Three-dimensional cutting and packing problems have a range of important applications and are of particular relevance to the transportation of cargo in the form of container loading problems. Recent years have seen a marked increase in the number of papers examining a variant of the container loading problem ranging from largely theoretical to implementations that focus on meeting the many critical real-world constraints. In this paper, we review the literature focusing on the solution methodologies employed by researchers, with the aim of providing insight into some of the critical algorithmic design issues. In addition, we provide an extensive comparison of algorithm performance across the benchmark literature.

The paper examines a new problem in the irregular packing literature that has existed in industry for decades; two-dimensional irregular (convex) bin packing with guillotine constraints. Due to the cutting process of certain materials, cuts are restricted to extend from one edge of the stock-sheet to another, called guillotine cutting. This constraint is common place in glass cutting and is an important constraints in two-dimensional cutting and packing problems. In the literature, various exact and approximate algorithms exist for finding the two dimensional cutting patterns that satisfy the guillotine cutting constraint. However, to the best of our knowledge, all of the algorithms are designed for solving rectangular cutting where cuts are orthogonal with the edges of the stock-sheet. In order to satisfy the guillotine cutting constraint using these approaches, when the pieces are non-rectangular, practitioners implement a two stage approach. First, pieces are enclosed within rectangle shapes and then the rectangles are packed. Clearly, imposing this condition is likely to lead to additional waste. This paper aims to generate guillotine-cutting layouts of irregular shapes using a number of strategies. The investigation compares two two-stage approaches; one approximates pieces by rectangles, the other approximates pairs of pieces by rectangles using phi-functions for optimal clustering. Both these approaches use state of the art rectangle bin packing with guillotine constraints. Further, we design and implement a one-stage approach using a self-adapted forest search algorithm. Experimental results show the one-stage strategy to produce good solutions in less time over the two-stage approach.