Monotonic optimization consists of minimizing or maximizing a monotonic objective function over a set of constraints defined by monotonic functions. Many optimization problems in economics and engineering often have monotonicity while lacking other useful properties, such as convexity. This thesis is concerned with the development and application of global optimization algorithms for monotonic optimization problems. First, we propose enhancements to an existing outer-approximation algorithm | called the Polyblock Algorithm | for monotonic optimization problems. The enhancements are shown to significantly improve the computational performance of the algorithm while retaining the convergence properties. Next, we develop a generic branch-and-bound algorithm for monotonic optimization problems. A computational study is carried out for comparing the performance of the Polyblock Algorithm and variants of the proposed branch-and-bound scheme on a family of separable polynomial programming problems. Finally, we study an important class of monotonic optimization problems | probabilistically constrained linear programs. We develop a branch-and-bound algorithm that searches for a global solution to the problem. The basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partitions and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires the solution of only linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.

The time/cost trade-off models in project management aim to compress the project completion time by accelerating the activity durations at an expense of additional resources. The budget problem in discrete time/cost trade-off scheduling selects the time/cost mode -among the discrete set of specified modes- for each activity so as to minimize the project completion time without exceeding the available budget. There may be alternative modes that solve the budget problem optimally, however each solution may have a different total cost value. In this study we aim to find the minimum cost solution among the optimal solutions of the budget problem. We analyze the structure of the problem together with its linear programming relaxation and derive some mechanisms for reducing the problem size. We solve the reduced problem by linear programming relaxation and branch and bound based approximation and optimization algorithms. We find that our branch and bound algorithm finds optimal solutions for medium-sized problem instances in reasonable times and the approximation algorithms produce high quality solutions. We also discuss the way our algorithms could be used to construct the time/cost trade-off curve.

In this thesis, we consider the Multi Resource Agent Bottleneck Generalized Assignment Problem. We aim to minimize the maximum load over all agents. We study the Linear Programming (LP) relaxation of the problem. We use the optimal LP relaxation solutions in our Branch and Bound algorithm while defining lower and upper bounds and branching schemes. We find that our Branch and Bound algorithm returns optimal solutions to the problems with up to 60 jobs when the number of agents is 5, and up to 30 jobs when the number of agents is 10, in less than 20 minutes. To find approximate solutions, we define a tabu search algorithm and an &amp / #945 / approximation algorithm. Our computational results have revealed that these procedures can find high quality solutions to large sized instances very quickly.

Resource Leveling Problem (RLP) aims to minimize undesired fluctuations in resource distribution curves which cause several practical problems. Many studies conclude that commercial project management software packages can not effectively deal with RLP. In this study a branch and bound algorithm is presented for solving RLP for single and multi resource, small size networks. The algorithm adopts a depth-first strategy and stores start times of non-critical activities in the nodes of the search tree. Optimal resource distributions for 4 different types of resource leveling metrics can be obtained via the developed procedure. To prune more of the search tree and thereby reduce the computation time, several lower bound calculation methods are employed. Experiment results from 20 problems showed that the suggested algorithm can successfully locate optimal solutions for networks with up to 20 activities. The algorithm presented in this study contributes to the literature in two points. First, the new lower bound improvement method (maximum allowable daily resources method) introduced in this study reduces computation time required for achieving the optimal solution for the RLP. Second, optimal solutions of several small sized problems have been obtained by the algorithm for some traditional and recently suggested leveling metrics. Among these metrics, Resource Idle Day (RID) has been utilized in an exact method for the first time. All these solutions may form a basis for performance evaluation of heuristic and metaheuristic procedures for the RLP. Limitations of the developed branch and bound procedure are discussed and possible further improvements are suggested.

