The alpha-shortest path problem

Many real-life applications, arising in transportation and telecommunication systems, can be mathematically represented as shortest path problems. The deterministic version of the problem, where a deterministic cost is associated to each arc and the configuration of the network (nodes and arcs) is assumed to be known in advance, is easy to solve and has been extensively studied. However, in real applications, costs are typically not known a priori and may be subject to significant uncertainty. In addition, due to failure, maintenance, natural disasters, weather conditions, etc., some arcs could not be available causing a change of the network configuration. In this paper we introduce a variant of the shortest path problem under uncertainty, that concerns the situation in which for each arc two different states are possible (i.e. operating and failed states) and the aim is to find the path connecting a given pair of nodes with a sufficiently large probability α and such that the total cost is minimized. The problem can be formulated as a large scale integer programming model with knapsack constraints. For its solution a heuristic approach has been designed and implemented. Preliminary numerical experiments have been carried out on a set of randomly generated test problems.