Interval linear programming under transformations: optimal solutions and optimal value range

Abstract

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interval program cannot always be converted into a desired form without affecting its properties, due to the so-called dependency problem.In this paper, we discuss the common transformations used in linear programming, such as imposing non-negativity on free variables or splitting equations into inequalities, and their effects on interval programs. Specifically, we examine changes in the set of all optimal solutions, optimal values and the optimal value range. Since some of the considered properties do not holds in the general case, we also study a special class of interval programs, in which uncertainty only affects the objective function and the right-hand-side vector. For this class, we obtain stronger results.

The Worst Case Finite Optimal Value in Interval Linear Programming

“…In Garajová et al [10], it was also observed that the first transformation may change finite optimal values in the case with interval A. Below, we show by an example that this is also true for the second transformation.…”

“…Nevertheless, in some cases, it is possible. Garajová et al [10] showed that provided A is real, finite optimal values (and therefore also f fin ) is not changed under the following transformations:…”

“…Basically, one of the following canonical forms is usually considered. As was repeatedly observed, in the interval setting, these forms are not equivalent to each other in general [10,12,17], so they have to be analyzed separately. We can consider a general form involving all the canonical forms together [13], but from the sake of exposition, it is better to consider the canonical forms separately.…”

“…Hladík [15] proposes approximation method for the intractable cases. Garajová et al [10] study what is the effect of transformations of the constraints on the optimal value range, among others.…”

Editorial

“…Linear programming is a classical approach in OR that provides new research challenges until today. The paper written by Garajová et al (2018) considers the interval linear programming that is a tool when the optimization problems are to be solved under interval-valued uncertainty. Tavakoli and Klavžar (2019) address a discrete optimization problem studying global defensive k-alliances.…”