# An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

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## Abstract

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an *L*-approach for packing rectangles into larger rectangles and *L*-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50 000 instances). It is also effective for solving the instances of problem set Cover III (almost 100 000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.

### Keywords

cutting and packing manufacturer's pallet loading problem woodpulp stowage problem non-guillotine cutting pattern dynamic programming raster points## Notes

### Acknowledgements

This work was partially supported by PRONEX-Optimization (PRONEX-CNPq/FAPERJ E-26/171.510/2006-APQ1), FAPESP (Grants 2006/53768-0, 2006/03496-3 and 2005/57984-6) and CNPq (Grants PROSUL 490333/2004-4 and 522973/1995-4). The authors would like to thank Dr W. F. Mascarenhas for his help to prove Lemmas A.1 and A.2 and an anonymous referee for his/her useful comments and suggestions.

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