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| [[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]
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| In [[computer science]] and [[operations research]], the '''ant colony optimization''' [[algorithm]] '''(ACO)''' is a [[probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[graph (mathematics)|graph]]s.
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| This algorithm is a member of the '''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,<ref>A. Colorni, M. Dorigo et V. Maniezzo, ''Distributed Optimization by Ant Colonies'', actes de la première conférence européenne sur la vie artificielle, Paris, France, Elsevier Publishing, 134-142, 1991.</ref><ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy, 1992.</ref> 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 [[ant colony|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.
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| ==Overview==
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| ===Summary===
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| In the natural world, ants (initially) wander [[random]]ly, 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|Ant communication]]).
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| 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.
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| 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 to 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.
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| ==Common extensions==
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| Here are some of most popular variations of ACO Algorithms
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| ===Elitist ant system===
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| The global best solution deposits pheromone on every iteration along with all the other ants.
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| ===Max-Min ant system (MMAS)===
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| Added Maximum and Minimum pheromone amounts [τ<sub>max</sub>,τ<sub>min</sub>]
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| Only global best or iteration best tour deposited pheromone <MAZ>.
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| All edges are initialized to τ<sub>max</sub> and reinitialized to τ<sub>max</sub> when nearing stagnation.
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| <ref name="T. Stützle et H.H. Hoos">T. Stützle et H.H. Hoos, ''MAX MIN Ant System'', Future Generation Computer Systems, volume 16, pages 889-914, 2000</ref>
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| ===Ant Colony System===
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| It has been presented above.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem'', IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 53-66, 1997.</ref>
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| ===Rank-based ant system (ASrank)===
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| 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.
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| ===Continuous orthogonal ant colony (COAC)===
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| 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.
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| 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.<ref>[http://eprints.gla.ac.uk/3894/ X Hu, J Zhang, and Y Li (2008). Orthogonal methods based ant colony search for solving continuous optimization problems. ''Journal of Computer Science and Technology'', 23(1), pp.2-18.]</ref>
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| ===Recursive Ant Colony Optimization===
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| It is a recursive form of Ant System which runs nested ant systems to increase the precision of output.
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| <ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology (RAIT), 2012 1st International Conference on , vol., no., pp.448,454, 15-17 March 2012</ref>
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| ==Convergence==
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| 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 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 [[metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence.
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| In 2004, Zlochin and his colleagues<ref name="Zlochin model-based search">M. Zlochin, M. Birattari, N. Meuleau, et M. Dorigo, ''Model-based search for combinatorial optimization: A critical survey'', Annals of Operations Research, vol. 131, pp. 373-395, 2004.</ref> | |
| showed that COA-type algorithms could be assimilated methods of [[stochastic gradient descent]], on the [[cross-entropy]] and [[estimation of distribution algorithm]]. They proposed these [[metaheuristic]]s as a "[[research-based model]]".
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| ==Example pseudo-code and formulae==
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| <source lang="Java">
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| procedure ACO_MetaHeuristic
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| while(not_termination)
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| generateSolutions()
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| daemonActions()
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| pheromoneUpdate()
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| end while
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| end procedure
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| </source>
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| ===Edge selection===
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| 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 <math>x</math> to state <math>y</math>, corresponding to a more complete intermediate solution. Thus, each ant <math>k</math> computes a set <math>A_k(x)</math> of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant <math>k</math>, the probability <math>p_{xy}^k</math> of moving from state <math>x</math> to state <math>y</math> depends on the combination of two values, viz., the ''attractiveness'' <math>\eta_{xy}</math> of the move, as computed by some heuristic indicating the ''a priori'' desirability of that move and the ''trail level'' <math>\tau_{xy}</math> of the move, indicating how proficient it has been in the past to make that particular move.
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| 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.
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| In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability
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| <math>
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| p_{xy}^k =
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| \frac
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| { (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }
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| { \sum_{y\in \mathrm{allowed}_y} (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }
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| </math>
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| where
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| <math>\tau_{xy}</math> is the amount of pheromone deposited for transition from state <math>x</math> to <math>y</math>, 0 ≤ <math>\alpha</math> is a parameter to control the influence of <math>\tau_{xy}</math>, <math>\eta_{xy}</math> is the desirability of state transition <math>xy</math> (''a priori'' knowledge, typically <math>1/d_{xy}</math>, where <math>d</math> is the distance) and <math>\beta</math> ≥ 1 is a parameter to control the influence of <math>\eta_{xy}</math>. <math>\tau_{xy}</math> and <math>\eta_{xy}</math> represent the attractiveness and trail level for the other possible state transitions.
