## 4 Case Study  
The printing data example used by Vining and Myers [6] and [PERSON] [9] was selected to demonstrate the application of the proposed methods. The printing experiment investigates the effects of speed (\(x_{1}\)), pressure (\(x_{2}\)), and distance (\(x_{3}\)) on a printing machine's ability to add colored ink to a package (\(y\)). The case study has a factorial design with three level three factors, so in total, the number of experimental runs is \(3^{3}\). To ensure the accuracy of the experiment, each treatment combination is repeated three times. In this case study, the target \(r_{2}=500\). The experimental data are given in Vining and Myers [6]. The estimated mean and standard deviation functions given by LSM-based RSM are  
\[\hat{\eta}_{\text{LSM}}(\mathbf{x}) =327.6296+\mathbf{x}^{T}\mathbf{a}_{LSM}+\mathbf{x}^{T}\mathbf{A }_{LSM}\mathbf{x} \tag{24}\] \[\hat{\sigma}_{\text{LSM}}(\mathbf{x}) =34.8832+\mathbf{x}^{T}\mathbf{b}_{LSM}+\mathbf{x}^{T}\mathbf{B} _{LSM}\mathbf{x} \tag{25}\]  
Figure 16: Criterion space of the estimated functions in simulation study 2. (**a**) LSM-based RSM. (**b**) Proposed FFNN. (**c**) Proposed CFNN. (**d**) Proposed RBFN. \[\text{where }\textbf{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right]\textbf{, }\textbf{a}_{LSM}=\left[ \begin{array}{c}177.0001\\ 109.4259\\ 131.4630\end{array} \right]\textbf{, }\textbf{A}_{LSM}=\left[ \begin{array}{c}32.0001\\ 66.0278\\ 75.4721\end{array} \right.\] \[\textbf{b}_{LSM}=\left[ \begin{array}{c}11.5268\\ 15.3231\\ 29.1904\end{array} \right]\textbf{, and }\textbf{B}_{LSM}=\left[ \begin{array}{c}4.2038\\ 7.7195\\ 7.7195\\ 5.1093\end{array} \right.\] \[\textbf{.}\]  
The information used for the RD modeling methods after training with the associated transfer functions, training functions, NN architectures (number of inputs, number of hidden neurons, number of outputs), and number of epochs in the case study are summarized in Tables 10 and 11. The specified weight and bias values of the proposed FFNN, DFNN, and RBFN used to approximate the process mean and standard deviation functions in the case study are shown in Appendix C (Tables A9-A14). The contour plots of the response functions for the process mean and standard deviation estimated by the LSM-based RSM, FFNN, CFNN, and RBFN robust design modeling methods are demonstrated in Figures 17-20, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Model** & **Transfer Function** & **Training Function** & **Architecture** & **\#Epoch** \\ \hline \multirow{2}{*}{FFNN} & Mean & Tansig-Purelin & Traingdx & 3-9-1 & 105 \\  & Standard deviation & Tansig-Purelin & Traingda & 3-10-1 & 19 \\ \hline \multirow{2}{*}{CFNN} & Mean & Tansig-Purelin & Traingda & 3-12-1 & 107 \\  & Standard deviation & Tansig-Purelin & Trainrp & 3-11-1 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Summary of the multilayer FFNN used in the case study. Figure 17: Response plots created using the conventional LSM-based RSM in the case study: (**a**) Mean (R\({}^{2}\) = 92.68%), (**b**) Standard deviation (R\({}^{2}\) = 45.42%). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Model** & **Transfer Function** & **Goal** & **Spread** \\ \hline \multirow{2}{*}{RBFN} & Mean & Radbas-Purelin & \(1\times 10^{-27}\) & 0.5 \\  & Standard deviation & Radbas-Purelin & \(1\times 10^{-27}\) & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Summary of the RBFN used in the case study. Table 12 presents the optimal factor settings, associated process mean, process bias, process variance, and EQL values obtained from the proposed NN-based modeling methods  
Figure 19: Response plots created using the proposed CFNN-based modeling method in the case study: (a) Mean (R\({}^{2}\) = 83.05%), (b) Standard deviation (R\({}^{2}\) = 69.47%). Figure 20: Response plots created using the proposed RBFN-based modeling method in the case study: (a) Mean (R\({}^{2}\) = 99.99%), (b) Standard deviation (R\({}^{2}\) = 99.99%).