Ant colony optimization algorithms
In computer science and operations research, the ant colony optimization algorithm ACO is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs
This algorithm is a member of 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, the first algorithm was aiming to search for an optimal path in a graph, based on the behavior of ants seeking a path between their colony and a source of food The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems have emerged, drawing on various aspects of the behavior of ants From a broader perspective, ACO performs a modelbased search and share some similarities with Estimation of Distribution Algorithms
Contents
 1 Overview
 2 Common extensions
 21 Elitist ant system
 22 Maxmin ant system MMAS
 23 Ant colony system
 24 Rankbased ant system ASrank
 25 Continuous orthogonal ant colony COAC
 26 Recursive ant colony optimization
 3 Convergence
 4 Example pseudocode and formula
 41 Edge selection
 42 Pheromone update
 5 Applications
 51 Scheduling problem
 52 Vehicle routing problem
 53 Assignment problem
 54 Set problem
 55 Device sizing problem in nanoelectronics physical design
 56 Image processing
 57 Others
 6 Definition difficulty
 7 Stigmergy algorithms
 8 Related methods
 9 History
 10 References
 11 Publications selected
 12 External links
Overview
In the natural world, ants of some species initially wander randomly, and upon finding food return to their colony while laying down pheromone trails If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food see Ant communication
Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones In that case, the exploration of the solution space would be constrained The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems
The overall result is that when one ant finds a good ie, 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
Common extensions
Here are some of the most popular variations of ACO algorithms
Elitist ant system
The global best solution deposits pheromone on every iteration along with all the other ants
Maxmin ant system MMAS
Added maximum and minimum pheromone amounts Only global best or iteration best tour deposited pheromone <MAZ> All edges are initialized to τmin and reinitialized to τmax when nearing stagnation
Ant colony system
It has been presented above
Rankbased ant system ASrank
All solutions are ranked according to their length The amount of pheromone deposited is then weighted for each solution, such that solutions with shorter paths deposit more pheromone than the solutions with longer paths
Continuous orthogonal ant colony COAC
The pheromone deposit mechanism of COAC is to enable ants to search for solutions collaboratively and effectively By using an orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently, with enhanced global search capability and accuracy
The orthogonal design method and the adaptive radius adjustment method can also be extended to other optimization algorithms for delivering wider advantages in solving practical problems
Recursive ant colony optimization
It is a recursive form of ant system which divides the whole search domain into several subdomains and solves the objective on these subdomains 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 illposed geophysical inversion problems and works well
Convergence
For some versions of the algorithm, it is possible to prove that it is convergent ie, 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 graphbased ant system algorithm, and then algorithms for ACS and MMAS Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence In 2004, Zlochin and his colleagues showed that COAtype algorithms could be assimilated methods of stochastic gradient descent, on the crossentropy and estimation of distribution algorithm They proposed these metaheuristics as a "researchbased model" A performance analysis of continuous ant colony algorithm based on its various parameter suggest its sensitivity of convergence on parameter tuning
Example pseudocode and formula
procedure ACO_MetaHeuristic whilenot_termination generateSolutions daemonActions pheromoneUpdate end while end procedureEdge selection
