Advanced Metaheuristics-Based Approach for Fuzzy Control Systems Tuning - Complex System Modelling and Control Through Intelligent Soft Computations - page 625

x i ; d
k

x i ; d
low

x i ; d
up

x i ; d
low

¼ þ rand 0 ðÞ ; 1 ð 25 Þ

where x i ; d
low

and x i ; d

up are the lower and upper ranges, respectively, for decision
variables in the dimension.
This behaviour of the arti ects a powerful mechanism to

escape the problem of trapping in local optima. The value of the

cial bee colony re fl

“ Limit for

abandonment ” control parameter of ABC algorithm is calculated as follows:

N
2

L ¼ D ð 26 Þ

Finally, the steps of the original version of ABC algorithm, as described in
Basturk and Karaboga ( 2006 ), Karaboga ( 2005 ), Karaboga and Basturk ( 2007 ,
2008 ), can be summarized as follows:

1. Initialize the ABC algorithm parameters: population size N, limit of abandon-
ment L, dimension of the search space D, …

2. Generate a random population equal to the speci ed number of employed bees,

where each of them contains the value of all the design variables.
3. Obtain the values of the objective function, de

ned as the amount of nectar for
the food source, for all the population members.
4. Update the position of employed bees using Eq. ( 23 ), obtain the value of
objective function and select the best solutions to replace the existing ones.
5. Run the onlooker bee phase: onlookers proportionally choose the employed bees
depending on the amount of nectar found by the employed bees, Eq. ( 24 ).
6. Update the value of onlooker bees using Eq. ( 23 ) and replace the existing
solution with the best new one.
7. Identify the abundant solutions using the limit value. If such solutions exist then
these are transformed into the scout bees and the solution is updated using
Eq. ( 25 ).
8. Repeat the steps 3 7 until the termination criterion is reached, usually chosen as

-

the speci ed number of generations.

3.4 Particle Swarm Optimization

The PSO technique is an evolutionary computation method developed in 1995 by
Kennedy and Eberhart ( 1995 ), Eberhart and Kennedy ( 1995 ). This recent meta-
heuristic technique is inspired by the swarming or collaborative behaviour of bio-
logical populations. The cooperation and the exchange of information between
population individuals allow solving various complex optimization problems. The
convergence and parameters selection of the PSO algorithm are proved using several
advanced theoretical analysis (Bouall

è gue et al. 2011 , 2012a , b ;Madiouni et al. 2013 ).