Multi-Objective Optimization of Planetary Gearbox with Adaptive Hybrid Particle Swarm Differential Evolution Algorithm  
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the volume of gears, shaft and bearings. More recently, authors in [4] proposed a method to minimize the volume of the planetary gear trains with the aim to reduce space and mass. The proposed method successfully obtained the optimal value in benchmark test functions and the best parameters regarding the design of planetary gear trains compared to the well-known designs in the literature. Majority of the above-mentioned papers deal with the problem of single-objective optimization (SOO). However, many practical engineering optimization problems have one or more conflicting objectives, which need to be simultaneously optimized. Regarding the gear train optimization, it is important to analyze the influence of the different contradictory objectives, such as weight or volume, power loss, center distance etc. Therefore, the multi-objective optimization (MOO) problems have received significant attention especially presently in the field of mechanical engineering applications, rotor-dynamic, electrical machine design and wireless communications, etc., [7; 8; 9; 10; 11; 12]. It is a challenging task, which motivated researchers to have a growing interest in developing optimization methods for solving these problems. In this regard, researchers considered the optimization problem of the multi-speed gearbox using a multi-objective evolutionary algorithm involving four conflicting objectives [13]. The author applied non-dominated sorting genetic algorithm (NSGA-II) and showed the efficiency of proposed method in solving mixed integer optimization problem under several non-linear constraints in obtaining the optimal Pareto curve. Recently, authors considered the multi-objective optimization of two-stage helical gearbox, considering two conflicting objectives the volume of the gearbox and the total power loss [14]. Furthermore, besides the constraints such as bending stress and pitting stress, in this study authors employed tribological constraints such as scuffing and wear. Similarly, authors consider the two-stage spur gearbox which is formulated as the multi-objective optimization problem with three objectives, e.g., volume, power-output and center distance under several geometric and design constraints [9]. The obtained results concluded that the proposed method obtains better design solutions compared to those reported in the literature. Recently, authors considered the multi-objective optimization of the planetary gearbox with the same objectives [15]. In addition, several different constraints have been considered, such as regular strength requirement as well as the bearing selection and scuffing. Generally, the MOO problems have been solved by converting them into the equivalent single-objective optimization problem using the conventional techniques, known as scalarization procedures, such as: weighted-sum and epsilon-constraint methods [16; 17]. The widely applied weighted-sum method has good convergence properties; however, it can only be applied to obtain convex parts of the Pareto optimal front [18]. On the other hand, epsilon-constraint method can be applied not only to obtain convex parts of the Pareto set, but also for the non-convex parts [17]. However, there is no unified strategy in the literature to deal with epsilon-constraints values that always lead to feasible problem instances. To overcome these drawbacks, recently several parameter-free methods have been proposed in the literature that work directly with the multi-objective problem, without scalarization [19; 20]. To successfully solve MOO problems, various optimization methods have been applied from conventional gradient-based methods, surrogate-based algorithms to nature-inspired evolutionary algorithms [20; 21; 22].