ant colony optimization algorithms for the traveling salesman problem aco 3.1-3.5 kristie simpson...
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Ant Colony Optimization Algorithms for the Traveling Salesman Problem
ACO 3.1-3.5Kristie SimpsonEE536: Advanced Artificial IntelligenceMontana State University
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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.
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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).
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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
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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.
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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:
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University of Heidelburg
NAME : att532TYPE : TSPCOMMENT : 532-city problem
(Padberg/Rinaldi)DIMENSION : 532EDGE_WEIGHT_TYPE : ATTNODE_COORD_SECTION1 7810 60532 7798 57093 7264 55754 7324 55605 7547 55036 7744 54767 7821 54578 7883 5408
att532 : 27686 http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/
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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.
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Tour Construction
1) Choose a start city.
2) Use pheromone and heuristic values to add cites until all have been visited.
3) Go back to the initial city.
Note: Tour may be improved with a local search (section 3.7).
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Skeleton for ACO algorithm
Set parameters, initialize pheromone trails. While termination condition not met
– ConstructAntSolutions– ApplyLocalSearch– UpdatePheromones
Only solution construction and pheromone updates considered.
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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
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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)
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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:
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Ant System (AS)
Pheromone update:
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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)
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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.
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Hyper-cube Framework for ACO
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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.
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Partially Asynchronous Parallel Implementation (PAPI)
Information exchanged at fixed intervals. Studies show it is better to exchange the best
solutions rather than all solutions.