Implement in pure python:

# Any Colony System

## Name

Ant Colony System (ACS), Ant Colony Optimization (ACO)

## Taxonomy

Ant Colony System is a metaheuristic optimization algorithm inspired by the foraging behavior of ants, belonging to the field of Swarm Intelligence, which is a subfield of Computational Intelligence. It is closely related to other Ant Colony Optimization algorithms such as Ant System (AS) and Max-Min Ant System (MMAS).

- Computational Intelligence
  - Biologically Inspired Computation
    - Swarm Intelligence
      - Ant Colony Optimization (ACO)
        - Ant System (AS)
        - Ant Colony System (ACS)
        - Max-Min Ant System (MMAS)

## Strategy

### Pheromone-based Optimization

In Ant Colony System, artificial ants construct solutions to an optimization problem by iteratively selecting components based on pheromone trails and heuristic information. Pheromone trails are updated by ants to reflect the quality of the solutions found, guiding the search process towards promising regions of the search space.

### Local Search

Ant Colony System incorporates a local search procedure to improve the solutions constructed by the ants. This local search is typically problem-specific and aims to find better solutions by exploring the neighborhood of the current solution.

### Pheromone Update

Pheromone trails are updated in two ways: local update and global update. The local update is performed by each ant after constructing a solution, slightly decreasing the pheromone level to encourage exploration. The global update is performed by the best ant, reinforcing the pheromone trails associated with the best solution found so far.

## Procedure

Data Structures:
- Pheromone Matrix: A matrix storing the pheromone levels associated with each component of the problem.
- Heuristic Information: Problem-specific information guiding the ants' decisions, often representing the desirability of choosing a particular component.

Parameters:
- Number of Ants: The number of artificial ants used in the algorithm.
- Alpha: The influence of pheromone trails on the ants' decisions.
- Beta: The influence of heuristic information on the ants' decisions.
- Evaporation Rate: The rate at which pheromone trails evaporate over time.
- Q: A constant used in the pheromone update process.

### Initialization
1. Set the parameters: number of ants, alpha, beta, evaporation rate, and Q.
2. Initialize the pheromone matrix with a small positive value.
3. Initialize the heuristic information based on the problem.

### Solution Construction
1. For each ant:
   1. Select a starting component based on the pheromone trails and heuristic information.
   2. While the solution is not complete:
      1. Select the next component based on the pheromone trails and heuristic information.
      2. Apply the local pheromone update to the selected component.
   3. Perform local search to improve the constructed solution (optional).

### Pheromone Update
1. Find the best solution among all the ants.
2. Update the pheromone matrix:
   1. Evaporate the pheromone trails by a factor of the evaporation rate.
   2. Reinforce the pheromone trails associated with the best solution.

### Termination
1. If the termination criteria are met (e.g., maximum number of iterations or satisfactory solution quality), stop the algorithm.
2. Otherwise, go back to the Solution Construction step.

## Considerations

Advantages:
- Ant Colony System is effective in solving combinatorial optimization problems, especially those with a graph structure, such as the Traveling Salesman Problem.
- It can adapt to dynamic problems where the optimal solution may change over time.
- Ant Colony System is scalable and can be parallelized to handle large-scale problems.

Disadvantages:
- The performance of Ant Colony System is sensitive to the values of its parameters, requiring careful tuning.
- It may converge prematurely to suboptimal solutions if the balance between exploration and exploitation is not properly maintained.
- The computational cost of Ant Colony System can be high, especially for problems with a large number of components.

## Heuristics

### Parameter Settings
- The number of ants should be set based on the problem size and complexity. A common heuristic is to use a number of ants equal to the number of components in the problem.
- The values of alpha and beta should be chosen to balance the influence of pheromone trails and heuristic information. Typical values range from 1 to 5.
- The evaporation rate should be set to a small value, typically between 0.01 and 0.1, to allow for gradual forgetting of old pheromone trails.
- The value of Q should be set based on the magnitude of the objective function values. It is often set to 1 for simplicity.

### Initialization
- The initial pheromone levels should be set to a small positive value to encourage initial exploration. A common heuristic is to set the initial pheromone levels to the inverse of the number of components.

### Local Search
- The choice of local search procedure depends on the problem being solved. Common local search methods include 2-opt, 3-opt, and nearest neighbor search for the Traveling Salesman Problem.
- The local search should be applied sparingly to avoid excessive computational cost. A common heuristic is to apply local search to a fraction of the constructed solutions, such as the best solution in each iteration.

### Termination Criteria
- The maximum number of iterations should be set based on the problem size and the available computational resources. A common heuristic is to use a number of iterations proportional to the square of the number of components.
- An alternative termination criterion is to stop the algorithm when the best solution found has not improved for a certain number of iterations, indicating convergence.