New Operations On Fuzzy Neutrosophic Soft Matrices Abstract: In this paper we define some new operations on fuzzy neutrosophic soft matrices and their propertieswhicharemorefunctionaltomaketheoreticalstudies.Wefinallyconstructadecision 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.SumathiDepartment of Mathematics, Nirmala College for Women, Coimbatore, Tamilnadu, India. I.ArockiaraniDepartment of Mathematics, Nirmala College for Women, Coimbatore, Tamilnadu, India. I SSN 2319-9725 December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 110 1.Introduction: In1999,[9]Molodtsovinitiatedthenovelconceptofsoftsettheorywhichisacompletely 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. Moreoverin [5],[6] Maji et al extended soft sets to intuitionistic fuzzy soft setsandNeutrosophicsoftsetsandtheconceptofNeutrosophicsetwasinitiatedby Smarandache[11]whichisageneralizationoffuzzylogicandseveralrelatedsystems. Neutrosophicsetsandlogicarethefoundationsformanytheorieswhicharemoregeneral than their classical counterparts in fuzzy, intuitionistic fuzzy and interval valued frameworks.Oneoftheimportanttheoryofmathematicswhichhasavastapplicationinscienceand 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 theintuitionisticfuzzysoftmatricesandappliedintheapplicationofmedicaldiagnosis.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.Apair(F,E)iscalledasoftsetoverUwhereFisamappinggivenbyF:EP(U). Clearly,asoftsetisamappingfromparameterstoP(U)anditisnotaset,buta parameterized family of subsets of the universe. Definition 2.2:[1]AFuzzy Neutrosophic set A on the universe of discourse X is defined asA=,

,

,

, where , , : [0, 1] and 3 ) ( ) ( ) ( 0 s + + s x F x I x TA A A December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 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 c 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 byF: 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 byRA = { (u, e) ; eeA, u e fA(e)} which is called a relation formof(fA,E).Themembershipfunction,indeterminacymembershipfunctionandnon membershipfunctionarewrittenby ] 1 , 0 [ : E U TAR, ] 1 , 0 [ : E U IARand ] 1 , 0 [ : E U FAR where ] 1 , 0 [ ) , ( e e u TAR , ] 1 , 0 [ ) , ( e e u IARand ] 1 , 0 [ ) , ( e e u FAR are the membership value, indeterminacy value andnon membership value respectively of u e U for each eeE.If [(Tij , Iij , Fij)] = (Tij(ui , ej) , Iij(ui , ej) , Fij(ui , ej)) we can define a matrix( ) | |((((((

|.|

\||.|

\||.|

\||.|

\||.|

\||.|

\||.|

\||.|

\||.|

\|=mnFmnImnTmFmImTmFmImTnFnInT F I T F I TnFnInT F I T F I TmxnijFijIijT, , .... .......... , , , ,....... .......... ...... .......... ...... .......... ....... .........., , ...... .......... , , , ,, , ...... .......... , , , ,, ,2 2 2 1 1 12 2 2 22 22 22 21 21 211 1 1 12 12 12 11 11 11 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}.LetA_E.Apair(F,A)beafuzzyneutrosophicsoftset.Thenfuzzy 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 whereDecember, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 112 ee|.|

\|=Aje ifAje ificjFicjIicjTija) 1 , 0 , 0 () ( ), ( , ) (,where ) (icjTrepresentthemembershipof ) (ic, ) (icjIrepresenttheindeterminacyof ) (icand ) (icjFrepresentthenon-membershipof ) (icinthefuzzyneutrosophicset ) (je F. Note: Fuzzy Neutrosophic soft matrix is denoted by FNSM 3.Fuzzy Neutrosophic Soft Matrices: Definition 3.1:Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e FNSMm x n . Then a) A~ is Fuzzy neutrosophic softsub matrix of B~ denoted by A~_B~ if AijT~s BijT~ , AijI~s BijI~ andAijF~> BijF~, i and j.b) A~ is Fuzzy neutrosophic soft super matrix of B~ denoted by A~_B~ if AijT~> BijT~ , AijI~> BijI~ andAijF~sBijF~, i and j.c) A~and B~aresaidtobeFuzzyneutrosophicsoftequalmatricesdenotedby A~=B~if AijT~= BijT~ , AijI~=BijI~ andAijF~= BijF~, i and j.Definition 3.2:If A~=[aij]eFNSMmxn, where[aij]= |.|

