Two-Stage Robust Network Flow and Design Under Demand Uncertainty

Abstract

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 singlestage 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 locationtransportation 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 twostage robust optimization approach with single-stage robust optimization as well as scenario-based two-stage stochastic optimization.

Robust Optimization for a Maritime Inventory Routing Problem

“…Constraints (12) require that the ship capacity is obeyed. Constraints (13) impose lower and upper limits on the loading and unloading quantities.…”

“…That is, the set of feasible solutions can be projected onto the space of the rst stage variables. This approach was followed in [12] for a simpler two-stage robust network ow and design problem, and in [8] robust approaches were used for the two dimensional spaces (with rst stage and with rst and second stage variables) for the robust VRPTW problem, albeit the projection has not been done explicitly. In [12] it was shown that even for simple graph structures the separation of the inequalities resulting from the projection is NP-hard.…”

“…This approach was followed in [12] for a simpler two-stage robust network ow and design problem, and in [8] robust approaches were used for the two dimensional spaces (with rst stage and with rst and second stage variables) for the robust VRPTW problem, albeit the projection has not been done explicitly. In [12] it was shown that even for simple graph structures the separation of the inequalities resulting from the projection is NP-hard. In [8] a path formulation was used for the formulation in the rst stage variables space and the corresponding approach was not considered preferable than the one working in the original space.…”

Histogram models for robust portfolio optimization

“…A similar approach has been developed in [EGL97], [EGOL98]. By now a large body of elegant work exists; the approach has been refined, extended and used in many applications, see [AZ05], [BGGN04], [BGNV05], [BS03], [BPS03], [BT06]. [BBN06] presents a framework for compensating for errors that fall outside of the uncertainty region being considered.…”

Robust Storage Assignment in Unit-Load Warehouses

“…Applications of the adjustable robust counterpart include Atamtürk and Zhang (2007) and Erera et al (2009). Unfortunately, adjustable robust counterpart models are generally NP-hard.…”