Lázaro Cánovas

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.

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.

On the facets of the simple plant location packing polytope

We introduce new classes of facet-defining inequalities for the polytope Ppd associated with the set packing formulation of the simple plant location problem (SPLP) with p plants and d destinations. The inequalities are obtained by identifying subgraphs of the intersection graph G(p,d) of SPLP that are facet-defining, and lifting their associated facets if it is necessary. To this end, we find subfamilies of previously known structured families of facet-defining graphs, like fans and wheels, inside G(p,d). We also characterize a class of facets of SPLP and summarize the previous polyhedral results on this problem.

On the convergence of the Weiszfeld algorithm

Abstract.In this work we analyze the paper “Brimberg, J. (1995): The Fermat-Weber location problem revisited. Mathematical Programming 71, 71–76” which claims to close the question on the conjecture posed by Chandrasekaran and Tamir in 1989 on the convergence of the Weiszfeld algorithm. Some counterexamples are shown to the proofs showed in Brimberg’s paper.

New facets for the set packing polytope

We introduce a family of graphs, named grilles, and a facet of the set packing polytope associated with a grille. The proof is based on a new facet generating procedure which is valid in a wider context. We also obtain new facets for the simple plant location polytope.

Algorithms for the decomposition of a polygon into convex polygons

Abstract Decomposing a non-convex polygon into simpler subsets has been a recurrent theme in the literature due to its many applications. In this paper, we present different algorithms for decomposing a polygon into convex polygons without adding new vertices as well as a procedure, which can be applied to any partition, to remove the unnecessary edges of a partition by merging the polygons whose union remains a convex polygon. Although the partitions produced by the algorithms may not have minimal cardinality, their easy implementation and their quick CPU times even for polygons with many vertices make them very suitable to be used when optimal decompositions are not required, as for instance, in constrained optimization problems having as feasible set a non-convex polygon (optimization problems are usually easier to solve in convex regions and making use of a branch and bound process or other techniques, it is not necessary to find the optimal solution in all the subsets, so finding convex decomposition with minimal cardinality may be time-wasting). Computational experiments are presented and analyzed.

Facet Obtaining Procedures for Set Packing Problems

New results concerning the facial structure of set packing polyhedra are presented. In particular, new methods are given to obtain facets from the subgraphs of the intersection graph associated with a set packing polyhedron that properly subsume several other methods in the literature. A new class of facet defining graphs, termed fans, is also introduced, as well as a procedure to link any graph to a certain claw K1,k in order to obtain a new graph and an associated facet.

A new assignment rule to improve seed points algorithms for the continuous k-center problem

Abstract Due to its complexity, the k-center problem cannot be solved exactly when the number of points is high enough. So, heuristic algorithms must be used to find good solutions for this location-allocation problem. The allocation part of the heuristics given in the literature consists of assigning each given point to its nearest center. In this paper, we propose a new assignment rule (NAR), from which a new class of seed points algorithms is obtained for the k-center problem in R n . The new class differs from the usual class of seed points algorithms in the way the points are assigned to each seed point. Different methods to generate seed points are shown from which different algorithms are given by using the two assignment rules, the classical and the new. Most of these algorithms have linear complexity in the number of points and generate solutions with objective values within two times the optimal value, and all of them run very quickly and can be used for any metric. It is shown by computational experience that the running times and the quality of the solutions generated are notably improved by using the NAR.

The Minimum Covering l_pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in ℝn. A hypersphere is a set S(c,r)={x ∈ ℝn : d(x,c) ≤ r}, where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l2(x-c). We consider the extension of this problem when d(x, c) is given by any lpb-norm, where 10, j=1,...,n, then S(c,r) is called an lpb-hypersphere, in particular for p=2 and bj=1, j=1,..., n, we obtain the l2-norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallest lpb-hypersphere enclosing a large set of points in ℝn.