# The Fuzzy Process Quality Evaluation Model for the STB Quality Characteristic of Machining
## 3 Fuzzy Testing Method  
The advantage of the confidence-interval-based fuzzy evaluation method is that the original measurement method is used to collect data, which can improve the timeliness of data collection [27, 28]. Next, the fuzzy number is constructed by means of the confidence interval, and then fuzzy evaluation rules are proposed according to the fuzzy number to increase the accuracy of evaluation. On the basis of this concept, this study directly used the measurement data of the control chart as the evaluation data. First, let \((x_{h,1},\ldots\ x_{h,j},\ldots\ x_{h,n})\) represent the observed values of \((X_{h,1},\ldots,X_{h,j},\ldots,X_{h,n})\) for \(h=1\),..., m, and then observed values of \(\overline{X}_{h}\) and \(S_{h}\) can be shown, respectively, as follows:  
\[\overline{x}_{h}=\sum_{j=1}^{n}x_{hj} \tag{22}\]  
and  
\[s_{h}=\sqrt{\frac{1}{n}{\sum_{j=1}^{n}}{\left(x_{hj}-\overline{x}_{h}\right) }^{2}}. \tag{23}\]  
Therefore, the observed values of \(\overline{X}\) and \(S\) are  
\[\overline{x}=\frac{1}{m}{\sum_{h=1}^{m}}\overline{x}_{h} \tag{24}\]  
and  
\[s^{2}=\frac{1}{m}{\sum_{h=1}^{m}}s_{h}^{2}. \tag{25}\]  
Obviously, the observed value of \(UQ_{\text{ISS}}\) can be denoted as follows:  
\[UQ_{\text{ISS0}}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\lambda_{1-\alpha/2,m( n-1)}^{2}}{m(n-1)}}+\frac{Z_{\alpha/2}}{\sqrt{mn}} \tag{26}\]  
where \(\hat{Q}_{\text{ISS0}}\) is the observed value of \(\hat{Q}_{\text{ISS}}\) as follows:  
\[\hat{Q}_{\text{ISS0}}=\frac{USL-\overline{\overline{x}}}{s}. \tag{27}\]  
Many studies have pointed out that a process capability evaluation is made based on the upper confidence limit of the index. Then, the null hypothesis \(H_{0}\) is \(Q_{\text{ISS}}\geq k\), and alternative hypothesis \(H_{a}\) is \(Q_{\text{ISS}}\)\(<\)k. The statistical testing rules can be expressed as follows [26, 29]:  
(1) If \(\hat{Q}_{\text{ISS0}}\geq k\), then \(H_{0}\) is not rejected, and \(Q_{\text{ISS}}\geq k\) is concluded. (2) If \(\hat{Q}_{\text{ISS0}}<k\), then \(H_{0}\) is rejected, and \(Q_{\text{ISS}}<k\) is concluded. According to the above two statistical testing rules and the method introduced by [PERSON] [29], a fuzzy testing method was developed on the basis of the upper confidence limit of the quality index. As suggested by [PERSON] [29], the \(\alpha-\)cuts of the triangular fuzzy number \(Q_{\text{ISS}}\) can be acquired below:  
\[\widetilde{\hat{Q}}_{\text{ISS0}}[\alpha]=\left\{ \begin{array}{ll}\left[\hat {Q}_{\text{ISS0}}(1),\hat{Q}_{\text{ISS0}}(\alpha)\right],&\text{for }0.01\leq\alpha\leq 1\\ \left[\hat{Q}_{\text{ISS0}}(1),\hat{Q}_{\text{ISS0}}(0.01)\right],&\text{for }0 \leq\alpha\leq 0.01\end{array} \right. \tag{28}\]  
\[UQ_{\text{ISS0}}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\lambda_{1-\alpha/2,m (n-1)}^{2}}{m(n-1)}}+\frac{Z_{\alpha/2}}{\sqrt{mn}} \tag{29}\]where  
\[\hat{Q}_{\text{ISS0}}(1)=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{0.5,m(n-1)} ^{2}}{m(n-1)}}\]  
\[\hat{Q}_{\text{ISS0}}(\alpha)=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{1- \alpha/2,m(n-1)}^{2}}{m(n-1)}}+\frac{Z_{\alpha/2}}{\sqrt{mn}}.\]  
Therefore, the half-triangular fuzzy number of \(Q_{\text{ISS}}\) is \(\widetilde{\hat{Q}}_{\text{ISS0}}=(Q_{IM},Q_{IR})\), where  
\[Q_{IM}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{0.5,m(n-1)}^{2}}{m(n-1)}} \tag{30}\]  
\[Q_{IR}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{0.995,m(n-1)}^{2}}{m(n-1)} }+\frac{Z_{0.005}}{\sqrt{mn}}. \tag{31}\]  
Therefore, the membership function of the fuzzy number \(Q\) is  
\[\eta(x)=\left\{ \begin{array}{ccc}0&if&x<Q_{IM}\\ 1&if&x=Q_{IM}\\ \alpha&if&Q_{IM}<x<Q_{IR}\\ 0&if&x\geq Q_{IR}\end{array} \right. \tag{32}\]  
where \(\alpha\) is determined by  
\[x=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{1-\alpha/2,m(n-1)}^{2}}{m(n-1) }}+\frac{Z_{\alpha/2}}{\sqrt{mn}}.\]  
In other words, when \(\alpha=a\), the corresponding value of \(x\) is \(x_{0}\), that is \(\eta(x_{0})=a\).