# The Fuzzy Process Quality Evaluation Model for the STB Quality Characteristic of Machining
## 4 Case Study  
In this section, this study takes the gear grinding process conducted by a mechanical processing factory in central Taiwan as an example. Roundness is one of the important quality characteristics of the gear grinding process; the requirement for process quality of USL = 0.01 \(\mu m\) must reach the five-sigma (\(k=5\)) quality level. Subsequently, this study used the fuzzy testing rules proposed in Section 3 to appraise whether the process quality met the requirements of the five-sigma quality level. Based on Section 3, the assumptions for performing fuzzy testing can be denoted as follows:  
Null Hypothesis \(H_{0}\):\(Q_{\text{LSS}}\geq 5\).  
Alternative Hypothesis \(H_{a}\):\(Q_{\text{LSS}}<5\).  
Because the subsample size of the gear grinding process \(\overline{X}-S\) control chart is \(n=11\), 20 sets of subsample data (\(m=20\)) were taken to perform fuzzy testing under the control of the statistical process. Then, the observed values of \(\overline{X}\) and \(S\) were calculated separately below:  
\[\overline{\overline{x}}=\frac{1}{20}\sum_{h=1}^{20}\overline{x}_{h}=0.0082\]  
\[s=\sqrt{\frac{1}{20}\sum_{h=1}^{20}s_{h}^{2}}=0.00041.\]  
Furthermore, the observed values of \(\hat{Q}_{\text{ISS}}\) and \(UQ_{\text{ISS}}\) were calculated as follows:  
\[\hat{Q}_{\text{ISS0}}=\frac{USI-\overline{x}}{s}=\frac{0.01-0.0082}{0.00041}=4.39\]  
\[UQ_{\text{ISS0}}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{0.999,200}^{2}}{ 200}}+\frac{Z_{0.005}}{\sqrt{220}}=4.39\times\sqrt{\frac{255.26}{200}}+\frac{2.576}{\sqrt{220}}=5.13.\]  
Based on Equations (30) and (31), we derived the observed values of \(Q_{IM}\) and \(Q_{IR}\) from the following equations:  
\[Q_{IM}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{0.5,200}^{2}}{200}}=4.39 \times\sqrt{\frac{199.33}{200}}=4.38\]  
\[Q_{IR}=\hat{Q}_{\text{ISS0}}\times\sqrt{\frac{\chi_{0.999,200}^{2}}{200}}+\frac {Z_{0.005}}{\sqrt{2.00}}=4.39\times\sqrt{\frac{255.26}{200}}+\frac{2.576}{ \sqrt{220}}=5.13.\]  
Consequently, the half-triangular fuzzy number of \(Q_{\text{ISS}}\) was \(\widetilde{\mathcal{Q}}_{\text{ISS0}}\)\(=\)(4.38,5.13), and the membership function of the fuzzy number \(\widetilde{\mathcal{Q}}_{\text{ISS0}}\) was  
\[\eta(x) = \left\{\begin{array}{ll}0&\text{if}&x<4.38\\ 1&\text{if}&x=4.38\\ \alpha&\text{if}&4.38<x<5.13\\ 0&\text{if}&x\geq 5.13\end{array}\right..\]  
Figure 2 presents a diagram of Membership function \(\eta(x)\) with vertical line \(x=5\).  
Figure 2: Membership function \(\eta(x)\) with vertical line \(x=5\).  
Obviously, according to the above statistical test rule (1), when the observed value \(UQ_{\text{ISS0}}=5.13\)\(\geq\)5 (the required quality level is five-sigma), then \(H_{0}\) was not rejected, and \(Q_{\text{ISS}}\geq\)5 was concluded. In fact, \(\hat{Q}_{\text{ISS0}}=4.39\) was much smaller than the five-sigma quality level. Next, based on Equations (37) and (38), we calculated the value of \(d_{R}\) and \(d_{T}\) as follows:  
\[d_{R}=4.39\times\sqrt{\frac{\chi_{0.995,200}^{2}}{200}}+\frac{Z_{0.005}}{\sqrt{ 220}}-5=0.13\]  
\[d_{T}=2\times\hat{Q}_{\text{ISS0}}\times\left(\sqrt{\frac{\chi_{0.995,200}^{2} }{200}}-\sqrt{\frac{\chi_{0.5,200}^{2}}{200}}\right)+2\times\frac{Z_{0.005}}{ \sqrt{220}}\]  
\[=2\times 4.39\times\left(\sqrt{\frac{255.26}{200}}-\sqrt{\frac{199.33}{200}} \right)+2\times\frac{2.576}{\sqrt{220}}\]  
\[=2\times 4.39\times 0.1314+2\times 0.1736\]  
\[=1.50.\]  
Therefore, \(d_{R}/d_{T}=0.087\). Take \(\phi_{1}=0.2\) and \(\phi_{2}=0.4\). Based on the decision rule (1) of fuzzy testing, when \(d_{R}/d_{T}=0.087<\phi_{1}\), then reject \(H_{0}\) and conclude that \(Q_{\text{ISS}}<5\). According to [PERSON] et al. [17] and [PERSON] [29], the fuzzy testing method suggested by this study seems to be more reasonable than statistical inference.