new operations on fuzzy neutrosophic soft matrices

New Operations On Fuzzy Neutrosophic Soft

Matrices

Abstract: In this paper we define some new operations on fuzzy neutrosophic soft matrices and their

properties which are more functional to make theoretical studies. We finally construct a decision

making method which can be applied to the problems that contain uncertainties.

Keywords: Soft sets, Fuzzy Neutrosophic soft set and Fuzzy Neutrosophic soft matrices.

I.R.Sumathi

Department of Mathematics, Nirmala College for Women,

Coimbatore, Tamilnadu, India.

I.Arockiarani

Department of Mathematics, Nirmala College for Women,

Coimbatore, Tamilnadu, India.

ISSN 2319-9725

December, 2014 www.ijirs.com Vol3 Issue 12

International Journal of Innovative Research and Studies Page 110

1. Introduction:

In 1999, [9] Molodtsov initiated the novel concept of soft set theory which is a completely

new approach for modeling vagueness and uncertainty. In [4] Maji et al. initiated the concept

of fuzzy soft sets with some properties regarding fuzzy soft union , intersection, complement

of fuzzy soft set. Moreover in [5],[6] Maji et al extended soft sets to intuitionistic fuzzy soft

sets and Neutrosophic soft sets and the concept of Neutrosophic set was initiated by

Smarandache [11] which is a generalization of fuzzy logic and several related systems.

Neutrosophic sets and logic are the foundations for many theories which are more general

than their classical counterparts in fuzzy, intuitionistic fuzzy and interval valued frameworks.

One of the important theory of mathematics which has a vast application in science and

engineering is the theory of matrices. In [12] Yong et al. initiated a matrix representation of

fuzzy soft set and applied it in decision making problems. In [8] Manoj Bora et al introduced

the intuitionistic fuzzy soft matrices and applied in the application of medical diagnosis. In

this paper we have introduced some new operations on fuzzy neutrosophic soft matrices and

some related properties have been established.

2. Preliminaries:

Definition 2.1: [9]

Suppose U is an universal set and E is a set of parameters, Let P(U) denote the power set of

U. A pair (F, E) is called a soft set over U where F is a mapping given by F: EP(U).

Clearly, a soft set is a mapping from parameters to P(U) and it is not a set, but a

parameterized family of subsets of the universe.

Definition 2.2:[1]

A Fuzzy Neutrosophic set A on the universe of discourse X is defined as

A= 𝑥,𝑇𝐴 𝑥 , 𝐼𝐴 𝑥 ,𝐹𝐴 𝑥 , 𝑥 ∈ 𝑋 where 𝑇, 𝐼,𝐹:𝑋 → [0, 1] and 3)()()(0 xFxIxT AAA



Definition 2.3:[1]

Let U be the initial universe set and E be a set of parameters. Consider a non-empty set A, A

E. Let P (U) denote the set of all fuzzy neutrosophic sets of U. The collection (F, A) is

termed to be the fuzzy neutrosophic soft set over U, where F is a mapping given by

F: A P(U).

Definition 2.4:[2]

Let U = { c1, c2, ………… cm} be the universal set and E be the set of parameters given by E

= {e1, e2, ……. en}. Let A E. A pair (F, A) be a Fuzzy Neutrosophic soft set over U. Then

the subset of U x E is defined by RA = { (u, e) ; eA, u fA(e)} which is called a relation

form of (fA ,E). The membership function , indeterminacy membership function and non –

membership function are written by ]1,0[: EUT

AR , ]1,0[: EUI

AR and

]1,0[: EUFAR

where ]1,0[),( euT

AR ,

]1,0[),( euIAR

and ]1,0[),( euF

AR are the

membership value, indeterminacy value and non membership value respectively of u U for

each eE.

If [(Tij , Iij , Fij)] = (Tij(ui , ej) , Iij(ui , ej) , Fij(ui , ej)) we can define a matrix

mnF

mnI

mnT

mF

mI

mT

mF

mI

mT

nF

nI

nTFITFIT

nF

nI

nTFITFIT

mxnijFijIijT

,,..............,,,,

..................................................................

,,................,,,,

,,................,,,,

,,

222111

222222222212121

111121212111111

which is called an m x n Fuzzy Neutrosophic Soft Matrix of the FNSS (fA , E) over U.

