The optimal solution set of the interval linear programming problems

Abstract

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ j , μ ij , μ i ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ ij = μ i = λ j = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ j , μ ij and μ i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.