HEghbalpouACO

8/11/2019 HEghbalpouACO

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Shahid Beheshti University of Tehran

Presented by:

Ant Colony Optimization

Requirements : .

+98 21 897 882 08

kourosh.eghbalpour

Hamid EghbalpourOperating Manager of Asia Peyman.Co

[email protected]

@IEEE.org1987H.Eghbalpour.

ir.academia.edu/HEghbalpour-https://sbu
mailto:[email protected]:[email protected]:[email protected]:[email protected]://sbu-ir.academia.edu/HEghbalpourhttps://sbu-ir.academia.edu/HEghbalpourhttps://sbu-ir.academia.edu/HEghbalpourhttps://sbu-ir.academia.edu/HEghbalpourhttps://sbu-ir.academia.edu/HEghbalpourhttps://sbu-ir.academia.edu/HEghbalpourmailto:[email protected]:[email protected]:[email protected]:[email protected]


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Lecture 01

AntColony Optimization

Swarm Intelligence


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3


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5


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6

Food

Nest


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7


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8


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9

NEST

FOOD


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Nest Food

10


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Nest Food

Obstacle

Interrupt The Flow

11


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The Path Thickens!

Nest Food

Obstacle

12


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The New Shortest Path

Nest Food

Obstacle

13


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Adapting to Environment Changes

NestFood

Obstacle

14


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Adapting to Environment Changes

NestFood

Obstacle

15


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History of Ant Algorithms

Goss et al. 1989,

Deneuborg et al. 1990,

experiments with

Argentine ants

Dorigo et al. 1991,

applications to shortest

path problems

Now: established methodfor various optimization

problems

16


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17

Natural Ant System

Initially, ants explore randomly

Leave behind pheromone when they travel back

to colony from food

Pheromone evaporates over time

Ants follow strong pheromone left behind by

other ants

Short routes to food have more pheromone (less

distance = less evaporation)


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18

DeneubourgsSimple Experiment


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Probabilistic Transition Function

is the probability that ant k will choose the link that goes from itoj

at time t

tpkij

is the amount of pheromone currently on the path that goes directly from ito j

at time t

)(tij

otherwise

allowedjift

t

tpk

allowedkk

ijij

ijij

k

ij

0

)(

)(

We let denote the intensity of trail on link(i,j) at time t.)(tij

19


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is the heuristic value of this linkin the classic TSP application, this is

chosen to be 1/distance(i,j) -- I.e. the shorter the distance, the higher the heuristic

value.

ij

, are parameters that we can call the heuristic strength

Where our antis at iandj is a point as yet unvisited on its tour, and the summation

is over all of ksunvisited points

20

otherwise

allowedjift

t

tpk

allowedkk

ijij

ijij

k

ij

0

)(

)(


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21

Three factors drive the probabilistic model:

1) Visibility,denoted ij, equals the quantity 1/dij

2) Trail,denoted ij(t)

3) Evaporation

These three factors play an essential role in the centralprobabilistic transition function of the Ant System.

In return, the weight of either factor in the transition

function is controlled by the variables and ,respectively. Significant study has been undertaken byresearchers to derive optimal :combinations.


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22

A high value for means that trail is very important

and therefore ants tend to choose edges chosen by

other ants in the past. On the other hand, low values

of make the algorithm very similar to a stochastic

multigreedy algorithm.


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Trail intensity is updated following the completion of each

algorithm cycle, at which time every ant will have

completed a tour. Each ant subsequently deposits trail ofquantity Q/Lkon 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 ij. Following trail deposition by all ants, the trailvalue is updated using

Where is the rate of trail decay per time interval and

ij = .

m

k

ij

1

ijijij tnt )()(

23

N

k

kijijij tt

1

)()1()1(


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ijijij tnt )()(Fixed : For all links is fixed

ij ij

otherwise0

by tabudescribedtour),( k,

jiifL

Q

kji

otherwise0pathselect0

, ji

is proportional to link which ant passes, the ant who passes

the short links, produce more pheromone

ij

24

Pheromoneleft on each pathShort tourHigh pheromoneLong TourLow Pheromone

Pheromone increaseat the end of tour

After each ant tour the trail intensity on each edge is updated using the following

formula


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Example 1

Here we have 10 cities as follow :

X Y

1 974.55 90.95

2 81.189 701.4

3 428 783.79

4 81.56 685.07

5 852.81 699.51

6 883.9 330.777 226.25 673.3

8 478.86 820.88

9 638.63 884.22

10 406.19 782.67


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26

The pheromone trail must not build unbounded.

