Information Technology Reference
In-Depth Information
r
t
and
(λ,δ)
r
j
are non-orthogonal, then
(λ,δ)
r
k
and
(λ,δ)
r
j
are non-orthogonal. In this
case,
y
k
and
y
j
are clustered into one class.
(3) (Heterogeneous principle) If
r
k
and
r
j
are non-orthogonal,
(2) (Indirect clustering principle) If
(λ,δ)
(λ,δ)
(λ,δ)
(
0
,
1
)
;
r
k
·
(λ,δ)
r
j
=
(2.133)
(λ,δ)
(
0
,
0
),
r
k
is orthogonal to
r
j
. In this case,
y
k
and
y
j
do not belong to one class.
then
(λ,δ)
(λ,δ)
Theorem 2.17
(Xu et al. 2011) (Dynamic clustering theorem) If the objects
y
k
and
y
j
are clustered into one class by the orthogonal principle under the confidence level
(λ
1
,δ
1
)
, then when
λ
2
<λ
1
,δ
2
>δ
1
,
y
k
and
y
j
are still clustered into one class
under the confidence level
(λ
2
,δ
2
)
.
Proof
Since the objects
y
k
and
y
j
are clustered into one class by the orthogonal
principle under the confidence level
r
k
and
(λ
1
,δ
1
)
, then two column vectors
(λ
,δ
)
1
1
r
j
of
(λ
1
,δ
1
)
-cutting matrix
(λ,δ)
R
are non-orthogonal, i.e., the inner product
(λ
1
,δ
1
)
r
k
and
r
j
is equal to
of
(
1
,
0
)
or
(
1
,
1
)
. Suppose that in the
i
th line, there
(λ
,δ
)
(λ
,δ
)
1
1
1
1
exist
μ
ik
>λ
1
and
μ
ij
>λ
1
. Then
μ
ik
=
1 and
μ
ij
=
1, and if
λ
2
<λ
1
,δ
2
>
λ
λ
1
1
δ
1
,μ
ik
>λ
2
and
μ
ij
>λ
2
, then
μ
ik
=
1 and
μ
ij
=
1 under the confidence level
λ
λ
2
2
. Thus, two column vectors
(λ
2
,δ
2
)
r
k
and
(λ
2
,δ
2
)
r
j
are also non-orthogonal,
i.e.,
y
k
and
y
j
are clustered into one class.
(λ
2
,δ
2
)
Based on the orthogonal principle, Xu et al. (2011) presented an orthogonal algo-
rithm for clustering intuitionistic fuzzy information:
Algorithm 2.6
Step 1
Let
Y
= {
y
1
,
y
2
,...,
y
n
}
and
G
= {
G
1
,
G
2
,...,
G
m
}
be defined as in
Sect.
2.1
, and assume that the characteristics of the objects
y
i
(
i
=
1
,
2
,...,
n
)
with
respect to the attributes
G
j
(
are represented as in Eq. (
2.131
).
Step 2
Construct the intuitionistic fuzzy similarity matrix
R
j
=
1
,
2
,...,
m
)
=
(
r
ij
)
n
×
n
by using
Eqs. (
2.125
) and (
2.131
), where
r
ij
is an IFV, and
r
ij
=
(
u
ij
,
v
ij
)
=
R
(
y
i
,
y
j
)
,
i
=
1
,
2
,...,
n
;
j
=
1
,
2
,...,
m
.
Step 3
Determine the
(λ, δ)
-cutting matrix
(λ,δ)
R
=
(
(λ,δ)
r
ij
)
n
×
n
of
R
=
(
r
ij
)
n
×
n
by using Eq. (
2.130
) under the confidence level
.
Step 4
Calculate the inner products of the column vectors of the
(λ, δ)
-cutting
matrix
(λ,δ)
R
, and then check whether each pair of the column vectors are orthogonal
or not.
Step 5
Cluster the objects
y
i
(
(λ, δ)
i
=
1
,
2
,...,
n
)
by the orthogonal principles.
Example 2.5
(Xu et al. 2011) In the supply chain management, supplier strategies
are to formulate the different levels of strategies considering the relationships among
the suppliers. From the procurement point of view, the supplier classification is to
Search WWH ::
Custom Search