Abstract

Abstract
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.

Abstract
The bi-dimensional irregular strip packing is a difficult problem in the cutting and packing field. Its main feature, and central source of complexity, is the irregularity of the shape of the items. Consequently, mathematical solvers are only able to obtain optimal solutions for small instances and heuristics are often employed in the literature. In such algorithms, it is not possible to guarantee that the optimum solution is found. In such cases, a restricted version of the problem can be adopted in order to improve the performance. One possible restriction is the adoption of pairwise clustering, i.e., elimination of items by joining two pieces. In this work, an automatic pairwise clustering method is proposed for the dotted board model, which limits the placement of items to equally distributed discrete points. The clustered problems are then used as input to an irregular strip packing solver. The results obtained in this paper can be used as an initial guideline for the use of clustering in a discrete grid, which was beneficial in some of the tested cases.

Abstract
Cutting and packing (C&P) is an important area of operational research and its problems arise in various industries such as: textile, wood, glass and shipbuilding. The main objective is to maximize the efficiency of a layout by rearranging and/or reassigning items inside containers in order to reduce costs and environmental impact. In this work, a raster solution to the bidimensional irregular strip packing problem, which consists of placing irregular shapes items inside a single rectangular container with variable length, is studied. In raster methods, the selection of the grid size is very important to the outcome of the algorithm. It influences the size of the search space, the overlap algorithm efficiency, as well as the memory requirements of the packing algorithm. An analysis of the impact of the choice of grid size is performed using 15 benchmark cases from the literature and, through careful observation of such test results, a simple rule to define the grid size is suggested.

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.

Abstract
Among the most complex problems in the field of 2-dimensional cutting & packing are irregular packing problems, in which items may have a more complex geometry. These problems are prominent in several areas, including, but not limited to, the textile, shipbuilding and leather industries. They consist in placing a set of items, whose geometry is often represented by simple polygons, into one or more containers such that there is no overlap between items and the utility rate of the container is maximized. In this work, the irregular strip packing problem, an irregular packing variant with a variable length container, is investigated. The placement space is reduced by adopting a rectangular grid and a full search is performed using preprocessed raster penetration maps to efficiently determine the new position of an item. Tests were performed using simple dotted board model cases and irregular strip packing benchmark instances. The comparison of our results with the state of the art solutions showed that the proposed algorithm is very competitive, achieving better or equal compaction in 9 out of 15 instances and improving the average density in 13 instances. Besides the contribution of the new best results, the proposed approach showed the advantage of adopting discrete placement, which can be potentially applied to other irregular packing problems.