Therefore, we need evaporation

Evaporation

)()1()( tnt ijij Remove links of poor paths


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Cities distribution


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Distance between cities

1 2 3 4 5 6 7 8 9 10

1 0.0 1082.0 882.5 1072.6 620.6 256.4 948.2 882.3 861.5 895.3

2 1082.0 0.0 356.5 16.3 771.6 884.1 147.8 415.2 586.7 335.03 882.5 356.5 0.0 360.2 433.1 642.7 230.0 62.9 233.3 21.8

4 1072.6 16.3 360.2 0.0 771.4 877.1 145.2 419.9 591.6 339.0

5 620.6 771.6 433.1 771.4 0.0 370.0 627.1 393.2 282.8 454.3

6 256.4 884.1 642.7 877.1 370.0 0.0 741.5 635.8 605.4 657.6

7 948.2 147.8 230.0 145.2 627.1 741.5 0.0 292.6 463.2 210.6

8 882.3 415.2 62.9 419.9 393.2 635.8 292.6 0.0 171.9 82.1

9 861.5 586.7 233.3 591.6 282.8 605.4 463.2 171.9 0.0 253.7

10 895.3 335.0 21.8 339.0 454.3 657.6 210.6 82.1 253.7 0.0


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Distribute ants on cities randomly

City'sNumber 1 2 3 4 5 6 7 8 9 10

Numberof ant (s) 2 0 2 1 1 1 1 0 1 1


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Initiating pheromone table:

0 0 &ij nn

M

C

AntsNumber

Length of

a random

tour


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1 2 3 4 5 6 7 8 9 10

1 0.000 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

2 0.001 0.000 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

3 0.001 0.001 0.000 0.001 0.001 0.001 0.001 0.001 0.001 0.001

4 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.001 0.001 0.001

5 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.001 0.001

6 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.001 0.001

7 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.001

8 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.001

9 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001

10 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.000

Pheromone distributions table


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Making taboo-list table and initiate

it depending on current Ants Position

1 2 3 4 5 6 7 8 9 10

1 2 0 0 0 0 0 0 0 0 0

2 2 0 0 0 0 0 0 0 0 0

3 0 0 2 0 0 0 0 0 0 04 0 0 2 0 0 0 0 0 0 0

5 0 0 0 2 0 0 0 0 0 0

6 0 0 0 0 2 0 0 0 0 0

7 0 0 0 0 0 2 0 0 0 0

8 0 0 0 0 0 0 2 0 0 0

9 0 0 0 0 0 0 0 0 2 0

10 0 0 0 0 0 0 0 0 0 2

Ants

Index


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Making Ant-city tour table

Ant-city is a matrix that shows tour ( in row)which a

ant passes and columns are cities that ant visits

them:

Ants

Index

First city and last city are the same.

1 1 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 1

3 3 0 0 0 0 0 0 0 0 0 3

4 3 0 0 0 0 0 0 0 0 0 3

5 4 0 0 0 0 0 0 0 0 0 4

6 5 0 0 0 0 0 0 0 0 0 5

7 6 0 0 0 0 0 0 0 0 0 6

8 7 0 0 0 0 0 0 0 0 0 7

9 9 0 0 0 0 0 0 0 0 0 9

10 10 0 0 0 0 0 0 0 0 0 10


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Making Ant-path length table

In this matrix we have total distance that any

ant passed until now!

In the first step all of the path-length are zero.

