View source for Ant colony optimization algorithms

{{Short description|Optimization algorithm}}
[[File:Safari ants.jpg|thumb|Ant behavior was the inspiration for the metaheuristic optimization technique]]
[[File:Artificial ants.jpg|thumb|400px|When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.<ref>{{cite book |last = Waldner  |first = Jean-Baptiste  |title = Nanocomputers and Swarm Intelligence |publisher = ISTE {{wipe|John Wiley & Sons}} |place = London |year = 2008 |isbn = 978-1-84704-002-2  | page = 225}}</ref>]]

In [[Computer science|computer science]] and [[Operations research|operations research]], the '''ant colony optimization''' [[Algorithm|algorithm]] ('''ACO''') is a [[Probability|probabilistic]] technique for solving computational problems which can be reduced to finding good paths through [[Graph (discrete mathematics)|graph]]s. Artificial ants stand for multi-agent methods inspired by the behavior of real [[Biology:Ant|ant]]s.
The pheromone-based communication of biological ants is often the predominant paradigm used.<ref>{{cite book |last = Monmarché Nicolas, Guinand Frédéric and Siarry Patrick  |title = Artificial Ants |publisher = Wiley-ISTE  |year = 2010 |isbn = 978-1-84821-194-0}}</ref>   Combinations of artificial ants and [[Local search (optimization)|local search]] algorithms have become a method of choice for numerous optimization tasks involving some sort of [[Graph (discrete mathematics)|graph]], e.g., [[Vehicle routing problem|vehicle routing]] and internet [[Routing|routing]].

As an example, ant colony optimization<ref>{{cite journal|last = Dorigo, Gambardella |first = M, L.M. |title = Learning Approach to the Traveling Salesman Problem  |journal= IEEE Transactions on Evolutionary Computation |volume=1 |issue=1 |page=214 |year = 1997|doi = 10.1109/4235.585892 }}</ref>  is a class of optimization [[Algorithm|algorithm]]s modeled on the actions of an [[Biology:Ant colony|ant colony]].<ref name="Birattari Pellegrini Dorigo 2007 pp. 732–742">{{cite journal | last1=Birattari | first1=M. | last2=Pellegrini | first2=P. | last3=Dorigo | first3=M. | title=On the Invariance of Ant Colony Optimization | journal=IEEE Transactions on Evolutionary Computation | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=11 | issue=6 | year=2007 | issn=1941-0026 | doi=10.1109/tevc.2007.892762 | pages=732–742| s2cid=1591891 }}</ref>  Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a [[Parameter space|parameter space]] representing all possible solutions.  Real ants lay down [[Biology:Pheromone|pheromone]]s directing each other to resources while exploring their environment.  The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.<ref>Ant Colony Optimization by Marco Dorigo and Thomas Stützle, MIT Press, 2004. {{ISBN|0-262-04219-3}}</ref>  One variation on this approach is [[Bees algorithm|the bees algorithm]], which is more analogous to the foraging patterns of the [[Biology:Honey bee|honey bee]], another social insect.

This algorithm is a member of the ant colony algorithms family, in [[Swarm intelligence|swarm intelligence]] methods, and it constitutes some [[Metaheuristic|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 [[Biology: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. From a broader perspective, ACO performs a model-based search<ref>{{cite journal|last1=Zlochin|first1=Mark|last2=Birattari|first2=Mauro|last3=Meuleau|first3=Nicolas|last4=Dorigo|first4=Marco|title=Model-Based Search for Combinatorial Optimization: A Critical Survey|journal=Annals of Operations Research|date=1 October 2004|volume=131|issue=1–4|pages=373–395|doi=10.1023/B:ANOR.0000039526.52305.af|language=en|issn=0254-5330|citeseerx=10.1.1.3.427|s2cid=63137}}</ref> and shares some similarities with [[Estimation of distribution algorithm|estimation of distribution algorithm]]s.

==Overview==

In the natural world, ants of some species (initially) wander randomly,  and upon finding food return to their colony while laying down [[Biology:Pheromone|pheromone]] trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see [[Biology:Ant#Communication|Ant communication]]).<ref>{{cite book |last1=Fladerer |first1=Johannes-Paul |last2=Kurzmann |first2=Ernst |title=WISDOM OF THE MANY : how to create self -organisation and how to use collective... intelligence in companies and in society from mana. |date=November 2019 |publisher=BOOKS ON DEMAND |isbn=9783750422421}}</ref>

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. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.<ref>Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p.12. 2004.</ref>

The overall result is that 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]] eventually leads to many 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.

===Ambient networks of intelligent objects===
New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. Ambient networks of intelligent objects and, sooner or later, a new generation of information systems that are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.<ref name="Waldner 2008 214">{{cite book |last = Waldner  |first = Jean-Baptiste  |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2  | page = 214}}</ref>

Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[Social:Collective intelligence|collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.<ref>{{cite book|last = Waldner  |first = Jean-Baptiste  |title = Inventer l'Ordinateur du XXIème Siècle |publisher = Hermes Science |place = London |year = 2007  | pages = 259–265 |isbn = 978-2-7462-1516-0}}</ref> They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.<ref>{{cite book |last = Waldner  |first = Jean-Baptiste  |author-link = Jean-Baptiste Waldner  |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2  | page = 215}}</ref>

