Ant Colony Optimisation - Unit...

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Ant Colony OptimisationAlexander Mathews, Angeline Honggowarsito & Perry Brown

Image Source: http://baynature.org/articles/the-ants-go-marching-one-by-one/

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Contents

● Introduction to Ant Colony Optimisation● Natural Inspiration● Ant System

● Advanced Ant Colony Optimisation● Extensions to Ant System● Max-Min Ant System● Best-Worst Ant System

● Applications of Ant Colony Optimisation● The Generalised Assignment Problem (Static)● Packet-switched network Routing (Dynamic)

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ACO:Introduction

●Originally proposed by Marco Dorigo in 1992●To solve the Travelling Salesman Problem (TSP)●By simulation the movements of ants

●The original ACO was not competitive with specialised TSP algorithms●Many improvements have been proposed

●ACO can be extended to solve generic optimisation problems●Most applications are to routing and scheduling problems

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Natural Inspiration

Model the behaviour of ants searching for food● Model based on experimental observations● Ants mark a path to food using pheromone● Other ants can detect the pheromone trail● Ants tend to follow paths with more

pheromone

This is location based communication

An ant trail. Source: Wikipedia, Ants, Retrieved 12/9/12

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Natural Inspiration Example● Ants leave the nest at

regular intervals

● 50% take the top path

● 50% take the lower path

● Ants which find food return to the nest by a random path

● Ants taking the top path return first

● The top path starts to accumulate pheromone at A

● Ants leaving the nest tend to take the top path

● Has more pheromone

Graphics adapted from: Dorigo et al. 2006

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Towards an Algorithm

Want to solve different problems those solved by real ants● Intermediate nodes: are components of a solution● Edges: are possible connections between components● Food Sources: are a complete solutions (with a fitness)

Virtual ants are allowed:● Perfect memory: ants can exactly retrace their steps● Variable pheromone strength: strength related to quality of

solution found

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Towards an Algorithm

Typical example Travelling Salesman Problem (TSP)

(Find the shortest cycle that visits every city once and only once)

● Intermediate nodes: are cities● Edges: are paths between cities● Food: A completed cycle of the graph

Note: that nodes are cities not partially constructed tours.

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Ant System (AS)

We have:

● A population of ants (starting at random cities)● Ants which construct one solution per iteration● A simple heuristic giving the 'fitness' of an edge ● Only pheromone information kept to the next

iteration

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Ant System: Overview

While not done:

randomise ant positions

For each ant: Construct a random tour

For each edge: Evaporate pheromone

For each ant: Lay pheromone on each edge in tour

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● Evaporate pheromone from every edge

● Increase the pheromone on edges in the random tour

● Construct a random tour

● Initialise the pheromone for every edge

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After a number of iterations we get the optimal tour.

Hopefully.

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Ant System: Ant Movement

Until a solution is completed:

For each edge to a node not in the tour:

Calculate heuristic for edge

Check pheromone strength for edge

Calculate attractiveness of edge

Randomly select edge with probability based on attractiveness

Attractiveness is calculated as:

(need normalise across outgoing edges to get a probability)

(Pheromone)α∗(Heuristic)β

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Ant System: Ant Movement

● Calculate Attractiveness

● A(A) = 28.3

● A(B) = 8.9

● Convert to Probability

● P(A) = 0.76

● P(B) = 0.24

● An edge to (A) is likely

● Discard edges to nodes already in the tour

● Get the Heuristic and Pheromone values

● H(A) = 300-100 = 200

● H(B) = 300-220 = 80

● P(A) = 2, P(B) = 1

● An ant is at the green point trying to complete the tour

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Parameters

N : Number of ants

α : Influence of the pheromone trail

β : Influence of the heuristic

ρ : Fraction of pheromone that evaporates

● Low evaporation leads to slower convergence & possibly better solutions

Q : scales the strength of the pheromone laid

(Pheromone)α∗(Heuristic)β

Pheromone Strength=Q

Length of Tour

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Termination Conditions

● Iteration/Time limit● Solution is good enough● Stagnation: Only one tour will be tried as the

pheromone is too strong.

