# Novel Service Efficiency Evaluation and Management Model
## 3 Upper Confidence Limit of Service Efficiency Index and Minimum Value  
Let \(Y_{h,1},\ldots,Y_{h,j},\ldots,Y_{h,n}\) be a random sample form \(N\big{(}\delta_{h},\gamma_{h}^{2}\big{)}\) with sample size \(n\), then the maximum likelihood estimators (_MLE_) of \(\delta_{h}\) and \(\gamma_{h}\) of the \(h\)th workstation can be obtained as follows:  
\[\delta_{h}^{*} =\frac{1}{n}{\sum_{j=1}^{n}Y_{h,j}}\text{ and }\gamma_{h}^{*}= \sqrt{\frac{1}{n}{\sum_{j=1}^{n}\left(Y_{h,j}-\delta_{h}^{*}\right)^{2}}} \tag{3}\]  
Therefore, the estimator of service efficiency index is indicated as follows:  
\[S_{Ih}^{*}=\frac{1-\delta_{h}^{*}}{\gamma_{h}^{*}} \tag{4}\]  
Under the assumption of normality, we let:  
\[T=\frac{\sqrt{n}\big{(}\delta_{h}^{*}-\delta_{h}\big{)}}{\gamma_{h}^{*}}\text { and }K=\frac{n{\gamma_{h}^{*}}^{2}}{\gamma_{h}^{2}}. \tag{5}\]  
Then, statistics \(T\) is distributed as t-distribution with \(n-1\) degree of freedom, denoted by \(T\sim t_{n-1}\). Furthermore, statistics \(K\) is distributed as chi-square distribution with \(n-1\) degree of freedom, denoted by \(K\sim\chi_{n-1}^{2}\). To derive the \((1-\alpha)\times 100\%\) upper confidence limit of service efficiency index \(S_{Ih}\), this paper defines two events as follows:  
\[L_{\delta} =\big{\{}T\leq t_{\alpha/2;n-1}\big{\}}{=}\left\{\delta_{h}\geq \delta_{h}^{*}-\frac{t_{\alpha/2;n-1}}{\sqrt{n}}\gamma_{h}^{*}\right\} \tag{6}\]  
and  
\[U_{\gamma}=\left\{K\leq\chi_{1-\alpha/2;N-m}^{2}\right\}=\left\{\gamma_{h} \geq\sqrt{\frac{n}{\chi_{1-\alpha/2;n-1}^{2}}}\gamma_{h}^{*}\right\}, \tag{7}\]  
Figure 1: The relationship between service efficiency index and service accomplishment rate. where \(\chi^{2}_{1-\alpha/2n-1}\) is the lower \(1-\alpha/2\) quintile of \(\chi^{2}_{n-1}\). In fact, \(p(L_{\delta})=p(U_{\gamma})=1-\alpha/2\) and \(p\big{(}L_{\delta}^{\zeta}\big{)}=p\big{(}U_{\gamma}^{\zeta}\big{)}=\alpha/2\). According to [PERSON]'s inequality and [PERSON]'s theorem, we have \(P(L_{\delta}\cap U_{\gamma})\geq 1-p\big{(}L_{\delta}^{\zeta}\big{)}-p \Big{(}U_{\gamma}^{\zeta}\big{)}\!=1-\alpha\). Then  
\[ \begin{split} p\Big{\{}T\leq t_{\alpha/2;N-m},K\leq\chi^{2}_{1- \alpha/2;N-m}\Big{\}}=1-\alpha\\ \Rightarrow p\bigg{\{}\delta_{h}\geq\delta_{h}^{*}-t_{\alpha/2n- 1}\frac{\gamma_{h}^{*}}{\sqrt{n}},\gamma_{h}\geq\sqrt{\frac{n}{\chi^{2}_{1- \alpha/2n-1}}}\gamma_{h}^{*}\bigg{\}}\geq 1-\alpha\\ \Rightarrow p\bigg{\{}\delta_{h}\geq\delta_{h}^{*}-t_{\alpha/2n- 1}\frac{\gamma_{h}^{*}}{\sqrt{n}},\gamma_{h}\geq\sqrt{\frac{n}{\chi^{2}_{1- \alpha/2n-1}}}\gamma_{h}^{*}\bigg{\}}\geq 1-\alpha\\ \Rightarrow p\bigg{\{}1-\delta_{h}\leq 1-\delta_{h}^{*}+t_{ \alpha/2n-1}\times\left(\frac{\gamma_{h}^{*}}{\sqrt{n}}\right),\frac{1}{\gamma _{h}}\leq\frac{1}{\gamma_{h}^{*}}\sqrt{\frac{\chi^{2}_{1-\alpha/2n-1}}{n}} \bigg{\}}\geq 1-\alpha.\end{split}  \tag{8}\]  
Therefore, the upper confidence limit of service efficiency index is:  
\[US_{\textit{II}}\!=\bigg{(}S_{\textit{II}}^{*}+\frac{t_{\alpha/2n-1}}{\sqrt{n }}\bigg{)}\sqrt{\frac{\chi^{2}_{1-\alpha/2n-1}}{n}} \tag{9}\]  
To determining whether the service efficiency meets the required value for the \(h\)th workstation, we adopt the following:  
* Null hypothesis \(H_{0}\): \(S_{\textit{II}}\geq k\);
* Alternative hypothesis \(H_{1}\): \(S_{\textit{II}}<k\).