## 3 Practical Application  
In this section, a numerical example is used to verify the proposed confidence interval based fuzzy evaluation model. As mentioned earlier, lead frame deformation (\(h=1\)), wire sweep (\(h=2\)), and package warpage (\(h=3\)) are three crucial STB quality characteristics associated with the IC packaging molding process. Based on [5,20], according to Table 2, if we want to test whether \(Q_{PU\mu}\) is bigger than 5.28\(\alpha\) (\(\omega^{\prime}\)= 5.28), then the null hypothesis \(H_{0}:Q_{PU\mu}\geq 5.28\) versus the alternative hypothesis will be \(H_{1}:Q_{PU\mu}<5.28,\quad h=1,\quad 2,\quad 3.\) The observed values of three indices with sample size \(n=36\) can be described as follows:  
\[Q_{PU\mu 10}^{*}=4.2,\;Q_{PU\mu 20}^{*}=5.1,\;\text{and}\;Q_{PU30}^{*}=5.3\]  
Based on Equations (19) and (20), the half-triangular fuzzy numbers of \(\widetilde{Q}_{PU\mu 10}\), \(\widetilde{Q}_{PU\mu 20}\), and \(\widetilde{Q}_{PU30}\) are as follows:  
\[\Delta(Q_{M1},\;Q_{R1})=\Delta(4.17,\;5.47),\;\Delta(Q_{M2},\;Q_{R2})=\Delta(5.07,\;6.37),\;\text{and}\;\Delta(Q_{M3},\;Q_{R3})=\Delta(5.27,\;6.57)\]  
In addition, based on Equation (21), the membership function of fuzzy number \(\widetilde{Q}_{PU\mu 10}\) is  
\[\eta_{1}(x)=\left\{\begin{array}{ll}0&if\;x<4.17\\ 1&if\;x=4.17\\ \alpha&if\;4.17<x\leq 5.47\\ 0&if\;5.47<x\end{array}\right.,\]  
where \(\alpha\) is determined by  
\[2.7\times\sqrt{\frac{\chi_{1-\alpha/2,35}^{2}}{35}}+\frac{Z_{\alpha/2}}{6}+1.5 =x.\]  
Similarly, the membership function of fuzzy number \(\widetilde{Q}_{PU\mu 20}\) is  
\[\eta_{2}(x)=\left\{\begin{array}{ll}0&if\;x<5.07\\ 1&if\;x=5.07\\ \alpha&if\;5.07<x\leq 6.37\\ 0&if\;6.37<x\end{array}\right.,\]  
where \(\alpha\) is determined by  
\[3.6\times\sqrt{\frac{\chi_{1-\alpha/2,35}^{2}}{35}}+\frac{Z_{\alpha/2}}{6}+1.5 =x.\]  
Figure 2: Flowchart of the fuzzy evaluation model for the IC packaging molding process.  
The membership function of fuzzy number \(\widetilde{Q}_{PU30}\) is  
\[\eta_{3}(x)=\left\{\begin{array}{ll}0&if\ x<5.27\\ 1&if\ x=5.27\\ \alpha&if\ 5.27<x\leq 6.57\\ 0&if\ 6.57<x\end{array}\right.,\]  
where \(\alpha\) is determined by  
\[3.8\times\sqrt{\frac{\chi_{1-\alpha/2,35}^{2}}{35}}+\frac{Z_{\alpha/2}}{6}+1.5 =x.\]  
Obviously, the upper confidence limits of these three indices were all larger than 5.28. Based on statistical testing rules, we do not reject \(H_{0}\). However, although the observed value of \(Q_{PU10}^{*}\) was 4.2, which is far less than 5.28, our conclusion was that \(Q_{PU1}\geq 5.28\). This was because the smaller sample size (\(n=36\)) led to a substantial error. We therefore applied the fuzzy evaluation model proposed in Section 2.3. Based on Equations (25) and (26), the values of \(d_{Rh}\) and \(d_{Th}\) for \(h=1\), 2, 3 are as follows:  
\[\begin{array}{ll}d_{R1}=Q_{R1}-5.28=5.47-5.28=0.19;\\ d_{R2}=Q_{R2}-5.28=6.37-5.28=1.09;\\ d_{R3}=Q_{R3}-5.28=6.57-5.28=1.29;\\ d_{T1}=Q_{R1}-Q_{M1}=5.47-4.17=1.30;\\ d_{T2}=Q_{R2}-Q_{M2}=6.37-5.07=1.30;\\ d_{T3}=Q_{R3}-Q_{M3}=6.57-5.27=1.30.\end{array}\]  
Therefore,  
\[d_{R1}/2d_{T1}=0.19/2.60=0.073d_{R2}/2d_{T2}=1.09/2.60=0.419d_{R3}/2d_{T3}=1.29 /2.60=0.496.\]  
In fact, \(\phi_{1}\) and \(\phi_{2}\) can be determined by industry characteristics. In this paper, we adopted [21] to set \(\phi_{1}\) and \(\phi_{2}\) as 0.2 and 0.4, respectively. Because \(d_{R1}/2d_{T1}=0.073<\phi_{1}\), then we reject \(H_{0}\) and conclude that \(Q_{PU1}<5.28\). However, since both \(d_{R2}/2d_{T2}\) and \(d_{R3}/2d_{T3}\) are larger than \(\phi_{2}\), we do not reject \(H_{0}\) and conclude that \(Q_{PU2}\geq 5.28\) and \(Q_{PU3}\geq 5.28\). Clearly, in this case, the quality level of lead frame deformation in the IC packaging molding process does not meet requirements; improvements are necessary.