===Artificial pheromone system===
Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as
bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical <ref>Lima, Danielli A., and Gina MB Oliveira. "[https://doi.org/10.1016/j.apm.2017.03.021 A cellular automata ant memory model of foraging in a swarm of robots]." Applied Mathematical Modelling 47, 2017: 551-572.</ref><ref>Russell, R. Andrew. "[https://ieeexplore.ieee.org/abstract/document/774005/ Ant trails-an example for robots to follow?]." Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference on. Vol. 4. IEEE, 1999.</ref><ref>Fujisawa, Ryusuke, et al. "[https://www.researchgate.net/profile/Shigeto_Dobata/publication/265053113_Designing_pheromone_communication_in_swarm_robotics_Group_foraging_behavior_mediated_by_chemical_substance/links/551500f60cf260a7cb2e39eb.pdf Designing pheromone communication in swarm robotics: Group foraging behavior mediated by chemical substance]." Swarm Intelligence 8.3 (2014): 227-246.</ref> or physical (RFID tags,<ref>Sakakibara, Toshiki, and Daisuke Kurabayashi. "[https://link.springer.com/article/10.1016/S1672-6529(07)60038-9 Artificial pheromone system using rfid for navigation of autonomous robots]." Journal of Bionic Engineering 4.4 (2007): 245-253.</ref> light,<ref>Arvin, Farshad, et al. "[http://eprints.lincoln.ac.uk/22466/7/Aggregation-Final.pdf Investigation of cue-based aggregation in static and dynamic environments with a mobile robot swarm]." Adaptive Behavior (2016): 1-17.</ref><ref>Farshad Arvin, et al. "[https://www.researchgate.net/profile/Masoud_Bekravi/publication/241683938_Imitation_of_Honeybee_Aggregation_with_Collective_Behavior_of_Swarm_Robots/links/546518320cf25b85d17d2587/Imitation-of-Honeybee-Aggregation-with-Collective-Behavior-of-Swarm-Robots.pdf Imitation of honeybee aggregation with collective behavior of swarm robots]." International Journal of Computational Intelligence Systems 4.4 (2011): 739-748.</ref><ref>Schmickl, Thomas, et al. "[http://swarmrobot.org/publications/Get_in_touch.pdf Get in touch: cooperative decision making based on robot-to-robot collisions]." Autonomous Agents and Multi-Agent Systems 18.1 (2009): 133-155.</ref><ref>Garnier, Simon, et al. "[http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002903 Do ants need to estimate the geometrical properties of trail bifurcations to find an efficient route? A swarm robotics test bed.]" PLoS Comput Biol 9.3 (2013): e1002903.</ref> sound<ref>Arvin, Farshad, et al. "[https://www.researchgate.net/profile/Farshad_Arvin/publication/273892103_Cue-based_aggregation_with_a_mobile_robot_swarm_A_novel_fuzzy-based_method/links/55e4b97a08ae6abe6e9031be/Cue-based-aggregation-with-a-mobile-robot-swarm-A-novel-fuzzy-based-method.pdf Cue-based aggregation with a mobile robot swarm: a novel fuzzy-based method]." Adaptive Behavior 22.3 (2014): 189-206.</ref>) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.

Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots.<ref>Garnier, Simon, et al. "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.716.1460&rep=rep1&type=pdf Alice in pheromone land: An experimental setup for the study of ant-like robots]." 2007 IEEE Swarm Intelligence Symposium. IEEE, 2007.</ref> Another study presented a system in which pheromones were implemented via a horizontal LCD screen on which the robots moved, with the robots having downward facing light sensors to register the patterns beneath them.<ref>Farshad Arvin et al. "[http://eprints.lincoln.ac.uk/17957/1/APH-colias.pdf COSΦ: artificial pheromone system for robotic swarms research]." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) 2015.</ref><ref>Krajník, Tomáš, et al. "[http://eprints.lincoln.ac.uk/13653/1/jint_2014_public.pdf A practical multirobot localization system]." Journal of Intelligent & Robotic Systems 76.3-4 (2014): 539-562.</ref>

==Algorithm and formula==
In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the [[Shortest path problem|shortest path]] on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared.  The last step consists of updating the pheromone levels on each edge.

 '''procedure''' ACO_MetaHeuristic '''is'''
     '''while not''' terminated '''do'''
         generateSolutions()
         daemonActions()
         pheromoneUpdate()
     '''repeat'''
 '''end procedure'''

===Edge selection===
Each ant needs to construct a solution to move through the graph.  To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step 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, 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. The ''trail level'' represents a posteriori indication of the desirability of that move.

In general, the <math>k</math>th ant moves from state <math>x</math> to state <math>y</math> with probability

:<math>
p_{xy}^k =
\frac
{ (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) }
{ \sum_{z\in \mathrm{allowed}_y} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) }
</math>

where <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_{xz}</math> and <math>\eta_{xz}</math> represent the trail level and attractiveness for the other possible state transitions.

===Pheromone update===

Trails are usually updated 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. An example of a global pheromone updating rule is

:<math>
\tau_{xy} \leftarrow
(1-\rho)\tau_{xy} + \sum_{k}^{m}\Delta \tau^{k}_{xy}
</math>

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'', <math>m</math> is the number of ants 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

:<math>
\Delta \tau^{k}_{xy} =
\begin{cases}
Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\
0 & \mbox{otherwise}
\end{cases}
</math>

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.

==Common extensions==
Here are some of the most popular variations of ACO algorithms.

===Ant system (AS)===
The ant system is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.<ref name="Ant system" />

===Ant colony system (ACS)===
In the ant colony system algorithm, the original ant system was modified in three aspects:

# The edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone);
# While building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule;
# At the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.<ref name="M. Dorigo et L.M. Gambardella">M. Dorigo et L.M. Gambardella, ''[http://www.idsia.ch/~luca/acs-ec97.pdf 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>

===Elitist ant system===
In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants. The elitist strategy has as its objective directing the search of all ants to construct a solution to contain links of the current best route.

===Max-min ant system (MMAS)===
This algorithm controls the maximum and minimum pheromone amounts on each trail. Only the global best tour or the iteration best tour are allowed to add pheromone to its trail. To avoid stagnation of the search algorithm, the range of possible pheromone amounts on each trail is limited to an interval [τ<sub>max</sub>,τ<sub>min</sub>]. All edges are initialized to τ<sub>max</sub> to force a higher exploration of solutions.  The trails are reinitialized to τ<sub>max</sub> when nearing stagnation.<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>

===Rank-based ant system (ASrank)===
All solutions are ranked according to their length. Only a fixed number of the best ants in this iteration are allowed to update their trials. The amount of pheromone deposited is weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths.

===Parallel ant colony optimization (PACO)===
An ant colony system (ACS) with communication strategies is developed. The artificial ants are partitioned into several groups. Seven communication
methods for updating the pheromone level between groups in ACS are proposed and work on the traveling salesman problem.
<ref>Chu S C, Roddick J F, Pan J S. Ant colony system with communication strategies[J]. Information sciences, 2004, 167(1-4): 63-76.</ref>

===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.<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>

===Recursive ant colony optimization===
It is a recursive form of ant system which divides the whole search domain into several sub-domains and solves the objective on these subdomains.<ref>Gupta, D.K.; Arora, Y.; Singh, U.K.; Gupta, J.P., "Recursive Ant Colony Optimization for estimation of parameters of a function", 1st International Conference on Recent Advances in Information Technology (RAIT), vol., no., pp. 448-454, 15–17 March 2012</ref> The results from all the subdomains are compared and the best few of them are promoted for the next level. The subdomains corresponding to the selected results are further subdivided and the process is repeated until an output of desired precision is obtained. This method has been tested on ill-posed geophysical inversion problems and works well.<ref>Gupta, D.K.; Gupta, J.P.; Arora, Y.; Shankar, U., "[http://nsg.eage.org/publication/download/?publication=68286 Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data]," Near Surface Geophysics, vol. 11, no. 3, pp.325-339</ref>

