To achieve high work performance for compliant mechanisms of motion scope, continuous work condition, and high frequency, we propose a new hybrid algorithm that could be applied to multi-objective optimum design. In this investigation, we use the tools of finite element analysis (FEA) for a magnification mechanism to find out the effects of design variables on the magnification ratio of the mechanism and then select an optimal mechanism that could meet design requirements. A poly-algorithm including the Grey-Taguchi method, fuzzy logic system, and adaptive neuro-fuzzy inference system (ANFIS) algorithm, was utilized mainly in this study. The FEA outcomes indicated that design variables have significantly affected on magnification ratio of the mechanism and verified by analysis of variance and analysis of the signal to noise of grey relational grade. The results are also predicted by employing the tool of ANFIS in MATLAB. In conclusion, the optimal findings obtained: Its magnification is larger than 40 times in comparison with the initial design, the maximum principal stress is 127.89 MPa, and the first modal shape frequency obtained 397.45 Hz. Moreover, we found that the outcomes obtained deviation error compared with predicted results of displacement, stress, and frequency are 8.76%, 3.6%, and 6.92%, respectively.