\|) ( ), ( , ) (icjFicjIicjTthen CA~iscalledfuzzy neutrosophic soft complement matrix if CA~ = [bij]m x n where December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 113 |.|

\| = ) ( ), ( 1 , ) ( ] [icjTicjIicjFijb Definition 3.3: Let| |AijAijAijF I T A~ ~ ~, ,~= e FNSMm x n . Then A~ is calledd)aAijT~-universal FNSM denoted by UT= [1,0,0] if AijT~ = 1 , AijI~ = 0 and AijF~ = 0, i and j.e)aAijI~-universal FNSM denoted by UI = [0,1,0] if AijT~ = 0 , AijI~ = 1 and AijF~ = 0, i and j. f)aAijF~-universal FNSM denoted by UF= [0,0,1] if AijT~ = 0 , AijI~ = 0 and AijF~ = 1, i and j.Definition 3.4: Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e FNSMmxn. Then FSNM | |CijCijCijF I T C~ ~ ~, ,~=is calleda)Union ofA~ and B~ denoted by A~

B~ if CijT~ = max (AijT~, BijT~) , CijI~ = max (AijI~, BijI~)and CijF~ = min (AijF~, BijF~) i and j. b)Intersection ofA~ and B~ denoted by A~ B~ if CijT~ = min (AijT~, BijT~) ,

CijI~ = min (AijI~, BijI~)and CijF~ = max (AijF~, BijF~) i and j. Definition 3.5: Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e FNSMmxn. Then FSNM | |CijCijCijF I T C~ ~ ~, ,~=is calleda)The * (product) operation ofA~ and B~ denoted by C~= A~ * B~ if CijT~ = AijT~. BijT~ ,

CijI~ = AijI~. BijI~and CijF~ = AijF~ + BijF~ AijF~. BijF~ i and j. December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 114 b)The +~ (probabilistic sum) operation ofA~ and B~ denoted by C~= A~ +~ B~ if

CijT~ = AijT~ + BijT~ AijT~. BijT~ , CijI~ = AijI~ + BijI~ AijI~. BijI~and CijF~ = AijF~. BijF~ i and j. c)The @ (Arithmetic mean) operation ofA~ and B~ denoted by C~= A~ @ B~ if

2~ ~~BijAijCijT TT+= ,2~ ~~BijAijCijI II+= and2~ ~~BijAijCijF FF+= i and j. d)The @W (weighted Arithmetic mean) operation ofA~ and B~ denoted by

C~= A~ @W B~ if 2 1~2~1~w wT w T wTBijAijCij++= ,2 1~2~1~w wI w I wIBijAijCij++= and 2 1~2~1~w wF w F wFBijAijCij++= i and j. e)The $ (Geometric mean) operation ofA~ and B~ denoted by C~= A~ $ B~ if

BijAijCijT T T~ ~ ~ = ,BijAijCijI I I~ ~ ~ = andBijAijCijF F F~ ~ ~ = i and j. f)The $W (Weighted geometric mean) operation ofA~ and B~ denoted by C~= A~ $W B~ if

( ) ( )2 1 2 11~ ~ ~w w wBijwAijCijT T T+((

=, ( ) ( )2 1 2 11~ ~ ~w w wBijwAijCijI I I+((

= and( ) ( )2 1 2 11~ ~ ~w w wBijwAijCijF F F+((

= i and j. g)The & (Harmonic mean) operation ofA~ and B~ denoted by C~= A~ & B~ if

BijAijBijAijCijT TT TT~ ~~ ~~2+ = ,BijAijBijAijCijI II II~ ~~ ~~2+ = andBijAijBijAijCijF FF FF~ ~~ ~~2+ = i and j. h)The &W (Weighted Harmonic mean) operation ofA~ and B~ denoted by C~= A~ &W B~

