0-1 Combinatorial Optimization Problems with Qualitative and Uncertain Profits

Abstract

Recent works have studied 0-1 combinatorial optimization problems where profits of items are measured on a qualitative scale such as “low”, “medium” and “high”. In this study, we extend this body of work by allowing these profits to be both qualitative and uncertain. In the first step, we use probability theory to handle uncertainty. In the second step, we use evidence theory to handle uncertainty. We combine their approaches with approaches in decision making under uncertainty that utilize the Maximum Expected Utility principle and generalized Hurwicz criterion, to compare solutions. We show that under probabilistic uncertainty and a special case of evidential uncertainty where the focal sets are rectangles, the task of identifying the non-dominated solutions can be framed as solving a multi-objective version of the considered problem. This result mirrors that of the case of qualitative profits with no uncertainty.