# A Bayesian Control Chart for Monitoring Process Variance
## 4 The Construction of the EWMA Variance Chart with the Bayesian Approach
### Performance Comparison of the EWMA Variance Control Chart
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Panel A:** & \multicolumn{6}{c}{**When (\(\alpha_{0}\), \(\beta_{0}\)) = (5, 10) and \(E_{0}(p)\) = 0.3333**} \\ \hline \multirow{2}{*}{(\(\alpha_{1}\), \(\beta_{1}\))} & \multirow{2}{*}{\(E_{1}(p)\)} & \(n=2\) & \(n=3\) & \(n=5\) & \(n=10\) & \(n=15\) & \(n=20\) & \(n=25\) \\  & & **(3.21, 2.76) b** & **(3.02, 2.91)** & **(3.04, 2.86)** & **(3.02, 2.95)** & **(3.01, 2.96)** & **(3.01, 2.94)** & **(3.03, 2.95)** \\ \hline \multirow{2}{*}{(\(10,10\)) a} & \multirow{2}{*}{0.5000} & 2.32 & 1.40 & 1.11 & 1.01 & 1.00 & 1.00 & 1.00 \\  & & 0.3750 & 101.42 & 49.93 & 33.33 & 18.01 & 12.90 & 10.99 & 9.86 \\ \multirow{2}{*}{(\(5,10\)) a} & \multirow{2}{*}{0.3333} & 378.79 & 371.75 & 373.13 & 374.53 & 380.23 & 383.14 & 378.79 \\  & & 220.75 & 130.72 & 96.90 & 57.67 & 46.69 & 38.52 & 37.84 \\ \multirow{2}{*}{(\(5,5\))} & \multirow{2}{*}{0.5000} & 2.29 & 1.41 & 1.13 & 1.01 & 1.00 & 1.00 & 1.00 \\ \multirow{2}{*}{(\(1,10\))} & \multirow{2}{*}{0.0909} & 1.01 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \hline \multicolumn{10}{c}{**Panel B:**} & \multicolumn{6}{c}{**When (\(\alpha_{0}\), \(\beta_{0}\)) = (3, 9) and \(E_{0}(p)\) = 0.2500**} \\ \hline \multirow{2}{*}{(\(\alpha_{1}\), \(\beta_{1}\))} & \multirow{2}{*}{\(E_{1}(p)\)} & \(n=2\) & \(n=3\) & \(n=5\) & \(n=10\) & \(n=15\) & \(n=20\) & \(n=25\) \\  & & **(3.14, 2.65) b** & **(3.25, 2.73)** & **(3.05, 2.83)** & **(3.08, 2.83)** & **(3.10, 2.79)** & **(3.08, 2.81)** & **(3.09, 2.83)** \\ \hline \multirow{2}{*}{(\(9,9\)) a} & \multirow{2}{*}{0.5000} & 1.07 & 1.01 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & & 29.39 & 25.41 & 11.20 & 6.56 & 4.98 & 4.11 & 3.73 \\ \multirow{2}{*}{(\(3,9\)) a} & \multirow{2}{*}{0.2500} & 377.36 & 383.14 & 370.37 & 378.79 & 373.13 & 375.94 & 375.94 \\  & & 273.28 & 174.83 & 83.40 & 57.84 & 49.48 & 41.67 & 39.62 \\ \multirow{2}{*}{(\(3,3\))} & \multirow{2}{*}{0.5000} & 1.08 & 1.02 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \multirow{2}{*}{(\(1,9\))} & \multirow{2}{*}{0.1000} & 1.43 & 1.15 & 1.03 & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \hline \multicolumn{10}{c}{**Panel C:**} & \multicolumn{6}{c}{**When (\(\alpha_{0}\), \(\beta_{0}\)) = (46, 100) and \(E_{0}(p)\) = 0.3151**} \\ \hline \multirow{2}{*}{(\(\alpha_{1}\), \(\beta_{1}\))} & \multirow{2}{*}{\(E_{1}(p)\)} & \(n=2\) & \(n=3\) & \(n=5\) & \(n=10\) & \(n=15\) & \(n=20\) & \(n=25\) \\  & & **(3.16, 2.75) b** & **(2.97, 2.89)** & **(3.08, 2.93)** & **(3.00, 2.94)** & **(3.03, 2.98)** & **(3.00, 2.99)** & **(3.00, 3.01)** \\ \hline \multirow{2}{*}{(\(100,100\)) a} & \multirow{2}{*}{0.3151} & \multirow{2}{*}{374.53} & \multirow{2}{*}{370.37} & \multirow{2}{*}{384.62} & \multirow{2}{*}{377.36} & \multirow{2}{*}{380.23} & \multirow{2}{*}{375.94} & \multirow{2}{*}{378.79} \\  & & 35.56 & 15.69 & 9.46 & 3.29 & 2.09 & 1.55 & 1.32 \\ \multirow{2}{*}{(\(46,100\)) a} & \multirow{2}{*}{0.3151} & \multirow{2}{*}{374.53} & \multirow{2}{*}{370.37} & \multirow{2}{*}{384.62} & \multirow{2}{*}{377.36} & \multirow{2}{*}{380.23} & \multirow{2}{*}{375.94} & \multirow{2}{*}{378.79} \\  & & 374.53 & 370.37 & 384.62 & 377.36 & 380.23 & 375.94 & 378.79 \\ \multirow{2}{*}{(\(46,46\))} & \multirow{2}{*}{0.3651} & \multirow{2}{*}{54.05} & \multirow{2}{*}{25.03} & \multirow{2}{*}{16.01} & \multirow{2}{*}{5.52} & \multirow{2}{*}{3.34} & \multirow{2}{*}{2.31} & \multirow{2}{*}{1.85} \\  & & 5000 & 1.60 & 1.13 & 1.01 & 1.00 & 1.00 & 1.00 \\ \multirow{2}{*}{(\(10,100\))} & \multirow{2}{*}{0.0909} & 1.01 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \hline \hline \end{tabular}  
* \({}^{\text{a}}\) denotes the in-control process; b denotes the coefficients of the control limits (\(k_{1},k_{2}\)).  
\end{table} 
Table 4: The \(ARL_{1}\)s of the proposed Bayesian EWMA variance chart (\(\lambda=0.05\)). are all smaller than those of the SS EWMA-V chart, showing that our proposed chart could detect a shift of variance more quickly than the SS EWMA-V chart.