Studyonsingle-valuedneutrosophicgraph withapplicationinshortestpathproblem
RuxiangLiu ✉
ISSN2468-2322
Receivedon3rdJune2020 Revisedon25thJune2020 Acceptedon14thJuly2020 doi:10.1049/trit.2020.0111 www.ietdl.org
DepartmentofElectronicandInformationEngineering,BozhouUniversity,Bozhou,Anhui236800,People’sRepublicofChina ✉ E-mail:kelton_bzsz@sina.cn
Abstract: Fuzzysetandneutrosophicsetaretwoefficienttoolstohandletheuncertaintiesandvaguenessofanyrealworldproblems.Neutrosophicsetismoreusefulthanfuzzyset(intuitionisticfuzzysets)tomanagetheuncertaintiesof areal-lifeproblem.Thisstudyintroducessomenewconc eptsofsingle-valuedneutrosophicgraph(SVNG).The authorshavediscussedthedefinitionofregularSVNG, completeSVNGandstrongSVNG.Theshortestpathproblem isawell-knowncombinatorialoptimisationproblemint hefieldofgraphtheoryduet oitsvariousapplications. Uncertaintyispresentinalmosteveryapplicationofsho rtestpathproblemwhichmakesitveryhardtodecidethe edgeweightproperly.Themainobjectivebehindtheworki nthisstudyistodetermineanalgorithmictechniquefor shortestpathproblemwhichwillbeveryeasyandefficientf oruseinreal-lifescenarios.Inthisstudy,theauthors considerneutrosophicnumbertodescribetheedgeweightsofaneutrosophicgraphforneutrosophicshortestpath problem.Analgorithmisintroducedtosolvethisproblem.TheuncertaintiesareincorporatedinBellman – Ford algorithmforshortestpathproblemusingneutrosophicnumberasarclength.Theyuseonenumericalexampleto illustratetheeffectivenessoftheproposedalgorithm.
1Introduction
In1965,Prof.Zadeh[1]describedtheideaoffuzzysettheorywhich hasbeenexpeditiouslyutilisedtomodeltheseveraldecision-making problemsinwhichuncertaintiesmayexist.Fuzzysetisamodified versionofsimpleset,wherealltheelementsofthefuzzyset havechangingdegreesofmembershipvalues.Thesimpleset (crispset)alwayshavetwotruthvalues,either0(indicatefalse)or 1(indicatetrue).Crispsetisunabletohandletheuncertaintiesof theproblems.However,thefuzzysetallowsforitsobjectstohave themembershipdegreewithin1and0whichprovidesmore beneficialresults,ratherthanconsideringonlysinglevalueof either1or0.Themembershipdegreeofafuzzysetisaspecific singlevaluewithin0and1.Expertsarenotabletohandlewith theuncertaintyofanydecision-makingproblemproperlyusing fuzzyset.Atanassov[2]hasintroducedtheideaofintuitionistic fuzzyset(IFS)[3, 4]byincludinganon-membershipgradeanda hesitancygradeofalltheelementsofthefuzzyset.IFSispresent todescribetheelements/objectsofthefuzzysetfromthree differentaspectsofinferiority,superiorityandhesitation,which aregenerallymodelledbytheintuitionisticfuzzynumbers(IFNs). Tohandlemoreusefulinformationofreal-lifeproblemunder imprecise,vagueanduncertainenvironment,Smarandache[5–7] haspresentedthenovelideaofneutrosophicset,bygeneralising theideaofIFS.Theneutrosophicsetcanbeusedtocapturethe uncertaintiesduetoinconsistent,vaguenessandindeterminatedata ofanyproblem.Itisnothingbutanextendededitionofsimple classicalset,fuzzysetandIFS.Inneutrosophicset,eachobject hasthreedifferenttypesofmembershipgrade:truth,falseand indeterminate.Thosethreemembershipgradesofneutrosophicset arenotdependentoneachotherandalwayswithin]0,1[.
