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

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.