Engineering Optimization: Theory and Practice, Fourth Edition

720 Modern Methods of Optimization

Step 4: Assuming that the ants return home and start again in search of food, we set

the iteration number asl=2. We need to update the pheromone array as

τ 1 (^2 j)=τ 1 ( ldoj )+

∑

k

τ(k) (E 1 )

where

∑

k

τ(k)is the pheromone deposited by the best antkand the summation

extends over all the best antsk(if multiple ants take the best path). In the

present case, there is only one best ant,k=1, which used the pathx 13. Thus

the value of

∑

k

τ(k)can be determined in this case as

∑

k

τ(k)= τ(k=^1 )=

ςfbest

fworst

=

( 2 )(− 12. 0 )

(− 8. 0 )

= 3. 0

where the scaling parameterςis assumed to be 2. Using a pheromone decay

factor ofρ= 0 .5 in Eq. (13.43),τ 1 ( ldoj )can be computed as

τ 1 ( ldjo )= ( 1 − 0. 5 )τ 1 (^1 j)= 0. 5 ( 1. 0 )= 0. 5 ;j= 1 , 2 , 4 , 5 , 6 , 7

Thus Eq. (E 1 ) gives

τ 1 (j^2 )= 1. 0 + 3. 0 = 4. 0 forj=3 andτ 1 (j^2 )= 0. 5 forj= 1 , 2 , 4 , 5 , 6 , 7

With this, we go to step 5.

Step 2: For any antk, the probability of selecting pathx 1 jin Fig. 13.4 is given by

p 1 j=

τ 1 j

∑^7

p= 1

τ 1 p

; j= 1 , 2 ,... , 7

whereτ 1 j= 0. 5 ;j= 1 , 2 , 4 , 5 , 6 , 7 andτ 13 =. This gives 4

p 1 j=

0. 5

7. 0

= 0. 0714 ;j= 1 , 2 , 4 , 5 , 6 , 7 ;p 13 =

4. 0

7. 0

= 0. 5714

To determine the discrete value or path selected by ant using a random num-

ber selected in the range (0, 1), cumulative probabilities are associated with

different paths as (roulette wheel selection process):

x 11 = ( 0 , 0. 0714 ), x 12 = ( 0. 0714 , 0. 1429 ), x 13 = ( 0. 1429 , 0. 7143 ),

x 14 = ( 0. 7143 , 0. 7857 ), x 15 = ( 0. 7857 , 0. 8571 ), x 16 = ( 0. 8571 , 0. 9286 ),

x 17 = ( 0. 9286 , 1. 0 )

Step 3: Generate four random numbers in the range (0, 1), one for each of the ants

asr 1 = 0. 3688 , r 2 = 0. 8577 , r 3 = 0. 0776 , r 4 = 0. 5 791. Using the cumulative

probability range (given in step 2) in which the value ofrifalls, the discrete