Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 173

y i ={ G j ,μ y i ( G j ), v y i ( G j ) | G j ∈

} , = , ,...,

G j 1 2 m (2.75)

where μ y i ( G j ) indicates the degree that the alternative y i satisfies the attribute

G j , v y i ( G j ) indicates the degree that the alternative y i does not satisfy the attribute

G j ,π y i ( indicates the uncertainty degree that the alterna-

tive y i to the attribute G j . By the intuitionistic fuzzy similarity degree formula ( 2.11 ),
we establish the intuitionistic fuzzy similarity matrix Z

G j ) = 1 − μ y i ( G j ) − v y i ( G j )

= ( z ij ) n × n , where

1

λ d ∗ ( λ d ∗ (

z ij = ϑ( y i , y j ) =

−

y i , y j ), y i , y j )

, , = , ,...,

i j 1 2 n (2.76)

β 1 | μ y i (

G k ) | λ + β 2 | G k ) | λ d ∗ ( y i , y j ) = min

k

G k ) − μ y j ( v y i ( G k ) − v y j (

G k ) | λ + β 3 | π y i ( G k ) − π y j ( (2.77)

mi k β 1 | μ y i ( G k ) | λ + β 2 | G k ) | λ d ∗ ( y i , y j ) = G k ) − μ y j ( v y i ( G k ) − v y j (

G k ) | λ

+ β 3 | π y i ( G k ) − π y j ( (2.78)

ma k β 1 | μ y i (

d ∗ ( G k ) | λ + β 2 | G k ) | λ y i , y j ) = G k ) − μ y j ( v y i ( G k ) − v y j (

G k ) | λ

+ β 3 | π y i ( G k ) − π y j ( (2.79)

and λ, β 1 ,β 2 ,β 3 are the predefined parameter, λ ≥ 1 ,β i ∈[ 0 , 1 ] , i = 1 , 2 , 3, and

i = 1 β i =

1.
Step 2 Check whether the intuitionistic fuzzy matrix Z is the intuitionistic fuzzy
equivalence matrix or not (i.e., check Z 2

⊆ Z or not); otherwise, do the composition

Z 2 l + 1 . Then Z 2 l is the
derived intuitionistic fuzzy equivalence matrix. For the sake of convenience, without
loss of generality, let Z ∗

Z 2 k , until Z 2 l

Z 2 Z 4

→ → → ··· → → ··· =

operation: Z

z ij ) n × n be the derived intuitionistic fuzzy equivalence

= (

matrix, where z ij = (μ z ij ,

v z ij ), i , j = 1 , 2 ,..., n .

Step 3 For the given confidence level λ ,byEq.( 2.33 ), we calculate the λ -cutting

matrix λ Z ∗ = ( λ z ij ) n × n of the intuitionistic fuzzy equivalence matrix Z ∗ .
Step 4 According to the -cutting matrix λ Z ∗ and Definition 2.8, we cluster the λ

given alternatives.

Example 2.1 (Zhang et al. 2007) Consider a car classification problem. There are
five new cars y i ( to be classified in the Guangzhou car market in

Guangdong, China, and six attributes: (1) G 1 : Fuel economy; (2) G 2 : Aerod. Degree;
(3) G 3 : Price; (4) G 4 : Comfort; (5) G 5 : Design; and (6) G 6 : Safety, are taken into
consideration in the classification problem. The characteristics of the ten new cars
y i (

i = 1 , 2 ,..., 5 )

i = 1 , 2 ,..., 5 ) under the six attributes G j ( j = 1 , 2 ,..., 6 ) are represented by

the IFSs, shown in Table 2.1 (Zhang et al. 2007).