ant colony optimisation
DESCRIPTION
Ant Colony Optimisation. Dr Jonathan Thompson, School of Mathematics, Cardiff University. Outline. Introduction to ant algorithms Overview of implementation for TSP (Dorigo et al.) Application to Dynamic Vehicle Routing (Wallace and Thompson) Application to Room Fitting (Thompson) - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/1.jpg)
SWORDS 2004
Ant Colony Optimisation
Dr Jonathan Thompson,
School of Mathematics, Cardiff University
![Page 2: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/2.jpg)
SWORDS 2004
Outline
Introduction to ant algorithms Overview of implementation for TSP (Dorigo et al.) Application to Dynamic Vehicle Routing (Wallace and
Thompson) Application to Room Fitting (Thompson) Application to Examination Timetabling / Graph
Colouring (Dowsland and Thompson) Conclusions
![Page 3: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/3.jpg)
SWORDS 2004
Classification of Meta-heuristics
Osman divided meta-heuristics into: Local search Population Construction
![Page 4: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/4.jpg)
SWORDS 2004
Solution SpaceFeasible regions
![Page 5: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/5.jpg)
SWORDS 2004
Local Search
Solution Space
![Page 6: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/6.jpg)
SWORDS 2004
Local Search
Solution Space
![Page 7: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/7.jpg)
SWORDS 2004
Local Search
Solution Space
![Page 8: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/8.jpg)
SWORDS 2004
Population Based
Solution Space
![Page 9: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/9.jpg)
SWORDS 2004
Population Based
Solution Space
![Page 10: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/10.jpg)
SWORDS 2004
Construction
Solution Space
![Page 11: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/11.jpg)
SWORDS 2004
Construction
Solution Space
![Page 12: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/12.jpg)
SWORDS 2004
Construction
Solution Space
![Page 13: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/13.jpg)
SWORDS 2004
Ant Algorithms
![Page 14: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/14.jpg)
SWORDS 2004
The Ant System in Nature
![Page 15: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/15.jpg)
SWORDS 2004
Ant Algorithms
Proposed by Dorigo et al. Probabilistic greedy construction heuristic Probabilities are adjusted according to information on
solution quality gained from previous solutions Initially use limited to routing problems but now being
applied to a broad range of problems.
![Page 16: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/16.jpg)
SWORDS 2004
Ant Systems for the TSP
No. ants = No. cities Each ant i produces a (probabilistic) solution, cost ci
Trail between each pair of cities in route i = trail + Q/ci
Trail is subject to evaporation i.e. trail = trail * σ, σ < 1 Each ant chooses next city with a probability proportional
to the distance and the trail.
![Page 17: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/17.jpg)
SWORDS 2004
Ants for TSP
Probability of next city from i being k is
P(ik) = [ik] [ik]
[jk] [jk] where
μik = Q / distance from i to k, ik = trail between i and k. Continues through generations until reaches
stagnation Has outperformed other meta-heuristics
![Page 18: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/18.jpg)
SWORDS 2004
The Static VRP
A set of geographically dispersed customers need to be serviced.
All routes start and end at the depot. A cost cij, is associated with travelling from customer i to
customer j. Customer i has demand Di, vehicles have capacity Q Construct a set of routes such that cost is minimised and
each customer is visited once and capacity constraints are satisfied.
![Page 19: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/19.jpg)
SWORDS 2004
VRP Routes
0
10
20
30
40
50
60
70
80
90
0 10 20 30 40 50 60 70 80
Series1
Series2
Series3
Series4
Example VRPVRP Routes
0
10
20
30
40
50
60
70
80
90
0 10 20 30 40 50 60 70 80
Series1
Series2
Series3
Series4
![Page 20: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/20.jpg)
SWORDS 2004
Adding Dynamism (DVRP)
Static VRP Customers are known a priori and do not change after
the routes have been constructed.
Dynamic VRP Customers can change after the initial routes have
been constructed.
![Page 21: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/21.jpg)
SWORDS 2004
Graphically ExplainedThe DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
New Customer Arrives
What happens now ?
![Page 22: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/22.jpg)
SWORDS 2004
Two Solution Approaches
Insertion Heuristics Insert the new customers in the current routes. Improvement heuristics when system is idle.
