operating, surface, and lubrication conditions. The importance of lubrication technologies in gear transmission systems for meeting the industrial challenges of increased load, speed, and temperature is discussed in [7]. To provide more comprehensive references for gear research and engineering, the authors discuss the effects of lubrication on efficiency, fatigue performance, and dynamics. In [4], a model for calculating gear efficiency that accounts for sliding and rolling friction forces is developed, and a complete planetary gearbox's appropriate efficiency is also presented. Moreover, in [6], researchers evaluated the efficiency of a planetary multiplier gearbox with helical gears using a back-to-back gearbox test rig with recirculating power and various gear oil formulations, which resulted in distinct stabilized operating temperatures and varied power loss results based on the employed oil type. For the purpose of determining how design parameters and lubricant qualities affect the performance of planetary gearboxes, a composite power loss model and design-sensitivity study were devised and applied to wind turbines [1]. Due to present-day engineering requirements, which stem from strict regulations and a highly competitive environment, which requires the successful fulfillment of a number of criteria, including power efficiency, reliability, and noise reduction, the optimization of gear parameters has become an important area of study. The primary objective is to reduce the volume (mass) of the construction and power loss while producing improved components with a longer service life. In this regard, a great deal of research has been devoted to the optimal design of gear trains, as discussed in a number of papers [8; 9; 10]. These researchers explored gear optimization using a genetic algorithm, where the module, face width, number of teeth, and profile shift coefficients were utilized as design factors; the gear pair volume served as a fitness function; and the tooth root bending strength and contact pressure calculations were employed as constraints [8]. They concluded that the design process for gearboxes can be enhanced by employing genetic algorithms to determine the optimal parameter values for gear pairs. The problem of the optimal spur gear tooth profile has been solved using a new evolutionary optimization algorithm called the adaptive mixed differential evolution, which is based on a self-adaptive approach [9]. The objectives are to equalize the maximum bending stresses and the specific sliding coefficients at contact path extremes, which were successfully solved using the proposed method. The volume minimization of a single stage planetary gearbox was considered in [11] utilizing a hybrid algorithm combining genetic and artificial immune system algorithms, with the obtained data demonstrating improved results when compared to the existing results. The study employed nine meta-heuristics to explore the optimal design of an automatic planetary gear train [10], considering nine mixed decision parameters. The findings indicate that the differential evolution algorithm exhibited superior statistical performance compared to other algorithms, while the firefly algorithm demonstrated the most favorable convergence behavior. The aforementioned works primarily concentrate on single-objective optimization, which is insufficient for many of the demands of contemporary engineering design problems, which frequently involve multiple conflicting objectives. Therefore, there is growing interest in the development of multi-objective optimization techniques that can handle such problems effectively. In recent times, there has been an expansion of meta-heuristic algorithms that draw inspiration from nature. These algorithms encompass a range of techniques such as the genetic algorithm [12], differential evolution [13], Particle Swarm Optimization (PSO) [14], and butterfly optimization algorithm (BOA) [15], among others.