An Inventory-Routing Problem with Pickups and Deliveries Arising in the Replenishment of Automated Teller Machines

The purpose of this paper is to introduce, model, and solve a rich multiperiod inventory-routing problem with pickups and deliveries motivated by the replenishment of automated teller machines in the Netherlands. Commodities can be brought to and from the depot, as well as being exchanged among customers to efficiently manage their inventory shortages and surpluses. A single customer can both provide and receive commodities at different periods, since its demand changes dynamically throughout the planning horizon and can be either positive or negative. In the case study, new technology provides these machines with the additional functionality of receiving deposits and reissuing banknotes to subsequent customers. We first formulate the problem as a very large-scale mixed-integer linear programming model. Given the size and complexity of the problem, we first decompose it into several more manageable subproblems by means of a clustering procedure, and we further simplify the subproblems by fixing some variables. The resulting subproblems are strengthened through the generation of valid inequalities and solved by branch and cut. We assess the performance of the proposed solution methodology through extensive computational experiments using real data. The results show that we are able to obtain good lower and upper bounds for this new and challenging practical problem.