Ant_Num 1 2 3 4 5 6 7 8 9 10

Total

Path_len 0 0 0 0 0 0 0 0 0 0


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For first ant in first city we determine available

cities-cities that we can go-from taboo-list.Suppose the next ant is in the city 1

In next step we compute probability of

Available cities to go by this formula:

Available

Cities 2 3 4 5 6 7 8 9 10


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( )ij tPheromone amount

between city I & j

ij 1/(distance between city I & j)

otherwise

sNCift

t

tp

p

ij

sNC

ilil

ijij

kij

pil

0

)()(

)(

)(

When ant k is in city i and has so far constructed the partial solution sp, the

probability of going to cityj is given by

is the set of feasible components, that is, edges

(i, l) where l is a city not yet visited by the ant k.

)( psN


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For the first ant we have:

Available

Cities 2 3 4 5 6 7 8 9 10

Available Cities 2 3 4 5 6 7 8 9 10Distance from

current city(1) 1082 882 1072 620 256 948 882 861 895

Available Cities 2 3 4 5 6 7 8 9 10

1/Distance from

current city(1) 9.2E-04 1.1E-03 9.3E-04 1.6E-03 3.9E-03 1.1E-03 1.1E-03 1.2E-03 1.1E-03

Pheromone between city 1 and available


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Available Cities 2 3 4 5 6 7 8 9 10

(pheromone

amount)^1 1.E-03 1.E-03 1.E-03 1.E-03 1.E-03 1.E-03 1.E-03 1.E-03 1.E-03

Pheromone between city 1 and available

cities

Available Cities 2 3 4 5 6 7 8 9 10(1/Distance from

current city(1))^4

9.2E-

04

1.1E-

03

9.3E-

04

1.6E-

03

3.9E-

03

1.1E-

03

1.1E-

03

1.2E-

03

1.1E-

03

2 3 4 5 6 7 8 9 10

1.1E-15 2.4E-15 1.1E-15 9.9E-15 3.4E-13 1.8E-15 2.4E-15 2.7E-15 2.3E-15


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And finally based on eq.1 we have:

City's Number 2 3 4 5 6* 7 8 9 10

Probability 0% 1% 0% 3% 93% 0% 1% 1% 1%


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93%

3%


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In next step we must select next city

from available list and depending onprobabilities in previous stage.

Normally city with more probability

has more chance to be selected!

Here in program we have city number

6 next city.


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Then we update tables:

1 2 3 4 5 6 7 8 9 10

1 1 0 0 0 0 2 0 0 0 02 2 0 0 0 0 0 0 0 0 0

3 0 0 2 0 0 0 0 0 0 04 0 0 2 0 0 0 0 0 0 0

5 0 0 0 2 0 0 0 0 0 0

6 0 0 0 0 2 0 0 0 0 0

7 0 0 0 0 0 2 0 0 0 0

8 0 0 0 0 0 0 2 0 0 09 0 0 0 0 0 0 0 0 2 0

10 0 0 0 0 0 0 0 0 0 2

Ants

Index

Taboo-list


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We do this operations to other ant at other

cities to move all ants to next city.

1 2 3 4 5 6 7 8 9 10

1 1 0 0 0 0 2 0 0 0 0

2 1 0 0 0 0 2 0 0 0 03 0 0 1 0 0 0 0 0 0 2

4 0 0 1 0 0 0 0 0 0 2

5 0 2 0 1 0 0 0 0 0 0

6 0 0 0 0 1 0 0 0 2 0

7 2 0 0 0 0 1 0 0 0 08 0 0 0 2 0 0 1 0 0 0

9 0 0 0 0 0 0 0 2 1 0

10 0 0 2 0 0 0 0 0 0 1

Taboo-list

Ants

Index


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Ant_Num 1 2 3 4 5 6 7 8 9 10

Path_Len 256.4 256.4 21.8 21.8 16.3 282.8 256.4 145.2 171.9 21.8

1 1 6 0 0 0 0 0 0 0 0 1

2 1 6 0 0 0 0 0 0 0 0 13 3 10 0 0 0 0 0 0 0 0 3

4 3 10 0 0 0 0 0 0 0 0 3

5 4 2 0 0 0 0 0 0 0 0 4

6 5 9 0 0 0 0 0 0 0 0 5

7 6 1 0 0 0 0 0 0 0 0 6

8 7 4 0 0 0 0 0 0 0 0 7

9 9 8 0 0 0 0 0 0 0 0 9

10 10 3 0 0 0 0 0 0 0 0 10

Ant-city table

Ants

Index

Ant-path length table


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At next step we come back to first ant

and move it to next city as same as last.Only different is that one city has

eliminated from available cities list. And

do same operation it to other ants until

a tour completed for all ants.