==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 finite time). The first evidence of convergence for an ant colony algorithm was made in 2000, the graph-based ant system algorithm, and later on for the ACS and MMAS algorithms. Like most [[Metaheuristic|metaheuristic]]s, it is very difficult to estimate the theoretical speed of convergence. A performance analysis of a continuous ant colony algorithm with respect to its various parameters (edge selection strategy, distance measure metric, and pheromone evaporation rate) showed that its performance and rate of convergence are sensitive to the chosen parameter values, and especially to the value of the pheromone evaporation rate.<ref>V.K.Ojha, A. Abraham and V. Snasel, [https://arxiv.org/abs/1707.01812 ACO for Continuous Function Optimization: A Performance Analysis], 14th International Conference on Intelligent Systems Design and Applications (ISDA), Japan, Page 145 - 150, 2017, 978-1-4799-7938-7/14 2014 IEEE.</ref> 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 COAC-type algorithms could be assimilated methods of [[Stochastic gradient descent|stochastic gradient descent]], on the [[Cross-entropy|cross-entropy]] and [[Estimation of distribution algorithm|estimation of distribution algorithm]]. They proposed these [[Metaheuristic|metaheuristic]]s as a "research-based model".

==Applications==
[[File:Knapsack ants.svg|thumb|[[Knapsack problem|Knapsack problem]]: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugar]]
Ant colony optimization algorithms have been applied to many [[Combinatorial optimization|combinatorial optimization]] problems, ranging from quadratic assignment to [[Biology:Protein|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 computing|parallel]] implementations.
It has also been used to produce near-optimal solutions to the [[Travelling salesman problem|travelling salesman problem]]. They have an advantage over [[Simulated annealing|simulated annealing]] and [[Genetic algorithm|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.

The first ACO algorithm was called the ant system<ref name="Ant system">M. Dorigo, V. Maniezzo, et A. Colorni, ''[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf 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> 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:

[[File:Ant Colony Algorihm applied to the Travelling Salesman Problem.gif|thumb|Visualization of the ant colony algorithm applied to the travelling salesman problem. The green lines are the paths chosen by each ant. The blue lines are the paths it may take at each point. When the ant finishes, the pheromone levels are represented in red.]]

# 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.

[[File:Aco TSP.svg|thumb|600px|center]]

===Scheduling problem===
* Sequential ordering problem (SOP) <ref>L.M. Gambardella, M. Dorigo, "An Ant Colony System Hybridized with a New Local Search for the Sequential Ordering Problem", INFORMS Journal on Computing, vol.12(3), pp. 237-255, 2000.</ref>
* Job-shop scheduling problem (JSP)<ref>D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, ''[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]'', IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.
</ref>
* [[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, "[http://lia.disi.unibo.it/Courses/SistInt/articoli/beam-aco.pdf Beam-ACO, Hybridizing ant colony optimization with beam search. An application to open shop scheduling]," Technical report TR/IRIDIA/2003-17, 2003.</ref>
* Permutation flow shop problem (PFSP)<ref>T. Stützle, "An ant approach to the flow shop problem," Technical report AIDA-97-07, 1997.</ref>
* Single machine total tardiness problem (SMTTP)<ref>A. Bauer, 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>
* 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, "[http://www.ccas.ru/orsot/library/An%20Ant%20Algorithm%20with%20a%20New%20Pheromone%20Evaluation%20Rule%20for%20Total%20Tardiness%20Problems.pdf 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>
* 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>
* Group-shop scheduling problem (GSP)<ref>C. Blum, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=28 ACO applied to group shop scheduling: a case study on intensification and diversification]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Proceedings of ANTS 2002, vol. 2463 of Lecture Notes in Computer Science, pp.14-27, 2002.</ref>
* Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)<ref>C. Gagné, W. L. Price and M. Gravel, "[https://link.springer.com/article/10.1057/palgrave.jors.2601390 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>
* 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>
* Assembly Sequence Planning (ASP) problems<ref>Han, Z., Wang, Y. & Tian, D. Ant colony optimization for assembly sequence planning based on parameters optimization. Front. Mech. Eng. 16, 393–409 (2021). https://doi.org/10.1007/s11465-020-0613-3</ref>

