Product packing and stacking under uncertainty: A robust approach

China, the world’s largest logistics market, is experiencing insatiable demand for logistics services. The boom in China’s e-commerce has led to massive growth in the need for product shipping and warehouse storage. Some of China’s largest retailers have invested tremendous amounts of money in warehouse development. Two important functions of a warehouse are packing and stacking: packing small products into medium-sized containers and stacking the containers on pallets. Often, the packing problem is solved independently from that of stacking or is ignored. Generally, the stacking problem is solved while treating the packing decisions as given. However, considering these two problems separately may result in suboptimal, i.e., less efficient operations. Therefore, we propose a mixed integer linear programming formulation that addresses these two problems as a whole and two relaxations for actual-size examples. Due to the inherent demand uncertainty, we present a two-stage robust model that employs interval-based and scenario-based uncertainty sets. We numerically verify that the integrated model achieves considerable economic benefits and efficient space utilization while exploiting spatial flexibility. The additional computational experiments suggest that two relaxations yield tiny optimality gaps of 0.3% on average (4.6% at maximum), and considerable computation time improvements of 207.9% on average (347.4% at maximum). Our numerical results also illustrate that the scenario-based robust approach outperforms the deterministic approach and interval-based robust approach. We verify the model with varying variability and sizes of containers. The computational results indicate that compared with single type of containers, two or three types would bring significant economic benefits.