ANewHybridDistance-BasedSimilarityMeasureforRefined NeutrosophicSetsanditsApplicationinMedicalDiagnosis
1 VakkasUlu¸cay, 2AdilKılı¸c, 3MemetS¸ahinand 4HarunDeniz
1K¨okl¨uceNeighbourhood,Araban,Gaziantep27310-Turkey
2,3,4DepartmentofMathematics,GaziantepUniversity,Gaziantep27310-Turkey ∗
Correspondingauthor:vulucay27@gmail.com
Articlehistory
Received:6March2018
Receivedinrevisedform:7August2018
Accepted:12November2018
Publishedonline:1April2019
Abstract Theaimofthispaperistointroduceanewdistance-basedsimilaritymeasure forrefinedneutrosophicsets.Thepropertiesoftheproposednewdistance-basedsimilarity measurehavebeenstudiedandthefindingsareappliedinmedicaldiagnosisofsome diseaseswithacommonsetofsymptoms.
Keywords Neutrosophicsets;refinedneutrosophicsets;similaritymeasuredecisionmaking;medicaldiagnosis.
MathematicsSubjectClassification 03B52
1Introduction
Thevaguenessoruncertaintyrepresentationofimperfectknowledgebecomesacrucialissuein theareasofcomputerscienceandartificialintelligence.Todealwiththeuncertainty,thefuzzy setproposedbyZadeh[1]allowstheuncertaintyofasetwith amembershipdegreebetween0 and1.Then,Atanassov[2]introducedanintuitionisticFuzzyset(IFS)asageneralizationof theFuzzyset.TheIFSrepresentstheuncertaintywithrespecttobothmembershipandnonmembership.However,itcanonlyhandleincompleteinformationbutnottheindeterminate andinconsistentinformationwhichexistscommonlyinreal situations.Therefore,Smarandache[3]proposedaneutrosophicset.Itcanindependently expresstruth-membershipdegree, indeterminacy-membershipdegree,andfalsemembershipdegreeanddealwithincomplete,indeterminate,andinconsistentinformation.Also,several generalizationofthesettheoriesmade suchasfuzzymulti-settheory[4,5],intuitionsiticfuzzy multi-settheory[6-10]andrefined neutrosophicsettheory[11-20].Manyresearchtreatingimprecisionanduncertaintyhavebeen developedandstudied.Sincethen,itisappliedtovariousareas,suchasdecisionmaking problems[21-44].
Anothergeneralizationofabovetheoriesthatisrelevantforourworkissinglevalued neutrosophicrefined(multi)settheorybyintroducedYe[45]whichcontainafewdifferentvalues.Asinglevaluedneutrosophicmultisettheoryhavetruth-membershipsequence 35:1(2019)83–96 | www.matematika.utm.my | eISSN0127-9602 |
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(µ1 A (u) ,µ2 A (u) ,...µp A (u)),indeterminacy-membershipsequence(v1 A (u) ,v2 A (u) ,...vp A (u))and falsity-membershipsequence(w1 A (u) ,w2 A (u) ,...wp A (u))oftheelement u ∈U
Thepaperisorganizedasfollows:Insection2,introducessomeconceptsandbasicoperationsarereviewed.Insection3,presentsanewdistance-basedsimilaritymeasureforrefined neutrosophicsetsandinvestigatestheirproperties.Insection4,thesimilaritymeasuresare appliedtomedicinediagnosis.Finally,Conclusionsandfurtherresearcharecontained
2Preliminaries
