Forthcoming in Operations Research TWO - STAGE ROBUST NETWORK FLOW AND DESIGN 1 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 singe-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 multi-commodity network flow and design, and give applications to lot-sizing and location-transportation problems. By projecting out secondstage 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. December 2004, revised February 2006 Address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720–1777, USA. Email: [email protected], [email protected]. 1 2 ALPER ATAMTÜRK AND MUHONG ZHANG