# Multi-Strategy-Improvement-Based Slime Mould Algorithm
## 4 Experimentation and Application
### Collaborative Multi-UAV Path Planning
\begin{aligned} & 0&&,\ d_{R}>d_{R_{max}}\\ &\frac{1}{d_{R}^{4}}&&,\ d_{R_{min}}\leq d_{R}\leq\ d_{R_{max}}\\ & 1&&,d_{R}<d_{R_{min}}\end{aligned} \right. \tag{26}\]  
Here, \(d_{R}\) is the distance between the UAV and the radar, and the fractions \(D_{Rmax}\) and \(D_{Rmin}\) are the maximum detection radius and effective detection radius of the radar. \[P(d_{M})=\left\{ \begin{aligned} & 0&&,\ d_{M}>d_{M_{max}}\\ &\frac{1}{d_{M}}&&,\ d_{M_{min}}\leq d_{M}\leq\ d_{M_{ max}}\\ & 1&&,d_{M}<d_{M_{min}}\end{aligned} \right. \tag{27}\]Here, \(d_{M}\) is the distance between the UAV and the missile, and \(D_{Mmax}\) and \(D_{Mmin}\) are the maximum kill radius and effective kill radius of the missile. \[P(d_{A})=\left\{ \begin{aligned} 0&&,\ d_{A}>d_{A_{ max}}\\ \frac{1}{d_{A}}&&,\ d_{A_{min}}\leq d_{A}\leq\ d_{A_{ max}}\\ 1&&,\ d_{A}<d_{A_{min}}\end{aligned} \right. \tag{28}\]  
Here, \(d_{A}\) is the distance between the UAV and the anti-aircraft gun, and the ratios of \(D_{Amax}\) and \(D_{Amin}\) are the maximum kill radius and effective kill radius of the anti-aircraft gun. \[P(d_{C})=\left\{ \begin{aligned} 0&&,\ d_{C}>d_{C_{ max}}\\ \frac{1}{d_{C}}&&,\ d_{C_{min}}\leq d_{C}\leq\ d_{C_{ max}}\\ 1&&,\ d_{C}<d_{C_{min}}\end{aligned} \right. \tag{29}\]  
Here, \(d_{C}\) is the distance between the UAV and the atmosphere, and the ratios of \(D_{Cmax}\) and \(D_{Cmin}\) are the maximum kill radius and effective kill radius of the atmosphere. Then, the threat function\(f_{B}\) is determined as shown in Equation (30). \[f_{B}=r\times P(d_{R})+m\times P(d_{M})+a\times P(d_{A})+c\times P(d_{C}) \tag{30}\]  
In the formula, \(r\), \(m\), \(a\), and \(c\) are the influence factors of the radar, missile, artillery, and atmosphere on the threat probability of the UAV, respectively. [PERSON] et al. set up a method of defining the time co-constraints with fewer steps and less computation [33], i.e., they set up the command time for the UAV to reach the end point and the time range to solve for its arrival at the target end point and determine the time cost function by judging the relationship between the two as shown in Equation (31). Assuming that the command time for the UAV to reach the target endpoint is \(tc\), the shortest time \(t^{i}_{min}\) and the longest time \(t^{i}_{max}\) for the UAV to reach the target endpoint are derived based on its speed range and trajectory length. \[f(t_{i})=\left\{ \begin{aligned} 0&&,\ t^{i}_{min}\leq\ t_{c}\leq\ t^{i}_{max}\\ |t_{i}-t_{c}|&&,\ t^{i}_{min}>t_{c}\ \ or\ \ t_{c}>t^{i}_{max} \end{aligned} \right. \tag{31}\]  
Here, \(t_{i}\) is the actual time for the ith UAV to reach the target endpoint. Summing the time cost function yields the total time cost function \(f_{T}\) as shown in Equation (32). \[f_{T}=\sum_{i=1}^{n}f(t_{i}) \tag{32}\]  
Based on the above, the following evaluation function for path planning is derived:  
\[f4=\omega_{1}\times f_{L}+\omega_{2}\times f_{H}+\omega_{3}\times f_{B}+\omega _{4}\times f_{T} \tag{33}\]  
Here, \(w1\), \(w2\), \(w3\), and \(w4\) are the weights of the functions \(f_{L}\), \(f_{H}\), \(f_{B}\), and \(f_{T}\), respectively, and satisfy the constraint \(\omega 1+\omega 2+\omega 3+\omega 4=1\). Figure 5: Threat areas and pathways planned by MSSMA. illustrates the established threat areas and the path planned by MSSMA. In order to test the performance of the MSSMA on collaborative multi-UAV path planning applications, the Whale Optimization Algorithm (WOA), K-mean Optimization Algorithm (KO), Beluga Optimization Algorithm (BWO), Ivy Optimization Algorithm (IVY), and SMA and MSSMA collaborative multi-UAV path planning algorithms were used. To ensure the fairness of the test, the number of populations of all algorithms was 50,the maximum number of iterations was 100, and each algorithm was tested independently 30 times, and the minimum, mean, and variance of each method were calculated, and the results are shown in Table 4. The minimum cost, mean cost, and variance of all the algorithms were significantly higher compared to those of the MSSMA.