Annals of Operations Research

, 172:405 | Cite as

Bidirectional best-fit heuristic for orthogonal rectangular strip packing

  • Önder Barış Aşık
  • Ender ÖzcanEmail author


In a non-guillotinable rectangular strip packing problem (RF-SPP), the best orthogonal placement of given rectangular pieces on a strip of stock sheet having fixed width and infinite height are searched. The aim is to minimize the height of the strip while including all the pieces in appropriate orientations. In this study, a novel bidirectional best-fit heuristic (BBF) is introduced for solving RF-SPPs. The proposed heuristic as a new feature considers the gaps in both horizontal and vertical directions during the placement process. The performance of BBF is compared to some previous approaches, including one of the best heuristics from the literature. BBF achieves better results than the existing heuristics and delivers a better or matching performance as compared to the most of the previously proposed meta-heuristics for solving RF-SPPs.


Heuristic Sequencing Cutting Packing 



bidirectional best-fit heuristic


rectangular strip packing problem


non-guillotinable, variable orientation subtype


non-guillotinable, variable orientation (orthogonal) SPP


genetic algorithm


simulated annealing


greedy randomized adaptive search procedure


best fit heuristic


bottom left heuristic


improved bottom left heuristic


bottom left fill heuristic


  1. Alvarez-Valdes, R., Parreno, F., & Tamarit, J. M. (2008). Reactive GRASP for the strip-packing problem. Computers & Operations Research, 35(4), 1065–1083. CrossRefGoogle Scholar
  2. Baker, B. S., Coffman, E. G., & Rivest, R. L., Jr. (1980). Orthogonal packings in two dimensions. SIAM Journal on Computing, 9(4), 808–826. CrossRefGoogle Scholar
  3. Beasley, J. E. (1985a). An exact two-dimensional non-guillotine cutting tree search procedure. Operations Research, 33(1), 49–64. CrossRefGoogle Scholar
  4. Beasley, J. E. (1985b). Algorithms for unconstrained two-dimensional guillotine cutting. Journal of Operations Research Society, 36(4), 297–306. Google Scholar
  5. Beltrán, J. D., Calderón, J. E., Cabrera, R. J., Moreno-Pérez, J. A., & Moreno-Vega, J. M. (2004). GRASP-VNS hybrid for the strip packing problem. In Proceeding of the workshop on hybrid metaheuristics (pp. 79–90). Google Scholar
  6. Bortfeld, A. (2006). A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces. European Journal of Operational Research, 172(3), 814–837. CrossRefGoogle Scholar
  7. Burke, E., Hart, E., Kendall, G., Newall, J., Ross, P., & Schulenburg, S. (2003). Hyper-heuristics: An emerging direction in modem search technology. In Handbook of metaheuristics (pp. 457–474). Dordrecht: Kluwer Academic. Google Scholar
  8. Burke, E. K., Kendall, G., & Whitwell, G. (2004). A new placement heuristic for the orthogonal stock-cutting problem. Operations Research, 52(4), 655–671. CrossRefGoogle Scholar
  9. Burke, E. K., Kendall, G., & Whitwell, G. (2006). Metaheuristic enhancements of the best-fit heuristic for the orthogonal stock cutting problem (Technical Report No NOTTCS-TR-2006-3). University of Nottingham, Computer Science. Google Scholar
  10. Burke, E. K., Kendall, G., & Whitwell, G. (2009). A simulated annealing enhancements of the best-fit heuristic for the orthogonal stock cutting problem. INFORMS Journal on Computing, 21(3), 505–516. CrossRefGoogle Scholar
  11. Chazelle, B. (1983). The bottom-left bin-packing heuristic: An efficient implementation. IEEE Transactions on Computers, C-32(8), 697–707. CrossRefGoogle Scholar
  12. Dagli, C. H., & Poshyanonda, P. (1997). New approaches to nesting rectangular patterns. Journal of Intelligent Manufacturing, 8(3), 177–190. CrossRefGoogle Scholar
  13. Dowsland, K. A., & Dowsland, W. B. (1992). Packing problems. European Journal of Operational Research, 56(1), 2–14. CrossRefGoogle Scholar
  14. Dyckhoff, H. (1990). A typology of cutting and packing problems. European Journal of Operational Research, 44(2), 145–159. CrossRefGoogle Scholar
  15. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability. New York: Freeman. Google Scholar
  16. Garrido, P., & Riff, M.-C. (2007). An evolutionary hyperheuristic to solve strip-packing problems. In Proceedings of the intelligent data engineering and automated learning—IDEAL 2007 (pp. 406–415) Google Scholar
  17. Hässler, R. W., & Sweeney, P. E. (1991). Cutting stock problems and solution procedures. European Journal of Operational Research, 54(2), 141–150. CrossRefGoogle Scholar
  18. Hinxman, A. I. (1980). The trim loss and assortment problems: A survey. European Journal of Operational Research, 5(1), 8–18. CrossRefGoogle Scholar
  19. Hopper, E. (2000). Two-dimensional packing utilising evolutionary algorithms and other meta-heuristic methods. PhD thesis, Cardiff University, School of Engineering. Google Scholar
