Two-Dimensional Packing: Expected Performance of Simple Level Algorithms

The packing of rectangles with both dimensions i.i.d. ∼U(0, 1) onto a semi-infinite fixed-width strip is considered. The expected efficiency, expressed in terms of unused area on the strip is calculated for three simple procedures, all of which are level-algorithms: Next Fit, Rotatable Next Fit (where pieces are possibly orientated before packing, so that their width always exceeds their height), and Next Fit Decreasing (pieces are presorted by their height). It becomes evident that the single most important determinant of this efficiency is the variance of level heights, and the procedures can be ranked by their success in keeping this variance down. It is indirectly demonstrated that worstcase behavior of simple packing algorithms is a poor predictor of their expected performance. Results were obtained via analysis, computer (symbolic) integration, and simulation.