Covering point sets with two disjoint disks or squares

Open archive-ElsevierWe study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n8/3 log2 n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares SR and SB instead of unit disks can be solved in O(nlog n) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n3 log n) time

On multimodality of obnoxious faclity location models

Obnoxious single facility location models are models that have the aim to find the best location for an undesired facility. Undesired is usually expressed in relation to the so-called demand points that represent locations hindered by the facility. Because obnoxious facility location models as a rule are multimodal, the standard techniques of convex analysis used for locating desirable facilities in the plane may be trapped in local optima instead of the desired global optimum. It is assumed that having more optima coincides with being harder to solve. In this thesis the multimodality of obnoxious single facility location models is investigated in order to know which models are challenging problems in facility location problems and which are suitable for site selection. Selected for this are the obnoxious facility models that appear to be most important in literature. These are the maximin model, that maximizes the minimum distance from demand point to the obnoxious facility, the maxisum model, that maximizes the sum of distance from the demand points to the facility and the minisum model, that minimizes the sum of damage of the facility to the demand points. All models are measured with the Euclidean distances and some models also with the rectilinear distance metric. Furthermore a suitable algorithm is selected for testing multimodality. Of the tested algorithms in this thesis, Multistart is most appropriate. A small numerical experiment shows that Maximin models have on average the most optima, of which the model locating an obnoxious linesegment has the most. Maximin models have few optima and are thus not very hard to solve. From the Minisum models, the models that have the most optima are models that take wind into account. In general can be said that the generic models have less optima than the weighted versions. Models that are measured with the rectilinear norm do have more solutions than the same models measured with the Euclidean norm. This can be explained for the maximin models in the numerical example because the shape of the norm coincides with a bound of the feasible area, so not all solutions are different optima. The difference found in number of optima of the Maxisum and Minisum can not be explained by this phenomenon

Localização de serviços semiobnóxios: uma proposta de sistema de apoio à decisão

A localização de serviços semiobnóxios é caracterizada por critérios contraditórios, que podem ser expressos como forças de atracção e repulsão. Os serviços semiobnóxios definem-se como sendo desejáveis para as comunidades pelos benefícios que geram, mas que por outro lado são indesejáveis pelos efeitos negativos (obnóxios) que lhes estão associados. Exemplos de serviços semiobnóxios são: aeroportos, estações de tratamento de resíduos sólidos, estações de tratamento de águas e certos tipos de indústrias. Os agentes de decisão são assim confrontados com a necessidade de localizar esses serviços de modo a servir melhor as comunidades, simultaneamente reduzindo os impactos negativos. Outros factores tidos em conta, na localização de serviços semiobnóxios, são os custos relacionados com a distância dos serviços às comunidades que se pretendem mínimos. Pretende-se então, maximizar a acessibilidade aos serviços conjuntamente com a minimização dos efeitos obnóxios. Os modelos desenvolvidos para a resolução deste tipo de problemas são frequentemente apresentados numa linguagem mais orientada para especialistas em Investigação Operacional (IO) e menos adequada para os decisores nas diferentes organizações, quer sejam governamentais, autárquicas ou empresariais. O objectivo deste trabalho é o desenvolvimento de um Sistema de Apoio à Decisão para Problemas de Localização de Serviços Semiobnóxios. Com transferência de algum conhecimento científico da IO para os agentes de decisão, facilitando a escolha das localizações dos serviços ao apresentar alternativas de qualidade, tendo presente a simplicidade e a facilidade de utilizaçãoThe location of semi-obnoxious services is characterized by opposing criteria, which can be expressed as attraction and repulsion forces. The semi- obnoxious services can be defined as being desirable to the communities for the benefits they generate, but on the other side, are undesirable for their associated obnoxious effects. Examples of semi-obnoxious services are: airports, solid waste treatment facilities, water or effluent treatment facilities and certain types of industries. Therefore, the decision agents are faced with the need to locate those services in a way that they serve best the communities, while simultaneously reducing the negative effects. Other factors to hold in account in the location of semi-obnoxious services, are the costs concerning the distances between the services and the communities. Those costs are to minimize, which is equivalent to maximize the accessibility. Consequently, the problem becomes to maximize the accessibility while minimizing the obnoxious effects. It is frequent that the developed models to solve this type of problems are presented in an “Operations Research” (OR) experts oriented language. This may not be suitable for the decision agents at governmental or private organizations. The main goal of this work is to develop a Decision Support System for the Problem of Locating Semi-obnoxious Services. This way some of the OR scientific knowledge is transferred to the decision agents, allowing a usable system that provides quality location alternatives, making easier to select the best locations for the servicesMestrado em Gestã