Mixed pallet loading optimisation with stability constraints

Boxes of mixed sizes can be loaded on a pallet automatically by robotized palletizing systems. The locations, orientations and loading order of the boxes can be determined reasonably well by heuristic optimization algorithms. However, most published algorithms have only very limited consideration of dynamic pallet stability, i.e., how well the pallet holds together when moved without external support. The objective of this work was to develop a practical pallet loading algorithm to minimize the number of pallets for the given set of boxes, with both hard and soft constraints on pallet stability. We developed heuristic measures of static and dynamic pallet stability, based on geometric considerations and the level of binding between successive layers. We implemented the stability measures in a multi-level algorithm based on an integer programming column generation heuristic, in which loading patterns for individual pallets were constructed by a beam search algorithm that performed box insertions at projected extreme points. The pallet construction algorithm also regularly performed local improvement by push operations. We customized the insertion and push operations to respect stability constraints and to maintain a robot-packable loading order. The top-level algorithm employed a separate physics-based simulation to validate the constructed pallets, re-planning any pallet that would fall over when moved in simulation. Test results are reported on a public benchmark data set.