Abstract
In this paper, we propose a variable sampling interval Shewhart control chart to monitor the coefficient of variation (CV) squared, denoted by VSI SHγ^{2}. The new model overcomes the ARLbiased (average run length) property of the control chart monitoring the CV in a previous study by designing two onesided charts rather than one twosided chart. Moreover, the effect of measurement error on the performance of the VSI SHγ^{2} control chart is investigated. The incorrect formula for the distribution of the CV in the presence of measurement error in a former study is fixed. Numerical simulations show that the precision errors and accuracy errors do have negative influences on the VSI SHγ^{2} chart. An appropriate strategy based on the obtained results is suggested to reduce these negative effects.
This is a preview of subscription content, access via your institution.
References
 1.
Bennet C (1954) Effect of measurement error on chemical process control. Ind Quality Control 10(4):17–20
 2.
Calzada M, Scariano SM (2013) A synthetic control chart for the coefficient of variation. J Stat Comput Simul 83(5):853–867
 3.
Castagliola P, Achoure A, Taleb H, Celano G, Psarakis S (2013) Monitoring the coefficient of variation using control charts with run rules. Quality Technol Quant Manag 10:75–94
 4.
Castagliola P, Achouri A, Taleb H, Celano G, Psarakis S (2013) Monitoring the coefficient of variation using a variable sampling interval control chart. Qual Reliab Eng Int 29(8):1135–1149
 5.
Castagliola P, Achouri A, Taleb H, Celano G, Psarakis S (2015) Monitoring the coefficient of variation using a variable sample size control chart. Int J Adv Manuf Technol 81(912):1561–1576
 6.
Castagliola P, Amdouni A, Taleb H, Celano G (2015) Onesided Shewharttype charts for monitoring the coefficient of variation in short production runs. Quality Technol Quant Manag 12(1):53–67
 7.
Castagliola P, Celano G, Psarakis S (2011) Monitoring the coefficient of variation using EWMA charts. J Qual Technol 43(3):249–265
 8.
Celano G, Castagliola P, Nenes G, Fichera S (2013) Performance of t control charts in short runs with unknown shift sizes. Comput Ind Eng 64:56–68
 9.
Costa A, Castagliola P (2011) Effect of measurement error and autocorrelation on the \(\bar {X}\) chart. J Appl Stat 38(4):661–673
 10.
Hong EP, Kang CW, Baek JW, Kang HW (2008) Development of CV control chart using EWMA technique. J Soc Korea Ind Syst Eng 31:114–120
 11.
Kang C, Lee M, Seong Y, Hawkins D (2007) A control chart for the coefficient of variation. J Qual Technol 39(2):151–158
 12.
Khaw K, Khoo M, Yeong W, Wu Z (2017) Monitoring the coefficient of variation using a variable sample size and sampling interval control chart. Commun Stat Simul Comput 46(7):5722–5794
 13.
Li Z, Zou C, Gong Z, Wang Z (2014) The computation of average run length and average time to signal: an overview. J Stat Comput Simul 84(8):1779–1802
 14.
Linna K, Woodall W (2001) Effect of measurement error on Shewhart control chart. J Qual Technol 33(2):213–222
 15.
Manual (2010) Measurement system analysis. Automotive industry action group, 4th edn
 16.
Maravelakis P (2012) Measurement error effect on the CUSUM control chart. J Appl Stat 39(2):323–336
 17.
Nguyen HD, Tran KP, Heuchenne C (2019) Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts. Qual Reliab Eng Int 35(1):439–460. https://doi.org/10.1002/qre.2412
 18.
Noorossana R, Zerehsaz Y (2015) Effect of measurement error on phase II monitoring of simple linear profiles. Int J Adv Manuf Technol 79(912):2031–2040
 19.
Reynolds M, Amin R, Arnold J, Nachlas J (1988) \(\bar {X}\) Charts with variable sampling intervals. Technometris 30(2):181–192
 20.
Reynolds M, Arnold J (1989) Optimal onesided Shewhart charts with variable sampling interval. Seq Anal 80(1):181–192
 21.
Tran K, Castagliola P, Celano G (2016) The performance of the ShewhartRZ control chart in the presence of measurement error. Int J Prod Res 54:7504–7522
 22.
Tran KP, Nguyen HD, Nguyen QT, Chattinnawat W (2018) Onesided synthetic control charts for monitoring the coefficient of variation with measurement errors. In: 2018 IEEE international conference on industrial engineering and engineering management (IEEM), pp 1667–1671. https://doi.org/10.1109/IEEM.2018.8607320
 23.