  20. Hopper, E., & Turton, B. C. H. (2001a). An empirical investigation of metaheuristic and heuristic algorithms for a 2D packing problem. European Journal of Operational Research, 128, 34–57. CrossRefGoogle Scholar
  21. Hopper, E., & Turton, B. C. H. (2001b). A review of the application of meta-heuristic algorithms to 2D strip packing problems. Artificial Intelligence Review, 16(4), 257–300. CrossRefGoogle Scholar
  22. Imahori, S., & Yagiura, M. (2010). The best-fit heuristic for the rectangular strip packing problem: An efficient implementation and the worst-case approximation ratio. Computers & Operations Research, 37, 325–333. CrossRefGoogle Scholar
  23. Imahori, S., Yagiura, M., & Ibaraki, T. (2005). Improved local search algorithms for the rectangle packing problem with general spatial costs. European Journal of Operational Research, 167, 48–67. CrossRefGoogle Scholar
  24. Jakobs, S. (1996). On genetic algorithms for the packing of polygons. European Journal of Operational Research, 88, 165–181. CrossRefGoogle Scholar
  25. Kenmochi, M., Imamichi, T., Nonobe, K., Yagiura, M., & Nagamochi, H. (2009). Exact algorithms for the 2-dimensional strip packing problem with and without rotations. European Journal of Operational Research, 198, 73–83. CrossRefGoogle Scholar
  26. Lesh, N., Marks, J., McMahon, A., & Mitzenmacher, M. (2004). Exhaustive approaches to 2D rectangular perfect packings. Information Processing Letters, 90, 7–14. CrossRefGoogle Scholar
  27. Lesh, N., Marks, J., McMahon, A., & Mitzenmacher, M. (2005). New heuristic and interactive approaches to 2D rectangular strip packing. ACM Journal of Experimental Algorithmics, 10, 1–18. Google Scholar
  28. Liu, D., & Teng, H. (1999). An improved bottom left algorithm for genetic algorithm of the orthogonal packing of rectangle. European Journal of Operational Research, 112, 413–419. CrossRefGoogle Scholar
  29. Lodi, A., Martello, S., & Vigo, D. (1999). Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems. INFORMS Journal on Computing, 11, 345–357. CrossRefGoogle Scholar
  30. Lodi, A., Martello, S., & Monaci, M. (2002). Two-dimensional packing problems: A survey. European Journal of Operational Research, 141(2), 241–252. CrossRefGoogle Scholar
  31. Martello, S., & Vigo, D. (1998). Exact solution of the two-dimensional finite bin packing problem. Management Science, 44, 388–399. CrossRefGoogle Scholar
  32. Martello, S., Monaci, M., & Vigo, D. (2003). An exact approach to the strip-packing problem. INFORMS Journal on Computing, 15(3), 310–319. CrossRefGoogle Scholar
  33. Özcan, E., Bilgin, B., & Korkmaz, E. E. (2008). A comprehensive analysis of hyper-heuristics. Intelligent Data Analysis, 12(1), 3–23. Google Scholar
  34. Pinto, E., & Oliveira, J. F. (2005). Algorithm based on graphs for the non-guillotinable two-dimensional packing problem. In 2nd ESICUP meeting, Southampton. Google Scholar
  35. Ramesh Babu, A., & Ramesh Babu, N. (1999). Effective nesting of rectangular parts in multiple rectangular sheets using genetic and heuristic algorithms. International Journal of Production Research, 37(7), 1625–1643. CrossRefGoogle Scholar
  36. Sarin, S. C. (1983). Two-dimensional stock cutting problems and solution methodologies. ASME Transactions Journal of Engineering for Industry, 104, 155–160. CrossRefGoogle Scholar
  37. Terashima-Marin, H., Flores-Alvarez, E. J., & Ross, P. (2005a). Hyper-heuristics and classifier systems for solving 2D-regular cutting stock problems. In Proceedings of the genetic and evolutionary computation conference (vol. 2, pp. 637–643). Google Scholar
  38. Terashima-Marin, H., Moran-Saavedra, A., & Ross, P. (2005b). Forming hyper-heuristics with GAs when solving 2D-regular cutting stock problems. In Proceedings of the IEEE congress on evolutionary computation (vol. 2, pp. 1104–1110). Google Scholar
  39. Valenzuela, C. L., & Wang, P. Y. (2001). Heuristics for large strip packing problems with guillotine patterns: An empirical study. In Proceedings of the 4th metaheuristics international conference (pp. 417–421). University of Porto, Porto, Portugal. Google Scholar
  40. Waescher, G., Haussner, H., & Schumann, H. (2007). An improved typology of cutting and packing problems. European Journal of Operational Research, 183, 1109–1130. CrossRefGoogle Scholar
  41. Zhang, D., Kang, Y., & Deng, A. (2006). A new heuristic recursive algorithm for the strip rectangular packing problem. Computers & Operations Research, 33, 2209–2217. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Computer EngineeringYeditepe UniversityKadıköy/İstanbulTurkey
  2. 2.ASAP Research Group, School of Computer ScienceUniversity of NottinghamNottinghamUK

Personalised recommendations