Modelling and forecasting industrial innovations via the transfer function S-shaped learning curve

  • M. M. Naim
  • D. R. Towill


Algorithms are derived for the four-parameter transfer function S-shaped curve, using a least-squared-error (LSE) method for an exponential function. The S-shaped curve is just one in a family of industrial dynamics learning-curve models of increasing complexity which may be used to replicate and forecast the start-up of industrial innovations.

Controlled experiments are undertaken, via simulation of “message” and “noise”, to test the modelling and forecasting capabilities of the algorithms. A number of strategies are introduced to improve forecasting performance, such as “boots-trapping”, sequential and parallel adaptation, and alternatively adopting the simplified three-parameter S-curve model.

Four examples of modelling industrial innovations via the transfer function learning curve models are presented. The paper concludes that although there is now the capability to model the general four-parameter S-curve, its applications are limited. This is because simpler (and hence less accurate) transfer function models tend to be more robust.


S-shaped learning curve Industrial innovations Modelling Forecasting Simulation modelling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. W. Bevis, “An exploratory study of industrial learning with special reference to work study standards,” MSc thesis, UWIST, Cardiff, Wales, 1970.Google Scholar
  2. 2.
    W. D. Spears, “Measurement of learning and transfer through curve fitting”,Human Factors,27(3), pp. 251–266, 1985.Google Scholar
  3. 3.
    R. H. Baran, “A modified Ainsworth measure of learning efficiency”, Proceedings, IEEE Conference on Systems, Man and Cybernetics, Atlanta, Ga., pp. 912–916, 14–17 Oct. 1986.Google Scholar
  4. 4.
    V. Mahajan and R. A. Peterson,Models for Innovation Diffusion, Sage Publications, Beverly Hills, 1985.Google Scholar
  5. 5.
    I. Tchijov and E. Norov, “Forecasting methods for CIM technologies”,Engineering Costs and Production Economics,17, pp. 323–329, 1989.Google Scholar
  6. 6.
    L. E. Yelle, “Industrial life cycles and learning curves: interaction of marketing and production”,Industrial Marketing Management,9, pp. 311–318, 1980.Google Scholar
  7. 7.
    C. Edquist and S. Jacobsson,Flexible automation: The Global Diffusion of New Technology in the Engineering Industry, Blackwell, Oxford, 1988.Google Scholar
  8. 8.
    J. E. Butler, “Theories of technological innovation as useful tools for corporate strategy”,Strategic Management Journal,9, pp. 15–29, 1988.Google Scholar
  9. 9.
    D. R. Towill, A. Davies and M. M. Naim, “The dynamics of capacity planning for flexible manufacturing system startup”,Engineering Costs and Production Economics,17, pp. 55–64, 1989.Google Scholar
  10. 10.
    M. M. Naim, D. R. Towill, D. Coward and J. Cherrington, “Resource availability for AMT installation and operation”, Extended Summaries, Xth International Conference on Production Research, Nottingham, UK, pp. 370–371, 14–18 Aug. 1989.Google Scholar
  11. 11.
    A. Davies, H. J. Lewis, M. M. Naim and D. R. Towill, “The introduction of advanced manufacturing technology: Triumph or trauma?” Proceedings, 6th Conference of the Irish Manufacturing Committee, Advanced Manufacturing Technology, pp. 874–890, 31 Aug-1 Sept, 1989, Dublin, Eire.Google Scholar
  12. 12.
    Z. A. Sabri and A. A. Husseiny, “Analytical modeling of nuclear power station operator reliability”,Annals of Nuclear Energy,6, pp. 309–325, 1979.Google Scholar
  13. 13.
    J. A. Sharp and D. H. R. Price, “Experience curves in the electricity supply industry”,International Journal of Forecasting,6(4), pp. 531–540, 1990.Google Scholar
  14. 14.
    M. M. Naim and D. R. Towill, “An engineering approach to LSE modelling of experience curves in the electricity supply industry”,International Journal of Forecasting,6(4), pp. 549–556, 1990.Google Scholar
  15. 15.
    C. T. Goddard, “On growth and forecasting in electronics”, 31st Electronic Components Conference, IEEE, Atlanta, GA., pp. 187–194, May 1981.Google Scholar
  16. 16.
    H. O. Hartley, “The modified Gauss-Newton method for the fitting of non-linear regression functions by least squares”,Technometrics,3(2), pp. 269–280, 1982.Google Scholar
  17. 17.
    J. E. Cherrington, “Modelling and prediction of performance standards for multi-operator industrial process task”, CNAA PhD thesis, City of Birmingham Polytechnic, Birmingham, UK, 1982.Google Scholar
  18. 18.
    H. Sriyananda and D. R. Towill, “Prediction of Human Operator Performance”,IEEE Transactions,R-22, pp. 148–158, 1973.Google Scholar
  19. 19.
    D. R. Towill, “How complex a learning curve model need we use?”The Radio and Electronic Engineer,52(7), pp. 331–338, July 1982.Google Scholar
  20. 20.
    B. Hitchings and D. R. Towill, Error analysis of the time constant learning curve model.International Journal of Production Research,13(2), pp. 105–135, 1975.Google Scholar
  21. 21.
    R. N. Foster,Innovation: The Attacker's Advantage. Macmillan, London, 1986.Google Scholar
  22. 22.
    G. F. Ray,The Diffusion of Mature Technologies, Cambridge University Press, Cambridge, 1984.Google Scholar
  23. 23.
    F. M. Bass,A new product growth model for consumer durables, Creating and Marketing New Products, Granada Publishing Ltd, Section 4.6, pp. 445–463, 1973.Google Scholar
  24. 24.
    E. A. Hackett, “Application of a set of learning curve models to repetitive tasks”,Radio and Electronic Engineer,53(1), pp. 25–32, 1983.Google Scholar
  25. 25.
    M. M. Naim, “A case history of the introduction of a flexible manufacturing system into a level 5 company, Technical Note 137, UWIST, Cardiff, May 1988.Google Scholar
  26. 26.
    B. James, “Process industry — a user's view”, Seminar, Condition Monitoring in Manufacturing — The Key to Reliable and Economic Performance, UWIST, Cardiff, UK, 16 June 1988.Google Scholar

Copyright information

© Springer-Verlag London Limited 1993

Authors and Affiliations

  • M. M. Naim
    • 1
  • D. R. Towill
    • 1
  1. 1.School of Electrical, Electronic and Systems EngineeringUniversity of Wales College of CardiffCardiffUK

Personalised recommendations