## 5 Multi-Objective Optimization of Seat Dynamic Comfort Parameters
### Multi-Objective Optimization Using the NSGA-II Algorithm  
NSGA-II is a powerful multi-objective optimization algorithm, renowned for efficiently balancing solution quality and diversity. It employs non-dominated sorting to prioritize Pareto-optimal solutions and uses crowding distance metrics to ensure a uniform spread across the Pareto front while maintaining computational efficiency and robustness in complex engineering and machine learning applications [19; 20; 21]. Figure 14: The result of comparing the predicted value with the actual value. (**a**) The RMSE between the predicted value and the actual value of the maximum transmissibility. (**b**) The RMSE between the predicted value and the actual value of the resonant frequency. This chapter establishes a nonlinear mathematical model with four design parameters as optimization variables: air spring stiffness (K\({}_{1}\)), damper damping coefficient (C\({}_{1}\)), equivalent stiffness of the seat cushion (K\({}_{2}\)), and equivalent damping coefficient of the seat cushion (C\({}_{2}\)). The optimization objectives are the maximum transmissibility ratio (TR) and the resonance frequency (f) of the seat surface response. Multi-objective optimization is performed using the Non-dominated Sorting Genetic Algorithm II (NSGA-II). The mathematical model of the optimization objectives can be expressed as (14) and (15):  
\[\text{min}f_{T_{R}}(K_{1},C_{1},K_{2},C_{2}) \tag{14}\]  
\[\text{min}f_{f}(K_{1},C_{1},K_{2},C_{2}) \tag{15}\]  
According to the range of mechanical parameters provided by the manufacturer of the seats studied in this paper, the optimization constraints can be expressed as (16):  
\[\left\{ \begin{array}{l}20\text{ N/mm}\leq K_{1}\leq 90\text{ N/mm}\\ 1\text{ N}\times\text{s/mm}\leq C_{2}\leq 15\text{ N}\times\text{s/mm}\\ 10\text{ N/mm}\leq K_{2}\leq 50\text{ N/mm}\\ 0.1\text{ N}\times\text{s/mm}\leq C_{2}\leq 3\text{ N}\times\text{s/mm}\end{array} \right. \tag{16}\]  
The NSGA-II multi-objective optimization algorithm was implemented based on the established GA-BP neural network model, with the initial population size set to 50, 100 iterations, a crossover probability of 0.90, and a mutation probability of 0.10. The optimized parameters for the seat suspension and cushion were obtained, with the resulting Pareto front shown in Figure 15. The optimal target parameter selected from the Pareto front is T\({}_{\text{R}}\) = 1.142, f = 1.32 Hz. The corresponding optimized seat structural parameters are listed in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{3}{c}{**Max Transmissibility**} & \multicolumn{3}{c}{**Resonant Frequency**} \\ \cline{2-7}  & **R\({}^{2}\)** & **MAE** & **RMSE** & **R\({}^{2}\)** & **MAE** & **RMSE** \\ \hline
1 & 0.933 & 0.213 & 0.246 & 0.955 & 0.162 & 0.287 \\
2 & 0.909 & 0.268 & 0.220 & 0.986 & 0.196 & 0.344 \\
3 & 0.932 & 0.248 & 0.193 & 0.924 & 0.238 & 0.270 \\
4 & 0. To validate the optimization results, dynamic simulations were conducted using the seat parameters corresponding to the selected Pareto optimal solutions. The optimized vibration response curves of the seat are illustrated in Figure 16.  
Figure 16(a),b depict the displacement response curves and acceleration input-output curves, respectively. The calculated results indicate that the maximum transmissibility is 1.18 and the resonance frequency is 1.26 Hz. These metrics demonstrate significant improvements in vibration isolation performance compared to the baseline configuration, aligning with the design objectives of minimizing resonant amplification and enhancing ride comfort. To validate the effectiveness of the dynamic comfort parameters optimized by NSGA-II, three groups of data with the most comprehensive optimal dynamic comfort metrics were selected from the sample dataset for comparison, as illustrated in Figure 17 and Table 6.  
Figure 16: Optimized seat vibration response. Figure 15: Optimal pareto front.