# Dynamic Weight and Mapping Mutation Operation-Based Salp Swarm Algorithm for Global Optimization
## 3 Improved Salp Swarm Algorithm
### The Mapping Mutation Operation for Followers  
As shown in the SSA, followers use Equation (7) to update their current position, which shows a classic movement ground on the [PERSON]'s law. However, this kind of movement also leads to the single movement of followers and low population diversity. Therefore, a mapping mutation operation, which is adopted in the improved SSA called MOSSA, is given by:  
\[X_{i,j}^{l+1}=\begin{cases}X_{i,j}^{l}\times(1+G(\beta)),&c_{4}\geq 0.5\\ lb_{j}+\beta\times(ub_{j}-lb_{j}),&c_{4}<0.5\end{cases} \tag{8}\]  
where \(c_{4}\) is a stochastic number from the boundary [0, 1], Suppose the probability density of continuous random variable \(\beta\) is \(G(\beta)\), and it can be seen from the characteristics of normal distribution that Gaussian variation is also focused on searching a local region near the original individual. Formula (8) \(G(\beta)\) is a Gaussian mutation operation [27] and \(\beta\) is a stochastic number produced by the Tent chaotic map [28].  
In the MOSSA, the density function of the normal distribution, which is utilized to advance the variety of followers during the search course, is shown as follows:  
\[G(\beta)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(\beta-\mu)^{2}}{2\sigma^{2}}} \tag{9}\]  
where \(\mu\) (\(\mu\) = 0) is a mean and \(\sigma^{2}\) (\(\sigma^{2}\) = 1) is the variance of candidate measure. Hence, the Gaussian mutation operation used to obtain a new position of follower is calculated as follows:  
\[X_{i,j}^{l+1}=X_{i,j}^{l}\times(1+G(\beta)) \tag{10}\]  
Due to the traversal and nonrepeatability of the chaos, it can achieve a more effective search process than the random search. Thus, MOSSA uses the Tent chaotic mapping to generate a stochastic number, which can boost the search capacity of the algorithm. The particular expression is as follows:  
\[\beta_{k+1}=c_{5}\times\beta_{k}\times(1-\beta_{k}) \tag{11}\]  
where \(c_{5}\) is control parameter (default = 4) [26]. Figure 2 shows the random number distribution generated by Tent Chaos after 200 iterations. In the optimization algorithm, the more evenly the initialized population is distributed in the search space, the more beneficial it is to improve the optimization efficiency and solving accuracy of the algorithm. The chaotic sequence generated by Tent mapping has the following advantages.  
1. Strong randomness: Tent chaos has a high degree of randomness and unpredictability, and can generate sequences with good randomness. This randomness can help the swarm intelligence algorithm to better explore the solution space in the search process and avoid falling into the local optimal solution.
2. Wide area search ability: Tent Chaos has a wide area search ability, which can carry out more uniform and comprehensive search in the solution space. This helps the swarm intelligence algorithm to fully explore the solution space in the search process and improve the global search ability of the algorithm.
3. Easy to implement and calculate: The calculation of Tent Chaos is simple, as only a simple mathematical operation is required. This makes Tent Chaos easy to implement and calculate, and can be easily integrated into swarm intelligence algorithms.
4. Adjustability: The Tent Chaos function has one parameter, the chaos control parameter. By adjusting this parameter, you can adjust the degree of chaos and random property of Tent chaos, and then affect the search behavior of the algorithm. This adjustability makes Tent Chaos suitable for different types of optimization problems and algorithmic requirements.  
Combining the Gaussian mutation operation and the Tent chaotic map, when followers cannot produce a better solution, the MOSSA can generate a new solution. By making use of the strengths of the chaotic mapping and mutation operation, they promote the search capacity and enhance the diversity of the population for followers.