# Optimal Design of One-Sided Exponential Adaptive EWMA Scheme Based on Median Run Length
## 6 Two Illustrative Examples
### Example 1  
Using the example from the study by [PERSON] et al. (see [37]), consider a production process for OLED components to illustrate the application of the optimized lower-sided exponential AEWMA scheme based on MRL. The dataset consists of 30 intervals of failure times from accelerated life tests in the OLED production process. Assuming an exponential distribution but with unknown parameters, the bootstrap method was employed to estimate the process mean using the first 10 data points (see Table 5). The remaining 20 data points (see Table 6) were reserved for testing purposes. To achieve a robust estimation of the process mean, the initial 10 data points were used to compute the maximum likelihood estimate (MLE) of the process mean \(\widehat{\theta}_{0}=1.245\). Subsequently, the bootstrap method [39; 40] was utilized to generate multiple resamples from these 10 data points. For each resample, the MLE of \(\widehat{\theta}_{0}\) was calculated, thereby constructing a distribution of estimates. Based on this distribution, the 95% confidence interval for \(\widehat{\theta}_{0}\) was determined as \([\widehat{\theta}_{0L},\widehat{\theta}_{0U}]=[0.853,1.706]\). This approach allowed for a more accurate and reliable estimation of the process mean while accounting for the variability inherent in the data. The factory employs an optimal lower-sided exponential AEWMA scheme to monitor downward shifts in the process mean. Suppose the process mean shift to be monitored is within \([c_{2},c_{1}]=[0.01,0.8]\), and the acceptable \(MRL_{0}\) is 300. The parameters \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(t\) & \(X_{t}\) & \(A_{t}^{-}\) & \(A_{tL}^{-}\) & \(Q_{t}^{-}\) & \(t\) & \(X_{t}\) & \(A_{t}^{-}\) & \(A_{tL}^{-}\) & \(Q_{t}^{-}\) \\ \hline
1 & 2.19 & 1.294 & 1.240 & 1.294 & 11 & 0.94 & 1.413 & 1.020 & 1.247 \\
2 & 1.10 & 1.284 & 1.233 & 1.284 & 12 & 1.08 & 1.395 & 1.023 & 1.239 \\
3 & 0.60 & 1.248 & 1.200 & 1.248 & 13 & 2.31 & 1.443 & 1.360 & 1.294 \\
4 & 0.97 & 1.234 & 1.188 & 1.234 & 14 & 3.11 & 1.723 & 1.706 & 1.389 \\
5 & 3.16 & 1.773 & 1.706 & 1.334 & 15 & 0.28 & 1.648 & 1.230 & 1.331 \\
6 & 2.93 & 1.833 & 1.706 & 1.417 & 16 & 0.66 & 1.597 & 1.200 & 1.296 \\
7 & 0.99 & 1.789 & 1.669 & 1.395 & 17 & 0.68 & 1.549 & 1.173 & 1.264 \\
8 & 1.28 & 1.763 & 1.649 & 1.389 & 18 & 0.75 & 1.507 & 1.151 & 1.237 \\
9 & 0.12 & 1.507 & 1.070 & 1.323 & 19 & 0.47 & 1.453 & 1.116 & 1.197 \\
10 & 0.19 & 1.438 & 1.024 & 1.264 & 20 & 1.82 & 1.473 & 1.152 & 1.230 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Phase II samples \(X_{t}\) (in minutes) and corresponding monitoring statistics \(A_{t}^{-}\), \(A_{tL}^{-}\) and \(Q_{t}^{-}\) of the optimal lower-sided exponential schemes for the OLED production process. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(t\) & \(Z_{t}\) & \(t\) & \(Z_{t}\) \\ \hline
1 & 1.07 & 6 & 1.26 \\
2 & 0.54 & 7 & 0.48 \\
3 & 0.54 & 8 & 1.78 \\
4 & 0.72 & 9 & 1.26 \\
5 & 2.53 & 10 & 2.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Phase I samples \(Z_{t}\) (in minutes) of the intervals of failure times from accelerated life tests in the OLED production line. \((\lambda^{-},k,LCL^{-})=(0.052,1.175,0.754)\) for the optimal lower-sided exponential AEWMA scheme can be obtained through the design steps in Section 4. Then, multiplying \(B^{-},k\), and \(LCL^{-}\) by \(\widehat{\theta}_{0}=1.245\) yields the reflection boundary \(B^{-}\), threshold parameter \(k^{\prime}\), and lower control limit \(LCL^{\prime}\) for the optimal lower-sided exponential AEWMA scheme:  
\[B^{-}{}^{\prime} = 2\qquad\times 1.245=2.490,\] \[k^{\prime} = 1.175\times 1.245=1.463,\] \[LCL^{\prime} = 0.754\times 1.245=0.9387.\]  
The monitoring statistic \(A_{t}^{-}\) for the optimal lower-sided exponential AEWMA scheme in the OLED production process can be calculated using Equation (9) (see Table 6).