Minimizing the Cross-Sectional Area and Maximizing Efficiency in Split-Input Two-Stage Gearboxes via NSGA-II with SAW-Guided Choice

Split-input two-stage helical gearboxes must balance a compact footprint with high transmission efficiency. This study proposes a preference-aware multi-objective framework that couples NSGA-II with Simple Additive Weighting (SAW) to design split-input two-stage gearboxes under constraints. A parameterized gearbox model is formulated using the key design variables, including the first-stage gear ratio u1 and the face-width coefficients of the first and second stages, Xba1 and Xba2. The model is optimized with two objectives: minimizing the gearbox cross-sectional area and maximizing the overall efficiency ηgb. For each target overall transmission ratio, uh, NSGA-II is employed to generate the corresponding Pareto front, after which the SAW method ranks the non-dominated solutions based on user-specified preferences to identify a best-compromise design. The post-processing identified simple design rules: u1 scales approximately linearly with uh (fit R2 > 0.95); Xba1 decreases with uh and plateaus at its lower bound for higher uh, while Xba2 remains nearly constant. The resulting fronts show the footprint–efficiency trade-off, and trend analysis demonstrates increasing area and slightly declining efficiency as uh grows—guiding early sizing. The proposed NSGA-II + SAW workflow delivers reproducible, preference-aware selections and interpretable parametric relations, providing practical decision support for compact, high-efficiency split-input two-stage gearbox design.