## 5 Process of Optimal Foil Width Determination and Results
### Results TP1
#### 5.1.2 Determination of Parameters \(a\), \(b\), and \(c\)
\[X_{new}=X_{current}+r\cdot\left(X_{teacher}-T_{F}\cdot\overline{X}\right) \tag{23}\]  
\(r\) is a random number and \(T_{\text{F}}\) is the teaching factor, which is commonly 1 or 2. The next step is a learning phase, written in (24). \[X_{new}=\left\{ \begin{array}{ll}X_{i}+r\cdot\left(X_{j}-X_{i}\right)&\text{ if $f\!\left(X_{j}\right)<f\!\left(X_{i}\right)$}\\ X_{i}+r\cdot\left(X_{i}-X_{j}\right)&\text{otherwise}\end{array} \right. \tag{24}\]  
The uppercase \(X\) represents a population vector. According to that, the number of objective function evaluations (OFEs) consumed can be determined as OFEs = (2 \(\times\) population size \(\times\) iterations). TLBO has no parameters. To carry out a fair comparison of methods, the same number of fitness evaluations (OFEs) was set as a stopping criterion for all of them, which was 900,000. The stopping criterion is determined based on a large number of test runs, and it was set to a large enough value to obtain correct results. The population number (NP), which, according to [73], is between 5 and 10 times the number of parameters, was set to 10 because the calculation of the objective function is not time-consuming. The number of iterations was thus the stopping criterion of OFEs/NP. The calculation parameters for all three evolutionary methods used are presented in Table 8.  
To determine the coefficients of the function, the objective function was used based on the three values of \(U_{\text{Smax}}\) for the three foil widths, marked as OF3, written in (25). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Number of** & **Population Number (NP)** & **Number of** & **Objective Function** \\  & **Parameters (P)** & & **Iterations** & **Evaluations** \\ \hline DE/rand/1/exp, & P & NP = 10 \(\times\) P & ITER & OFEs = NP \(\times\) ITER \\ DE/rand/2/exp, & 3 & 30 & 30,000 & 900,000 \\ DE/best/1/bin & & & & \\ \hline TLBO & P & NP = 10 \(\times\) P & ITER & OFEs = NP \(\times\) 2 \(\times\) ITER \\  & 3 & 30 & 15,000 & 900,000 \\ \hline ABC & 5 & NP = 10 \(\times\) P & ITER & OFEs = NP \(\times\) ITER + scouts \\  & & 30 & \(\leq\)30,000 & Max. 900,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Calculation parameters using different methods. \[\text{OF3}=\sum_{i=1}^{3}\left(\frac{U_{\text{Smax, calculated,i}}-U_{\text{Smax, FEM,i}}}{U_{\text{Smax, FEM,i}}}\right)^{2} \tag{25}\]  
However, the actual quality was determined by taking into account the \(U_{\text{Smax}}\) for all foil width values, marked as OF7, written in (26), which is similar to the cross-validation procedure. It is important that the function written with three values fits as well as possible with the other values. \[\text{OF7}=\sum_{i=1}^{7}\left(\frac{U_{\text{Smax, calculated,i}}-U_{\text{Smax, FEM,i}}}{U_{\text{Smax, FEM,i}}}\right)^{2} \tag{26}\]  
Table 9 presents the results of determining the parameters for function F2 for the purpose of comparing evolutionary methods. Based on the results presented in Table 9, it can be seen that all methods were adequate. DE/rand/1/exp and DE/rand/2/exp were very robust for all 30 runs, and the calculated parameters in all iterations were the same down to the last decimal value. Since DE/rand/1/exp was minimally faster, further analyses are performed using the DE/rand/1/exp method. Tables 10-14 show the results of the calculations for the evaluation of functions from F1 to F5. For each of the functions, three combinations of foil widths were used, namely the following: \(U_{\text{Smax}}\) for 2, 4, and 8 m; \(U_{\text{Smax}}\) for 2, 5, and 8 m; and \(U_{\text{Smax}}\) for 2, 6, and 8 m. For each combination, OF7 was calculated, and the functions were valued according to the sum of OF7\({}_{\text{all}}\) = OF7\({}_{\text{2,4,8m}}\) + OF7\({}_{\text{2,5,8m}}\) + OF7\({}_{\text{2,6,8m}}\). OF3 is also shown in the tables, which we expected to be zero or very close to zero.