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Ant Colony Optimization - A Novel Approach to Solve NP-Hard
Problems
PRAGATI ENGINEERING COLLEGE
Keywords: Ant colony, Swarm, TSP
Abstract
The complex social behaviours of ants have been much studied, and computer scientists are now
finding that these behaviour patterns can provide models for solving difficult combinatorial
optimization problems. The attempt to develop algorithms inspired by one aspect of ant
behavior, the ability to find shortest paths, has become the field of ant colony optimization
(ACO). The ant colony optimization algorithm (ACO), introduced by Marco Dorigo, in the year
1992, is a probabilistic technique for solving computational problems which can be reduced to
finding good paths through graphs. This paper presents an overview of this rapidly growing
field, from its theoretical inception to practical applications, including descriptions of many
available ACO algorithms and their various applications to solve NP-Hard Problems. In this
paper, we also discuss the Travelling Salesman Problem.
1. Introduction.
1. Behaviour of Social Animals
Insects are the most diverse group of animals on the earth. Many insects possess very sensitive
organs of perception. perception. Swarm Intelligence(SI) is an artificial technique based on the
study of collective, self-organized systems. SI systems
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Many of numerous species of insects that live in colonies manifest group integration and division
of labour. Swarm intelligence are typically made up of a population of simple agents interacting
locally with one another and with their environment. Although there is normally no centralized
control structure dictating how individual agents should behave, local interactions between such
agents often lead to the emergence of global behaviour.
The rest of the paper is divided as follows: Sections 1.2 and 1.3 discuss the Ant Colony and other
Social Insect Behaviour approaches. Section 2 gives a theoretical overview and an example of
ACO. Variants of ACO are briefly explained in Section 3. In Section 4, the application aspects of
ACO to real world problems is discussed with emphasis on the TSP. The next section gives the
merits and demerits of ACO. Finally the conclusion and references are given.
2.
Ant Colonies
Ant Communication is accomplished primarily through chemicals called pheromones. Ants
communicate to one another by laying down pheromones along their trails. Other ants perceive
the presence of pheromone and tend to follow paths where pheromone concentration is higher.
Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive
strength. The more time it takes for an ant to travel down the path and back again, the more time
the pheromones have to evaporate. A short path gets marched over faster, and thus the
pheromone density remains high as it is laid on the path as fast as it can evaporate.
Figure 1:Ant Behaviour
A. Ants in a pheromone trail between nest and food; B. An obstacle interrupts the trail; C. Ants find two paths to go
around the obstacle; D. A new pheromone trail is formed along the shorter path.
Pheromone evaporation has also the advantage of avoiding the convergence to a locally optimal
solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be
excessively attractive to the following ones. In that case, the exploration of the solution space
would be constrained. Thus, when one ant finds a good path to a food source, other ants are more
likely to follow that path, and positive feedback eventually leaves all the ants following a single
path. Figure 1 describes this ant behaviour.
3.
Other Social Insects for
Optimization
Examples of other optimisation systems which can be found in nature include bird flocking,
animal herding, bacteria moulding and fish schooling, pedestrians, traffic. The beauty of the
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entire phenomenon lies in the seemingly intelligent group behaviour that emerges from multiple
individual animals.
2. Ant Colony Optimization.
2.1 The Theory of Ant Colonies
The distinctive behaviour of ants has been
extensively studied and has inspired a number
of methods and techniques among which the most successful is the general purpose optimization
technique known as Ant Colony Optimization (ACO). ACO exploits a similar mechanism
similar to that of the foraging behaviour of some ant species. From the early nineties, when the
first ant colony optimization algorithm was first proposed, ACO attracted the attention of
increasing numbers of researchers and many successful applications are now available.
Moreover, a substantial corpus of theoretical results is becoming available that provides useful
guidelines to researchers and practitioners in further applications of ACO
2.2 Characteristics of ACO
Stigmergy is a method of communication in systems in which the individual parts of the
system communicate with one another by modifying their local environment. The two main
characteristics of stigmergy that differentiate it from other forms of communication are:
i.
Stigmergy is an indirect, non-symbolic form of communication mediated by the
environment: insects exchange information by modifying the environment; and
ii.
