Abstract
Cutting and packing problems are challenging combinatorial optimization problems that have many rel-evant industrial applications and arise whenever a raw material has to be cut into smaller parts while minimizing waste, or products have to be packed, minimizing the empty space. Thus, the optimal solution to these problems has a positive economic and environmental impact. In many practical applications, both the raw material and the cut parts have a rectangular shape, and cut-ting plans are generated for one raw material rectangle (also known as plate) at a time. This is known in the literature as the (two-dimensional) rectangular cutting problem. Many variants of this problem may arise, led by cutting technology constraints, raw-material characteristics, and different planning goals, the most relevant of which are the guillotine cuts. The absence of the guillotine cuts imposition makes the problem harder to solve to optimality.Based on the Floating-Cuts paradigm, a general and flexible mixed-integer programming model for the general rectangular cutting problem is proposed. To the best of our knowledge, it is the first mixed inte-ger linear programming model in the literature for both non-guillotine and guillotine problems. The basic idea of this model is a tree search where branching occurs by successive first-order non-guillotine-type cuts. The exact position of the cuts is not fixed, but instead remains floating until a concrete small rect-angle (also known as item) is assigned to a child node. This model does not include decision variables either for the position coordinates of the items or for the coordinates of the cuts. Under this framework, it was possible to address various different variants of the problem.Extensive computational experiments were run to evaluate the model's performance considering 16 dif-ferent problem variants, and to compare it with the state-of-the-art formulations of each variant. The results confirm the power of this flexible model, as, for some variants, it outperforms the state-of-the-art approaches and, for the other variants, it presents results fairly close to the best approaches. But, even more importantly, this is a new way of looking at these problems which may trigger even better approaches, with the consequent economic and environmental benefits.

Abstract
AbstractIn the pallet building problem, we aim at loading a given set of items into one or more pallets, by satisfying specific constraints and minimizing the number of pallets used. In this paper, we address a practical case of this problem that originates from a real-world robotized application, subject to some non-trivial operational constraints. In practice, items are grouped into families and must be packed into horizontal layers. To facilitate loading/unloading operations, items of the same type packed into the same layer should be contiguous and at least one of them should be visible from the outside. We present a formal mathematical description for layer and pallet creation subproblems and then we propose heuristic, metaheuristic, matheuristic algorithms to solve the overall problem. The performance of the algorithms is assessed through extensive computational tests on real-world instances.

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