Definition1 [2]Let U beauniverse. A neutrosophicsets A over U isdefinedby A = {≺, (µA(µ),vA(µ),wA : u ∈U}
where, µA (u), vA (u)and wA (u)arecalledtruth-membershipfunction,indeterminacy-membership functionandfalsity-membershipfunction,respectively. Theyarerespectivelydefinedby µA : U→] 0, 1+ [ ,vA : U→] 0, 1+ [ ,wA : U→] 0, 1+ [
suchthat0 ≤ µA (u)+vA (u)+wA (u) ≤ 3+
Definition2 [19]Let U beauniverse.Asinglevaluedneutrosophicset(SVN-set)over U isaneutrosophicsetover U ,butthetruth-membershipfunction,indeterminacy-membership functionandfalsity-membershipfunctionarerespectivelydefinedby
µA : U→ [0, 1],vA : U → [0, 1] ,wA : U→ [0, 1]
Suchthat0 ≤ µA (u)+vA (u)+wA (u) ≤ 3
Definition3 [32]Let U beauniverse.Aneutrosophicmultisetset(Nms) A on U canbe definedasfollows:
A = { ≺ u µ 1 A (u) ,µ 2 A (u) ,...µp A (u) v 1 A (u) ,v 2 A (u) ,...vp A (u) w 1 A (u) ,w 2 A (u) ,...wp A (u) : u ∈U} where, µ 1 A (u) µ 2 A (u) ,...µp A (u): U→ [0, 1] v 1 A (u) v 2 A (u) ,...vp A (u): U→ [0, 1] and w 1 A (u) w 2 A (u) ,...wp A (u): U→ [0, 1] suchthat 0 ≤ sup µi A (u)+sup vi A (u)+sup wi A (u) ≤ 3 (i =1, 2,...,P )and µ 1 A (u) ,µ 2 A (u) ,...,µp A (u) v 1 A (u) ,v 2 A (u) ,...,vp A (u) and w 1 A (u) ,w 2 A (u) ,...,wp A (u)
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isthetruth-membershipsequence,indeterminacy-membershipsequenceandfalsity-membershipsequenceoftheelement u respectively.Also,Piscalledthedimension(cardinality)of NmsA denoted d(A)Wearrangethetruth-membershipsequenceindecreasingorderbutthe correspondingindeterminacy-membershipandfalsity-membershipsequencemaynotbeindecreasingorincreasingorder.
ThesetofallNeutrosophicmultisetson U isdenotedbyNMS(U ). Definition4 [11,19,38]Let A, B∈ NMS(U ).Then,
(1) A issaidtobeNm-subsetof B isdenotedby A⊆B if µi A (u) ≤ µi B (u) vi A (u) ≥ vi B (u) ,wi A (u) ≥ wi B (u) , ∀ u ∈U and i =1, 2,...P
(2) A issaidtobeneutrosophicequalof B isdenotedby A = B if µi A (u)= µi B (u) vi A (u)= vi B (u) wi A (u)= wi B (u) , ∀ u ∈U and i =1, 2,...P
(3)Thecomplementof A denotedby Ac andisdefinedby Ac = ≺ u w 1 A (u) ,w 2 A (u) ,...,wp A (u) v 1 A (u) ,v 2 A (u) ,...vp A (u) µ 1 A (u) ,µ 2 A (u) ,...µp A (u) : u ∈U
(4)If µi A (u)=0and vi A (u)= wi A (u)=1forall u ∈U and i =1, 2,...P then A iscallednullns-setanddenotedbyΦ
(5)If µi A (u)=1and vi A (u)= wi A (u)=0forall u ∈U and i =1, 2,...P then A iscalleduniversalns-setanddenotedby U
(6)Theunionof A and B isdenotedby A∪B = C andisdefinedby C = {
≺ u µ 1 C (u) ,µ 2 C (u) ,...µp C (u) v 1 C (u) ,v 2 C (u) ,...vp C (u) w 1 C (u) ,w 2 C (u) ,...wp C (u) : u ∈U}
Where µi C = µi A (u) ∨ µi B (u), vi C = vi A (u) ∧ vi B (u), wi C = wi A (u) ∧ wi B (u) ∀ u ∈U and i =1, 2,...P
(7)Theintersectionof A and B isdenotedby A∩B = D andisdefinedby D = {
≺ u µ 1 D (u) ,µ 2 D (u) ,...µp D (u) v 1 D (u) ,v 2 D (u) ,...vp D (u) w 1 D (u) ,w 2 D (u) ,...wp D (u) : u ∈U}