Stigmergic information is local: it can only be accessed by those insects that visit the
locus in which it was released (or its immediate neighbourhood)
2.3 The Double Bridge Experiment
Deneubourg et al. thoroughly investigated the behaviour of ants. In an experiment known as the
“double bridge experiment”( Figure 2) the nest of a colony of Argentine ants was connected to a
food source by two bridges of equal lengths. In such a setting, ants start to explore the
surroundings of the nest and eventually reach the food source. Along their paths, the ants deposit
pheromone. Initially, each ant randomly chooses one of the two bridges. However, due to
random fluctuations, after some time, one of the two bridges presents a higher concentration of
pheromones than the other and , thus, attracts more ants. This brings a further amount of
pheromone on that bridge making it more attractive with the result that after some time the
whole colony converges towards the use of the same bridge.
This colony–level behaviour based on autocatalysis , that is , on the exploitation of positive
feedback , can be used by ants to find the shortest path between a food source and their nests. a
variant of the “double bridge experiment”, in which one bridge is significantly longer than the
other was considered. In this case, the stochastic fluctuations in the initial choice of a bridge
are much reduced and a second mechanism.
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Figure 2: Experimental Setup for the double bridge experiment (a) branches have equal
lengths;(b) branches have different lengths.
plays an important role: the ants choosing by chance the short bridge are the first to reach the
nest. The short bridge, therefore, receives pheromone earlier than the long one and this fact
increases the probability that further ants select it rather than the long one. A model of this
observed behaviour was developed.
2.4 ACO Procedure - Pseudo Algorithm
The ACO pseudo-algorithm is shown below. After initialization, the algorithm iterates over
three phases: at each iteration, a number of solutions are constructed by the ants; these solutions
are then improved through a local search (this step is optional), and finally the pheromone is
updated.
(Initialization)
Set parameters, initialize pheromone trails.
Initialize τιψ, for each ι,ψ
2. (Construction of AntSolution)
For each ant k do
Repeat
Compute ηιψ for each ι,ψ.
Choose in probability the state to
move into
Append the chosen move to move
the kth ant’s set tabu.
Until ant k has completed its solution.
[apply a local search(optional)]
3. (Pheromone 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.
1.
3. Variants of the ACO Algorithm.
Different ant colony optimization algorithms have been proposed .The original ACO algorithm
is known as Ant System and was proposed in the early nineties. It’s main characteristic is that at
each iteration, the pheromone values are updated by all the ants that have built a solution in the
iteration itself.
Since then a number of other ACO algorithms, variants of the original Ant System, were
presented, of which the two most successful algorithms were MAXMIN and Ant Colony system
(ACS).
MAXMIN is an improvement over the original AntSystem. Its characterizing elements are that
only the best ant updates the pheromone trails and that the value of the pheromone is bound.
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The most interesting contribution of ACS is the introduction of a local pheromone update in
addition to the pheromone update performed at the end of the construction process. The local
update done diversifies the search performed by subsequent ants during an iteration: by
decreasing the pheromone concentration on the traversed edges, ants encourage subsequent ants
to choose other edges an, hence, to produce different solutions. This makes it less likely that
several ants produce identical solutions during one iteration.
4. Applications of ACO in NP-Hard Problems.
In recent years, the interest of the scientific community in ACO has risen sharply. In fact,
several successful applications of ACO to a wide range of different discrete optimization
problems are now available. The large majority of these applications are to NP-Hard problems;
that is, to problems for which the best known algorithms that guarantee to identify an optimal
solution have exponential time worst case complexity. The use of such algorithms is often
infeasible in practice, and ACO algorithms can be useful for quickly finding high quality
solution. Other popular applications are to dynamic shortest path problems arising in
telecommunication networks problems. The number of successful applications to academic
problems has motivated people to adopt ACO for the solution of industrial problems, proving
that this computation of intelligence technique is also useful in real world applications.
4.1 Application to telecommunication networks
ACO algorithms have shown to be a very effective approach for routing problems in
telecommunication networks where the properties of the system, such as the cost of using links
or the availability of the nodes, vary over time. ACO algorithms were first applied to routing
problems in circuit-switched networks (such as telephone networks) and then in packet-switched
networks (such as local area networks or the internet). A well known example is AntNet which
has been extensively tested, in simulation, on different networks and under different traffic
patterns, proving to be highly adaptable and robust. A comparison with state-of-the-art routing
algorithms has shown that, in most of the considered situations, AntNet outperforms its
competitors.
Ant-based algorithms have given rise to several other routing algorithms, enhancing
performance in a variety of wired network scenarios.
1.
