## 4 Development of Multi-Objective Optimization Functions for Comprehensive Driving Performance
### Multi-Objective Optimization Functions Considering Uncertainty of Driving Conditions  
Based on the interval number optimization method, for any design vector \(\mathbf{X}\), when the uncertainty vector \(\mathbf{U}\) is constantly changing, the possible values of the regression functions \(f_{1}(\mathbf{X},\mathbf{U})\), \(f_{2}(\mathbf{X},\mathbf{U})\) can form intervals as follows:  
\[f_{1}^{I}(\mathbf{X})=\left\langle f_{1}^{C}(\mathbf{X}),f_{1}^{W}(\mathbf{X}) \right\rangle=\left\{\left.f_{1}(\mathbf{X})\middle|f_{1}^{C}(\mathbf{X})-f_{1 }^{W}(\mathbf{X})\leq f_{1}(\mathbf{X})\leq f_{1}^{C}(\mathbf{X})+f_{1}^{W}( \mathbf{X})\right\}\right. \tag{15}\]  
\[f_{2}^{I}(\mathbf{X})=\left\langle f_{2}^{C}(\mathbf{X}),f_{2}^{W}(\mathbf{X}) \right\rangle=\left\{\left.f_{2}(\mathbf{X})\middle|f_{2}^{C}(\mathbf{X})-f_{2 }^{W}(\mathbf{X})\leq f_{2}(\mathbf{X})\leq f_{2}^{C}(\mathbf{X})+f_{2}^{W}( \mathbf{X})\right.\right\} \tag{16}\]  
where \(f_{1}(\mathbf{X})\), \(f_{2}(\mathbf{X})\) are all possible values of \(f_{1}(\mathbf{X},\mathbf{U})\), \(f_{2}(\mathbf{X},\mathbf{U})\), respectively; \(f_{1}^{I}(\mathbf{X})\), \(f_{2}^{I}(\mathbf{X})\) are the intervals formed by \(f_{1}(\mathbf{X})\), \(f_{2}(\mathbf{X})\), respectively; \(f_{1}^{C}(\mathbf{X})\), \(f_{1}^{W}(\mathbf{X})\) are the midpoint and radius of the \(f_{1}^{I}(\mathbf{X})\) interval, respectively; and \(f_{2}^{C}(\mathbf{X})\), \(f_{2}^{W}(\mathbf{X})\) are the midpoint and radius of the \(f_{2}^{I}(\mathbf{X})\) interval, respectively.  
In order to form objective functions representing the uncertainty of driving conditions, the following functions are used [26]:  
\[F_{1}(\mathbf{X})=\frac{0.5f_{1}^{C}(\mathbf{X})}{\psi_{1}}+\frac{0.5f_{1}^{W} (\mathbf{X})}{\zeta_{1}} \tag{17}\]  
\[F_{2}(\mathbf{X})=\frac{0.5f_{2}^{C}(\mathbf{X})}{\psi_{2}}+\frac{0.5f_{2}^{W} (\mathbf{X})}{\zeta_{2}} \tag{18}\]  
where \(F_{1}(\mathbf{X})\) and \(F_{2}(\mathbf{X})\) are the evaluation functions of the intervals \(f_{1}^{I}(\mathbf{X})\) and \(f_{2}^{I}(\mathbf{X})\), respectively; and \(\psi_{i}\) and \(\zeta_{i}\) are the minimum values of \(f_{i}^{C}(\mathbf{X})\), \(f_{i}^{W}(\mathbf{X})\) in the full factorial experiments, respectively. The values of \(\psi_{i}\) and \(\zeta_{i}\) at different spring heights are listed in Table 3.  
Figure 2: (**a**) Comparisons of dynamic load-sharing coefficient (\(DLSC\)) between the two models at the spring height of \(H_{0}\); (**b**) Comparisons of \(a\) between the two models at the spring height of \(H_{0}\); (**c**) Comparisons of _DLSC_ between the two models at the spring height of \(H_{1}\); (**d**) Comparisons of \(a\) between the two models at the spring height of \(H_{1}\); (**e**) Comparisons of _DLSC_ between the two models at the spring height of \(H_{2}\); (**f**) Comparisons of \(a\) between the two models at the spring height of \(H_{2}\); (**g**) Comparisons of _DLSC_ between the two models at the spring height of \(H_{3}\); (**h**) Comparisons of \(a\) between the two models at the spring height of \(H_{3}\).  
Then, the multi-objective optimization functions can be written as  
\[\left\{\begin{array}{l}\min_{X}(F_{1}(\mathbf{X}))\\ \min_{X}(F_{2}(\mathbf{X}))\\ s.t.\ x_{1}>x_{2},x_{i}^{\min}\leq x_{i}\leq x_{i}^{\max},i=1,2.\end{array}\right. \tag{19}\]