# An Improved Pied Kingfisher Optimizer for Maritime UAV Path Planning
## 3 Performance Validation of IPKO  
In order to rigorously evaluate the performance of the IPKO algorithm, a meticulous selection of 23 benchmark test functions was conducted to assess both the accuracy and stability of its outcomes. The effectiveness of the improved algorithm was verified from multiple perspectives through comparisons with other state-of-the-art optimization algorithms, including Grey Wolf Optimizer (GWO), Snake Optimization (SO) [27], Whale Optimization Algorithm (WOA) [28], Dingo Optimization Algorithm (DOA) [29], and Particle Swarm Optimization (PSO). The parameter settings for each algorithm are shown in Table 1. To ensure a fair comparison among these algorithms, the initial population size for each was uniformly set to 30, while the maximum number of iterations was established at 1000. The algorithms were systematically evaluated using three distinct metrics derived from the benchmark test functions: the optimal value, mean value, and standard deviation. The optimal values are highlighted in bold for emphasis. Furthermore, the effectiveness of the algorithms was validated using statistical tests including the Wilcoxon rank-sum test and Friedman test, with a significance level set at 5%. In this context, P represents the value obtained from comparing IPKO with each algorithm using the Wilcoxon test; NaN indicates a high similarity between the algorithms. The comprehensive ranking of each algorithm subsequent to the Friedman test is represented by their respective ranks. Detailed information about the 23 benchmark functions can be found in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameter** \\ \hline DOA & \(NA\in[2,\,M/2]\) \\ GWO & A was linearly decreased from 2 to 0 \\ WOA & \(\alpha\in[0,\,2]\), \(a_{2}\in[-2,\,-1]\), \(b=1\), \(l=[-1,\,1]\) \\ SCA & \(r_{1}=2.(1-t/T)\), \(r_{2}\in[0,\,2\pi]\), \(r_{3}\in[0,\,2]\), \(r_{4}\in[0,\,1]\) \\ SO & \(C_{1}=0.5\) \\ PKO & \(PE_{\max}=0.5\), \(PE_{\min}=0\) \\ IPKO & \(PE_{\max}=0.5\), \(PE_{\min}=0\) \\ PSO & \(c_{1}=0.5\), \(c_{2}=1.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameter settings for each algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Expression** & **Dim** & **Range** & **Min** \\ \hline \(f_{3}(x)=\sum_{i=1}^{n}\left\{\sum_{i=1}^{4}y_{i}^{2}\right\}^{2}\) & 30 & \([-100,\,100]\) & 0 \\ \(f_{3}(x)=\sum_{i=1}^{n}[100(x_{i+1}-x_{i})^{2}+(x_{i}-1)^{2}]\) & 30 & \([-30,\,30]\) & 0 \\ \(f_{4}(x)=\sum_{i=1}^{n}[x_{i}^{2}-100\cos(2\pi x_{i})^{-10}]\) & 30 & \([-512,\,512]\) & 0 \\ \(f_{11}(x)=\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-1\sum_{i=1}^{n}\cos(\frac{x_{ i}}{\sqrt{t}})+1\) & 30 & \([-600,\,600]\) & 0 \\ \(f_{30}(x)=\begin{array}[t]{c}0.1\left(\sin^{2}(3\pi x_{1})+\sum_{i=1}^{n}(x_{ i}-1)^{2}\left[1+\sin^{2}(3\pi x_{i}+1)\right]\right.\\ +(x_{i}-1)^{2}\left[1+\sin^{2}(2\pi x_{i})\right]+\sum_{i=1}^{n}(x_{i},5,100,4 )\end{array}\) & 30 & \([-50,\,50]\) & 0 \\ \(f_{4}(x)=\left[\frac{1}{900}+\sum_{i=1}^{5}\left[1-\frac{x_{i}}{1-x_{i}} \right]^{2}\right.\\ \left.f_{53}(x)=\sum_{i=1}^{51}\left[a_{i}-\frac{x_{i}(b_{i}^{2}+\pi x_{i})}{b _{i}^{2}+b\pi x_{i}}\right]\right.\\ f_{54}(x)=\begin{array}[t]{c}f_{72}(x)=(x_{72}-\frac{51}{34}x_{72}^{2}+5x_{72}- 6)^{2}+1001-\frac{x_{72}}{34}\cos x_{1}+10\\ \times[30+(2x_{1}-3x_{2})^{2}\times(18-32x_{1}+12x_{1}^{2}+48x_{2}-36x_{12}+2 \pi x_{2}^{2})]\\ f_{23}(x)=-\sum_{i=1}^{51}\left((X-a_{i})(X-a_{i}^{T})^{-t}\right)^{-t}\end{array}\) & 4 & \([0,\,10]\) & \(-10.5363\)  
\end{table} 
Table 2: Partial benchmark test function formulas. As illustrated by the experimental data presented in Table 3, IPKO outperforms most functions in the CEC2005 test [30], particularly excelling in f1-f4, f6-f14, f16-f18, and f21-f23. IPKO ranks first in all three metrics with the minimum values achieved. Although some other algorithms may have a slight advantage over IPKO in certain functions, they do not exhibit the same level of consistent optimization capability.