Published

2016-07-01

A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework

Un estudio de cartas de control de cantidades acumuladas por mixturas de modelos Rayleigh bajo un enfoque Bayesiano

DOI:

https://doi.org/10.15446/rce.v39n2.58915

Keywords:

Quality control, Inverse Transformation Method, Loss Functions and Bayes Estimators, MRCQC-Chart, SRCQC-Char (en)
Control de calidad, Método de la transformación, Función de pérdida, Estimador bayesiano, Carta MRCQC, Carta SRCQC (es)

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Authors

  • Muhammad Aslam Department of Basic Sciences, Riphah International University, Islamabad, Pakistan
  • Muhammad Riaz Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals, Dhahran, Saudi Arabia
  • Tabassum Naz Sindhu Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan
  • Zaheer Ahmed Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, Pakistan

This study deals with the cumulative charting technique based on a simple and a mixture of Rayleigh models. The respective charting schemes are referred as the SRCQC-chart and the MRCQC-chart. These are stimulated from existing statistical control charts in this direction i.e. the cumulative quantity control (CQC) chart, based on exponential and Weibull models, and the cumulative count control (CCC) chart, based on the simple geometricmodel. Another motivation for this study is the mixture cumulative count control (MCCC) chart based on the two component geometric model. The use of mixture cumulative quantity is an attractive approach for process monitoring. The design structure of the proposed control chart is derived by using the cumulative distribution function of simple, and two components of mixture distribution(s). We observed that the proposed charting structure 
is efficient in detecting the changes in process parameters. The application of the proposed scheme is illustrated using a real dataset.

Este estudio trata con cartas de control acumuladas basadas en distribuciones Rayleigh y en mixturas de estas mismas. Las cartas se denominan SRCQC y MRCQC, respectivamente. Estas se fundamentan en cartas existentes como la carta de control de cantidades acumuladas (CQC), basada en modelos exponencial y Weibull en la carta de control de conteos acumulados (CCC), soportada en un modelo geométrico. Otra propuesta del estudio es la carta de control de mixtura de conteos acumulados (MCCC). Esta última es muy atractiva en procesos de monitoreo. La estructura de diseño de las cartas propuestas se deriva usando la función de distribución acumulada simple y la mixtura de dos distribuciones acumuladas. Se observa que las cartas propuestas son eficientes para detectar cambios en los parámetros del proceso. La aplicación del esquema propuesto es ilustrada usando un conjunto
de datos reales.

https://doi.org/10.15446/rce.v39n2.58915

A Study of Cumulative Quantity Control Chart for a Mixture of RayleighModel under a Bayesian Framework

Un estudio de cartas de control de cantidades acumuladas por mixturas de modelos Rayleigh bajo un enfoque Bayesiano

TABASSUM NAZ SINDHU1, MUHAMMAD RIAZ2, MUHAMMAD ASLAM3, ZAHEER AHMED4

1Quaid-i-Azam University, Department of Statistics, Islamabad, Pakistan. Professor. Email: sindhuqau@gmail.com
2King Fahad University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran, Saudi Arabia. Professor. Email: riaz76qau@yahoo.com
3Riphah International University, Department of Basic Sciences, Islamabad, Pakistan. Professor. Email: aslamsdqu@yahoo.com
4Institute of Space Technology, Department of Applied Mathematics and Statistics, Islamabad, Pakistan. Professor. Email: zaheerqau77@gmail.com

Abstract

This study deals with the cumulative charting technique based on a simple and a mixture of Rayleigh models. The respective charting schemes are referred as the SRCQC-chart and the MRCQC-chart. These are stimulated from existing statistical control charts in this direction i.e. the cumulative quantity control (CQC) chart, based on exponential and Weibull models, and the cumulative count control (CCC) chart, based on the simple geometric model. Another motivation for this study is the mixture cumulative count control (MCCC) chart based on the two component geometric model. The use of mixture cumulative quantity is an attractive approach for process monitoring. The design structure of the proposed control chart is derived by using the cumulative distribution function of simple, and two components of mixture distribution(s). We observed that the proposed charting structure is efficient in detecting the changes in process parameters. The application of the proposed scheme is illustrated using a real dataset.

Key words: Quality control, Inverse transformation method, Lossfunctions and bayes estimators, MRCQC-Chart, SRCQC-Char.

