Interval Tolerance Solution for Adjusted Problem of Optimistic Interval Linear Program

摘要

An optimistic solution to an interval linear program is a real-valued solution derived from the best-case deterministic linear program. However, relying solely on this best-case solution can be overly simplistic, as the actual realization of parameters lies within specified intervals. Instead, it is more appropriate to provide an interval vector solution that remains near the optimistic solution, particularly when the decision-maker prefers proximity to the best-case scenario. In this paper, we establish the equivalence between the weak feasible solution set of an interval equality system and the union of basic feasible solutions across all scenarios of an interval linear program with an interval inequality system, where the interval inequalities require the left-hand side to be lower than the right-hand side, but not excessively so. Furthermore, we demonstrate that, under positive variables, this set coincides with the union of basic optimal solution sets. This result enables the use of a tolerance-based approach to identify an interval vector solution near the optimistic solution. Specifically, we modify the interval linear program so that the optimistic solution becomes a tolerance solution for the adjusted problem. We then propose a method to derive the interval tolerance vector solution for the modified problem, with the goal of maximizing the total sum of the dimensions of the interval tolerance vector hyper-box. Our proposed method differs from most existing methods for finding interval solutions, as those methods typically yield interval solutions that merely include weak solutions without specifying the solution type. Even though there are existing methods for obtaining interval tolerance solutions, none of them consider interval tolerance solutions that are close to the optimistic solution.