Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 6

Obviously, the distance measure ( 1.9 ) takes into account all three parameters of
IFVs. Szmidt andKacprzyk (2000) showed that the third parameter cannot be omitted
when calculating distance between two IFVs. From Eq. ( 1.10 ), we can know that the
IFV which has the smaller membership degree and the larger hesitancy degree has
the larger value of L in the sense of

the amount and reliability of information (Szmidit and Kacprzyk 2009a, b, 2010).
Especially, from Eq. ( 1.10 ), we can see that if L

(α) . In general, the lower L (α) , the better α

(α) = 0, then we get the largest IFV

α ∗ = ( α ∗ = (

in the sense
of the reliability of the information (we have no information at all, which means the
situation with 100% lack of knowledge, clearly, this result is very different from the
smallest IFVs

1 , 0 , 0 ) ;if L (α) = 1, then we get the “smallest” IFV 0 , 0 , 1 )

derived by the other ranking methods), and the “quality”
measured by the distance from

α ∗ = ( 0 , 1 , 0 )

α ∗ = ( (here, the distance is the biggest).

In addition, in some situations, the formula Eq. ( 1.10 ) is also not enough in ranking
IFVs. Let's see an example below:

1 , 0 , 0 )

Example 1.3 (Zhang and Xu 2012) Let α 1 = ( 0 . 2 , 0 . 3 , 0 . 5 ) and α 2 = ( 0 , 0 . 8 , 0 . 2 ) be

two IFVs. Obviously, the IFVs α 1 and α 2 are intuitively different. But by Eq. ( 1.10 ),

we have

1
2 (

1
2 (

L (α 1 ) = 1 + 0 . 5 ) × 0 . 8 = 0 . 6 , L (α 2 ) = 1 + 0 . 2 ) × 1 = 0 . 6

then L (α 1 ) = L (α 2 ) , and thus, in this case the formula ( 1.10 ) cannot distinguish the

IFVs α 1 and α 2 .

1.1.2.3 The Method for Ranking IFVs by Using the Intuitionistic
Fuzzy Point Operators

Liu and Wang (2007) proposed a new score function by using the intuitionistic fuzzy
point operators (Atanassov 1999; Burillo and Bustince 1996):

n − 1

J n (α) = μ α + σπ α + σ( 1 − σ − θ)π α +···+ σ( 1 − σ − θ) π α , n = 1 , 2 ,...

(1.11)

σ
σ + θ π α

J ∞ (α) = μ α + (1.12)

where ,

the more priority should be given in ranking. From Eqs. ( 1.11 ) and ( 1.12 ), we can
infer that the hesitancy degree of the IFV

σ, θ ∈[ 0 , 1 ] and σ + θ ≤ 1. In this way, the larger the value of J n (α)

α is divided into three parts: σπ α , θπ α

and ( 1 − σ − θ)π α , where σπ α matches μ α ,θπ α matches v α , and ( 1 − σ − θ)π α

is uncertain. In particular, if reduces to a fuzzy value

μ α + σπ α . In practical applications, the decision maker can choose the suitable
parameters

σ + θ = 1, then the IFV α

σ θ

and according to the actual demands.