## 3 Experiment and Application
### Performance Analysis: Classic Benchmark Functions

Table 6: Best, worst,According to Tables 5 and 6, the effectiveness of the APO algorithm can be verified. In very few test functions, the results of APO are slightly inferior to other swarm intelligence algorithms, but still meet the discussion of the no free lunch theorem. In 70% of the test functions, APO performs significantly better than other algorithms. In addition, in the 22% function, the APO result is not significantly different from other functions. Therefore, the performance of APO is obviously superior to other algorithms, and the APO has a higher priority in the choice of algorithm when solving optimization problems. [PERSON] [69] reported that mutation occurs during particle movement in the initial iteration process, and this mutation is conducive to the extensive and comprehensive exploration of particles in the search space. As the number of iterations increases, the mutation should be reduced accordingly to be able to fully demonstrate the local development ability of the algorithm. Column 3 in Figure 3 lists the search history and track of the first particle in the first dimension. Mutation occurs in the initial stage of the iteration. As the iteration continues, the mutation gradually decreases. According to Reference [69], this behavior ensures that the anas platyrhynchos optimizer eventually converges to a point in the search space. Meanwhile, Column 2 in Figure 3 shows the search history of the particles. The particles of the anas platyrhynchos optimizer tend to search for a prospective area in the search space to develop a better area. In this problem, we set the population number to 50 and the maximum number of iterations to 100 and used different dimensions from those in Tables 2 and 3. Figure 3: _Cont._  
From a graphical point of view, Figures 4 and 5 present the partly converged curves abstained by the five algorithms with 30 independent runs. When Figures 4 and 5 are examined, APO converges rapidly for most test functions. Especially for the unimodal benchmark functions, APO has a faster convergence rate compared to the other algorithms. For the most multimodal functions, APO also presents the promising performance in convergence rate as shown in Figures 4 and 5. Figure 4: _Cont._  
Figure 3: Search history and track of the first particle in the first dimension (part). Figure 4: Evolution curves of the fitness value for some of the functions in Table 2.  
Figure 5: Evolution curves of the fitness value for some of the functions in Table 3.  
For detailed comparison, the Wilcoxon signed tanks test [70], in which [PERSON]'s procedure was used as a post-hoc procedure, was firstly used to make a pairwise comparison so as to illustrate the superiority of APO on the 23 test functions. The statistical results are summarized in Table 7. The \(p\)-values in Table 7 additionally present the superiority of the APO because most of the \(p\)-values are much smaller than 0.05. In summary, the results of this section experimentally prove that the APO shows very competitive results and outperforms other well-known algorithms on the classic benchmark functions, and we state that the APO has merit among the other five algorithms. The next section inspects the performance of the APO in solving the problem of multi-UAVs cooperative path planning, and multi-UAVs cooperative path planning has high-dimension, multi-constraint, and space-time cooperation characteristics which are a challenging problem.