(3) Pheromone Update

According to the expected value of each capacity of each node and the pheromone concentration value on the path, the transition probability of the *n*-th ant at each capacity point on the node is calculated, and then the path is selected.

In order to prevent excessive heuristic pheromone and flooding the heuristic information, the residual pheromone is updated after the ant completes an ergodic operation on all *m* position capacities. According to Equations (27) and (28), adjusting the amount of information on the path (*i*, *j*) at time *t*+1

$$
\pi\_{\vec{l}\vec{\jmath}}(t+1) = (1-\rho)\tau\_{\vec{l}\vec{\jmath}}(t) + \Delta\tau\_{\vec{l}\vec{\jmath}}(t) \tag{27}
$$

$$
\Delta \tau\_{ij}(t) = \sum\_{k=1}^{m} \Delta \tau\_{ij}^{k}(t) \tag{28}
$$

where *ρ* is the pheromone volatilization coefficient, 1 − *ρ* is the pheromone residual coefficient, and *ρ* is in the range of (0, 1). ∆*τij*(*t*) represents the pheromone increment on the path (*i*, *j*) during the current cycle, ∆*τij*(*t*)=0 at the initial time. ∆*τ k ij*(*t*) represents the number of pheromones left by the *k*-th ant in the path (*i*, *j*).

$$
\Delta \tau\_{ij}^k(t) = \begin{cases}
\ \mathcal{Q}/L\_k & \text{if the kth passes the path (i,j)} \\
\ 0 & \text{else}
\end{cases}
\tag{29}
$$

where *Q* represents the pheromone strength, and its value affects the convergence speed of the algorithm. *L<sup>k</sup>* represents the total cost of the path taken by the *k*-th ant in this cycle.