An ant is a simple computational agent in the ant colony optimization algorithm It iteratively constructs a solution for the problem at hand The intermediate solutions are referred to as solution states At each iteration of the algorithm, each ant moves from a state x to state y , corresponding to a more complete intermediate solution Thus, each ant k computes a set A k x x of feasible expansions to its current state in each iteration, and moves to one of these in probability For ant k , the probability p x y k ^ of moving from state x to state y depends on the combination of two values, viz, the attractiveness η x y of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level τ x y of the move, indicating how proficient it has been in the past to make that particular move
The trail level represents a posteriori indication of the desirability of that move Trails are updated usually when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively
In general, the k th ant moves from state x to state y with probability
p x y k = τ x y α η x y β ∑ z ∈ a l l o w e d x τ x z α η x z β ^=^\eta _^ _\tau _^\eta _^
where
τ x y is the amount of pheromone deposited for transition from state x to y , 0 ≤ α is a parameter to control the influence of τ x y , η x y is the desirability of state transition x y a priori knowledge, typically 1 / d x y , where d is the distance and β ≥ 1 is a parameter to control the influence of η x y τ x z and η x z represent the attractiveness and trail level for the other possible state transitions
Pheromone update
When all the ants have completed a solution, the trails are updated by τ x y ← 1 − ρ τ x y + ∑ k Δ τ x y k \leftarrow 1\rho \tau _+\sum _\Delta \tau _^
where τ x y is the amount of pheromone deposited for a state transition x y , ρ is the pheromone evaporation coefficient and Δ τ x y k ^ is the amount of pheromone deposited by k th ant, typically given for a TSP problem with moves corresponding to arcs of the graph by
Δ τ x y k = ^=Q/L_&kxy\\0&\end
where L k is the cost of the k th ant's tour typically length and Q is a constant
Applications
Knapsack problem: The ants prefer the smaller drop of honey over the more abundant, but less nutritious, sugarAnt colony optimization algorithms have been applied to many combinatorial optimization problems, ranging from quadratic assignment to protein folding or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multitargets and parallel implementations It has also been used to produce nearoptimal solutions to the travelling salesman problem They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time This is of interest in network routing and urban transportation systems
The first ACO algorithm was called the ant system and it was aimed to solve the travelling salesman problem, in which the goal is to find the shortest roundtrip to link a series of cities The general algorithm is relatively simple and based on a set of ants, each making one of the possible roundtrips along the cities At each stage, the ant chooses to move from one city to another according to some rules:
 It must visit each city exactly once;
 A distant city has less chance of being chosen the visibility;
 The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;
 Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
 After each iteration, trails of pheromones evaporate
Scheduling problem
 Jobshop scheduling problem JSP
 Openshop scheduling problem OSP
 Permutation flow shop problem PFSP
 Single machine total tardiness problem SMTTP
 Single machine total weighted tardiness problem SMTWTP
 Resourceconstrained project scheduling problem RCPSP
 Groupshop scheduling problem GSP
 Singlemachine total tardiness problem with sequence dependent setup times SMTTPDST
 Multistage flowshop scheduling problem MFSP with sequence dependent setup/changeover times
Vehicle routing problem
 Capacitated vehicle routing problem CVRP
 Multidepot vehicle routing problem MDVRP
 Period vehicle routing problem PVRP
 Split delivery vehicle routing problem SDVRP
 Stochastic vehicle routing problem SVRP
 Vehicle routing problem with pickup and delivery VRPPD
 Vehicle routing problem with time windows VRPTW
 Time dependent vehicle routing problem with time windows TDVRPTW
 Vehicle routing problem with time windows and multiple service workers VRPTWMS
Assignment problem
 Quadratic assignment problem QAP
 Generalized assignment problem GAP
 Frequency assignment problem FAP
 Redundancy allocation problem RAP
Set problem
 Set cover problem SCP
 Partition problem SPP
 Weight constrained graph tree partition problem WCGTPP
 Arcweighted lcardinality tree problem AWlCTP
 Multiple knapsack problem MKP
 Maximum independent set problem MIS
Device sizing problem in nanoelectronics physical design
 Ant colony optimization ACO based optimization of 45 nm CMOSbased sense amplifier circuit could converge to optimal solutions in very minimal time
