## 2 Problem Formulation  
The development of planetary gearboxes that are both reliable and lightweight while maintaining high efficiency is a significant challenge in the present field of mechanical engineering. The creation of a multi-objective optimization model that integrates design variables, objectives, and constraints is crucial for attaining the intended performance of planetary gearboxes. The objective of this paper is to develop the appropriate multi-objective optimization model of a planetary gearbox and investigate the possibilities for solving this complex optimization problem, with the aim of identifying gear design parameters that can lead to weight reduction, decreased power loss, and improved reliability of components. In this context, we examine a single-stage planetary gearbox, schematically illustrated in Figure 1. The planetary gearbox under consideration comprises three planet gears and two central gears. One of the central gears features external gearing and is referred to as the sun gear, which is directly linked to the input shaft. The other central gear has internal gearing and is stationary, serving to secure the entire assembly in position. The system under consideration receives power through the input shaft and sun gear, which is subsequently distributed among the planet gears and transmitted to the carrier. The output shaft is connected to the carrier to complete the transfer of power. The analysis maintains a constant input speed of \(n_{a}=2750\) min\({}^{-1}\) and a power of \(P_{a}=175\) kW throughout. The remainder of the parameters used in the design of the planetary gearbox are given in Table 1.  
The multi-objective optimization implies the simultaneous optimization of multiple objectives \(f_{1},f_{2},\ldots,f_{M}\), which are often conflicting in nature. This type of optimization problem often arises in different fields of engineering, since a number of these problems involve multiple objectives and a number of constraints that need to be satisfied to ensure the physical feasibility of the solution. In general, the multi-objective optimization problem can be stated as  
\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 & \(\upmu\)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: Schematic illustration of the planetary gearbox.  
\[\begin{split}\min f_{m}(\mathbf{x}_{i})& m=1,\ldots,M \wedge i=1,\ldots N_{P}\\ s.t.&\\ & g_{k}(\mathbf{x}_{i})\geq 0,\quad k=1,\ldots,K\\ & h_{l}(\mathbf{x}_{i})=0,\quad l=1,\ldots,L\\ & x_{i,j}\in\left[x_{j}^{Lower},x_{j}^{Upper}\right],\quad j=1, \ldots,n\end{split} \tag{1}\]  
where \(M\) is the number of objectives taken for optimization; \(M\geq 2\), \(g_{k}(\mathbf{x}_{i})\) denotes the inequality constraint, with the \(K\) being the total number of inequality constraints in the considered problem; \(h_{l}(\mathbf{x}_{i})\) denotes the equality constraint, where \(L\) is the total number of equality constraints; and \(N_{P}\) is the number of solutions in the population. Here, the boundaries of decision variables are denoted as \(x_{j}^{Lower}\)and \(x_{j}^{Upper}\).  
Therefore, in formulating the MOO problem of planetary gear optimization, we start firstly with objective functions that will be taken into account, and then we present a number of constraints that are developed in order to provide a planetary gearbox with long service life.