## 5 Discussion  
In the literature, there are notable instances of studies delving into the enhancement and progression of the PEWMA control chart, which was originally introduced by [PERSON] et al. [9]. One of the studies offering suggestions for obtaining a more effective control chart was found in ref. [18]. PEWMA has argued that it is more effective by applying a square root transformation to control chart data.  
Overdispersion in process data has been handled by categorizing a certain data rate as \"noisy data\" in SPC. This approach was adopted as a precautionary measure to mitigate the effects of overdispersion. In view of this concept, [PERSON] et al. [21] suggested in their study that handling overdispersion with the Poisson mixture distribution would eliminate the problem and increase the ARL performance. However, the charts they deal with are not EWMA charts as similar to [PERSON] et al. [27], but for the \(c\) and \(u\) charts developed for attribute data. As far as we have seen in the literature, our work is a novel approach for PEWMA control charts, taking into account the Poisson mixture distribution to deal with overdispersed data.  
The outcomes of our simulation experiments, conducted using the generated Poisson mixture data and a range of design parameter combinations, illuminated several significant findings. Notably, we observed that the selection of suitable values for the parameters played a pivotal role in shaping the performance of the PEWMA control charts.  
We embarked on an exploration of the performance of PEWMA control charts when confronted with process data fitted to a Poisson mixture distribution. The key objective was to gain insights into how the PEWMA control charts respond under varying conditions of process mean shifts induced by the incorporation of nuisance data. Our investigation aimed to shed light on the optimal selection of design parameters to ensure efficient and effective detection of ARLs across different shifts in mean. To achieve this, we employed a method that involved the generation of nuisance data by mixing data from a Poisson distribution characterized by diverse parameters into a dataset, subsequently resembling another Poisson distribution. This approach endowed the new dataset with a Poisson mixture distribution while introducing noticeable shifts in the process mean. This allowed us to simultaneously investigate the influence of distribution characteristics and mean shifts on the PEWMA control chart performance.  
In the simulation experiments conducted with this Poisson mixture data and different combinations of design parameters, we observed that selecting a large value for \(\lambda\) resulted in better performance for almost all values of \(A\) when nuisance data caused small shifts in the process mean. Conversely, when nuisance data caused large shifts, selecting a small value for \(\lambda\) shows better performance for almost all values of \(A\). Informative graphical representations and tables have been meticulously crafted to facilitate users in the optimal parameter selection. These images serve as valuable tools to quickly identify ARLs, enabling fast and effective decision making based on average shift magnitudes.  
As a result of this study, Table 5 representing the parameter combinations \(A\) and \(\lambda\) for some shift levels (multiple of \(\sigma\)) in the mean, taking into account the Poisson mixture distribution, to determine the ARLs in different PEWMA control schemes. This template will be useful for selecting parameters before monitoring the process. For example, \(\lambda\) should be chosen small to detect smaller shifts. However, for larger shifts, it is observed that the ARL values do not differ much from each other when \(A<2.90\) and \(\lambda<0.4\).