The capacitated mobile facility location problem

The capacitated mobile facility location problem (CMFLP) arises in logistics planning of community outreach programs delivered via mobile facilities. It adds capacity restrictions to the mobile facility location problem introduced previously by Demaine et al. (2009), thereby extending the problem to a practical setting. In the problem, one seeks to relocate (or move) a set of existing facilities and assign clients to these facilities while respecting capacities so that the weighted sum of facility movement costs and the client travel costs (each to its assigned facility) is minimized. We provide two integer programming formulations for the CMFLP. The first is on a layered graph, while the second is a set partitioning formulation. We prove that the linear relaxation of the set partitioning formulation provides a tighter lower bound to the CMFLP than the linear relaxation of the layered graph formulation. We then develop a branch-and-price algorithm on the set partitioning formulation. We find that the branch-and-price procedure is particularly effective both in terms of solution quality and running time, when the ratio of the number of clients to the number of facilities is small and the facility capacities are tight. Finally, we present two heuristic approaches for the CMFLP. One is a linear programming rounding heuristic, and the other is based on a natural decomposition of the problem on the layered graph.