December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 115 if BijAijCijTwTww wT~2~12 1~++= ,BijAijCijIwIww wI~2~12 1~++= andBijAijCijFwFww wF~2~12 1~++= i and j. Proposition 3.6: Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e 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 {AijT~, BijT~} , min {AijI~, BijI~}, max {AijF~, BijF~}]+~ [max {AijT~, BijT~} , max {AijI~, BijI~}, min {AijF~, BijF~}] = {[min {AijT~, BijT~}+ max {AijT~, BijT~} min {AijT~, BijT~}. max {AijT~, BijT~}] ,[min {AijI~, BijI~} + max {AijI~, BijI~} min {AijI~, BijI~} . max {AijI~, BijI~}], [max {AijF~, BijF~}. min {AijF~, BijF~}]} = {(AijT~ + BijT~ AijT~. BijT~ ,AijI~ + BijI~ AijI~. BijI~ AijF~. BijF~)} [since min (a,b) + max(a,b) = a+ b& min(a,b) . max(a,b) = a.b] = A~ +~ B~ December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 116 Similarly we can prove the other results. Example 3.7: Let | |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e FNSMm x n where 2 2) 2 . 0 , 4 . 0 , 7 . 0 ( ) 3 . 0 , 7 . 0 , 5 . 0 () 5 . 0 , 6 . 0 , 4 . 0 ( ) 2 . 0 , 4 . 0 , 3 . 0 (~xA((

= and 2 2) 4 . 0 , 5 . 0 , 3 . 0 ( ) 2 . 0 , 6 . 0 , 7 . 0 () 2 . 0 , 5 . 0 , 6 . 0 ( ) 3 . 0 , 5 . 0 , 4 . 0 (~xB((

= 2 2) 4 . 0 , 4 . 0 , 3 . 0 ( ) 3 . 0 , 6 . 0 , 5 . 0 () 5 . 0 , 5 . 0 , 4 . 0 ( ) 3 . 0 , 4 . 0 , 3 . 0 (~ ~xB A((

= and 2 2) 2 . 0 , 5 . 0 , 7 . 0 ( ) 2 . 0 , 7 . 0 , 7 . 0 () 2 . 0 , 6 . 0 , 6 . 0 ( ) 2 . 0 , 5 . 0 , 4 . 0 (~ ~xB A((

= ( ) ( )2 2) 08 . 0 , 7 . 0 , 79 . 0 ( ) 06 . 0 , 88 . 0 , 85 . 0 () 1 . 0 , 8 . 0 , 76 . 0 ( ) 06 . 0 , 7 . 0 , 58 . 0 (~ ~~~ ~xB A B A((

=+ = A~ +~ B~ Definition 3.8:Let| |AijAijAijF I T A~ ~ ~, ,~= e FNSMm x n a)The necessity operation ofA~denoted by A~ and is defined as | |AijAijAijT I T A~ ~ ~1 , ,~ = i and j. b)The possibility operation ofA~denoted by o A~ and is defined as o| |AijAijAijF I T A~ ~ ~, 1 ,~ = i and j. a)The falsity operation ofA~denoted by ^ A~ and is defined as ^| |AijAijAijF I F A~ ~ ~, , 1~ = i and j. Proposition 3.9: Let| |AijAijAijF I T A~ ~ ~, ,~= e FNSMm x n , then (i) [(CA~)]C = ^ A~ December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 117 (ii) [o(CA~)]C = A~ (iii) [^(CA~)]C = A~ (iv)^ A~=^ A~ (v) ^ A~= A~ (vi) o A~= o A~ (vii) o ^ A~=^ o A~ (viii) A~= A~ (ix) ^ ^ A~=^ A~ (x) o o A~= A~ Proof: (i) [(CA~)]C =[ | |AijAijAijT I F~ ~ ~, 1 , ]C = | |CAijAijAijF I F~ ~ ~1 , 1 , = | |AijAijAijF I F~ ~ ~, , 1 =^ A~ Similarly we can prove the other results. Example 3.10: Let | |AijAijAijF I T A~ ~ ~, ,~= e FNSMm x n where2 2) 2 . 0 , 9 . 0 , 7 . 0 ( ) 3 . 0 , 3 . 0 , 5 . 0 () 5 . 0 , 7 . 0 , 4 . 0 ( ) 2 . 0 , 4 . 0 , 3 . 0 (~xA((

= then2 2) 7 . 0 , 1 . 0 , 2 . 0 ( ) 5 . 0 , 7 . 0 , 3 . 0 () 4 . 0 , 3 . 0 , 5 . 0 ( ) 3 . 0 , 6 . 0 , 2 . 0 (~xCA((

=

2 2) 8 . 0 , 1 . 0 , 2 . 0 ( ) 7 . 0 , 7 . 0 , 3 . 0 () 5 . 0 , 3 . 0 , 5 . 0 ( ) 8 . 0 , 6 . 0 , 2 . 0 (~ xCA((

