## 2 Modeling Theory and Methods  
### Chaos-Enhanced Accelerated Particle Swarm Optimization Algorithm  
Particle swarm optimization (PSO) was proposed by [PERSON] and [PERSON] in 1995 [39], but the performance of the standard PSO algorithm was not high enough and showed some defects, such as parameter sensitivity, premature convergence, and slow local search. In recent years, a variant PSO called accelerated PSO (APSO) has attracted wide attention from scholars [38; 40; 41; 42]. Although the APSO improves the convergence speed, it may also lead to premature convergence and omit some extreme values. Therefore, in this study we propose a new chaos-enhanced accelerated particle swarm optimization algorithm (CAPSO) by integrating chaos theory into the improvement of the APSO algorithm. In the APSO algorithm, the influence of the inertial weight factor or cognitive factor on the particle is not considered and the algorithm is only improved by the global exploration factor [43]. The main idea of the algorithm is to fully attribute the power to the variable that is responsible for global search and to consider the update of the particle with the exploration factor. In the whole search process, the particle is only constrained by the global extreme value. The position update formula is:  
\[X_{i,d}^{K+1}=(1-C_{2})X_{i,d}^{K}+C_{2}p_{g,d}^{K}+C_{1}r \tag{1}\]  
where \(C_{1}\) and \(C_{2}\) are learning factors; \(r\) is the random number between 0 and 1; \(X_{i,d}^{K+1}\) is the position of particle \(i\) in \(d\)-dimensional \(k\)-th iteration; and \(p_{g,d}^{K}\) is the position of the global extremum of the whole population in the \(d\)-th dimension. Compared with the standard PSO algorithm, APSO adds two parameters, \(C_{1}\) and \(C_{2}\), to reduce the randomness in the iterative process. In this paper, \(C_{1}\) represents the monotonically decreasing function: \(C_{1}=\delta^{t}\), where \(0<\delta<1\) and \(t\) is the current iteration number. Therefore, the performance of the APSO algorithm is mainly affected by parameter \(C_{2}\). For common problems, the value is (0.2,0.7). When \(C_{2}\) is 1, the particle can converge at any time to the current global value and does not change any more. Moreover, the global value may not be the real global value at all. When \(C_{2}\) is 0, the global search speed of the algorithm is extremely slow. Therefore, the optimization of \(C_{2}\) is important in the APSO algorithm. A chaotic system refers to a deterministic system involving random irregular movements, whose behavior is uncertain, unrepeatable, and unpredictable. In a chaotic system, when the initial conditions are slightly changed, the system will be greatly different after continuous amplification. In the process of the APSO algorithm, the value of learning factor \(C_{2}\) is uncertain and unpredictable and has partial chaotic characteristics. Therefore, in order to simulate the chaotic characteristics of \(C_{2}\), the classical logistic equation is used to realize the evolution of chaotic variables and optimize the parameters in this paper. The iterative formula is provided as follows:  
\[X_{i}^{K+1}=4X_{i}^{K}(1-X_{i}^{K}) \tag{2}\]  
when \(0<X_{i}^{K}<1\), the logistic equation is in a completely chaotic state. The CAPSO algorithm involves the following steps:  
Step 1: To initialize the particle group. The particles in the PSO algorithm are initialized. The optimal value of the individual extremum is selected as the global optimal value to generate chaotic values;  
Step 2: To calculate the adaptive value of group particles;  
Step 3: The adaptive value of each particle is compared with that of the particle at the best position. If the adaptive value is better, the best position is updated;  
Step 4: The learning factor \(C_{2}\) is obtained from the chaotic sequence (generated by Equation (2)) and the position of the particle is updated with Equation (1);  
Step 5: If the end condition of the algorithm is satisfied, the global optimum position is the optimal solution.