A biobjective approach to robustness based on location planning

Abstract

Finding robust solutions of an optimization problem is an important issue in practice, and various concepts on how to define the robustness of a solution have been suggested. The idea of recoverable robustness requires that a solution can be recovered to a feasible one as soon as the realized scenario becomes known. The usual approach in the literature is to minimize the objective function value of the recovered solution in the nominal or in the worst case. As the recovery itself is also costly, there is a trade-off between the recovery costs and the solution value obtained; we study both, the recovery costs and the solution value in the worst case in a biobjective setting. To this end, we assume that the recovery costs can be described by a metric. We demonstrate that this leads to a location planning problem, bringing together two fields of research which have been considered separate so far. We show how weakly Pareto efficient solutions to this biobjective problem can be computed by minimizing the recovery costs for a fixed worst-case objective function value and present approaches for the case of linear and quasiconvex problems for finite uncertainty sets. We furthermore derive cases in which the size of the uncertainty set can be reduced without changing the set of Pareto efficient solutions.