Graphisanefficienttooltomodelthereal-lifeproblems. Bymodellingthegraph,theobjectsandtheirrelationsare symbolisedbynodesandarcs.Thereexistsmanydifferenttypes ofinformationinreal-lifeproblemsandweneedseveraltypesof graphstomodelthoseproblemssuchasfuzzygraph,intuitionistic fuzzygraphsandneutrosophicgraphtheory[8–15].Shannonand Atanassov[16]presentedtheconceptofrelationshipbetweenIFS. Thentheyhaveintroducedtheconceptofintuitionisticfuzzy
graphsandpresentedmanytheoremsin[16].Parvathi etal. [17–19]proposedsomeoperationsbetweentwointuitionisticfuzzy graphs.In[20],Rashmanlou etal. proposedmanyproducts operationssuchaslexicographic,directproduct,strongproduct, semi-strongproductonintuitionisticfuzzygraphs.Theyhave describedtheCartesianproduction,join,compositionandunion onintuitionisticfuzzygraphsintheirpaper.Forfurtherstudyon intuitionisticfuzzygraphs,pleasereferto[21–27].Akram etal. [28–32]haveintroducedtheideaofpythagoreanfuzzygraph. Theyhavedescribedtheseveralapplicationsofpythagoreanfuzzy graphintheirpaper.Neutrosophicgraph[33]isusedtomodel manyreal-worldproblemswhichconsistofinconsistent information.Recently,manyscientistshaveresearchedongraphin neutrosophicenvironment[34–41],forinstance,Yang etal. [9], Arkam[12, 14],Ye[8],Naz etal. [10],DasandEdalatpanah [42],Dey etal. [43–47]andBroumi[48–51].In2020,Prof. Smarandacheintroducedtheideaof n-superhyper-graph[52]with super-nodesandhyper-arcsforneutrosophicgraph.
Theshortestpathproblem(SPP)isawell-knownnetworkoptimisationproblemintheareaofoperationresearch.Inthisproblem,decisionmakerfocusesondeterminingashortestpathbetweena specifiedstartingnodeandothernodes.TheSPPhasbeenconsidered tomodelinmanyreal-lifeproblems,e.g.economics,telecommunications,transportation,scheduling,routingandsupplychainmanagement.ManyresearchershavestudiedintensivelyontheSPPswith deterministicedgecosts.TheseSPPsarereferredtoasstandard SPPs.DecisionmakercansolvethestandardSPPsefficientlyusing severalwell-knownalgorithmsintroducedbysomeexcellentresearchers.AlthoughinstandardSPP,thecostsofthearcsareconsideredreal numbers([ R),mostreal-lifescenarios,however,havemanyparametersthatmaynotbealwaysprecise(i.e.travellingdemands,travellingcosts,travellingcapacities,travellingtimeetc.).Severaltypesof uncertaintyaregenerallyencounteredinpracticalapplicationsofSPP duetoimperfectdata,maintenance,failureorotherreasons.Insuch scenarios,thearccostsarenon-deterministicinnature.Someresearchers[53]usetype-1fuzzynumbersforhandlingtheuncertaintiesin standardSPPandthistypeofSPPisdefinedasfuzzySPP(FSPP). TheFSPPcannotmanagetheseveraltypesofuncertaintiesbecause themembershipdegreeoftype1fuzzynumbersissimplyreal
CAAITrans.Intell.Technol.,2020,Vol.5,Iss.4,pp.308–313 308ThisisanopenaccessarticlepublishedbytheIET,ChineseAssociationforArtificialIntelligenceand ChongqingUniversityofTechnologyundertheCreativeCommonsAttribution-NonCommercial-NoDerivsLicense(http:// creativecommons.org/licenses/by-nc-nd/3.0/)
ResearchArticle
number.Tosolvethisproblem,fewresearchersworkedonintuitionisticFSPP.InintuitionisticFSPP,thearclengthsareconsideredas IFNs.Itcanworkwithuncertaininformationaboutthearclength whichconsistsofmembershipgradeandnon-membershipgradesimultaneously.ThemaindisadvantageofintuitionisticFSPPisthatit cannothandletheuncertaininformationofarclengthifthesumof non-membershipandmembershipisbiggerthan1.Insuchreal-life scenarios,anappropriatemodellingtechniquemayjustifiably employtheneutrosophicset,andsodoesthenameneutrosophic SPP[48–51]appearsintheareaofgraphtheory.Theneutrosophic