## 5 Process of Optimal Foil Width Determination and Results
### Results TP1
#### 5.1.2 Determination of Parameters \(a\), \(b\), and \(c\)  
Determining the parameters of the F1 function is very easy using the Vandermond matrix written in (14). \[ \begin{bmatrix}a\\ b\\ c\end{bmatrix} = \begin{bmatrix}1&fw_{1}&fw_{1}^{2}\\ 1&fw_{2}&fw_{2}^{2}\\ 1&fw_{3}&fw_{3}^{2}\end{bmatrix} ^{-1} \begin{bmatrix}U_{S\max}(fw_{1})\\ U_{S\max}(fw_{2})\\ U_{S\max}(fw_{3})\end{bmatrix}  \tag{14}\]  
Functions F2 to F5 are nonlinear, and the coefficients can be determined, e.g., by the Newton-Raphson iterative procedure described in (15) for F2. \[ \begin{bmatrix}a^{(k+1)}\\ b^{(k+1)}\\ c^{(k+1)}\end{bmatrix} = \begin{bmatrix}a^{(k)}\\ b^{(k)}\\ c^{(k)}\end{bmatrix} - \begin{bmatrix}e^{-b\cdot fw_{1}}&-fw_{1}\cdot a\cdot e^{-b \cdot fw_{1}}&1\\ e^{-b\cdot fw_{2}}&-fw_{2}\cdot a\cdot e^{-b\cdot fw_{2}}&1\\ e^{-b\cdot fw_{3}}&-fw_{3}\cdot a\cdot e^{-b\cdot fw_{3}}&1\end{bmatrix} ^{- 1} \begin{bmatrix}a\cdot e^{-b\cdot fw_{1}}+c-U_{S\max}(fw_{1})\\ a\cdot e^{-b\cdot fw_{2}}+c-U_{S\max}(fw_{2})\\ a\cdot e^{-b\cdot fw_{3}}+c-U_{S\max}(fw_{3})\end{bmatrix}  \tag{15}\]  
The \(k\) symbol represents the iteration of the calculation. We used Matlab for the calculation, but we did not obtain a result. A warning was issued: \"The Matrix is close to singular or badly scaled. Results may be inaccurate\". As is known for the Newton-Raphson method, the initial values should be close enough to the result, which we did not achieve despite the choice of different initial values. For this reason, it was decided to determine the coefficients using evolutionary methods. ABC [43; 44; 45; 46; 47; 48; 49; 50; 51; 52] was used. A new candidate is generated by (16) (foragers phase). \[v_{ij}=x_{ij}+\phi_{ij}\cdot\left(x_{ij}-x_{kj}\right) \tag{16}\]  
\(k\
eq i\) and \(\phi_{\bar{\eta}}\) is a random number between \(-1\) and \(1\). After the foragers phase, the onlooker phase is made, which represents the probability of selecting a solution, written in (17). \[P_{i}=\frac{f_{i}}{\sum\limits_{j=1}^{n}f_{j}} \tag{17}\]  
The last are the abandonment and scout phases. If a solution is not improved after the stopping criteria, they are abandoned and replaced. Lowercase letters \(v\) and \(x\) represent population members. Since different evolutionary methods are suitable for different problems, other methods have also been tested, such as differential evolution (DE) [19; 30; 31] and teaching-learning-based optimization (TLBO) [32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. For DE, three different strategies were used, which were DE/rand/1/exp, DE/rand/2/exp, and DE/best/1/bin. DE has two internal parameters. Based on our experiences using DE [69], the step size (F) was set at 0.6 and the crossover probability (CR) was set at 0.8. DE stars with variation, which is written in (18) for DE/rand/1/exp, in (19) for DE/rand/2/exp, and in (20) for DE/best/1/bin. \[v_{i}^{(g)}=x_{r_{1}}^{(g)}+F\cdot\left(x_{r_{2}}^{(g)}-x_{r_{3}}^{(g)}\right) \tag{18}\]  
\[v_{i}^{(g)}=x_{r_{1}}^{(g)}+F\cdot\left(x_{r_{2}}^{(g)}-x_{r_{3}}^{(g)}\right)+ F\cdot\left(x_{r_{4}}^{(g)}-x_{r_{5}}^{(g)}\right) \tag{19}\]  
\[v_{i}^{(g)}=x_{best}^{(g)}+F\cdot\left(x_{r_{2}}^{(g)}-x_{r_{3}}^{(g)}\right) \tag{20}\]Crossover follows variation, and it is made using Equation (21). \[u_{i}^{(g)}=\left\{ \begin{array}{ll}v_{i,j}^{(g)}&\text{if $r_{j}\leq CR$ or $j=j_{rand}$}\\ x_{i,j}^{(g)}&\text{otherwise}\end{array} \right. \tag{21}\]  
The last step is selection, written in (22), and a new population is obtained. \[x_{i}^{(g+1)}=\left\{ \begin{array}{ll}u_{i}^{(g)}&\text{if $f\!\left(u_{i}^{(g)} \right)\leq f\!\left(x_{i}^{(g)}\right)$}\\ x_{i}^{(g)}&\text{otherwise}\end{array} \right. \tag{22}\]  
Lowercase letters \(u\), \(v\), and \(x\) represent population members. TLBO implementation with no duplicate elimination phase was used, which means that the determination of a new population was made in two phases, which were the teaching and learning phases. The teacher phase is written in (23).