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.