The multi-criteria decision-making (MCDM) method was applied in a novel way in this study to the multi-objective optimization problem (MOOP) of designing a two-stage helical gearbox with double gear-sets in the second stage. Finding the best fundamental components to increase gearbox efficiency and decrease gearbox cross-section area was the aim of this study. Three main design factors were chosen for investigation in this work: the first stage gear ratio and the first and second stage coefficients of wheel face width (CWFW). Phase 1 solves the single-objective optimization problem to reduce the gap between variable levels, and phase 2 solves the MOOP to determine the optimal critical design factors. This additionally splits the MOOP into two phases. Additionally, the TOPSIS method was used as an MCDM approach to address the multi-objective optimization issue, and the entropy approach was used to compute the weight criteria. In this study, gearbox efficiency is calculated by considering power losses during idle motion. The multi-objective optimization of a helical gearbox with second stage double gear-sets is addressed using the TOPSIS technique for the first time.

This study describes the outcomes of employing the Simple Additive Weighting (SAW) approach to address the Multi-Objective Optimization Problem (MOOP) of a two-stage helical gearbox with two gear sets at the first stage. Its objective is to determine the key design variables that can reduce the volume of the gearbox while simultaneously maximizing its efficiency. For this investigation, three key design parameters were selected, namely the coefficients of the wheel face width of the first and second stages (Xba1 and Xba2), and the gear ratio of the first stage u1. In addition, the SAW technique was deployed to deal with the problem of Multi-Criteria Decision Making (MCDM), while the Method based on the Removal Effects of Criteria (MEREC) was employed to determine the weight criterion for addressing the MOOP. The obtained results are valuable for defining the optimal values for three primary design factors, which are essential for the development of a two-stage helical gearbox with two gear sets at the first stage.

This study presents a novel approach for the optimal design of a two-stage helical gear reducer with a split output stage, to achieve a balance between the mechanical compactness and transmission performance. The optimization problem considers two conflicting objectives: minimizing the cross-sectional area of the gearbox structure and maximizing its mechanical efficiency. To solve this Multi-Objective Optimization Problem (MOOP), the Non-dominated Sorting Genetic Algorithm II (NSGA-II) is employed, aiming to generate a set of Pareto-optimal solutions, capturing the trade-offs between the two design goals. Subsequently, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is applied to rank and identify the most preferable solution among the Pareto front based on decision-makers' preferences. The proposed hybrid framework enables a systematic exploration of the design space, providing engineering insights into how the gear ratio distribution and geometric parameters influence the gearbox’s performance. The results demonstrate that the integration of NSGA-II and TOPSIS effectively supports the optimal design of compact and high-efficiency gear reducers with split-stage configurations.

This study proposes a hybrid Multi-Objective Optimization (MOO) approach that integrates the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and the Evaluation by an Area-based Method of Ranking (EAMR) to optimize the design of a two-stage helical gearbox featuring a split input stage. The optimization simultaneously targets two conflicting objectives: minimizing the base area of the gearbox housing and maximizing the transmission efficiency. A parametric design model was developed to capture the key geometrical and performance parameters, while ensuring compliance with the practical constraints related to gear strength and manufacturing limitations. The NSGA-II algorithm was employed to generate a set of Pareto-optimal solutions, which were subsequently ranked using the EAMR method to support decision-making. The results demonstrate the effectiveness of the hybrid approach in identifying high-performance design alternatives that offer an optimal balance between compactness and efficiency. This is the first study to apply a combination of NSGA-II and a Multi-Criteria Decision-Making (MCDM) method to the dual-objective optimization of a two-stage gearbox with a split input stage. In particular, the proposed approach successfully identifies the optimal design parameters across a wide range of transmission ratios, thereby facilitating the initial selection of appropriate gearbox transmission ratios in the early-stage design. This study also provides valuable design guidelines for engineers seeking to improve space utilization and performance in gear transmission systems.