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v
min
(
z
ik
→
z
jk
,
z
jk
→
z
ik
)
(2.210)
Equation (
2.210
) has the following desirable properties (Wang et al. 2012):
n
μ
min
(
z
ik
→
z
jk
,
z
jk
→
z
ik
)
,
Z
−
1
j
(
y
i
,
y
j
)
=
(
Z
i
)
ij
=
sim
min
1
≤
k
≤
(1)
sim
(
y
i
,
y
j
)
is an IFV.
(2)
sim
(
y
i
,
y
i
)
=
(
1
,
0
)(
i
=
1
,
2
,...,
n
)
.
(3)
sim
(
y
i
,
y
j
)
=
sim
(
y
j
,
y
i
)(
i
,
j
=
1
,
2
,...,
n
)
.
Proof
(1) Let's prove that
sim
(
y
i
,
y
j
)
is an IFV:
Since the results of
z
ik
→
z
jk
and
z
jk
→
z
ik
are all IFVs as proven previously,
then
μ
min
(
z
ik
→
z
jk
,
z
jk
→
z
ik
)
,
v
min
(
z
ik
→
z
jk
,
z
jk
→
z
ik
)
is an IFV, for any
k
.
(2) Since
n
μ
min
(
z
ik
→
z
ik
,
z
ik
→
z
ik
)
,
v
min
(
z
ik
→
z
ik
,
z
ik
→
z
ik
)
Z
−
1
i
sim
(
y
i
,
y
i
)
=
(
Z
i
)
ii
=
min
1
≤
k
≤
(
y
i
,
y
i
)
=
(
,
)
and with Definition 2.4, we can easily know that
sim
1
0
.
(3) Since
μ
min
(
z
ik
→
z
jk
,
z
jk
→
z
ik
)
,
v
min
(
z
ik
→
z
jk
,
z
jk
→
z
ik
)
Z
−
1
j
sim
(
y
i
,
y
j
)
=
(
Z
i
)
ij
=
min
1
≤
k
≤
n
n
μ
min
(
z
jk
→
z
ik
,
z
ik
→
z
jk
)
,
v
min
(
z
jk
→
z
ik
,
z
ik
→
z
jk
)
=
min
1
≤
k
≤
Z
−
1
i
=
(
Z
j
)
ji
=
sim
(
y
j
,
y
i
)
then
sim
(
y
i
,
y
j
)
=
sim
(
y
j
,
y
i
)(
i
,
j
=
1
,
2
,...,
n
)
.
From the analysis above, we can know that Eq. (
2.210
) satisfies the conditions
of intuitionistic fuzzy similarity relation, and thus, we can use it to construct an
intuitionistic fuzzy similarity matrix.
2.9.5 A Direct Intuitionistic Fuzzy Cluster Analysis Method
After we have gotten an intuitionistic fuzzy similarity matrix
R
, with this method,
there is no need to seek for its equivalent matrix before doing cluster analysis. Starting
with an intuitionistic fuzzy similarity matrix, we may get the wanted cluster analysis
results as with an intuitionistic fuzzy equivalent matrix, which has been proven
strictly (Luo 1989). Luo (1989) introduced a direct method for clustering fuzzy sets
which can only consider the membership degrees of fuzzy sets. In this section, we
shall introduce a direct intuitionistic fuzzy cluster analysis method, which can take
into account both the membership degrees and the non-membership degrees of IFVs
under intuitionistic fuzzy environments. The method involves the following steps
(Wang et al. 2012):
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