=

[2 2) 2 . 0 , 9 . 0 , 8 . 0 ( ) 3 . 0 , 3 . 0 , 7 . 0 () 5 . 0 , 7 . 0 , 5 . 0 ( ) 2 . 0 , 4 . 0 , 8 . 0 (] ]~[xC CA((

== ^ A~. December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 118 Proposition 3.11: Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e 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) [o [CA~+~ CB~]]C = o( A~ * B~) (vi) [o [CA~* CB~]]C = o( 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) o [ A~ @ B~] = o A~@ oB~ Proof: (i) [ A~ B~] = [ min {AijT~, BijT~} , min {AijI~, BijI~}, max {AijF~, BijF~}] =[ min {AijT~, BijT~} , min {AijI~, BijI~}, 1 min {AijT~, BijT~}]---------(1) | |AijAijAijT I T A~ ~ ~1 , ,~ = , | |BijBijBijT I T B~ ~ ~1 , ,~ =

A~ B~ = [min {AijT~, BijT~} , min {AijI~, BijI~}, max {1AijT~, 1BijT~}] =[ min {AijT~, BijT~} , min {AijI~, BijI~}, 1 min {AijT~, BijT~}] ----------(2) December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 119 From (1) & (2) we have [ A~ B~] = A~ B~. Similarly we can prove the other results. Definition 3.12:Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~=e FNSMm x n . Then the operation (Implication) denoted by A~B~ is defined asA~B~ = [(max (AijF~, BijT~) , max(AijI~1, BijI~), min (AijT~, BijF~)]. Proposition 3.13: Let| |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~= , | |CijCijCijF I T C~ ~ ~, ,~=e 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 {AijT~, BijT~} , min {AijI~, BijI~}, max {AijF~, BijF~}] [ A~ B~] C~ =[max((max (AijF~, BijF~)),CijT~ ) , max((AijI~min( 1, BijI~)),CijI~), min((min (AijT~, BijT~))CijF~)]. -------(1) December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 120 A~C~ = [(max (AijF~, CijT~) , max(AijI~1, CijI~), min (AijT~, CijF~)]. B~C~ = [(max (BijF~,CijT~) , max(BijI~1,CijI~), min (BijT~,CijF~)]. [ A~C~] [B~C~] = [max{(max (AijF~, CijT~),(max (BijF~,CijT~)},max { max(AijI~1, CijI~), max(BijI~1,CijI~)}, min {min (AijT~, CijF~), min (BijT~,CijF~)}] =[max((max (AijF~, BijF~)),CijT~ ) , max((AijI~min( 1, BijI~)),CijI~), min((min (AijT~, BijT~))CijF~)]. -------(2) From (1) & (2) [ A~ B~] C~ = [ A~C~] [B~C~] Similarly we can prove the other results. Example 3.14:Let | |AijAijAijF I T A~ ~ ~, ,~= , | |BijBijBijF I T B~ ~ ~, ,~= ,| |CijCijCijF I T C~ ~ ~, ,~=eFNSMm x n where2 2) 2 . 0 , 4 . 0 , 7 . 0 ( ) 3 . 0 , 7 . 0 , 5 . 0 () 5 . 0 , 6 . 0 , 4 . 0 ( ) 2 . 0 , 4 . 0 , 3 . 0 (~xA((

=,2 2) 4 . 0 , 5 . 0 , 3 . 0 ( ) 2 . 0 , 6 . 0 , 7 . 0 () 2 . 0 , 5 . 0 , 6 . 0 ( ) 3 . 0 , 5 . 0 , 4 . 0 (~xB((

=and2 2) 4 . 0 , 4 . 0 , 5 . 0 ( ) 3 . 0 , 7 . 0 , 6 . 0 () 6 . 0 , 5 . 0 , 4 . 0 ( ) 6 . 0 , 4 . 0 , 3 . 0 (~xC((

= [ A~+~B~] C~ = 2 2) 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~] 2 2) 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) December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 121 From (1) and (2) [ A~+~B~] C~_ [ A~C~]+~ [B~C~] 4.ApplicationOfWeightedArithmeticMean(WAMA~)OfFuzzyNeutrosophicSoft Matrix In Decision Making: Inthissectionwedefinearithmeticmean(AMA~)andweightedarithmeticmean(WAMA~)of fuzzy neutrosophic soft matrix. Definition 4.1: Let | |AijAijAijF I T A~ ~ ~, ,~=eFNSMmxn ,thentheweightedarithmeticmeanoffuzzy neutrosophicsoftmatrix A~denotedby WAMA~isdefinedas, (((((

