tailieunhanh - Solving fuzzy linear programming problems with linear membership functions

In this paper, we concentrate on two kinds of fuzzy linear programming problems: Linear programming problems with only fuzzy technological coefficients and linear programming problems in which both the right-hand side and the technological coefficients are fuzzy numbers. We consider here only the case of fuzzy numbers with linear membership functions. | Turk J Math 26 (2002) , 375 – 396. ¨ ITAK ˙ c TUB Solving Fuzzy Linear Programming Problems with Linear Membership Functions Rafail N. Gasimov, K¨ ur¸sat Yenilmez Abstract In this paper, we concentrate on two kinds of fuzzy linear programming problems: linear programming problems with only fuzzy technological coefficients and linear programming problems in which both the right-hand side and the technological coefficients are fuzzy numbers. We consider here only the case of fuzzy numbers with linear membership functions. The symmetric method of Bellman and Zadeh [2] is used for a defuzzification of these problems. The crisp problems obtained after the defuzzification are non-linear and even non-convex in general. We propose here the “modified subgradient method” and use it for solving these problems. We also compare the new proposed method with well known “fuzzy decisive set method”. Finally, we give illustrative examples and their numerical solutions. Key Words: Fuzzy linear programming; fuzzy number; modified subgradient method; fuzzy decisive set method. 1. Introduction In fuzzy decision making problems, the concept of maximizing decision was proposed by Bellman and Zadeh [2]. This concept was adopted to problems of mathematical programming by Tanaka et al. [13]. Zimmermann [14] presented a fuzzy approach to multiobjective linear programming problems. He also studied the duality relations in fuzzy linear programming. Fuzzy linear programming problem with fuzzy coefficients was formulated by Negoita [8] and called robust programming. Dubois and Prade [3] investigated linear fuzzy constraints. Tanaka and Asai [12] also proposed a formulation of fuzzy linear programming with fuzzy constraints and gave a method for its solution which bases on inequality relations between fuzzy numbers. Shaocheng [11] considered 2000 Mathematical Subject Classification: 90C70, 90C26. 375 ˙ GASIMOV, YENILMEZ the fuzzy linear programming problem with fuzzy constraints and .