A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 56

is randomly produced. At each iteration, the inertia weight, the learning factors, and
the operators probabilities are computed. Afterward, ! pbest i and ! gbest are speci

ed

when the fitness values of all members are evaluated. The genetic algorithm

operators, that is, traditional crossover, multiple-crossover and mutation operators
are used to adjust the chromosomes (which are chosen randomly). Each chromo-
some is a particle and the group of chromosome is regarded as a swarm. Further-
more, the chromosomes which are not chosen for genetic operations will be
appointed to particles and improved via PSO. Until the user-de ned stopping cri-

terion is satis ed, this cycle is repeated. Figures 2 and 3 illustrate the pseudo code

and fl flow chart of the technique respectively.

5.1.3 The Results of Single-Objective Optimization

To evaluate the performance of the hybrid of particle swarm optimization and the
genetic algorithm, nine prominent benchmark problems are utilized regarding a
single-objective optimization problem. Essential information about these functions
is summarized in Table 2 (Yao et al. 1999 ). Some of these functions are unimodal
and the others are multimodal. Unimodal functions have only one optimal point
while multimodal functions have some local optimal points in addition to one
global optimal point.
The hybrid of particle swarm optimization and the genetic algorithm is applied to
all the test functions with 30 dimensions (Mahmoodabadi et al. 2013 ).

The mean and standard deviation of the best solution for thirty runs are presented in
Table 4 . In this regard, the results are contrasted to the results of three other
algorithms [i.e., GA with traditional crossover (Chang 2007 ), GA with multiple-
crossover (Chang 2007 ; Chen and Chang 2009 ), standard PSO (Kennedy and
Eberhart 1995 )]. Table 3 illustrates the list of essential parameters to run these

ð n ¼ 30 Þ

Initialize population and determine the algorithm configuration of the hybrid method.
While stopping criterion is satisfied
Determine

t P , m P , P based on the hybrid formulations.

→ →

Find fitness values of each member and store and

pbest gbest

i

If

rand <
Select randomly two chromosomes from population and update them using traditional crossover

t P

operator;
Elseif

rand <

Select randomly three chromosomes from population and update them using multiple-crossover

P

mc

operator;
Elseif

rand <

Select randomly a chromosome from population and update it using mutation operator;

P

Else

Select randomly a particle from swarm and update its position based on PSO formula;

End.

Fig. 2 The pseudo code of the hybrid algorithm for single-objective optimization