Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd W. Easton / Integer programming is an important discipline in operation research that positively impacts society. Unfortunately, no algorithm currently exists to solve IP's in polynomial time. Researchers are constantly developing new techniques, such as cutting planes, to help solve IPs faster. For example, DeLissa discovered the existence of equality cuts limited to zero and one coefficients for the multiple knapsack equality problem (MKEP). An equality cut is an improper cut because every feasible point satisfies the equality. However, such a cut always reduces the dimension of the linear relaxation space by at least one. This thesis introduces lifted equality cuts, which can have coefficients greater than or equal to two. Two main theorems provide the conditions for the existence of lifted equalities. These theorems provide the foundation for The Algorithm of Lifted Equality Cuts (ALEC), which finds lifted equality cuts in quadratic time. The computational study verifies the benefit of lifted equality cuts in random MKEP instances. ALEC generated millions of lifted equality cuts and reduced the solution time by an average of 15%. To the best of the author's knowledge, ALEC is the first algorithm that has found over 30.7 million cuts on a single problem, while reducing the solving time by 18%.

Cette thèse s’intéresse aux problèmes d’emploi du temps d’universités. Ces problèmes sont rencontrés chaque année par les utilisateurs. Nous proposons des bornes inférieures, des méthodes heuristiques et des modèles de programmation mixte en nombres entiers et de programmation par contraintes. Nous traitons le problème d’emploi du temps d’examens et celui d’affectation des étudiants. Nous proposons de nouvelles méthodes et formulations et les comparons aux approches existantes. Nous proposons, pour le problème d’emploi du temps d’examens, une amélioration d’un modèle mathématique en nombres entiers qui permettra d’obtenir des solutions optimales. Ensuite, des bornes inférieures, une formulation plus compacte des contraintes et un modèle de programmation par contraintes sont proposés. Pour le problème d’emploi du temps d’examens à l’Université de Technologie de Compiègne, nous proposons une approche mémétique. Enfin, nous présentons un modèle mathématique pour le problème d’affectation des étudiants et nous étudions sa performance sur un ensemble d’instances réelles. / This thesis investigates university timetabling problems. These problems occur across universities and are faced each year by the practitioners. We propose new lower bounds, heuristic approaches, mixed integer and constraint programming models to solve them. We address the exam timetabling and the student scheduling problem. We investigate new methods and formulations and compare them to the existing approaches. For exam timetabling, we propose an improvement to an existing mixed integer programming model that makes it possible to obtain optimal solutions. Next, lower bounds, a more compact reformulation for constraints and a constraint programming model are proposed. For the exam timetabling problem at Université de Technologie de Compiègne, we designed a memetic approach. Finally, we present a new formulation for the student scheduling problem and investigate its performance on a set of real-world instances.

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.

Scheduling is a practical optimisiation problem which can be solved by means of integer or binary programming methods. This paper focuses on university scheduling, in particular the schedule of the University of Economics in Prague, it is however possible to apply the results to schedules of other universities. We begin with the basics of linear programming, focusing on integer and binary programming as well as selected methods used to solve these problems. We then construct an optimisation model for the schedule of a subset of subjects based on real requirements (data 2009) and we compare the results with the actual schedule of the University of Economics in Prague. In conclusion we discuss some of the assumptions made during model development. The model is then generalised to include the entire set of subjects of the university. For the conversion of the software results into a more legible format, we include two MS Excel macros as part of this paper.

Crew scheduling is very important, especially in continuous operating environments running 24 hours a day, 7 days a week, more so if the demand for staff is varying over each hour of the day. This thesis focuses on staff optimization in a fast food chain where special conditions for scheduling like flexible starting-times and shift lengths or heterogeneous crew are present. Two new models based on a mixed integer programming approach were designed, dealing with data from a particular restaurant with the aim of improving schedules and saving time spent on the creation of those schedules. At the end of the thesis the empiric schedules and results obtained are compared and the computational efficiency of both models is discussed.

This master's thesis describes modified Chinese Postman Problems. These Problems are solved by (mixed) integer linear programming. The modified problems and also used approach (integer programming) belong at least to the NP complexity class. The thesis analyzes, compares and estimates computational complexity of each model. Based on this analysis, usability of described models for solving real-life problems is deduced. The models are focused on problems in urban environment. Therefore, it is possible to apply these models on problems like optimization of a waste collection or road maintenance. Graph and problem generator is programmed for purposes of this thesis.

