New Operations On Fuzzy Neutrosophic Soft Matrices ISSN

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1 Ne Operatios O uzzy Neutrosophic Soft Matrices SSN RSumathi Departmet of Mathematics Nirmala ollege for Wome oimbatore amiladu dia rockiarai Departmet of Mathematics Nirmala ollege for Wome oimbatore amiladu dia bstract: this paper e defie some e operatios o fuzzy eutrosophic soft matrices their properties hich are more fuctioal to make theoretical studies We fially costruct a decisio makig method hich ca be applied to the problems that cotai ucertaities Keyords: Soft sets uzzy Neutrosophic soft set uzzy Neutrosophic soft matrices

2 December 204 irscom Vol3 ssue 2 troductio: 999 [9] Molodtsov iitiated the ovel cocept of soft set theory hich is a completely e approach for modelig vagueess ucertaity [4] Mai et al iitiated the cocept of fuzzy soft sets ith some properties regardig fuzzy soft uio itersectio complemet of fuzzy soft set Moreover i [5][6] Mai et al exteded soft sets to ituitioistic fuzzy soft sets Neutrosophic soft sets the cocept of Neutrosophic set as iitiated by Smarache [] hich is a geeralizatio of fuzzy logic several related systems Neutrosophic sets logic are the foudatios for may theories hich are more geeral tha their classical couterparts i fuzzy ituitioistic fuzzy iterval valued frameorks Oe of the importat theory of mathematics hich has a vast applicatio i sciece egieerig is the theory of matrices [2] Yog et al iitiated a matrix represetatio of fuzzy soft set applied it i decisio makig problems [8] Mao ora et al itroduced the ituitioistic fuzzy soft matrices applied i the applicatio of medical diagosis this paper e have itroduced some e operatios o fuzzy eutrosophic soft matrices some related properties have bee established 2 Prelimiaries: Defiitio 2: [9] Suppose U is a uiversal set E is a set of parameters Let P(U) deote the poer set of U pair ( E) is called a soft set over U here is a mappig give by : EP(U) learly a soft set is a mappig from parameters to P(U) it is ot a set but a parameterized family of subsets of the uiverse Defiitio 22:[] uzzy Neutrosophic set o the uiverse of discourse X is defied as x x x x x X here : X [0 ] 0 ( x) ( x) ( x) 3 teratioal Joural of ovative Research Studies Page 0

3 December 204 irscom Vol3 ssue 2 Defiitio 23:[] Let U be the iitial uiverse set E be a set of parameters osider a o-empty set E Let P (U) deote the set of all fuzzy eutrosophic sets of U he collectio termed to be the fuzzy eutrosophic soft set over U here is a mappig give by : P(U) Defiitio 24:[2] ( ) is Let U { c c 2 c m } be the uiversal set E be the set of parameters give by E {e e 2 e } Let E pair ( ) be a uzzy Neutrosophic soft set over U he the subset of U x E is defied by R { (u e) ; e u f (e)} hich is called a relatio form of (f E) he membership fuctio idetermiacy membership fuctio o membership fuctio are ritte by R : U E [0] here R ( u e) [0] R R : U E [0] ( u e) [0] R R : U E [0] ( u e) [0] are the membership value idetermiacy value o membership value respectively of u U for each ee f [( )] ( (u i e ) (u i e ) (u i e )) e ca defie a matrix mx m m m m2 m2 m m m m hich is called a m x uzzy Neutrosophic Soft Matrix of the NSS (f E) over U Defiitio 25:[2] Let U { c c 2 c m } be the uiversal set E be the set of parameters give by E {e e 2 e } Let E pair ( ) be a fuzzy eutrosophic soft set he fuzzy eutrosophic soft set ( ) i a matrix form as m x [a ] m x or [a ] i2 m 2 here teratioal Joural of ovative Research Studies Page

4 December 204 irscom Vol3 ssue 2 a ( c ) ( c ) ( c ) i i i (0 0) if if e e here ( c ) i ( e ( c i ) ) ( c i represet the membership of ) ( c ) i represet the o-membership of ( c i ( c ) i ) represet the idetermiacy of i the fuzzy eutrosophic set Note: uzzy Neutrosophic soft matrix is deoted by NSM 3 uzzy Neutrosophic Soft Matrices: Defiitio 3: Let NSM m x he a) is uzzy eutrosophic soft sub matrix of deoted by if i b) is uzzy eutrosophic soft super matrix of deoted by if i c) are said to be uzzy eutrosophic soft equal matrices deoted by if Defiitio 32: i f [a ] NSM m x here [a ] eutrosophic soft complemet matrix if ( c ) ( c ) ( c ) i i i [b ] m x here the is called fuzzy teratioal Joural of ovative Research Studies Page 2

