## 3 Optimization of SM-OGFC by Response Surface Methodology (RSM)
### Build ISM Model  
Model Fit Summary tab is a statistical appraisement used to suggest the most appropriate model. Take \(-15\)\({}^{\circ}\)C splitting strength response model as an example, the details of Model Fit Summary tab are summarized in Table 6. The proposed quadratic model is suggested, due to the \(p\)-value of the model is less than 0.0001 and the F-value is equal to 44.51.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Source**} & **Sequential** & **Lack of Fit** & **Adjusted** & **Predicted** \\  & \(p\)**-Value** & **\(p\)**-Value** & **R-Squared** & **R-Squared** \\ \hline Linear & 0.0345 & 0.0050 & 0.2225 & 0.0708 \\
2 Fl & 0.9022 & 0.0030 & 0.0783 & \(-0.1258\) \\ \hline Quadratic & \(<\)0.0001 & 0.8855 & 0.9546 & 0.9240 \\ \hline Cubic & 0.7924 & 0.9204 & 0.9382 & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Different statistical values form Model Fit Summary tab.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & **Units** & **Lower Limit (\(-1\))** & **Upper Limit (\(+1\))** \\ \hline Asphalt-aggregate ratio & \% & 4.3 & 5.7 \\ Silane coupling agent content & \% & \(1\) & 3 \\ \hline Mixing temperature & \({}^{\circ}\)C & 140 & 160 \\ \hline Compaction times & n & 50 & 70 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The coded level of affected variable.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & **Asphalt-** & **Silane Coupling** & **Mixing** & **Compaction** & **Marshall** & \(-15\)\({}^{\circ}\)C & **Spring-Thawing** \\  & **Aggregate** & **Agent Content** & **Tem-Perture** & **Times (D)** & **Stability** & **Splitting** & **Stability (KN)** \\  & **Ratio (A)** & **(B)** & **(C)** & **(K)** & **Strength (Mpa)** & **Stability (KN)** \\ \hline
1 & 1 & \(-1\) & \(1\) & \(-1\) & \(9.01\) & \(3.06\) & \(7.62\) \\
2 & \(-1\) & \(1\) & \(-1\) & \(1\) & \(8.49  
Based on the experimental data of Table 5, the regression equation of \(-15\)\({}^{\circ}\)C splitting strength is calculated by the Equation (2):  
\[\begin{array}{c}\mathrm{R}_{T}=3.79+0.49\times\mathrm{A}-0.08\times\mathrm{B} +0.05\times\mathrm{C}+0.097\times\mathrm{D}+0.084\times\mathrm{AB}\\ -0.073\times\mathrm{AC}+0.086\times\mathrm{AD}+0.006\times\mathrm{BC}-0.16\times \mathrm{BD}+0.032\times\mathrm{CD}-0.56\times\mathrm{A}^{2}\\ -0.13\times\mathrm{B}^{2}-0.1\times\mathrm{C}^{2}-0.018\times\mathrm{D}^{2}. \end{array} \tag{2}\]  
Moreover, it is found that A, B, D, AB, AC, AD, BD, A\({}^{2}\), B\({}^{2}\), and C\({}^{2}\) terms are significant factors with \(p\)-values less than 0.0500 through ANOVA Analysis. The details of ANOVA Analysis are summarized in Table 7. Based on the analysis above, the original regression is revised. Equation (3) presents the modified regression equation, which deletes the non-significant model terms:  
\[\begin{array}{c}\mathrm{R}_{T}=3.79+0.49\times\mathrm{A}-0.08\times\mathrm{B }+0.097\times\mathrm{D}+0.084\times\mathrm{AB}\\ -0.073\times\mathrm{AC}+0.086\times\mathrm{AD}-0.16\times\mathrm{BD}-0.56\times \mathrm{A}^{2}-0.13\times\mathrm{B}^{2}-0.1\times\mathrm{C}^{2}.\end{array} \tag{3}\]  
Credibility analysis of modified regression equation is summarized in Table 8. According to Table 8, the value of R\({}^{2}\) is almost one, and therefore the experimental results are in good agreement with the predicted values. In addition, the difference between R\({}^{2}\) and R\({}_{\mathrm{Adj}}\)\({}^{2}\) is 0.02, and there is a good accordance between R\({}_{\mathrm{Pred}}\)\({}^{2}\) and R\({}_{\mathrm{Adj}}\)\({}^{2}\) indicating the accuracy of the proposed model.