Minimax Multivariate Control Chart Using a Polynomial Function

Author(s)
Johnson Ademola Adewara,
Kayode Samuel Adekeye,
Osebekwin Ebenezer Asiribo,
Samuel Babatope Adejuyigbe

ABSTRACT

Minimax control chart uses the joint probability distribution of the maximum and minimum standardized sample means to obtain the control limits for monitoring purpose. However, the derivation of the joint probability distribution needed to obtain the minimax control limits is complex. In this paper the multivariate normal distribution is integrated numerically using Simpson’s one third rule to obtain a non-linear polynomial (NLP) function. This NLP function is then substituted and solved numerically using Newton Raphson method to obtain the control limits for the minimax control chart. The approach helps to overcome the problem of obtaining the joint probability distribution needed for estimating the control limits of both the maximum and the minimum statistic for monitoring multivariate process.

Minimax control chart uses the joint probability distribution of the maximum and minimum standardized sample means to obtain the control limits for monitoring purpose. However, the derivation of the joint probability distribution needed to obtain the minimax control limits is complex. In this paper the multivariate normal distribution is integrated numerically using Simpson’s one third rule to obtain a non-linear polynomial (NLP) function. This NLP function is then substituted and solved numerically using Newton Raphson method to obtain the control limits for the minimax control chart. The approach helps to overcome the problem of obtaining the joint probability distribution needed for estimating the control limits of both the maximum and the minimum statistic for monitoring multivariate process.

Cite this paper

nullJ. Adewara, K. Adekeye, O. Asiribo and S. Adejuyigbe, "Minimax Multivariate Control Chart Using a Polynomial Function,"*Applied Mathematics*, Vol. 2 No. 12, 2011, pp. 1539-1545. doi: 10.4236/am.2011.212219.

nullJ. Adewara, K. Adekeye, O. Asiribo and S. Adejuyigbe, "Minimax Multivariate Control Chart Using a Polynomial Function,"

References

[1] H. Hotelling, “Multivariable Quality Control—Illustrated By The Air Testing Of Sample Bombsights,” In: C. Eisenhart, M. W. Hastay and W. A. Wallis, Eds., Techniques of Statistical Analysis, McGraw Hill, New York, 1947, pp. 111-184.

[2] W. H. Woodall and M. M. Ncube, “Multivariate Cusum Quality Control Procedures,” Technometrics, Vol. 27, No. 3, 1985, pp. 285-292. doi:10.2307/1269710

[3] C. A. Lowry, W. H. Woodall, C. W. Champ and S. E. Rigdon, “A Multivariate Exponentially Weighted Moving Average Control Chart,” Technometrics, Vol. 34, No. 1, 1992, p. 46. doi:10.2307/1269551

[4] G. C. Runger, J. B. Keats, D. C. Montgomery and R. D. Scranton, “Improving The Performance Of The Multivariate Exponentially Weighted Moving Average Control Chart,” Quality and Reliability International, Vol. 15, No. 3, 1996, pp. 161-166. doi:10.1002/(SICI)1099-1638(199905/06)15:3<161::AID-QRE215>3.0.CO;2-V

[5] C. M. Mastrangelo, G. C. Runger, and D.C. Montgomery, “Statistical Process Monitoring with Principal Components,” Quality and Reliability International, Vol. 12, No. 3, 1996, pp. 203-210. doi:10.1002/(SICI)1099-1638(199605)12:3<203::AID-QRE12>3.0.CO;2-B

[6] A. Sepulveda, “The Minimax Control Chart for Multivariate Quality Control,” Dissertation, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, 1996.

[7] N. M. Patrikalakis and T. Maekawa, “Shape Interrogation for Computer Aided Design and Manufacturing,” Springer-Verlag, Heidelberg, 2002.

[8] E.C. Sherbrooke and N. M. Patrikalakis, “Computation of the Solutions of Nonlinear Polynomial Systems,” Computer Aided Geometric Design, Vol. 10, No. 5, 1993, pp. 379-405. doi:10.1016/0167-8396(93)90019-Y

[9] S. M. Rump, “Ten Methods To Bound Multiple Roots of Polynomials,” Journal of Computational and Applied Mathematics, Vol. 156, No. 2, 2003, pp. 403-432. doi:10.1016/S0377-0427(03)00381-9

[10] .M. McNamee, “A Bibliography On Roots of Polynomials,” Journal of Computational and Applied Mathematics, Vol. 47, No. 3, 1993, pp. 391-394. doi:10.1016/0377-0427(93)90064-I

[11] H. S. Wilf, “A Global Bisection Algorithm for Computing the Zeros of Polynomials in the Complex Plane,” Journal of the Association for Computing Machinery, Vol. 25, No. 3, 1978, pp. 415-420. doi:10.1145/322077.322084

[12] A. C. David, L. John and O. Donal, “Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra,” Springer, 2nd Edition, Berlin, 1996.

