Ant Colony Optimization

Ant Colony Optimization

Swarm Intelligence “Swarm intelligence (SI) is artificial intelligence based on the collective behaviour of decentralized, self-organized systems”

Swarm Intelligence General Characteristics • Composed of many individuals • Individuals are homogeneous • Local interaction based on simple rules • Self-organization • Constituting a natural model particularly suited to distributed problem solving

Swarm Intelligence General Characteristics • Collective system capable of accomplishing difficult tasks in dynamic and varied environments without any external guidance or control and with no central coordination • Achieving a collective performance which could not normally be achieved by an individual acting alone • Constituting a natural model particularly suited to distributed problem solving

http://www.scs.carleton.ca/~arpwhite/courses/95590Y/notes/SI%20Lecture%203.pdfhttp://www.scs.carleton.ca/~arpwhite/courses/95590Y/notes/SI%20Lecture%203.pdf

Ant Colony System • First introduced by Marco Dorigo in 1992 as a method for solving hard combinatorial optimization problems (COPs). • Progenitor to “Ant Colony System,” later discussed • Result of research on computational intelligence approaches to combinatorial optimization • Originally applied to Traveling Salesman Problem • Applied later to various hard optimization problems

Inherent Features • Inherent parallelism • Stochastic nature • Adaptivity • Use of positive feedback • Autocatalytic in nature

Real Ant actual scenario • Almost blind. • Incapable of achieving complex tasks alone. • Rely on the phenomena of swarm intelligencefor survival. • Capable of establishing shortest-route paths from their colony to feeding sources and back. • Use stigmergic communication via pheromone trails. • Follow existing pheromone trails with high probability. • What emerges is a form of autocatalyticbehavior: the more ants follow a trail, the more attractive that trail becomes for being followed.

Real Ant actual scenario • The process is thus characterized by a positive feedback loop, where the probability of a discrete path choice increases with the number of times the same path was chosen before.

Natural behavior of an ant :Foraging modes • Wander mode • Search mode • Return mode • Attracted mode • Trace mode • Carry mode

Behavior of Ant colony • regulation of nest temperature within 1 degree celsius range; • forming bridges; • raiding specific areas for food; • building and protecting nest; • sorting brood and food items; • cooperating in carrying large items; • emigration of a colony; • finding shortest route from nest to food source; • preferentially exploiting the richest food source available.

Autocatalyzation Autocatalysisis a positive feedback loopthat drives the ants to explore promising aspects of the search space over less promising areas.

A key concept: Stigmergy Stigmergy is: indirect communication via interaction with the environment. • A problem gets solved bit by bit .. • Individuals communicate with each other in the above way, affecting what each other does on the task. • Individuals leave markers or messages – these don’t solve the problem in themselves, but they affect other individuals in a way that helps them solve the problem.

Stigmergy in Ants Ants are behaviourally unsophisticated, but collectively they can perform complex tasks. Ants have highly developedsophisticatedsign-basedstigmergy. • They communicate using pheromones; • They lay trails of pheromone that can be followed by other ants.

Food source Nest Pheromone Trails • Individual ants lay pheromone trails while travelling from the nest, to the nest or possibly in both directions. • The pheromone trail gradually evaporates over time. • But pheromone trail strength accumulate with multiple ants using path.

Naturally Observed Ant Behavior All is well in the world of the ant.

Naturally Observed Ant Behavior An obstacle has blocked our path

Naturally Observed Ant Behavior

ACO Algorithms: Basic Ideas • Ants are agents that: Move along between nodes in a graph. • They choose where to go based on pheromone strength (and maybe other things) • An ant’s path represents a specific candidate solution. • When an ant has finished a solution, pheromone is laid on its path, according to quality of solution. • This pheromone trail affects behaviour of other ants by `stigmergy’

Artificial Ants • artifcial ants may simulate pheromone laying by modifying appropriate pheromone variables associated with problem states they visit while building solutions to the optimization problem. Also, according to the stigmergic communication model, the artifcial ants would have only local access tothese pheromone variables.

Artificial Ants Main characteristics of stigmergy can be extended to artificial agents by • Associating state variables with different problem states; and • Giving the agents only local access to these variables. • Coupling between the autocatalytic mechanism and the implicit evaluation of solutions • Just like real ants, artificial ants create their solutions sequentially by moving from one problem state to another

Differences between real and artificial ants: • Artificial ants live in a discrete world| they move sequentially through a finite set of problem states. • The pheromone update (i.e., pheromone depositing and evaporation) is not accomplished in exactly the same way by artificial ants as by real ones. Sometimes the pheromone update is done only by some of the artificial ants, and often only after a solution has been constructed. • Some implementations of artificial ants use additional mechanisms that do not exist in the case of real ants. Examples include look-ahead, local search, backtracking, etc.