 Ant colony optimization ACO based reversible circuit synthesis could improve efficiency significantly
Image processing
ACO algorithm is used in image processing for image edge detection and edge linking
 Edge detection:
The graph here is the 2D image and the ants traverse from one pixel depositing pheromoneThe 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:
Step1: Initialization:
Randomly place
K
ants on the image
I
M
1
M
2
M_
where
K
=
M
1
∗
M
2
1
2
M_^
Pheromone matrix
τ
i
,
j
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 i,j
η i , j = 1 Z ∗ V c ∗ I i , j =VcI_
Where
I
is the image of size
M
1
∗
M
2
M_
Z
=
∑
i
=
1
:
M
1
∑
j
=
1
:
M
2
V
c
I
i
,
j
\sum _VcI_
,which is a normalization factor
V c I i , j = f  I i − 2 , j − 1 − I i + 2 , j + 1  +  I i − 2 , j + 1 − I i + 2 , j − 1  +  I i − 1 , j − 2 − I i + 1 , j + 2  +  I i − 1 , j − 1 − I i + 1 , j + 1  +  I i − 1 , j − I i + 1 , j  +  I i − 1 , j + 1 − I i − 1 , j − 1  +  I i − 1 , j + 2 − I i − 1 , j − 2  +  I i , j − 1 − I i , j + 1  VcI_=&f\left\vert I_I_\right\vert +\left\vert I_I_\right\vert \\&+\left\vert I_I_\right\vert +\left\vert I_I_\right\vert \\&+\left\vert I_I_\right\vert +\left\vert I_I_\right\vert \\&+\left\vert I_I_\right\vert +\left\vert I_I_\right\vert \end
f
⋅
can be calculated using the following functions:
f
x
=
λ
x
,
for x ≥ 0; 1
f
x
=
λ
x
2
,
for x ≥ 0; 2
,\quad
f
x
=
\sin,&\lambda \\0,&\end
f
x
=
\pi x\sin,&\lambda \\0,&\end
The parameter
λ
in each of above functions adjusts the functions’ respective shapes
Step 2 Construction process:
The ant's movement is based on 4connected pixels or 8connected pixels The probability with which the ant moves is given by the probability equation
P
x
,
y
Step 3 and Step 5 Update process:
The pheromone matrix is updated twice in step 3 the trail of the ant given by
τ
x
,
y
is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation
τ
n
e
w
←
1
−
ψ
τ
o
l
d
+
ψ
τ
0
\leftarrow 1\psi \tau _+\psi \tau _
, where
ψ
is the pheromone decay coefficient
0
<
τ
<
1
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 above images are generated using different functions given by the equation 1 to 4
 Edge linking:
ACO has also been proven effective in edge linking algorithms too
Others
 Classification
 Connectionoriented network routing
 Connectionless network routing
 Data mining
 Discounted cash flows in project scheduling
 Distributed information retrieval
 Grid workflow scheduling problem
 Intelligent testing system
 System identification
 Protein folding
 Power electronic circuit design
 bankruptcy prediction
Definition difficulty
This section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed January 2010 Learn how and when to remove this template message 
With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses Broadly speaking, ant colony algorithms are regarded as populated metaheuristics with each solution represented by an ant moving in the search space Ants mark the best solutions and take account of previous markings to optimize their search They can be seen as probabilistic multiagent algorithms using a probability distribution to make the transition between each iteration In their versions for combinatorial problems, they use an iterative construction of solutions According to some authors, the thing which distinguishes ACO algorithms from other relatives such as algorithms to estimate the distribution or particle swarm optimization is precisely their constructive aspect In combinatorial problems, it is possible that the best solution eventually be found, even though no ant would prove effective Thus, in the example of the Travelling salesman problem, it is not necessary that an ant actually travels the shortest route: the shortest route can be built from the strongest segments of the best solutions However, this definition can be problematic in the case of problems in real variables, where no structure of 'neighbours' exists The collective behaviour of social insects remains a source of inspiration for researchers The wide variety of algorithms for optimization or not seeking selforganization in biological systems has led to the concept of "swarm intelligence", which is a very general framework in which ant colony algorithms fit
Stigmergy algorithms
There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies COA In practice, the use of an exchange of information between ants via the environment a principle called "stigmergy" is deemed enough for an algorithm to belong to the class of ant colony algorithms This principle has led some authors to create the term "value" to organize methods and behavior based on search of food, sorting larvae, division of labour and cooperative transportation
