1. Ant Colony Optimization Andriy Baranov andriybaranov@yahoo.com

2. 2 Brief Facts Swarm intelligence - the collective behaviour of decentralized, self-organized systems. As a rule, the system consists of agents that interact with each other and the environment. Ant Colony Optimization Algorithm (ACO) - a technique for solving problems which can be reduced to finding good paths through graphs. It is based on the behavior of ants seeking a path between their colony and a source of food. It was proposed by Belgian Marco Dorigo in 1991. (first published in 1992)

3. 3 Overview First, ants wander randomly looking for food. When ant finds food it returns to the colony. Anywhere ants go, they lay down the pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but to instead follow the trail.

4. 4 The more pheromone, the more attractive the path is. Over time, the pheromone trail starts to evaporate, thus reducing its attractive strength. In the end only the shortest path will remain due to evaporation.

5. 5 Types of ACO Algorithm Elitist Ant System Ant-Q Ant Colony System Max-min Ant System ASrank

6. 6 Algorithm Classical ACO Algorithm: 1.Initialization. 2.Find solution. 3.Update trails. 4.End.

7. 7 Initialization Creating an agents (“ants”). –Our agents are quite simple. Each “ant” has its own tabu list (a list of vertices where he was) and choses the next vertex by its attractiveness and pheromone amount. Initialize “attractiveness” and “trail level”. At the beginning set equal quantity of pheromone on each vertex and set length of each move (attractiveness. The less length the more attractive). Set  and  parameters.

8. 8 Finding Solution For each ant k : 1.Choose the vertex to move to. 2.Add vertex into tabu list. 3.Repeat 1 and 2 until ant k won’t complete its path.

9. 9 Finding Solution: Detailed 1. Choose the vertex to move. How ant does it ? For ant k, the probability P k xy of moving from vertex x to vertex y depends on the combination of two values : attractiveness η xy and trail level τ xy. In general the probability of move from x to y is calculated by such formula. p k j f

10. 10 if V xy Є N 0else N – set of feasible vertices. α – is the parameter to control the influence of the pheromone. β – is the parameter to control the influence η xy. (η xy = 1/d xy, where d usually the distance between x and y) Note: 0< α, β < 1 and α+β=1.

11. 11 Update trails After each iteration of the algorithm, when all ants have completed the solution, trails are updated by the means of such formula: τ xy ← (1-ρ) * τ xy + τ k xy ρ – the evaporation rate. Δ τ xy – the amount of pheromone deposited. Q/L k if ant k used path xy. Δ τ k xy = 0if not where Q is constant, and L k is length of the tour of k ant.

12. 12 References 1. Marco Dorigo, Mauro Birattari, and Thomas Stützle, “Ant Colony Optimization. Artificial Ants as a Computational Intelligence Technique”, Université Libre de Bruxelles, BELGIUM. 2. Vittorio Maniezzo, Luca Maria Gambardella, Fabio de Luigi, “ Ant Colony Optimization”.

13. 13 Example

14. 14 Thanks for your attention Ant Colony Optamization Andriy Baranov andriybaranov@yahoo.com