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Hong and Hwang (1995) further considered the case where the set
X
is infinite
and defined another association coefficient of A-IFSs as follows:
X
(μ
A
(
x
)
·
μ
B
(
x
)
+
v
A
(
x
)
·
v
B
(
x
))
dx
c
2
(
A
,
B
)
=
X
μ
(2.158)
)
dx
·
X
μ
)
dx
2
v
A
(
2
v
B
(
A
(
)
+
B
(
)
+
x
x
x
x
where
c
1
(
A
,
B
)
and
c
2
(
A
,
B
)
satisfy the three conditions: (1) 0
≤
c
(
A
,
B
)
≤
1;
(2)
c
. But they cannot guarantee the
necessity in the condition (2). Hong and Hwang (1995) and Mitchell (2004) pointed
out that if association coefficients don't guarantee the necessity in the condition (2),
then some situations where the obtained results are counter-intuitive will appear,
although in most cases the association coefficient may give reasonable result. For
this reason, Xu et al. (2008) proposed an axiomatic definition for the association
measure of IFSs, which is an improved version of Gerstenkorn and Mafiko (1991)
and Hong and Hwang (1995):
(
A
,
B
)
=
1if
A
=
B
; and (3)
c
(
A
,
B
)
=
c
(
B
,
A
)
2
Definition 2.31
(Xu et al. 2008) Let
c
be a mapping
c
:
(
IFS
(
X
))
→
[0
,
1], then the
(
,
)
association coefficient between two IFSs
A
and
B
is defined as
c
A
B
, which has
≤
(
,
)
≤
(
,
)
=
=
the following properties: (1) 0
c
A
B
1; (2)
c
A
B
1 if and only if
A
B
;
and (3)
c
(
A
,
B
)
=
c
(
B
,
A
)
.
Furthermore, Szmidt and Kacprzyk (2000) pointed out that omitting any one of
the three parameters may lead to incorrect results, and therefore, we should take the
three parameters into account when computing the association coefficients between
IFSs.
Based on the two ideas above when constructing an association coefficient
between IFSs, Zhao et al. (2012b) improved Eq. (
2.155
) to a new form, satisfy-
ing all the conditions proposed by Hong and Hwang (1995), Mitchell (2004) and
Szmidt and Kacprzyk (2000):
c
3
(
A
,
B
)
j
=
1
μ
A
(
x
j
)
x
j
)
·
μ
B
(
x
j
)
+
v
A
(
x
j
)
·
v
B
(
x
j
)
+
π
A
(
x
j
)
·
π
B
(
=
j
=
1
μ
x
j
)
·
j
=
1
μ
x
j
)
(2.159)
2
v
A
(
2
2
v
B
(
2
A
(
x
j
)
+
x
j
)
+
π
A
(
B
(
x
j
)
+
x
j
)
+
π
B
(
It is clear that
c
3
(
,
)
takes the third parameter of an IFS (the hesitancy degree)
into consideration, moreover, wewill prove that it also satisfies all the three conditions
of Definition 2.31:
A
B
Proof
Because
A
,
B
∈
IFS(
X
)
, then from the concept of IFS and Eq. (
2.159
), we
know that
c
3
(
A
,
B
)
≥
0. To prove the inequality
c
3
(
A
,
B
)
≤
1, we can use the famous
Cauchy-Schwarz inequality:
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