Ants coony optimiztion problem

Ant Colony Optimization:

an introduction

2

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic

4

Ant colonies

• Distributed systems of social insects • Consist of simple individuals

6

Ant Cooperation

• Stigmergy – indirect communication

between individuals (ants)

Denebourg’s double bridge

experiments

• Studied Argentine ants I. humilis

8

Double bridge experiments:

equal lengths (2)

• Run for a number of trials

10

Double bridge experiments:

different lengths (2)

12

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems

3. The ACO metaheuristic

14

Combinatorial optimization

• Find values of discrete variables

Combinatorial optimization

Π = (S, f, Ω) – problem instance • _{S – set of candidate solutions} • f – objective function

• Ω – set of constraints

• – set of feasible solutions (with respect to Ω)

• Find globally optimal feasible solution s*

16

NP-hard combinatorial problems

• Cannot be exactly solved in polynomial time

• Approximate methods – generate

near-optimal solutions in reasonable time • No formal theoretical guarantees

Approximate methods

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Constructive algorithms

• Add components to solution incrementally • Example – greedy heuristics:

Local search

• Explore neighborhoods of complete solutions

• Improve current solution by local changes

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What is a metaheuristic?

• A set of algorithmic concepts

Examples of metaheuristics

• Simulated annealing • Tabu search

• Iterated local search

• Evolutionary computation

• _{Ant colony optimization}

22

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems

3. The ACO metaheuristic

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ACO metaheuristic

• A colony of artificial ants cooperate in finding good solutions

• Each ant – simple agent

• Ants communicate indirectly using

Combinatorial optimization

problem mapping (1)

• Combinatorial problem (S, f, Ω(t)) • Ω(t) – time-dependent constraints

 Example – dynamic problems

• Goal – find globally optimal feasible solution s*

• Minimization problem

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Combinatorial optimization

problem mapping (2)

• C = {c₁, c₂, …, c_Nc} – finite set of

components

• States of the problem:

X = {x = <c_i, c_j, …, c_h, …>, |x| < n < +∞} • Set of candidate solutions:

Combinatorial optimization

problem mapping (3)

• Set of feasible states:

• We can complete into a solution satisfying Ω(t)

• Non-empty set of optimal solutions:

X

X

~



X x  ~

S

X

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Combinatorial optimization

problem mapping (4)

S*

X

~

S

X

• X – states

• S – candidate solutions • – feasible states

• _{S* – optimal solutions}

Combinatorial optimization

problem mapping (5)

• Cost g(s, t) for each

• In most cases – g(s, t) ≡ f(s, t)

• G_C = (C, L) – completely connected graph • C – set of components

• L – edges fully connecting the components (connections)

• G – construction graph

S

s



S

s



~

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Combinatorial optimization

problem mapping (last



)

Construction graph

• Each component c_i or connection l_ij have associated:

 heuristic information

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Heuristic information

• A priori information about the problem • Does not depend on the ants

• On components c_i – η_i

• On connections l_ij – η_ij

• Meaning:

Pheromone trail

• Long-term memory about the entire search process

• On components c_i – τ_i

• On connections l_ij – τ_ij

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Artificial ant (1)

• Stochastic constructive procedure • Builds solutions by moving on G_C

• Has finite memory for:

– Implementing constraints Ω(t) – Evaluating solutions

Artificial ant (2)

• Has a start state x

• Has termination conditions ek

• From state x_r moves to a node from the neighborhood – Nk(x

r)

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Artificial ant (3)

• Selects a move with a probabilistic rule depending on:

 Pheromone trails and heuristic information of neighbor components and connections

 Memory

Artificial ant (4)

• Can update pheromone on visited components (nodes)

• and connections (edges) • Ants act:

 Concurrently

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The ACO metaheuristic

While not doStop():

ConstructAntSolutions

• A colony of ants build a set of solutions • Solutions are evaluated using the

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UpdatePheromones

• Two opposite mechanisms:

 Pheromone deposit

UpdatePheromones: pheromone

deposit

• Ants increase pheromone values on

visited components and/or connections • Increases probability to select visited

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UpdatePheromones: pheromone

evaporation

• Decrease pheromone trails on all

components/connections by a same value • Forgetting – avoid rapid convergence to

DaemonActions

• Optional centralized actions, e.g.:

 Local optimization

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ACO applications

• Traveling salesman

• Quadratic assignment • Graph coloring

• Multiple knapsack • Set covering

• Maximum clique • Bin packing

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic

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Traveling salesman problem

• N – set of nodes (cities), |N| = n

• A – set of arcs, fully connecting N

• Weighted graph G = (N, A)

• Each arc has a weight d_ij – distance • Problem:

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TSP: construction graph

• Identical to the problem graph • _{C = N}

• L = A

TSP: constraints

• All cities have to be visited

• Each city – only once

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TSP: pheromone trails

TSP: heuristic information

• η_ij = 1 / d_ij

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TSP: solution construction

• Select random start city

ACO algorithms for TSP

• Ant System

• Elitist Ant System

• Rank-based Ant System • Ant Colony System

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Ant System: Pheromone

initialization

• Pheromone initialization

Τ_ij = m / Cnn,

where:

 m – number of ants

 Cnn – path length of nearest-neighbor

Ant System: Tour construction

• Ant k is located in city i

• is the neighborhood of city i

• Probability to go to city :









p

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Tour construction:

comprehension

• α = 0 – greedy algorithm

• β = 0 – only pheromone is at work

 _{quickly leads to stagnation}









p

Ant System: update pheromone

trails – evaporation

Evaporation for all connections∀(i, j) ∈ L:

τ_ij ← (1 – ρ) τ_ij,

ρ ∈[0, 1] – evaporation rate

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Ant System: update pheromone

trails – deposit

• Tk – path of ant k

• Ck – length of path Tk

• _{Ants deposit pheromone on visited arcs:}

 

i

j

L

m

k

k ij ij

ij















,

,







 

Ck k

Elitist Ant System

• Best-so-far ant deposits pheromone on each iteration: best ij m k k ij ij

ij





e

















 







_





otherwise

T

j

i

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Rank-based Ant System

• Rank all ants

MAX-MIN Ant System

1. Only iteration-best or best-so-far ant deposits pheromone

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Ant Colony System

• Differs from Ant System in three points:

 More aggressive tour construction rule

 Only best ant evaporates and deposits pheromone

Ant Colony System

 









_



otherwise

System

Ant

like

q

q

if

j

,

,

max

arg





1.Tour Construction

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State of the art in TSP

• CONCORDE

http://www.tsp.gatech.edu/concorde.html • Solved an instance of 85900 cities

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Current ACO research activity

Further reading

• M. Dorigo, T. Stützle. Ant Colony Optimization. MIT Press, 2004.

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Next time…

Thank you!

Any questions?