# Intelligent Design of Pavement Concrete Based on RSM-NSGA-III-CRITIC-VIKOR
## 3 Results and Analysis
### Comprehensive Evaluation Using CRITIC-VIKOR  
To achieve objective weight assignment, the CRITIC method was employed. Prior to weight determination, correlation analysis was conducted to assess the interdependence among evaluation parameters, as shown in Table 8. The correlation matrix reveals that \(\mathrm{R_{f}}/\mathrm{R_{c}}\) and porosity demonstrate a moderate positive correlation coefficient (0.40), indicating partial covariance between these performance indicators. Conversely, shrinkage exhibits minimal correlation with both \(\mathrm{R_{f}}/\mathrm{R_{c}}\) (0.18) and porosity (0.21), suggesting its relative independence as an evaluation criterion. These correlation coefficients were subsequently integrated with the standard deviation of each parameter to calculate the information content (C), which quantifies the degree of distinctive information conveyed by individual indicators and serves as the foundation for objective weight derivation. The resulting parameter values and weights presented in Table 9. Among the response indicators, \(\mathrm{R_{f}}/\mathrm{R_{c}}\) displayed the highest weight (0.39), underscoring its critical importance in the overall performance evaluation. Porosity and shrinkage followed, with weights of 0.32 and 0.29, respectively, signifying their significant but relatively lower influence.  
Weights were then incorporated into the VIKOR method to rank and identify the optimal mix proportions, effectively balancing the trade-offs among the three indicators. These results highlight the pivotal influence of \(\mathrm{R_{f}}/\mathrm{R_{c}}\) in guiding mix design, while also emphasizing the necessity of controlling porosity and shrinkage to enhance concrete performance.  
The CRITIC analysis results reveal that \(\mathrm{R_{f}}/\mathrm{R_{c}}\) holds the highest weight at 39%, while shrinkage carries the lowest weight at 29%. By integrating CRITIC with VIKOR, we systematically ranked the Pareto-optimal solutions to identify the most balanced mix design. As summarized in Table 10, the optimal mix proportions consist of a W/C of 0.35, a sand ratio of 36.0%, and an AEA of 0.200%.  
This optimized framework significantly reduced the experimental workload, generating 73 diverse Pareto-optimal solutions from just 17 mix designs, dramatically improving efficiency compared to traditional trial-and-error methods. When comparing the optimal mix to the baseline mix (W/C = 0.39, sand ratio = 39%, AEA = 0.2%), substantial improvements were achieved: \(\mathrm{R_{f}}/\mathrm{R_{c}}\) increased by 4.5% (from 0.135 to 0.141), shrinkage decreased by 13.0% (from 0.0515% to 0.0446%), and porosity reduced by 13.5% (from 3.24% to 2.82%). These performance enhancements directly contribute to concrete pavements with greater structural integrity and durability under the harsh environmental conditions of arid regions. The optimized mix's lower shrinkage rate significantly reduces crack formation potential, while the improved microstructure from reduced porosity enhances long-term resistance to weathering and traffic loads, resulting in fewer maintenance interventions and extended service life for infrastructure projects in challenging environments.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Shrinkage** & **Porosity** & \(\mathrm{R_{f}}/\mathrm{R_{c}}\) \\ \hline Shrinkage & 1.00 & 0.21 & 0.18 \\ Porosity & 0.21 & 1.00 & 0.40 \\ \(\mathrm{R_{f}}/\mathrm{R_{c}}\) & 0.18 & 0.40 & 1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Correlation coefficients calculated by CRITIC.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Standard Deviation (S\({}_{\mathrm{i}}\))** & **Information Content (C\({}_{\mathrm{j}}\))** & **Weight (\(W_{\mathrm{j}}\))** \\ \hline Shrinkage & 0.22 & 1.32 & 0.29 \\ Porosity & 0.16 & 2.00 & 0.32 \\ \(\mathrm{R_{f}}/\mathrm{R_{c}}\) & 0.21 & 1.86 & 0.39 \\ \hline \hline \end{tabular}
\end{table}
Table 9: CRITIC parameter values and weights.