# Fuzzy Quality Evaluation Model Constructed by Process Quality Index
## 5 A Practical Application  
When the customer's requirement for the quality level of the process is 4-sigma (\(k=4\)), then we adopt the following hypotheses:  
null hypothesis \(H_{0}\): \(P_{Q1}\geq 4\)  
versus  
alternative hypothesis \(H_{1}\):\(P_{Q1}<4\). In order to prove the practical application of our methodology, we have the observed value \((y_{h,1},y_{h,2},\cdots,y_{h,11})\) of \(h\)th subsample (\(h\ =\ 1,2,\ldots,25\)) with \(n=11\), \(m=25\), and \(N=275\) from in-control chart data. Then, the critical value can be denoted as follows:  
\[C_{0}\ =\ \frac{t_{0.01,250}\Big{(}\Delta\ =\ 4\sqrt{275}\Big{)}}{\sqrt{275}}\ =\ 3.599 \tag{37}\]  
Thus, the half-triangular shaped fuzzy number of \(\widetilde{C}_{0}\) is denoted as \(\Delta\widetilde{C}_{0}\ =\ (3.599,\ 4.197)\), where  
\[C_{M}\ =\ C_{0}\ =\ 3.599 \tag{38}\]and  
\[C_{R}\ =\ \left(C_{0}+\frac{t_{0.005;N-m}}{\sqrt{mn}}\right)\sqrt{\frac{\chi_{0.99 5;N-m}^{2}}{\chi_{0.5;N-m}^{2}}}\ =\ 4.197. \tag{39}\]  
Then, the fuzzy membership function of \(\widetilde{C}_{0}\) is defined as follows:  
\[\eta(x)\ =\ \left\{ \begin{array}{lll}0&if&x<3.599\\ 1&if&x\ =\ 3.599\\ \alpha&if&3.599<x\leq 4.197\\ 0&if&4.197<x\end{array} \right.. \tag{40}\]  
Based on Equation (31), \(\alpha\) is determined by  
\[\left(3.599+\frac{t_{\alpha/2250}}{\sqrt{275}}\right)\sqrt{\frac{\chi_{1- \alpha/2;250}^{2}}{\chi_{0.5;250}^{2}}}\ =\ x. \tag{41}\]  
In addition, \(\delta_{0}^{*}\) and \(\gamma_{0}^{*}\) are the observed values of \(\delta^{*}\) and \(\gamma^{*}\), respectively, as follows:  
\[\delta_{0}^{*}\ =\ \frac{1}{m}\sum_{h\ =\ 1}^{m}\overline{y}_{h}\ =\ \frac{1}{25}\sum_{h\ =\ 1}^{25} \overline{y}_{h}\ =\ 0.691 \tag{42}\]  
and  
\[\gamma_{0}^{*}\ =\ \sqrt{\frac{1}{N-m}\sum_{h\ =\ 1}^{m}(n-1)s_{h}^{2}}\ =\ \sqrt{\frac{1}{250}\sum_{h\ =\ 1}^{25}10\times s_{h}^{2}}\ =\ 0.085 \tag{43}\]  
Therefore, the observed value of \(P_{QI0}^{*}\) is as follows:  
\[P_{QI0}^{*}\ =\ \frac{1-\delta_{0}^{*}}{\gamma_{0}^{*}}\ =\ 3.635 \tag{44}\]  
Thus, the observed value of the upper confidence limit of index \(P_{QI}\) with \(\alpha\ =\ 0.01\) can be shown as follows:  
\[x_{R}\ =\ \left(P_{QI0}^{*}+\frac{t_{0.005;250}}{\sqrt{275}}\right)\sqrt{\frac{ \chi_{0.995;250}^{2}}{\chi_{0.5;250}^{2}}}\ =\ \left(3.635+\frac{2.596}{\sqrt{275}}\right)\sqrt{\frac{311.346}{249.334}}\ =\ 4.237 \tag{45}\]  
The half-triangle-shaped fuzzy number of \(\widetilde{x}\) is \(\Delta\widetilde{x}\ =\ (\ 3.635,\ 4.237)\). Then, the fuzzy membership function of \(\widetilde{x}\) is illustrated below:  
\[\eta_{0}(x)=\ \left\{ \begin{array}{lll}0&if&x<3.635\\ 1&if&x\ =\ 3.635\\ \alpha&if&3.635<x\leq 4.237\\ 0&if&4.237<x\end{array} \right., \tag{46}\]  
where \(\alpha\) is determined by  
\[\left(3.635+\frac{t_{\alpha/2250}}{\sqrt{275}}\right)\sqrt{\frac{\chi_{1- \alpha/2;250}^{2}}{\chi_{0.5;250}^{2}}}\ =\ x. \tag{47}\]  
Figure 3 is a diagram of membership function \(\eta_{0}(x)\) with \(\Delta\widetilde{x}\ =\ (\ 3.635,\ 4.237)\) and \(\eta(x)\) with \(\Delta\widetilde{C}_{0}\ =\ (3.599,\ 4.197)\). Then,  
\[d_{R}\ =\ \left(C_{0}+\frac{t_{0.005;250}}{\sqrt{275}}\right)\sqrt{\frac{\chi_{0.9 95;250}^{2}}{\chi_{0.5;250}^{2}}}-P_{QI0}^{*}\ =\ 4.197-3.635\ =\ 0.562 \tag{48}\]  
and  
\[d_{T}\ =\ \left(C_{0}+\frac{t_{0.005;250}}{\sqrt{275}}\right)\sqrt{\frac{\chi_{0.9 95;250}^{2}}{\chi_{0.5;250}^{2}}}-C_{0}\ =\ 4.197-3.599\ =\ 0.598 \tag{49}\]  
Then,  
\[\frac{d_{R}}{2d_{T}}\ =\ \frac{\left(P_{QI0}^{*}+\frac{t_{0.005;250}}{\sqrt{275 }}\right)\sqrt{\frac{\chi_{0.995;250}^{2}}{\chi_{0.5;250}^{2}}}-C_{0}}{2\Big{(} \left(P_{QI0}^{*}+\frac{t_{0.005;N-n}}{\sqrt{mn}}\right)\sqrt{\frac{\chi_{0.99 5;N-m}^{2}}{\chi_{0.5;N-m}^{2}}}-x_{M}\Big{)}}\ =\ \frac{0.562}{2\times 0.598}\ =\ 0.469 \tag{50}\]  
According to [PERSON] and [PERSON] [32], this paper takes \(\phi_{1}\ =\ 0.2\) and \(\phi_{2}\ =\ 0.4\). Based on the fuzzy test rules in Section 4, the value of \(d_{R}/(2d_{T})\) is \(0.469\), so that \(H_{0}\) will be rejected and \(P_{QI}<4\) will be concluded.