Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 225

d ( y 1 , y 2 ) = d ( y 2 , y 1 ) = ( 0 . 141 , 0 . 784 ), d ( y 1 , y 3 ) = d ( y 3 , y 1 ) = ( 0 . 059 , 0 . 892 )

d 1 ( y 1 , y 4 ) = d 1 ( y 4 , y 1 ) = ( 0 , 0 . 808 ), d ( y 1 , y 5 ) = d ( y 5 , y 1 ) = ( 0 . 123 , 0 . 837 )

d ( y 1 , y 6 ) = d ( y 6 , y 1 ) = ( 0 . 057 , 0 . 892 ), d ( y 2 , y 3 ) = d ( y 3 , y 2 ) = ( 0 . 059 , 0 . 892 )

d ( y 2 , y 4 ) = d ( y 4 , y 2 ) = ( 0 . 033 , 0 . 859 ), d ( y 2 , y 5 ) = d ( y 5 , y 2 ) = ( 0 . 071 , 0 . 918 )

d ( y 2 , y 6 ) = d ( y 6 , y 2 ) = ( 0 . 108 , 0 . 829 ), d ( y 3 , y 4 ) = d ( y 4 , y 3 ) = ( 0 . 071 , 0 . 914 )

d ( y 3 , y 5 ) = d ( y 5 , y 3 ) = ( 0 . 071 , 0 . 929 ), d ( y 3 , y 6 ) = d ( y 6 , y 3 ) = ( 0 , 0 . 894 )

d ( y 4 , y 5 ) = d 1 ( y 5 , y 4 ) = ( 0 . 049 , 0 . 878 ), d ( y 4 , y 6 ) = d ( y 6 , y 4 ) = ( 0 . 057 , 0 . 914 )

d ( y 5 , y 6 ) = d ( y 6 , y 5 ) = ( 0 . 071 , 0 . 829 )

Accordingly, we get the intuitionistic fuzzy distance matrix as follows:

⎛
⎝

⎞
⎠

( 0 , 1 ) ( 0 . 141 , 0 . 784 )( 0 . 059 , 0 . 892 )( 0 , 0 . 808 )( 0 . 123 , 0 . 837 )( 0 . 057 , 0 . 892 )

( 0 . 141 , 0 . 784 ) ( 0 , 1 ) ( 0 . 059 , 0 . 892 )( 0 . 033 , 0 . 859 )( 0 . 071 , 0 . 918 )( 0 . 108 , 0 . 829 )

( 0 . 059 , 0 . 892 )( 0 . 059 . 0 . 892 ) ( 0 , 1 ) ( 0 . 071 , 0 . 914 )( 0 . 071 , 0 . 929 )( 0 , 0 . 894 )

D =

( 0 , 0 . 808 )( 0 . 033 , 0 . 859 )( 0 . 071 , 0 . 914 ) ( 0 , 1 ) ( 0 . 049 , 0 . 878 )( 0 . 057 , 0 . 914 )

( 0 . 123 , 0 . 837 )( 0 . 071 , 0 . 918 )( 0 . 071 , 0 . 929 )( 0 . 049 , 0 . 878 ) ( 0 , 1 ) ( 0 . 071 , 0 . 829 )

( 0 . 057 , 0 . 892 )( 0 . 108 , 0 . 829 )( 0 , 0 . 894 )( 0 . 057 , 0 . 914 )( 0 . 071 , 0 . 829 ) ( 0 , 1 )

(2) Draw the intuitionistic fuzzy graph ( V , D ) with 6 nodes associated to the

samples y i ( i 6) to be clustered and every edge E ij between y i and y j

having the weight d ij , which is an element of the intuitionistic fuzzy distance matrix
D

= 1 , 2 , ...,

d ij ) 6 × 6 and denotes the dissimilarity degree between the samples y i and y j (see
Fig. 2.5 ) (Zhao et al. 2012a).
Step 2 Compute the intuitionistic fuzzy MST of the intuitionistic fuzzy graph by
Kruskal method (Kruskal 1956):
(1) Arrange the edges of ( V

= (

D ) in order from the smallest weight to the largest
one. Because the weight of each edge is an IFV, we can first use the scores and the
accuracy degrees of each IFV in the intuitionistic fuzzy distance matrix to sort all
the intuitionistic fuzzy weights (based on Definition 2.28) as follows:

,

S ( d 12 ) = 0 . 141 − 0 . 784 =− 0 . 643 , S ( d 13 ) = 0 . 059 − 0 . 892 =− 0 . 833

S ( d 14 ) = 0 − 0 . 808 =− 0 . 808 , S ( d 15 ) = 0 . 123 − 0 . 837 =− 0 . 714

S ( d 16 ) = 0 . 057 − 0 . 892 =− 0 . 835 , S ( d 23 ) = 0 . 059 − 0 . 892 =− 0 . 833

S ( d 24 ) = 0 . 033 − 0 . 859 =− 0 . 826 , S ( d 25 ) = 0 . 071 − 0 . 918 =− 0 . 847

S ( d 26 ) = 0 . 108 − 0 . 829 =− 0 . 721 , S ( d 34 ) = 0 . 071 − 0 . 914 =− 0 . 843

S ( d 35 ) = 0 . 071 − 0 . 929 =− 0 . 858 , S ( d 36 ) = 0 − 0 . 894 =− 0 . 894

S ( d 36 ) = 0 . 049 − 0 . 878 =− 0 . 829 , S ( d 46 ) = 0 . 057 − 0 . 914 =− 0 . 857

S ( d 56 ) = 0 . 071 − 0 . 829 =− 0 . 758

Thus

d 36 < d 35 < d 46 < d 25 < d 34 < d 16 < d 13

= d 23 < d 45 < d 24 < d 14 < d 56 < d 26 < d 15 < d 12