![The Algorithm
• After each ant k has moved to the next
node, the pheromones evaporate by the
following equation to all the arcs:
• where is a parameter. An
iteration is a complete cycle involving
ants’ movement, pheromone evaporation,
and pheromone deposit.
(1 ) , ( , )
ij ij
p i j A
(0,1]
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Ant Colony Optimization algorithms in ADSA
- 2. Overview
- 3. ACO Concept • Ants (blind) navigate from nest to food source • Shortest path is discovered via pheromone trails – each ant moves at random – pheromone is deposited on path – ants detect lead ant’s path, inclined to follow – more pheromone on path increases probability of path being followed
- 4. ACO System • Virtual “trail” accumulated on path segments • Starting node selected at random • Path selected at random – based on amount of “trail” present on possible paths from starting node – higher probability for paths with more “trail” • Ant reaches next node, selects next path • Continues until reaches starting node • Finished “tour” is a solution
- 5. ACO System, cont. • A completed tour is analyzed for optimality • “Trail” amount adjusted to favor better solutions – better solutions receive more trail – worse solutions receive less trail – higher probability of ant selecting path that is part of a better-performing tour • New cycle is performed • Repeated until most ants select the same tour on every cycle (convergence to solution)
- 6. ACO System, cont. • Often applied to TSP (Travelling Salesman Problem): shortest path between n nodes • Algorithm in Pseudocode: – Initialize Trail – Do While (Stopping Criteria Not Satisfied) – Cycle Loop • Do Until (Each Ant Completes a Tour) – Tour Loop • Local Trail Update • End Do • Analyze Tours • Global Trail Update – End Do
- 7. Background • Discrete optimization problems difficult to solve • “Soft computing techniques” developed in past ten years: – Genetic algorithms (GAs) • based on natural selection and genetics – Ant Colony Optimization (ACO) • modeling ant colony behavior
- 8. Background, cont. • Developed by Marco Dorigo (Milan, Italy), and others in early 1990s • Some common applications: – Quadratic assignment problems – Scheduling problems – Dynamic routing problems in networks • Theoretical analysis difficult – algorithm is based on a series of random decisions (by artificial ants) – probability of decisions changes on each iteration
- 10. Ant Algorithms
- 11. Ant Algorithms
- 12. Implementation • Can be used for both Static and Dynamic Combinatorial optimization problems • Convergence is guaranteed, although the speed is unknown – Value – Solution
- 13. The Algorithm • Ant Colony Algorithms are typically use to solve minimum cost problems. • We may usually have N nodes and A undirected arcs • There are two working modes for the ants: either forwards or backwards. • Pheromones are only deposited in backward mode.
- 14. The Algorithm • The ants memory allows them to retrace the path it has followed while searching for the destination node • Before moving backward on their memorized path, they eliminate any loops from it. While moving backwards, the ants leave pheromones on the arcs they traversed.
- 15. The Algorithm • The ants evaluate the cost of the paths they have traversed. • The shorter paths will receive a greater deposit of pheromones. An evaporation rule will be tied with the pheromones, which will reduce the chance for poor quality solutions.
- 16. The Algorithm • At the beginning of the search process, a constant amount of pheromone is assigned to all arcs. When located at a node i an ant k uses the pheromone trail to compute the probability of choosing j as the next node: • where is the neighborhood of ant k when in node i. 0 k i ij k i k il ij l N k i if j N p if j N k i N
- 17. The Algorithm • When the arc (i,j) is traversed , the pheromone value changes as follows: • By using this rule, the probability increases that forthcoming ants will use this arc. k ij ij
- 18. The Algorithm • After each ant k has moved to the next node, the pheromones evaporate by the following equation to all the arcs: • where is a parameter. An iteration is a complete cycle involving ants’ movement, pheromone evaporation, and pheromone deposit. (1 ) , ( , ) ij ij p i j A (0,1] p
- 19. • where nij is the heuristic visibility of edge (i, j), generally it is a value of 1/dij, where dij is the distance between city i and city j. City J is a set of cities which remain to be visited when the ant is at city i. α and β are two adjustable positive parameters that control the relative weights of the pheromone trail and of the heuristic visibility. If α=0, the closed vertex i more likely to be selected. This is responding to a greedy algorithm. If β=0, only pheromone amplification is at work: This method will lead the system to a stagnation situation, i.e. a situation in which all the ants generate a sub-optimal tour. So the trade-off between edge length and pheromone intensity is necessary.
