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In this work, we rst de ne a relation on neutrosophic soft sets which allows to compose two neutrosophic soft sets. It is devised to derive useful information through the composition of two neutrosophic soft sets. Then, we examine symmetric, transitive and re exive neutrosophic soft relations and many related concepts such as equivalent neutrosophic soft set relation, partition of neutrosophic soft sets, equivalence classes, quotient neutrosophic soft sets, neutrosophic soft composition are given and their propo- sitions are discussed. Finally a decision making method on neutrosophic soft sets is presented.
Citation preview
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
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 111
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
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 112
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
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 113
)(),(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.
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 114
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~
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 115
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~
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 116
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
~
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 117
(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
~
Similarly we can prove the other results.
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~
.
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 118
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)
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 119
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)
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 120
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
~]
Similarly we can prove the other results.
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)
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International Journal of Innovative Research and Studies Page 121
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~
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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
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 123
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.
December, 2014 www.ijirs.com Vol3 Issue 12
International Journal of Innovative Research and Studies Page 124
References:
1. I.Arockiarani,I.R.Sumathi & J.Martina Jency “Fuzzy Neutrosophic soft topological
spaces”IJMA – 4 (10) 2013., 225-238.
2. I.Arockiarani and I.R.Sumathi, A fuzzy neutrosophic soft Matrix approach in
decision making, JGRMA, Vol 2, No.2, 14-23, Feb 2014.
3. Chetia.B,Das.P.K, “Some results of intuitionistic fuzzy soft matrix theory”Advances
in applied science research 3(1) : 412 – 423 (2012).
4. Maji.P.K, Biswas.R and Roy.A.R , “Fuzzy soft set” The journal of fuzzy
mathematics, Vol 9(3), pp – 589 - 602 (2001).
5. Maji.P.K, Biswas.R and Roy.A.R , “Intuitionistic fuzzy soft sets” Journal of fuzzy
mathematics, 12, pp - 669 – 683 (2004).
6. Pabitra Kumar Maji , Neutrosophic soft set , Annals of Fuzzy Mathematics and
Informatics,Volume 5, No.1, pp.157-168 (2013).
7. Manash Jyoti Borah, Tridiv Jyotic Neog, Dusmenta Kumar Sut , “Fuzzy soft matrix
theory and its decision making” IJMER Vol.2.,pp – 121 – 127 (2012).
8. Manoj Bora, Banashree Bora, Tridiv Jyotic Neog, Dusmenta Kumar Sut,
“Intuitionistic fuzzy soft matrix theory and its application in medical diagnosis”,
Annals of Fuzzy Mathematics and Informatics,Volume x, No.x, pp.1 - xx (2013).
9. Molodtsov D., “Soft set theory first results” , Computer and mathematics with
applications, 37 ,pp- 19 – 31(1999).
10. F.Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference
On Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics University of
New Mexico, Gallup, NM 87301, USA (2002).
11. F.Smarandache, Neutrosophic set, a generialization of the intuituionistics fuzzy sets,
Inter.J.Pure Appl.Math.24 287 – 297. (2005).
12. Yong Yang and Chenli Ji., “Fuzzy soft matrices and their application” part I, LNAI
7002; pp:618 – 627(2011).