SPP,involvingadditionoperationandcomparisonoperationofneutrosophicset,isdifferentfromthestandardSPP(FSPP),whichonly usescrispnumbers(fuzzynumber).InaneutrosophicSPP,thearc lengthbeingneutrosophicnumbers,themainobjectiveofdetermining apathbetweentwonodesbeingsmallerthanalltheotherpathsisnot easy,astherankingofneutrosophicsetasacomparisonoperationcan bedescribedinseveralways.TheBellman–Fordalgorithmisa commonandefficientalgorithmtosolvethestandardSPP.TheclassicalBellman–FordalgorithmiseasytoimplementforstandardSPP. Inthismanuscript,anextendedBellman–Fordalgorithmisdesigned tosolvetheneutrosophicSPP.Inthisalgorithm,weneedtoaddress twokeyissuesto findthesolutionofSPPwithneutrosophicparameters.The firstissueishowto findthesummingoperationoftwo edges,i.e.neutrosophicnumbers.Itisneededtocalculatethepath length.Thesecondoneisthathowtocomparethetwodifferentneutrosophicpathswiththeirarccostsdescribedbyneutrosophic numbers.Tosolvetheseproblems,therankingmethodofneutrosophicsetisadoptedtoextendtheclassicalBellman–Fordalgorithm. Thisresearchpaperintroducessomenewconceptsof single-valuedneutrosophicgraph(SVNG).Wehavediscussedthe definitionofregularSVNG,completeSVNGandstrongSVNG. Inthismanuscript,weconsiderneutrosophicnumbertodescribe theedgeweightsofaneutrosophicgraphforneutrosophicSPP. Analgorithmisintroducedtosolvethisproblem.The uncertaintiesareincorporatedinBellman–FordalgorithmforSPP usingneutrosophicnumberasarclength.Weuseonenumerical exampletoillustratetheeffectivenessoftheproposedalgorithm.
2Preliminary
Inthissection,wedefineneutrosophicgraphandintroducedifferent typesofregularneutrosophicgraph,strongneutrosophicgraph, completeneutrosophicgraphandcomplementneutrosophicgraph.
Definition1: Let U beaclassicaluniversalset.Aneutrosophicset [54] D onthe U isdescribedbythreeindependentmembership functions:truemembershipfunction TD (x),indeterminate membershipfunction ID (x)andfalsemembershipfunction FD (x) 0 ≤ sup TD (x) + sup ID (x) + sup FD (x) ≤ 3+ (1)
Definition2: Let U beauniversalset.Thesingle-valued neutrosophicset[55] D ontheuniversal U isdenotedasfollows: A = {kx:TD (x), ID (x), FD (x)|x [ U l}(2)
Thefunctions TD (x) [ [0,1], IA (x) [ [0,1]and FA (x) [ [0,1]are namedasdegreeoftruth,indeterminacyandfalsitymembershipof x in A,satisfythefollowingcondition: 0 ≤ sup TD (x) + sup ID (x) + sup FD (x) ≤ 3+ (3)
Definition3: Let A = (TD , ID , FD )beasingle-valuedneutrosophic set.Ascorefunction S [34]isdefinedasfollows: S (D) = (1 + (TD 2ID FD )(2 TD FD )) 2 (4)
Definition4::Let G∗ =V , E () beasimplegraph.Apair G=C , D () isaneutrosophicgraphon G∗ where C= TC , IC , FC isapicture
fuzzyseton j and D= TD , ID , FD isapicturefuzzyseton
# V×V suchthatforeacharc ij [ E
TD i, j ≤ min TC i (), TC j , ID i, j ≤ min IC i (), IC j , FD i, j ≥ max FC i (), FC j (5)
Definition5: Aneutrosophicgraph G=C , D () issaidtoberegular neutrosophicgraphif
TD i, j = constant, i i=j
i i=j
ID i, j = constant, i i=j
FD i, j = constant, ∀i, j [ E
(6)
Definition6: Aneutrosophicgraph G=C , D () isdefinedasstrong neutrosophicgraphif
TD i, j = TC i () ^ TC j , ID i, j = IC i () ^ IC j ,
FD i, j = FC i () _ FC j ∀i, j [ E (7)
Definition7: Aneutrosophicgraph G=C , D () isdefinedas completeneutrosophicgraphif
TD i, j = TC i () ^ TC j , ID i, j = IC i () ^ IC j , FD i, j = FC i () _ FC j ∀i, j [ V (8)
Definition8: Apath p inaneutrosophicgraph G=C , D () isa sequenceofdifferentvertices p0 , p1 , p2 , , pk suchthat TD pi 1 pi ID pi 1 pi FD pi 1 pi . 0, i = 1,2, , k (9)
Here, k representsthelengthofpath.
