## 5 Multi-Objective Optimization

Table 4: Experimental plans and output responses for u\({}_{t}\) = 15.  
the min and max values of \(\Delta\)i(k); \(\zeta\) is the characteristic coefficient, \(0\leq\zeta\leq 1\). In this work \(\zeta\) = 0.5. (+) Calculating the mean of the grey relational coefficient:  
The degree of grey relation is determined by calculating the mean of the grey relational coefficients associated with the output objectives. \[\overline{y_{i}}=\frac{1}{k}\sum_{j=0}^{k}y_{ij}(k) \tag{52}\]  
where \(y_{\overline{y_{i}}}\) is the grey relation value of the jth output targets in the ith experiment. Table 6 displays the calculated results of the grey relation value (y\({}_{\rm i}\)) and the average grey relation value of all experiments. To ensure harmony among the output parameters, a higher average grey relation value is desirable. As a result, the objective function of the multi-objective problem can be transformed into a single-objective optimization problem, with the mean grey relation value serving as the output. The impact of the main design factors on the average grey relation value (\(\overline{y}\)) was analyzed using the ANOVA method, and the corresponding results are presented in Table 7. From the results in Table 7, AS\({}_{2}\) has the most influence on \(\overline{y}\) (57.35%), followed by the influence of u\({}_{1}\) (25.16%), \(X_{\rm ba1}\) (5.38%), \(X_{\rm ba2}\) 2.50%), and AS\({}_{1}\) (0.84%). The order of influence of the main design factors on \(\overline{y}\) through ANOVA analysis is described in Table 8.  
(+) Evaluation of experimental model:  
The adequacy of the proposed model is assessed using the Anderson-Darling method, and the results are presented in Figure 6. From the graph, it is evident that the data points corresponding to the experimental observations (represented by blue dots) fall within the region bounded by upper and lower limits with a 95% standard deviation. Furthermore, the \(p\)-value of 0.226 significantly exceeds the significance level \(\alpha\) = 0.05. These findings indicate that the empirical model employed in this study is appropriate and suitable for the analysis. Continuing from the previous discussion, the optimal values for the main design parameters corresponding to the remaining \(\mathrm{u_{t}}\) values of 10, 20, 25, 30, and 35 are presented in Table 10.  
Based on the data in Table 10, the following conclusions can be drawn:  
The optimal values for \(\mathrm{X_{bal}}\) are mostly their minimum values (\(\mathrm{X_{ba1}}\) = 0.25) and for \(\mathrm{X_{ba2}}\) are the maximum values (\(\mathrm{X_{ba2}}\) = 0.4). This is due to the desire for a low gearbox mass, and because the second gear stage is more loaded than the first, a larger \(\mathrm{Xba}\) is required to reduce the diameter and thus the total gear mass. This is also consistent with the instructions in [23] for determining the factor \(\mathrm{X_{ba}}\) for a two-stage helical gearbox. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{6}{c}{\(\mathbf{u_{t}}\)} \\ \cline{2-7}  & **10** & **15** & **20** & **25** & **30** & **35** \\ \hline \(\mathrm{u_{1}}\) & 1 & 1.5 & 2.1 & 2.7 & 3.2 & 3.8 \\ \(\mathrm{X_{ba1}}\) & 0.25 & 0.2875 & 0.2875 & 0.25 & 0.25 & 0.25 \\ \(\mathrm{X_{ba2}}\) & 0.4 & 0.3625 & 0.4 & 0.4 & 0.4 & 0.4 \\ \(\mathrm{AS_{1}}\) & 350 & 367.5 & 367.5 & 367.5 & 367.5 & 420 \\ \(\mathrm{AS_{2}}\) & 420 & 420 & 420 & 420 & 420 & 420 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Optimum main design factors finding by traditional method. Figure 5: Influence of main design factors on \(\mathrm{S/N}\) ratios of \(\overline{\mathrm{y}}\). Similarly, the optimal values for AS\({}_{1}\) and AS\({}_{2}\) are also their maximum values. This is because minimizing the gearbox mass requires maximizing the values of AS\({}_{1}\) and AS\({}_{2}\). By increasing these values, the center distance of gear stage i (as represented by Equation (19)) can be minimized.