Reformulation of capacitated facility location problems:How redundant information can help

Abstract

When solving hard combinatorial optimization problems by branch-and-bound, obtaininga good lower bound (considering a minimization problem) from the linear relaxation iscrucial for the performance of the algorithm. On the other hand, we want to avoid an initialformulation that is too large. This requires careful modeling of the problem. One way ofobtaining a good linear formulation is by applying a cutting plane algorithm where strongcutting planes are added if they violate the current fractional solution. By “strong” cuttingplanes, we mean linear inequalities that define high-dimensional faces of the convex hull offeasible solutions. For some classes of inequalities, effective algorithms for identifyingviolated inequalities belonging to these classes have been implemented as standard featuresin commercial branch-and-bound packages. Such classes are for instance the knapsack coverinequalities and the flow cover inequalities that were originally developed for the knapsackproblem and the single-node flow problem. These problems form relaxations of severalcapacitated combinatorial optimization problems such as various capacitated facility locationproblems. If, however, we consider traditional models for location problems, then theknapsack and single-node flow relaxations are not explicitly stated in the models, andunless we modify the models, the mentioned classes of inequalities will not be generated“automatically” by the systems. The extra variables and constraints that we need to add tothe traditional models in order to make the various relaxations explicit are redundant, notonly to the integer formulation but also to the linear relaxation. Computational experimentsdo, however, indicate that the inequalities that are generated based on the relaxations arevery effective and that the gain from the stronger linear relaxation far outweighs the drawbackof expanding the traditional models.