A Branch-and-Price Algorithm for the Vehicle Routing Problem with Stochastic Demands and Probabilistic Duration Constraints

In many routing applications, it is necessary to place limits on the duration of the individual routes. When demands are stochastic and restocking during route execution is allowed, the durations of the resulting routes are also stochastic. In this paper, we consider the vehicle routing problem with stochastic demands and probabilistic duration constraints (VRPSD-PDC). We assume optimal restocking, which means that, during the route execution, replenishment trips to the depot are performed in an optimal way. The resulting optimization problem calls for a set of routes with minimal total expected cost for visiting all customers, such that the duration of each route, with a given probability, does not exceed a prescribed limit. We solve the VRPSD-PDC with a novel branch-and-price algorithm. An orienteering-based completion bound is proposed to control the growth of labels in the pricing algorithm. Feasibility of a priori routes is verified by applying Chebyshev’s bounds, by Monte Carlo simulation and statistical inference, or by analytically deriving the distribution of the route duration. Consistency checks are incorporated into the branch-and-price framework to detect statistical errors. Computational experiments are performed with demands following binomial, Poisson, or negative binomial probability distributions and with duration constraints enforced at the levels of 90%, 95%, and 98%. Optimal solutions to the VRPSD-PDC may contain routes that serve an expected demand that is larger than the capacity of the vehicle. These solutions actively employ optimal restocking to reduce traveling costs and the number of required vehicles. Sensitivity analyses indicate that high demand variability negatively impacts the solution, both in terms of total expected cost and the number of routes employed.