A parametric approach to fuzzy linear programming
This paper is devoted to the study of linear programming problems with imprecise values of the involved parameters. Say, the parameters are unknown with varying grades of precision. The imprecision is traditionally evaluated through post-optimization analysis or by the use of stochastic programming. The fuzzy set approach is different, not traditional, but possibly better for studying some of the problems arising when the parameters are subject to error. Generally the decision-maker is able to fix intervals for the parameters of the problem or to give some ranking of the possible values of them. Then the Bellman-Zadeh principle may be used and strict optimization and constraints may be replaced by a gradual attainment of aspiration levels \((z_ 0,b_ 0)\). An overview of this problem is given in section 2 and the handling of the imprecision by the different approaches is analyzed. In section 3 a theorem states that the solution of an LP problem is an increasing function of the parameters. Then using the knowledge of the intervals where they should be located and assuming parametric dependent and monotonically decreasing membership functions ways for obtaining the parameters for the LP problem are developed. Linear parametric programming techniques provide the tools for solving the corresponding applications. Section 4 is devoted to the solution of a numerical example and the behaviour of the proposed approach is illustrated.
scientific article; zbMATH DE number 3974728