Annals of Fuzzy Mathematics and Informatics
ISSN: 2093–9310 (print version)
ISSN: 2287–6235 (electronic version) http:// www.afmi.or.kr
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c KyungMoonSaCo. http://www.kyungmoon.com
Cosinesimilaritymeasuresofneutrosophicsoftset
I.R.SumathiandI.Arockiarani
Received8January2016; Revised12February2016; Accepted15May2016
Abstract. Inthispaperwehaveintroducedtheconceptofcosinesimilaritymeasuresforneutrosophicsoftsetandintervalvaluedneutrosophic softset.Anapplicationisgiventoshowitspracticalityandeffectiveness.
2010AMSClassification: 03B99,03E99
Keywords: Neutrosophicsoftset,CosinesimilarityofNeutrosophicsoftset, IntervalvaluedNeutrosophicsoftset.
CorrespondingAuthor: I.R.Sumathi(sumathi raman2005@yahoo.co.in)
1. Introduction
Neutrosophyisabranchofphilosophy,whichemphasizestheoriginandnature ofneutralities,alongwiththeirinteractionwithdifferentconceptivedomains.Fuzzy logicextendsclassicallogicbyassigningamembershipfunctionrangingindegreebetween0and1tothevariables.Asageneralizationoffuzzylogic,neutrosophiclogic introducesanewcomponentcalledindeterminacyandcarriesmoreinformationthan fuzzylogic.Theapplicationofneutrosophiclogicwouldleadtobetterperformance thanfuzzylogic.Neutrosophiclogicissonewthatitsuseinmanyfieldsmerit exploration.Thewords”neutrosophy”and”neutrosophic”wereintroducedbyF. Smarandacheinhis1998book[23],.Etymologically,”neutro-sophy”(noun)[French neutre < Latin neuter,neutral,andGreek sophia,skill/wisdom]meansknowledge ofneutralthought.Neutrosophicset[24],[25]isamathematicaltoolforhandling problemsinvolvingimprecise,indeterminacyandinconsistentdata.Inneutrosophic logicapropositionhasadegreeoftruth(T),adegreeofindeterminacy(I),anda degreeoffalsity(F),whereT,I,Farestandardornon-standardsubsetsof]-0,1+[. A.A.Salama[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]studiedthe variousnotionsofneutrosophicsets.
Inseveralapplicationitisoftenneededtocomparetwosetsandweareinterestedtoknowwhethertwopatternsorimagesareidenticalorapproximately identicalofatleasttowhatdegreetheyareidentical.SeveralresearcherslikeHung
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[3],JunYe[4, 5],P.Majumdar[6],C.Wang[26]andmanyauthorshavestudiedthe problemofsimilaritymeasuresbetweenintuitionisticfuzzysets,neutrosophicsets andvaguesoftsets.Inthispaperwehaveintroducedsomenewcosinesimilarity measuresforneutrosophicsoftsetsandderivedsomeoftheirproperties.Adecision makingmethodbasedonthissimilaritymeasureisconstructed.[7],[8],[9]
2. Preliminaries
Definition2.1 ([1]). AneutrosophicsetAontheuniverseofdiscourse X is definedas A = x,TA(x),IA(x),FA(x) ,x ∈ X where T,I,F : X → [0, 1]and 0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3,where T representsthetruthvalue, I represents theindeterministicvalueand F representsthefalsevalue.
Definition2.2. [1]LetXbeanonemptyset,andlet A = x,TA(x),IA(x),FA(x) and B = x,TB (x),IB (x),FB (x) beneutrosophicsets.ThenAisasubsetofBif ∀ x ∈ X, TA(x) ≤ TB (x),IA(x) ≤ IB (x)and FA(x) ≥ FB (x).
Definition2.3 ([1]). LetXbeanonemptyset,andlet A = x,TA(x),IA(x),FA(x) and B = x,TB (x),IB (x),FB (x) beneutrosophicsets.Then
A ∪ B =<x,max(TA(x),TB (x)),max(IA(x),IB (x)),min(FA(x),FB (x)) >,
A ∩ B =<x,min(TA(x),TB (x)),min(IA(x),IB (x)),max(FA(x),FB (x)) >
Definition2.4 ([1]). LetUbeatheinitialuniversalsetandEbeasetofparameters. Consideranon-emptysetA,A ⊆ E.LetP(U)denotethesetofallneutrosophicsets ofU.Thecollection(F,A)istermedtobetheneutrosophicsoftsetoverU,where FisamappinggivenbyF:A−→ P(U).(NeutrosophicsoftsetisdenotedbyNSS).
