Applications of Interval Computations pp 91-131 | Cite as

# Quality Improvement via Optimization of Tolerance Intervals During the Design Stage

## Abstract

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.

## Keywords

Quality Improvement Noise Factor Interval Computation Tolerance Interval Estimation Stage## Preview

Unable to display preview. Download preview PDF.

## References

- [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]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]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]G. Cagnac, E. Ramis and J. Commeau,
*Nouveau cours de mathématiques spéciales*, Masson, Paris, 1965, Vol. 3, pp. 17–19.Google Scholar - [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]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]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]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]P. Combettes, “The foundations of set theoretic estimation”,
*Proc. of the IEEE*, 1993, Vol. 81, pp. 182–208.CrossRefGoogle Scholar - [10]J. R. Deller, “Set membership identification in digital signal processing”,
*IEEE ASSP Magazine*, 1989, Vol. 6, pp. 4–20.CrossRefGoogle Scholar - [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]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]E. R. Hansen, “Global optimization using interval analysis — the multidimensional case”,
*Numer. Math*., 1980, Vol. 34, pp. 247–270.zbMATHCrossRefMathSciNetGoogle Scholar - [14]K. Ishikawa, “Quality and standardization: progress for economic success”,
*Quality Progress*, 1984, Vol. 1, pp. 16–20.Google Scholar - [15]J. M. Lucas, “Optimum composite designs”,
*Technometrics*, 1974, Vol. 16, pp. 561–567.zbMATHCrossRefMathSciNetGoogle Scholar - [16]Y. A. Merkuryev, “Identification of objects with unknown bounded disturbances”,
*Int. J. Control*, 1989, Vol. 50, pp. 2333–2340.zbMATHCrossRefMathSciNetGoogle Scholar - [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]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]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]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]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]R. E. Moore,
*Methods and Applications of Interval Analysis*, SIAM Studies in Applied Mathematics, Philadelphia, 1979.zbMATHGoogle Scholar - [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]J. P. Norton, “Identification and application of bounded-parameter models”,
*Automatica*, 1987, Vol. 23, pp. 497–507.zbMATHCrossRefGoogle Scholar - [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 - 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]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 - [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 - [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 - [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 - [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 - [31]F. S. Schweppe,
*Uncertain Dynamic Systems*, Prentice-Hall, Englewood Cliffs, 1973.Google Scholar - [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 - [33]G. Taguchi,
*Introduction to Quality Engineering*, APO, Tokyo, 1986.Google Scholar - [34]G. Taguchi and M. S. Phadke, “Quality engineering through design optimization”,
*IEEE Global Telecommunications Conference*, Atlanta, GA, 1984, pp. 1106–1113Google Scholar - [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 - [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 - [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 - [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 - [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 - [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 - [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 - [42]H. P. Wynn and A. Winterbottom,
*Lectures on Experimental Design and Off-line Quality Control (Taguchi methods)*, City University, London, 1986.Google Scholar