Determining the set of all optimal solutions of a linear program with interval data is one of the main problems discussed in interval optimization. We review two methods based on duality in linear programming, which are used to approximate the optimal set. Additionally, another decomposition method based on complementary slackness is proposed. This method provides the exact description of the optimal set for problems with a fixed coefficient matrix. The second part of the thesis is focused on studying the topological and geometric properties of the optimal set. We examine sufficient conditions for closedness, boundedness, connectedness and convexity. We also prove that testing boundedness is co- NP-hard for inequality-constrained problems with free variables. Stronger results are derived for some special classes of interval linear programs, such as problems with a fixed coefficient matrix. Furthermore, we study the effect of transformations commonly used in linear programming on interval problems, which allows for a direct generalization of some results to different types of interval linear programs. Powered by TCPDF (www.tcpdf.org)

As we are currently in the information age, people expect access to information to exist by default. In order to facilitate the communication of knowledge, efficient networks must be built. In particular, the networks must be built satisfying some constraints while minimizing cost or time. These constraints often make these problems NP-hard. In this thesis, we investigate two different sets of problems: communicating information quickly and building cheap networks. In the communication problems, the goal is to minimize the number of rounds of communication. In the network design problems the goal is to construct a network of minimum cost.

This dissertation addresses the problem of optimally scheduling the hoists of a twin rock winder system in a demand side management context. The objective is to schedule the hoists at minimum energy cost taking into account various physical and operational constraints and production requirements as well as unplanned system delays. The problem is solved by first developing a static linear programming model of the rock winder system. The model is built on a discrete dynamic winder model and consists of physical and operational winder system constraints and an energy cost based objective function. Secondly a model predictive control based scheduling algorithm is applied to the model to provide closed-loop feedback control. The scheduling algorithm first solves the linear programming problem before applying an adapted branch and bound integer solution methodology to obtain a near optimal integer schedule solution. The scheduling algorithm also compensates for situations resulting in infeasible linear programming solutions. The simulation results show the model predictive control based scheduling algorithm to be able to successfully generate hoist schedules that result in steady state solutions in all scenarios studied, including where delays are enforced. The energy cost objective function is proven to be very effective in ensuring minimal hoisting during expensive peak periods and maximum hoisting during low energy cost off-peak periods. The algorithm also ensures that the hoist target is achieved while controlling all system states within or around their boundaries for a sustainable and continuous hoist schedule. Copyright / Dissertation (MEng)--University of Pretoria, 2010. / Electrical, Electronic and Computer Engineering / Unrestricted

As the "network of network", the Internet has been playing a central and crucial role in modern society, culture, knowledge, businesses and so on in a period of over two decades by supporting a wide variety of network technologies and applications. However, due to its popularity and multi-provider nature, the future development of the Internet is limited to simple incremental updates. To address this challenge, network virtualization has been propounded as a potential candidate to provide the essential basis for the future Internet architecture. Network virtualization is capable of providing an open and flexible networking environment in which service providers are allowed to dynamically compose multiple coexisting heterogeneous virtual networks on a shared substrate network. Such a flexible environment will foster the deployment of diversified services and applications. A major challenge in network virtualization area is the Virtual Network Embedding (VNE), which aims to statically or dynamically allocate virtual nodes and virtual links on substrate resources, physical nodes and paths. Making effective use of substrate resources requires high-efficient and survivable VNE techniques. The main contribution of this thesis is two high-performance p-Cycle-based survivable virtual network embedding approaches. These approaches take advantage of p-Cycle-based protection techniques that minimize the backup resources while providing a full VN protection scheme against link and node failures.

The thesis describes a way of searching the best bill of fare for certain patient in hospital. The target is to set up and then to solve an optimization model. Criterion of performance is to minimize spending on selected foodstuff. The first part describes a theory of optimization models, the second is about nutrition and the third aims to solve a model and to interpret the results.

Municipalities in the Czech Republic deal with their budget, which among others consists of granted subsidies from the region, State, European Union or other organizations. Nowadays the budget transactions are not being under the control. In the future, it is appropriate to introduce some external view to control their spending. Establishment of an audit service for each municipality would be financially inevitable. Therefore it is suggested to provide the State audit services only in some municipalities and to share them with more municipalities within one region. Deployment of the audit centers and assigning municipalities lead to solving a linear problem which falls under the covering problem class. The establishment of audit centers is only illustrative, the employment of more shared state services could follow a similar principle.

This thesis deals with a creation of a timetable on Business College. In the first part the areas in which we can encounter scheduling are introduced, because timetabling undoubtedly belongs to scheduling tasks. Then the linear programming theory is explained. With its help a model for a timetable is made. The assignment problem, which forms a basis for timetable creation model, will be mentioned. We cannot imagine all that without the support of computers. That is why one chapter is dedicated to useable computer programs. The next chapter discusses didactics and its requirements for a good timetable. These requirements are then taken into account in an application chapter. Fundamental part of the thesis is the creation of the timetable for real Business College, taking into account all important conditions. At the end of this thesis a mathematical model is introduced which is suitable for creating timetables for twenty-four classes of this school.

