A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 52

operator is to change the value of the number of chromosomes in the population.
This number is calculated via P m N, in which P m and N are the probability of
mutation and population size, correspondingly. In this regard, a variety in popu-
lation and a decrease in the possibility of convergence toward local optima are
gained through this operation. By regarding a randomly chosen chromosome, the
mutation formula is obtained as (Mahmoodabadi et al. 2013 ):

x ! ð t þ 1 Þ ¼x min ð t Þþtð x max ð t Þ x min ð t ÞÞ ð 3 Þ

in which x ! ð present the randomly chosen chromosome, upper

bound and lower bound with regard to search domain, correspondingly and

t Þ , ~ x max ð t Þ and ~ x min ð t Þ

t 2

½ 0 ; 1 is a random value. When Eq. ( 3 ) is calculated, between ~ x ð t Þ and ~ x ð t þ 1 Þ ,

whichever has the fewer fitness should be chosen.

The second optimization algorithm used for the hybrid algorithm is particle
swarm optimization and this algorithm and its details involving the inertia weight
and learning factors will be presented in the following section.

3 Particle Swarm Optimization (PSO)

Particle swarm optimization introduced by Kennedy and Eberhart ( 1995 )isa
population-based search algorithm based upon the simulation of the social behavior
of

first used to balance the weights in
neural networks (Eberhart et al. 1996 ), it is now a very popular global optimization
algorithm for problems where the decision variables are real numbers (Engelbrecht
2002 , 2005 ).
In particle swarm optimization, particles are

fl flocks of birds. While this algorithm was

flying through hyper-dimensional
search space and the changes in their way are based upon the social-psychological
tendency of individuals to mimic the success of other individuals. Here, the PSO
operator adjusted the value of positions of particles which are not chosen for genetic
operators (Mahmoodabadi et al. 2013 ). In fact, the positions of these particles are
adjusted based upon their own and neighbors

fl

experience. x ! ð ' t Þ represents the posi-

tion of a particle and it is adjusted through adding a velocity v ! ð

t Þ to it, that is to say:

x ! ð x ! ð v ! ð

t þ 1 Þ ¼ t Þþ t þ 1 Þ ð 4 Þ

The socially exchanged information is presented by a velocity vector de ned as

follows:

v ! ð Wv ! ð x ! ð x ! ð t þ 1 Þ ¼ t Þþ C 1 r 1 ð~ x pbest i t ÞÞ þ C 2 r 2 ð~ x gbest t ÞÞ ð 5 Þ

where C 1 represents the cognitive learning factor and denotes the attraction that a
particle has toward its own success. C 2 is the social learning factor and represents
the attraction that a particle has toward the success of the entire swarm. W is the