O conteúdo deste trabalho trata da formulação para solução do problema de otimização estrutural com minimização de massa em estruturas unifilares, sujeitas a restrição de tensão, flambagem das barras isoladas e fadiga. São considerados três casos de otimização: paramétrica, de forma e dimensional. Os problemas de singularidades nas restrições de tensão e flambagem são evitados através de uma formulação que faz uso de programação inteira para solução do problema. Outra singularidade encontrada na otimização topológica é a singularidade na matriz de rigidez da estrutura. Este problema foi evitado através de uma formulação que considera a existência de matriz de rigidez regular como restrição do problema. O método de solução utilizado para resolver problema de otimização é o método dos algoritmos genéticos. As restrições do problema são impostas através da penalização da função objetivo. O método de solução mostrou-se adequado para solução dos problemas estudados. A formulação implementada é validada através da solução de problemas clássicos de otimização estrutural. Os resultados obtidos são comparados com a literatura onde verificou-se a coerência dos mesmos. Após realizar a validação, a formulação é utilizada em um estudo que tem como base uma estrutura real: uma torre de queima de gases (flare) oriundos do processo de extração e armazenagem de petróleo em uma unidade flutuante. Para o problema da torre as restrições foram determinadas com base em critérios de falha estabelecido na norma DNV. A otimização do flare permitiu minimizar a massa da estrutura sem que os critérios de falha fossem violados. Verificou-se que a metodologia proposta é adequada para solução com grande número de restrições e com diversos casos de carregamento. / The purpose of this work is the development of a methodology to solve the structural optimization problem of frame structures subject to stress, buckling of isolated members, and fatigue constraints. Three types of structural optimization problems are considered: sizing, shape and topological. The stress and buckling singularity problems are avoided by an integer design variable formulation, using integer programing to obtain the optimization problem solution. Another issue found in optimization problems is the stiffness matrix singularity. The proposed formulations include the linear system stability as a constraint in the optimization problem. A genetic algorithm is used to solve the general optimization problem. All constraints of the problem are included with a penalization equation. The results show that genetic algorithm is a good approach to solve the proposed formulation. The proposed formulation is tested for solving classical optimization problems. The obtained results are consistent with the literature. A real engineering problem is solved with proposed methodology: a gas burning tower (flare). In this problem, all constraints are based on failure criteria recommended by DNV standards. The structural optimization of this problem shows that structural mass minimization is possible without violating the failure criteria. It is observed that solution methodology deals successfully with problems with multiple constraints and load cases

Nesta dissertação, discute-se o problema de otimização de carteiras de investimento para estratégia passiva de Index Tracking. Os objetivos principais são (i) apresentar um modelo de otimização de Index Tracking e (ii) a solucionar esse modelo com uso do método heurístico de Algoritmos Genéticos (AG) para formação de carteiras com número reduzido de ativos. O índice de referência utilizado é o Ibovespa, para o período de Janeiro/2009 a Julho/2012, com um total de 890 observações diárias de preços. A partir de uma amostra de 67 ativos, são formadas carteiras sem limite de ativos e limitadas a 40, 30, 20, 10 e 05 ativos; os intervalos de rebalanceamento das carteiras são 20, 40 e 60 períodos (dias úteis), ou seja, rebalanceamento mensal, bimestral e trimestral. É verificado que, para essa amostra, não é possível formar carteiras de 20 ou menos ativos via otimização direta com o solver Cplex com menos de 1 hora de processamento e gap abaixo de 5%. Com uso da heurística de Algoritmos Genéticos, são formadas carteiras de 10 e 05 ativos com tempo de processamento em torno de 5 minutos; nesse caso, o gap médio fica abaixo de 10% para ambos os tipos de carteira. E, com tempo de processamento do AG um pouco maior, em torno de 8 minutos, o algoritmo fornece soluções para carteiras de 10 e 05 ativos com gap médio abaixo de 5%. / In this master’s thesis it is discussed the portfolio optimization problem using the passive investment strategy of Index Tracking. The main goals are (i) to present an optimization model for the Index Tracking problem and (ii) to solve this model using the heuristic approach of Genetic Algorithms (GA) to create portfolios with reduced amount of stocks. The benchmark used is the Ibovespa Index (main reference for the Brazilian Stock Market), during the period from January/2009 to July/2012 (using a total of 890 daily stock prices). The sample contains 67 assets, and the model is used to build portfolios without limit in the amount of assets and portfolios limited to 40, 30, 20, 10 and 05 assets; the ranges of time to rebalance the portfolios are 20, 40, and 60 trading days, which means to rebalance monthly, bimonthly and quarterly. The results show that, considering this sample, it is not possible to build portfolios with 20 stocks (or less than 20) through direct optimization using the solver Cplex with computational processing time less than 1 hour and results with gap below 5%. On the other hand, using the Genetic Algorithms heuristic approach, portfolios limited to 10 and 05 stocks are built with computational time close to 5 minutes; for both types of portfolio, the solutions provided by the GA have average gap below 10%. Also, with a computational time slightly bigger, close to 8 minutes, the algorithm provides solutions with average gap below 5% for portfolios limited to 10 and 05 stocks.

