ant colony optimization for solving travelling sp

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.
– Defined a stochastic model for real ants, and then
modified the definition for artificial ants.
– Discussed the Simple-ACO algorithm.

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).

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

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.

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:

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/

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.

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).

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

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

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)

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:

Ant System (AS)
 Pheromone update:

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.

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.

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.

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.

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.

 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.

 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.

Approximate Nondeterministic Tree
Search (ANTS)
 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.

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)

Hyper-cube Framework for ACO
 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.

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.

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

ant colony optimization for solving travelling sp

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