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t-conorm and t-norm, Xia and Xu (2011), and Xia et al. (2012c) presented some
intuitionistic fuzzy aggregation operators and their generalizations. In this chapter,
we shall introduce these newly developed aggregation operators for IFVs, and their
applications in decision making.
1.1 Rankings of Intuitionistic Fuzzy Values
1.1.1 Intuitionistic Fuzzy Values
Atanassov (1986) introduced the concept of intuitionistic fuzzy set (IFS):
Definition 1.1
(Atanassov 1986) Let
X
be a fixed set, then
A
={
x
,μ
A
(
x
),
v
A
(
x
)
|
x
∈
X
}
(1.1)
is called an intuitionistic fuzzy set (IFS), which assigns to each element
x
a
membership degree
μ
A
(
)
and a non-membership degree
v
A
(
)
x
x
, with the condi-
μ
A
(
),
v
A
(
)
≥
≤
μ
A
(
)
+
v
A
(
)
≤
,
∀
∈
tions
x
x
0 and 0
x
x
1
x
X
. Furthermore,
π
A
(
)
=
−
μ
A
(
)
−
v
A
(
)(
∀
∈
)
x
1
x
x
x
X
is called a hesitancy degree or an intuitionis-
tic index of
x
to
A
.
A
c
={
x
,
v
A
(
x
), μ
A
(
x
)
|
x
∈
X
}
is called the complement of
A
.
In the special case
π
A
(
x
)
=
0, i.e.,
μ
A
(
x
)
+
v
A
(
x
)
=
1, the IFS
A
reduces to a
fuzzy set (Zadeh 1965).
Xu and Yager (2006) called each triple
an intuitionistic
fuzzy value (IFV) (or an intuitionistic fuzzy number (IFN)), and for convenience,
denoted an IFV by
(μ
A
(
x
),
v
A
(
x
), π
A
(
x
))
α
=
(μ
α
,
v
α
,π
α
)
, where
μ
α
,
v
α
≥
0
,μ
α
+
v
α
≤
1
,π
α
=
1
−
μ
α
−
v
(1.2)
α
Each IFV has a physical interpretation, for example, if
α
=
(
0
.
6
,
0
.
3
,
0
.
1
)
, then
μ
α
=
1, which can be interpreted as “the vote for
resolution is 6 in favor, 3 against, and 1 abstention”.
0
.
6,
v
α
=
0
.
3 and
π
α
=
0
.
1.1.2 Methods for Ranking IFVs
In the process of applying IFVs to practical problems, one key step is to rank
IFVs. Clearly, there are two basic principles we should follow in ranking IFVs:
the first is that the IFV which has the larger membership degree and the smaller non-
membership degree should be given priority; the second is that the IFV which has a
smaller hesitancy degree should be ranked first. When we use these two principles
to rank IFVs, the first one is top-priority. If it is not applicable individually, then
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