Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 5

n

1
2 n

d 1 ( A 1 , A 2 ) =

1 ( | μ A 1 (

x j ) − μ A 2 ( x j ) |+| v A 1 ( x j ) − v A 2 ( x j ) | ) (1.6)

j =

Szmidt and Kacprzyk (2000) extended Eq. ( 1.6 ) by adding the hesitancy degrees:

n

1
2 n

d 2 ( A 1 , A 2 ) =

1 ( | μ A 1 (

x j ) − μ A 2 ( x j ) |+| v A 1 ( x j ) − v A 2 ( x j ) |

j =

+| π A 1 ( x j ) − π A 2 ( x j ) | ) (1.7)

Motivated by Eqs. ( 1.6 ) and ( 1.7 ), respectively, we can calculate the distances
between the IFV α ∗ α and the positive ideal point (i.e., the largest IFV) (Xu and

Yager 2008):

1
2 ( | μ α −

d 1 (α, α ∗ ) = 1 |+| v

α −

0 | )

1
2 (

= 1 − μ α + v α )

1
2 (

= − ( μ α −

1 v α ))

1
2 (

= 1 − S (α)) (1.8)

1
2 ( | μ α −

d 2 (α, α ∗ ) = 1 |+| v α − 0 |+| π α − 0 | )

1
2 (

= 1 − μ α + v α + π α )

1
2 (

= 1 − μ α + v

α +

1 − μ α − v

α )

= − μ α

1 (1.9)

We can infer fromEq. ( 1.8 ) that it has the similar result with Chen and Tan (1994)'s
method, which may produce the same score values even if the two IFVs are different;
while we can infer from Eq. ( 1.9 ) that the result only relies on the value of ,so

it produces the loss of information and cannot distinguish the IFVs which have the
same membership degrees and the different non-membership degrees.
Later, Szmidt and Kacprzyk (2009a, b, 2010) further improved the distance mea-
sure Eq. ( 1.9 ) by considering the hesitancy degrees simultaneously:

μ α

1
2 (

d 2 (α, α ∗ ) L (α) = 1 + π α )

1
2 (

= 1 + π α )( 1 − μ α ) (1.10)