And update all table at any step seehere all table at the end of first tour:


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1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 2 1 1 1 1 1

2 1 1 1 1 1 1 1 1 2 1

3 1 1 1 1 1 1 1 1 2 1

4 1 1 1 1 1 1 1 1 2 1

5 1 1 1 1 2 1 1 1 1 1

6 1 1 1 1 1 1 1 2 1 1

7 1 1 1 1 1 1 1 1 2 1

8 1 1 1 1 1 1 1 1 2 1

9 1 1 1 1 1 1 1 1 1 2

10 1 1 1 1 1 1 1 1 2 1

Taboo-list

Ants

Index


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Ant-city table

Ants

Index

Ant_Num 1 2 3 4 5 6 7* 8 9 10

Path_Len 3116.34051.53053.24653.03341.43163.6 3046.54041.33620.94121.4


1 1 6 2 4 7 9 3 10 8 5 1

2 1 6 8 3 10 2 4 5 7 9 1

3 3 10 8 7 2 4 5 6 1 9 3

4 3 10 8 1 4 2 6 5 7 9 3

5 4 2 7 1 6 3 10 8 9 5 4

6 5 9 1 6 7 2 4 3 10 8 57 6 1 5 2 4 7 8 3 10 9 6

8 7 4 2 6 3 10 8 1 5 9 7

9 9 8 5 2 4 7 1 6 3 10 9

10 10 3 8 1 5 2 4 7 6 9 10

At fi l t i fi t it ti t


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At final step in first iteration we must

save best tour(tour with shortest

length tour).

Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 6 1 5 2 4 7 8 3 10 9 6

Best tour in Itr.1 belongs to ant number 7

Length of best tour at Itr.1 : 3046.5

Ant Number 7 is winner7


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After each iteration we must

update pheromone table by usingthis formula:

Pheromone evaporation coefficient= 0.5

N

k

kijijij tt

1

)()1()1(


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( , )

( , )0

kk

kij

k

Qwhere edge i j T t

Lt if edge i j T t

Q Constant value =1

kLTours length of ant k

Tour of ant kkT


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For ant number k=1 to 10 :

/ kQ L

Ant

_number 1 2 3 4 5 6 7 8 9 10

1/L3.2E-

042.5E-

043.3E-

042.1E-

043.0E-

043.2E-

043.3E-

042.5E-

042.8E-

042.4E-

04


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1 2 3 4 5 6 7 8 9 10

1 0 0 0 0 3.2 3.2 0 0 0 0

2 0 0 0 3.2 0 3.2 0 0 0 0

3 0 0 0 0 0 0 0 0 3.2 3.2

4 0 3.2 0 0 0 0 3.2 0 0 05 3.2 0 0 0 0 0 0 3.2 0 0

6 3.2 3.2 0 0 0 0 0 0 0 0

7 0 0 0 3.2 0 0 0 0 3.2 0

8 0 0 0 0 3.2 0 0 0 0 3.2

9 0 0 3.2 0 0 0 3.2 0 0 010 0 0 3.2 0 0 0 0 3.2 0 0

Delta-pheromone table (*e-4)

updated by ant number 1


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1 2 3 4 5 6 7 8 9 10

1 0 0 0 2.1 11.4 14.6 0 0 5.7 02 0 0 0 20.6 0 4.6 3.0 0 0 0

3 0 0 0 0 0 0 0 2.4 5.4 25.8

4 0 7.6 3.2 0 8.7 0 11.7 0 0 0

5 0 8.5 0 0 0 3.3 4.6 3.2 5.6 0

6 6.6 3.2 8.2 0 2.1 0 3.2 2.5 5.7 07 5.8 6.4 0 2.5 0 2.4 0 3.3 10.3 0

8 7.0 0 5.8 0 6.0 0 3.3 0 3.0 0

9 3.2 0 3.2 0 3.0 0 0 2.8 0 2.8

10 0 2.5 2.4 0 0 0 0 17.3 5.7 0

Delta-pheromone table(*e-4)