where µi D = µi A (u) ∨ µi B (u), vi D = vi A (u) ∧ vi B (u), wi D = wi A (u) ∧ wi B (u), ∀ u ∈U and i =1, 2,...P
(8)Theadditionof A and B isdenotedby A+B = U1 andisdefinedby
U1 = {
≺ u µ 1 U1 (u) ,µ 2 U1 (u) ,...µp U1 (u) v 1 U1 (u) ,v 2 U1 (u) ,...vp U1 (u) w 1 U1 (u) ,w 2 U1 (u) ,...wp U1 (u) : u ∈U
where µi U1 = µi A (u)+µi B (u) µi A (u) µi B (u) vi U1 = vi A (u) vi B (u), wi U1 = wi A (u) wi B (u) ∀ u ∈U and i =1, 2,...P
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(9)Themultiplicationof A and B isdenotedby AXB = U2 andisdefinedby U2 = { ≺ u µ 1 U2 (u) ,µ 2 U2 (u) ,...µp U2 (u) v 1 U2 (u) ,v 2 U2 (u) ,...vp U2 (u) w 1 U2 (u) ,w 2 U2 (u) ,...wp U2 (u) : u ∈U where µi U2 = µi A (u) µi B (u), vi U2 = vi A (u)+ vi B (u) vi A (u) vi B (u) wi U2 = wi A (u)+ wi B (u) wi A (u) wi B (u) ∀ u ∈U and i =1, 2,...P Here ∨, ∧, +,., denotesmaximum,minimum,addition,multiplication,subtractionof realnumbersrespectively.
Definition5 [12]Let A = { ≺ u µ 1 A (u) ,µ 2 A (u) ,...µp A (u) v 1 A (u) ,v 2 A (u) ,...vp A (u) w 1 A (u) ,w 2 A (u) ,...wp A (u) : u ∈U and B = { ≺ u µ 1 B (u) ,µ 2 B (u) ,...µp B (u) v 1 B (u) ,v 2 B (u) ,...vp B (u) w 1 A (u) ,w 2 A (u) ,...wp A (u) : u ∈U
AndbetwoNMSs,thenthenormalizedhammingdistancebetween A and B canbedefinedas follows: dp (A,B)= 1 3 p i=1 ωi |µA (ui) µB (ui)|p + |vA (ui) vB (ui)|p + |wA (ui) wB (ui)|p 1/p where A, B aretwoSVNSs p> 0,wi (i =1, 2,...,p)aretheweightoftheelement xi (i =1, 2,...,p) with wi ≥ 0and n i=1 wi =1
Definition6 Fortworefinedneutrosophicsets A and B inauniverseofdiscoursewhichare denotedby A = { ≺ u µ 1 A (u) ,µ 2 A (u) ,...µp A (u) v 1 A (u) ,v 2 A (u) ,...vp A (u) w 1 A (u) ,w 2 A (u) ,...wp A (u) : u ∈U
VakkasUlu¸cay etal. /MATEMATIKA35:1(2019)83–96 87 and B = {
≺ u µ 1 B (u) ,µ 2 B (u) ,...µp B (u) v 1 B (u) ,v 2 B (u) ,...vp B (u) w 1 B (u) ,w 2 B (u) ,...wp B (u) : u ∈U
µi A (u) µi B (u) vi A (?) vi B (u) wi A (u) wi B (u) ∈ [0, 1]forevery i =1, 2,...P .Letusconsiderthe weight ωi i =1, 2,...P with ωi ≥ 0,i =1, 2,...P and P i=1 ωi =1
Then,wedefinetherefinedgeneralizedneutrosophicweighteddistancemeasure: dP (A,B)= 1 3
P i=1 ωi µ i A (ui) µ i B (ui) P + v i A (ui) v i B (ui) P + w i A (ui) w i B (ui) P 1/p , (1) where P> 0
AstheHammingdistanceandEuclideandistance,whicharetwotypicaldistancemeasures, areusuallyusedinpracticalapplicationswhen P =1, 2wecanobtaintherefinedneutrosophicweightedHammingdistanceandtherefinedneutrosophicweightedEuclideandistance, respectively,asfollows: d (A,B)= 1 3
P i=1 ωi[ µ i A (ui) µ i B (ui) + v i A (ui) v i B (ui) + w i A (ui) w i B (ui) , (2) d2 (A,B)= 1 3
P i=1 ωi µ i A (ui) µ i B (ui) 2 + v i A (ui) v i B (ui) 2 + w i A (ui) w i B (ui) 2 1/p . (3)
Therefore,equations(2)and(3)arethespecialcasesofequation(1).Then,forthedistance measure,wehavethefollowingproposition.
