## 1 Introduction
Furthermore, the total power loss is reduced by half when multi-goal optimization is used instead of single-objective optimization. Using the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) approach, [PERSON] et al. [29] carried out an optimization analysis of a planetary gearbox while taking into account considerable regular mechanical and tribological constraints. When comparing the results of single-objective optimization with and without tribological limits, using the study's recommendations led to a significant reduction in the weight and power loss. With the aid of the NSGA-II method, [PERSON] et al. [30] performed a multi-objective optimization of the gear pair characteristics with the aim of lowering the transmission volume and power losses. This method was also used in [31] to solve a MOOP, including the reduction of both the gearing mass and the flank adhesive wear speed, and in [32] to solve an optimization problem involving the design of a multi-speed gearbox that has four competing objectives. The NSGA-II was combined with the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) approach in [33] to solve a multi-objective optimization problem of a small gearbox with three objectives: the gearbox volume, the power output, and the center distance. Three distinct techniques were used in the multi-objective optimization of a two-stage helical gear train by [PERSON] et al. [34]: genetic algorithms (GAs), NSGA-II, and the MATLAB optimization toolbox. The study had two aims, which were to determine the minimal gearbox volume and the load-carrying capacity. The comparison of the findings shows that the NSGA-II approach produced superior outcomes than the other techniques for both objectives. [PERSON], [PERSON], et al. [35] lookedinto the multi-objective design of the transmission using helical gear pairs. The opposing number of overlap ratios and gear volumes served as indicators of the goal functions. The study employed the Parameter Adaptive Harmony Search Algorithm (PAHS) to resolve the optimization problem. Using a mono-objective self-adaptive algorithm, the optimization of tooth modifications for spur and helical gears was resolved in [36]. The particle swarm optimization (PSO) technology served as the foundation for this tactic. The multi-objective optimization enhanced the maximal contact pressures and the transmission error signal's root mean square values. A multi-objective micro-geometry optimization of the gear tooth with response surface methods was conducted by [PERSON] et al. [37]. [PERSON] et al. [38] also employed the response surface methodology approach to multi-objectively optimize the helical gear in a centrifugal compressor. [PERSON], et al. [39] presented a MOOP work using two-stage helical gear sets in a helical hydraulic rotary actuator. The optimization was performed via the NSGA-II method, incorporating four permutations of the three objectives: volume, transmission efficiency, and maximum contact stress. The findings suggest that there is a trade-off between the maximal contact stress and the volume in the ideal region. Also, taking into account various combinations enhances the durability of the optimization outcomes. [PERSON], and [PERSON], [40] utilized a Discrete Non-Dominated Sorting Genetic Algorithm III (NSGA-III) to optimize a two-stage gearbox. The study investigated three objectives: volume, center distance, and power loss. It also analyzed several design characteristics, such as the number of gear teeth, module, and helix angle. Gray relation analysis (GRA) and Taguchi techniques were recently applied by [PERSON] and [PERSON] [41] to the MOOP of designing a TSHG. The lowest gearbox mass and the highest gearbox efficiency were the two objectives selected for this investigation. The optimal values for the five crucial design components that contribute to making a TSHG were established using the study's findings.