This diploma focuses on the real use of operations research methods in conjunction with logistics technologies. The aim is to show the way of optimization and simulation methods application and to evaluate if these methods are suitable tools for logistics technology application. Following logistic technologies were analysed: Just in Time, Kanban, Cross-Docking and Hub and Spoke. Used mathematical tools include: mixed linear programming models, distribution models, methods of multi-criteria evaluation and simulation models. On the basis of mathematical models 4 examples of new technologies introduction or analysis of already functioning systems have been solved.

Orientadores: Aurélio Ribeiro Leite de Oliveira, Marta Ines Velazco Fontova / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T12:49:39Z (GMT). No. of bitstreams: 1 Coelho_AlessandroFonsecaEsteves_M.pdf: 2659631 bytes, checksum: fc1bec925179612ee07a4aaef7092d8a (MD5) Previous issue date: 2011 / Resumo: Um método frequentemente utilizado para a solução de problemas de programação linear é o método de pontos interiores. Nestes métodos precisamos resolver sistemas lineares para calcular a direção de Newton a cada iteração. A solução desses sistemas consiste no passo de maior esforço computacional nos métodos de pontos interiores. A fatoração de Cholesky é a opção mais utilizada para resolver estes sistemas. Contudo, quando trabalhamos com problemas de grande porte, esta fatoração pode ser densa e torna-se inviável trabalhar com esses métodos. Nestes casos, uma boa opção consiste no uso de métodos iterativos precondicionados. Estudos anteriores utilizam o método dos gradientes conjugados precondicionado para obter uma solução destes sistemas. Particularmente, os sistemas originados dos métodos de pontos interiores, são, naturalmente, sistemas de equações normais. Porém, a versão padrão do método dos gradientes conjugados, não considera a estrutura de equações normais do sistema. Neste trabalho propomos a utilização de duas versões do método de gradientes conjugados precondicionado que consideram a estrutura de equações normais destes sistemas. Estas versões serão comparadas com a versão de gradientes conjugados precondicionada que não considera a estrutura de equações normais do sistema. Resultados numéricos com problemas de grande porte mostram que uma dessas versões é competitiva em relação à versão padrão / Abstract: An often used method for solving linear programming problems is the interior point method. In these methods we need to solve linear systems to compute the Newton search direction at each iteration. The solution of these systems is the procedure of most computational effort in interior point methods. The Cholesky factorization is the most often used method to solve these systems. However, when dealing with large scale problems, this factorization can be dense and it become impossible to apply such methods. In such cases, a good option is the use of preconditioned iterative methods. Previous studies have used the preconditioned conjugate gradient method to find the solution of these systems. Particularly, the systems arising from interior point methods are, naturally, systems of normal equations type. Nevertheless, the standard version of the conjugate gradient method, does not take into account the normal equations system structure. This study proposes the use of two versions of preconditioned conjugate gradient method considering the normal equations structure of these systems. These versions are compared with the preconditioned conjugate gradient version that does not consider that structure. Numerical results with large scale problems show that one of these versions is competitive with the standard one / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

Orientador: Aurelio Ribeiro Leite de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-24T06:40:02Z (GMT). No. of bitstreams: 1 SunaguaSalgado_Porfirio_D.pdf: 2121161 bytes, checksum: 6d961fd2da8ded7cbc0733d9f190497a (MD5) Previous issue date: 2014 / Resumo: Uma abordagem do método preditor-corretor de Mehrotra para resolver problemas de programação linear de grande porte, que utiliza uma classe do precondicionador separador, para resolver sistemas lineares envolvidos por métodos iterativos precondicionados, necessita de uma base que é encontrada por um sofisticado processo de fatoração LU retangular da matriz de restrições. O precondicionador separador tem bom desempenho perto de uma solução ótima, onde as matrizes envolvidas ficam muito mal condicionadas. Neste trabalho, primeiro desenvolvemos o método preditor-corretor com o parâmetro de penalização a fim de reduzir o mau condicionamento da matriz de equações normais. O sucesso desta abordagem é garantido pela demonstração do teorema de convergência de penalização mista com o parâmetro de barreira. Em seguida, implementamos uma nova abordagem para encontrar uma base para o precondicionador separador mediante um processo de fatoração LU retangular padrão aplicada à matriz transposta de restrições escalada. Na maioria das vezes, esta base encontrada é melhor condicionada que a base do método de fatoração retangular anterior. Testes computacionais comprovam uma redução da média do número de iterações do método de gradientes conjugados precondicionado. Também, a eficácia e a robustez da nova abordagem é comprovada por conseguir uma melhor curva de desempenho / Abstract: The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra's predictor-corrector method for large-scale linear programming problems needs to find a base by a sophisticated process based upon applying a rectangular LU factorization. The class of splitting preconditioners works better near a solution of the linear programming problems when the matrices are highly ill-conditioned. In this work, we develop the penalty parameter in Mehrotra's predictor-corrector method in order to reduce the ill-conditioning of the normal equations matrix. The success of this approach is guaranteed by the proof of the theorem of convergence of mixed penalty with the barrier parameter. In addition, we develop and implement a new approach to find a basis for the splitting preconditioner, based upon standard rectangular LU factorization with partial permutation of the transpose of the scaled linear programming constraint matrix. In most cases, the basis is better conditioned than the existing one. Computational tests show a reduction in the average number of iterations of the preconditioned conjugate gradient method. Also, the efficiency and robustness of the new approach is demonstrated by achieving better performance profile / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada