## 2 Case Introduction
### Performance Evaluation Index of Manufacturing Time Performance  
According to [PERSON] et al. [17], process quality performance and manufacturing time performance are two pivotal indices for evaluating manufacturing process quality. Therefore, we set and created a manufacturing time performance index according to the concept proposed by [PERSON] et al. [17]. We hypothesized that manufacturing time performance is \(T\), the conforming mean is \(\mu_{T}\), and the variance is the normal distribution of \(\sigma_{T}\)2; that is, \(T\sim N\big{(}\mu_{T},\sigma_{T}\big{)}\). To account for both manufacturing process quality performance and manufacturing time performance, a manufacturer usually sets the upper limit of the manufacturing time performance as \(U_{T}\) and the lower limit of the manufacturing time performance as \(L_{T}\). Consequently, if the chance of exceeding the upper limit of manufacturing time performance is \(P_{u}\) and the chance of exceeding the lower limit of manufacturing time performance is \(P_{l}\), the disapproval rate of manufacturing time performance can be described by the following:  
\[P_{u}=P(T>U_{T})=P\bigg{(}Z>\frac{U_{T}-\mu_{T}}{\sigma_{T}}\bigg{)}=1-\[PERSON] \left(\frac{U_{T}-\mu_{T}}{\sigma_{T}}\right)=1-\[PERSON](Z_{U}) \tag{8}\]  
\[P_{l}=P(T<L_{T})=P\bigg{(}Z>\frac{L_{T}-\mu_{T}}{\sigma_{T}}\bigg{)}=1-\[PERSON] \bigg{(}\frac{\mu_{T}-L_{T}}{\sigma_{T}}\bigg{)}=1-\[PERSON](Z_{L}) \tag{9}\]  
Subsequently, we directly used the approval rate of manufacturing time (\(I_{MT}\)) as the performance index of manufacturing time, and the equation is as follows:  
\[I_{MT}=\Phi(Z_{U})+\Phi(Z_{L})-1 \tag{10}\]  
Combining \(P_{u}\) and \(P_{l}\), the range of the approval rate of manufacturing time performance (\(I_{MT}\)) becomes \(0\leq\ I_{MT}\leq 1\). The disapproval rate of manufacturing time performance (\(J_{MT}\)) is defined as the following:  
\[J_{MT}=1-I_{MT}=P_{u}+P_{l}=2-\left[\Phi(Z_{U})+\Phi(Z_{L})\right] \tag{11}\]  
when \(I_{MT}\geq I_{0}\), \(J_{MT}\leq J_{0}\) and \(J_{0}=1-I_{0}\). Per the description of sampling in Section 3.2, statistical process control was conducted on manufacturing time. The statistical data of the sample \(J_{MTi}\) were used, yielding the following:  
\[\overline{J}_{MT}=\frac{1}{n}\sum_{i=1}^{n}J_{MTi},\ S_{J_{MT}}^{2}=\frac{1}{ n-1}\sum_{i=1}^{n}\left(J_{MTi}-\overline{J}_{MT}\right)^{2}\]  
where \(\overline{J}_{MT}\) is the sample mean, and \(S_{J_{MT}}\) is the sample standard deviation. Therefore, the estimator of index \(J_{MT}\) is as follows:  
\[\overline{J}_{MT}=2-\left[\Phi\big{(}Z_{U}\big{)}+\Phi\big{(}Z_{L}\big{)}\right] \tag{12}\]  
Construction of Evaluation Model of Manufacturing Process Quality and Manufacturing Time Performance  
[PERSON] et al. [24] and [PERSON] et al. [25] suggested that a product usually has several quality characteristics. Only when all quality characteristics meet the required level can the product be regarded as satisfactory. The manufacturing process performance evaluation chart proposed by [PERSON] et al. [11] can be used to simultaneously evaluate nominal-the-best, the-larger-the-better, and the-smaller-the-better quality characteristics. \(\delta\) is used as the x-coordinate and \(\gamma\) is used as the y-coordinate to analyze whether the accuracy and precision of process quality characteristics meet the required level. Accordingly, process quality characteristics with poor performance are improved. The \(X\)-axis and \(Y\) -axis are first defined according to the following:  
\[X\ axis:|\delta|\leq\frac{1.5}{k}\]\[Y\;axis:\gamma=\frac{1}{k}\]  
A conversion table of the manufacturing process quality index \(Q_{pbh}\) and the \(k-sigma\) quality level can be developed using Equation (3). For example, when the quality level of the manufacturing process reaches \(3\sigma\), \(Q_{pbh}(3)=3.785\) and the yield rate is 93.31894%. When the quality level of the manufacturing process reaches \(4\sigma\), \(Q_{pbh}(4)=4.580\) and the yield rate is 99.37903%.