===Vehicle routing problem===
* Capacitated vehicle routing problem (CVRP)<ref>{{Cite journal |doi = 10.1016/S0166-218X(01)00351-1|title = Models, relaxations and exact approaches for the capacitated vehicle routing problem|journal = Discrete Applied Mathematics|volume = 123|issue = 1–3|pages = 487–512|year = 2002|last1 = Toth|first1 = Paolo|last2 = Vigo|first2 = Daniele|doi-access = free}}</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>
* Multi-depot vehicle routing problem (MDVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(96)00253-6|title = A multi-level composite heuristic for the multi-depot vehicle fleet mix problem|journal = European Journal of Operational Research|volume = 103|pages = 95–112|year = 1997|last1 = Salhi|first1 = S.|last2 = Sari|first2 = M.}}</ref>
* Period vehicle routing problem (PVRP)<ref>{{Cite journal |doi = 10.1016/S0377-2217(01)00206-5|title = The periodic vehicle routing problem with intermediate facilities|journal = European Journal of Operational Research|volume = 137|issue = 2|pages = 233–247|year = 2002|last1 = Angelelli|first1 = Enrico|last2 = Speranza|first2 = Maria Grazia|}}</ref>
* Split delivery vehicle routing problem (SDVRP)<ref>{{Cite journal |citeseerx = 10.1.1.8.7096|title = A Tabu Search Heuristic for the Vehicle Routing Problem with Time Windows and Split Deliveries|journal = Computers and Operations Research|volume = 31|issue = 12|pages = 1947–1964|year = 2002|last1 = Ho|first1 = Sin C.|last2 = Haugland|first2 = Dag|doi = 10.1016/S0305-0548(03)00155-2}}</ref>
* Stochastic vehicle routing problem (SVRP)<ref>{{Cite journal |citeseerx = 10.1.1.392.4034|title = Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands|journal = Computers & Operations Research|pages = 2000|last1 = Secomandi|first1 = Nicola}}</ref>
* Vehicle routing problem with pick-up and delivery (VRPPD)<ref>W. P. Nanry and J. W. Barnes, "[https://web.archive.org/web/20180902151637/https://pdfs.semanticscholar.org/f5a3/fffdfb26ead53680a5f9d3334e556181317b.pdf 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, "[https://pdfs.semanticscholar.org/3952/105ddd7477f04ab1225cf2821021fefeab50.pdf 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>
* Vehicle routing problem with time windows (VRPTW)<ref name="L.M. Gambardella, E. Taillard pp. 63-76">L.M. Gambardella, E. Taillard, G. Agazzi, "MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows", In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, London, UK, pp. 63-76, 1999.</ref><ref>{{Cite journal |doi = 10.1016/0166-218X(95)00027-O|title = The simulated trading heuristic for solving vehicle routing problems|journal = Discrete Applied Mathematics|volume = 65|issue = 1–3|pages = 47–72|year = 1996|last1 = Bachem|first1 = A.|last2 = Hochstättler|first2 = W.|last3 = Malich|first3 = M.|doi-access = free}}</ref><ref>{{Cite journal |doi = 10.1016/S0925-5273(98)00250-3|title = A heuristic for bi-objective vehicle routing with time window constraints|journal = International Journal of Production Economics|volume = 62|issue = 3|pages = 249–258|year = 1999|last1 = Hong|first1 = Sung-Chul|last2 = Park|first2 = Yang-Byung}}</ref><ref>{{Cite journal |doi = 10.1016/j.ejor.2004.08.018|title = Scatter search for the vehicle routing problem with time windows|journal = European Journal of Operational Research|volume = 169|issue = 2|pages = 606–622|year = 2006|last1 = Russell|first1 = Robert A.|last2 = Chiang|first2 = Wen-Chyuan}}</ref>
* Time dependent vehicle routing problem with time windows (TDVRPTW)<ref>A. V. Donati, R. Montemanni, N. Casagrande, A. E. Rizzoli, L. M. Gambardella, "[ftp://ftp.idsia.ch/pub/andrea/ASP_Aprile07/EJOR2007.pdf 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>
* Vehicle routing problem with time windows and multiple service workers (VRPTWMS)

===Assignment problem===
* [[Quadratic assignment problem]] (QAP)<ref>{{Cite journal |citeseerx = 10.1.1.47.5167|title = MAX-MIN Ant System for Quadratic Assignment Problems|year = 1997}}</ref>
* [[Generalized assignment problem]] (GAP)<ref>R. Lourenço and D. Serra "[https://upcommons.upc.edu/bitstream/handle/2099/3627/4-ramalhinho.pdf 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, "[http://leeds-faculty.colorado.edu/glover/Publications/TS%20-%20PR%20-%20GAP%20in%20INFORMS%20JOC.pdf An ejection chain approach for the generalized assignment problem]," INFORMS Journal on Computing, vol. 16, no. 2, pp. 133–151, 2004.</ref>
* 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>
* Redundancy allocation problem (RAP)<ref>Y. C. Liang and A. E. Smith, "[http://www.eng.auburn.edu/sites/personal/aesmith/files/antcolonyIEEETrRel%20paper.pdf An ant colony optimization algorithm for the redundancy allocation problem (RAP)]{{Dead link|date=September 2019 |bot=InternetArchiveBot |fix-attempted=yes }}," IEEE Transactions on Reliability, vol.53, no.3, pp.417-423, 2004.</ref>

===Set problem===
* [[Set cover problem]] (SCP)<ref>G. Leguizamon and Z. Michalewicz, "[https://cs.adelaide.edu.au/users/zbyszek/Papers/as4.pdf 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>
* [[Partition problem]] (SPP)<ref>V Maniezzo and M Milandri, "[https://link.springer.com/chapter/10.1007/3-540-45724-0_19 An ant-based framework for very strongly constrained problems]," Proceedings of ANTS2000, pp.222-227, 2002.</ref>
* Weight constrained graph tree partition problem (WCGTPP)<ref>R. Cordone and F. Maffioli,"[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/A/Applications%20of%20Evolutionary%20Computing,%20EvoWorkshops%202001..%20EvoCOP(LNCS2037,%20Springer,%202001)(ISBN%203540419209)(529s)_CsLn_.pdf#page=74 Colored Ant System and local search to design local telecommunication networks]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol.2037, pp.60-69, 2001.</ref>
* Arc-weighted l-cardinality tree problem (AWlCTP)<ref>C. Blum and M.J. Blesa, "[https://upcommons.upc.edu/bitstream/handle/2117/97393/R03-1.ps Metaheuristics for the edge-weighted k-cardinality tree problem]," Technical Report TR/IRIDIA/2003-02, IRIDIA, 2003.</ref>
* 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>
* Maximum independent set problem (MIS)<ref>G. Leguizamon, Z. Michalewicz and Martin Schutz, "[http://sedici.unlp.edu.ar/bitstream/handle/10915/23384/Documento_completo.pdf?sequence=1 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>

===Device sizing problem in nanoelectronics physical design===
* Ant colony optimization (ACO) based optimization of 45&nbsp;nm CMOS-based sense amplifier circuit could converge to optimal solutions in very minimal time.<ref>O. Okobiah, S. P. Mohanty, and E. Kougianos, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf Ordinary Kriging Metamodel-Assisted Ant Colony Algorithm for Fast Analog Design Optimization] {{webarchive |url=https://web.archive.org/web/20160304110324/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2012/Mohanty_ISQED2012_Kriging-ACO.pdf |date=March 4, 2016 }}", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design (ISQED), pp. 458--463, 2012.</ref>
* Ant colony optimization (ACO) based reversible circuit synthesis could improve efficiency significantly.<ref>M. Sarkar, P. Ghosal, and S. P. Mohanty, "[http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf Reversible Circuit Synthesis Using ACO and SA based Quinne-McCluskey Method] {{webarchive |url=https://web.archive.org/web/20140729081848/http://www.cse.unt.edu/~smohanty/Publications_Conferences/2013/Mohanty_MWSCAS2013_Reversible-Circuit.pdf |date=July 29, 2014 }}", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems (MWSCAS), 2013, pp. 416--419.</ref>

