Information Technology Reference
In-Depth Information
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 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
. 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 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 }
 
Search WWH ::




Custom Search