Ant Colony Optimization:
an overview
Vittorio Maniezzo
University of Bologna, Italy
Metaheuristics
Metaheuristics include:
• Simulated Annealing
• Tabu Search
• GRASP
• Genetic Algorithms
• Variable Neighborhood Search
• …
• ACO
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Ant System
Ant System (AS) was the first ACO algorithm
presented [CDM91].
The idea it was to modify a constructive heuristic so
that the ordering of the components could be
recalculated at each iteration taking into account
• the a priori expectation, ηi, of the usefulness of a
particular component ci as in standard
constructive approaches,
• but also an a posteriori measure, τi, of the
goodness of solutions constructed using that
particular component.
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Ant System
• general purpose heuristic inspired by nature;
• parallel distributed search effort;
• combines a problem-specific constructive
heuristic (visibility) and a general-purpose
adaptive problem representation
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ACO
Ant Colony Optimization is the name given by
M.Dorigo [D98] to a class of algorithms whose first
member was AS.
The main underlying idea is that of parallelizing
search over several constructive computational
threads, all based on a dynamic memory structure
with information on the effectiveness of previously
obtained results and in which the behavior of each
single agent is inspired by the behavior of real ants.
Web page:
http://iridia.ulb.ac.be/~mdorigo/ACO/ACO.html
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The biological metaphor
E
H
E
E
D
D
Obstacle
H
C
B
A
A
t=0
E
d=1
D
d=0.5
H
C
15 ants
B
A
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C
B
E
30 ants
D
15 ants
A
t=1
E
30 ants
D
10 ants
τ=15
H
C
d=0.5
d=1
Obstacle
H
C
τ=15
15 ants
B
15 ants
10 ants
τ=30
B
30 ants
A
20 ants
τ=30
20 ants
30 ants
A
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Ant agents
An ant is defined to be a simple computational
agent, which iteratively constructs a solution for the
problem to solve.
Partial problem solutions are seen as states; each
ant moves from a state ι to another one ψ,
corresponding to a more complete partial solution.
At each step σ, each ant k computes a set of
feasible expansions to its current state, and moves
to one of these in probability, according to a
probability distribution specified as follows.
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Probability of a move
The formula for defining the probability distribution
at each move makes use of a set tabuk which
indicates a set of infeasible moves for ant k.
Probabilities are computed as follows:
α ⋅ τιψ + (1 − α ) ⋅ ηιψ