Nos travaux portent sur la résolution de problèmes d'optimisation en ordonnancement d'ateliers de production, et plus particulièrement ceux relatifs à l'ordonnancement dynamique dans les industries agroalimentaires. <br />Les contraintes et les critères considérés sont spécifiques à ce type d'industrie qui présente certaines particularités, dues à la nature des produits manipulés et fabriqués, dont les durées de vie assez courtes. Ils concernent aussi le respect des dates de validité des composants primaires formant les opérations, des produits semi-finis et des produits finis. Les critères retenus sont aussi liés à ces particularités. On a distingué le coût des produits périmés, le coût du discount de distribution et la date de fin de l'ordonnancement, le makespan. Une méthode exacte et deux méthodes approchées ont été retenues et mises en œuvre, avec succès, pour les problèmes à une machine. <br />La méthode exacte, branch & bound, est appliquée pour la minimisation de la fonction de coût total. Les algorithmes génétiques, dotés d'un nouveau codage et hybridés avec l'approche Pareto-optimale, sont proposés pour la recherche de la solution optimale et pour aider le décideur de prendre une décision. Les algorithmes d'optimisation par colonie de fourmis, constituant la deuxième méthode approchée, est un processus stochastique qui, malgré la difficulté de paramétrage de l'algorithme correspondant, nous a permis de construire des solutions, en ajoutant des composants aux solutions temporaires.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

Zugl.: Clausthal, Techn. Universiẗat, Diss., 2008.

A diferença entre estações de trabalho é considerada desprezível em linhas de montagem tradicionais. Por outro lado, linhas de montagem heterogêneas consideram o problema de indústrias nas quais os tempos das tarefas variam de acordo com alguma característica a ser selecionada para a tarefa. No Problema de Balanceamento e Atribuição de Trabalhadores em Linhas de Montagem (do inglês Assembly Line Worker Assignment and Balancing Problem, ALWABP), os trabalhadores são responsáveis por estações de trabalho e de acordo com as suas habilidades, eles executam as tarefas em diferentes quantidades de tempo. Em alguns casos, os trabalhadores podem até ser incapazes de executar algumas tarefas. No Problema de Balanceamento de Linhas de Montagem Robóticas (do inglês Robotic Assembly Line Balancing Problem, RALBP), há diferentes tipos de robôs e o conjunto de tarefas de cada estação deve ser executada por um robô. Robôs do mesmo tipo podem ser usados múltiplas vezes. Nós propomos métodos exatos e heurísticos para a minimização do tempo de ciclo destes dois problemas, para um número fixo de estações. Os problemas têm características similares que são exploradas para produzir limitantes inferiores, métodos inferiores, models de programação inteira mista, e regras de redução e dominância. Para a estratégia de ramificação do método de branch-and-bound, entretanto, as diferenças entre os problemas forçam o uso de dois algoritmos diferentes. Uma estratégia orientada a tarefas tem os melhores resultados para o ALWABP-2, enquanto uma estratégia orientada a estações tem os melhores resultados para o RALBP-2. Nós mostramos que os limitantes inferiores, heurísticas, modelos de programação inteira mista e algoritmos de branch-and-bound para estes dois problemas são competitivos com os métodos do estado da arte da literatura. / The difference among workstations is assumed to be negligible in traditional assembly lines. Heterogeneous assembly lines consider the problem of industries in which the task times vary according to some property to be selected for the task. In the Assembly Line Worker Assignment and Balancing Problem (ALWABP), workers are assigned to workstations and according to their abilities, they execute tasks in different amounts of time. In some cases they can even be incapable of executing some tasks. In the Robotic Assembly Line Balancing Problem (RALBP) there are different types of robots and each station must be executed by a robot. Multiple robots of the same type may be used. We propose exact and heuristic methods for minimizing the cycle time of these two problems, for a fixed number of stations. The problems have similar characteristics that are explored to produce lower bounds, heuristic methods, mixed-integer programming models, and reduction and dominance rules. For the branching strategy of the branch-and-bound method, however, the differences among the problem force the use of two different algorithms. A task-oriented strategy has the best results for the ALWABP-2 while a station-oriented strategy has the best results for the RALBP-2. The lower bounds, heuristics, MIP models and branch-and-bound algorithms for these two problems are shown to be competitive with the state-of-the-art methods in the literature.

