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Section Steel usually is a order-oriented production and not easy to resell. The material cost is large percentages of overall production cost. Hence, the key to boost efficient management is to increase the material output rate. In other words, we need to publish a efficient and reasonable cutting plan before production. And the cutting plan can cope with change to meet the market demand. The cutting plan designing is a one-dimensional cutting stock problem, and also is a typical bin packing problem. In this study we examine a combined heuristic approach for this problem. The proposed approach combines two themes of solving method¡Ga neighborhood search algorithm with threshold accepting techniques, and a Branch and Bound method. The performance of the combined heuristic approach was verified by running several benchmarking problems and the results were reported. Experimental results indicate that the proposed solving process can effectively search the feasible region and avoid being trapped in local optimal.

In this study, the two-dimensional cutting stock problem (2DCSP) is considered. This is a problem that occurs in the cutting of a number of smaller rectangular pieces or items from a set of large stock rectangles. It is assumed that the set of large objects is sufficient to accommodate all the small items. A heuristic procedure is developed to solve the two-dimensional single stock-size cutting stock problem (2DSSSCSP). This is the special case where the large rectangles are all of the same size. The major objective is to minimize waste and the number of stock sheets utilized. The heuristic procedures developed to solve the 2DSSSCSP are based on the generation of cutting pattern. The Wang algorithm and a specific commercial software package are made use of to generate these patterns. The commercial software was chosen from a set of commercial software packages available in the market. A combinatoric process is applied to generate sets of cutting patterns using the Wang algorithm and the commercial software. The generated cutting patterns are used to formulate an integer linear programming model which is solved using an optimization solver. Empirical experimentation is carried out to test the heuristic procedures using data obtained from both small and real world application problem instances. The results obtained shows that the heuristic procedures developed produce good quality results for both small and real life problem instances. It is quite clear that the heuristic procedure developed to solve the 2DSSSCSP produces cutting patterns which are acceptable in terms of waste generated and may offer useful alternatives to approaches currently available. Broadly stated, this study involves investigating available software (commercial) in order to assess, formulate and investigate methods to attempt to benchmark software systems and algorithms and to employ ways to enhance solutions obtained by using these software systems. / Thesis (M.Sc. (Computer Science))--North-West University, Potchefstroom Campus, 2011.

In this study, the two-dimensional cutting stock problem (2DCSP) is considered. This is a problem that occurs in the cutting of a number of smaller rectangular pieces or items from a set of large stock rectangles. It is assumed that the set of large objects is sufficient to accommodate all the small items. A heuristic procedure is developed to solve the two-dimensional single stock-size cutting stock problem (2DSSSCSP). This is the special case where the large rectangles are all of the same size. The major objective is to minimize waste and the number of stock sheets utilized. The heuristic procedures developed to solve the 2DSSSCSP are based on the generation of cutting pattern. The Wang algorithm and a specific commercial software package are made use of to generate these patterns. The commercial software was chosen from a set of commercial software packages available in the market. A combinatoric process is applied to generate sets of cutting patterns using the Wang algorithm and the commercial software. The generated cutting patterns are used to formulate an integer linear programming model which is solved using an optimization solver. Empirical experimentation is carried out to test the heuristic procedures using data obtained from both small and real world application problem instances. The results obtained shows that the heuristic procedures developed produce good quality results for both small and real life problem instances. It is quite clear that the heuristic procedure developed to solve the 2DSSSCSP produces cutting patterns which are acceptable in terms of waste generated and may offer useful alternatives to approaches currently available. Broadly stated, this study involves investigating available software (commercial) in order to assess, formulate and investigate methods to attempt to benchmark software systems and algorithms and to employ ways to enhance solutions obtained by using these software systems. / Thesis (M.Sc. (Computer Science))--North-West University, Potchefstroom Campus, 2011.

