Planning approximations to the average length of vehicle routing problems with time window constraints

This paper studies approximations to the average length of vehicle routing problems (VRP) with time window, route duration, and capacity constraints. The approximations are valuable for the strategic and planning analysis of transportation and logistics problems. Using asymptotic properties of vehicle routing problems and the average probability of successfully sequencing a customer with time windows a new expression to estimate VRP distances is developed. The increase in the number of routes when time constraints are added is modeled probabilistically. This paper introduces the concept of the average probability of successfully sequencing a customer with time windows. It is proven that this average probability is a unique characteristic of a vehicle routing problem. The approximation proposed is tested in instances with different customer spatial distributions, depot locations and number of customers. Regression results indicate that the proposed approximation is not only intuitive but also predicts the average length of VRP problems with a high level of accuracy.