Research Article
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Year 2025, Volume: 54 Issue: 4, 1588 - 1621, 29.08.2025

Rashid Mehmood , Muhammad Naveed Khan , Iftikhar Ali , Tajammal Imran , Babar Zaman , Kassimu Mpungu , Fawwad Qureshi , Maysaa Elmahi Abdelwahab

https://doi.org/10.15672/hujms.1541132

Abstract

References

  • [1] W. A. Shewhart, Economic control of quality of manufactured product, ASQ Qual. Press, 1931.
  • [2] H. Hotelling, Multivariate 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, pp. 113–184, 1947.
  • [3] J. Park and C.-H. Jun, A new multivariate EWMA control chart via multiple testing, J. Process Control 26, 51–55, 2015.
  • [4] D. S. Moore, G. P. McCabe, L. C. Alwan, B. A. Craig, and W. M. Duckworth, The practice of statistics for business and economics, W. H. Freeman, 2016.
  • [5] Y. Zhao, X. He, M. G. Pecht, J. Zhang, and D. Zhou, Detection and detectability of intermittent faults based on moving average $T^2$ control charts with multiple window lengths, J. Process Control 92, 296–309, 2020.
  • [6] B. N. de Oliveira, M. Valk, and D. Marcondes Filho, Fault detection and diagnosis of batch process dynamics using ARMA-based control charts, J. Process Control 111, 46–58, 2022.
  • [7] E. C. Western, Statistical quality control handbook, Western Electr. Co., Indianapolis, 1956.
  • [8] C. W. Champ and W. H. Woodall, Exact results for Shewhart control charts with supplementary runs rules, Technometrics 29(4), 393–399, 1987.
  • [9] M. Klein, Two alternatives to the Shewhart X control chart, J. Qual. Technol. 32(4), 427–431, 2000.
  • [10] M. B. Khoo, Design of runs rules schemes, Qual. Eng. 16(1), 27–43, 2003.
  • [11] O. A. Adeoti and J.-C. Malela-Majika, Double exponentially weighted moving average control chart with supplementary runs-rules, Qual. Technol. Quant. Manage., 1–24, 2019.
  • [12] S. Shongwe and J.-C. Malela-Majika, Shewhart-type monitoring schemes with supplementary w-of-w runs-rules to monitor the mean of autocorrelated samples, Commun. Stat. Simul. Comput., 1–30, 2019.
  • [13] J. Oh and C. H. Weiß, On the individuals chart with supplementary runs rules under serial dependence, Methodol. Comput. Appl. Probab., 1–17, 2020.
  • [14] C. Chong and M. Lee, The bivariate generalized variance $|S|$ control chart with runs rules, in: Proc. IEEE Int. Conf. Ind. Eng. Eng. Manage., 1448–1452, 2013.
  • [15] M. Riaz, R. Mehmood, and R. J. Does, On the performance of different control charting rules, Qual. Reliab. Eng. Int. 27(8), 1059–1067, 2011.
  • [16] D. C. Montgomery, Introduction to statistical quality control, John Wiley & Sons, New York, 2009.
  • [17] R. Mehmood, M. Riaz, M. H. Lee, I. Ali, and M. Gharib, Exact computational methods for univariate and multivariate control charts under runs rules, Comput. Ind. Eng. 163, 107821, 2022.
  • [18] R. Mehmood, M. S. Qazi, and M. Riaz, On the performance of X-bar control chart for known and unknown parameters supplemented with runs rules under different probability distributions, J. Stat. Comput. Simul. 88(4), 675–711, 2018.
  • [19] R. Mehmood, M. Riaz, and R. J. Does, Efficient power computation for r out of m runs rules schemes, Comput. Stat. 28(2), 667–681, 2013.
  • [20] R. Mehmood, M. Riaz, and R. J. M. M. Does, Quality quandaries: on the application of different ranked set sampling schemes, Qual. Eng. 26(3), 370–378, 2014.
  • [21] M. Riaz, R. Mehmood, N. Abbas, and S. A. Abbasi, On effective dual use of auxiliary information in variability control charts, Qual. Reliab. Eng. Int. 32(4), 1417–1443.
