An Exact Algorithm for Multicommodity Network Design Under Stochastic Interdictions

In this paper, we study the multicommodity network design problem by considering the effects of disruptions under an uncertain interdiction budget. The goal is to install links between nodes to satisfy the demand for different commodities with minimum installation cost and the weighted sum of flow costs before and after interdictions. Using the designer-interdictor-designer framework, we present a trilevel mixed-integer stochastic network design model. In the first level, the designer selects a subset of links to install and route flows under normal conditions. Most studies in the literature assume that the interdiction budget is known to the decision maker (network designer) with certainty; however, in practice, the designer is not aware of interdiction capabilities. Therefore, the designer’s objective is to minimize the installation cost and the weighted sum of pre-interdiction and expected post-interdiction costs. In the second level, the interdictor interdicts a subset of installed arcs with a limited interdiction budget. In the third level, the designer optimizes the flow over the surviving links in the residual network. Furthermore, we extend the model to consider the uncertainty in the demand besides uncertain interdiction budget. We present a branch-and-Benders-cut algorithm to solve the proposed model. The algorithm is enhanced through the use of several features such as multicut reformulation, warm start, variable fixing, cut selection, penalty reformulation, generation of strong Pareto-optimal cuts, and supervalid and valid inequalities. Extensive computational experiments are performed to evaluate the efficiency and robustness of the proposed algorithmic refinements. We compare the performance of our algorithm with a state-of-the-art, general-purpose stochastic mixed-integer bilevel linear optimization solver and show that our algorithm is faster by orders of magnitude. Our results demonstrate that the branch-and-Benders-cut algorithm combined with some of these acceleration techniques solves large-scale instances with up to 20 nodes, 220 arcs, and 200 commodities. Furthermore, we present a sensitivity analysis to highlight the advantages of stochastic design over deterministic design when the interdiction budget is uncertain.