Methods of Expert Information Formalization Based on Complete Orthogonal Semantic Spaces - Expert Fuzzy Information Processing - page 63

b

⎧

0 0 ≤ x ≤ −

m − 1

;

⎪

a

⎪

m − 1

b 1 − b

⎪

a x + b , −

m − 1

< x ≤

m − 1

;

⎪

m − 1 m − 1

a a

m − 1 m − 1

⎪ ⎪

1 − b b

()

m − 1 m

μ x = 1 < x ≤ − ;

⎨ m − 1

a a

⎪

m − 1 m

b 1 − b

⎪

1 − a x − b , −

m

< x ≤

m

;

⎪

m m

a a

m m

⎪

1 − b

⎪

0

m

< x ≤ 1

⎪ a

⎩

m

if this condition is not satisfied, it is supposed that

b

− a

m

+ b = 1

m − 1 m − 1

a

m

a The unknown coefficient is obtained from the condition

m − 1

v

( )

∑
=

2

F = a y + b − ω → min .

( )

m − 1 m − 1 j + 1 m − 1 m − 1 i

In this case we obtain membership function of a normal triangular number

i 1

⎧

⎛ a + a b ⎞

0 0 ≤ x ≤

⎜ ⎜
⎝

−

m m − 1 m

⎟ ⎟
⎠

;

⎪

a a

⎪

m m − 1

⎪

b ⎛ a + a b ⎞ b

a x 1 a

m

,

⎜ ⎜
⎝

m m − 1 m

⎟ ⎟
⎠

x

m

;

+ + − < ≤ −

⎪ ⎪

m − 1 m − 1

a a a a

()

μ x =

⎨

m m m − 1 m

m − 1

b 1 − b

⎪

1 − a x − b , −

m

< x ≤

m

;

⎪

m m

a a

m m

⎪

1 − b

⎪

0

m

< x ≤ 1 .

⎪

a

⎩

m

Definition of the left boundary of membership function of term yields an

X

m − 1

() μ x

X unambiguous definition of its right boundary of term , i.e. at m − 2

m − 2

b 1 − b

m − 1

x

m − 1

− < ≤

a a

m − 1 m − 1

() μ

x = 1 − a x − b .

m − 2 m − 1 m − 1

()

μ x

The membership functions ; is constructed as described above.

l = 3 m − 2

l