# Process Quality Evaluation Model with Taguchi Cost Loss Index
## 2 Find Confidence Intervals Using the Mathematical Programming Method
Then,  
\[p\left\{\delta-t_{\alpha/4\text{N}-m}\times\sqrt{\frac{\hat{\gamma}^{2}}{N}} \leq\delta\leq\delta+t_{\alpha/4\text{N}-m}\times\sqrt{\frac{\hat{\gamma}^{2}} {N}},\frac{(N-m)\hat{\gamma}^{2}}{\chi^{2}_{1-\alpha/4\text{N}-m}}\leq\gamma^ {2}\leq\frac{(N-m)\hat{\gamma}^{2}}{\chi^{2}_{\alpha/4\text{N}-m}}\right\}=1-\alpha. \tag{18}\]  
Let \((y_{i1},y_{i2},\cdots,y_{i\eta})\) represent the observed value of \((Y_{i1},Y_{i2},\cdots,Y_{i\eta})\). \(\delta_{0}\) and \(\hat{\gamma}^{2}_{0}\) are the observed values of \(\delta\) and \(\hat{\gamma}^{2}\) respectively as follows:  
\[\delta_{0}=\overline{\hat{y}}=\frac{1}{m}\sum_{i=1}^{m}\overline{y}_{i}, \tag{19}\]  
and  
\[\hat{\gamma}^{2}_{0}=\frac{1}{m}\sum_{i=1}^{m}s_{i}^{2} \tag{20}\]  
Therefore, the confidence region can be displayed as:  
\[CR=\left\{ \begin{array}{c}\delta_{0}-t_{\alpha/4\text{N}-m}\times\sqrt{\frac {\hat{\gamma}^{2}}{N}}\leq\delta\leq\delta_{0}+t_{\alpha/4\text{N}-m}\times \sqrt{\frac{\hat{\gamma}^{2}_{0}}{N}},\frac{(N-m)\hat{\gamma}^{2}}{\chi^{2}_{1 -\alpha/4\text{N}-m}}\leq\gamma^{2}\leq\frac{(N-m)\hat{\gamma}^{2}_{0}}{\chi^ {2}_{\alpha/4\text{N}-m}}\end{array} \right\} \tag{21}\]  
According to [PERSON] et al. [13], the mathematical program model can be presented as follows:  
\[\left\{ \begin{array}{c}LC_{PM}=Min\frac{1}{3\sqrt{\delta^{2}+\gamma^{2}}}\\ subject to\\ \delta_{L}\leq\delta\leq\delta_{U}\\ \gamma^{2}_{L}\leq\gamma^{2}\leq\gamma^{2}_{U}\end{array} \right.\text{and} \left\{ \begin{array}{c}UC_{PM}=Max\frac{1}{3\sqrt{\delta^{2}+\gamma^{2}}}\\ subject to\\ \delta_{L}\leq\delta\leq\delta_{U}\\ \gamma^{2}_{L}\leq\gamma^{2}\leq\gamma^{2}_{U}\end{array} \right.,\]  
where  
\[\delta_{L}=\delta_{0}-t_{\alpha/4\text{N}-m}\times\sqrt{\frac{\hat{\gamma}^{2} _{0}}{N}},\,\delta_{U}=\delta_{0}+t_{\alpha/4\text{N}-m}\times\sqrt{\frac{ \hat{\gamma}^{2}_{0}}{N}}\]  
\[\gamma^{2}_{L}=\frac{(N-m)\hat{\gamma}^{2}_{0}}{\chi^{2}_{1-\alpha/4\text{N}-m }}, \tag{22}\]  
\[\gamma^{2}_{U}=\frac{(N-m)\hat{\gamma}^{2}_{0}}{\chi^{2}_{\alpha/4\text{N}-m }}. \tag{23}\]  
Let \(\Delta(\delta,\gamma)\!=\!\sqrt{\delta^{2}+\gamma^{2}}\), where \(\Delta(\delta,\gamma)\) is the function of \((\delta,\gamma)\) and represents the distance from the punctuation point \((\delta,\gamma)\) to the origin coordinate (0, 0). Obviously, the closer the punctuation point \((\delta,\gamma)\) to the origin is, the greater the index value, whereas the farther the punctuation point \((\delta,\gamma)\) from the origin, the smaller the index value. According to this concept, this study solves the values of lower confidence limit \(LC_{PM}\) and upper confidence limit \(UC_{PM}\) in three situations as follows:  
Situation 1: \(\delta_{U}\!\!<\!\!0\)  
In this situation, we can conclude that \(\delta<0\) and for any \((\delta,\gamma)\in\ CR\), \(\Delta(\delta_{U},\gamma_{L})\leq\Delta(\delta,\gamma)\leq\Delta(\delta_{L}, \gamma_{U})\). Therefore,  
\[LC_{PM}=\frac{1}{3\sqrt{\delta^{2}_{L}+\gamma^{2}_{U}}}=1/3\sqrt{\left(\hat{ \delta}_{0}-t_{\alpha/4\text{N}-m}\times\sqrt{\frac{\hat{\gamma}^{2}_{0}}{N}} \right)^{2}+\frac{(N-m)\hat{\gamma}^{2}_{0}}{\chi^{2}_{\alpha/4\text{N}-m}}} \tag{24}\]and  
\[UC_{PM}=\frac{1}{3\sqrt{\delta_{U}^{2}+\gamma_{L}^{2}}}=1/3\sqrt{\left(\hat{\delta }_{0}+t_{\alpha/4;N-m}\times\sqrt{\frac{\hat{\gamma}_{0}^{2}}{N}}\right)^{2}+ \frac{(N-m)\hat{\gamma}_{0}^{2}}{\chi_{1-\alpha/4;N-m}^{2}}}. \tag{25}\]  
Situation 2: \(\delta_{L}>0\)  
In this situation, we can conclude that \(\delta>0\) and for any \((\delta,\gamma)\in CR\), \(\Delta(\delta_{L},\gamma_{L})\leq\Delta(\delta,\gamma)\leq\Delta(\delta_{U}, \gamma_{U})\).