Definition9: Let G=C , D () beaneutrosophicgraph.Then, G issaid tobeconnectedneutrosophicgraphifforeveryvertices i, j [ V , T 1 B (i, j ) . 0or I 1 B (i, j ) . 0or F 1 B (i, j ) , 1.
Definition10: Thecomplementofaneutrosophicgraph G=C , D () isaneutrosophicgraph G′ =C ′ , D′ ifandonlyiffollowsthe followingequation:
(10)
Definition11: Aneutrosophicgraph G issaidtobe
(i)The G isself-complementaryneutrosophicgraphthen G= G ′ . (ii)The G isself-weakcomplementneutrosophicgraphthen G is weakisomorphicto G′
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E
(T ′ C , I ′ C , F ′ C ) = (TC , IC , FC ) T ′ D i, j = TC i () ^ TC j TB i, j , I ′ D i, j = IC i () ^ IC j IB i, j , F ′ D i, j = FC i () _ FC j FB i, j ∀i, j [ V
3Operationsonneutrosophicgraph
Inthissection,weintroducesixoperationsonneutrosophicgraph, viz.,Cartesianproduct,composition,join,directproduct, lexicographicandstrongproduct.
Definition12: Let G1 =C 1 , D1 and G2 =C 2 , D2 aretwo neutrosophicgraphsof G∗ 1 =V 1 , E 1 and G∗ 2 =V 2 , E 2 , respectively.TheCartesianproduct G1 ×G2 ofneutrosophic graph G1 and G2 isdefinedby C , D (),where C= TC , IC , FC and D= TD , ID , FD aretwoneutrosophicsetson V=V 1 ×V 2 , and E= { i, i2 , i, j2 |i [ V 1 , i2 j2 [ E 2 }< { i1 , k , j1 , k |k [ V 2 , i1 j1 [ E 1 },respectively,whichsatisfiesthefollowing:
(i) ∀(i1 , i2 ) [ V 1 ×V 2 ,
(a) TC ((i1 , i2 )) = TC 1 (i1 ) ^ TC 2 (i2 ) (b) IC ((i1 , i2 )) = IC 1 (i1 ) ^ IC 2 (i2 ) (c) FC ((i1 , i2 )) = FC 1 (i1 ) _ FC 2 (i2 )
(ii) ∀i [ V 1 and ∀(i2 , j2 ) [ E 2 ,
(a) TD ((i, i2 )(i, j2 )) = TC 1 (i1 ) ^ TD2 (i2 j2 ) (b) ID ((i, i2 )(i, j2 )) = IC 1 (i1 ) ^ ID2 (i2 j2 ) (c) FD ((i, i2 )(i, j2 )) = FC 1 (i1 ) _ FD2 (i2 j2 )
(iii) ∀k [ V 2 and ∀(i1 , i2 ) [ E 1 ,
(a) TD ((i1 , k )(j1 , k )) = TD1 (i1 j1 ) ^ TC 2 (k ) (b) ID ((i1 , k )(j1 , k ) = ID1 (i1 j1 ) ^ IC 2 (k ) (c) FD ((i1 , k )(j1 , k )) = FD1 (i1 j1 ) _ FC 2 (k )
