## 5 Conclusions  
The present paper examines the multi-objective optimization of the planetary gearbox, taking into account various objective functions, including the center distance, gear volume, gearbox efficiency, and the surface and bending stresses. A multi-objective optimization model was developed for a planetary gearbox, incorporating various objectives and several constraints, such as safety and reliability constraints, mounting constraints, and a constraint ensuring uniform wear of the tooth profile. This study presents a novel hybrid method, namely HMOBPSO, which integrates the PSO and BOA algorithms to address a challenging MOO problem. The proposed approach integrates the PSO velocity update scheme into the equations governing butterfly movement. Additionally, it combines the use of crowding distance to optimize the selection of individuals. Furthermore, it suggests the inclusion of an adaptive parameter for individual position updates and the substitution of switch probability.  
The algorithm under consideration has successfully generated the non-convex Pareto frontier, which serves as a graphical representation of the interplay between opposing objectives and is of importance for the designer in facilitating the selection of suitable building parameters. The obtained findings were subjected to validation and subsequently compared with the parameters of the reference industrial gearbox. The analysis revealed a significant reduction in center distance, with a maximum decrease of 27.5%. Additionally, the volume of the gearbox exhibited a notable improvement of 35%. Furthermore, the efficiency of the gearbox displayed an enhancement of 2.56%. In addition, an assessment of the optimization performance of the HMOBPSO algorithm was conducted by comparing it to other established algorithms across several ZDT and DTLZ benchmark problems. The efficacy of the HMOBPSO algorithm in the optimization of the planetary gearbox model is evidenced by the improved solutions it yields in comparison to the NSGA-II method. This observation underscores the efficacy and resilience of the algorithm proposed for identifying optimal solutions.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Algorithm** & **DTLZ1** & **DTLZ2** & **DTLZ3** & **DTLZ4** & **DTLZ5** & **DTLZ6** & **DTLZ7** & **ZDT1** & **ZDT2** & **Mean Ranking** & **Rank** \\ \hline HMOBPSO & **1** & 2.67 & **1** & 2.55 & **2.22** & **1.667** & **1.78** & 1.89 & **1.89** & 1.85 & 1 \\ NSGA-II & 3 & 2.55 & **2** & **2.11** & 3.11 & 1.889 & 2.33 & **1.78** & 2. & 2.35 & 2 \\ CMOPSO & 4 & **2.22** & 4 & 2.67 & 2.67 & 2.44 & 1.89 & 2.33 & 2.11 & 2.78 & 3 \\ MOPSO & 2 & 2.67 & 3 & 2.67 & 3 & 5 & 4.79 & 4.44 & 4.89 & 3.45 & 4 \\ MOBOA & 5 & 4.89 & 5 & 5 & 4 & 4 & 4.22 & 4.55 & 4.11 & 4.58 & 5 \\ Friedman \(p\) value & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\ \hline \hline \end{tabular}
\end{table}
Table 8: The results of the Friedman statistical test for all algorithms across all ZDT and DTLZ test problems, with a significance level of \(\alpha=0.05\).  
Future work can focus on examining the algorithm's effectiveness across other issue domains and exploring its capacity for addressing more intricate optimization challenges.  
Conceptualization, Methodology and Validation, M.S. and M.R.; Formal Analysis, Software, Investigation, Data Curation, M.S. and M.R.; Resources, M.S.; Writing--Original Draft Preparation, M.S.; Writing--Review and Editing, M.S.; Visualization, M.S. and M.R.; Supervision, M.S. All authors have read and agreed to the published version of the manuscript.  
The research of [PERSON] was supported by the Serbian Ministry of Education and Science under Grant No. TR35006. The research of [PERSON] was supported by the Serbian Ministry of Education and Science under Grant TR35029.  
Not applicable.  
No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The authors