## 3 Simulation Studies
### Simulation Study 1  
Simulation study 1 was conducted by randomly adding a number of small standard deviation values \(\sigma_{noise\_small}=\sqrt{\phi^{0.35x_{1}-0.35x_{2}+2}}\) to the true response to give 50 replicates of the output response. Experimental data from simulation study 1 are presented in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Run** & \(\mathbf{x}_{1}\) & \(\mathbf{x}_{2}\) & \(\mathbf{y}_{true}\) \\ \hline
1 & 0.5 & \(-1.0\) & 113.5914 \\
2 & 1.0 & \(-0.5\) & 112.9478 \\
3 & \(-1.0\) & \(-0.5\) & 104.7632 \\
4 & 1.0 & 0.0 & 100.0000 \\
5 & 0.5 & 0.5 & 78.6806 \\
6 & 0.0 & 0.0 & 100.0000 \\
7 & 0.5 & 0.0 & 100.0000 \\
8 & \(-1.0\) & 0.5 & 89.9294 \\
9 & \(-1.0\) & 1.0 & 97.0000 \\
10 & 1.0 & \(-1.0\) & 105.0000 \\
11 & 0.0 & 1.0 & 91.8451 \\
12 & \(-1.0\) & 0.0 & 100.0000 \\
13 & 1.0 & 1.0 & 99.5939 \\
14 & \(-0.5\) & 0.5 & 64.8503 \\
15 & 0.0 & \(-0.5\) & 158.0282 \\
16 & \(-0.5\) & \(-1.0\) & 105.0000 \\
17 & \(-0.5\) & \(-0.5\) & 127.4106 \\
18 & 0.5 & 1.0 & 97.0000 \\
19 & \(-1.0\) & \(-1.0\) & 100.6766 \\
20 & 0.0 & \(-1.0\) & 113.5914 \\
21 & 0.0 & 0.5 & 54.8669 \\
22 & 1.0 & 0.5 & 96.2952 \\
23 & \(-0.5\) & 0.0 & 100.0000 \\
24 & \(-0.5\) & 1.0 & 91.8451 \\
25 & 0.5 & \(-0.5\) & 145.1924 \\ \hline \end{tabular}
\end{table} 
Table 1: Experimental data and true function response value. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Run** & \(\mathbf{x}_{1}\) & \(\mathbf{x}_{2}\) & \(\mathbf{y}_{true}\) \\ \hline
1 & 0.5 & \(-1.0\) & 112.5910 & 5.9420 & 35.3060 \\
2 & \(1.00\) & \(-0.50\) & 111.1480 & 6.3660 & 40.5310 \\
3 & \(-1.00\) & \(-0.50\) & 105.1830 & 4.9330 & 24.3300 \\
4 & \(1.00\) & 0.00 & 100.6400 & 5.2010 & 27.0510 \\
5 & \(0.50\) & 0.50 & 77.4810 & 5.3530 & 28.6530 \\
6 & \(0.00\) & 0.00 & 99.7600 & 5.1250 & 26.2680 \\
7 & \(0.50\) & 0.00 & 101.0600 & 5.0650 & 25.6490 \\
8 & \(-1.00\) & 0.50 & 89.9690 & 3.9950 & 15.9580 \\
9 & \(-1.00\) & 1.00 & 96.6600 & 2.9110 & 8.4740 \\
10 & \(1.00\) & \(-1.00\) & 105.8000 & 6.6580 & 44.3270 \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental data from simulation study 1. The approximated mean and standard deviation functions shown in Equations (20) and (21) were estimated using the LSM-based RSM method in MATLAB software. \[\hat{\mu}_{LSM}(\mathbf{x})=103.5282+3.3709x_{1}-15.4188x_{2}-2.7589x_{1}^{2}-0.4 410x_{2}^{2}-0.6280x_{1}x_{2} \tag{20}\]  
\[\hat{\sigma}_{LSM}(\mathbf{x})=4.7239+0.8883x_{1}-0.7607x_{2}+0.1752x_{1}^{2}- 0.0442x_{2}^{2}+0.0713x_{1}x_{2} \tag{21}\]  
Information about the trained multilayer feed-forward NN model, including the transfer function and NN architecture (number of inputs, number of hidden neurons, number of outputs), is given in Tables 3 and 4. Furthermore, the weight and bias values associated with each layer of the three proposed NN models (FFNN, CFNN, and RBFN) for both the process mean and standard deviation function estimations are exhibited in Appendix A (Tables A1-A4, respectively). The conventional LSM-based RSM method is compared to the proposed FFNN-, CFNN-, and RBFN-based RD modeling methods in Figures 7-10 in contour plot form. Furthermore, Table 5 shows the ultimate solution set and corresponding best response values, while the comparison is done in terms of EQL. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Model**} & \multicolumn{1}{c}{**Transfer Function**} & \multicolumn{1}{c}{**Machine Function**} & \multicolumn{1}{c}{**Architecture**} & \multicolumn{1}{c}{**\#Epoch**} \\ \hline \multirow{2}{*}{FFNN} & Mean & Tansig-Purelin & Trainlm & 2-17-1 & 3 \\  & Standard deviation & Tansig-Purelin & Trainrp & 2-6-1 & 44 \\ \hline \multirow{2}{*}{CFNN} & Mean & Tansig-Purelin & Trainlm & 2-17-1 & 5 \\  & Standard deviation & Tansig-Purelin & Trainrp & 2-5-1 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of RBFN information for simulation \(1\).