The Setting of Shelf Heights and the Distribution of Box Sizes in Two-dimensional Shelf Packing

In this paper we explore two factors which have been relatively neglected in the study of shelf packing algorithms. Boxes arrive one by one at a bin that is shelved over its width and are placed into the shelves side by side, left-justified. We consider the setting of the shelf heights, which needs to be decided upon before packing starts. We investigate the relationship between the number of boxes to be packed and the number of shelf heights that leads to minimal space wastage in the resulting packings. We also consider the distribution of the box sizes, for which we distinguish three types: (a) the continuous uniform distribution U(0, 1); (b) the discrete uniform distribution where M sizes {1/M, 2/M,……,1} have equal probability of being chosen; and (c) an alternative discrete uniform distribution where M sizes {s₁, s₂,……,s_M} are drawn from U(0, 1), which then each have equal probability of being chosen. In simulation studies we illustrate important differences between the three types of distribution.