Proponents of robust optimization typically make three claims concerning the value of the robust approach. The first is that robust optimization models lead to solutions whose expected cost is close to the minimum expected cost in cases where the optimum can be computed. The second is that robust optimization models in special cases lead to closed form solutions which provide analytical insight into the underlying problems. The third is that such models do not suffer the curse of dimensionality which plagues expected-cost optimization models: in many cases, the dimension of the robust optimization formulation is no larger than the size of a corresponding deterministic model.
We investigate these claims as we apply the robust optimization approach to a classical problem in inventory management: multi-period stock allocation. This is an important problem in commercial, industrial, military, and public health settings: A central authority must ration a limited supply of material to multiple locations over time in the face of uncertain demand. Holding back inventory centrally in order to re-balance stock positions in subsequent periods is a critical strategy which exploits the phenomenon of risk pooling. Because of the curse of dimensionality, the expected-cost minimization form of the problem is computationally intractable.
We propose a robust optimization formulation of this problem which appears to meet most of the claims listed above. It gives good solutions, where optimal solutions can indeed be computed. It also leads to a nice closed form solution in one special case. The third claim is partially met: the method is computationally tractable, but the number of constraints used to represent the risk pooling phenomenon increases with the square of the number of locations considered. This research suggests that large-scale optimization problems involving risk pooling strategies are now within reach of solution.
Based on joint work with Jack Muckstadt.