Rafael Blanquero

Continuous location problems and Big Triangle Small Triangle: constructing better bounds

The Big Triangle Small Triangle method has shown to be a powerful global optimization procedure to address continuous location problems. In the paper published in J. Global Optim. (37:305–319, 2007), Drezner proposes a rather general and effective approach for constructing the bounds needed. Such bounds are obtained by using the fact that the objective functions in continuous location models can usually be expressed as a difference of convex functions. In this note we show that, exploiting further the rich structure of such objective functions, alternative bounds can be derived, yielding a significant improvement in computing times, as reported in our numerical experience.

A D.C. biobjective location model

In this paper we address the biobjective problem of locating a semiobnoxious facility, that must provide service to a given set of demand points and, at the same time, has some negative effect on given regions in the plane. In the model considered, the location of the new facility is selected in such a way that it gives answer to these contradicting aims: minimize the service cost (given by a quite general function of the distances to the demand points) and maximize the distance to the nearest affected region, in order to reduce the negative impact. Instead of addressing the problem following the traditional trend in the literature (i.e., by aggregation of the two objectives into a single one), we will focus our attention in the construction of a finite ε-dominating set, that is, a finite feasible subset that approximates the Pareto-optimal outcome for the biobjective problem. This approach involves the resolution of univariate d.c. optimization problems, for each of which we show that a d.c. decomposition of its objective can be obtained, allowing us to use standard d.c. optimization techniques.

Locating Objects in the Plane Using Global Optimization Techniques

We address the problem of locating objects in the plane such as segments, arcs of circumferences, arbitrary convex sets, their complements or their boundaries. Given a set of points, we seek the rotation and translation for such an object optimizing a very general performance measure, which includes as a particular case the classical objectives in semi-obnoxious facility location. In general, the above-mentioned model yields a global optimization problem, whose resolution is dealt with using difference of convex (DC) techniques such as outer approximation or branch and bound.

Inferring Efficient Weights from Pairwise Comparison Matrices

Several multi-criteria-decision-making methodologies assume the existence of weights associated with the different criteria, reflecting their relative importance.One of the most popular ways to infer such weights is the analytic hierarchy process, which constructs first a matrix of pairwise comparisons, from which weights are derived following one out of many existing procedures, such as the eigenvector method or the least (logarithmic) squares. Since different procedures yield different results (weights) we pose the problem of describing the set of weights obtained by “sensible” methods: those which are efficient for the (vector-) optimization problem of simultaneous minimization of discrepancies. A characterization of the set of efficient solutions is given, which enables us to assert that the least-logarithmic-squares solution is always efficient, whereas the (widely used) eigenvector solution is not, in some cases, efficient, thus its use in practice may be questionable.

On Covering Methods for D.C. Optimization

Covering methods constitute a broad class of algorithms for solving multivariate Global Optimization problems. In this note we show that, when the objective f is d.c. and a d.c. decomposition for f is known, the computational burden usually suffered by multivariate covering methods is significantly reduced. With this we extend to the (non-differentiable) d.c. case the covering method of Breiman and Cutler, showing that it is a particular case of the standard outer approximation approach. Our computational experience shows that this generalization yields not only more flexibility but also faster convergence than the covering method of Breiman-Cutler.

Optimization of the norm of a vector-valued DC function and applications

In this paper, we show that a DC representation can be obtained explicitly for the composition of a gauge with a DC mapping, so that the optimization of certain functions involving terms of this kind can be made by using standard DC optimization techniques. Applications to facility location theory and multiple-criteria decision making are presented.

Locating a competitive facility in the plane with a robustness criterion

A new model for locating a competitive facility in the plane in a robust way is presented and embedded in the literature on robustness in facility location. Its mathematical properties are investigated and new sharp bounds for a deterministic method that guarantees the global optimum are derived and evaluated.

On minimax-regret Huff location models

We address the following single-facility location problem: a firm is entering into a market by locating one facility in a region of the plane. The demand captured from each user by the facility will be proportional to the users buying power and inversely proportional to a function of the user-facility distance. Uncertainty exists on the buying power (weight) of the users. This is modeled by assuming that a set of scenarios exists, each scenario corresponding to a weight realization. The objective is to locate the facility following the Savage criterion, i.e., the minimax-regret location is sought. The problem is formulated as a global optimization problem with objective written as difference of two convex monotonic functions. The numerical results obtained show that a branch and bound using this new method for obtaining bounds clearly outperforms benchmark procedures.

Solving the median problem with continuous demand on a network

Where to locate one or several facilities on a network so as to minimize the expected users-closest facility transportation cost is a problem well studied in the OR literature under the name of median problem.In the median problem users are usually identified with nodes of the network. In many situations, however, such assumption is unrealistic, since users should be better considered to be distributed also along the edges of the transportation network. In this paper we address the median problem with demand distributed along edges and nodes. This leads to a global-optimization problem, which can be solved to optimality by means of a branch-and-bound with DC bounds. Our computational experience shows that the problem is solved in short time even for large instances.

p-facility Huff location problem on networks

The p-facility Huff location problem aims at locating facilities on a competitive environment so as to maximize the market share. While it has been deeply studied in the field of continuous location, in this paper we study the p-facility Huff location problem on networks formulated as a Mixed Integer Nonlinear Programming problem that can be solved by a branch-and-bound algorithm. We propose two approaches for the initialization and division of subproblems, the first one based on the straightforward idea of enumerating every possible combination of p edges of the network as possible locations, and the second one defining sophisticated data structures that exploit the structure of the combinatorial and continuous part of the problem. Bounding rules are designed using DC (difference of convex) and Interval Analysis tools.