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.

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 out-of-control region, respectively. According to the subdivision of control interval, the formulae to calculate p_S,p_L, 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 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

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

(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)