The document describes ant colony optimization (ACO), a swarm intelligence technique inspired by how ants find the shortest paths between their nest and food sources. ACO is used to find near-optimal solutions to discrete optimization problems. It works by having "ants" indirectly communicate using pheromone trails, where more intense trails increase the probability that other ants will follow that path. Over iterations, the shortest paths become reinforced by pheromones while longer paths evaporate, eventually leading ants to converge on an optimal or near-optimal solution.

1. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• A search technique used in computing to find near optimal
solutions to discrete optimization problems.
• ACO is a swarm intelligence inspired from the way that ants
indirectly communicate directions to each other.
• The most interesting aspect of the collaborative behaviour of ant
species is their ability to find the shortest paths between the ants'
nest and the food sources.
• Finding the shortest path between their nest and a food source by
chase pheromone trails exposed by other ants. The more intense the
trail, the higher probability that an ant will follow it and thus enrich
the trail with its own pheromone. The pheromone trails of longest
paths evaporate.

2. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• A search technique used in computing to find near optimal
solutions to discrete optimization problems.
• ACO is a swarm intelligence inspired from the way that ants
indirectly communicate directions to each other.
• The most interesting aspect of the collaborative behaviour of ant
species is their ability to find the shortest paths between the ants'
nest and the food sources.
• Finding the shortest path between their nest and a food source by
chase pheromone trails exposed by other ants. The more intense the
trail, the higher probability that an ant will follow it and thus enrich
the trail with its own pheromone. The pheromone trails of longest
paths evaporate.

3. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• A search technique used in computing to find near optimal
solutions to discrete optimization problems.
• ACO is a swarm intelligence inspired from the way that ants
indirectly communicate directions to each other.
• The most interesting aspect of the collaborative behaviour of ant
species is their ability to find the shortest paths between the ants'
nest and the food sources.
• Finding the shortest path between their nest and a food source by
chase pheromone trails exposed by other ants. The more intense the
trail, the higher probability that an ant will follow it and thus enrich
the trail with its own pheromone. The pheromone trails of longest
paths evaporate.

4. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• A search technique used in computing to find near optimal
solutions to discrete optimization problems.
• ACO is a swarm intelligence inspired from the way that ants
indirectly communicate directions to each other.
• The most interesting aspect of the collaborative behaviour of ant
species is their ability to find the shortest paths between the ants'
nest and the food sources.
• Finding the shortest path between their nest and a food source by
chase pheromone trails exposed by other ants. The more intense the
trail, the higher probability that an ant will follow it and thus enrich
the trail with its own pheromone. The pheromone trails of longest
paths evaporate.

5. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• A search technique used in computing to find near optimal
solutions to discrete optimization problems.
• ACO is a swarm intelligence inspired from the way that ants
indirectly communicate directions to each other.
• The most interesting aspect of the collaborative behaviour of ant
species is their ability to find the shortest paths between the ants'
nest and the food sources.
• Finding the shortest path between their nest and a food source by
chase pheromone trails exposed by other ants. The more intense the
trail, the higher probability that an ant will follow it and thus enrich
the trail with its own pheromone. The pheromone trails of longest
paths evaporate.

6. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

7. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

8. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

9. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

10. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

11. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

12. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

13. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Key Terms
 Ants 𝑘: Any possible solution.
 Population 𝑁- Group of all ants.
 Search Space [𝑙𝑏, 𝑢𝑏]- All possible solutions to the problem.
 Search Space is divided by step size ℎ
 Pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ

14. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Initialization
 Assume a suitable number of ants in the colony (population 𝑁)
 Assume a set of permissible discrete values 𝑚 for each of the design variables (step size ℎ).
 Initialize all discrete values of design variables equal amounts of pheromone 𝜏.

15. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Initialization
 Assume a suitable number of ants in the colony (population 𝑁)
 Assume a set of permissible discrete values 𝑚 for each of the design variables (step size ℎ).
 Initialize all discrete values of design variables equal amounts of pheromone 𝜏.

16. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Initialization
 Assume a suitable number of ants in the colony (population 𝑁)
 Assume a set of permissible discrete values 𝑚 for each of the design variables (step size ℎ).
 Initialize all discrete values of design variables equal amounts of pheromone 𝜏.

17. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Initialization
 Assume a suitable number of ants in the colony (population 𝑁)
 Assume a set of permissible discrete values 𝑚 for each of the design variables (step size ℎ).
 Initialize all discrete values of design variables equal amounts of pheromone 𝜏.

18. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Build tours
 From the home node, ants start travelling through the various paths and end
at the destination node in each iteration (discrete values of design variables).
 The probability to select discrete values of design variables is pj
k
=
τj
τj
m
j=1
 Find the cumulative probability ranges associated with different discrete
values based on its probabilities.
 The specific discrete values chosen by ant k will be determined using the
roulette-wheel selection.
Roulette-wheel

19. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Build tours
 From the home node, ants start travelling through the various paths and end
at the destination node in each iteration (discrete values of design variables).
 The probability to select discrete values of design variables is pj
k
=
τj
τj
m
j=1
 Find the cumulative probability ranges associated with different discrete
values based on its probabilities.
 The specific discrete values chosen by ant k will be determined using the
roulette-wheel selection.
Roulette-wheel

20. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Build tours
 From the home node, ants start travelling through the various paths and end
at the destination node in each iteration (discrete values of design variables).
 The probability to select discrete values of design variables is pj
k
=
τj
τj
m
j=1
 Find the cumulative probability ranges associated with different discrete
values based on its probabilities.
 The specific discrete values chosen by ant k will be determined using the
roulette-wheel selection.
Roulette-wheel

21. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Build tours
 From the home node, ants start travelling through the various paths and end
at the destination node in each iteration (discrete values of design variables).
 The probability to select discrete values of design variables is pj
k
=
τj
τj
m
j=1
 Find the cumulative probability ranges associated with different discrete
values based on its probabilities.
 The specific discrete values chosen by ant k will be determined using the
roulette-wheel selection.
Roulette-wheel

22. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Build tours
 From the home node, ants start travelling through the various paths and end
at the destination node in each iteration (discrete values of design variables).
 The probability to select discrete values of design variables is pj
k
=
τj
τj
m
j=1
 Find the cumulative probability ranges associated with different discrete
values based on its probabilities.
 The specific discrete values chosen by ant k will be determined using the
roulette-wheel selection.
Roulette-wheel

23. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
2
7
4
11
9
6
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel

24. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
design variables

25. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
design variables

26. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
design variables

27. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables

28. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables

29. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1

30. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3

31. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3
0.1
0.1
0.3
0.1

32. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3
0.1
0.1
0.3
0.1

33. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3
0.1
0.1
0.3
0.1

34. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3
0.1
0.1
0.3
0.1

35. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
4
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3
0.1
0.1
0.3
0.1
0.48

36. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Select Path (solution)
 Generate N random numbers r in the range (0, 1), one for each ant.
 Determine the discrete value by ant k for variable as the one for which the
cumulative probability range includes the random numbers r.
𝐴𝑛𝑡1 → 𝑟 = 0.48 → 𝑥 = 4
Roulette-wheel
2
7
𝟒
11
9
6
design variables
cumulative probability for
discrete values of design
variables
0.1
0.3
0.1
0.1
0.3
0.1
0.48

37. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Deposit and update trail
 Evaluate the objective function values of each ant
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update the pheromone
 Best ants: reinforcement the pheromone of the best path by: 𝜏𝑗
𝑛𝑒𝑤
← 𝜏𝑗
𝑜𝑙𝑑
+ ∆𝜏𝑗
𝑘
𝑘
 Other ants: evaporates the pheromone of other paths by: 𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
∆𝜏𝑗
𝑘
=
𝜁𝑓𝑏𝑒𝑠𝑡
𝑓𝑤𝑜𝑟𝑠𝑡
𝜁 → scaling parameter
𝑓𝑏𝑒𝑠𝑡 → best objective function
𝑓𝑤𝑜𝑟𝑠𝑡 → worst best objective function
ρ → evaporate rate

38. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Deposit and update trail
 Evaluate the objective function values of each ant
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update the pheromone
 Best ants: reinforcement the pheromone of the best path by: 𝜏𝑗
𝑛𝑒𝑤
← 𝜏𝑗
𝑜𝑙𝑑
+ ∆𝜏𝑗
𝑘
𝑘
 Other ants: evaporates the pheromone of other paths by: 𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
∆𝜏𝑗
𝑘
=
𝜁𝑓𝑏𝑒𝑠𝑡
𝑓𝑤𝑜𝑟𝑠𝑡
𝜁 → scaling parameter
𝑓𝑏𝑒𝑠𝑡 → best objective function
𝑓𝑤𝑜𝑟𝑠𝑡 → worst best objective function
ρ → evaporate rate

39. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Deposit and update trail
 Evaluate the objective function values of each ant
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update the pheromone
 Best ants: reinforcement the pheromone of the best path by: 𝜏𝑗
𝑛𝑒𝑤
← 𝜏𝑗
𝑜𝑙𝑑
+ ∆𝜏𝑗
𝑘
𝑘
 Other ants: evaporates the pheromone of other paths by: 𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
∆𝜏𝑗
𝑘
=
𝜁𝑓𝑏𝑒𝑠𝑡
𝑓𝑤𝑜𝑟𝑠𝑡
𝜁 → scaling parameter
𝑓𝑏𝑒𝑠𝑡 → best objective function
𝑓𝑤𝑜𝑟𝑠𝑡 → worst best objective function
ρ → evaporate rate

40. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Deposit and update trail
 Evaluate the objective function values of each ant
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update the pheromone
 Best ants: reinforcement the pheromone of the best path by: 𝜏𝑗
𝑛𝑒𝑤
← 𝜏𝑗
𝑜𝑙𝑑
+ ∆𝜏𝑗
𝑘
𝑘
 Other ants: evaporates the pheromone of other paths by: 𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
∆𝜏𝑗
𝑘
=
𝜁𝑓𝑏𝑒𝑠𝑡
𝑓𝑤𝑜𝑟𝑠𝑡
𝜁 → scaling parameter
𝑓𝑏𝑒𝑠𝑡 → best objective function
𝑓𝑤𝑜𝑟𝑠𝑡 → worst best objective function
ρ → evaporate rate

41. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Deposit and update trail
 Evaluate the objective function values of each ant
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update the pheromone
 Best ants: reinforcement the pheromone of the best path by: 𝜏𝑗
𝑛𝑒𝑤
← 𝜏𝑗
𝑜𝑙𝑑
+ ∆𝜏𝑗
𝑘
𝑘
 Other ants: evaporates the pheromone of other paths by: 𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
∆𝜏𝑗
𝑘
=
𝜁𝑓𝑏𝑒𝑠𝑡
𝑓𝑤𝑜𝑟𝑠𝑡
𝜁 → scaling parameter
𝑓𝑏𝑒𝑠𝑡 → best objective function
𝑓𝑤𝑜𝑟𝑠𝑡 → worst best objective function
ρ → evaporate rate

42. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Deposit and update trail
 Evaluate the objective function values of each ant
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update the pheromone
 Best ants: reinforcement the pheromone of the best path by: 𝜏𝑗
𝑛𝑒𝑤
← 𝜏𝑗
𝑜𝑙𝑑
+ ∆𝜏𝑗
𝑘
𝑘
 Other ants: evaporates the pheromone of other paths by: 𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
∆𝜏𝑗
𝑘
=
𝜁𝑓𝑏𝑒𝑠𝑡
𝑓𝑤𝑜𝑟𝑠𝑡
𝜁 → scaling parameter
𝑓𝑏𝑒𝑠𝑡 → best objective function
𝑓𝑤𝑜𝑟𝑠𝑡 → worst best objective function
ρ → evaporate rate

43. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Termination
 The steps of ACO algorithm are iteratively repeated until the maximum number of iteration
is reached or a termination criterion is met.
 Convergence: is the case where the positions of all particles converge to the same set of
values, the method is assumed to have converged.

44. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Termination
 The steps of ACO algorithm are iteratively repeated until the maximum number of iteration
is reached or a termination criterion is met.
 Convergence: is the case where the positions of all particles converge to the same set of
values, the method is assumed to have converged.

45. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Procedure
 Termination
 The steps of ACO algorithm are iteratively repeated until the maximum number of iteration
is reached or a termination criterion is met.
 Convergence: is the case where the of all ants converge to the same set of values, the
method is assumed to have converged.

46. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Parameters required from user:
 Population size 𝑁
 Set of permissible discrete values 𝑚 for each of the design variables
 Step size ℎ
 Initial pheromone trail 𝜏
 Scaling parameter 𝜁
 Evaporate rate ρ
 Termination criteria (i.e. number of iteration 𝑇)

47. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Pseudo code
1. Input
 Objective function (fitness function), upper bound (𝑢𝑏) and lower bound (𝑙𝑏), population size (𝑁), number of iteration 𝑇, scaling parameter 𝜁, evaporate rate ρ,
step size ℎ (or number of discrete value 𝑚)
2. Initialization
 Assume a set of permissible discrete values 𝑚 for each of the design variables
 Initialize all discrete values 𝑚 of design variables equal amounts of pheromone 𝜏
3. Loop:
 For 𝑡 = 1: 𝑇
 Find probability to select discrete values of design variables is 𝑝𝑗
𝑘
=
𝜏𝑗
𝜏𝑗
𝑚
𝑗=1
 Find the cumulative probability ranges associated with different discrete values based on its probabilities.(design roulette-wheel)
 For 𝑖 = 1: 𝑁 
 Generate a random numbers 𝑟, Find corresponding discrete value , Evaluate the objective function 𝑓𝑥𝑗
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update best path by: 𝜏𝑗
𝑛𝑒𝑤 ← 𝜏𝑗
𝑜𝑙𝑑 + ∆𝜏𝑗
𝑘
𝑘 and other paths by:𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
 If there is no convergence of the current solution & if 𝑡 > 𝑇 go to Loop
4. Print 𝑥𝑓𝑏𝑒𝑠𝑡 and 𝑓𝑏𝑒𝑠𝑡

48. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Pseudo code
1. Input
 Objective function (fitness function), upper bound (𝑢𝑏) and lower bound (𝑙𝑏), population size (𝑁), number of iteration 𝑇, scaling parameter 𝜁, evaporate rate ρ,
step size ℎ (or number of discrete value 𝑚)
2. Initialization
 Assume a set of permissible discrete values 𝑚 for each of the design variables
 Initialize all discrete values 𝑚 of design variables equal amounts of pheromone 𝜏
3. Loop:
 For 𝑡 = 1: 𝑇
 Find probability to select discrete values of design variables is 𝑝𝑗
𝑘
=
𝜏𝑗
𝜏𝑗
𝑚
𝑗=1
 Find the cumulative probability ranges associated with different discrete values based on its probabilities.(design roulette-wheel)
 For 𝑖 = 1: 𝑁 
 Generate a random numbers 𝑟, Find corresponding discrete value , Evaluate the objective function 𝑓𝑥𝑗
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update best path by: 𝜏𝑗
𝑛𝑒𝑤 ← 𝜏𝑗
𝑜𝑙𝑑 + ∆𝜏𝑗
𝑘
𝑘 and other paths by:𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
 If there is no convergence of the current solution & if 𝑡 > 𝑇 go to Loop
4. Print 𝑥𝑓𝑏𝑒𝑠𝑡 and 𝑓𝑏𝑒𝑠𝑡

49. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Pseudo code
1. Input
 Objective function (fitness function), upper bound (𝑢𝑏) and lower bound (𝑙𝑏), population size (𝑁), number of iteration 𝑇, scaling parameter 𝜁, evaporate rate ρ,
step size ℎ (or number of discrete value 𝑚)
2. Initialization
 Assume a set of permissible discrete values 𝑚 for each of the design variables
 Initialize all discrete values 𝑚 of design variables equal amounts of pheromone 𝜏
3. Loop:
 For 𝑡 = 1: 𝑇
 Find probability to select discrete values of design variables is 𝑝𝑗
𝑘
=
𝜏𝑗
𝜏𝑗
𝑚
𝑗=1
 Find the cumulative probability ranges associated with different discrete values based on its probabilities.(design roulette-wheel)
 For 𝑖 = 1: 𝑁 
 Generate a random numbers 𝑟, Find corresponding discrete value , Evaluate the objective function 𝑓𝑥𝑗
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update best path by: 𝜏𝑗
𝑛𝑒𝑤 ← 𝜏𝑗
𝑜𝑙𝑑 + ∆𝜏𝑗
𝑘
𝑘 and other paths by:𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
 If there is no convergence of the current solution & if 𝑡 > 𝑇 go to Loop
4. Print 𝑥𝑓𝑏𝑒𝑠𝑡 and 𝑓𝑏𝑒𝑠𝑡

50. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Pseudo code
1. Input
 Objective function (fitness function), upper bound (𝑢𝑏) and lower bound (𝑙𝑏), population size (𝑁), number of iteration 𝑇, scaling parameter 𝜁, evaporate rate ρ,
step size ℎ (or number of discrete value 𝑚)
2. Initialization
 Assume a set of permissible discrete values 𝑚 for each of the design variables
 Initialize all discrete values 𝑚 of design variables equal amounts of pheromone 𝜏
3. Loop:
 For 𝑡 = 1: 𝑇
 Find probability to select discrete values of design variables is 𝑝𝑗
𝑘
=
𝜏𝑗
𝜏𝑗
𝑚
𝑗=1
 Find the cumulative probability ranges associated with different discrete values based on its probabilities.(design roulette-wheel)
 For 𝑖 = 1: 𝑁 
 Generate a random numbers 𝑟, Find corresponding discrete value , Evaluate the objective function 𝑓𝑥𝑗
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update best path by: 𝜏𝑗
𝑛𝑒𝑤 ← 𝜏𝑗
𝑜𝑙𝑑 + ∆𝜏𝑗
𝑘
𝑘 and other paths by:𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
 If there is no convergence of the current solution & if 𝑡 > 𝑇 go to Loop
4. Print 𝑥𝑓𝑏𝑒𝑠𝑡 and 𝑓𝑏𝑒𝑠𝑡

51. Optimization Techniques
Swarm Optimization Techniques
 Ant Colony Optimization
• Pseudo code
1. Input
 Objective function (fitness function), upper bound (𝑢𝑏) and lower bound (𝑙𝑏), population size (𝑁), number of iteration 𝑇, scaling parameter 𝜁, evaporate rate ρ,
step size ℎ (or number of discrete value 𝑚)
2. Initialization
 Assume a set of permissible discrete values 𝑚 for each of the design variables
 Initialize all discrete values 𝑚 of design variables equal amounts of pheromone 𝜏
3. Loop:
 For 𝑡 = 1: 𝑇
 Find probability to select discrete values of design variables is 𝑝𝑗
𝑘
=
𝜏𝑗
𝜏𝑗
𝑚
𝑗=1
 Find the cumulative probability ranges associated with different discrete values based on its probabilities.(design roulette-wheel)
 For 𝑖 = 1: 𝑁 
 Generate a random numbers 𝑟, Find corresponding discrete value , Evaluate the objective function 𝑓𝑥𝑗
 Determine the best 𝑓𝑏𝑒𝑠𝑡 and worst 𝑓𝑤𝑜𝑟𝑠𝑡 objective function of the discrete value among ants
 Update best path by: 𝜏𝑗
𝑛𝑒𝑤 ← 𝜏𝑗
𝑜𝑙𝑑 + ∆𝜏𝑗
𝑘
𝑘 and other paths by:𝜏𝑗
𝑛𝑒𝑤
← 1 − ρ 𝜏𝑗
𝑜𝑙𝑑
 If there is no convergence of the current solution & if 𝑡 > 𝑇 go to Loop
4. Print 𝑥𝑓𝑏𝑒𝑠𝑡 and 𝑓𝑏𝑒𝑠𝑡