Uma revisão dos algoritmos e conjuntos de instâncias presentes na literatura do Problema da Mochila com Repetições (PMR) é apresentada nessa dissertação de mestrado. Os algoritmos e conjuntos de instâncias usados são brevemente descritos nesse trabalho, afim de que o leitor tenha base para entender as discussões. Algumas propriedades bem conhecidas e específicas do PMR, como a dominância e a periodicidade, são explicadas com detalhes. O PMR é também superficialmente estudado no contexto de problemas de avaliação gerados pela abordagem de geração de colunas aplicada na relaxação contínua do Bin Packing Problem (BPP) e o Cutting Stock Problem (CSP). Múltiplos experimentos computacionais e comparações são realizadas. Para os conjuntos de instâncias artificiais mais recentes da literatura, um simples algoritmo de programação dinâmica, e uma variante do mesmo, parecem superar o desempenho do resto dos algoritmos, incluindo aquele que era estado-da-arte. O modo que relações de dominância é aplicado por esses algoritmos de programação dinâmica têm algumas implicações para as relações de dominância previamente estudadas na literatura. O autor dessa dissertação defende a tese de que a escolha dos conjuntos de instâncias artificiais definiu o que foi considerado o melhor algoritmo nos trabalhos anteriores. O autor dessa dissertação disponibilizou publicamente todos os códigos e conjuntos de instâncias referenciados nesse trabalho. / A review of the algorithms and datasets in the literature of the Unbounded Knapsack Problem (UKP) is presented in this master's thesis. The algorithms and datasets used are brie y described in this work to provide the reader with basis for understanding the discussions. Some well-known UKP-speci c properties, such as dominance and periodicity, are described. The UKP is also super cially studied in the context of pricing problems generated by the column generation approach applied to the continuous relaxation of the Bin Packing Problem (BPP) and Cutting Stock Problem (CSP). Multiple computational experiments and comparisons are performed. For the most recent arti cial datasets in the literature, a simple dynamic programming algorithm, and its variant, seems to outperform the remaining algorithms, including the previous state-of-the-art algorithm. The way dominance is applied by these dynamic programming algorithms has some implications for the dominance relations previously studied in the literature. In this master's thesis we defend that choosing sets of arti cial instances has de ned what was considered the best algorithm in previous works. We made available all codes and datasets referenced in this master's thesis.

Uma revisão dos algoritmos e conjuntos de instâncias presentes na literatura do Problema da Mochila com Repetições (PMR) é apresentada nessa dissertação de mestrado. Os algoritmos e conjuntos de instâncias usados são brevemente descritos nesse trabalho, afim de que o leitor tenha base para entender as discussões. Algumas propriedades bem conhecidas e específicas do PMR, como a dominância e a periodicidade, são explicadas com detalhes. O PMR é também superficialmente estudado no contexto de problemas de avaliação gerados pela abordagem de geração de colunas aplicada na relaxação contínua do Bin Packing Problem (BPP) e o Cutting Stock Problem (CSP). Múltiplos experimentos computacionais e comparações são realizadas. Para os conjuntos de instâncias artificiais mais recentes da literatura, um simples algoritmo de programação dinâmica, e uma variante do mesmo, parecem superar o desempenho do resto dos algoritmos, incluindo aquele que era estado-da-arte. O modo que relações de dominância é aplicado por esses algoritmos de programação dinâmica têm algumas implicações para as relações de dominância previamente estudadas na literatura. O autor dessa dissertação defende a tese de que a escolha dos conjuntos de instâncias artificiais definiu o que foi considerado o melhor algoritmo nos trabalhos anteriores. O autor dessa dissertação disponibilizou publicamente todos os códigos e conjuntos de instâncias referenciados nesse trabalho. / A review of the algorithms and datasets in the literature of the Unbounded Knapsack Problem (UKP) is presented in this master's thesis. The algorithms and datasets used are brie y described in this work to provide the reader with basis for understanding the discussions. Some well-known UKP-speci c properties, such as dominance and periodicity, are described. The UKP is also super cially studied in the context of pricing problems generated by the column generation approach applied to the continuous relaxation of the Bin Packing Problem (BPP) and Cutting Stock Problem (CSP). Multiple computational experiments and comparisons are performed. For the most recent arti cial datasets in the literature, a simple dynamic programming algorithm, and its variant, seems to outperform the remaining algorithms, including the previous state-of-the-art algorithm. The way dominance is applied by these dynamic programming algorithms has some implications for the dominance relations previously studied in the literature. In this master's thesis we defend that choosing sets of arti cial instances has de ned what was considered the best algorithm in previous works. We made available all codes and datasets referenced in this master's thesis.

