Benders Decomposition for Production Routing Under Demand Uncertainty

The production routing problem (PRP) is a generalization of the inventory routing problem and concerns the production and distribution of a single product from a production plant to multiple customers using capacitated vehicles in a discrete and finite time horizon. In this study, we consider the stochastic PRP with demand uncertainty in two-stage and multi-stage decision processes. The decisions in the first stage include production setups and customer visit schedules, while the production and delivery quantities are determined in the subsequent stages. We introduce formulations for the two problems which can be solved by a branchand-cut algorithm. To handle a large number of scenarios, we propose a Benders decomposition approach which is enhanced through lower bound lifting inequalities, scenario group cuts and Pareto-optimal cuts. For the multi-stage problem, we also use a warm-start procedure that relies on the solution of the simpler two-stage problem. Finally, we exploit the reoptimization capabilities of Benders decomposition in a sample average approximation method for the two-stage problem and in a rollout algorithm for the multi-stage problem. Computational experiments show that instances of realistic size can be solved to optimality for both the two-stage and multi-stage problems, and that Benders decomposition provides significant speedups compared to a classical branch-and-cut algorithm.