1. Ant Colony Optimization

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

3. 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

4. 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

6. http://www. scs. carleton

7. http://www. scs. carleton

8. http://www. scs. carleton

9. Ant Colony System
First introduced by Marco Dorigo in 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

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

11. Real Ant actual scenario
Almost blind.
Incapable of achieving complex tasks alone.
Rely on the phenomena of swarm intelligence for 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 autocatalytic behavior: the more ants follow a trail, the more attractive that trail becomes for being followed.

12. 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.

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

14. 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.

15. Autocatalyzation
Autocatalysis is a positive feedback loop that drives the ants to explore promising aspects of the search space over less promising areas.

16. 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.

17. Stigmergy in Ants
Ants are behaviourally unsophisticated, but collectively they can perform complex tasks.
Ants have highly developed sophisticated sign-based stigmergy.
They communicate using pheromones;
They lay trails of pheromone that can be followed by other ants.

18. 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.
Food source
Nest

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

20. Naturally Observed Ant Behavior
An obstacle has blocked our path

21. Naturally Observed Ant Behavior

22. Naturally Observed Ant Behavior

23. 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’

24. 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.

25. 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

26. 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.

27. Nature/Technology

28. 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.

29. Minimum Cost Path

30. 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

31. 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.

32. 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

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

34. 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

35. 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

36. ACO Parameters

37. ACO Parameters

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

39. 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

40. 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.

41. 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.

42. 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:

43. 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.

44. 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*

45. 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

46. 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:

47. Combinatorial optimization problem mapping

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

49. 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)

50. 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.

51. ACO for Traveling Salesman Problem
TSP PROBLEM : Given N cities, and a distance function d between cities, find a tour that:
1. Goes through every city once and only once
2. Minimizes the total distance.

52. HOW TO IMPLEMENT IN A PROGRAM
Ants: Simple computer agents
Move ant: Pick next component in the const. solution
Pheromone:
Memory: MK or TabuK
Next move: Use probability to move ant
Graph (N,E): where N = cities/nodes, E = edges
= the tour cost from city i to city j (edge weight)
Ant move from one city i to the next j with some transition probability.

53. A simple TSP example B C A D dAB =8;dBC = 4;dCD =15;dDA =6 [] [] [] []2 [] B C 3 [] A 4 [] D
dAB =8;dBC = 4;dCD =15;dDA =6 53

54. Iteration 1 3
[C] 2
[B] C B 1
[A] A 4
[D] D 54

55. How to build next sub-solution?1
[A] B C A D 55

56. Iteration 2 2
[B,C] 1
[A,B] C B 4
[D,A] A 3
[C,D] D 56

57. Iteration 3 1
[A,B,C] 4
[D,A,B] C B 3
[B,C,A] A 2
[B,C,D] D 57

58. Iteration 4 4
[D,A,B,C] 3
[C,D,A,B] C B 2
[B,C,D,A] A 1
[A,B,C,D] D 58

59. Path and Pheromone Evaluation1
[A,B,C,D
L1 =27 2
[B,C,D,A]
L2 =25 3
[C,D,A,B]
L3 =29 4
[D,A,B,C]
Best tour
L4 =18 59

60. MAX–MIN Ant System
MAX–MIN Ant System (MMAS) (Stu¨ tzle & Hoos, 1997, 2000; Stu¨ tzle, 1999) introduces four main modifications with respect to AS. First, it strongly exploits the best tours found: only either the iteration-best ant, that is, the ant that produced the best tour in the current iteration, or the best-so-far ant is allowed to deposit pheromone. Unfortunately, such a strategy may lead to a stagnation situation in which all the ants follow the same tour, because of the excessive growth of pheromone trails on arcs of a good, although suboptimal, tour.

61. MAX–MIN Ant System
To counteract this effect, a second modification introduced by MMAS is that it limits the possible range of pheromone trail values to the interval [Ʈmin; Ʈmax]. Third, the pheromone trails are initialized to the upper pheromone trail limit, which, together with a small pheromone evaporation rate, increases the exploration of tours at the start of the search. Finally, in MMAS, pheromone trails are reinitialized each time the system approaches stagnation or when no improved tour has been generated for a certain number of consecutive iterations.

62. Greedy Search Algorithm
A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage[1] with the hope of finding a global optimum. In many problems, a greedy strategy does not in general produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a global optimal solution in a reasonable time.

63. Greedy Search Algorithm
For example, a greedy strategy for the traveling salesman problem (which is of a high computational complexity) is the following heuristic: "At each stage visit an unvisited city nearest to the current city". This heuristic need not find a best solution, but terminates in a reasonable number of steps; finding an optimal solution typically requires unreasonably many steps. In mathematical optimization, greedy algorithms solve combinatorial problems having the properties of matroids.

64. Constructive Heuristics
Start from an “empty solution”
Repeatedly, extend the current solution until a complete solution is constructed
Use heuristics to try to extend in such a way that the final solution is a good one
It is essential to know the difference between:
Constructive methods
Extend empty solution until get complete solution
Local search
Take complete solution and try to improve it via local moves