# Fuzzy Evaluation Model of Process Improvement Capability with Costs Consideration
## 2 Upper Confidence Limit  
Suppose \((X_{1},\ldots,X_{i},\ldots,X_{n})\) is a random sample from a normal random variable \(X\) with mean \(\mu\) and variance \(\sigma^{2}\), i.e., \(X\sim N\big{(}\mu,\sigma^{2}\big{)}\). Let \(Y=(X-T)/d\), distributed as \(N\big{(}\delta,\gamma^{2}\big{)}\), where \(\delta=(\mu-T)/d\) is the mean of \(Y\) and \(\gamma=\sigma/d\) is the standard deviation of \(Y\). Then, the sample mean and sample standard deviation can be obtained as follows:  
\[\delta^{*}=\frac{1}{n}{\sum_{i=1}^{n}Y_{i}}, \tag{4}\]  
\[\gamma^{*}=\sqrt{\frac{1}{n-1}{\sum_{i=1}^{n}}\big{(}Y_{i}-\overline{Y}\big{)} ^{2}}. \tag{5}\]  
Figure 1: Process loss distribution chart. Therefore, the estimator of \(C_{PIM}\) is \(C_{PIM}^{*}=\frac{1}{3\sqrt{\delta^{*2}+R\gamma^{*2}}}\). Furthermore, we let  
\[Z=\frac{\sqrt{n}(\delta^{*}-\delta)}{\gamma} \tag{6}\]  
and  
\[K=\frac{(n-1)(\gamma^{*})^{2}}{\gamma^{2}}. \tag{7}\]  
Under the assumption of normality, \(Z\) and \(K\) are distributed as \(N(0,1)\) and \(\chi_{n-1}^{2}\), respectively. Therefore,  
\[p\{-Z_{\alpha^{\prime}/2}\leq Z\leq Z_{\alpha^{\prime}/2}\}=1-\alpha^{\prime}. \tag{8}\]  
and  
\[p\Big{\{}\chi_{\alpha^{\prime}/2n-1}^{2}\leq K\leq\chi_{1-\alpha^{\prime}/2n-1 }^{2}\Big{\}}=1-\alpha^{\prime}, \tag{9}\]  
where \(\alpha^{\prime}=1-\sqrt{1-\alpha}\). \(Z_{\alpha^{\prime}/2}\) is the upper \(\alpha^{\prime}/2\) quintile of \(N(0,1)\), \(\chi_{\alpha^{\prime}/2n-1}^{2}\) is the lower \(\alpha^{\prime}/2\) quintile of \(\chi_{n-1}^{2}\), and \(1-\alpha^{\prime}\) represents the confidence level. Since \(\delta^{*}\) and \(\left(\gamma^{*}\right)^{2}\) are mutually independent, then \(Z\) and \(K\) are also mutually independent. From these relationships, we can further obtain the following equation:  
\[p\Big{\{}-Z_{\alpha^{\prime}/2}\leq Z\leq Z_{\alpha^{\prime}/2},\chi_{\alpha^ {\prime}/2n-1}^{2}\leq K\leq\chi_{1-\alpha^{\prime}/2n-1}^{2}\Big{\}}=1-\alpha \tag{10}\]  
Equivalently,  
\[p\Bigg{\{}\delta^{*}-Z_{\alpha^{\prime}/2}\times\left(\frac{\gamma}{\sqrt{n} }\right)\leq\delta\leq\delta^{*}+Z_{\alpha^{\prime}/2}\times\left(\frac{ \gamma}{\sqrt{n}}\right),\sqrt{\frac{n-1}{\chi_{\alpha^{\prime}/2n-1}^{2}}} \gamma^{*}\leq\gamma\leq\sqrt{\frac{n-1}{\chi_{1-\alpha^{\prime}/2n-1}^{2}}} \gamma^{*}\Bigg{\}}=1-\alpha. \tag{11}\]  
Let \((y_{1},\ldots,y_{i},\ldots,y_{n})\) be the observed value of \((Y_{1},\ldots,Y,\ldots,Y_{n})\). Then, \(\delta_{0}^{*}\) and \(\gamma_{0}^{*}\) are the observed values of \(\delta^{*}\) and \(\gamma^{*}\), respectively, as follows:  
\[\delta_{0}^{*}=\frac{1}{n}{\sum_{i=1}^{n}}y_{i}, \tag{12}\]  
and  
\[\gamma^{*}=\sqrt{\frac{1}{n-1}{\sum_{i=1}^{n}}\big{(}Y_{i}-\overline{Y}\big{)} ^{2}}. \tag{5}\]0  
Therefore, the confidence region can be shown as follows:  
\[\gamma^{*}=\sqrt{\frac{1}{n-1}{\sum_{i=1}^{n}}\big{(}Y_{i}-\overline{Y}\big{)} ^{2}}. \tag{5}\]1  
Obviously, \(C_{PIM}\) is the function of \((\delta,\gamma)\). Based on the above results, this study refers to the mathematical programming (MP) method of [PERSON] et al. (2017). Based on two conditions, \(0\in CI\) and \(0\
otin CI\), \(C_{PIM}\) is seen as the objective function, and the confidence region (\(CR\)) is regarded as the feasible solution region to obtain the upper confidence limit \(UC_{PIM}\) of \(C_{PIM}\), as illustrated below:  
\[\left\{ \begin{array}{c}UC_{PIM}=Max\ \frac{1}{3\sqrt{\delta^{2}+R\gamma^{2}}} \\ s.t.\ (\delta,\gamma)\in CR\end{array} \right., \tag{15}\]  
where  
\[CI=[CI_{1},CI_{2}]=[\delta_{0}^{*}-E_{0}(\alpha)\times\gamma_{0}^{*}.\delta_{0 }^{*}+E_{0}(\alpha)\times\gamma_{0}^{*}]. \tag{16}\]and  
\[E_{0}(\alpha)=\frac{Z_{0.5-\sqrt{1-\alpha}/2}}{\sqrt{\chi_{0.5+\sqrt{1-\alpha}/2 \pi-1}^{2}}}\times\sqrt{\frac{n-1}{n}}. \tag{17}\]  
Case 1: \(0\in CI\)  
In case 1, we can conclude \(\delta=0\).