# Cutting ellipses from area-minimizing rectangles

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## Abstract

A set of ellipses, with given semi-major and semi-minor axes, is to be cut from a rectangular design plate, while minimizing the area of the design rectangle. The design plate is subject to lower and upper bounds of its widths and lengths; the ellipses are free of any orientation restrictions. We present new mathematical programming formulations for this ellipse cutting problem. The key idea in the developed non-convex nonlinear programming models is to use separating hyperlines to ensure the ellipses do not overlap with each other. For small number of ellipses we compute feasible points which are globally optimal subject to the finite arithmetic of the global solvers at hand. However, for more than 14 ellipses none of the local or global NLP solvers available in GAMS can even compute a feasible point. Therefore, we develop polylithic approaches, in which the ellipses are added sequentially in a strip-packing fashion to the rectangle restricted in width, but unrestricted in length. The rectangle’s area is minimized in each step in a greedy fashion. The sequence in which we add the ellipses is random; this adds some GRASP flavor to our approach. The polylithic algorithms allow us to compute good, near optimal solutions for up to 100 ellipses.

## Keywords

Global optimization Non-convex nonlinear programming Mixed integer programming Cutting stock problem Packing problem Shape constraints Non-overlap constraints Design problem Polylithic solution approach Computational geometry## Notes

### Acknowledgments

Thanks are directed to Prof. Dr. Siegfried Jetzke (Ostfalia Hochschule, Salzgitter, Germany) for his interest in this work, comments on the manuscript and discussion about the usefulness of ellipses and ellipsoids in real world problems.

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