# Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming
## 3 Application in GP Programming  
Performance of MGGP can be determined by fitness, maximum error, minimum error, and standard deviation. These are created after training value and test value. Table 3 shows the simulation results of Model 1 and Model 2. In Model 1, best fitness value \(\left(\sigma_{\text{eqv}}\right)_{1}\), maximum error value \(\left(\sigma_{\text{eqv}}\right)_{2}\), minimum error value \(\left(\sigma_{\text{eqv}}\right)_{1}\) and standard deviation \(\left(\sigma_{\text{eqv}}\right)_{4}\) are found in training results. In Model 2, best fitness value \(\left(\sigma_{\text{eqv}}\right)_{4}\), maximum error value \(\left(\sigma_{\text{eqv}}\right)_{1}\), minimum error value \(\left(\sigma_{\text{eqv}}\right)_{4}\) and standard deviation \(\left(\sigma_{\text{eqv}}\right)_{3}\) are found in training results. In Model 1, best fitness value \(\left(\sigma_{\text{eqv}}\right)_{4}\), maximum error value \(\left(\sigma_{\text{eqv}}\right)_{2}\), minimum error value \(\left(\sigma_{\text{eqv}}\right)_{1}\) and standard deviation \(\left(\sigma_{\text{eqv}}\right)_{4}\) are found in test results. In Model 2, best fitness value \(\left(\sigma_{\text{eqv}}\right)_{4}\), maximum error value \(\left(\sigma_{\text{eqv}}\right)_{1}\), minimum error value \(\left(\sigma_{\text{eqv}}\right)_{4}\) and standard deviation \(\left(\sigma_{\text{eqv}}\right)_{3}\) are found in test results. When we look at these values, we see that best produced equivalent stress value is \(\left(\sigma_{\text{eqv}}\right)_{1}\) for Model 1 and \(\left(\sigma_{\text{eqv}}\right)_{4}\) for Model 2. Figure 5: Flow diagram of optimization. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Model 1**} & \multicolumn{2}{c}{**Model 2**} \\ \hline
**Equivalent Stress** & **Criteria** & **Training** & **Test** & **Training** & **Test** \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{1}\)} & Fitness & 0.143 & 0.157 & 15.486 & 20.877 \\  & Max & 0.287 & 0.321 & 26.223 & 62.125 \\  & Min & 0.063 & 0.073 & 6.254 & 6.816 \\  & S.D & 0.047 & 0.054 & 3.864 & 8.614 \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{2}\)} & Fitness & 0.925 & 0.916 & 1.860 & 1.979 \\  & Max & 2.000 & 2.087 & 3.889 & 4.173 \\  & Min & 0.237 & 0.260 & 0.934 & 0.898 \\  & S.D & 0.391 & 0.377 & 0.647 & 0.767 \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{3}\)} & Fitness & 1.082 & 1.156 & 0.668 & 0.780 \\  & Max & 1.343 & 1.380 & 0.800 & 0.929 \\  & Min & 0.763 & 0.815 & 0.328 & 0.400 \\  & S.D & 0.113 & 0.106 & 0.093 & 0.104 \\ \hline \multirow{4}{*}{\(\left(\sigma_{\text{eqv}}\right)_{4}\)} & Fitness & 0.218 & 0.143 & 0.617 & 0.658 \\  & Max & 0.692 & 0.287 & 1.284 & 1.393 \\ \cline{1-1}  & Min & 0.162 & 0.165 & 0.299 & 0.285 \\ \cline{1-1}  & S.D & 0.066 & 0.047 & 0.191 & 0.206 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation results. Table 4 contains information on tree structures of the best models of Model 1 and Model 2. When all stress values are examined, gene and tree depths are reached at maximum values. When tree depth investigated by complex structure and node number, least node and minimum complex structure are seen in \(\left(\sigma_{\text{eqv}}\right)_{3}\) for Model 1 and in \(\left(\sigma_{\text{eqv}}\right)_{4}\) for Model 2. When all the models are studied, the most complex structure and maximum number of nodes occurred in \(\left(\sigma_{\text{eqv}}\right)_{1}\) of Model 2. In our study, the best equations of MGGP are shown in Table 5. As we have explained in the other tables, the most complex equations are seen in \(\left(\sigma_{\text{eqv}}\right)_{1}\) in Model 2. All equations can be used as a guide in optimization for researchers.