## 3 Methodology
### Ant Colony Optimization  
Ant Colony Optimization (ACO) algorithms are inspired by the foraging behavior of ants, particularly how they locate food and communicate this information to their colony through pheromone trails. This approach is suitable for combinatorial optimization problems, quadratic assignments, derivative methods, and stochastic problems. Initially, ants wander randomly; when one locates food, it returns to the colony while depositing pheromones, marking the path for others. Ants that follow this path reinforce the pheromone trail, making it more attractive. Over time, pheromone marks fade, giving shorter and more frequently traveled paths an advantage. In ACO algorithms, artificial ants traverse a search space to locate optimal solutions. These artificial ants track their locations and the candidate solutions they encounter, using this information to guide subsequent iterations. This method is often combined with local search techniques activated after identifying a promising area. ACO's combined optimization ability and robustness make it highly effective. It was introduced ACO to the rough set theory, converting minimal-feature subset problems into optimal path combination problems, thus proposing a feature selection method based on ACO [23; 24]. The ACO algorithm can be outlined through several key steps. Parameter Initialization involves setting the pheromone intensity \(\tau_{ij}(t)\), and parameters \(\alpha\) (pheromone intensity control), \(\beta\) (visibility control), \(\rho\) (pheromone evaporation rate), and \(k\) (number of ants). The Transmission State Rule determines the probability that an ant at one point will move to another, defined by Equation (28). Here, \(S_{k}\) is the set of objects to be scheduled, \(\tau_{i,j}\) is the pheromone amount between points \(i\) and \(j\), and \(\eta_{i,j}=\frac{1}{t_{i,j}}\) represents the visibility of connection \((i,j)\). The parameters \(\alpha\) and \(\beta\) control the relative weights of pheromone and visibility, respectively, both ranging from 0 to 1. Pheromone Update occurs after ants traverse a path, updating pheromones via Equation (29). The variable \(\tau_{i,j}\) represents the pheromone intensity at time \(t\), with \(\rho\) as the evaporation rate (\(0<\rho<1\)). \(\Delta\tau_{i,j}(t)\) is the probability that an ant at point \(i\) will choose point \(j\), defined by Equation (30). A Global Pheromone Update is performed after ants complete their paths, marking the best routes based on the highest pheromone concentration. This update uses Equation (31), where \(\Delta\tau_{i,j}(t)\) is defined in Equation (32). \[P_{i,j}^{k}(t)= \begin{cases}\frac{[\tau_{i,j}(t)]^{a}[\eta_{i,j}]^{b}}{\sum_{i=1}^ {n}[\tau_{i,j}(t)]^{a}[\eta_{i,j}(t)]^{b}},&\text{if}(i,j)\in S_{k}\\ 0,&\text{if}(i,j)\
otin S_{k}\end{cases}  \tag{28}\]  
\[\tau_{i,j}(t+1)=(1-\rho)\tau_{i,j}(t)+\Delta\tau_{i,j}(t) \tag{29}\]  
\[\Delta\tau_{i,j}(t)= \begin{cases}\frac{[\tau_{i,j}(t)]^{a}[\eta_{i,j}]^{b}}{ \sum_{i=1}^{n}[\tau_{i,j}(t)]^{a}[\eta_{i,j}(t)]^{b}},&\text{if}(i,j)\text{ is passed by ants}\\ 0,&\text{otherwise}\end{cases}  \tag{30}\]  
\[\tau_{i,j}(t+1)=(1-\rho)\tau_{i,j}(t)+\Delta\tau_{i,j}(t) \tag{31}\]  
\[\Delta\tau_{i,j}(t)= \begin{cases}\zeta T,&\text{if}(i,j)\text{ is passed by ant-k}\\ 0,&\text{otherwise}\end{cases}  \tag{32}\]  
The suitability or quality of a path (solution) in ACO is linked to the pheromone concentration on that path. Higher pheromone concentrations indicate better paths, which are more likely to be chosen. The roulette-wheel selection technique selects candidates based on their probability \(P_{i,j}\), which is influenced by the heuristic value \(\eta_{i,j}\). This method balances probabilistic selection with the exploration of new paths, enhancing the robustness of the ACO algorithm in finding optimal solutions for discrete optimization problems.