===Antennas optimization and synthesis===
[[File:ANT Antenna 1.jpg|thumb|Loopback vibrators 10×10, synthesized by means of ACO algorithm<ref name="slyusarant1">Ermolaev S.Y., Slyusar V.I. Antenna synthesis based on the ant colony optimization algorithm.// Proc. ICATT’2009, Lviv, Ukraine 6 - 9 Octobre, 2009. - Pages 298 - 300 [http://slyusar.kiev.ua/298_300_ICATT_2009.pdf]</ref>]]
[[File:ANT antenna 2.jpg|thumb|Unloopback vibrators 10×10, synthesized by means of ACO algorithm<ref name=slyusarant1/>]]
To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),<ref>Marcus Randall, Andrew Lewis, Amir Galehdar, David Thiel. Using Ant Colony Optimisation to Improve the Efficiency of Small Meander Line RFID Antennas.// In 3rd IEEE International e-Science and Grid Computing Conference [http://www98.griffith.edu.au/dspace/bitstream/10072/17063/1/47523_1.pdf], 2007</ref> loopback and unloopback vibrators 10×10<ref name=slyusarant1/>

===Image processing===
The ACO algorithm is used in image processing for image [[Edge detection|edge detection]] and edge linking.<ref>S. Meshoul and M Batouche, "[https://web.archive.org/web/20180902151639/https://pdfs.semanticscholar.org/bdd2/61ab1f5a0c90009c6d84dbe4121a87dd4d31.pdf Ant colony system with extremal dynamics for point matching and pose estimation]," Proceedings of the 16th International Conference on Pattern Recognition, vol.3, pp.823-826, 2002.</ref><ref>H. Nezamabadi-pour, S. Saryazdi, and E. Rashedi, "[https://www.researchgate.net/profile/Esmat_Rashedi/publication/220176122_Edge_detection_using_ant_algorithms/links/5743d1ab08ae9ace841b4063.pdf Edge detection using ant algorithms]", Soft Computing, vol. 10, no.7, pp. 623-628, 2006.</ref>
* '''Edge detection:'''
The graph here is the 2-D image and the ants traverse from one pixel depositing pheromone. The movement of ants from one pixel to another is directed by the local variation of the image's intensity values. This movement causes the highest density of the pheromone to be deposited at the edges.

The following are the steps involved in edge detection using ACO:<ref>{{cite book|last1=Tian|first1=Jing|title=2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)|pages=751–756|last2=Yu|first2=Weiyu|last3=Xie|first3=Shengli|chapter=An ant colony optimization algorithm for image edge detection |doi=10.1109/CEC.2008.4630880|year=2008|isbn=978-1-4244-1822-0|s2cid=1782195}}</ref><ref>{{cite web|last1=Gupta|first1=Charu|last2=Gupta|first2=Sunanda|title=Edge Detection of an Image based on Ant ColonyOptimization Technique|url=https://www.academia.edu/4688002}}</ref><ref>{{Cite book|pages = 2193–2198|doi = 10.1109/ICSMC.2009.5345922|first1 = A.|last1 = Jevtić|first2 = J.|last2 = Quintanilla-Dominguez|first3 = M.G.|last3 = Cortina-Januchs|first4 = D.|last4 = Andina| title=2009 IEEE International Conference on Systems, Man and Cybernetics | chapter=Edge detection using ant colony search algorithm and multiscale contrast enhancement |year = 2009|isbn = 978-1-4244-2793-2|s2cid = 11654036}}</ref>

''Step 1: Initialization.'' Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.

There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:
the local statistics at the pixel position <math>(i,j)</math>.

:<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)},</math>

where <math>I</math> is the image of size <math>M_1*M_2</math>,

:<math>Z =\sum_{i=1:M_1}  \sum_{j=1:M_2} Vc(I_{i,j})</math>

is a normalization factor, and

:<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math>

<math>f(\cdot)</math> can be calculated using the following functions:
:<math>f(x) = \lambda x, \quad \text{for x ≥ 0;  (1)} </math>
:<math>f(x) = \lambda x^2, \quad \text{for x ≥ 0;  (2)} </math>
:<math>f(x) =
\begin{cases}
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (3)} \\
0, & \text{else}
\end{cases}</math>
:<math>f(x) =
\begin{cases}
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (4)} \\
0, & \text{else}
\end{cases}</math>
The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.

''Step 2: Construction process.'' The ant's movement is based on 4-connected [[Pixel|pixel]]s or 8-connected [[Pixel|pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math>

''Step 3 and Step 5: Update process.'' The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by:
:<math>
\tau_{new} \leftarrow
(1-\psi)\tau_{old} + \psi \tau_{0}
</math>,
where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>

''Step 7: Decision process.'' Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].

Image edge detected using ACO: The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[Software:MATLAB|MATLAB]] Central|date=21 July 2023 |url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref>
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]
* '''Edge linking:'''<ref>{{cite book|last1 = Jevtić|first1 = A.|title = 2009 35th Annual Conference of IEEE Industrial Electronics|last2 = Melgar|first2 = I.|last3 = Andina|first3 = D.|pages = 3353–3358|year = 2009|location = 35th Annual Conference of IEEE Industrial Electronics, 2009. IECON '09.|doi = 10.1109/IECON.2009.5415195|isbn = 978-1-4244-4648-3|s2cid = 34664559}}</ref> ACO has also proven effective in edge linking algorithms.

