Exact algorithms for routing problems under vehicle capacity constraints

The solution of a vehicle routing problem calls for the determination of a set of routes, each performed by a single vehicle which starts and ends at its own depot, such that all the requirements of the customers are fulfilled and the global transportation cost is minimized. The routes have to satisfy several operational constraints which depend on the nature of the transported goods, on the quality of the service level, and on the characteristics of the customers and of the vehicles. One of the most common operational constraint addressed in the scientific literature is that the vehicle fleet is capacitated and the total load transported by a vehicle cannot exceed its capacity. This paper provides a review of the most recent developments that had a major impact in the current state-of-the-art of exact algorithms for vehicle routing problems under capacity constraints, with a focus on the basic Capacitated Vehicle Routing Problem (CVRP) and on heterogeneous vehicle routing problems. The most important mathematical formulations for the problem together with various relaxations are reviewed. The paper also describes the recent exact methods and reports a comparison of their computational performances.