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Ant behavior was the inspiration for the metaheuristic optimization technique

The ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs.

This algorithm is a member of ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis , 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.

Overview

Summary

In the real world, ants (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but to instead follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).

Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over faster, and thus the pheromone density remains high as it is laid on the path as fast as it can evaporate. Pheromone evaporation has also the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained.

Thus, when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads all the ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.

Detailed

File:Aco branches.svg

The original idea comes from observing the exploitation of food resources among ants, in which ants’ individually limited cognitive abilities have collectively been able to find the shortest path between a food source and the nest.

1. The first ant finds the food source (F), via any way (a), then returns to the nest (N), leaving behind a trail pheromone (b)
2. Ants indiscriminately follow four possible ways, but the strengthening of the runway makes it more attractive as the shortest route.
3. Ants take the shortest route, long portions of other ways lose their trail pheromones.

In a series of experiments on a colony of ants with a choice between two unequal length paths leading to a source of food, biologists have observed that ants tended to use the shortest route. A model explaining this behaviour is as follows:

1. An ant (called "blitz") runs more or less at random around the colony;
2. If it discovers a food source, it returns more or less directly to the nest, leaving in its path a trail of pheromone;
3. These pheromones are attractive, nearby ants will be inclined to follow, more or less directly, the track;
4. Returning to the colony, these ants will strengthen the route;
5. If two routes are possible to reach the same food source, the shorter one will be, in the same time, traveled by more ants than the long route will;
6. The short route will be increasingly enhanced, and therefore become more attractive;
7. The long route will eventually disappear, pheromones are volatile;
8. Eventually, all the ants have determined and therefore "chosen" the shortest route.

Ants use the environment as a medium of communication. They exchange information indirectly by depositing pheromones, all detailing the status of their "work". The information exchanged has a local scope, only an ant located where the pheromones were left has a notion of them. This system is called "Stigmergy" and occurs in many social animal societies (it has been studied in the case of the construction of pillars in the nests of termites). The mechanism to solve a problem too complex to be addressed by single ants is a good example of a self-organized system. This system is based on positive feedback (the deposit of pheromone attracts other ants that will strengthen it themselves) and negative (dissipation of the route by evaporation prevents the system from thrashing). Theoretically, if the quantity of pheromone remained the same over time on all edges, no route would be chosen. However, because of feedback, a slight variation on an edge will be amplified and thus allow the choice of an edge. The algorithm will move from an unstable state in which no edge is stronger than another, to a stable state where the route is composed of the strongest edges.

Разширения на понятието

Следват някои от най-популярните вариации на алгоритмите за оптимизация по метода на мравките.

Оптимизация чрез елитни мравки

Това е първата по време модификация на алгоритъма. При нея мравките, които към всяка отделна итерация дават най-добро решение, се обявяват за елитни мравки и те получават право да полагат специален феромон по маршрута си.

Минимаксна мравчена оптимизация

Моделът се променя, като се добавят максимални и минимални количества феромон [τ_max,τ_min] Феромон се полага само по маршрута, отговарящ на глобално най-доброто решение или най-доброто решение за конкретната итерация. Всички дъги се инициализират с τ_max и повторно получават стойността τ_max при наближаване на ситуация на стагнация.

Proportional pseudo-random rule

Представен е по-горе.

Мравчена система, базирана на рангове

Всички решения се ранжират по отношение на фитнес функцията им. После количеството отложен феромон се претегля за всяко решение, по такъв начин, че решенията с по-добри показатели на фитнес функцията да получат повече феромон от тези с по-слаби показатели.

Непрекъсната ортогонална мравчена система

Механизмът за полагане на феромон при непрекъснатата ортогонална мравчена система е такъв, че позволява на мравките на търсят решения по един колаборативен и ефективен начин. Използвайки orthogonal design method, мравките могат да изследват избраните от тях региони от допустимата област бързо и ефикасно с подобрени възможности за глобално търсене и с по-висока точност.

The orthogonal design method и adaptive radius adjustment method също могат да бъдат разширени към други оптимизационни алгоритми, предоставяйки по-добри възможности за решаване на задачи от практиката.

Сходимост

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 a finite time). The first evidence of a convergence ant colony algorithm was made in 2000, the graph-based ant system algorithm, and then algorithms for ACS and MMAS. Like most metaheuristics, it is very difficult to estimate the theoretical speed of convergence. In 2004, Zlochin and his colleagues have shown COA type algorithms could be assimilated methods of stochastic gradient descent, on the cross-entropy and Estimation of distribution algorithm. They proposed that these metaheuristics as a "research-based model".

Приложения

File:Knapsack ants.svg

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 problems, ranging from quadratic assignment to fold protein or routing vehicles and a lot of derived methods have been adapted to dynamic problems in real variables, stochastic problems, multi-targets and parallel implementations. It has also been used to produce near-optimal solutions to the travelling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches of similar problems when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing and urban transportation systems.

As a very good example, ant colony optimization algorithms have been used to produce near-optimal solutions to the travelling salesman problem. 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 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:

1. It must visit each city exactly once;
2. A distant city has less chance of being chosen (the visibility);
3. The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen;
4. Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short;
5. After each iteration, trails of pheromones evaporate.

