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Friday, September 24, 2010

Simplex Method for Solving Linear Programming Problems with Fuzzy Numbers

Abstract — T he fuzzy set theory has been applied in many fields, such as operations research, control theory, and management sciences, etc. In particular, an application of this …… theory in decision making problems is linear programming problems with fuzzy numbers. In this study, we present a new method for solving fuzzy number linear programming problems, by use of linear ranking function. In fact, our method is similar to simplex method that was used for solving linear programming problems in crisp environment before. number linear programming problems, by use of linear rankingfunction. In fact, our method is similar to simplex method that was used for solving linear …

Abstract — The fuzzy set theory has been applied in many fields, such as operations research, control theory, and management sciences, etc. In particular, an application of this theory in decision making problems is linear programming problems with fuzzy numbers. In this study, we present a new method for solving fuzzy number linear programming problems, by use of linear ranking function. In fact, our method is similar to simplex method that was used for solving linear programming problems in crisp environment before. Keywords — Fuzzy number linear programming, ranking function, simplex method. I. INTRODUCTION UZZY linear programming first formulated by Zimmermann [10]. Recently, these problems are considered in several kinds, that is, it is possible that some coefficients of the problem in the objective function, technical coefficients, the right-hand side coefficients or decision making variables be fuzzy number [3], [4], [5], [6], [7], [8], [9]. In this work, we focus on the linear programming problems with fuzzy numbers in the objective function. Verdegay and et al [4], [9] proposed the equivalent parametric linear programming problems for these problems by use of a certain membership function and proposed a dual method for fuzzy number linear programming problems. Here, we first explain the concept of the comparison of fuzzy numbers by introducing a linear ranking function. Moreover, we describe basic feasible solution for the FNLP problems and state optimality conditions for these problems. Finally, we provide some important results for FNLP problems and we propose simplex algorithm for solving these problems.

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