Image Processing Reference
In-Depth Information
The characteristic information of all the alternatives is represented by intu-
itionistic fuzzy values
a
ij
(
i
= 1, 2, 3, …,
m
;
j
= 1, 2, 3, …,
n
). If
a
ij
= (μ
ij
, ν
ij
), then
A
i
= {〈
B
j
,
a
ij
〉,
B
j
∈
B
} where μ
ij
denotes the degree to which the alternative
A
i
satisfies the attribute
B
j
and ν
ij
denotes the degree to which the alternative
A
i
does not satisfy the attribute
B
j
with μ
ij
∈ [0, 1], ν
ij
∈ [0, 1], respectively.
To get the best alternative, one can use GIFWA, GIFOWA and GIFHA to
derive
a
i
a
i
= (μ
ai
, ν
ai
) of the alternatives
A
i
as
a
=
GIFWA( ,
a aa a
,
,
…
,
),
i
= …
123
,,,
,
m
i
w
i
123
i
i
in
a
=
GIFOWA
(,
a aa a
123
…
,
,
, ,
i
=
123
, ,, ,
…
m
i
w
i
i
i
in
a
=
GIFHA
(,
a aa a
123
…
,
,
, ,
i
=
1 23
, ,, ,
…
m
i
w
i
i
i
in
with
w
= (
w
1
,
w
2
, …,
w
n
)
T
as the weight vector related to the operators with
w
j
∑
n
=
1
, which can be derived from the normal distribution-based
j
=
1
method.
Scores
s
(
a
i
) (
i
= 1, 2, 3, …,
m
) of the overall
a
i
(
i
= 1, 2, 3, …,
m
) are calculated.
Then the alternatives are ranked according to the score values, and the best
one is selected.
3.7 Intuitionistic Fuzzy Triangular Norms
and Triangular Conorms
Intuitionistic fuzzy
t
-norms and
t
-conorms can be constructed using
t
-norms
and
t
-conorms [4]. IFSs can also be visualized as
L
-fuzzy sets according to
Deschrijver and Kerre [3]. If
L
*
is a set and ≤
L
*
is the operation,
{
}
*
2
La aaa
=
(,)|(,)[,]
∈ ∧+≤
01
a
a
1
12 12
1
2
*
(,)
aa bb abab aa bb L
L
≤ ⇔≤∧≥
(
, )
forall
(, ),(, )
∈
*
12 12 1
1
2
2
12 12
Then (
L
*
, ≤
L
*
) is a complete lattice.
Using the lattice (
L
*
, ≤
L
*
), triangular norms and conorms can be extended
to intuitionistic fuzzy case.
An intuitionistic fuzzy triangular norm is a binary operation ℑ which is
increasing, commutative, associative and satisfying,
ℑ
. It is a map-
(, )
*
x
1
=
x
ping
ℑ
:(
*
)
L
2
→
such that
L
*
1. Commutativity: ℑ(
x
,
y
) = ℑ(
y
,
x
)
2. Associativity: ℑ(
x
, ℑ(
y
,
z
)) = ℑ(ℑ(
x
,
y
),
z
)
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