Updating delta pheromone will be

continued by other ants and finally we

have:


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1 2 3 4 5 6 7 8 9 10

1 0 5 5 5 5 5 5 5 5 5

2 5 0 5 5 5 5 5 5 5 5

3 5 5 0 5 5 5 5 5 5 5

4 5 5 5 0 5 5 5 5 5 5

5 5 5 5 5 0 5 5 5 5 5

6 5 5 5 5 5 0 5 5 5 5

7 5 5 5 5 5 5 0 5 5 58 5 5 5 5 5 5 5 0 5 5

9 5 5 5 5 5 5 5 5 0 5

10 5 5 5 5 5 5 5 5 5 0

Pheromone table must be evaporated

at first then add with delta pheromone:

Evaporated pheromone table(*e-4)


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1 2 3 4 5 6 7 8 9

1

01 0 5 5 5 5 5 5 5 5 5

2 5 0 5 5 5 5 5 5 5 5

3 5 5 0 5 5 5 5 5 5 5

4 5 5 5 0 5 5 5 5 5 5

5 5 5 5 5 0 5 5 5 5 5

6 5 5 5 5 5 0 5 5 5 5

7 5 5 5 5 5 5 0 5 5 5

8 5 5 5 5 5 5 5 0 5 5

9 5 5 5 5 5 5 5 5 0 5

1

0 5 5 5 5 5 5 5 5 5 0

1 2 3 4 5 6 7 8 9 10

1 0 0 0 2 11 14 0 0 5 0

2 0 0 0 20 0 4 3 0 0 0

3 0 0 0 0 0 0 0 2 5 25

4 0 7 3 0 8 0 11 0 0 0

5 0 8 0 0 0 3 4 3 5 06 6 3 8 0 2 0 3 2 5 0

7 5 6 0 2 0 2 0 3 10 0

8 7 0 5 0 6 0 3 0 3 0

9 3 0 3 0 3 0 0 2 0 2

10 0 2 2 0 0 0 0 17 5 0

Evaporated pheromone table(*e-4) Delta-pheromone table(*e-4)


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1 2 3 4 5 6 7 8 9 10

1 0 7 7 10 19 22 7 7 13 7

2 7 0 7 28 7 12 10 7 7 7

3 7 7 0 7 7 7 7 10 13 33

4 7 15 11 0 16 7 19 7 7 75 7 16 7 7 0 11 12 11 13 7

6 14 11 16 7 10 0 11 10 13 7

7 13 14 7 10 7 10 0 11 18 7

8 14 7 13 7 13 7 11 0 10 7

9 11 7 11 7 10 7 7 10 0 1010 7 10 10 7 7 7 7 25 13 0

pheromone table(*e-4)


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For next iteration taboo-list and ant-city list set to initial

values. delta-pheromone and ant-path length table set to

be zero.

1 2 3 4 5 6 7 8 9 10

1 2 0 0 0 0 0 0 0 0 0

2 2 0 0 0 0 0 0 0 0 0

3 0 0 2 0 0 0 0 0 0 04 0 0 2 0 0 0 0 0 0 0

5 0 0 0 2 0 0 0 0 0 0

6 0 0 0 0 2 0 0 0 0 0

7 0 0 0 0 0 2 0 0 0 0

8 0 0 0 0 0 0 2 0 0 09 0 0 0 0 0 0 0 0 2 0

10 0 0 0 0 0 0 0 0 0 2

Ants

Index


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1 2 3 4 5 6 7 8 9 101 2 0 0 0 0 0 0 0 0 0