Related methods
 Genetic algorithms GA maintain a pool of solutions rather than just one The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded
 Estimation of Distribution Algorithm EDA is an Evolutionary Algorithm that substitutes traditional reproduction operators by modelguided 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 or generated from guidedcrossover
 Simulated annealing SA is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution A superior neighbor is always accepted An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter The temperature parameter is modified as the algorithm progresses to alter the nature of the search
 Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to selftune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution
 Tabu search TS is similar to simulated annealing in that both traverse the solution space by testing mutations of an individual solution While simulated annealing generates only one mutated solution, tabu search generates many mutated solutions and moves to the solution with the lowest fitness of those generated To prevent cycling and encourage greater movement through the solution space, a tabu list is maintained of partial or complete solutions It is forbidden to move to a solution that contains elements of the tabu list, which is updated as the solution traverses the solution space
 Artificial immune system AIS algorithms are modeled on vertebrate immune systems
 Particle swarm optimization PSO, a swarm intelligence method
 Intelligent water drops IWD, a swarmbased optimization algorithm based on natural water drops flowing in rivers
 Gravitational search algorithm GSA, a swarm intelligence method
 Ant colony clustering method ACCM, a method that make use of clustering approach,extending the ACO
 Stochastic diffusion search SDS, an agentbased probabilistic global search and optimization technique best suited to problems where the objective function can be decomposed into multiple independent partialfunctions
History
Chronology of COA algorithmsChronology of ant colony optimization algorithms
 1959, PierrePaul Grassé invented the theory of stigmergy to explain the behavior of nest building in termites;
 1983, Deneubourg and his colleagues studied the collective behavior of ants;
 1988, and Moyson Manderick have an article on selforganization among ants;
 1989, the work of Goss, Aron, Deneubourg and Pasteels on the collective behavior of Argentine ants, which will give the idea of ant colony optimization algorithms;
 1989, implementation of a model of behavior for food by Ebling and his colleagues;
 1991, M Dorigo proposed the ant system in his doctoral thesis which was published in 1992 A technical report extracted from the thesis and coauthored by V Maniezzo and A Colorni was published five years later;
 1994, Appleby and Steward of British Telecommunications Plc published the first application to telecommunications networks
 1996, publication of the article on ant system;
 1996, Hoos and Stützle invent the maxmin ant system;
 1997, Dorigo and Gambardella publish the ant colony system;
 1997, Schoonderwoerd and his colleagues published an improved application to telecommunication networks;
 1998, Dorigo launches first conference dedicated to the ACO algorithms;
 1998, Stützle proposes initial parallel implementations;
 1999, Bonabeau, Dorigo and Theraulaz publish a book dealing mainly with artificial ants
 2000, special issue of the Future Generation Computer Systems journal on ant algorithms
 2000, first applications to the scheduling, scheduling sequence and the satisfaction of constraints;
 2000, Gutjahr provides the first evidence of convergence for an algorithm of ant colonies
 2001, the first use of COA algorithms by companies Eurobios and AntOptima;
 2001, IREDA and his colleagues published the first multiobjective algorithm
 2002, first applications in the design of schedule, Bayesian networks;
 2002, Bianchi and her colleagues suggested the first algorithm for stochastic problem;
 2004, Zlochin and Dorigo show that some algorithms are equivalent to the stochastic gradient descent, the crossentropy method and algorithms to estimate distribution
 2005, first applications to 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
References
 ^ 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, 134142, 1991
 ^ a b M Dorigo, Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italy, 1992