  • [22] M. Riaz, R. Mehmood, M. R. Iqbal, and S. A. Abbasi, On efficient skewness correction charts under contamination and non-normality, Qual. Reliab. Eng. Int. 32(3), 837–854.
  • [23] R. Mehmood, M. H. Lee, S. Hussain, and M. Riaz, On efficient construction and evaluation of runs rules-based control chart for known and unknown parameters under different distributions, Qual. Reliab. Eng. Int. 35(2), 582–599, 2019.
  • [24] R. Mehmood, M. Riaz, I. Ali, and M. H. Lee, Generalized Hotelling $T^2$ control chart based on bivariate ranked set techniques with runs rules, Trans. Inst. Meas. Control 43(10), 2180–2195, 2021.
  • [25] R. Mehmood, M. H. Lee, A. Iftikhar, and R. Muhammad, Comparative analysis between FAR and ARL-based control charts with runs rules, Hacet. J. Math. Stat., 1–14, 2021.
  • [26] R. Mehmood, M. H. Lee, M. Riaz, B. Zaman, and I. Ali, Hotelling $T^2$ control chart based on bivariate ranked set schemes, Commun. Stat. Simul. Comput. 0(0), 1–28, 2019.
  • [27] S. Hussain, L. Song, R. Mehmood, and M. Riaz, New dual auxiliary information-based EWMA control chart with an application in physicochemical parameters of ground water, Iran. J. Sci. Technol. Trans. A: Sci., 1–20, 2018.
  • [28] Y. Ou, Z. Wu, and F. Tsung, A comparison study of effectiveness and robustness of control charts for monitoring process mean, Int. J. Prod. Econ. 135(1), 479–490, 2012.
  • [29] T. Nawaz, M. A. Raza, and D. Han, A new approach to design efficient univariate control charts to monitor the process mean, Qual. Reliab. Eng. Int. 34(8), 1732–1751, 2018.
  • [30] A. Tang, P. Castagliola, J. Sun, and X. Hu, Optimal design of the adaptive EWMA chart for the mean based on median run length and expected median run length, Qual. Technol. Quant. Manage., 1–20, 2018.
  • [31] Z. Wu, M. Yang, W. Jiang, and M. B. Khoo, Optimization designs of the combined Shewhart-CUSUM control charts, Comput. Stat. Data Anal. 53(2), 496–506, 2008.
  • [32] J. J. Pignatiello Jr. and G. C. Runger, Comparisons of multivariate CUSUM charts, J. Qual. Technol. 22(3), 173–186, 1990.
  • [33] F. Aparisi, C. W. Champ, and J. C. García-Díaz, A performance analysis of Hotelling’s $\chi^2$ control chart with supplementary runs rules, Qual. Eng. 16(3), 359–368, 2004.
  • [34] A. C. Rakitzis and D. L. Antzoulakos, Control charts with switching and sensitizing runs rules for monitoring process variation, J. Stat. Comput. Simul. 84(1), 37–56, 2014.
  • [35] R. Mehmood, M. H. Lee, I. Ali, M. Riaz, and S. Hussain, Multivariate cumulative sum control chart and measure of process capability based on bivariate ranked set schemes, Comput. Ind. Eng. 150, 106891, 2020.
  • [36] E. Santos-Fernández, Multivariate statistical quality control using R, Springer, 2012.
  • [37] R. Mehmood, K. Mpungu, I. Ali, B. Zaman, F. H. Qureshi, and N. Khan, A new approach for designing the Shewhart-type control charts with generalized sensitizing rules, Comput. Ind. Eng. 182(1), 109389, 2023.
  • [38] A. N. Philippou, C. Georghiou, and G. N. Philippou, A generalized geometric distribution and some of its properties, Stat. Probab. Lett. 1(4), 171–175, 1983.
  • [39] F. B. Oppong and S. Y. Agbedra, Assessing univariate and multivariate normality, a guide for non-statisticians, Math. Theory Model. 6(2), 26–33, 2016.
  • [40] E. Santos-Fernández, Multivariate statistical quality control using R, Springer, 2012.

Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules

Year 2025, Volume: 54 Issue: 4, 1588 - 1621, 29.08.2025

Rashid Mehmood , Muhammad Naveed Khan , Iftikhar Ali , Tajammal Imran , Babar Zaman , Kassimu Mpungu , Fawwad Qureshi , Maysaa Elmahi Abdelwahab

https://doi.org/10.15672/hujms.1541132

Abstract

In this study, efficient techniques are utilized to design the multivariate Hotelling control chart with sensitizing rules for detecting small-to-moderate variations. The control limit of the proposed chart is derived relative to probability of a single point and number of process characteristics. To calculate probability of a single point for sustained in-control average run length, a generalized single polynomial equation is derived. For evaluation, performance measures are considered based on the average, the median, and the percentile run length. These measures are calculated using Monte Carlo simulation and numerical integration. The results indicate that the proposed control chart has consistent behavior when a process is in control. The in-control average run length is obtained equal to prefixed level which remains valid for all choices of sensitizing rules. This implies that the proposed control chart can resolve the issue of existing control chart in terms of sustained behavior. The effectiveness of sensitizing rules is dependent on process characteristics and variations of mean vector. A comparative analysis of different choices of sensitizing rules is conducted to locate optimal choices of process characteristics. Real-life example, dowel-pin manufacturing, shows that proposed control chart with sensitizing rules is efficient for diagnosing small variations.

Keywords

Average run length Hotelling median run length percentile run length performance measures sensitizing rules single polynomial approach

Supporting Institution

Dr. Maysaa recieved funding through supporting project number (PNURSP2025R913),Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