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| ===Pheromone update===
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| When all the ants have completed a solution, the trails are updated by
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| <math>
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| \tau_{xy} \leftarrow
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| (1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy}
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| </math>
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| where <math>\tau_{xy}</math> is the amount of pheromone deposited for a state transition <math>xy</math>, <math>\rho</math> is the ''pheromone evaporation coefficient'' and <math>\Delta \tau^{k}_{xy}</math> is the amount of pheromone deposited by <math>k</math>th ant, typically given for a [[Travelling salesman problem|TSP]] problem (with moves corresponding to arcs of the graph) by
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| <math>
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| \Delta \tau^{k}_{xy} =
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| \begin{cases}
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| Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\
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| 0 & \mbox{otherwise}
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| \end{cases}
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| </math>
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| where <math>L_k</math> is the cost of the <math>k</math>th ant's tour (typically length) and <math>Q</math> is a constant.
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| ==Applications==
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| [[File:Knapsack ants.svg|thumb|[[Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]
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| Ant colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to [[protein]] folding or [[Vehicle routing problem|routing vehicles]] and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations.
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| It has also been used to produce near-optimal 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.
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| The first ACO algorithm was called the Ant system
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| <ref name="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.</ref>
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| 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.
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| 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:
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| # It must visit each city exactly once;
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| # A distant city has less chance of being chosen (the visibility);
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| # The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;
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| # Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
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| # After each iteration, trails of pheromones evaporate.
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| [[File:Aco TSP.svg|thumb|600px|center]]
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| ===Scheduling problem===
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| *Job-shop scheduling problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''Classification with Ant Colony Optimization'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.
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| </ref>
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| *Open-shop scheduling problem (OSP)<ref>B. Pfahring, "Multi-agent search for open scheduling: adapting the Ant-Q formalism," Technical report TR-96-09, 1996.</ref><ref>C. Blem, "Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling," Technical report TR/IRIDIA/2003-17, 2003.</ref>
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| *Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref>
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| *Single machine total tardiness problem (SMTTP)<ref>A. Baucer, B. Bullnheimer, R. F. Hartl and C. Strauss, "Minimizing total tardiness on a single machine using ant colony optimization," Central European Journal for Operations Research and Economics, vol.8, no.2, pp.125-141, 2000.</ref>
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| *Single machine total weighted tardiness problem (SMTWTP)<ref>M. den Besten, "Ants for the single machine total weighted tardiness problem," Master’s thesis, University of Amsterdam, 2000.</ref><ref>M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of Lecture Notes in Computer Science, pp.611-620, 2000.</ref><ref>D. Merkle and M. Middendorf, "An ant algorithm with a new pheromone evaluation rule for total tardiness problems," Real World Applications of Evolutionary Computing, vol. 1803 of Lecture Notes in Computer Science, pp.287-296, 2000.</ref>
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| *Resource-constrained project scheduling problem (RCPSP)<ref>D. Merkle, M. Middendorf and H. Schmeck, "Ant colony optimization for resource-constrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2000), pp.893-900, 2000.</ref>
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| *Group-shop scheduling problem (GSP)<ref>C. Blum, "ACO applied to group shop scheduling: a case study on intensification and diversification," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref>
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| *Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequence-dependent setup times," Journal of the Operational Research Society, vol.53, pp.895-906, 2002.</ref>
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| *Multistage Flowshop Scheduling Problem (MFSP) with sequence dependent setup/changeover times<ref>A. V. Donati, V. Darley, B. Ramachandran, "An Ant-Bidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, ISBN 978-3-540-72959-4, pp.111-138, 2008.</ref>
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| ===Vehicle routing problem===
