## 4 Development of Multi-Objective Optimization Functions for Comprehensive Driving Performance
### Fitting Models of Evaluation Indices Based on Support Vector Regression  
For a certain spring height, \(IRI\) (international roughness index), vehicle speed, and load vary within a certain range. Besides this, during the process of manufacturing, the inner diameter of the air pipe is no smaller than that of the connector. Table 2 shows the ranges of the variables at different spring heights.  
A full factorial experiment method was employed for the simulation of the vehicle-road coupling model described in Section 2.1. For each spring height, each uncertain variable was set to three levels, and each design variable was set to four levels [36]. Eliminating the experimental schemes in which the inner diameter of the air pipe is smaller than that of the connector, a total of 270 schemes remained. Simulation results of 220 schemes, which were randomly selected from the 270 schemes, were used for training of the fitting model, while the simulation results of the other 50 schemes were employed for the performance testing of the fitting model. Since support vector regression (SVR) features favorable fitting precision, fitting efficiency, and wide application range [37], it was used to fit the regression models of \(DLSC\) and \(a\) with the changes of the uncertain and design variables, respectively, based on the simulation results of the training schemes. The regression models are  
\[f_{1}(\mathbf{X},\mathbf{U})=\sum_{i=1}^{l}{(\mathbf{\alpha}_{1i}^{*}-\mathbf{ \alpha}_{1i})K(\mathbf{x}_{1i},\mathbf{x})}+b_{1} \tag{13}\]  
\[f_{2}(\mathbf{X},\mathbf{U})=\sum_{i=1}^{l}{(\mathbf{\alpha}_{2i}^{*}-\mathbf{ \alpha}_{2i})K(\mathbf{x}_{2i},\mathbf{x})}+b_{2} \tag{14}\]  
where \(f_{1}(\mathbf{X},\mathbf{U})\), \(f_{2}(\mathbf{X},\mathbf{U})\) are the expressions of \(DLSC\) and \(a\) after fitting via SVR, respectively; \(\mathbf{X}~{}=~{}[x_{1},x_{2}]^{T}\) is the design vector; \(\mathbf{U}=[u_{1},u_{2},u_{3}]^{T}\) is the uncertain vector; \(l\) = 220; \(\mathbf{\alpha}_{ji}\) and \(a_{ji}^{*}\) are Lagrange multipliers, \(j\) = 1, 2; \(K(\mathbf{x}_{ji},\mathbf{x})\) is the kernel function; \(\mathbf{x}_{ji}\) is the support vector; \(\mathbf{x}=[x_{1},x_{2},u_{1},u_{2},u_{3}]^{T}\); and \(b_{j}\) is the bias term.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Variable Type** & **Description** & **Dimension** & \begin{tabular}{c} **Range of Variable at** \\ **Spring Height \(H_{0}\)** \\ \end{tabular} & \begin{tabular}{c} **Range of Variable at** \\ **Spring Height \(H_{1}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Range of Variable at** \\ **Spring Height \(H_{2}\)** \\ \end{tabular} &
\begin{tabular}{c} **Range of Variable at** \\ **Spring Height \(H_{3}\)** \\ \end{tabular} \\ \hline Uncertain variable \(u_{1}\) & _IRI_ & m/km & \([1,10]\) & \([1,5]\) & \([6,10]\) & \([1,5]\) \\ Uncertain variable \(u_{2}\) & Speed & km/h & \((0,90]\) & \((0,50]\) & \((0,50]\) & \([70,90]\) \\ Uncertain variable \(u_{3}\) & Load & kg & \([0,3735]\) & \([3735,7455]\) & \([3735,7455]\) & \([3735,7455]\) \\ Design variable \(x_{1}\) & Inner diameter of air line & mm & \([5,100]\) & \([5,100]\) & \([5,100]\) & \([5,100]\) \\ Design variable \(x_{2}\) & Inner diameter of connector & mm & \([5,100]\) & \([5,100]\) & \([5,100]\) & \([5,100]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Ranges of variables at different spring heights.  
Figure 2 shows the comparisons between the simulation values of Simulink models and the fitted values of SVR models using the test schemes. The coefficient of determination (R\({}^{2}\)) was chosen to test the goodness of fit of the SVR models. It is shown in the figure that the values of R\({}^{2}\) are close to 1 for both the _DLSC_ and \(a\), which means that the accuracy of the SVR models is desirable. Thus, the SVR models can be used as the internal particle swarm fitness functions of the subsequent DL-MOPSO algorithm.  
Figure 2: _Cont._