Solving the Problem of Stacking Goods: Mathematical Model, Heuristics and a Case Study in Container Stacking in Ports

Stacking goods or items is one of the most common operations in everyday life. It happens abundantly in not only transportation applications such as container ports, container ships, warehouses, factories, sorting centers, freight terminals, etc., but also computing systems, supermarkets, and so on. We investigate the problem of stacking a sequence of items into a set of capacitated stacks, subject to stacking constraints. In every stack, items are accessed in the last-in-first-out order. So at retrieval time, getting any lower item requires reshuffling all upper items that are blocking the way (called blocking items). These reshuffles are redundant and expensive. The challenge is to prevent reshuffles from happening. For this purpose, we aim at assigning items to stacks to minimize the number of blocking items with respect to the retrieval order. We provide some mathematical analyses on the feasibility of this problem and lower bounds. Besides, we provide a mathematical model and a two-step heuristic framework. We illustrate the applications of these models and heuristic framework in the real cargo handling process in an Asian port. Experimental results on real scenarios show that the proposed model can eliminate almost all reshuffles, and thus decrease the number of stacking violations from 62.6 % to 0.9 %. We also provide an empirical analysis of variants of the heuristic framework.