A biobjective approach to recoverable robustness based on location planning

Abstract

Finding robust solutions of an optimization problem is an important issue in practice, and various con- cepts on how to define the robustness of a solution have been suggested. The idea of recoverable robust- ness 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-offbetween 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 show that in this case the recovery robust problem can be reduced to a location problem. We show how weakly Pareto efficient solutions to this biobjective problem can be computed by minimiz- ing 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.