## 3 Improved Adaptive Ant Colony Algorithm
### SA-Based Adaptive Adjustment Strategy of Pheromones  
In the traditional ant colony algorithm, the pheromone is often a fixed value and will not automatically adjust with the iteration situation. Ants search blindly in the early stage due to the low concentration of pheromones. In the later stage, pheromones will accumulate in a large amount, which will reduce the diversity of trajectory selection. Therefore, the fixed increase in the pheromones cannot meet the requirements of search efficiency and will slow down the convergence speed. Based on the temperature update function of the simulated annealing algorithm, this paper proposes a new annealing strategy to dynamically update the pheromone volatilization factor. The temperature \(T\) is an important control parameter in the simulated annealing algorithm, which determines the annealing direction and the running speed of the algorithm. As the iteration makes the de-temperature function have linear characteristics, the running speed of the de-temperature function is further improved. Moreover, a dynamic pheromone volatilization factor \(\rho\) is designed to realize the adaptive update of pheromone. \(\rho\) is set by Equations (10) and (11). \[\rho=l-\exp(-(\mu/T_{i})) \tag{10}\]  
\[T(i)=\left\{ \begin{array}{ll}T_{\text{end}}+\frac{(T_{\text{start}}-T_{\text {end}})*(I_{\text{max}}-l_{i})}{I_{\text{max}}},&I_{i}\leq\frac{I_{\text{max}} }{2}\\ T_{\text{end}}+\frac{(T_{\text{start}}-T_{\text{end}})*(I_{\text{max}}-(I_{ \text{max}}-I_{i}))}{I_{\text{max}}},&I_{i}>\frac{I_{\text{max}}}{2}\end{array} \right. \tag{11}\]  
where \(\mu\) is a constant and \(T_{i}\) is the current temperature, which is calculated by the de-temperature function \(T(i)\). \(I_{\text{max}}\) is the number of maximum iterations. \(T_{\text{start}}\) is the initial temperature, and \(T_{end}\) is the final temperature. As the number of iterations increases, the de-temperature function linearly decreases and then rises to further increase the running speed of the de-temperature function. To achieve better experimental results, the value of parameter \(\mu\) is discussed in this paper. Other parameters are set to: \(T_{\text{start}}=100\), \(T_{end}=0.1\), \(I_{\text{max}}=50\). In the previous work, many experiments were performed to determine the value of \(\mu\), and finally three possible and suitable values were selected and are discussed in this paper. The values of \(\mu\) were set to 33, 50, and 90, respectively, to calculate \(\rho\) according to Equation (10). The adaptive change curve of \(\rho\) is shown in Figure 3. In the early stage, the volatilization factor \(\rho\) is small, which is conducive to accumulating pheromones and improving the directionality of the ants' search. In the mid-term, \(\rho\) becomes larger to speed up the iteration speed of the algorithm and to avoid falling into the local optimum. In the later stage, \(\rho\) becomes small to accelerate convergence. To further test the influence of the value of \(\mu\) on the convergence of the improved adaptive ant colony algorithm (IAACA) proposed in this paper, three benchmark functions were applied for verification. Other swarm intelligence algorithms including ACO, PSO, Grey Wolf Optimizer (GWO) [32] were chosen to compare with IAACA in terms of convergence performance. Table 1 presents the details about the different types of benchmark functions. The three benchmark functions had a minimum value of 0, the particle dimension was set to 10, and the value range of x and y was (\(-5.12\), \(5.12\)). Sphere is a unimodal function, while Rastrigin and Ackley are multimodal functions with many local values and difficult to solve. These three benchmark functions were suitable for testing the convergence performance of the algorithm. Figure 3: Adaptive change of the pheromone volatilization factor.