Definition 2.5:[2]

Let U = { c1, c2, ………… cm} be the universal set and E be the set of parameters given by E

= {e1, e2, ……. en}. Let A E. A pair (F, A) be a fuzzy neutrosophic soft set. Then fuzzy

neutrosophic soft set (F, A) in a matrix form as Am x n = [aij]m x n or A = [aij] , i=1,2,….m, j =

1,2, ….. n where



A

jeif

Aj

eifi

cj

Fi

cj

Ii

cj

T

ija)1,0,0(

)(),(,)(

,

where )(

icjT

represent the membership of )(

ic

, )(

icjI

represent the indeterminacy of

)(i

c and

)(i

cj

F represent the non-membership of

)(i

c in the fuzzy neutrosophic set

)(j

eF.

Note: Fuzzy Neutrosophic soft matrix is denoted by FNSM

3. Fuzzy Neutrosophic Soft Matrices:

Definition 3.1:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n . Then

a) A~

is Fuzzy neutrosophic soft sub matrix of B~

denoted by A~B

~ if

A

ijT~

B

ijT~

, A

ijI~

B

ijI~

and A

ijF~

B

ijF~

, i and j.

b) A~

is Fuzzy neutrosophic soft super matrix of B~

denoted by A~B

~ if

A

ijT~

B

ijT~

, A

ijI~

B

ijI~

and A

ijF~

B

ijF~

, i and j.

c) A~

and B~

are said to be Fuzzy neutrosophic soft equal matrices denoted by A~

= B~

if

A

ijT~

= B

ijT~

, A

ijI~

=B

ijI~

and A

ijF~

= B

ijF~

, i and j.

Definition 3.2:

If A~

= [aij] FNSMm x n , where [aij] =

)(),(,)(i

cj

Fi

cj

Ii

cj

T then

CA~

is called fuzzy

neutrosophic soft complement matrix if CA

~ = [bij]m x n where



)(),(1,)(][i

cj

Ti

cj

Ii

cj

Fij

b

Definition 3.3:

Let A

ij

A

ij

A

ij FITA~~~

,,~

FNSMm x n . Then A~

is called

d) a A

ijT~

-universal FNSM denoted by UT= [1,0,0] if A

ijT~

= 1 , A

ijI~

= 0 and A

ijF~

= 0, i and j.

e) a A

ijI~

-universal FNSM denoted by UI = [0,1,0] if A

ijT~

= 0 , A

ijI~

= 1 and A

ijF~

= 0, i and j.

f) a A

ijF~

-universal FNSM denoted by UF= [0,0,1] if A

ijT~

= 0 , A

ijI~

= 0 and A

ijF~

= 1, i and j.

Definition 3.4:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n. Then FSNM C

ij

C

ij

C

ij FITC~~~

,,~

is

called

a) Union of A~

and B~

denoted by A~

B~

if C

ijT~

= max (A

ijT~

, B

ijT~

) , C

ijI~

= max (A

ijI~

, B

ijI~

)

and C

ijF~

= min (A

ijF~

, B

ijF~

) i and j.

b) Intersection of A~

and B~

denoted by A~

B~

if C

ijT~

= min (A

ijT~

, B

ijT~

) ,

C

ijI~

= min (A

ijI~

, B

ijI~

) and C

ijF~

= max (A

ijF~

, B

ijF~

) i and j.

Definition 3.5:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n. Then FSNM C

ij

C

ij

C

ij FITC~~~

,,~

is

called

a) The “*” (product) operation of A~

and B~

denoted by C~

= A~

* B~

if C

ijT~

= A

ijT~

. B

ijT~

,

C

ijI~

= A

ijI~

. B

ijI~

and C

ijF~

= A

ijF~

+ B

ijF~

– A

ijF~

. B

ijF~

i and j.



b) The “~

” (probabilistic sum) operation of A~

and B~

denoted by C~

= A~

~

B~

if

C

ijT~

= A

ijT~

+ B

ijT~

– A

ijT~

. B

ijT~

, C

ijI~

= A

ijI~

+ B

ijI~

–A

ijI~

. B

ijI~

and C

ijF~

= A

ijF~

. B

ijF~

i and j.