Definition2.5 ([1]). Aneutrosophicsoftset(F,A)overtheuniverseUissaid tobeemptyneutrosophicsoftsetwithrespecttotheparameterAif TF (e) =0, IF (e) =0,FF (e) =1, ∀ x ∈ U, ∀ e ∈ A.Itisdenotedby 0.
Definition2.6 ([1]). Aneutrosophicsoftset(F,A)overtheuniverseUissaidto beuniverseneutrosophicsoftsetwithrespecttotheparameterAif TF (e) =1, IF (e) =1, FF (e) =0, ∀ x ∈ U, ∀ e ∈ A.Itisdenotedby 1.
Definition2.7 ([1]). Aneutrosophicsoftset(F,A)issaidtobeasubsetofneutrosophicsoftset(G,B)ifA⊆ BandF(e)⊆ G(e) ∀ e ∈ E,u ∈ U.Wedenoteitby (F,A)⊆ (G,B).
Definition2.8 ([1]). Thecomplementofneutrosophicsoftset(F,A)denotedby (F,A)c andisdefinedas(F,A)c =(F c , ¬A)where F c : ¬A −→ P (U )isamapping givenby F c(α)=<x,TF c (x)= FF (x),IF c (x)=1 IF (x),FF c (x)= TF (x) >
Definition2.9. [1]Theunionoftwoneutrosophicsoftsets(F,A)and(G,B)over (U,E)isneutrosophicsoftsetwhereC=A∪B, ∀e ∈ C
H(e)=
F (e)ife ∈ A B G(e)ife ∈ B A F (e) ∪ G(e)ife ∈ A ∩ B 2
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andiswrittenas(F,A)∪(G,B)=(H,C).
Definition2.10 ([1]). Theintersectionoftwoneutrosophicsoftsets(F,A)and (G,B)over(U,E)isneutrosophicsoftsetwhereC=A∩B,∀e ∈ CH(e)= F (e)∩G(e) andiswrittenas(F,A)∩(G,B)=(H,C).
Definition2.11 ([2]). Anintervalvaluedneutrosophicset(IVNSinshort)ona universeXisanobjectoftheform,where TA(x)= X → Int([0, 1]), IA(x)= X → Int([0, 1])and FA(x)= X → Int([0, 1]),Int([0,1])standsforthesetofallclosed subintervalof[0,1]satisfiesthecondition: ∀x ∈ X, supTA(x)+ supIA(x)+ supFA(x) ≤ 3.
Definition2.12 ([2]). Foranarbitraryset A ⊆ [0, 1], wedefine A = infA and A = supA
Definition2.13 ([2]). LetUbeaninitialuniverseandEbeasetofparameters. IVNS(U)denotesthesetofallintervalvaluedneutrosophicsetsofU.LetA ⊆E.A pair(F,A)isanintervalvaluedneutrosophicsoftsetoverU,whereFisamapping givenbyF:A → IVNS(U).
Note:Intervalvaluedneutrosophicsoftset/setsisdenotedbyIVNSS/IVNSSs.
Definition2.14 ([2]). LetUbeaninitialuniverseandEbeasetofparameters. SupposethatA,B ⊆E,(F,A)and(G,B)betwoIVNSSs,wesaythat(F,A)isan intervalvaluedneutrosophicsoftsubsetof(G,B)ifandonlyif (i)A ⊆ B, (ii)e ∈A,F(e)isanintervalvaluedneutrosophicsoftsubsetofG(e),thatis,for allx ∈Uande ∈ A, T F (e)(x) ≤ T G(e)(x), T F (e)(x) ≤ T G(e)(x), I F (e)(x) ≤ I G(e)(x), I F (e)(x) ≤ I G(e)(x), F F (e)(x) ≥ F G(e)(x), F F (e)(x) ≥ F G(e)(x). Anditisdenotedby(F,A) ⊆ (G,B).
Similarly(F,A)issaidtobeanintervalvaluedneutrosophicsoftsupersetof (G,B),if(G,B)isanintervalvaluedneutrosophicsoftsubsetof(F,A),wedenoteit by(F,A)⊇ (G,B).
