# Dynamic Weight and Mapping Mutation Operation-Based Salp Swarm Algorithm for Global Optimization
## 3 Improved Salp Swarm Algorithm
### A Dynamic Weight for Leaders  
There is a control parameter \(c_{1}\) in SSA, which can effectively balance the transition from the global exploration to the local exploitation. Thus, \(c_{1}\) can also be called a transition parameter. Through the observation of Equation (4), the result shows that \(c_{1}\) adopts a nonlinear model, and it adaptively attenuates from the constant 2 to 0 with the increase of iterations. When the transition parameter is greater than 1, the algorithm performs the global exploration to find the desired search area; when it is less than 1, the algorithm starts to exploit the local area to obtain an accurate estimate. However, after the use of the mapping mutation operation for followers, we observed that the transition parameter \(c_{1}\) used in the classic SSA is not too suitable for the algorithm. Therefore, the fine-tuned transition parameter \(c_{1}\) is specifically expressed as:  
\[c_{1}=a\times 3^{-(\frac{b\times l}{max})^{2}} \tag{12}\]  
where \(a\) (\(default=2\)) and \(b\) (\(default=5\)) are constants. Besides, the algorithm that uses the mapping mutation operation and the fine-tuned transition parameter can be called IMOSSA. Even though SSA has improved, as mentioned above, a lot of problems still require more precise nonlinear changes to avoid the locally optimal solution. Therefore, this paper proposes to use a nonlinear dynamic weight to adaptively control the transition parameter \(c_{1}\), which is inspired by the inertia weight in the particle swarm optimization [29]. The dynamic weight is as follows:  
\[\omega=\omega_{max}-(\omega_{max}-\omega_{min})\times(\frac{l}{l_{max}})^{(1/l)} \tag{13}\]  
Figure 2: Distribution of random numbers produced by Tent chaotic with 200 iterations. where \(\omega_{max}\) and \(\omega_{min}\) are expressed as the upper bound and lower bound of the dynamic weight \(\omega\), respectively. Furthermore, \(\omega_{max}\) and \(\omega_{min}\) are constant values (\(\omega_{max}=1\), \(\omega_{min}=0.0001\)). Therefore, according to Equations (12) and (13), the improved position update of leaders is defined as:  
\[X_{i,j}= \begin{cases}F_{j}+\omega\cdot c_{1}((ub_{j}-lb_{j})c_{2}+lb_{j}),&c_{ 3}\geq 0.5\\ F_{j}-\omega\cdot c_{1}((ub_{j}-lb_{j})c_{2}+lb_{j}),&c_{3}<0.5\end{cases}  \tag{14}\]  
Based on the dynamic weight strategy, the algorithm has a larger dynamic weight at initial stage of the search and increases the global exploration capacity; at a later stage, the dynamic weight is smaller and heightens the local exploitation capacity of the algorithm. By precisely controlling the fine-tuned transition parameter \(c_{1}\) again, the improved algorithm not only improves the transition effect from the exploration to exploitation, but also accelerates the rate of convergence. The comparison of the fine-tuned transition parameter and the dynamic weight with the original transition parameter \(c_{1}\) is presented in Figure 3.  
Equation (12) and Figure 3 show that compared with the original control parameter, the fine-tuned transition parameter still presents a smooth transition from the exploration to exploitation as a whole, but the curve will transfer from the exploration to exploitation at a faster rate of convergence. In addition, according to Equation (13), the attenuation curve in Figure 3 shows that initially, the value of the dynamic weight is high and after a few iterations, its value decreases in a nonlinear tendency to a smaller value. The tendency of nonlinear transition is beneficial to promote the algorithm to transfer from the global exploration to local exploitation. The implementation of the improved algorithm called WMSSA ground on all above proposed tactics is presented in Figure 4.  
After adding nonlinear dynamic weights in the exploration phase and mapping variation operations in the development phase of the SSA algorithm, as shown in Figure 4, Formulas (12)-(14) updates the leader and Formulas (7) and (8) updates the follower until the last iteration of the algorithm is reached.