A block-building constraint programming model for the container loading problem

The container loading problem involves packing a set of given rectangular boxes into a larger rectangular container of fixed size, with the objective of maximizing the volume of the loaded boxes. Most of the literature on the container loading problem and its variants proposes heuristic approaches that can find good solutions quickly. Current exact methods are mostly limited to mixed-integer programming (MIP) formulations, which often struggle to obtain good solutions for large problem instances.

In this paper, we introduce two exact constraint programming models for the container loading problem. The first model uses integer and binary variables to assign boxes to valid positions and orientations within the container. The second model enhances this by incorporating the concept of block-building, commonly used in heuristic methods. Extensive computational experiments on classical benchmark instances from the literature show that the solutions obtained with the proposed models significantly outperform those achieved with existing MIP models. We also perform an instance space analysis of the proposed models to map the models’ performances across problem instances, providing deeper insights into the strengths and weaknesses of the block-building approach.