=======njjnjAij jnjjnjAij jnjjnjAij jWwF wwI wwT wAAM11~11~11~, ,~ andwjfor j = 1, 2, n. Definition 4.2: Let | |AijAijAijF I T A~ ~ ~, ,~=eFNSMmxn ,thenthearithmeticmeanoffuzzyneutrosophicsoft matrix A~denoted by AMA~ is defined as ,(((((

= = = =nFnInTAnjAijnjAijnjAijAM1~1~1~, ,~ when weights are equal. Definition 4.3: The score value of Si for uieU is defined as i i i iF I T S . =. Algorithm: 1.Construct an FNSS (F,A) over U 2.Constructthe Fuzzy Neutrosophic soft matrix of (F,A) 3.Compute arithmetic mean AMA~ and weighted arithmetic mean WAMA~ December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 122 4.Compute the score value of Si for AMA~ andWAMA~ 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 = theabilityofthemanagementande3=levelofservice.Supposeamanufacturingcompany whichwantstoselectthebestsupplieraccordingtothecorecompetencies.Afuzzy 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 byDecember, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 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 (~WAMA The score of (((((((

=4067 . 04755 . 04944 . 03664 . 0~0.5422AMAand the score of (((((((

=2977 . 05276 . 05076 . 02904 . 0~0.7536WAMA. Formtheaboveresults,itisclearthatifwegiveequalpreferencefortheparameters,we haveu4isthebestalternativeandwhenwegivemorepreferenceonleveloftechnology innovation, then we have same u2 is the best alternative. 5.Conclusion: Softsettheoryisageneralmethodforsolvingproblemsofuncertainty.Inthispaper,we defineFuzzyNeutrosophicsoftmatricesandtheiroperationswhicharemorefunctionalto improveseveralnewresults.Finallywehavegivenoneapplicationfordecisionmaking based on arithmetic mean. December, 2014www.ijirs.comVol3 Issue 12 International Journal of Innovative Research and StudiesPage 124 References: 1.I.Arockiarani,I.R.Sumathi&J.MartinaJencyFuzzyNeutrosophicsofttopologicalspacesIJMA 4 (10)2013., 225-238. 2.I.ArockiaraniandI.R.Sumathi,AfuzzyneutrosophicsoftMatrixapproachin decisionmaking, JGRMA, Vol 2, No.2, 14-23, Feb 2014. 3.Chetia.B,Das.P.K,SomeresultsofintuitionisticfuzzysoftmatrixtheoryAdvances inapplied scienceresearch 3(1) : 412 423 (2012). 4.Maji.P.K,Biswas.RandRoy.A.R,FuzzysoftsetThejournaloffuzzy mathematics,Vol 9(3), pp 589 - 602 (2001). 5.Maji.P.K,Biswas.RandRoy.A.R,IntuitionisticfuzzysoftsetsJournaloffuzzy mathematics, 12, pp - 669 683 (2004). 6.PabitraKumarMaji,Neutrosophicsoftset,AnnalsofFuzzyMathematicsand Informatics,Volume 5, No.1, pp.157-168 (2013). 7.ManashJyotiBorah,TridivJyoticNeog,DusmentaKumarSut,Fuzzysoftmatrix theory and its decision making IJMER Vol.2.,pp 121 127 (2012). 8.ManojBora,BanashreeBora,TridivJyoticNeog,DusmentaKumarSut, Intuitionisticfuzzysoftmatrixtheoryanditsapplicationinmedicaldiagnosis, Annals of Fuzzy Mathematics and Informatics,Volume x, No.x, pp.1 - xx (2013). 9.MolodtsovD.,Softsettheoryfirstresults,Computerandmathematicswith applications, 37 ,pp- 19 31(1999). 10. F.Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference OnNeutrosophy,NeutrosophicLogic,Set,ProbabilityandStatisticsUniversityof New Mexico, Gallup, NM 87301, USA (2002). 11. F.Smarandache,Neutrosophicset,agenerializationoftheintuituionisticsfuzzysets, Inter.J.Pure Appl.Math.24 287 297. (2005). 12. YongYangandChenliJi.,FuzzysoftmatricesandtheirapplicationpartI,LNAI 7002; pp:618 627(2011).