Optimal location of single and multiple obnoxious facilities: Algorithms for the maximin criterion under different norms.

This thesis investigates the computational problem of locating obnoxious (undesirable) facilities in a way that minimizes their effect on a given set of clients (e.g. population centres). Supposing that the undesirable effects of such a facility on a given client are a decreasing function of the distance between them the objective is to locate these facilities as far away as possible from the given set of clients, subject to constraints that prevent location at infinity. Emphasis is given to the MAXIMIN criterion which is to maximize the minimum client-to-facility distance. Distances are measured either in the Euclidean or the rectilinear metric. The properties of the optimal solution to the single facility problem are viewed from different, seemingly unrelated, perspectives ranging from plane geometry to duality theory. In particular, duality results from a mixed integer programming model are used to derive new properties of the optimal solution to the rectilinear problem. A new algorithm is developed for the rectilinear problem where the feasible region is a convex polygon. Unlike previous approaches, this method does not require linear programming at all. In addition to this, an interactive graphical approach is proposed as a site-generation tool used to identify potential locations in realistic problems. Its main advantages are that it requires minimal user intervention and makes no assumptions regarding the feasible region. It has been applied in large scale problems with up to 1000 clients, whereas the largest reported application so far involved 10 clients. Alternative models are presented for the multi-facility problem as well. Each of them is based on different assumptions and is applicable to specific situations. Moreover, an algorithm is established for the two-facility problem based on the properties of the optimal solution. To the best of our knowledge this is the first attempt to address this problem in the plane. Finally, a number of unresolved issues, especially in the multi-facility problem, are outlined and suggested as further research topics

Minisum and minimax transfer point location problem with random demands points

This paper is concerned with analyzing some models of the weighted transfer point location problem under the minisum and minimax criterions when demand points are randomly distributed over regions of the plane and the location of the service facility is known. In case of minisum objective with rectilinear distance, an iterative procedure was constructed for estimating the optimal transfer point location using the hyperbolic application procedure. Exact analytic solution was obtained when the random demand points follow uniform distributions. A unified analytic optimal solution was provided for all types of probability distributions of the random demand points when the distance is the squared Euclidean distance. For minimax objective with squared Euclidean distance, an iterative procedure based on Karush-Kuhn-Tucker conditions was developed to produce an approximate solution to the optimal solution. Illustrative numerical examples were provided

Minimax and Maximin Fitting of Geometric Objects to Sets of Points

This thesis addresses several problems in the facility location sub-area of computational geometry. Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function curve of size k \u3c n, i.e., by an x-monotone orthogonal polyline ℜ with k \u3c n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error ε. After O(n) preprocessing time, we solve instances of the latter in O(min{k log n, n}) time per instance. We can then solve the former problem in O(min{n2, nk log n}) time. Both algorithms require O(n) space. The second contribution is a heuristic for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k - 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O(n log n) time and O(n) space. Then, we present an exact algorithm for the weighted version of this problem that runs in O(n2) time and generalize the heuristic to handle weights at the expense of an additional log n factor. At this point, a randomized algorithm that runs in O(n log2 n) expected time for the unweighted version is presented. It easily generalizes to the weighted case, though at the expense of an additional log n factor. Finally, we treat the maximin problem and present an O(n3 log n) solution to the problem of finding the furthest separating line through a set of weighted points. We conclude with solutions to the obnoxious wedge problem: an O(n2 log n) algorithm for the general case of a wedge with its apex on the boundary of the convex hull of S and an O(n2) algorithm for the case of the apex of a wedge coming from the input set S

A review of network location theory and models

Cataloged from PDF version of article.In this study, we review the existing literature on network location problems. The study has a broad scope that includes problems featuring desirable and undesirable facilities, point facilities and extensive facilities, monopolistic and competitive markets, and single or multiple objectives. Deterministic and stochastic models as well as robust models are covered. Demand data aggregation is also discussed. More than 500 papers in this area are reviewed and critical issues, research directions, and problem extensions are emphasized.Erdoğan, Damla SelinM.S