Re-Optimization i.e. solve static VRPs Re-planning from scratch when new information is
received / after each time period.
![Page 23: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/23.jpg)
SWORDS 2004
Graphically Explained – Re-OptThe DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
New Customer Arrives
If capacity is not exceeded then a new route is created.
Current route is discarded.
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
The DVRP
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
If exceeded then new routes are constructed
![Page 24: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/24.jpg)
SWORDS 2004
ACS-DVRP Pseudo-code
Overnight, an initial solution is created and commit drivers to customers for first time period + small commitment time.
Each time period is then considered in turn. Customers = existing customers who have not been
committed to + new customers who arrived during the previous time slot.
Solve problem during current time period and commit drivers to customers if start time is within the subsequent time period
Continue until all customers are committed to.
![Page 25: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/25.jpg)
SWORDS 2004
Roulette Wheel ChoiceA schedule is being created, truck is at customer i.A roulette wheel type selection is made to decide where to go next.
Size of the slot depends on attractiveness of the move.
kiFl
ijij
ijijkijP
![Page 26: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/26.jpg)
SWORDS 2004
Ants for the DVRP (ACS-DVRP)
Better solutions obtained by: Diversification: local trail update ij = (1 – ρ) ij + ρ 0
Select best choice with probability q, probabilistically with probability (1 – q).
Elitism – only use best ant solution to update trail globally
Maintain trails from one problem to another ij = (1 – γ) ij + γ 0
Apply descent to each ant solution Full parameter optimisation
![Page 27: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/27.jpg)
SWORDS 2004
Preliminary Results
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
1 2 3
Min, Ave, Max
3 Data Sets
![Page 28: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/28.jpg)
SWORDS 2004
Problems with the ACS-DVRP
Sequential route construction: Each route is completed before the next is created. Customers may be scheduled in an early truck,
though may have been better placed in a later one.
High variance within results: Difficult to interpret results. Minimized through use of trials.
![Page 29: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/29.jpg)
SWORDS 2004
Extended Roulette Wheel
Consider more than 1 truck at a time.
Arcs that would have been ignored previously are now considered.
Consider 3 trucks, 1 at customer i, 1 at customer j and 1 at the depot
![Page 30: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/30.jpg)
SWORDS 2004
Extended Roulette Wheel Results
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
1 2 3
Why ? An increase in the number of trucks used.
![Page 31: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/31.jpg)
SWORDS 2004
Truck Reduction Techniques
Capacity Utilization. Encourages the use of existing trucks over the
selection of a new truck. Let kij=(Qi+qj)/Q
Advanced construction formula
kiFl
ijijij
ijijijkij
k
kP
![Page 32: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/32.jpg)
SWORDS 2004
Truck Reduction Results
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
1 2 3
Reduction in average cost for each data set.
![Page 33: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/33.jpg)
SWORDS 2004
Truck Reduction GraphicRelationship Between Cost and No. of Trucks (k)
R2 = 0.7132
900
950
1000
1050
1100
1150
1200
1250
1300
1350
7 8 9 10 11 12 13 14 15
No. of Trucks (k)
Co
st
![Page 34: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/34.jpg)
SWORDS 2004
The Room Fitting Problem
Allocate examinations to rooms such that :- no rooms are overfull exams of different length are allocated to different
rooms
Additional objectives:-
Minimise the number of split exams Allocate exams to adjacent rooms
![Page 35: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/35.jpg)
SWORDS 2004
Simple Solutions
Randomly allocate exams to rooms Evaluate cost function = overflow + number of pairs of
exams of different duration in the same room Randomly move a single exam to a new room or swap
two exams Apply local search - SA / TS etc.
![Page 36: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/36.jpg)
SWORDS 2004
Ants for Room Fitting
Number of ants = number of exams Exams ordered according to size / duration Initially, each ant i produces a solution, with cost ci. Possible trail: Between pairs of exams? (1) Or Between exams and rooms (2) 1(j,k)= 1 (j,k) + 100/ci if ant i places exams j and k in the
same room 2 (j,l) = 2 (j,l) + 100/ci if ant i places exam j in room l All trails are reduced by an evaporation rate
![Page 37: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/37.jpg)
SWORDS 2004
Ants for Room Fitting
Each ant now produces another solution, by considering each exam in turn and choosing which room to place it in according to the two trails and the immediate cost.
