## 1 Introduction  
Optimizing gearboxes is a critical aspect of mechanical engineering as it has a direct impact on their efficiency, durability, and reliability, as well as the performance of other machinery and equipment. Multi-objective optimization of gearboxes involves simultaneously optimizing various performance parameters, such as load-carrying capacity, noise, mass, size, and efficiency, making it a complex and challenging task. To address these challenges, various optimization techniques have been developed in recent years. Helical gearboxes, one of many gearbox types, are widely used in industrial applications due to their superior load-carrying capacity, smooth operation [1], simple structure, and low cost. However, designing a helical gearbox involves numerous design parameters, making it difficult to optimize for multiple objectives. Numerous studies on the optimal design of helical gearboxes have been conducted thus far. This research looked into both single-objective and multi-objective optimization problems. Many authors are also interested in the single-objective optimization problem. [PERSON] and [PERSON] [2] presented formulas for calculating gear ratios for two-, three-, and four-stage helical gearboxes in order to obtain the smallest gear mass. [PERSON] et al. [3] presented a practical approach for reducing the mass of a two-stage helical gearbox in their paper. The method entailed analyzing data from gearbox manufacturers. Their findings suggested a center distance ratio (\(\text{aw}_{2}/\text{aw}_{1}\)) between the second and first stages in the range of 1.4 to 1.6 to achieve the minimum mass. Once the optimal center distance ratio has been determined, the corresponding partial gear ratios are obtained using a lookup table. Various objective functions are utilized to solve the single-objective optimization problem of determining the optimal gear ratio of helical gearboxes. These objective functions include minimizing gearbox length [4; 5; 6; 7], minimizing gearbox cross section area [8; 6; 9; 10],minimizing gearbox mass [6; 11], minimizing gearbox volume [12], and minimizing gearbox cost [13; 14; 15; 16]. All of these studies share a common feature in that only one design parameter, the partial gear ratio, is defined in the form of an explicit model. The multi-objective optimization problem for designing helical gearboxes has also been of interest to researchers. [PERSON] et al. [1] conducted a study in which a two-stage helical gearbox was subjected to multi-objective optimization with a broad range of constraints using a specially formulated discrete version of non-dominated sorting genetical algorithm II (NSGA-II). The study formulated two objective functions, namely the minimum gearbox volume and minimum total gearbox power loss. Moreover, the study considered constraints, such as bending stress, pitting stress, and tribological factors. [PERSON], [PERSON] and [PERSON] [17] introduced a new microgeometry modification for helical gear pairs, leading to substantial enhancements in performance with regards to noise and vibration. In their research, [PERSON] and [PERSON] [18] utilized a genetic algorithm (GA) to optimize the volume of a two-stage helical gearbox. To handle the design constraints, such as bending stress, contact stress, number of teeth on pinion and gear, module, and face width of gear, the objective function was subject to static and dynamic penalty functions. The results from the GA were compared to those of a deterministic design procedure, with the GA being found to be the superior method. [PERSON] and colleagues [19] presented a generalized optimization technique for reducing the volume of two-stage and three-stage helical gearboxes, while considering tradeoffs with surface fatigue life. In their study, [PERSON] and [PERSON] [20] explored multi-objective optimization by simultaneously considering three objectives: volume, power output, and center-distance. They employed the NSGA-II evolutionary algorithm to generate Pareto frontiers in their research.