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2.9.2 The Applications of Two Intuitionistic Fuzzy Products
In this subsection, we shall apply the intuitionistic fuzzy triangle product to com-
pare any two alternatives in multi-attribute decision making with intuitionistic fuzzy
information, and then use the intuitionistic fuzzy square product to construct an intu-
itionistic fuzzy similarity matrix which is used as a basis for further investigating
intuitionistic fuzzy clustering technique.
Consider a multi-attribute decision making problem, let
Y
and
G
be as defined
previously. The characteristic (or called attribute value) of each alternative
y
i
under
all the attributes
G
j
(
j
=
1
,
2
,...,
m
)
is represented as an IFS:
y
i
={
G
j
,μ
y
i
(
G
j
),
v
y
i
(
G
j
)
|
G
j
∈
G
}
,
i
=
1
,
2
,...,
n
;
j
=
1
,
2
,...,
m
(2.207)
where
μ
y
i
(
G
j
)
denotes the membership degree of
y
i
to
G
j
and
v
y
i
(
G
j
)
denotes
the non-membership degree of
y
i
to
G
j
. Obviously,
π
y
i
(
G
j
)
=
1
−
μ
y
i
(
G
j
)
−
v
y
i
(
G
j
)
is the uncertainty (or hesitation) degree of
y
i
to
G
j
.Ifwelet
z
ij
=
(μ
ij
,
v
ij
)
=
(μ
y
i
(
G
j
),
v
y
i
(
G
j
))
, which is an IFV, then based on these IFVs
z
ij
(
i
=
1
,
2
,...,
n
;
j
=
1
,
2
,...,
m
)
, we can construct an
n
×
m
intuitionistic fuzzy decision
matrix
Z
=
(
z
ij
)
n
×
m
.
2.9.3 The Application of the Intuitionistic Fuzzy Triangle Product
For the above problem, the characteristic vectors of any two alternatives
y
i
and
y
j
are
expressed as
Z
i
respectively. The
implication degree of the alternatives
y
i
and
y
j
can be calculated with the following
intuitionistic fuzzy triangle product:
=
(
z
i
1
,
z
i
2
,...,
z
im
)
and
Z
j
=
(
z
j
1
,
z
j
2
,...,
z
jm
)
1
m
m
m
1
m
Z
−
1
j
(
Z
i
)
ij
=
1
μ
z
ik
→
z
jk
,
v
z
ik
→
z
jk
(2.208)
k
=
k
=
1
which shows the degree that howmuch the alternative
y
j
is preferred to the alternative
y
i
, where
Z
−
1
Z
−
1
j
denotes the inverse of
Z
j
, which is defined as
(
)
kj
=
(
Z
j
)
jk
=
j
z
jk
,μ
z
ik
→
z
jk
and
v
z
ik
→
z
jk
are respectively as shown in Eq. (
2.199
) for any
k
.
Similarly, we can calculate
1
m
k
=
1
μ
z
jk
→
z
ik
,
m
m
1
m
Z
−
1
i
(
Z
j
)
ji
=
v
z
jk
→
z
ik
(2.209)
k
=
1
which shows the degree that how much the alternative
y
i
is preferred to the alterna-
tive
y
j
.
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