On one approach for solving of some class of multiobjective optimization problems with interval coefficients

Abstract

In present paper a multicriterion linear programming problem under interval indeterminacy is investigated, when the coefficients of the models' significant constraints can take any value from the specified intervals, at that significance of criteria are not known a priori. We use the weighted sum method to reduce the original multicriterion problem to single-criterion interval problem, in which the criterion is a strictly convex combination of the original criteria. It was revealed that widely known weights coefficients' estimation algorithm using the mechanism of doubling and bisection can lead to cycling that has led to necessity of its modification. Received interval single-criterion linear programming problem is intrinsically a parametric family of deterministic linear programming problems. Proposed approach consists in finding such common solution (so-called universal solution) to all problems of family that would satisfy the constraints of the problem accurate within the minimal residuals norm. With the help of the duality theory, a solvability of the reduced interval single-criterion problem is proved. For practical application of the foregoing approach for solving of the reduced interval single-criterion problem, the Big M Simplex method is used. It was concluded that solvability condition/Big M number lower estimate for the interval linear programming general problem, discovered in the literature are doubtful enough. In addition, we develop the appropriate software for computer implementation of the obtained theoretical results of this paper.