c) The “@” (Arithmetic mean) operation of A~

and B~

denoted by C~

= A~

@ B~

if

2

~~

~B

ij

A

ijC

ij

TTT

, 2

~~

~B

ij

A

ijC

ij

III

and 2

~~

~B

ij

A

ijC

ij

FFF

i and j.

d) The “@W

” (weighted Arithmetic mean) operation of A~

and B~

denoted by

C~

= A~

@W

B~

if 21

~

2

~

1~

ww

TwTwT

B

ij

A

ijC

ij

, 21

~

2

~

1~

ww

IwIwI

B

ij

A

ijC

ij

and

21

~

2

~

1~

ww

FwFwF

B

ij

A

ijC

ij

i and j.

e) The “$” (Geometric mean) operation of A~

and B~

denoted by C~

= A~

$ B~

if

B

ij

A

ij

C

ij TTT~~~

,

B

ij

A

ij

C

ij III~~~

and

B

ij

A

ij

C

ij FFF~~~

i and j.

f) The “$W

” (Weighted geometric mean) operation of A~

and B~

denoted by C~

= A~

$W

B~

if

2121

1

~~~ wwwB

ij

wA

ij

C

ij TTT

,

2121

1

~~~ wwwB

ij

wA

ij

C

ij III

and

2121

1

~~~ wwwB

ij

wA

ij

C

ij FFF

i and j.

g) The “&” (Harmonic mean) operation of A~

and B~

denoted by C~

= A~

& B~

if

B

ij

A

ij

B

ij

A

ijC

ijTT

TTT ~~

~~

~

2

,B

ij

A

ij

B

ij

A

ijC

ijII

III ~~

~~

~

2

and B

ij

A

ij

B

ij

A

ijC

ijFF

FFF ~~

~~

~

2

i and j.

h) The “&W

” (Weighted Harmonic mean) operation of A~

and B~

denoted by C~

= A~

&W

B~



if B

ij

A

ij

C

ij

T

w

T

w

wwT

~2

~1

21~

,B

ij

A

ij

C

ij

I

w

I

w

wwI

~2

~1

21~

and B

ij

A

ij

C

ij

F

w

F

w

wwF

~2

~1

21~

i and j.

Proposition 3.6:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n , then

(i) ( A~

B~

) ~

( A~

B~

) = A~

~

B~

(ii) ( A~

B~

) * ( A~

B~

) = A~

* B~

(iii) ( A~

B~

) @ ( A~

B~

) = A~

@ B~

(iv) ( A~

B~

) $ ( A~

B~

) = A~

$ B~

(v) ( A~

B~

) & ( A~

B~

) = A~

& B~

(vi) ( A~~ B

~) @ ( A

~ * B

~) = A

~ @ B

~

Proof:

(i) ( A~

B~

) ~

( A~

B~

) = [min {A

ijT~

, B

ijT~

} , min {A

ijI~

, B

ijI~

}, max {A

ijF~

, B

ijF~

}]~

[max {A

ijT~

, B

ijT~

} , max {A

ijI~

, B

ijI~

}, min {A

ijF~

, B

ijF~

}]

= {[min {A

ijT~

, B

ijT~

}+ max {A

ijT~

, B

ijT~

}– min {A

ijT~

, B

ijT~

}. max {A

ijT~

, B

ijT~

}] ,

[min {A

ijI~

, B

ijI~

} + max {A

ijI~

, B

ijI~

}– min {A

ijI~

, B

ijI~

} . max {A

ijI~

, B

ijI~

}],

[max {A

ijF~

, B

ijF~

}. min {A

ijF~

, B

ijF~

}]}

= {(A

ijT~

+ B

ijT~

– A

ijT~

. B

ijT~

,A

ijI~

+ B

ijI~

–A

ijI~

. B

ijI~

A

ijF~

. B

ijF~

)} [since min (a,b) + max(a,b) = a+ b

& min(a,b) . max(a,b) = a.b]

= A~

~

B~



Similarly we can prove the other results.

Example 3.7:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n where

22)2.0,4.0,7.0()3.0,7.0,5.0(

)5.0,6.0,4.0()2.0,4.0,3.0(~

x

A

and 22)4.0,5.0,3.0()2.0,6.0,7.0(

)2.0,5.0,6.0()3.0,5.0,4.0(~

x

B

22)4.0,4.0,3.0()3.0,6.0,5.0(

