## 5 Multi-Objective Optimization  
To conduct simulation experiments, a computer program has been developed. For the purpose of the investigation, the gearbox ratios of 10, 15, 20, 25, 30, 35, and 40 are all considered. The solutions to this u\({}_{\text{t}}\) = 35 problem are shown below. For the first 125 testing cycles, this total gearbox ratio is employed (as stated in Section 3.1). The gearbox mass and efficiency, which are the experiment's output values, will be entered into the MAIRCA as input parameters in order to solve the MOOP. With each MAIRCA step, the gap between the two levels of each variable will decrease. For example, when u\({}_{\text{g}}\) = 35 in step 1, u\({}_{\text{1}}\) rises from 4.58 to 8.31 (Table 2). Consequently, the distance between the two levels of u1 is \((8.31-4.58)/4=0.933\). Until there is less than 0.02 between the two levels of the variable, this process will be repeated. Table 3 displays the main design parameters and output responses for u\({}_{\text{t}}\) = 35 in the MAIRCA experiment's fourth and final iteration. The weights of the criteria have been determined through the application of the entropy approach (see Section 3.3). To begin with, obtain the normalized values of p\({}_{\text{ij}}\) using Equation (43). The entropy value of each indicator (me) is found using Equation (44). Lastly, determine the weight of the criteria w\({}_{\text{j}}\) using Equation (45). For the latest MAIRCA experiment, the weights of V\({}_{\text{gb}}\) and n\({}_{\text{gb}}\) are found to be 0.4508 and 0.5492, respectively. Section 3.2 provides guidelines for using the MAIRCA approach for multi-objective decision-making. Adopting the measures recommended in Section 2.1, in particular, after the starting matrix is put up, the priority, or criterion PAj, is calculated by applying Equation (2). As a result, V\({}_{\text{gb}}\) and n\({}_{\text{gb}}\) have the same priority, which is \(1/125=0.008\). In addition, the value of parameter t\({}_{\text{pji}}\) is determined using Equation (38) and the weight of the criterion is determined in Section 2.2. The tp\({}_{\text{ij}}\) values for n\({}_{\text{gb}}\) and V\({}_{\text{gb}}\) are 0.0036 and 0.0044, respectively. The values of tr\({}_{\text{ij}}\) are then obtained using Equations (39) and (40), and the values of g\({}_{\text{ij}}\) are then determined using Equation (41). In the end, the values of the criteria functions Qi are determined using Equation (42). The results of the option ranking and the different parameters that the MAIRCA technique computed (for the last run of MAIRCA work) are displayed in Table 4. The table shows that option 55 is the best option out of all the options provided. Accordingly, u\({}_{\text{1}}\) = 7.02, \(\psi_{ba1}\) = 0.25, and \(\psi_{ba2}\) = 0.4 are the ideal values for the primary design features (see Table 3). \begin{table}
\begin{tabular}{c c c} \hline \hline
**u\({}_{\text{t}}\)** & \multicolumn{2}{c}{**u\({}_{\text{1}}\)**} \\ \cline{2-3}  & **Lower Limit** & **Upper Limit** \\ \hline
10 & 2.2 & 2.36 \\
15 & 2.79 & 4.76 \\
20 & 3.32 & 5.75 \\
25 & 3.76 & 6.66 \\
30 & 4.19 & 7.51 \\
35 & 4.58 & 8.31 \\
40 & 4.93 & 8.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: New constraints of u\({}_{\text{1}}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Trial.** & **u\({}_{\text{1}}\)** & \(\psi_{\text{ba1}}\) & \(\psi_{\text{ba2}}\) & **V\({}_{\text{gb}}\) (dm\({}^{\text{3}}\))** & \(\eta_{\text{gb}}\) **(\%)** \\ \hline
1 & 6.99 & 0.25 & 0.25 & 21.23 & 93.44 \\
2 & 6.99 & 0.25 & 0.29 & 20.82 & 93.38 \\
3 & 6.99 & 0.25 & 0.33 & 20.52 & 93.32 \\
4 & 6.99 & 0.25 & 0.36 & 20.30 & 93.26 \\
5 & 6.99 & 0.25 & 0.40 & 20.12 & 93.21 \\
6 & 6.99 & 0.29 & 0.25 & 21.37 & 92.26 \\
7 & 6.99 & 0.29 & 0.29 & 20.93 & 92.21 \\
26 & 7.01 & 0.25 & 0.25 & 21.21 & 93.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main design factors and output results for u\({}_{\text{t}}\) = 35 in the 4 th run of the MAIRCA. The background color describes the best option.