Algunas aportaciones de la investigación operativa a los problemas de localización

La toma de decisiones sobre localizaciones atrae, por su impacto social y económico, creciente interés de geógrafos, economistas y matemáticos. En las páginas que siguen describimos algunas aportaciones que se están realizando desde las Matemáticas (más concretamente, desde la Investigación Operativa), tanto en el modelado, como en la resolución de los problemas de Análisis de Localizaciones

Problemas de clasificación y optimización

El desarrollo de técncias que permitan clasificar entes (seres vivos, elementos, problemas, ... ) en distintas categorías ha sido recurrente en diversas ramas del saber. La creación y difusión de grandes bases de datos y la consiguiente necesidad de extraer conocimiento de las mismas han revitalizado el interés de la comunidad científica por tales técnicas. En estas páginas se ilustra cómo la Programación Matemática puede contribuir al diseño de métodos autoináticos de clasificación y profundizar en el conociiniento teórico de los mismos. Nuestra intención no es hacer una revisión cotnpleta del estado del arte en el tema, sino más bien describir someramente las aportaciones que en esle campo se están realizando en el seno del grupo PAi FQM-809 y en el proyecto de investigación BFM2002-04525-C02-02 del MCYT.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia y Tecnologí

Binarized support vector machines

The widely used Support Vector Machine (SVM) method has shown to yield very good results in Supervised Classification problems. Other methods such as Classification Trees have become more popular among practitioners than SVM thanks to their interpretability, which is an important issue in Data Mining. In this work, we propose an SVM-based method that automatically detects the most important predictor variables, and the role they play in the classifier. In particular, the proposed method is able to detect those values and intervals which are critical for the classification. The method involves the optimization of a Linear Programming problem, with a large number of decision variables. The numerical experience reported shows that a rather direct use of the standard Column-Generation strategy leads to a classification method which, in terms of classification ability, is competitive against the standard linear SVM and Classification Trees. Moreover, the proposed method is robust, i.e., it is stable in the presence of outliers and invariant to change of scale or measurement units of the predictor variables. When the complexity of the classifier is an important issue, a wrapper feature selection method is applied, yielding simpler, still competitive, classifiers

The Stop Location Problem with Realistic Traveling Time

In this paper we consider the location of stops along the edges of an already existing public transportation network. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The positive effect of new stops is given by the better access of the customers to the public transport network, while the traveling time increases due to the additional stopping activities of the trains which is a negative effect for the customers. Our goal is to locate new stops minimizing a realistic traveling time which takes acceleration and deceleration of the vehicles into account. We distinguish two variants: in the first (academic) version we locate $p$ stops, in the second (real-world applicable) version the goal is to cover all demand points with a minimal amount of realistic traveling time. As in other works on stop location, covering may be defined with respect to an arbitrary norm. For the first version, we present a polynomial approach while the latter version is NP-hard. We derive a finite candidate set and an IP formulation. We discuss the differences to the model neglecting the realistic traveling time and provide a case study showing that our procedures are applicable in practice and do save in average more than 3% of traveling time for the passengers

Binarized support vector machines

The widely used Support Vector Machine (SVM) method has shown to yield very good results in Supervised Classification problems. Other methods such as Classification Trees have become more popular among practitioners than SVM thanks to their interpretability, which is an important issue in Data Mining. In this work, we propose an SVM-based method that automatically detects the most important predictor variables, and the role they play in the classifier. In particular, the proposed method is able to detect those values and intervals which are critical for the classification. The method involves the optimization of a Linear Programming problem, with a large number of decision variables. The numerical experience reported shows that a rather direct use of the standard Column-Generation strategy leads to a classification method which, in terms of classification ability, is competitive against the standard linear SVM and Classification Trees. Moreover, the proposed method is robust, i.e., it is stable in the presence of outliers and invariant to change of scale or measurement units of the predictor variables. When the complexity of the classifier is an important issue, a wrapper feature selection method is applied, yielding simpler, still competitive, classifiers.Supervised classification, Binarization, Column generation, Support vector machines

Heliostat field cleaning scheduling for Solar Power Tower plants: a heuristic approach

Soiling of heliostat surfaces due to local climate has a direct impact on their optical efficiency and therefore a direct impact on the productivity of the Solar Power Tower plant. Cleaning techniques applied are dependent on plant construction and current schedules are normally developed considering heliostat layout patterns, providing sub-optimal results. In this paper, a method to optimise cleaning schedules is developed, with the objective of maximising energy generated by the plant. First, an algorithm finds a cleaning schedule by solving an integer program, which is then used as a starting solution in an exchange heuristic. Since the optimisation problems are of large size, a p-median type heuristic is performed to reduce the problem dimensionality by clustering heliostats into groups to be cleaned in the same period.Ministerio de Economía y Competitivida

Supervised Classification and Mathematical Optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data

Location and design of a competitive facility for profit maximisation

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits. It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favor of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in ill-posed problems.Dirección General de Enseñanza Superio

A dissimilarity-based approach for Classification

The Nearest Neighbor classifier has shown to be a powerful tool for multiclass classification. In this note we explore both theoretical properties and empirical behavior of a variant of such method, in which the Nearest Neighbor rule is applied after selecting a set of so-called prototypes, whose cardinality is fixed in advance, by minimizing the empirical mis-classification cost. With this we alleviate the two serious drawbacks of the Nearest Neighbor method: high storage requirements and time-consuming queries. The problem is shown to be NP-Hard. Mixed Integer Programming (MIP) programs are formulated, theoretically compared and solved by a standard MIP solver for problem instances of small size. Large sized problem instances are solved by a metaheuristic yielding good classification rules in reasonable time.operations research and management science;