## 3 Mixed-Strategy Harris Hawk Optimization
### Gaussian Random Walk Strategy
### Quantitative Analysis for MSHHO

#### 4.3.1 Scalability Testing

In this section, MSHHO is compared with 11 other high-quality algorithms. To ensure accurate and stable results, all algorithms were run 30 times, and the mean values and standard deviations were recorded. The results are shown in Table 4.

In modern computer theory, merely considering the iterative process is insufficient when comparing different algorithms [56]. It is essential to analyze the statistical significance of the improvements. Therefore, the Friedman test is necessary. The Friedman test results for the algorithm comparison on the 12 functions F1 to F12 are displayed in Table 4.

The last three rows of Table 4 are labeled (W! T! L), Ave, and Ranking. (W! T! L) indicates the best, average, and worst performance, respectively. Ave represents the Friedman mean ranking, and Ranking shows the ranking values of all competitors. The results indicate that MSHHO achieves six wins and zero losses across the 12 functions. The Friedman mean ranking of MSHHO is 1.58, placing it first. Additionally, the convergence curves of all the algorithms are shown in Figure 10, demonstrating that MSHHO iterates until it finds an optimal solution. It highlights MSHHO's superior global exploration and local exploitation capabilities, confirming its effectiveness and advantages. \[ppc(t)=\frac{\sum_{i=1}^{N}X_{ij}(t)}{dim} \tag{26}\]0 Exploration and exploitation curves. \[ppc(t)=\frac{\sum_{i=1}^{N}X_{ij}(t)}{dim} \tag{26}\]1 _Cont._

In Figure 10, the convergence curves of C1, C3, C4, C5, C8, and C9 show that MSHHO has higher precision and converges to the global optimum faster than the other 10 algorithms. Due to the principle that positive charges repel each other during initialization, C6, C7, C10, C11, and C12 do not have the lowest initial function values in the early iteration. However, they eventually escape local optima in later iterations, achieving good convergence results. C2 converges the fastest initially but loses iterative superiority in the final phase, making MSHHO's precision slightly lower than that of the HHO algorithm. A comprehensive analysis of Table 4 and Figure 10 reveals that the MSHHO algorithm outperforms the other 10 algorithms in optimization performance. It demonstrates superior ability to escape local optima, higher convergence precision, and faster convergence speed. #### 4.3.2 Statistical Testing

In this section, the Wilcoxon rank sum test is used to compare the performances of the algorithms. The above analysis confirms MSHHO's robustness and effectiveness on CEC2005 functions. The significance level for the Wilcoxon rank sum test is set at 0.05. Table 5 shows that the difference between MSHHO and the other algorithms in 30 dimensions is substantial. This indicates a significant difference between MSHHO and other algorithms in most cases. Therefore, based on this analysis, MSHHO performs well, and its overall performance surpasses other algorithms. \[ppc(t)=\frac{\sum_{i=1}^{N}X_{ij}(t)}{dim} \tag{26}\]2 The convergence curves. ## 5 UAV Path Planning and Real-World Engineering Optimization Issues

### MSHHO for 3D Path Planning of UAV

In this paper, MSHHO is applied to three-dimensional path planning for UAVs to verify its practical effectiveness. The MSHHO algorithm's potential for solving current hot path planning challenges is demonstrated by simulating UAV flight paths in complex mountainous terrain models. #### 5.1.1 Cost Functions for UAV Path Planning

In UAV path planning, three main cost functions are used: flying distance cost function, flying altitude cost function, and flying risk cost function. Flying distance cost is a key indicator for UAV performance, primarily related to fuel consumption. Assuming the UAV maintains a constant speed during a mission, its fuel consumption is directly proportional to the total flying distance. The flying distance cost function is presented below.