This article presents an exact algorithm for the multi-depot vehicle routing problem (MDVRP) under capacity and route length constraints. The MDVRP is formulated using a vehicle-flow and a set-partitioning formulation, both of which are exploited at different stages of the algorithm. The lower bound computed with the vehicle-flow formulation is used to eliminate non-promising edges, thus reducing the complexity of the pricing subproblem used to solve the set-partitioning formulation. Several classes of valid inequalities are added to strengthen both formulations, including a new family of valid inequalities used to forbid cycles of an arbitrary length. To validate our approach, we also consider the capacitated vehicle routing problem (CVRP) as a particular case of the MDVRP, and conduct extensive computational experiments on several instances from the literature to show its effectiveness. The computational results show that the proposed algorithm is competitive against stateof-the-art methods for these two classes of vehicle routing problems, and is able to solve to optimality some previously open instances. Moreover, for the instances that cannot be solved by the proposed algorithm, the final lower bounds prove stronger than those obtained by earlier methods.

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Vehicle routing problems (VRPs) are among the most studied problems in operations research. Nowadays, the leading exact algorithms for solving many classes of VRPs are branch-price-and-cut algorithms. In this survey paper, we highlight the main methodological and modeling contributions made over the years on branch-and-price (branch-price-and-cut) algorithms for VRPs, whether they are generic or specific to a VRP variant. We focus on problems related to the classical VRP—that is, problems in which customers must be served by several capacitated trucks and which are not combinations of a VRP and another optimization problem.