if (ιψ ) ∉ tabu k

 ∑ (α ⋅ τιν + (1 − α ) ⋅ ηιν )
k
p ιψ = 
 (ιν )∉tabu k

0
otherwise
Parameter α defines the relative importance of trail
with respect to attractiveness.
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Trail update
Trails are updated at each iteration, increasing the
level of those related to moves that were part of
"good" solutions, while decreasing all others.
After each iteration t of the algorithm, trails are
updated using the following formula.
τ ιψ (t ) = ρτιψ (t − 1) + ∆τ ιψ
where ρ is a user-defined coefficient and ∆τιψ
represents the sum of the contributions of all ants
that used move (ιψ) to construct their solution.
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Attractiveness and trail
For ant k, the probability pιψ of moving from state ι
to state ψ depends on the combination of two
values:
• the attractiveness η of the move, as computed by
some heuristic indicating the a priori desirability
of that move;
• the trail level τ of the move, indicating how
proficient it has been in the past to make that
particular move, the a posteriori the
desirability of that move.
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CCA0 example
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10
7
1
9
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1
5
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ACO pseudo code
1
(Initialization)
Initialize τιψ, ∀ι,ψ
2 (Construction)
For each ant k do
repeat
compute ηιψ ∀ι,ψ
choose in probability the state to move into
append the chosen move to the k-th ant’s set tabuk
until ant k has completed its solution
[apply a local optimization procedure]
enddo
3. (Trail update)
For each ant move (ι,ψ) do
compute ∆τιψ and update the trail values
4. (Terminating condition)
If not(end_condition) go to step 2.
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Ant System
1. Initialize a set A of partial solutions
ai=∅, i=1, … , m.
2. For i=1 to m
choose a component cj to append to solution ai with
probability given as a function of ai, ηj, τj.
3. If the solutions in A are not complete, go to step 2.
4. Evaluate z(ai), i=1, … , m and update τj, j=1, … , n
accordingly.
5. If not (end condition) go to step 1.
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ANTS conferences
The ACO has been specified in different ways by
different authors.
This variety was represented in the ANTS series of
international workshops (ANTS'98, ANTS‘2000 and
ANTS'2002), conferences entirely devoted to
algorithms inspired by the observation of ants
behavior.
Many applications: from plan merging to routing
problems, from driver scheduling to search space
sharing, from set covering to nurse scheduling, from
graph coloring to dynamic multiple criteria
balancing problems, etc.
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ACO approaches
ABC
ACS
Bonabeau et al. [B98], van der Put, Rothkrantz
[vdPR98]
network routing
Dorigo, Ganbardella [DG97]
TSP, VRP
AntNet Di Caro, Dorigo [DCD97], [DCD98]
ANTS Maniezzo [M98] , Maniezzo, Carbonaro [MC99]
AS
Colorni, Dorigo, Maniezzo [CDM91], [D92],
[CDMT94], [DMC96]
ASrank Bullnheimer, Hartl, Strauss [BHS97]
network routing
HAS
Gambardella, Taillard, Dorigo [GTD99]
QAP, VRP, SOP
MMAS
Stuetzle, Hoos [SH98,SD99]
TSP, QAP
QAP, FAP
TSP, QAP, JSP
TSP, VRP
AS-SCS Michel, Middendorf [MM98]
Costa, Hertz [CH97]
SCS
-
Merkle, Middendorf, Schmeck [MM00,MMS00]
Scheduling Problems
-
Gambardella, Dorigo [GD00]
SOP
-
Kawamura et al. [KYSO00]
TSP
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GCP
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Max-min ant system
Stuetzle and Hoos [SH97] introduced Max-Min Ant
System (MMAS), a modification of AS.
They use two parameters, a maximum and minimum
trail levels, in order to restrict possible trail values to
the interval [τmin, τmax].
Moreover, MMAS controls the trail levels (initialized to
their maximum value τmax, only allowing the best ant
at each iteration to update trails.
Trails that receive very rare reinforcements will lower
their strength and will be selected more and more
rarely by the ants, until they reach the τmin value.
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AS-rank
Bullnheimer, Hartl and Strauss [BHS97] proposed yet
another modification of AS, called AS-rank,
introducing a rank-based version of the probability
distribution to limit the danger of over-emphasized
trails caused by many ants using sub-optimal
solutions.
At each iteration the ants are sorted by solution
quality and the contribution of an ant to the trail
level update is weighted according to the rank of
the ant, considering only the Ω best ants.
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Ant Colony System
Gambardella and Dorigo [GD95, 97] proposed ACS,
where trails are updated with (constant) values which
predict the quality of solution using the relative edges.
Trail values are added offline only to the arcs
belonging to the best tour so far, while ants perform
online step-by-step trail updates to favor the
emergence of other solutions.
Each ant uses a pseudo-random proportional rule to
choose the next node to move to based on a q0 ∈ [0,1]
parameter that modulates the exploration inclinacy.
ACS also uses a data structure associated to vertices
called candidate list, as in GRASP.
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ANTS
1
(Initialization)
Compute a (linear) lower bound LB to the problem to solve
Initialize τιψ, ∀ι,ψ with the primal variable values
2 (Construction)
For each ant k do
repeat
compute ηιψ ∀ι,ψ, as a lower bound to the cost of
completing a solution containing ι,ψ
choose in probability the state to move into
append the chosen move to the k-th ant’s set tabuk
until ant k has completed its solution
apply a local optimization procedure
enddo
3. (Trail update)
For each ant move (ι,ψ) do
compute ∆τιψ and update the trail values
4. (Terminating condition)
If not(end_test) go to step 2.
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Convergence
W.Gutjahr [W00] proposed a convergence proof for
AS.
For a restricted version of AS and using suitable
parameter values (many ants or little evaporation),
the propobability of AS achieving an optimal
solution can be made arbitrarily close to 1.
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Current applications …
Applications are listed by class of problems and in
chronological order (from Dorigo, Gambardella, 2003).
Problem name
Traveling salesman
Authors
Dorigo, Maniezzo, Colorni
Gambardella, Dorigo
Dorigo, Gambardella
Stuetzle, Hoos
Bullnheimer, Hartl, Strauss
Cordòn, et al.
Quadratic assignment Maniezzo, Colorni, Dorigo
Gambardella, Taillard, Dorigo
Stuetzle, Hoos
Maniezzo
Maniezzo, Colorni
Scheduling problems Colorni, Dorigo, Maniezzo
Stuetzle
Bauer et al.
den Besten, Stuetzle, Dorigo
Merkle, Middendorf, Schmeck
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Algorithm name Year
AS
1991
Ant-Q
1995
ACS, ACS-3-opt 1996
MMAS
1997
ASrank
1997
BWAS
2000
AS-QAP
1994
HAS-QAP
1997
MMAS-QAP
1997
ANTS
1998
AS-QAP
1999
AS-JSP
1994
AS-FSP
1997
ACS-SMTTP
1999
ACS-SMTWTP 1999
ACO-RCPS
2000
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Current applications (cont.)
Problem name
Vehicle routing
Authors
Algorithm name
Bullnheimer, Hartl, Strauss AS-VRP
Gambardella, Taillard, Agazzi HAS-VRP
Connection-oriented
Schoonderwoerd et al.
ABC
network routing
White, Pagurek, Oppacher
ASGA
Di Caro, Dorigo
AntNet-FS
Bonabeau et al.
ABC-smart ants
Connection-less
Di Caro, Dorigo
AntNet, AntNet-FA
network routing
Subramanian, Druschel, Chen Regular ants
Heusse et al.
CAF
van der Put, Rothkrantz
ABC-backward
Sequential ordering
Gambardella, Dorigo
HAS-SOP
Graph coloring
Costa, Hertz
ANTCOL
Shortest common supersequence Michel, Middendorf AS-SCS
Frequency assignment Maniezzo, Carbonaro
ANTS-FAP
Generalized assignment Ramalhinho Lourenco, Serra MMAS-GAP
Multiple knapsack
Leguizamòn, Michalewicz
AS-MKP
Optical networks routing Navarro Varela, Sinclair
ACO-VWP
Redundancy allocation Liang, Smith
ACO-RAP
Constraint satisf.
Solnon
Ant-P-solver
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Year
1997
1999
1996
1998
1998
1998
1997
1997
1998
1998
1997
1997
1998
1998
1998
1999
1999
1999
2000
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Reality bites
Real-world ACO applications
An ACO-based company!
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Car pooling
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Stochastic VRP
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Urban solid waste collection
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Civil protection
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Traffic flow optimization
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