The multi-period location routing problem with locker boxes

When optimal parcel delivery plans have to be made, a logistic system with locker boxes as alternative delivery locations can greatly increase efficiency. An important decision in such a system is where to locate the locker box stations. Hence, we extend the vehicle routing problem with locker boxes to a location routing problem with multiple planning periods. In every period, decisions have to be made about which customers are served at home and which are served at locker boxes as well as how the used delivery locations are routed. On the other hand, the decision of which stations are opened is made only once. The objective function is minimizing total cost, which comprises travel costs, costs for compensating locker box customers, and site costs for realized stations. We provide a mixed integer programming formulation of the problem and propose a metaheuristic solution method. We use self-generated instances to compare the performance of the two approaches and can show that the metaheuristic method yields, on average, very good solutions - in most cases, the optimal solution - in a short computational time. We present and test different strategies for distributing the locker box stations, including based on customer clusters obtained by k-means, which works very well for a variety of geographic customer settings. Moreover, we analyze the utilization of locker box delivery in relation to the length of customer time windows and observe that tight time windows lead to a higher utilization.