 ^ Zlochin, Mark; Birattari, Mauro; Meuleau, Nicolas; Dorigo, Marco 1 October 2004 "ModelBased Search for Combinatorial Optimization: A Critical Survey" Annals of Operations Research 131 14: 373–395 doi:101023/B:ANOR000003952652305af ISSN 02545330
 ^ Marco Dorigo and Thomas Stültze, Ant Colony Optimization, p12 2004
 ^ a b T Stützle et HH Hoos, MAX MIN Ant System, Future Generation Computer Systems, volume 16, pages 889914, 2000
 ^ a b M Dorigo et LM Gambardella, Ant Colony System : A Cooperative Learning Approach to the Traveling Salesman Problem, IEEE Transactions on Evolutionary Computation, volume 1, numéro 1, pages 5366, 1997
 ^ 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, 231, pp218
 ^ Gupta, DK; Arora, Y; Singh, UK; Gupta, JP, "Recursive Ant Colony Optimization for estimation of parameters of a function," Recent Advances in Information Technology RAIT, 2012 1st International Conference on , vol, no, pp448454, 15–17 March 2012
 ^ Gupta, DK; Gupta, JP; Arora, Y; Shankar, U, "Recursive ant colony optimization: a new technique for the estimation of function parameters from geophysical field data," Near Surface Geophysics , vol 11, no 3, pp325339
 ^ a b M Zlochin, M Birattari, N Meuleau, et M Dorigo, Modelbased search for combinatorial optimization: A critical survey, Annals of Operations Research, vol 131, pp 373395, 2004
 ^ VKOjha, A Abraham and V Snasel, ACO for Continuous Function Optimization: A Performance Analysis, 14th International Conference on Intelligent Systems Design and Applications ISDA, Japan, Page 145  150 9781479979387/14 2014 IEEE
 ^ a b c M Dorigo, V Maniezzo, et A Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and CyberneticsPart B , volume 26, numéro 1, pages 2941, 1996
 ^ 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
 ^ B Pfahring, "Multiagent search for open scheduling: adapting the AntQ formalism," Technical report TR9609, 1996
 ^ C Blem, "BeamACO, Hybridizing ant colony optimization with beam search An application to open shop scheduling," Technical report TR/IRIDIA/200317, 2003
 ^ T Stützle, "An ant approach to the flow shop problem," Technical report AIDA9707, 1997
 ^ 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, vol8, no2, pp125141, 2000
 ^ M den Besten, "Ants for the single machine total weighted tardiness problem," Master's thesis, University of Amsterdam, 2000
 ^ M, den Bseten, T Stützle and M Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSNVI, Sixth International Conference on Parallel Problem Solving from Nature, vol 1917 of Lecture Notes in Computer Science, pp611620, 2000
 ^ 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, pp287296, 2000
 ^ D Merkle, M Middendorf and H Schmeck, "Ant colony optimization for resourceconstrained project scheduling," Proceedings of the Genetic and Evolutionary Computation Conference GECCO 2000, pp893900, 2000
 ^ 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, pp1427, 2002
 ^ C Gagné, W L Price and M Gravel, "Comparing an ACO algorithm with other heuristics for the single machine scheduling problem with sequencedependent setup times," Journal of the Operational Research Society, vol53, pp895906, 2002
 ^ A V Donati, V Darley, B Ramachandran, "An AntBidding Algorithm for Multistage Flowshop Scheduling Problem: Optimization and Phase Transitions", book chapter in Advances in Metaheuristics for Hard Optimization, Springer, ISBN 9783540729594, pp111138, 2008
 ^ P Toth, D Vigo, "Models, relaxations and exact approaches for the capacitated vehicle routing problem," Discrete Applied Mathematics, vol123, pp487512, 2002
 ^ J M Belenguer, and E Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol30, no5, pp705728, 2003
 ^ T K Ralphs, "Parallel branch and cut for capacitated vehicle routing," Parallel Computing, vol29, pp607629, 2003
 ^ S Salhi and M Sari, "A multilevel composite heuristic for the multidepot vehicle fleet mix problem," European Journal for Operations Research, vol103, no1, pp95112, 1997
 ^ E Angelelli and M G Speranza, "The periodic vehicle routing problem with intermediate facilities," European Journal for Operations Research, vol137, no2, pp233247, 2002
 ^ S C Ho and D Haugland, "A tabu search heuristic for the vehicle routing problem with time windows and split deliveries," Computers & Operations Research, vol31, no12, pp19471964, 2004
 ^ N Secomandi, "Comparing neurodynamic programming algorithms for the vehicle routing problem with stochastic demands," Computers & Operations Research, vol27, no11, pp12011225, 2000
