Page 562 - Mathematical Techniques of Fractional Order Systems

540 Mathematical Techniques of Fractional Order Systems

N
where X i is the i-th grasshopper position, S i 5 P sðd ij Þd ij is the social
b
j51;j6¼i
interaction between grasshoppers, G i is the gravity force, and A i is the wind
advection (Saremi et al., 2017). d ij 5 jx j 2 x i j is the distance between i-th and
j-th grasshopper and d ij 5 jx j 2 x i j is the unit vector of the distance. The func-
b
d ij
2r 2r
tion s 5 fe l 2 e defines social forces between grasshoppers. f shows the
intensity of attraction and l indicates the attractive length scale (Saremi
et al., 2017).
Eq. 18.18 is considered to be a controlled equation of the GOA algo-
rithm. However, this mathematical equation cannot be employed directly to
solve the optimization problems, because the grasshoppers quickly arrive at
the local zone. Therefore, the controlled equation Eq. 18.18 of GOA algo-
rithm will be modified as in the following Eq. 18.19 (Saremi et al., 2017).
!
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X ub d 2 lb d jx j 2 x i j
d
X 5 c c sðjx j 2 x i jÞ b ð18:19Þ
i 2 1 T d
j51;j6¼i d ij
where ub d and lb d are the upper and lower boundaries in the D th dimension.
T d is the value best solution obtained so far, and c is a decreasing coefficient

c max 2 c min
from 1 to zero over the course of iteration by this relation c max 2 jð Þ ,
J
j is the current iteration, and J is the maximum number of iterations. c max , c min
are 1 and 0.00001, respectively (Saremi et al., 2017).
!
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P jx j 2 x i j
The first term of Eq. 18.19, c c ub d 2 lb d sðjx j 2 x i jÞ simulates
2 d ij
j51;j6¼i
the interaction of grasshoppers in nature, whereas, the second term, T d ,
implements their tendency to move towards the source of food (Saremi
et al., 2017). Accordingly, characteristics of grasshoppers in natural have
been implemented in this Eq. 18.19 (Saremi et al., 2017). The flowchart of
the GOA algorithm is shown in Fig. 18.4.
18.4.3.2 Chaotic Grasshopper Optimizer Strategy (CGOA)
As illustrated in Eq. (18.19), there are three important parameters c, f, and l
that control the behavior of the GOA technique. The parameter c is the most
essential one that helps in balancing between exploration and exploitation
phases, c value linearly decreases from 1 to 0 over the course of iterations.
While, both f and l are responsible for managing the social interaction
between the grasshoppers to prevent them from sticking in local minima. In
GOA, f and l values are selected as 0.5 and 1.5 from the intervals [0 1] and
[1 2], respectively. While, in the developed CGOA technique, the values of
c, f, and l are changed based on the chaos maps distribution as illustrated in
the following Eqs. (18.20) (18.21). Where c will chaotically deceased from