Definition2.15 ([2]). ThecomplementofanIVNSS(F,A)isdenotedby(F,A)c andisdefinedas(F,A)c =(F c , ¬A)where F c : ¬A → IVNSS(U )isamapping givenby F c(e)=<FF (¬e)(x), (IF (¬e)(x))c,TF (¬e)(x)forall x ∈ U and e ∈¬A, where(IF (¬e)(x))c =[1 I F (e)(x), 1 I F (e)(x)
3. Cosinesimilaritymeasureneutrosophicsoftsetandinterval valuedneutrosophicsoftset
Definition3.1. LetU= {x1,x2,.......xn} betheuniversalsetofelementsandE = {e1,e2,.......em} betheuniversalsetofparameters.Let(F,A)and(G,B)betwo neutrosophicsoftsetsoverU.Thenwedefinethecosinesimilaritymeasuresbetween (F,A)and(G,B)asfollows.
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CS1((F,A), (G,B)) = 1 mn m i=1
n j=1 cos π 2 (|TF (ei )(xj ) TG(ei )(xj )|) ∨ (|IF (ei )(xj ) IG(ei )(xj )|)∨ (|FF (ei )(xj ) FG(ei )(xj )|) ,
CS2((F,A), (G,B)) = 1 mn m i=1
n j=1 cos π 6 (|TF (ei )(xj ) TG(ei )(xj )|)+(|IF (ei )(xj ) IG(ei )(xj )|)+ (|FF (ei )(xj ) FG(ei )(xj )|) , wherethesymbol ∨ isthemaximumoperation.Thetwosimilaritymeasuressatisfy theaxiomaticrequirementsofsimilaritymeasures.
Proposition3.2. LetU= {x1,x2,.......xn} betheuniversalsetofelementsandE = {e1,e2,.......em} betheuniversalsetofparameters.Thenfortwoneutrosophic softsets(F,A)and(G,B)overUthecosinesimilaritymeasure CSk((F,A), (G,B)) (k=1,2)shouldsatisfythefollowingproperties(1)–(4).
(1)0 ≤ CSk((F,A), (G,B)) ≤ 1 (2) CSk((F,A), (G,B))=1 ifandonlyif(F,A)=(G,B). (3) CSk((F,A), (G,B))= CSk((G,B)(F,A))
(4) If(H,C)isaneutrosophicsoftsetoverUand(F,A) ⊆(G,B)⊆(H,C),then CSk((F,A), (H,C)) ≤ CSk((F,A)(G,B)) and CSk((F,A), (H,C)) ≤ CSk((G,B), (H,C))
Proof. (1)Sincethetruthmembershipdegree,indeterminacy-membershipdegreee andfalsity-membershipdegreeinneutrosophicsetandthevalueofcosinefunction arewithin[0,1],thesimilaritymeasurebasedoncosinefunctionisalsowithin[0,1].
Thus0 ≤ CSk((F,A), (G,B)) ≤ 1fork=1,2.
(2)Foranytwoneutrosophicsoftsets,(F,A)=(G,B)implies
TF (ei )(xj )= TG(ei )(xj ), IF (ei )(xj )= IG(ei )(xj ), FF (ei )(xj )= FG(ei )(xj ) foralli=1,2,...mandj=1,2,...n.Thus
|TF (ei )(xj ) TG(ei )(xj )| = |IF (ei )(xj ) IG(ei )(xj )| = |FF (ei )(xj ) FG(ei )(xj )| =0. So CSk((F,A), (G,B))=1fork=1,2.
Converselyif CSk((F,A), (G,B))=1fork=1,2,thisimplies
|TF (ei )(xj ) TG(ei )(xj )| =0, |IF (ei )(xj ) IG(ei )(xj )| =0, |FF (ei )(xj ) FG(ei )(xj )| =0 sincecos(0)=1.Thentheseequalitiesindicate
TF (ei )(xj )= TG(ei )(xj ), IF (ei )(xj )= IG(ei )(xj ),FF (ei )(xj )= FG(ei )(xj ) foralli=1,2,...mandj=1,2,...n.Hence(F,A)=(G,B).
(3)Clearly CSk((F,A), (G,B))= CSk((G,B)(F,A)).
