Heuristic algorithms for on-line packing in three dimensions

Abstract

In this paper, we investigate a three-dimensional packing problem, i.e., packing a list of rectangular boxes into a column with a unit-square cross section so that the height of the packing is minimized. Several on-line packing algorithms, called level-strip algorithms, are proposed and analyzed. Each of these algorithms has a separate heuristic in each dimension. In the height dimension, a packing consists of levels of height r k for some k ≥ 0, where r is a parameter, 0 < r < 1. In the length dimension, a level is further divided into strips of width one and lengths in a pre-defined set { l 1, l 2,…}, where 1 = l 1 > l 2 > …. Boxes b with height r k+1 < h( b) <- r k and length l m+1 < l( b) <- l m are packed into strips of length l m in levels of height r k by using well-known bin packing algorithms such as Next-Fit, First-Fit, and Harmonic. It is shown that the overall performance ratio of each algorithm is a product of the performance ratios of each individual heuristic used independently in different dimensions. It is also proven that there exists an algorithm which has an asymptotic performance bound that can be made arbitrarily close to 2.89 by choosing certain parameters appropriately.