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Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications

In: Optimization Methods and Applications

Author

Listed:
  • Yuriy Stoyan

    (Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine)

  • Alexandr Pankratov

    (Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine)

  • Tatiana Romanova

    (Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine)

Abstract

We describe our methodology for solving NP-hard irregular placement problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their free rotations within a container. We formulate a basic irregular placement problem (IRPP), which covers a wide spectrum of practical packing, cutting, nesting, clustering, and layout problems. We provide a nonlinear programming (NLP) model of the problem, employing the phi-function technique. Our model involves a large number of inequalities with nonsmooth functions. We describe a solution tree for our placement problem and evaluate the number of its terminal nodes. We reduce IRPP problem to a sequence of NLP-subproblems with smooth functions. Our solution strategy is based on combination of discrete and continuous optimization methods. We employ two approaches to solve IRPP problem: a branching scheme algorithm and an efficient optimization algorithm, which involves a feasible starting point and local optimization procedures. To show the benefits of our methodology we present computational results for a number of new challenger and the best known benchmark instances.

Suggested Citation

  • Yuriy Stoyan & Alexandr Pankratov & Tatiana Romanova, 2017. "Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications," Springer Optimization and Its Applications, in: Sergiy Butenko & Panos M. Pardalos & Volodymyr Shylo (ed.), Optimization Methods and Applications, pages 521-559, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-68640-0_25
    DOI: 10.1007/978-3-319-68640-0_25
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    Citations

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    Cited by:

    1. Andreas Fischer & Igor Litvinchev & Tetyana Romanova & Petro Stetsyuk & Georgiy Yaskov, 2023. "Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container," Mathematics, MDPI, vol. 11(9), pages 1-19, April.
    2. Alexander Pankratov & Tatiana Romanova & Igor Litvinchev, 2020. "Packing Oblique 3D Objects," Mathematics, MDPI, vol. 8(7), pages 1-17, July.
    3. Lastra-Díaz, Juan J. & Ortuño, M. Teresa, 2024. "Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts," European Journal of Operational Research, Elsevier, vol. 313(1), pages 69-91.
    4. Germán Pantoja-Benavides & David Álvarez-Martínez & Francisco Parreño Torres, 2024. "The Normalized Direct Trigonometry Model for the Two-Dimensional Irregular Strip Packing Problem," Mathematics, MDPI, vol. 12(15), pages 1-25, August.
    5. 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.
    6. 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.

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