# Optimization Dubins Path of Multiple UAVs for Post-Earthquake Rapid-Assessment
## 3 HGSA Algorithm
### Mutation
### Numerical Experiments

The TOP can be regarded as a special case of MD-RDTOP-VP while meeting the following five conditions at the same time: (i) all UAVs are launched from the same depot and return to the same destination, which may be different from the launching depot; (ii) each potential target can be accessed at most once; (iii) the sensor detection error probability is 0; (iv) all UAVs are homogeneous; and (v) the UAV kinematic constraint is omitted, i.e., the turning radius of UAV is 0. Therefore, we can use the TOP benchmark instances which is downloaded from [[https://www.mech.kuleuven.be/en/cib/op](https://www.mech.kuleuven.be/en/cib/op)]([https://www.mech.kuleuven.be/en/cib/op](https://www.mech.kuleuven.be/en/cib/op)) to test the model and algorithm proposed in this paper. We select six sets of TOP benchmark instances [40] for two numerical experiments. In the first numerical experiment, we selected Set 6 with 64 vertices to show the impact of the UAV turning radius on the collected profit. To further demonstrate the impact of heading angle dispersion on the collected profit, numerical experiment 2 was performed with six sets of TOP benchmark instances. #### 4.1.1 Numerical Experiment 1

To evaluate the influence of the turning radius of UAV on the routing scheme, we select 10 benchmark instances with a routing number of \(k\) = 2 for Set 6. The number of vertices of these benchmark instances is \(N\) = 64, and the vertex positions and weights are shown in Figure 6.

Since the distance between any two nodes is assumed to be greater than \(2\rho\) in Section 2.3, where \(\rho\) is the minimum turning radius of the UAV. Since the minimum distance between any two vertices in the above figure is \(\sqrt{2}\), we select the typical turning radii \(\rho\in\{0,\ 0.1,\ 0.3,\ 0.5,\ 0.7\}\) to perform numerical experiments on the above data set, where \(\rho=0\) is a solution of the traditional TOP. We set the heading angle discretization \(N_{\theta}=8\) in other cases of turning radii. The results for the proposed method are presented in Table 3; \(R_{BEST}\) represents the known traditional TOP optimal solutions of these 10 instances. Table 3 records the best results for 10 instances of Set 6, which was run 10 times under the same experimental conditions. The experimental results in Table 3 show that the HGSA algorithm can find the known optimal solution of the 10 TOP benchmark instances when \(\rho=0\), which demonstrates that the HGSA algorithm which is specially designed for the MD-RDTOP-VP model in this paper can also obtain the optimal solution of the classic TOP benchmark instances. We also find that for the turning radius \(\rho=0.1\), the optimal solution of all 14 instances is as same as that for \(\rho=0\). For other turning radius the collected profit decreases gradually with increasing \(\rho\). Figure 7 shows the routing schemes of maximum profit under two different turning radii of the p6.2.e instance. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multirow{2}{*}{\(N\)} & \multirow{2}{*}{\(K\)} & \multirow{2}{*}{\(T_{MAX}\)} & \multirow{2}{*}{\(R_{BEST}\)} & \multicolumn{6}{c}{**HGSA**} \\ \cline{5-10}  & & & & & & \(\rho\) = 0 & \(\rho\) = 0.1 & \(\rho\) = 0.3 & \(\rho\) = 0.5 & \(\rho\) = 0.7 \\ \hline p6.2.a & 64 & 2 & 7.5 & 0 & 0 & 0 & 0 & 0 & 0 \\ p6.2.b & 64 & 2 & 10 & 0 & 0 & 0 & 0 & 0 & 0 \\ p6.2.c & 64 & 2 & 12.5 & 0 & 0 & 0 & 0 & 0 & 0 \\ p6.2.d & 64 & 2 & 15 & 192 & 192 & 192 & 192 & 192 & 192 \\ p6.2.e & 64 & 2 & 17.5 & 360 & 360 & 360 & 360 & 360 & 354 \\ p6.2.f & 64 & 2 & 20 & 588 & 588 & 588 & 552 & 540 & 504 \\ p6.2.g & 64 & 2 & 22.5 & 660 & 660 & 660 & 660 & 594 & 522 \\ p6.2.h & 64 & 2 & 25 & 780 & 780 & 780 & 780 & 744 & 672 \\ p6.2.i & 64 & 2 & 27.5 & 888 & 888 & 888 & 840 & 840 & 840 \\ p6.2.j & 64 & 2 & 30 & 948 & 948 & 948 & 936 & 936 & 876 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Best results for the 10 instances of Set 6

Figure 6: Vertex position and weight diagram of the TOP benchmark instance Set 6.