Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 10

μ α >

Case 2 v α

1 − v 1 − v 1

2

α α

⇒

α ) > α ) =

( 1 − μ α ) + ( 1 − v ( 1 − v

α ) + (

1 − v

1
2

⇒ ϑ(α) >

Case 3 μ α < v

α

1 − v α 1 − v α 1

2

⇒

v α ) < v α ) = ( 1 − μ α ) + ( 1 − ( 1 − v α ) + ( 1 −

1
2

⇒ ϑ(α) <

μ α and non-
membership degree v α change in specific ranges, whichwe can see from the contours.
In the following, we shall demonstrate it:
Let

In fact, for each value derived by Eq. ( 1.15 ), its membership degree

α = (μ α , v α ,π α ) be an IFV, then

(1) If ϑ(α) ≤ 0 . 5, then

(a) When π α = 0, μ α and v α get the maximums, i.e.,

π α = 0 ⇒ μ α + v α = 1

and

1 − v

α

ϑ(α) = = 1 − v

α ⇒

v

α =

1 − ϑ(α) ⇒ μ α = ϑ(α)

1 + π α

(b) When μ α = 0, v gets the minimum, i.e.,

α

1 − v 1 − v 1 − v 1 − 2 ϑ(α)

α α α

ϑ(α) = = = ⇒ v

α =

1 + π α 1 + 1 − μ α − v 2 − v 1 − ϑ(α)

α α

then the ranges of μ α and v α are as follows, respectively:

1 − 2 ϑ(α)

0 ≤ μ α ≤ ϑ(α), ≤ v α ≤ 1 − ϑ(α)

1 − ϑ(α)

(2)If0 . 5 <ϑ(α) ≤ 1, then

(a) When π α = 0 ,μ α and v α get the maximums, i.e., μ α = ϑ(α) and

v α = ;

(b) When v α =

1 − ϑ(α)

0 ,μ α gets the minimum, i.e.,

1 − v α 1 − v α 1 2 ϑ(α) − 1

ϑ(α) = = = ⇒ μ α =

1 + π α 1 + 1 − μ α − v α 2 − μ α ϑ(α)