Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 171

or there exists a positive integer s , such that

1
2

min { λ z is , λ z sj }= (2.56)

Case 1 If for any k ,wehave

min { λ z ik , λ z kj }= 0 (2.57)

Then for any λ ∈[ 0 , 1 ] ,ityields

max { v z ik , v z kj } > 1 − λ (2.58)

hence

mi k { max { v z ik , v z kj }} ≥ v z ij > 1 − λ (2.59)

Considering the arbitrary of λ , when λ tends to be infinitely small, we get

ma k { min { μ z ik ,μ z kj }} ≤ 1 − mi k { max { v z ik , v z kj }} = 0 (2.60)

As a result,

ma k { min { μ z ik ,μ z kj }} ≤ μ z ij (2.61)

Case 2 If there exists a positive integer k 1 , such that

1
2

min { λ z ik 1 , λ z k 1 j }= (2.62)

and for any k = k 1 ,let

min { λ z ik , λ z kj }= 0 (2.63)

Then according to Case 1, we have

k 1 {

{

v z ik , v z kj }} ≥

min
k

max v z ij (2.64)

=

max
k k 1 {

min { μ z ik ,μ z kj }} ≤ μ z ij (2.65)

=

and when k = k 1 , suppose that

min { μ z ik 1 ,μ z k 1 j }= μ z ik 1 ,μ z ik 1 >μ z ij (2.66)

Then let λ = (μ z ij + μ z ik 1 )/ 2, and thus, μ z ij ≤ λ ≤ μ z ik 1 . Accordingly,

z ik 1 = λ z k 1 j = 1 , min { λ z ik 1 , λ z k 1 j }= 1 (2.67)

λ