(4)If(F,A)⊆(G,B)⊆(H,C),then
TF (ei )(xj ) ≤ TG(ei )(xj ) ≤ TH(ei )(xj ), IF (ei )(xj ) ≤ IG(ei )(xj ) ≤ IG(ei )(xj ), FF (ei )(xj ) ≥ FG(ei )(xj ) ≥ FH(ei )(xj ) 4
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foralli=1,2,...mandj=1,2,...n. Thenwehavethefollowinginequalites:
|TF (ei )(xj ) TG(ei )(xj )|≤|TF (ei )(xj ) TH(ei )(xj )|,
|TG(ei )(xj ) TH(ei )(xj )|≤|TF (ei )(xj ) TH(ei )(xj )|,
|IF (ei )(xj ) IG(ei )(xj )|≤|IF (ei )(xj ) IH(ei )(xj )|,
|IG(ei )(xj ) IH(ei )(xj )|≤|IF (ei )(xj ) IH(ei )(xj )|,
|FF (ei )(xj ) FG(ei )(xj )|≤|FF (ei )(xj ) FH(ei )(xj )|,
|FG(ei )(xj ) FH(ei )(xj )|≤|FF (ei )(xj ) FH(ei )(xj )|
Thus
CSk((F,A), (H,C))) ≤ CSk((F,A), (G,B))) and
CSk((F,A), (H,C))) ≤ CSk((G,B), (H,C)))
fork=1,2,sincecosinefunctionisadecreasingfunctionwithintheinterval 0, π 2
Definition3.3. LetU= {x1,x2,.......xn} betheuniversalsetofelementsandE = {e1,e2,.......em} betheuniversalsetofparametersand(F,A)and(G,B)betwo neutrosophicsoftsetsoverU.Nowweconsidertheweights wj of xj (j=1,2,...n) thentheweightedcosinesimilaritymeasuresbetween(F,A)and(G,B)isdefinedas follows. WCS1((F,A), (G,B))
= 1 m m i=1
n j=1 wj cos π 6 (|TF (ei )(xj ) TG(ei )(xj )|)+(|IF (ei )(xj ) IG(ei )(xj )|)+
n j=1 wj cos π 2 (|TF (ei )(xj ) TG(ei )(xj )|) ∨ (|IF (ei )(xj ) IG(ei )(xj )|)∨ (|FF (ei )(xj ) FG(ei )(xj )|) , WCS2((F,A), (G,B)) = 1 m m i=1
(|FF (ei )(xj ) FG(ei )(xj )|) where wj ∈ [0, 1],j=1,2,...nand n j=1 wj =1.
Inparticularifwetake wj = 1 n ,j=1,2,3....n,then WCSk((F,A), (G,B))= CSk((F,A), (G,B)), k=1,2.
Definition3.4. LetU= {x1,x2,.......xn} betheuniversalsetofelementsandE = {e1,e2,.......em} betheuniversalsetofparameters,and(F,A)and(G,B)betwo intervalvaluedneutrosophicsoftsetsoverU.Thenthecosinesimilaritymeasures between(F,A)and(G.B)isdefinedas
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CS3((F,A), (G,B))
= 1 mn m i=1
n j=1 cos π 4 ((|T F (ei )(xj ) T G(ei )(xj )|) ∨ (|I F (ei )(xj ) I G(ei )(xj )|)∨
(|F F (ei )(xj ) F G(ei )(xj )|)+(|T F (ei )(xj ) T G(ei )(xj )|)∨ (|I F (ei )(xj ) I G(ei )(xj )|) ∨ (|F F (ei )(xj ) F G(ei )(xj )|))
CS4((F,A), (G,B))
= 1 mn m i=1
n j=1 cos π 12 ((|T F (ei )(xj ) T G(ei )(xj )|)+(|I F (ei )(xj ) I G(ei )(xj )|)+ (|F F (ei )(xj ) F G(ei )(xj )|)+(|T F (ei )(xj ) T G(ei )(xj )|)+ (|I F (ei )(xj ) I G(ei )(xj )|)+(|F F (ei )(xj ) F G(ei )(xj )|)) wherethesymbol ∨ isthemaximumoperation.Thecosinesimilaritymeasures CSk((F,A), (G,B)),k=3,4satisfythefollowingproperties.
Proposition3.5. LetU= {x1,x2,.......xn} betheuniversalsetofelementsandE = {e1,e2,.......em} betheuniversalsetofparameters.Thenfortwointervalvaluedneutrosophicsoftsets(F,A)and(G,B)overUthecosinesimilaritymeasure CSk((F,A), (G,B)) (k=3,4)shouldsatisfythefollowingproperties(1)–(4).