=== Other applications ===
* [[Finance:Bankruptcy prediction|Bankruptcy prediction]]<ref>{{cite journal|last1=Zhang|first1=Y.|title=A Rule-Based Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm|journal=Mathematical Problems in Engineering|date=2013|volume=2013|page=753251|doi=10.1155/2013/753251|doi-access=free}}</ref>
* [[Classification]]<ref name="D. Martens, M pages 651">D. Martens, M. De Backer, R. Haesen, J. Vanthienen, M. Snoeck, B. Baesens, "[https://ieeexplore.ieee.org/abstract/document/4336122/ Classification with Ant Colony Optimization]", IEEE Transactions on Evolutionary Computation, volume 11, number 5, pages 651—665, 2007.</ref>
* Connection-oriented network routing<ref>G. D. Caro and M. Dorigo, "Extending AntNet for best-effort quality-of-service routing," Proceedings of the First International Workshop on Ant Colony Optimization (ANTS’98), 1998.</ref>
* Connectionless network routing<ref>G.D. Caro and M. Dorigo "[http://www.idsia.ch/~gianni/Papers/tech-rep-iridia-97-12.pdf 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, "[https://www.researchgate.net/profile/Gianni_Di_Caro/publication/2328604_Two_Ant_Colony_Algorithms_For_Best-Effort_Routing_In_Datagram_Networks/links/0deec52909f32c7e6d000000/Two-Ant-Colony-Algorithms-For-Best-Effort-Routing-In-Datagram-Networks.pdf 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>
* [[Data mining]]<ref name="D. Martens, M pages 651"/><ref>D. Martens, B. Baesens, T. Fawcett "[https://link.springer.com/content/pdf/10.1007/s10994-010-5216-5.pdf 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, "[http://neuro.bstu.by/ai/To-dom/My_research/Paper-0-again/For-courses/Ants/heuristic-dm-bk.pdf 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, "[https://www.academia.edu/download/31181466/datamining070.pdf Data mining with an ant colony optimization algorithm]{{dead link|date=July 2022|bot=medic}}{{cbignore|bot=medic}}," IEEE Transactions on Evolutionary Computation, vol.6, no.4, pp.321-332, 2002.</ref>
* Discounted cash flows in project scheduling<ref>W. N. Chen, J. ZHANG and H. Chung, "[http://webdelprofesor.ula.ve/economia/gsfran/Asignaturas/EvaluacionFinEconProyec/2%20OptimizingDiscounted.pdf 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>
* [[Distributed computing|Distributed]] [[Information retrieval|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, "[http://hal.upmc.fr/docs/00/65/63/63/PDF/manuscript.pdf Image Retrieval over Networks : Active Learning using Ant Algorithm]", IEEE Transactions on Multimedia, vol. 10, no. 7, pp. 1356--1365 - nov 2008</ref>
* Energy and electricity network design<ref name="warner-and-vogel-2008">
{{cite conference
 | last1 = Warner | first1 = Lars
 | last2 = Vogel | first2 = Ute
 | title = Optimization of energy supply networks using ant colony optimization
 | date = 2008
 | conference = Environmental Informatics and Industrial Ecology — 22nd International Conference on Informatics for Environmental Protection
 | publisher = Shaker Verlag
 | location = Aachen, Germany
 | isbn = 978-3-8322-7313-2
 | url = http://enviroinfo.eu/sites/default/files/pdfs/vol119/0327.pdf
 | access-date = 2018-10-09
}}
</ref>
* 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>
* Inhibitory peptide design for protein protein interactions<ref name=":0">{{Cite journal|last1=Zaidman|first1=Daniel|last2=Wolfson|first2=Haim J.|date=2016-08-01|title=PinaColada: peptide–inhibitor ant colony ad-hoc design algorithm|journal=Bioinformatics|volume=32|issue=15|pages=2289–2296|doi=10.1093/bioinformatics/btw133|pmid=27153578|issn=1367-4803|doi-access=free}}</ref>
* Intelligent testing system<ref>Xiao. M.Hu, J. ZHANG, and H. Chung, "[https://ieeexplore.ieee.org/abstract/document/5061647/ 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>
* Power [[Engineering:Electronic circuit design|electronic circuit design]]<ref>J. ZHANG, H. Chung, W. L. Lo, and T. Huang, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.140.4340&rep=rep1&type=pdf 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>
* [[Biology:Protein folding|Protein folding]]<ref>X. M. Hu, J. ZHANG，J. Xiao and Y. Li, "[http://eprints.gla.ac.uk/5306/1/5306.pdf 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, "[ftp://nozdr.ru/biblio/kolxo3/Cs/CsLn/Ant%20Algorithms,%203%20conf.,%20ANTS%202002(LNCS2463,%20Springer,%202002)(ISBN%203540441468)(318s).pdf#page=54 An ant colony optimization algorithm for the 2D HP protein folding problem]{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}," 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 book |author1=M. Nardelli |author2=L. Tedesco |author3=A. Bechini |title=Proceedings of the 28th Annual ACM Symposium on Applied Computing |chapter=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|isbn=9781450316569 |s2cid=1216890 |url=https://zenodo.org/record/3445092 }}</ref>
* 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, "[https://www.ars.usda.gov/arsuserfiles/20360500/pdf_pubs/P1797.pdf Estimating unsaturated soil hydraulic parameters using ant colony optimization]," Advances In Water Resources, vol. 24, no. 8, pp. 827-841, 2001.</ref>

==Definition difficulty==
[[File:Aco shortpath.svg|thumb]]
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths.<ref>{{Cite journal | doi=10.1186/1471-2105-6-30| pmid=15710037| pmc=555464|year = 2005|last1 = Shmygelska|first1 = Alena| title=An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem| journal=BMC Bioinformatics| volume=6| pages=30| last2=Hoos| first2=Holger H.| doi-access=free}}</ref> 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.<ref>Fred W. Glover, Gary A. Kochenberger, ''Handbook of Metaheuristics'', [https://books.google.com/books?id=P-HpBwAAQBAJ&dq=aco+algorithms+with+guaranteed+convergence+to+the+optimal+solution+metaheuristics&pg=PA276], Springer (2003)</ref> 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]] to make the transition between each [[Iteration|iteration]].<ref>{{Cite web|url=http://www.multiagent.fr/extensions/ICAPManager/pdf/LauriCharpillet2006.pdf|title=Ciad-Lab &#124;}}</ref> In their versions for combinatorial problems, they use an iterative construction of solutions.<ref>WJ Gutjahr, ''ACO algorithms with guaranteed convergence to the optimal solution'', [https://homes.di.unimi.it/cordone/courses/2016-ae/Lez07-Materiali/ACOAlgoithmsWithGuaranteedConvergenceToTheOptimalSolution.pdf], (2002)</ref> 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]]",<ref name="Waldner 2008 214"/> 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.<ref>Santpal Singh Dhillon, ''Ant Routing, Searching and Topology Estimation Algorithms for Ad Hoc Networks'', [https://books.google.com/books?id=j5fOJqhwcJoC&dq=Stigmergy+algorithms&pg=PA33], IOS Press, (2008)</ref> In practice, the use of an exchange of information between ants via the environment (a principle called "[[Stigmergy|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>

