# Construct the Optimum Process Model for Transistor Gaskets with Six-Sigma DMAIC
## 2 Research Methods
### Measure
To derive the \(100(1-\alpha)\%\) joint confidence region of index \(\delta_{h}\) and index \(\gamma_{h}\), this paper defines \(E_{\delta h}=\{\delta_{Lh}\leq\delta_{Llh}\}\) and \(E_{\gamma h}=\{\gamma_{Lh}\leq\gamma_{h}\leq\gamma_{UH}\}\), where  
\[\delta_{Lh}=\delta_{h}-\frac{t_{\alpha/4\eta-1}}{\sqrt{n}}\hat{\gamma}_{h},\ \delta_{UH}=\delta_{h}+\frac{t_{\alpha/4\eta-1}}{\sqrt{n}}\hat{\gamma}_{h} \tag{8}\]  
\[\gamma_{Lh}=\sqrt{\frac{n}{\chi_{1-\alpha/4\eta-1}^{2}}}\hat{\gamma}_{h},\ \gamma_{UH}=\sqrt{\frac{n}{\chi_{\alpha/4\eta-1}^{2}}}\hat{\gamma}_{h} \tag{9}\]  
In fact, \(p(E_{\delta h})=p\big{(}E_{\gamma h}\big{)}=1-\alpha/2\) and \(p\big{(}E_{\delta h}^{\mathrm{C}}\big{)}=p\Big{(}E_{\gamma h}^{\mathrm{C}} \big{)}=\alpha/2\). On the basis of Boole's inequality and DeMorgan's rules, we have \(p\big{(}E_{\delta h}\cap E_{\gamma h}\big{)}\geq 1-p\big{(}E_{\delta h}^{ \mathrm{C}}\big{)}-p\big{(}E_{\gamma h}^{\mathrm{C}}\big{)}=\alpha/2\). Therefore, the \(100(1-\alpha)\%\) joint confidence region of index \(\delta_{h}\) and index \(\gamma_{h}\) is exhibited as:  
\[CR=\{\delta_{Lh}\leq\delta_{h}\leq\delta_{UH},\gamma_{LH}\leq\gamma_{h}\leq \gamma_{UH}\}\]  
Figure 1: The MPCAC of transistor gaskets. In fact, the \(100(1-a)\%\) joint confidence region CR is a rectangle, and the coordinate points of the four corners of the rectangle are expressed as follows:  
\[\text{upper}-\text{left coordinate point}:\ (\delta_{Lh},\gamma_{UH}) =\left(\delta_{h}-\frac{t_{a/4m-1}}{\sqrt{n}}\hat{\gamma}_{h},\ \sqrt{\frac{n}{\lambda_{a/4m-1}^{2}}}\hat{\gamma}_{h}\right); \tag{10}\]  
\[\text{lower}-\text{left coordinate point}:\ (\delta_{Lh},\gamma_{Lh}) =\left(\delta_{h}-\frac{t_{a/4m-1}}{\sqrt{n}}\hat{\gamma}_{h},\ \sqrt{\frac{n}{\lambda_{1-a/4m-1}^{2}}}\hat{\gamma}_{h}\right); \tag{11}\]  
\[\text{upper}-\text{right coordinate point}:\ (\delta_{UH},\gamma_{UH}) =\left(\delta_{h}+\frac{t_{a/4m-1}}{\sqrt{n}}\hat{\gamma}_{h},\ \sqrt{\frac{n}{\lambda_{a/4m-1}^{2}}}\hat{\gamma}_{h}\right); \tag{12}\]  
\[\text{lower}-\text{right coordinate point}:\ (\delta_{UH},\gamma_{Lh}) =\left(\delta_{h}+\frac{t_{a/4m-1}}{\sqrt{n}}\hat{\gamma}_{h},\ \sqrt{\frac{n}{\lambda_{1-a/4m-1}^{2}}}\hat{\gamma}_{h}\right). \tag{13}\]  
Based on the concept of [PERSON] et al. [13], the coordinate point corresponding to the upper confidence limit of process capability index \(C_{pkh}\) is the measurement coordinate point \((x_{h},y_{h})\), \(h=1,2,3,4,5\) as follows:  
\[(x_{h},y_{h})=\left\{ \begin{array}{l}(\delta_{Lh},\gamma_{Lh}),\delta_{Lh} >0\\ (0,\gamma_{Lh}),\delta_{Lh}\leq 0\leq\delta_{UH}\\ (\delta_{UH},\gamma_{Lh}),\delta_{UH}<0\end{array} \right. \tag{14}\]  
Then, based on the location of the coordinate point, the process quality of quality characteristic h is measured, and the measurement rules are listed as follows:  
When \((x_{h},y_{h})\)\(\in\)\(A\), then the value of process capability index \(C_{pkh}\) is larger than 1.61, and the quality level of quality characteristic h reaches six-sigma. When \((x_{h},y_{h})\)\(\
otin\)\(A\), the value of process capability index \(C_{pkh}\) is smaller than 1.61, and the quality level of quality characteristic h does not attain six-sigma. Thus, the process quality of quality characteristic h must be boosted. This paper calls this quality characteristic critical to quality, CTQ.