Relations between capacity utilization, minimal bin size and bin number

We consider the two-dimensional bin packing problem (BPP): given a set of rectangular items, find the minimal number of rectangular bins needed to pack all items. Rotation of the items is not permitted. We show for any integer k ≥ 3 that at most k − 1 bins are needed to pack all items if every item fits into a bin and if the total area of items does not exceed k/4-times the bin area. Moreover, this bound is tight. Furthermore, we show that only two bins are necessary to pack all items if the total area of items is not larger than the bin area, and if the height of each item is not larger than a third bin height and the width of every item does not exceed a half bin width.