Ant colony optimization
Encyclopedia
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
and operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...
, the ant colony optimization algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
(ACO) is a probabilistic
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
technique for solving computational problems which can be reduced to finding good paths through graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
s.
This algorithm is a member of ant colony algorithms family, in swarm intelligence
Swarm intelligence
Swarm intelligence is the collective behaviour of decentralized, self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence...
methods, and it constitutes some metaheuristic
Metaheuristic
In computer science, metaheuristic designates a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Metaheuristics make few or no assumptions about the problem being optimized and can search very large spaces...
optimizations. Initially proposed by Marco Dorigo
Marco Dorigo
Marco Dorigo is a research director for the Belgian Funds for Scientific Research , a professor in the computer science department of the University of Paderborn and a co-director of , the artificial intelligence lab of the Université Libre de Bruxelles.He is the proponent of the "ant colony...
in 1992 in his PhD thesis, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony
Ant colony
An ant colony is an underground lair where ants live, eat and mate. Colonies consist of a series of underground chambers, connected to each other and the surface of the earth by small tunnels. There are rooms for nurseries, food storage, and mating...
and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants.
Summary
In the natural world, ants (initially) wander randomly, and upon finding food return to their colony while laying down pheromonePheromone
A pheromone is a secreted or excreted chemical factor that triggers a social response in members of the same species. Pheromones are chemicals capable of acting outside the body of the secreting individual to impact the behavior of the receiving individual...
trails. If other ants find such a path, they are likely not to keep travelling at random, but to instead follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).
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, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has 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 (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback
Positive feedback
Positive feedback is a process in which the effects of a small disturbance on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn produces more of A. In contrast, a system that responds to a perturbation in a way that reduces its effect is...
eventually leads all the ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.
Detailed
- The first ant finds the food source (F), via any way (a), then returns to the nest (N), leaving behind a trail pheromone (b)
- Ants indiscriminately follow four possible ways, but the strengthening of the runway makes it more attractive as the shortest route.
- Ants take the shortest route, long portions of other ways lose their trail pheromones.
In a series of experiments on a colony of ants with a choice between two unequal length paths leading to a source of food, biologists have observed that ants tended to use the shortest route.
A model explaining this behaviour is as follows:
- An ant (called "blitz") runs more or less at random around the colony;
- If it discovers a food source, it returns more or less directly to the nest, leaving in its path a trail of pheromone;
- These pheromones are attractive, nearby ants will be inclined to follow, more or less directly, the track;
- Returning to the colony, these ants will strengthen the route;
- If there are two routes to reach the same food source then, in a given amount of time, the shorter one will be traveled by more ants than the long route;
- The short route will be increasingly enhanced, and therefore become more attractive;
- The long route will eventually disappear because pheromones are volatile;
- Eventually, all the ants have determined and therefore "chosen" the shortest route.
Ants use the environment as a medium of communication. They exchange information indirectly by depositing pheromones, all detailing the status of their "work". The information exchanged has a local scope, only an ant located where the pheromones were left has a notion of them. This system is called "Stigmergy
Stigmergy
Stigmergy is a mechanism of indirect coordination between agents or actions. The principle is that the trace left in the environment by an action stimulates the performance of a next action, by the same or a different agent...
" and occurs in many social animal societies (it has been studied in the case of the construction of pillars in the nests of termites).
The mechanism to solve a problem too complex to be addressed by single ants is a good example of a self-organized system. This system is based on positive feedback (the deposit of pheromone attracts other ants that will strengthen it themselves) and negative (dissipation of the route by evaporation prevents the system from thrashing). Theoretically, if the quantity of pheromone remained the same over time on all edges, no route would be chosen. However, because of feedback, a slight variation on an edge will be amplified and thus allow the choice of an edge. The algorithm will move from an unstable state in which no edge is stronger than another, to a stable state where the route is composed of the strongest edges.
The basic philosophy of the algorithm involves the movement of a colony of ants through the different states of the problem influenced by two local decision policies, viz.,
and 
. Thereby, each such ant incrementally constructs a solution to the problem. When an ant completes a solution, or during the construction phase, the ant evaluates the solution and modifies the trail value on the components used in its solution. This pheromone information will direct the search of the future ants. Furthermore, the algorithm also includes two more mechanisms, viz., 
and 
. 
reduces all trail values over time thereby avoiding any possibilities of getting stuck in local optima. The 
are used to bias the search process from a non-local perspective.Elitist ant system
The global best solution deposits pheromone on every iteration along with all the other ants.Max-Min ant system (MMAS)
Added Maximum and Minimum pheromone amounts [τmax,τmin]Only global best or iteration best tour deposited pheromone.
All edges are initialized to τmax and reinitialized to τmax when nearing stagnation.
