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Januar 2013
Mohsen Afsharian
The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with Defects
This thesis deals with a two-dimensional cutting problem in which small rectangular items of given types are to be cut from a rectangular large object which contains several defects. It is assumed that the number of pieces of each small item type, which can be cut from the large object, is not limited. In addition, all cuts are restricted to be of the guillotine-type and the number of stages, which are necessary to perform all cuts, can either be limited or not. Furthermore, no small item must overlap with a defective region. The objective is to maximize the value of the cut small items.
In the presence of defects, the definition of discretization sets and the identification of duplicated patterns are revisited. For the exact solution of the above-described problem, dynamic programming algorithms are presented. Moreover, the computational complexity of the algorithms is analyzed and the factors which affect the running time of the algorithms are identified.
The exact algorithms are able to solve the problem instances of small and medium sizes. For large problem instances for which optimal solutions cannot be computed efficiently, the dynamic programming algorithms are modified into algorithms in which an adaptation of a variable beam search approach equipped with several heuristic rules is applied.
The proposed algorithms are evaluated by means of a series of detailed numerical experiments which are performed on problem instances extracted from the literature, as well as on randomly generated instances. The experiments do not only illustrate how the proposed method can identify optimal solutions of the test problem instances, but they also explain why already existing methods fail to do so. Furthermore, the computational results indicate that the exact and heuristic recursive-based methods are able to overcome efficiently some structural and computational limitations for solving problem instances of realistic sizes.
In the presence of defects, the definition of discretization sets and the identification of duplicated patterns are revisited. For the exact solution of the above-described problem, dynamic programming algorithms are presented. Moreover, the computational complexity of the algorithms is analyzed and the factors which affect the running time of the algorithms are identified.
The exact algorithms are able to solve the problem instances of small and medium sizes. For large problem instances for which optimal solutions cannot be computed efficiently, the dynamic programming algorithms are modified into algorithms in which an adaptation of a variable beam search approach equipped with several heuristic rules is applied.
The proposed algorithms are evaluated by means of a series of detailed numerical experiments which are performed on problem instances extracted from the literature, as well as on randomly generated instances. The experiments do not only illustrate how the proposed method can identify optimal solutions of the test problem instances, but they also explain why already existing methods fail to do so. Furthermore, the computational results indicate that the exact and heuristic recursive-based methods are able to overcome efficiently some structural and computational limitations for solving problem instances of realistic sizes.
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