# Metaheuristic Approaches to Solve a Complex Aircraft Performance Optimization Problem
## 5 Case Study  
Both GA and GP techniques were applied to solve the above MAO problem for an aircraft conceptual design. Six responses related to stiffness, wingtip displacement, wingtip torsion, aileron efficiency, flutter and drag were built using GP. A cross-validation check of GP and GA approaches was carried out by comparison of the obtained optimal designs.  
Figure 6: GP metamodel-assisted optimization flowchart.  
To construct the metamodels using GP, 1997 sampling points were selected as training data by the optimal Latin hypercube DoE to maximally represent information in the whole design space. Two fit indices were computed to evaluate the accuracy of predictive models constructed by GP, namely root mean square error (RSME) and coefficient of determination (R2) in Equations (6) and (9), respectively. In this paper, main parameters used in GP metamodel building are shown in Table 2. As an example, the mathematical expression for the wingtip torsion is given as below:  
\[F_{torsion}=-0.238478+0.04X_{1}-0.71435X_{10}+0.14171X_{11}- \frac{0.252627X_{11}}{X_{12}}- \tag{12}\] \[0.0317857X_{13}+\frac{0.12126X_{13}}{X_{12}}-762.093X_{15}+0.19778 6X_{18}-0.01X_{2}+0.682686X_{3}-\] \[0.297X_{4}+\frac{0.511492X_{4}}{X_{12}}-0.014X_{5}-\frac{0.007X_{ 5}}{X_{11}}+\frac{0.00172534X_{11}X_{5}}{X_{12}}+\frac{0.00003781X_{4}X_{5}{}^{2 }}{X_{11}X_{12}}+\] \[0.0071\bigg{(}-\frac{16.6482X_{13}}{X_{12}}+0.106X_{15}-110.845X_ {3}+1.73842X_{5}\bigg{)}-0.00245223X_{7}-0.261464X_{18}X_{9}+\] \[(0.194348X_{12}+762.093X_{15}+\frac{0.000126113X_{4}X_{5}{}^{2}}{ X_{11}X_{3}})\text{Tanh}\Big{(}64742.1X_{7}{}^{3}\Big{)}.\]  
where \(X_{1}\) to \(X_{19}\) are design variables detailed in Table 1. It should be noted that results of two fit indices, for example, \(\text{RSME}=1.27\times 10^{-2}\) and \(\text{R}^{2}=0.995\) for the predicted wingtip torsion response, indicate the metamodels by GP are quite accurate to estimate the nonlinear responses of the interests. Furthermore, design variables regarding horizontal bending stiffness and torsional stiffness have slight effects on the wingtip torsion, and this can be proved by sensitivity analysis performed in Section 5.2. To validate the accuracy of generated metamodels, a validation data set of 100 sampling points was also used to check the overall performance capacity of the metamodels. In this paper, the relative error was used to measure the discrepancy between the actual response and the predicted model,  
\[e=\big{|}r_{a}-r_{p}\big{|}/r_{a} \tag{13}\]  
where \(r_{a}\) is the actual response value and \(r_{p}\) is the prediction value. The relative errors of 100 validation points are given in Figure 7 and their relative mean errors for all six responses were 0.083%, 0.008%, 0.024%, 0.119%, 0.106%, and 0.598%.  
On average, the predictions of all the responses by GP technique were quite accurate. However, the most limitation of GP methodology was its complexity, discontinuity, and consumption of the model construction. The higher the dimension of the design space was, the more complicated the constructed GP expression was, the higher the number of step discontinuities of the response functions occurred, and more RAM was required. For the design space composed of 19 variables, it took more  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Population size & 500 \\ Generations & 100 \\ Number of fitness cases & 1997 \\ Initialisation & 50\% full + 50\% grow \\ Maximum depth initialised trees & 4 \\ Minimum depth initialised trees & 2 \\ Maximal depth for a tree & 50 \\ Reproduction rate & 20\% \\ Crossover rate & 40\% \\ Mutation rate & 40\% \\ Termination of the evolution if RMSE is less than & \(1.0\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main evolution parameters used in GP metamodel building.  
than three days for the GP methodology to construct the six responses on a personal computer with eight cores (E5-2670, 2.6 GHz) and 24 G RAM.