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Solving nesting problems involves the waste minimisation in cutting processes, and therefore it is not only economically relevant for many industries but has also an important environmental impact, as the raw materials that are cut are usually a natural resource. However, very few exact approaches have been proposed in the literature for the nesting problem (also known as irregular packing problem), and the majority of the known approaches are heuristic algorithms, leading to suboptimal solutions. The few mathematical programming models known for this problem can be divided into discrete and continuous models, based on how the placement coordinates of the pieces to be cut are dealt with. In this paper, we propose an innovative semi-continuous mixed-integer programming model for two-dimensional cutting and packing problems with irregular shaped pieces. The model aims to exploit the advantages of the two previous classes of approaches and discretises the y-axis while keeping the x-coordinate continuous. The board can therefore be seen as a set of stripes. Computational results show that the model, when solved by a commercial solver, can deal with large problems and determine the optimal solution for smaller instances, but as it happens with discrete models, the optimal solution value depends on the discretisation step that is used.

Agradeço primeiramente a Deus por ter me abençoado em todas as etapas necessárias deste trabalho e em todos os momentos da minha vida. A minha mãe, a quem eu amo muito, pelo carinho, paciência e pela preocupação. A minha irmã Carla pelo apoio e paciência. Ao meu pai pela torcida. Aos professores Arenales e Maristela pela orientação, amizade e paciência. A minha avó Maria, ao tio Hélio e ao tio Osvaldo pelo carinho, preocupação e pelas orações. A minha madrinha Cida pelo carinho, pelas orações e pela confiança que sempre depositou em mim. À professora Silvely pela amizade e por ter me incentivado na pesquisa. A todos os meus amigos que sempre estiveram do meu lado. Aos colegas da sala de estudo dos alunos de pós-graduação, pelos momentos de estudo e pelas muitas risadas. Ao pessoal do laboratório pela convivência nos momentos de estudo e de descontração. A todos que contribuíram direta ou indiretamente para este trabalho.

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The Three-dimensional Single Large Object Placement Problem consists of a set of weakly heterogeneous items that must be placed inside a single larger object without overlapping each other. Many constraints can be considered depending on the practical specifications of the problem being solved, such as orientation, stability, weight limit and positioning. Although this is a well-known problem which has received considerable academic attention, most of the research limits itself to considering only three basic constraints: non-overlap, orientation and stability of the placed items. Recent literature concerning the problem has indicated that there is a pressing need for solution methods which consider a more realistic number of sets of practical constraints given that it is very uncommon to find real-world situations where only a few of these constraints are considered together. Therefore, this paper introduced a well-performing matheuristic framework which considers multiple practical constraints: orientation, load balance, loading priorities, positioning, stability, stacking and weight limit. An extension of the developed method considering multiple containers is also discussed. Results are compared against the state of the art from the literature and demonstrate the robustness of the proposed matheuristic with respect to different combinations of constraints.

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