This paper addresses the irregular strip packing problem, a particular two-dimensional cutting and packing problem in which convex/nonconvex shapes (polygons) have to be packed onto a single rectangular object. We propose an approach that prescribes the integration of a metaheuristic engine (i.e., genetic algorithm) and a placement rule (i.e., greedy bottom-left). Moreover, a shrinking algorithm is encapsulated into the metaheuristic engine to improve good quality solutions. To accomplish this task, we propose a no-fit polygon based heuristic that shifts polygons closer to each other. Computational experiments performed on standard benchmark problems, as well as practical case studies developed in the ambit of a large textile industry, are also reported and discussed here in order to testify the potentialities of proposed approach.

The constant competitiveness between the modern industries requires that a part of the investments has to be directed to the optimization of the production processes. In garment, glass, paper, sheet metal, textile, and wood industries, for instance, the main concern is to avoid the excessive expenditure of raw material required to meet a particular demand.

In this scenario, the two-dimensional irregular strip packing problem is included. Concisely, the irregular strip packing problem is a combinatorial optimization problem that consists of finding the most efficient design for packing irregular shaped items onto a single rectangular object with minimum waste material. More precisely, the problem can be defined as follows. Assume a rectangular object that has a constant width and infinite length. Consider also a collection of irregular items grouped in

A specialization of this problem is the placement of irregular figures with characteristics similar to regular cut, but dealing with irregular figures, the nesting problems [

The irregular strip packing problem is known to be NP-hard even without rotation [

To have better analysis, the remainder of this paper is organized as follows. In Section

Problems involving irregular shapes comprise the most difficult class of packing problems. Whatever the constraints or secondary objectives, there are basically three approaches to find suitable layouts: (1) the polygons may be considered one at a time and packed onto the rectangular object according to the sorting criteria or (2) may be nested either singly or in groups into a set of enclosing polygons which are then packed onto the rectangular object, or (3) an initial allocation is improved iteratively.

Jakobs [

In recent papers, mathematical programming techniques have been adopted for solving one of the following subproblems:

Compaction procedure (Gomes and Oliveira [

Separation procedure (Gomes and Oliveira [

By other means, a successful approach that combines a local search method with a guided local search to deal with two- and three-dimensional irregular packing problems was proposed by Egeblad et al. [

To describe our proposed methodology, we first explain essential concepts related to its behavior, involving genetic algorithm, greedy bottom-left heuristic, and no-fit polygon.

Introduced by Holland [

An implementation of a typical genetic algorithm begins with a population of (generally random)

A traditional constructive algorithm for solving any two-dimensional cutting or packing problem aims to order pieces and then place them in turn, choosing the leftmost feasible position and breaking ties by selecting the lowest, as illustrated by Figure

Greedy bottom-left procedure for an input piece.

Some papers have considered placement algorithms based on the greedy bottom-left rule in the field of two-dimensional cutting and in this work the aforementioned heuristic was chosen as placement policy.

The

No-fit polygon generated by polygons

For our implementation, the construction of no-fit polygon was performed by using the Minkowski sum, whose concept involves two arbitrary point sets

The proposed methodology prescribes the integration of the distinct components described in Section

Allocation of the first piece.

Best position computed by no-fit polygon.

Final location of pieces.

Shrinking heuristic.

The methodology flow chart is presented in Figure

Methodology flow chart.

In this section, we discuss the parameters that control many aspects of the proposed methodology; some of these parameters represent tradeoffs between opposite goals, while others simply allow the fine-tuning of the effectiveness and/or efficiency of the methodology for particular problem instances. However, the set of control parameters discussed here allows the fine tuning of the methodology for a large range of optimization problems. We will see that several parameters were actually set to constant values during the course of our computational experiments, suggesting that certain parameter-value pairs tend to work well on a wide variety of problem instances. On what concerns the meta-heuristic engine, for the sake of computational performance, a simple genetic algorithm instance has been used, which is configured with uniform order-based crossover, swap mutation, fitness proportional parent selection, and age-based population replacement (whereby the parents are replaced by the entire set of offspring). Important control parameters related are described as follows.

