## 3 Mixed-Strategy Harris Hawk Optimization
### Gaussian Random Walk Strategy
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Risk Area** & **Central Coordinate** & **Risk Radius** \\ \hline
1 & (10, 60) & 5 \\
2 & (40, 50) & 6 \\
3 & (60, 50) & 5 \\
4 & (85, 88) & 5 \\
5 & (100, 30) & 8 \\
6 & (125, 55) & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: 2D coordinate parameters of risk area. Figure 11: Three-dim threat topographic map. Figure 12: Two-dimensional path plan for HHO and MSHHO. Under the same test conditions, 30 independent simulation experiments were conducted for both HHO and MSHHO algorithms. The relevant statistical data are shown in Table 7. All competitors have significantly higher mean costs and standard deviations compared to the improved MSHHO. This indicates that MSHHO performs better in terms of optimization stability during route planning. ### Typical Engineering Applications

To confirm the universality and effectiveness of MSHHO, MSHHO was utilized to solve three engineering design issues: three-bar truss design, welded beam design, and tension/compression spring design. Their structure diagrams are shown in Figure 13. A comparison was made with seven algorithms--DE, WOA, CDO, OMA, AOA, GWO, and HHO--to verify the effectiveness of MSHHO. All the optimization algorithms were tested with the same common parameters: the maximum volume of iterations was 500, and the population size was fixed to 30. All the comparison algorithms in the experiment were independently run 30 times on each engineering design problem to reduce the deviation caused by random factors in the experiment. The extremum values of their optimization results were counted to \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ID** & **Minimum Cost** & **Maximum Cost** & **Mean Cost** & **Std** \\ \hline DE & 75.9856 & 78.8234 & 77.4617 & 1.06 \\ GA & 84.2317 & 86.6925 & 85.4881 & 0.92 \\ WOA & 73.5892 & 75.7243 & 74.6290 & 0.78 \\ CDO & 75.0881 & 77.0971 & 76.1465 & 0.79 \\ PSO & 72.7223 & 74.8324 & 73.7888 & 0.81 \\ OMA & 79.3685 & 81.5637 & 80.4180 & 0.85 \\ AOA & 72.2127 & 73.7554 & 73.1234 & 0.61 \\ DA & 100.1379 & 123.8629 & 111.5393 & 9.50 \\ GWO & 75.8564 & 77.6214 & 76.6927 & 0.70 \\ HHO & 72.6193 & 74.1397 & 73.3736 & 0.61 \\ MSHHO & 72.3854 & 73.7528 & 73.0708 & 0.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: UAV three-dimensional path planning statistics results. Figure 13: Three engineering optimization issues: (**a**) three-bar truss design; (**b**) welded beam design; (**c**) design of tension/compression spring. evaluate the universal applicability and effectiveness of the algorithms in engineering application issues. The weight of a three-bar truss directly impacts the cost of the structure and the convenience of transportation and installation. Optimal design can reduce the weight of the members, resulting in the lightest possible weight for the structure while meeting strength and stiffness requirements [57; 58]. The design variables in this truss are the cross-sectional area \(A_{1}\) for rod 1, the cross-sectional area \(A_{2}\) for rod 2, and the cross-sectional area \(A_{3}\) for rod 3, with \(A_{1}=A_{3}\). To minimize weight to constraints on stress, deflection, and buckling, it is necessary to manipulate the lengths of the two bars (\(A_{1}\) and \(A_{2}\)) [59]. The design goal of a welded beam is to minimize manufacturing costs while satisfying the constraints [60]. The design variables in this problem are weld thickness \(h1\), length \(l\), height \(t1\), and reinforcement thickness \(b1\). The constraints to be met include bearing load, shear stress, maximum bending stress, and maximum end deflection. For the tension/compression spring, the design goal is to minimize the mass of the spring to save production costs while satisfying the constraint conditions. The main design variables are spring wire diameter \(d1\), average spring coil diameter \(D1\), and the number of active spring coils \(N1\). The constraints to be met include minimum deflection, shear stress, natural frequency, and outer diameter limit.