## 4 Surrogate-Assisted Uncertain Optimization for Pavement Strategy  
### Surrogate-Assisted RBDO  
RBDO is the synthesis of reliability analysis and optimization design [35]. In contrast to standard deterministic optimization, RBDO considers the uncertainties as a kind of random variable that makes up probabilistic constraints, which are assessed in the reliability routine. Therefore, compared to deterministic optimization, RBDO can offer a more accurate and competitive solution to an optimization problem [36]. The RBDO problem is as follows:  
\[\begin{array}{ll}find&D=\left[d_{1},d_{2},\ldots,d_{k}\right]^{T}\\ obj.&\min f_{obj}(D,U)\\ s.t.&\beta[G_{i}(D,U)]\geq\beta_{i}^{T}\\ &d_{k}^{L}\leq d_{k}\leq d_{k}^{U}\end{array}\qquad i=1,2,\cdots,m\\ &k=1,2,\cdots,n\end{array} \tag{10}\]  
where \(D\) is the vector of design variables, \(U\) is the vector of random variables, \(d^{L}\) and \(d^{U}\) are the lower limit and upper limit of the design vector, respectively, and \(\beta^{T}\) is the required target reliability index. The limit state function \(G(X)\) is defined in Formula (1).  
The traditional solution process of RBDO usually focuses on the optimization iteration of the outer layer on the design variables. When the reliability index is required by the objective functions or constraint functions, the outer layer optimization is suspended, and the reliability analysis module is called to calculate the corresponding reliability index. After the reliability analysis, the outer layer iterative optimization is continued. The process is repeated until convergence. The disadvantages of this method are low computational efficiency and poor convergence. Therefore, the key problem restricting the calculation efficiency of RBDO is how to obtain the structural reliability index quickly and accurately.  
Therefore, as shown in Figure 6, this paper proposes a surrogate-assisted RBDO, which used a GRNN-based surrogate model to replace the complex nonlinear calculation of traditional finite element analysis and reliability analysis. On this basis, NSGA-II [37] was adopted to further improve the optimization speed and obtain the Pareto optimal sets. The detailed steps are as follows:Step 1: The objective functions, constraint functions, and design variables are determined according to the construction status and sensitivity analysis results;  
Step 2: The optimization interval of design variables \([X^{0}-\lambda\,|\,X^{0}\,|\,,\,X^{0}+\lambda\,|\,X^{0}\,|\,]\) (\(0\leq\lambda\leq 0.5\)) is initialized based on considering the updated structural parameters, and the design values \(X^{0}\) are chosen as the center;  
Step 3: The optimization interval of the design variables is sampled uniformly using the Latin hypercube sampling technique (LHST) [38], and the corresponding sample data for the objective functions and constraint functions are produced using the FEM and reliability analysis.  
Step 4: The explicit functional relation between input side samples and output side samples is generated by the GRNN. The reliability optimization model is established when the GRNN models meet the accuracy requirements.  
Step 5: The multiobjective optimization problem is solved by NSGA-II with respect to the design variables defined in step (2).  
Step 6: The optimization results are output when they fall into the optimization interval and satisfy the discrimination condition; otherwise, steps (2-6) are iterated using the results as a new optimization interval center.