A Column Generation-Based Heuristic for the Split Delivery Vehicle Routing Problem with Time Windows

The vehicle routing problem with time windows (VRPTW) is one of the most studied variants of routing problems. We consider the split delivery VRPTW (SDVRPTW), an extension in which customers can be visited multiple times, if advantageous. While this additional flexibility can result in significant cost reductions, it also results in additional modeling and computational challenges. Indeed, the branch-and-price algorithms used to successfully solve VRPTW instances require substantial modifications before they can be used to solve SDVRPTW instances, and then often exhibit inferior performance when compared with branch-and-cut algorithms for solving SDVRPTW instances (whereas branch-and-price algorithms tend to perform better than branch-and-cut algorithms for the VRPTW). We propose a new route-based formulation for the SDVRPTW that differs fundamentally from others presented in the literature. It is the first formulation in which the number of decision variables related to delivery quantities as well as the number of constraints is polynomial in the number of customers. We use this formulation as the basis for a column generation-based heuristic that produces high-quality solutions for a wide range of benchmark instances with 50 and 100 customers and vehicle capacity equal to 50 and 100. It finds many new best known solutions, and, for the first time, we report upper bounds for all 100-customer instances.