On Minmax Robustness for Multiobjective Optimization with Decision or Parameter Uncertainty

Abstract

Multiobjective optimization problems (MOPs) are problems with two or more objective functions. Two types of uncertainty in MOPs are distinguished, namely decision uncertainty and parameter uncertainty. Decision uncertainty means that solutions cannot be implemented exactly as targeted and parameter uncertainty means that a part of the problem data is unknown. In the first publication of this cumulative thesis, a minmax robustness concept for MOPs with decision uncertainty is introduced and decision robust efficient solutions are defined as the solutions to a set-valued optimization problem. Continuity properties of the set-valued objective functions are investigated. Specific solution approaches are shown for uncertain MOPs whose objective functions are linear, Lipschitz continuous or monotonic. Decision robust efficiency is also compared to further robustness concepts in the literature. In the second publication, a multiobjective robustness gap is presented. The robustness gap quantifies the difference between the conservative approach of minmax robustness and the optimistic approach of choosing the efficient solutions for some scenario of the uncertainty set. The multiobjective robustness gap is defined as the distance between the robust Pareto set and the Pareto sets of the scenarios. For convex MOPs, a lower and an upper bound for the gap are given. In the third and fourth publications, the concept of decision robust efficiency is applied to an uncertain MOP that arises in horticulture. The problem is solved analytically, and the solutions for a case study are determined.