Rank-based ant system (ASrank)
All solutions are ranked according to their length. The amount of pheromone deposited is then weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.Continuous orthogonal ant colony (COAC)
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively. By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy.The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems.
Convergence
For some versions of the algorithm, it is possible to prove that it is convergent (i.e. it is able to find the global optimum in a finite time). The first evidence of a convergence ant colony algorithm was made in 2000, the graph-based ant system algorithm, and then algorithms for ACS and MMAS. Like most metaheuristicMetaheuristic
In computer science, metaheuristic designates a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Metaheuristics make few or no assumptions about the problem being optimized and can search very large spaces...
s, it is very difficult to estimate the theoretical speed of convergence.
In 2004, Zlochin and his colleagues
showed that COA-type algorithms could be assimilated methods of stochastic gradient descent
Stochastic gradient descent
Stochastic gradient descent is an optimization method for minimizing an objective function that is written as a sum of differentiable functions.- Background :...
, on the cross-entropy and estimation of distribution algorithm. They proposed these metaheuristic
Metaheuristic
In computer science, metaheuristic designates a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Metaheuristics make few or no assumptions about the problem being optimized and can search very large spaces...
s as a "research-based model".
Example pseudo-code and formulae
procedure ACO_MetaHeuristic
while(not_termination)
generateSolutions
daemonActions
pheromoneUpdate
end while
end procedure
Edge selection
An ant is a simple computational agent in the ant colony optimization algorithm. It iteratively constructs a solution for the problem at hand. The intermediate solutions are referred to as solution states. At each iteration of the algorithm, each ant moves from a state
to state 
, corresponding to a more complete intermediate solution. Thus, each ant 
computes a set 
of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant 
, the probability 
of moving from state 
to state 
depends on the combination of two values, viz., the attractiveness 
of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level 
of the move, indicating how proficient it has been in the past to make that particular move.The trail level represents a posteriori indication of the desirability of that move. Trails are updated usually when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively.
In general, the
th ant moves from state 
to state 
with probability
where
is the amount of pheromone deposited for transition from state 
to 
, 0 ≤ 
is a parameter to control the influence of 
, 
is the desirability of state transition 
(a priori knowledge, typically 
, where 
is the distance) and 
≥ 1 is a parameter to control the influence of 
.Pheromone update
When all the ants have completed a solution, the trails are updated by
where
is the amount of pheromone deposited for a state transition 
, 
is the pheromone evaporation coefficient and 
is the amount of pheromone deposited, typically given for a TSP problem (with moves corresponding to arcs of the graph) by
where
is the cost of the 
th ant's tour (typically length) and 
is a constant.Applications
Protein
Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...
folding or routing vehicles
Vehicle routing problem
The vehicle routing problem is a combinatorial optimization and integer programming problem seeking to service a number of customers with a fleet of vehicles. Proposed by Dantzig and Ramser in 1959, VRP is an important problem in the fields of transportation, distribution and logistics...
and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.
It has also been used to produce near-optimal solutions to the travelling salesman problem
Travelling salesman problem
The travelling salesman problem is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find the shortest possible tour that visits each city exactly once...
. They have an advantage over simulated annealing
Simulated annealing
Simulated annealing is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete...
and genetic algorithm
Genetic algorithm
A genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems...
approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.
As a very good example, ant colony optimization algorithms have been used to produce near-optimal solutions to the travelling salesman problem.
The first ACO algorithm was called the Ant system
M. Dorigo, V. Maniezzo, et A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics--Part B , volume 26, numéro 1, pages 29-41, 1996.
and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest round-trip to link a series of cities.
The general algorithm is relatively simple and based on a set of ants, each making one of the possible round-trips along the cities. At each stage, the ant chooses to move from one city to another according to some rules:
- It must visit each city exactly once;
- A distant city has less chance of being chosen (the visibility);
- The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;
- Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
- After each iteration, trails of pheromones evaporate.
