Optimization techniques: Ant Colony Optimization: Bee Colony Optimization: Traveling Salesman Problem

Optimization Techniques
09/29/19 1
Prepared by:
Soumen santra
10/18/19 Soumen Santra 1

Sections
•Introduction (Swarm intelligence)
•Natural behavior of ants
•First Algorithm: Ant System
•Improvements to Ant System
•Applications
•Bee Colony Optimization
•Applications
2

Swarm intelligence
• 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
3

Swarms
• Natural phenomena as inspiration
• A flock of birds sweeps across the Sky.
• How do ants collectively forage for food?
• How does a school of fish swims, turns
together?
• They are so ordered.

What made them to be so ordered?
• There is no centralized controller
• But they exhibit complex global behavior.
• Individuals follow simple rules to interact
with neighbors .
• Rules followed by birds
– collision avoidance
– velocity matching
– Flock Centering

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

Characteristics of Swarms
• Composed of many individuals
• Individuals are homogeneous
• Local interaction based on simple rules
• Self-organization

09/29/19 10

Ant Colony Optimization
•Real ants
•Stigmergy
•Autocatalyzation
•Ant System
•Ant Colony System

Overview
• Ant colony optimization
• TSP
• Bees Algorithms
• Comparison between bees and ants
• Conclusions

• The way ants find their food in shortest path
is interesting.
• Ants secrete pheromones to remember their
path.
• These pheromones evaporate with time.

Ant Colony Optimization..
• Whenever an ant finds food , it marks its
return journey with pheromones.
• Pheromones evaporate faster on longer
paths.
• Shorter paths serve as the way to food for
most of the other ants.

• The shorter path will be reinforced by the
pheromones further.
• Finally , the ants arrive at the shortest path.

Optimizations of SI
• Swarms have the ability to solve problems
• Ant Colony Optimization (ACO) , a meta-
heuristic
• ACO can be used to solve hard problems like
TSP, Quadratic Assignment Problem(QAP)
• We discuss ACO meta-heuristic for TSP

ACO-TSP
• Given a graph with n nodes, should give the
shortest Hamiltonian cycle
• m ants traverse the graph
• Each ant starts at a random node

Transitions
• Ants leave pheromone trails when they
make a transition
• Trails are used in prioritizing transition

Transitions
Suppose ant k is at u.
Nk(u) be the nodes not visited by k
Tuv be the pheromone trail of edge (u,v)
k jumps from u to a node v in Nk(u) with
probability
puv(k) = Tuv ( 1/ d(u,v))

Iteration of AOC-TSP
• m ants are started at random nodes
• They traverse the graph prioritized on trails
and edge-weights
• An iteration ends when all the ants visit all
nodes
• After each iteration, pheromone trails are
updated.

Updating Pheromone trails
New trail should have two components
Old trail left after evaporation and
Trails added by ants traversing the edge during
the iteration
T'uv = (1-p) Tuv + ChangeIn(Tuv)
Solution gets better and better as the
number of iterations increase

Performance of TSP with ACO heuristic
• Performs better than state-of-the-art TSP
algorithms for small (50-100) of nodes
• The main point to appreciate is that Swarms
give us new algorithms for optimization

Inherent features
• Inherent parallelism
• Stochastic nature
• Adaptivity
• Use of positive feedback
• Autocatalytic in nature
25

Natural behavior of an ant
Foraging modes
•Wander mode
•Search mode
•Return mode
•Attracted mode
•Trace mode
•Carry mode
26

Ant Colony Optimization (ACO)
Naturally Observed Ant Behavior
All is well in the world of the ant.

Oh no! An obstacle has blocked our path!

Where do we go? Everybody, flip a coin.

Shorter path reinforced.

“Stigmergic?”
•Stigmergy, a term coined by French biologist Pierre-
Paul Grasse, is interaction through the environment.
•Two individuals interact indirectly when one of them
modifies the environment and the other responds to
the new environment at a later time. This is
stigmergy.

