Information Technology Reference
In-Depth Information
α k w i w j w k
1
p + q + r
1
n
1
p
q
r
μ
α i μ
α j μ
i
,
j
,
k
=
1
r w i w j w k
1
p + q + r
n
1
1
p
q
+
1
(
1
v
)
(
1
v
)
(
1
v
)
α
α
α
i
j
k
i , j , k = 1
r w i w j w k
1
p + q + r
n
1
1
p
q
1
+
(
1
v
)
(
1
v
)
(
1
v
)
α
α
α
i
j
k
i , j , k =
1
r w i w j w k
1
p + q + r
n
1
1
p
q
(
1
v α i )
(
1
v α j )
(
1
v α k )
=
1
i
,
j
,
k
=
1
(1.164)
which completes the proof of the theorem.
Moreover, the GIFWBM also has the following properties (Xia et al. 2012b):
Theorem 1.17 If all
α i
(
i
=
1
,
2
,...,
n
)
are equal, i.e.,
α i
= α
, for all i , then
GIFWBM p , q , r
GIFWBM p , q , r
1 2 ,...,α n ) =
(α,α,...,α) = α
(1.165)
Theorem 1.18 Let
β i
= β i ,
v
β i β i )(
i
=
1
,
2
,...,
n
)
be a collection of IFVs,
if
μ α i
μ β i and v
v
β i , for all i , then
α i
GIFWBM p , q , r
GIFWBM p , q , r
1 2 ,...,α n )
1 2 ,...,β n )
(1.166)
Proof Since
μ α i
μ β i and v
v
β i , for all i , then
α i
p
β i μ
q
β j μ
p
q
r
α k
r
β k
μ
α i μ
α j μ
μ
n
1
k w i w j w k
n
1
k w i w j w k
p
β i μ
q
β j μ
p
α
q
α
r
α
r
β
μ
μ
μ
μ
i
j
i
,
j
,
k
=
1
i
,
j
,
k
=
1
n
1
k w i w j w k
n
1
k w i w j w k
p
β i μ
q
β j μ
p
α
q
α
r
α
r
β
1
μ
μ
μ
1
μ
i
j
i
,
j
,
k
=
1
i
,
j
,
k
=
1
α k w i w j w k
1
1
p
+
q
+
r
n
1
p
q
r
μ
α i μ
α j μ
i
,
j
,
k
=
1
β k w i w j w k
1
1
p
+
q
+
r
n
p
β
q
β
1
r
μ
i μ
j μ
(1.167)
i
,
j
,
k
=
1
 
Search WWH ::




Custom Search