# Process Quality Evaluation Model with Taguchi Cost Loss Index
## 2 Find Confidence Intervals Using the Mathematical Programming Method  
Let random variable \(Y=(X-T)/d\). Then, \(Y\) is distributed as a normal distribution with mean \(\delta\) and standard deviation \(\gamma\), where \(\delta\)= \((\mu-T)/d\) and \(\gamma\)= \(\sigma/d\). Thus, the Taguchi cost loss index \(C_{PM}\) can be rewritten as follows:  
\[C_{PM}=\frac{1}{3\sqrt{\delta^{2}+\gamma^{2}}}, \tag{6}\]  
where \(\delta^{2}+\gamma^{2}\) is the expected value of Taguchi cost loss. As noted above, random variable \(Y\) is distributed as a normal distribution. In the statistical control, the subsamples are obtained over a period of time. Each subsample contains \(n\) observations related to the quality characteristics, and \(m\) subsamples are available. Consequently, for each subsample, let \(\overline{Y}_{j}\) and \(S_{j}^{2}\) respectively, represent the sample mean and sample variance of the \(i\)th subsample, and let \(N=m\ \times\ n\) represent the total number of observations as follows:  
\[\overline{Y}_{i}=\frac{1}{n}{\sum\limits_{j=1}^{n}Y_{ij}} \tag{7}\]  
and  
\[S_{j}^{2}=\frac{1}{n}{\sum\limits_{j=1}^{n}\bigl{(}Y_{ij}-\overline{Y}_{i} \bigr{)}^{2}} \tag{8}\]  
The overall sample mean and the pooled sample variance, unbiased estimators of \(\delta\) and \(\gamma^{2}\), are respectively expressed as follows:  
\[\delta=\overline{Y}=\frac{1}{m}{\sum\limits_{i=1}^{m}\overline{Y}_{i}} \tag{9}\]  
and  
\[\hat{\gamma}^{2}=\frac{1}{m}{\sum\limits_{i=1}^{m}S_{i}^{2}} \tag{10}\]  
Obviously, the estimator \(\hat{\delta}\) is distributed as a normal distribution with mean \(\delta\) and variance \(\gamma^{2}/N\), while the estimator \(\hat{\gamma}^{2}\) is distributed as \(\gamma^{2}\chi_{N-m}^{2}/\bigl{(}N-m\bigr{)}\). Furthermore, we let  
\[T_{N}=\frac{\sqrt{N}\bigl{(}\hat{\delta}-\delta\bigr{)}}{\hat{\gamma}} \tag{11}\]  
and  
\[K_{N}=\frac{(N-m)\hat{\gamma}^{2}}{\gamma^{2}}. \tag{12}\]  
Then, \(T_{N}\) and \(K_{N}\) are distributed as \(t_{N-m}\) and \(\chi_{N-m}^{2}\) respectively. \(t_{N-m}\) is \(t\) distribution with \(N-m\) degree of freedom and \(\chi_{N-m}^{2}\) is chi-square distribution with \(N-m\) degree of freedom. Thus, we have  
\[ \begin{array}{c}1-\frac{\delta}{2}=\left\{-t_{\alpha/4;N-m}\leq T_{N}\leq t _{\alpha/4;N-m}\right\}\\ =\left\{-t_{\alpha/4;N-m}\leq\frac{\sqrt{N}(\delta-\delta)}{\gamma}\leq t_{ \alpha/4;N-m}\right\}\\ =\left\{\hat{\delta}-t_{\alpha/4;N-m}\Bigl{(}\frac{\gamma}{\sqrt{N}}\Bigr{)} \leq\delta\leq\hat{\delta}+t_{\alpha/4;N-m}\Bigl{(}\frac{\gamma}{\sqrt{N}} \Bigr{)}\right\}\end{array}  \tag{13}\]  
and  
\[ \begin{array}{c}1-\frac{\delta}{2}=\left\{\chi_{\alpha/4;N-m}^{2}\leq K_{N} \leq\chi_{1-\alpha/4;N-m}^{2}\right\}\\ =\left\{\chi_{\alpha/4;N-m}^{2}\leq\frac{(N-m)\hat{\gamma}^{2}}{\gamma^{2}} \leq\chi_{1-\alpha/4;N-m}^{2}\right\}\\ =\left\{\frac{(N-m)\hat{\gamma}^{2}}{\chi_{1-\alpha/4;N-m}^{2}}\leq\gamma^{2} \leq\frac{(N-m)\hat{\gamma}^{2}}{\chi_{\alpha/4;N-m}^{2}}\right\}\end{array}  \tag{14}\]  
To derive the \((1-\alpha)\times 100\%\) confidence interval of the Taguchi cost loss index, this study defines the event \(E_{T}\) and event \(E_{K}\) as follows:  
\[E_{T}=\left\{\hat{\delta}-t_{\alpha/4;N-m}\Bigl{(}\frac{\gamma}{\sqrt{N}} \Bigr{)}\leq\delta\leq\delta+t_{\alpha/4;N-m}\Bigl{(}\frac{\gamma}{\sqrt{N}} \Bigr{)}\right\} \tag{15}\]  
and  
\[E_{T}=\left\{\frac{(N-m)\hat{\gamma}^{2}}{\chi_{1-\alpha/4;N-m}^{2}}\leq\gamma ^{2}\leq\frac{(N-m)\hat{\gamma}^{2}}{\chi_{\alpha/4;N-m}^{2}}\right\}, \tag{16}\]where \(t_{\alpha/4\text{N}-m}\) is the upper \(\alpha/4\) quantile of \(t_{\text{N}-m}\), and \(\chi^{2}_{\alpha;N-m}\) is the upper \(a\) quantile of \(\chi^{2}_{N-m}\). Based on [PERSON]'s inequality and [PERSON]'s theorem, we have  
\[P(E_{T}\cap E_{K})\geq 1-P\left(E_{T}^{C}\right)-P\left(E_{K}^{C}\right), \tag{17}\]  
where \(p(E_{T})=p(E_{K})=1-\alpha/2\) and \(p\left(E_{T}^{C}\right)=p\left(E_{K}^{C}\right)=\alpha/2\).