 ^ W P Nanry and J W Barnes, "Solving the pickup and delivery problem with time windows using reactive tabu search," Transportation Research Part B, vol34, no 2, pp107121, 2000
 ^ R Bent and PV Hentenryck, "A twostage hybrid algorithm for pickup and delivery vehicle routing problems with time windows," Computers & Operations Research, vol33, no4, pp875893, 2003
 ^ A Bachem, W Hochstattler and M Malich, "The simulated trading heuristic for solving vehicle routing problems," Discrete Applied Mathematics, vol 65, pp4772, 1996
 ^ S C Hong and Y B Park, "A heuristic for biobjective vehicle routing with time window constraints," International Journal of Production Economics, vol62, no3, pp249258, 1999
 ^ R A Rusell and W C Chiang, "Scatter search for the vehicle routing problem with time windows," European Journal for Operations Research, vol169, no2, pp606622, 2006
 ^ 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, vol185, no3, pp1174–1191, 2008
 ^ T Stützle, "MAXMIN Ant System for the quadratic assignment problem," Technical Report AIDA974, FB Informatik, TU Darmstadt, Germany, 1997
 ^ R Lourenço and D Serra "Adaptive search heuristics for the generalized assignment problem," Mathware & soft computing, vol9, no23, 2002
 ^ 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
 ^ 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, vol1, no4, pp261317, 2001
 ^ Y C Liang and A E Smith, "An ant colony optimization algorithm for the redundancy allocation problem RAP," IEEE Transactions on Reliability, vol53, no3, pp417423, 2004
 ^ G Leguizamon and Z Michalewicz, "A new version of ant system for subset problems," Proceedings of the 1999 Congress on Evolutionary ComputationCEC 99, vol2, pp14581464, 1999
 ^ R Hadji, M Rahoual, E Talbi and V Bachelet "Ant colonies for the set covering problem," Abstract proceedings of ANTS2000, pp6366, 2000
 ^ V Maniezzo and M Milandri, "An antbased framework for very strongly constrained problems," Proceedings of ANTS2000, pp222227, 2002
 ^ R Cordone and F Maffioli,"Colored Ant System and local search to design local telecommunication networks," Applications of Evolutionary Computing: Proceedings of Evo Workshops, vol2037, pp6069, 2001
 ^ C Blum and MJ Blesa, "Metaheuristics for the edgeweighted kcardinality tree problem," Technical Report TR/IRIDIA/200302, IRIDIA, 2003
 ^ S Fidanova, "ACO algorithm for MKP using various heuristic information", Numerical Methods and Applications, vol2542, pp438444, 2003
 ^ 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, vol2, pp10271040, 2001
 ^ O Okobiah, S P Mohanty, and E Kougianos, "Ordinary Kriging MetamodelAssisted Ant Colony Algorithm for Fast Analog Design Optimization Archived March 4, 2016, at the Wayback Machine", in Proceedings of the 13th IEEE International Symposium on Quality Electronic Design ISQED, pp 458463, 2012
 ^ M Sarkar, P Ghosal, and S P Mohanty, "Reversible Circuit Synthesis Using ACO and SA based QuinneMcCluskey Method Archived July 29, 2014, at the Wayback Machine", in Proceedings of the 56th IEEE International Midwest Symposium on Circuits & Systems MWSCAS, 2013, pp 416419
 ^ 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, vol3, pp823826, 2002
 ^ H Nezamabadipour, S Saryazdi, and E Rashedi, " Edge detection using ant algorithms", Soft Computing, vol 10, no7, pp 623628, 2006
 ^ Tian, Jing; Yu, Weiyu; Xie, Shengli "An Ant Colony Optimization Algorithm For Image Edge Detection"
 ^ Gupta, Charu; Gupta, Sunanda "Edge Detection of an Image based on Ant ColonyOptimization Technique"
 ^ Jevtić, A; QuintanillaDominguez, J; CortinaJanuchs, MG; Andina, D 2009 "Edge detection using ant colony search algorithm and multiscale contrast enhancement" IEEE International Conference on Systems, Man and Cybernetics, 2009 SMC 2009: 2193–2198 doi:101109/ICSMC20095345922
 ^ "File Exchange – Ant Colony Optimization ACO" MATLAB Central
 ^ Jevtić, A; Melgar, I; Andina, D 2009 "Ant based edge linking algorithm" 35th Annual Conference of IEEE Industrial Electronics, 2009 IECON '09 pp 3353–3358
 ^ a b 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
 ^ G D Caro and M Dorigo, "Extending AntNet for besteffort qualityofservice routing," Proceedings of the First International Workshop on Ant Colony Optimization ANTS’98, 1998
 ^ GD Caro and M Dorigo "AntNet: a mobile agents approach to adaptive routing," Proceedings of the ThirtyFirst Hawaii International Conference on System Science, vol7, pp7483, 1998
 ^ G D Caro and M Dorigo, "Two ant colony algorithms for besteffort routing in datagram networks," Proceedings of the Tenth IASTED International Conference on Parallel and Distributed Computing and Systems PDCS’98, pp541546, 1998
