Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 170

From the known condition, it follows that

ma k { min { μ z ik ,μ z kj }} ≤ μ z ij <λ (2.48)

which indicates that the case

λ ≤ ma k { min { μ z ik ,μ z kj }} (2.49)

does not exist. Therefore, when μ z ij <λ ≤ 1 − v z ij ,wehave

ma k { min { λ z ik , λ z kj }} ≤ λ z ij (2.50)

Hence, Z λ = ( λ z ij ) n × n satisfies the transitivity property.
(Sufficiency)

(1) (Reflexivity) Since λ z ii = 1, then for any λ ∈[ 0 , 1 ] , λ ≤ μ z ii , and then let

λ = .

(2) (Symmetry) Since for any i

1. Then z ii = ( 1 , 0 )

, = λ z ki , if there exists z ik =

k , z ik z ki , i.e.,

λ

μ z ik = μ z ki or v z ij = μ z ij <μ z ji , and let

v z ji , without loss of generality, suppose that

λ = (μ z ij + μ z ji )/ 2, then μ z ij <λ<μ z ji , z ik = 0or1 / 2, and z ki = 1, z ik = λ z ki ,

λ λ λ

which contradicts the known condition. Therefore, Z = ( z ij ) n × n is symmetry.

(3) (Transitivity) Since for any i , j ,wehave

ma k { min { λ z ik , λ z kj }} ≤ λ z ij (2.51)

and each element in Z takes its value from { 0 , 1 / 2 , 1 } . Then

λ

(a) When z ij = 1, for any λ ∈[ 0 , 1 ] ,wehave μ z ij ≥ λ , taking λ = 1, it can be

λ

obtained that μ z ij = 1 and v z ij = 0. Consequently,

ma k { min { μ z ik ,μ z kj }} ≤ μ z ij , mi k { max { v z ik , v z kj }} ≥ v z ij (2.52)

(b) When λ z ij = 1 / 2, for any λ ∈[ 0 , 1 ] ,wehave μ z ij <λ ≤ 1 − v z ij . Also since

1
2

ma k { min { λ z ik , λ z kj }} ≤ (2.53)

then

0or 1
2

ma k {

{ λ z ik , λ z kj }} =

min (2.54)

Thus, for any k , we get

{ λ z ik , λ z kj }=

min 0 (2.55)