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Packing ovals in optimized regular polygons

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Abstract

We present a model development framework and numerical solution approach to the general problem-class of packing convex objects into optimized convex containers. Specifically, we discuss the problem of packing ovals (egg-shaped objects, defined here as generalized ellipses) into optimized regular polygons in \( {\mathbb{R}}^{2} \). Our solution strategy is based on the use of embedded Lagrange multipliers, followed by nonlinear optimization. Credible numerical results are attained using randomized starting solutions, refined by a single call to a local optimization solver. We obtain visibly good quality packings for packing 4 to 10 ovals into regular polygons with 3 to 10 sides in all 224 test problems presented here. Our modeling and solution approach can be extended towards handling other difficult packing problems.

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Authors and Affiliations

  1. Physicist at Large Consulting LLC, Bryn Mawr, PA, USA

    Frank J. Kampas

  2. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA

    János D. Pintér

  3. Lazaridis School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, Canada

    Ignacio Castillo

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  1. Frank J. Kampas
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  2. János D. Pintér
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  3. Ignacio Castillo
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Correspondence to Ignacio Castillo.

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Kampas, F.J., Pintér, J.D. & Castillo, I. Packing ovals in optimized regular polygons. J Glob Optim 77, 175–196 (2020). https://doi.org/10.1007/s10898-019-00824-8

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