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

Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving $(SDP)$ that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm

Robustness in facility location

Facility location concerns the placement of facilities, for various objectives, by use of mathematical models and solution procedures. Almost all facility location models that can be found in literature are based on minimizing costs or maximizing cover, to cover as much demand as possible. These models are quite efficient for finding an optimal location for a new facility for a particular data set, which is considered to be constant and known in advance. In a real world situation, input data like demand and travelling costs are not fixed, nor known in advance. This uncertainty and uncontrollability can lead to unacceptable losses or even bankruptcy. A way of dealing with these factors is robustness modelling. A robust facility location model aims to locate a facility that stays within predefined limits for all expectable circumstances as good as possible. The deviation robustness concept is used as basis to develop a new competitive deviation robustness model. The competition is modelled with a Huff based model, which calculates the market share of the new facility. Robustness in this model is defined as the ability of a facility location to capture a minimum market share, despite variations in demand. A test case is developed by which algorithms can be tested on their ability to solve robust facility location models. Four stochastic optimization algorithms are considered from which Simulated Annealing turned out to be the most appropriate. The test case is slightly modified for a competitive market situation. With the Simulated Annealing algorithm, the developed competitive deviation model is solved, for three considered norms of deviation. At the end, also a grid search is performed to illustrate the landscape of the objective function of the competitive deviation model. The model appears to be multimodal and seems to be challenging for further research

Modelos bicritério para a localização de serviços semiobnóxios com restrições de capacidade por níveis

Tese de mestrado em Investigação Operacional, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2010O problema de localização de serviços é uma das componentes mais importantes no planeamento estratégico aplicado quer no sector público quer no sector privado. Assume maior relevância quando se trata de decidir onde instalar serviços que, tais como, estações de tratamento de águas ou resíduos, exercem um impacto negativo sobre o ambiente. Estes serviços, designados por serviços semiobnóxios são simultaneamente de utilidade pública e ao mesmo tempo prejudiciais para as comunidades que usufruem dos seus serviços. Neste trabalho apresentam-se quatro modelos, em programação linear inteira, para o problema de localização de serviços semiobnóxios com restrições de capacidade por níveis. Estes modelos diferem nos objectivos e nas funções usadas para calcular o efeito obnóxio. Em dois dos modelos minimiza-se os custos totais e o efeito obnóxio total, nos outros dois modelos minimiza-se os custos totais e o máximo efeito obnóxio sobre um indivíduo. Para calcular o efeito obnóxio é considerada uma função inversamente proporcional ao cubo da distância euclidiana ou uma função da distância euclidiana linear por partes. As soluções não dominadas foram obtidas usando um método interactivo desenvolvido por Ferreira, Clímaco, & Paixão, (1994) para problemas bicritério. Os resultados computacionais foram obtidos para exemplos gerados com base em informação relativa ao território continental português. As coordenadas geográficas e população de cada município correspondem a dados reais, sendo todos os outros dados gerados aleatoriamente. Os quatro modelos considerados para o problema em estudo foram comparados em termos de tempos médios de execução e de equidade das soluções não dominadas obtidas.The facility location problem is one of the most important components of strategic planning in the public sector or in the private sector. It has higher relevance when deciding where facilities such as waste disposals and water treatment facilities will be installed, that have a more prominent negative impact on the environment will be installed. These facilities, called semi-obnoxious facilities are simultaneously of public use and at the same time prejudicial to the communities who use these services. In this project four models in integer linear programming are presented, for the modular semi-obnoxious facility location problem. These models differ in the objectives and functions used to calculate the obnoxious effect. In two of the models the total cost and total obnoxious effect are minimized, in the other two models, the total cost and maximum obnoxious effect on one individual are minimized. The obnoxious effect is calculated either by a function inversely proportional to the cube of the euclidean distance or by a piecewise linear function of the euclidean distance. The non dominated solutions were obtained using an interactive method developed by (Ferreira, et al.) for bicriteria problems. The computational results were obtained for examples generated on the basis of information relative to the Portuguese territory. The geographical coordinates and population of each municipality correspond to real data, while all the other data were randomly generated. The four models presented for the study case were compared in terms of average CPU times and equity of the non dominated solutions obtained

Discrete location problems with push-pull objectives

AbstractThe models within operational research concerned with locational decisions mostly either consider only the positive effects, pulling the facilities towards demand, or only negative effects, pushing the facilities away from the places affected by the facilities nearness. In real-world situations both of these opposing forces are at work. We give an overview of a number of push–pull models, yielding alternative ways to incorporate both types of effects simultaneously. The discussion is restricted to models of combinatorial optimisation and includes indications of reduction to standard models and/or algorithmic approaches where possible

Location of an agribusiness enterprise with respect to economic viability: a risk analysis

This study analyzes the economic and geographic effects of alternative locations on risky investment decisions in a probabilistic framework. Historically, alternative locations for multi-million dollar investments are often evaluated with deterministic models that rely on expected values or best case/worst case scenarios. Stochastic simulation was used to estimate the probability distribution for select key output variables, including net present value (NPV), of a proposed biomass to ethanol production facility in three alternative regions in Texas. The simulated NPV probability distributions were compared using Stochastic Efficiency with Respect to a Function (SERF) to predict the location preference of decision makers with alternative levels of risk aversion. Risk associated with input availability and costs were analyzed for the proposed plant locations so each location resulted in different levels of economic viability and risk that would not have been observed with a traditional deterministic analysis. For all analyzed scenarios, the projected financial feasibility results show a positive NPV over the 16 year planning horizon with a small probability of being negative. The SERF results indicate the Central Region of Texas is preferred for risk averse decision makers compared to the Panhandle and Coastal Bend Regions. Risk premiums were calculated for the alternative locations and are consistent for all risk averse decision makers, indicating the ranking of alternative locations are robust. Positive community impacts and sensitivity elasticities for key variables were estimated in the model. The estimated positive economic gains for the local economy are quite large and indicate locating a production facility in the region could substantially impact the local economy. The calculated sensitivity elasticities show ethanol price, ethanol yield, and hydrogen price are the three variables that have the greatest affect on the feasibility of a biomass to ethanol production facility

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