## 3 Simulation Studies
### Simulation Study 2  
Simulation study 2 was conducted by randomly adding a number of large standard deviation values \(\sigma_{noise\_j\_large}=\sqrt{e^{0.35x_{1}-0.35x_{2}+5}}\) to the true response to give 50 replicates. The experimental data of simulation study 2 are presented in Table 6. The estimated mean and standard deviation functions given by LSM-based RSM in simulation study 2 are:  
\[\hat{\mu}_{LSM}(\mathbf{x})=103.7401+2.8205x_{1}-16.0324x_{2}-4.7497x_{1}^{2}+ 2.4012x_{2}^{2}+2.6455x_{1}x_{2} \tag{22}\]  
\[\hat{\sigma}_{LSM}(\mathbf{x})=21.5666+3.4305x_{1}-3.5459x_{2}-0.8637x_{1}^{2} +0.2259x_{2}^{2}-0.7160x_{1}x_{2} \tag{23}\]  
### Simulation Study 2  
Simulation study 2 was conducted by randomly adding a number of large standard deviation values \(\sigma_{noise\_j\_large}=\sqrt{e^{0.35x_{1}-0.35x_{2}+5}}\) to the true response to give 50 replicates. The experimental data of simulation study 2 are presented in Table 6. The estimated mean and standard deviation functions given by LSM-based RSM in simulation study 2 are:  
\[\hat{\mu}_{LSM}(\mathbf{x})=103.7401+2.8205x_{1}-16.0324x_{2}-4.7497x_{1}^{2}+ 2.4012x_{2}^{2}+2.6455x_{1}x_{2} \tag{24}\]  
\[\hat{\sigma}_{LSM}(\mathbf{x})=21.5666+3.4305x_{1}-3.5459x_{2}-0.8637x_{1}^{2} +0.2259x_{2}^{2}-0.7160x_{1}x_{2} \tag{25}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Run** & \(x_{1}\) & \(x_{2}\) & \(\mathbf{y}\) & \(\mathbf{s}\) & \(\mathbf{s}^{2}\) \\ \hline
1 & 0.5 & \(-1.0\) & 117.0310 & 28.4560 & 809.7210 \\
2 & 1.0 & \(-0.5\) & 108.9480 & 28.5790 & 816.7760 \\
3 & \(-1.0\) & \(-0.5\) & 107.1230 & 17.1750 & 294.9700 \\
4 & 1.0 & 0.0 & 100.5800 & 22.3820 & 500.9420 \\
5 & 0.5 & 0.5 & 83.9410 & 22.0650 & 486.8490 \\
6 & 0.0 & 0.0 & 98.3400 & 20.3830 & 415.4530 \\
7 & 0.5 & 0.0 & 98.0000 & 23.8250 & 567.6330 \\
8 & \(-1.0\) & 0.5 & 90.1890 & 16.3130 & 266.1150 \\
9 & \(-1.0\) & 1.0 & 92.2200 & 14.7930 & 218.8280 \\
10 & 1.0 & \(-1.0\) & 100.1200 & 26.8040 & 718.4340 \\
11 & 0.0 & 1.0 & 91.2050 & 17.0730 & 291.5000 \\
12 & \(-1.0\) & 0.0 & 97.5800 & 18.1380 & 328.9830 \\
13 & 1.0 & 1.0 & 104.2340 & 21.8010 & 475.2960 \\
14 & \(-0.5\) & 0.5 & 68.7300 & 18.4750 & 341.3320 \\
15 & 0.0 & \(-0.5\) & 157.4680 & 23.2770 & 541.8020 \\
16 & \(-0.5\) & \(-1.0\) & 115.1800 & 21.6910 & 470.5180 \\
17 & \(-0.5\) & \(-0.5\) & 128.1910 & 21.9950 & 483.7670 \\
18 & 0.5 & 1.0 & 98.3600 & 18.7680 & 352.2350 \\
19 & \(-1.0\) & \(-1.0\) & 104.5970 & 20.9930 & 440.6870 \\
20 & 0.0 & \(-1.0\) & 114.6310 & 26.1640 & 684.5700 \\
21 & 0.0 & 0.5 & 53.3870 & 20.1200 & 404.8260 \\
22 & 1.0 & 0.5 & 93.0950 & 20.7210 & 429.3470 \\
23 & \(-0.5\) & 0.0 & 97.8600 & 19.4930 & 379.9600 \\
24 & \(-0.5\) & 1.0 & 95.2650 & 16.5790 & 274.8610 \\
25 & 0.5 & \(-0.5\) & 147.8720 & 25.1310 & 631.5690 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Experimental data for simulation 2. Figure 11: Criterion space of the estimated functions in simulation study 1. (**a**) LSM-based RSM. (**b**) Proposed FFNN. (**c**) Proposed CFNN. (**d**) Proposed RBFN. Table 9 presents the optimal input factor settings, associated process mean, process bias, process variance, and EQL values obtained from the proposed NN-based modeling methods and LSM-based RSM in simulation study 2. The proposed NN-based modeling methods again produced significantly smaller EQL values than the LSM-based RSM. The plot of squared process bias vs. the variance for LSM-based RSM and the proposed FFNN-, CFNN-, and RBNN-based modeling methods are illustrated in Figure 16. The optimal settings are marked with green stars in each figure. Figure 14: Response plots created using the proposed CFNN-based modeling method in simulation study 2: (**a**) Mean (R2 = 80.98%), (**b**) Standard deviation (R2 = 89.32%). Figure 15: Response plots created using the proposed RBFN-based modeling method in simulation study 2: (**a**) Mean (R2 = 99.99%), (**b**) Standard deviation (R2 = 99.99%).