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Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

Author

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  • N. Chernov
  • Yu. Stoyan
  • T. Romanova
  • A. Pankratov

Abstract

We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples.

Suggested Citation

  • N. Chernov & Yu. Stoyan & T. Romanova & A. Pankratov, 2012. "Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs," Advances in Operations Research, Hindawi, vol. 2012, pages 1-26, May.
    • Handle: RePEc:hin:jnlaor:346358
    DOI: 10.1155/2012/346358
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    Cited by:

    1. 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.
    2. Masoud Hekmatfar & M. R. M. Aliha & Mir Saman Pishvaee & Tomasz Sadowski, 2023. "A Robust Flexible Optimization Model for 3D-Layout of Interior Equipment in a Multi-Floor Satellite," Mathematics, MDPI, vol. 11(24), pages 1-41, December.
    3. 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.
    4. Qiang Luo & Yunqing Rao, 2022. "Improved Sliding Algorithm for Generating No-Fit Polygon in the 2D Irregular Packing Problem," Mathematics, MDPI, vol. 10(16), pages 1-18, August.

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