Scheduling problem
- Job-shop scheduling problem (JSP)
- Open-shop scheduling problem (OSP)
- Permutation flow shop problem (PFSP)
- Single machine total tardiness problem (SMTTP)
- Single machine total weighted tardiness problem (SMTWTP)
- Resource-constrained project scheduling problem (RCPSP)
- Group-shop scheduling problem (GSP)
- Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)
- Multistage Flowshop Scheduling Problem (MFSP) with sequence dependent setup/changeover times
Vehicle routing problem
- Capacitated vehicle routing problem (CVRP)
- Multi-depot vehicle routing problem (MDVRP)
- Period vehicle routing problem (PVRP)
- Split delivery vehicle routing problem (SDVRP)
- Stochastic vehicle routing problem (SVRP)
- Vehicle routing problem with pick-up and delivery (VRPPD)
- Vehicle routing problem with time windows (VRPTW)
- Time Dependent Vehicle Routing Problem with Time Windows (TDVRPTW)
Assignment problem
- Quadratic assignment problem (QAP)
- Generalized assignment problem (GAP)
- Frequency assignment problem (FAP)
- Redundancy allocation problem (RAP)
Set problem
- Set covering problem(SCP)
- Set partition problem (SPP)
- Weight constrained graph tree partition problem (WCGTPP)
- Arc-weighted l-cardinality tree problem (AWlCTP)
- Multiple knapsack problem (MKP)
- Maximum independent set problem (MIS)
Others
- Classification
- Connection-oriented network routing
- Connectionless network routing
- Data mining
- Discounted cash flows in project scheduling
- Distributed Information Retrieval
- Grid Workflow Scheduling Problem
- Image processing
- Intelligent testing system
- System identification
- Protein Folding
- Power Electronic Circuit Design
Definition difficulty
Broadly speaking, ant colony algorithms are regarded as populated
People
People is a plurality of human beings or other beings possessing enough qualities constituting personhood. It has two usages:* as the plural of person or a group of people People is a plurality of human beings or other beings possessing enough qualities constituting personhood. It has two usages:*...
metaheuristics with each solution represented by an ant moving in the search space. Ants mark the best solutions and take account of previous markings to optimize their search.
They can be seen as probabilistic multi-agent algorithms using a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
to make the transition between each iteration
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
. In their versions for combinatorial problems, they use an iterative construction of solutions.
According to some authors, the thing which distinguishes ACO algorithms from other relatives (such as algorithms to estimate the distribution or particle swarm optimization) is precisely their constructive aspect. In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective. Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions. However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists.
The collective behaviour of social insects remains a source of inspiration for researchers. The wide variety of algorithms (for optimization or not) seeking self-organization in biological systems has led to the concept of "swarm intelligence
Swarm intelligence
Swarm intelligence is the collective behaviour of decentralized, self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence...
", which is a very general framework in which ant colony algorithms fit.
Stigmergy algorithms
There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies (COA). In practice, the use of an exchange of information between ants via the environment (a principle called "StigmergyStigmergy
Stigmergy is a mechanism of indirect coordination between agents or actions. The principle is that the trace left in the environment by an action stimulates the performance of a next action, by the same or a different agent...
") is deemed enough for an algorithm to belong to the class of ant colony algorithms. This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation.
Related methods
- Genetic algorithmGenetic algorithmA genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems...
s (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded. - Simulated annealingSimulated annealingSimulated annealing is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete...
(SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search. - Reactive search optimizationReactive search optimizationReactive search optimization defines local-search heuristics based on machine learning, a family of optimization algorithms based on the local search techniques...
focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution. - Tabu searchTabu searchTabu search is a mathematical optimization method, belonging to the class of trajectory based techniques. Tabu search enhances the performance of a local search method by using memory structures that describe the visited solutions: once a potential solution has been determined, it is marked as...
(TS) is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution. While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated. To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions. It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space. - Artificial immune systemArtificial immune systemIn computer science, Artificial immune systems are a class of computationally intelligent systems inspired by the principles and processes of the vertebrate immune system...
(AIS) algorithms are modeled on vertebrate immune systems. - Particle swarm optimizationParticle swarm optimizationIn computer science, particle swarm optimization is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality...
(PSO), a Swarm intelligenceSwarm intelligenceSwarm intelligence is the collective behaviour of decentralized, self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence...
method - Intelligent Water Drops (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers
- Gravitational Search Algorithm (GSA), a Swarm intelligenceSwarm intelligenceSwarm intelligence is the collective behaviour of decentralized, self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence...
method - Ant colony clustering method (ACCM), a method that make use of clustering approach,extending the ACO.
- Stochastic diffusion searchStochastic Diffusion SearchStochastic diffusion search was first described in 1989 as a population-based, pattern-matching algorithm [Bishop, 1989]. It belongs to a family of swarm intelligence and naturally inspired search and optimisation algorithms which includes ant colony optimization, particle swarm optimization and...
(SDS), an agent-based probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partial-functions
History
Chronology of Ant colony optimization algorithms.- 1959, Pierre-Paul GrasséPierre-Paul GrasséPierre-Paul Grassé, born on November 27, 1895 in Périgueux and died on July 9, 1985, was a French zoologist, author of over 300 publications including the influential 35-volume Traité de Zoologie. He was an expert on termites.- Studies :...
invented the theory of StigmergyStigmergyStigmergy is a mechanism of indirect coordination between agents or actions. The principle is that the trace left in the environment by an action stimulates the performance of a next action, by the same or a different agent...
