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

TL;DR

This paper introduces a successive refinement algorithm for two-stage stochastic programs with decision-dependent random capacities, improving solution efficiency by dynamically tightening bounds and outperforming benchmark methods in computational experiments.

Contribution

The paper proposes a novel successive refinement algorithm that effectively handles decision-dependent uncertainty in stochastic programs, enhancing solution accuracy and computational performance.

Findings

The algorithm significantly outperforms benchmark approaches.

It finds optimal solutions before the state space becomes too large.

Structural bounds are key to improving solution efficiency.

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.