1 Department of Biomedical, Industrial and Human Factors Engineering Ant Colony Systems and the Ant Algorithm
1Department of Biomedical, Industrial and Human Factors Engineering
Ant Colony Systems
and the
Ant Algorithm
2Department of Biomedical, Industrial and Human Factors Engineering
REAL ANT BEHAVIOR
3Department of Biomedical, Industrial and Human Factors Engineering
Natural behavior of ants have inspired scientists to mimic insect
operational methods to solve real-life complex problems
By observing ant behavior, scientists have begun to understand
their means of communication
Ant-based behavioral patterns to address combinatorial
problems - first proposed by Marco Dorigo
REAL ANT BEHAVIOR
Ants secrete pheromone while
traveling from the nest to food,
and vice versa in order to
communicate with one another to
find the shortest path
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EXPERIMENTAL STUDY OF ANTS
The more ants follow a trail, the more attractive that trail
becomes for being followed
NEST FOODNEST FOODNEST FOOD
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ANT Behavior
The more ants follow a trail, the more attractive that trail
becomes for being followed
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ANT Behavior
Even when the tracks are equal the behavior will encourage one
over the other--convergence (Deneubourg et al)
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ROUTE SELECTION
Ants are forced to decide whether they should go left or right, and
the choice that is made is a random decision
Pheromone accumulation is faster on the shorter path
The difference in pheromone content between the two paths over
time makes the ants choose the shorter path
Positive feedback mechanism to arrive at the shortest route while
foraging
Stygmergy or stigmergetic model of communication
Different optimization problems have been explored using a
simulation of this real ant behavior
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TRAVELING SALESMAN PROBLEM
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PROBLEM DEFINITION
OBJECTIVE
Given a set of n cities, the Traveling
Salesman Problem requires a salesman
to find the shortest route between the
given cities and return to the starting
city, while keeping in mind that each city
can be visited only once
10Department of Biomedical, Industrial and Human Factors Engineering
WHY IS TSP DIFFICULT TO SOLVE?
Finding the best solution may entail an exhaustive search for all
combinations of cities. This can be prohibitive as “n” gets very
large
Heuristics like a “greedy” route doesn’t guarantee optimal
solutions
d
e
gfb
c
h
a d
e
gfb
c
h
a
11Department of Biomedical, Industrial and Human Factors Engineering
TSP Applications
• Lots of practical applications
• Routing such as in trucking, delivery, UAVs
• Manufacturing routing such as movement of parts along manufacturing floor or the amount of solder on circuit board
• Network design such as determining the amount of cabling required
• Two main types
– Symmetric
– Asymmetric
12Department of Biomedical, Industrial and Human Factors Engineering
General Formulation - Symmetric
iji ij ijxd∑∑>min
ixxij ij jiij ∀≥+∑∑< > 2,,
32,, ≥∋∀≥+∑∑∑∑∈>∉ ∉>∈SSxxSi ijSj Si ijSj
jiji
+∈Bxji,
13Department of Biomedical, Industrial and Human Factors Engineering
General Formulation - Asymmetric
iji ij ijxd∑∑≠min
ixjij ∀=∑1,
ixiji ∀=∑1,
21, ≥∋∀≥∑∑∈∉ SSxSi Sj ji
+∈Bxji,
14Department of Biomedical, Industrial and Human Factors Engineering
TSP Heuristics
• Variety of heuristics used to solve the TSP
• The TSP is not only theoretically difficult it is also difficult in practical application since the tour breaking contraints get quite numerous
• As a result there have been a variety of methods proposed for the TSP
• Nearest Neighbor is a typical greedy approach
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Simple Examples
2 3 4 5 6 7 8 9 10
1 Chicago 96 105 50 41 86 46 29 56 70
2 Dallas 78 49 94 21 64 63 41 37
3 Denver 60 84 61 54 86 76 51
4 KC 45 35 20 26 17 18
5 Minn 80 36 55 59 64
6 OK City 46 50 28 8
7 Omaha 45 37 30
8 St Louis 21 45
9 Sprngfld 25
10 Wichita
1
28
3
9
7
6
5
4
6
21 2
6
22 2
2 1
621
2
2
16Department of Biomedical, Industrial and Human Factors Engineering
Nearest Neighbor Solution
Iteration Node Arc Cost Total
1 1
2 8 1,8 29
3 9 8,9 21 50
4 4 9,4 17 67
5 10 4,10 18 85
6 6 10,6 8 93
7 2 6,2 21 114
8 7 2,7 64 178
9 5 7,5 36 214
10 3 5,3 84 298
11 3,1 105 403
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0
50
100
150
200
250
300
350
400
450
500
0 50 100 150 200 250 300 350 400 450 500
Way Point
Depot
Obj Fx: min d
Larger TSP Example
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0
50
100
150
200
250
300
350
400
450
500
0 50 100 150 200 250 300 350 400 450 500
Way Point
Depot
d=3138
Initial Order Solution
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0
50
100
150
200
250
300
350
400
450
500
0 50 100 150 200 250 300 350 400 450 500
Way Point
Depot
d=2108
Nearest Neighbor Solution
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0
50
100
150
200
250
300
350
400
450
500
0 50 100 150 200 250 300 350 400 450 500
Way Point
Depot
d=1830
Tabu Search Solution
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THE ANT COLONY OPTIMIZATION METAHEURISTIC
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GOAL OF ACO HEURISTIC
Artificial ants form a multi-agent system performing the
