Exact and Heuristic Methods for the Split Delivery Vehicle Routing Problem

This paper describes an exact branch-and-cut (B&C) algorithm for the split delivery vehicle routing problem. The underlying model is based on a previously proposed two-index vehicle flow formulation that models a relaxation of the problem. We dynamically separate two well-known classes of valid inequalities, namely capacity and connectivity cuts, and use an in-out algorithm to improve the convergence of the cutting phase. We generate no-good cuts from feasible integer solutions to the relaxation using a recently proposed single-commodity flow formulation in the literature. The exact methodology is complemented by a very effective adaptive large neighborhood search (ALNS) heuristic that provides high-quality upper bounds to initiate the B&C algorithm. Key ingredients in the design of the heuristic include the use of a tailored construction algorithm, which can exploit the situation in which the ratio of the number of customers to the minimum number of vehicles needed is low, and the use of a route-based formulation to improve the solutions found before, during, and after the ALNS procedure. An earlier version of this work was submitted to the DIMACS (Center for Discrete Mathematics and Theoretical Computer Science) implementation challenge, where it placed third. On sets of well-known benchmark instances for limited and unlimited fleet variants of the problem, we demonstrate that the heuristic provides very competitive solutions, with respective average gaps of 0.19% and 0.18% from best-known values. Furthermore, the exact B&C framework is also highly competitive with state-of-the-art methods, providing solutions with an average optimality gap of 1.82%.