## 2 Case Introduction
### Manufacturing Process Quality Evaluation Index  
External diameter, internal diameter, and threads were used as important references. Using variables to convert important process quality characteristics, they were subsequently standardized to determine the manufacturing process quality of bearing connectors. If a random variable \(X_{h}\) is hypothesized to indicate the distribution of the manufacturing process of external diameter (\(h=1\)), internal diameter (\(h=2\)), and threads (\(h=3\)), respectively, accuracy index \(\delta\) and precision index \(\gamma\) are first defined as follows:  
\[\delta=\frac{\mu-T}{d}\,\gamma=\frac{\sigma}{d}\]  
where \(\delta\) is used to judge the mean deviation of the process from the target value, \(\gamma\) is used to determine the degree of process variation, and \(d\) represents the tolerance of process quality characteristics. When the process quality level of the product reaches \(k\sigma\), \(\delta\) and \(\gamma\) can be represented as follows:  
\[|\delta|=\frac{\left|\mu-T\right|}{d}\leq\frac{1.5}{k} \tag{1}\]  
\[\gamma=\frac{\sigma}{d}=\frac{1}{k} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline
**Quality Characteristic** & **Tolerance** & **Manufacturing Time** \\ \hline External diameter & \(19\pm 0.04\) & \(300\pm 40\)/s \\ Internal diameter & \(10^{+0.03}_{+0.01}\) & \(300\pm 40\)/s \\ Threads & \(8.125\pm 0.1\) & \(600\pm 40\)/s \\ \hline \end{tabular}
\end{table} 
Table 1: Process quality characteristic specifications. Figure 3: Product diagram. Apparently, the SSQIs proposed by [PERSON] et at. [PERSON] et al. (2019) can be seen as the function of accuracy index and precision index; therefore, they can adequately display the causes of poor process quality (i.e., insufficient accuracy or insufficient precision) to determine the factors of process performance. Based on this, the indices can be re-defined as the following:  
\[Q_{puh}=\frac{1-\delta_{h}}{\gamma_{h}}+1.5 \tag{3}\]  
\[Q_{pth}=\frac{1+\delta_{h}}{\gamma_{h}}+1.5 \tag{4}\]  
\[Q_{pth}=\min\Bigl{(}Q_{puh},Q_{pth}\Bigr{)}=\left(\frac{1-\delta_{h}}{\gamma_{ h}}+1.5,\ \frac{1+\delta_{h}}{\gamma_{h}}+1.5\right) \tag{5}\]  
Many studies indicate that if a product has multiple quality characteristics, the product can satisfy customers and be viewed as a good product only when all quality characteristics of the process quality meet the required quality level (15; 16; 18; 19; 20; 21; 22). According to [PERSON] et al. (2019), the required quality level of quality characteristics is \(k^{\prime}\sigma\) and the required quality level of bearing connector is \(k\sigma\). That is, the quality level of the manufacturing process concerning all process quality characteristics must be increased to ensure the reliability and overall production process quality of bearing connectors. If any of a product's process quality characteristics fail to meet the specifications, the product is regarded as defective. Under the hypothesis that the manufacturing process quality has a normal distribution, this study defined index events to explain the corresponding yield rate as the following:  
\(E_{h}\): Event of quality characteristic \(h\) meeting the specifications: \(P(E_{h})\geq p_{h}=2\Phi\Bigl{(}Q_{pth}-1.5\Bigr{)}-1\)  
\(E_{h}^{c}\): Event of quality characteristic \(h\) not meeting the specifications: \(P\Bigl{(}E_{h}^{C}\Bigr{)}\leq 1-P_{h}\leq 2\times\Bigl{[}1-\Phi\Bigl{(}Q_{pth}-1.5 \Bigr{)}\Bigr{]}\). The relationship between bearing connectors' crucial quality characteristics and yield rate \(P_{T}\) can be represented by the following:  
\[P\Bigl{(} \begin{subarray}{c}3\\ h=1\end{subarray} E_{h}^{C}\Bigr{)}\leq\sum_{h=1}^{3}P\Bigl{(}E_{h}^{C}\Bigr{)} \leq 6\times[1-\Phi(k^{\prime}-1.5)]\]  
\[P_{T}=P\Bigl{(} \begin{subarray}{c}3\\ h=1\end{subarray}