The price of multiobjective robustness: Analyzing solution sets to uncertain multiobjective problems

Defining and finding robust efficient solutions to uncertain multiobjective optimization problems has been an issue of growing interest recently. Different concepts have been published defining what a “robust efficient” solution is. Each of these concepts leads to a different set of solutions, but it is difficult to visualize and understand the differences between these sets. In this paper we develop an approach for comparing such sets of robust efficient solutions, namely we analyze their outcomes under the nominal scenario and in the worst case using the upper set-less order from set-valued optimization. Analyzing the set of nominal efficient solutions, the set of minmax robust efficient solutions and different sets of lightly robust efficient solutions gives insight into robustness and nominal objective function values of these sets of solutions. Among others we can formally prove that lightly robust efficient solutions are good compromises between nominal efficient solutions and minmax robust efficient solutions. In addition, we also propose a measure to quantify the price of robustness of a single solution. Based on the measure, we propose two strategies which can be used to support a decision maker to find solutions to a multiobjective optimization problem under uncertainty. All our results are illustrated by examples.