Annals of Operations Research

, Volume 57, Issue 1, pp 147–173 | Cite as

The nesting problem in the leather manufacturing industry

  • Jörg Heistermann
  • Thomas Lengauer


The part-nesting problem is the problem of arranging a set of plane irregularly shapedparts on a plane irregularly shapedsurface, such that no parts overlap and as much of the surface is covered as possible. This problem occurs, e.g., in the textile and clothing industry, and one of the most challenging applications appears in the manufacturing of leather, e.g., in the furniture, car, clothing, and shoe industry. This application is characterized by a high degree of inhomogeneity of the surface as well as the parts and by severe restrictions on run time. We present an algorithmic method for computer-aided nesting in this context. The algorithm is characterized by selective data reduction, sequential part placement, a topological part fitting process, and a carefully tuned evaluation function for partial placements. Experiments show that the method is competitive with human nesters, for relatively nicely behaved part sets and surfaces.


Evaluation Function Data Reduction Shoe Sequential Part Manufacturing Industry 
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 and A. Albano, Nesting two-dimensional shapes in rectangular modules, Comp. Aided Design 8(1976)27–33.Google Scholar
  2. [2]
    Y. Akeda and M. Hori, On random sequential packing in two and three dimensions, Biometrika 63(1976)361–366.Google Scholar
  3. [3]
    A. Albano, A method to improve two-dimensional layout, Comp. Aided Design 9(1977)48–52.Google Scholar
  4. [4]
    A. Albano and G. Sapuppo, Optimal allocation of two-dimensional irregular shapes using heuristic search methods, IEEE Trans. Syst., Man Cybern. 10(1980) 242–248.Google Scholar
  5. [5]
    J. Bailleul, R. Soenen and K. Tiaibia, Nesting two dimensional irregular shapes in anisotropic material, in:Advances in CAD/CAM-Proc. 5th Int. IFIP/IFAC Conf. on Programming Research and Operations Logistics in Advanced Manufacturing Technology, ed. T. Ellis and O. Semenkov (North-Holland, Amsterdam, 1983) pp. 191–201.Google Scholar
  6. [6]
    B.S. Baker, D.J. Brown and H.P. Katseff, A 5/4 algorithm for two-dimensional packing, J. Algor. 2(1981)348–368.Google Scholar
  7. [7]
    B.S. Baker, D.J. Brown and H.P. Katseff, Lower bounds for the two-dimensional packing algorithm, Acta Inf. 18(1982)207–225.Google Scholar
  8. [8]
    B.S. Baker, E. Coffman and R.L. Rivest, Orthogonal packing in two dimensions, SIAM J. Comp. 9(1980)847–855.Google Scholar
  9. [9]
    B.S. Baker and J.S. Schwarsz, Shelf algorithms for two-dimensional packing problems, SIAM J. Comp. 12(1983)508–525.Google Scholar
  10. [10]
    J.J. Bartholdi, J.H. Vate and J. Zhang, Expected performance of the shelf heuristic for 2 dimensional packing, Oper. Res. Lett. ORSA J. 8(1989)11–16.Google Scholar
  11. [11]
    M. Ben-Bassat and D. Dori, Efficient nesting of congruent convex figures, Commun. ACM 27(1984)228–235.Google Scholar
  12. [12]
    J. Berkey and P. Wang, Two-dimensional finite bin-packing algorithms, J. Oper. Res. Soc. 38(1987)423–429.Google Scholar
  13. [13]
    J. Bernardo and J. Fernandes, Geometric algorithms and heuristic methods for automatic packing of irregular shapes — An object oriented approach, in:ORSA/TIMS Bulletin Number 34 (Program of the 34th Joint National Meeting, San Francisco) (1992).Google Scholar
  14. [14]
    B. Chazelle, The bottom-left bin packing heuristic: An efficient implementation, IEEE Trans. Comp. C-32(1983)697–707.Google Scholar
  15. [15]
    N. Christofides and C. Whitlock, An algorithm for two-dimensional cutting problems, Oper. Res. (ORSA J.) 25(1977)30–44.Google Scholar
  16. [16]
    F. Chung, M. Garey and D. Johnson, On packing two-dimensional bins, SIAM J. Algor. Discr. Meth. 3(1982)66–76.Google Scholar
  17. [17]
    J. Chung, D.J. Hillman and D. Scott, An intelligent nesting system on 2-D highly irregular resources, in:Applications of Artificial Intelligence VIII - Proc. 8th Int. Conf. of the International Society for Optical Engineering (SPIE), (1990) pp.472–483.Google Scholar
