## 3 Optimization Design
### Reliability-Based Multi-Objective NSGA-II Optimization
\begin{table}
\begin{tabular}{c c} \hline \hline \(1\). Objective functions: & \\ \hline \(f_{1}=Max(f_{0})\) & (43) \\ \(f_{2}=Max\Big{(}\frac{f_{out}}{f_{in}}\Big{)}\) & (44) \\ \hline \hline \(2\). Limit state function (*): & \\ \hline \(\{G(F_{i},u_{x},\sigma_{m})-D_{0}\leq 0\}\geq\Phi(\beta),\ i=x,y,\ z\) & (45) \\ Reliability \(R_{0}\geq 98\%\) & \\ \hline \hline \(3\). DVs: \(T_{1},\ T_{2},\ T_{3}\) & \\ \hline \hline \(4\). Subjected to: & \\ \hline \(\quad\) & \(0.4\leq T_{1},\ T_{2},\ T_{3}\leq 1.5\) & (46) \\ \(\quad\) & \(\quad\) & (47) \\ \(\quad\) & \(\quad\) & (48) \\ \(\quad\) & \(\quad\) & (49) \\ \(\quad\) & \(\quad\) & (50) \\ \(\quad\) & \(\quad\) & (51) \\ \(\quad\) & \(\quad\) & (52) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Multi-objective design optimization formulation. the beginning for global computation, this research still suggests using this engineering computational tool after the theoretical PRB model. This analysis sequence is beneficial to thoroughly understand a new design before one can relinquish the entire process to commercial engineering software. The material used to fabricate the mechanism is aluminum alloy (Al 7075-T6). Its mechanical properties are [PERSON]'s modulus \(E=71.7\) GPa, [PERSON]'s ratio \(\theta=0.33\), yield strength \(\sigma_{y}=503\) MPa, and density \(\rho=2810\)kg/m\({}^{3}\). The total number of plane elements in the FEM model is 22,638 elements. In the RBDO-NSGA-II optimization process in Figure 7, the number of generations is preliminarily set as 100 and the population as 20. The optimization process can also be terminated by another convergence criteria when the population has no further evolution. Figure 9 shows a Pareto front of this RBDO-NSGA-II optimization (Pareto P2) with two objective functions. In this research, another determined optimization (DO), without considering any provided limit state functions, is also implemented to evaluate the reliability analysis contribution in the RBDO-NSGA-II optimization. This optimization result (Pareto P1) is also presented in Figure 9 for comparison with its counterpart. To further elaborate on the difference between the DO and the RBDO results, this section will use several solutions lying on the two Pareto fronts (P1 and P2) in Figure 9 for comparison. Table 4 presents the benchmarking results, where points (C, E) are two solutions of Pareto P2, and points (A, B, D) are three solutions of Pareto P1. In Table 4, the reliability index \(\beta\) is the ratio of the mean of a random variable to its standard deviation. The purpose of this table is to illustrate that \(\beta\) significantly influences the optimal results. The FORM analysis evaluates \(\beta\) in a calculation loop. Generally, after three iterations, \(\beta\) is converged and will be used to access the reliability level \((R_{0})\). For instance, at the extreme of the amplification and natural frequency objectives, the optimal results of the RBDO problem at point C (3.104; 1500) are smaller than those of the DO problem at point A (3.405; 1500). However, when using the FORM assessment method to calculate the reliability index at point A, the \(\beta\) values for \((\beta_{1},\beta_{2},\beta_{3},\text{and}\ \beta_{4})\) are (2.25, 10.88, 0.0, and 0.0), respectively. The safety level of a solution is evaluated via the \(\beta_{3}\) index. This value of \(\beta_{3}\) is \(R_{0}=42.91\%\) for point A. At point C, the \(\beta\) values for \((\beta_{1},\beta_{2},\beta_{3},\text{ and}\ \beta_{4})\) are (4.11, 14.20, 3.0, and 3.0), respectively. The \(\beta_{3}\) value for point C has a reliability level at \(R_{0}=99.87\%\), which is much higher than that of point A. Another  
Figure 9: Pareto diagrams of DO and RBDO solutions. comparison between point E (3.101; 1499) and point B (3.13; 1499), with approximate values of objective functions, also shows that the level of safety at point E is \(R_{0}=99.87\%\); meanwhile, point B only has a safety level at \(R_{0}=48.26\%\).