# Intelligent Design of Pavement Concrete Based on RSM-NSGA-III-CRITIC-VIKOR
## 3 Results and Analysis
### Analysis and Discussion Based on the RSM Model  
#### 3.2.1 Response Surface Analysis of \(\rm R_{f}/R_{c}\)  
Figure 7 presents the response trajectories of \(\rm R_{f}/R_{c}\) as influenced by W/C, sand ratio, and AEA. The analysis shows that \(\rm R_{f}/R_{c}\) increases with W/C in the range of 0.36 to 0.39 but decreases beyond 0.39. The sand ratio shows a negative correlation with \(\rm R_{f}/R_{c}\), where higher sand ratios lead to lower \(\rm R_{f}/R_{c}\). For AEA, \(\rm R_{f}/R_{c}\) decreases slightly in the range of 0.100-0.170% but increases significantly in the range of 0.170-0.300%. Judging by the curvature, AEA has the most significant effect on \(\rm R_{f}/R_{c}\), followed by W/C, with the sand ratio exhibiting a relatively smaller impact.  
Figure 8a-d illustrate the interactions between key mix design parameters and their effects on \(\rm R_{f}/R_{c}\). The relationship between W/C and AEA is explored in Figure 8a, where the contour gradient shows that AEA has a slightly greater impact than W/C. Within the W/C range of 0.37-0.39 and AEA range of 0.280-0.300%, the combination of the two factors synergistically enhances \(\rm R_{f}/R_{c}\). Figure 8c highlights the interaction between the sand ratio and AEA, where the effect of AEA significantly outweighs that of the sand ratio. The most pronounced interaction occurs when the sand ratio is between 40.0% and 42.0% and AEA is between 0.270% and 0.300%, optimizing \(\rm R_{f}/R_{c}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Term**} & \multicolumn{2}{c}{\(\boldsymbol{y_{1}}\)} & \multicolumn{2}{c}{\(\boldsymbol{y_{2}}\)} & \multicolumn{2}{c}{\(\boldsymbol{y_{3}}\)} \\ \cline{2-7}  & \(\boldsymbol{F}\  
The results can be attributed to the following single-factor effects: an optimal sand ratio effectively fills voids between coarse aggregates, increasing concrete compactness and improving both compressive and flexural strength [11]. However, an excessive sand ratio weakens the coarse aggregate skeleton, leading to strength reduction. An increase in W/C raises porosity, which significantly decreases compressive strength, thereby increasing R\({}_{\rm f}\)/R\({}_{\rm c}\)[7]. Similarly, a moderate AEA introduces microbubbles that slightly enhance flexural strength but reduce compressive strength due to decreased compactness, ultimately increasing R\({}_{\rm f}\)/R\({}_{\rm c}\)[7].  
Under the combined influence of W/C and AEA, as illustrated in Figure 8a,b, a low W/C results in reduced hydration products, making the pore structure heavily reliant on AEA. At this stage, a higher AEA enhances microbubble formation, effectively optimizing crack propagation paths and improving flexural strength. However, as AEA increases further, the density of the concrete decreases, reducing compressive strength and leading to a higher R\({}_{\rm f}\)/R\({}_{\rm c}\).  
As shown in Figure 8c,d, at relatively high sand ratios, the filling effect enhances the density of the aggregate skeleton, creating a more robust structural foundation. Concurrently, the introduction of AEA optimizes stress distribution within the concrete matrix by generating microbubbles, which slightly enhances flexural strength. However, an excessive AEA leads to a decline in compressive strength due to reduced density, increasing the R\({}_{\rm f}\)/R\({}_{\rm c}\).