Definition13: Thecomposition G1 G2 oftwoneutrosophicgraphs G1 =C 1 , D1 and G2 =C 2 , D2 definedasapair C , D (),where C= TC , IC , FC and D= TD , ID , FD aretwoneutrosophicsets on V=V 1 ×V 2 ,and E= { i, i2 , i, j2 |i [ V 1 , i2 j2 [ E 2 }< { i1 , k , j1 , k |k [ V 2 , i1 j1 [ E 1 } < { i1 , i2 , j1 , j2 |i2 j2 [ V 2 = j2 , i1 j1 [ E 1 },respectively,whichsatisfiesthefollowing:
(i) ∀(i1 , i2 ) [ V 1 ×V 2 ,
(a) TC ((i1 , i2 )) = TC 1 (i1 ) ^ TC 2 (i2 ) (b) IC ((i1 , i2 )) = IC 1 (i1 ) ^ IC 2 (i2 ) (c) FC ((i1 , i2 )) = FC 1 (i1 ) _ FC 2 (i2 )
(ii) ∀i [ V 1 and ∀(i2 , j2 ) [ E 2 ,
(a) TD ((i, i2 )(i, j2 )) = TC 1 (i1 ) ^ TD2 (i2 j2 ) (b) ID ((i, i2 )(i, j2 )) = IC 1 (i1 ) ^ ID2 (i2 j2 ) (c) FD ((i, i2 )(i, j2 )) = FC 1 (i1 ) _ FD2 (i2 j2 )
(iii) ∀k [ V 2 and ∀(i1 , i2 ) [ E 1 ,
(a) TD ((i1 , k )(j1 , k )) = TD1 (i1 j1 ) ^ TC 2 (k ) (b) ID ((i1 , k )(j1 , k ) = ID1 (i1 j1 ) ^ IC 2 (k ) (c) FD ((i1 , k )(j1 , k )) = FD1 (i1 j1 ) _ FC 2 (k )
(iv) ∀i2 j2 [ V 2 , i2 = j2 and ∀(i1 j1 ) [ E 1 ,
(a) TD ((i1 , i2 )(j1 , j2 )) = TC 2 (i2 ) ^ TC 2 (j2 ) ^ TD1 (i1 j1 ) (b) ID ((i1 , i2 )(j1 , j2 )) = IC 2 (i2 ) ^ IC 2 (j2 ) ^ ID1 (i1 j1 ) (c) FD ((i1 , i2 )(j1 , j2 )) = FC 2 (i2 ) ^ FC 2 (j2 ) ^ FD1 (i1 j1 )
Definition14: Theunion G1 < G2 oftwoneutrosophicgraph G1 = C 1 , D1 and G2 =C 2 , D2 isdefinedas C , D (),where C= TC , IC , FC isaneutrosophicseton V=V 1 < V 2 and D= TD , ID , FD isananotherneutrosophicseton E=E 1 < E 2 , whichsatisfiesthefollowing:
(i)
(a) TC (i) = TC 1 (i)if i [ V 1 and i V 2 (b) TC (i) = TC 2 (i)if i [ V 2 and i V 1 (c) TC (i) = TC 1 (i) ^ TC 2 (i)if i [ V 1 > V 2
(ii) (a) IC (i) = IC 1 (i)if i [ V 1 and i V 2 (b) IC (i) = IC 2 (i)if i [ V 2 and i V 1 (c) IC (i) = IC 1 (i) ^ IC 2 (i)if i [ V 1 > V 2
(iii)
(a) FC (i) = FC 1 (i)if i [ V 1 and i V 2 (b) FC (i) = FC 2 (i)if i [ V 2 and i V 1 (c) FC (i) = FC 1 (i) ^ IC 2 (i)if i [ V 1 > V 2 (iv)
(a) TD (ij ) = TD1 (ij )if ij [ E 1 and ij E 2 (b) TD (ij ) = TD2 (ij )if ij [ E 2 and ij E 1 (c) TD (ij ) = TD1 (ij ) ^ TD2 (ij )if ij [ E 1 > E 2
(v)
(a) ID (ij ) = ID1 (ij )if ij [ E 1 and ij E 2 (b) ID (ij ) = ID2 (ij )if ij [ E 2 and ij E 1 (c) ID (ij ) = ID1 (ij ) ^ ID2 (ij )if ij [ E 1 > E 2
(vi)