Nesse trabalho, estudamos um problema de otimização combinatorial conhecido por Problema da Mochila Compartimentada, que é uma extensão do clássico Problema da Mochila. O problema consiste em determinar as capacidades adequadas de vários compartimentos que podem vir a ser alocados em uma mochila e como esses compartimentos devem ser carregados, respeitando as restrições de capacidades dos compartimentos e da mochila. Busca-se maximizar o valor de utilidade total. O problema é muito pouco estudado na literatura, apesar de surgir naturalmente em aplicações práticas. Nesse estudo, propomos uma modelagem matemática não linear para o problema e verificamos algumas heurísticas para sua resolução. / In this work, we studied a combinatorial optimization problem called the Clustered Knapsack Problem, that is an extension of the standard Knapsack Problem. The problem is to determine the right capacities of several clusters which can be allocated in a knapsack and how these clusters should be placed so as to respect the constraints on the capacities of the clusters and the knapsack. The objective is to maximize a total utility value. The problem has seldom been studied in the literature, even though it appears naturally in practical applications. In this study, we propose a non-linear model for the problem and we insert some heuristics for its resolution.

Orientador: Eduardo Candido Xavier / Dissertação (mestrado) - Universidade Estadual de Campinas, Insituto de Computação / Made available in DSpace on 2018-08-17T16:49:45Z (GMT). No. of bitstreams: 1 DelValle_AlineMarques_M.pdf: 1217777 bytes, checksum: 66f10d1b6b4533727cbe82431f97660d (MD5) Previous issue date: 2010 / Resumo: Nesta dissertação, estudamos problemas de empacotamento com itens irregulares. Estamos particularmente interessados no Problema da Mochila Bidimensional: dados um recipiente de tamanho W x H e uma lista de itens bidimensionais, o objetivo é empacotar um subconjunto dos itens de forma a maximizar a área dos itens empacotados. Existem diversos trabalhos que lidam com problemas para itens e recipientes bidimensionais com forma regular (retangular). No entanto, são poucos os estudos que tratam de itens com formas irregulares. Nós propomos algoritmos de empacotamento para itens irregulares em recipientes limitados baseados no uso de No-Fit-Polygon (NFP). Este trabalho apresenta uma heurística GRASP para a versão restrita do Problema da Mochila: uma solução inicial gulosa é gerada e, em seguida, utiliza-se um algoritmo de busca local para melhorar solução atual. Uma estratégia híbrida também foi proposta para versão irrestrita do Problema da Mochila. Ela divide-se em passos de empacotamento de itens irregulares e empacotamento de itens regulares. Testamos os algoritmos com instâncias adaptadas do problema de Strip Packing. O GRASP obteve empacotamentos ótimos para várias instancias testadas e, mesmo para as instâncias em que o algoritmo não obteve resultados ótimos, os empacotamentos obtidos tiveram boa taxa de ocupação, com valores relativamente próximos do ótimo. O tempo de execução do algoritmo foi razoável. Na estratégia híbrida, obtiveram-se empacotamentos bons para a maioria das instâncias, com taxa de ocupação acima de 90% e tempos de execução relativamente baixos / Abstract: In this work, we study packing problems dealing with two dimensional irregular items. We are particularly interested in the knapsack version of the problem: given a container with size W x H and a list of two dimensional items, the goal is to pack a subset of items such that the total area of packed items is maximized. There are several works that deal with problems for the case where items and containers have regular shapes (rectangular). However, only a few studies deal with items with irregular shapes. We propose algorithms for packing irregular items in limited containers based on the use of No-Fit-Polygon (NFP). This work presents a GRASP algorithm for the restricted version of the Knapsack Problem: first, a greedy initial solution is generated, then, the local search algorithm is used to improve the current solution. A hybrid strategy has also been proposed for the unrestricted version of the Knapsack Problem. It is divided into steps of packing irregular items and packing regular items. We tested the algorithms using adapted instances for the Strip Packing problem. The GRASP algorithm achieved optimal packings for several of the tested instances, and, even for those that the algorithm did not, the obtained packings had a good occupancy rate, with values relatively close to the optimum. The runtime of the algorithm was reasonable. In the hybrid strategy, we obtained good packings for most of the instances, with occupancy rates above 90% and relatively low execution times / Mestrado / Teoria da Computação / Mestre em Ciência da Computação

