A. Miguel Gomes

Solving Irregular Strip Packing problems by hybridising simulated annealing and linear programming

Abstract In this paper a hybrid algorithm to solve Irregular Strip Packing problems is presented. The metaheuristic simulated annealing is used to guide the search over the solution space while linear programming models are solved to generate neighbourhoods during the search process. These linear programming models, which are used to locally optimise the layouts, derive from the application of compaction and separation algorithms. Computational tests were run using instances that are commonly used as benchmarks in the literature. The best results published so far have been improved by this new hybrid packing algorithm.

A 2-exchange heuristic for nesting problems

Abstract This paper describes a new heuristic for the nesting problem based on a 2-exchange neighbourhood generation strategy. This mechanism guides the search through the solution space consisting of the sequences of pieces and relies on a low-level placement heuristic to actually convert one sequence in a feasible layout. The placement heuristic is based on a bottom-left greedy procedure with the ability to fill holes in the middle of the layout at a later stage. Several variants of the 2-exchange nesting heuristic were implemented and tested with different initial solution ranking criteria, different strategies for selecting the next solution, and different neighbourhood sizes. The computational experiments were based on data sets published in the literature. In most of the cases, the 2-exchange nesting algorithm generated better solutions than the best known solutions.

TOPOS – A new constructive algorithm for nesting problems

Abstract. In this paper we present a new constructive algorithm for nesting problems. The layout is built by successively adding a new piece to a partial solution, i.e. to the set of pieces previously nested. Several criteria to choose the next piece to place and its orientation are proposed and tested. Different objective functions are also proposed to evaluate and compare partial solutions. A total of 126 variants of the algorithm, generated by the complete set of combinations of criteria and objective functions, are computationally tested.The computational experiments are based on data sets published in the literature or provided by other authors. In some cases this new algorithm generates better solutions than the best known (published) solutions.Zusammenfassung. In dem vorliegenden Beitrag entwickeln die Autoren ein neues Konstruktionsverfahren für das „Nesting-Problem”, d.h. für ein zweidimensionales Zuschneideproblem mit unregelmäßigen Objekten. Das Schnittmuster wird dadurch gebildet, dass sukzessive zuzuschneidende Objekte (Teile) einer Teillösung angegliedert und damit neue Teillösungen gebildet werden. Verschiedene Kriterien zur Auswahl des jeweils anzuordnenden Teils und seiner Orientierung werden vorgestellt. Außerdem werden verschiedene Zielfunktionen zur Bewertung der Teillösungen herangezogen. Insgesamt ergeben sich so 126 Varianten des Konstruktionsverfahrens, die systematisch anhand von Datensätzen aus der Literatur getestet werden. Für einige Testprobleme stellt das neue Verfahren Lösungen bereit, die besser sind als die besten bisher in der Literatur beschriebenen Lösungen.

Heuristic approaches to large-scale periodic packing of irregular shapes on a rectangular sheet

The nesting problem is a two-dimensional cutting and packing problem where the small pieces to cut have irregular shapes. A particular case of the nesting problem occurs when congruent copies of one single shape have to fill, as much as possible, a limited sheet. Traditional approaches to the nesting problem have difficulty to tackle with high number of pieces to place. Additionally, if the orientation of the given shape is not a constraint, the general nesting approaches are not particularly successful. This problem arises in practice in several industrial contexts such as footwear, metalware and furniture. A possible approach is the periodic placement of the shapes, in a lattice way. In this paper, we propose three heuristic approaches to solve this particular case of nesting problems. Experimental results are compared with published results in literature and additional results obtained from new instances are also provided.

Circle Covering Representation for Nesting problems with continuous rotations

Abstract This paper analyses distinct methods to represent a polygon through circle covering, which satisfy specific requirements, that impact primarily the feasibility and the quality of the layout of final solution. The trade-off between the quality of the polygonal representation and its derived number of circles is also discussed, showing the impact on the resolution of the problem, in terms of computational efficiency. The approach used to tackle the Nesting problem in strip packing uses a Non-Linear Programming model. Addressing these problems allows to tackle real world problems with continuous rotations.

Circle Covering Using Medial Axis

A good representation of a simple polygon, with a desired degree of approximation and complexity, is critical in many applications. This paper presents a method to achieve a complete Circle Covering Representation of a simple polygon, through a topological skeleton, the Medial Axis. The aim is to produce an efficient circle representation of irregular pieces, while considering the approximation error and the resulting complexity, i.e. the number of circles. This will help to address limitations of current approaches to some problems, such as Irregular Placement problems, which will, in turn, provide a positive economic and environmental impact where similar problems arise.

