## 2 Materials and Methods  
### Six Sigma Quality Index for an Integrated-Circuit (IC) Packaging Molding Process  
As previously mentioned, the molding process in IC packaging has three crucial STB quality characteristics: (A) lead frame deformation, (B) wire sweep, and (C) package warpage. For these STB quality characteristics, the target value \(T\) is zero [19]. Thus, the tolerance \(d=USL-T=USL\), and let  
\[Y=\frac{X-T}{d}=\frac{X}{USL}, \tag{1}\]  
where \(USL\) is the upper specification limit. Then, \(Y\) is distributed normally with mean \(\delta=\mu/USL\) and standard deviation \(\gamma=\sigma/USL\left(\text{ }i.e.,\ Y\sim N(\delta,\gamma^{2})\right)\). Therefore, the transformed \(USL\) is 1, and an SSQI for STB quality characteristics can be defined as follows [19]:  
\[Q_{PU}=\frac{USL-\mu}{\sigma}+1.5=\frac{1-\delta}{\gamma}+1.5. \tag{2}\]  
When \(USL-\mu\geq(\omega-1.5)\sigma\), then the process reaches \(\omega-sigma\) quality level [19]. If we let \(Q_{PU}(\omega)\) denote the value of \(Q_{PU}\) with \(USL-\mu=(\omega-1.5)\sigma\), then  
\[Q_{PU}\geq Q_{PU}(\omega)=\frac{(\omega-1.5)\sigma}{\sigma}+1.5=\omega. \tag{3}\]  
Obviously, when the process quality level is \(\omega-sigma\), then the value of \(Q_{PU}\) is at least equal to \(\omega\). Furthermore, under the assumption of normality, there exists a one-to-one mathematical relationship between index \(Q_{PU}\) and process yield, which can be shown as follows:  
\[p=\int_{-\infty}^{1}N(\delta,\gamma^{2})d_{y}=\int_{-\infty}^{Q_{PU}-1.5}\ 1\ \ d_{\Phi(z)}=\Phi(Q_{PU}-1.5), \tag{4}\]  
where \(\Phi(z)\) is the cumulative function of standard normal distributions (i.e., \(Z\sim N(0,1)\)). Table 1 displays the upper specification limit \(USL_{h}\) of these three quality characteristics (\(h=1,\,2,\) and \(3\)). Assume a random variable \(Y_{h}\) is normally distributed with mean \(\delta_{h}\) and standard deviation \(\gamma_{h}\)(_i.e.,_\(Y_{h}\sim N(\delta_{h},\gamma_{h}^{2})\)). The SSQI for these three quality characteristics can, therefore, be defined as follows:  
\[Q_{PUh}=\frac{1-\delta_{h}}{\gamma_{h}}+1.5. \tag{5}\]  
As in the work of [1], the SSQI for the IC packaging molding process can be defined as follows:  
\[Q_{PU}^{T}=\Phi^{-1}\Bigg{\{}1-\sum\limits_{h=1}^{3}[1-\Phi(Q_{PUh}-1.5)] \Bigg{\}}+1.5. \tag{6}\]  
Let the event \(E_{h}=\{Y_{h}\leq 1\}\) and the compliment event of \(E_{h}\) be \(E_{h}^{C}=\{Y_{h}>1\}\), \(h=1,\,2,\) and \(3\). Then, based on Equation (4),  
\[p_{h}=p[Y_{h}\leq 1]=p\Bigg{\{}Z\leq\frac{1-\delta_{h}}{\gamma_{h}}\Bigg{\}}= \int_{-\infty}^{Q_{PUh}-1.5}\frac{1}{\sqrt{2\pi}}exp\Bigg{\{}\frac{z^{2}}{2} \Bigg{\}}dz_{z}=\Phi(Q_{PUh}-1.5); \tag{7}\]  
\[q_{h}=1-p_{h}=p\Big{(}E_{h}^{C}\Big{)}=1-\Phi(Q_{PUh}-1.5). \tag{8}\]  
Let \(E=E_{1}\cap E_{2}\cap E_{3}\). Then, based on the [PERSON] theorem and [PERSON]'s inequality, the yield \(p_{T}\) of the IC packaging molding process is as follows:  
\[p_{T}=P(E)=P(E_{1}\cap E_{2}\cap E_{3})\geq 1-P\Big{(}E_{1}^{C}\Big{)}-P \Big{(}E_{2}^{C}\Big{)}-P\Big{(}E_{3}^{C}\Big{)}=1-\sum\limits_{h=1}^{3}[1- \Phi(Q_{PUh}-1.5)]=\Phi\Big{(}Q_{PU}^{T}-1.5\Big{)}.\]  
Thus, there is an inequality between index \(Q_{PU}^{T}\) and process yield \(p_{T}\). For example, when \(Q_{PU}^{T}=5.0\), then we can guarantee the product yield \(p_{T}\geq\Phi(3.5)=99.977\%\). Let \(Q_{PU}^{T}=\omega\), then  
\[p_{T}\geq p_{T}(\omega)=\Phi(\omega-1.5). \tag{9}\]  
Obviously, \(p_{T}\Big{(}Q_{PU}^{T}\Big{)}\) is an increasing function of \(Q_{PU}^{T}\). When the required quality level of the IC packaging molding process is \(\omega-sigma\), then the quality levels of the three quality characteristics must reach \(\omega^{\prime}-sigma\) and \(\omega\leq\omega^{\prime}\)[5; 20]. Based on Equation (6),  
\[Q_{PU}=\frac{USL-\mu}{\sigma}+1.5=\frac{1-\delta}{\gamma}+1.5. \tag{2}\]0  
Equivalently,  
\[Q_{PU}=\frac{USL-\mu}{\sigma}+1.5=\frac{1-\delta}{\gamma}+1.5. \tag{2}\]1  
Table 2 presents a comparison of quality levels for \(\omega\) and \(\omega^{\prime}\).