Depuis le début des années 2000, le secteur social et médico-social connait des évolutions et des mutations importantes. D’un côté, le nombre de personnes prises en charge est en perpétuelle augmentation. D’un autre côté, les finances et les budgets mis à disposition des établissements ne cessent de se réduire, ce qui oblige les décideurs à s’adapter et à trouver de nouvelles solutions pour faire plus avec moins de moyens. Dans cette thèse, nous étudions un problème pratique auquel sont souvent confrontés les directeurs des établissements qui est l’élaboration de plans d’actions optimaux. Un plan d’actions est un ensemble d’actions qui sont mises en place afin d’améliorer à la fois les performances de l’établissement et la qualité de prise en charge de ses résidents.Élaborer un plan d’actions optimal consiste à identifier et choisir les meilleures actions qui améliorent tous les objectifs du plan tout en respectant quelques contraintes. Après la présentation du contexte pratique et théorique, nous fournissons une modélisation formelle du problème sous forme d’un problème de sac-à-dos multi-objectif.Puis nous présentons quelques méthodes de résolution à base d’indicateurs de qualité et de la dominance de Lorenz. Nous montrons que la méthode IBMOLS combinée avec l’indicateur de qualité R2 permet d’obtenir des solutions efficaces et d’intégrer facilement les préférences du décideur. Nous montrons également que dans un contexte où les préférences du décideur sont inconnues ou les objectifs ont tous la même importance, la dominance de Lorenz est un outil très efficace qui permet, d’un côté, d’intégrer l’équité dans le processus de recherche et, d’un autre côté, de réduire le nombre de solutions non dominées ainsi que le temps d’exécution. / Since the early 2000s, the social and medico-social sector is experiencing significant evolutions and mutations. On the one hand, the number of persons taken over is constantly increasing. On the other hand, the finances and budgets available to the structures are constantly decreasing. This forces decision-makers to adapt and find new solutions to do more with fewer resources. In this thesis, we study a practical problem that is often faced by the decision-makers, which is the elaboration of optimal action plans. An action plan is a set of actions that are realized to improve both the performance of the structure and the quality of service offred to its residents. Elaborating an optimal action plan consists of identifying and selecting the best actions that improve all the objectives of the plan while respecting some constraints. After presenting the practical and theoretical context, we provide a formal modeling of the problem as a multi-objective knapsack problem. Then, we present a number of solution methods based on quality indicators and Lorenz dominance. We show that combining IBMOLS method with R2 indicator allows obtaining efficient solutions and easily integrating the decision-maker preferences. We also show that in a context where decision-maker preferences are not known or all the objectives are considered equals, Lorenz dominance is a very efficient tool to incorporate equity into the search process and reduce the number of non-dominated solutions as well as the algorithm runtime.

Title: The knapsack and its applications Author: Romana Linkeová Department: Department of Algebra Supervisor: doc. Mgr. Pavel Příhoda, Ph.D., Department of Algebra Abstract: This thesis is focused on various aspects of cryptosystems based on NP (non-deterministic polynomial) complete knapsack problem. From the theory of complexity point of view, the less known parts of the proof of knapsack problem NP completeness are shown in detail. From the cryptographical point of view, a demonstration of breaking of the Merkle-Hellman cryptosystem (the basic de- sign of knapsack-type cryptosystems) is provided, showing that poor parameters choice can lead to easy obtaining of the whole private key. Another contribution of this thesis consists in a presented proposal of a new cryptosystem concept based on the matrix 0-1 knapsack problem. This concept was developed in order to prevent known attacks, however, in the thesis we provide a proof analogous to J. C. Lagarias and A. M. Odlyzko, 1985, which shows that an attack based on the LLL algorithm will be successful on the majority of the matrix 0-1 kna- psack problem cryptosystems. Finally, a list of modern cryptosystems based on the knapsack problem is provided and a cryptanalysis thereof is given. Keywords: knapsack problem, NP complete problems, LLL algorithm 1

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programming (IP) has been and continues to be widely used by industries to minimize cost and effectively manage resources. Faster computers and innovative IP techniques have enabled the solution to many large-scale IPs. However, IPs are NP-hard and many IPs require exponential time to solve. Lifting is one of the most widely used techniques that helps to reduce computational time and is widely applied in today's commercial IP software. Lifting was first introduced by Gomory for bounded integer programs and a theoretical and computationally intractible technique to simultaneously lift sets of variables was introduced by Zemel in 1978. This thesis presents a new algorithm called the Maximal Simultaneous Lifting Algorithm (MSLA), to simultaneously uplift sets of binary integer variables into a cover inequality. These lifted inequalities result in strong inequalities that are facet defining under fairly moderate assumptions. A computational study shows that this algorithm can find numerous strong inequalities for random Knapsack (KP) instances. The pre-processing time observed for these instances is less than 1/50th of a second, which is negligible. These simultaneously lifted inequalities are easy to find and incorporating these cuts to KP instances reduced the solution time by an average of 41%. Therefore, implementing MSLA should be highly beneficial for large real-world problems.

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%.

The thesis describes the principles of dynamic programming and it's application to concrete problems. (The travelling salesman problem, the knapsack problem, the shortest path priblem,the set covering problem.)

The diploma thesis focuses on tracing in layout by handlers between assembly lines in new plant for corporation Continental Automotive Czech Republic ltd, where boosters are produced. The theoretical part involves definitions of logistics, supply chain, material flow and handling equipment. Furthermore, methods of mathematic programming and software equipment are described, such as quadratic assignment problem, knapsack problem, travelling salesman problem from graph theory. In the practical part the situation in corporation has been analyzed and the data prepared for further examination. Then layout of plant and internal processes are evaluated and an appropriate model or concept of solution is selected. Subsequently, application in MS Excel is created with support of VBA scripts (3 kinds of layouts). The user manipulates with application followed by Solver for implementation of a new solution into practice. Finally, the models are interpreted and verified by Lingo. The focus of the thesis is the design of a layout change of a new plant including the description of tracing.