E
D
C
H
B
A
(b )
3 0 a n ts
3 0 a n ts
1 5 a n ts
1 5 a n ts
1 5 a n ts
1 5 a n ts
t = 0
d = 0 .5
d = 0 .5
d = 1
d = 1
E
D
C
H
B
A
(a )
E
D
C
H
B
A
(c )
3 0 a n ts
3 0 a n ts
2 0 a n ts
2 0 a n ts
1 0 a n ts
1 0 a n ts
t = 1
τ = 30
τ = 30
τ = 15
τ = 15
Initial state:
no ants

The Ant System (AS)

Ant System
•First introduced by Marco Dorigo in 1992
•Result of research on computational intelligence
approaches to combinatorial optimization
•Originally applied to Traveling Salesman Problem
•Applied later to various hard optimization problems
•Variations later developed (you are only responsible
for the basic Ant System: item 1 in Section II.C of
Dorigo et al. 2006)

We use dij to denote the distance between any two
cities in the problem.

We let τij(t) denote the intensity of trail on edge (i,j) at
time t. Trail intensity is updated subsequently
deposits trail of quantity Q/Lk on every edge (i,j)
visited in its individual tour. Notice how this method
would favor shorter tour segments. The sum of all
newly deposited trail is denoted by ∆ τij. Following
trail deposition by all ants, the trail value is updated
using τij(t + 1) = р × τij(t) + ∆ τij, where p is the rate of
trail decay per time interval and ∆ τij =



m
k
ij
1


Two factors drive the probabilistic model:
1) Visibility, denoted ηij, equals the quantity 1/dij
2) Trail, denoted τij(t)
These two factors play an essential role in the central
probabilistic transition function of the Ant System.
In return, the weight of either factor in the transition
function is controlled by the variables α and β,
respectively. Significant study has been undertaken by
researchers to derive optimal α:β combinations.

Probabilistic Transition Function

pij
k
t
 
ij (t)
 

 ij
 

ij (t)
 

 ij
 

kallowedk

if k  allowedk
0 otherwise







The subscripts in denominator should be ik, not ij

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
k
j
,
i


39

A simple TSP example
40
A
E
D
C
B
1
[]
4
[]
3
[]
2
[]
5
[]
dAB =100;dBC = 60…;dDE =150

Iteration 1
41
A
E
D
C
B
1
[A]
5
[E]
3
[C]
2
[B]
4
[D]

How to build next sub-solution?
42
A
E
D
C
B
1
[A]
1
[A]
1
[A]
1
[A]
1
[A,D]
otherwise
0
allowed
j
if k










 k
allowed
k
ik
ik
ij
ij
k
ij
]
[
)]
t
(
[
]
[
)]
t
(
[
)
t
(
p









Iteration 2
43
A
E
D
C
B
3
[C,B]
5
[E,A]
1
[A,D]
2
[B,C]
4
[D,E]

44
Iteration 3
A
E
D
C
B
4
[D,E,A]
5
[E,A,B]
3
[C,B,E]
2
[B,C,D]
1
[A,D,C]

Iteration 4
45
A
E
D
C
B
4
[D,E,A,B]
2
[B,C,D,A]
5
[E,A,B,C]
1
[A,DCE]
3
[C,B,E,D]

Iteration 5
46
A
E
D
C
B
1
[A,D,C,E,B]
3
[C,B,E,D,A]
4
[D,E,A,B,C]
2
[B,C,D,A,E]
5
[E,A,B,C,D]

Path and Pheromone Evaluation
47
1
[A,D,C,E,B]
5
[E,A,B,C,D]
L1 =300







otherwise
0
tour
)
j
,
i
(
if
L
Q
k
k
j
,
i


L2 =450
L3 =260
L4 =280
L5 =420
2
[B,C,D,A,E]
3
[C,B,E,D,A]
4
[D,E,A,B,C]
5
B
,
A
4
B
,
A
3
B
,
A
2
B
,
A
1
B
,
A
total
B
,
A 