The trails are combined so that the trail associated with allocating exam j to room k is combined with the trails between exam j and any other exams already allocated to room k.
![Page 38: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/38.jpg)
SWORDS 2004
The Trails
How should we combine the trails? Average of all trails. Average of the two trails. Alter the factor applied to each trail according to the cost
function. When producing trails, just use part of cost function
concerning each trail.
![Page 39: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/39.jpg)
SWORDS 2004
Experiments
Data Set One - Artificial, Optimal cost = 0, one optimal solution only. Size varies from 5 rooms, 13 exams to 20 rooms, 70 exams
Data Set Two - Real, Optimal cost is unknown. Size varies from 12 exams to 90 exams.
Five random runs Best parameters found by experimentation with artificial
data, then used on real data set.
![Page 40: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/40.jpg)
SWORDS 2004
Results from Artificial Data
0
2
4
6
8
10
12
14
16
18
Heuristic Descent SA Ants
![Page 41: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/41.jpg)
SWORDS 2004
Results from Real Data
0
20
40
60
80
100
120
Heuristic Descent SA Ants
![Page 42: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/42.jpg)
SWORDS 2004
Graph Colouring
Given a Graph with a set of vertices and a set of edges, assign to each vertex a colour so that no adjacent vertices have the same colour.
NP-Complete Applications include timetabling, frequency assignment
and register allocation.
![Page 43: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/43.jpg)
SWORDS 2004
Ants for Graph Colouring
Costa and Hertz, Ants Can Colour Graphs (1997) Each ant produces a solution by choosing vertices
randomly in proportion to some definition of cost. Allocate to some colour.
A trail records the quality of solution when pairs of vertices are in the same colour class.
Subsequent ants construct solutions, considering cost (visibility) and trail.
Trails are subject to evaporation.
![Page 44: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/44.jpg)
SWORDS 2004
Ants for Graph Colouring Normal Probabilistic selection: Trail update:
tij = ρtij + 1/q(s) for each solution s in which vertices i and j are in the same colour class
Trail calculation:
jk = tij / |Vk| where Vk = no. vertices already coloured k.
![Page 45: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/45.jpg)
SWORDS 2004
Ants for Graph Colouring
Eight variants based on RLF and DSatur: W = uncoloured vertices than can be included in current
colour class B = uncoloured vertices that cannot be included in current
colour class degX(i) = degree of vertex i in subgraph of vertices X RLF(i,j), i = visibility, j = choice of first vertex i = 1 degB(i) j = 1 max degW(i) i = 2 |W| - degW(i) j = 2 randomly i = 3 degBUW(i)
![Page 46: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/46.jpg)
SWORDS 2004
Ants for Graph Colouring
DSatur(i). Visibility = saturation degree i = 1. First available colour. i = 2. Vertex chosen wrt visibility. Colour chosen
probabilistically wrt trail. Results are encouraging. However Vesel and Zerovnik (2000) question quality of
results.
![Page 47: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/47.jpg)
SWORDS 2004
Improvements - Reward Function 40 colour graph No. Colours Trail Ratio
41 1
42 0.976
43 0.953
44 0.932
10 colour graph No. Colours Trail Ratio
11 1
12 0.917
13 0.846
14 0.785
![Page 48: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/48.jpg)
SWORDS 2004
Improved Reward Function
Instead reward function
= 1
q(s) - r
where r = best solution so far
N colour graph No. Extra Trail Ratio
Colours
1 1
2 0.5
3 0.33
4 0.25
![Page 49: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/49.jpg)
SWORDS 2004
Further Improvements
Solution 1 r colours rth colour class contains
several vertices
Solution 2 r colours rth colour class contains
one vertex
Instead limit number of colours to target value and use number of uncoloured vertices u(s) as the reward function.
![Page 50: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/50.jpg)
SWORDS 2004
Potential Problems
No trail from uncoloured vertices Instead trail between uncoloured vertices and all other
vertices is increased by 1/u(s) This is a diversification function.
