## 6 Conclusions  
In this study, we analyzed the actuator's operational performance, i.e., the effectiveness of the design parameters for the electromagnetic force performance and electromagnetic characteristics, and investigated the methods for magnetic circuit optimization and stochastic prediction of the performance of design parameter tolerance. As the research methods, we used a full factorial design for verifying the effectiveness and screening the characteristics of the design parameters and performed an experimental verification by applying the response surface design and multi-purpose functions for optimization. We also performed statistical analyses using the Markov chain Monte Carlo method to investigate the effects of the tolerances in the production process of design parameters on performance and verified its usefulness. In conclusion, the effectiveness factors for the electromagnetic force in the vertical magnetization magnetic circuit of a permanent magnet, which is the research object of this study, were identified to be the permanent magnet thickness, magnetic force collecting plate thickness, yoke position, and yoke thickness. With the effective factors showing the second-order nonlinear curvature characteristics including interactions, a response surface design was judged to be the most suitable model to attain optimization by modeling the curved surface characteristics that can consider the nonlinearity. As a result of optimizing the multi-objective function to minimize the boundary conditions of the design parameters and the magnetic circuit thickness and maximize the electromagnetic force, the magnetic circuit thickness was estimated at 19.40 mm, and the electromagnetic force was estimated at 20.02 N. It was found that the electromagnetic force performance can reach a range of 20.01-20.09 N, and the magnetic circuit slimming can reach 19.39-19.46 mm, in terms of the design tolerance and the design production tolerance at the 3\(\sigma\) level based on a reliability of 99.73%. A design tolerance optimization analysis revealed that minimizing the variations in yoke thickness and yoke position is advantageous for electromagnetic force performance, and minimizing the variations of yoke thickness and yoke position in the magnetic circuit thickness is advantageous for slimming an actuator. Regarding the vertical magnetization structure, an increase in yoke thickness (YT) and yoke position (YP) was found to have a positive effect on the electromagnetic field performance, which allowed the conclusion that slimming is possible up to 19.39 mm, even considering the 3\(\sigma\) process conditions. Taking these results together, it may be concluded that optimization using response surface design and optimization of tolerance management based on stochastic performance prediction using the Monte Carlo method are highly viable and a useful method for the optimum design of an AVAS.  
Furthermore, we plan to analyze the mass production performance and perform ongoing data verification based on the accumulated data for tolerance management optimization and to pursue research that applies the AI-based optimization approach.  
All participating researchers discussed the idea, agreed to the result, and contributed to the thesis writing. In particular, [PERSON] planned and organized the research contents as a whole, and contributed to the analysis of the results and improvement of the manuscript. [PERSON] performed the tolerance design optimization analysis, and [PERSON] contributed to the actuator performance optimization by the experimental design method. All authors have read and agreed to the published version of the manuscript.  
This work was supported by a Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government, MOTIE (No. 20172010000420).  
Not applicable.  
The authors declare no conflict of interest.