## 3 The Improved Strategy of the ACO Algorithm  
### Overview of ACO  
The ACO algorithm is widely used to find an optimal path between a source node and a destination node. When searching for the destination node, artificial ants deposit a chemical substance called a pheromone on the path that they pass [56]. The pheromone is the medium that ants used to communicate and it guides other ants. Ants are more likely to follow a path with a higher pheromone concentration, and the following ants also release pheromones on the path, which increases the pheromone concentration. The increased pheromone concentration attracts more ants, which forms a positive feedback loop [57]. The pheromone matrix is a two-dimensional matrix used to record the pheromone values on every partial path. We use \(\tau_{ij}(t)\) to denote the pheromone concentration between node \(i\) and node \(j\) at time \(t\). Additionally, \(t\) is the iteration counter. Moreover, the pheromone volatilizes with time. After all ants have completed a path search, the pheromone matrix should be updated. The global pheromone update rule is presented as follows:  
\[\tau_{ij}(t+1)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}(t) \tag{6}\]  
\[\Delta\tau_{ij}(t)=\sum_{k=1}^{q}\Delta\tau_{ij}^{k}(t) \tag{7}\]  
\[\Delta\tau_{ij}^{k}(t)=\left\{\begin{array}{ll}Q/L_{k}&,(i,j)\in\text{tour byant}\;k\\ 0&,\text{otherwise}\end{array}\right. \tag{8}\]  
where \(\rho\) (\(0<\rho<1\)) is the evaporation parameter, \(q\) is the total number of ants, \(Q\) is the total amount of pheromone, and \(L_{k}\) is the total length of the path that the \(k\)th ant passes during this time. Nevertheless, a too high pheromone concentration may cause a local optimum of the algorithm and a too lowpheromone concentration may not attract other ants. Thus, we employ the method introduced in the max-min ant system (MMAS) to limit the pheromone value [58], which is presented as follows:  
\[\tau_{ij}(t+1)=\left\{\begin{array}{c}\tau_{\max},\tau_{ij}(t+1)>\tau_{\max}\\ (1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}(t),\text{otherwise}\\ \tau_{\min},\tau_{ij}(t+1)<\tau_{\min}\end{array}\right. \tag{9}\]  
where \(\tau_{\max}\) and \(\tau_{\min}\) represent the maximum and the minimum of the pheromone values, respectively. The limitation of the pheromone values could avoid the stagnation of the searching process and improve the global convergence of the algorithm.  
In ACO, the transition probability from node \(i\) to node \(j\) for the \(k\)th ant can be given by:  
\[p_{ij}^{k}=\left\{\begin{array}{c}\frac{(\tau_{ij}(t))^{n}(\eta_{ij})^{ \beta}}{\sum\limits_{m\in U_{k}}(\tau_{im}(t))^{\alpha}(\eta_{im})^{\beta}},j \in U_{k}\\ 0,\text{ otherwise}\end{array}\right. \tag{10}\]  
where \(U_{k}\) represents the set of next hop nodes available to the ants, \(\eta_{ij}\) is the heuristic information, \(\alpha\) is the pheromone parameter, and \(\beta\) denotes the heuristic information parameter.  
Ants transfer to the next hop node according to (10) until they arrive at the destination node. After all \(q\) ants have reached the destination node, the pheromone matrix is updated. It is decreased by evaporation, and \(\rho\) ranging from 0 to 1 is the evaporation parameter. The evaporation process contributes to avoiding unrestrained accumulation of the pheromone concentration. If one partial path is not selected by ants, its pheromone concentration decreases gradually, which makes ants not choose this bad path over time. The pheromone value is increased if the ants deposit pheromone on the path. The better paths receive more pheromone released by ants, which are more likely to be selected in future. Every pheromone value in the pheromone matrix is updated according to (7), (8), and (9). After the pheromone matrix is updated, the next iteration begins.