 Ant colony optimization algorithms

In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs.
This algorithm is a member of ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis,^{[1]}^{[2]} 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 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.
Contents
Overview
Summary
In the natural world, ants (initially) wander randomly, and upon finding food return to their colony while laying down pheromone 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 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 original idea comes from observing the exploitation of food resources among ants, in which ants’ individually limited cognitive abilities have collectively been able to find the shortest path between a food source and the nest.
 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. ^{[3]} ^{[4]} 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" 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 selforganized 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., trails and attractiveness. 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., trail evaporation and daemon actions. Trail evaporation reduces all trail values over time thereby avoiding any possibilities of getting stuck in local optima. The daemon actions are used to bias the search process from a nonlocal perspective.
Common extensions
Here are some of most popular variations of ACO Algorithms
Elitist ant system
The global best solution deposits pheromone on every iteration along with all the other ants.
MaxMin ant system (MMAS)
Added Maximum and Minimum pheromone amounts [τ_{max},τ_{min}] Only global best or iteration best tour deposited pheromone. All edges are initialized to τ_{max} and reinitialized to τ_{max} when nearing stagnation. ^{[5]}
Ant Colony System
It has been presented above.^{[6]}
Rankbased 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.^{[7]}
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 graphbased ant system algorithm, and then algorithms for ACS and MMAS. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. In 2004, Zlochin and his colleagues^{[8]} showed that COAtype algorithms could be assimilated methods of stochastic gradient descent, on the crossentropy and estimation of distribution algorithm. They proposed these metaheuristics as a "researchbased model".
Example pseudocode 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 x to state y, corresponding to a more complete intermediate solution. Thus, each ant k computes a set A_{k}(x) of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant k, the probability of moving from state x to state y depends on the combination of two values, viz., the attractiveness η_{xy} of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level τ_{xy} 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 kth ant moves from state x to state y with probability
where
τ_{xy} is the amount of pheromone deposited for transition from state x to y, 0 ≤ α is a parameter to control the influence of τ_{xy}, η_{xy} is the desirability of state transition xy (a priori knowledge, typically 1 / d_{xy}, where d is the distance) and β ≥ 1 is a parameter to control the influence of η_{xy}.
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 xy, ρ 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 L_{k} is the cost of the kth ant's tour (typically length) and Q is a constant.
Applications
Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multitargets and parallel implementations. It has also been used to produce nearoptimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm 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 nearoptimal solutions to the travelling salesman problem. The first ACO algorithm was called the Ant system ^{[9]} and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest roundtrip 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 roundtrips 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
 Jobshop scheduling problem (JSP)^{[10]}
 Openshop scheduling problem (OSP)^{[11]}^{[12]}
 Permutation flow shop problem (PFSP)^{[13]}
 Single machine total tardiness problem (SMTTP)^{[14]}
 Single machine total weighted tardiness problem (SMTWTP)^{[15]}^{[16]}^{[17]}
 Resourceconstrained project scheduling problem (RCPSP)^{[18]}
 Groupshop scheduling problem (GSP)^{[19]}
 Singlemachine total tardiness problem with sequence dependent setup times (SMTTPDST)^{[20]}
 Multistage Flowshop Scheduling Problem (MFSP) with sequence dependent setup/changeover times^{[21]}
Vehicle routing problem
 Capacitated vehicle routing problem (CVRP)^{[22]}^{[23]}^{[24]}
 Multidepot vehicle routing problem (MDVRP)^{[25]}
 Period vehicle routing problem (PVRP)^{[26]}
 Split delivery vehicle routing problem (SDVRP)^{[27]}
