## 5 Conclusions  
In this research, a design framework exploiting a DR approach, machine learning-based surrogate modeling, structural optimization, and reliability assessment was proposed to handle a multivariate and multidisciplinary engineering system. The efficiency of the proposed design framework was addressed by a stretchable MSP antenna-based strain sensor. Compared with the conventional rigid antenna, the efficiency of the proposed stretchable antenna design was highlighted because it contains careful consideration for all possible mechanical flexibility, which weights realistic system prediction and estimation. Such consideration demands meticulous examination on the flexible substrate that includes non-uniform thickness. In this research, therefore, the non-uniformity was regarded as the DVs to represent geometric uncertainty captured by a Gaussian copula function. With the copula function, a non-uniform substrate thickness was represented by 202 subparts, taking the mean of 0.9705 mm and the COV of 0.05. However, a huge number of DVs escalate the redundancy and complexity of data, inhibiting precise system prediction and estimation. In order to resolve the issue, the proposed framework employed a DR process, particularly FE and FS. In the DR process, ECC (\(e\)) was exploited as a clear guideline that suggests a better DR method grounded on data behavior. On the basis of the ECC estimation, FE was employed to reduce data redundancy regarding \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(P_{f}\) **of Original Data** & \(P_{f}\) **of New Data by IFT** & \(P_{f}\) **Difference of Original/New Data** \\  & **(2.5 GHz/5 GHz)** & **(2.5 GHz/5 GHz)** & **(2.5 GHz/5 GHz)** \\ \hline PNN & 0.278/0.313 & 0.291/0.325 & 4.67\%/4.06\% \\ MCS & 0.298/0.334 & 0.327/0.342 & 9.48\%/9.41\% \\ \(P_{f}\) difference of & 6.46\%/6.91\% & 8.05\%/5.14\% & \\ PNN/MCS & \multicolumn{1}{c}{} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Probability of failure conducted by probabilistic neural network (PNN) and Monte Carlo simulation (MCS). the substrate non-uniformity in which \(e\) is greater than 0.5, whereas FS eliminated unnecessary coordinates of the patch within the specific case (0.5 \(\leq e<\) 1). Both FE (PCA and AE) and FS (IFT) derived good redundancy reduction of 50.21%, 44.57%, and 32.11%, respectively. With the criterion, engineers can make a decision on the applicable DR process and management of multivariate data. Moreover, owing to the limitation to formulate a relationship between structural deformation and electrical antenna's response, the ANN-based surrogate model employing the multivariate data purified by the DR process was developed to predict a complex engineering system. As the second drawback of the stretchable MSP strain sensor, the narrow bandwidth that restricts the functionality of the sensor was handled by a structural shape optimization. Compared with the conventional antenna shape, the proposed optimum shape drew the antenna's performance improvement of 5.77% at 2.5 GHz and 29.03% at 5 GHz within the reliable frequency range. A new optimum design implementing the DR process also escalated the performance improvement of 29.05% at 2.5 GHz and 35.08% at 5 GHz. Thus, it is confirmed that the developed optimal design maximized the frequency stability within a reliable bandwidth and antenna performance under structural deformation. In the final step, the reliability of the stretchable antenna for the strain sensor was assessed by PNN and validated by MCS. PNN evaluated the efficiency of the proposed design by showing the classification accuracy of 96%. The accuracy of PNN was validated by that of MCS. The results obtained for the stretchable strain MSP antenna show that the proposed design framework with the uncertainty characterization and dimension reduction is effective on multi-physics-based and multi-objective design processes. Conceptualization, S.-K.C. ; methodology, S.H. ; software, S.H. ; validation, S.-K.C. and R.M.G. ; formal analysis, S.H. ; investigation, S.H.