## 5 Multi-Objective Optimization
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Ranking**} \\ \hline
**Trial.** & **MAIRCA** & **EAMR** & **MARCOS** \\ \hline
1 & 54 & 55 & 65 \\
2 & 30 & 35 & 38 \\
3 & 20 & 20 & 20 \\
4 & 10 & 10 & 10 \\
5 & 3 & 3 & 3 \\
6 & 75 & 77 & 85 \\
7 & 55 & 52 & 55 \\... & 26 & 51 & 54 & 64 \\
27 & 29 & 33 & 36 \\
28 & 19 & 19 & 19 \\... & & & \\
54 & 7 & 7 & 7 \\
**55** & 1 & 1 & 1 \\
56 & 73 & 74 & 83 \\... & & & \\
71 & 123 & 123 & 123 \\
72 & 118 & 118 & 115 \\
73 & 108 & 108 & 108 \\... & & & \\
101 & 46 & 48 & 59 \\
102 & 26 & 27 & 31 \\
103 & 16 & 16 & 16 \\... & & & \\
123 & 106 & 106 & 106 \\
124 & 91 & 91 & 91 \\
125 & 83 & 81 & 71 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Rankings of the options when using the MAIRCA, EAMR, and MARCOS methods. The background color describes the best option. total gearbox ratio is 7.5, the gearbox efficiency will be 99.33-99.71% (determined from input power of 10 kW, maximum output power of 9.971 kW and minimum output power of 9.933 kW). In addition, in this work, when \(\text{u}_{\text{t}}\) changes from 10 to 40, the gearbox efficiency will have the values of 87.42-93.44% (Table 3). In fact, the efficiency of a helical gear train is 0.93-0.98 (93-98%) and the efficiency of a pair of bearings is 0.99-0.995 (99-99.5%) [51]. From these data, the efficiency of a TSHG can be determined as follows:  
\[\text{n}_{\text{gb}}=\text{n}_{\text{g}}^{2}\cdot\text{n}_{\text{b}}^{3}=0.93 ^{2}\cdot 0.99^{3}\div 0.98^{2}\cdot 0.995^{3}=0.83\div 0.95\text{ or }\text{n}_{\text{gb}}=83\div 95 \tag{46}\]  
By comparing the above computed values with (1), it is evident that the gearbox efficiency in this work (87.42-93.44%), is fairly close to reality (83-95%). Moreover, the gearbox efficiency in [33] (99.33-99.71%) is extremely high and not consistent with reality. This proves that using the formula to calculate the power loss in the idle motion in this work is a valuable new point and should be applied. To evaluate the accuracy of finding the optimal values of the proposed method with other methods, like the Taguchi and Gray Relational Analysis (GRA) methods, the result of finding the optimal gear ratio of the first stage \(\text{u}_{1}\) in this work is compared with that in [41]. In this work, the solutions to the optimization problem, for example, \(\text{u}1\), are found by directly comparing (over multiple steps) 22,500 values (with \(\text{u}_{1}\) = 1-9) to choose from (as mentioned in Section 3.1). In this case, the step between the variables of \(\text{u}_{1}\) is only 0.02. Meanwhile, when using the Taguchi and GRA methods, the total number of options to choose the optimal value of \(\text{u}_{1}\) is only 25 (Table 3 [41]). In this case, the step between the variables is 0.775 (much larger than when using the proposed method). As a result, the value \(\text{u}_{1}\) found using the proposed method will be more accurate than using the Taguchi and GRA methods. In other words, the proposed method is better than the Taguchi and GRA methods. The association between the ideal values of \(\text{u}_{1}\) and \(\text{u}_{\text{t}}\) is depicted in Figure 9. Additionally, it is discovered that the ideal values of \(\text{u}_{1}\) may be determined using the regression equation that follows (with R\({}^{2}\) = 0.9974):  
\[u_{1}=3.7098\cdot ln(u_{t})-6.0709 \tag{47}\]  
After finding \(\text{u}_{1}\), the optimal value of \(\text{u}_{2}\) is calculated by the following formula:  
\[u_{2}=u_{t}/u_{1} \tag{48}\]