(1)0 ≤ CSk((F,A), (G,B)) ≤ 1
(2) CSk((F,A), (G,B))=1 ifandonlyif(F,A)=(G,B).
(3) CSk((F,A), (G,B))= CSk((G,B)(F,A)).
(4) If(H,C)isaintervalvaluedneutrosophicsoftsetoverUand (F,A)⊆(G,B)⊆(H,C),then
CSk((F,A), (H,C)) ≤ CSk((F,A)(G,B)) and
CSk((F,A), (H,C)) ≤ CSk((G,B), (H,C)).
Proof. (1)Sincethetruthmembershipdegree,indeterminacy-membershipdegreee andfalsity-membershipdegreeinneutrosophicsetandthevalueofcosinefunction arewithin[0,1],thesimilaritymeasurebasedoncosinefunctionisalsowithin[0,1]. Then0 ≤ CSk((F,A), (G,B)) ≤ 1fork=3,4.
(2)Foranytwointervalvaluedneutrosophicsoftsets,(F,A)=(G,B)implies
TF (ei )(xj )= TG(ei )(xj ), IF (ei )(xj )= IG(ei )(xj ),FF (ei )(xj )= FG(ei )(xj ) foralli=1,2,...mandj=1,2,...n.Then
|T F (ei )(xj ) T G(ei )(xj )| = |I F (ei )(xj ) I G(ei )(xj )| = |F F (ei )(xj ) F G(ei )(xj )| =0 and |T F (ei )(xj ) T G(ei )(xj )| = |I F (ei )(xj ) I G(ei )(xj )| = |F F (ei )(xj ) F G(ei )(xj )| =0.
Thus CSk((F,A), (G,B))=1fork=3,4. Converselyif CSk((F,A), (G,B))=1fork=3,4,then,sincecos(0)=1, |T F (ei )(xj ) T G(ei )(xj )| =0, |I F (ei )(xj ) I G(ei )(xj )| =0, |F F (ei )(xj ) F G(ei )(xj )| =0, 6
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|T F (ei )(xj ) T G(ei )(xj )| = |I F (ei )(xj ) I G(ei )(xj )| = |F F (ei )(xj ) F G(ei )(xj )| =0.
Thentheseequalitiesindicate
TF (ei )(xj )= TG(ei )(xj ), IF (ei )(xj )= IG(ei )(xj ), FF (ei )(xj )= FG(ei )(xj ) foralli=1,2,...mandj=1,2,...n.Thus(F,A)=(G,B).
(3)Clearly CSk((F,A), (G,B))= CSk((G,B)(F,A)). (4)If(F,A)⊆(G,B)⊆(H,C),then
T F (ei )(xj ) ≤ T G(ei )(xj ) ≤ T H(ei )(xj ),
T F (ei )(xj ) ≤ T G(ei )(xj ) ≤ T H(ei )(xj ),
I F (ei )(xj ) ≤ I G(ei )(xj ) ≤ I H(ei )(xj ),
I F (ei )(xj ) ≤ I G(ei )(xj ) ≤ I H(ei )(xj ),
F F (ei )(xj ) ≥ F G(ei )(xj ) ≥ F H(ei )(xj ),
F F (ei )(xj ) ≥ F G(ei )(xj ) ≥ F H(ei )(xj ), foralli=1,2,...mandj=1,2,...n.Thuswehavethefollowinginequalites:
|T F (ei )(xj ) T G(ei )(xj )|≤|T F (ei )(xj ) T H(ei )(xj )|,
|T G(ei )(xj ) T H(ei )(xj )|≤|T F (ei )(xj ) T H(ei )(xj )|,
|T F (ei )(xj ) T G(ei )(xj )|≤|T F (ei )(xj ) T H(ei )(xj )|,
|T G(ei )(xj ) T H(ei )(xj )|≤|T F (ei )(xj ) T H(ei )(xj )|,
|I F (ei )(xj ) I G(ei )(xj )|≤|I F (ei )(xj ) I H(ei )(xj )|,
|I G(ei )(xj ) I H(ei )(xj )|≤|I F (ei )(xj ) I H(ei )(xj )|,
|I F (ei )(xj ) I G(ei )(xj )|≤|I F (ei )(xj ) I H(ei )(xj )|,
|I G(ei )(xj ) I H(ei )(xj )|≤|I F (ei )(xj ) I H(ei )(xj )|,
|F F (ei )(xj ) F G(ei )(xj )|≤|F F (ei )(xj ) F H(ei )(xj )|,
|F G(ei )(xj ) F H(ei )(xj )|≤|F F (ei )(xj ) F H(ei )(xj )|,
|F F (ei )(xj ) F G(ei )(xj )|≤|F F (ei )(xj ) F H(ei )(xj )|,
|F G(ei )(xj ) F H(ei )(xj )|≤|F F (ei )(xj ) F H(ei )(xj )| Thus
CSk((F,A), (H,C)) ≤ CSk((F,A), (G,B)) and
CSk((F,A), (H,C)) ≤ CSk((G,B), (H,C)) 7
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fork=3,4,sincecosinefunctionisadecreasingfunctionwithintheinterval 0, π 2 .