Proposition7 Thedistancemeasuredp (A,B ) for p> 0 satisfiesthefollowingproperties: (H1)0 ≤ dp (A, B ) ≤ 1; (H2) dp (A, B )=0 ifandonlyif A = B ; (H3) dp (A, B )= dp (B, A ); (H4)If ⊆B⊆C, A3 isarefinedneutrosophicin U ,then dp(A, B ) ≤ dp(AC)anddp(B, C) ≤ dp(A, C)
Proof: Itiseasytoseethat dp (A, B)satisfiestheproperties(H1) (H3)Therefore,we onlyprove(H4).Let A⊆B⊆C,then µ i A (u) ≤ µ i B (u) ≤ µ i C (u) ,v i A (u) ≥ v i B (u) ≥ µ i C (u) ,w i A (u) ≥ w i B (u) ≥ µ i C (u) , ∀ ui ∈U and i =1, 2,...P weobtainfollowingrelations: µi A (ui) µi B (ui) P ≤ µi A (ui) µi C (ui) P ; µi B (ui) µi C (ui) P ≤ µi A (ui) µi C (ui) P v i A (ui) v i B (ui) P ≤ v i A (ui) v i C (ui) P ; v i B (ui) v i C (ui) P ≤ v i A (ui) v i C (ui) P w i A (ui) w i B (ui) P ≤ w i A (ui) w i C (ui) P ; w i B (ui) w i C (ui) P ≤ w i A (ui) w i C (ui) P
VakkasUlu¸cay etal. /MATEMATIKA35:1(2019)83–96 88 Hence, µ i A (ui) µ i B (ui) P + v i A (ui) v i B (ui) P + w i A (ui) w i B (ui) P ≤ µ i A (ui) µ i C (ui) P + v i A (ui) v i C (ui) P + w i A (ui) w i C (ui) P µi B (ui) µi C (ui) P + vi B (ui) vi C (ui) P + wi B (ui) wi C (ui) P ≤ µ i A (ui) µ i C (ui) P + v i A (ui) v i C (ui) P + w i A (ui) w i C (ui) P dλ(A, C) ≥ dλ(A, B)and dλ(A, C) ≥ dλ(B, C)for λ> 0 ✷
1 3 [ω1 |µ1 A (u1) µ1 B (u1)|2 + |v1 A (u1) v1 B (u1)|2 + |w1 A (u1)
P i=1 ωi [ µi A (ui) µi B (ui) 2 + vi A (ui) vi B (ui) 2 + wi A (ui) wi B (ui) 2] 1/2
= 1/2 = 1 3 [0 2 |0 8 0 5|2 + |0 3 0 2|2 + |0 2 0 1|2 +0.4 |0.5 0.7|2 + |0.1 0.3|2 + |0.3 0.3|2 +0 4(|0 6 0 6|2 + |0 5 0 4|2 + |0 4 0 2|2)]
1/2 = 1 3 {(0, 2(0, 09+0, 01+0, 01))+(0, 4(0, 04+0, 04))+(0, 4(0, 01+0, 04))} 1/2 d2 (A, B)=0, 157
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Definition9 Let A = { ≺ u µ 1 A (u) ,µ 2 A (u) ,...µp A (u) v 1 A (u) ,v 2 A (u) ,...vp A (u) w 1 A (u) ,w 2 A (u) ,...wp A (u) : u ∈U} and B = { ≺ u µ 1 B (u) ,µ 2 B (u) ,...µp B (u) v 1 B (u) ,v 2 B (u) ,...vp B (u) w 1 B (u) ,w 2 B (u) ,...wp B (u) : u ∈U} betworefinedneutrosophicsets.Then,hybridsimilaritymeasurebetweenrefinedneutrosophic sets A and B denoted Hybd (A, B)= ϕ 1 3
P i=1 ωi µ i A (ui) µ i B (ui) + v i A (ui)+ v i B (ui) + w i A (ui) w i B (ui) +(1 ϕ) 1 3