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| *Capacitated vehicle routing problem (CVRP)<ref>P. Toth, D. Vigo, "Models, relaxations and exact approaches for the capacitated vehicle routing problem," Discrete Applied Mathematics, vol.123, pp.487-512, 2002.</ref><ref>J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.</ref><ref>T. K. Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol.29, pp.607-629, 2003.</ref>
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| *Multi-depot vehicle routing problem (MDVRP)<ref>S. Salhi and M. Sari, "A multi-level composite heuristic for the multi-depot vehicle fleet mix problem," European Journal for Operations Research, vol.103, no.1, pp.95-112, 1997.</ref>
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| *Period vehicle routing problem (PVRP)<ref>E. Angelelli and M. G. Speranza, "The periodic vehicle routing problem with intermediate facilities," European Journal for Operations Research, vol.137, no.2, pp.233-247, 2002.</ref>
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| *Split delivery vehicle routing problem (SDVRP)<ref>S. C. Ho and D. Haugland, "A tabu search heuristic for the vehicle routing problem with time windows and split deliveries," Computers & Operations Research, vol.31, no.12, pp.1947-1964, 2004.</ref>
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| *Stochastic vehicle routing problem (SVRP)<ref>N. Secomandi, "Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands," Computers & Operations Research, vol.27, no.11, pp.1201-1225, 2000.</ref>
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| *Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "Solving the pickup and delivery problem with time windows using reactive tabu search," Transportation Research Part B, vol.34, no. 2, pp.107-121, 2000.</ref><ref>R. Bent and P.V. Hentenryck, "A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.</ref>
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| *Vehicle routing problem with time windows (VRPTW)<ref>A. Bachem, W. Hochstattler and M. Malich, "The simulated trading heuristic for solving vehicle routing problems," Discrete Applied Mathematics, vol. 65, pp.47-72, 1996..</ref><ref>[57] S. C. Hong and Y. B. Park, "A heuristic for bi-objective vehicle routing with time window constraints," International Journal of Production Economics, vol.62, no.3, pp.249-258, 1999.</ref><ref>R. A. Rusell and W. C. Chiang, "Scatter search for the vehicle routing problem with time windows," European Journal for Operations Research, vol.169, no.2, pp.606-622, 2006.</ref>
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| *Time Dependent Vehicle Routing Problem with Time Windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "Time Dependent Vehicle Routing Problem with a Multi Ant Colony System", European Journal of Operational Research, vol.185, no.3, pp.1174–1191, 2008.</ref>
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| *Vehicle Routing Problem with Time Windows and Multiple Service Workers (VRPTWMS)
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| ===Assignment problem===
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| *[[Quadratic assignment problem]] (QAP)<ref>T. Stützle, "MAX-MIN Ant System for the quadratic assignment problem," Technical Report AIDA-97-4, FB Informatik, TU Darmstadt, Germany, 1997.</ref>
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| *[[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "Adaptive search heuristics for the generalized assignment problem," Mathware & soft computing, vol.9, no.2-3, 2002.</ref><ref>M. Yagiura, T. Ibaraki and F. Glover, "An ejection chain approach for the generalized assignment problem," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref>
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| *[[Frequency assignment problem]] (FAP)<ref>K. I. Aardal, S. P. M.van Hoesel, A. M. C. A. Koster, C. Mannino and Antonio. Sassano, "Models and solution techniques for the frequency assignment problem," A Quarterly Journal of Operations Research, vol.1, no.4, pp.261-317, 2001.</ref>
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| *[[Redundancy allocation problem]] (RAP)<ref>Y. C. Liang and A. E. Smith, "An ant colony optimization algorithm for the redundancy allocation problem (RAP)," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref>
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| ===Set problem===
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| *[[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "A new version of ant system for subset problems," Proceedings of the 1999 Congress on Evolutionary Computation(CEC 99), vol.2, pp.1458-1464, 1999.</ref><ref>R. Hadji, M. Rahoual, E. Talbi and V. Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp.63-66, 2000.</ref>
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| *[[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "An ant-based framework for very strongly constrained problems," Proceedings of ANTS2000, pp.222-227, 2002.</ref>
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| *Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"Colored Ant System and local search to design local telecommunication networks," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref>
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| *Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "Metaheuristics for the edge-weighted k-cardinality tree problem," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref>
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| *Multiple knapsack problem (MKP)<ref>[http://parallel.bas.bg/~stefka/heuristic.ps S. Fidanova, "ACO algorithm for MKP using various heuristic information"], Numerical Methods and Applications, vol.2542, pp.438-444, 2003.</ref>
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| *Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "An ant system for the maximum independent set problem," Proceedings of the 2001 Argentinian Congress on Computer Science, vol.2, pp.1027-1040, 2001.</ref>
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| ===Others===
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| *Classification<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "Classification with Ant Colony Optimization", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref>