Application to Industrial Problems
The success of ACO on academic problems has raised the attention of a number of companies
that have started to sue ACO algorithms for real-world applications. They have applied ACO to
a number of scheduling problems such as a continuous two-stage flow shop problem with finite
reservoirs. The problems modeled included various real world constraints such as setup times,
capacity restrictions, resource compatibilities and maintenance calendars. Some researchers have
developed a set of tools for the solution of vehicle-routing problems whose optimization
algorithms are based on ACO. Examples of companies who have successfully used ACO are
EuroBios, AntOptima, DYVOIL, AntRoute, Migros (the main Swiss supermarket chain) and
Barilla (the main Italian pasta maker).
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2.
Conventional ACO for the Travelling Salesman Problem (TSP)
The traveling salesman problem (TSP) is a problem in discrete or combinatorial optimization. It
is a prominent illustration of a class of problems in computational complexity theory which are
hard to solve. The problem states that : Given a number of cities and the costs of traveling from
any city to any other city, what is the cheapest round-trip route that visits each city exactly once
and then returns to the starting city?
In ant colony optimization, the problem is tackled by simulating a number of artificial ants
moving on a graph that encodes the problem itself: each vertex represents a city and each edge
represents a connection between two cities. A variable called pheromone is associated with each
edge and can be read and modified by ants. Ant colony optimization is an iterative algorithm. At
each iteration , a number of artificial ants are considered. Each of them builds a solution by
walking from vertex to vertex on the graph with the constraint of not visiting any vertex that she
has already visited in her walk. At each step of the solution construction, an ant selects the
following vertex to be visited according to a stochastic mechanism that is biased by the
pheromone: when in vertex i, the following vertex is selected stochastically
among the previously unvisited ones. In particular, if j has not been previously visited, it can be
selected with a probability that is proportional to the pheromone associated with the edge (i,j). At
the end of the iteration, on the basis of the quality of the solution constructed by the ants, the
pheromone values are modified in order to bias ants in future iterations to construct solutions
similar to the best ones previously constructed. In Figure 3, ACO-TSP is given.
Figure 3 :An ant in city i chooses the next city to visit via a stochastic mechanism: If j has not been previously
visited, it can be selected with a probability that is proportional to the pheromone associated with edge (i.j).
Many of the tackled NP-Hard problems can be considered as falling into one of the following
categories: routing problems as they arise for example in the distribution of goods; assignment
problems, where a set of items (objects, activities etc.) has to be assigned to a given number of
resources(location, agents etc.)subject to some constraints; scheduling problems , which –in the
widest sense- concerned with the allocation of scare resources to tasks over time; and subset
problems , where a solution to a problem is considered to be a selection of a subset of available
items. In addition, ACO has been successfully applied to other problems emerging in fields such
as machine learning and bioinformatics.
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5. The Merits and Demerits of ACO.
Some of the inherent advantages of ACO algorithms are as follows :
● The Positive Feedback in ACO accounts for rapid discovery of good solutions
● ACO employs Distributed computation, which avoids premature convergence.
● The greedy heuristic used in ACO helps find an acceptable solution in the early solution
in the early stages of the search process.
Some of the disadvantages in ACO algorithms are:
● These algorithms have slower convergence than other Heuristics.
● They performed poorly for TSP problems larger than 75 cities.
● There is no centralized processor to guide the Ant System towards good solutions.
6. Conclusion
Ant Colony Optimization has been and continues to be a fruitful paradigm for designing
effective combinatorial optimization solution algorithms. After more than ten years of studies,
both its application effectiveness and its theoretical groundings have been demonstrated, making
ACO one of the most successful paradigm in the metaheuristic area.
Fifteen years ago, when the first ACO algorithm was introduced, taking inspiration from ants for
designing optimization algorithms seemed a crazy idea. The many successful applications have
changed our perspective: what seemed a far-out idea is now considered one of the most
promising approaches to the approximate solution of difficult optimization problems.
This paper is a brief write-up of the ideas and inspirations behind ACO. The effectiveness of
ACO is justified by the variety of applications of ACO. This being an emerging field of the
Swarm Intelligence area, much more exploration and experimentation can be done in future.
7. References
[1] Marco Dorigo, Mauro Birattari, and Thomas Stiitzli, “Ant Colony Optimization: Artificial Ants as a
Computational Intelligence Technique”, IEEE Transaction on Evolutionary Computation, November
2006, pp. 28-36.
[2] J.-L.Deneubourg, S.Aron, S.Goss, and J.-M.Pasteels, “The Self Organizing Exploratory Pattern of the
Argentine Ant”, Journal of Insect Behaviour, Vol.3, p.159, 1990.
[3] Official website of the Ant Colony Metaheuristic: www.aco-metaheuristic.org
[4] Other Related links:
www.en.wikipedia.org/wiki/Swarm_intelligence
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