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Two-Stage Robust Network Flow and Design Under Demand Uncertainty

Department of Industrial Engineering and Operations Research, University of California, Berkeley, California 94720
Department of Industrial Engineering and Operations Research, University of California, Berkeley, California 94720
atamturk{at}ieor.berkeley.edu
mhzhang{at}ieor.berkeley.edu

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 -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.

Subject classifications: network/graphs; applications; programming; integer.
History: Received February 2005; revision received March 2006; accepted March 2006.

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