Locker box location planning under uncertainty in demand and capacity availability

In this paper, we address the location of locker boxes in the last-mile delivery context under uncertainty in demand and capacity. The problem is modelled as an extension of the capacitated facility location problem, in which a fixed number of facilities has to be opened, choosing among a set of potential locations. Facilities are characterized by a homogeneous capacity, but a capacity reduction may occur with a given probability. The uncertainty in demand and capacity is incorporated through a set of discrete scenarios. Each customer can be assigned only to compatible facilities, i.e., to facilities located within a given radius from the individual location. The goal is to first maximize the total number of customers assigned to locker boxes, while, in case of a tie on this primary objective, a secondary objective intervenes aiming at minimizing the average distance covered by customers to reach their assigned locker box. A stochastic mathematical model as well as three matheuristics are presented. We provide an extensive computational study in order to analyze the impact of different parameters on the complexity of the problem. The importance of considering uncertainty in input data is discussed through the usage of general stochastic indicators from the literature as well as of problem specific indicators. A real-world case related to the City of Turin in Italy is analyzed in detail. The benefit achievable by optimizing locker box locations is discussed and a comparison with the current configuration is provided.