The transition from plastic packaging to renewable materials, a growing population and increased e-commerce, have resulted in a greater need for paper packaging. Furthermore, increasingly stringent regulations for shelf-ready packaging and food safety mean that the market, for fresh fiber-based packaging that are chemical free and have good strength and printability, is expanding. Given these trends, SCA decided in 2019 to expand the paper mill in Obbola, to be able to meet the rising demand. The planned expansion will see SCA Obbola go from operating Europe’s largest Kraftliner machine, to the world’s largest Kraftliner machine. With the new paper machine going into service, there will be an increased pressure on internal transports of products from the Obbola factory to the terminal in Holmsund, where they are stored in preparation for the onward journey to deliver them to customers. These internal transports work according to two set goals. On the one hand, it is desirable to achieve a high degree of filling in the trucks to reduce transport costs. On the other hand, one wants to avoid the trucks traveling long distances in the terminal, again in order to minimize costs. However, the current storage system is set up to optimize the terminal’s bearing surfaces rather than the resulting mileage for the trucks. SCA Logistics therefore saw a need for investigating how internal transports could be optimized, with the aim of minimizing total costs. This was the starting point of this project, which was carried out in five phases. First, a feasibility study was conducted, where the focus was on gathering information about the intended problem area. Secondly, the warehouse logistics at SCA Logistics Umeå, the terminal operated by SCA Logistics in Holmsund, were investigated by means of a thorough data analysis. In a third phase, a model was created to simulate future paper production at SCA Obbola. In a fourth phase, it was possible to simulate the flow of internal transports and design tests to investigate the outcome of different strategies. The final phase of the project included continuous verification and validation of first four phases. The results confirm the company management’s fear, that a one-sided optimization of the degree of filling will lead to long and time-consuming driving segments for the trucks. It also demonstrates that all components of the flow must be considered, in order to minimize the total costs associated with internal transports. Accordingly, this report proposes a new strategy for internal transports, which take the entire flow into account.

Nesta tese é abordado o problema da mochila compartimentada e o problema de corte de estoque unidimensional acoplado ao problema dimensionamento de lotes. Para o problema da mochila compartimentada é apresentada a versão unidimensional e proposta a versão bidimensional, denominados como problema da mochila compartimentada unidimensional e problema da mochila compartimentada bidimensional, respectivamente. Para o problema de corte de estoque acoplado ao dimensionamento de lotes são apresentadas três variações: uma máquina para produzir um tipo de objeto; uma máquina para produzir vários tipos de objetos; múltiplas máquinas para produzir vários tipos de objetos. Algumas formulações matemáticas de programação inteira e inteira-mista, decomposições dos problemas em problema mestre e subproblemas e heurísticas baseadas no método geração de colunas são propostas para os problemas da mochila compartimenta e o problema acoplado. Em específico, para o problema acoplado são aplicadas decomposições Dantzig-Wolfe, que podem ser por período, por máquina ou por período e máquina. Além disso, uma heurística baseada em grafo E/OU é proposta para o problema da mochila compartimentada bidimensional / In this thesis we present the constrained compartmentalized knapsack problem and the one dimensional cutting stock problem integrated with the capacitated lot sizing problem. For the constrained compartmentalized knapsack problem, the one dimensional version is presented and the two dimensional version is proposed, called one-dimensional compartmentalized knapsack problem and two-dimensional compartmentalized knapsack problem, respectively. For the cutting stock problem integrated with the capacitated lot sizing problem three variations are considered: one machine to produce one type of object; one machine to produce multiple types of objects; multiple machines to produce multiple types of objects. Some integer and mixed programming formulations, decompositions of the problems in master problem and subproblems and heuristics based on column generation method are proposed for the compartmentalized knapsack problem and the cutting stock problem integrated with the capacitated lot sizing problem. In particular, the period, the machine, and the period and machine Dantzig- Wolfe decompositions are applied for the integrated problem. Moreover, a heuristic based on the graph AND/OR is proposed for the two-dimensional compartmentalized knapsack problem. Computational results show that these mathematical formulations and methods provide good solutions