)5.0,5.0,4.0()3.0,4.0,3.0(~~

x

BA

and 22)2.0,5.0,7.0()2.0,7.0,7.0(

)2.0,6.0,6.0()2.0,5.0,4.0(~~

x

BA

22

)08.0,7.0,79.0()06.0,88.0,85.0(

)1.0,8.0,76.0()06.0,7.0,58.0(~~~~~

x

BABA

= A~

~

B~

Definition 3.8:

Let A

ij

A

ij

A

ij FITA~~~

,,~

FNSMm x n

a) The necessity operation of A~

denoted by A~

and is defined as

A

ij

A

ij

A

ij TITA~~~

1,,~

i and j.

b) The possibility operation of A~

denoted by A~

and is defined as

A

ij

A

ij

A

ij FITA~~~

,1,~

i and j.

a) The falsity operation of A~

denoted by A~

and is defined as

A

ij

A

ij

A

ij FIFA~~~

,,1~

i and j.

Proposition 3.9:

Let A

ij

A

ij

A

ij FITA~~~

,,~

FNSMm x n , then

(i) [(CA

~)]

C = A

~



(ii) [(CA

~)]

C = A

~

(iii) [(CA

~)]

C = A

~

(iv) A~

= A~

(v) A~

= A~

(vi) A~

= A~

(vii) A~

= A~

(viii) A~

= A~

(ix) A~

= A~

(x) A~

= A~

Proof:

(i) [(CA

~)]

C =[

A

ij

A

ij

A

ij TIF~~~

,1, ]

C =

CA

ij

A

ij

A

ij FIF~~~

1,1, = A

ij

A

ij

A

ij FIF~~~

,,1 = A

~


Example 3.10:

Let A

ij

A

ij

A

ij FITA~~~

,,~

FNSMm x n where

22)2.0,9.0,7.0()3.0,3.0,5.0(

)5.0,7.0,4.0()2.0,4.0,3.0(~

x

A

then 22)7.0,1.0,2.0()5.0,7.0,3.0(

)4.0,3.0,5.0()3.0,6.0,2.0(~

x

CA

22)8.0,1.0,2.0()7.0,7.0,3.0(

)5.0,3.0,5.0()8.0,6.0,2.0(~

x

CA

[ 22)2.0,9.0,8.0()3.0,3.0,7.0(

)5.0,7.0,5.0()2.0,4.0,8.0(]]

~[

x

CCA

= A~

.



Proposition 3.11:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n , then

(i) [ A~

B~

] = A~ B

~

(ii) [ A~

B~

] = A~ B

~

(iii) [[CA

~~

CB~

]]C = A

~ * B

~

(iv) [[CA

~*

CB~

]]C = A

~ ~

B~

(v) [ [CA

~~

CB~

]]C

= ( A~

* B~

)

(vi) [ [CA

~*

CB~

]]C

= ( A~

~

B~

)

(vii) [ [CA

~~

CB~

]]C = A

~ * B

~

(viii) [ [CA

~*

CB~

]]C = A

~ ~

B~

(ix) [ A~

@ B~

] = A~

@ B~

(x) [ A~

@ B~

] = A~

@ B~

(xi) [ A~

@ B~

] = A~

@ B~

Proof:

(i) [ A~

B~

] = [ min {A

ijT~

, B

ijT~

} , min {A

ijI~

, B

ijI~

}, max {A

ijF~

, B

ijF~

}]

=[ min {A

ijT~

, B

ijT~

} , min {A

ijI~

, B

ijI~

}, 1 – min {A

ijT~

, B

ijT~

}] ---------(1)

A

ij

A

ij

A

ij TITA~~~

1,,~

,

B

ij

B

ij

B

ij TITB~~~

1,,~

A~ B

~ = [min {

A

ijT~

, B

ijT~

} , min {A

ijI~

, B

ijI~

}, max {1–A

ijT~

, 1–B

ijT~

}]

=[ min {A

ijT~

, B

ijT~

} , min {A

ijI~

, B

ijI~

}, 1 – min {A

ijT~

, B

ijT~

}] ----------(2)


International Journal of Innovative Research and Studies 9

From (1) & (2) we have [ A~

B~

] = A~ B

~. Similarly we can prove the other results.

Definition 3.12:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

FNSMm x n . Then the operation „‟(Implication)

denoted by A~ B

~ is defined as

A~ B

~ = [(max (

A

ijF~

, B

ijT~

) , max(A

ijI~

1,

B

ijI~

), min (A

ijT~

