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 provi...

The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure. The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with constant capacity K for all plants. These facet inequalities depend on K and thus differ fundamentally from the valid inequalities for the uncapacitated version of the problem. We also introduce a second formulation for a model with indivisible cus-tomer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.

We introduce a generalization of the well-known Uncapacitated Facility Location Problem, in which clients can be served not only by single facilities but also by sets of facilities. The problem, called Generalized Uncapacitated Facility Location Problem (GUFLP), was inspired by the Index Selection Problem in physical database design. We formulate GUFLP as a Set Packing Problem, showing that our model contains all the clique inequalities (in polynomial number). Moreover, we describe an exact separation procedure for odd-hole inequalities, based on the particular structure of the problem. These results are used within a branch-and-cut algorithm for the exact solution of GUFLP. Computational results on two different classes of test problems are given.

The Index Selection Problem (ISP) is a phase of fundamental importance in the physical design of databases, calling for a set of indexes to be built in a database so as to minimize the overall execution time for a given database workload. The problem is a generalization of the well-known Uncapacitated Facility Location Problem (UFLP). In [6], we formulate ISP as a set packing problem, showing that our mathematical model contains all the clique inequalities, and describe a branch-and-cut algorithm based on the separation of odd-hole inequalities. In this paper, we describe an effective exact separation procedure for a suitably-defined family of lifted odd-hole inequalities, obtained by applying a Chvátal-Gomory derivation to the clique inequalities. Our analysis goes in the direction of determining a new class of inequalities over which ecient separation is possible, rather than introducing new classes of (facet-de ning) inequalities that later turn out to be difficult to separate. Our separation procedure is embedded within our branch-and-cut algorithm for the exact solution of ISP. Computational results on two different classes of instances are given, showing the e ectiveness of the new approach. We also test our algorithm on UFLP instances both taken from the literature and randomly generated.

We examine the feasibility polyhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal delivery services. In particular we determine the dimension and derive some classes of facets for this polyhedron. We develop a general rule about lifting facets from the uncapacitated facility location (UFL) problem to UHL. Using this lifting procedure we obtain a new class of facets for UHL which dominates the inequalities in the original formulation.

In this paper we describe properties of a particular transformation of the simple plant location problem into a vertex packing problem on a corresponding undirected graph. We characterize the undirected graphs that arise from simple plant location problems and we give a necessary and sufficient condition for such graphs to be perfect. This allows us to find a new family of polynomially solvable instances of the simple plant location problem. 3. 1. Introduction We assume familiarity with basic notions of graph theory (see, for instance, [3]). Let G be an undirected graph; a clique in G is any set of pairwise adjacent vertices; a stable set in G is any set of pairwise nonadjacent vertices. A clique or a stable set are called maximal if there exists no other clique or stable set containing them. The largest size of a clique and of a stable set in G is denoted by !(G) and ff(G), respectively. The vertex packing problem on an undirected graph G, with weights on its vertices, is to find i...

A good strategy for the solution of a large-scale problem is its division into small ones. In this context, this work explores the lagrangean relaxation with clusters (LagClus) that can be applied to combinatorial problems modeled by conflict graphs. By partitioning and removing the edges that connect the clusters of vertices, the conflict graph is divided in subgraphs with the same characteristics of the whole problem. When relaxing the removed edges in the lagrangean way, subproblems are solved and better limits than the traditional lagrangean relaxation are obtained. This work applies the LagClus to the Uncapacitated Facility