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| *Connection-oriented network routing<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First Internation Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref>
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| *Connectionless network routing<ref>G.D. Caro and M. Dorigo "AntNet: a mobile agents approach to adaptive routing," Proceedings of the Thirty-First Hawaii International Conference on System Science, vol.7, pp.74-83, 1998.</ref><ref>G. D. Caro and M. Dorigo, "Two ant colony algorithms for best-effort routing in datagram networks," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS’98), pp.541-546, 1998.</ref>
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| *Data mining <ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "Editorial Survey: Swarm Intelligence for Data Mining," Machine Learning, volume 82, number 1, pp. 1-42, 2011</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "An ant colony algorithm for classification rule discovery," Data Mining: A heuristic Approach, pp.191-209, 2002.</ref><ref>R. S. Parpinelli, H. S. Lopes and A. A Freitas, "Data mining with an ant colony optimization algorithm," IEEE Transaction on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref>
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| *Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "Optimizing Discounted Cash Flows in Project Scheduling--An Ant Colony Optimization Approach", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews Vol.40 No.5 pp.64-77, Jan. 2010.</ref>
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| *Distributed Information Retrieval<ref>D. Picard, A. Revel, M. Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010</ref><ref>D. Picard, M. Cord, A. Revel, "Image Retrieval over Networks : Active Learning using Ant Algorithm", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref>
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| *Grid Workflow Scheduling Problem<ref>W. N. Chen and J. ZHANG "Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 31, No. 1,pp.29-43,Jan 2009.</ref>
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| *Image processing<ref>S. Meshoul and M Batouche, "Ant colony system with extremal dynamics for point matching and pose estimation," Proceeding of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, " Edge detection using ant algorithms", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref>
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| *Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "An Intelligent Testing System Embedded with an Ant Colony Optimization Based Test Composition Method", IEEE Transactions on Systems, Man, and Cybernetics--Part C: Applications and Reviews, Vol. 39, No. 6, pp. 659-669, Dec 2009.</ref>
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| *System identification<ref>L. Wang and Q. D. Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp.401-406, 2001.</ref><ref>K. C. Abbaspour, R. Schulin, M. T. Van Genuchten, "Estimating unsaturated soil hydraulic parameters using ant colony optimization," Advances In Water Resources, vol.24, no.8, pp.827-841, 2001.</ref>
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| *Protein Folding<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "Protein Folding in Hydrophobic-Polar Lattice Model: A Flexible Ant- Colony Optimization Approach ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp. 469-477.</ref><ref>A. Shmygelska, R. A. Hernández and H. H. Hoos, "An ant colony algorithm for the 2D HP protein folding problem," Proceedings of the 3rd International Workshop on Ant Algorithms/ANTS 2002, Lecture Notes in Computer Science, vol.2463, pp.40-52, 2002.</ref><ref>{{cite journal |author= M. Nardelli, L. Tedesco, and A. Bechini |title= Cross-lattice behavior of general ACO folding for proteins in the HP model |journal= Proc. of ACM SAC 2013|year=2013|pages=1320-1327 |doi= 10.1145/2480362.2480611}}</ref>
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| *Power Electronic Circuit Design<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "Extended Ant Colony Optimization Algorithm for Power Electronic Circuit Design", IEEE Transactions on Power Electronic. Vol.24,No.1, pp.147-162, Jan 2009.</ref>
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| ==Definition difficulty==
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| {{unreferenced section|date=January 2010}}
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| [[File:Aco shortpath.svg|thumb|]]
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| 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.
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| Broadly speaking, ant colony algorithms are regarded as [[people|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.
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| They can be seen as [[probabilistic]] [[multi-agent]] 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.
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| 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.
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| 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]]", which is a very general framework in which ant colony algorithms fit.
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| == Stigmergy algorithms ==
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| 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.<ref>A. Ajith; G. Crina; R. Vitorino (éditeurs), ''Stigmergic Optimization'', Studies in Computational Intelligence , volume 31, 299 pages, 2006. ISBN 978-3-540-34689-0</ref>
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| ==Related methods==
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| *[[Genetic algorithm]]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.
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| *[[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.
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| * [[Reactive search optimization]] 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.
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| *[[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.
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| *[[Artificial immune system]] (AIS) algorithms are modeled on vertebrate immune systems.
