## 2 Problem Formulation  
This paper considers the problem of MOO of the planetary gearbox with the aim to obtain the gear design parameters, which lead to construction with lower weight and volume while simultaneously having increased efficiency and service life. In this regard, Figure 1 illustrates the planetary gearbox considered for this MOO problem, which consists of a carrier, three planet gears (g) and a single sun gear with external gearing (a), input shaft, output shaft and a ring gear with internal gearing (b). The power comes thought the input shaft, on which the sun gear is mounted, then it is transferred by meshing engagement between gears to planet gears, which are mounted on the carrier, while the ring gear is kept fixed. Then the power is outputted thought the output shaft, which is directly connected to the carrier, as illustrated in Figure 1.  
The materials, input power and speed, etc. and other parameters used in the design of the planetary gearbox in this study are outlined in Table 1.  
In this regard, in this paper the following objective functions have been formulated for the considered MOO optimization problem: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Units** & **Symbol** & **Value** \\ \hline Input Power & [kW] & \(P_{a}\) & 175 \\ Input speed & [min\({}^{-1}\)] & \(n_{a}\) & 2750 \\ Pressure angle & [\({}^{\circ}\)] & \(\alpha_{n}\) & 20 \\ Gear material & & _18 CrNi8_ & \\ Gear surface Roughness & [\(\mu\)m] & \(R_{a}\) & 0.8 \\ Factor of safety against bending & [-] & \(S_{F\min}\) & 1.2 \\ Factor of safety against pitting & [-] & \(S_{H\min}\) & 1.25 \\ Number of planet gears & [-] & \(n_{w}\) & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the planetary gearbox considered in the paper. Figure 1: 3D illustration of the planetary gearbox considered in this paper. Center distance \[f_{1}=a=\frac{m_{n}z_{a}}{\cos\beta}\Big{(}1+u_{a-g}^{H}\Big{)}\frac{\cos\alpha_{t }}{\cos\alpha_{wt}},\] (1) where the involute of the pressure angle is defined as \[\text{inv}\alpha_{wt}=2\frac{\chi_{a}+x_{b}}{z_{a}+z_{b}}\tan\alpha_{n}+\text {inv}\alpha_{t}.\] (2)
* Contact ratio \[f_{2}=\varepsilon_{a(a-g)}=\frac{0.5\Big{(}\sqrt{d_{a(a)}^{2}-d_{a(b)}^{2}}+ \sqrt{d_{a(g)}^{2}-d_{a(g)}^{2}}\Big{)}-a\sin\alpha_{wt}}{\pi m_{t}\cos\alpha_{ t}}.\] (3)
* Total volume of gears \[f_{3}=V(x)=\frac{\pi}{4}b\Big{[}d_{(a)}^{2}+n_{w}\Big{(}d_{a(b)}^{2}-D^{2} \Big{)}+\Big{(}d_{(g)}^{2}-d_{s}^{2}\Big{)}\Big{]}.\] (4)
* Safety factor for bending stress \[f_{4}=S_{F(a)}(x)=\frac{\left[\sigma_{F}\right]_{M(a)}}{\sigma_{F(a)}}.\] (5)
* Safety factor for contact stress \[f_{5}=S_{H(a)}(x)=\frac{\left[\sigma_{H}\right]_{M(a)}}{\sigma_{H(a)}}.\] (6)
* Safety factor for bending stress for ring gear \[f_{6}=S_{F(b)}(x)=\frac{\left[\sigma_{F}\right]_{M(b)}}{\sigma_{F(b)}}.\] (7) In the above equations, the tooth root stress for the sun gear is determined according to \[\sigma_{F,a}=\frac{F_{t}}{bm_{m}}Y_{F_{a}}Y_{S_{a}}Y_{E}K_{A}K_{b}K_{F\text{ air}},\] (8) while the critical root stress is obtained as Figure 1:0 (9) In this regard, the safety factor against breakage can be obtained according to the equation Figure 1:1 (10) where the \(S_{F\text{min}}\) is the minimum allowed safety factor value, which is given in Figure 1:2  
Effective contact stress is determined according to the expression  
Figure 1:3  
where the contact stress for sun gear - planet gear pair is obtained as the minimum value, according to  
Figure 1:4The critical contact stress is determined as  
Figure 1:5  
while the safety factor form pitting is calculated as  
Figure 1:6  
where the \(S_{H\mathrm{{min}}}\) is the minimum allowed safety factor value, which is given in Figure 1:7 The different factors used in the above equations are given in the Appendix A.