*2.6. Ant Colony Optimization (ACO)*

Ant colony optimization (ACO) is also known as a swarm intelligence technique, which is inspired by the foraging pattern of ant colonies. Ants can find the shortest path from a food source to their nest, without having to see it directly. The ants have a unique and very advanced solution, namely using a pheromone trail on a path to communicate and building a solution, the more pheromone traces are left, the other ants will follow that path. These pheromones too relate to the previous good element solutions formed by the ants. Equation (9) describes the probability of an ant move from one node (*i*) to another (*j*):

$$P\_{i,j} = \frac{\left(\tau^a\_{i,j}\right)\left(\mathbf{n}^\beta\_{i,j}\right)}{\sum \left(\tau^a\_{i,j}\right)\left(\mathbf{n}^\beta\_{i,j}\right)}\tag{9}$$

where *τi*,*<sup>j</sup>* indicates the amount of pheromone on edge *i,j* (refer to Equation (10)) and the *α* symbolizes the factors selected to regulate the impact of *τi*,*j*. *τi*,*j*. Meanwhile, *ni*,*<sup>j</sup>* is the desirability of edge *i,j* (commonly 1/d*i,j*) and the *β* implies the factors selected to regulate the impact of *ni*,*j*.

$$
\pi\_{i,\bar{j}} = 1 - \rho \pi\_{i,\bar{j}} + \Delta \pi\_{i,\bar{j}} \tag{10}
$$

where *ρ* is the rate of pheromone evaporation and Δ*τi*,*<sup>j</sup>* is the amount of pheromone deposited.

If ant *k* travels on edge *i,j*, the amount of pheromone deposited is given by Equation (11):

$$
\Delta \tau\_{i,j}^k = \begin{cases}
\frac{1}{\tau\_k}, & \text{if ant k travels on edge } i, j \\
0, & \text{Otherwise}
\end{cases}
\tag{11}
$$

where *Lk* is the cost of the *k*th ant's tour (typically length).

## *2.7. Statistical Evaluation of the Developed Models*

Different statistical measures, including correlation coefficient (R), coefficient of determination (R*2*), root mean absolute error (RMSE), standard error of prediction (SEP), mean absolute error (MAE) and Chi-square were used to test the developed models, as described in Equations (12)–(17) [39]:

$$\mathcal{R} = \frac{\sum\_{i=1}^{n} \left( M\_p - M\_{p, \text{av}\,\text{\textdegree\text\text\text\text\text\textdegree\text\text\text\text\textdegree}} \right) \times \left( M\_\text{\text{\textell\text\textquotesingle}} - M\_{\text{\textell\text\textquotesingle}} \right)}{\sqrt{\left[ \sum\_{i=1}^{n} \left( M\_p - M\_{p, \text{av}\,\text{\textgreater\text\textquotesingle}} \right)^2 \right] \left[ \sum\_{i=1}^{n} \left( M\_\text{\textell} - M\_{\text{\textell\text\textquotesingle}} \right)^2 \right]}} \tag{12}$$

$$\mathbf{R}^2 = 1 - \frac{\sum\_{i=1}^n \left(M\_{\mathbf{c}} - M\_p\right)^2}{\sum\_{i=1}^n \left(M\_{\mathbf{c}, \text{avg}} - M\_p\right)^2} \tag{13}$$

$$\text{RMSE} = \sqrt{\frac{1}{n}} \sum\_{i=1}^{n} \left( M\_c - M\_p \right)^2 \tag{14}$$

$$\text{SEP} = \frac{\sqrt{\frac{1}{n}\sum\_{i=1}^{n} \left(M\_{\text{f}} - M\_{\text{f}}\right)^{2}}}{M\_{\text{c}\_{\text{c}}\text{avg}}} \times 100\tag{15}$$

$$\text{MAE} = \frac{1}{n} \left( \sum\_{i=1}^{n} |\left( M\_i - M\_p \right)| \right) \tag{16}$$

$$\text{Chi}-\text{square} = \sum\_{i=1}^{n} \frac{\left(M\_{\varepsilon} - M\_{\mathbb{P}}\right)^{2}}{M\_{\varepsilon}} \tag{17}$$

where *n* shows the number of points, and *Mp*, *Me*, *Mp*,*avg* and *Me*,*avg* are the predicted value, experimental value and the average of the predicted and experimental values, respectively.
