# Construct the Optimum Process Model for Transistor Gaskets with Six-Sigma DMAIC
## 2 Research Methods
### Measure  
Next, in this paper, we built a MPCAC for transistor gaskets using \(\delta_{h}\) as the x-axis and \(\gamma_{h}\) as the y-axis to simultaneously measure the process capabilities of the five important quality characteristics of transistor gaskets. First, let \(C_{pbh}=\)1.61, \(\delta_{h}=x\), and \(\gamma_{h}=y\), \(h=1,2,3,4,5\). Then, the process capability acceptance zone \(A\) is expressed in the following equation:  
\[A=\{\,(x,y)|x+4.83y\leq 1,x-4.83y\geq-1,y\geq 0\} \tag{4}\]  
Furthermore, the MPCAC of transistor gaskets is displayed in Figure 1 below:  
This paper derived the \(100(1-\alpha)\%\) joint confidence region of index \(\delta_{h}\) and index \(\gamma_{h}\) based on sample data. Let \(Y_{h,1},\ldots,Y_{h,j},\ldots,Y_{h,n}\) be a random sample from quality characteristic \(h\). Then the estimators of \(\delta_{h}\) and \(\gamma_{h}\) can be expressed as follows:  
\[\delta_{h}=\frac{1}{n}{\sum_{j=1}^{n}Y_{j}}\text{ and }\hat{\gamma}_{h}= \sqrt{\frac{1}{n}{\sum_{j=1}^{n}\left(Y_{h,j}-\overline{Y}_{h}\right)^{2}}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Item** & \begin{tabular}{c} **Quality** \\ **Characteristics** \\ \end{tabular} & \(\mathbf{Y_{h}}\) & \(\delta_{h}\) & \(\gamma_{h}\) & \(\mathbf{C_{pbh}}\) \\ \hline \(h=1\) & \begin{tabular}{c} Diameter of fitting \\ assembly parts \\ \end{tabular} & \(\frac{X_{1}-4.5}{0.2}\) & \(\frac{\mu_{1}-4.5}{0.2}\) & \(\frac{\sigma_{1}}{0.2}\) & \(\frac{1-|\delta_{1}|}{3\gamma_{1}}\) \\ \(h=2\) & \begin{tabular}{c} Diameter of fitting \\ assembly hole \\ \end{tabular} & \(\frac{X_{2}-9.8}{0.2}\) & \(\frac{\mu_{2}-9.8}{0.2}\) & \(\frac{\sigma_{2}}{0.2}\) & \(\frac{1-|\delta_{1}|}{3\gamma_{2}}\) \\ \(h=3\) & \begin{tabular}{c} Height of fitting \\ assembly parts \\ \end{tabular} & \(\frac{X_{3}-10.3}{0.3}\) & \(\frac{\mu_{3}-10.3}{0.3}\) & \(\frac{\sigma_{3}}{0.3}\) & \(\frac{1-\delta_{3}}{3\gamma_{3}}\) \\ \(h=4\) & \
\begin{tabular}{c} Spacing \\ Eccentric control \\ \end{tabular} & \(\frac{X_{4}-2}{0.2}\) & \(\frac{\mu_{4}-2}{0.2}\) & \(\frac{\sigma_{4}}{0.2}\) & \(\frac{1-|\delta_{1}|}{3\gamma_{4}}\) \\ \(h=5\) &
\begin{tabular}{c} Eccentric control \\ \end{tabular} & \(\frac{X_{6}}{0.1}\) & \(\frac{\mu_{5}}{0.1}\) & \(\frac{\mu_{5}}{0.1}\) & \(\frac{1-|\delta_{3}}{3\gamma_{5}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation indicators of five important quality characteristics. Considering normality, let \(T=\sqrt{n}\left(\delta_{h}-\delta_{h}\right)/\hat{\gamma}_{h}\) and \(K=n\hat{\gamma}_{h}^{2}/\gamma_{h}^{2}\); then, \(T\) is distributed as \(t\)-distribution with \(n-1\) degree of freedom, denoted by \(t_{n-1}\), and \(K\) is distributed as chi-square distribution with \(n-1\) degree of freedom, denoted by \(\chi_{n-1}^{2}\). Therefore,  
\[ \begin{split}& 1-\alpha/2=p(-t_{\alpha/4\eta-1}\leq T\leq t_{\alpha/4 \eta-1})\\ &=p\left(-t_{\alpha/4\eta-1}\leq\frac{\sqrt{n}\left(\hat{\delta}_ {h}-\delta_{h}\right)}{\hat{\gamma}_{h}}\leq t_{\alpha/4\eta-1}\right)\\ &\qquad=p\left(\delta_{h}-e_{h}\leq\delta_{h}\leq\delta_{h}+e_{h} \right),\end{split}  \tag{6}\]  
where \(e_{h}=t_{\alpha/4\eta-1}\hat{\gamma}_{h}/\sqrt{n}\) and \(t_{\alpha/4\eta-1}\) is the upper \(\alpha/4\) quintile of \(t_{n-1}\). Similarly,  
\[ \begin{split}& 1-\alpha/2=p\left(\chi_{\alpha/4\eta-1}^{2}\leq K \leq\chi_{1-\alpha/4\eta-1}^{2}\right)\\ &\qquad=p\left(\chi_{\alpha/4\eta-1}^{2}\leq\frac{n\hat{\gamma}_ {h}^{2}}{\gamma_{h}^{2}}\leq\chi_{1-\alpha/4\eta-1}^{2}\right)\\ &\qquad=p\left(\sqrt{\frac{n}{\chi_{1-\alpha/4\eta-1}^{2}}}\hat{ \gamma}_{h}\leq\gamma_{h}\leq\sqrt{\frac{n}{\chi_{\alpha/4\eta-1}^{2}}}\hat{ \gamma}_{h}\right),\end{split}  \tag{7}\]  
where \(\chi_{\alpha/4\eta-1}^{2}\) is the lower \(\alpha/4\) quintile of \(\chi_{n-1}^{2}\) and \(\chi_{1-\alpha/4\eta-1}^{2}\) is the lower \(1-\alpha/4\) quintile of \(\chi_{n-1}^{2}\).