Genetic Algorithms and Heuristic Rules for Solving the Nesting Problem in the Package Industry

  • Roberto Selow
  • Flávio NevesJr.
  • Heitor S. Lopes
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 6)

The cutting/nesting problem in the packing industry can be stated as finding the maximum number of packages that can be arranged in a paper sheet of known size, in such a way as to minimize the loss of material. Figure 14.1 illustrates an example of six packages that will be drawn from a standard paper sheet and turned into a box. This problem is commonly found in many industrial areas that deal with cutting out shapes from raw stock, such as fabric, steel plate, paper, and so on.

An important factor in the search for the optimal solution for this problem is the number of parts that will be manipulated in the mounting settle. This is discussed later. There is a combinatorial explosion as the number of parts increases, leading to infeasible computational costs. For real-world problems, the number of parts is usually not larger than 20.

Genetic algorithms (GA) [10] have been used successfully in the last decades for several complex combinatorial problems and also for problems similar to the above-mentioned one [5, 12]. Therefore, the objective of this work is to propose a new method that uses genetic algorithms and heuristic rules to solve the problem.


Genetic Algorithm Search Space Search Region Heuristic Rule Paper Sheet 
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.
    M. Adamowicz, A. Albano, Nesting two-dimensional shapes in rectangular modules, Computer Aided Design, 1976, vol. 8, no. 1, pp. 27–33.CrossRefGoogle Scholar
  2. 2.
    Albano, G. Sapuppo, Optimal allocation of two-dimensional irregular shapes using heuristic search methods, IEEE Transactions on Systems, Man, and Cybernetics, 1980, vol. 10, no. 5, pp. 242–248.CrossRefGoogle Scholar
  3. 3.
    P. András, A. András, S. Zsuzsa, A genetic solution for the cutting stock problem, Proceedings of the First On-Line Workshop on Soft Computing, 1996, Nagoya University, pp. 87–92.Google Scholar
  4. 4.
    P. Chen, Z. Fu, A. Lim, B. Rodrigues, Two-dimensional packing for irregular shaped objects, Hawaii International Conference on Information Sciences (HICSS-36, Hawaii, USA), 2003.Google Scholar
  5. 5.
    P.C. Chu, J.E. Beasley, A genetic algorithm for the generalized assignment problem, Computers in Operations Research, 1997, vol. 24, no. 1, pp. 17–23.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    K. Fujita, S. Gakkai, Approach for optimal nesting algorithm using genetic algorithm and local minimization algorithm, Transactions of the Japanese Society of Mechanical Engineers, 1993, part C, vol. 59, no. 564, pp. 2576–2583.Google Scholar
  7. 7.
    G.C. Han, S.J. Na, Two-stage approach for nesting in two-dimensional cutting problems using neural network and simulated annealing, Proceedings of the Institution of Mechanical Engineering Part B Journal of Engineering Manufacture, 1996, vol. 210, no. 6, pp. 509–519.CrossRefGoogle Scholar
  8. 8.
    P.C. Gilmore, R.E. Gomory, Multistage cutting stock problems of two and more dimensons, Operations Research, 1965, vol. 13, pp. 94–120.MATHCrossRefGoogle Scholar
  9. 9.
    P.C. Gilmore, R.E. Gomory, The theory and computation of knapsack functions, Operations Research, 1966, vol. 14, no. 61, pp. 1045–1074.CrossRefMathSciNetGoogle Scholar
  10. 10.
    D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Reading, MA: Addison-Wesley, 1989.MATHGoogle Scholar
  11. 11.
    M.J. Haims, H. Freeman, Amultistage solution of the template layout problem, IEEE Transactions on Systems Science and Cybernetics, 1970, vol. 6, no. 2, pp. 145–151.CrossRefGoogle Scholar
  12. 12.
    E. Hopper, B. Turton, A genetic algorithm for a 2D industrial packing problem, Computers & Industrial Engineering, 1999, vol. 37, pp. 375–378.CrossRefGoogle Scholar
  13. 13.
    H.S. Ismail, K.K.B. Hon, New approaches for the nesting of two-dimensional shapes for press tool design, International Journal of Production Research, 1992, vol. 30, no. 4, pp. 825–837.MATHGoogle Scholar
  14. 14.
    A.Y.C. Nee, A heuristic algorithm for optimum layout of metal stamping blanks, Annals of CIRP, 1984, vol. 33, no. 1, pp. 317–320.CrossRefGoogle Scholar
  15. 15.
    V. Petridis, S. Kazarlis, A. Bazarlis, Varying fitness functions in genetic algorithm constrained optimization: The cutting stock and unit commitment problems, IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics, 1998, vol. 28, no. 5, pp. 629–639.CrossRefGoogle Scholar
  16. 16.
    R. Selow, Optimized arrangement of packages using genetic algorithms, M.Sc. Thesis, 2001, UTFPR, Brazil [in Portuguese].Google Scholar
  17. 17.
    W. Siedlecki, W. Sklanski, Constrained genetic optimization via dynamic reward-penalty balancing and its use in pattern recognition, Proceedings of Third International Conference on Genetic Algorithms, Ed., San Mateo, CA: Morgan Kaufmann, 1989, pp. 141–150.Google Scholar
  18. 18.
    Uday, E. Goodman, A. Debnath, Nesting of Irregular Shapes Using Feature Matching and Parallel Genetic Algorithms, Genetic and Evolutionary Computation Conference Late-Breaking Papers, E. Goodman, Ed., San Francisco: ISGEC Press, 2001, pp. 429–494.Google Scholar
  19. 19.
    H. Wang, Z. Ma, K. Nakayama, Effectiveness of penalty function in solving the subset sum problem, Proceedings of Third IEEE Conference on Evolutionary Computation, 1996, pp. 422–425.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Roberto Selow
    • 1
  • Flávio NevesJr.
    • 2
  • Heitor S. Lopes
    • 2
  1. 1.Electrical Engineering DepartmentCentro Universitário PositivoCuritibaBrazil
  2. 2.CPGEIUniversidade Tecnológica Federal do Paraná (UTFPR)CuritibaBrazil

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