Image Processing Reference
In-Depth Information
The following cases hold from the GIFOWA operator:
Case 1: If λ
= 1, then GIFOWA reduces to intuitionistic fuzzy ordered
weighted averaging (IFOWA):
GIFOWA
w
(,,
aaa
,
, )(
a
=
wa wa wa
⊕
⊕
⊕ ⊕
wa
)
…
123
n
1
σ
()
1
2
σ
()
2
3
σ
()
3
nn
σ
()
⎛
⎞
⎛
⎞
n
n
(
)
∏
∏
w
(
)
w
(
)
j
⎜
j
⎟
⎟
⎜
⎜
⎟
⎟
==− −
1
1
μ
,
11 11
−− −−
ν
a
a
⎜
σ
()
j
σ
()
j
⎝
⎠
⎝
⎠
j
=
1
j
=
1
⎛
⎞
n
n
∏∏
(
)
w
⎜
⎜
j
w
⎟
⎟
=− −
1
1
μ
,
ν
j
a
a
σ
()
j
σ
()
j
⎝
⎠
j
=
1
j
=
1
T
111 1
, , , ,…
and λ = 1, then GIFOWA reduces to the
intuitionistic fuzzy averaging operator:
=
⎛
⎞
⎟
Case 2: If
w
⎜
nnn n
1
(
)
IFA(
aaa
,
,
,
,
a
)
= ⊕⊕⊕
n
aaa
a
…
123
n
1
2
3
n
Case 3: If λ → ∞, then GIFWA reduces to the intuitionistic fuzzy maximum
operator:
IFMAX
w
(,,
aaa
123
…
=
,
, )max(
a
a
)
n
j
j
An example is given to calculate GIFOWA of the four intuitionistic fuzzy
values.
Example 3.2
Let us consider five intuitionistic fuzzy values
a
1
= (0.2, 0.6),
a
2
= (0.4, 0.3),
a
3
= (0.6, 0.2),
a
4
= (0.7, 0.1),
a
5
= (0.1, 0.7) with
weight vector be
w
= (0.112, 0.236, 0.304, 0.236, 0.112)
T
of
a
j
(
j
= 1, 2, 3, 4, 5)
and λ = 2
Solution
From the intuitionistic fuzzy values, we have
μ
=
02
.,
μ
=
04
.,
μ
=
06
.,
μ
=
07
.,
μ
=
01
.
a
a
a
a
a
1
2
3
4
5
ν
=
06
.,
ν
=
03
.,
ν
=
02
.,
ν
=
01
.,
ν
=
07
.
a
a
a
a
a
1
2
3
4
5
Now, the scores of
a
j
(
j
= 1, 2, 3, 4, 5) are computed as
sa
() .
=−=−
02 06 04
.
., () .
sa
=−= =−=
04 03 01
.
., () .
sa
06 02 04
.
.,
1
2
3
sa
(
)
=−=
07 01 06
.
.
.
, ()
sa
= −=−
01 07 06
.
.
.
4
5
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