*2.1. Ant System*

TSP can be described as finding the shortest route for a salesman who needs to visit each city at least once and no more than once [45]. TSP is a classical combination optimization problem which is employed to test ACO algorithms, and, therefore, TSP is used here as an example as well. The TSP includes symmetric TSP and asymmetric TSP. We only discuss symmetric TSP in this paper.

There are two primary steps in the AS algorithm, path construction, and pheromone updating [2]. During the first step, a solution is established according to the random proportion rule, and it can be described in detail as follows.

In the beginning, *m* ants are randomly assigned to *n* cities. At the *t*-th iteration, the probability, called the state transition probability, for the *k*-th ant to travel from the city *i* to *j* is

$$p\_{ij}^k(t) = \begin{cases} \frac{[\tau\_{ij}(t)]^a [\eta\_{ij}(t)]^b}{\sum\_{s \in [l\_k(i)]} [\eta\_{is}(t)]^b} & \text{if } j \in J\_k(i) \\\ 0, & \text{otherwise} \end{cases} \tag{1}$$

where *τij* is the pheromone trail and *ηij* is the heuristic information, accordingly, while *α* and *β* are parameters deciding their relative influences, respectively. Generally, *ηij* = 1/*dij* and *dij* is the distance of the path (*i*, *j*). *Jk* (*i*) is the feasible neighborhood of *k*-th ant at the *i*-th city.

When all the ants finish touring around each city, pheromone updating is as follows:

$$
\pi\_{i\bar{j}}\left(t+1\right) = \left(1-\rho\right)\pi\_{i\bar{j}}\left(t\right) + \sum\_{k=1}^{m} \Delta \tau\_{i\bar{j}}^{k} \tag{2}
$$

where *ρ* (0 < *ρ* ≤ 1) is the evaporation rate, <sup>Δ</sup>*τkij* represents the extra pheromone left in the path (*i*, *j*) by the *k*-th ant. <sup>Δ</sup>*τkij*could be decided through

$$
\Delta \tau\_{ij}^k(t) = \begin{cases}
\begin{array}{c}
\frac{Q}{L\_k} \\
0, \\
\end{array} \begin{array}{c}
\text{if ant } k \text{ passes the path } (i, j) \\
0, \\
\end{array}
\tag{3}
$$

where *Q* is the pheromone enhancement coefficient and *Lk* is the total path length for the *k*-th ant.
