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A customized branch-and-bound approach for irregular shape nesting

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Abstract

We study the Nesting Problem, which aims to determine a configuration of a set of irregular shapes within a rectangular sheet of material of fixed width, such that no overlap among the shapes exists, and such that the length of the sheet is minimized. When both translation and rotation of the shapes are allowed, the problem can be formulated as a nonconvex quadratically constrained programming model that approximates each shape by a set of inscribed circles and enforces that circle pairs stemming from different shapes do not overlap. However, despite many recent advances in today’s global optimization solvers, solving this nonconvex model to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we propose a customized branch-and-bound approach to address the Nesting Problem to guaranteed optimality. Our approach utilizes a novel branching scheme to deal with the reverse convex quadratic constraints in the quadratic model and incorporates a number of problem-specific algorithmic tweaks. Our computational studies on a suite of 64 benchmark instances demonstrate the customized algorithm’s effectiveness and competitiveness over the use of general-purpose global optimization solvers, including for the first time the ability to find global optimal nestings featuring five polygons under free rotation.

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Notes

  1. In fact, one needs only impose these constraints for the vertices that constitute the convex hull of polygon \(p \in P\).

  2. For compatibility with constraints (15) and (16), the symmetry center must also be moved to point \(\left( 0,0\right) \).

  3. We prefer to use \(h_p + v_p \le h_q + v_q\), because it is likely to break symmetry in more instances.

  4. In the original implementation of Jones [14], the values 5 and 13 were proposed for parameter \(N^{\text {init}}\). However, in most instances used in this study, we found that using 5 or more circles per each and every polygon leads to initial QCP models that cannot be solved within the time limit, i.e., the GO solvers could not even complete the first (and also easiest) iteration. In contrast, using the setting \(N^{\text {init}}= 3\) and allowing the circle sets to grow more judiciously as overlaps are identified led to much better performance of the QP-Nest approach.

  5. For the sake of better numerical stability, in our implementations the width was normalized down to the value of 1, and all coordinates were scaled correspondingly.

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Acknowledgements

This work is dedicated to the memory of our academic father and grandfather, Professor Christodoulos A. Floudas. While his untimely passing has left us all with sorrow, his lasting impact on the field of Global Optimization shall remain a source of inspiration.

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  1. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Akang Wang, Christopher L. Hanselman & Chrysanthos E. Gounaris

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  1. Akang Wang
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  2. Christopher L. Hanselman
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  3. Chrysanthos E. Gounaris
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Correspondence to Chrysanthos E. Gounaris.

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Wang, A., Hanselman, C.L. & Gounaris, C.E. A customized branch-and-bound approach for irregular shape nesting. J Glob Optim 71, 935–955 (2018). https://doi.org/10.1007/s10898-018-0637-y

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