## 1 Introduction  
Research on modularization has primarily been aimed at consolidating multiple functions into a single component to achieve lightweight vehicle designs [1, 2, 3]. In particular, automobile bearing hubs have become a prominent example of this approach, evolving from first-generation structures that focused on reducing initial friction to third-generation structures that integrate the inner ring and hub, outer ring, and steering knuckle flange to enhance assembly accuracy [4, 5] (Figure 1). However, these modular components exhibit considerable structural complexity, necessitating a multi-stage forging process rather than a single-step operation. This multi-stage forging process typically involves the stages of upsetting, blocking, and finishing, all originating from the initial billet. The preform is produced through a hot forging process, as shown in Figure 2. The billet is first cut, taking into account the processing environment and the final form [Figure 2a]. The raw material is then heated to the target temperature for hot forging [Figure 2b], after which the heated raw material undergoes the hot forging process according to the shape of the preform [Figure 2c,d]. Finally, the raw material undergoes a finishing process to complete the bearing hub hot forging [Figure 2e]. The bearing hub preform is created through a two-step process involving upsetting and blocking. By developing an appropriate preform blocking process, complex shapes can be formed, and productivity can be enhanced by streamlining the process and reducing the total manufacturing time [6; 7; 8]. The current preform design has traditionally relied on empirical knowledge to address issues such as underfill and flash formation; however, new advancements in optimal design methodologies have recently been proposed to effectively resolve these issues [9; 10; 11]. Despite these advancements, previous studies have largely employed deterministic design optimization (DDO) methods that only consider ideal scenarios and do not account for manufacturing tolerances. DDO fails to consider the stringent allowable tolerances specified in standards such as the KS or JJS [12; 13] and the inherent manufacturing tolerances found in the manufacturer's internal quality control and production processes [14]. As a result, the optimal values of design variables obtained through DDO may not be suitable for actual manufacturing processes. To overcome these limitations, a reliability-based design optimization (RBDO) approach has been introduced as an extension of the current optimal design methodology [15]. Because the objective of reliability-based optimization is to minimize the probability of system failure, it must account for uncertainties such as manufacturing tolerances within the system as design variables. Numerous researchers have conducted studies aimed at reducing the likelihood of system failure by employing the RBDO technique. For example, [PERSON] et al. [16] introduced the local sampling with variable radius (LSVR) approach, which adjusts the sampling radius based on the nonlinearity and reliability of constraints. Consequently, this sampling method is an optimized sampling method for nonlinear constraints, though its efficacy may be less pronounced in linear or other scenarios. [PERSON] et al. [17] introduced a dynamic Kriging-based approximation model sampling technique optimized for nonlinear systems, which demonstrated high accuracy for such cases. However, the computational time for data acquisition was excessively long due to the method's high dependence on the accuracy of the approximation model. Additionally, these studies have been highly reliant on certain  
Figure 1: Third generation of bearing hub. Figure 2: Hot forging process of bearing hub: (**a**) billet cutting; (**b**) heating; (**c**) upsetting; (**d**) blocker; (**e**) finishing. types of constraints. [PERSON] et al.