![Page 51: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/51.jpg)
SWORDS 2004
Experiments
Initially used examination scheduling data Known number of colours (number of timeslots) Some structure to graphs Larger number of large cliques than random graphs
![Page 52: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/52.jpg)
SWORDS 2004
Examination Scheduling Data
7 datasets from Carter Problem sizes vary between 80 and 487 vertices Density varies between 6% and 29% Maximum clique between 10 and 21 Clique vertices have high degree Optimal colouring between 10 and 21
![Page 53: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/53.jpg)
SWORDS 2004
Results for Sample Dataset
No. optimal completions - HEC dataset
0
20
40
60
80
100
120
RLF(1,1) RLF(2,1) RLF(3,1) RLF(1,2) RLF(2,2) RLF(3,2) Dsat(1) Dsat(2)
ORIG EXTRA UNCOL DIVERS
![Page 54: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/54.jpg)
SWORDS 2004
Results for all Datasets
No. optimal completions
0
100200
300
400
500600
700
800
RLF(1,1) RLF(2,1) RLF(3,1) RLF(1,2) RLF(2,2) RLF(3,2) Dsat(1) Dsat(2)
NO TRL ORIG EXTRA UNCOL DIVERS
![Page 55: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/55.jpg)
SWORDS 2004
Results for Best Parameters
No. optimal completions for best /
0
5
10
15
20
25
30
35
RLF(1,1) RLF(2,1) RLF(3,1) RLF(1,2) RLF(2,2) RLF(3,2) Dsat(1) Dsat(2)
NO TRL ORIG EXTRA UNCOL DIVERS
![Page 56: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/56.jpg)
SWORDS 2004
Best Ant Implementation
RLF(1,2) Reward function of 1/u(s) Diversification included Alpha = 2, higher beta values (4 or 5) Evaporation rate = 0.5 Results confirmed on larger datasets Results comparable to those of other researchers
![Page 57: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/57.jpg)
SWORDS 2004
Random Graphs
Graphs used by Costa and Hertz (from Ferland and Fleurent)
20 G(100,0.5), 10 G(300,0.5), 5 G(500,0.5) and 2 G(1000,0.5)
Size C & H D & T
100 15.2 15.15
300 35.7 34.5
500 55.6 53.0
1000 111.0 100.5
![Page 58: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/58.jpg)
SWORDS 2004
Random Graphs
Graphs used by Costa and Hertz (from Ferland and Fleurent)
20 G(100,0.5), 10 G(300,0.5), 5 G(500,0.5) and 2 G(1000,0.5)
Size C & H D & T C et al.
100 15.2 15.15 14.95
300 35.7 34.5 33.3
500 55.6 53.0 49.5
1000 111.0 100.5 85.0
![Page 59: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/59.jpg)
SWORDS 2004
Random Graphs
% Improvement
80
85
90
95
100
105
100 300 500 1000
C & H
Power=1
Power=2
Power=3
Emphasise diversity function by raising to power
![Page 60: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/60.jpg)
SWORDS 2004
Improvements
Improve Construction – make several attempts at each colour class and select the one that minimises the no. edges in remaining graph
Change evaluation function to no. clashing edges if vertex inserted into best colour class
Add local search
![Page 61: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/61.jpg)
SWORDS 2004
Local Search
Shuffle – Reorder colours and move vertices into earlier colours if possible. Focus on likely colour classes i.e. consider them last.
Tree. Depth first search where uncoloured vertices are coloured into colours with least clashes, and clashing vertices are moved out, and we try to colour them elsewhere.
Tabu Search, considering clashing vertices. Tabu list contains vertex colour pairs
![Page 62: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/62.jpg)
SWORDS 2004
Best Results
Graph G & H Ants Time (secs)
Le450_15 15.4 15.0 43.0
Le450_25 26.0 25.8 365.0
Flat300 31.4 31.0 361.4
DSJC500 48.5 49.0 2007.6
DSJC250 28.1 28.0 440.2
G500 49.0* 49.0 1811.1
* = Costa et al.
![Page 63: Ant Colony Optimisation](https://reader033.vdocuments.site/reader033/viewer/2022061605/56814e00550346895dbb6bba/html5/thumbnails/63.jpg)
SWORDS 2004
Conclusions
Ant colony optimisation is a competitive meta-heuristic for a wide variety of problems
Often needs extra assistance e.g. local search Need the correct balance between intensification and
diversification (exploitation and exploration) Randomness is important Parameter heavy Future – consider more ill-structured / dynamic /
stochastic problems