## 3 Methodology
### Iajs-Hkrvm Model
#### 3.1.2 Improved Artificial Jellyfish Search Algorithm
In this paper, Fuch chaotic mapping is used to generate chaotic sequences, and then the chaotic sequences are mapped into the design parameter space to  
Figure 2: Diagram of artificial jellyfish search algorithm [35]. generate the initial jellyfish population. The Fuch chaotic mapping expression is shown in Equation (5). \[x(t+1)=\cos\Bigl{[}1/x(t)^{2}\Bigr{]} \tag{5}\]  
where \(x(t)\
eq 0\), \(x\in Z+\), \(t=1,2,\cdots,T\). In order to further illustrate the differences between the two chaotic mapping methods, the scatterplot and histogram of the two chaotic mapping sequences are compared, respectively. Figure 3 shows the histogram and scatter diagram of logistic chaotic mapping and Fuch chaotic mapping. As shown in Figure 3, variables generated by logistic chaotic mapping are distributed centrally in the edge region of the design space, while variables generated by Fuch chaotic mapping are distributed centrally in a small range and dispersed in a large range. Therefore, using Fuch chaotic mapping to generate the initial population can make the population quickly concentrate in the optimal solution region, reduce the number of algorithm iterations, and improve the optimization efficiency of the population. Lens-imaging reverse learning is a reverse-learning strategy inspired by the principle of convex lens imaging. Its scaling factor changes nonlinearly and dynamically with the number of iterations. The equations are as follows:  
\[x_{j}^{\prime*}=\frac{U_{bj}+L_{bj}}{2}+\frac{U_{bj}+L_{bj}}{2k}-\frac{x_{j}^ {*}}{k} \tag{6}\]  
\[k=k_{\max}-(k_{\max}-k_{\min})\cdot(t/T)^{2} \tag{7}\]  
where \(k\) is the scaling factor, \(U_{bj}\) and \(L_{bj}\) are, respectively, the upper and lower bounds of the \(j\)-dimensional components in the search space, \(x_{j}^{*}\) is the \(j\)-dimensional component of the jellyfish, \(x_{j}^{\prime*}\) is the \(j\)-dimensional component of the jellyfish by lens-imaging reverse learning, \(k_{\max}\) and \(k_{\min}\) are the maximum scaling factor and minimum scaling factor, respectively, and their values are 10 and 1, respectively. Figure 3: (**a**) Histogram of logistic chaotic mapping sequence; (**b**) scatter diagram of logistic chaotic mapping sequence; (**c**) histogram of Fuch chaotic mapping sequence; (**d**) scatter diagram of Fuch chaotic mapping sequence. In this paper, an adaptive hybrid mutation strategy is adopted to mutate the position of the optimal jellyfish obtained in each iteration, and its parameters are dynamically updated with the number of iterations to better balance the ability of the global exploration and local development of the algorithm. The equation for the adaptive hybrid mutation is shown in Equation (8). \[u_{best}^{t}=x_{best}^{t}(1+\lambda_{1}Cauchy(0,1)+\lambda_{2}Gauss(0,1)) \tag{8}\]  
where \(u_{best}^{t}\) is the location of the current optimal jellyfish after mutation, \(x_{best}^{t}\) is the location of the current optimal jellyfish, \(Cauchy(0,1)\) is the random variable that satisfies the Cauchy distribution, \(Gauss(0,1)\) is the random variable that satisfies the Gauss distribution, and \(\lambda_{1}=1-(t/T)^{2}\) and \(\lambda_{2}=(t/T)^{2}\) are the dynamic parameters that change adaptively with the number of iterations. In order to validate the performance of the improved artificial jellyfish search algorithm, four benchmark test functions are chosen to compare the optimization performance of the IAJS algorithm, the AJS algorithm, Grey wolf optimization algorithm (GWO), Whale optimization algorithm (WOA), and Harris hawks optimization algorithm (HHO). Among them, F1 and F2 are unimodal functions, F3 is a multimodal function, and F4 is a fixed-dimensional multimodal function. The basic parameter settings of each algorithm are shown in Table 1. The other parameters remain the same, the optimization dimension is 30, the population size is 30, and the maximum number of iterations is 500.