Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 9

( )

Fig. 1.1 The results derived by Eq. ( 1.15 ) and their contours

But sometimes we will face the situations where the considered two IFVs have
the same value derived by Eq. ( 1.15 ), which can be shown in Fig. 1.1 (Zhang and Xu
2012).
In the bottom of Fig. 1.1 , we have given the contours of the results derived by
Eq. ( 1.15 ), from which we can get the following conclusions:

Theorem 1.1 (Zhang and Xu 2012)

(1) If the membership degree of an IFV is the same as the non-membership degree,
then the value is 1

2 .
(2) If the membership degree of an IFV is larger than the non-membership degree,
then the

ϑ

value is larger than 1

2 .
(3) If the membership degree of an IFV is smaller than the non-membership degree,
then the

ϑ

value is smaller than 1 ϑ

2 .

Proof Let α = (μ α , v α ,π α ) be an IFV, then we calculate the ϑ value of α :

1 − v 1 − v 1 − v

α α α

ϑ(α) = =

α ) =

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

α )

which should be discussed in three cases:

μ α =

Case 1 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) =