Nesta tese é abordado o problema da mochila compartimentada e o problema de corte de estoque unidimensional acoplado ao problema dimensionamento de lotes. Para o problema da mochila compartimentada é apresentada a versão unidimensional e proposta a versão bidimensional, denominados como problema da mochila compartimentada unidimensional e problema da mochila compartimentada bidimensional, respectivamente. Para o problema de corte de estoque acoplado ao dimensionamento de lotes são apresentadas três variações: uma máquina para produzir um tipo de objeto; uma máquina para produzir vários tipos de objetos; múltiplas máquinas para produzir vários tipos de objetos. Algumas formulações matemáticas de programação inteira e inteira-mista, decomposições dos problemas em problema mestre e subproblemas e heurísticas baseadas no método geração de colunas são propostas para os problemas da mochila compartimenta e o problema acoplado. Em específico, para o problema acoplado são aplicadas decomposições Dantzig-Wolfe, que podem ser por período, por máquina ou por período e máquina. Além disso, uma heurística baseada em grafo E/OU é proposta para o problema da mochila compartimentada bidimensional / In this thesis we present the constrained compartmentalized knapsack problem and the one dimensional cutting stock problem integrated with the capacitated lot sizing problem. For the constrained compartmentalized knapsack problem, the one dimensional version is presented and the two dimensional version is proposed, called one-dimensional compartmentalized knapsack problem and two-dimensional compartmentalized knapsack problem, respectively. For the cutting stock problem integrated with the capacitated lot sizing problem three variations are considered: one machine to produce one type of object; one machine to produce multiple types of objects; multiple machines to produce multiple types of objects. Some integer and mixed programming formulations, decompositions of the problems in master problem and subproblems and heuristics based on column generation method are proposed for the compartmentalized knapsack problem and the cutting stock problem integrated with the capacitated lot sizing problem. In particular, the period, the machine, and the period and machine Dantzig- Wolfe decompositions are applied for the integrated problem. Moreover, a heuristic based on the graph AND/OR is proposed for the two-dimensional compartmentalized knapsack problem. Computational results show that these mathematical formulations and methods provide good solutions

Problemas de corte de estoque consistem em arranjar peças menores, em tamanhos e quantidades especificados, dentro de peças maiores. Tais problemas têm sido investigados intensamente nas últimas décadas, acrescidos de novas características e novos métodos de solução. Nesta tese abordamos o problema de corte de estoque multiperíodo que surge imerso no planejamento e programação da produção em empresas que têm um estágio de produção caracterizado pelo corte de peças. As demandas dos itens ocorrem em períodos diversos de um horizonte de planejamento finito, sendo possível antecipar ou não a produção de itens. Os objetos disponíveis em estoque não utilizados em um período ficam disponíveis no próximo período, juntamente com novos objetos adquiridos ou produzidos pela própria empresa. Um modelo de otimização linear inteira de grande porte é proposto, cujo objetivo pondera o custo das perdas nos cortes, os custos de estocagem de objetos e itens. O método simplex com geração de colunas foi especializado para resolver a relaxação linear do modelo proposto. Foram realizados experimentos computacionais com problemas de corte de estoque unidimensional e bidimensional. Tais experimentos mostram que ganhos efetivos podem ser obtidos usando-se o modelo de corte de estoque multiperíodo, quando comparado com a solução lote-por-lote, tipicamente utilizada na prática. Porém, na prática, a solução relaxada é de pouca, ou nenhuma, utilidade. Assim, nesta tese, desenvolvemos dois procedimentos de arredondamento da solução do problema multiperíodo, baseado em horizonte rolante, ou seja, determinamos uma solução inteira factível apenas para o primeiro período, a qual será, de fato, implementada. Enfim, concluímos que o modelo para o problema de corte de estoque multiperíodo permite flexibilidade na análise de uma solução a ser implementada e, portanto, é uma ferramenta que permite ao gerente de produção uma visão global do problema para auxiliá-lo na tomada de decisões / Cutting stock problems consist of cutting a set of available stock objects in order to produce smaller ordered items. Such problems have been intensively researched over the last decades, together with additional characteristics and new methods for solving them. In this thesis, we address the multiperiod cutting stock problem, which arises in the production planning and programming in many industries that have a cutting process as an important stage. Ordered items have different due date over a finite planning horizon. An integer linear optimization model of large scale is proposed. The model makes possible to anticipate or not the production of items. Unused objects in inventory in a period become available to the next period, added to new inventory, which are acquired or produced by the own company. The mathematical model\'s objective is to minimize the cost of waste in the cutting process and costs for holding objects and fInal items. The simplex method with column generation was specialized to solve its linear relaxation. Computational experiments were carried out to solve one-dimensional and two-dimensional cutting stock problems. Such experiments showed that the multiperiod model could obtain effective gains when compared with the lot-for-lot solution, which is typically used in practice. However, in practical problems, the fractional solution is useless. So, in this thesis, two rounding procedures are developed to determine integer solutions for multiperiod cutting stock problems. Such procedures are based on a rolling horizon scheme, which roughly means, find an integer solution only for the first period, since this is the solution to be, in fact, carried out. Finally, we conclude that the proposed model for multiperiod cutting stock problems allows flexibility on analyzing a solution to be put in practice. The multiperiod cutting problem can be a tool that provides the decision maker a wide view of the problem and it may help him/her on making decisions

