A New Approach to Packing Non-Convex Polygons Using the No Fit Polygon and Meta-Heuristic and Evolutionary Algorithms

  • Edmund Burke
  • Graham Kendall
Conference paper


Earlier work by the authors has presented a method for packing convex polygons, using a construction known as the no fit polygon. Although the method was shown to be successful, it could only deal with convex polygons. This paper addresses the issue by showing how a non-convex, no fit polygon algorithm can (and should) produce better quality solutions. We demonstrate the approach on two test problems that the authors have used in previous work. The new algorithm does, however, have an increased computational complexity. We tackle this by presenting an algorithm that allows the non-convex, no fit polygon to be approximated. This means that much more of the search space can be considered and computational results show that better quality solutions can be achieved, in less time than when using the full no fit polygon algorithm.


Tabu Search Convex Polygon Memetic Algorithm Partial Evaluation Good Quality Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Burke E.K, and Kendall, G. 1999. Applying Ant Algorithms and the No Fit Polygon to the Nesting Problem, Proceedings of 12th Australian Joint Conference on Artificial Intelligence, Sydney, Australia, 6-10 December 1999, Lecture Notes in Artificial Intelligence (1747), Foo, N. (Ed), pp 453–464.Google Scholar
  2. 2.
    Burke E.K, and Kendall, G. 1999. Applying Simulated Annealing and the No Fit Polygon to the Nesting Problem, Proceedings of WMC ′99: World Manufacturing Congress, Durham, UK, 27-30 September, 1999, pp 70–76.Google Scholar
  3. 3.
    Burke E.K, and Kendall, G. 1999. Applying Evolutionary Algorithms and the No Fit Polygon to the Nesting Problem, in proceedings of IC-AI′99: The 1999 International Conference on Artificial Intelligence, Las Vegas, Nevada, USA, 28 June-1 July 1999, pp 51–57.Google Scholar
  4. 4.
    Bennell J.A., Dowsland K.A, and Dowsland W.B. 2001. The irregular cutting stock problem — a new procedure for deriving the no-fit polygon, Computers and Operations Research 28 pp 271–287.CrossRefGoogle Scholar
  5. 5.
    Kendall G. 2000. Applying Meta-Heuristic Algorithms to the Nesting Problem Utilising the No Fit Polygon PhD Thesis, Department of Computer Science & IT, The University of Nottingham, UK.Google Scholar
  6. 6.
    Karp, R.M. 1972. Reducibility Among Combinatorial Problems. Complexity of Computer Computations, Miller, R.E., Thatcher, J.W. (eds.), Plenum Press, New York, pp 85–103.CrossRefGoogle Scholar
  7. 7.
    Art, R.C., 1966. An Approach to the Two-Dimensional Irregular Cutting Stock Problem. Technical Report 36.008, IBM Cambridge Centre.Google Scholar
  8. 8.
    Adamowicz, M., Albano, A. 1976. Nesting Two-Dimensional Shapes in Rectangular Modules. Computer Aided Design, 8, 27–33.CrossRefGoogle Scholar
  9. 9.
    Albano, A., Sappupo, G. 1980. Optimal Allocation of Two-Dimensional Irregular Shapes Using Heuristic Search Methods. IEEE Trans. Syst., Man and Cybernetics, SMC-10, pp 242–248.Google Scholar
  10. 10.
    Oliveira, J.F., Gomes, A.M., Ferreira, S. 1998. TOPOS A new constructive algorithm for nesting problems. OR Spektrum 22 (2000) 2, 263–284.MathSciNetGoogle Scholar
  11. 11.
    Cunninghame-Green, R. 1989. Geometry, Shoemaking and the Milk Tray Problem. New Scientist, 12, August 1989, 1677, pp 50–53.Google Scholar
  12. 12.
    Cunninghame-Green R., Davis, L.S. 1992. Cut Out Waste! O.R. Insight,. Vol 5, iss 3, pp 4–7.Google Scholar
  13. 13.
    Mahadevan, A. 1984. Optimisation in Computer-Aided Pattern Packing. PhD thesis, North Caroline State University.Google Scholar
  14. 14.
    Canny, J. 1987. The Complexity of Robot Motion Planning. MIT Press, Cambridge, MA.Google Scholar
  15. 15.
    Ghosh, P. K. 1993. A Unified Computational Framework for Minkowski Operations. Computers and Graphics, Vol. 17, No. 4, pp 357–378.CrossRefGoogle Scholar
  16. 16.
    Lozano-Pérez, T., Wesley, M. ?. 1979. An Algorithm for planning collision-free paths among polyhedral obstacles. Communications of the ACM 22, pp 560–570.CrossRefGoogle Scholar
  17. 17.
    O’Rourke, J. 1998. Computational Geometry in C. Cambridge University Press.Google Scholar
  18. 18.
    Ramkumar, G. D. 1996. An Algorithm to Compute the Minkowski Sum Outer-Face of Two Simple Polygons. In proceedings of 12th Annual ACM Symposium of Computational Geometry, pp 234–241.Google Scholar
  19. 19.
    Schwartz, J. T., Sharir, M. 1990. Algorithmic Motion Planning in Robotics, in J. van Leeuwen (ed), Algorithms and Complexity, Handbook of Theoretical Computer Science, Vol A, Elsevier, Amsterdam, pp 391–430.Google Scholar
  20. 20.
    Serra, J. 1982. Image Analysis and Mathematical Morphology, Vol. 1, Academic Press, New York.MATHGoogle Scholar
  21. 21.
    Zhenyu, Li. 1994. Compaction Algorithms for Non-Convex Polygons and Their Applications. PhD Thesis, Computer Science, Harvard University, Cambridge, Massachusetts.Google Scholar
  22. 22.
    Burke, K. E. and Kendall, G. 1999d. Evaluation of Two Dimensional Bin Packing Problem using the No Fit Polygon, Proceedings of the 26th International Conference on Computers and Industrial Engineering, Melbourne, Australia, 15-17 December 1999, pp 286–291.Google Scholar
  23. 23.
    Falkenauer, E. 1998. Genetic Algorithms and Grouping Problems. John Wiley and Sons.Google Scholar
  24. 24.
    Hopper E. and Turton B.C.H. 2000. An Empirical Investigation of Meta-Heuristics and Heuristic Algorithms or 2D Packing Problem. European Journal of Operational Research (EJOR) 128(1), 34–57.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2002

Authors and Affiliations

  1. 1.The University of Nottingham, School of Computer Science & ITNottinghamUK

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