Intuitionistic Fuzzy Set and Type II Fuzzy Set - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 9

Also for Yager's fuzzy complement,

() /

1 α

⎣

α

⎦

Nx x (()) μ = ϕ μ (()) = 1 − μ x

λ

At α > 1, the condition for intuitionistic fuzzy generator does not hold.
At these conditions, it follows that μ( x ) + N (μ( x )) > 1, which is not true.
For intuitionistic fuzzy generator, the condition of λ, α is changed.
For Yager-type intuitionistic fuzzy generator, the condition is

αα 1 /

ϕμ (()) x =− [ 1 μ ( )] , x 0 << α 1

and for Sugeno type,

1 −

+⋅

μ
λμ

()
() ,

x

ϕμ (()) x = λ ≥ 0

1 x

Thus, with the help of the Sugeno-type intuitionistic fuzzy complement, IFS
becomes

⎧ ⎫

⎪
⎪

( 1 −

+⋅

μ
λμ

( )) x

⎪
⎪

IFS A

Ax = , μ ( ), x

x xX

∈

λ A

( 1 ( ))

A

with hesitation degree

( 1 −

+⋅

μ
λμ

( )) x

A

π () x =− − 1 μ () x

A A

( 1 ( )) x

A

Since the denominator, 1 + λ ⋅ μ A ( x ), in the non-membership term (1 − μ A ( x ))/
(1 + λ ⋅ μ A ( x )) is greater than 1, the non-membership term is less than 1 − μ( x )
for all x ∈ X .
Likewise, with Yager's intuitionistic fuzzy generator, IFS becomes

IFS αα 1 /

Ax = {, (),( μ x 1 − μ ())| } x xX ∈

λ A A

Chaira [7] also suggested an intuitionistic fuzzy generator as follows:

1
11

x
e x

() 1 () x

−
−−

μ −

+−

μ

Nx (()) μ = =

) () ,

λ > 0 (1.10)

λ λ

( ) () μ 1 ( e 1 μ x