Solving a dynamic facility location problem with partial closing and reopening

• We provide a new formulation for a complex multi-period facility location that allows for the partial closing and reopening of facilities.
• The new formulation provides integrality gaps that are, on average, 29 times smaller than those of traditional formulations.
• We develop a Lagrangian relaxation based heuristic, able to solve large-scale problem instances for which state-of-the-art mixed integer programming solvers do not find feasible solutions.

Motivated by an industrial application, we consider a recently introduced multi-period facility location problem with multiple commodities and multiple capacity levels. The problem allows for the relocation of facilities, as well as for the temporary closing of parts of the facilities, while other parts remain open. In addition, it uses particular capacity constraints that involve integer rounding of the allocated demands. In this paper, we propose a strong formulation for the problem, as well as a hybrid heuristic that first applies Lagrangian relaxation and then constructs a restricted mixed-integer programming model based on the previously obtained Lagrangian solutions. Computational results for large-scale instances emphasize the usefulness of the heuristic in practice. While general-purpose mixed-integer programming solvers do not find feasible solutions for about half of the instances, the heuristic consistently provides high-quality solutions in short computing times, as well as tight bounds on their optimality.