2 2 0 0 0 0 0 0 0 0 0

3 0 0 2 0 0 0 0 0 0 0

4 0 0 2 0 0 0 0 0 0 0

5 0 0 0 2 0 0 0 0 0 06 0 0 0 0 2 0 0 0 0 0

7 0 0 0 0 0 2 0 0 0 0

8 0 0 0 0 0 0 2 0 0 0

9 0 0 0 0 0 0 0 0 2 0

10 0 0 0 0 0 0 0 0 0 2

Ants

Index

Taboo list


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1 1 0 0 0 0 0 0 0 0 0 1

2 1 0 0 0 0 0 0 0 0 0 1

3 3 0 0 0 0 0 0 0 0 0 3

4 3 0 0 0 0 0 0 0 0 0 3

5 4 0 0 0 0 0 0 0 0 0 4

6 5 0 0 0 0 0 0 0 0 0 5

7 6 0 0 0 0 0 0 0 0 0 68 7 0 0 0 0 0 0 0 0 0 7

9 9 0 0 0 0 0 0 0 0 0 9

10 10 0 0 0 0 0 0 0 0 0 10

Ant-city table

Ants

Index

Ant_Num 1 2 3 4 5 6 7 8 9 10

Path_len 0 0 0 0 0 0 0 0 0 0



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We move ants to next city as same as last

iteration and fill taboo-list ,.

Until complete a tour. At the end of tour we

update pheromone table and find best tourand compare it to previous best tour and if

new best tour is better best tour is replace

with new best tour.

d


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second iteration:Ant-city table

Ants

Index

Ant_Num 1 2 3 4 5 6* 7 8 9 10

Path_Len 2915.12617.24450.12926.95043.52610.63108.43707.53573.43451.4


1 1 6 5 7 2 4 3 10 8 9 1

2 1 6 5 9 8 3 10 4 2 7 1

3 3 10 8 4 2 1 5 7 6 9 3

4 3 10 8 9 1 6 5 2 4 7 3

5 4 2 1 3 10 8 5 7 6 9 4

6 5 9 8 3 10 2 4 7 1 6 5

7 6 1 4 2 7 3 10 8 5 9 6

8 7 2 4 9 3 10 8 1 5 6 7

9 9 10 3 8 4 2 1 6 5 7 9

10 10 3 8 9 2 4 6 1 5 7 10


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Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5


Length of best tour in Itr.2 : 2610.6

Length of new best tour is shorter than old one then we

replace it to pervious best tour.

Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5

Best tour

Ant Number 6 is winner 6


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1 2 3 4 5 6 7 8 9 10

1 0 0.000 0.001 0.001 0.002 0.003 0.001 0.000 0.001 0.000

2 0.001 0 0.000 0.003 0.000 0.001 0.001 0.000 0.000 0.000

3 0.000 0.000 0 0.000 0.000 0.000 0.001 0.001 0.001 0.004

4 0.000 0.002 0.001 0 0.001 0.001 0.002 0.000 0.001 0.0005 0.000 0.001 0.000 0.000 0 0.001 0.002 0.001 0.002 0.000

6 0.001 0.001 0.001 0.000 0.002 0 0.001 0.000 0.001 0.000

7 0.001 0.001 0.001 0.000 0.000 0.001 0 0.001 0.001 0.000

8 0.001 0.000 0.001 0.001 0.001 0.000 0.001 0 0.001 0.000

9 0.001 0.001 0.001 0.000 0.001 0.000 0.001 0.001 0 0.00110 0.000 0.001 0.001 0.001 0.000 0.000 0.001 0.003 0.001 0

pheromone table in Itr.2

Thi d i i


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Third iteration:Ant-city table

