Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 169

z ij ) n × n and the compo-
sition matrix of itself is an intuitionistic fuzzy matrix, it yields

= (

Also since the intuitionistic fuzzy equivalence matrix Z

ma k { min { μ z ik ,μ z kj }} ≥ mi k { max { v z ik , v z kj }} (2.36)

(a) When λ ≤ μ z ij and z ij = 1, also since

λ

ma k { min { λ z ik , λ z kj }} ∈ [ 0 , 1 ] (2.37)

then

ma k { min { λ z ik , λ z kj }} ≤ λ z ij = 1 (2.38)

−

v z ij <λ z ij =

(b) When 1 and 0, also since

λ

mi k { max { v z ik , v z kj }} ≥ v z ij > 1 − λ (2.39)

then, for any k ,wehavemax { v z ik , v z kj } > 1 − λ , i.e., for any k , it can be obtained that

min { λ z ik , λ z kj }= 0 (2.40)

Then

ma k { min { λ z ik , λ z kj }= 0 (2.41)

Thus

ma k { min { λ z ik , λ z kj }≤ λ z ij (2.42)

(c) When μ z ij <λ ≤ 1 − v z ij ,wehave λ z ij = 1 / 2. In this case, if

mi k {

{

v z ik , v z kj }} ≥ v z ij >

− λ

max 1 (2.43)

then by (b), we get

ma k { min { λ z ik , λ z kj }} = 0 (2.44)

Therefore

ma k { min { λ z ik , λ z kj }} ≤ λ z ij (2.45)

If

ma k { min { μ z ik ,μ z kj }} ≤ λ ≤ 1 − mi k { max { v z ik , v z kj }} (2.46)

then

1
2 ,

ma k { min { λ z ik , λ z kj }} = ma k { min { λ z ik , λ z kj }} = λ z ij (2.47)