## 5 Multi-Objective Optimization
The MOO problem is handled using two additional MCDM techniques, Evaluation by an Area-based Method of Ranking (EAMR) and Measurement of Alternatives and Ranking According to COmpromise Solution (MARCOS), in order to evaluate the reliability of the outcomes that are found (option 55 is the best). To determine the weights, the entropy approach is also used. When both of these methods are used to address the MOO problem, option 55 finally performs the best (see Table 5). Using all three MCDM methods, the best (see Table 5) is the best (see Table 5). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Trial.**} & \multicolumn{2}{c}{\(\mathbf{tr_{ij}}\)} & \multicolumn{2}{c}{\(\mathbf{g_{ij}}\)} & \multirow{2}{*}{**Qi**} & \multirow{2}{*}{**Rank**} \\ \cline{2-2} \cline{5-6}  & \(\mathbf{V_{gb}}\) & \(\mathbf{n_{gb}}\) & & \(\mathbf{V_{gb}}\) & \(\mathbf{n_{gb}}\) & \\ \hline  
& 0.0006 & 0.0036 & 0.00422 & 83 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculated results and ranking of the options via the MAIRCA method for \(\mathbf{u_{t}}=35\). The background color describes the best option. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Trial. **} & \(\mathbf{u_{1}}\) & \(\mathbf{\Psi_{ba1}}\) & \(\mathbf{\Psi_{ba2}}\) & \(\mathbf{V_{gb}}\) (\(\mathbf{dm^{3}}\)) & \(\mathbf{\eta_{gb}}\) (\%) \\ \hline
27 & 7.01 & 0.25 & 0.29 & 20.81 & 93.36 \\
28 & 7.01 & 0.25 & 0.33 & 20.51 & 93.30 \\
54 & 7.02 & 0.25 & 0.36 & 20.27 & 93.23 \\
55 & 7.02 & 0.25 & 0.40 & 20.10 & 93.18 \\
56 & 7.02 & 0.29 & 0.25 & 21.35 & 92.22 \\
71 & 7.02 &best option, 55, is produced, demonstrating that the best option is independent of the decision-making process selected. Continuing with the preceding discussion, Table 6 displays the ideal values for the primary design parameters that match the remaining \(\mathbf{u_{t}}\) values of 10, 20, 25, 30, 35, and 40. With the data in this table, the following conclusions can be drawn. Whereas \(\psi_{ba2}\) chooses the greatest value (\(\psi_{ba2}=0.4\)), \(\psi_{ba1}\) chooses the lowest value (\(\psi_{ba1}=0.25\)). This is because the smallest gearbox volume can only be produced by a small box with a cross-sectional area (LxH). This requires \(\mathrm{d}_{w21}\) and \(\mathrm{d}_{w22}\) to be roughly equal in order to be accomplished [55]. In addition, a larger \(\psi_{ba2}\) is needed to reduce the diameter of \(\mathrm{d}_{w22}\) due to the higher torque of the second stage. On the other hand, because the first stage has a lower torque, a smaller \(\psi_{ba1}\) needs to be selected in order to raise \(\mathrm{d}_{w21}\). In order to evaluate the effectiveness of applying the formula for the power loss in the idle motion when calculating power loss in gears, the calculated values of the gearbox efficiency in this work are compared with the results in [33]. From Table 4 [33], when the \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** & **40** \\ \hline \(\mathrm{u_{1}}\) & 2.4 & 4.12 & 5.02 & 5.79 & 6.63 & 7.02 & 7.66 \\ \(\psi_{ba1}\) & 0.25 & 0.25 & 0.25 & 0.25 & 0.25 & 0.25 & 0.25 \\ \(\psi_{ba2}\) & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimum values of the main design factors.