P i=1 ωi µ i A (ui) µ i B (ui) 2 + v i A (ui)+ v i B (ui) 2 + w i A (ui) w i B (ui) 2 1/2 for i =1, 2,...,P .Notethatsimilarityanddistance(dissimilarity)measuresarecomplementary:whenthefirstincreases,theseconddecreases.Normalizeddistancemeasureand similaritymeasurebelowaredualconcepts.Thus, δ (A, B)=1 Hybd(A, B) andviceversa.Thepropertiesofdistancemeasuresbelowarecomplementarytothoseof similaritymeasures. Remark: Then,tocomparehybridsimilaritymeasures,thepositiveidealrefinedneutrosophic solutionandnegativeidealrefinedneutrosophicsolutionaredefinedas; ϕ+ i = max µ1 A (u) ,µ2 A (u) ,...µp A (u) +min v1 A (u) ,v2 A (u) ,...vp A (u) +min w1 A (u) ,w2 A (u) ,...wp A (u) 3 ϕi = min µ1 A (u) ,µ2 A (u) ,...µp A (u) +max v1 A (u) ,v2 A (u) ,...vp A (u) +max w1 A (u) ,w2 A (u) ,...wp A (u) 3 respectively,for i =1, 2,...,n
Proposition10 Thesimilaritymeasure δp (A, B) for p> 0 satisfiesthefollowingproperties; (HD1)0 ≤ δp (A, B) ≤ 1; (HD2) δp (A, B)=1 ifandonlyif A = B; (HD3) δp (A, B)= δp (B, A); (HD4) If ⊆B⊆C, C isarefinedneutrosophicin U ,then δp (A, C) ≤ δp (A, B) and δp (A, C) ≤ δp (B, C)
VakkasUlu¸cay etal. /MATEMATIKA35:1(2019)83–96 90 Assumethattherearetworefinedneutrosophic A = { ≺ u µ 1 A (u) ,µ 2 A (u) ,...µp A (u) v 1 A (u) ,v 2 A (u) ,...vp A (u) w 1 A (u) ,w 2 A (u) ,...wp A (u) : u ∈U} and B = { ≺ u µ 1 B (u) ,µ 2 B (u) ,...µp B (u) v 1 B (u) ,v 2 B (u) ,...vp B (u) w 1 B (u) ,w 2 B (u) ,...wp B (u) : u ∈U}
inauniverseofdistance ∀ ui ∈U .Thus,accordingtotherelationshipbetweenthedistanceand thesimilaritymeasure,wecanobtainthefollowingrefinedneutrosophicsimilaritymeasure: δ (A, B)=1 Hybd (A, B) =1 ϕ 1 3
P i=1 ωi µ i A (ui) µ i B (ui) + v i A (ui)+ v i B (ui) + w i A (ui) w i B (ui) + (1 ϕ) 1 3
P i=1 ωi µi A (ui) µi B (ui) 2 + vi A (ui)+ vi B (ui) 2 + wi A (ui) wi B (ui) 2 1/2 (4)
Obviously,wecaneasilyprovethat δ1 (A,B)satisfiedtheproperties(HD1) (HD4)inproposition10bytherelationshipbetweenthedistanceandthesimilaritymeasureandtheproofof proposition7,whichisomittedhere. Furthermore,wecanalsoproposeanotherrefinedneutrosophicsimilaritymeasure: δ2 (A, B)= 1 Hybd (A, B) 1+ Hybd (A, B) =
1 ϕ 1 3 P i=1 ωi µi A (ui) µi B (ui) + vi A (ui)+ vi B (ui) + wi A (ui) wi B (ui) + (1 ϕ) 1 3 P i=1 ωi µi A (ui) µi B (ui) 2 + vi A (ui)+ vi B (ui) 2 + wi A (ui) wi B (ui) 2 1/2 1+ ϕ 1 3 P i=1 ωi µi A (ui) µi B (ui) + vi A (ui)+ vi B (ui) + wi A (ui) wi B (ui) + (1 ϕ) 1 3 P i=1 ωi µi A (ui) µi B (ui) 2 + vi A (ui)+ vi B (ui) 2 + wi A (ui) wi B (ui) 2 1/2