Orientadores: Pedro Jussieu de Rezende, Cid Carvalho de Souza / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-26T19:09:30Z (GMT). No. of bitstreams: 1 Zambon_MauricioJosedeOliveira_M.pdf: 2774684 bytes, checksum: 1d0ed1f5c1ae01b7646e4bffea6a3736 (MD5) Previous issue date: 2014 / Resumo: Nesta dissertação, apresentamos a primeira abordagem algorítmica e os primeiros resultados experimentais da literatura para tratamento do Problema Cromático Discreto da Galeria de Arte (DCAGP). Trata-se de um problema de natureza geométrica que consiste de uma variante do clássico Problema da Galeria de Arte. Neste, deseja-se encontrar um conjunto de guardas com cardinalidade mínima que consiga vigiar toda uma dada galeria. Já no DCAGP temos por objetivo obter um conjunto de observadores que cubra a galeria e que admita uma coloração válida com o menor número de cores. Uma coloração é válida se dois observadores que veem um mesmo ponto recebem cores distintas. Abordamos a resolução deste problema através de duas abordagens: uma exata e uma heurística. Inicialmente, apresentamos uma heurística primal que fornece limitantes superiores de boa qualidade e, em seguida, um modelo de programação linear inteira para resolução exata do DCAGP. Este método foi capaz de resolver todas as instâncias de um extenso conjunto de galerias, representadas por polígonos simples aleatoriamente gerados, de até 2500 vértices, em menos de um minuto. Já num outro conjunto de instâncias onde a representação inclui polígonos com buracos e polígonos fractais de von Koch com até 800 vértices, o método encontrou soluções comprovadamente ótimas para 80% das instâncias em menos de 30 minutos. No contexto dessas soluções, discutimos o uso de lazy-constraints e de técnicas de fortalecimento do modelo, assim como uma breve análise da dificuldade das instâncias. Reportamos ainda resultados da utilização de relaxação Lagrangiana, para obtenção de bons limitantes, principalmente superiores, e também resultados obtidos por meio de uma variação da técnica relax-and-fix. Finalmente, discutimos um processo de branch-and-price para resolução exata do DCAGP / Abstract: In this dissertation, we present the first algorithmic approach and the first experimental results in the literature for solving the Discrete Chromatic Art Gallery Problem (DCAGP). This problem is geometric in nature and consists of a variation of the classic Art Gallery Problem. In the latter, we want to find a minimum cardinality guard set that is able to watch over a given gallery. On the other hand, in the DCAGP, the objective is to find a set of watchers that covers the gallery and admits a valid coloring with a minimum number of colors. A coloring is valid if two watchers that observe a same point are assigned different colors. To solve this problem we apply two approaches: an exact and a heuristic one. Firstly, we present a primal heuristic able to provide good quality upper bounds, and subsequently an integer programming model that yields exact solutions for the DCAGP. This method was able to solve all instances from an extensive set of galleries, represented by randomly generated simple polygons, of up to 2500 vertices, in less than one minute. On another set of instances, where the representation includes polygons with holes and fractal von Koch polygons, with up to 800 vertices, this method found proven optimal solutions for 80% of the instances in less than 30 minutes. In the context of these solutions, we discuss the use of lazy constraints and techniques for strengthening the model, besides a brief analysis of the hardness of the instances. Moreover, we report on results obtained through a Lagrangian relaxation, mainly as a means to obtain good upper bounds, as well as from a variation of the relax-and-fix technique. Lastly, we discuss a branch-and-price process for solving the DCAGP to exactness / Mestrado / Ciência da Computação / Mestre em Ciência da Computação