==Related methods==
;[[Genetic algorithm]]s (GA)
: These 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.
;[[Estimation of distribution algorithm]] (EDA)
:An [[Evolutionary algorithm|evolutionary algorithm]] that substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing machine learning techniques and represented as probabilistic graphical models, from which new solutions can be sampled<ref>{{cite book|last1=Pelikan|first1=Martin|last2=Goldberg|first2=David E.|last3=Cantú-Paz|first3=Erick|title=BOA: The Bayesian Optimization Algorithm|journal=Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 1|date=1 January 1999|pages=525–532|url=http://dl.acm.org/citation.cfm?id=2933973|isbn=9781558606111|series=Gecco'99}}</ref><ref>{{cite book|last1=Pelikan|first1=Martin|title=Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms|date=2005|publisher=Springer|location=Berlin [u.a.]|isbn=978-3-540-23774-7|edition=1st}}</ref> or generated from guided-crossover.<ref>{{cite book|last1=Thierens|first1=Dirk|title=Parallel Problem Solving from Nature, PPSN XI |s2cid=28648829|chapter=The Linkage Tree Genetic Algorithm|date=11 September 2010|pages=264–273|doi=10.1007/978-3-642-15844-5_27|language=en|isbn=978-3-642-15843-8}}</ref><ref>{{cite journal|last1=Martins|first1=Jean P.|last2=Fonseca|first2=Carlos M.|last3=Delbem|first3=Alexandre C. B.|title=On the performance of linkage-tree genetic algorithms for the multidimensional knapsack problem|journal=Neurocomputing|date=25 December 2014|volume=146|pages=17–29|doi=10.1016/j.neucom.2014.04.069}}</ref>
;[[Simulated annealing]] (SA)
: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 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 search]] (TS)
: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)
:Modeled on vertebrate immune systems.
;[[Particle swarm optimization]] (PSO)
:A [[Swarm intelligence|swarm 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 intelligence|swarm intelligence]] method.
;Ant colony clustering method (ACCM)
:A method that make use of clustering approach, extending the ACO.
;[[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.

==History==

{{image frame|content=
<timeline>
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  from:1989  till:1989 shift:($dx,$dy)    text:studies of collective behavior
  from:1991  till:1992 shift:($dx,$dy)    text:ant system (AS)
  from:1995  till:1995 shift:($dx,$dy)    text:continuous problem (CACO)
  from:1996  till:1996 shift:($dx,$dy)    text:ant colony system (ACS)
  from:1996  till:1996 shift:($dx,$dy2)   text:max-min ant system (MMAS)
  from:2000  till:2000 shift:($dx,$dy)   text:proof to convergence (GBAS)
  from:2001  till:2001 shift:($dx,$dy)   text:multi-objective algorithm

</timeline>|caption=Chronology of COA algorithms
}}

Chronology of ant colony optimization algorithms.
* 1959, Pierre-Paul Grassé invented the theory of [[Stigmergy|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>
* 1983, Deneubourg and his colleagues studied the [[Social:Collective behavior|collective behavior]] of ants;<ref>J.L. Denebourg, J.M. Pasteels et J.C. Verhaeghe, ''[https://web.archive.org/web/20181006194827/https://pdfs.semanticscholar.org/caac/71608a5e6b0907a8a99bba30d72d9a304152.pdf Probabilistic Behaviour in Ants : a Strategy of Errors?]'', Journal of Theoretical Biology, numéro 105, 1983.</ref>
* 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>
* 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, ''[https://www.researchgate.net/profile/Serge_Aron/publication/301232811_Self-Organized_Shortcuts_in_the_Argentine_Ant/links/59967b5daca27283b11d9070/Self-Organized-Shortcuts-in-the-Argentine-Ant.pdf Self-organized shortcuts in the Argentine ant]'', Naturwissenschaften, volume 76, pages 579-581, 1989</ref>
* 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>
* 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" />
* 1994, Appleby and Steward of British Telecommunications Plc published the first application to telecommunications networks<ref>Appleby, S. & Steward, S. Mobile software agents for control in telecommunications networks, BT Technol. J., 12(2):104–113, April 1994</ref>
* 1995, Gambardella and Dorigo proposed '''ant-q''',<ref>L.M. Gambardella and M. Dorigo, "Ant-Q: a reinforcement learning approach to the traveling salesman problem", Proceedings of ML-95, Twelfth International Conference on Machine Learning, A. Prieditis and S. Russell (Eds.), Morgan Kaufmann, pp. 252–260, 1995</ref> the preliminary version of ant colony system as first extension of ant system;.<ref name="Ant system" />
* 1996, Gambardella and Dorigo proposed '''ant colony system''' <ref>L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, May 20–22, pp. 622-627, 1996;</ref>
* 1996, publication of the article on ant system;<ref name="Ant system" />
* 1997, Dorigo and Gambardella proposed ant colony system hybridized with local search;<ref name="M. Dorigo et L.M. Gambardella" />
* 1997, Schoonderwoerd and his colleagues published an improved application to [[Engineering:Telecommunication|telecommunication]] networks;<ref>R. Schoonderwoerd, O. Holland, J. Bruten et L. Rothkrantz, ''[https://web.archive.org/web/20170902102027/https://pdfs.semanticscholar.org/f09e/03c5d759c7ca04e443d496e23c981f1b4a5d.pdf Ant-based load balancing in telecommunication networks]'', Adaptive Behaviour, volume 5, numéro 2, pages 169-207, 1997</ref>
* 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>
* 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>
* 1999, Gambardella, Taillard and Agazzi proposed ''' macs-vrptw''', first multi ant colony system applied to vehicle routing problems with time windows,<ref name="L.M. Gambardella, E. Taillard pp. 63-76"/>
* 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>
* 2000, special issue of the Future Generation Computer Systems journal on ant algorithms<ref>M. Dorigo, G. Di Caro et T. Stützle, ''[https://www.academia.edu/download/30765111/FGCS-Editorial-final.pdf Special issue on "Ant Algorithms]{{dead link|date=July 2022|bot=medic}}{{cbignore|bot=medic}}"'', Future Generation Computer Systems, volume 16, numéro 8, 2000</ref>
* 2000, Hoos and Stützle invent the '''max-min ant system''';<ref name="T. Stützle et H.H. Hoos" />
* 2000, first applications to the scheduling, scheduling sequence and the [[Constraint satisfaction|satisfaction of constraints]];
* 2000, Gutjahr provides the first evidence of [[Limit of a sequence|convergence]] for an algorithm of ant colonies<ref>W.J. Gutjahr, ''[http://iridia.ulb.ac.be/~mdorigo/ACO/downloads/ants5.pdf A graph-based Ant System and its convergence]'', Future Generation Computer Systems, volume 16, pages 873-888, 2000.</ref>
* 2001, the first use of COA algorithms by companies ([http://www.eurobios.com/ Eurobios] and [http://www.antoptima.com/ AntOptima]);
* 2001, Iredi and his colleagues published the first '''multi-objective''' algorithm<ref>S. Iredi, D. Merkle et M. Middendorf, ''[https://link.springer.com/chapter/10.1007/3-540-44719-9_25 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>
* 2002, first applications in the design of schedule, Bayesian networks;
* 2002, Bianchi and her colleagues suggested the first algorithm for [[Stochastic|stochastic]] problem;<ref>L. Bianchi, L.M. Gambardella et M.Dorigo, ''[http://hcot.ir/wp-content/uploads/2015/03/An-Ant-Colony-Optimization-Approach-to-the-Probabilistic-Traveling-Salesman-Problem.pdf 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>
* 2004, Dorigo and Stützle publish the Ant Colony Optimization book with MIT Press <ref>M. Dorigo and T. Stützle, ''Ant Colony Optimization'', MIT Press, 2004.</ref>
* 2004, Zlochin and Dorigo show that some algorithms are equivalent to the [[Stochastic gradient descent|stochastic gradient descent]], the [[Cross-entropy method|cross-entropy method]] and algorithms to estimate distribution<ref name="Zlochin model-based search"/>
* 2005, first applications to [[Biology:Protein folding|protein folding]] problems.
* 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>
* 2016, first application to peptide sequence design.<ref name=":0" />
* 2017, successful integration of the multi-criteria decision-making method PROMETHEE into the ACO algorithm ([[HUMANT (HUManoid ANT) algorithm|HUMANT algorithm]]).<ref>{{cite journal|last1=Mladineo|first1=Marko|last2=Veza|first2=Ivica|last3=Gjeldum|first3=Nikola|title=Solving partner selection problem in cyber-physical production networks using the HUMANT algorithm|journal=International Journal of Production Research|date=2017|volume=55|issue=9|pages=2506–2521|doi=10.1080/00207543.2016.1234084|s2cid=114390939}}</ref>

