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.
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References
Bennet C (1954) Effect of measurement error on chemical process control. Ind Quality Control 10(4):17–20
Calzada M, Scariano SM (2013) A synthetic control chart for the coefficient of variation. J Stat Comput Simul 83(5):853–867
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
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
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
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
Castagliola P, Celano G, Psarakis S (2011) Monitoring the coefficient of variation using EWMA charts. J Qual Technol 43(3):249–265
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
Costa A, Castagliola P (2011) Effect of measurement error and autocorrelation on the \(\bar {X}\) chart. J Appl Stat 38(4):661–673
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
Kang C, Lee M, Seong Y, Hawkins D (2007) A control chart for the coefficient of variation. J Qual Technol 39(2):151–158
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
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
Linna K, Woodall W (2001) Effect of measurement error on Shewhart control chart. J Qual Technol 33(2):213–222
Manual (2010) Measurement system analysis. Automotive industry action group, 4th edn
Maravelakis P (2012) Measurement error effect on the CUSUM control chart. J Appl Stat 39(2):323–336
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
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
Reynolds M, Amin R, Arnold J, Nachlas J (1988) \(\bar {X}\) Charts with variable sampling intervals. Technometris 30(2):181–192
Reynolds M, Arnold J (1989) Optimal onesided Shewhart charts with variable sampling interval. Seq Anal 80(1):181–192
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
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
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
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
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
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
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
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
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
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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
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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
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DOI: https://doi.org/10.1007/s00170019033527