On the two-level uncapacitated facility location problem

We study the two-level uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call y-facilities and z-facilities, the problem is to decide which facilities of both types to open, and to which pair of y- and z-facilities each client should be assigned, in order to satisfy the demand at maximum profit. We first present two multi-commodity flow formulations of TUFL and investigate the relationship between these formulations and similar formulations of the one-level uncapacitated facility location (UFL) problem. In particular, we show that all nontrivial facets for UFL define facets for the two-level problem, and derive conditions when facets of TUFL are also facets for UFL. For both formulations of TUFL, we introduce new families of facets and valid inequalities and discuss the associated separation problems. We also characterize the extreme points of the LP-relaxation of the first formulation. While the LP-relaxation of a multi-commodity formulation provides good bounds in general, the number of variables and constraints grows rapidly with the size of the problem instance. An alternative model of TUFL is a single-commodity fixed-charge network flow problem. Rardin and Wolsey showed that by projecting a so-called multi-commodity extended formulation of fixed-charge network flow problems onto the space of flow variables used in the weaker flow formulation, a broad class of valid inequalities can be obtained. We discuss a subclass of these inequalities for TUFL that seems particularly useful for computational purposes.