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

Most facility location problems are computationally hard to solve. The standard technique for solving these problems is branch-and-bound. To keep the size of the branch-and-bound tree as small as possible it is important to obtain a good lower bound on the optimal solution by deriving strong linear relaxations. One way of strengthening the linear relaxation is by adding inequalities that de ne facets of the convex hull of feasible solutions. Here we describe some simple, but computationally very useful classes of inequalities that were originally developed for relaxations of the facility location problems. Algorithms for generating violated inequalities belonging to the described classes have been implemented as system features in various branchand-bound software packages, so as long as the software can recognize the relaxations for which the inequalities are developed, the inequalities will be generated \automatically". Here we explicitly add the variables and constraints that are necessary to describe the relaxations which means that we actually add information that is redundant both with respect to to the integer formulation and the linear relaxation. We present computational results indicating that the reformulations are useful as the generated inequalities close a substantial part of the duality gap. Due to the smaller branch-and-bound tree, the time needed by the branch-andbound algorithm to solve the problem to optimality is reduced by up to ninety percent. Research partially supported by by ESPRIT Long Term Research Project No. 20244 (Project ALCOM-IT: Algorithms and Complexity in Information Technology), and by NSF grant CCR-9307391 throughDavid B. Shmoys, Cornell University.