 Stochastic vehicle routing problem (SVRP)^{[28]}
 Vehicle routing problem with pickup and delivery (VRPPD)^{[29]}^{[30]}
 Vehicle routing problem with time windows (VRPTW)^{[31]}^{[32]}^{[33]}
 Time Dependent Vehicle Routing Problem with Time Windows (TDVRPTW)^{[34]}
Assignment problem
 Quadratic assignment problem (QAP)^{[35]}
 Generalized assignment problem (GAP)^{[36]}^{[37]}
 Frequency assignment problem (FAP)^{[38]}
 Redundancy allocation problem (RAP)^{[39]}
Set problem
 Set covering problem(SCP)^{[40]}^{[41]}
 Set partition problem (SPP)^{[42]}
 Weight constrained graph tree partition problem (WCGTPP)^{[43]}
 Arcweighted lcardinality tree problem (AWlCTP)^{[44]}
 Multiple knapsack problem (MKP)^{[45]}
 Maximum independent set problem (MIS)^{[46]}
Others
 Classification^{[47]}
 Connectionoriented network routing^{[48]}
 Connectionless network routing^{[49]}^{[50]}
 Data mining ^{[51]}^{[47]}^{[52]}^{[53]}
 Discounted cash flows in project scheduling^{[54]}
 Distributed Information Retrieval^{[55]}^{[56]}
 Grid Workflow Scheduling Problem^{[57]}
 Image processing^{[58]}^{[59]}
 Intelligent testing system^{[60]}
 System identification^{[61]}^{[62]}
 Protein Folding^{[63]}^{[64]}
 Power Electronic Circuit Design^{[65]}
Definition difficulty
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths. It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses. Broadly speaking, ant colony algorithms are regarded as populated 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 multiagent algorithms using a probability distribution to make the transition between each iteration. 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 selforganization in biological systems has led to the concept of "swarm 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 "Stigmergy") 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.^{[66]}
Related methods
 Genetic algorithms (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 annealing (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 optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to selftune 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 search (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 system (AIS) algorithms are modeled on vertebrate immune systems.
 Particle swarm optimization (PSO), a Swarm intelligence method
 Intelligent Water Drops (IWD), a swarmbased optimization algorithm based on natural water drops flowing in rivers
 Gravitational Search Algorithm (GSA), a Swarm intelligence method
 Ant colony clustering method (ACCM), a method that make use of clustering approach,extending the ACO.
 Stochastic diffusion search (SDS), an agentbased probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partialfunctions
History
Chronology of Ant colony optimization algorithms.
 1959, PierrePaul Grassé invented the theory of Stigmergy to explain the behavior of nest building in termites;^{[67]}
 1983, Deneubourg and his colleagues studied the collective behavior of ants;^{[68]}
 1988, and Moyson Manderick have an article on selforganization among ants;^{[69]}
 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;^{[3]}
 1989, implementation of a model of behavior for food by Ebling and his colleagues;^{[70]}
 1991, M. Dorigo proposed the Ant System in his doctoral thesis (which was published in 1992^{[2]}). A technical report extracted from the thesis and coauthored by V. Maniezzo and A. Colorni ^{[71]} was published five years later;^{[9]}
 1996, publication of the article on Ant System;^{[9]}
 1996, Hoos and Stützle invent the MAXMIN Ant System;^{[5]}
 1997, Dorigo and Gambardella publish the Ant Colony System;^{[6]}
 1997, Schoonderwoerd and his colleagues developed the first application to telecommunication networks;^{[72]}
 1998, Dorigo launches first conference dedicated to the ACO algorithms;^{[73]}
 1998, Stützle proposes initial parallel implementations;^{[74]}
 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants ^{[75]}
 2000, special issue of the Future Generation Computer Systems journal on ant algorithms^{[76]}
 2000, first applications to the scheduling, scheduling sequence and the satisfaction of constraints;
 2000, Gutjahr provides the first evidence of convergence for an algorithm of ant colonies^{[77]}
 2001, the first use of COA Algorithms by companies (Eurobios and AntOptima);
 2001, IREDA and his colleagues published the first multiobjective algorithm ^{[78]}
 2002, first applications in the design of schedule, Bayesian networks;
 2002, Bianchi and her colleagues suggested the first algorithm for stochastic problem;^{[79]}
 2004, Zlochin and Dorigo show that some algorithms are equivalent to the stochastic gradient descent, the crossentropy and algorithms to estimate distribution ^{[8]}
 2005, first applications to protein folding problems.
