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()
()
⎡
q
q
⎤
−
1
−
1
1
1
θ
=
b
x
z
+
x
z
+
z
x
+
x
z
−
⎢
⎣
⎥
⎦
~
~
~
q
q
q
M
q
M
M
M
a
X
6
6
12
q
q
q
q
()
()
⎡
q
q
⎤
1
−
1
−
1
1
−
b
x
z
+
x
z
+
z
x
+
x
z
;
⎢
⎣
⎥
⎦
L
q
q
q
M
q
M
M
M
6
12
12
20
q
q
q
q
()
()
⎡
r
r
⎤
−
1
−
1
1
2
θ
=
b
x
z
+
x
z
+
z
x
+
x
z
+
⎢
⎣
⎥
⎦
~
~
~
r
r
r
M
r
M
M
M
a
X
Z
6
6
12
r
r
r
r
()
()
⎡
r
r
⎤
1
−
1
−
1
1
+
b
x
z
+
x
z
+
z
x
+
x
z
,
⎢
⎣
⎥
⎦
R
r
r
r
M
r
M
M
M
6
12
12
20
r
r
r
r
1
b
−
b
≥
0
L
,
q
=
1
⎩
⎨
⎧
⎩
⎨
⎧
L
q
=
0
M
=
2
q
2
b
+
b
<
R
,
q
=
R
2
b
−
b
≥
0
L
,
r
=
1
⎨
⎧
⎩
⎨
⎧
L
r
=
0
M
=
2
r
1
b
+
b
<
R
,
r
=
⎩
The proof
. Let us consider a fuzzy number which is product of fuzzy number
(
R
~
~
)
(
)
X
≡
x
,
x
,
x
L
x
,
Z
≡
z
,
z
,
z
L
z
,
by fuzzy number
, and let us denote it with
1
2
R
1
2
R
. Let us write out
α
-level sets
~
and
~
[
~
~
~
G
=
X
×
Z
]
[
(
)
(
)
]
1
2
X
=
X
,
X
=
x
−
1
−
α
x
,
x
+
1
−
α
x
;
α
α
α
1
L
2
R
[
(
)
(
)
]
[
]
Z
=
Z
1
,
Z
2
=
z
−
1
−
α
z
,
z
+
1
−
α
z
.
α
α
α
1
L
2
R
As
~
and
~
are nonnegative fuzzy numbers, the
-level set of fuzzy number
G
α
[
]
1
,
2
G
=
G
G
looks like
, where
(
α
α
α
)
(
)
(
)
2
1
G
=
x
z
−
1
−
α
x
z
−
1
−
α
z
x
+
1
−
α
x
z
;
α
1
1
1
L
1
L
L
L
(
)
(
)
(
)
2
1
G
=
x
z
+
1
−
α
x
z
+
1
−
α
z
x
+
1
−
α
x
z
.
α
2
2
2
R
2
R
R
R
a
~
[
]
~
~
(
)
1
,
2
If
a
≡
b
,
b
L
b
,
is nonnegative number, the
α
-level set
looks like
G
G
,
R
α
α
where
[
]
−
(
)
(
)
(
)
2
G
1
=
b
x
z
−
1
−
α
x
z
−
1
−
α
z
x
+
1
−
α
x
z
α
1
1
1
L
1
L
L
L
[
]
(
)
(
)
(
)
(
)
2
2
3
−
b
1
−
α
x
z
−
1
−
α
x
z
−
1
−
α
z
x
+
1
−
α
x
z
;
1
1
1
L
1
L
L
L
[
]
+
(
)
(
)
(
)
2
2
G
=
b
x
z
+
1
−
α
x
z
+
1
−
α
z
x
+
1
−
α
x
z
α
2
2
2
R
2
R
R
R
[
]
(
)
(
)
(
)
(
)
2
2
3
+
b
1
−
α
x
z
+
1
−
α
x
z
+
1
−
α
z
x
+
1
−
α
x
z
.
R
2
2
2
R
2
R
R
R
Then
`
(
)
1
1
1
⎛
⎞
=
∫
θ
1
bx
z
+
G
1
α
d
α
=
b
x
z
−
x
z
−
z
x
+
x
z
−
⎜
⎝
⎟
⎠
~
~
~
1
1
α
1
1
1
L
1
L
L
L
a
X
6
6
12
0