# Process Quality Evaluation Model with Taguchi Cost Loss Index
## 4 A Practical Application  
As mentioned earlier, the process quality evaluation model developed by the Taguchi loss index in this paper is applicable to the machine tool industry, semiconductor manufacturing processes, the IC packaging industry, and the electronics industry, among others. Central Taiwan is a place of strategic importance for the machine tool industry, which is formed due to the division of labor. The improvement of process quality enhances the competitiveness of the industrial chain. The shaft is an important component of many mechanical products. The outer diameter of the shaft is an important quality characteristic. The outer diameter tolerance of the shaft of a fan motor is 1.1 \(\pm\) 0.05. Therefore, the lower specification limit is \(LSL=\) 1.05, while the upper specification limit is \(USL=\) 1.15. Then, the second-generation capability index can be exhibited as follows:  
\[C_{PM}=\frac{0.05}{3\sqrt{\left(\mu-1.1\right)^{2}+\sigma^{2}}} \tag{36}\]  
Let random variable \(Y=(X-1.1)/0.05\). Then, the new tolerance and the Taguchi cost loss index can be rewritten as follows:  
\[C_{PM}=\frac{1}{3\sqrt{\delta^{2}+\gamma^{2}}} \tag{37}\]To test whether the Taguchi cost loss index value is equal to 1.00 with \(\alpha\) = 0.01, the null hypothesis and alternative hypothesis can be defined as follows:  
\[\begin{array}{c}\text{null hypothesis }H_{0}:\ C_{PM}=1.00\\ \text{and}\\ \text{alternative hypothesis }H_{1}:\ C_{PM}\
eq 1.00.\end{array}\]  
We collected 20 subsamples (\(m\) = 20) of a control chart data with sample size \(n\) = 11, where the dates \(\left(y_{i1},\ldots,y_{ij},\ldots,y_{i11}\right)\) are the \(i\)th sub-sample (\(i\) = 1, 2, \(\ldots\), 20). Then,  
\[\hat{\delta}_{0}=\frac{1}{20}\underset{i=1}{\overset{20}{\sum}}\overline{y}_{ i}=0.16 \tag{38}\]  
and  
\[\hat{\gamma}_{0}^{2}=\frac{1}{20}\underset{i=1}{\overset{20}{\sum}}s_{i}^{2}=0.11 \tag{39}\]  
Therefore, we have  
\[\delta_{L}= \hat{\delta}_{0}-t_{0.025;200}\times\sqrt{\frac{\hat{\gamma}_{0}^{ 2}}{220}}=0.16-1.972\times\sqrt{\frac{0.11}{220}}=0.16-0.001=0.159, \tag{40}\] \[\delta_{U}= \hat{\delta}_{0}+t_{0.025;200}\times\sqrt{\frac{\hat{\gamma}_{0}^{ 2}}{220}}=0.25+1.972\times\sqrt{\frac{0.11}{220}}=0.16+0.001=0.161,\] (41) \[\gamma_{L}^{2}=\frac{200\times\hat{\gamma}_{0}^{2}}{\chi_{0.975;200}^{2}}= \frac{200\times 0.11}{241.06}=0.091, \tag{42}\]  
and  
\[\gamma_{U}^{2}=\frac{200\times\hat{\gamma}_{0}^{2}}{\chi_{0.025;200}^{2}}= \frac{200\times 0.11}{162.73}=0.135. \tag{43}\]  
Obviously, when \(\delta_{L}\)=0.159\(>\)0, we can conclude that \(\delta\)\(>\)0,  
\[LC_{PM}=1/3\sqrt{\left(\hat{\delta}_{0}+t_{0.025;200}\times\sqrt{\frac{\hat{ \gamma}_{0}^{2}}{220}}\right)^{2}+\frac{200\hat{\gamma}_{0}^{2}}{\chi_{0.025;20 0}^{2}}}=1/3\sqrt{\left(0.161\right)^{2}+0.135}=0.83, \tag{44}\]  
and  
\[UC_{PM}=1/3\sqrt{\left(\hat{\delta}_{0}-t_{0.025;200}\times\sqrt{\frac{\hat{ \gamma}_{0}^{2}}{220}}\right)^{2}+\frac{200\hat{\gamma}_{0}^{2}}{\chi_{0.975;2 00}^{2}}}=1/3\sqrt{\left(0.159\right)^{2}+0.091}=0.98. \tag{45}\]  
Based on the statistical testing rule (2), if \(UC_{PM}\)=0.98\(<\)1.00, then reject \(H_{0}\) and conclude that \(C_{PM}\)\(<\)1.00, which means the process needs to improve. According to the previous research, the assessments of process capability indices were performed under the statistical process control condition [32], and the data of statistical process control used to estimate process capability indices resulting in some advantages:  
1. Because of large volume of data collected in long term of statistical process control condition, the estimated confidence interval of indices would be smaller and result in greater accuracy of estimation [33; 34].
2. The confidence interval of indices derived from large volume of data used as assessment tools suggest the risk of misjudgments by sampling error would decline [35; 36; 37; 38].
3. Assessment of process capability and monitoring the process with a control chart helps with parameter tuning and keeps process capability stable [39].