The Floating-Cuts model: a general and flexible mixed-integer programming model for non-guillotine and guillotine rectangular cutting problems
•A general MIP model for non-guillotine and guillotine rectangular cutting problems•Allows for the incorporation of many and diverse cutting technological constraints•Outperforms problem tailored state-of-the-art approaches for some problem variants•For the other problem variants presents results fa...
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| Published in: | Omega (Oxford) Vol. 114; p. 102738 |
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| Format: | Journal Article |
| Language: | English |
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01.01.2023
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| ISSN: | 0305-0483, 1873-5274 |
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| Abstract | •A general MIP model for non-guillotine and guillotine rectangular cutting problems•Allows for the incorporation of many and diverse cutting technological constraints•Outperforms problem tailored state-of-the-art approaches for some problem variants•For the other problem variants presents results fairly close to the best approaches
Cutting and packing problems are challenging combinatorial optimization problems that have many relevant 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 cutting 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 integer 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 rectangle (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 different 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. |
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| AbstractList | •A general MIP model for non-guillotine and guillotine rectangular cutting problems•Allows for the incorporation of many and diverse cutting technological constraints•Outperforms problem tailored state-of-the-art approaches for some problem variants•For the other problem variants presents results fairly close to the best approaches
Cutting and packing problems are challenging combinatorial optimization problems that have many relevant 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 cutting 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 integer 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 rectangle (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 different 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. |
| ArticleNumber | 102738 |
| Author | Carravilla, Maria Antónia Silveira, Tiago Mundim, Leandro Oliveira, José Fernando Silva, Elsa |
| Author_xml | – sequence: 1 givenname: Elsa orcidid: 0000-0002-6274-8095 surname: Silva fullname: Silva, Elsa organization: INESC TEC - Instituto de Engenharia de Sistemas e Computadores, Portugal – sequence: 2 givenname: José Fernando orcidid: 0000-0002-4061-1311 surname: Oliveira fullname: Oliveira, José Fernando email: jfo@fe.up.pt organization: INESC TEC, Faculdade de Engenharia, Universidade do Porto, Portugal – sequence: 3 givenname: Tiago surname: Silveira fullname: Silveira, Tiago organization: Department of Engineering and Architecture, University of Parma, Italy – sequence: 4 givenname: Leandro surname: Mundim fullname: Mundim, Leandro organization: Optimized Decision Making (ODM), São Carlos, Brazil – sequence: 5 givenname: Maria Antónia orcidid: 0000-0002-9245-2674 surname: Carravilla fullname: Carravilla, Maria Antónia organization: INESC TEC, Faculdade de Engenharia, Universidade do Porto, Portugal |
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| Keywords | non-guillotine and guillotine cutting and packing problems cutting mixed-integer linear programming model tree search |
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