## 4 Results and Discussion
### First Application: Tilt and Sidelodes Control
#### 4.2.2 Analysis of the Simulations  
The results of 10 simulations for each desired tilt angle, \(\theta_{0}\), are given in Table 1, which contains the number of tilt errors, \(\mathrm{NE}_{\theta}\), the average value of the calculated tilt angles, \(\theta_{\mathrm{C,Ave}}\), the number of RSLL errors, \(\mathrm{NE}_{\mathrm{RSLL}}\), the average value of RSLL, \(\mathrm{RSLL}_{\mathrm{Ave}}\), and the maximum value of RSLL, \(\mathrm{RSLL}_{\mathrm{Max}}\)[13]. The test results were satisfactory, since all simulations converged properly (according to Section 3.2.9), showing no tilt errors (\(\mathrm{NE}_{\theta}=0\%\) in all cases) and the desired RSLL was achieved in almost all simulations. Thus, the desirable operating conditions were met in more than 90% of the tests, far greater than the minimum established (70% according to Section 4.1), validating the GA-MMC effectiveness. This indicates an excellent result for a probabilistic method. On the other hand, for the region of interest of a PAR with four faces (Figure 7b), \(\theta_{0}\geq 45\)\({}^{\circ}\), the results would be even better, meeting the operating conditions in more than 97% of simulations of the tests total. The RSLL has high values (\(\mathrm{RSLL}_{\mathrm{Max}}\) between 30.2 and 34.7 dB), with little difference between \(\mathrm{RSLL}_{\mathrm{Ave}}\) and \(\mathrm{RSLL}_{\mathrm{Max}}\), indicating stability in results. As expected, the largest number of errors occurred at RSLL with low values of \(\theta_{0}\) (10 \({}^{\circ}\)and 30 \({}^{\circ}\)), near endfire condition, since it is very difficult to control the radiation pattern under such circumstance. In these cases, values for RSLL could be higher, but the objective function prioritizes \(\theta_{0}\), due to the importance of tilt angle in PARs. This reduces the chance of getting even higher RSLL values, but it ensures achieving the operating requirements throughout the E-plane. Table 2 contains the values of \(A_{n}\) and \(\beta\) for individuals with best fitness obtained in each test simulations [13]. As each individual in the population contains only one side of PA (\(M=20\) elements), due to the amplitude symmetrical distribution, one notes that the value of \(A_{n}\) is higher in the core elements, \(A_{1}\) to \(A_{10}\), and the absolute value of \(\beta\) increases as \(\theta_{0}\) decreases. Thus, GA-MMC reduced the value of \(A_{n}\) in the side elements, in order to raise RSLL, while increased \(\beta\) to decrease \(\theta_{0}\) (90 \({}^{\circ}\)-10 \({}^{\circ}\)). This was expected since \(\beta\) usually controls the main lobe tilt angle and \(A_{n}\) controls the sidelobes level [7,13]. To illustrate, Figure 13 contains the polar and Cartesian forms in the E-plane of the radiation pattern optimized by GA-MMC for \(\theta_{0}=45\)\({}^{\circ}\)in Table 2.  
In order to validate the optimization performed by GA-MMC, the simulation results are compared with data from references. In the first reference, there is a linear array of 40 isotropic elements symmetrically distributed, aiming at optimizing RSLL and \(\theta_{0}\), obtaining RSLL of 32.9 dB to \(\theta_{0}=60\)\({}^{\circ}\). Now, there is a GA with 200 individuals and binary coding of eight levels to \(A_{n}\) and 216 levels to \(\beta\)[20]. In the second reference, there is a linear array of 40 isotropic elements symmetrically distributed, whose optimization objective is to increase RSLL, achieving results on the order of 20 dB to \(\theta_{0}=90\)\({}^{\circ}\). This setting used a GA with 100 individuals, binary coding and two levels for \(A_{  
Even though each of those references has their own peculiarities [20, 22], their results confirm the success of the proposed optimization method. As comparison, GA-MMC reached RSLL = 34.7 dB, to \(\theta_{0}=90\)\({}^{\circ}\), and RSLL = 32.6 dB, to \(\theta_{0}=60\)\({}^{\circ}\)(Table 1), values that are comparable or even larger than those cited in the references.