Intuitionistic Fuzzy IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 37

2.5 Level Operators Over EIFIMs

K ∗ , L ∗ , { μ k i , l j ,ν k i , l j }]

Let the EIFIM A =[ be given.

Let for i be fixed numbers.

In [7,13], several level operators are defined. One of them, for a given IFS

= 1 , 2 , 3 : ρ i ,σ i ,ρ i + σ i ∈[ 0 , 1 ]

X ={ x ,μ X ( x ), ν X ( x ) | x ∈ E }

is

N

α,β (

X ) ={ x ,μ X ( x ), ν X ( x ) | x ∈ E & μ X ( x ) ≥ α & ν X ( x ) ≤ β } ,

where 1.

Here, its analogues are introduced. They are three: N 1
ρ

α, β ∈[ 0 , 1 ] are fixed and α + β ≤

N 2
ρ

N 3
ρ

, , and

,σ ,σ ,σ 1 1 2 2 3 3

affect the K -, L -indices and -elements, respectively. The three operators

can be applied over an EIFIM A either sequentially, or simultaneously. In the first
case, their forms are

μ k i , l j ,ν k i , l j

N 1
ρ

K ∗ ), L ∗ , { ϕ k i , l j ,ψ k i , l j }] , ( A ) =[ N ρ 1 ,σ 1 (

,σ 1 1

where

ϕ k i , l j ,ψ k i , l j = μ k i , l j ,ν k i , l j

k k

K ∗ )

l l

L ∗ ;

only for k i ,α

i ,β i ∈

N

ρ 1 ,σ 1 (

and for each l j ,α

j ,β j ∈

N 2 K ∗ , L ∗ ), { ϕ k i , l j ,ψ k i , l j }] ,

ρ 2 ,σ 2 (

A ) =[ N (

ρ ,σ 2 2

where

ϕ k i , l j ,ψ k i , l j = μ k i , l j ,ν k i , l j

k k

K ∗ and only for

l l

L ∗ ) for each k i ,α

i ,β i ∈

l j ,α

j ,β j ∈

N ρ 2 ,σ 2 ( ;

N 3 K ∗ , L ∗ , { ϕ k i , l j ,ψ k i , l j }] ,

ρ 3 ,σ 3 (

A ) =[

where

⎧
⎨
⎩

μ k i , l j ,ν k i , l j , if μ k i , l j ≥ ρ 3 & ν k i , l j ≤ σ 3

ϕ k i , l j ,ψ k i , l j = ,

0 , 1 , otherwise

In the second case, their form is

N 1 N 2 N 3 K ∗ ), L ∗ ), { ϕ k i , l j ,ψ k i , l j }] , (

ρ 1 ,σ 1 , ρ 2 ,σ 2 , ρ 3 ,σ 3 )(

A ) =[ N ( N (

ρ ,σ ρ ,σ 1 1 2 2