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Packing ellipses in an optimized convex polygon

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Abstract

Packing ellipses with arbitrary orientation into a convex polygonal container which has a given shape is considered. The objective is to find a minimum scaling (homothetic) coefficient for the polygon still containing a given collection of ellipses. New phi-functions and quasi phi-functions to describe non-overlapping and containment constraints are introduced. The packing problem is then stated as a continuous nonlinear programming problem. A solution approach is proposed combining a new starting point algorithm and a new modification of the LOFRT procedure (J Glob Optim 65(2):283–307, 2016) to search for locally optimal solutions. Computational results are provided to demonstrate the efficiency of our approach. The computational results are presented for new problem instances, as well as for instances presented in the recent paper (http://www.optimization-online.org/DB_FILE/2016/03/5348.pdf, 2016).

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Acknowledgements

Our interest in packing continuously rotating ellipses in an optimized convex polygonal container was largely motivated by the paper [2]. We would like to thank all authors of the paper and personally János Pintér for sharing with us their experience and for many fruitful discussions. The work of the third author was partially supported by the CONACYT Grants #167019, 293403.

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Authors and Affiliations

  1. Department of Mathematical Modelling and Optimal Design, Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine

    A. Pankratov & T. Romanova

  2. Faculty of Mechanical and Electrical Engineering, Nuevo Leon State University (UANL), San Nicolas de los Garza, Mexico

    I. Litvinchev

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  1. A. Pankratov
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  2. T. Romanova
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  3. I. Litvinchev
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Correspondence to T. Romanova.

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Pankratov, A., Romanova, T. & Litvinchev, I. Packing ellipses in an optimized convex polygon. J Glob Optim 75, 495–522 (2019). https://doi.org/10.1007/s10898-019-00777-y

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  • DOI: https://doi.org/10.1007/s10898-019-00777-y

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