## 3 Proposal and Verification of Improved LoTFWA
### Classic Algorithm of LoTFWA  
The LoTFWA has improved the equation for generating the number of explosive sparks in the FWA. The power distribution equation is used to generate the number of explosive sparks, and its generation equation is shown in Equation (1).  
\[\lambda_{i}=\hat{\lambda}.\frac{r_{i}^{-\alpha}}{\sum_{i=1}^{m}r_{i}^{- \alpha}}, \tag{1}\]  
where \(r_{i}\) is the fitness value ranking of the \(i\)th fireworks; \(m\) is the total number of fireworks; \(\alpha\) is the parameters of explosive spark distribution shape, and a better explosion effect is obtained when \(\alpha\) is larger; and \(\hat{\lambda}\) is a parameter that controls the number of sparks. In previous improved FWAs, only the best explosion amplitude of fireworks was dynamically controlled. However, the dynamic control is added to the explosion amplitude of each firework in the LoTFWA, and its explosion amplitude improvement publicity is shown in Equation (2).  
\[A_{i}^{g}=\begin{cases}A_{i}^{1}&g=1\\ C_{r}A_{i}^{g-1}&f\left(x_{i}^{g}\right)\geq f\left(x_{i}^{g-1}\right)\\ C_{\alpha}A_{i}^{g-1}&f\left(x_{i}^{g}\right)<f\left(x_{i}^{g-1}\right)\end{cases} \tag{2}\]  
where \(A_{i}^{g}\) is the explosion amplitude of the \(i\)th fireworks in the g generation and \(x_{i}^{g}\) is the position of generation g of the \(i\)th fireworks. When g generation fireworks are better than generation \(g-1\) fireworks, multiply by the amplification factor \(C_{\alpha}>1\) (\(C_{\alpha}=1.2\)); otherwise, multiply by the reduction factor \(C_{r}\) (\(C_{r}=0.9\)) so that the best fireworks enter the next generation.  
The explosion spark generation equation of the LoTFWA is shown in Equation (3).  
\[S_{ij}^{(k)}\gets X_{i}^{k}+\eta\cdot A_{i}, \tag{3}\]  
where \(X_{i}^{k}\) is the position of the \(i\)th fireworks \(X_{i}\) in the \(k\)th dimension of the explosion space; \(S_{ij}^{(k)}\) is the position of the \(j\)th explosion spark generated by fireworks \(X_{i}\) in the \(k\)th dimension; Ai is the explosion amplitude of the \(i\)th fireworks; and \(\eta\) is a uniformly distributed random number in \((-1,1)\).  
When the explosion sparks and Gaussian variation sparks generated in the explosion operation exceed the boundary of the feasible region, the explosion sparks and Gaussian sparks are mapped into the feasible region by mapping rules. The equation is shown in Equation (4).  
\[\overline{X}_{i}^{k}=LB+rand*(UB-LB), \tag{4}\]  
where \(\overline{X}_{i}^{k}\) is the new location of the explosion space; \(UB\) and \(LB\) are the upper and lower boundaries of the explosion space; and rand means to generate a random number from 0 to 1.  
The traditional firework selection framework is to add the current fireworks, the generated explosion sparks and the Gaussian variation sparks into a candidate pool from which n solutions are selected as the next generation of fireworks. The LoTFWA uses the independent selection fireworks framework; that is, each firework has an independent candidate pool, and the next-generation fireworks are also selected from their respective candidate pools. The LoTFWA adopts a combination of greedy selection and elite selectionmechanisms, and selects m fireworks from the current generation of fireworks, explosive sparks, and Gaussian variation sparks as the next generation of fireworks.