Then,thesimilaritymeasure δ2 (A, B)alsosatisfiedtheproperties(HD1) (HD4)inProposition2. Proof: Itiseasytoseethat δ2 (A, B)satisfiestheproperties(HD1) (HD3).Therefore,we onlypropertyhavetoprove(HD4) Asweobtain δp (A, B) ≤ δp (A, C)and δp (B, C) ≤ δp (A, C)for p> 0fromtheproperty (H4)inproposition1,thereare1 δp (A, B) ≥ 1 δp (A, C) , 1 δp (B, C) ≥ 1 δp (A, C). 1+ δp (A, B) ≤ 1+ δp (A, C)and1+ δp (B, C) ≤ 1+ δp (A , C).Then,therearethefollowing inequalities: 1 dp(A , B) 1+ dp(A , B) ≥ 1 dp (A, C) 1+ dp (A, C) and 1 dp (B, C) 1+ dp (B, C) ≥ 1 dp (A, C) 1+ dp (A, C)
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Then,thereare δp (A, C) ≤ δp (A, B)and δp (A, C) ≤ δp (B, C).Hence,theproperty(HD4)is satisfied. ✷ Example11: Assumethatwehavethefollowingtworefinedneutrosophichybridsimilarity measureinauniverseofdiscourse u ∈U ;let ω1 =0.2,ω2 =0.3,ω3 =0.5 A = u, (0 5, 0 5, 0 3), (0 8, 0 1, 0 2), (0 2, 0 8, 0 9) B = u, (0 5, 0 1, 0 6), (0 2, 0 3, 0 9), (0 6, 0 3, 0 2)
µ3 B (u3) 2 + v3 A (u3 ) v3 B (u3 ) 2 + w3 A (u3 ) w3 B (u3 ) 2)]
0 2(|0 5 0 5| + |0 8 0 2| + |0 2 0 6|) +0 3(|0 5 0 1| + |0 1 0 3| + |0 8 0 3|) +0 5(|0 3 0 6| + |0 2 0 9| + |0 9 0 2|)
1 3 [0 2 |0 5 0 5|2 + |0 8 0 2|2 + |0 2 0 6|2 +0 3 |0 5 0 1|2 + |0 1 0 3|2 + |0 8 0 3|2 +0.5(|0.3 0.6|2 + |0.2 0.9|2 + |0.9 0.2|2 )]
=0.3 1 3 [(0, 2(0, 6+0, 4))+(0, 3(0, 4+0, 2+0, 5))+(0, 5(0, 3+0, 7+0, 7))] +0 7 1 3 {(0, 2(0, 36+0, 16))+(0, 3(0, 16+0, 04+0, 25))+(0, 5(0, 09+0, 49+0, 49))} 1/2 Hybd (A, B)=0 4935 δ (A, B)=1 Hybd (A, B) δ (A, B)=0 5065
4MedicalDiagnosisUsingtheHybridSimilarityMeasure
Weconsideramedicaldiagnosisproblemfrompracticalpointofviewforillustrationofthe proposedapproach.Medicaldiagnosiscomprisesofuncertaintiesandincreasedvolumeofinformationavailabletophysiciansfromnewmedicaltechnologies.Theprocessofclassifying differentsetofsymptomsunderasinglenameofadiseaseisaverydifficulttask.Insome practicalsituations,thereexistspossibilityofeachelementwithinalowerandanupperapproximationofrefinedneutrosophicsets.Itcandealwiththemedicaldiagnosisinvolvingmore indeterminacy.Actuallythisapproachismoreflexibleandeasytouse.Theproposedsimilarity measureamongthepatientsversussymptomsandsymptomsversusdiseaseswillprovidethe propermedicaldiagnosis.Themainfeatureofthisproposed approachisthatitconsiderstruth membership,indeterminateandfalsemembershipofeachelementbetweentwoapproximations ofrefinedneutrosophicsetsbytakingonetimeinspectionfordiagnosis.
