The Vehicle Routing Problem with Probabilistic Customers (VRP-PC) is a fundamental building block within the broad family of stochastic routing models, and has two decision stages. In the first stage, a dispatcher determines a set of vehicle routes serving all potential customer locations, before 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. 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. Compared to branch-and-cut algorithms for the VRP-PC using arc-based formulations, our framework can more readily incorporate sequence-dependent constraints. We provide a priori and a posteriori performance guarantees for these algorithms, and demonstrate their effectiveness via a computational study on instances with realization probabilities ranging from 0.5 to 0.9.
Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, May 2018