# Intelligent Design of Pavement Concrete Based on RSM-NSGA-III-CRITIC-VIKOR
## 2 Materials and Methods
### Multi-Objective Optimization Framework
#### 2.2.2 Multi-Objective Optimization and Decision-Making Method  
The study introduces a multi-objective optimization and decision-making framework based on NSGA-III and the CRITIC-VIKOR method to address the trade-offs among indicators in concrete performance optimization. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MgO** & **SO\({}_{3}\)** & **3 CaO-SiO\({}_{2}\)** & **3 CaO-Al\({}_{2}\)O\({}_{3}\)** & **LOI** & **Insoluble** \\ \hline
2.89 & 1.80 & 47.18 & 1.85 & 0.56 & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chemical composition of cement (%). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Coding**} & \multicolumn{4}{c}{**Factor**} \\ \cline{2-4}  & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\)/\% & \(\mathbf{x_{3}}\)/\% \\ \hline \(-1\) & 0.35 & 36 & 0.1 \\
0 & 0.39 & 39 & 0.2 \\
1 & 0.43 & 42 & 0.3 \\ \hline \hline \end{tabular} Note: \(x_{1}\) represents W/C; \(x_{2}\) represents: sand ratio; \(x_{3}\) represents AEA.  
\end{table} 
Table 2: Levels and codings of factors. As shown in Figure 1, NSGA-III employs a reference point distribution mechanism to improve coverage and uniformity within the solution set for high-dimensional objective optimization, making it suitable for complex multi-objective problems. Traditional evolutionary algorithms typically suffer from deteriorating performance when handling more than two objective functions, which is particularly problematic for concrete performance optimization where porosity, shrinkage, and R\({}_{\mathrm{f}}\)/R\({}_{\mathrm{c}}\) must be simultaneously considered. NSGA-III addresses this limitation through its structured reference point mechanism, which systematically partitions the normalized objective space to maintain solution diversity. This approach ensures comprehensive exploration of the Pareto front by associating candidate solutions with uniformly distributed reference points, thereby preserving diversity throughout the evolutionary process. This approach is particularly advantageous for concrete optimization problems where balanced trade-offs are critical for overall performance. In concrete performance optimization, NSGA-III effectively balances key indicators such as porosity, R\({}_{\mathrm{f}}\)/R\({}_{\mathrm{c}}\), and dry shrinkage, providing reliable solutions for complex multi-objective problems. The decision-making process, illustrated in Figure 2, integrates the CRITIC and VIKOR methods to prioritize the Pareto solution set. The CRITIC method quantifies the objective importance of indicators through a dual consideration of contrast intensity and conflict criteria. The standard deviation of each normalized criterion measures the intensity of contrast, while correlation coefficients between criteria pairs measure the conflict between criteria. This mathematical approach proves particularly valuable for concrete optimization, where performance indicators exhibit complex interdependencies that subjective weighting methods often fail to capture accurately. Figure 1: The detailed process of NSGA-III. Meanwhile, VIKOR implements a systematic compromise ranking based on distance functions from the ideal solution. The method calculates utility measure (S\({}_{i}\)) representing \"group utility\" and regret measure (R\({}_{i}\)) representing maximum individual regret, then combines these measures into a comprehensive index (Q\({}_{i}\)). For concrete mix design, this approach is particularly suitable as it prevents solutions with excellent average performance but critical deficiencies in individual properties--a significant advantage when optimizing for extreme environments where weakness in any single property could lead to premature failure. The adjustable compromise parameter in VIKOR provides additional flexibility to balance between maximum group utility and minimum individual regret strategies [31; 32; 33; 34]. Traditional research methods applied in isolation demonstrate significant limitations for concrete optimization in extreme environments.