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Ant colony optimization (ACO) is a heuristic search algorithm for solving optimization problems that are inspired by the foraging behaviour of ants. In ACO, a set of artificial ants cooperate to find good solutions to an optimization problem. Each ant builds a partial solution by moving through the problem space. The moves of the individual ants are influenced by pheromones deposited on the edges of the problem graph by other ants; more attractive edges (with more pheromones) are more likely to be traversed by an ant. The pheromone concentration on an edge decays over time, encouraging the ants to explore new parts of the problem space. When an ant completes a tour (i.e., visits all nodes in the problem graph), it deposits pheromones on the edges of the tour. The pheromone concentration on an edge is then increased by a constant factor. The best tour found by any ant is called the global best tour.

The process of building a solution with the artificial ants can be described as follows:

1. Initialization: Each ant is placed at a random node in the problem graph.

2. Construction: Each ant builds a partial solution by moving from node to node. The moves are governed by a transition rule, which is usually a probabilistic function that takes into account the pheromone concentration on the edges.

3. Pheromone Update: When an ant completes a tour, it deposits the pheromone on the edges of the tour. The pheromone concentration on an edge is then increased by a constant factor.

4. Termination: The algorithm terminates when a predefined stopping criterion is met, such as a maximum number of iterations or the desired solution quality.

The pseudo-code for the ant colony optimization algorithm is shown below.

Ant colony optimization pseudo-code
Input:

• G = (V, E): A graph representing the problem space.
• n: The number of ants.
• ρ: The pheromone decay rate.
• Q: The pheromone deposition rate.
• stopping criterion: A criterion for terminating the algorithm.

Output:

A solution to the optimization problem.

1. Initialize the pheromone concentration on all edges to ρ.
2. Place n ants at random nodes in the graph.
3. While the stopping criterion is not met:
4. For each ant:
5. Construct a solution by moving from node to node.6. Update the pheromone concentration on the edges in the solution.
7. Select the best global solution.
8. Return the best global solution.

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