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| *[[Particle swarm optimization]] (PSO), a [[Swarm intelligence]] method
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| *[[Intelligent Water Drops]] (IWD), a swarm-based optimization algorithm based on natural water drops flowing in rivers
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| *Gravitational Search Algorithm (GSA), a [[Swarm intelligence]] method
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| *Ant colony clustering method (ACCM), a method that make use of clustering approach,extending the ACO.
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| * [[Stochastic diffusion search]] (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
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| ==History==
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| {{image frame|content=
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| <timeline>
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| ImageSize = width:210 height:300
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| PlotArea = width:170 height:280 left:40 bottom:10
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| DateFormat = yyyy
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| Period = from:1985 till:2005
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| TimeAxis = orientation:vertical
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| ScaleMajor = unit:year increment:5 start:1985
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| Colors=
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| id:fond value:white #rgb(0.95,0.95,0.98)
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| id:marque value:rgb(1,0,0)
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| id:marque_fond value:rgb(1,0.9,0.9)
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| BackgroundColors = canvas:fond
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| Define $dx = 7 # décalage du texte à droite de la barre
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| Define $dy = -3 # décalage vertical
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| Define $dy2 = 6 # décalage vertical pour double texte
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| PlotData=
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| bar:Leaders color:marque_fond width:5 mark:(line,marque) align:left fontsize:S
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| from:1989 till:1989 shift:($dx,$dy) text:comportement collectifs
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| from:1991 till:1992 shift:($dx,$dy) text:Ant System (AS)
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| from:1995 till:1995 shift:($dx,$dy) text:continuous problem (CACO)
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| from:1996 till:1996 shift:($dx,$dy) text:Ant Colony System (ACS)
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| from:1996 till:1996 shift:($dx,$dy2) text:MAX-MIN Ant System (MMAS)
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| from:2000 till:2000 shift:($dx,$dy) text:proof to convergence (GBAS)
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| from:2001 till:2001 shift:($dx,$dy) text:multi-objectif
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| </timeline>|caption=Chronology of COA algorithms
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| }}
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| Chronology of Ant colony optimization algorithms.
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| * 1959, [[Pierre-Paul Grassé]] invented the theory of [[Stigmergy]] to explain the behavior of nest building in [[termites]];<ref>P.-P. Grassé, ''La reconstruction du nid et les coordinations inter-individuelles chez Belicositermes natalensis et Cubitermes sp. La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs'', Insectes Sociaux, numéro 6, p. 41-80, 1959.</ref>
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| * 1983, Deneubourg and his colleagues studied the [[collective behavior]] of [[ants]];<ref>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.</ref>
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| * 1988, and Moyson Manderick have an article on '''self-organization''' among ants;<ref name="F. Moyson, B. Manderick">F. Moyson, B. Manderick, ''The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism'', Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988.</ref>
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| * 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;<ref name="S. Goss">S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels, ''Self-organized shortcuts in the Argentine ant'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref>
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| * 1989, implementation of a model of behavior for food by Ebling and his colleagues;<ref>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</ref>
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| * 1991, M. Dorigo proposed the '''Ant System''' in his doctoral thesis (which was published in 1992<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms" />). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni <ref>Dorigo M., V. Maniezzo et A. Colorni, ''Positive feedback as a search strategy'', rapport technique numéro 91-016, Dip. Elettronica, Politecnico di Milano, Italy, 1991</ref> was published five years later;<ref name="Ant system" />
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| * 1996, publication of the article on Ant System;<ref name="Ant system" />
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| * 1996, Hoos and Stützle invent the '''MAX-MIN Ant System''';<ref name="T. Stützle et H.H. Hoos" />
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| * 1997, Dorigo and Gambardella publish the '''Ant Colony System''';<ref name="M. Dorigo et L.M. Gambardella" />
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| * 1997, Schoonderwoerd and his colleagues developed the first application to [[telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''Ant-based load balancing in telecommunication networks'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref>
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| * 1998, Dorigo launches first conference dedicated to the ACO algorithms;<ref>M. Dorigo, ''ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98'', Bruxelles, Belgique, octobre 1998.</ref>
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| * 1998, Stützle proposes initial '''parallel implementations''';<ref>T. Stützle, ''Parallelization Strategies for Ant Colony Optimization'', Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, Springer-Verlag, volume 1498, pages 722-731, 1998.</ref>
