A fast and scalable algorithm to solve nesting problems using a semi-discrete representation

We present a fast algorithm to solve nesting problems based on a semi-discrete representation of both the 2D non-convex pieces and the strip. The pieces and the strip are represented by a set of equidistant vertical line segments. The discretization algorithm uses a sweep-line method and applies minimal extensions to the line segments of a piece to ensure that non-overlapping placement of the segments, representing two pieces, cannot cause overlap of the original pieces. We implemented a bottom-left-fill greedy placement procedure, using an optimized ordering of the segment overlap tests. The C++ implementation of our algorithm uses appropriate data structures that allow fast execution. It executes the bottom-left-fill algorithm for typical ESICUPdata sets in a few milliseconds, even when the rotation of the pieces is considered, and thus provides a suitable basis for integration in metaheuristics. Moreover, we show that the algorithm scales well when the number of pieces increases.