## 2 Model Establishment
### Fireworks Algorithm and Improved Fireworks Algorithm
The sparks and the mutation sparks are selected as the fireworks of next generation explosion. The unselected fireworks disappear. This paper assumes the total number of fireworks, explosions sparks, and mutation sparks is \(M\). The size of fireworks is \(N\). The spark with best fitness should be chosen as the next generation firework. Then, \(N-1\) sparks are chosen from the remaining \(M-1\) fireworks by using the roulette method. The probability of the candidate sparks chosen as the next explosion fireworks is as follows. \[p(x_{ik})=\frac{R(x_{i})}{\sum\limits_{x_{j}\in k}x_{j}} \tag{6}\]  
where \(R(x_{i})=\sum\limits_{x_{j}\in k}d(x_{i}-x_{j})=\sum\limits_{x_{j}\in k}\|x_{ i}-x_{j}\|\). When the selection strategy is used, the fewer sparks there are around is an individual, the greater the probability that the spark is selected as the next explosion fireworks. The explosion and selection processes are repeated until the termination conditions are met. The original mapping rules of the FWA algorithm cause the algorithm to converge slowly. In view of the deficiencies of the traditional fireworks algorithm, this paper proposes a new improved fireworks algorithm. The best position is at or near the origin. For the solution that is out of bounds in the FWA, according to the mapping rule of Equation (5), the value of one of the firework positions in one dimension is mapped to a position close to the origin. The optimal solution of optimization problems is at the origin, which can accelerate the convergence process of the algorithm to a certain extent. To avoid the illusion of the premature convergence, a new mapping rule is applied, which is expressed as follows. \[\hat{x}_{ik}=x_{LowB,\ k}+U(0,\ 1)(x_{UpB,\ k}-x_{LowB,\ k}) \tag{7}\]where \(U(0,\ 1)\) represents a random number of uniform distribution on the interval [0,1]. Gaussian mutation is applied to generate new sparks in the FWA to avoid the fireworks population of the explosion being too simple, which is shown in Equation (4). This Gaussian mutation has flaws. For example, when the \(e\) is close to 0, the position of the Gaussian mutation fireworks is close to 0. The optimal solutions are close to the origin in optimization problems, which may converge to the optimal point after a small number of iteration times. The traditional Gauss mutation method not only enlarges the diversity of fireworks, but also reduces the convergence speed of the algorithm. To solve this problem, the optimal fireworks explosion position is introduced into the improved fireworks, which makes the fireworks ensure the diversity and accelerate the algorithm convergence speed. A new Gaussian mutation spark \(\hat{x}_{ik}\) equation is proposed to solve this problem, which is expressed as follows. \[\hat{x}_{ik}=x_{ik}+(x_{Bk}-x_{ik})\times e \tag{8}\]  
where \(e\sim N(0,\ 1)\); \(N(0,\ 1)\) represents a Gaussian distribution whose mean value and variance is 0 and 1. \(x_{Bk}\) is the position of the fireworks with the best fitness of the current fireworks population in the \(k_{th}\) dimension. The traditional FWA controls the radius and the number of sparks generated from fireworks explosion using Equations (1) and (2). The better the original fireworks fitness is, the smaller the spark radius is. The poorer the fitness is, the larger the spark radius is. A better solution is obtained by expanding the search scope. The entire fireworks are treated variates during the fireworks explosion. Because of the randomness of the explosion, the dimension change in the fireworks may explode in the direction of deviation from the optimal solution, which may result that the local search cannot be completed. An improved method is proposed to enhance the local search. The specific process is as follows. (1) Ten sparks with better fitness are chosen. Suppose that each spark \(x_{i}\) is divided into \(p\)-dimensional variates \(c_{i1},\ c_{i2},\ \cdots,\ c_{ip}\). The mean value \(\bar{c}_{p}\) and standard deviation \(S_{p}\) of each dimension variate are calculated.