Solving facility location models with modern optimization software: the weak and strong formulations revisited

It is well known that choosing a good formulation for a mixed-integer optimization model can drastically reduce its solution time with a linear-programming based branch-and-bound algorithm. Integer optimization software such as CPLEX and XPRESS-MP has progressed rapidly in recent years and implements powerful branchand-cut algorithms, combining branch-and-bound with cutting plane generation. In this article, we test the effectiveness of modern optimization software to solve classic facility location models with different formulations. We report experiments with the uncapacitated, capacitated, and single-sourcing capacitated facility location models using the strong and weak formulations, together with additional redundant constraints to help cut generation. These experiments lead to the conclusion that the strong formulation is still preferable for uncapacitated models, but the weak formulation is much more effective to solve capacitated models, thanks to the effectiveness of cutting planes.