# A fully general, exact algorithm for nesting irregular shapes

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## Abstract

This paper introduces a fully general, exact algorithm for nesting irregular shapes. Both the shapes and material resource can be arbitrary nonconvex polygons. Moreover, the shapes can have holes and the material can have defective areas. Finally, the shapes can be arranged using both translations and *arbitrary* rotations (as opposed to a finite set of rotation angles, such as 0\(^\circ \)and 180\(^\circ \)). The insight that has made all this possible is a novel way to relax the constraint that the shapes not overlap. The key idea is to inscribe a few circles in each irregular shape and then relax the non-overlap constraints for the *shapes* by replacing them with non-overlap constraints for the *inscribed circles*. These relaxed problems have the form of quadratic programming problems (QPs) and can be solved to optimality to provide valid lower bounds. Valid upper bounds can be found via local search with strict non-overlap constraints. If the shapes overlap in the solution to the relaxed problem, new circles are inscribed in the shapes to prevent this overlapping configuration from recurring, and the QP is then resolved to obtain improved lower bounds. Convergence to any fixed tolerance is guaranteed in a finite number of iterations. A specialized branch-and-bound algorithm, together with some heuristics, are introduced to find the initial inscribed circles that approximate the shapes. The new approach, called “QP-Nest,” is applied to three problems as proof of concept. The most complicated of these is a problem due to Milenkovic that has four nonconvex polygons with 94, 72, 84, and 74 vertices, respectively. QP-Nest is able prove global optimality when nesting the first *two* or the first *three* of these shapes. When all *four* shapes are considered, QP-Nest finds the best known solution, but cannot prove optimality.

## Keywords

Fabric nesting Leather nesting Nonconvex polygon Irregular shape Inscribed circle Inner approximation Global optimization Quadratic programming Cutting and packing Translation Rotation Guaranteed optimality## Notes

### Acknowledgments

Lee Bristol of Delphi Interior and Lighting Systems introduced me to the fabric nesting problem, and Bill Crawford of Delphi Interior and Lighting Systems provided Fig. 1 and shared his knowledge on the nature the fabric nesting problem and the relative merits of different nesting approaches. Karen Daniels graciously guided me through the work she has done with Milenkovic and Li. Victor Milenkovic provided me with the data for his four-part test problem. Mary Dalzell of the GM Research library carried out a computerized literature search on the subject of nesting. Paul Sweeney and Bob Haessler of the University of Michigan also helped identify the relevant literature. Special thanks go to Chris Floudas of Princeton University for giving me an account on the Princeton computer system for carrying out the numerical experiments. Thanks also to Ruth Misener (co-creator of GloMIQO), who provided me with an updated literature search and helped me understand how different problem formulations would affect the performance of GloMIQO. Santosh Tiwari of GM installed LaTeX and BibTeX on my computer and showed me how to use them. Finally, I would like to thank the two anonymous reviewers for their comments and suggestions that pushed me to add clarifications, extend the literature review, and do additional numerical work, all of which substantially improved the paper.

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