An analysis of minimax facility location problems with area demands /

The unconstrained model, and its solution technique can be easily modified to solve the limiting case where all facilities are fixed points, and also the case when metric constraints are added.Examples are solved to show the impact of assuming area demands, the conflicting nature of the minimax and minisum criteria and to illustrate the solutions techniques developed.A minimax objective function constrained by a bound on the total average cost of servicing all existing facilities (minisum function) is then discussed. Using duality properties, this problem is shown to be equivalent to another model which minimizes the minisum function subject to a bound on the same minimax function. This last problem proves to be easier to solve, and a specialized solution technique is developed. The resulting solutions are nondominated solutions in relation to the two criteria involved. Another way to generate nondominated solutions is by combining the two functions into a weighted sum. The constrained criterion method is shown to be superior both analytically and practically.Most probabilistic facility location problems investigated to date were variations of the generalized Weber formulation. In this research, several single facility minimax location models are analyzed, where both the weights and the locations of the existing facilities are random variables. The demand points are uniformly distributed over rectangular areas, the rectilinear metric is used and the weights are assumed to be independently distributed random variables. Two unconstrained probabilistic models are analyzed and compared to the centroid formulation, it is seen that the probabilistic models are sensitive to deviations from optimal solutions. An expected value criterion formulation is also presented along with lower and upper bound approximating functions

Multifacility location problems on a sphere

A unified approach to multisource location problems on a sphere is presented. Euclidean, squared Euclidean and the great circle distances are considered. An algorithm is formulated and its convergence properties are investigated

Planar p-center problem with Tchebychev distance

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent Univ., 1994.Thesis (Master's) -- Bilkent University, 1994.Includes bibliographical references leaves 83-85.The p-center problem is a model for locating p facilities to serve clients so that the distance between a farthest client and its closest facility is minimized. Emergency service facilities such as fire stations, hospitals and police stations are most of the time located in this manner. In this thesis, the planar p-center problem with Tchebychev distance is studied. The problem is known to be NPHard. We identify certain polynomial time solvable cases and give an efficient branching method which makes use of polynomial time methods in subproblem solutions whenever possible. In addition, a dual problem is posed in light of the existing duality theory on tree networks.Yılmaz, DilekM.S

Problemas de localização-distribuição de serviços semiobnóxios: aproximações e apoio à decisão

Doutoramento em Gestão IndustrialA presente tese resulta de um trabalho de investigação cujo objectivo se centrou no problema de localização-distribuição (PLD) que pretende abordar, de forma integrada, duas actividades logísticas intimamente relacionadas: a localização de equipamentos e a distribuição de produtos. O PLD, nomeadamente a sua modelação matemática, tem sido estudado na literatura, dando origem a diversas aproximações que resultam de diferentes cenários reais. Importa portanto agrupar as diferentes variantes por forma a facilitar e potenciar a sua investigação. Após fazer uma revisão e propor uma taxonomia dos modelos de localização-distribuição, este trabalho foca-se na resolução de alguns modelos considerados como mais representativos. É feita assim a análise de dois dos PLDs mais básicos (os problema capacitados com procura nos nós e nos arcos), sendo apresentadas, para ambos, propostas de resolução. Posteriormente, é abordada a localização-distribuição de serviços semiobnóxios. Este tipo de serviços, ainda que seja necessário e indispensável para o público em geral, dada a sua natureza, exerce um efeito desagradável sobre as comunidades contíguas. Assim, aos critérios tipicamente utilizados na tomada de decisão sobre a localização destes serviços (habitualmente a minimização de custo) é necessário adicionar preocupações que reflectem a manutenção da qualidade de vida das regiões que sofrem o impacto do resultado da referida decisão. A abordagem da localização-distribuição de serviços semiobnóxios requer portanto uma análise multi-objectivo. Esta análise pode ser feita com recurso a dois métodos distintos: não interactivos e interactivos. Ambos são abordados nesta tese, com novas propostas, sendo o método interactivo proposto aplicável a outros problemas de programação inteira mista multi-objectivo. Por último, é desenvolvida uma ferramenta de apoio à decisão para os problemas abordados nesta tese, sendo apresentada a metodologia adoptada e as suas principais funcionalidades. A ferramenta desenvolvida tem grandes preocupações com a interface de utilizador, visto ser direccionada para decisores que tipicamente não têm conhecimentos sobre os modelos matemáticos subjacentes a este tipo de problemas.This thesis main objective is to address the location-routing problem (LRP) which intends to tackle, using an integrated approach, two highly related logistics activities: the location of facilities and the distribution of materials. The LRP, namely its mathematical formulation, has been studied in the literature, and several approaches have emerged, corresponding to different real-world scenarios. Therefore, it is important to identify and group the different LRP variants, in order to segment current research and foster future studies. After presenting a review and a taxonomy of location-routing models, the following research focuses on solving some of its variants. Thus, a study of two of the most basic LRPs (capacitated problems with demand either on the nodes or on the arcs) is performed, and new approaches are presented. Afterwards, the location-routing of semi-obnoxious facilities is addressed. These are facilities that, although providing useful and indispensible services, given their nature, bring about an undesirable effect to adjacent communities. Consequently, to the usual objectives when considering their location (cost minimization), new ones must be added that are able to reflect concerns regarding the quality of life of the communities impacted by the outcome of these decisions. The location-routing of semi-obnoxious facilities therefore requires to be analysed using multi-objective approaches, which can be of two types: noninteractive or interactive. Both are discussed and new methods proposed in this thesis; the proposed interactive method is suitable to other multi-objective mixed integer programming problems. Finally, a newly developed decision-support tool to address the LRP is presented (being the adopted methodology discussed, and its main functionalities shown). This tool has great concerns regarding the user interface, as it is directed at decision makers who typically don’t have specific knowledge of the underlying models of this type of problems

