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 yand z-facilities each client should be assigned, in order to satisfy the demand at maximum profit. We first present two multi-commodity How 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 How problem. Rardin and Wolsey showed that by projecting a so-called multicommodity extended formulation of fixed-charge network How problems onto the space of How variables used in the weaker How 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. Subject classification: Facility Location: Discrete; Integer Programming: Cutting planes/Facets; Transportation: Location Models. The two-level uncapacitated facility location (TUFL) problem involves two types of facilities, which we call y-facilities and z-facilities. Decisions have to be made simultaneously on which facilities of both types to open, and to which pair of yand z-facility each client should be assigned. The objective is to maximize profit under the constraint that the demand of all clients has to be met. The two-level problem is a natural extension of the well-known uncapacitated facility location (UFL) problem. This two-level problem can be used to model a hierarchical structure with a set of major facilities connected to minor facilities, which in turn are connected to clients. Many practical location situations have a clear two-level structure. For example, the distribution networks of many companies often involve major (central) as well as minor (regional) depots. The central facilities supply the regional ones and such shipment quantities are typically large, whereas each client is served from a regional depot where the transport is usually carried out using smaller vehicles. Other applications involve two types of non-hierarchical facilities. For example, in garbage collection, a truck travels from a depot to the client and then to a disposal plant, so each client is assigned to a truck depot and a disposal plant. In situations where spent products are recycled, each client have to be assigned to a supply facility and a recycling facility. Numerous other applications of TUFL exist within areas such as telecommunication and computer network design. Some applications are described in Barros and Labbe (1992). Very little is known about the structural properties of TUFL, except that it is an NP-hard problem, since it generalizes UFL which is NP-hard (see Cornuejols, Fisher and Nemhauser, 1977, and Cornuejols, Nemhauser and Wolsey, 1990). Only a few algorithms for TUFL have been developed. Kaufman et aL (1977) developed a branch-and-bound algorithm generalizing an algorithm for UFL developed by Efroymson and Ray (1966). Tcha and Lee (1984) also developed a branch-and-bound method where a dual ascent procedure, similar to the one developed by Erlenkotter (1978), is used together with a primal descent method to get good lower and upper bounds. Barros and Labbe (1992) studied various formulations of a problem related to TUFL, and suggested a Lagrangean relaxation and a primal heuristic to derive bounds in a branch-and-bound algorithm. For the capacitated two-level problem, Aardal (1992) studied the cutting plane approach. Balakrishnan and Graves (1989) consider a generalization of TUFL and develop algorithms for finding good lower and upper bounds on the optimal solution value. We investigate some structural properties of TUFL. In Section 1, we introduce the necessary notation and a multi-commodity flow formulation of TUFL, and characterize the extreme points of its LP-relaxation. We investigate the relationship between TUFL and UFL in Section 2. Since UFL is a relaxation of TUFL, any inequality valid for UFL is also valid for TUFL. We show that certain inequalities that define facets for TUFL induce facets for UFL, and that basically all facet-defining inequalities for UFL can be extended to facet-defining inequalities for TUFL. Consequently, we obtain several classes of facets for TUFL from previously-known facet-defining inequalities for UFL (see Cho et al., 1983a,b,