Irregular Packing Problems: Industrial Applications and New Directions Using Computational Geometry

Cutting and Packing problems are hard Combinatorial Optimization problems that naturally arise in all industries and services where raw-materials or space must be divided into smaller non-overlapping items, so that waste is minimized. All the Cutting and Packing problems have in common the existence of a geometric sub-problem, originated by the natural item non-overlapping constraints. An important class of Cutting and Packing problems are the Irregular Packing problems that occur when raw materials have to be cut into items with irregular shapes. Irregular Packing problems, also known as Nesting problems, naturally arises in the garment, footwear, tools manufacturing and shipbuilding industries, among others. Each industrial application has its owns particular issues mainly related to the raw materials specific characteristics. Several challenges remain open in the Irregular Packing problems field. Some are due to the combinatorial nature of these problems. Others are of geometric nature, due to the non-convex and non-regular geometry of the items involved. Moreover these geometric challenges do not allow the combinatorial ones being properly tackled. This paper is mainly focused on presenting and discussing efficient tools and representations to tackle the geometric layer of nesting algorithms that capture the needs of the real-world applications of Irregular Packing problems.

Constraint Aggregation in Non-linear Programming Models for Nesting Problems

The Nesting problem is a complex problem that arises in industries where sets of pieces or space must be efficiently placed or allocated in order to minimize wasted space or wasted raw materials, without overlaps between pieces and fully contained inside a container. This paper analyses the impact that aggregating constraints can achieve in the reduction of computational cost of a Non-Linear Programming model for Nesting problems with continuous rotations. This is achieved by aggregating non-overlapping constraints and using spatial partition and hierarchical overlap detection methods. When aggregating constraints there is also an effect of reducing the sensitivity of the solver, which may reduce the quality of the final layout. Analyzing the trade-off between constraints aggregation and the impact on the quality of the final solution is an important issue to handle nesting instances with a large number of pieces. Computational experiments show that aggregating non-overlapping constraints allows the Non-Linear Programming model for Nesting problems to scale well to tackle large size real world problems with continuous rotations.

Data mining based framework to assess solution quality for the rectangular 2D strip-packing problem

Abstract In this paper, we explore the use of reference values (predictors) for the optimal objective function value of hard combinatorial optimization problems, instead of bounds, obtained by data mining techniques, and that may be used to assess the quality of heuristic solutions for the problem. With this purpose, we resort to the rectangular two-dimensional strip-packing problem (2D-SPP), which can be found in many industrial contexts. Mostly this problem is solved by heuristic methods, which provide good solutions. However, heuristic approaches do not guarantee optimality, and lower bounds are generally used to give information on the solution quality, in particular, the area lower bound. But this bound has a severe accuracy problem. Therefore, we propose a data mining-based framework capable of assessing the quality of heuristic solutions for the 2D-SPP. A regression model was fitted by comparing the strip height solutions obtained with the bottom-left-fill heuristic and 19 predictors provided by problem characteristics. Random forest was selected as the data mining technique with the best level of generalisation for the problem, and 30,000 problem instances were generated to represent different 2D-SPP variations found in real-world applications. Height predictions for new problem instances can be found in the regression model fitted. In the computational experimentation, we demonstrate that the data mining-based framework proposed is consistent, opening the doors for its application to finding predictions for other combinatorial optimisation problems, in particular, other cutting and packing problems. However, how to use a reference value instead of a bound, has still a large room for discussion and innovative ideas. Some directions for the use of reference values as a stopping criterion in search algorithms are also provided.

The Two-Dimensional Strip Packing Problem: What Matters?

This paper presents an exploratory approach to study and identify the main characteristics of the two-dimensional strip packing problem (2D-SPP). A large number of variables was defined to represent the main problem characteristics, aggregated in six groups, established through qualitative knowledge about the context of the problem. Coefficient correlation are used as a quantitative measure to validate the assignment of variables to groups. A principal component analysis (PCA) is used to reduce the dimensions of each group, taking advantage of the relations between variables from the same group. Our analysis indicates that the problem can be reduced to 19 characteristics, retaining most part of the total variance. These characteristics can be used to fit regression models to estimate the strip height necessary to position all items inside the strip.