References

  • [1] W. A. Shewhart, Economic control of quality of manufactured product, ASQ Qual. Press, 1931.
  • [2] H. Hotelling, Multivariate 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, pp. 113–184, 1947.
  • [3] J. Park and C.-H. Jun, A new multivariate EWMA control chart via multiple testing, J. Process Control 26, 51–55, 2015.
  • [4] D. S. Moore, G. P. McCabe, L. C. Alwan, B. A. Craig, and W. M. Duckworth, The practice of statistics for business and economics, W. H. Freeman, 2016.
  • [5] Y. Zhao, X. He, M. G. Pecht, J. Zhang, and D. Zhou, Detection and detectability of intermittent faults based on moving average $T^2$ control charts with multiple window lengths, J. Process Control 92, 296–309, 2020.
  • [6] B. N. de Oliveira, M. Valk, and D. Marcondes Filho, Fault detection and diagnosis of batch process dynamics using ARMA-based control charts, J. Process Control 111, 46–58, 2022.
  • [7] E. C. Western, Statistical quality control handbook, Western Electr. Co., Indianapolis, 1956.
  • [8] C. W. Champ and W. H. Woodall, Exact results for Shewhart control charts with supplementary runs rules, Technometrics 29(4), 393–399, 1987.
  • [9] M. Klein, Two alternatives to the Shewhart X control chart, J. Qual. Technol. 32(4), 427–431, 2000.
  • [10] M. B. Khoo, Design of runs rules schemes, Qual. Eng. 16(1), 27–43, 2003.
  • [11] O. A. Adeoti and J.-C. Malela-Majika, Double exponentially weighted moving average control chart with supplementary runs-rules, Qual. Technol. Quant. Manage., 1–24, 2019.
  • [12] S. Shongwe and J.-C. Malela-Majika, Shewhart-type monitoring schemes with supplementary w-of-w runs-rules to monitor the mean of autocorrelated samples, Commun. Stat. Simul. Comput., 1–30, 2019.
  • [13] J. Oh and C. H. Weiß, On the individuals chart with supplementary runs rules under serial dependence, Methodol. Comput. Appl. Probab., 1–17, 2020.
  • [14] C. Chong and M. Lee, The bivariate generalized variance $|S|$ control chart with runs rules, in: Proc. IEEE Int. Conf. Ind. Eng. Eng. Manage., 1448–1452, 2013.
  • [15] M. Riaz, R. Mehmood, and R. J. Does, On the performance of different control charting rules, Qual. Reliab. Eng. Int. 27(8), 1059–1067, 2011.
  • [16] D. C. Montgomery, Introduction to statistical quality control, John Wiley & Sons, New York, 2009.
  • [17] R. Mehmood, M. Riaz, M. H. Lee, I. Ali, and M. Gharib, Exact computational methods for univariate and multivariate control charts under runs rules, Comput. Ind. Eng. 163, 107821, 2022.
  • [18] R. Mehmood, M. S. Qazi, and M. Riaz, On the performance of X-bar control chart for known and unknown parameters supplemented with runs rules under different probability distributions, J. Stat. Comput. Simul. 88(4), 675–711, 2018.
  • [19] R. Mehmood, M. Riaz, and R. J. Does, Efficient power computation for r out of m runs rules schemes, Comput. Stat. 28(2), 667–681, 2013.
  • [20] R. Mehmood, M. Riaz, and R. J. M. M. Does, Quality quandaries: on the application of different ranked set sampling schemes, Qual. Eng. 26(3), 370–378, 2014.
  • [21] M. Riaz, R. Mehmood, N. Abbas, and S. A. Abbasi, On effective dual use of auxiliary information in variability control charts, Qual. Reliab. Eng. Int. 32(4), 1417–1443.
  • [22] M. Riaz, R. Mehmood, M. R. Iqbal, and S. A. Abbasi, On efficient skewness correction charts under contamination and non-normality, Qual. Reliab. Eng. Int. 32(3), 837–854.
  • [23] R. Mehmood, M. H. Lee, S. Hussain, and M. Riaz, On efficient construction and evaluation of runs rules-based control chart for known and unknown parameters under different distributions, Qual. Reliab. Eng. Int. 35(2), 582–599, 2019.
  • [24] R. Mehmood, M. Riaz, I. Ali, and M. H. Lee, Generalized Hotelling $T^2$ control chart based on bivariate ranked set techniques with runs rules, Trans. Inst. Meas. Control 43(10), 2180–2195, 2021.
  • [25] R. Mehmood, M. H. Lee, A. Iftikhar, and R. Muhammad, Comparative analysis between FAR and ARL-based control charts with runs rules, Hacet. J. Math. Stat., 1–14, 2021.
  • [26] R. Mehmood, M. H. Lee, M. Riaz, B. Zaman, and I. Ali, Hotelling $T^2$ control chart based on bivariate ranked set schemes, Commun. Stat. Simul. Comput. 0(0), 1–28, 2019.
  • [27] S. Hussain, L. Song, R. Mehmood, and M. Riaz, New dual auxiliary information-based EWMA control chart with an application in physicochemical parameters of ground water, Iran. J. Sci. Technol. Trans. A: Sci., 1–20, 2018.
  • [28] Y. Ou, Z. Wu, and F. Tsung, A comparison study of effectiveness and robustness of control charts for monitoring process mean, Int. J. Prod. Econ. 135(1), 479–490, 2012.
  • [29] T. Nawaz, M. A. Raza, and D. Han, A new approach to design efficient univariate control charts to monitor the process mean, Qual. Reliab. Eng. Int. 34(8), 1732–1751, 2018.
  • [30] A. Tang, P. Castagliola, J. Sun, and X. Hu, Optimal design of the adaptive EWMA chart for the mean based on median run length and expected median run length, Qual. Technol. Quant. Manage., 1–20, 2018.
  • [31] Z. Wu, M. Yang, W. Jiang, and M. B. Khoo, Optimization designs of the combined Shewhart-CUSUM control charts, Comput. Stat. Data Anal. 53(2), 496–506, 2008.
  • [32] J. J. Pignatiello Jr. and G. C. Runger, Comparisons of multivariate CUSUM charts, J. Qual. Technol. 22(3), 173–186, 1990.
  • [33] F. Aparisi, C. W. Champ, and J. C. García-Díaz, A performance analysis of Hotelling’s $\chi^2$ control chart with supplementary runs rules, Qual. Eng. 16(3), 359–368, 2004.
  • [34] A. C. Rakitzis and D. L. Antzoulakos, Control charts with switching and sensitizing runs rules for monitoring process variation, J. Stat. Comput. Simul. 84(1), 37–56, 2014.
  • [35] R. Mehmood, M. H. Lee, I. Ali, M. Riaz, and S. Hussain, Multivariate cumulative sum control chart and measure of process capability based on bivariate ranked set schemes, Comput. Ind. Eng. 150, 106891, 2020.
  • [36] E. Santos-Fernández, Multivariate statistical quality control using R, Springer, 2012.
  • [37] R. Mehmood, K. Mpungu, I. Ali, B. Zaman, F. H. Qureshi, and N. Khan, A new approach for designing the Shewhart-type control charts with generalized sensitizing rules, Comput. Ind. Eng. 182(1), 109389, 2023.
  • [38] A. N. Philippou, C. Georghiou, and G. N. Philippou, A generalized geometric distribution and some of its properties, Stat. Probab. Lett. 1(4), 171–175, 1983.
  • [39] F. B. Oppong and S. Y. Agbedra, Assessing univariate and multivariate normality, a guide for non-statisticians, Math. Theory Model. 6(2), 26–33, 2016.
  • [40] E. Santos-Fernández, Multivariate statistical quality control using R, Springer, 2012.
There are 40 citations in total.

