# Fuzzy Evaluation Model of Process Improvement Capability with Costs Consideration
## 2 Upper Confidence Limit
Then, the objective function is \(C_{PIM}=\left(3\sqrt{R}\gamma\right)^{-1}\), and  
\[CR=\Bigg{\{}\left(\delta,\gamma\right)\lvert\delta=0,\sqrt{\frac{n-1}{\chi_{0. 5-\sqrt{1-\alpha}/2\pi-1}^{2}}}\gamma_{0}^{*}\leq\gamma\leq\sqrt{\frac{n-1}{ \chi_{0.5+\sqrt{1-\alpha}/2\pi-1}^{2}}}\gamma_{0}^{*}\Bigg{\}}. \tag{18}\]  
Then, the \(MP\) model can be rewritten as follows:  
\[\left\{ \begin{array}{l}UC_{PIM}=Max\bigg{\{}\left(3\sqrt{R}\gamma\right)^{ -1}\bigg{\}}\\ \text{\emph{subject to}}\\ \sqrt{\frac{n-1}{\chi_{0.5-\sqrt{1-\alpha}/2\pi-1}^{2}}}\gamma_{0}^{*}\leq \gamma\leq\sqrt{\frac{n-1}{\chi_{0.5+\sqrt{1-\alpha}/2\pi-1}^{2}}}\gamma_{0}^ {*}\end{array} \right.. \tag{19}\]  
Apparently, when \(0\in CI\), the maximum value will be generated at the coordinate point. \[\left(\delta,\gamma\right)=\left(0,\gamma_{0}^{*}\sqrt{\frac{n-1}{\chi_{0.5- \sqrt{1-\alpha}/2\pi-1}^{2}}}\right). \tag{20}\]  
The maximum value \(UC_{PIM}\) of \(C_{PIM}\) at this time is expressed as follows:  
\[UC_{PIM}=\left(3\gamma_{0}^{*}\sqrt{\frac{n-1}{\chi_{0.5-\sqrt{1-\alpha}/2\pi- 1}^{2}}}\right)^{-1}. \tag{21}\]  
Case 2: \(0\
otin CI\)  
In case 2, similarly to case 1, we can conclude \(\delta\
eq 0\). The \(MP\) model can be rewritten as follows:  
\[\left\{ \begin{array}{l}UC_{PIM}=Max\bigg{\{}\frac{1}{3\sqrt{\delta^{2}+R \gamma^{2}}}\bigg{\}}\\ \text{\emph{Subject to}}\\ \delta_{0}^{*}-Z_{0.5-\sqrt{1-\alpha}/2}\times\left(\frac{\gamma}{\sqrt{n}} \right)\leq\delta\leq\delta_{0}^{*}+Z_{0.5-\sqrt{1-\alpha}/2}\times\left(\frac {\gamma}{\sqrt{n}}\right)\\ \sqrt{\frac{n-1}{\chi_{0.5-\sqrt{1-\alpha}/2\pi-1}^{2}}}\gamma_{0}^{*}\leq \gamma\leq\sqrt{\frac{n-1}{\chi_{0.5+\sqrt{1-\alpha}/2\pi-1}^{2}}}\gamma_{0}^ {*}\end{array} \right.. \tag{22}\]  
Obviously, the feasible region is a trapezoid-like convex set with a wide top and a narrow bottom. The closer \(\left(\delta,\gamma\right)\) is to the origin, the greater the value of \(C_{PIM}\) is. Therefore, when \(CI_{1}>0\), the maximum value (\(UC_{PIM}\)) will be generated in the lower-left corner. Its coordinate point is seen as follows:  
\[\text{Lower-left corner}:\ (\delta,\gamma)=\left(\delta_{0}^{*}-E_{0}(\alpha) \times\gamma_{0}^{*}\sqrt{\frac{n-1}{\chi_{0.5-\sqrt{1-\alpha}/2\pi-1}^{2}}} \times\gamma_{0}^{*}\right). \tag{23}\]  
When \(CI_{2}<0\), the maximum value (\(UC_{PIM}\)) will be generated in the lower-right corner. Its coordinate point is seen as follows:  
\[\text{Lower-right corner}:\ (\delta,\gamma)=\left(\delta_{0}^{*}+E_{0}(\alpha) \times\gamma_{0}^{*}\sqrt{\frac{n-1}{\chi_{0.5-\sqrt{1-\alpha}/2\pi-1}^{2}}} \times\gamma_{0}^{*}\right). \tag{24}\]Therefore,  
\[UC_{PIM}(\alpha)=\left(3\gamma_{0}^{*}\sqrt{Min\left\{\left(\frac{\delta_{0}^{*}}{ \gamma_{0}^{*}}-E_{0}(\alpha)\right)^{2},\left(\frac{\delta_{0}^{*}}{\gamma_{0}^ {*}}+E_{0}(\alpha)\right)^{2}\right\}+\frac{(n-1)R}{\chi_{0.5-\sqrt{1-\alpha}/ 2n-1}^{2}}}\right)^{-1}. \tag{25}\]  
According to Equations (17) and (22), then  
\[UC_{PIM}(\alpha)=\left\{ \begin{array}{ll}\left(3\gamma_{0}^{*}\sqrt{\frac{(n -1)R}{\chi_{0.5-\sqrt{1-\alpha}/2n-1}^{2}}}\right)^{-1}\text{ if }0\in CI\\ \left(3\gamma_{0}^{*}\sqrt{Min\left\{\left(\frac{\delta_{0}^{*}}{\gamma_{0}^ {*}}-E_{0}(\alpha)\right)^{2},\left(\frac{\delta_{0}^{*}}{\gamma_{0}^{*}}+E_{0 }(\alpha)\right)^{2}\right\}+\frac{(n-1)R}{\chi_{0.5-\sqrt{1-\alpha}/2n-1}^{2} }\right)}^{-1}\text{ if }0\
otin CI\end{array} \right. \tag{26}\]