[13] J. Rehmert, “A Performance Analysis of the Minimax Multivariate Quality Control Chart,” M.Sc. Dissertation, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, 1997.

[14] A. J. Hayter and K. L. Tsui, “Identification and Quantification in Multivariate Quality Control Problems,” Journal of Quality Technology, Vol. 26, 1994, pp. 197-208.

[15] N. H. Timm, “Multivariate Quality Control Using Finite Intersection Tests,” Journal of Quality Technology, Vol. 28, 1996, pp. 233-243.

[1] H. Hotelling, “Multivariable Quality Control—Illustrated By The Air Testing Of Sample Bombsights,” In: C. Eisenhart, M. W. Hastay and W. A. Wallis, Eds., Techniques of Statistical Analysis, McGraw Hill, New York, 1947, pp. 111-184.

[2] W. H. Woodall and M. M. Ncube, “Multivariate Cusum Quality Control Procedures,” Technometrics, Vol. 27, No. 3, 1985, pp. 285-292. doi:10.2307/1269710

[3] C. A. Lowry, W. H. Woodall, C. W. Champ and S. E. Rigdon, “A Multivariate Exponentially Weighted Moving Average Control Chart,” Technometrics, Vol. 34, No. 1, 1992, p. 46. doi:10.2307/1269551

[4] G. C. Runger, J. B. Keats, D. C. Montgomery and R. D. Scranton, “Improving The Performance Of The Multivariate Exponentially Weighted Moving Average Control Chart,” Quality and Reliability International, Vol. 15, No. 3, 1996, pp. 161-166. doi:10.1002/(SICI)1099-1638(199905/06)15:3<161::AID-QRE215>3.0.CO;2-V

[5] C. M. Mastrangelo, G. C. Runger, and D.C. Montgomery, “Statistical Process Monitoring with Principal Components,” Quality and Reliability International, Vol. 12, No. 3, 1996, pp. 203-210. doi:10.1002/(SICI)1099-1638(199605)12:3<203::AID-QRE12>3.0.CO;2-B

[6] A. Sepulveda, “The Minimax Control Chart for Multivariate Quality Control,” Dissertation, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, 1996.

[7] N. M. Patrikalakis and T. Maekawa, “Shape Interrogation for Computer Aided Design and Manufacturing,” Springer-Verlag, Heidelberg, 2002.

[8] E.C. Sherbrooke and N. M. Patrikalakis, “Computation of the Solutions of Nonlinear Polynomial Systems,” Computer Aided Geometric Design, Vol. 10, No. 5, 1993, pp. 379-405. doi:10.1016/0167-8396(93)90019-Y

[9] S. M. Rump, “Ten Methods To Bound Multiple Roots of Polynomials,” Journal of Computational and Applied Mathematics, Vol. 156, No. 2, 2003, pp. 403-432. doi:10.1016/S0377-0427(03)00381-9

[10] .M. McNamee, “A Bibliography On Roots of Polynomials,” Journal of Computational and Applied Mathematics, Vol. 47, No. 3, 1993, pp. 391-394. doi:10.1016/0377-0427(93)90064-I

[11] H. S. Wilf, “A Global Bisection Algorithm for Computing the Zeros of Polynomials in the Complex Plane,” Journal of the Association for Computing Machinery, Vol. 25, No. 3, 1978, pp. 415-420. doi:10.1145/322077.322084

[12] A. C. David, L. John and O. Donal, “Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra,” Springer, 2nd Edition, Berlin, 1996.

[13] J. Rehmert, “A Performance Analysis of the Minimax Multivariate Quality Control Chart,” M.Sc. Dissertation, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, 1997.

[14] A. J. Hayter and K. L. Tsui, “Identification and Quantification in Multivariate Quality Control Problems,” Journal of Quality Technology, Vol. 26, 1994, pp. 197-208.

[15] N. H. Timm, “Multivariate Quality Control Using Finite Intersection Tests,” Journal of Quality Technology, Vol. 28, 1996, pp. 233-243.