A fully general, exact algorithm for nesting irregular shapes

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<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$^\circ $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow/> <mo>∘</mo> </msup> </math> </EquationSource> </InlineEquation>and 180<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$^\circ $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow/> <mo>∘</mo> </msup> </math> </EquationSource> </InlineEquation>). 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. Copyright Springer Science+Business Media New York 2014