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() ()
{
}
,,
AA
Ax x xxX
µν
=∈ (2)
Where
[
]
:0,1
A
X
µ
→ and
[
]
:0,1
A
X
ν
→ , with the condition
()
(
)
01
AA
xx
µν
≤+≤,
x
X∀∈
The numbers
()
A
x
µ
and
()
A
x
ν
represent, respectively, the membership degree and non-
membership degree of the element
x
to the set A .
Definition 3. For each IFS
A in
X
, if
(
)
(
)
(
)
1
AAA
x
xx
πµν
=− − ,
x
X∀∈ . (3)
Then
()
A
x
π
is called the degree of indeterminacy of
x
to
A
[1,2].
Definition 4. Let
()
,a
µ
ν
=
%
be an intuitionistic fuzzy number, a score function S of an
intuitionistic fuzzy value can be represented as follows [32]:
(
)
Sa
µ
ν
=
−
%
，
(
)
[
]
1,1Sa∈−
%
. (4)
Definition 5. Let
()
,a
µ
ν
=
%
be an intuitionistic fuzzy number, a accuracy function H of an
intuitionistic fuzzy value can be represented as follows [33]:
(
)
Ha
µ
ν
=
+
%
，
(
)
[
]
0,1Ha∈
%
. (5)
to evaluate the degree of accuracy of the intuitionistic fuzzy value
()
,a
µ
ν
=
%
, where
()
[
]
0,1Ha∈
%
. The larger the value of
(
)
Ha
%
, the more the degree of accuracy of the
intuitionistic fuzzy value a
%
.
As presented above, the score function
S
and the accuracy function H are, respectively,
defined as the difference and the sum of the membership function
()
A
x
µ
%
and the non-
membership function
()
A
x
ν
%
. Hong and Choi [33] showed that the relation between the score
function
S
and the accuracy function H is similar to the relation between mean and variance in
statistics. Based on the score function S and the accuracy function
H , in the following, Xu[13]
give an order relation between two intuitionistic fuzzy values, which is defined as follows:
Definition 6. Let
()
111
,a
µ
ν
=
%
and
(
)
222
,a
µ
ν
=
%
be two intuitionistic fuzzy values,
()
111
sa
µ
ν
=−
%
and
()
222
sa
µ
ν
=−
%
be the scores of
a
%
and
b
%
, respectively, and let
(
)
111
Ha
µ
ν
=+
%
and
()
222
Ha
µ
ν
=+
%
be the accuracy degrees of a
%
and b
%
, respectively, then if
()
()
Sa Sb<
%
%
, then
a
%
is smaller than b
%
, denoted by ab<
%
%
; if
()
(
)
Sa Sb=
%
%
, then
(1) if
()
()
Ha Hb=
%
%
, then a
%
and b
%
represent the same information, denoted by ab=
%
%
; (2) if
()
()
Ha Hb<
%
%
, a
%
is smaller than b
%
, denoted by ab
<
%
%
.
Definition 7. Let
()
()
,1,2,,
jjj
ajn
µν
==
%
L be a collection of intuitionistic fuzzy values,
and let IFWG:
n
QQ→ , if
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