Shortest path network interdiction under uncertainty

Abstract

This research considers three extensions of the shortest path network interdiction problem to protect against parameter uncertainty. The shortest path interdiction problem is a game of two players with conflicting agendas and capabilities: an evader, who traverses the arcs of a network from a source node to a sink node using the path of shortest length, and an interdictor, who maximizes the length of the evader's shortest path by interdicting arcs on the network. It is usually assumed that the parameters defining the network are known exactly by both players. In the first variant, we consider the situation where the evader assumes the nominal parameter values while the interdictor uses robust optimization techniques to account for parameter uncertainty or sensor degradation. Solving the shortest path interdiction problem with asymmetric uncertainty protects the interdictor from investing in the obvious strategy if that strategy hinges on key interdictions performing as promised. It also provides an alternate strategy that mitigates the risk of these worst-case possibilities. In the second variant, we extend past the previous model to allow the interdictor to interdict an arc multiple times or bolster an arc to further combat parameter uncertainty. We formulate these problems as nonlinear mixed integer trilevel programs and show that they can be converted into mixed integer linear programs with second order cone constraints. The third variant extends an existing variant of the shortest path network interdiction problem where the evader has asymmetric knowledge of the network parameters. The interdictor knows exactly what the evader's misassumptions are and can leverage that information for an improved outcome. Our extension allows for interdictor uncertainty as to exactly what the evader assumes as the network parameters. We develop a decomposition algorithm to solve this model. For all models, we use random geometric networks and transportation networks to perform computational studies and demonstrate the unique decision strategies that our variants produce.