 





E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
An ant is placed at a random node

E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
The ant decides where to go from that node,
based on probabilities
calculated from:
- pheromone strengths,
- next-hop distances.
Suppose this one chooses BC

E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
The ant is now at C, and has a `tour memory’ = {B, C} – so he cannot
visit B or C again.
Again, he decides next hop
(from those allowed) based
on pheromone strength
and distance;
suppose he chooses
CD

E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
The ant is now at D, and has a `tour memory’ = {B, C, D}
There is only one place he can go now:

E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
So, he has nearly finished his tour, having gone over the links:
BC, CD, and DA.

E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
So, he has nearly finished his tour, having gone over the links:
BC, CD, and DA. AB is added to complete the round trip.
Now, pheromone on the tour
is increased, in line with the
fitness of that tour.

E.g. A 4-city TSP
A B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
Next, pheromone everywhere
is decreased a little, to model
decay of trail strength over
time

E.g. A 4-city TSP
B
C
D
Pheromone
Ant
AB: 10, AC: 10, AD, 30, BC, 40, CD 20
We start again, with another ant in a random position.
Where will he go?
Next , the actual algorithm
and variants.
A

The ACO algorithm for the TSP
[a simplified version with all essential details]
We have a TSP, with n cities.
1. We place some ants at each city. Each ant then does this:
• It makes a complete tour of the cities, coming back to its starting city,
using a transition rule to decide which links to follow. By this rule, it
chooses each next-city stochastically, biased partly by the pheromone
levels existing at each path, and biased partly by heuristic information.

The ACO algorithm for the TSP
[a simplified version with all essential details]
• It makes a complete tour of the cities, coming back to its starting city,
using a transition rule to decide which links to follow. By this rule, it
chooses each next-city at random, but biased partly by the pheromone
levels existing at each path, and biased partly by heuristic information.
2. When all ants have completed their tours.
Global Pheromone Updating occurs.
• The current pheromone levels on all links are reduced (I.e. pheromone
levels decay over time).
• Pheromone is lain (belatedly) by each ant as follows: it places pheromone
on all links of its tour, with strength depending on how good the tour was.
Then we go back to 1 and repeat the whole process many
times, until we reach a termination criterion.

A very common variation, which gives the best results
• It makes a complete tour of the cities, coming back to its starting city, using a transition rule
to decide which links to follow. By this rule, it chooses each next-city at random, but biased
partly by the pheromone levels existing at each path, and biased partly by heuristic
information.
2. When all ants have completed their tours.
Apply some iterations of LOCAL SEARCH to the completed tour; this finds a
better solution, which is now treated as the ant’s path. Then continue the next
steps as normal.
Global Pheromone Updating occurs.
• The current pheromone levels on all links are reduced (I.e. pheromone levels decay over
time).
• Pheromone is lain (belatedly) by each ant as follows: it places pheromone on all links of its
tour, with strength depending on how good the tour was.
Then we go back to 1 and repeat the whole process many times, until we reach a
termination criterion.

The transition rule
T(r,s) is the amount of pheromone currently on the path that goes
directly from city r to city s.
H(r,s) is the heuristic value of this link – in the classic TSP
application, this is chosen to be 1/distance(r,s) -- I.e. the shorter
the distance, the higher the heuristic value.
is the probability that ant k will choose the link that goes
from r to s
is a parameter that we can call the heuristic strength
)
,
( s
r
pk
The rule is:
Where our ant is at city r, and s is a city as yet unvisited on its
tour, and the summation is over all of k’s unvisited cities

 


c
k
c
r
H
c
r
T
s
r
H
s
r
T
s
r
p
cities
unvisited
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(



Global pheromone update
Ak(r,s) is amount of pheromone added to the (r, s) link by ant k.
m is the number of ants
is a parameter called the pheromone decay rate.
Lk is the length of the tour completed by ant k
T(r, s) at the next iteration becomes:
Where





m
k
k s
r
A
s
r
T
1
)
,
(
)
,
(

k
k L
s
r
A /
1
)
,
( 
Study these – they’re not that hard.
How do you think the parameters m, beta, rho etc … affect the search?