Resumen

Este estudio trata con cartas de control acumuladas basadas en distribuciones Rayleigh y en mixturas de estas mismas. Las cartas se denominan SRCQC y MRCQC, respectivamente. Estas se fundamentan en cartas existentes como la carta de control de cantidades acumuladas (CQC), basada en modelos exponencial y Weibull en la carta de control de conteos acumulados (CCC), soportada en un modelo geométrico. Otra propuesta del estudio es la carta de control de mixtura de conteos acumulados (MCCC). Esta última es muy atractiva en procesos de monitoreo. La estructura de diseño de las cartas propuestas se deriva usando la función de distribución acumulada simple y la mixtura de dos distribuciones acumuladas. Se observa que las cartas propuestas son eficientes para detectar cambios en los parámetros del proceso. La aplicación del esquema propuesto es ilustrada usando un conjunto de datos reales.

Palabras clave: control de calidad, método de la transformación, función de pérdida, estimador bayesiano, carta MRCQC, carta SRCQC.

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References

1. Aslam, M. (2003), 'An application of prior predictive distribution to elicit the prior density', Journal of Statistical Theory and Applications 2(1), 70-83.

2. Banjevic, D., Jardine, A. K. S., Makis, V. & Ennis, M. (2001), 'A control-limit policy and software for condition-based maintenance', INFOR 39, 32-50.

3. Calvin, T. (1983), 'Quality control techniques for zero defects', IEEE Transactions on Components Hybrids and Manufacturing Technology 6(3), 323-328.

4. Chan, L. Y., Lin, D. K. J., Xie, M. & Goh, T. N. (2002), 'Cumulative probability control charts for geometric and exponential process characteristics', International Journal of Production Research 40(1), 133-150.

5. Chan, L. Y., Xie, M. & Goh, T. N. (2000), 'Cumulative quantity control charts for monitoring production processes', International Journal of Production Research 38(2), 397-408.

6. Degroot, M. H. (1970), Optimal statistical decision, McGraw-Hill Inc..

7. Grimshaw, S. D., Collings, B. J., Larsen, W. A. & Hurt, C. R. (2001), 'Eliciting factor importance in a designed experiment', Technometrics 43(2), 133-146.

8. Jenkinson, D. J. (2005), The elicitation of probabilities a review of the statistical literature, BEEP working paper , University of Sheffield, United Kingdom.

9. Leon, J. C., Vazquez-Polo, J. F. & Gonzalez, L. (2003), 'Environmental and resource economics', Kluwer Academic Publishers 26, 199-210.

10. Majeed, M. Y., Aslam, M. & Riaz, M. (2012), 'Mixture cumulative count control chart for mixture geometric process characteristics', Quality & Quantity 26, 1-19.

11. Mendenhall, W. & Hadar, R. J. (1958), 'Estimation of parameters of mixed exponentially distributed failure time distribution from censored life test data', Biometrika 45(3-4), 504-520.

12. Montgomery, D. C. (2009), Introduction to Statistical Quality Control, 6 edn, John Wiley & Sons, New York.

13. Norstrom, J. G. (1996), 'The use of precautionary loss functions in risk analysis', IEEE Transactions on Reliability 45(3), 400-403.

14. O'Hagan, A., Buck, C. E., Daneshkhah, A., Eiser, J. E., Garthwaite, P. H., Jenkinson, D. J., Oakley, J. E. & Rakow, T. (2006), Uncertain Judgements: Eliciting expert probabilities, John Wiley & Sons, New York.

15. Quesenberry, C. P. (2007), SPC Methods for Quality Improvement, John Wiley & Sons, Toronto.

16. Sun, F. B., Yang, J., del Rosario, R. & Murphy, R. (2001), A conditional-reliability control-chart for the post-production extended reliability-test, 'Proceedings Annual Reliability and Maintainability Symposium', p. 64-9.

17. Xie, M. & Goh, T. N. (1992), 'Some procedures for decision making in controlling high yield processes', Quality and Reliability Engineering International 8(4), 355-360.

18. Xie, M., Goh, T. N. & Kuralmani, V. (2000), 'On optimal setting of control limits for geometric chart', International Journal of Reliability Quality and Safety Engineering 7(1), 17-25.

