Gray Wolf Optimization
Gray wolf optimization is presented in the following subsections based on the work in [13].
1) Inspiration: Grey wolves are considered as apex predators, meaning that they are at the top of the food chain. Grey wolves mostly prefer to live in a pack. The group size is 512 on average.They have a very strict social dominant hierarchy.
The leaders are a male and a female, called alphas. The alpha is mostly responsible for making decisions about hunting, sleeping place, time to wake, and so on. The alphas decisions are dictated to the pack.
The second level in the hierarchy of grey wolves is beta. The betas are subordinate wolves that help the alpha in decisionmaking or other pack activities. The beta wolf can be either
male
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Omega wolves always have to submit to all the other dominant wolves. They are the last wolves that are allowed to eat.
The fourth class is called subordinate (or delta in some references). Delta wolves have to submit to alphas and betas, but they dominate the omega. Scouts, sentinels, elders, hunters, and caretakers belong to this category. Scouts are responsible for watching the boundaries of the territory and warning the pack in case of any danger. Sentinels protect and guarantee the safety of the pack. Elders are the experienced wolves who used to be alpha or beta. Hunters help the alphas and betas when hunting prey and providing food for the pack. Finally, the caretakers are responsible for caring for the weak, ill, and wounded wolves in the pack.
2) Mathematical model: In the mathematical model for the
GWO the fittest solution is called the alpha (α). The second and third best solutions are named beta (β) and delta (δ) respectively. The rest of the candidate solutions are assumed to be omega (ω). The hunting is guided by α, β, and δ and the ω follow these three candidates.
In order for the pack to hunt a prey they first encircling it. In order to mathematically model encircling behavior the following equations are used
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−→
D (11) where −→
D is as defined in 12 and t is the iteration number,
−→
A ,
−→
C are coefficient vectors,
−→
X p is the prey position, . is dot product and
−→
X is the gray wolf position.
−→
D = |
−→
C .
−→
X p
(t) −
−→
X (t)| (12)
The
−→
A ,
−→
C vectors are calculated as in equations 13 and
14
−→
A = 2
−→
A .
−→
r
1 −
−→
a (13)
−→
C = 2
−→
r
2 (14) where components of
−→
a are linearly decreased from 2 to 0 over the course of iterations and r
1
, r2 are random vectors in [0, 1]. The hunt is usually guided by the alpha. The beta and delta might also participate in hunting occasionally. In order to mathematically simulate the hunting behavior of grey wolves, the alpha (best candidate solution) beta, and delta are assumed to have better knowledge about the potential location of prey. The first three best solutions obtained so far and oblige the other search agents (including the omegas) to update their positions according to the position of the best search agents.
So the updating for the wolves positions is as in equations
15,16,17.
−→
Dα = |
−→
C1.
−→
Xα −
−→
X |,
−→
Dβ = |
−→
C2.
−→
Xβ −
−→
X |,
−→
Dδ = |
−→
C3.
−→
Xδ −
−→
X |
(15)
−→
X1 = |
−→
Xα−
−→
A1.
−→
Dα
|,
−→
X2 = |
−→
Xβ −
−→
A2.
−→
Dβ
|,
−→
X3 =
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