Ants

Index

Ant_Num 1 2 3 4* 5 6 7 8 9 10

Path_Len 3174.73217.23382.92771.53399.13273.53525.33913.03609.83365.0


1 1 6 5 2 4 7 9 8 3 10 1

2 1 6 5 9 2 4 3 10 8 7 1

3 3 10 8 6 1 2 4 7 5 9 3

4 3 10 8 2 4 7 1 6 5 9 3

5 4 2 7 3 10 8 9 1 5 6 4

6 5 9 1 6 4 2 7 8 3 10 5

7 6 1 9 3 10 8 5 4 2 7 6

8 7 2 4 1 5 3 10 8 9 6 7

9 9 5 7 3 10 8 2 4 1 6 9

10 10 3 8 9 1 5 6 2 4 7 10


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Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 3 10 8 2 4 7 1 6 5 9 3

Best tour in Itr.3 belong to ant number 4


Length of new best tour is longer than old one then we dont


Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5

Best tour



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1 2 3 4 5 6 7 8 9 10

1 0 0.000 0.000 0.000 0.002 0.003 0.001 0.000 0.001 0.000

2 0.001 0 0.000 0.004 0.000 0.000 0.001 0.000 0.000 0.000

3 0.000 0.000 0 0.000 0.000 0.000 0.000 0.001 0.001 0.005

4 0.001 0.002 0.001 0 0.000 0.001 0.002 0.000 0.000 0.0005 0.000 0.001 0.000 0.000 0 0.001 0.001 0.000 0.002 0.000

6 0.001 0.001 0.000 0.000 0.002 0 0.001 0.000 0.001 0.000

7 0.001 0.001 0.001 0.000 0.000 0.001 0 0.001 0.001 0.000

8 0.000 0.001 0.001 0.000 0.001 0.000 0.001 0 0.002 0.000

9 0.001 0.001 0.001 0.000 0.001 0.001 0.000 0.001 0 0.00010 0.000 0.000 0.001 0.000 0.000 0.000 0.001 0.004 0.000 0


F th it ti


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Fourth iteration:Ant-city table

Ants

Index

Ant_Num 1 2 3 4 5* 6 7 8 9 10

Path_Len 3358.43455.83805.04020.62970.7 4862.63219.74973.13297.43465.3


1 1 6 3 10 8 4 2 7 5 9 1

2 1 6 7 2 4 9 5 3 10 8 1

3 3 10 8 9 5 6 2 4 7 1 3

4 3 10 8 9 1 2 4 7 5 6 3

5 4 2 7 3 10 8 5 6 1 9 4

6 5 3 10 8 9 6 2 4 1 7 5

7 6 1 2 4 7 8 3 10 5 9 6

8 7 6 8 3 10 1 2 4 5 9 7

9 9 8 3 10 2 4 7 1 5 6 9

10 10 3 8 9 4 2 5 1 6 7 10


69/79

Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 4 2 7 3 10 8 5 6 1 9 4





Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5

Best tour



70/79

1 2 3 4 5 6 7 8 9 10

1 0 0.001 0.000 0.000 0.001 0.002 0.001 0.000 0.001 0.000

2 0.000 0 0.000 0.004 0.000 0.000 0.001 0.000 0.000 0.000

3 0.000 0.000 0 0.000 0.000 0.000 0.000 0.001 0.001 0.005

4 0.001 0.002 0.000 0 0.000 0.000 0.002 0.000 0.001 0.0005 0.000 0.000 0.001 0.000 0 0.002 0.001 0.000 0.002 0.000

6 0.001 0.001 0.000 0.000 0.001 0 0.001 0.000 0.001 0.000

7 0.001 0.001 0.001 0.000 0.001 0.001 0 0.001 0.001 0.000

8 0.000 0.000 0.001 0.001 0.001 0.000 0.000 0 0.002 0.000

9 0.001 0.000 0.000 0.000 0.001 0.001 0.000 0.001 0 0.00010 0.000 0.001 0.001 0.000 0.000 0.000 0.001 0.003 0.000 0