, B

ijF~

)].

Proposition 3.13:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

, C

ij

C

ij

C

ij FITC~~~

,,~

FNSMm x n. Then

(i) [ A~

B~

] C~

= [ A~C

~] [ B

~C

~]

(ii) A~

CA~

=CA

~

(iii) [ A~

B~

] C~

[ A~C

~] [ B

~C

~]

(iv) [ A~ B

~] C

~ [ A

~C

~] [ B

~C

~]

(v) [ A~~ B

~] C

~ [ A

~C

~]~

[ B~C

~]

(vi) [ A~

* B~

] C~

[ A~C

~] * [ B

~C

~]

(vii) A~ [ B

~~ C

~] [ A

~ B

~]~

[ A~C

~]

(viii) A~ [ B

~* C

~] [ A

~ B

~]* [ A

~C

~]

Proof:

[ A~

B~

] = [min {A

ijT~

, B

ijT~

} , min {A

ijI~

, B

ijI~

}, max {A

ijF~

, B

ijF~

}]

[ A~

B~

] C~

= [max((max (A

ijF~

, B

ijF~

)),C

ijT~

) , max((A

ijI~

min(1,

B

ijI~

)),C

ijI~

),

min((min (A

ijT~

, B

ijT~

))C

ijF~

)]. -------(1)


International Journal of Innovative Research and Studies 0

A~C

~ = [(max (

A

ijF~

, C

ijT~

) , max(A

ijI~

1,

C

ijI~

), min (A

ijT~

, C

ijF~

)].

B~C

~ = [(max (

B

ijF~

,C

ijT~

) , max(B

ijI~

1,

C

ijI~

), min (B

ijT~

,C

ijF~

)].

[ A~C

~] [ B

~C

~] = [max{(max (

A

ijF~

, C

ijT~

),(max (B

ijF~

,C

ijT~

)},

max { max(A

ijI~

1,

C

ijI~

), max(B

ijI~

1,

C

ijI~

)},

min {min (A

ijT~

, C

ijF~

), min (B

ijT~

,C

ijF~

)}]

= [max((max (A

ijF~

, B

ijF~

)),C

ijT~

) , max((A

ijI~

min(1,

B

ijI~

)),C

ijI~

),

min((min (A

ijT~

, B

ijT~

))C

ijF~

)]. -------(2)

From (1) & (2) [ A~

B~

] C~

= [ A~C

~] [ B

~C

~]


Example 3.14:

Let A

ij

A

ij

A

ij FITA~~~

,,~

, B

ij

B

ij

B

ij FITB~~~

,,~

, C

ij

C

ij

C

ij FITC~~~

,,~

FNSMm x n where

22)2.0,4.0,7.0()3.0,7.0,5.0(

)5.0,6.0,4.0()2.0,4.0,3.0(~

x

A

, 22)4.0,5.0,3.0()2.0,6.0,7.0(

)2.0,5.0,6.0()3.0,5.0,4.0(~

x

B

and

22)4.0,4.0,5.0()3.0,7.0,6.0(

)6.0,5.0,4.0()6.0,4.0,3.0(~

x

C

[ A~~ B

~] C

~ = 22

)4.0,4.0,5.0()3.0,7.0,6.0(

)6.0,5.0,4.0()58.0,4.0,3.0(

x

……………. (1)

[ A~C

~]~

[ B~C

~] 22

)12.0,8.0,75.0()09.0,91.0,84.0(

)24.0,75.0,7.0()12.0,8.0,51.0(

x

….………… (2)



From (1) and (2) [ A~~ B

~] C

~ [ A

~C

~]~

[ B~C

~]

4. Application Of Weighted Arithmetic Mean (W

AMA~

) Of Fuzzy Neutrosophic Soft

Matrix In Decision Making:

In this section we define arithmetic mean( AMA~

) and weighted arithmetic mean(W

AMA~

) of

fuzzy neutrosophic soft matrix.

Definition 4.1:

Let A

ij

A

ij

A

ij FITA~~~

,,~

FNSMm x n , then the weighted arithmetic mean of fuzzy

neutrosophic soft matrix A~

denoted by W

AMA~

is defined as,

n

j

j

n

j

A

ijj

n

j

j

n

j

A

ijj

n

j

j

n

j

A

ijj

W

w

Fw

w

Iw

w

Tw

AAM

1

1

~

1

1

~

1

1

~

,,~

and wj for j = 1, 2, …n.

Definition 4.2:

Let A

ij

A

ij

A

ij FITA~~~

,,~

FNSMm x n , then the arithmetic mean of fuzzy neutrosophic soft

