## 1 Introduction  
A gearbox is a crucial part of a mechanical drive system. It is employed to improve the torque and decrease the speed from the input to the output of the gearbox. In reality, there are a wide variety of forms of gearboxes: worm, planetary, bevel, and cylindrical gearboxes (with spur or helical gears). Because they operate more smoothly and silently than spur-gear gearboxes, helical gearboxes are highly common and exceed other types of gearboxes in terms of efficiency [1]. This is the rationale behind the efforts of numerous academics to optimize the helical gearbox. Up to now, there has been quite a lot of research on optimizing a helical gearbox. Research on helical gearbox optimization has been fairly extensive up to this point on both single-objective and multi-objective optimization problems. The single-objective optimization problems have been conducted with different objectives, including increasing gearbox efficiency [2, 3], reducing radiated noise [4, 5, 6], gear box length [7, 8, 9, 10], gearbox height [11], gearbox volume [8, 9, 10, 11, 12], gear mass [13, 14, 15], gearbox cross-section area [16, 17, 18, 15],and gearbox cost [19; 20; 21]. The research was conducted for two-stage [2; 3; 7; 8; 9; 10], three-stage [14; 18; 22], and four-stage gearboxes [9; 21]. Solving the MOOP for the helical gearbox, in particular, is highly studied as, in practice, a gearbox can only be satisfied by a large number of objective functions. Various techniques have been employed to address the multi-objective optimization issue with a gearbox. [PERSON], [PERSON], et al. [23] provided an optimal multi-objective analysis of a cycloid pin gear planetary reducer. The study focused on analyzing the reducer volume, turning arm bearing force, and pin maximum bending stress using Pareto optimal solutions in order to minimize all three of these objectives. The study's findings demonstrate that the modified algorithm can generate Pareto optimal solutions that surpass those generated by the regular design. [PERSON], [PERSON], et al. [24] conducted a study on multi-objective optimization problem (MOOP) to reduce the size of the gears and minimize the vibration caused by their meshing in cylindrical gear pairs. The design method minimized the size and vibration of cylindrical gear pairs by taking into account the strength and geometric limitations. The method's validity will be verified by the design findings presently employed in an elevator reduction drive. [PERSON]. [25] reported the outcomes of their study on addressing the MOOP for reducing helical gear noise. The study employed the design of an experiment and the response surface method as the approach to address the MOOP. The MCDM method was utilized by [PERSON], [PERSON], et al. [26] to determine the optimal gear material for a gearbox with the goal of improving the wear resistance and surface fatigue. The objective of this study is to improve the efficiency of the surface fatigue resistance when it is applied to gearboxes. [PERSON] and [PERSON] [27] developed an approach for spur gear set design optimization that aims to reduce the gear weight, contact stress throughout the contact path, and suitable film thickness at the contact point. This work combined particle swarm optimization, particle swarm optimization-based teaching learning optimization, and Jaya method to ensure a significant decrease in the weight and contact stress of a profile-modified spur gear set with sufficient film thickness at the site of contact. The results of the study show that the gear design is significantly better with optimum addendum coefficient values inside the design space than with traditional designs. [PERSON] et al. presented a novel multi-objective optimization of a two-stage spur gearbox using the NSGA-II approach [28]. This attempt involved two aim functions: minimum gearbox power losses and minimum gearbox volume. The results of the study indicate that there is a high probability of wear failure in solutions that are obtained using single-objective minimization.