## 3 Enhanced RDO Procedure
Since the reliability probability result of the constraint according to the reliability analysis method has a large effect on the design safety, the reliability analysis method suitable for the RDO of the seat frame design with respect to the seat belt anchorage strength was reviewed by comparing the evaluated reliability probability results using the MCS. The conventional RBDO method generally handled the reliability probability of the constraint with only one approximate reliability analysis method, whereas the proposed RDO method not only enhanced the reliability probability of the constraint via both the approximate and sampling reliability analysis methods, but also included the design robustness in the side constraints. The MVRM approximates the mean and the standard deviation of the probability constraint function using Taylor series expansion. In the MVRM, the accuracy of the reliability analysis result depends on the Taylor series expansion method. The reliability index of the MVRM is defined as follow [27]:  
\[\beta=\frac{\mu_{x}}{\sigma_{x}} \tag{2}\]  
The failure probability (\(p_{f}\)) and the reliability probability (\(R\)) are estimated by substituting the reliability index of Equation (2) into the following Equations:  
\[p_{f}=\Phi(-\beta) \tag{3}\]  
\[R=1-p_{f} \tag{4}\]  
where \(\Phi\) is a standard normal distribution function. Figure 7: Enhanced RDO procedure. In the AISM, the failure region for the constraint function is approximated as a second order polynomial using principal curvature, as shown in Figure 8. The curvature value is repeatedly updated to include the fracture region based on the number of samples existing in the fracture region. In Figure 8, the reliability index is denoted as \(\beta\). MCS is a reliability analysis method that can actually calculate the uncertainty of the probabilistic performance function due to the random variable in cases wherein the number of samples conforming to the failure probability, or the reliability probability, to be evaluated at the target confidence level is applied. In the reliability analysis using MCS, if the probabilistic performance function \(y\) is calculated by the number of samples \(N\), the following average \(\mu\), variance \(\sigma^{2}\), and failure probability \(p_{f}\) can be calculated [29]. \[\mu=\frac{1}{N}\ \sum_{i=1}^{N}y_{i} \tag{5}\]  
\[Minimizef\left\{\mu_{y}(x),\;\sigma_{y}(x)\right\} \tag{1}\]0  
\[Minimizef\left\{\mu_{y}(x),\;\sigma_{y}(x)\right\} \tag{1}\]1  
where \(X=(X_{1},\ X_{2},\ \cdots,\ X_{n})\) and \(x=(X_{1},\ X_{2},\ \cdots,\ X_{n})\)  
In Equation (5), the failure probability is defined using the indicator function _I(x)_ as follows:  
\[Minimizef\left\{\mu_{y}(x),\;\sigma_{y}(x)\right\} \tag{1}\]2  
\[\text{where }I(x)=\left\{ \begin{array}{c}1\ \text{if}\ G(x)\leq 0\\ 0\ \text{otherwise}\end{array} \right.\]  
The failure probability is expressed using the average of the indicator function as follows:  
\[p_{f}=\overline{I}(x)=\frac{1}{N}\ \sum_{i=1}^{N}I(x)=\frac{N_{f}}{N} \tag{7}\]  
where \(N_{f}\) is the number of samples in the case of \(G\leq 0\). The reliability probability \(R\) is defined as follows:  
\[R=1-p_{f}=\frac{N-N_{f}}{N} \tag{8}\]  
Figure 8: Failure region in AISM [28]. Cumulative distribution function (CDF) _F(y)_ and probability density function (PDF) _f(y)_ can be calculated as follows:  
\[F(y)=P\{G\leq y\}=\frac{1}{N}\,\sum_{i=1}^{N}I^{\prime}(y_{i}) \tag{9}\]  
\[\text{where }I(x)=\left\{ \begin{array}{c}1\text{ if }G(x)\leq y\\ 0\text{ otherwise}\end{array} \right.\]  
\[f(y)=\left[F(y)\right]^{\prime}\]  
As represented in Equation (9), the PDF is obtained from the derivative of the CDF.