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,
. It is a map-
(, )
*
x
1
=
x
ping :( * )
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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