Worst-case analysis for split delivery vehicle routing problems

In the vehicle routing problem (VRP) the objective is to construct a minimum cost set of routes serving all customers where the demand of each customer is less than or equal to the vehicle capacity and where each customer is visited once. In the split delivery vehicle routing problem the restriction that each customer is visited once is removed. We show that the cost savings that can be realized by allowing split deliveries is at most 50%. We also study the variant of the VRP in which the demand of a customer may be larger than the vehicle capacity, but where each customer has to be visited a minimum number of times. We show that the cost savings that can be realized by allowing more than the minimum number of required visits is again at most 50%. Furthermore, we analyze the performance of simple heuristics that handle customers with demands larger than the vehicle capacity by employing full load out-and-back trips to these customers until the demands become less than or equal to the vehicle capacity. Finally, we investigate situations in which demands are discrete and vehicle capacities are small.