Worst-case demand distributions in vehicle routing

•We derive the worst-case demand distributions for the vehicle routing problem.•When vehicles face time window constraints, there are two worst-case distributions.•We study the trade-off between two kinds of routing costs, “radial” and “circular”.•We conduct numerical simulations that use our result to solve a practical problem.

A recent focal point in research on the vehicle routing problem (VRP) is the issue of robustness in which customer demand is uncertain. In this paper, we conduct a theoretical analysis of the demand distributions whose induced workloads are as undesirable as possible. We study two common variations of VRP in a continuous approximation setting: the first is the VRP with time windows, and the second is the capacitated VRP, in which regular returns to the vehicle’s point of origin are required.