A New Method For Solving Fuzzy Linear Programming Problems Based on The Fuzzy Linear Complementary Problem (FLCP)

Abstract

Linear programming (LP) is one of the most widely used methods in the area of optimization. Dealing with the formulation of LP problems, the parameters of objective function and constraints should be assigned by experts. In most cases, precise data have been used, but in most of the real-life situations, these parameters are imprecise and ambiguous. In order to deal with the problem of ambiguity and imprecision, fuzzy numbers can be appropriate. By replacing precise numbers with fuzzy numbers, LP problems change to fuzzy linear programming (FLP) problems. So FLPs can be considered as a broader category in comparison to LPs. Considering the above-mentioned points, FLP problems play an important rule in operational researches hence there is a need to investigate these problems. In this paper, a new method for solving the FLP problems is presented in which the coefficients of the objective function and the values of the right-hand side are represented by fuzzy numbers, while the elements of the coefficient matrix are represented by real numbers. To this end, we develop the Karush–Kuhn–Tucker (KKT) optimality conditions for FLP problems. Then, every FLP problem is converted to a fuzzy linear complementary problem (FLCP) by considering KKT conditions. In order to solve the FLCP problems, ranking functions and Lemke’s algorithm are used. Consequently, the solution to primal and dual problems of FLP is obtained. In addition to simplicity in calculations and feasibility, this method solves the primal and dual problems of FLP simultaneously. In order to illustrate the proposed method, some numerical examples are considered.