Nature/Technology

SHORTEST PATH • Ants deposit pheromones on ground that form a trail. The trail attracts other ants. • Pheromones evaporate faster on longer paths. • Shorter paths serve as the way to food for most of the other ants.

Minimum Cost Path

General ACO • A stochastic construction procedure • Probabilistically build a solution • Iteratively adding solution components to partial solutions - Heuristic information - Trace/Pheromone trail • Reinforcement Learning reminiscence • Modify the problem representation at each iteration • Ants work concurrently and independently • Collective interaction via indirect communication leads to good solutions

Some inherent advantages • Positive Feedback accounts for rapid discovery of good solutions • Distributed computation avoids premature convergence • The greedy heuristic helps find acceptable solution in the early solution in the early stages of the search process. • The collective interaction of a population of agents.

Disadvantages in Ant Systems • Slower convergence than other Heuristics • Performed poorly for TSP problems larger than 75 cities. • No centralized processor to guide the AS towards good solutions

Ant System (AS) Algorithm • Initialization • Randomly place ants • Build tours • Deposit trail • Update trail • Loop or exit

Ant with Binary Bridge • Let the amount of pheromone on a branch be proportional to the number of ants that used the branch in the past and let ms(t) and ml(t) be the numbers of ants that have used the short and the long branches after a total of t ants have crossed the bridge, with ms(t) þ ml(t) =t.The probability ps(t) with which the (t+1) th ant chooses the short branch can then be written as

Ant with Binary Bridge The number of ants choosing the short branch is given by The number of ants choosing the long branch by where q is a uniform random number drawn from the interval [0; 1]. Mote Carlo Simulation method will give good solution

ACO Parameters

Application • Traveling Salesman Problem • Quadratic Assignment Problem • Network Model Problem • Vehicle routing • Graph coloring

ACO system -PSEUDOCODE • Often applied to TSP (Travelling Salesman Problem): shortest path between n nodes • Algorithm in Pseudocode: • Initialize Trail • Do While (Stopping Criteria Not Satisfied) – Cycle Loop • Do Until (Each Ant Completes a Tour) – Tour Loop • Local Trail Update • End Do • Analyze Tours • Global Trail Update • End Do

ACO Algorithm • Ant Colony Algorithms are typically use to solve minimum cost problems. • We may usually have N nodes and A undirected arcs • There are two working modes for the ants: either forwards or backwards • The ants memory allows them to retrace the path it has followed while searching for the destination node • Before moving backward on their memorized path, they eliminate any loops from it. While moving backwards, the ants leave pheromones on the arcs they traversed.

ACO Algorithm • At the beginning of the search process, a constant amount of pheromone is assigned to all arcs. When located at a node i an ant k uses the pheromone trail to compute the probability of choosing j as the next node: where is the neighborhood of ant k when in node i.

ACO Algorithm • When the arc (i,j) is traversed , the pheromone value changes as follows: • By using this rule, the probability increases that forthcoming ants will use this arc. • After each ant k has moved to the next node, the pheromones evaporate by the following equation to all the arcs:

Steps for Solving a Problem by ACO • Represent the problem in the form of sets of components and transitions, or by a set of weighted graphs, on which ants can build solutions • Define the meaning of the pheromone trails • Define the heuristic preference for the ant while constructing a solution • If possible implement a efficient local search algorithm for the problem to be solved. • Choose a specific ACO algorithm and apply to problem being solved • Tune the parameter of the ACO algorithm.

Combinatorial optimization • Find values of discrete variables • Optimizing a given objective function Π = (S, f, Ω) – problem instance • S – set of candidate solutions • f – objective function • Ω – set of constraints • set of feasible solutions (with respect to Ω) • Find globally optimal feasible solution s*

Combinatorial optimization problem mapping • Combinatorial problem (S, f, Ω(t)) • Ω(t) – time-dependent constraints • Example – dynamic problems • Goal – find globally optimal feasible solution s* • Minimization problem • Mapped on another problem

Combinatorial optimization problem mapping • C = {c1, c2, …, cNc} – finite set of components • States of the problem: X = {x = <ci, cj, …, ch, …>, |x| < n < +∞} • Set of candidate solutions:

Combinatorial optimization problem mapping

Combinatorial optimization problem mapping • X – states • S – candidate solutions • – feasible states • S* – optimal solutions

Combinatorial optimization problem mapping • Cost g(s, t) for each • In most cases – g(s, t) ≡ f(s, t) • GC = (C, L) – completely connected graph • C – set of components • L – edges fully connecting the components (connections) • GC – construction graph • Artificial ants build solutions by performing randomized walks on GC(C, L)

ACO for Traveling 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. 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.

Ant Colony Optimization