8th it ti


71/79

8th iteration:Ant-city table

Ants

Index

Ant_Num 1* 2 3 4 5 6 7 8 9 10

Path_Len 2876.7 2925.54339.93499.52881.24747.83108.43458.73651.43421.4


1 1 6 5 9 7 4 2 3 10 8 1

2 1 6 5 8 3 10 2 4 7 9 1

3 3 10 8 9 1 2 4 6 5 7 3

4 3 10 8 9 1 6 2 4 7 5 3

5 4 2 7 3 10 8 1 6 5 9 4

6 5 9 8 3 10 6 4 2 1 7 5

7 6 1 4 2 7 3 10 8 5 9 6

8 7 1 6 5 8 3 10 2 4 9 7

9 9 8 3 10 1 5 6 2 4 7 9

10 10 3 8 9 4 2 5 6 1 7 10


72/79

Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 1 6 5 9 7 4 2 3 10 8 1





Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5

Best tour



73/79

1 2 3 4 5 6 7 8 9 10

1 0 0.001 0.000 0.000 0.001 0.003 0.001 0.001 0.000 0.000

2 0.000 0 0.000 0.003 0.000 0.000 0.002 0.000 0.000 0.000

3 0.000 0.000 0 0.000 0.000 0.000 0.000 0.001 0.000 0.005

4 0.000 0.003 0.000 0 0.000 0.001 0.002 0.000 0.001 0.0005 0.000 0.000 0.000 0.000 0 0.001 0.001 0.001 0.002 0.000

6 0.001 0.001 0.000 0.000 0.003 0 0.000 0.000 0.001 0.000

7 0.001 0.000 0.001 0.000 0.001 0.000 0 0.000 0.001 0.000

8 0.001 0.000 0.002 0.000 0.000 0.000 0.000 0 0.002 0.000

9 0.001 0.000 0.000 0.001 0.000 0.000 0.001 0.001 0 0.000

10 0.000 0.001 0.001 0.000 0.000 0.000 0.001 0.003 0.000 0


last(10th) iteration


74/79

last(10th) iteration:Ant-city table

Ants

Index

Ant_Num 1 2* 3 4 5 6 7 8 9 10

Path_Len 2771.5 2610.63605.53924.32929.23339.02885.34073.33998.04665.2


1 1 6 5 9 3 10 8 2 4 7 1

2 1 6 5 9 8 3 10 2 4 7 1

3 3 10 8 9 2 4 7 1 5 6 3

4 3 10 8 9 6 4 2 7 1 5 3

5 4 2 7 3 10 8 9 1 6 5 4

6 5 2 4 7 1 6 3 10 8 9 5

7 6 1 5 9 4 2 7 3 10 8 6

8 7 2 4 5 1 9 8 3 10 6 7

9 9 8 3 10 1 2 4 7 5 6 9

10 10 3 8 1 2 4 5 6 7 9 10


75/79

Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 1 6 5 9 8 3 10 2 4 7 1


Length of best tour in Itr.10 :2610.62030

Length of new best tour is the same as old one then we dont


Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5

Best tour



76/79

1 2 3 4 5 6 7 8 9 10

1 0 0.001 0.000 0.000 0.001 0.003 0.001 0.000 0.000 0.000

2 0.000 0 0.000 0.004 0.000 0.000 0.002 0.000 0.000 0.000

3 0.000 0.000 0 0.000 0.000 0.000 0.000 0.000 0.000 0.005

4 0.000 0.002 0.000 0 0.001 0.000 0.002 0.000 0.000 0.0005 0.000 0.000 0.000 0.000 0 0.001 0.001 0.000 0.002 0.000

6 0.001 0.000 0.000 0.000 0.002 0 0.000 0.000 0.001 0.000

7 0.001 0.000 0.001 0.000 0.001 0.000 0 0.000 0.001 0.000

8 0.001 0.001 0.002 0.000 0.000 0.000 0.000 0 0.002 0.000

9 0.001 0.001 0.001 0.001 0.000 0.001 0.000 0.001 0 0.000

10 0.000 0.001 0.000 0.000 0.000 0.001 0.000 0.004 0.000 0

pheromone table in last Itr


77/79

Finally we find best-tour as:

Steps 1 2 3 4 5 6 7 8 9 10 11

City

Number 5 9 8 3 10 2 4 7 1 6 5

2610.62030


78/79


79/79

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Date post:	02-Jun-2018
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