Tran KP, Heuchenne C, Balakrishnan N (2019) On the performance of coefficient of variation charts in the presence of measurement errors. Qual Reliab Eng Int 35(1):329–350. https://doi.org/10.1002/qre.2402
 24.
Tran P, Tran K, Rakitzis A (2019) A synthetic median control chart for monitoring the process mean with measurement errors. Quality and Reliability Engineering International. https://doi.org/10.1002/qre.2447
 25.
Tran P, Tran KP (2016) The efficiency of CUSUM schemes for monitoring the coefficient of variation. Appl Stoch Model Bus Ind 32(6):870–881
 26.
Yeong W, Khoo M, Lim S, Lee M (2017) A direct procedure for monitoring the coefficient of variation using a variable sample size scheme. Commun Stat Simul Comput 46(6):4210–4225
 27.
Yeong WC, Khoo MBC, Lim SL, Teoh WL (2017) The coefficient of variation chart with measurement error. Qual Technol Quant Manag 14(4):353–377
 28.
Yeong WC, Lim SL, Khoo MBC, Castagliola P (2018) Monitoring the coefficient of variation using a variable parameters chart. Qual Eng 30(2):212–235
 29.
You H, Khoo BM, Castagliola P, Haq A (2016) Monitoring the coefficient of variation using the side sensitive group runs chart. Qual Reliab Eng Int 32(5):1913–1927
Acknowledgements
The authors thank the anonymous referees for their insightful and valuable suggestions which helped to improve the quality of the final manuscript.
Author information
Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Let p_{S},p_{L}, and q be the probability that a monitored sample point drops into the central region, the warning region and the outofcontrol region, respectively. According to the subdivision of control interval, the formulae to calculate p_{S},p_{L}, and q in VSI SHγ^{2} without considering the measurement errors are as follows.

For downward chart,
$$\begin{array}{@{}rcl@{}} p_{L} &=& P(\hat{\gamma}^{2}\!\ge \!LWL^{}) = 1  F_{\hat{\gamma}^{2}}(LWL^{  } \mid \!n,\!\gamma^{2} ), \end{array} $$(10)$$\begin{array}{@{}rcl@{}} p_{S} &=& P(LCL^{}\leqslant \hat{\gamma}^{2} \leqslant LWL^{}) \end{array} $$(11)$$\begin{array}{@{}rcl@{}} &=&F_{\hat{\gamma}^{2}}(LWL^{} n,\gamma^{2})F_{\hat{\gamma}^{2}}(LCL^{}n,\gamma^{2} ),\\ q &=& P(\hat{\gamma}^{2}< LCL^{})= 1p_{S}p_{L}. \end{array} $$(12) 
For upward chart,
$$\begin{array}{@{}rcl@{}} p_{L} &=& P(\hat{\gamma}^{2} \leqslant UWL^{+})=F_{\hat{\gamma}^{2}}(UWL^{+}n,\gamma^{2} ), \end{array} $$(13)$$\begin{array}{@{}rcl@{}} p_{S} &=& P(UWL^{+}\leqslant \hat{\gamma}^{2} \leqslant UCL^{+}) \end{array} $$(14)$$\begin{array}{@{}rcl@{}} &=&F_{\hat{\gamma}^{2}}(UCL^{+}n,\gamma^{2} )F_{\hat{\gamma}^{2}}(UWL^{+} n,\gamma^{2}),\\ q &=& P(\hat{\gamma}^{2}> UCL^{+})= 1p_{S}p_{L}. \end{array} $$(15)
The c.d.f \(F_{\hat {\gamma }^{2}}(.n,\gamma ^{2})\) in this case is defined in (6).
In the VSI SHγ^{2} control charts considering the presence of measurement errors, the formulae for p_{S},p_{L}, and q are the same in Eqs. 10–15 for both charts, but the distribution \(F_{\gamma ^{2}}(.n,\gamma ^{2})\) in these equations are replaced by \(F_{\gamma ^{*2}}(.n,\gamma ^{*2})\), defined in Eq. 8.
From its definition, the ASI is calculated by
The formula of ATS is given by [20] and adopted by [4] as
Rights and permissions
About this article
Cite this article
Nguyen, H.D., Nguyen, Q.T., Tran, K.P. et al. On the performance of VSI Shewhart control chart for monitoring the coefficient of variation in the presence of measurement errors. Int J Adv Manuf Technol 104, 211–243 (2019). https://doi.org/10.1007/s00170019033527
Received:
Accepted:
Published:
Issue Date:
Keywords
 VSI control chart
 Coefficient of variation
 Measurement errors