19. Xie, M., Goh, T. N. & Ranjan, P. (2002), 'Some effective control chart procedures for reliability monitoring', Reliability Engineering and System Safety 19(77), 143-150.

20. Xie, M., Goh, T. N. & Tang, X. Y. (2000), 'Data transformation for geometrically distributed quality characteristics', Quality and Reliability Engineering International 16(1), 9-15.

[Recibido en noviembre de 2014. Aceptado en julio de 2015]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv39n2a03,
    AUTHOR  = {Sindhu, Tabassum Naz and Riaz, Muhammad and Aslam, Muhammad and Ahmed, Zaheer},
    TITLE   = {{A Study of Cumulative Quantity Control Chart for a Mixture of RayleighModel under a Bayesian Framework}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2016},
    volume  = {39},
    number  = {2},
    pages   = {185-205}
}

References

Aslam, M. (2003), ‘An application of prior predictive distribution to elicit the prior density’, Journal of Statistical Theory and Applications 2(1), 70–83.

Banjevic, D., Jardine, A. K. S., Makis, V. & Ennis, M. (2001), ‘A control-limit policy and software for condition-based maintenance’, INFOR 39, 32–50.

Calvin, T. (1983), ‘Quality control techniques for zero defects’, IEEE Transactions on Components Hybrids and Manufacturing Technology 6(3), 323–328.

Chan, L. Y., Lin, D. K. J., Xie, M. & Goh, T. N. (2002), ‘Cumulative probability control charts for geometric and exponential process characteristics’, International Journal of Production Research 40(1), 133–150.

Chan, L. Y., Xie, M. & Goh, T. N. (2000), ‘Cumulative quantity control charts for monitoring production processes’, International Journal of Production Research 38(2), 397–408.

Degroot, M. H. (1970), Optimal statistical decision, McGraw-Hill Inc.

Grimshaw, S. D., Collings, B. J., Larsen, W. A. & Hurt, C. R. (2001), ‘Eliciting factor importance in a designed experiment’, Technometrics 43(2), 133–146.

Jenkinson, D. J. (2005), The elicitation of probabilities a review of the statistical literature, Beep working paper, University of Sheffield, United Kingdom.

Leon, J. C., Vazquez-Polo, J. F. & Gonzalez, L. (2003), ‘Environmental and resource economics’, Kluwer Academic Publishers 26, 199–210.

Majeed, M. Y., Aslam, M. & Riaz, M. (2012), ‘Mixture cumulative count control chart for mixture geometric process characteristics’, Quality & Quantity 26, 1–19.

Mendenhall, W. & Hadar, R. J. (1958), ‘Estimation of parameters of mixed exponentially distributed failure time distribution from censored life test data’, Biometrika 45(3-4), 504–520.

Montgomery, D. C. (2009), Introduction to Statistical Quality Control, 6 edn, John Wiley & Sons, New York.

Norstrom, J. G. (1996), ‘The use of precautionary loss functions in risk analysis’, IEEE Transactions on Reliability 45(3), 400–403.

O’Hagan, A., Buck, C. E., Daneshkhah, A., Eiser, J. E., Garthwaite, P. H., Jenkinson, D. J., Oakley, J. E. & Rakow, T. (2006), Uncertain Judgements: Eliciting expert probabilities, John Wiley & Sons, New York.

Quesenberry, C. P. (2007), SPC Methods for Quality Improvement, John Wiley & Sons, Toronto.

Sun, F. B., Yang, J., del Rosario, R. & Murphy, R. (2001), A conditional-reliability control-chart for the post-production extended reliability-test, in ‘Proceedings Annual Reliability and Maintainability Symposium’, pp. 64–9.

Xie, M. & Goh, T. N. (1992), ‘Some procedures for decision making in controlling high yield processes’, Quality and Reliability Engineering International 8(4), 355–360.

Xie, M., Goh, T. N. & Kuralmani, V. (2000), ‘On optimal setting of control limits for geometric chart’, International Journal of Reliability Quality and Safety Engineering 7(1), 17–25.

Xie, M., Goh, T. N. & Ranjan, P. (2002), ‘Some effective control chart procedures for reliability monitoring’, Reliability Engineering and System Safety 19(77), 143–150.