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Now,anexampleofamedicaldiagnosisispresented.Let P = {Erol, Harun, Deniz} be asetofpatients, D = {ViralFever, Tuberculosis, Throat} diseasebeasetofdiseasesand S={Throatpain,headache,bodypain}beasetofsymptoms.Oursolutionistoexaminethe patientatdifferenttimeintervals(threetimesaday),whichinturngivearisetodifferenttruth membership,indeterminateandfalsemembershipfunctionforeachpatientinTable1-Table8. Let ω1 =0 2,ω2 =0 5,ω3 =0 3.Letthesamplesbetakenatthreedifferenttimingsinaday (in07:00,15:00and23:00)
Table1:Q(TheRelationBetweenPatientandSymptoms)
,
,
8)(0 9, 0 3, 0 7) (0 7, 0 2, 0 1), (0 5, 0 6, 0 3)(0 2, 0 3, 0 4) (0 8, 0 2, 0 9), (0 7, 0 4, 0 4)(0 2, 0 7, 0 5) (0 5, 0 5, 0 5), (0 6, 0 3, 0 5)(0 4, 0 3, 0 5) (0 5, 0 5, 0 3), (0 3, 0 4, 0 5)(0 3, 0 2, 0 2) (0 9, 0 5, 0 6), (0 3, 0 3, 0 5)(0 2, 0 3, 0 4) (0 6, 0 7, 0 8), (0 5, 0 3, 0 5)(0 5, 0 4, 0 1) (0 1, 0 3, 0 1), (0 6, 0 9, 0 3)(0 2, 0 4, 0 4)
Harun (0 6, 0 5, 0 5), (0 5, 0 4, 0 5)(0 4, 0 3, 0 4) (0 1, 0 5, 0 6), (0 6, 0 7, 0 5)(0 3, 0 3, 0 4) (0 8, 0 6, 0 6), (0 5, 0 6, 0 4)(0 4, 0 3, 0 5) (0 3, 0 5, 0 6), (0 6, 0 9, 0 4)(0 5, 0 3, 0 9) (0 3, 0 3, 0 6), (0 6, 0 7, 0 5)(0 3, 0 3, 0 9) (0 5, 0 2, 0 3), (0 9, 0 6, 0 5)(0 9, 0 3, 0 4) (0 8, 0 7, 0 1), (0 8, 0 7, 0 5)(0 3, 0 7, 0 4) (0 8, 0 5, 0 6), (0 5, 0 8, 0 5)(0 8, 0 5, 0 4) (0 8, 0 5, 0 7), (0 6, 0 5, 0 4)(0 5, 0 3, 0 4)
Deniz (0 8, 0 6, 0 6), (0 6, 0 6, 0 6)(0 6, 0 7, 0 5) (0 1, 0 5, 0 6), (0 9, 0 6, 0 8)(0 9, 0 5, 0 7) (0 4, 0 5, 0 6), (0 7, 0 6, 0 6)(0 7, 0 8, 0 6) (0 1, 0 3, 0 6), (0 8, 0 8, 0 7)(0 8, 0 5, 0 9) (0 7, 0 4, 0 2), (0 9, 0 8, 0 8)(0 6, 0 8, 0 5) (0 5, 0 2, 0 3), (0 8, 0 9, 0 7)(0 9, 0 5, 0 5) (0 8, 0 5, 0 6), (0 6, 0 7, 0 9)(0 6, 0 9, 0 5) (0 9, 0 5, 0 7), (0 6, 0 8, 0 7)(0 5, 0 8, 0 6) (0 7, 0 2, 0 1), (0 8, 0 6, 0 9)(0 5, 0 9, 0 9)
Table2:R(TheRelationAmongSymptomsandDiseases)
R ViralFever Tuberculosis Typhoid Throatpain (0 2, 0 5, 0 6), (0 3, 0 1, 0 6)(0 2, 0 8, 0 4) (0 6, 0 5, 0 7), (0 2, 0 1, 0 5)(0 4, 0 3, 0 4) (0 1, 0 5, 0 6), (0 3, 0 1, 0 2)(0 5, 0 3, 0 7) Headache (0 4, 0 5, 0 6), (0 2, 0 7, 0 5)(0 7, 0 4, 0 1) (0 7, 0 2, 0 1), (0 5, 0 7, 0 3)(0 3, 0 2, 0 4) (0 3, 0 2, 0 3), (0 3, 0 1, 0 8)(0 9, 0 3, 0 4) BodyPain (0 9, 0 5, 0 7), (0 6, 0 9, 0 1)(0 5, 0 9, 0 4) (0 7, 0 5, 0 6), (0 3, 0 1, 0 5)(0 5, 0 3, 0 9) (0