functions as observed in the real ant system
Exploit stigmergistic communication
The ACO meta-heuristic relies on the co-operation of a
group of artificial ants to obtain a good solution to a
discrete optimization problem such as the TSP
Artificial ants are mutants of a real ant system
The resulting shortest route mapping determined by the agents
can be applied to the optimization problem
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ACO CHARACTERISTICS
Exploit a positive feedback mechanism
Demonstrate a distributed computational architecture
Exploit a global data structure that changes dynamically as each ant transverses the route
Has an element of distributed computation to it involving the population of ants
Involves probabilistic transitions among states or rather between nodes
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REAL vs. ARTIFICIAL ANTS
• Discrete time steps
• Memory Allocation
• Quality of Solution
• Time of Pheromone deposition
• Distance Estimation
REAL ANT ARTIFICIAL ANT
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FLOWCHART OF ACO
Have all cities been
visited
Have the maximum
Iterations been performed
START ACO
Locate ants randomly in cities across the grid and store the
current city in a tabu list
Determine probabilistically as to which city to visit next
Move to next city and place this city in the
tabu list
Record the length of tour and clear tabu list
Determine the shortest tour till now and
update pheromone
NO
YES
STOPACO
YESNO
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KEY PARAMETERS
Trail intensity is given by value of ij which indicates the intensity of the pheromone on the trail segment, (ij)
Trail visibility is ij = 1/dij
The importance of the intensity in the probabilistic transition is
The importance of the visibility of the trail segment is
The trail persistence or evaporation rate is given as
Q is a constant and the amount of peromone laid on a trail segment employed by an Ant; this amount may be modified
in various manners
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PROBABILISTIC CITY SELECTION
Helps determine the city to visit next while the ant is in a tour
Determined by variables such as the pheromone content in an
edge (i,j) at time instant t, heuristic function of the desirability of
adding edge, and their control parameters
[ ] [ ][ ] [ ]
)(
)(
0
)(
)(
)()(
iJjf
iJjift
t
tp
k
kilil
ijij
kij
ikJl
∉
∈
⎪⎪⎩
⎪⎪⎨
⎧
= ∑∈
βα
βα
ητ
ητ
28Department of Biomedical, Industrial and Human Factors Engineering
PHEROMONE UPDATING
Using the tour length for the k-th Ant, Lk, the quantity of
pheromone added to each edge belonging to the completed tour is
given by
( ) ( )( )tTjiedgeif
tTjiedgewhereL
Qt
k
k
k
kij ∉
∈
⎪⎩
⎪⎨⎧
=Δ),(
),(
0τ
)()()1()1( ttt ijijij Δ+−=+
The pheromone decay in each edge of a tour is given by
29Department of Biomedical, Industrial and Human Factors Engineering
ACTUALLY 3 ALGORITHMS
The ant-cycle is the approach discussed so far
Information is updated at the end of each tour as such function of tour length
The ant-density is an approach wherein the pheromone quantity Q is deposited once the segment is transversed
Pretty much a greedy approach (local information) and not really providing relative information
The ant-quantity is an approach wherein the pheromone quantity Q/dij is deposited once the segment is transversed
Also a greedy approach but providing some relative information by scaling Q by the length of the segment
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Consider the Case Studies in Papers
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EXTENSIONS
Found that communication among the ants via the intensity factor is important; makes sense
since it provides some global insight
Found that a good number of ants is about equal to the number of cities
Found that the initial distribution of the ants among the cities does not really matter
Found that an elitist strategy in which the segments on the best solution(s) is(are) continually
reinforced work well so long as there were not too few or too many elitist solutions
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APPLICATIONS
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APPLICATIONS
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Population-Based Incremental Learning
Lots of similarities to the ACO
Actually inspired by genetic algorithms
Generate members of a population randomly based on probability of selection functions that are
increased or decreased based on the quality of past solutions involving the member variables
Once a population is generated, evaluate and then increase or lower the probability used by the
generating vector to encourage better solutions.
35Department of Biomedical, Industrial and Human Factors Engineering
Population-Based Incremental Learning
Benefits
Will converge to solutions under correct circumstances
Efficient in terms of storage
Computationally pretty cheap
Involves learning
Disadvantages
Keeps primarily a local focus
Cannot handle interdependence among parameters very well
Will need to involve penalty functions for constraints
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Population-Based Incremental Learning
A solution to overcome disadvantages proposed by Miagkiky
and Punch
Combine reinforcement with population generation
37Department of Biomedical, Industrial and Human Factors Engineering
QUESTIONS??