IEEE Systems, Man and Cybernetics Magazine - July 2022 - 54

Therefore:
q
Ay aX
1
i =|
j =
q
Ay aX
1
i =|r j
j =
q
By bX
1
i =|
j =
q
i =
j
j
j
^h
j
i
.
^h
j
i
.
^h
j
i
.
Cy |cXj
j =1
q
Ay aX
1
i =|
j =
i
j
i
^h
i
.
^h
j
i
.
6rri
If YA i ,, ,, ,
ri
u =66 @@i 6
i
,,
@@ the degree of the fitting of Yi
u
i yAyByCyCy @ and u =y pp q ,, ,
is measured
i
to yi
u
by the h-level sets of the actual and the forecasted fuzzy
numbers, respectively, where the h-level has been
described according to Definition 8. The vital notion is to
acquire A ,j
u
parameters a ,j
a ,j
r
Yy ,.ih yYiii h
u 11
uu
b ,j
u
(4)
We utilized the necessity for LMFs and the possibility
model for UMFs. Since it is sufficient to close the membership
functions of the observed and predicted values as
much as possible, an h-cut of the observed value is included
in the predicted value [18].
The objective function is described by
II II I1234
77AA (5a)
Ic .aX
n
1
j
q
Ia aXj
i 11
=- +==
||
2
===
i
11
n
q
6
h
3
===
q
i
11
n
< i
q
j
Ic .aX
j
4
===
i
11
n
||7^ jj j
h
^ h 2
i
A
rr
^^
j
j
i 11
==
|| rr^ jj j
j
j
^h
j
i
^ h
i
@
Iq .bX|| F
2
2
hh
hh
^^i
j
i22
j
n
q
|| cc Xj
j
=+-+
.
.
c ,j
c ,j
r
such that:
66
i iir @
(3a)
(3b)
(3c)
(3d)
(3e)
By 11 h^
h^
ii i ii i
-- -- -^
By
+- -+ --rii
h^
i 11i ii
By 11
h^
By +- -- -- i
ii i ii i
-- -- -- r
^
i 11i ii
h#
$ ^^
$ ^^
^
i
Our problem can be rewritten as the following:
.
MinII II I1234
=+-+
hByAyq hq p .
^ hCyByq hr q
hByAyq hq p .
^ hCyByq hr q h .
h^
h#
^
h
.
h hh
h hh
h^
(7)
General Type-2 Regression Model
The suggested model is a developed version of the
Tanaka's model [9], [26], whose underlying idea is to
minimize the model's fuzziness by minimizing the general
spread of fuzzy coefficients, subject to including
all the provided data. In the proposed model, vagueness
in primary and secondary fuzzy sets should be
minimized. Also, a specified h-plane of the observed
value should be included in the same h-plane of the
predicted value.
Since the h-plane builds a FOU-like area, it should be
discussed similarly to the IT2FR model. Figure 4 shows
the H-cut that we employed in this article. In this method,
H-cut is employed in two stages: first in the secondary
membership function and then in the primary membership
function.
After this h-plane representation, we calculate a new
(5b)
(5c)
(5d)
This four-objective function works as follows.
1) the ambiguity of the secondary membership function
minimized by I1
2) the ambiguity of LMF of Yu
by I2
that is a T1FS minimized
3) the distance between point with the biggest membership
rate in a predicted and the point with the highest membership
value in the corresponding observed value minimized
by I3
4) the necessity problem, which is met by I4
a ,j
r
b ,j
c ,j
r
and hence
should be maximized.
The final goal is to find out the fuzzy parameters
c ,j
Min I2
subject to
54 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2022
and Max I3
a ,j
which are the solution of the following
two quadratic programming problems and all objective
function [18]:
(6)
If Y 0=
Therefore, with two points ^hq ,1 and ,,
line equation can be derived, and x2h
(x2h
xy2lh
is the point that Y ).0=
written as:
Y 1
-= - l Xq .^h1-
2h
qx
y
(11)
^h the 2h
has been found
The line equation can be
where: a=Thus:
2
parameter
in a new FOU. Figures 4 and 5 clearly show
the support parameters for the LMF and UMF of a new
FOU. Line equations of (1) and (2) that are legs of LMF
and UMF are achievable (Figure 5). Hence, we can get
two points (y1
and ).y2
B =
With symmetric secondary
membership function (Figure 6), B is given next:
.
yy
2
12
+
To find x1h
FOU, y1h
and y2h
(8)
and x ,2h which are in support of a new
should be found first. The following
relation can be derived from from the secondary triangular
membership function:
1 h k kh
1
- == -^h ,
a & a 1
yB .
yB hy B^^hh
h
221=+ -- .
(10)
(9)
IEEE Systems, Man and Cybernetics Magazine - July 2022

Table of Contents for the Digital Edition of IEEE Systems, Man and Cybernetics Magazine - July 2022
