# Intelligent Design of Pavement Concrete Based on RSM-NSGA-III-CRITIC-VIKOR
## 3 Results and Analysis
### Analysis and Discussion Based on the RSM Model
#### 3.2.3 Response Surface Analysis of Porosity  
Figure 11 illustrates the response patterns of porosity to variations in W/C, sand ratio, and AEA. A positive correlation is observed between W/C and porosity. Within the range of 36.0-38.0% sand ratio, porosity decreases, while it increases as the sand ratio extends from 38.0% to 42.0%. Porosity decreases with the increase in AEA from 0.100% to 0.225% but continues to rise as the AEA exceeds the threshold. By analyzing the curvature, AEA exerts the most significant effect on porosity, exhibiting a pronounced non-linear trend, followed by the sand ratio and W/C.  
Figure 12a-d collectively illustrate the interactions between W/C, sand ratio, and AEA on porosity. Figure 12a,b demonstrate that the sand ratio slightly surpasses W/C in influencing porosity, with both factors working together to minimize porosity within the ranges of W/C (0.35-0.36) and the sand ratio (36.0-37.0%). Figure 12c,d emphasize  
Figure 11: The responses trace for porosity.  
the dominant impact of AEA over the sand ratio, where the combination of sand ratio (36.0-38.0%) and AEA (0.140-0.200%) leads to the most substantial porosity reduction.  
The results can be attributed to the following single-factor effects: (1) porosity increases with higher W/C because excessive water in the paste cannot fully participate in the hydration reaction, reducing the production of hydration products [9]. (2) Porosity initially decreases and then increases with higher sand ratios. An appropriate sand ratio effectively fills voids between coarse aggregates, reducing porosity; however, an excessive sand ratio diminishes the structural support of coarse aggregates, causing porosity to rebound [11]. (3) Porosity increases with higher AEA, as the introduction of uniform air bubbles increases the total pore volume and reduces the number of interconnected pores, complicating the pore structure [15].  
In Figure 12a,b, the lowest porosity is observed when both the W/C and sand ratios are low. The explanation lies in the fact that, although hydration is not fully completed with a low W/C, the amount of hydration products is relatively well matched to the available water. Hydration products effectively encapsulate the fine and coarse aggregates, filling most of the voids. Additionally, low W/C minimizes excess water retention, limiting the formation of additional pores. The lower connectivity of pores under the conditions results in a more closed pore structure, which further reduces porosity. As the sand ratio increases, however, the cement paste is no longer sufficient to fully coat the aggregates, leading to higher porosity.  
Figure 12: Porosity Contour chart and 3D Surface chart. (**a**) Effect of W/C and Sand ratio on Porosity. Contour chart; (**b**) Effect of W/C and Sand ratio on Porosity. 3D Surface chart; (**c**) Effect of Sand ratio and AEA on Porosity. Contour chart; (**d**) Effect of Sand ratio and AEA on Porosity. 3D Surface chart.  
In Figure 12c,d, the relationship between the sand ratio and AEA reveals a clear trend. The lowest porosity occurs when the sand ratio is between 36.0% and 38.0%, and AEA is between 0.160% and 0.180%. The optimal sand ratio efficiently fills the gaps between the coarse aggregates, improving the overall compactness of the concrete. Simultaneously, a moderate AEA introduces fine air bubbles that refine the pore structure without overwhelming it, maintaining a uniform pore distribution and minimizing porosity. When the sand ratio exceeds the optimal range, the proportion of coarse aggregates decreases, weakening the aggregate structure and causing the porosity to increase. Similarly, when AEA is excessively high, excessive air bubbles lead to a more complex pore structure, resulting in increased porosity.