Image Processing Reference
In-Depth Information
3. ℑ(
x
,
z
) = ℑ(
y
,
z
),
x
≤
L
*
y
4. Border condition: ℑ(
x
, (1, 0)) =
x
An intuitionistic fuzzy triangular conorm is a binary operation
S
which is
increasing, commutative, associative and satisfying,
S
(, )
=
It is a map-
x
0
x
.
*
L
ping such that
1. Commutativity:
S
(,)
xy
=
S
( ,)
yx
2. Associativity:
SS SS
(,(,)) ((, ,)
xyz
=
x yz
3.
S
(,)
xz
=
S
( ,),
yzxy
L
≤
*
4. Border condition:
S
(,(,))
x
01
=
x
In the earlier formulation, 1
L
*
= (1, 0), 0
L
*
, (0, 1).
As in fuzzy set theory, fuzzy
t
-norm and
t
-conorm are modelled as union
and intersection, and intuitionistic fuzzy
t
-norm and
t
-conorm can be mod-
elled as union and intersection, respectively.
Let
T
be the
t
-norm and
S
be the
t
-conorm. If
ℑ
(,)
for
xy Sxy
≤− −−
111
(
,
)
all
x
,
y
∈ [0, 1], then
ℑ=
11 22
is an intuitionistic fuzzy
t
-norm
(,)((,), (,))
xy Tx ySxy
=
11 22
is an intuitionistic fuzzy
t
-conorm
S
(,)((,), (,))
xy Sx yTxy
For two elements (
x
1
,
x
2
) and (
y
1
,
y
2
), some examples of intuitionistic fuzzy
t
-norms are
1.
ℑ=
(,) max( ,
0
+−
1
),min(,
1
+
))
xy
xy
xy
1
1
2
2
2.
ℑ=
(,)( ,
xy xy xyxy
+−
)
11 2
2 22
3.
ℑ=
(,) min(
xy
xy xy
, ,max(,))
11 22
3.8 Summary
This chapter summarizes various types of operators such as fuzzy aggre-
gation operators, namely, WA and OWA operators, and fuzzy triangular
operators, namely,
t
-norms and
t
-conorms. Different types of
t
-norms and
t
-conorms are also discussed. Extension of fuzzy aggregation operators to
intuitionistic fuzzy case such as intuitionistic fuzzy aggregation opera-
tors, namely, the IFWA, IFOWA and IFHA operators, is discussed with
examples. Application of intuitionistic fuzzy operator in multi-attribute
decision-making is also provided as well as extension of
t
-norms and
t
-conorms to IFSs to intuitionistic fuzzy triangular
t
-norms and
t
-conorms.
Search WWH ::
Custom Search