## 3 Mixed-Strategy Harris Hawk Optimization
### Gaussian Random Walk Strategy
The design objective of the tension/compression spring is to minimize its mass, thereby reducing production costs. The primary design variables are the spring wire diameter \(d1\), the average spring coil diameter \(D1\), and the number of active spring coils \(N1\). The design must also adhere to constraints including minimum deflection, shear stress, natural frequency, and an outer diameter limit. According to the experimental results in Tables 8-10, the MSHHO method's optimization result ranks first and is the closest to the engineering problem's ideal value. This suggests that the MSHHO algorithm can offer an excellent solution for the three-bar truss, the welded beam, and the tension/compression spring. The simulation results show that MSHHO is not only better than standard HHO but also better than other commonly used MAs, providing a good solution for engineering application issues. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ID** & \(\mathbf{A1}\) & \(\mathbf{A2}\) & **Extremum** & **Rank** \\ \hline DE & 0.7886 & 0.4084 & 264.0640 & 5 \\ WOA & 0.7738 & 0.4520 & 264.0668 & 6 \\ CDO & 0.7812 & 0.4300 & 263.9385 & 4 \\ OMA & 0.7891 & 0.4071 & 264.5564 & 8 \\ AOA & 0.8051 & 0.3683 & 263.9096 & 3 \\ GWO & 0.7857 & 0.4166 & 263.9026 & 2 \\ HHO & 0.7691 & 0.4667 & 264.1986 & 7 \\ MSHHO & 0.8040 & 0.3681 & 263.8960 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Extremum solutions for the three-bar truss design. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ID** & \(\mathbf{h1}\) & \(\mathbf{l}\) & \(\mathbf{t1}\) & \(\mathbf{b1}\) & **Extremum** & **Rank** \\ \hline DE & 0.1944 & 3.5110 & 8.9846 & 0.2138 & 1.7203 & 4 \\ WOA & 0.1275 & 9.7770 & 9.5890 & 0.1971 & 2.3371 & 8 \\ CDO & 0.1847 & 4.1371 & 9.5663 & 0.1997 & 1.7135 & 3 \\ OMA & 0.1916 & 3.6981 & 8.8353 & 0.2152 & 1.7690 & 5 \\ AOA & 0.1250 & 6.3618 & 10.0000 & 0.1975 & 2.0441 & 7 \\ GWO & 0.1953 & 3.4111 & 9.1985 & 0.1988 & 1.6757 & 2 \\ HHO & 0.1284 & 5.6418 & 9.2286 & 0.2024 & 1.8679 & 6 \\ MSHHO & 0.1988 & 3.3374 & 9.1920 & 0.1988 & 1.6702 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Extremum solutions for the welded beam design. ### Discussion

MSHHO has proven its application potential across multiple fields through a series of experiments. On the CEC2005 benchmark functions, MSHHO's convergence curves show balanced dynamic performance: quick exploration in early iterations and gradual convergence towards the global optimum. This efficient mechanism is primarily attributed to the following: (1) mutual benefit and symbiosis strategies; and (2) Gaussian random walk strategy. In three engineering application tests, MSHHO achieved optimal or near-optimal results, demonstrating its effectiveness and reliability. Compared to HHO, MSHHO plans shorter, smoother flight paths and avoids risky areas, promising broad application prospects in UAVs. ## 6 Conclusions

As a novel swarm intelligence approach, HHO has garnered significant attention for its ability to tackle complex optimization challenges by emulating the predatory behavior of Harris hawks. However, its effectiveness is limited by an imbalance between exploration and exploitation, often leading to suboptimal solutions in the face of complex, multi-peak optimization landscapes. To overcome these limitations, MSHHO was introduced. MSHHO enhances the convergence capability of HHO through innovative strategies such as the positive charge repulsion strategy for initializing the hawk population and the incorporation of a nonlinear decreasing operator. These enhancements promote population diversity and improve the algorithm's ability to explore diverse solution spaces effectively. Moreover, MSHHO integrates Gaussian random walk and mutually beneficial position update strategies to further enhance its robustness in escaping local optima and finding global optima.