## 5 Conclusions  
This study evaluated the dimensional tolerances of the raw materials and preforms of bearing hubs as probabilistic variables with inherent uncertainty, conducting a reliability-based optimization design of preforms to enhance product quality and minimize defects.  
1. An index was developed to assess the quality and defects of the final form based on the raw materials used in the hot forging process of bearing hubs and the initial process design. This allowed for predicting potential quality issues that may arise when implementing dimensional tolerances.
2. A variety of modeling techniques were employed to create an approximation model for interpreting bearing hub hot forging. The predicted R\({}^{2}\) values were compared, and the Kriging technique, which demonstrated the highest prediction accuracy, was selected. This choice facilitated the rapid and precise derivation of results without further analysis.
3. A comparison of the initial design, DDO, and RBDO revealed that the performance index representing quality was somewhat lower in RBDO. However, this was attributable to the more stringent application of constraints regarding the defect quantity and minimum forming load.
4. Although DDO was assessed as a superior design plan for quality due to its lower effective strain compared to RBDO, its reliability was relatively low. In contrast, RBDO significantly reduced the probability of defect occurrence, achieved a reliability exceeding 99%, and presented a more stable design plan.  
This study introduces a methodology for assessing the dimensional tolerances of raw materials and preforms using reliability-based optimization design. It is expected to facilitate the development of designs that enhance quality and substantially decrease the likelihood of defects in future multi-stage forging processes. However, this study primarily focuses on computational simulations, necessitating validation by comparing the analytical results with actual experimental data. In future research, experiments and validation will be conducted to ensure the reliability of the results derived from reliability-based optimization design.  
M.O.: Model design, methodology, software, writing--original draft. [PERSON] and J.C.: Resources, model design. M.K. and M.J.: Writing--modification of draft; writing--review and editing. S.H.: Supervision; writing--modification of draft; writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This work was partly supported by the Technology development Program of MSS [RS-2024-00446241] and the Regional Innovation Strategy (RIS) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (MOE) [(2021 RIS-004)].  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
[PERSON] and [PERSON] affiliate with Dong Eun Forging Research Center, [PERSON] affiliate with Metal Forming Research Corporation (MFRC) Research Center. The affiliated institutions and funding sponsors had no role in the design of study, in the collection, analyses or interpretation of data, and in the decision to publish the results. The other authors have no conflicts of interests to declare.