Not just for TSP of course
ACO is naturally applicable to any
sequencing problem, or indeed any problem
All you need is some way to represent
solutions to the problem as paths in a
network.

E.g.
Single machine scheduling with due-dates
These jobs have to be done; their length
represents the time they will take.
A
B
C
D
E

E.g.
A
B
C
D
E
Each has a `due date’, when it needs to be finished
3pm
3:30pm
5pm
4pm
4:30pm

E.g.
A
B
C
D
E
Each has a `due date’, when it needs to be finished
3pm
3:30pm
5pm
4pm
4:30pm
Only one `machine’ is
available to process these jobs,
so can do just one at a time.
[e.g. machine might be
human tailor, photocopier,
Hubble Space Telescope,
Etc …]

An example schedule
A due 3pm B – 3:30 C - 5pm D – 4pm E -4:30
2 pm 3 pm 5 pm
4 pm 6 pm
A is 10min late Fitness might be average lateness;
B is 40min late in this case 46min
C is 20min early (lateness = 0)
D is 90min late or fitness could be Max lateness,
E is 90min late in this case 90min

Another schedule
A due 3pm
B – 3:30 C - 5pm
D – 4pm E -4:30
2 pm 3 pm 5 pm
4 pm 6 pm
A is 70min late Fitness might be average lateness;
B is 30min early (0 lateness) in this case again 46min
C is 60min late
D is 50min late or fitness could be Max lateness,
E is 50min late in this case 70min

Applying ACO to this problem
Just like with the TSP, each ant finds paths in
a network, where, in this case, each job is a
node. Also, no need to return to start node –
path is complete when every node is visited.

A B
C D
Initially, random levels of pheromone are scattered on the edges,
an ant starts at a Start node (so the first link it chooses defines
the first task to schedule on the machine); as before it uses a
transition ruleto take one step at a time, biased by pheromone levels,
and also a heuristic score, each time choosing the next machine
to schedule. What heuristic might you use in this case?
E Start
Why is it sensible to have a Start node for this problem

Example table from a research paper comparing ACO with
other things on some scheduling problems

• http://iridia.ulb.ac.be/dorigo/ACO/ACO.html
See here if you’re very interested in ACO:
ACO is a thriving and maturing research area – it has its own
conferences. It gets very good results on some difficult problems.
Following the above link will help you find examples.
ACO research and practice tends to concentrate on:
• hybridisation with other methods; e.g. it is common to
improve an individual ant’s solution by local search, and then
lay pheromone.
• New and adaptive ways to control the relative influence of
heuristics, pheromone strength and pheromone decay.

Ant System (Ant Cycle) Dorigo [1] 1991
otherwise
0
allowed
j
if k










 k
allowed
k
ik
ik
ij
ij
k
ij
]
[
)]
t
(
[
]
[
)]
t
(
[
)
t
(
p















otherwise
0
by tabu
described
tour k
)
j
,
i
(
if
L
Q
k
k
j
,
i


ij
ij
ij )
t
(
)
n
t
( 


 


71
t = 0; NC = 0; τij(t)=c for ∆τij=0
Place the m ants on the n nodes
Update tabuk(s)
Compute the length Lk of every ant
Update the shortest tour found
=
For every edge (i,j)
Compute
For k:=1 to m do
Initialize
Choose the city j to move
to. Use probability
Tabu list management
Move k-th ant to town j.
Insert town j in tabuk(s)
Set t = t + n; NC=NC+1; ∆τij=0 NC<NCmax
&& not stagn.
Yes
End
No
Yes
ij
ij
ij )
t
(
)
n
t
( 


 









otherwise
0
by tabu
described
tour k
)
j
,
i
(
if
L
Q
k
k
j
,
i


k
ij
ij
ij : 




