## 3 Design Optimization
### Metadmodeling  
Based on the DOE results described in the previous section, we created a metamodel for the objective function and constraint. Most of the existing studies have proposed metamodels, but accuracy evaluations were not performed. Furthermore, searches for optimal values of design variables have been conducted. Therefore, to evaluate the metamodel accuracy, the best metamodel was selected by comparing the RMSE test outcomes according to the efficiency and cogging torque characteristics.  
The Kriging model [28], an interpolation model type, passes the test points accurately and is suitable for approximation without random errors. Therefore, a numerically robust model was provided. The estimation equation for the Kriging model was defined to eliminate bias, thus minimizing error variance.  
MLP [29] is a type of deep learning algorithm and has the advantage of expressing a nonlinear relationship between input and output variables. In particular, metamodels can be created even when there are plenty of data. However, if there are many parameters that depend on the know-how of the user, accuracy may be deteriorated and training time may be required.  
The EDT [30] has a feature that allows the generation of multiple decision trees (DTs) and the prediction of an output value as the average value of all the outputs predicted by each DT for a specific input value. The instability and performance variance of the DT model were reduced compared with the use of a single DT. Additionally, the predictive power was improved and the performance was excellent in large datasets. However, parameters (depth, number of DTs, etc.) must be determined in advance. As the depth of DT becomes deeper, the DT model becomes complicated and leads to an increase of the calculation complexity. Figure 6 shows the conceptual diagram of the Kriging model, MLP, and EDT used for metamodeling in this study.  
Given that the predictive performance of the metamodel affects the reliability of the optimal design, the predicted performance was compared based on the RMSE test. The RMSE values should be used to evaluate the accuracy of the interpolation model. The predictive performance of the metamodel was evaluated based on the RMSE test and was calculated according to Equation (3) [32].  
\[\text{RMSE}=\sqrt{\frac{1}{n_{test}}\sum_{i=1}^{n_{test}}[y(X_{i})-\hat{y}(X_{ i})]^{2}} \tag{3}\]  
where \(n_{test}\) is the number of test points for metamodel validation, \(y(X_{i})\) is the value of the real function, and \(y^{\prime}(X_{i})\) is the value of the metamodel.  
The predicted performance of the metamodel was evaluated as an output variable of efficiency and cogging torque through the RMSE test. The comparison of the RMSE test results of the metamodels for the objective function is shown in Figure 7. The lower the value of the RMSE is, the better the predictive performance will be. As shown in Figure 7, the Kriging model yielded the best prediction performance as a metamodel of efficiency and cogging torque. However, if the number of design  
Figure 6: Conceptual diagram: (**a**) Kriging; (**b**) multilayer perceptron (MLP); (**c**) ensemble of decision tree (EDT).  
variables, objective functions, and constraints are different, different metamodels for each characteristic may yield the best predictive performances. In this study, Kriging metamodels that yielded the best predictive performances for each of the output variables were selected for optimal design.