 ^ D Martens, B Baesens, T Fawcett "Editorial Survey: Swarm Intelligence for Data Mining," Machine Learning, volume 82, number 1, pp 142, 2011
 ^ R S Parpinelli, H S Lopes and A A Freitas, "An ant colony algorithm for classification rule discovery," Data Mining: A heuristic Approach, pp191209, 2002
 ^ R S Parpinelli, H S Lopes and A A Freitas, "Data mining with an ant colony optimization algorithm," IEEE Transaction on Evolutionary Computation, vol6, no4, pp321332, 2002
 ^ W N Chen, J ZHANG and H Chung, "Optimizing Discounted Cash Flows in Project SchedulingAn Ant Colony Optimization Approach", IEEE Transactions on Systems, Man, and CyberneticsPart C: Applications and Reviews Vol40 No5 pp6477, Jan 2010
 ^ D Picard, A Revel, M Cord, "An Application of Swarm Intelligence to Distributed Image Retrieval", Information Sciences, 2010
 ^ D Picard, M Cord, A Revel, "Image Retrieval over Networks : Active Learning using Ant Algorithm", IEEE Transactions on Multimedia, vol 10, no 7, pp 13561365  nov 2008
 ^ 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 CyberneticsPart C: Applications and Reviews, Vol 31, No 1,pp2943,Jan 2009
 ^ Xiao MHu, 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 CyberneticsPart C: Applications and Reviews, Vol 39, No 6, pp 659669, Dec 2009
 ^ L Wang and Q D Wu, "Linear system parameters identification based on ant system algorithm," Proceedings of the IEEE Conference on Control Applications, pp 401406, 2001
 ^ 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 827841, 2001
 ^ X M Hu, J ZHANG，J Xiao and Y Li, "Protein Folding in HydrophobicPolar Lattice Model: A Flexible Ant Colony Optimization Approach ", Protein and Peptide Letters, Volume 15, Number 5, 2008, Pp 469477
 ^ 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, vol2463, pp4052, 2002
 ^ M Nardelli; L Tedesco; A Bechini 2013 "Crosslattice behavior of general ACO folding for proteins in the HP model" Proc of ACM SAC 2013: 1320–1327 doi:101145/24803622480611
 ^ 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 Vol24,No1, pp147162, Jan 2009
 ^ Zhang, Y 2013 "A RuleBased Model for Bankruptcy Prediction Based on an Improved Genetic Ant Colony Algorithm" Mathematical Problems in Engineering 2013: 753251
 ^ A Ajith; G Crina; R Vitorino éditeurs, Stigmergic Optimization, Studies in Computational Intelligence , volume 31, 299 pages, 2006 ISBN 9783540346890
 ^ Pelikan, Martin; Goldberg, David E; CantúPaz, Erick 1 January 1999 "BOA: The Bayesian Optimization Algorithm" Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation  Volume 1 Morgan Kaufmann Publishers Inc: 525–532
 ^ Pelikan, Martin 2005 Hierarchical Bayesian optimization algorithm : toward a new generation of evolutionary algorithms 1st ed ed Berlin : Springer ISBN 9783540237747 CS1 maint: Extra text link
 ^ Thierens, Dirk 11 September 2010 "The Linkage Tree Genetic Algorithm" Parallel Problem Solving from Nature, PPSN XI Springer Berlin Heidelberg: 264–273 doi:101007/9783642158445_27
 ^ Martins, Jean P; Fonseca, Carlos M; Delbem, Alexandre C B 25 December 2014 "On the performance of linkagetree genetic algorithms for the multidimensional knapsack problem" Neurocomputing 146: 17–29 doi:101016/jneucom201404069
 ^ PP Grassé, La reconstruction du nid et les coordinations interindividuelles chez Belicositermes natalensis et Cubitermes sp La théorie de la Stigmergie : Essai d’interprétation du comportement des termites constructeurs, Insectes Sociaux, numéro 6, p 4180, 1959
 ^ JL Denebourg, JM Pasteels et JC Verhaeghe, Probabilistic Behaviour in Ants : a Strategy of Errors, Journal of Theoretical Biology, numéro 105, 1983
 ^ F Moyson, B Manderick, The collective behaviour of Ants : an Example of SelfOrganization in Massive Parallelism, Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988
 ^ S Goss, S Aron, JL Deneubourg et JM Pasteels, Selforganized shortcuts in the Argentine ant, Naturwissenschaften, volume 76, pages 579581, 1989
 ^ M Ebling, M Di Loreto, M Presley, F Wieland, et D Jefferson,An Ant Foraging Model Implemented on the Time Warp Operating System, Proceedings of the SCS Multiconference on Distributed Simulation, 1989
 ^ Dorigo M, V Maniezzo et A Colorni, Positive feedback as a search strategy, rapport technique numéro 91016, Dip Elettronica, Politecnico di Milano, Italy, 1991
 ^ Appleby, S & Steward, S Mobile software agents for control in telecommunications networks, BT Technol J, 122:104–113, April 1994
