# A Bayesian Control Chart for Monitoring Process Variance
## 3 The Construction of the Shewhart-Type Variance Chart with the Bayesian Approach
### The Parameters of the Prior Distribution of \(p\) and Process Variance Are Known
#### 3.1.2 The Out-of-Control ARL of the Shewhart-Type Bayesian Variance Chart  
When the process is out-of-variance-control and parameters \(\alpha_{0}\) and \(\beta_{0}\) are changed to \(\alpha_{1}\) and \(\beta_{1}\), the chance that the proposed chart alarms a true signal is calculated by:  
\[ALARM_{1}=P(\,M_{t}\leq LCL_{M}\text{ or }M_{t}\geq UCL_{M}|M_{t}\sim Beta-Binomial(n,\ \alpha_{1},\ \beta_{1})). \tag{18}\]  
So, \(ARL_{1}=1/ALARM_{1}\). Table 2 presents the \(ARL_{1}\)s of the proposed Bayesian variance chart for various combinations of \(n\), \(\alpha_{1}\), and \(\beta_{1}\) under the in-control parameters (\(\alpha_{0}\), \(\beta_{0}\)) = (5, 10), (3, 9), (46,100), and (32, 100), respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline (\(\alpha_{0}\), \(\beta_{0}\)) & \(E_{0}\)(\(p\)) & \(n=2\) & \(n=3\) & \(n=5\) & \(n=10\) & \(n=15\) & \(n=20\) & \(n=25\) \\ \hline (1, 1) & 0.5000 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ (1, 2) & 0.3333 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ (1, 3) & 0.2500 & \(\infty\) & \(\infty\) & \(\infty\) & 286.00 & 204.00 & 177.10 & 163.80 \\ (1, 4) & 0.2000 & \(\infty\) & \(\infty\) & 126.00 & 200.20 & 110.74 & 84.33 & 113.10 \\ (1, 5) & 0.1667 & \(\infty\) & 56.00 & 42.00 & 143.00 & 123.05 & 67.08 & 71.18 \\ (2, 2) & 0.5000 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ (2, 3) & 0.4000 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ (2, 4) & 0.3333 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & 969.00 & 526.04 & 1131.00 \\ (2, 5) & 0.2875 & \(\infty\) & \(\infty\) & \(\infty\) & 728.00 & 180.28 & 222.23 & 257.80 \\ (2, 6) & 0.2500 & \(\infty\) & \(\infty\) & 132.00 & 273.92 & 151.19 & 240.66 & 174.65 \\ (3, 3) & 0.5000 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ (3, 4) & 0.4286 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) \\ (3, 5) & 0.3750 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & 1254.00 & 693.23 & 1703.37 \\ (3, 6) & 0.3333 & \(\infty\) & \(\infty\) & \(\infty\) & 663.00 & 572.80 & 567.09 & 577.14 \\ (3, 7) & 0.3000 & \(\infty\) & \(\infty\) & \(\infty\) & 204.83 & 333.89 & 468.32 & 270.28 \\ (10, 10) & 0.5000 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & 3441.00 & 1607.81 \\ (8, 10) & 0.4444 & \(\infty\) & \(\infty\) & \(\infty\) & \(\infty\) & 3317.17 & 2130.37 & 1798.07 \\ (6, 10) & 0.3750 & \(\infty\) & \(\infty\) & \(\infty\) & 1088.50 & 1177.06 & 399.88 & 562.37 \\ (4, 10) & 0.2857 & \(\infty\) & \(\infty\) & 153.00 & 460.20 & 270.31 & 214.86 & 404.65 \\ (2, 10) & 0.1666 & 26.00 & 91.00 & 78.00 & 149.08 & 91.49 & 140.96 & 106.46 \\ (100, & 0.5000 & \(\infty\) & \(\infty\) & \(\infty\) & 413.00 & 709.04 & 1426.63 & 589.89 \\ (80, & & & & & & & \\
100) & 0.4444 & \(\infty\) & \(\infty\) & \(\infty\) & 2473.90 & 609.75 & 424.17 & 573.01 \\ (60, & 100) & 0.3750 & \(\infty\) & \(\infty\) & \(\infty\) & 757.85 & 642.55 & 655.36 & 725.14 \\ (40, & 100) & 0.2857 & \(\infty\) & \(\infty\) & 442.91 & 618.83 & 258.22 & 620.79 & 418.48 \\ (20, & 100) & 0.1666 & 34.57 & 191.71 & 242.62 & 270.86 & 406.05 & 174.27 & 297.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The in-control average run lengths (\(ARL_{0}\)s) of the proposed Bayesian variance chart.