# Optimal Design of One-Sided Exponential Adaptive EWMA Scheme Based on Median Run Length
## 5 Comparisons of the Performance  
This section employs MRL as the performance metric to evaluate and compare the effectiveness of one-sided exponential AEWMA, EWMA, and Shewhart schemes. Tables 1 and 2 provide the optimized parameters and corresponding \(MRL_{1}\) for lower-sided exponential monitoring schemes designed to monitor downward shifts, with \(MRL_{0}\) set at \(\{100,200,300,500\}\). For the lower-sided exponential AEWMA scheme, the parameters \((\lambda^{-},k,LCL^{-})\) are optimized for the shift range \([c_{2},c_{1}]\), where smaller process mean shifts \(c_{1}\in\{0.6,0.8\}\) and larger shifts \(c_{2}=0.01\). Similarly, the parameters \((\lambda^{-},LCL^{-})\) for the lower-sided exponential EWMA scheme are optimized for the shift size \(c_{1}\), while the parameter \(LCL^{-}\) for the lower-sided exponential Shewhart scheme is constrained to achieve the desired \(MRL_{0}\). For instance, in Table 2, when \(MRL_{0}=300\), \(c_{1}=0.6\), and \(c_{2}=0.01\), the optimized parameters \((\lambda^{-},k,LCL^{-})=(0.089,1.025,0.597)\) for the lower-sided exponential AEWMA scheme are derived using the optimization procedure outlined in Section 4. The tables demonstrate that the lower-sided exponential AEWMA scheme consistently delivers robust performance across various shift sizes. For smaller mean shifts in the production process, the lower-sided exponential AEWMA scheme performs comparably to the EWMA scheme but significantly outperforms the Shewhart scheme. For example, in Table 1, when \(MRL_{0}=200\), \(c_{1}=0.8\), and \(c_{2}=0.01\), if the current process mean shift \(c=0.8\), the \(MRL_{1}\) of the lower-sided exponential Shewhart scheme is 160, while the \(MRL_{1}\) for the lower-sided exponential AEWMA and EWMA schemes are both 43. This indicates that both exponential EWMA-type schemes can detect process mean shifts much more rapidly than the Shewhart scheme. In contrast, for moderate shifts such as \(c=0.65\), the \(MRL_{1}\) of the lower-sided exponential AEWMA and EWMA schemes are 17 and 23, respectively, highlighting the AEWMA scheme's superior effectiveness over the EWMA scheme. Only for substantial shifts (\(c<0.05\)) does the exponential Shewhart scheme marginally outperform the AEWMA and EWMA schemes in detecting downward shifts. The robustness of the AEWMA scheme is particularly notable, as it maintains consistent performance across a wide range of shift sizes. Unlike the Shewhart scheme, which performs better only for very large shifts, the AEWMA scheme exhibits superior sensitivity for both small and moderate shifts, making it a more dependable choice for practical applications. Additionally, Tables 3 and 4 provide analogous comparisons for upper-sided exponential monitoring schemes, further emphasizing the AEWMA scheme's robust performance in detecting upward shifts. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(MRL_{0}=100\)} & \multicolumn{4}{c}{\(MRL_{0}=200\)} & \multicolumn{4}{c}{\(MRL_{0}=300\)} & \multicolumn{4}{c}{\(MRL_{0}=500\)} \\ \cline{2-13}  & \multicolumn{2}{c}{**AEWMA**} & **EWMA** & **Sheewhart** & **AEWMA** & **EWMA** & **Sheewhart** & **AEWMA** & **EWMA** & **Sheewhart** & **AEWMA** & **EWMA** & **Sheewhart** \\  & \(\lambda^{+}\) & **0.050** & **0.050** & **-** & **0.052** & **0.052** & **-** & **0.050** & **0.050** & **-** & **0.050** & **0.050** & **-** \\  & \(k\) & **1.336** & **-** & **-** & **1.432** & **-** & **-** & **1.452** & **-** & **-** & **1.497** & **-** & **-** \\ \(\epsilon\) & \(h^{+}\) & **4.325** & **1.277** & **4.9651** & **4.975** & **1.369** & **5.6616** & **5.500** & **1.401** & **6.0681** & **6.100** & **1.454** & **6.5798** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: The optimal parameters and \(MRL_{1}\) for the upper-sided exponential AEWMA, EWMA, Shewhart scheme when \(c_{1}=1.02,c_{2}=5,MRL_{0}\in(100,200,300,500)\).