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

TL;DR

This paper introduces a fast, scalable bottom-left-fill algorithm for nesting problems using a semi-discrete representation, enabling quick placement of non-convex pieces with minimal extensions, suitable for metaheuristics.

Contribution

The paper presents a novel semi-discrete representation and an optimized bottom-left-fill algorithm that significantly improves speed and scalability in solving nesting problems.

Findings

Algorithm executes in a few milliseconds on typical data sets.

Scales well with increasing number of pieces.

Effective even when considering piece rotation.

Abstract

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 optimised ordering of the segments 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 ESICUP data sets in a few milliseconds, even when rotation of the pieces is considered, and thus provides a suitable `building block' for integration in metaheuristics. Moreover, we show that the algorithm scales well when the number of pieces increases.