A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 53

inertia weight utilized to control the impact of the previous history of velocities on
the current velocity of a given particle. x pbest i denotes the personal best position of

the particle i. ~ x gbest represents the position of the best particle of the entire swarm.

r 1 ; are random values. Moreover, in this paper, a uniform probability

distribution is assumed for all the random parameters (Mahmoodabadi et al. 2013 ).
The trade-off between the global and local search abilities of the swarm is adjusted
by using the parameter W. An appropriate value of the inertia weight balances
between global and local search abilities by regarding the fact that a large inertia
weight helps the global search and a small one helps the local search. Based upon
experimental results, linearly decreasing the inertia weight over iterations enhances
the ef

r 2 2 ½ 0 ; 1

ciency of particle swarm optimization (Eberhart and Kennedy 1995 ). The
particles approach to the best particle of the entire swarm ( x gbest ) via using a small

value of C 1 and a large value of C 2 . On the other hand, the particles converge into
their personal best position ( ! pbest i ) through employing a large value of C 1 and a
small value of C 2 . Furthermore, it was obtained that the best solutions were gained
via a linearly decreasing C 1 and a linearly enhancing C 2 over iterations (Ratnaweera
et al. 2004 ). Thus, the following linear formulation of inertia weight and learning
factors are utilized as follows:

~

t
maximum iteration Þ

W 1 ¼ W 1 ð W 1 W 2 Þð ð 6 Þ

t
maximum iteration Þ

C 1 ¼ C 1i ð C 1i C 1f Þð ð 7 Þ

t
maximum iteration Þ

C 2 ¼ C 2i ð C 2i C 2f Þð ð 8 Þ

in which W 1 and W 2 represent the initial and final values of the inertia weight,

correspondingly. C 1i and C 2i denote the initial values of the learning factors C 1 and
C 2 , correspondingly. C 1f and C 2f represent the final values of the learning factors

C 1 and C 2 , respectively. t is the current number of iteration and maximum iteration
is the maximum number of allowable iterations. The mutation probabilities at each
iteration which is based on fuzzy rules will be presented in the next section.

4 The Mutation Probabilities Based on Fuzzy Rules

The mutation probability at each iteration is calculated via using the following
equation:

P m ¼ f m Limit ð 9 Þ