A series of experiments were carried out on a desktop machine with a 3.60 GHz Intel i5 CPU and 4 GB of RAM. The genetic algorithm was implemented in Java and configured with those parameters described in Section

To evaluate the potentialities behind the proposed methodology, we conducted a series of experiments on benchmark problems available on the EURO Special Interest Group on Cutting and Packing (ESICUP) at

Data set characteristics.

Data set | Number of different pieces | Total number of pieces | Orientations (degrees) |
---|---|---|---|

DIGHE1 | 16 | 16 | 0 |

DIGHE2 | 10 | 10 | 0 |

JAKOBS1 | 25 | 25 | 0 |

JAKOBS2 | 25 | 25 | 0 |

TROUSERS | 17 | 64 | 0, 180 |

SHAPES0 | 4 | 43 | 0 |

SHAPES1 | 4 | 43 | 0, 180 |

MARQUES | 8 | 24 | 90 incremental |

MAO | 9 | 20 | 90 incremental |

ALBANO | 8 | 24 | 0, 180 |

SWIM | 10 | 48 | 0, 180 |

Table

Computational Results.

Data set | Length | Utilization (%) | Time (seconds) |
---|---|---|---|

DIGHE1 | 100.00 | 100.00 | 4113 |

DIGHE2 | 100.00 | 100.00 | 3751 |

JAKOBS1 | 12.22 | 84.46 | 13498 |

JAKOBS2 | 26.11 | 79.61 | 11985 |

TROUSERS | 245.45 | 88.90 | 14775 |

SHAPES0 | 63.275 | 65.17 | 4303 |

SHAPES1 | 61.301 | 69.93 | 5032 |

MARQUES | 81.89 | 87.80 | 6871 |

MAO | 1840.37 | 82.69 | 6348 |

ALBANO | 10247.28 | 88.40 | 10.230 |

SWIM | 6099.00 | 72.63 | 15.204 |

In Table

Comparative analysis.

Data set | SAHA | BLF | 2DNest | BS | AM | Dif |
---|---|---|---|---|---|---|

DIGHE1 | 100.00 | 77.40 | 99.86 | 100.00 | 100.00 | 100 |

DIGHE2 | 100.00 | 79.40 | 99.95 | 100.00 | 100.00 | 100 |

JAKOBS1 | 78.89 | 82.60 | 89.07 | 85.96 | 84.46 | 95 |

JAKOBS2 | 77.28 | 74.80 | 80.41 | 80.40 | 79.61 | 99 |

TROUSERS | 89.96 | 88.50 | 89.84 | 90.38 | 88.90 | 98 |

SHAPES0 | 66.50 | 60.50 | 67.09 | 64.35 | 65.17 | 97 |

SHAPES1 | 71.25 | 66.50 | 73.83 | 71.25 | 69.93 | 95 |

MARQUES | 88.14 | 86.50 | 89.17 | 88.92 | 87.80 | 98 |

MAO | 82.54 | 79.50 | 85.15 | 84.07 | 82.69 | 97 |

ALBANO | 89.96 | 84.60 | 86.96 | 87.88 | 88.40 | 98 |

SWIM | 74.36 | 71.60 | 71.53 | 75.04 | 72.63 | 96 |

Mean | 83.53 | 77.45 | 84.81 | 84.39 | 83.60 | 98 |

We denote by Dif the relation, in percentage, between the solution found by AM and the best solution of the row:

To sum up, regarding the utilization percentage of the rectangular object, it can be stated that AM has presented promising results. Taking as reference the scores achieved by BLF and SAHA, the proposed methodology presents better solutions in most of cases, as well as the average solutions. Regarding the 2DNest algorithm, we obtained better results in three problem instances, namely, ALBANO, DIGHE1, DIGHE2, and SWIM. Moreover, AM yielded better scores compared to the BS in two data sets, namely, SHAPES0 and ALBANO. Furthermore, even without applying the same rotation variants allowed by other studies (90 incremental), Table

In Figures

Dynamics of the genetic algorithm evolutionary process: a step-by-step improvement-DIGHE1.

Dynamics of the genetic algorithm evolutionary process: a step-by-step improvement-ALBANO.