File:Aco TSP.svg

Примерен псевдокод и формули

 procedure ACO_MetaHeuristic
   while(not_termination)
      generateSolutions()
      daemonActions()
      pheromoneUpdate()
   end while
 end procedure

Edge Selection:

An ant will move from node [math]\displaystyle{ i }[/math] to node [math]\displaystyle{ j }[/math] with probability

[math]\displaystyle{ p_{i,j} = \frac { (\tau_{i,j}^{\alpha}) (\eta_{i,j}^{\beta}) } { \sum (\tau_{i,j}^{\alpha}) (\eta_{i,j}^{\beta}) } }[/math]

where

[math]\displaystyle{ \tau_{i,j} }[/math] is the amount of pheromone on edge [math]\displaystyle{ i,j }[/math]

[math]\displaystyle{ \alpha }[/math] is a parameter to control the influence of [math]\displaystyle{ \tau_{i,j} }[/math]

[math]\displaystyle{ \eta_{i,j} }[/math] is the desirability of edge [math]\displaystyle{ i,j }[/math] (a priori knowledge, typically [math]\displaystyle{ 1/d_{i,j} }[/math], where d is the distance)

[math]\displaystyle{ \beta }[/math] is a parameter to control the influence of [math]\displaystyle{ \eta_{i,j} }[/math]

Обновяване на феромона

[math]\displaystyle{ \tau_{i,j} = (1-\rho)\tau_{i,j} + \Delta \tau_{i,j} }[/math]

where

[math]\displaystyle{ \tau_{i,j} }[/math] is the amount of pheromone on a given edge [math]\displaystyle{ i,j }[/math]

[math]\displaystyle{ \rho }[/math] is the rate of pheromone evaporation

and [math]\displaystyle{ \Delta \tau_{i,j} }[/math] is the amount of pheromone deposited, typically given by

[math]\displaystyle{ \Delta \tau^{k}_{i,j} = \begin{cases} 1/L_k & \mbox{if ant }k\mbox{ travels on edge }i,j \\ 0 & \mbox{otherwise} \end{cases} }[/math]

where [math]\displaystyle{ L_k }[/math] is the cost of the [math]\displaystyle{ k }[/math]th ant's tour (typically length).

Други примери

The ant colony algorithm was originally used mainly to produce near-optimal solutions to the travelling salesman problem and, more generally, the problems of combinatorial optimization.

Job-shop scheduling problem

• Job-shop scheduling problem (JSP)
• Open-shop scheduling problem (OSP)
• Permutation flow shop problem (PFSP)
• Single machine total tardiness problem (SMTTP)
• Single machine total weighted tardiness problem (SMTWTP)
• Resource-constrained project scheduling problem (RCPSP)
• Group-shop scheduling problem (GSP)
• Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)

Vehicle routing problem

• Capacitated vehicle routing problem (CVRP)
• Multi-depot vehicle routing problem (MDVRP)
• Period vehicle routing problem (PVRP)
• Split delivery vehicle routing problem (SDVRP)
• Stochastic vehicle routing problem (SVRP)
• Vehicle routing problem with pick-up and delivery (VRPPD)
• Vehicle routing problem with time windows (VRPTW)

Assignment problem

• Quadratic assignment problem (QAP)
• Generalized assignment problem (GAP)
• Frequency assignment problem (FAP)
• Redundancy allocation problem (RAP)

Set problem

• Set covering problem(SCP)
• Set partition problem (SPP)
• Weight constrained graph tree partition problem (WCGTPP)
• Arc-weighted l-cardinality tree problem (AWlCTP)
• Multiple knapsack problem (MKP)
• Maximum independent set problem (MIS)

Други

• Classification
• Connection-oriented network routing
• Connectionless network routing
• Data mining
• Discounted cash flows in project scheduling
• Grid Workflow Scheduling Problem
• Image processing
• Intelligent testing system
• System identification
• Protein Folding
• Power Electronic Circuit Design

A difficulty in definition

File:Aco shortpath.svg

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 multi-agent algorithms using a probability distribution to make the transition between each iteration. In their versions for combinatorial problems, they use an iterative construction of solutions. 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", 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. .

Свързани подходи

• Genetic algorithms (GA) maintain a pool of solutions rather than just one. The process of finding superior solutions mimics that of evolution, with solutions being combined or mutated to alter the pool of solutions, with solutions of inferior quality being discarded.
• Simulated annealing (SA) is a related global optimization technique which traverses the search space by generating neighboring solutions of the current solution. A superior neighbor is always accepted. An inferior neighbor is accepted probabilistically based on the difference in quality and a temperature parameter. The temperature parameter is modified as the algorithm progresses to alter the nature of the search.
• 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 swarm-based 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.

История

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 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-objectif

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Chronology of COA Algorithms.

Chronology of Ant colony optimization algorithms.

• 1959, Pierre-Paul Grass 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 self-organization 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 co-authored by V. Maniezzo and A. Colorni was published five years later;
• 1996, publication of the article on Ant System;
• 1996, Hoos and Stützle invent the MAX-MIN Ant System ;
• 1997, Dorigo and Gambardella publish the Ant Colony System ;
• 1997, Schoonderwoerd and his colleagues developed the first 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 multi-objective 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 cross-entropy and algorithms to estimate distribution
• 2005, first applications to folding protein problems.

Избрани публикации

• 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 0-19-513159-2
• M. Dorigo & T. Stützle, 2004. Ant Colony Optimization, MIT Press. ISBN 0-262-04219-3
• M. Dorigo, 2007. "Ant Colony Optimization". Scholarpedia.
• C. Blum, 2005 "Ant colony optimization: Introduction and recent trends". Physics of Life Reviews, 2: 353-373
• M. Dorigo, M. Birattari & T. Stützle, 2006 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. 163-181, December 2008, ISSN0128-3790.

Въъншни препратки

• Ant Colony Optimization Home Page
• 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)
• Department of Electronics & Information Engineering, JBNU, Jeonju, Korea, Ant Colony Optimization