A decision-tree framework to select optimal box-sizes for product shipments

In package-handling facilities, boxes of varying sizes are used to ship products. Improperly sized boxes with box dimensions much larger than the product dimensions create wastage and unduly increase the shipping costs. Since it is infeasible to make unique, tailor-made boxes for each of the $N$ products, the fundamental question that confronts e-commerce companies is: How many $K << N$ cuboidal boxes need to manufactured and what should be their dimensions? In this paper, we propose a solution for the single-count shipment containing one product per box in two steps: (i) reduce it to a clustering problem in the $3$ dimensional space of length, width and height where each cluster corresponds to the group of products that will be shipped in a particular size variant, and (ii) present an efficient forward-backward decision tree based clustering method with low computational complexity on $N$ and $K$ to obtain these $K$ clusters and corresponding box dimensions. Our algorithm has multiple constituent parts, each specifically designed to achieve a high-quality clustering solution. As our method generates clusters in an incremental fashion without discarding the present solution, adding or deleting a size variant is as simple as stopping the backward pass early or executing it for one more iteration. We tested the efficacy of our approach by simulating actual single-count shipments that were transported during a month by Amazon using the proposed box dimensions. Even by just modifying the existing box dimensions and not adding a new size variant, we achieved a reduction of $4.4\%$ in the shipment volume, contributing to the decrease in non-utilized, air volume space by $2.2\%$. The reduction in shipment volume and air volume improved significantly to $10.3\%$ and $6.1\%$ when we introduced $4$ additional boxes.