Two-stage robust network flow and design under demand uncertainty

We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization, one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to single-stage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than single-stage optimization. For network flow and design under demand uncertainty, we give a characterization of the first-stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is NP-hard even for a network flow problem on a bipartite graph. We show, however, that if the second-stage network topology is totally ordered or an arborescence, then the separation problem is tractable. Unlike single-stage robust optimization under demand uncertainty, two-stage robust optimization allows one to control conservatism of the solutions by means of an allowed ‘budget for demand uncertainty’. Using a budget of uncertainty, we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector. We generalize the approach to multicommodity network flow and design, and give applications to lot-sizing and location–transportation problems. By projecting out second-stage flow variables, we define an upper bounding problem for the two-stage min–max–min optimization problem. Finally, we present computational results comparing the proposed two-stage robust optimization approach with single-stage robust optimization as well as scenario-based two-stage stochastic optimization.