Information Technology Reference
In-Depth Information
n
(3) Transitivity: Z 2
Z , i.e.,
1 (
z ik
z kj )
z ij ,
i
,
j
=
1
,
2
,...,
n .
k
=
Then Z is called an intuitionistic fuzzy equivalence matrix.
In order to save computation, motivated by the idea of Wang (1983), we have the
following conclusion:
Theorem 2.6 (Zhang et al. 2007) Let Z be an intuitionistic fuzzy similarity matrix.
Then after the finite times of compositions:
Z 2 k
Z 2
Z 4
Z
→ ··· →
→ ···
There must exist a positive integer k such that Z 2 k
Z 2 ( k + 1 ) , and Z 2 k
=
is an intuition-
istic fuzzy equivalence matrix.
Definition 2.6 (Zhang et al. 2007) Let Z
= (
z ij ) n × n be an intuitionistic fuzzy simi-
larity matrix, where z ij = z ij ,
v z ij )
, i
,
j
=
1
,
2
,...,
n . Then Z λ = ( λ z ij ) n × n is called
the
λ
-cutting matrix of Z , where
0
,
if
λ>
1
v z ij ,
1
2 ,
z ij =
(2.33)
if
μ z ij
1
v z ij ,
λ
1
,
if
μ z ij λ.
Definition 2.7 (Wang 1983) If the matrix Z
= ( ˙
z ij ) n × n satisfies the following con-
ditions:
(1) Reflexivity:
z ii =
˙
1, i
=
1
,
2
,...,
n , and for any
z ij ∈[
˙
0
,
1
]
, i
,
j
=
1
,
2
,...,
n .
(2) Symmetry:
z ji .
(3) Transitivity: ma k {
˙
z ij
min
z ik , ˙
z kj }} ≤ ˙
z ij , for all i
,
j
=
1
,
2
,...,
n .
Then Z is called a fuzzy equivalence matrix.
Theorem 2.7 (Zhang et al. 2007) Z
= (
z ij ) n × n is an intuitionistic fuzzy equivalence
matrix if and only if its
λ
-cutting matrix Z
λ = ( λ
z ij ) n × n is a fuzzy equivalence matrix,
where z ij = z ij ,
v z ij )
, i
,
j
=
1
,
2
,...,
n .
Proof
(Necessity)
(1)
(Reflexivity) Since z ii = (
1
,
0
)
,
λ ∈[
0
,
1
]
, then
λ μ z ii =
1, λ z ii =
1.
(Symmetry) Since z ij =
μ z ij = μ z ji , v z ij =
v z ji , thus λ z ij = λ z ji .
(2)
z ji , i.e.,
= (
z ij ) n × n is an intuitionistic fuzzy equivalence matrix,
(3)
(Transitivity) Since Z
we have
ma k {
min
{ μ z ik z kj }} ≤ μ z ij
(2.34)
mi k {
max
{
v z ik ,
v z kj }} ≤
v z ij
(2.35)
 
Search WWH ::




Custom Search