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.

Problemas de corte de estoque consistem em arranjar peças menores, em tamanhos e quantidades especificados, dentro de peças maiores. Tais problemas têm sido investigados intensamente nas últimas décadas, acrescidos de novas características e novos métodos de solução. Nesta tese abordamos o problema de corte de estoque multiperíodo que surge imerso no planejamento e programação da produção em empresas que têm um estágio de produção caracterizado pelo corte de peças. As demandas dos itens ocorrem em períodos diversos de um horizonte de planejamento finito, sendo possível antecipar ou não a produção de itens. Os objetos disponíveis em estoque não utilizados em um período ficam disponíveis no próximo período, juntamente com novos objetos adquiridos ou produzidos pela própria empresa. Um modelo de otimização linear inteira de grande porte é proposto, cujo objetivo pondera o custo das perdas nos cortes, os custos de estocagem de objetos e itens. O método simplex com geração de colunas foi especializado para resolver a relaxação linear do modelo proposto. Foram realizados experimentos computacionais com problemas de corte de estoque unidimensional e bidimensional. Tais experimentos mostram que ganhos efetivos podem ser obtidos usando-se o modelo de corte de estoque multiperíodo, quando comparado com a solução lote-por-lote, tipicamente utilizada na prática. Porém, na prática, a solução relaxada é de pouca, ou nenhuma, utilidade. Assim, nesta tese, desenvolvemos dois procedimentos de arredondamento da solução do problema multiperíodo, baseado em horizonte rolante, ou seja, determinamos uma solução inteira factível apenas para o primeiro período, a qual será, de fato, implementada. Enfim, concluímos que o modelo para o problema de corte de estoque multiperíodo permite flexibilidade na análise de uma solução a ser implementada e, portanto, é uma ferramenta que permite ao gerente de produção uma visão global do problema para auxiliá-lo na tomada de decisões / Cutting stock problems consist of cutting a set of available stock objects in order to produce smaller ordered items. Such problems have been intensively researched over the last decades, together with additional characteristics and new methods for solving them. In this thesis, we address the multiperiod cutting stock problem, which arises in the production planning and programming in many industries that have a cutting process as an important stage. Ordered items have different due date over a finite planning horizon. An integer linear optimization model of large scale is proposed. The model makes possible to anticipate or not the production of items. Unused objects in inventory in a period become available to the next period, added to new inventory, which are acquired or produced by the own company. The mathematical model\'s objective is to minimize the cost of waste in the cutting process and costs for holding objects and fInal items. The simplex method with column generation was specialized to solve its linear relaxation. Computational experiments were carried out to solve one-dimensional and two-dimensional cutting stock problems. Such experiments showed that the multiperiod model could obtain effective gains when compared with the lot-for-lot solution, which is typically used in practice. However, in practical problems, the fractional solution is useless. So, in this thesis, two rounding procedures are developed to determine integer solutions for multiperiod cutting stock problems. Such procedures are based on a rolling horizon scheme, which roughly means, find an integer solution only for the first period, since this is the solution to be, in fact, carried out. Finally, we conclude that the proposed model for multiperiod cutting stock problems allows flexibility on analyzing a solution to be put in practice. The multiperiod cutting problem can be a tool that provides the decision maker a wide view of the problem and it may help him/her on making decisions