A Necessity Measure of Fuzzy Inclusion Relation in Linear Programming Problems

A programming problem with linear equality constraints can be generalised to the one with linear inclusions when coefficients are imprecisely given as possible ranges. In the problem with linear inclusions, the possible ranges of linear function values should always fluctuate within given ranges. In this paper, we investigate the programming problem with linear inclusions with coefficients being triangular fuzzy sets. To treat it, we introduce a new necessity measure, a linear extension of the one defined by the Dienes implication function, for the degree of the inclusions, and formulate a necessity measure maximisation problem. We propose a solution method based on the trade-off ratio and show that the maximisation problem becomes a regular linear programming problem in a particular condition. In general conditions, we also propose an algorithm utilising its properties and give a numerical example to demonstrate the solution procedure.