## 3 Project Overview
### Establishment Model  
As illustrated in Figure 2, the GA-BP model was established as follows:  
(1) Use the trained BP prediction model as the GA's fitness function. Based on the optimization model, an objective function is established to minimize the cost per linear meter of the reinforced soil embankment:  
\[minf=\frac{hc_{1}}{x_{1}}(x_{4}+x_{1}+2)+\frac{hc_{2}}{2}(2x_{4}+\frac{x_{1}} {x_{2}}) \tag{4}\]  
In this equation, \(h\) is the slope height, set at 24 m; \(c_{1}\) and \(c_{2}\) represent the unit prices of the reinforcement and fill, respectively, valued at 30/m\({}^{2}\) and 9.49/m\({}^{3}\); \(x_{1}\) denotes the wrapping thickness; \(x_{2}\) is the slope rate; and \(x_{4}\) signifies the reinforcement strip length. The remaining parameters have minimal relevance to the embankment cost and are therefore excluded from the objective function.  
(2) In setting constraints, the value ranges for the design parameters are presented in Table 3, and the constraint ranges for the evaluation indices are defined by the following equation:  
\[\left\{\begin{array}{l}0\geq y(1)=y_{pred}(1)\geq 1\%H\\ 0\geq y(2)=y_{pred}(2)\geq 0.3\%H\\ 0\geq y(3)=y_{pred}(3)\geq-115\\ 0\leq y(4)=y_{pred}(4)\leq 64\\ y(5)=y_{pred}(5)\geq 1.45\end{array}\right. \tag{5}\]  
In this equation, \(y(i)\) represents the evaluation indices, including the maximum lateral displacement, maximum vertical displacement, maximum XZ stress, maximum shear strain increment, and the safety factor. The slope height \(h\) is set at 24 m. According to the construction drawings, the maximum lateral displacement and maximum vertical displacement should not exceed 1% and 0.3% of the slope height, respectively. For the maximum XZ stress and maximum shear strain increment, the upper limits are based on the highest values observed during training, as the design drawings and specifications do not provide explicit values. The safety factor, according to current Chinese highway design specifications, must be at least 1.45 in the most unfavorable conditions.  
These parameter upper and lower bounds are transformed into normalized data ranges, which serve as constraints for the genetic algorithm model.  
(3) To ensure the algorithm identifies a solution that meets the constraints during the search process, a penalty function is constructed using the interior point penalty function method. This approach transforms the constrained minimization problem into an unconstrained minimization problem. The penalty function is as follows:  
\[\phi(x)=f(x)-10\sum_{j=1}^{n}\left(abs(y_{i}(x))\right) \tag{6}\]  
(4) Set the GA parameters (Table 6).  
Traditional embankment design involves selecting multiple solutions based on specifications and design experience, followed by finite element modeling to choose the optimal solution. This approach is influenced by the time required for finite element modeling and the individual subjectivity of the designer, limiting the number of samples for solution selection. The GA-BP model established in this paper offers two key advantages: it can quickly output evaluation indices based on design parameters, facilitating the comparison of different schemes, and it can provide a more accurate solution tailored to the decision maker's requirements in terms of cost and embankment stability. This significantly enhances design efficiency.