## 2 Methodology
### Algorithm Improvement
#### 2.3.5 Simplified Charge and Particle Resultant Force Calculations  
The formula for calculating the charge in the classical algorithm is complicated, especially when the difference between the maximum and the minimum value of the objective function is large, which significantly increases the amount of calculation and decreases the optimization efficiency of the algorithm. From the perspective of reducing the computational load, we define the standardized benchmark of charge quantity as: \(\tau=f(x_{worst})-f(x_{best})\), then, the improved charge calculation formula is  
\[Q_{i}=exp\Big{\{}-n*\frac{f(x_{i})-f(x_{best})}{\tau}\Big{\}},\forall i. \tag{9}\]  
For formula \(\frac{f(x_{i})-f(x_{best})}{\tau}\), when \(x_{i}=x_{worst}\), take the maximum value 1; when \(x_{i}=x_{best}\), take the minimum value 0, i.e., the value domain is [0, 1]. Compared with the standard formula, the amount of calculation is significantly reduced after the improvement. At the same time, the improved charge calculation formula still satisfies the relationship of \"the smaller the value of the objective function, the larger the charge carried by the particle\".  
In the particle resultant force formula of the standard algorithm, when the two particles are positioned infinitely close to each other, the denominator part of the resultant force formula tends to 0 infinitely, which leads to the overflow of the calculation result. Therefore, we remove the distance factor from the calculation formula, and the modified resultant force formula is  
\[F_{i}=\sum_{j\
eq i}^{m}\left\{\begin{array}{ll}(x_{j}-x_{i})Q_{i}Q_{j},f(x _{j})<f(x_{i})\\ (x_{i}-x_{j})Q_{i}Q_{j},f(x_{j})\geq f(x_{i})\end{array}\right.\forall i. \tag{10}\]  
When calculating the resultant force for each particle, the component forces exerted on it by the other (m \(-\) 1) particles need to be calculated. However, the calculation process of the component forces is relatively complex, which means that the population size m cannot be too large. Otherwise, it will lead to an excessively long calculation time. Except for the optimal particle \(x_{best}\), the resultant force needs to be calculated for all particles in the population. Therefore, in each iteration, there are \((m-1)^{2}\) calculations of component forces. To reduce the number of particles involved in the calculation of the resultant force and simplify the calculation process, the idea of the mutation strategy in the DE algorithm is introduced. The calculation of all particles in the resultant force calculation formula is modified to the calculation of three particles in the population, including two different particles randomly selected and the optimal particle. After this modification, the computational complexity has changed significantly, from the original \(O(m^{2}\times n)\) to \(O(m\times n)\); n is the dimension of the problem. This improves the computational performance to a certain extent, especially for the case of large-scale populations, which can effectively shorten the computation time and improve the execution efficiency of the algorithm. Figure 3 visualizes the improvement of this resultant force calculation process. Eventually, the formula for the calculation of the resultant force is simplified as  
\[F_{i}=\sum_{j=R_{1},R_{2},Best}\left\{\begin{array}{ll}(x_{j}-x_{i})Q_{i}Q_{j },f(x_{j})<f(x_{i})\\ (x_{i}-x_{j})Q_{i}Q_{j},f(x_{j})\geq f(x_{i})\end{array}\right.,\forall i. \tag{11}\]