Quality Improvement via Optimization of Tolerance Intervals During the Design Stage

  • Sevgui Hadjihassan
  • Eric Walter
  • Luc Pronzato
Part of the Applied Optimization book series (APOP, volume 3)


A deterministic model-based approach to quality improvement is proposed, along Taguchi’s ideas for off-line quality control. This new approach takes into account fluctuations in the factors; these fluctuations are characterized in terms of tolerance intervals.

The case of an a priori known model (i.e., known dependency of the performance characteristics on the factors) is considered first. The optimal factor design is chosen by worst-case optimization.

If the model’s parameters are unknown, a bounded-error approach is used to characterize their uncertainty. A min-max optimization is then performed, taking into account the fluctuations of the quality of the product components as well as the uncertainty on the model parameters.


Quality Improvement Noise Factor Interval Computation Tolerance Interval Estimation Stage 
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. [1]
    G. Belforte, B. Bona and V. Cerone, “Parameter estimation algorithms for a set-membership description of uncertainty”, Automatica, 1990, Vol. 26, pp. 887–898.zbMATHCrossRefGoogle Scholar
  2. [2]
    G. Belforte and T. T. Tay, “Recursive estimation for linear models with set membership measurement errors”, Prep. 9th IFAC/IFORS Symp. Identification and System Parameter Estimation, Budapest, 8-12 July, 1991, pp. 872–877.Google Scholar
  3. [3]
    V. Broman and M. J. Shensa, “A compact algorithm for the intersection and approximation of N—dimensional polytopes”, Math, and Comput. in Simulation, 1990, Vol. 32, pp. 469–480.CrossRefMathSciNetGoogle Scholar
  4. [4]
    G. Cagnac, E. Ramis and J. Commeau, Nouveau cours de mathématiques spéciales, Masson, Paris, 1965, Vol. 3, pp. 17–19.Google Scholar
  5. [5]
    V. Cerone, “Feasible parameter sets for linear models with bounded errors in all variables”, Automatica, 1993, Vol. 29, pp. 1551–1555.zbMATHCrossRefGoogle Scholar
  6. [6]
    L. Chisci, A. Garulli and G. Zappa, “Recursive set membership state estimation via parallelotopes”, Prep. 10th IF AC Symp. on System Identification, Copenhagen, 4–6 July, 1994, Vol. 3, pp. 383–388.Google Scholar
  7. [7]
    T. Clement and S. Gentil, “Reformulation of parameter identification with unknown but bounded errors”, Math, and Comput. in Simulation, 1988, Vol. 30, pp. 257–270.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    T. Clément and S. Gentil, “Recursive membership set estimation for output-error models”, Math. Comput. in Simulation, 1990, Vol. 32, pp. 505–513.CrossRefGoogle Scholar
  9. [9]
    P. Combettes, “The foundations of set theoretic estimation”, Proc. of the IEEE, 1993, Vol. 81, pp. 182–208.CrossRefGoogle Scholar
  10. [10]
    J. R. Deller, “Set membership identification in digital signal processing”, IEEE ASSP Magazine, 1989, Vol. 6, pp. 4–20.CrossRefGoogle Scholar
  11. [11]
    J. R. Deller, M. Nayeri and S. F. Odeh, “Least-square identification with error bounds for real-time signal processing and control”, Proc. of the IEEE, 1993, Vol. 81, pp. 813–849.CrossRefGoogle Scholar
  12. [12]
    E. Fogel and Y. F. Huang, “On the value of information in system identification — bounded noise case”, Automatica, 1982, Vol. 18, pp. 229–238.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    E. R. Hansen, “Global optimization using interval analysis — the multidimensional case”, Numer. Math., 1980, Vol. 34, pp. 247–270.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    K. Ishikawa, “Quality and standardization: progress for economic success”, Quality Progress, 1984, Vol. 1, pp. 16–20.Google Scholar
  15. [15]
    J. M. Lucas, “Optimum composite designs”, Technometrics, 1974, Vol. 16, pp. 561–567.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Y. A. Merkuryev, “Identification of objects with unknown bounded disturbances”, Int. J. Control, 1989, Vol. 50, pp. 2333–2340.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    H. Messaoud and G. Favier, “Recursive determination of parameter uncertainty intervals for linear models with unknown but bounded errors”, Prep. 10th IFAC Symp. on System Identification, Copenhagen, 4–6 July, 1994, Vol. 3, pp. 365–369.Google Scholar
  18. [18]
    H. Messaoud, G. Favier and R. Santos Mendes, “Adaptative robust pole placement by connecting identification and control”, Prep. 4th IFAC Int. Symp. Adaptative Systems in Control and Signal Processing, Grenoble, 1992, pp. 41–46.Google Scholar
  19. [19]
    M. Milanese and G. Belforte, “Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: linear families of models and estimators”, IEEE Trans. Autom. Control, 1982, Vol. 27, pp. 408–414.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    M. Milanese and A. Vicino, “Optimal estimation theory for dynamic systems with set membership uncertainty: an overview”, Automatica, 1991, Vol. 27, pp. 997–1009.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    S. H. Mo and J. P. Norton, “Fast and robust algorithms to compute exact polytope parameter bounds”, Math. and Comput. in Simulation, 1990, Vol. 32, pp. 481–493.CrossRefMathSciNetGoogle Scholar