  18. [18]
    E. Coffman, M. Garey, D. Johnson and R. Tarjan, Performance bounds for level-oriented two-dimensional packing algorithms, SIAM J. Comp. 9(1980)808–826.Google Scholar
  19. [19]
    E. Coffman and E. Gilbert, Dynamic, first-fit packings in two or more dimensions, Inf. Contr. 61(1984)1–14.Google Scholar
  20. [20]
    E. Coffman and P. Shor, Average-case analysis of cutting and packing in two dimensions, Euro. J. Oper. Res (EJOR) 44(1990)134–144.Google Scholar
  21. [21]
    E. Coffman and P. Shor, Packings in two dimensions: Asymptotic average-case analysis of algorithms, Algorithmica 9(1993)253–277.Google Scholar
  22. [22]
    R. Cuninghame-Green, Geometry, shoemaking and the milk tray problem, New Scientist, no. 12(1989)50–53.Google Scholar
  23. [23]
    C.H. Dagli, A hybrid approach to composite stock cutting using artificial neural networks and knowledge-based systems, in:ORSA/TIMS Bulletin Number 34 (Program of the 34th Joint National Meeting, San Francisco) (1992).Google Scholar
  24. [24]
    C.H. Dagli and M. Tatoglu, An approach to two-dimensional cutting stock problems, Int. J. Prod. Res. 25(1987)175–190.Google Scholar
  25. [25]
    K. Daniels, Z. Li and V. Milenkovic, Automatic marker making, in:Proc. 3rd Canadian Conf. on Computational Geometry, ed. T. Shermer (August 1991).Google Scholar
  26. [26]
    K. Daniels, Z. Li and V. Milenkovic, Placement and compaction of non-convex polygons for clothing manufacture, in:Proc. 4th Canadian Conf. on Computational Geometry, ed. C. Wang (August 1992).Google Scholar
  27. [27]
    K. Daniels, Z. Li and V. Milenkovic, Automatic marker making, Progress report, Center for Research in Computing Technology, Harvard University, Cambridge, MA 02138, USA (January 1993).Google Scholar
  28. [28]
    J. Doenhardt and T. Lengauer, Algorithmic aspects of one-dimensional layout compaction, IEEE Trans. Comp. Aided Design CAD-6(1987)863–878.Google Scholar
  29. [29]
    K.A. Dowsland and W.B. Dowsland, Packing problems, Euro. J. Oper. Res. (EJOR) 56(1992)2–14.Google Scholar
  30. [30]
    L. Dutg and C. Fabian, Solving cutting-stock problems through the Monte-Carlo method, Econ. Comp. Econ. Cybern. Studies Res. 19(1984)35–54.Google Scholar
  31. [31]
    H. Dyckhoff and U. Finke,Cutting and Packing in Production and Distribution (Physica-Verlag, Heidelberg, Germany, 1992).Google Scholar
  32. [32]
    H. Dyckhoff and G. Wäscher (eds), Special issue onCutting and Packing, Euro. J. Oper. Res. (EJOR) 44, no 2 (1990).Google Scholar
  33. [33]
    A.A. Farley, Mathematical programming models for cutting-stock problems in the clothing industry, J. Oper. Res. Soc. 39(1988)41–53.Google Scholar
  34. [34]
    A.A. Farley, The cutting stock problem in the canvas industry, Euro. J. Oper. Res. (EJOR) 44(1990)247–255.Google Scholar
  35. [35]
    A.A. Farley, Selection of stockplate characteristics and cutting style for two dimensional cutting stock situations, Euro. J. Oper. Res. (EJOR) 44(1990)239–246.Google Scholar
  36. [36]
    J.S. Ferreira and J.F. Oliveira, An improved version of Wang's algorithm for two-dimensional cutting problems, Euro. J. Oper. Res. (EJOR) 44(1990)209–223.Google Scholar
  37. [37]
    J.S. Ferreira and J.F. Oliveira, An application of simulated annealing to the nesting problem, in:ORSA/TIMS Bulletin Number 34 (Program of the 34th Joint National Meeting, San Francisco) (1992).Google Scholar
  38. [38]
    R.J. Fowler, M.S. Paterson and S.L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inf. Proc. Lett. (IPL) 12(1981)133–137.Google Scholar
  39. [39]
    G.N. Frederickson, Probalistic analysis for simple one and two-dimensional bin packing, Inf. Proc. Lett. (IPL) 11(1980)156–161.Google Scholar
  40. [40]
    H. Freeman and M.J. Haims, A multistage solution of the template-layout problem, IEEE Trans. Syst. Sci. Cybern. 4(1970)145–151.Google Scholar
  41. [41]
