## 3 Enhanced RDO Procedure  
The robust design method has the object of minimizing the variation in the response performance by treating the variation source so as to reduce the sensitivity, rather than eliminating the variation source in the design process, in order to improve the quality and reliability of the design [16]. Generally, the Taguchi robust design using the orthogonal array method handles a small number of constraints in a discrete design space, whereas the RDO based on mathematical formulation can accommodate many constraints in a continuous design space [17]. The RDO is similar to a reliability-based design optimization (RBDO) in terms of ensuring the robustness of the constraint, but the RDO is able to guarantee the robustness of both the objective function and the constraint through an optimization process that considers, respectively or simultaneously, the robustness of the objective function, including the existing design and tolerance [18,19]. The evaluation of the feasibility and reliability of the constraints is the most important matter in reducing the risk in the design process. However, since the conventional RDO method could not provide information on whether the robustness of the constraint is secured quantitatively, it is difficult to clearly ensure the design safety [20]. In this study, an enhanced RDO method that is able to compute the reliability probability of the constraint in the RDO process was developed, and the proposed RDO procedure is shown in Figure 7. As represented in Figure 7, in order to quantitatively guarantee the robustness of the constraint, the robustness of the objective function and the constraint were considered using not only mean and standard deviation, but reliability analysis methods such as MVRM and AISM were also applied to the constraint. Additionally, the probability of reliability was evaluated for the converged optimum results using the MCS to verify the actual reliability. The RDO formulation was defined as follows [21,22]:  
\[Minimizef\left\{\mu_{y}(x),\;\sigma_{y}(x)\right\} \tag{1}\]  
\[s.t.G_{i}\left\{\mu_{y}(x),\;\sigma_{y}(x)\right\}\;\leq 0,\;\;i=1,\;2\]  
\[x_{L}+n\sigma_{x_{j}}\;\leq\;\mu_{x}\;\leq x_{U}-n\sigma_{x_{j}},\;\;j=1,\; \cdots,\;9\]  
where \(f\) is the objective function, \(G\) is the probabilistic constraint, \(\mu\) is the mean, \(\sigma\) is the standard deviation, \(x_{L}\) is the lower limit of the design range, \(x_{U}\) is the upper limit of the design range, and \(n\) is the sigma level. In the RDO, the probabilistic design solutions are determined by minimizing the objective function subject to both the probabilistic constraint and the side constraints of the robustness, considering the uncertainties of the design variables and the sigma level, which is the target design's reliability. In this study, an evolutionary optimization algorithm was [23] was applied to explore the optimum solution in the constraint optimization problem. The evolutionary optimization algorithm is an evolution strategy that mutates designs by adding a normally distributed random value to each design variable. The mutation strength (standard deviation of the normal distribution) is self-adaptive and changes during the optimization process [24,25,26]. In the evolutionary optimization algorithm, the most appropriate learning technique and parameter values are increasingly self-adapted according to the learning experience gained during evolution [24,26]. The algorithm efficiently solves design problems with low numbers of variables and with some noise in the design space. The probability constraint functions were evaluated through the reliability analysis, and in this study, MVRM and AISM were each applied for the reliability analysis.