## 4 Results and Discussions  
### The Influence of Input Parameters and Their Interactions  
The evolution of the optimum gear ratio of the second step (\(u_{2}\)) as functions of each input parameter is presented in Figure 4. It is observed that \(u_{2}\) increases when Total gearbox ratio \((u_{t})\), Allowable contact stress of stage 2 (\(AS_{2}\)), and Cost of shafts (\(C_{s}\)) increase also. Nevertheless, for this tendency it is realized that Total gearbox ratio \((u_{t})\) has greater influence than that of other factors. Conversely, \(u_{2}\) decreases with the growth of Allowable contact stress of stage 1 and 3 (\(AS_{1}\) and \(AS_{3}\)), and Output torque (\(T_{out}\)). Moreover, it is shown that Coefficients of wheel face width of stage 1, 2, and 3 (\(X_{bn1}\), \(X_{bn2}\) and \(X_{ba3}\)) do not have influence on \(u_{2}\). Regarding to the case of \(u_{3}\) (c.f. Figure 4b), the experimental results reveal that Cost of gears \(\left(C_{g}\right)\) and Cost of shafts (\(C_{s}\)) have a significant effect on the value of \(u_{3}\). It means that \(u_{3}\) develops when Cost of shafts rises or Cost of gears declines. Furthermore, the first five input parameters mentioned in Table 2 do not have impact on the evolution of \(u_{3}\). It is noticed that the influence investigation of the input factors on response as previously mentioned do not take their interactions into account. This will be considered in the next part of the current study. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Real Factor** & **Minitab®19** & **Name** & **Unit** & **Low** & **High** \\ \hline Total gearbox ratio & A & \(u_{t}\) & - & 10 & 100 \\ Coefficient of wheel face width of stage 1 & B & \(X_{bn1}\) & - & 0.3 & 0.35 \\ Coefficient of wheel face width of stage 2 & C & \(X_{bn2}\) & - & 0.33 & 0.38 \\ Coefficient of wheel face width of stage 3 & D & \(X_{bn3}\) & - & 0.35 & 0.4 \\ Allowable contact stress of stage 1 & E & \(AS_{1}\) & MPa & 350 & 420 \\ Allowable contact stress of stage 2 & F & \(AS_{2}\) & MPa & 350 & 420 \\ Allowable contact stress of stage 3 & G & \(AS_{3}\) & MPa & 350 & 420 \\ Output torque & H & \(T_{out}\) & Nm & 1000 & 10000 \\ Cost of gearbox housing & I & \(C_{gh}\) & USD/kg & 1 & 5 \\ Cost of gears & J & \(C_{g}\) & USD/kg & 2 & 9 \\ Cost of shafts & K & \(C_{s}\) & USD/kg & 1.5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Run Order** & **Center Pt** & **Blocks** & \(\mathbf{u_{t}}\) & \(\mathbf{X_{bn1}}\) & \(\mathbf{X_{bn2}}\) & \(\mathbf{X_{bn3}}\) & \(AS_{1}\) & \(AS_{2}\) & \(AS_{3}\) & \(T_{out}\) & \(C_{gh}\) & \(C_{g}\) & \(\mathbf{C_{s}}\) & \(\mathbf{u_{2}}\) & \(\mathbf{u_{3}}\) \\ \hline
1 & 1 & 1 & 30 & 0.35 & 0.38 & 0.4 & 350 & 420 & 350 & 1000 & 1 & 2 & 1.5 & 4.02 & 3.64 \\
2 & 1 & 1 & 100 & 0.35 & 0.33 & 0.36 & 420 & 350 & 420 & 1000 & 5 & 2 & 1.5 & 4.08 & 4.24 \\
3 & 1 & 1 & 100 & 0.3 & 0.33 & 0.4 & 420 & 350 & 350 & 1000 & 5 & 9 & 5 & 4.11 & 3.73 \\
4 & 1 & 1 & 30 & 0.35 & 0.33 & 0.4 & 350 & 350 & 350 & 1000 & 5 & 2 & 1.5 & 3.63 & 3.58 \\
5 & 1 & 1 & 100 & 0.3 & 0.33 & 0.36 & 350 & 420 & 350 & 1000 & 5 & 9 & 1.5 & 5.22 & 2.59 \\
6 & 1 & 1 & 30 & 0.35 & 0.38 & 0.4 & 420 & 350 & 350 & 10000 & 5 & 9 & 1.5 & 3.03 & 3.28 \\ \multicolumn{1}{c}{...} \\ \multicolumn{1}{c}{(Appendix A)} & & & & & & & & & & & & & & & & \\
127 & 1 & 1 & 30 & 0.3 & 0.38 & 0.4 & 420 & 420 & 10000 & 1 & 2 & 5 & 4.11 & 5.83 \\
128 & 1 & 1 & 100 & 0.3 & 0.38 & 0.36 & 420 & 350 & 350 & 10000 & 5 & 9 & 5 & 4.02 & 3.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental plans and output responses. Figure 5a displays the interactions between the input parameters on the response of \(u_{2}\).