Abstract

Abstract

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. © 2018 Elsevier Ltd

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. © 2019 Elsevier B.V.

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Abstract