A Tight Lagrangean Relaxation Bound for the Manufacturer's Pallet Loading Problem

In this study we analyze the performance of a Lagrangean relaxation upper bound for the problem of packing the maximum number of rectangles of sizes (l; w) and (w; l) into a larger rectangle (L;W ) without overlapping. This problem appears in the loading of identical boxes on pallets, namely the manufacturer’s pallet loading problem (MPL), as well as in package design and truck or rail car loading. Although apparently easy to be optimally solved, the MPL is claimed to be NP-complete and several authors have proposed heuristic methods to deal with it. The present upper bound can be useful for these methods, since it is very tight and was able to provide an optimality certificate for almost all examples analyzed. It is based on Beasley’s 0-1 integer programming formulation of the problem and it is obtained by a standard subgradient optimization procedure enhanced by problem reduction tests. We also discuss a simple Lagrangean heuristic to derive feasible solutions along the subgradient iterations. Computational results are presented for examples of the literature as well as actual examples from a Brazilian carrier.