matrix A~

denoted by AMA~

is defined as ,

n

F

n

I

n

T

A

n

j

A

ij

n

j

A

ij

n

j

A

ij

AM

1

~

1

~

1

~

,,~

when weights are

equal.

Definition 4.3:

The score value of Si for uiU is defined as iiii FITS ..

Algorithm:

1. Construct an FNSS (F,A) over U

2. Construct the Fuzzy Neutrosophic soft matrix of (F,A)

3. Compute arithmetic mean AMA~

and weighted arithmetic mean W

AMA~



4. Compute the score value of Si for AMA~

andW

AMA~

5. Find Sj = max(Si), then we conclude that ui is suitable.

Example:

Suppose there are five suppliers U={u1, u2, u3, u4, u5} whose core competencies are evaluated

by means of parameters E = A = {e1, e2, e3} where e1 = level of technology innovation, e2 =

the ability of the management and e3 = level of service. Suppose a manufacturing company

which wants to select the best supplier according to the core competencies. A fuzzy

neutrosophic soft set is given by

(F,A) =

{{F(e1) = {(u1,0.5,0.6,0.4), (u2,0.9,0.4,0.1), (u3,0.6,0.4,0.2), (u4,0.6,0.4,0.2), (u5,0.4,0.5,0.3)}

{F(e2) = {(u1,0.6,0.7,0.2), (u2,0.5,0.7,0.1), (u3,0.5,0.4,0.4), (u4,0.8,0.6,0.2), (u5,0.6,0.4,0.2)}

{F(e3) = {(u1,0.5,0.6,0.2), (u2,0.4,0.8,0.3), (u3,0.7,0.6,0.2), (u4,0.6,0.4,0.4), (u5,0.5,0.5,0.1)}}

The matrix representation for the above is given as

)1.0,5.0,5.0()2.0,4.0,6.0()3.0,5.0,4.0(

)4.0,4.0,6.0()2.0,6.0,8.0()2.0,4.0,6.0(

)2.0,6.0,7.0()4.0,4.0,5.0()2.0,4.0,6.0(

)3.0,8.0,4.0()1.0,7.0,5.0()1.0,4.0,9.0(

)2.0,6.0,5.0()2.0,7.0,6.0()4.0,6.0,5.0(

~A

)2.0,4667.0,5.0(

)2667.0,4667.0,6667.0(

)2667.0,4667.0,6.0(

)1667.0,6333.0,6.0(

)2667.0,6333.0,5333.0(

~AMA

If we prefer to give weights 0.8, 0.1 and 0.1 on the parameters e1, e2 and e3 respectively then

the weighted arithmetic mean is given by



27.0,49.0,43.0(

)22.0,42.0,62.0(

)22.0,42.0,6.0(

)12.0,47.0,81.0(

)36.0,61.0,51.0(

~W

AMA

The score of

4067.0

4755.0

4944.0

3664.0

~

0.5422

AMA

and the score of

2977.0

5276.0

5076.0

2904.0

~0.7536

W

AMA

.

Form the above results , it is clear that if we give equal preference for the parameters , we

have u4 is the best alternative and when we give more preference on level of technology

innovation, then we have same u2 is the best alternative.

5. Conclusion:

Soft set theory is a general method for solving problems of uncertainty. In this paper, we

define Fuzzy Neutrosophic soft matrices and their operations which are more functional to

improve several new results. Finally we have given one application for decision making

based on arithmetic mean.



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new operations on fuzzy neutrosophic soft matrices

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