## 3 Genetic Algorithm with Maximum-Minimum Crossover (GA-MMC)
### Structure of Genetic Algorithm with Maximum-Minimum Crossover (GA-MMC)
#### 3.2.4 Evaluation  
The quality of each individual in the population is considered by its fitness value, \(f_{i}\), obtained from the objective function, which uses a weighting between optimization objectives. GA-MMC was then tested in two applications. In the first one, the objectives are to control the tilt angle of the main lobe and to maximize the RSLL, with the objective function given by Equation (5), in which \(A_{M}\) is the main lobe amplitude, \(A_{S}\) is the greater sidelobe amplitude, \(\theta_{C}\) is the calculated tilt angle, \(\theta_{0}\) is the desired tilt angle and \(K_{M}\) is the main lobe adjustment factor [20]. The values \(A_{M}\) and \(A_{S}\) are obtained from \(AF\) in Equation (4), considering \(\theta_{0}\) varying from 0 rad (0 ) until \(\pi\) rad (180 ) with resolution, \(\Delta\theta\), equal to 0.01 rad (0,573 ), in a total of 315 scanning angles. Due to the importance of the tilt angle in PARs, the objective function prioritizes \(\theta_{0}\). Hence, the term containing RSLL (\(A_{M}\)/\(A_{S}\)) just weighs the fitness value, while the \(\theta_{0}\) is inserted into a non-linear term, producing a fitness peak to the desired angle, \(\theta_{C}=\theta_{0}\), and a rapid reduction in fitness as it moves away from \(\theta_{0}\). \(K_{M}\) enables to control the influence of \(\theta_{0}\) in the fitness value, and the algorithm tests indicated \(K_{M}=1\):  
\[f_{i}=\frac{A_{M}}{A_{S}}\cdot\frac{I}{I+K_{M}\cdot\left(\theta_{C}-\theta_{0} \right)^{2}} \tag{5}\]  
In the second application of GA-MMC, besides controlling \(\theta_{0}\) and maximizing RSLL, the objective function aims at positioning nulls of the radiation pattern to reject two interfering signals (e.g., jammers), \(A_{11}\) and \(A_{12}\), according to Equation (6). The null value is given by the signal-to-interference ratio, \(SIR\), between main lobe (\(A_{M}\)) and interference (maximum value between \(A_{11}\) and \(A_{12}\)). Null position in the directions of interferences is essential, so that the calculated \(SIR\) should be greater than or equal to the desired value. Therefore, the term involving \(SIR\) in Equation (6) is nonlinear, so that: (i) it does not alter fitness, if \(A_{M}\)/max (\(A_{11}\), \(A_{12}\)) \(\geq SIR\), (ii) it reduces fitness by 75% if \(A_{M}\)/\(A_{II}\)\(\leq SIR\) or \(A_{M}\)/\(A_{12}\)\(\leq SIR\), and (iii) it reduces fitness in 93.75% if \(A_{M}\)/\(A_{11}\)\(\leq SIR\) and \(A_{M}\)/\(A_{12}\)\(\leq SIR\). Thus, \(\theta_{0}\) has priority, as in Equation (5), because one must have \(\theta_{C}=\theta_{0}\) to prevent fitness reduction. On the other hand, a \(SIR\) higher than desired does not alter the results, because, in current use, right positioning of the main beam is more important than positioning nulls towards the interferences direction.  
\[\begin{split} f_{i}=\frac{A_{M}}{A_{S}}\cdot\frac{I}{I+K_{M} \cdot\left(\theta_{T}-\theta_{0}\right)^{2}}\cdot\left(\frac{x}{abs(X)}^{-I} \right)\cdot\left(\frac{y}{abs(Y)}^{-I}\right)\\ X=\frac{A_{M}}{A_{1}}-SIR\quad\& Y=\frac{A_{M}}{A _{2}}-SIR\end{split} \tag{6}\]