Lakhasly

Jang [23] introduced ANFIS, which is a hybrid model that combines fuzzy logic and NNs.The adaptation methods of most fuzzy inference systems rely on the back-propagation algorithm that is applied to deal with parameter optimization in general.(4) The node in layer 4 is an adaptive node, and its output is computed as O4,i = wi fi = wi (pix + qiy + ri ), where pi , qi , and ri are the consequent parameters of the node i. In the last layer, there exists only one node whose output is computed by using the following equation: O5 = ?The crisp inputs x and y to the node of the first layer and the output O1i of this node are defined as O1i = uAi (x), i = 1, 2, O1i = uBi-2 (y), i = 3, 4, (1) where Ai and Bi are the membership values of the generalized Gaussian membership function defined as [23] u(x) = e

( x- ?i ?i )2 , (2) where pi and ?i are the premise parameters.Nevertheless, these methods could not achieve the promised results in all experimental cases and need much computation time; therefore, we use the GWO algorithm to determine the optimal weights of ANFIS and reduce the time complexity.One of these hybrid learning algorithms is the hybrid between the back-propagation algorithm and the LSM.

Jang [23] introduced ANFIS, which is a hybrid model that combines fuzzy logic and NNs. ANFIS is based on
the Takagi-Sugeno inference model that creates a non-linear mapping from input to the output space by using
fuzzy IF-THEN rules. The ANFIS model contains five layers, as shown in Figure 1.
The crisp inputs x and y to the node of the first layer and the output O1i of this node are defined as
O1i = µAi
(x), i = 1, 2, O1i = µBi−2
(y), i = 3, 4, (1)
where Ai and Bi are the membership values of the generalized Gaussian membership function defined as [23]
µ(x) = e
−
(︁
x−
ρi
αi
)︁2
, (2)
where pi and σi are the premise parameters. In the second layer, the node’s output is the firing strength of a
rule, as
O2i = µAi
(x) × µBi−2
(y). (3)
The node’s output in the third layer is the normalized firing strength as
O3i = wi =
ωi
∑︀2
(i=1) ωi
. (4)
The node in layer 4 is an adaptive node, and its output is computed as
O4,i = wi
fi = wi
(pix + qiy + ri
),
where pi
, qi
, and ri are the consequent parameters of the node i. In the last layer, there exists only one node
whose output is computed by using the following equation:
O5 =
∑︁
i
wi
fi
. (6)
There are two sets of adjustable parameters of the ANFIS model: the premise and the consequent parame-
ters. The least square method (LSM) can be used to determine the fitness values of the consequent parameters.
However, if the premise parameters are not steady, the search space will be wider and the convergence of
training will be slower. Therefore, the hybrid learning techniques can be used to overcome this problem. One
of these hybrid learning algorithms is the hybrid between the back-propagation algorithm and the LSM. This
algorithm involves a two-step process [23].
In the first one, if the premise parameters are steady, the functional signals will propagate to layer 4,
where the LSM specifies the consequent parameters. Thereafter, the consequent parameters will be kept fixed.
The adaptation methods of most fuzzy inference systems rely on the back-propagation algorithm that
is applied to deal with parameter optimization in general. This traditional optimization technique can get
trapped in a local optimum. To fix this problem, evolutionary methods like GA have been widely applied.
Nevertheless, these methods could not achieve the promised results in all experimental cases and need
much computation time; therefore, we use the GWO algorithm to determine the optimal weights of ANFIS
and reduce the time complexity.