The classical multiple knapsack problem selects a set of items and assigns each to one of the knapsacks so as to maximize the total profit. The knapsacks have limited capacities. The cardinality constrained multiple knapsack problem assumes limits on the number of items that are to be put in each knapsack, as well. Despite many efforts on the classical multiple knapsack problem, the research on the cardinality constrained multiple knapsack problem is scarce. In this study we consider the cardinality constrained multiple knapsack problem. We propose heuristic and optimization procedures that rely on the optimal solutions of the linear programming relaxation problem. Our computational results on the large-sized problem instances have shown the satisfactory performances of our algorithms.

Made available in DSpace on 2016-12-12T20:22:53Z (GMT). No. of bitstreams: 1 LEANDERSON ANDRE.pdf: 2236080 bytes, checksum: a52e91a8b1a8e6a12497786254e94344 (MD5) Previous issue date: 2015-08-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nature offers a wide range of phenomena that inspire the development of new technologies. The researchers from the area of Natural Computing abstracts the concept of optimization from various biological processes such as the evolution of species, the behavior of social groups, the search for food, among others. Such computer systems that have a similarity to natural biological systems are called biologically plausible. The development of biologically plausible algorithms gets interesting by the fact that biological systems are able to handle extremely complex problems. In this way, symbiotic relationships are one of several phenomena that can be observed in nature. These relationships consist of interactions that organisms carry out with each other resulting in benefit or disadvantage to those involved. In an optimization context, symbiotic relationships can be used to perform exchange of information between populations of candidate solutions to a given problem. Thus, this work highlights the concepts involving symbiotic relationships that may be important for the development of computer systems to solve complex problems. The main discussion presented in this study refers to the use of symbiotic relationships between populations of candidate solutions co-evolving in an ecological context. According to the analogy, populations interact with each other according to a specific symbiotic relationship in order to evolve their solutions. The proposed model is applied to several continuous benchmark functions with a high number of dimensions (D = 200) and in several benchmark instances of the multiple knapsack problem. The results obtained so far were promising concerning the application of symbiotic relationships. Finally, the conclusions are presented and some future directions for research are suggested. / A Natureza apresenta uma grande variedade de fenômenos que inspiram o desenvolvimento de novas tecnologias. Os pesquisadores da área de Computação Natural abstraem o conceito de otimização de vários processos biológicos, tais como a evolução das espécies, comportamento de grupos sociais, busca por comida, dentre outros. Tais sistemas computacionais que apresentam uma semelhança com os sistemas biológicos naturais são chamados de biologicamente plausíveis. O desenvolvimento de algoritmos biologicamente plausíveis se torna interessante pelo fato de que os sistemas biológicos são capazes de lidar com problemas extremamente complexos. As relações simbióticas são um dos vários fenômenos que podem ser observados na natureza. Essas relações consistem de interações que organismos realizam entre si resultando em benefícios ou prejuízos para os envolvidos. Em um contexto de otimização, as relações simbióticas podem ser utilizadas para realizar a troca de informação entre populações de soluções candidatas para um dado problema. Desta forma, este trabalho destaca os conceitos que envolvem as relações simbióticas que podem ser importantes para o desenvolvimento de sistemas computacionais para a resolução de problemas complexos. A principal discussão apresentada nesse trabalho refere-se a utilização de relações simbióticas entre populações de soluções candidatas, coevoluindo em um contexto ecológico. Com essa analogia, cada população interage com uma outra de acordo com uma relação simbiótica específica, com o objetivo de evoluir suas soluções. O modelo apresentado é aplicado a várias funções benchmark contínuas com um número alto de dimensões (D = 200) e várias instâncias benchmark do problema da mochila múltipla. Os resultados obtidos se mostraram promissores considerando a aplicação das relações simbióticas. Por fim, as conclusões são apresentadas e algumas direções para pesquisas futuras são sugeridas.

This paper deals with various approaches to solving optimization tasks. In prolog some examples from real life that show the application of optimization methods are given. Then term optimization task is defined and introducing of term fitness function which is common to all optimization methods follows. After that approaches by particle swarm optimization, ant colony optimization, simulated annealing, genetic algorithms and reinforcement learning are theoretically discussed. For testing we are using two discrete (multiple knapsack problem and set cover problem) and two continuous tasks (searching for global minimum of Ackley's and Rastrigin's function) which are presented in next chapter. Description of implementation details follows. For example description of solution representation or how current solutions are changed. Finally, results of measurements are presented. They show optimal settings for parameters of given optimization methods considering test tasks. In the end are given test tasks, which will be used for finding optimal settings of given approaches.