Dynamics of the genetic algorithm evolutionary process: a step-by-step improvement-JAKOBS2.

Best solution found for DIGHE1 instance.

Best solution found for ALBANO instance.

Best solution found for JAKOBS2 instance.

Brazil is the sixth largest textile producer country of the world and the main power, according to the Brasilian Textile Industry Association (ABIT). Among the Brazilian poles, we can highlight the state of Ceara. The success of cotton in this state, until the mid-1980s, stimulated the establishment of a solid textile and apparel park, which soon had to adapt to the new reality.

The aggregation method has also been applied in the ambit of a large textile industry. This particular industrial unit prints soccer team logos (Figure

Soccer team logos.

A significant difference in the production of shells is to define a feasible layout and has a prominent advantage. The acquisition of the points of the figures is performed by means of discretization, or some points are selected manually or automatically, in a manner that represent them.

The greater the number of points to describe the polygon is, the better the resolution is; however, all the processing routines consume high computational time. The width of a rectangular strip is 210 mm and each logo is composed by a set of points (see input data for each instance at

Moreover, Table

Features of the figures of the case study.

Data set | Number of different pieces | Total number of pieces | Orientations (degrees) | Length |
---|---|---|---|---|

Soccer Logos-I | 2 | 40 | 45 incremental | 210 mm |

Soccer Logos-II | 1 | 20 | 45 incremental | 210 mm |

Soccer Logos-III | 1 | 30 | 45 incremental | 210 mm |

In Table

Results obtained from the practical case studied—empirical methods (EM).

Data set | Length | Improvement (%) | Time (seconds) |
---|---|---|---|

Soccer Logos-I | 637.617 | 78.01 | 5100 |

Soccer Logos-II | 307.474 | 79.21 | 1200 |

Soccer Logos-III | 499.758 | 76.21 | 4800 |

The layout produced by applying the empirical methods (EM) to Soccer Logos-I, Soccer Logos-II, and Soccer Logos-III instances is displayed in Figures

Best solution found for Soccer Logos-I instance.

Best solution found for Soccer Logos-II instance.

Best solution found for Soccer Logos-III instance.

In addition, the layout produced by applying the aggregation method (AM) to Soccer Logos-I, Soccer Logos-II, and Soccer Logos-III instances is displayed in Figures

Results obtained from the practical case studied—aggregation method (AM).

Data set | Length | Improvement (%) | Time (seconds) |
---|---|---|---|

Soccer Logos-I | 633.58 | 78.51 | 5220 |

Soccer Logos-II | 300.95 | 80.94 | 3800 |

Soccer Logos-III | 453.39 | 84.01 | 3690 |

Best solution found for Soccer Logos-I instance: (AM).

Best solution found for Soccer Logos-II instance: (AM).

Best solution found for Soccer Logos-III instance: (AM).

The highlighted variation in a small increase in the number of generations from the compression method is applied to the methodology. Moreover, it is observed that the application of aggregation method for the experiments of the textile industry presented has made significant progress.

The main objective is to minimize the length of the layouts, while the width remains fixed. For this, we developed a strategy based on hybridization of genetic algorithms and a heuristic placement that applies the concepts of calculating the no-fit polygons methodology and bottom-left heuristic. In this initial study, we have introduced an aggregation methodology to cope with the irregular strip packing problem, which is based on a kind of hybridization between a genetic algorithm greedy bottom-left heuristic.

For specific types of figures, certain approaches will produce best computational results. However, depending on the shape of polygons, the same may be a decrease in efficiency. The criteria for selection of various positioning method is an outstanding solution to the problem; however, the computational complexity involved in such methods is high. The computational results were good cheer, compared to other approaches, and especially according to the inherent difficulty of this problem.

Overall, the optimization performance achieved with the novel methodology has been promising, taking as reference the results achieved by other approaches and taking into account the inherent difficulties associated with this particular cutting and packing problem.

As future work, we plan to investigate the performance of the biased random-key genetic algorithm [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The first and third authors are thankful to Coordination for the Improvement of Higher Level or Education Personnel (CAPES) and the second and fourth authors are thankful to National Counsel of Technological and Scientific Development (CNPq) via Grants no. 475239/2012-1.