==References==
{{Reflist|30em}}

==Publications (selected)==
* M. Dorigo, 1992. ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italy.
* M. Dorigo, V. Maniezzo & A. Colorni, 1996. "[http://www.cs.unibo.it/babaoglu/courses/cas05-06/tutorials/Ant_Colony_Optimization.pdf Ant System: Optimization by a Colony of Cooperating Agents]", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 (1): 29–41.
* M. Dorigo & [[Biography:Luca Maria Gambardella|L. M. Gambardella]], 1997. "[http://www.idsia.ch/~luca/acs-ec97.pdf 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. "[http://people.idsia.ch/~gianni/Papers/ArtificialLife-original.pdf Ant Algorithms for Discrete Optimization]". Artificial Life, 5 (2): 137–172.
* E. Bonabeau, 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. [http://www.scholarpedia.org/article/Ant_Colony_Optimization  "Ant Colony Optimization"]. Scholarpedia.
* C. Blum, 2005 "[http://aisii.azc.uam.mx/mcbc/Cursos/IntCompt/Lectura16.pdf Ant colony optimization: Introduction and recent trends]". Physics of Life Reviews, 2: 353-373
* M. Dorigo, M. Birattari & T. Stützle, 2006 ''[https://web.archive.org/web/20120222061542/http://iridia.ulb.ac.be/IridiaTrSeries/IridiaTr2006-023r001.pdf 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.&nbsp;163–181, December 2008, {{ISSN|0128-3790}}.
* N. Monmarché, F. Guinand & P. Siarry (eds), "Artificial Ants", August 2010 Hardback 576 pp.&nbsp;{{ISBN|978-1-84821-194-0}}.
* A. Kazharov, V. Kureichik, 2010. "[https://www.researchgate.net/profile/Asker_Kazharov/publication/225549674_Ant_colony_optimization_algorithms_for_solving_transportation_problems/links/56e0268e08aec4b3333d0039.pdf Ant colony optimization algorithms for solving transportation problems]", Journal of Computer and Systems Sciences International, Vol. 49. No. 1. pp.&nbsp;30–43.
* C-M. Pintea, 2014, [https://www.springer.com/la/book/9783642401787 Advances in Bio-inspired Computing for Combinatorial Optimization Problem], Springer {{ISBN|978-3-642-40178-7}}
* K. Saleem, N. Fisal, M. A. Baharudin, A. A. Ahmed, S. Hafizah and S. Kamilah, "Ant colony inspired self-optimized routing protocol based on cross layer architecture for wireless sensor networks", WSEAS Trans. Commun., vol. 9, no. 10, pp.&nbsp;669–678, 2010. {{ISBN|978-960-474-200-4}}
* K. Saleem and N. Fisal, "Enhanced Ant Colony algorithm for self-optimized data assured routing in wireless sensor networks", Networks (ICON) 2012 18th IEEE International Conference on, pp.&nbsp;422–427. {{ISBN|978-1-4673-4523-1}}
* Abolmaali S, Roodposhti FR. ''[https://www.researchgate.net/profile/Saina_Abolmaali2/publication/312523236_Portfolio_Optimization_using_ant_colony_method_a_case_study_on_Tehran_stock_exchange/links/5ad5cd24a6fdcc29358108d4/Portfolio-Optimization-using-ant-colony-method-a-case-study-on-Tehran-stock-exchange.pdf Portfolio Optimization Using Ant Colony Method a Case Study on Tehran Stock Exchange.]'' Journal of Accounting. 2018 Mar;8(1).

==External links==
* [http://www.scholarpedia.org/article/Ant_colony_optimization Scholarpedia Ant Colony Optimization page]
* [https://web.archive.org/web/20131011081948/http://www.aco-metaheuristic.org/ Ant Colony Optimization Home Page]
* [http://vk.com/ant_colony_optimization "Ant Colony Optimization" - Russian scientific and research community]
* [https://web.archive.org/web/20080616044645/http://www.nightlab.ch/antsim/ AntSim - Simulation of Ant Colony Algorithms]
* [http://www.midaco-solver.com/ MIDACO-Solver] General purpose optimization software based on ant colony optimization (Matlab, Excel, VBA, C/C++, R, C#, Java, Fortran and Python)
* [https://web.archive.org/web/20110719105224/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)
* [http://www.djoh.net/inde/ANTColony/applet.html Ant algorithm simulation (Java Applet)]
* [https://github.com/ugochirico/Java-Ant-Colony-System-Framework Java Ant Colony System Framework]
* [https://www.kaggle.com/code/jamesmcguigan/ant-colony-optimization-algorithm Ant Colony Optimization Algorithm Implementation (Python Notebook)]

{{Optimization algorithms}}
{{collective animal behaviour}}

{{DEFAULTSORT:Ant Colony Optimization}}
[[Category:Nature-inspired metaheuristics]]
[[Category:Optimization algorithms and methods]]

{{Sourceattribution|Ant colony optimization algorithms}}