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Table5:TheWeightedHybridDistanceRefinedNeutrosophicSets Q and R with ϕ+ 1 =0.43
Weightedhybrid
ViralFever Tuberculosis Typhoid Erol 0,105623 0,133542 0,196243 Harun 0,10078 0,12956 0,206745 Deniz 0,190548 0,123797 0,221704
Optimal Erol(ViralFever);Harun(ViralFever);Deniz(Tuberculosis)
Table6:TheWeightedSimilarityMeasureHybridDistanceRefinedNeutrosophicSets Q and R with ϕ+ 1 =0.43.
WeightedsimilarityMeasurehybrid
ViralFever Tuberculosis Typhoid Erol 0,894377 0,866458 0,803757 Harun 0,89922 0,87044 0,793255 Deniz 0,809452 0,876203 0,778296
Optimal Erol(ViralFever);Harun(ViralFever);Deniz(Tuberculosis)
Table7:TheWeightedSimilarityMeasureHybridDistanceRefinedNeutrosophicSets Q and R with ϕ+ 2 =0 5
WeightedsimilarityMeasurehybrid
ViralFever Tuberculosis Typhoid Erol 0,878966 0,880894 0,824446 Harun 0,911212 0,885525 0,815309 Deniz 0,829983 0,890417 0,801533
Optimal Erol(Tuberculosis);Harun(ViralFever);Deniz(Tuberculosis)
Table8:TheWeightedSimilarityMeasureHybridDistanceRefinedNeutrosophicSets Q and R with ϕ+ 2 =0 67
WeightedsimilarityMeasurehybrid
ViralFever Tuberculosis Typhoid Erol 0,84154 0,915953 0,874691 Harun 0,940335 0,922159 0,86887 Deniz 0,879843 0,924937 0,857965
Optimal Erol(Tuberculosis);Harun(ViralFever);Deniz(Tuberculosis)
5Conclusion
Inthispaper,anewhybridsimilaritymeasureandaweighted hybridsimilaritymeasurefor refinedneutrosophicsetsarepresentedandsomeofitsbasic propertiesarediscussed.The
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proposedhybridsimilaritymeasureenrichesthetheoriesandtechniquesformeasuringthe degreeofhybridsimilaritybetweenrefinedneutrosophicsets.Thismeasuregreatlyreduces theinfluenceofimprecisemeasuresandprovidesanextremelyintuitivequantification.The effectivenessoftheproposedhybridsimilaritymeasureisdemonstratedinanumericalexample withthehelpofmeasureofperformanceandmeasureoferror(refertoTable1-8).Moreover, medicaldiagnosisproblemshavebeenexhibitedthroughahypotheticalcasestudybyusing thisproposedhybridsimilaritymeasure.Theauthorshopethattheproposedconceptcanbe appliedinsolvingrealisticmulti-criteriadecisionmakingproblems.
CompliancewithEthicalStandards
Theauthorsdeclarethatthereisnoconflictofinterestsregardingthepublicationofthispaper.
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