A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 51

2.1 The Crossover Operator

The role of crossover operator is to generate new individuals, that is, offspring from
parents in the mating pool. Afterward, two offspring are generated based upon the
selected parents and will be put in the place of the parents. Moreover, this operator
is used for a number of pair of parents to mate (Chang 2007 ). This number is
calculated by using the formula as P tc 2 , where P tc and N denote the probability of
traditional crossover and population size, correspondingly. By regarding x ! ð t Þ and

x ! ð t Þ as two random selected chromosomes in such a way that x ! ð t Þ has a smaller

fitness value than x ! ð

Þ

t , the traditional crossover formula is as follows

x ! ð t þ Þ ¼x ! ð t Þþc 1 ð x ! ð t Þ x ! ð t ÞÞ 1

ð 1 Þ

x ! ð t þ Þ ¼x ! ð t Þþc 2 ð x ! ð t Þ x ! ð t ÞÞ 1

where c 1 and c 2 2 ½ 0 ; 1 represent random values. When Eq. ( 1 ) is calculated,

between fitness should be chosen.

Another crossover operator called multiple-crossover operator is employed in this
paper (Mahmoodabadi et al. 2013 ). This operator was presented in (Ker-Wei and
Shang-Chang 2006 ) for the

~ x ð t Þ and ~ x ð t þ 1 Þ , whichever has the fewer

first time. The multiple-crossover operator consists of
three chromosomes. The number of P mc 3 chromosomes is chosen for adjusting in
which P mc denotes the probability of multiple-crossover. Furthermore, x ! ð

, x ! ð

t Þ t Þ

and x ! ð denote three random chosen chromosomes in which x ! ð

t Þ t Þ has the smallest

fitness value among these chromosomes. Multiple-crossover is computed as follows

x ! ð x ! ð 2 x ! ð x ! ð x ! ð

ÞÞ
x ! ð t þ 1 Þ ¼x ! ð t Þþk 2 ð 2 x ! ð t Þ x ! ð t Þ x ! ð t ÞÞ
x ! ð t þ

t þ 1 Þ ¼ t Þþk 1 ð t Þ t Þ t

ð 2 Þ

Þ ¼x ! ð t Þþk 3 ð 2 x ! ð t Þ x ! ð t Þ x ! ð t ÞÞ

1

where k 1 ; k 2 ; and k 3 2 ½ 0 ; 1 represent random values. When Eq. ( 2 ) is computed,

and ! ð

between ~ x ð t Þ t þ 1 Þ , whichever has the fewer fitness should be selected.

2.2 The Mutation Operator

According to the searching behavior of GA, falling into the local minimum points is
unavoidable when the chromosomes are trying to find the global optimum solution.

In fact, after several generations, chromosomes will gather in several areas or even
just in one area. In this state, the population will stop progressing and it will become
unable to generate new solutions. This behavior could lead to the whole population
being trapped in the local minima. Here, in order to allow the chromosomes '
exploration in the area to produce more potential solutions and to explore new
regions of the parameter space, the mutation operator is applied. The role of this