    P. Gilmore and R. Gomery, Multistage cutting stock problems of two and more dimensions, Oper. Res. (ORSA J.) 13(1965)94–120.Google Scholar
  42. [42]
    I. Golan, Performance bounds for orthogonal oriented two-dimensional packing algorithms, SIAM J. Comp. 10(1981)571–582.Google Scholar
  43. [43]
    S.G. Hahn, On the optimal cutting of defective sheets, Oper. Res. (ORSA J.) 16(1968)1100–1114.Google Scholar
  44. [44]
    J. Heistermann, Effizientes rechnergestütztes Nesting, Ph.D Thesis, Department of Computer Science, University of Bonn, Römerstrasse 164, 53117 Bonn, Germany (July 1993).Google Scholar
  45. [45]
    J. Heistermann and T. Lengauer. Efficient automatic part nesting on irregular and inhomogeneous surfaces, in:Proc. 4th ACM-SIAM Symp. on Discrete Algorithms (SODA '93), Austin, Texas, USA (January 1993) pp. 251–259.Google Scholar
  46. [46]
    M. Hofri, Two-dimensional packing: Expected performance of simple level algorithms, Inf. Contr. 45(1980)1–17.Google Scholar
  47. [47]
    R.W. Holtkamp, Lokale Polygonplazierung bei der Erzeugung von Nesting-Schnittbildern, Diploma thesis, Department of Mathematics and Computer Science, University of Paderborn, Warburger Strasse 100, 33 095 Paderborn, Germany (May 1993).Google Scholar
  48. [48]
    T. Lengauer,Combinatorial Algorithms for Integrated Circuit layout, Applicable Theory in Computer Science — A Wiley — Teubner Series (Wiley, New York, 1990).Google Scholar
  49. [49]
    Z. Li and V. Milenkovic, A compaction algorithm for non-convex polygons and its application, in:Proc. 9th Annual ACM Symp. on Computational Geometry (May 1993).Google Scholar
  50. [50]
    R. Lindemann, G. Scheithauer and J. Terno, Zuschnittprobleme und ihre praktische lösung (Harri Deutsch, Frankfurt am Main, Germany, 1987).Google Scholar
  51. [51]
    W. Lu, Computer aided nesting of two dimensional polygonal components, UK thesis, Department of Production Research, University of Nottingham, Nottingham, Great Britain (1987).Google Scholar
  52. [52]
    V. Marques, Packing of irregular patterns based on simulated annealing, in:ORSA/TIMS Bulletin Number 34 (Program of the 34th Joint National Meeting, San Francisco) (1992).Google Scholar
  53. [53]
    W. Medetz, ANIS — A software package for automatic nesting of irregular shapes, in:ORSA/TIMS Bulletin Number 34 (Program of the 34th Joint National Meeting, San Francisco) (1992).Google Scholar
  54. [54]
    F.P. Preparata and M.I. Shamos,Computational Geometry (Springer, New York 2nd ed., 1988).Google Scholar
  55. [55]
    W. Qu and J.L. Sanders, A nesting algorithm for irregular parts and parts affecting trim loss, Int. J. Prod. Res. 25(1987)381–397.Google Scholar
  56. [56]
    M. Rarey, Optimierung von Nesting-Schnittbildern, Diploma thesis, Department of Mathematics and Computer Science, University of Paderborn, Warburger Strasse 100, 33 095 Paderborn, Germany (September 1992).Google Scholar
  57. [57]
    E. Ridenour Paternoster and P.E. Sweeney, Cutting and packing problems: A categorized application-orientated research bibliography, J. Oper. Res. Soc. 43(1992)691–706.Google Scholar
  58. [58]
    S. Roberts, Application of heuristic techniques to the cutting-stock problem for worktops, J. Oper. Res. Soc. 35(1984)369–377.Google Scholar
  59. [59]
    D. Sleator, A 2.5 times optimal algorithm for packing in two dimensions, Inf. Proc. Lett. (IPL) 10(1980)37–40.Google Scholar
  60. [60]
    P. Wang, Two algorithms for constrained two-dimensional cutting stock problems, Oper. Res. (ORSA J.) 31(1983)575–586.Google Scholar
  61. [61]
    G. Wäscher, An LP-based approach to cutting stock problems with multiple objectives, Euro. J. Oper. Res. (EJOR) 44(1990)175–184.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1995

Authors and Affiliations

  • Jörg Heistermann
    • 2
  • Thomas Lengauer
    • 1
    • 2
  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany
  2. 2.Institute for Algorithms and Scientific ComputingGerman National Research Center for Computer Science (GMD)Sankt AugustinGermany

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