## 4 Procedure of Meta-Modeling
### Design of Experiments  
The meta-model of this study was designed based on the response surface methodology. Several experimental design methods exist, such as the three-level factorial design, central composite design (CCD), and Box-Behnken design [48]. In this study, simulation results were obtained according to the CCD experimental design method. The CCD is useful for building a second-order (quadratic) model with various input variables as it does not require a complete three-level factorial experiment. The CCD can be designed by adding center point and axial point experiments to the two-level factorial design method. Hence, it has a relatively small number of experiments compared with other methods for estimating the quadratic polynomial model. If the number of input variables is 12, then the total number of experiments of the three-level full factorial design method is \(3^{12}\) = 531,441; however, the total number of experiments of the CCD is \(2^{(12-p)}+2\times 12+C_{p}\), where \(p\) and \(C_{p}\) are the size of fraction of the two-level full factorial design and the number of center points, respectively. We set \(p=4\) and \(C_{p}=8\); therefore, the total number of experiments was 288. In addition, the CCD has five levels of analysis, i.e., \((-\alpha,-1,0,1,\) and \(\alpha)\), for one variable. We set the value of \(\alpha\), which is the distance from the center point to the axis point, to 2. The number of experiments for the fractional factorial 2-level designs of 12 factors is 128 with resolution level IV, but we set the number of experiments to 256 with resolution level VI for increasing the accuracy of the meta-model [49]. The design of experiments for 256 points was selected according to the plan recommended in RcmdrPlugin.DoE package in R package [50]. Eight experiments for center points were conducted by changing the random number streams. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Input Variables** & & & & **Range of Levels (\(\alpha\) = 2)** & \\ \hline
**Notation** & **Definition** & **Min (\(a_{i}\))** & **Low** & **Medium** & **High** & **Max (\(b_{i}\))** \\ \hline \(z_{1}\) & Buffer capacity of group 1 (\(B_{1}\), \(B_{2}\), \(B_{3}\), \(B_{4}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{2}\) & Buffer capacity of group 2 (\(B_{5}\), \(B_{6}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{3}\) & Buffer capacity of group 3 (\(B_{7}\), \(B_{9}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{4}\) & Buffer capacity of group 4 (\(B_{8}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{5}\) & Buffer capacity of group 5 (\(B_{10}\), \(B_{12}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{6}\) & Buffer capacity of group 6 (\(B_{11}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{7}\) & Buffer capacity of group 7 (\(B_{13}\)) & 1 & 4 & 7 & 10 & 13 \\ \(z_{8}\) & Buffer capacity of group 8 (\(B_{14}\)) & 1 & 2 & 3 & 4 & 5 \\ \(z_{9}\) & Efficiency of each station (\(\epsilon\)) & 0.94 & 0.95 & 0.96 & 0.97 & 0.98 \\ \(z_{10}\) & Mean time to failure (\(1/f\)) & 160 & 200 & 240 & 280 & 320 \\ \(z_{11}\) & Product mix & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\ \(z_{12}\) & The process time of each station for electric cars (\(PT_{2}\)) & 5.6 & 6.2 & 6.8 & 7.4 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Definitions of input variables and the ranges of levels. The level of each factor was determined within the ranges considered in practice. \(z_{1}\)-\(z_{8}\) denote the buffer capacities of \(B_{1}\)-\(B_{14}\). \(z_{9}\) and \(z_{10}\) are variables related to the isolated efficiency of each workstation defined in Equation (3), where \(z_{9}\) is the efficiency (\(e\)) in Equation (3), and \(x_{10}\) denotes the mean time-to-failure (MTTF). When the values of \(z_{9}\) and \(z_{10}\) are determined, the mean time-to-repair (MTTR) can be calculated easily using Equation (3). In a mixed production environment, \(z_{11}\) denotes the production volume ratio of an electric car when the total production amount of two cars is fixed as 1; hence, the production volume ratio of an engine car is \((1-z_{11})\).