Complete Orthogonal Semantic Spaces in Problems of the Fuzzy Regression Analysis - Expert Fuzzy Information Processing - page 193

~

( ) ( )

If bb , then according to

multiplication operation for fuzzy numbers, the α -level set

a ≡ b , b

L b

, is negative fuzzy number + < 0

R R

a ~

~

looks like

[ ]

1 2

B , B

, where

α α

( ) ( ) ( )

2

1

B = bx + 1 − α bx − 1 − α b x − 1 − α b x ;

α 2 R L 2 L R

( ) ( ) ( )

2

2

B = bx − 1 − α bx + 1 − α b x − 1 − α b x .

α 1 L R 1 R L

Then

1

1 1 1

( )

= ∫ θ

1 ~
~

bx + B

1

α d α = bx + bx − b x − b x =

2 α 2 R L 2 L R

a X

6 6 12

0

1 1 1

⎛ ⎞ ⎛ ⎞

= b x + x − b x + x ;

⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

2 R L 2 R

6 6 12

1

1 1 1

( )

2 ~
~

= ∫

2

θ bx + B α d α = bx − bx + b x − b x =

1 α 1 L R 1 R L

a X

6 6 12

0

⎛ 1 ⎞ ⎛ 1 1 ⎞

= b x − x + b x − x

⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

1 L R 1 L

6 6 12

or

1 1 1

⎡ ⎤ ⎡ ⎤

() ()

q q

1 ~
~

θ = b x + − 1 x − b x + − 1 x ;

⎢ ⎣ ⎥ ⎦ ⎢ ⎣ ⎥ ⎦

q M L q M

a X

6 6 12

q q

1 1 1

⎡ ⎤ ⎡ ⎤

() ()

r r

2 ~
~

θ = b x + − 1 x + b x + − 1 x ;

⎢ ⎣ ⎥ ⎦ ⎢ ⎣ ⎥ ⎦

r M R r M

a X

6 6 12

r r

⎨ ⎧

1 b − b ≥ 0 L , q = 1

⎩ ⎨ ⎧

L

q = M =

q

2 b + b < 0 R , q = 2

⎩

R

2 b − b ≥ 0 L , r = 1

⎩ ⎨ ⎧ ⎩ ⎨ ⎧

L

r = M = r

1 b + b < 0 R , r = 2

R

The proposition 6.4 is proved.
According to the proposition 6.4, if ( )

~

j j

L

j
R

a ≡ b , b , b

,

j = 1 m

is nonnegative

j

( )

fuzzy number

j j

L

, then

b + b ≥ 0

1 1 1

( )

⎛ ⎞ ⎛ ⎞

j j

L

j
R

j ji ji

L

j
L

ji ji

L

θ

1

b , b , b = b x − x − b x − x ;

⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

~

~

i

1 1

a X

6 6 12

j j

1 1 1

( )

⎛ ⎞ ⎛ ⎞

j j

L

j
R

j ji ji

R

j
R

ji ji

R

θ

2

b , b , b = b x + x + b x + x .

⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

~

~

i
j

2 2

a X

6 6 12

j

( )

~

( )

j j

L

j
R

a ≡ b , b , b b

j

+ b

j
R

< 0 If ,

j = 1 m

is negative fuzzy number , then

j