Understanding the structure of bin packing problems through principal component analysis

This paper uses a knowledge discovery method, Principal Component Analysis (PCA), to gain a deeper understanding of the structure of bin packing problems and how this relates to the performance of heuristic approaches to solve them. The study considers six heuristics and their combination through an evolutionary hyper-heuristic framework. A wide set of problem instances is considered, including one-dimensional and two-dimensional regular and irregular problems. A number of problem features are considered, which are reduced to the subset of nine features that more strongly relate with heuristic performance. PCA is used to further reduce the dimensionality of the instance features and produce 2D maps. The performance of the heuristics and hyper-heuristics is then super-imposed into these maps to visually reveal relationships between problem features and heuristic behavior. Our analysis indicates that some instances are clearly harder to solve than others for all the studied heuristics and hyper-heuristics. The PCA maps give a valuable indication of the combination of features characterizing easy and hard to solve instances. We found indeed correlations between instance features and heuristic performance. The so-called DJD heuristics are able to best solve a large proportion of instances, but simpler and faster heuristics can outperform them in some cases. In particular when solving 1D instances with low number of pieces, and, more surprisingly, when solving some difficult 2D instances with small areas with low variability. This analysis can be generalized to other problem domains where a set of features characterize instances and several problem solving heuristics are available.