to explain the behavior of nest building in termites; - 1983, Deneubourg and his colleagues studied the collective behaviorCollective behaviorThe expression collective behaviour was first used by Robert E. Park, and employed definitively by Herbert Blumer, to refer to social processes and events which do not reflect existing social structure , but which emerge in a "spontaneous" way.Collective behavior might also be defined as action...
of ants; - 1988, and Moyson Manderick have an article on self-organization among ants;
- 1989, the work of Goss, Aron, Deneubourg and Pasteels on the collective behavior of Argentine ants, which will give the idea of Ant colony optimization algorithms;
- 1989, implementation of a model of behavior for food by Ebling and his colleagues;
- 1991, M. Dorigo proposed the Ant System in his doctoral thesis (which was published in 1992). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni was published five years later;
- 1996, publication of the article on Ant System;
- 1996, Hoos and Stützle invent the MAX-MIN Ant System;
- 1997, Dorigo and Gambardella publish the Ant Colony System;
- 1997, Schoonderwoerd and his colleagues developed the first application to telecommunicationTelecommunicationTelecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...
networks; - 1998, Dorigo launches first conference dedicated to the ACO algorithms;
- 1998, Stützle proposes initial parallel implementations;
- 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants
- 2000, special issue of the Future Generation Computer Systems journal on ant algorithms
- 2000, first applications to the scheduling, scheduling sequence and the satisfaction of constraints;
- 2000, Gutjahr provides the first evidence of convergenceLimit of a sequenceThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
for an algorithm of ant colonies - 2001, the first use of COA Algorithms by companies (Eurobios and AntOptima);
- 2001, IREDA and his colleagues published the first multi-objective algorithm
- 2002, first applications in the design of schedule, Bayesian networks;
- 2002, Bianchi and her colleagues suggested the first algorithm for stochasticStochasticStochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
problem; - 2004, Zlochin and Dorigo show that some algorithms are equivalent to the stochastic gradient descentStochastic gradient descentStochastic gradient descent is an optimization method for minimizing an objective function that is written as a sum of differentiable functions.- Background :...
, the cross-entropy and algorithms to estimate distribution - 2005, first applications to protein foldingProtein foldingProtein folding is the process by which a protein structure assumes its functional shape or conformation. It is the physical process by which a polypeptide folds into its characteristic and functional three-dimensional structure from random coil....
problems.
Publications (selected)
- M. DorigoMarco DorigoMarco Dorigo is a research director for the Belgian Funds for Scientific Research , a professor in the computer science department of the University of Paderborn and a co-director of , the artificial intelligence lab of the Université Libre de Bruxelles.He is the proponent of the "ant colony...
, 1992. Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italy. - M. Dorigo, V. Maniezzo & A. Colorni, 1996. "Ant System: Optimization by a Colony of Cooperating Agents", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.
- M. Dorigo & L. M. GambardellaLuca Maria GambardellaLuca Maria Gambardella has been co-director of the Swiss AI lab IDSIA since 1995. Together with former IDSIA senior researcher Marco Dorigo and others he made substantial contributions to the rapidly growing research field of ant colony optimization, a set of multi-agent methods inspired by the...
, 1997. "Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem". IEEE Transactions on Evolutionary Computation, 1 (1): 53–66. - M. Dorigo, G. Di Caro & L. M. Gambardella, 1999. "Ant Algorithms for Discrete Optimization". Artificial Life, 5 (2): 137–172.
- S. Ghosh, M. Dorigo et G. Theraulaz, 1999. Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press. ISBN 0-19-513159-2
- M. Dorigo & T. Stützle, 2004. Ant Colony Optimization, MIT Press. ISBN 0-262-04219-3
- M. Dorigo, 2007. "Ant Colony Optimization". Scholarpedia.
- C. Blum, 2005 "Ant colony optimization: Introduction and recent trends". Physics of Life Reviews, 2: 353-373
- M. Dorigo, M. Birattari & T. Stützle, 2006 Ant Colony Optimization: Artificial Ants as a Computational Intelligence Technique. TR/IRIDIA/2006-023
- Mohd Murtadha Mohamad,"Articulated Robots Motion Planning Using Foraging Ant Strategy",Journal of Information Technology - Special Issues in Artificial Intelligence, Vol.20, No. 4 pp. 163–181, December 2008, ISSN0128-3790.
- N. Monmarché, F. Guinand & P. Siarry (eds), "Artificial Ants", August 2010 Hardback 576 pp. ISBN 9781848211940.
External links
- Ant Colony Optimization Home Page
- AntSim - Simulation of Ant Colony Algorithms
- MIDACO-Solver General purpose optimization software based on Ant Colony Optimization (Matlab, Excel, C/C++, Fortran, Python)
- University of Kaiserslautern, Germany, AG Wehn: Ant Colony Optimization Applet Visualization of Traveling Salesman solved by Ant System with numerous options and parameters (Java Applet)
- Ant Farm Simulator