5 December 204 irscom Vol3 ssue 2 [ b ] ( c ) ( c ) ( c ) i i i Defiitio 33: NSM m x he is called Let d) a e) a f) a -uiversal NSM deoted by U [00] if -uiversal NSM deoted by U [00] if -uiversal NSM deoted by U [00] if i 0 i i Defiitio 34: Let NSM m x he SNM called is a) Uio of deoted by if max ( ) max ( ) mi ( ) i b) tersectio of deoted by if mi ( ) mi ( ) max ( ) i Defiitio 35: Let NSM m x he SNM called is a) he * (product) operatio of deoted by * if + i teratioal Joural of ovative Research Studies Page 3

6 December 204 irscom Vol3 ssue 2 b) he (probabilistic sum) operatio of deoted by if + + i c) (rithmetic mea) operatio of deoted if i d) W (eighted rithmetic mea) operatio of deoted by W if i e) he $ (Geometric mea) operatio of deoted by $ if i f) he $ W (Weighted geometric mea) operatio of deoted by $ W if i g) he & (Harmoic mea) operatio of deoted by & if i h) he & W (Weighted Harmoic mea) operatio of deoted by & W teratioal Joural of ovative Research Studies Page 4

7 December 204 irscom Vol3 ssue 2 if i Propositio 36: Let (i) ( ) ( ) (ii) ( ) * ( ) * (iii) ( ( (iv) ( ) $ ( ) $ (v) ( ) & ( ) & (vi) ( ( * Proof: NSM m x the (i) ( ) ( ) [mi { } mi { } max { }] [max { } max { } mi { }] {[mi { }+ max { } mi { } max { }] [mi { } + max { } mi { } max { }] [max { } mi { }]} {( + + )} [sice mi (ab) + max(ab) a+ b & mi(ab) max(ab) ab] teratioal Joural of ovative Research Studies Page 5

8 December 204 irscom Vol3 ssue 2 Similarly e ca prove the other results Example 37: Let NSM m x here (030402) (050703) (040605) (070402) 2x2 (040503) (060502) (070602) (030504) 2x2 (030403) (050603) (040505) (030404) 2x2 (040502) (060602) (070702) (070502) 2x2 Defiitio 38: Let ( ) ( ) NSM m x (076080) ( ) 2x2 a) he ecessity operatio of deoted by is defied as i b) he possibility operatio of deoted by is defied as i a) he falsity operatio of deoted by is defied as Propositio 39: Let (i) [( i NSM m x the )] teratioal Joural of ovative Research Studies Page 6

9 December 204 irscom Vol3 ssue 2 (ii) [( (iii) [( )] )] (iv) (v) (vi) (vii) (viii) (ix) (x) Proof: (i) [( )] [ Similarly e ca prove the other results Example 30: NSM m x here Let ] (030402) (050303) (040705) (070902) 2x2 (020603) (050304) (030705) (02007) the 2x2 (020608) (030707) (050305) (02008) 2x2 (080402) (050705) [ ]] (070303) (080902) [ 2x2 teratioal Joural of ovative Research Studies Page 7

10 December 204 irscom Vol3 ssue 2 Propositio 3: Let (i) [ ] (ii) [ ] (iii) [[ (iv) [[ ]] * * ]] NSM m x the (v) [ [ ]] ( * ) (vi) [ [ (vii) [ [ (viii) [ [ * ]] ( ) ]] * * ]] (ix) (x) (xi) Proof: (i) [ ] [ mi { } mi { } max { }] [ mi { } mi { } mi { }] () [mi { } mi { } max { }] [ mi { } mi { } mi { }] (2) teratioal Joural of ovative Research Studies Page 8

11 December 204 irscom Vol3 ssue 2 rom () & (2) e have [ ] Similarly e ca prove the other results Defiitio 32: Let deoted by is defied as NSM m x he the operatio (mplicatio) [(max ( ) max( ) mi ( )] Propositio 33: Let NSM m x he (i) [ ] [ ] [ ] (ii) (iii) [ ] [ ] [ ] (iv) [ ] [ ] [ ] (v) [ ] [ ] [ ] (vi) [ * ] [ ] * [ ] (vii) [ ] [ ] [ ] (viii) [ * ] [ ]* [ ] Proof: [ ] [mi { } mi { } max { }] [ ] [max((max ( )) ) max(( mi( )) ) mi((mi ( )) )] () teratioal Joural of ovative Research Studies Page 9

12 December 204 irscom Vol3 ssue 2 [(max ( ) max( ) mi ( )] [(max ( ) max( ) mi ( )] [ ] [ ] [max{(max ( )(max ( )} max { max( ) max( )} mi {mi ( ) mi ( )}] [max((max ( )) ) max(( mi( )) ) mi((mi ( )) )] (2) rom () & (2) [ ] [ ] [ ] Similarly e ca prove the other results Example 34: Let NSM m x here (030402) (050703) (030406) (060703) (040605) (070402) (040506) (050404) (040503) (070602) (060502) (030504) 2x2 2x2 2x2 ( ) (040506) [ ] (060703) (050404) 2x2 () (050802) ( ) [ ] [ ( ) ( ) ] 2x2 (2) teratioal Joural of ovative Research Studies Page 20