 ^ R Schoonderwoerd, O Holland, J Bruten et L Rothkrantz, Antbased load balancing in telecommunication networks, Adaptive Behaviour, volume 5, numéro 2, pages 169207, 1997
 ^ M Dorigo, ANTS’ 98, From Ant Colonies to Artificial Ants : First International Workshop on Ant Colony Optimization, ANTS 98, Bruxelles, Belgique, octobre 1998
 ^ T Stützle, Parallelization Strategies for Ant Colony Optimization, Proceedings of PPSNV, Fifth International Conference on Parallel Problem Solving from Nature, SpringerVerlag, volume 1498, pages 722731, 1998
 ^ É Bonabeau, M Dorigo et G Theraulaz, Swarm intelligence, Oxford University Press, 1999
 ^ M Dorigo , G Di Caro et T Stützle, Special issue on "Ant Algorithms", Future Generation Computer Systems, volume 16, numéro 8, 2000
 ^ WJ Gutjahr, A graphbased Ant System and its convergence, Future Generation Computer Systems, volume 16, pages 873888, 2000
 ^ S Iredi, D Merkle et M Middendorf, BiCriterion Optimization with Multi Colony Ant Algorithms, Evolutionary MultiCriterion Optimization, First International Conference EMO’01, Zurich, Springer Verlag, pages 359372, 2001
 ^ L Bianchi, LM Gambardella et MDorigo, An ant colony optimization approach to the probabilistic traveling salesman problem, PPSNVII, Seventh International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, Springer Verlag, Berlin, Allemagne, 2002
 ^ B Prabhakar, K N Dektar, D M Gordon, "The regulation of ant colony foraging activity without spatial information ", PLOS Computational Biology, 2012 URL: http://wwwploscompbiolorg/article/info%3Adoi%2F101371%2Fjournalpcbi1002670
Publications selected
 M Dorigo, 1992 Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italy
 M Dorigo, V Maniezzo & A Colorni, 1996 "Ant System: Optimization by a Colony of Cooperating Agents", IEEE Transactions on Systems, Man, and Cybernetics–Part B, 26 1: 29–41
 M Dorigo & L M Gambardella, 1997 "Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem" IEEE Transactions on Evolutionary Computation, 1 1: 53–66
 M Dorigo, G Di Caro & L M Gambardella, 1999 "Ant Algorithms for Discrete Optimization" Artificial Life, 5 2: 137–172
 E Bonabeau, M Dorigo et G Theraulaz, 1999 Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press ISBN 0195131592
 M Dorigo & T Stützle, 2004 Ant Colony Optimization, MIT Press ISBN 0262042193
 M Dorigo, 2007 "Ant Colony Optimization" Scholarpedia
 C Blum, 2005 "Ant colony optimization: Introduction and recent trends" Physics of Life Reviews, 2: 353373
 M Dorigo, M Birattari & T Stützle, 2006 Ant Colony Optimization: Artificial Ants as a Computational Intelligence Technique TR/IRIDIA/2006023
 Mohd Murtadha Mohamad,"Articulated Robots Motion Planning Using Foraging Ant Strategy",Journal of Information Technology  Special Issues in Artiﬁcial Intelligence, Vol20, No 4 pp 163–181, December 2008, ISSN 01283790
 N Monmarché, F Guinand & P Siarry eds, "Artificial Ants", August 2010 Hardback 576 pp ISBN 9781848211940
 A Kazharov, V Kureichik, 2010 "Ant colony optimization algorithms for solving transportation problems", Journal of Computer and Systems Sciences International, Vol 49 No 1 pp 30–43
 K Saleem, N Fisal, M A Baharudin, A A Ahmed, S Hafizah and S Kamilah, "Ant colony inspired selfoptimized routing protocol based on cross layer architecture for wireless sensor networks", WSEAS Trans Commun, vol 9, no 10, pp 669–678, 2010 ISBN 9789604742004
 K Saleem and N Fisal, "Enhanced Ant Colony algorithm for selfoptimized data assured routing in wireless sensor networks", Networks ICON 2012 18th IEEE International Conference on, pp 422–427 ISBN 9781467345231
External links
 Ant Colony Optimization Home Page
 "Ant Colony Optimization"  Russian scientific and research community
 AntSim  Simulation of Ant Colony Algorithms
 MIDACOSolver General purpose optimization software based on ant colony optimization Matlab, Excel, VBA, C/C++, R, C#, Java, Fortran and Python
 University of Kaiserslautern, Germany, AG Wehn: Ant Colony Optimization Applet Visualization of Traveling Salesman solved by ant system with numerous options and parameters Java Applet
 Ant Farm Simulator
 Ant algorithm simulation Java Applet
 Java Ant Colony System Framework



Biological swarming 


Animal migration 


Swarm algorithms 


Collective motion 


Swarm robotics 


Related topics 

Ant colony optimization algorithms Information about
Ant colony optimization algorithms
Ant colony optimization algorithms
Ant colony optimization algorithms Information Video
Ant colony optimization algorithms viewing the topic.
There are excerpts from wikipedia on this article and video