Definition3.6. LetU= {x1,x2,.......xn} betheuniversalsetofelementsandE = {e1,e2,.......em} betheuniversalsetofparameters,and(F,A)and(G,B)betwo intervalvaluedneutrosophicsoftsetsoverU.Thentheweightedcosinesimilarity measuresbetween(F,A)and(G.B)isdefinedas
CS3((F,A), (G,B))
= 1 m m i=1
n j=1 wj cos π 4 ((|T F (ei )(xj ) T G(ei )(xj )|) ∨ (|I F (ei )(xj ) I G(ei )(xj )|)∨
(|F F (ei )(xj ) F G(ei )(xj )|)+(|T F (ei )(xj ) T G(ei )(xj )|)∨ (|I F (ei )(xj ) I G(ei )(xj )|) ∨ (|F F (ei )(xj ) F G(ei )(xj )|)) , CS4((F,A), (G,B))
= 1 m m i=1
n j=1 wj cos π 12 ((|T F (ei )(xj ) T G(ei )(xj )|)+(|I F (ei )(xj ) I G(ei )(xj )|)+ (|F F (ei )(xj ) F G(ei )(xj )|)+(|T F (ei )(xj ) T G(ei )(xj )|)+ (|I F (ei )(xj ) I G(ei )(xj )|)+(|F F (ei )(xj ) F G(ei )(xj )|)) where wj ∈ [0, 1],j=1,2,...nand n j=1 wj =1.
Inparticularifwetake wj = 1 n ,j=1,2,3....n,then WCSk((F,A), (G,B))= CSk((F,A), (G,B)),k=3,4.
4. Applicationofcosinesimilaritymeasureofneutrisophicsoftset
Example4.1. Considerareasofastateareaffectedbyflood.Ateamofthreemembers U = {m1,m2,m3} fromRehabilitationdepartmentinspectthefloodaffected areasandthereliefmeasuresrecommendedbythemaredescribedbytheparameterset E = {e1,e2,e3,e4} where e1 =reliefforcroploss, e2 =reliefforecological damage, e3 =reliefforlivestockand e4 =provisionforalternatelivelihood.Based ontheserecommendationsthegovernmenthastoallocatefundsaccordingtothe levelofdamage.Algorithm:
Step1:Anintervalvaluedneutrosophicsoftset(F,A)overUisconstructedbased onpreviousrecordsofreliefmeasuresinsimilarsituation.
Step2:Anintervalvaluedneutrosophicsoftset(G,B)overUbasedonthe recommendationsoftheteamvisitingAreaIisconstructed.
Step3:Cosinesimilaritymeasuresbetween(F,A)and(G,B)iscalculated.
Step4:Anintervalvaluedneutrosophicsoftset(H,C)overUbasedonthe recommendationsoftheteamvisitingAreaIIisconstructed.
Step5:Cosinesimilaritymeasuresbetween(F,A)and(H,C)iscalculated.
Step6:Estimateresultbyusingthesimilarityvalue.Anintervalvalued(F,A) overUwithA=Ebasedonthepreviousrecordsofreliefmeasuresinsimilarsituationisgivenby.
