Intuitionistic Fuzzy IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 33

Addition-(min,max)

T ∗ , V ∗ , { ϕ t u ,v w ,ψ t u ,v w }] ,

A ⊕ ( min , max ) B =[

where T ∗ , V ∗ ,α

t u ,β w ,

have the above forms but

⎧
⎨

μ k i , l j ,ν k i , l j , if t u = k i ∈ K

and v w = l j ∈ L − Q

or t u = k i ∈ K − P

and v w = l j ∈ L ;

ρ p r , q s ,σ p r , q s , if t u = p r ∈ P

v w =

q s ∈

−

and Q L

ϕ t u ,v w ,ψ t u ,v w =

or t u =

∈ −

⎩ p r P K

and v w = q s ∈ Q ;

min (μ k i , l j ,ρ p r , q s ), if t u = k i = p r ∈ K ∩ P

max (ν k i , l j ,σ p r , q s ) , and v w = l j = q s ∈ L ∩ Q

0 , 1 , otherwise

Termwise multiplication-(max,min)

T ∗ , V ∗ , { ϕ t u ,v w ,ψ t u ,v w }] , A ⊗ ( max , min ) B =[

where

T ∗ = K ∗ ∩ P ∗ ={

t t

t u ,α

u ,β u |

t u ∈ K ∩ P } ,

V ∗ = L ∗ ∩ Q ∗ ={ v w ,α w ,β w | v w ∈

L ∩ Q } ,

r

t k

α

u =

min (α

i ,α

), for t u = k i = p r ∈ K ∩ P ,

q
s

β w =

l

min (β

j ,β

), for v w = l j = q s ∈ L ∩ Q

and

ϕ t u ,v w ,ψ t u ,v w = max (μ k i , l j ,ρ p r , q s ), min (ν k i , l j ,σ p r , q s ) .

Termwise multiplication-(min,max)

T ∗ , V ∗ , { ϕ t u ,v w ,ψ t u ,v w }] , A ⊗ ( min , max ) B =[

where T ∗ , V ∗ ,α

t u ,β w ,

have the above forms but

ϕ t u ,v w ,ψ t u ,v w = min (μ k i , l j ,ρ p r , q s ), max (ν k i , l j ,σ p r , q s ) .