Multi-period capacitated facility location under delayed demand satisfaction

We address an extension of the classical multi-period facility location problem in which customers are sensitive to delivery lead times. Accordingly, two customer segments are considered. The first segment comprises customers that require timely demand satisfaction, whereas customers accepting delayed deliveries make up the second segment. Each customer belonging to the latter segment specifies a maximum delivery time. A tardiness penalty is incurred to each unit of demand that is not satisfied on time. In the problem that we study, a network is already in place with a number of facilities being operated at fixed locations. The network can be expanded by establishing new facilities at a finite set of potential sites and selecting their capacity levels from a set of available discrete sizes. In addition, existing facilities may be closed over the time horizon. Two mixed-integer linear programming formulations are proposed to re-design the network at minimum cost and a theoretical comparison of their linear relaxations is provided. We also extend the mathematical models to the case in which each customer accepting delayed demand satisfaction requires late shipments to occur at most once over the delivery lead time. To gain insight into the complexity of the problems at hand, a computational study is performed with randomly generated instances that are solved with a general-purpose solver. Useful insights are derived from analyzing the impact of different delivery lead time restrictions on the network structure and cost.