Or: run forever to maintain a good solution to a dynamic problem.

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Contents




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Elitist Ant System

• First improvement on Ant System

• Also called Elitist Strategy for Ant System

• Provide strong additional reinforcement to the arcs belonging to the best tour found

• Pheromone update rule

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Elitist Ant System

• Appropriate value for parameter e allows EAS to:

• Find better tours

• Find them in a lower number of iterations

• Evaporation rule is implemented as in Ant System

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Rank-Based Ant System

• Introduced by Bullnheimer, Hartl and Strauss in 1997

• Ranking of pheromone update

• The m ants are ascending ranked based on solutions’ quality . (S’1, S’2, ....... , S’m)

• Each ants deposit an amount of pheromone that decrease with its rank

• Experimental evaluation suggests that this system performs slightly better than EAS and significantly better than AS.

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Ant Colony System

• Another extensions of Ant System

• Three major improvements

• Different transition rule, pseudo-random proportional rule

• Probability Distribution

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ACS - Improvement

• Daemon triggers pheromone update

• Ant with global best solution, Sglobal-best

• Pheromone evaporation before update

• Online step-by-step pheromone trail update

• Rule:

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Max-Min Ant System (MMAS)

• Developed by Stuzzle and Hoos in 1996

• AS extension

• Similar to ACS, offline pheromone trail update

• Best Ant chosen to add pheromone

• Iteration-best ant or best-so-far ant

• Improve ant’s solutions using local optimisers

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MMAS - Extensions

• Pheromone trails range [ Tmin, Tmax ]

• Tmax :

• Tmin : constant factor lower than Tmax

• Pheromone trails are initialised to maximum allowed value

• Re-initialization of pheromone trails

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Best-Worst Ant System

• Proposed by Cordon et al. in 1999

• Incorporates Evolutionary Computation

• Transition rule and pheromone evaporation

• Similar with MMAS, use local optimisers to improve ant’s solutions

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Best-Worst Ant System

• Improvements:

• The best-worst pheromone trail update rule.

• Penalise all connection of the worst solution. Scurrent-worse

• Update rule:

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BWAS Improvement• Pheromone trail mutation for diversity in search

process.

• Pheromone is mutated with probability Pm.

• Adding or substracting the same amount in each iteration

• mut(it,Tthreshold) depend on the average of pheromone trails

• Re-initialization of the pheromone trail, setting every pheromone to T0

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Contents




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Further applications of ACO• Two major types of problem that ACO is particularly suited to

• (Static) NP-hard problems, including:

– Routing problems (e.g. TSP)

– Assignment problems (e.g. university timetabling)

– Scheduling problems (e.g. scheduling project tasks)

– Subset problems (e.g. multiple knapsack problem)

• Dynamic shortest path problems

– e.g. routing in packet-switched networks

• N.B.: The terms static and dynamic here refer to whether or not the problem instance changes at run-time

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Defining the pheromone trails• Key consideration for applications: decide what the pheromone

trails should signify

• Relative positioning

– Trails represent the desirability of choosing a particular solution component j directly after some other component i

– Intuitively suited to problems for which the position of components in the overall sequence is inconsequential (e.g. in TSP, no preference as to which cities get visited sooner or later)

• Absolute positioning

– Trails represent the desirability of choosing a particular solution component j at the ith position in the sequence

– Intuitively suited to problems for which the position of components in the overall sequence is significant (e.g. in scheduling project tasks, different tasks are due by different dates)

• Either approach is possible for any problem, but the intuitive choices described above generally give much better performance

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Example application: Generalised Assignment Problem (GAP)

• Example of a static NP-hard problem

• A number of tasks to be distributed among a number of agents, each with a limited capacity

