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Ant Colony Optimization Algorithms for the Traveling Salesman Problem

Ant Colony Optimization Algorithms for the Traveling Salesman Problem. ACO 3.1-3.5 Kristie Simpson EE536: Advanced Artificial Intelligence Montana State University. ACO Review . Chapter 1: From Real to Artificial Ants (Dr. Paxton) Looked at real ants and the double bridge experiment.

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Ant Colony Optimization Algorithms for the Traveling Salesman Problem

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  1. Ant Colony Optimization Algorithms for the Traveling Salesman Problem ACO 3.1-3.5 Kristie Simpson EE536: Advanced Artificial Intelligence Montana State University

  2. ACO Review • Chapter 1: From Real to Artificial Ants (Dr. Paxton) • Looked at real ants and the double bridge experiment. • Defined a stochastic model for real ants, and then modified the definition for artificial ants. • Discussed the Simple-ACO algorithm.

  3. ACO Review • Chapter 2: The ACO Metaheuristic (Chris, Shen) • Introduced combinatorial optimization problems. • Discussed exact and approximate solutions to NP-hard problems. • Discussed the ACO Metaheuristic and example applications (TSP presented in section 2.3.1).

  4. Chapter 3: ACO Algorithms for TSP • “But you’re sixty years old. They can’t expect you to keep traveling every week.” –Linda in act I, scene I of Death of a Salesman, Authur Miller, 1949

  5. Why use TSP? • NP-Hard (permutation problem, N!). • Easy application of ACO. • Easy to understand. • Ant System (the first ACO alogrithm) was tested on TSP. • Solutions tend to be most efficient for other applications.

  6. What is TSP? • Starting from his hometown, a salesman wants to find a shortest tour that takes him through a given set of customer cities and then back home, visiting each customer city exactly once. • Represented by a weighted graph G = (N,A). • The goal in TSP is to find a minimum length Hamiltonian circuit of the graph. • An optimal solution is:

  7. University of Heidelburg NAME : att532 TYPE : TSP COMMENT : 532-city problem (Padberg/Rinaldi) DIMENSION : 532 EDGE_WEIGHT_TYPE : ATT NODE_COORD_SECTION 1 7810 6053 2 7798 5709 3 7264 5575 4 7324 5560 5 7547 5503 6 7744 5476 7 7821 5457 8 7883 5408 att532 : 27686 http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/

  8. ACO Algorithms for the TSP • G = (C, L) is equal to G = (N, A). • All cities have to be visited and that each city is visited at most once. • Pheromone trail: the desirability of visiting city j directly after i. • Heuristic: inversely proportional to the distance between two cities i and j.

  9. Tour Construction • Choose a start city. • Use pheromone and heuristic values to add cites until all have been visited. • Go back to the initial city. Note: Tour may be improved with a local search (section 3.7).

  10. Skeleton for ACO algorithm • Set parameters, initialize pheromone trails. • While termination condition not met • ConstructAntSolutions • ApplyLocalSearch • UpdatePheromones • Only solution construction and pheromone updates considered.

  11. ACO Algorithms • Ant System (AS) • Elitist Ant System (EAS) • Rank-Based Ant System (ASrank) • Min-Max Ant System (MMAS) • Ant Colony System (ACS) • Approximate Nondeterministic Tree Search (ANTS) • Hyper-Cube Framework for ACO

  12. Ant System (AS) • m ants concurrently build tour. • Pheromone initialized to m/Cnn. • Ants initially in randomly chosen sites. • Random proportional rule used to decide which city to visit next. (see Box 3.1 for good parameter values)

  13. Ant System (AS) • Each ant k maintains a memory Mk for its neighborhood. • After all ants have constructed their tours, the pheromone trails are updated. • Pheromone evaporation:

  14. Ant System (AS) • Pheromone update:

  15. Elitist Ant System (EAS) • First improvement on AS. • Provide strong additional reinforcement to the arcs belonging to the best tour found since the start of the algorithm.

  16. Rank-Based Ant System (ASrank) • Another improvement over AS. • Each ant deposits an amount of pheromone that decreases with its rank. • In each iteration, only the best (w-1) ranked ants and the best-so-far ant are allowed to deposit pheromone.

  17. Min-Max Ant System (MMAS) • Four modifications with respect to AS. • Strongly exploits the best tours found. • This may lead to stagnation. So… • Limits the possible range of pheromone values. • Pheromone values initialized to upper limit. • Pheromone values are reinitialized when system approaches stagnation.

  18. Min-Max Ant System (MMAS) • After all ants construct a solution, pheromone values are updated. (Evaporation is the same as in AS) • Lower and upper limits on pheromones limit the probability of selecting a city. • Initial pheromone values are set to the upper limit, resulting in initial exploration. • Occasionally pheromones are reinitialized.

  19. Ant Colony System (ACS) • Uses ideas not included in the original AS. • Differs from AS in three main points: • Exploits the accumulated search experience more strongly than AS. • Pheromone evaporation and deposit take place only on the best-so-far tour. • Each time an ant uses an arc, some pheromone is removed from the arc.

  20. Ant Colony System (ACS) • Pseudorandom proportional rule used to decide which city to visit next. • Only best-so-far ant adds pheromone after each iteration. Evaporation and deposit only apply to best-so-far.

  21. Ant Colony System (ACS) • The previous pheromone update was global. Each ant in ACS also uses a local update that is applied after crossing an arc. • Makes arc less desirable for following ants, increasing exploration.

  22. Approximate Nondeterministic Tree Search (ANTS) • Uses ideas not included in the original AS. • Not applied to TSP. • Computes lower bounds on the completion of a partial solution to define the heuristic information that is used by each ant during the solution construction. • Creates a dynamic heuristic where the lower the estimate the more attractive the path.

  23. Approximate Nondeterministic Tree Search (ANTS) • Two modifications with respect to AS: • Use of a novel action choice rule. • Modified pheromone trail update rule. (No explicit pheromone evaporation)

  24. Hyper-cube Framework for ACO • Uses ideas not included in the original AS. • Not applied to TSP. • Automatically rescales the pheromone values for them to lie always in the interval [0,1]. • Decision variables {0, 1} typically correspond to the components used by the ants for construction. • A solution problem then corresponds to one corner of the n-dimensional hyper-cube, where n is the number of decision variables.

  25. Hyper-cube Framework for ACO

  26. Parallel Implementation • Fine-grained – few individuals per processor, frequent information exchange. • Can lead to major communication overhead. • Coarse-grained – larger subpopulations per processor, information exchange is rare. • Much more promising for ACO. • p colonies on p processors.

  27. Partially Asynchronous Parallel Implementation (PAPI) • Information exchanged at fixed intervals. • Studies show it is better to exchange the best solutions rather than all solutions.

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