Localización simple de servicios deseados y no deseados en redes con múltiples criterios

Análisis y desarrollo de varios modelos de localización de servicios deseados y no deseados en redes con múltiples criterios. Asimismo, se han propuesto algunas mejoras en modelos de localización de servicios no deseados en redes con un solo criterio. Por consiguiente, con respecto a la localización de servicios deseados sobre redes, se propone un algoritmo polinomial para solucionar el problema del cent-dian biobjetivo. También se ha estudiado la localización de un servicio en una red con múltiples objetivos tipo mediana. Asimismo, se ha desarrollado un algoritmo polinomial para solucionar el problema cent-dian multicriterio en redes con múltiples pesos por nodo y múltiples longitudes por arista. Con respecto a los problemas de localización de servicios no deseados, primero tratamos el problema de localización del 1-centro no deseado en redes. Demostramos que las cotas superiores ya propuestas en trabajos anteriores pueden ser ajustadas. Por medio de una formulación más adecuada del problema, se ha desarrollado un nuevo algoritmo polinomial el cual es más sencillo y computacionalmente más rápido que los ya divulgados en la literatura. También se ha analizado el problema de localizar una mediana no deseada en una red, obteniendo una nueva y mejor cota superior. Se presenta un nuevo algoritmo para solucionar este problema. Por otra parte, siguiendo la resolución del problema maxian, también se ha propuesto un nuevo algoritmo para solucionar el problema del anti-cent-dian en redes. Finalmente, se han estudiado los problemas del centro no deseado y de la mediana no deseada en redes multicriterio, estableciendo nuevas propiedades y reglas para eliminar aristas ineficientes. También se presenta el modelo anti-cent-dian como combinación convexa de los dos últimos problemas. Se propone una regla eficaz para quitar aristas que contienen puntos ineficientes, así como un algoritmo polinomial. Además, este modelo se puede modificar ligeramente para generalizar otros modelos presentados en la literatura

Bottling plant location of microbreweries in East Midlands area, UK

Facility location decisions are critical in real-life projects, which impact on profitability of investment and service levels from demand side. In this paper, a project-based facility location problem should be resolved which refers to the establishment of a centralized bottling plant to serve microbreweries in East Midlands area of UK. This problem will be structured by firstly finding a mathematically theoretical location using the centre-of-gravity method and then formulate the problem as a multi-criteria decision making problem applying Analytical Hierarchy Process based on selection of the optimal location out of the four candidate locations where three of those have been given. The second part is modeled by considering several criteria related to both the activities before and after bottling and also issues of surrounding area of the location where the prioritization of those criteria are based on the preferences of the project investor. The final result is obtained by applying EXPERT CHOICE to approach Eigenvalue methods to enhance Analytical Hierarchy Process. The outcome can be clarified with illustration of the sensitivities resulted from the weight changes of criteria and the pull-out of certain criteria. Key Words: Facility Location, center-of-gravity method, Multi-criteria decision making, Analytical Hierarchy Proces