# Novel Service Efficiency Evaluation and Management Model
## 2 Service Efficiency Index  
Without loss of generality, this paper assumes that the service operation must go through the service process of w workstations to complete. As mentioned earlier, the service operation efficiency of each workstation will affect the overall service operation efficiency. Let  
* \(X_{h}\) represent the service operation time of the \(h\)th workstation and;
* \(U_{h}\) represent the upper limit of the service operation time of the \(h\)th workstation.  
Let random variable \(Y_{h}=X_{h}/U_{h}\), h = 1,..., w. The value of \(U_{h}\) is usually determined by the self-regulation of the performance appraisal department or the operating unit. Then, \(Y_{h}\) represents the relative service operation time of the \(h\)th workstation, and the upper limit of the relative service operation time is one. Suppose random variable \(X_{h}\) is distributed as normal distribution with mean \(\mu_{h}\) and standard deviation \(\sigma_{h}\). Then, random variable \(Y_{h}\) is distributed as normal distribution with mean \(\delta_{h}\) and standard deviation \(\gamma_{h}\), where \(\delta_{h}=\mu_{h}/U_{h}\) and \(\gamma_{h}=\sigma_{h}/U_{h}\). The service efficiency index is denoted as follows:  
\[S_{lh}=\frac{1-\delta_{h}}{\gamma_{h}} \tag{1}\]  
where we can call \(\delta_{h}\) related mean and \(\gamma_{h}\) related standard deviation.  
According to [PERSON] et al. [7], when related mean \(\delta_{h}\) and related standard deviation \(\gamma_{h}\) are smaller, the service time is stable and the efficiency is better for workstation \(h\). Based on the above discussion, the larger the service efficiency index, the better it can indicate objective service efficiency. If the index value is zero, the job is finished on time. If it is a negative number, the job cannot be finished on time. If it is a positive number, the job has better service efficiency.  
Furthermore, under the assumption of normality, there is a one-to-one mathematical relationship between service efficiency index \(S_{lh}\) and service accomplishment rate \(p\), which can be shown as follows:  
\[p=p\{Y_{h}\leq 1\}=p\bigg{\{}Z\leq\frac{1-\delta_{h}}{\gamma_{h}}\bigg{\}}= \Phi(S_{lh}), \tag{2}\]  
where \(Z=(Y_{h}-\delta_{h})/\gamma_{h}\) is distributed as standard normal distribution denoted by \(Z\sim N(0,1)\). Let \(S_{lh}=\) z, then \(\Phi(z)=\int_{-\infty}^{z}\Big{(}1/\sqrt{2\pi}\Big{)}exp\big{(}-z^{2}/2\big{)} d_{z}\) is the cumulative function of standard normal distribution. For example, when index \(S_{lh}=\) 3, then we can guarantee the service accomplishment rate \(p=\Phi(3)=\) 99.865%. In Figure 1, the x-coordinate is index \(S_{lh}\), and the y-coordinate is service accomplishment rate \(p=\Phi(3)=\) 99.865%. Obviously, when the value of index \(S_{lh}\) is large, the value of service accomplishment rate \(p\) is also large.  
Apparently, service efficiency index \(S_{Ih}\) can not only measure the service efficiency of workstation \(h\), but it also has a one-to-one mathematical relationship with the service accomplishment rate. Therefore, this paper will use the service efficiency index to propose a service efficiency evaluation model, which is to evaluate whether the service operation efficiency of each workstation can meet the required level and is used as a decision-making basis for improvement.