multiple objective linear programming problems [MOLP]. In this research, we shall show a revised two-phase approach to the case of the fuzzy multiple objectives linear programming problems [FMOLP]. This revised model can improve the optimal decision obtained from min operator. Moreover, a compromise model embedded two-phase approach and average operator will be proposed to yield a fuzzy-efficient solution between non-compensatory and fully compensatory. More precisely, one compromise index to the membership function can be adjusted by decision-maker to reveal the way of change on degree of satisfaction for each objective and fuzzy constraint. One numerical example is employed to illustrate our assertions.

Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint

This paper aims to solve the problem of minimizing a linear objective function subject to a max-T fuzzy relational equation constraint, with T a continuous t-norm. This study makes two contributions. First, two extensions are given based on the results in Fang et al., where the max-min or max-product fuzzy relational equation constraint is considered. Second, a procedure for solving optimization problems with the max-Archimedean t-norm fuzzy relational equation constraint is provided. A numerical example is provided to illustrate the procedure.

Reducing the search space of a linear fractional programming problem under fuzzy relational equations with max-Archimedean t-norm composition

Prior studies have demonstrated that one of the minimal solutions of a fuzzy relational equation with the max-Archimedean t-norm composition is an optimal solution of a linear objective function with positive coefficients. However, this property cannot be adopted to optimize the problem of a linear fractional objective function. This study presents an efficient method to optimize such a linear fractional programming problem. First, some theoretical results are developed based on the properties of max-Archimedean t-norm composition. The result is used to reduce the feasible domain. The problem can thus be simplified and converted into a traditional linear fractional programming problem, and eventually optimized in a small search space. A numerical example is provided to illustrate the procedure.

Multi-objective optimization with a max-t-norm fuzzy relational equation constraint

In this paper, we consider minimizing multiple linear objective functions under a max-t-norm fuzzy relational equation constraint. Since the feasible domain of a max-Archimedean t-norm relational equation constraint is generally nonconvex, traditional mathematical programming techniques may have difficulty in yielding efficient solutions for such problems. In this paper, we apply the two-phase approach, utilizing the min operator and the average operator to aggregate those objectives, to yield an efficient solution. A numerical example is provided to illustrate the procedure.