## 3 Optimization Design
### Reliability-Based Multi-Objective NSGA-II Optimization
The second optimization step facilitates observation of the output responses of the mechanism, including input stiffness, natural frequency, displacement amplification, and stress, which are sensitive to any variation in the DVs. Figure 8 illustrates the sensitivity analysis results for the DVs. The analysis shows that the input stiffness of the mechanism increases proportionally to the thickness, as seen in Figure 8a. Since the natural frequency in Figure 8b is a function of the input stiffness, it also increases accordingly. Figure 8c of the displacement amplification confirms, again, that the larger the stiffness, the more complex the mechanism deformation. Apart from the apparent effect of \(T_{2}\) and \(T_{3}\), the impact of \(T_{1}\) shows a contrary forecast, since it supports slight displacement amplification. In this figure, it is also apparent that the \(A_{amp}\) is heavily affected by the thickness of the hinges, whereas Equation (4) does not involve the parameter \(T\). This second optimization step will compensate for any disadvantages of the previous model. Finally, the sensitivity of the DVs in Figure 8d appears inversely and nonlinearly proportional to the concentrated stress value to differing degrees. Figure 8: DV sensitivity analysis to the objective functions: (**a**) input stiffness; (**b**) natural frequency; (**c**) amplification ratio; (**d**) maximum stress. Table 3 describes the boundary of the DVs, constraint, and objective functions of this problem. The objective function \(f_{1}\) in Equation (43) aims to maximize the first natural frequency to prevent any resonant phenomena during the operation of the mechanism. The second objective function \(f_{2}\) in Equation (44) aims to maximize the displacement amplification ratio \(A_{amp}\). At first glance, it looks contradictory to the optimization target in the previous section in Equation (35), in which the aim is to maintain \(A_{amp}\) close to a given value of 3.0. Maximizing the function \(f_{2}\) ensures that it should not fall below 3.0. Due to some simplification assumptions in the PRBM, when more authentic parameters of elastic structures are considered, it is hard to maintain the \(A_{amp}\) at the expected ratio. To eliminate infeasible designs during the process of solving the multi-objective design optimization problem, the candidates must fulfill the constraints of stress concentration requirements for the elastic hinges in Equation (52). With the reliability analysis step integrated into the NSGA-II optimization process, constraint equations with an asterisk are supplemented. In this application, to assure that the structure has micron resolution, adequate rigidity, and allowable displacement error, the parasitic motion in the \(x\)-direction in Equation (51) must be satisfied, and treated as a limit state function. In addition, to ensure that the CFDM is machineable, limit state functions of cutting force are established, satisfying the Equations (48)-(50). It is chosen for this analysis task since the current objectives are implicit functions, and given the relative advantages of the FORM in dealing with such problems compared with the Monte Carlo and SORM methods, as analyzed in Section 1. The NSGA-II optimization in this step is implemented in MATLAB using the ANSYS ADPL as the medium to convey the calculation process. The plane strain element (PLANE82) is used in the analysis to evaluate the constraint functions (stiffness, maximum stress, parasitic motion) and objective functions (amplification ratio and natural frequency). Although the algorithm can directly use the finite element method (FEM) at