## 4 Numerical Results
### Linear Antenna Array Design
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Benchmark**} & \multicolumn{6}{c}{**Algorithm**} \\ \cline{2-7}  & **GWO** & **SSA** & **WOA** & **PSO** & **ABC** & **FA** \\ \hline Sphere & \(1.51\times 10^{-55}\) & \(4.65\times 10^{-08}\) & \(\mathbf{2.89\times 10^{-165}}\) & \(2.64\times 10^{-05}\) & \(1.42\times 10^{-07}\) & \(5.42\times 10^{-07}\) \\ \hline Rosenbrock & \(5.64\times 10^{+01}\) & \(5.79\times 10^{+01}\) & \(5.30\times 10^{+01}\) & \(\mathbf{5.17\times 10^{+01}}\) & \(5.85\times 10^{+01}\) & \(1.65\times 10^{+02}\) \\ \hline Ackley & \(2.54\times 10^{-14}\) & \(2.45\times 10^{+00}\) & \(\mathbf{3.84\times 10^{-15}}\) & \(6.23\times 10^{+00}\) & \(4.64\times 10^{-03}\) & \(2.07\times 10^{+00}\) \\ \hline Generalized Griewank & \(1.10\times 10^{-03}\) & \(5.62\times 10^{-03}\) & \(1.20\times 10^{-03}\) & \(1.73\times 10^{-01}\) & \(\mathbf{2.73\times 10^{-04}}\) & \(2.21\times 10^{-02}\) \\ \hline Weierstrass & \(1.71\times 10^{-15}\) & \(2.52\times 10^{+01}\) & \(\mathbf{0.00\times 10^{+00}}\) & \(2.56\times 10^{+01}\) & \(2.96\times 10^{-02}\) & \(2.23\times 10^{+01}\) \\ \hline Generalized Rastrigin & \(4.18\times 10^{-01}\) & \(6.43\times 10^{+01}\) & \(\mathbf{0.00\times 10^{+00}}\) & \(1.45\times 10^{+02}\) & \(1.59\times 10^{+01}\) & \(3.56\times 10^{+02}\) \\ \hline Noncont. & \(1.48\times 10^{+00}\) & \(9.94\times 10^{+01}\) & \(\mathbf{0.00\times 10^{+00}}\) & \(1.21\times 10^{+02}\) & \(1.88\times 10^{+01}\) & \(3.74\times 10^{+02}\) \\ \hline Schewfel & \(1.45\times 10^{+04}\) & \(1.03\times 10^{+04}\) & \(\mathbf{8.90\times 10^{+02}}\) & \(1.09\times 10^{+04}\) & \(4.68\times 10^{+03}\) & \(1.21\times 10^{+04}\) \\ \hline Generalized Penalized 1 & \(5.59\times 10^{-02}\) & \(4.98\times 10^{+00}\) & \(\mathbf{3.60\times 10^{-01}}\) & \(7.03\times 10^{-01}\) & \(4.42\times 10^{-03}\) & \(8.44\times 10^{+00}\) \\ \hline Generalized Penalized 2 & \(1.44\times 10^{+00}\) & \(2.04\times 10^{+01}\) & \(\mathbf{1.40\times 10^{-02}}\) & \(3.22\times 10^{-02}\) & \(7.41\times 10^{-02}\) & \(4.16\times 10^{+00}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical results (average values) of the emerging SI algorithms (GWO, SSA, WOA) and PSO, ABC, FA for several benchmark functions (\(D=60\), best values are indicated in bold). Taking into account all the above, let us consider an \(N\)-element linear array, where its elements are symmetrically placed along the \(x\)-axis (see Figure 5). Along the \(xz\) plane, the array factor can be expressed as  
\[AF(\vec{y},\vec{\phi},\vec{A},\theta)=\sum_{n=1}^{N}A_{n}e^{j\left(\frac{2\pi}{ \lambda}y_{n}\sin\theta+\phi_{n}\right)} \tag{8}\]  
where \(\lambda\) is the wavelength of the incident field, \(\theta\) is the steering angle that is computed along the \(z\)-axis (towards positive direction), \(\vec{y}\), \(\vec{A}\), and \(\vec{\phi}\) are the position, amplitude, and phase vectors of the linear array, and \(y_{n}\), \(A_{n}\), and \(\phi_{n}\) are the corresponding scalar quantities of the aforementioned vectors of the \(n\)-th element, accordingly. Assuming a symmetrical and uniformly excited array, the array factor in the \(xz\) plane can be rewritten as  
\[AF(\vec{y},\vec{\phi},\theta)=2\sum_{m=1}^{M}\cos\left[\frac{2\pi}{\lambda}y_ {m}\sin\theta+\phi_{m}\right] \tag{9}\]  
Equation (9) can be written in dB as  
\[AF_{\text{dB}}\big{(}\vec{y},\vec{\phi},\theta\big{)}=20\cdot\log_{10}\left| \frac{AF(\vec{y},\vec{\phi},\theta)}{AF(\vec{y},\vec{\phi},\theta_{o})}\right| \tag{10}\]  
where \(\theta_{o}\) is the direction of the maximum. The position of the elements in this array meets the following criteria  
\[y_{1}\geq\frac{d_{\text{min}}}{2},\quad y_{1}\leq\frac{d_{\text{max}}}{2} \tag{11}\]  
where \(y_{1}\) is the position of the first element, \(y_{m}\) is the position of the \(m\) element, \(y_{m+1}\) is the position of the \(m+1\) element, and \(d_{\text{min}}\), \(d_{\text{max}}\) are the minimum and maximum inter-element spacing, accordingly.