# Intelligent Design of Pavement Concrete Based on RSM-NSGA-III-CRITIC-VIKOR
## 3 Results and Analysis
### Analysis and Discussion Based on the RSM Model
#### 3.2.2 Response Surface Analysis of Shrinkage  
Figure 9 presents the response trajectories of shrinkage as influenced by W/C, sand ratio, and AEA. Shrinkage showed a positive correlation with W/C, with higher W/C  
Figure 8: R\({}_{\rm f}\)/R\({}_{\rm c}\) Contour chart and 3D Surface chart. (**a**) Effect of W/C and AEA on R\({}_{\rm f}\)/R\({}_{\rm c}\) Contour chart; (**b**) Effect of W/C and AEA on R\({}_{\rm f}\)/R\({}_{\rm c}\) 3D Surface chart; (**c**) Effect of Sand ratio and AEA on R\({}_{\rm f}\)/R\({}_{\rm c}\) Contour chart; (**d**) Effect of Sand ratio and AEA on R\({}_{\rm f}\)/R\({}_{\rm c}\) 3D Surface chart. leading to increased shrinkage, though the rate of increase diminished at higher levels, suggesting a non-linear behavior. For the sand ratio, shrinkage initially decreased as the ratio increased from 36.0% to 39.0%, but this trend reversed with further increases from 39.0% to 42.0%, resulting in a sharp rise in shrinkage. In contrast, higher AEA consistently led to higher shrinkage values. Among the factors studied, W/C had the most significant impact, followed by the sand ratio and AEA. Figure 10a-f illustrate the influence of key parameters on shrinkage. In Figure 10a,b, the interaction between W/C and the sand ratio reveals that W/C has a slightly stronger impact on shrinkage. The minimum shrinkage is observed when the W/C is around 0.35 and the sand ratio is below 39%. Figure 10c,d demonstrate that W/C exerts a slightly greater effect on shrinkage than AEA, with the lowest shrinkage occurring at a W/C of 0.35 and an AEA of 0.100%. Figure 10e,f highlight the dominant role of the sand ratio in shrinkage, with the lowest shrinkage observed at a sand ratio between 36.0% and 37.0%, and an AEA close to 0.100%. The results can be summarized into the following single-factor effects: (1) an increase in W/C raises porosity and free water content, leading to increased shrinkage [8]. (2) Variations in the sand ratio affect aggregate packing and pore distribution, causing shrinkage to first decrease and then increase [13]. (3) When AEA increases, excessive and unevenly distributed air bubbles compromise the structural integrity of the cement paste, weaken the bond between aggregates and the paste, and accelerate moisture migration and pore pressure release due to enhanced bubble connectivity. These combined effects lead to a significant increase in concrete shrinkage [14]. Figure 10a,b show a contour plot illustrating the non-linear interaction between W/C and the sand ratio on shrinkage. When the W/C is reduced to 0.35 and the sand ratio is below 39%, the shrinkage rate is minimized. This is due to the dense microstructure of the cement paste at lower W/C, which reduces moisture evaporation, combined with the moderate sand ratio (36-39%) that creates a stable aggregate framework. This framework helps limit the shrinkage of the paste, while the paste fills the gaps between the aggregates, reducing internal stress. On the other hand, when the W/C increases to 0.42-0.43 and the sand ratio is 36%, the shrinkage rate is maximized. This results from higher W/C increasing capillary porosity, leading to greater shrinkage from moisture evaporation, while a lower sand ratio means a higher proportion of paste, which reduces the restraining effect of the aggregates and creates a weaker, paste-dominated structure, intensifying shrinkage. Additionally, when the sand ratio exceeds 39%, the effect of aggregate gaps becomes more noticeable, leading to uneven paste distribution and an increase in shrinkage. Figure 9: The responses trace for shrinkage. Figure 10c,d show the combined effect of W/C and AEA on concrete shrinkage. When the W/C is 0.35 and the AEA is 0.100%, the shrinkage rate is at its lowest. This is due to the optimized interaction between the two factors: the lower W/C (0.35) creates a denser cement paste microstructure, reducing capillary porosity and minimizing moisture evaporation during the drying process.