Details

Primary Language English
Subjects Applied Statistics
Journal Section Research Article
Authors

Rashid Mehmood 0000-0002-0993-4665

Muhammad Naveed Khan 0000-0001-8961-1358

Iftikhar Ali 0000-0002-0371-2973

Tajammal Imran 0000-0002-8119-2432

Babar Zaman 0000-0003-1259-1429

Kassimu Mpungu 0000-0002-8669-6621

Fawwad Qureshi 0009-0006-6961-335X

Maysaa Elmahi Abdelwahab 0009-0000-7059-405X

Early Pub Date July 19, 2025
Publication Date August 29, 2025
Submission Date September 11, 2024
Acceptance Date July 4, 2025
Published in Issue Year 2025 Volume: 54 Issue: 4

Cite

APA Mehmood, R., Khan, M. N., Ali, I., … Imran, T. (2025). Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules. Hacettepe Journal of Mathematics and Statistics, 54(4), 1588-1621. https://doi.org/10.15672/hujms.1541132
AMA Mehmood R, Khan MN, Ali I, et al. Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules. Hacettepe Journal of Mathematics and Statistics. August 2025;54(4):1588-1621. doi:10.15672/hujms.1541132
Chicago Mehmood, Rashid, Muhammad Naveed Khan, Iftikhar Ali, Tajammal Imran, Babar Zaman, Kassimu Mpungu, Fawwad Qureshi, and Maysaa Elmahi Abdelwahab. “Efficient Techniques for Designing and Evaluating Multivariate Hotelling Control Chart With Generalized Sensitizing Rules”. Hacettepe Journal of Mathematics and Statistics 54, no. 4 (August 2025): 1588-1621. https://doi.org/10.15672/hujms.1541132.
EndNote Mehmood R, Khan MN, Ali I, Imran T, Zaman B, Mpungu K, Qureshi F, Elmahi Abdelwahab M (August 1, 2025) Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules. Hacettepe Journal of Mathematics and Statistics 54 4 1588–1621.
IEEE R. Mehmood, M. N. Khan, I. Ali, T. Imran, B. Zaman, K. Mpungu, F. Qureshi, and M. Elmahi Abdelwahab, “Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 4, pp. 1588–1621, 2025, doi: 10.15672/hujms.1541132.
ISNAD Mehmood, Rashid et al. “Efficient Techniques for Designing and Evaluating Multivariate Hotelling Control Chart With Generalized Sensitizing Rules”. Hacettepe Journal of Mathematics and Statistics 54/4 (August2025), 1588-1621. https://doi.org/10.15672/hujms.1541132.
JAMA Mehmood R, Khan MN, Ali I, Imran T, Zaman B, Mpungu K, Qureshi F, Elmahi Abdelwahab M. Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules. Hacettepe Journal of Mathematics and Statistics. 2025;54:1588–1621.
MLA Mehmood, Rashid et al. “Efficient Techniques for Designing and Evaluating Multivariate Hotelling Control Chart With Generalized Sensitizing Rules”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 4, 2025, pp. 1588-21, doi:10.15672/hujms.1541132.
Vancouver Mehmood R, Khan MN, Ali I, Imran T, Zaman B, Mpungu K, et al. Efficient techniques for designing and evaluating multivariate Hotelling control chart with generalized sensitizing rules. Hacettepe Journal of Mathematics and Statistics. 2025;54(4):1588-621.

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