Methods of Expert Information Formalization Based on Complete Orthogonal Semantic Spaces - Expert Fuzzy Information Processing - page 70

Let us assume that the fuzzy numbers corresponding to COSS terms are Λ -
numbers; and for functions

() ()

R the side condition (1 *) is satisfied (see
paragraph 2.2]. Membership functions of terms

,

L x

X :

with even m

⎧

2
1

1 a ≤ x ≤ x ;

⎪ ⎪
⎨

⎛ −

2
1

⎞

x x

() ⎪ ⎪

μ x = L

⎜ ⎜
⎝

⎟ ⎟
⎠

, x

2
1

< x ≤ x

1
2

;

1

1
2

2
1

x − x

1
2

0 x < x ≤ b ;

⎩

⎧

2
1

0 a ≤ x ≤ x ;

⎪

⎛

1
2

⎞

x − x

⎪

⎜ ⎜
⎝

⎟ ⎟
⎠

2
1

1
2

L , x < x ≤ x ;

1
2

2
1

⎪ ⎪

x − x

()

μ x =

1
2

2
2

1 x < x ≤ x ;

⎨

2

⎪

⎛

2
2

⎞

x − x

⎜ ⎜
⎝

⎟ ⎟
⎠

2
2

1
3

⎪ R , x < x ≤ x ;

1
3

−

2
2

x x

⎪

⎪

1
3

0 x < x ≤ b ;

⎩

⎧ 0 a x x

2

;

≤ ≤

m − 2

⎪

⎛ x

1

x ⎞

−

⎪

⎜ ⎜
⎝

m − 1

⎟ ⎟
⎠

2 1

R , x < x ≤ x ;

m − 2 m − 1

1 2

⎪ ⎪

x − x

m − 1 m − 2

()

μ x =

1 2

1 x < x ≤ x ;

⎨ m − 1

m − 1 m − 1

⎪

⎛

2

⎞

x − x

⎜ ⎜
⎝

⎟ ⎟
⎠

2 1

⎪ L

m − 1

, x < x ≤ x ;

m − 1 m

x

1

x

2

−

⎪ m m − 1

⎪

0 x

1

x b ;

< ≤

⎩

m

⎧

2

0 a ≤ x ≤ x ;

m − 1

⎪ ⎪
⎨

⎛

1

⎞

x − x

() ⎪ ⎪

⎜ ⎜
⎝

⎟ ⎟
⎠

2 1

μ x = L

m

, x < x ≤ x ;

m 1 2 m − 1 m

x − x

m m − 1

1 x

1

< x ≤ b

⎩

m

() ()

With odd m two last membership functions

, interchange their

L x R x

positions.
Constructed COSS reflects knowledge and subjective judgment of the expert
taking part in inquiry.
If deriving of the additional information on the values of membership functions
in the points of universal set which lies between typical intervals of the adjacent
terms is possible, then it is also possible to clarify a form of these functions.