## 5 Multi-Objective Reliability Optimization of Suspension Control Arm
### Multi-Objective Reliability-Based Optimization  
The fatigue life of the control arm is sensitive to the uncertainties of the six design variables. Therefore, a multi-objective reliability-based optimization of the control arm was developed based on deterministic optimization by considering the reliability constraint of fatigue life, in which the Monte Carlo simulation was applied to calculate the reliability. The deterministic optimization results of design variables were determined as the mean values of the design parameters of the control arm, which were taken as probability random variables subjected to normal distribution, as shown in Table 5. Considering the reliability of fatigue life more than \(10^{6}\) cycles, the multi-objective reliability optimization can be formulated as:  
\[\left\{ \begin{array}{c}\min(m(\mathbf{x}),-N(\mathbf{x}))\\ \text{find}\,\mathbf{x}=(x_{1},\ldots,x_{2})^{\text{T}}\\ \text{s.t.}R\left[N(\mathbf{x})\geq 10^{6}\right]\geq R_{j}\\ |k_{x}(\mathbf{x})-k_{x0}|\leq 0.1k_{x0}\\ |k_{y}(\mathbf{x})-k_{y0}|\leq 0.1k_{y0}\\ |f_{t}(\mathbf{x})-f_{t0}|\leq 0.1f_{t0}\\ \mathbf{x}\in(\mathbf{x}_{L},\mathbf{x}_{U})\end{array} \right. \tag{22}\]  
where \(R\) is the reliability; and \(R_{j}\) represents the expected reliability, set to 95%. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variables** & **Optimal Value** & **Corrected Value** & **Responses** & **Value** \\ \hline \(x_{1}/\text{mm}\) & \(-4.9711\) & \(-5.0\) & Mass/kg & 2.88 \\ \(x_{2}\) & \(0.94625\) & \(0.95\) & Life/cycles & 1,080,000 \\ \(x_{3}\) & \(1.0955\) & \(1.09\) & First order natural & 333.84 \\ \(x_{4}\) & \(0.90005\) & \(0.90\) & Longitudinal stiffness/N\(\cdot\text{mm}^{-1}\) & 11,765 \\ \(x_{5}\) & \(0.98121\) & \(0.98\) & Transverse stiffness/N\(\cdot\text{mm}^{-1}\) & 23,810 \\ \(x_{6}/\text{mm}\) & \(4.9934\) & \(5.0\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Deterministic multi-objective optimization results. Figure 12: Pareto front. There were 111 Pareto solutions obtained by solving the reliability-based optimization based on NSGA-II algorithm and Monte Carlo simulations. Then, the entropy weighted grey relational analysis was employed to determine the optimal design of the control arm. The Pareto solutions were defined as the decision-making matrix, which were used to calculate the grey relational coefficients, and some of them are listed in Table 6. Then, the weights of performance indexes were obtained using the entropy method, as listed in Table 6. Finally, the grey relational grade was calculated, as shown in Figure 13.  
It can be seen from Figure 13 that the grey relational grade of the third Pareto solution was the maximum, that is, this grey relational grade was the closest to the expected value. Therefore, the third Pareto solution was selected as the optimal design of the control arm after multi-objective reliability optimization. The design variables obtained by the reliability-based optimization were corrected according to practical engineering, as shown in Table 7. The finite element model of the control arm was modified using these design parameters, and the fatigue life contour is given in Figure 14. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Grey Relational Coefficient**} & \multirow{2}{*}{**Weight**} \\ \cline{2-2} \cline{4-6}  & **1** & **2** & **...** & **110** & **111** \\ \hline Mass & 0.6775 & 0.4280 &... & 0.1730 & 0.0954 & 0.2837 \\ Fatigue life & 0.3534 & 0.5154 &... & 0.0454 & 0.1197 & 0.1210 \\ First order natural & & & & & \\ frequency & 0.3678 & 0.7201 &... & 0.0206 & 0.0408 & 0.0450 \\ Longitudinal stiffness & 0.3774 & 0.7262 &... & 0.0227 & 0.0446 & 0.0529 \\ Transverse stiffness & 0.4079 & 0.5795 &... & 0.0365 & 0.0932 & 0.0946 \\ Reliability & 0.3520 & 0.3600 &... & 0.2364 & 0.1450 & 0.4027 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Grey relational coefficient. Figure 13: Grey relational grade.