## 6 Conclusions  
The structure of control charts is based on the assumption that process observations come from a specific distribution, such as the normal distribution. One of the criteria for evaluating their performance is the ARL when the fundamental distributional assumption is violated. In the literature, many studies have assessed the performance of control charts by using different distributions instead of the fundamental one. In the case of the PEWMA control chart, it is assumed that observations follow a Poisson distribution.  
In this study, we assessed the performance of the PEWMA control chart in the presence of Poisson mixture data instead of Poisson data. The Poisson mixture data were generated using simulation, with parameters such as sample mean and the proportion of data for each component. Subsequently, the simulation was executed to obtain the ARL while calculating the PEWMA statistics and control limits. The results were analyzed for various parameters of PEWMA control charts, and we provided necessary interpretations to enhance their performance. Additionally, we presented a real-world application for process control in fastener production. We analyzed the data using the EM algorithm to determine the number of components and distribution parameters of the Poisson mixture distribution. Finally, we constructed PEWMA charts with the desired parameters.  
The fact that there is more variability in process data, especially in real cases, makes the use of mixture distributions important. In future studies, the exploring efficacy of existing control charts for different mixture distributions will shed light on how to deal with over-dispersed data. This examination will also provide a foundation for the creation of more robust control charts specifically designed to address data characterized by various mixture distributions.  
Conceptualization, M.K. and N.U.; methodology, M.K. and N.U.; software, M.K. and N.U.; validation, M.K. and N.U.; formal analysis, M.K. and N.U.; investigation, M.K. and N.U.; writing--original draft preparation, M.K. and N.U.; writing--review and editing, M.K. and N.U.; visualization, N.U.; supervision, M.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
All datasets are publicly available.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} EWMA & Exponentially Weighted Moving Average \\ PEWMA & Poisson-Exponentially Weighted Moving Average \\ CUSUM & Cumulative Sum \\ SPC & Statistical Process Control \\ RL & Run Length \\ ARL & Average Run Length \\ \end{tabular}  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Shift in Mean** & \(A\) = 3.50 & \(A\) = 3.20 & \(A\) = 2.90 & \(A\) = 2.85 & \(A\) = 2.70 & \(A\) = 2.50 \\
**Multiple of \(\sigma\)** & \(\lambda\) = 0.75 & \(\lambda\) = 0.5 & \(\lambda\) = 0.4 & \(\lambda\) = 0.3 & \(\lambda\) = 0.2 & \(\lambda\) = 0.1 \\ \hline
0.00 & 419.33 & 360.49 & 245.72 & 237.14 & 223.07 & 125.27 \\
0.25 & 247.24 & 147.58 & 88.77 & 77.22 & 71.30 & 61.33 \\
0.52 & 72.04 & 49.99 & 32.80 & 31.29 & 26.65 & 20.39 \\
0.76 & 37.16 & 15.47 & 12.7 & 8.91 & 8.61 & 8.02 \\ \hline \hline \end{tabular}
\end{table}
Table 5: ARLs for different PEWMA control schemes with Poisson mixture data.