## 4 Procedure of Meta-Modeling
### Determination of the Meta-Model  
The data collected according to the experimental design in Table 5 were analyzed to estimate the coefficients \(\beta_{o}\), \(\beta_{i}\), \(\beta_{ii}\), and \(\beta_{ij}\) of the meta-model by variance analysis. Some ineffective terms were removed individually; the remaining terms and the analysis results are listed in Table 6. Considering traditional statistical principles, we should discard terms when the p-values are greater than 0.05. In this process, \(x_{12}\) is a candidate for removal (\(p\)-value-0.094), but we retained it. [PERSON] et al. (2017) described that \"There is no clear justification to the use of fixed significance, except by tradition\". Hence, we pre-determined that all first-order terms (\(x_{i}\)) would not be discarded, because they must be important and effective on the performance measure, and helpful for the expansion of the model. Similarly,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Data Set**} & \multicolumn{8}{c}{Original Levels} & \multicolumn{8}{c}{**Normalized Levels**} & \multicolumn{8}{c}{**PR\({}_{sim}\)**} \\ \cline{2-13}  & \(z_{1}\) & \(z_{2}\) & \(z_{3}\) & \(z_{4}\) & \(z_{5}\) & \(z_{6}\) & \(\gamma\) & \(\gamma\) & \(\gamma_{9}\) & \(z_{10}\)  
the meta-models for reconfiguration strategies 1 and 2 were developed individually. Only the analysis data of strategy 3 are described herein. The meta-models for estimating the production rates of strategies 1, 2, and 3 are denoted by \(f_{1}(\mathbf{X})\), \(f_{2}(\mathbf{X})\), and \(f_{3}(\mathbf{X})\), respectively, and the detailed functions are shown in Equations (5), (6) and (7), respectively. Table 7 shows the result of variance analysis for the meta-model of strategy 3. \[ \begin{array}{llll}PR_{1}=f_{1}(\mathbf{X})=&0.4316+0.0156x_{1}+0.01 51x_{2}+0.0100x_{3}+0.0095x_{4}+0.0021x_{5}+0.0018x_{6}+0.0057x_{7}\\ &+0.0003x_{8}+0.1080x_{9}-0.0283x_{10}+0.0061x_{11}-0.0002x_{12}-0.0033x_{1}^{2 }-0.0025x_{2}^{2}\\ &-0.0024x_{3}^{2}-0.0013x_{4}^{2}-0.0016x_{7}^{2}+0.0145x_{3}^{2}+0.0027x_{10}^{ 2}-0.0009x_{11}^{2}-0.0021x_{1}x_{2}\\ &+0.0012x_{1}x_{3}+0.0006x_{1}x_{4}+0.0004x_{1}x_{5}+0.0003x_{1}x_{6}+0.0006x_{1 }x_{7}+0.0011x_{1}x_{9}\\ &-0.0003x_{1}x_{10}+0.0010x_{1}x_{11}+0.0007x_{2}x_{3}+0.0002x_{2}x_{5}+0.0009x_ {2}x_{9}+0.0005x_{2}x_{11}\\ &-0.0009x_{3}x_{4}+0.0002x_{3}x_{5}+0.0002x_{3}x_{6}+0.0004x_{3}x_{7}+0.0008x_ {3}x_{9}-0.0005x_{3}x_{10}\\ &+0.0006x_{3}x_{11}+0.0006x_{4}x_{9}-0.0003x_{4}x_{10}+0.0004x_{4}x_{11}-0.0003x _{5}x_{6}-0.0005x_{5}x_{7}\\ &-0.0009x_{5}x_{11}-0.0003x_{6}x_{7}-0.0008x_{6}x_{11}+0.0003x_{7}x_{10}-0.0010 x_{7}x_{11}+0.0005x_{10}x_{11}\\ \end{array}  \tag{5}\]  
\[ \begin{array}{llll}PR_{2}=f_{2}(\mathbf{X})=&0.4296+0.0153x_{1}+0.0148x_{2}+0.00 99x_{3}+0.0093x_{4}+0.0025x_{5}+0.0022x_{6}+0.0060x_{7}\\ &+0.0003x_{8}+0.1063x_{9}-0.0278x_{10}+0.0066x_{11}-0.0002x_{12}-0.0032x_{1}^{ 2}-0.0021x_{2}^{2}\\ &-0.0021x_{3}^{2}-0.0013x_{4}^{2}-0.0007x_{5}^{2}-0.0015x_{7}^{2}+0.0143x_{9}^ {2}+0.0029x_{10}^{2}-0.0021x_{1}x_{2}\\ &+0.0012x_{1}x_{3}+0.0006x_{1}x_{4}+0.0005x_{1}x_{5}+0.0007x_{1}x_{7}+0.0008x_ {1}x_{9}+0.0009x_{1}x_{11}\\ &+0.0006x_{2}x_{3}+0.0002x_{2}x_{6}+0.0006x_{2}x_{9}+0.0007x_{2}x_{11}-0.0009 x_{3}x_{4}+0.0005x_{3}x_{7}\\ &+0.0006x_{3}x_{9}-0.0004x_{3}x_{10}+0.0007x_{3}x_{11}+0.0005x_{4}x_{9}+0.0004 x_{4}x_{11}-0.0005x_{5}x_{7}\\ &-0.0009x_{5}x_{11}-0.0007x_{6}x_{11}+0.0005x_{7}x_{9}-0.0011x_{7}x_{11}+0.000 2x_{9}x_{11}+0.0004x_{10}x_{11}\end{array}  \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Term** & **Coef. ** & **SE Coef. ** & \(\mathbf{t}\) & \(\mathbf{p}\) & **Term** & **Coef. ** & **SE Coef. ** & \(\mathbf{t}\) & \(\mathbf{p}\) \\ \hline Const. & 0.4340 & 0.0004 & 1070.01 & \(<\)0.001 & \(x_{1}x_{7}\) & 0.0005 & 0.0001 & 4.07 & \(<\)0.001 \\ \(x_{1}\) & 0.0160 & 0.0001 & 143.57 & \(<\)0.001 & \(x_{1}x_{9}\) & 0.0009 & 0.0001 & 8.01 & \(<\)0.001 \\ \(x_{2}\) & 0.0152 & 0  
\[