Irregular packing problems: A review of mathematical models

Highlights

• We review and categorize mathematical models for nesting problems.

• Linear, non-linear, integer, and constraint programming models are described.

• Geometric tools are described according to their use in the models.

• Similarities and differences among the models are highlighted.

• Research opportunities are identified.

Abstract

Irregular packing problems (also known as nesting problems) belong to the more general class of cutting and packing problems and consist of allocating a set of irregular and regular pieces to larger rectangular or irregular containers, while minimizing the waste of material or space. These problems combine the combinatorial hardness of cutting and packing problems with the computational difficulty of enforcing the geometric non-overlap and containment constraints. Unsurprisingly, nesting problems have been addressed, both in the scientific literature and in real-world applications, by means of heuristic and metaheuristic techniques. However, more recently a variety of mathematical models has been proposed for nesting problems. These models can be used either to provide optimal solutions for nesting problems or as the basis of heuristic approaches based on them (e.g. matheuristics). In both cases, better solutions are sought, with the natural economic and environmental positive impact. Different modeling options are proposed in the literature. We review these mathematical models under a common notation framework, allowing differences and similarities among them to be highlighted. Some insights on weaknesses and strengths are also provided. By building this structured review of mathematical models for nesting problems, research opportunities in the field are proposed.

Introduction

Cutting and packing problems arise in many industrial and logistics applications, as problems to be solved or as part of more complex optimization problems. They share a nice underlying geometric structure and, on top of it, many different settings and specific characteristics define a wide variety of particular problems. Therefore, it is not surprising they have recently attracted a great deal of attention in the scientific literature, and a typology for cutting and packing problems was introduced in Wäscher, Haußner, and Schumann (2007). Apart from the obvious classification criterion based on dimensionality, a second element that divides the cutting and packing field into two clearly different parts is the type of pieces being considered, whether they are regular or irregular. Regular, and more specifically rectangular, pieces appear in many cutting problems and in the vast majority of packing problems, in which products are usually packaged into rectangular boxes. Other regular, non-rectangular pieces, such as circles, have also been studied in specific applications. Nevertheless, there are also many applications in which the pieces to be cut or packed are irregularly shaped. Typical examples appear in the clothing industry, furniture, leather, glass, or sheet metal cutting. Fig. 1 shows an example of nesting problem found in garment industries. It illustrates a set of 64 pieces with regular and irregular shapes feasibly packed into a rectangular board.

Irregular cutting and packing problems, aka nesting problems, consist of packing a set of irregular-shaped pieces into a board or a set of boards. The pieces must be placed completely inside the boards in such a way that they do not overlap each other, while the waste material is minimized. Minimizing waste has not only economic, but also environmental impact if the use of raw materials can be kept to a minimum. The geometric structure of nesting problems is much more complex than the case of rectangular pieces. Checking if pieces are included in the board and do not overlap is a hard problem when irregular pieces are involved. As a consequence, nesting problems are much harder than their counterparts that deal with rectangular pieces and the results obtained so far are much more limited regarding the size (number of pieces) of the problems tackled. Their difficulty and their many applications make nesting problems the most challenging problems in the field of cutting and packing.

The objectives of this paper are to review the mathematical models proposed in the literature for nesting problems and to serve as a starting point for a new wave of research on this field. Several types of models have been proposed in the last decades: mixed-integer linear programming, non-linear programming, and constraint programming models. The characteristics of the models are strongly related to the geometric methodologies employed. Bennell and Oliveira (2008) provide a tutorial that covers the geometric tools used in irregular cutting and packing problems. Raster point, direct trigonometry, no-fit polygon and phi-functions have been applied to the modeling. In this study, we carefully review all these modeling approaches and their corresponding geometric tools. We focus on two-dimensional nesting problems, since this is the version addressed by most of the models proposed so far, although some references to the three-dimensional case are added when an extension to the 3D case is mentioned. References and surveys on heuristic and metaheuristic algorithms developed for nesting problems can be found in Bennell and Oliveira (2009), Hopper and Turton (2001), Dowsland and Dowsland (1995). We also focus on problems with no restrictive assumption on the shape of the pieces and the cutting process. Nevertheless, a few references of models developed for general convex pieces are given. For solution methods addressing problems with convex pieces and guillotine cuts, we refer to Han, Bennell, Zhao, and Song (2013) and Bennell, Cabo, and Mart Anez Sykora (2018).

The remainder of this paper is organized as follows. In Section 2, nesting problems are defined. Section 3 revises the geometric tools according to their usage in the mathematical models. Mixed-integer linear programming models, non-linear programming models and constraint programming models are described in 4 Mixed-integer linear programming models, 5 Non-linear programming models and Section 6, respectively. In Section 7 an outlook on future research is presented and in Section 8 we conclude with a categorization of the mathematical models.