• Assigning a particular task to a particular agent involves using a certain amount of that agent’s capacity

• Each task-to-agent assignment also incurs a specific cost

• Goal is to minimise the total cost of assigning all tasks within the agents’ capacity constraints

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Representing the problem graphically• Both tasks and agents represented by nodes

• Ants must walk from task to agent to signify assignment, then back to next task until all tasks have been visited and assigned

Tasks

Agents

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Tasks

Agents

Assignment

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Tasks

Agents

Assignment

Move on to next task

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Tasks

Agents

Assignment

Move on to next task

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Aspects of the problem representation• Two types of pheromone deposits

– One to determine which agent to assign a particular task to (relative)

– Other to determine best order in which to assign tasks (absolute)

• Heuristic information (static)

– Our pre-existing knowledge of this particular problem suggests that assignments with low cost and/or capacity usage should be preferred

• Range of feasible solutions limited by capacity constraints

– Strategy: assign a task to an agent with sufficient remaining capacity whenever possible

– Ants that produce an infeasible solution (i.e. one that violates capacity constraints) may have their pheromone contribution reduced

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Example application: packet-switched network routing (dynamic)

• Problem involves a network with links connecting pairs of nodes, reaching one node from another may involve traversing a number of links in sequence

• Each link has a cost associated with using it

• Goal to find minimum cost path between every pair of nodes

• Normally an easy problem, except that in this instance the link costs are subject to change over time

• ACO compared with several state-of-the-art algorithms for this application, performed superior to all of them

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Aspects of the problem representation• Two types of pheromone deposits

– One to determine which agent to assign a particular task to (relative)

– Other to determine best order in which to assign tasks (absolute)

• Heuristic information (static)

– Our pre-existing knowledge of this particular problem suggests that assignments with low cost and/or capacity usage should be preferred

• Range of feasible solutions limited by capacity constraints

– Strategy: assign a task to an agent with sufficient remaining capacity whenever possible

– Ants that produce an infeasible solution (i.e. one that violates capacity constraints) may have their pheromone contribution reduced

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References

• M. Dorigo and T. Stutzle, "The ant colony optimization metaheuristic: algorithms, applications, and advances," in Handbook of Metaheuristics, Eds. F. Glover and G. Kochenberger, New York: Springer, 2003, pp. 250-285.

• M. Dorigo et al., "Ant colony optimization," IEEE Comput. Intell. Mag, vol. 1, (4), pp. 28-39, Nov. 2006.

• G. Di Caro and M. Dorigo, "AntNet: distributed stigmergetic control for communications networks," J. Artif. Int. Res., vol. 9, (1), pp. 317-365, Aug. 1998.

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References

• B. Bullnheimer, R. F. Hartl and C. Strauss, A New Rank Based Version of the Ant System: A Computational Study. Central European Journal for Operations Research and Economics, 7(1):25-38, 1999.

• O. Cordon, F. Herrera, and T. Stützle, A Review on the Ant Colony Optimization Metaheuristic: Basis, Models and New Trends. Mathware and Soft Computing, 9(2-3), pp. 141--175, 2002h

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References

• M.Dorigo and L. M. Gambardella. Ant Colony System: A cooperative learning approach to the travelling salesman problem. IEEE Transaction on Evolutionary Computation, 1(1):53-66, 1997.

• T. Stützle and H. H. Hoos, MAX-MIN Ant System. Future Generation Computer Systems

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References

• O.Cordon, I. Fernandez de Viana and F. Herrera and L. Moreno. A new ACO model integration evolutionary computation concepts: The best-worst Ant System. In M. Dorigo, M. Middendorf, and T.Stutzle, editors, Abstract proceedings of ANTS2000 - From Ant Colonies to Artificial Ants: A Series of International Workshops on Ant Algorithms, pages 22 - 29, IRIDIA, Universite Libre de Bruxelles, Belgium, 2000.

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