Cutting Stock by Iterated Matching

  • Andreas Fritsch
  • Oliver Vornberger
Part of the Operations Research Proceedings book series (ORP, volume 1994)


The combinatorial optimization problem considered in this paper is a special two dimensional cutting stock problem arising in the wood, metal and glass industry. Given a demand of non oriented small rectangles and a theoretically infinite set of large stock rectangles of given lengths and widths, our aim is to generate slicing trees specifying how to cut the demand out of the stock rectangles. Only guillotine cuts are permitted. We are looking for layouts whose waste is minimal. We developed an iterative algorithm for solving this problem heuristically. By a maximum weight matching we match in every iteration step suitable rectangles and consider the matched pairs as new, so called meta rectangles. These meta rectangles can be treated in the same way as ordinary rectangles. Furthermore, we use shape functions, so that the orientations of the demand rectangles are not fixed until the layout has been computed. The algorithm, programmed in C, has been tested with several instances, containing between 52 and 161 rectangles, taken from real demands of a glass factory. The resulting layouts, calculated within a few minutes (on a 486-PC), have an average waste of less than 5 percent.


Shape Function Iteration Step Combinatorial Optimization Problem Glass Industry Real Demand 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Can 1979]
    P. de Cani, “Packing problems in theory and practice”, Department of Engineering Production, University of Birmingham, Marz 1979.Google Scholar
  2. [Cof 1980]
    E. G. Coffman, M. R. Garey, D. S. Johnson and R. E. Tarjan, “Performance Bounds for Level-Oriented Two-Dimensional Packing Algorithms”, SIAM Journal on Computing 9, 4 (1980), pp. 808–826.CrossRefGoogle Scholar
  3. [Cof 1990]
    E. G. Coffman and P. W. Shor, “Average-Case Analysis of Cutting and Packing in Two Dimensions”, European Journal of Operational Research 44, 2 (1990), pp. 134–145.CrossRefGoogle Scholar
  4. [Gab 1973]
    H.Gabow, “Implementation of Algorithms for Maximum Matching on Nonbipartite Graphs”, Ph.D.Thesis, Stanford University, 1973.Google Scholar
  5. [Gar 1981]
    M. R. Garey and D. S. Johnson, “Approximation Algorithms for Bin Packing Problems: A Survey”, in Analysis and Design of Algorithms in Combinatorial Optimization, Vol. 266, G. Ausiello and N. Lucertini, eds., Springer Verlag, Berlin, 1981, pp. 147–172.Google Scholar
  6. [Gil 1961]
    P. C. Gilmore and R. E. Gomory, “A Linear Programming Approach to the Cutting-Stock Problem”, Operations Research, Vol. 9 (1961), pp. 849–859.CrossRefGoogle Scholar
  7. [Gil 1965]
    P. C. Gilmore and R. E. Gomory, “Multistage cutting stock problems of two and more dimensions”, Operations Research, Vol. 13 (1965), pp. 94–120.CrossRefGoogle Scholar
  8. [Her 1972]
    J. C. Herz, “Recursive Computational Procedure for Two-Dimensional Stock Cutting”, IBM Journal of Research and Development 16 (1972), pp. 462–469.CrossRefGoogle Scholar
  9. [Whi 1977]
    C. Whitlock and N. Christofides, “An Algorithm for Two-Dimensional Cutting Problems”, Operations Research, Vol$125, Nr. 1, Januar-Februar 1977, pp. 30–44.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Andreas Fritsch
    • 1
  • Oliver Vornberger
    • 1
  1. 1.OsnabrückGermany

Personalised recommendations