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With the extensive application of laser cutting, the proper planning of the cutting path has a significate impact on industrial production, like clothing, metalware, and furniture. This paper proposes heuristic approaches for the two-dimensional cutting path problem, which aims to find the shortest path to cut pieces from the master plate. Two cutting methods, the arc routing method, and the node routing method, are studied based on whether the piece must be cut separately. A two-step heuristic approach is proposed for the arc routing method. A model based on a generalized traveling salesman problem (GTSP) and a variable neighborhood search (VNS) is introduced for the node routing method. The results show that the proposed two-step heuristic quickly finds the optimal or near-optimal solution. CPLEX using the GTSP model can solve small-size instances, while the VNS can find the solution with good quality in a reasonable time for medium-size instances. In addition, the model and approaches are also tested on larger instances with more than 250 pieces and 20,000 nodes. The results show that the proposed heuristics can handle these instances within a reasonable time. • A two-step heuristic is proposed for the arc routing method. • A new model is proposed for the node routing method. • A variable neighborhood search is introduced for the node routing method. • The two-step heuristic quickly finds the optimal or near-optimal solution. [ABSTRACT FROM AUTHOR]

In this paper, we address the Constrained Three-dimensional Guillotine Cutting Problem (C3GCP), which consists of cutting a larger cuboid block (object) to produce a limited number of smaller cuboid pieces (items) using orthogonal guillotine cuts only. This way, all cuts must be parallel to the object's walls and generate two cuboid sub-blocks, and there is a maximum number of copies that can be manufactured for each item type. The C3GCP arises in industrial manufacturing settings, such as the cutting of steel and foam for mattresses. To model this problem, we propose a new compact mixed-integer non-linear programming (MINLP) formulation by extending its two-dimensional version, and develop a mixed-integer linear programming (MILP) version. We also propose a new model for a particular case of the problem which considers 3-staged patterns. As a solution method, we extend the algorithm of Wang (1983) to the three-dimensional case. We emphasise that the C3GCP is different from 3D packing problems, namely from the Container Loading Problem, because of the guillotine cut constraints. All proposed approaches are evaluated through computational experiments using benchmark instances. The results show that the approaches are effective on different types of instances, mainly when the maximum number of copies per item type is small, a situation typically encountered in practical settings with low demand for each item type. These approaches can be easily embedded into existing expert systems for supporting the decision-making process. • We address the constrained three-dimensional guillotine placement problem. • We present ILP formulations for non-staged and 3-staged patterns. • We present a binary tree-search algorithm for non-staged and 3-staged patterns. • Optimal or near-optimal solutions are obtained in reasonable processing times. • These approaches can be useful for different industrial cutting processes. [ABSTRACT FROM AUTHOR]