Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 7

1.1.2.4 The Method for Ranking IFVs by Using the Similarity
Measure and the Accuracy Degree

Szmidt and Kacprzyk (2004) proposed a similarity measure of IFSs:

d ( A 1 , A 2 )

ϑ 1 ( A 1 , A 2 ) = (1.13)

A 2 ) d ( A 1 ,

A 2 ) where the distances d ( A 1 , A 2 ) and d ( A 1 , can be calculated by using Eq. ( 1.6 )

or Eq. ( 1.7 ).
Eq. ( 1.13 ) not only considers the distance between two IFSs but also reflects if
the compared IFSs are more similar or more dissimilar to each other. However, in
practical applications, it is generally expected that the degree of similarity would
describe to what extent the IFSs are similar, so the most similar IFSs should have the
largest degree of similarity (Hwang and Yoon 1981), which cannot be reflected by
Eq. ( 1.13 ). To solve this issue, Xu and Yager (2009) improved Szmidt and Kacprzyk
(2004)'s result according to Hwang and Yoon (1981)'s idea of technique for order
preference by similarity to ideal solution (TOPSIS) and developed the following
similarity measure:

A 2 ) d ( A 1 , A 2 ) d ( A 1 ,

ϑ 2 ( A 1 , A 2 ) = 1 −

A 2 ) =

(1.14)

A 2 ) d ( A 1 , A 2 ) + d ( A 1 , d ( A 1 , A 2 ) + d ( A 1 ,

The similarity measure ( 1.14 ) can not only overcome the disadvantages of
Eq. ( 1.13 ), but also examine if the compared values are more similar or more dissim-
ilar to each other so as to avoid drawing conclusions about strong similarity between
two IFSs on the basis of the small distances between these sets (Xu and Yager 2009).
Motivated by Eq. ( 1.14 ) and the idea of positive ideal point, Zhang and Xu (2012)
proposed a new method for ranking IFVs. First we give the definition of similarity
function

ϑ :

Definition 1.2 (Zhang and Xu 2012) Let α = (μ α , v α ,π α ) be an IFV, then the

similarity function ϑ about this IFV is defined as:

))
d 2 (α, ( 1 , 0 , 0 )) + d 2 (α, ( 0 , 1 , 0 ))

d 2 (α, ( 1 , 0 , 0

ϑ(α) = 1 −

1
2 ( | μ α −

1 |+| v α − 0 |+| π α − 0 | )

= 1 −

1
2 ( | μ α − 1 |+| v α − 0 |+| π α − 0 | ) +

1
2 ( | μ α − 0 |+| v α − 1 |+| π α − 0 | )

1 − μ α + v α + π α
2

= 1 −

+ 2 π α

1 + π α + μ α − v α

=

2 + 2 π α