# Intelligent Design of Pavement Concrete Based on RSM-NSGA-III-CRITIC-VIKOR
## 3 Results and Analysis
### Development and Validation of the RSM Model  
RSM encompasses a collection of statistical and mathematical techniques employed in experimental design, model fitting, regression analysis, and the examination of interactions between independent variables. RSM constructs the relationship between objective functions and design variables, enabling the prediction of system performance. The relationship between the objective function \(y\) and design variables \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{\mathrm{k}}\) can be expressed using the following linear or quadratic models:  
Linear model:  
\[y=a_{0}+\sum_{i=1}^{k}a_{i}x_{i}+\varepsilon \tag{1}\]  
Quadratic model:  
\[y=a_{0}+\sum_{i=1}^{k}a_{i}x_{i}+\sum_{i=1}^{k}a_{ii}x_{i}^{2}+\sum_{i<j}a_{ij} x_{i}x_{j}+\varepsilon \tag{2}\]  
where \(a_{0}\) is the intercept, \(a_{i}\) are the linear regression coefficients, and \(\varepsilon\) is the residual error. Where \(a_{ii}\) represents the quadratic effect of \(x_{i}\), and \(a_{ij}\) represents the interaction effect between \(x_{i}\) and \(x_{j}\). RSM is commonly employed to model non-linear relationships between objective functions and design variables using quadratic polynomials. The polynomial regression model enables the prediction of target output or system performance based on the given parameters. The BBD approach in Design-Expert 13 software, along with the least squares method, was used to perform non-linear fitting on the test results summarized in Table 4. The process led to the development of response surface models for the 28 d R\({}_{\mathrm{f}}\)/R\({}_{\mathrm{c}}\), shrinkage, and porosity, as expressed in Equations (3)-(5). \[y_{1}=-2.02+6.16x_{1}+2.39\times 10^{-2}x_{2}+11.17x_{3}-1.77x_{1}x_{3} \tag{3}\]  
\[ \begin{array}{c}y_{2}=-1.79+3.68x_{1}+0.06x_{2}-0.45x_{3}-5.67x_{1}x_{2}+2.7 3x_{1}x_{3}\\ \hskip 14.226378 pt+3.65x_{2}x_{3}-5.17x_{1}{}^{2}-7.26\times 10^{-4}x_{2}{}^{2}-0.48x _{3}{}^{2}\end{array}  \tag{4}\]  
\[ \begin{array}{c}y_{3}=-0.26+1.75x_{1}+3.82\times 10^{-3}x_{2}+28.53x_{3}-0.02x _{1}x_{2}-40.39x_{1}x_{3}\\ \hskip 14.226378 pt-0.28x_{2}x_{3}-1.01x_{1}{}^{2}+1.58\times 10^{-4}x_{2}{}^{2}-53.59 x_{3}{}^{2}\end{array}  \tag{5}\]  
In the equations, \(x_{1}\) represents W/C; \(x_{2}\) represents the sand ratio; \(x_{3}\) represents AEA; \(y_{1}\) represents R\({}_{\mathrm{f}}\)/R\({}_{\mathrm{c}}\); \(y_{2}\) represents the shrinkage; and \(y_{3}\) represents the porosity. Ensuring the reliability of the RSM model is essential for obtaining accurate optimization results. The study assessed the model's performance through goodness-of-fit analysis and ANOVA, focusing on correlation and reliability. Table 5 presents key metrics for model evaluation, including \(R^{2}\), \(R_{\mathrm{a}}{}^{2}\), \(R_{\mathrm{p}}{}^{2}\), \(R_{\mathrm{a}}{}^{2}\)-\(R_{\mathrm{p}}{}^{2}\), and precision. All models achieved \(R^{2}\) values above 0.8, confirming the ability to effectively represent non-linear interactions between influencing factors and response variables. Moreover, the \(R_{\mathrm{a}}{}^{2}\)-\(R_{\mathrm{p}}{}^{2}\) values were consistently below 0.1, indicating strong agreement between the training and testing datasets and reflecting stable predictive performance. Precision values exceeding four further demonstrate close alignment between the predicted and observed data. Figure 5 compares the predicted and actual values for the quadratic regression models, along with the regression lines. The alignment of data points for shrinkage, porosity, and \(\rm R_{f}/R_{c}\) with the regression lines highlights the strong predictive accuracy and goodness-of-fit of the models. Figure 6 illustrates the normal distribution of the residuals from the response surface model. Residual analysis is a statistical technique used to evaluate the reliability of a model by examining the residuals and associated statistical metrics.