The document discusses the traveling salesman problem (TSP) and how ant colony optimization (ACO) algorithms can be used to find optimal or near-optimal solutions. It provides an overview of ACO, including how artificial ants deposit and follow pheromone trails to probabilistically construct solutions. The ACO algorithm is described and an example TSP problem with 4 cities (A, B, C, D) is shown across 4 iterations to demonstrate the algorithm. Advantages are noted such as efficiency for small problems and ability to adapt to changes, while disadvantages include slow convergence time for large problems.

1. Travelling Salesman Problem

2. Agenda:
 Introduction
 Overview
 ACO Algorithm
 A Simple TSP Example
 Pros and Cons

3. Introduction:
 The concept of Travelling Salesman Problem TSP is
simple, it reflects a salesman's problems that has to
pass through all the cities given and return to its origin
with the shortest distance to be travel.

4. ANT Colony Optimization in TSP
Example:

5. Overview of Ant Colony Optimization:
 Optimization Technique Proposed by Marco
Dorigo in the early ’90
 Multi-agent approach for solving difficult
combinatorial optimization problems
 Originally applied to Traveling Salesman
Problem
 Has become new and fruitful research area

6. The Ants:
 Can explore vast areas without global view of
the ground.
 Can find the food and bring it back to the nest.
 Will converge to the shortest path.

7. How can they manage such great tasks ?
Ants are ,
essentially blind, deaf and dumb.
social creatures – behavior directed to survival of colony
Question: how can ants find the short path to food sources?

8. 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.

9. 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.

10. 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.

11. 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:

12. Algorithm for TSP(AS)
Initialize
Place each ant in a randomly chosen city
Choose NextCity(For Each Ant)
more cities
to visit
For Each Ant
Return to the initial cities
Update pheromone level using the tour cost for each ant
Print Best tour
yes
No
Stopping
criteria
yes
No

13. A
D
CB
1
[]
4
[]
3
[]
2
[]
dAB =8;dBC = 4;dCD =15;dDA =613
A Simple TSP Example:

14. A
D
C
B
1
[A]
3
[C]
2
[B]
4
[D]
14
Iteration 1:

15. 1
[A]
A
D
CB
15
How to build next sub-solution?

16. A
D
C
B
2
[B,C]
3
[C,D]
1
[A,B]
4
[D,A]
16
Iteration 2:

17. A
D
C
B
2
[B,C,D]
3
[B,C,A]
4
[D,A,B]
1
[A,B,C]
17
Iteration 3:

18. A
D
C
B
1
[A,B,C,D]
2
[B,C,D,A]
3
[C,D,A,B]
4
[D,A,B,C]
18
Iteration 4:

19. 1
[A,B,C,D
L1 =27
L2 =25
L3 =29
L4 =18
2
[B,C,D,A]
3
[C,D,A,B]
4
[D,A,B,C]
19
Best tour
Path and Pheromone Evaluation

20. 2. End of First Run
4. All ants are die
5. New ants are born
3. Save Best Tour (Sequence and length)
CALCULATION
1. Path and Pheromone Evaluation

21. Advantages and Disadvantages
 For TSPs (Traveling Salesman Problem), relatively efficient
 for a small number of nodes, TSPs can be solved by exhaustive search
 for a large number of nodes, TSPs are very computationally difficult to solve
exponential time to convergence
 Performs better against other global optimization techniques such as neural net,
genetic algorithms, simulated annealing
 Can be used in dynamic applications (adapts to changes such as new distances,
etc.)
 Convergence is guaranteed, but time to convergence uncertain
21