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| * 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants <ref>É. Bonabeau, M. Dorigo et G. Theraulaz, ''Swarm intelligence'', Oxford University Press, 1999.</ref>
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| * 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo , G. Di Caro et T. Stützle, ''Special issue on "Ant Algorithms"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref>
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| * 2000, first applications to the [[Scheduling algorithm|scheduling]], scheduling sequence and [[the satisfaction of constraints]];
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| * 2000, Gutjahr provides the first evidence of [[limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''A graph-based Ant System and its convergence'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref>
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| * 2001, the first use of COA Algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);
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| * 2001, IREDA and his colleagues published the first '''multi-objective''' algorithm <ref>S. Iredi, D. Merkle et M. Middendorf, ''Bi-Criterion Optimization with Multi Colony Ant Algorithms'', Evolutionary Multi-Criterion Optimization, First International Conference (EMO’01), Zurich, Springer Verlag, pages 359-372, 2001.</ref>
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| * 2002, first applications in the design of schedule, Bayesian networks;
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| * 2002, Bianchi and her colleagues suggested the first algorithm for [[stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''An ant colony optimization approach to the probabilistic traveling salesman problem'', PPSN-VII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002.</ref>
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| * 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[stochastic gradient descent]], the [[cross-entropy]] and [[algorithms to estimate distribution]] <ref name="Zlochin model-based search"/>
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| * 2005, first applications to [[protein folding]] problems.
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| * 2012, Prabhakar and colleagues publish research relating to the operation of individual ants communicating in tandem without pheromones, mirroring the principles of computer network organization. The communication model has been compared to the [[Transmission Control Protocol]]. <ref>B. Prabhakar, K. N. Dektar, D. M. Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012. URL: http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002670</ref>
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| == References ==
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| {{reflist|colwidth=30em}}
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| ==Publications (selected)==
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| * [[Marco Dorigo|M. Dorigo]], 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.
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| * 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.
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| * M. Dorigo & [[Luca Maria Gambardella|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.
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| * M. Dorigo, G. Di Caro & L. M. Gambardella, 1999. "Ant Algorithms for Discrete Optimization". Artificial Life, 5 (2): 137–172.
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| * E. Bonabeau, M. Dorigo et G. Theraulaz, 1999. ''Swarm Intelligence: From Natural to Artificial Systems'', Oxford University Press. ISBN 0-19-513159-2
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| * M. Dorigo & T. Stützle, 2004. ''Ant Colony Optimization'', MIT Press. ISBN 0-262-04219-3
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| * M. Dorigo, 2007. [http://www.scholarpedia.org/article/Ant_Colony_Optimization "Ant Colony Optimization"]. Scholarpedia.
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| * C. Blum, 2005 "Ant colony optimization: Introduction and recent trends". Physics of Life Reviews, 2: 353-373
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| * M. Dorigo, M. Birattari & T. Stützle, 2006 ''[http://iridia.ulb.ac.be/IridiaTrSeries/IridiaTr2006-023r001.pdf Ant Colony Optimization: Artificial Ants as a Computational Intelligence Technique]''. TR/IRIDIA/2006-023
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| * 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.
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| * N. Monmarché, F. Guinand & P. Siarry (eds), "Artificial Ants", August 2010 Hardback 576 pp. ISBN 978-1-84821-194-0.
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| == External links ==
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| *[http://www.aco-metaheuristic.org/ Ant Colony Optimization Home Page]
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| *[http://vk.com/ant_colony_optimization "Ant Colony Optimization" - Russian scientific and research community]
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| *[http://www.nightlab.ch/antsim AntSim - Simulation of Ant Colony Algorithms]
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| *[http://www.midaco-solver.com/ MIDACO-Solver] General purpose optimization software based on ant colony optimization (Matlab, Excel, C/C++, Fortran and Python)
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| * [http://ems.eit.uni-kl.de/index.php?id=156 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)
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| *[http://webspace.webring.com/people/br/raguirre/hormigas/antfarm/ Ant Farm Simulator]
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| *[http://www.djoh.net/inde/ANTColony/applet.html Ant algorithm simulation (Java Applet)]
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| {{collective animal behaviour}}
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| {{DEFAULTSORT:Ant Colony Optimization}}
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| [[Category:Stochastic optimization]]
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| [[Category:Articles which contain graphical timelines]]
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| {{Link FA|fr}}
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| {{Link FA|ca}}
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