A Successive Refinement for Solving Stochastic Programs with Decision-Dependent Random Capacities

Abstract: We study a class of two-stage stochastic programs in which the second stage includes a set of components with uncertain capacity, and the expression for the distribution function of the uncertain capacity includes first-stage variables. Thus, this class of problems has the characteristics of a stochastic program with decision-dependent uncertainty. A natural way to formulate this class of problems is to enumerate the scenarios and express the probability of each scenario as a product of the first-stage decision variables; unfortunately, this formulation results in an intractable model with a large number of variable products with high-degree. After identifying structural results related to upper and lower bounds and how to improve these bounds, we present a successive refinement algorithm that successively and dynamically tightens these bounds. Implementing the algorithm within a branch-and-cut method, we report the results of computational experiments that indicate that the successive refinement algorithm significantly outperforms a benchmark approach. Specifically, results show that the algorithm finds an optimal solution before the refined state space become too large.