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Ant Colony Optimization Algorithm
The ant colony optimization algorithm (ACO) is an algorithm that finds the most optimal path in a graph, from point A to point B, that consists of multiple paths in between the points. The concept stems from the behaviors of an ant colony foraging for food.

Background
An ant travels from its nest in search of food. As it walks around and finds a food source, the ant returns to its nest, leaving a trail of pheromones. The trail is now the path that connects the nest (point A) and the food source (point B). If the path from the nest and the food were too long, the intensity of the pheromones decreases, and other ants are not likely to approach it. When a second ant senses the strong trail, it might follow it, knowing that there is a chance that it leads to food. Once the second ant travels along the same path the first ant left behind, the pheromones become stronger and attract more ants. If the path was short and more ants are traveling along the path, other ants are more likely to follow that path because it has the most intense pheromones.

Formulas
The probability that an ant will move from node $$i$$ to node $$j$$ is $$p_{i,j} = \frac { (\tau_{i,j}^{\alpha}) (\eta_{i,j}^{\beta}) } { \sum_{} (\tau_{i,j}^{\alpha}) (\eta_{i,j}^{\beta}) }$$.

The amount of pheromone is updated without vapoization is $$\tau_{i,j}^k =\sum_{k=0}^m\Delta \tau^{k}_{i,j} $$.

The amount of pheromone is updated with vapoization is $$\tau_{i,j}^k = (1-\rho)\tau_{i,j} + \sum_{k=0}^m\Delta \tau^{k}_{i,j}$$. where

$$\tau_{i,j}$$ is the amount of pheromone on edge $${i,j}$$,

$$\alpha$$ is a parameter to control the influence of $$\tau_{i,j}$$,

$$\eta_{i,j}$$ is the desirability of edge $$i, j$$ depicted as $$\eta_{i,j}=\frac{2}{L_{i,j}} $$,

$$\beta$$ is a parameter to control the influence of $$\eta_{i,j}$$,

$$\rho$$ is the rate of pheromone evaporation,

$$m $$ is the number of ant,

$$\Delta \tau_{i,j}$$ is the amount of pheromone deposited depicted as $$\Delta \tau^{k}_{i,j} = \begin{cases} \frac{1}{L_k} & \text{if }k\text{th ant travels on edge }i\text{, }j \\ 0 & \text{otherwise} \end{cases}

$$,

$$L_{k} $$ is the tour length of the $$k $$th ant.

Vehicle Routing Problem
The vehicle routing problem asks for the optimal routes for multiple vehicles visiting a set of locations.

Traveling Salesman Problem
The traveling salesman problem asks for the least total distance that a salesman, starting from the first city A, can travel through to n cities. The salesman must start at city A, stop at all the other cities only once, and end at city A. It is essentially a simplified version of the vehicle routing problem.