## 3 Results
### Impact of Parameter \(\beta\)  
To balance the object number in each beam with a forceful method, we introduce parameter \(\beta\). This helps to determine the minimum number \(N_{BL}\) of objects in a single beam as shown in Equation (4). In Step 6 of the proposed algorithm, the smallest available objects \(N_{BL}\) in \(D_{j}\) are obligatorily allocated to cluster \(j\). In this subsection, the optimization results are illustrated when parameter \(\beta\) equals 0.2, 0.5 and 0.8, and the other variable \(\alpha\) remains at 0.5. Figure 9 illustrates the results of the beam allocation based on the modified K-means clustering when parameter \(\alpha\) equals 0.5 and \(\beta\) equals 0.2. Figure 9a shows the plot of the allocated 40 beams. The crossing indicates the centroid of each beam. The neighboring hexagonal cells in the same beam are marked with the same color. Figure 9b shows the statistics of the beam occupation. The number of objects in each beam ranges from 15 to 75 with the average number of 40.95 and standard deviation of 12.55. Figure 9c shows the statistics of the beam width. The half-cone angle of beam ranges from 1.2\({}^{\circ}\) to 3\({}^{\circ}\) with the average number of 1.95\({}^{\circ}\) and standard deviation of 0.46\({}^{\circ}\). Figure 8: Optimization results of the beam allocation based on the modified K-means clustering when parameter \(\alpha\) equals 0 and \(\beta\) equals 0: (**a**) Plot of the allocated 40 beams. The crossing indicates the geometrical center of each beam. The neighboring hexagonal cells in the same beam are marked with the same color; (**b**) Statistics of the beam occupation; (**c**) Statistics of the beam width. Figure 9: Optimization results of the beam allocation based on the modified K-means clustering when parameter \(\alpha\) equals 0.5 and \(\beta\) equals 0.2: (**a**) Plot of the allocated 40 beams. The crossing indicates the geometrical center of each beam. The neighboring hexagonal cells in the same beam are marked with the same color; (**b**) Statistics of the beam occupation; (**c**) Statistics of the beam width. Figure 10 illustrates the results of the beam allocation when parameter \(\alpha\) equals 0.5 and \(\beta\) equals 0.5. Figure 10a shows the plot of the allocated 40 beams. Figure 10b shows the statistics of the beam occupation. The number of objects in each beam ranges from 20 to 75 with the average number of 40.95 and standard deviation of 12.03. Figure 10c shows the statistics of the beam width. The half-cone angle of beam ranges from \(1^{\circ}\) to \(3.2^{\circ}\) with the average number of \(1.96^{\circ}\) and standard deviation of \(0.49^{\circ}\). Figure 11 illustrates the results of the beam allocation when parameter \(\alpha\) equals 0.5 and \(\beta\) equals 0.8. Figure 11a shows the plot of the allocated 40 beams. Figure 11b shows the statistics of the beam occupation. The number of objects in each beam ranges from 30 to 80 with the average number of 40.95 and standard deviation of 9.87. Figure 11c shows the statistics of the beam width. The half-cone angle of beam ranges from \(1.1^{\circ}\) to \(3.5^{\circ}\) with the average number of \(1.98^{\circ}\) and standard deviation of \(0.56^{\circ}\). All of the aforementioned results are summarized in Table 1 and are discussed in the next section. Figure 11: Optimization results of the beam allocation based on the modified K-means clustering when parameter \(\alpha\) equals 0.5 and \(\beta\) equals 0.8: (**a**) Plot of the allocated 40 beams. The crossing indicates the geometrical center of each beam. The neighboring hexagonal cells in the same beam are marked with the same color; (**b**) Statistics of the beam occupation; (**c**) Statistics of the beam width. Figure 10: Optimization results of the beam allocation based on the modified K-means clustering when parameter \(\alpha\) equals 0.5 and \(\beta\) equals 0.5: (**a**) Plot of the allocated 40 beams. The crossing indicates the geometrical center of each beam.