Branch-and-Price for Probabilistic Vehicle Routing

The Vehicle Routing Problem with Probabilistic Customers (VRP-PC) is a fundamental building block within the broad family of a priori routing models and has two decision stages. In the first stage, the dispatcher determines a set of vehicle routes serving all potential customer locations before the actual requests for service realize. In the second stage, vehicles are dispatched after observing the subset of customers requiring service; a customer not requiring service is skipped from its planned route at execution. The objective is to minimize the expected vehicle travel cost assuming known customer realization probabilities. We propose a column generation framework to solve the VRP-PC to a given optimality tolerance; as with other branch-and-price VRP schemes, our framework can handle sequencedependent constraints such as time windows. Specifically, we present two novel algorithms, one that under-approximates a solution’s expected cost, and another that uses its exact expected cost. Each algorithm is equipped with a route pricing mechanism that iteratively improves the approximation precision of a route’s reduced cost; this produces fast route insertions at the start of the algorithm and reaches termination conditions at the end of the execution. We provide a priori and a posteriori performance guarantees for these algorithms and test their performance on VRP-PC instances with time windows. Our results suggest that both algorithms can numerically optimize instances with up to 40 customers and realization probabilities of 0.5, 0.7 and 0.9.