- 20. Steps for Solving a Problem by ACO 1. Represent the problem in the form of sets of components and transitions, or by a set of weighted graphs, on which ants can build solutions 2. Define the meaning of the pheromone trails 3. Define the heuristic preference for the ant while constructing a solution 4. If possible implement a efficient local search algorithm for the problem to be solved. 5. Choose a specific ACO algorithm and apply to problem being solved 6. Tune the parameter of the ACO algorithm.
- 21. Applications Efficiently Solves NP hard Problems • Routing – TSP (Traveling Salesman Problem) – Vehicle Routing – Sequential Ordering • Assignment – QAP (Quadratic Assignment Problem) – Graph Coloring – Generalized Assignment – Frequency Assignment – University Course Time Scheduling 4 3 5 2 1
- 22. Applications • Scheduling – Job Shop – Open Shop – Flow Shop – Total tardiness (weighted/non-weighted) – Project Scheduling – Group Shop • Subset – Multi-Knapsack – Max Independent Set – Redundancy Allocation – Set Covering – Weight Constrained Graph Tree partition – Arc-weighted L cardinality tree – Maximum Clique
- 23. Applications • Other – Shortest Common Sequence – Constraint Satisfaction – 2D-HP protein folding – Bin Packing • Machine Learning – Classification Rules – Bayesian networks – Fuzzy systems • Network Routing – Connection oriented network routing – Connection network routing – Optical network routing
- 25. Advantages and Disadvantages • For TSPs (Traveling Salesman Problem), relatively efficient – for a small number of nodes, TSPs can be solved by exhaustive search – for a large number of nodes, TSPs are very computationally difficult to solve (NP-hard) – exponential time to convergence • Performs better against other global optimization techniques for TSP (neural net, genetic algorithms, simulated annealing) • Compared to GAs (Genetic Algorithms): – retains memory of entire colony instead of previous generation only – less affected by poor initial solutions (due to combination of random path selection and colony memory)
- 26. Advantages and Disadvantages, cont. • Can be used in dynamic applications (adapts to changes such as new distances, etc.) • Has been applied to a wide variety of applications • As with GAs, good choice for constrained discrete problems (not a gradient-based algorithm)
- 27. Advantages and Disadvantages, cont. • Theoretical analysis is difficult: – Due to sequences of random decisions (not independent) – Probability distribution changes by iteration – Research is experimental rather than theoretical • Convergence is guaranteed, but time to convergence uncertain
- 28. Advantages and Disadvantages, cont. • Tradeoffs in evaluating convergence: – In NP-hard problems, need high-quality solutions quickly – focus is on quality of solutions – In dynamic network routing problems, need solutions for changing conditions – focus is on effective evaluation of alternative paths • Coding is somewhat complicated, not straightforward – Pheromone “trail” additions/deletions, global updates and local updates – Large number of different ACO algorithms to exploit different problem characteristics
- 29. Questions?
- 30. • ൌ ఛ ఎ ∑ሺఛ ఎ ሻ (6) where τi - indicates the attractiveness of transition in the past ηi - adds to transition attractiveness for ants, Ni - set of nodes connected to point i, without the last visited point before i, α, - parameters found by simulation.
- 33. • reverse path according to formula (7). t t ij t ij 1 (7) where τijt - value of pheromone in step t, Δτ- value by ants saved pheromones in step t. Values Δτ can be constant or they can be
- 34. Assignment • Apply the ant colony optimization technique for solving travelling salesperson problem. • Ant Colony Optimization is a new meta-heuristic technique used for solving different combinatorial optimization problems. ACO is based on the behaviors of ant colony and this method has strong robustness as well as good distributed calculative mechanism. ACO has very good search capability for optimization problems. Travelling salesman problem is one of the most famous combinatorial optimization problems.