Multiobjective solution of the uncapacitated plant location problem

In this paper we consider the discrete multiobjective uncapacitated plant location problem. We present an exact and an approximate approach to obtain the set of non-dominated solutions. The two approaches resort to dynamic programming to generate in an efficient way the non-dominated solution sets. The solution methods that solve the problems associated with the generated states are based on the decomposition of the problem on two nested subproblems. We define lower and upper bound sets that lead to elimination tests that have shown to have a high performance. Computational experiments on a set of test problems show the good performance of the proposal

Two-phase strategies for the bi-objective minimum spanning tree problem

This paper presents a new two-phase algorithm for the bi-objective minimum spanning tree (BMST) prob-lem. In the first phase, it computes the extreme supported efficient solutions resorting to both mathematicalprogramming and algorithmic approaches, while the second phase is devoted to obtaining the remaining ef-ficient solutions (non-extreme supported and non-supported). This latter phase is based on a new recursiveprocedure capable of generating all the spanning trees of a connected graph through edge interchanges basedon increasing evaluation of non-zero reduced costs of associated weighted linear programs. Such a procedureexploits a common property of a wider class of problems to which the minimum spanning tree (MST) prob-lem belongs, that is the spanning tree structure of its basic feasible solutions. Computational experimentsare conducted on different families of graphs and with different types of cost. These results show that thisnew two-phase algorithm is correct, very easy to implement and it allows one to extract conclusions on thedifficulty of finding the entire set of Pareto solutions of the BMST problem depending on the graph topologyand the possible correlation of the edge cost

Partial Gröbner bases for multiobjective integer linear optimization

This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gr¨obner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gr¨obner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “´a la Buchberger” is developed to compute partial Gr¨obner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.Ministerio de Educación y Cienci

New models for the location of controversial facilities: A bilevel programming approach

Motivated by recent real-life applications in Location Theory in which the location decisions generate controversy, we propose a novel bilevel location model in which, on the one hand, there is a leader that chooses among a number of fixed potential locations which ones to establish. Next, on the second hand, there is one or several followers that, once the leader location facilities have been set, chooses his location points in a continuous framework. The leader’s goal is to maximize some proxy to the weighted distance to the follower’s location points, while the follower(s) aim is to locate his location points as close as possible to the leader ones. We develop the bilevel location model for one follower and for any polyhedral distance, and we extend it for several followers and any ℓp-norm, p ∈ Q, p ≥ 1. We prove the NP-hardness of the problem and propose different mixed integer linear programming formulations. Moreover, we develop alternative Benders decomposition algorithms for the problem. Finally, we report some computational results comparing the formulations and the Benders decompositions on a set of instances.Fonds de la Recherche Scientique - FNRSMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Averaging the k largest distances among n: k-centra in Banach spaces

Given a Banach space X let A ⊂ X containing at least k points. In location theory, reliability analysis, and theoretical computer science, it is useful to minimize the sum of distances from the k furthest points of A: this problem has received some attention for X a finite metric space (a network), see, e.g., [Discrete Appl. Math. 109 (2001) 293]; in the case X = En, k = 2 or 3, and A compact some results have been given in [Math. Notes 59 (1996) 507]; also, in the field of theoretical computer science it has been considered in [T. Tokuyama, Minimax parametric optimization problems in multidimensional parametric searching, in: Proc. 33rd Annu. ACM Symp. on Theory of Computing, 2001, pp. 75–84]. Here we study the above problem for a finite set A ⊂ X, generalizing—among others things—the results in [Math. Notes 59 (1996) 507].Ministerio de Ciencia y Tecnologí

Extensive facility location problems on networks with equity measures

AbstractThis paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case

A discretization result for some optimization problems in framework spaces with polyhedral obstacles and the Manhattan metric

In this work we consider the shortest path problem and the single facility Weber location problem in any real space of finite dimension where there exist different types of polyhedral obstacles or forbidden regions. These regions are polyhedral sets and the metric considered in the space is the Manhattan metric. We present a result that reduce these continuous problems into problems in a “add hoc” graph, where the original problems can be solved using elementary techniques of Graph Theory. We show that, fixed the dimension of the space, both the reduction and the resolution can be done in polynomial time.Ministerio de Economía and CompetitividadFondo Europeo de Desarrollo Regiona

An application of integer programming to the decomposition of numerical semigroups

This paper addresses the problem of decomposing a numerical semigroup into mirreducible numerical semigroups. The problem originally stated in algebraic terms is translated, introducing the so-called Kunz-coordinates, to resolve a series of several discrete optimization problems. First, we prove that finding a minimal m-irreducible decomposition is equivalent to solve a multiobjective linear integer problem. Then, we restate that problem as the problem of finding all the optimal solutions of a finite number of single objective integer linear problems plus a set covering problem. Finally, we prove that there is a suitable transformation that reduces the original problem to find an optimal solution of a compact integer linear problem. This result ensures a polynomial time algorithm for each given multiplicity m. We have implemented the different algorithms and have performed some computational experiments to show the efficiency of our methodology.Ministerio de Educación y CienciaJunta de AndalucíaFondo Europeo de Desarrollo Regiona

Modelos de localización continua

En este trabajo se revisan tres modelos de problemas de localización continua: (1) un problema general de localización con respecto a regiones de demanda; (2) el problema de la mediana ordenada y (3) un problema de localización multiobjetivo. Con ellos se pretende dar una amplia muestra de los problemas que aparecen en el ámbito de la Teoría de Localización continua, así como estudiar propiedades que permitan caracterizar los conjuntos de soluciones. El trabajo incluye una larga lista de referencias que facilitarán al lector adentrarse más profundamente en éstos y otros modelos de la Teoría de Localización.Ministerio de Ciencia y Tecnologí