 

otherwise
0
allowed
j
if k










 k
allowed
k
ik
ik
ij
ij
k
ij
]
[
)]
t
(
[
]
[
)]
t
(
[
)
t
(
p









General ACO
•A stochastic construction procedure
•Probabilistically build a solution
•Iteratively adding solution components to partial
solutions
- Heuristic information
- Pheromone trail
•Reinforcement Learning reminiscence
•Modify the problem representation at each iteration
72

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.
73

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
74

Improvements to AS
•Daemon actions are used to apply centralized actions
• Local optimization procedure
• Bias the search process from global information
75

Dynamic Optimization Problems
• ABC (circuit switched networks)
• AntNet (routing in packet-switched networks)
76

Applications
•Traveling Salesman Problem
•Quadratic Assignment Problem
•Network Model Problem
•Vehicle routing
77

Travelling Salesman Problem (TSP)
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.
• Problem is NP-hard
• Classical combinatorial
optimization problem to
test.
78

ACO for the Traveling Salesman Problem
The TSP is a very important problem in the context of
Ant Colony Optimization because it is the problem to
which the original AS was first applied, and it has later
often been used as a benchmark to test a new idea
and algorithmic variants.
The TSP was chosen for many reasons:
• It is a problem to which the ant colony metaphor
• It is one of the most studied NP-hard problems in the combinatorial optimization
• it is very easily to explain. So that the algorithm behavior is not obscured by
too many technicalities.
79

Search Space
Discrete Graph
To each edge is associated a static value
returned by an heuristic function  (r,s)
based on the edge-cost
Each edge of the graph is augmented with a
pheromone trail  (r,s) deposited by ants.
Pheromone is dynamic and it is learned at run-ime
80

Ant Systems (AS)
Ant Systems for TSP
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.
ij
d
A
D
C
B
81

Ant Systems Algorithm for TSP
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
82

Rules for Transition Probability
1. Whether or not a city has been visited
Use of a memory(tabu list): : set of all cities that are to be visited
2. = visibility:Heuristic desirability of choosing city j when in city i.
ij
N ij
d
1
k
i
J
3.Pheromone trail: This is a global type of information
Transition probability for ant k to go from city i to city j while building its route.
)
(t
Tij
a = 0: closest cities are selected
83

Trail pheromone in AS
After the completion of a tour, each ant lays some pheromone
for each edge that it has used. depends on how well the ant
has performed.
)
(t
ij
K


Trail pheromone decay =
84

ACS : Ant Colony System for TSP
85

ACO State Transition Rule
Next city is chosen between the not visited cities according to a
probabilistic rule
Exploitation: the best edge is chosen
Exploration: each of the edges in proportion to its value
86

ACS State Transition Rule : Formulae
87

ACS State Transition Rule : example
•with probability exploitation
(Edge AB = 15)
0
q
•with probability (1- )exploration
0
q
AB with probability 15/26
AC with probability 5/26
AD with probability 6/26
15
/
1
)
,
(
90
)
,
(
7
/
1
)
,
(
35
)
,
(
10
/
1
)
,
(
150
)
,
(






B
A
B
A
B
A
B
A
B
A
B
A






88

ACS Local Trail Updating
… similar to evaporation
89

ACS Global Trail Updating
At the end of each iteration the best ant is allowed to
reinforce its tour by depositing additional pheromone
inversely proportional to the length of the tour
90

Effect of the Local Rule
Local rule: learnt desirability of edges changes
dynamically
Local update rule makes the edge pheromone level
diminish.
Visited edges are less & less attractive as they are
visited by the various ants.
Favors exploration of not yet visited edges.
This helps in shuffling the
cities so that cities visited early in one ants tours are
being
visited later in another ants tour.
91

ACO vs AS
Pheromone trail update
Deposit pheromone after completing a tour in AS
Here in ACO only the ant that generated the best tour from the beginning
of the trial is allowed to globally update the concentrations of pheromone
on the branches (ants search at the vicinity of the best tour so far)
In AS pheromone trail update applied to all edges
Here in ACO the global pheromone trail update is applied only to the
best tour since trial began.
92

