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Irregular packing problems: A review of mathematical models

Author

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  • Leao, Aline A.S.
  • Toledo, Franklina M.B.
  • Oliveira, José Fernando
  • Carravilla, Maria Antónia
  • Alvarez-Valdés, Ramón

Abstract

Irregular packing problems (also known as nesting problems) belong to the more general class of cutting and packing problems and consist of allocating a set of irregular and regular pieces to larger rectangular or irregular containers, while minimizing the waste of material or space. These problems combine the combinatorial hardness of cutting and packing problems with the computational difficulty of enforcing the geometric non-overlap and containment constraints. Unsurprisingly, nesting problems have been addressed, both in the scientific literature and in real-world applications, by means of heuristic and metaheuristic techniques. However, more recently a variety of mathematical models has been proposed for nesting problems. These models can be used either to provide optimal solutions for nesting problems or as the basis of heuristic approaches based on them (e.g. matheuristics). In both cases, better solutions are sought, with the natural economic and environmental positive impact. Different modeling options are proposed in the literature. We review these mathematical models under a common notation framework, allowing differences and similarities among them to be highlighted. Some insights on weaknesses and strengths are also provided. By building this structured review of mathematical models for nesting problems, research opportunities in the field are proposed.

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  • Leao, Aline A.S. & Toledo, Franklina M.B. & Oliveira, José Fernando & Carravilla, Maria Antónia & Alvarez-Valdés, Ramón, 2020. "Irregular packing problems: A review of mathematical models," European Journal of Operational Research, Elsevier, vol. 282(3), pages 803-822.
    • Handle: RePEc:eee:ejores:v:282:y:2020:i:3:p:803-822
    DOI: 10.1016/j.ejor.2019.04.045
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    3. Chehrazad, Sahar & Roose, Dirk & Wauters, Tony, 2022. "A fast and scalable bottom-left-fill algorithm to solve nesting problems using a semi-discrete representation," European Journal of Operational Research, Elsevier, vol. 300(3), pages 809-826.
    4. Gahm, Christian & Uzunoglu, Aykut & Wahl, Stefan & Ganschinietz, Chantal & Tuma, Axel, 2022. "Applying machine learning for the anticipation of complex nesting solutions in hierarchical production planning," European Journal of Operational Research, Elsevier, vol. 296(3), pages 819-836.
    5. Umetani, Shunji & Murakami, Shohei, 2022. "Coordinate descent heuristics for the irregular strip packing problem of rasterized shapes," European Journal of Operational Research, Elsevier, vol. 303(3), pages 1009-1026.
    6. Josef Kallrath & Tatiana Romanova & Alexander Pankratov & Igor Litvinchev & Luis Infante, 2023. "Packing convex polygons in minimum-perimeter convex hulls," Journal of Global Optimization, Springer, vol. 85(1), pages 39-59, January.
    7. Jie Fang & Yunqing Rao & Xusheng Zhao & Bing Du, 2023. "A Hybrid Reinforcement Learning Algorithm for 2D Irregular Packing Problems," Mathematics, MDPI, vol. 11(2), pages 1-17, January.
    8. Romanova, Tatiana & Stoyan, Yurij & Pankratov, Alexander & Litvinchev, Igor & Plankovskyy, Sergiy & Tsegelnyk, Yevgen & Shypul, Olga, 2021. "Sparsest balanced packing of irregular 3D objects in a cylindrical container," European Journal of Operational Research, Elsevier, vol. 291(1), pages 84-100.
    9. Kimms, Alf & Király, Hédi, 2023. "An extended model formulation for the two-dimensional irregular strip packing problem considering general industry-relevant aspects," European Journal of Operational Research, Elsevier, vol. 306(3), pages 1202-1218.
    10. Frank J. Kampas & János D. Pintér & Ignacio Castillo, 2023. "Model Development and Solver Demonstrations Using Randomized Test Problems," SN Operations Research Forum, Springer, vol. 4(1), pages 1-15, March.

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