## 5 Predictive Simulation and Analysis  
### Particle Swarm Optimization (PSO)  
Particle swarm optimization (PSO), introduced by [PERSON] and [PERSON] [34,35], is a group intelligence algorithm that has been successful in both continuous and discrete optimization. The feasible solution of each optimization problem is abstracted into a \"particle\" without volume and mass in the search space. The PSO algorithm generates the initial population and the population of each particle motion in the feasible solution space, decides its own speed position and orientation of each particle movement following the current optimum particles, and obtains an optimal solution. In the PSO algorithm, if it selects particles as the random initial population, the position of each particle in the dimensional space can be expressed as the solution of the optimization problem; the degree of its performance depends on the fitness value determined by the objective function of the optimization problem and, through constant iteration, adjusts its direction change and speed size, finally finding the global optimal solution. The velocity of the particle is as follows [36]:  
\[\mathrm{v_{i}(t+1)=w\cdot v_{i}(t)+c_{i}r_{i}(pbest_{i}(t)+x_{i}(t))+c_{2}r_{2} (gbest(t)+x_{i}(t)),} \tag{3}\]  
\[x_{i}(t+1)=x_{i}(t)+v_{i}(t+1), \tag{4}\]  
In Equations (3) and (4), \(i=1,2,\cdots,m\); \(j=1,2,\cdots,n\); \(t\) is the number of iterations; \(c_{1}\), \(c_{2}\) is the learning factor, which measures the individual learning ability of \(pbest_{i}(t)\) and the global learning ability of \(gbest(t)\), respectively. \(r_{1},\ r_{2}\) is the random number transformed within the range of [0,1], \(x_{i}(t)\) is the position information of the particle i at time t, \(v_{i}(t)\) is the velocity information of the particle i at time t, \(pbest_{i}(t)\) is the current optimal position of the particle i (namely, the optimal solution), \(gbest(t)\) is the best location found by the whole population (namely, the global optimal solution), and \(w\) is the inertia factor, which  
Figure 4: Comparison of fatigue equations of 10% SMC and asphalt aggregate ratio 4.3 in the SCB fatigue test: (**a**) stress ratio; (**b**) stress amplitude.  
is calculated according to Equation (5) and used to control the influence of the last velocity on the current state.  
\[\mathrm{w}=(\mathrm{w}_{1}-\mathrm{w}_{2}).\frac{\mathrm{H}_{\mathrm{axgen}}- \mathrm{G}_{\mathrm{generation}}}{\mathrm{H}_{\mathrm{axgen}}} \tag{5}\]  
where, \(\mathrm{w}_{1}\), \(\mathrm{w}_{2}\) is the weight factor; \(\mathrm{H}_{\mathrm{axgen}}\) is the maximum number of iterations; and \(\mathrm{G}_{\mathrm{generation}}\) is the number of current iterations.  
Equations (3) and (4) are used to update the velocity and position of each particle. In order to limit the particle velocity and reduce some poor results, the contraction coefficient is applied to the PSO algorithm [37]. In the PSO algorithm with the contraction coefficient, the update velocity is changed as follows:  
\[\mathrm{v}_{i}^{\prime}(\mathrm{t}+1)=\mathrm{x}\Big{(}\mathrm{v}_{i}( \mathrm{t})+\mathrm{c}_{1}\mathrm{r}_{1}\big{(}\mathrm{pbest}_{i}(\mathrm{t})+ \mathrm{x}_{i}(\mathrm{t})\big{)}+\mathrm{c}_{2}\mathrm{r}_{2}\big{(}\mathrm{ gbest}(\mathrm{t})+\mathrm{x}_{i}(\mathrm{t})\big{)}\Big{)}, \tag{6}\]  
where, the contraction factor \(\mathrm{x}\) is as follows:  
\[\mathrm{x}=\frac{2}{\Big{|}2-\phi-\sqrt{\phi^{2}-4\phi}\Big{|}},\phi=\mathrm{ c}_{1}+\mathrm{c}_{2}>4 \tag{7}\]