A Multi-Objective Optimization and Sensitivity-Driven Decision Framework for Split-Output Two-Stage Helical Gearboxes Using NSGA-II and MCDM Methods

Duc Binh Vu; Van Thanh Dinh; Van Tung Nguyen; Thi Thu Huong Truong; Thi Phuong Thao Tran; Thanh Hien Bui

doi:10.48084/etasr.14407

Authors

Duc Binh Vu Viet Tri University of Industry, Tien Son Street, Thanh Mieu Ward, Viet Tri City, Vietnam
Van Thanh Dinh East Asia University of Technology, Trinh Van Bo Street, Hanoi City, Vietnam
Van Tung Nguyen Thai Nguyen University of Technology, Tich Luong Ward, Thai Nguyen City, Vietnam
Thi Thu Huong Truong Thai Nguyen University of Technology, Tich Luong Ward, Thai Nguyen City, Vietnam
Thi Phuong Thao Tran Thai Nguyen University of Technology, Tich Luong Ward, Thai Nguyen City, Vietnam
Thanh Hien Bui Thai Nguyen University of Technology, Tich Luong Ward, Thai Nguyen City, Vietnam

Volume: 16 | Issue: 1 | Pages: 30984-30990 | February 2026 | https://doi.org/10.48084/etasr.14407

Received: 30 August 2025 | Revised: 22 October 2025 | Accepted: 9 November 2025 | Online: 9 February 2026

Corresponding author: Thanh Hien Bui

Abstract

This study presents an optimization and decision-support framework for the design of split-output two-stage helical gearboxes, integrating multi-objective evolutionary optimization with Multi-Criteria Decision-Making (MCDM) techniques. The proposed framework employs the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to simultaneously minimize the gearbox length (L_gb) and maximize the gearbox efficiency (η_gb), generating a Pareto-optimal set of design alternatives. Three MCDM methods, MAIRCA, MARCOS, and Evaluation based on Average Ranking (EAMR) are utilized to rank the Pareto solutions based on varying weight distributions between the cost and benefit criteria. A sensitivity-driven analysis is conducted by systematically perturbing the weighting factors to evaluate the stability and robustness of each decision method under different trade-off conditions. The results reveal that while all three methods produce consistent trends, their sensitivity levels differ significantly. EAMR demonstrates the highest robustness, maintaining stable rankings across both local (per-transmission ratio uₕ) and global analyses, whereas MARCOS shows greater responsiveness to weight changes and MAIRCA demonstrates intermediate sensitivity. Heatmap-based visualization of global Top-1 frequencies and stability indices further confirms EAMR's superiority in delivering reliable, reproducible decisions. Overall, the proposed framework not only ensures an effective balance between compactness and efficiency in gearbox design but also provides a quantitative means to assess decision robustness across multiple MCDM strategies. The proposed methodology can be extended to other mechanical systems requiring trade-off optimization between geometric, energetic, and performance objectives.

Keywords:

two-stage helical gearbox, split output stage, multi-objective optimization, NSGA-II, MAIRCA, MARCOS, EAMR, sensitivity analysis, gearbox efficiency, gearbox length

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