(F,A)=
{F (e1)(m1)=([0 6, 0 7], [0 3, 0 4], [0 2, 0 3]),F (e2)(m1)=([0 4, 0 5], [0 3, 0 4], [0 2, 0 3]), F (e3)(m1)=([0 5, 0 6], [0 4, 0 5], [0 1, 0 2]),F (e4)(m1)=([0 5, 0 6], [0 2, 0 3], [0 3, 0 4]), 8
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F (e1)(m2)=([0.5, 0.6], [0.3, 0.4], [0.2, 0.3]),F (e2)(m2)=([0.4, 0.5], [0.2, 0.3], [0.5, 0.6]), F (e3)(m2)=([0.5, 0.6], [0.1, 0.2], [0.3, 0.4]),F (e4)(m2)=([0.5, 0.6], [0.3, 0.5], [0.4, 0.5]), F (e1)(m3)=([0.7, 0.8], [0.4, 0.5], [0.2, 0.3]),F (e2)(m3)=([0.5, 0.6], [0.3, 0.4], [0.2, 0.3]), F (e3)(m3)=([0 6, 0 7], [0 4, 0 5], [0 3, 0 4]),F (e4)(m3)=([0 6, 0 7], [0 2, 0 3], [0 4, 0 5])}
Anintervalvalued(G,B)overUwithB=EbasedonrecommendationsoftheexpertteamvisitingAreaI.
(G,B)=
{G(e1)(m1)=([0 6, 0 7], [0 5, 0 6], [0 4, 0 5]),G(e2)(m1)=([0 3, 0 4], [0 5, 0 6], [0 4, 0 5]), G(e3)(m1)=([0 3, 0 4], [0 4, 0 5], [0 5, 0 6]),G(e4)(m1)=([0 2, 0 3], [0 3, 0 4], [0 5, 0 6]), G(e1)(m2)=([0 8, 0 9], [0 4, 0 5], [0 2, 0 3]),G(e2)(m2)=([0 5, 0 6], [0 1, 0 2], [0 3, 0 4]), G(e3)(m2)=([0.5, 0.6], [0.1, 0.2], [0.2, 0.3]),G(e4)(m2)=([0.3, 0.4], [0.3, 0.4], [0.2, 0.3]), G(e1)(m3)=([0.5, 0.6], [0.4, 0.5], [0.3, 0.4]),G(e2)(m3)=([0.4, 0.5], [0.2, 0.3], [0.5, 0.6]), G(e3)(m3)=([0.2, 0.3], [0.4, 0.5], [0.4, 0.5]),G(e4)(m3)=([0.2, 0.3], [0.5, 0.6], [0.3, 0.4])}.
Anintervalvalued(H,C)overUwithC=Ebasedonrecommendationsoftheexpert teamvisitingAreaII.
(H,C)=
{H(e1)(m1)=([0 4, 0 5], [0 2, 0 3], [0 1, 0 2]),H(e2)(m1)=([0 3, 0 4], [0 2, 0 3], [0 1, 0 2]), H(e3)(m1)=([0 6, 0 7], [0 2, 0 3], [0 1, 0 2]),H(e4)(m1)=([0 1, 0 2], [0 3, 0 4], [0 6, 0 7]), H(e1)(m2)=([0 4, 0 5], [0 2, 0 3], [0 5, 0 6]),H(e2)(m2)=([0 2, 0 3], [0 3, 0 4], [0 4, 0 5]), H(e3)(m2)=([0 1, 0 2], [0 5, 0 6], [0 4, 0 5]),H(e4)(m2)=([0 5, 0 6], [0 3, 0 4], [0 1, 0 2]), H(e1)(m3)=([0.2, 0.3], [0.4, 0.5], [0.3, 0.4]),H(e2)(m3)=([0.1, 0.2], [0.3, 0.4], [0.4, 0.5]), H(e3)(m3)=([0.2, 0.3], [0.4, 0.5], [0.4, 0.5]),H(e4)(m3)=([0.3, 0.4], [0.5, 0.6], [0.6, 0.7])}. UsingDefinition 3.4,weget
CS3((F,A), (G,B))=0.904 and
CS3((F,A), (H,C))=0.871. Wehave CS3((F,A), (G,B)) >CS3((F,A), (H,C)).HenceweconcludethatArea Iisseverelyaffectedbyflood.
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I.R.Sumathi (sumathi raman2005@yahoo.co.in)
DepartmentofMathematics,KumaraguruCollegeofTechnology,Coimbatore,Tamilnadu,India
I.Arockiarani (stell11960@gmail.com)
DepartmentofMathematics,NirmalaCollegeforwomen,Coimbatore,Tamilnadu, India
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