## 5 Case Study and Algorithm Validation
### Effects of Iterations
It seems that it jumps around a possible solution but will never reach it. Actually, GA is suitable to solve the discrete problems instead of the continuous ones, such as path planning considering the ground risk. Especially for this  
Figure 18: Effects of iterations on path risk. kind of multi-objective optimization problem in a given solution space, the performances of GA will be quite poor and cannot satisfy all the constraint conditions. The path lengths of these different algorithms with the effects of the iterations are presented in Figure 20. It is obvious that the proposed BSO method is the best, with the shortest path length under all kinds of iteration settings. Also, its rate of convergence is the fastest, and it can complete around 30 iterations. PSO is similar, but the rate of convergence is lower than the proposed one. Generally, 70 iterations are needed for PSO based on the data provided in Figure 20. The final normalized path length of PSO is longer than that of the proposed BSO method under all the conditions. Standard BAS is worse than the aforementioned two but better than GA and FA. A lot of ups and downs exist in the beginning, and after that, it becomes stable until the convergence, but with a much longer path in the final iteration. FA and GA are the worst, with the longest paths. For FA, once it starts, the path length would remain the same without any changes. It proves that FA cannot locate the optimum solutions, and the first generated one will be seen as the global best, which is then kept in the whole optimization process. As a matter of fact, it is far away from the true optimum values. GA will fluctuate all the time, trying to obtain the global optimum solutions. However, because of the existence of the strong randomness, it cannot reach but only moves around it. As presented in Figure 20, the normalized path lengths undulate from iteration to iteration for GA. The final obtained waypoints may also not be the global best. Compared with PSO, GA, FA, and standard BAS, the proposed BSO is 18.59%, 59.66%, 58.30%, and 32.64% shorter in the normalized path length, respectively. Figure 19: Effects of iterations on turning angles. The overall path cost of the five algorithms can be found in Figure 21. There is no doubt that more iterations would make both of the proposed BSO and PSO algorithms become more smooth. That is because by evolving more generations, the individuals in a swarm would move closer and closer to the true optimum solutions in the search space. However, the total cost of the path generated by PSO would be higher than that of the proposed BSO method under all the conditions in Figure 21. Also, more iterations can smooth the curves of standard BAS because, by going through more iterations, the beetle would move more steps towards the best position. Correspondingly, the path cost would become more stable. But the values are higher than the proposed BSO method as well as PSO. As shown in Figure 21, GA and FA are the worst of the five. Ultimately, the total path costs of the two are the highest out of all options. GA jumps all the time from the first iteration to the last one, no matter how many iterations there are. It always fails to locate an optimum solution in this scenario. From the curves, we can see that GA would fluctuate a lot around a certain value but will never reach it. That is because GA belongs to a kind of discrete algorithm in that the cost values would be separated during two consecutive iterations. It will be more obvious when there are more iterations, as shown in Figure 21c,d. From the normalized path cost given in the figure, it is a straight line for FA, which is because a fake optimum value is found and is then used as the global best. The local best is the problem that cannot be overcome by FA when it is trying to locate an optimum solution in the search space. Even though there are 350 iterations for FA, as shown in Figure 21d, the path cost of FA still remains the same in the end, which is determined by itself.