Scatter Search for Vehicle Routing Problem with Time Windows and Split Deliveries

Abstract

The classical vehicle routing problem (VRP) aims to find a set of routes at a minimal cost (finding the shortest path, minimizing the number of vehicles, etc) beginning and ending the route at the depot, so that the known demand of all nodes are fulfilled. Each node is visited only once, by only one vehicle, and each vehicle has a limited capacity. Some formulations also present constraints on the maximum traveling time. The VRPSD is a variation of the classical VRP, where each customer can be served by more than one vehicle. Thus, for the VRPSD, besides the delivery routes, the amount to be delivered to each customer in each vehicle must also be determined. The option of splitting a demand makes it possible to service a customer whose demand exceeds the vehicle capacity. Splitting may also allow decreasing costs. The vehicle routing problem with time windows and split deliveries (VRPTWSD) is an extension of the VRPSD, adding to it the time window restraints. Lenstra and Rinnooy Kan (1981) have analyzed the complexity of the vehicle routing problem and have concluded that practically all the vehicle routing problems are NP-hard (among them the classical vehicle routing problem), since they are not solved in polynomial time. According to Solomon and Desrosiers (1988), the vehicle routing problem with time windows (VRPTW) is also NP-hard because it is an extension of the VRP. Although the vehicle routing problem with split deliveries (VRPSD) is a relaxation of the VRP, it is still NP-hard (Dror and Trudeau, 1990, Archetti et al., 2005). Therefore, the VRPTWSD is NP-hard, since it is a combination of the vehicle routing problem with time windows (VRPTW) and the vehicle routing problem with split delivery (VRPSD), and that makes a strong point for applying heuristics and metaheuristic in order to solve the problem. This work develops a scatter search (SS) algorithm to solve a vehicle routing problem with time windows and split deliveries (VRPTWSD). To generate the initial solutions of SS we propose an adaptation of the sequential insertion heuristic of Solomon (1987). Ho and Haugland (2004) modified the customers’ demands of the Solomon’s test problems in order to perform split deliveries. Numerical results of SS are reported as well as comparisons with the Ho and Haugland algorithm. O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m