## 4 Application in Fastener Quality Control  
In order to demonstrate the applicability of the Poisson mixture distribution in the fields of statistical quality control and reliability, the quality control processes from the automotive supplier sector were taken into account. The company produces a diverse range of fastener types. Among these various fasteners, we specifically focused on the data related to large flange rivets and square nuts because our tests indicated that they fit the Poisson mixture distribution. A square nut is a nut with four sides. They are commonly  
Figure 6: ARLs for various values of \(A\) and mixture proportions (\(\pi\))-shift in mean with \(\lambda=0.1\), \(\alpha_{1}=5\), \(\alpha_{2}=5.5\). Figure 7: ARLs for various values of \(A\) and mixture proportions (\(\pi\))-shift in mean with \(\lambda=0.1\), \(\alpha_{1}=5\), \(\alpha_{2}=7\). used alongside a bolt to join two or more objects together. A large flange river is ideal for high-pressure applications that require a seal to prevent the ingress of liquids, solids, or moisture. The pictures of square nut and large flange river are given in Figure 8.  
As part of its quality control policy, the company conducts a count of defective products within 500 randomly sampled units from the production process. The defects recorded during quality control of the products include cracks, wrinkles, shear burst, imperfections, damage, and coating flaws. In Figure 9, some examples of the defects are shown for each product. First, we applied Kolmogorov-Smirnov and chi-square tests to determine whether these data fit a discrete distribution such as Poisson, binomial and geometric distribution. Figure 8: Products: (**a**) Square nut; (**b**) Large flange river. Figure 9: Defects: (**a**) Shear on the square nut (**b**) Crack on the large flange river. This result led us to consider that these data might come from a mixture distribution. We used the package Flexmix in R version 4.3.0 to determine whether these data fit the Poisson mixture distribution. This package calculates the AIC, BIC and LL values using the Equations (22) and (23) to determine the number of components of the Poisson mixture distribution. To determine the number of components, run parameters are set at 10 replications and 1000 iterations per replication. In the replication where AIC and BIC are the smallest and LL is the largest, the \(k\) value is determined as the optimal number of components. Accordingly, the \(k\) values for the square nut and large flange river are selected as 4 and 3, respectively. The results are given in Tables 2 and 3 for the square nut and large flange river, respectively. Once the number of Poisson mixture components was determined for each set of product sample data, we used the Flexmix package in the R programming environment to compute the proportions and parameters for each Poisson mixture distribution. The proportions and parameters corresponding to each product sample data are given in Table 4.  
To verify whether the fitted Poisson mixture distributions represent the actual data distribution visually, the histograms of real data and generated data are illustrated in Figures 10 and 11. Next, we employed the Mann-Whitney U test to determine whether the generated data accurately represents the real data. The test hypotheses are formulated as follows: \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Product**} & \multicolumn{4}{c}{**Proportions**} & \multicolumn{4}{c}{**Parameters**} \\ \cline{2-7}  & \(\mathbf{\pi_{1}}\) & \(\mathbf{\pi_{2}}\) & \(\mathbf{\pi_{3}}\) & \(\mathbf{\pi_{4}}\) & \(\mathbf{\pi_{1}}\) & \(\mathbf{\pi_{2}}\) & \(\mathbf{\pi_{3}}\) & \(\mathbf{\pi_{4}}\) \\ \hline Square Nut & 0.14 & 0.39 & 0.09 & 0.38 & 11.165 & 60.348 & 163.485 & 29.883 \\ Large Flange Rivet & 0.06 & 0.53 & 0.41 & - & 115.11 & 0.697 & 21.782 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Poisson mixture distribution parameters for each product data.