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 ^ P.P. Grassé, La reconstruction du nid et les coordinations interindividuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs, Insectes Sociaux, numéro 6, p. 4180, 1959.
 ^ J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, Probabilistic Behaviour in Ants : a Strategy of Errors?, Journal of Theoretical Biology, numéro 105, 1983.
 ^ F. Moyson, B. Manderick, The collective behaviour of Ants : an Example of SelfOrganization in Massive Parallelism, Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.
 ^ M. Ebling, M. Di Loreto, M. Presley, F. Wieland, et D. Jefferson,An Ant Foraging Model Implemented on the Time Warp Operating System, Proceedings of the SCS Multiconference on Distributed Simulation, 1989
 ^ Dorigo M., V. Maniezzo et A. Colorni, Positive feedback as a search strategy, rapport technique numéro 91016, Dip. Elettronica, Politecnico di Milano, Italy, 1991
 ^ R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, Antbased load balancing in telecommunication networks, Adaptive Behaviour, volume 5, numéro 2, pages 169207, 1997
 ^ M. Dorigo, ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98, Bruxelles, Belgique, octobre 1998.
 ^ T. Stützle, Parallelization Strategies for Ant Colony Optimization, Proceedings of PPSNV, Fifth International Conference on Parallel Problem Solving from Nature, SpringerVerlag, volume 1498, pages 722731, 1998.
 ^ É. Bonabeau, M. Dorigo et G. Theraulaz, Swarm intelligence, Oxford University Press, 1999.
 ^ M. Dorigo , G. Di Caro et T. Stützle, Special issue on "Ant Algorithms", Future Generation Computer Systems, volume 16, numéro 8, 2000
 ^ W.J. Gutjahr, A graphbased Ant System and its convergence, Future Generation Computer Systems, volume 16, pages 873888, 2000.
 ^ S. Iredi, D. Merkle et M. Middendorf, BiCriterion Optimization with Multi Colony Ant Algorithms, Evolutionary MultiCriterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359372, 2001.
 ^ L. Bianchi, L.M. Gambardella et M.Dorigo, An ant colony optimization approach to the probabilistic traveling salesman problem, PPSNVII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.
Publications (selected)
 M. Dorigo, 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. Gambardella, 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 0195131592
 M. Dorigo & T. Stützle, 2004. Ant Colony Optimization, MIT Press. ISBN 0262042193
 M. Dorigo, 2007. "Ant Colony Optimization". Scholarpedia.
 C. Blum, 2005 "Ant colony optimization: Introduction and recent trends". Physics of Life Reviews, 2: 353373
 M. Dorigo, M. Birattari & T. Stützle, 2006 Ant Colony Optimization: Artificial Ants as a Computational Intelligence Technique. TR/IRIDIA/2006023
 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, ISSN01283790.
 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
 MIDACOSolver 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
Swarming Swarm algorithms  Agentbased models
 Ant colony optimization
 Ant robotics
 Artificial Ants
 Bees algorithm
 Bee colony optimization
 Boids
 Crowd simulation
 Firefly algorithm
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 Selfpropelled particles
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 Swarm (simulation)
Biological swarming  Agentbased model in biology
 Bait ball
 Collective animal behavior
 Feeding frenzy
 Flock
 Flocking
 Herd
 Herd behavior
 Mixedspecies foraging flock
 Mobbing behavior
 Pack hunter
 Patterns of selforganization in ants
 Sardine run
 Shoaling and schooling
 Sort sol
 Swarming behaviour
 Swarming (honey bee)
 Swarming motility
Animal migration Swarm robotics Related topics Categories:
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