O uso de sistemas quâticos como parte de sistemas de comunicação tem sido fonte de interessantes problemas muitos ainda sem solução. No presente trabalho, apresentamos os conceitos básicos em teoria da informação e mecânica quântica necessários ao entendimento do problema do cálculo da informação acessível, cuja solução maximiza a informação mútua de Shannon para um canal definido por um ensemble de estados quâticos dados a priori. Propomos o uso do método de otimização global Branch and Bound aliado à aritmética intervalar para a estimação de limites mais precisos que os teóricos disponíveis para a informação acessível. Experimentos numéricos e resultados relacionados são apresentados. / The use of quantum systems as part of the communication systems has been source of interesting problems many without solution. In the present work, we show the basic concepts of information theory and quantum mechanics necessary to understand the accessible information problem, whose solution maximizes the Shannon mutual information for a channel defined by an ensemble of quantum states given a priori. In order to estimate more precise bounds for accessible information, we propose the use of Branch and Bound method with interval arithmetic. Numerical experiments and related results are exhibited.

A diferença entre estações de trabalho é considerada desprezível em linhas de montagem tradicionais. Por outro lado, linhas de montagem heterogêneas consideram o problema de indústrias nas quais os tempos das tarefas variam de acordo com alguma característica a ser selecionada para a tarefa. No Problema de Balanceamento e Atribuição de Trabalhadores em Linhas de Montagem (do inglês Assembly Line Worker Assignment and Balancing Problem, ALWABP), os trabalhadores são responsáveis por estações de trabalho e de acordo com as suas habilidades, eles executam as tarefas em diferentes quantidades de tempo. Em alguns casos, os trabalhadores podem até ser incapazes de executar algumas tarefas. No Problema de Balanceamento de Linhas de Montagem Robóticas (do inglês Robotic Assembly Line Balancing Problem, RALBP), há diferentes tipos de robôs e o conjunto de tarefas de cada estação deve ser executada por um robô. Robôs do mesmo tipo podem ser usados múltiplas vezes. Nós propomos métodos exatos e heurísticos para a minimização do tempo de ciclo destes dois problemas, para um número fixo de estações. Os problemas têm características similares que são exploradas para produzir limitantes inferiores, métodos inferiores, models de programação inteira mista, e regras de redução e dominância. Para a estratégia de ramificação do método de branch-and-bound, entretanto, as diferenças entre os problemas forçam o uso de dois algoritmos diferentes. Uma estratégia orientada a tarefas tem os melhores resultados para o ALWABP-2, enquanto uma estratégia orientada a estações tem os melhores resultados para o RALBP-2. Nós mostramos que os limitantes inferiores, heurísticas, modelos de programação inteira mista e algoritmos de branch-and-bound para estes dois problemas são competitivos com os métodos do estado da arte da literatura. / The difference among workstations is assumed to be negligible in traditional assembly lines. Heterogeneous assembly lines consider the problem of industries in which the task times vary according to some property to be selected for the task. In the Assembly Line Worker Assignment and Balancing Problem (ALWABP), workers are assigned to workstations and according to their abilities, they execute tasks in different amounts of time. In some cases they can even be incapable of executing some tasks. In the Robotic Assembly Line Balancing Problem (RALBP) there are different types of robots and each station must be executed by a robot. Multiple robots of the same type may be used. We propose exact and heuristic methods for minimizing the cycle time of these two problems, for a fixed number of stations. The problems have similar characteristics that are explored to produce lower bounds, heuristic methods, mixed-integer programming models, and reduction and dominance rules. For the branching strategy of the branch-and-bound method, however, the differences among the problem force the use of two different algorithms. A task-oriented strategy has the best results for the ALWABP-2 while a station-oriented strategy has the best results for the RALBP-2. The lower bounds, heuristics, MIP models and branch-and-bound algorithms for these two problems are shown to be competitive with the state-of-the-art methods in the literature.