We consider a reliable network design problem under uncertain edge failures. Our goal is to select a minimum-cost subset of edges in the network to connect multiple terminals together with high probability. This problem can be seen as a stochastic variant of the Steiner tree problem. We propose a scenario-based Steiner cut formulation, and a formulation based on a probabilistic variant of Steiner cuts. We study the strength of the proposed valid inequalities, and develop a branch-and-cut solution method. We also propose an LP-based separation for scenario-based directed Steiner cut inequalities using Benders feasibility cuts, leveraging the success of directed Steiner cuts for the deterministic Steiner tree problem. Mixing inequalities are also applied to further strengthen the formulation. In our computational study, we show the performance of our branch-and-cut method on the test instances to demonstrate the strength of the scenario-based formulations and the benefit from adding the Benders feasibility cuts and mixing inequalities.
Accepted in Naval Research Logistics, 2015