## Preliminaries
### Aco  
As a representative bio-inspired meta-heuristic algorithm, ACO was firstly put forward by [PERSON] in the 1990s [19] to solve the travel salesman problem (TSP). Till now, ACO has been proven to be a more common framework for various optimization problems in a wide range of fields [23; 24; 25], such as job-shop scheduling, data mining, routing problems, and other complex optimization problems. When observing the foraging behavior of real ant colonies, researchers discovered that real ants can deposit a chemical substance, called a pheromone, while walking. The pheromone can be accumulative and evaporative, through which the ant colony can carry on indirect communication and finally achieve the cooperative goal. Ants can smell the pheromone and choose their way, in a probabilistic way, based on the amount of pheromone. The larger the amount of pheromone deposited on a route, the greater the probability that ants select the route. Meanwhile, on the shorter route that ants travel, the pheromone accumulates faster than on the longer routes. Thus, the faster the amount of pheromone increases on the shorter route, then the greater the probability that the ants travel this route. In the initial stage when the pheromone is absent, ants choose their routes fully randomly, but after a transitory period the shortest routes will be more and more frequently visited and pheromone will accumulate faster and faster on them, which in turn will attract more and more ants to choose these routes. The mathematical model of ACO is described as follows. Let \(M_{\mathrm{ant}}\) be the number of ants, and the matrix \(\boldsymbol{\tau}(t)=\{\tau_{ij}(t)\}\) be the pheromone, of which the element \(\tau_{ij}(t)\) is the level of pheromone deposited on the arc from node \(i\) to node \(j\), at time \(t\). The initial level of pheromone on each directed arc is a constant value, i.e., \(\tau_{ij}(0)=\tau_{0}\). Each ant builds a possible solution to the problem by moving through a finite sequence of neighbor nodes, and these moves are directed by the ant's internal state, problem-specific local information, and the shared information about the pheromone [19]. For the \(k\)-th ant located at the \(i\)-th node, it will move to the next \(j\)-th node with the transition probability:  
\[p_{ij}^{k}(t)=\left\{ \begin{array}{ll}\frac{[\tau_{ij}(t)]^{a}[\eta_{ij}]^{ \beta}}{\sum\limits_{u\in allowed_{k}}[\tau_{ii}(t)]^{a}[\eta_{ii}]^{\beta}}& \text{if }j\in allowed_{k}\\ 0&\text{otherwise}\end{array} \right. \tag{5}\]  
where \(\eta_{ij}\) represents the heuristic information about the problem, \(allowed_{k}\) denotes the feasible domain of the \(k\)-th ant at the \(i\)-th node, and \(\alpha\) and \(\beta\) are parameters that determine the relative importance of the pheromone with respect to the heuristic information. In addition, in order to achieve a trade-off between exploitation and exploration [2], a different transition rule is introduced and the next node \(j\) is selected as:  
\[j=\left\{ \begin{array}{ll}\arg\max\limits_{u\in allowed_{k}}\Big{\{}\ [\tau_{iu}(t)][\eta_{iu}]^{\beta}\Big{\}}&\text{if }q\leq q_{0}\\ J&\text{otherwise}\end{array} \right. \tag{6}\]  
where \(q\) is a random number uniformly distributed in \([0,1]\); \(q_{0}\)\(\in\)[0,1] is the parameter that determines the relative importance of exploitation versus exploration; and \(j\) is a node randomly selected according to the transition probability in (1) with \(\alpha=1\). As the ants move and build the possible solutions, the pheromone matrix is updated according to both the global updating and local updating processes [20]. As to the local updating process when building the solution, if an ant moves from node \(i\) to node \(j\), then the pheromone level on the corresponding arc \(ij\) is updated as follows:  
\[\tau_{ij}\leftarrow(1-\psi)\cdot\tau_{ij}+\psi\cdot\tau_{0} \tag{7}\]  
where \(\tau_{0}\) is the initial pheromone level on all arcs, and \(\psi\)\(\in\)(0,1] denotes the parameter that can control the pheromone evaporation.