Xie, M., Goh, T. N. & Tang, X. Y. (2000), ‘Data transformation for geometrically distributed quality characteristics’, Quality and Reliability Engineering International 16(1), 9–15.

How to Cite

APA

Aslam, M., Riaz, M., Sindhu, T. N. and Ahmed, Z. (2016). A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework. Revista Colombiana de Estadística, 39(2), 185–204. https://doi.org/10.15446/rce.v39n2.58915

ACM

[1]
Aslam, M., Riaz, M., Sindhu, T.N. and Ahmed, Z. 2016. A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework. Revista Colombiana de Estadística. 39, 2 (Jul. 2016), 185–204. DOI:https://doi.org/10.15446/rce.v39n2.58915.

ACS

(1)
Aslam, M.; Riaz, M.; Sindhu, T. N.; Ahmed, Z. A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework. Rev. colomb. estad. 2016, 39, 185-204.

ABNT

ASLAM, M.; RIAZ, M.; SINDHU, T. N.; AHMED, Z. A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework. Revista Colombiana de Estadística, [S. l.], v. 39, n. 2, p. 185–204, 2016. DOI: 10.15446/rce.v39n2.58915. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/58915. Acesso em: 19 apr. 2024.

Chicago

Aslam, Muhammad, Muhammad Riaz, Tabassum Naz Sindhu, and Zaheer Ahmed. 2016. “A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework”. Revista Colombiana De Estadística 39 (2):185-204. https://doi.org/10.15446/rce.v39n2.58915.

Harvard

Aslam, M., Riaz, M., Sindhu, T. N. and Ahmed, Z. (2016) “A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework”, Revista Colombiana de Estadística, 39(2), pp. 185–204. doi: 10.15446/rce.v39n2.58915.

IEEE

[1]
M. Aslam, M. Riaz, T. N. Sindhu, and Z. Ahmed, “A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework”, Rev. colomb. estad., vol. 39, no. 2, pp. 185–204, Jul. 2016.

MLA

Aslam, M., M. Riaz, T. N. Sindhu, and Z. Ahmed. “A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework”. Revista Colombiana de Estadística, vol. 39, no. 2, July 2016, pp. 185-04, doi:10.15446/rce.v39n2.58915.

Turabian

Aslam, Muhammad, Muhammad Riaz, Tabassum Naz Sindhu, and Zaheer Ahmed. “A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework”. Revista Colombiana de Estadística 39, no. 2 (July 1, 2016): 185–204. Accessed April 19, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/58915.

Vancouver

1.
Aslam M, Riaz M, Sindhu TN, Ahmed Z. A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework. Rev. colomb. estad. [Internet]. 2016 Jul. 1 [cited 2024 Apr. 19];39(2):185-204. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/58915

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CrossRef citations6

1. Anum Shafiq, Andaç Batur Çolak, Chetan Swarup, Tabassum Naz Sindhu, Showkat Ahmad Lone. (2022). Reliability Analysis Based on Mixture of Lindley Distributions with Artificial Neural Network. Advanced Theory and Simulations, 5(8) https://doi.org/10.1002/adts.202200100.

2. Tabassum Naz Sindhu, Zawar Hussain, Muhammad Aslam. (2019). Parameter and reliability estimation of inverted Maxwell mixture model. Journal of Statistics and Management Systems, 22(3), p.459. https://doi.org/10.1080/09720510.2018.1552412.

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4. Ya Sun, Ying-Ying Zhang, Ji Sun. (2022). The empirical Bayes estimators of the parameter of the uniform distribution with an inverse gamma prior under Stein’s loss function. Communications in Statistics - Simulation and Computation, , p.1. https://doi.org/10.1080/03610918.2022.2093904.

5. Anum Shafiq, Tabassum Naz Sindhu, Naif Alotaibi. (2022). A novel extended model with versatile shaped failure rate: Statistical inference with Covid -19 applications. Results in Physics, 36, p.105398. https://doi.org/10.1016/j.rinp.2022.105398.

6. Anum Shafiq, S.A. Lone, Tabassum Naz Sindhu, Youssef El Khatib, Qasem M. Al-Mdallal, Taseer Muhammad. (2021). A new modified Kies Fréchet distribution: Applications of mortality rate of Covid-19. Results in Physics, 28, p.104638. https://doi.org/10.1016/j.rinp.2021.104638.

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