  22. [22]
    R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia, 1979.zbMATHGoogle Scholar
  23. [23]
    J. P. Norton, “Identification of parameter bounds for ARMAX models from records with bounded noise”, Int. J. Control, 1987, Vol. 45, pp. 375–390.zbMATHCrossRefGoogle Scholar
  24. [24]
    J. P. Norton, “Identification and application of bounded-parameter models”, Automatica, 1987, Vol. 23, pp. 497–507.zbMATHCrossRefGoogle Scholar
  25. [25]
    J. P. Norton (Ed.), “Special issue on bounded-error estimation”, Issues I and II, Int. J. of Adapt. Contr. and Signal Proc., 1994, Vol. 8, and 1995, Vol. 9.Google Scholar
  26. J. P. Norton (Ed.), “Special issue on bounded-error estimation”, Issues I and II, Int. J. of Adapt. Contr. and Signal Proc., 1994, Vol. 8, and 1995, Vol. 9.Google Scholar
  27. [26]
    H. Piet-Lahanier and E. Walter, “Polyhedric approximation and tracking for bounded-error models”, Proc. IEEE Int. Symp. Circuits and Systems, Chicago, 1993, pp. 782–785.Google Scholar
  28. [27]
    L. Pronzato and E. Walter, “Minimum-volume ellipsoids containing compact sets: application to parameter bounding”, Automatica, 1994, Vol. 30, pp. 1731–1739.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [28]
    L. Pronzato, E. Walter and H. Piet-Lahanier, “Mathematical equivalence of two ellipsoidal algorithms for bounded-error estimation”, Proc. 28th IEEE Conference on Decision and Control, Tampa, 1989, pp. 1952–1955.CrossRefGoogle Scholar
  30. [29]
    L. Pronzato, E. Walter, A. Venot and J. -F. Lebruchec, “A general purpose global optimizer: implementation and applications”, Math, and Computers in Simulation, 1984, Vol. 26, pp. 412–422.CrossRefMathSciNetGoogle Scholar
  31. [30]
    B. Pschenichnyy and V. S. Pokotilo, “A minmax approach to the estimation of linear regression parameters”, Engrg. Cybernetics, 1983, pp. 77–85.Google Scholar
  32. [31]
    F. S. Schweppe, Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs, 1973.Google Scholar
  33. [32]
    K. Shimitzu and E. Aiyoshi, “Necessary conditions for min-max problems and algorithm by a relaxation procedure”, IEEE Trans. Autom. Control, 1980, Vol. 25, pp. 62–66.CrossRefGoogle Scholar
  34. [33]
    G. Taguchi, Introduction to Quality Engineering, APO, Tokyo, 1986.Google Scholar
  35. [34]
    G. Taguchi and M. S. Phadke, “Quality engineering through design optimization”, IEEE Global Telecommunications Conference, Atlanta, GA, 1984, pp. 1106–1113Google Scholar
  36. [35]
    S. M. Veres and J. P. Norton, “Parameter-bounding algorithms for linear errors in variables models”, Prep. 9th IFAC/IFORS Symp. on Identification and System Parameter Estimation, Budapest, 1991, pp. 1038–1043.Google Scholar
  37. [36]
    I. N. Vuchkov and L. N. Boyadjieva, “The robustness against tolerances of performance characteristics described by second order polynomials”, in Optimal design and analysis of experiments (Eds. Y. Dodge, V. Fedorov and H. Wynn ), North Holland, Amsterdam, 1988, pp. 293–309.Google Scholar
  38. [37]
    I. N. Vuchkov and L. N. Boyadjieva, “A model-based approach to the robustness against tolerances”, Proc. 33rd EOQC Annual Conference, Vienna, 1989, pp. 585–592.Google Scholar
  39. [38]
    I. N. Vuchkov and L. N. Boyadjieva, “Quality improvement through design of experiments with both product parameters and external noise factors”, in Model Oriented Data Analysis. A Survey of Recent Methods (Eds. V. Fedorov, W. G. Müller and I. N. Vuchkov ), Physica Verlag, Heildelberg, 1992.Google Scholar
  40. [39]
    E. Walter and H. Piet-Lahanier, “Exact recursive polyhedral description of the feasible parameter set for bounded-error models”, IEEE Trans. Autom. Control, 1989, Vol. 34, No. 8, pp. 911–915.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [40]
    E. Walter and H. Piet-Lahanier, “Estimation of parameter bounds from bounded — error data: a survey”, Math. Comput. in Simulation, 1990, Vol. 32, pp. 449–468.CrossRefMathSciNetGoogle Scholar
  42. [41]
    E. Walter and L. Pronzato, “Characterizing sets defined by inequalities”, Prep. 10th IFAC Symp. on System Identification, Copenhagen, 4–6 July, 1994, Vol. 2, pp. 15–26.Google Scholar
  43. [42]
    H. P. Wynn and A. Winterbottom, Lectures on Experimental Design and Off-line Quality Control (Taguchi methods), City University, London, 1986.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Sevgui Hadjihassan
    • 1
  • Eric Walter
    • 1
  • Luc Pronzato
    • 2
  1. 1.Laboratoire des Signaux et SystèmesCNRS/ESEGif sur YvetteFrance
  2. 2.Laboratoire I3SCNRS URA-1376ValbonneFrance

Personalised recommendations