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On the performance of VSI Shewhart control chart for monitoring the coefficient of variation in the presence of measurement errors

Abstract

In this paper, we propose a variable sampling interval Shewhart control chart to monitor the coefficient of variation (CV) squared, denoted by VSI SH-γ2. The new model overcomes the ARL-biased (average run length) property of the control chart monitoring the CV in a previous study by designing two one-sided charts rather than one two-sided chart. Moreover, the effect of measurement error on the performance of the VSI SH-γ2 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 SH-γ2 chart. An appropriate strategy based on the obtained results is suggested to reduce these negative effects.

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Acknowledgements

The authors thank the anonymous referees for their insightful and valuable suggestions which helped to improve the quality of the final manuscript.

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Correspondence to Quoc Thong Nguyen.

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Appendix

Appendix

Let pS,pL, and q be the probability that a monitored sample point drops into the central region, the warning region and the out-of-control region, respectively. According to the subdivision of control interval, the formulae to calculate pS,pL, and q in VSI SH-γ2 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)
    $$\begin{array}{@{}rcl@{}} &=&F_{\hat{\gamma}^{2}}(LWL^{-} |n,\gamma^{2})-F_{\hat{\gamma}^{2}}(LCL^{-}|n,\gamma^{2} ),\\ q &=& P(\hat{\gamma}^{2}< LCL^{-})= 1-p_{S}-p_{L}. \end{array} $$
    (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)
    $$\begin{array}{@{}rcl@{}} &=&F_{\hat{\gamma}^{2}}(UCL^{+}|n,\gamma^{2} )-F_{\hat{\gamma}^{2}}(UWL^{+} |n,\gamma^{2}),\\ q &=& P(\hat{\gamma}^{2}> UCL^{+})= 1-p_{S}-p_{L}. \end{array} $$
    (15)

The c.d.f \(F_{\hat {\gamma }^{2}}(.|n,\gamma ^{2})\) in this case is defined in (6).

In the VSI SH-γ2 control charts considering the presence of measurement errors, the formulae for pS,pL, and q are the same in Eqs. 1015 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

$$ ASI = E(h)=\frac{h_{S}p_{S}+h_{L}p_{L}}{1-q}. $$
(16)

The formula of ATS is given by [20] and adopted by [4] as

$$ ATS = \frac{h_{S}p_{S}+h_{L}p_{L}}{q(1-q)}. $$
(17)

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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/s00170-019-03352-7

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Keywords

  • VSI control chart
  • Coefficient of variation
  • Measurement errors