Ant colony optimization algorithms
ant colony optimization algorithms on matrix, ant colony optimization algorithmsIn 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
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With an ACO algorithm, the shortest path in a graph, between two points A and B, is built from a combination of several paths It is not easy to give a precise definition of what algorithm is or is not an ant colony, because the definition may vary according to the authors and uses 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
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 ^ 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
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 ^ 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
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 ^ 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
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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



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