Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 34

) =[ μ − A ( ), μ + A (

v − A ( v + A (

μ A ( ) ]

v A (

˜ ) =[ ), ) ]

where x x x and x x x are interval ranges,

in A is called an interval-valued intuitionistic
fuzzy value (IVIFV) or called an interval-valued intuitionistic fuzzy number (IVIFN)
(Xu andChen 2007b), where

and each triple ( μ A ( x ), ˜ v

A (

x ), π A ( x ))

) =[ π A ( ), π A ( π A ( − μ + A ( v + A ( π A ( x x x ) ] , x ) = 1 x ) − x )

π A ( − μ − A ( v − A ( and x ) = 1 x ) − x ) , for all x ∈ X . For convenience, we denote an IVIFV

by are the membership degree range, the

non-membership degree range and the hesitancy degree (or the uncertainty degree)
range respectively, and satisfy

α = ( μ α , ˜ v

α , π α )

, where μ α , ˜ v and π α

α

μ α =[ μ α ,μ α ]⊂[ v α , v α ]⊂[ ] ,μ α + v α ≤ 0 , 1 ] , ˜ v

α =[

0 , 1 1 ,

π α =[ π α ,π α ]⊂[ ] ,π α = − μ α −

v α ,π α =

− μ α −

v α

,

0 1 1 1 (1.65)

Similar to the comparison method of IFVs, Xu and Chen (2007b) defined the
score function and the accuracy degree of IVIFS α as follows:

1
2 (μ α −

v α + μ α − v α ) S ( α) = (1.66)

1
2 (μ α +

v α + μ α + v α ) H ( α i ) = (1.67)

and they gave the following definition to compare two IVIFVs:

Definition 1.4 (Xu and Chen 2007b) Let α i = ( μ α i , ˜ v

α i , π α i )(

i = 1 , 2 ) be any two

IVIFVs, S ( α i )( i = 1 , 2 ) and H ( α i )( i = 1 , 2 ) the scores and accuracy degrees of

α i ( i = 1 , 2 ) respectively, then

(1) If S ( α 1 )> S ( α 2 ) , then α 1 is larger than α 2 , denoted by α 1 > α 2 .

(2) If S ( α 1 ) = S ( α 2 ) , then

(a) If H ( α 1 ) = H ( α 2 ) , then there is no difference between α 1 and α 2 , denoted

α 1 ∼ α 2 .
(b) If H

by

( α 1 )> H ( α 2 ) , then α 1 is larger than α 2 , denoted by α 1 > α 2 .

Later, Wang et al. (2009) gave another two indices called the membership uncer-
tainty index:

g 1 ( α) = μ α + v α − μ α − v α (1.68)

and the hesitation uncertainty index:

g 2 ( α) = μ α + v α − μ α − v α

(1.69)

respectively to supplement the ranking procedure in Definition 1.4. In the case where
S ( α 1 ) = S ( α 2 ) and H ( α 1 ) = H ( α 2 ) , one can further consider these two indices:

(1) If g 1 ( α 1 )<g 1 ( α 2 ) , then α 1 is larger than α 2 , denoted by α 1 > α 2 .