13 December 204 irscom Vol3 ssue 2 rom () (2) [ ] [ ] [ ] 4 pplicatio Of Weighted rithmetic Mea ( Matrix Decisio Makig: this sectio e defie arithmetic mea( fuzzy eutrosophic soft matrix Defiitio 4: Let M W M ) Of uzzy Neutrosophic Soft ) eighted arithmetic mea( W M ) of NSM m x the the eighted arithmetic mea of fuzzy eutrosophic soft matrix deoted by W M is defied as W M for 2 Defiitio 42: NSM m x the the arithmetic mea of fuzzy eutrosophic soft Let matrix deoted by M is defied as equal Defiitio 43: M he eights are he score value of S i for u i U is defied as S i i i i lgorithm: ostruct a NSS () over U 2 ostruct the uzzy Neutrosophic soft matrix of () 3 ompute arithmetic mea M eighted arithmetic mea teratioal Joural of ovative Research Studies Page 2 W M

14 December 204 irscom Vol3 ssue 2 4 ompute the score value of S i for M W M 5 id S max(s i ) the e coclude that u i is suitable Example: Suppose there are five suppliers U{u u 2 u 3 u 4 u 5 } hose core competecies are evaluated by meas of parameters E {e e 2 e 3 } here e level of techology iovatio e 2 the ability of the maagemet e 3 level of service Suppose a maufacturig compay hich ats to select the best supplier accordig to the core competecies fuzzy eutrosophic soft set is give by () {{(e ) {(u ) (u ) (u ) (u ) (u )} {(e 2 ) {(u ) (u ) (u ) (u ) (u )} {(e 3 ) {(u ) (u ) (u ) (u ) (u )}} he matrix represetatio for the above is give as (050604) (09040) (060402) (060402) (040503) (060702) (05070) (050404) (080602) (060402) (050602) (040803) (070602) (060404) (05050) M ( ) ( ) ( ) ( ) ( ) f e prefer to give eights o the parameters e e 2 e 3 respectively the the eighted arithmetic mea is give by teratioal Joural of ovative Research Studies Page 22

15 December 204 irscom Vol3 ssue 2 W M ( ) ( ) ( ) ( ) ( he score of M the score of W M orm the above results it is clear that if e give equal preferece for the parameters e have u 4 is the best alterative he e give more preferece o level of techology iovatio the e have same u 2 is the best alterative 5 oclusio: Soft set theory is a geeral method for solvig problems of ucertaity this paper e defie uzzy Neutrosophic soft matrices their operatios hich are more fuctioal to improve several e results ially e have give oe applicatio for decisio makig based o arithmetic mea teratioal Joural of ovative Research Studies Page 23

16 December 204 irscom Vol3 ssue 2 Refereces: rockiarairsumathi & JMartia Jecy uzzy Neutrosophic soft topological spaces JM 4 (0) rockiarai RSumathi fuzzy eutrosophic soft Matrix approach i decisio makig JGRM Vol 2 No eb hetiadaspk Some results of ituitioistic fuzzy soft matrix theory dvaces i applied sciece research 3() : (202) 4 MaiPK isasr RoyR uzzy soft set he oural of fuzzy mathematics Vol 9(3) pp (200) 5 MaiPK isasr RoyR tuitioistic fuzzy soft sets Joural of fuzzy mathematics 2 pp (2004) 6 Pabitra Kumar Mai Neutrosophic soft set als of uzzy Mathematics formaticsvolume 5 No pp57-68 (203) 7 Maash Jyoti orah ridiv Jyotic Neog Dusmeta Kumar Sut uzzy soft matrix theory its decisio makig JMER Vol2pp 2 27 (202) 8 Mao ora aashree ora ridiv Jyotic Neog Dusmeta Kumar Sut tuitioistic fuzzy soft matrix theory its applicatio i medical diagosis als of uzzy Mathematics formaticsvolume x Nox pp - xx (203) 9 Molodtsov D Soft set theory first results omputer mathematics ith applicatios 37 pp- 9 3(999) 0 Smarache Neutrosophy Neutrosophic Logic irst teratioal oferece O Neutrosophy Neutrosophic Logic Set Probability Statistics Uiversity of Ne Mexico Gallup NM 8730 US (2002) Smarache Neutrosophic set a geerializatio of the ituituioistics fuzzy sets terjpure pplmath (2005) 2 Yog Yag heli Ji uzzy soft matrices their applicatio part LN 7002; pp:68 627(20) teratioal Joural of ovative Research Studies Page 24