Alfredo Marín

The Tree of Hubs Location Problem

This paper presents the Tree of Hubs Location Problem. It is a network hub location problem with single assignment where a fixed number of hubs have to be located, with the particularity that it is required that the hubs are connected by means of a tree. The problem combines several aspects of location, network design and routing problems. Potential applications appear in telecommunications and transportation systems, when set-up costs for links between hubs are so high that full interconnection between hub nodes is prohibitive. We propose an integer programming formulation for the problem. Furthermore, we present some families of valid inequalities that reinforce the formulation and we give an exact separation procedure for them. Finally, we present computational results using the well-known AP and CAB data sets.

The return plant location problem : Modelling and resolution

Abstract The formulation and analysis of a new plant location problem is presented. The problem studied, herein referred to as the Return Plant Location Problem (RPLP), is that of cost minimization in a system of suppliers and customers in which there exists a return product from each customer. Lagrangian decomposition based heuristic and exact solution methods are given. The methods are applied to test problems with different structures and compared with the classical subgradient optimization approach.

New formulations for the uncapacitated multiple allocation hub location problem

In this paper we review the integer linear formulations of the uncapacitated multiple allocation hub location problem, we study the scope of validity of these formulations and give new ones that generalize the older formulations. Our formulations allow one or two visits to hubs and include a more general cost structure that needs not satisfy the triangle inequality. We prove that the constraints defined by cliques of a related (intersection) graph are tighter constraints than the classical ones. We also discuss a pre-processing of the problem, which is very useful for reducing its size. Finally, we check the strength of the new formulations and compare them with others in the literature by solving instances of two commonly used data sets: the CAB (Civil Aeronautics Board) and AP (Australian Post), and also randomly generated instances. Our computational results clearly show that our formulations outperform all others previously used for small and medium problems.

Solving the uncapacitated multiple allocation hub location problem by means of a dual-ascent technique

This paper deals with the uncapacitated multiple allocation hub location problem. The dual problem of a four-indexed formulation is considered and a heuristic method, based on a dual-ascent technique, is designed. This heuristic, which is reinforced with several specifical subroutines and does not require any external linear problem solver, is the core tool embedded in an exact branch-and-bound framework. Besides, the heuristic provides the branch-and-bound algorithm with good lower bounds for the nodes of the branching tree. The results of the computational experience (with the classical CAB and AP data sets) are included, showing the great effectiveness of this approach: instances with up to 120 nodes are solved.

Formulating and solving splittable capacitated multiple allocation hub location problems

It is only recently that good formulations and properties for the basic versions of the hub location problem have become available. Now, versions closer to reality can be tackled with greater guarantees of success. This article deals with the case in which the capacity of the hubs is limited. The focus is on the following interpretation of this capacity: there is, for each hub, an upper bound on the total flow coming directly from the origins. Our problem has the so-called multiple allocation possibility, i.e., there is no hub associated to each node; on the contrary, flows with, say, the same origin but different destinations, can be sent through different routes. Moreover, it is assumed that the flow between a given origin-destination pair can be split into several routes; if this is not the case, the problem becomes quite different and cannot be approached by means of the techniques used in this paper. Tight integer linear programming formulations for the problem are presented, along with some useful properties of the optimal solutions which can be used to speed up the resolution. The computational experience shows that instances of medium size can be solved very efficiently using the new method, which outperforms other methods given in the literature.

Tight bounds from a path based formulation for the tree of hub location problem

This paper considers the tree of hub location problem. We propose a four index formulation which yields much tighter LP bounds than previously proposed formulations, although at a considerable increase of the computational burden when obtained with a commercial solver. For this reason we propose a Lagrangean relaxation, based on the four index formulation, that exploits the structure of the problem by decomposing it into independent subproblems which can be solved quite efficiently. We also obtain upper bounds by means of a simple heuristic that is applied at the inner iterations of the method that solves the Lagrangean dual. As a consequence, the proposed Lagrangean relaxation produces tight upper and lower bounds and enable us to address instances up to 100 nodes, which are notably larger than the ones previously considered in the literature, with sizes up to 20 nodes. Computational experiments have been performed with benchmark instances from the literature. The obtained results are remarkable. For most of the tested instances we obtain or improve the best known solution and for all tested instances the deviation between our upper and lower bounds, never exceeds 10%.

Solving Large p-Median Problems with a Radius Formulation

By means of a model based on a set covering formulation, it is shown how the p-median problem can be solved with just a column generation approach that is embedded in a branch-and-bound framework based on dynamic reliability branching. This method is more than competitive in terms of computational times and size of the instances that have been optimally solved. In particular, problems of a size larger than the largest ones considered in the literature up to now are solved exactly in this paper.

A flexible model and efficient solution strategies for discrete location problems

Flexible discrete location problems are a generalization of most classical discrete locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer, which depends on the position of that cost relative to the costs of fulfilling the demand of other customers. In this paper a covering type of model for the discrete ordered median problem is presented. For the solution of this model two sets of valid inequalities, which reduces the number of binary variables tremendously, and several variable fixing strategies are identified. Based on these concepts a specialized branch & cut procedure is proposed and extensive computational results are reported.

A strengthened formulation for the simple plant location problem with order

The simple plant location problem with order, a generalization of the well-known simple plant location problem where preferences for the customers are considered, is studied here. Some valid inequalities are introduced as well as a basic preprocessing analysis. A computational study shows the efficiency of this approach.

Applying Lagrangian relaxation to the resolution of two-stage location problems

A family of two‐stage location problems is considered. These problems involve a systemproviding a choice of depots andyor plants, each with an associated location cost, and a setof demand points which must be supplied, in such a way that the total cost is minimized. Byusing Lagrangian relaxation, lower bounds and heuristic solutions are obtained for two kindsof formulation involving two or three indexes for each transportation variable. The goodnessof both formulations is compared by means of a computational study, and the results forseveral instances are presented.