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ma
k
{
λ
z
ik
,
λ
z
kj
}=
>
λ
z
ij
1
(2.68)
which contradicts the known condition. Therefore,
max
k
k
1
{
min
{
μ
z
ik
,μ
z
kj
}} ≤
μ
z
ij
(2.69)
=
Similarly, we get
mi
k
{
max
{
v
z
ik
,
v
z
kj
}} ≥
v
z
ij
(2.70)
(c) When
z
ij
=
0, i.e., for any
λ
∈[
0
,
1
]
,wehave1
−
v
z
ij
<λ
. Then by
λ
ma
k
{
min
{
λ
z
ik
,
λ
z
kj
}} ≤
λ
z
ij
=
0
(2.71)
It can be seen that for any
k
, we get
min
{
λ
z
ik
,
λ
z
kj
}=
0
(2.72)
max
{
v
z
ik
,
v
z
kj
}
>
1
−
λ
(2.73)
Considering the arbitrary of
λ
,ityields
μ
z
ij
=
1
−
v
z
ij
=
0, and
ma
k
{
min
{
μ
z
ik
,μ
z
kj
}} ≤
1
−
mi
k
{
max
{
v
z
ik
,
v
z
kj
}} =
0
(2.74)
Thus
Z
satisfies the transitivity property.
From the above analysis, the sufficiency of Theorem 2.7 holds. The proof is
completed.
Definition 2.8
(Zhang et al. 2007) Let
A
i
(
i
=
1
,
2
,...,
n
)
be a collection of IFSs,
z
ij
)
n
×
n
is the intuitionistic fuzzy similarity matrix derived by Eq. (
2.11
),
Z
∗
=
Z
=
(
z
ij
)
n
×
n
is the intuitionistic fuzzy equivalence matrix of
Z
, and
Z
∗
=
(
λ
z
ij
)
n
×
n
is the
(
λ
-cutting matrix of
Z
∗
. If the corresponding elements in both the
i
th line (column)
and the
j
th line (column) of
λ
Z
∗
are equal, then
A
i
and
A
j
are classified into one type.
λ
-cutting matrix
λ
Z
∗
has the transitivity property, then if
A
i
and
A
k
are of the same type, while
A
k
and
A
j
are of the same type, then
A
i
and
A
j
are of the
same type.
On the basis of the above theory, Zhang et al. (2007) introduced a clustering
algorithm for IFSs, which involves the following steps:
Algorithm 2.1
Note
: Since
λ
Step 1
For a multi-attribute decision making problem, let
Y
={
y
1
,
y
2
,...,
y
n
}
be
a finite set of alternatives, and
G
the set of attributes. Suppose
that the characteristic information on the alternative
y
i
is expressed in IFSs:
={
G
1
,
G
2
,...,
G
m
}
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