Robust flows with adaptive mitigation

We consider an adjustable robust optimization problem arising in the area of supply chains: given sets of suppliers and demand nodes, we wish to find a flow that is robust with respect to failures of the suppliers. The objective is to determine a flow that minimizes the amount of shortage in the worst-case after an optimal mitigation has been performed. An optimal mitigation is an additional flow in the residual network that mitigates as much shortage at the demand sites as possible. For this problem we give a mathematical formulation, yielding a robust flow problem with three stages where the mitigation of the last stage can be chosen adaptively depending on the scenario. We show that already evaluating the robustness of a solution is $\mathrm{NP}$-hard. For optimizing with respect to this $\mathrm{NP}$-hard objective function, we compare three algorithms. Namely an algorithm based on iterative cut generation that solves medium-sized instances efficiently, a simple Outer Linearization Algorithm and a Scenario Enumeration algorithm. We illustrate the performance by numerical experiments. The results show that this instance of fully adjustable robust optimization problems can be solved exactly with a reasonable performance. We also describe possible extensions to the model and the algorithm.