ACO : Candidate List
Use of a candidate list
A list of preferred cities to visit: instead of
examining all cities, unvisited cities are examined first.
Cities are ordered by increasing distance & list is scanned sequentially.
• Choice of next city from those in the candidate list.
• Other cities only if all the cities in the list have been visited.
93

• Algorithm found best solutions on small problems
(75 city)
• On larger problems converged to good solutions –
but not the best
• On “static” problems like TSP hard to beat specialist
algorithms
• Ants are “dynamic” optimizers – should we even
expect good performance on static problems
• Coupling ant with local optimizers gave world
class results….
Performance
94

ROUTING IN COMM. NETWORKS
Problem statement
• Dynamic Routing
At any moment the pathway of a message must be
as small as possible. (Traffic conditions and the
structure of the network are constantly changing)
• Load balancing
Distribute the changing load over the system and
minimize lost calls.
95

Objective:
Minimize: Lost calls by avoiding congestion,
Minimize: Pathway
Dynamic Optimization Problem
+
Multi-Objectives Optimization Problem
96

97
Traditional way:
“Central Controllers”
Disadvantage:
•Communication overhead.
•Fault tolerance ~ Controller Failure.
•Scalability
•Dynamic ~ Uncertainty
•Authority.

Conclusions
• ACO is a recently proposed metaheuristic approach for
solving hard combinatorial optimization problems.
• Artificial ants implement a randomized construction
heuristic which makes probabilistic decisions.
• The a cumulated search experience is taken into account by
the adaptation of the pheromone trail.
• ACO Shows great performance with the “ill-structured”
problems like network routing.
• In ACO Local search is extremely important to obtain good
results.
98

References
• Dorigo M. and G. Di Caro (1999). The Ant Colony Optimization Meta-Heuristic. In D. Corne, M. Dorigo and F.
Glover, editors, New Ideas in Optimization, McGraw-Hill, 11-32.
• M. Dorigo and L. M. Gambardella. Ant colonies for the traveling salesman problem. BioSystems, 43:73–81,
1997.
• M. Dorigo and L. M. Gambardella. Ant Colony System: A cooperative learning approach to the traveling
salesman problem. IEEE Transactions on Evolutionary Computation, 1(1):53–66, 1997.
• G. Di Caro and M. Dorigo. Mobile agents for adaptive routing. In H. El-Rewini, editor, Proceedings of the 31st
International Conference on System Sciences (HICSS-31), pages 74–83. IEEE Computer Society Press, Los
Alamitos, CA, 1998.
• M. Dorigo, V. Maniezzo, and A. Colorni. The Ant System: An autocatalytic optimizing process. Technical
Report 91-016 Revised, Dipartimento di Elettronica,Politecnico di Milano, Italy, 1991.
• L. M. Gambardella, ` E. D. Taillard, and G. Agazzi. MACS-VRPTW: A multiple ant colony system for vehicle
routing problems with time windows. In D. Corne, M. Dorigo, and F. Glover, editors, New Ideas in
Optimization, pages 63–76. McGraw Hill, London, UK, 1999.
• L. M. Gambardella, ` E. D. Taillard, and M. Dorigo. Ant colonies for the quadratic assignment problem. Journal
of the Operational Research Society,50(2):167–176, 1999.
• V. Maniezzo and A. Colorni. The Ant System applied to the quadratic assignment problem. IEEE Transactions
on Data and Knowledge Engineering, 11(5):769–778, 1999.
• Gambardella L. M., E. Taillard and M. Dorigo (1999). Ant Colonies for the Quadratic Assignment Problem.
Journal of the Operational Research Society, 50:167-176.
09/29/19 99

Thank you
100

Optimization techniques: Ant Colony Optimization: Bee Colony Optimization: Traveling Salesman Problem

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