The Resource Constrained Shortest Path Problem with uncertain data: A robust formulation and optimal solution approach

Abstract

The Resource Constrained Shortest Path Problem (RCSPP) models several applications in the fields of transportation and communications. The classical problem supposes that the resource consumptions and the costs are certain and looks for the cheapest feasible path. These parameters are however hardly known with precision in real applications, so that the deterministic solution is likely to be infeasible or suboptimal. We address this issue by considering a robust counterpart of the RCSPP. We focus here on resource variation and model its variability through the uncertainty set defined by Bertismas and Sim (2003, 2004), which can model the risk aversion of the decision maker through a budget of uncertainty. We solve the resulting problem to optimality through the well-known three phase approach dealing with bounds computation, network reduction and gap closing. In particular, we compute robust bounds on the resource consumption and cost by solving the robust shortest path problem and the dual robust Lagrangian relaxation, respectively. Dynamic programming is used to close the duality gap. Upper and lower bounds are used to reduce the dimension of the network and incorporated in the dynamic programming in order to fathom unpromising states. An extensive computational phase is carried out in order to assess the behavior of the defined strategy comparing its performance with the state-of-the-art. The results highlight the effectiveness of our approach in solving to optimality benchmark instances for RCSPP when Γ is not too large, tailored for the robust counterpart. For larger values of Γ, we show that the most efficient method combines deterministic preprocessing with the iterative algorithm from Bertsimas and Sim (2003). We also illustrate the failure probability of the robust solutions through Monte Carlo sampling.