(a) FD (ij ) = FD1 (ij )if ij [ E 1 and ij E 2 (b) FD (ij ) = FD2 (ij )if ij [ E 2 and ij E 1 (c) FD (ij ) = FD1 (ij ) ^ FD2 (ij )if ij [ E 1 > E 2
Definition15: Thejoining G1 +G2 oftwoneutrosophicgraphs G1 =C 1 , D1 and G2 =C 2 , D2 isde finedas C , D (), where C= TC , IC , FC isaneutrosophicseton V=V 1 < V 2 and D= TD , ID , FD isananotherneutrosophicseton E= E 1 < E 2 < E ′ ((E ′ representsalledgesjoiningthevertexof V 1 and V 2 ),whichsatis fiesthefollowing:
(i)
(a) TC (i) = TC 1 (i)if i [ V 1 and i V 2 (b) TC (i) = TC 2 (i)if i [ V 2 and i V 1 (c) TC (i) = TC 1 (i) ^ TC 2 (i)if i [ V 1 > V 2
(ii)
(a) IC (i) = IC 1 (i)if i [ V 1 and i V 2 (b) IC (i) = IC 2 (i)if i [ V 2 and i V 1 (c) IC (i) = IC 1 (i) ^ IC 2 (i)if i [ V 1 > V 2
(iii)
(a) FC (i) = FC 1 (i)if i [ V 1 and i V 2 (b) FC (i) = FC 2 (i)if i [ V 2 and i V 1 (c) FC (i) = FC 1 (i) ^ IC 2 (i)if i [ V 1 > V 2
CAAITrans.Intell.Technol.,2020,Vol.5,Iss.4,pp.308–313 310ThisisanopenaccessarticlepublishedbytheIET,ChineseAssociationforArtificialIntelligenceand ChongqingUniversityofTechnologyundertheCreativeCommonsAttribution-NonCommercial-NoDerivsLicense(http:// creativecommons.org/licenses/by-nc-nd/3.0/)
(iv)
(a) TD (ij ) = TD1 (ij )if ij [ E 1 and ij E 2 (b) TD (ij ) = TD2 (ij )if ij [ E 2 and ij E 1 (c) TD (ij ) = TD1 (ij ) ^ TD2 (ij )if ij [ E 1 > E 2
(v)
(a) ID (ij ) = ID1 (ij )if ij [ E 1 and ij E 2 (b) ID (ij ) = ID2 (ij )if ij [ E 2 and ij E 1 (c) ID (ij ) = ID1 (ij ) ^ ID2 (ij )if ij [ E 1 > E 2
(vi)
(a) FD (ij ) = FD1 (ij )if ij [ E 1 and ij E 2 (b) FD (ij ) = FD2 (ij )if ij [ E 2 and ij E 1 (c) FD (ij ) = FD1 (ij ) ^ FD2 (ij )if ij [ E 1 > E 2
(vii)
(a) TD (ij ) = TD1 (i) _ TD2 (j )if ij [ E ′ (b) ID (ij ) = ID1 (i) _ ID2 (j )if ij [ E ′ (c) FD (ij ) = FD1 (i) ^ FD2 (j )if ij [ E ′
Definition16: Thedirectproduct G1 ∗ G2 oftwoneutrosophic graph G1 and G2 isdefinedasapair C , D (),where C= TC , IC , FC isaneutrosophicseton V=V 1 ×V 2 and D= TD , ID , FD isananotherneutrosophicseton E= { i1 , i2 j1 , j2 |i1 j1 [ E 1 , i2 j2 [ E 2 },whichsatisfiesthefollowing:
(i) ∀(i1 , i2 ) [ V 1 ×V 2
(a) TC (i1 , i2 ) = TC 1 (i1 ) _ TC 2 (i2 ) (b) IC (i1 , i2 ) = IC 1 (i1 ) _ IC 2 (i2 ) (c) FC (i1 , i2 ) = FC 1 (i1 ) ^ FC 2 (i2 )
(ii) ∀(i1 j1 ) [ E 1 , ∀(i2 j2 ) [ E 2
(a) TD (i1 , i2 )(j1 , j2 ) = TD1 (i1 j1 ) _ TD2 (i2 j2 ) (b) ID (i1 , i2 )(j1 , j2 ) = ID1 (i1 j1 ) _ ID2 (i2 j2 ) (c) FD (i1 , i2 )(j1 , j2 ) = FD1 (i1 j1 ) ^ TD2 (i2 j2 )
Definition17: Thelexicographicproduct G1 ·G2 oftwoneutrosophic graph G1 =C 1 , D1 and G2 =C 2 , D2 isdefinedasapair C , D (), where C= TC , IC , FC isaneutrosophicseton V=V 1 ×V 2 and D= TD , ID , FD isananotherneutrosophicseton E= { i, i2 × i, j2 |i [ V 1 , i2 j2 [ E 2 } < { i1 , i2 j1 , j2 |i1 j1 [ E 1 , i2 j2 [ E 2 } whichsatisfiesthefollowing:
(i) ∀(i1 , i2 )
(a) TC (i1 , i2 ) = TC 1 (i1 ) _ TC 2 (i2 ) = IC 1 (i1 ) _ IC 2 (i2 ) = FC 1 (i1 )^ FC 2 (i2 )
(ii) ∀i [ V 1 , ∀(i2 j2 ) [ E 2
(a) TD (i, i2 )(i, j2 ) = TC 1 (i) _ TD2 (i2 j2 )
(b) ID (i, i2 )(i, j2 ) = IC 1 (i) _ ID2 (i2 j2 )
(c) FD (i, i2 )(i, j2 ) = FC 1 (i) ^ FD2 (i2 j2 )
(iii) ∀i1 j1 [ E 1 , ∀(i2 j2 ) [ E 2
(a) TD (i1 , i2 )(j1 , j2 ) = TD1 (i1 j1 ) _ TD2 (i2 j2 )
(b) ID (i1 , i2 )(j1 , j2 ) = ID1 (i1 j1 ) _ ID2 (i2 j2 ) (c) FD (i1 , i2 )(j1 , j2 ) = FD1 (i1 j1 ) ^ FD2 (i2 j2 )
Definition18: Thestrongproduct G1 G2 oftwoneutrosophicgraphs G1 =C 1 , D1 and G2 =C 2 , D2 isdefinedasapair C , D (),where C= TC , IC , FC isaneutrosophicseton V=V 1 ×V 2 and D=
TD , ID , FD isananotherneutrosophicseton E= { i, i2 i, j2 |i [ V 1 , i2 j2 [ E 2 } < { i1 , k j1 , k |k [ V 2 , i1 j1 [ E 1 } < { i1 , i2 × j1 , j2 |i1 j1 [ E 1 , i2 j2 [ E 2 }whichsatisfiesthefollowing:
(i) ∀(i1 , i2 ) [ V 1 ×V 2
(a) TC (i1 , i2 ) = TC 1 (i1 ) _ TC 2 (i2 ) (b) IC (i1 , i2 ) = IC 1 (i1 ) _ IC 2 (i2 ) (c) FC (i1 , i2 ) = FC 1 (i1 ) ^ FC 2 (i2 )
(ii) ∀i [ V 1 , ∀(i2 j2 ) [ E 2
(a) TD (i, i2 )(i, j2 ) = TC 1 (i) _ TD2 (i2 j2 ) (b) ID (i, i2 )(i, j2 ) = IC 1 (i) _ ID2 (i2 j2 ) (c) FD (i, i2 )(i, j2 ) = FC 1 (i) ^ FD2 (i2 j2 )
(iii) ∀i1 j1 [ E 1 , ∀(i2 j2 ) [ E 2 (a) TD (i1 , i2 )(j1 , j2 ) = TD1 (i1 j1 ) _ TD2 (i2 j2 ) (b) ID (i1 , i2 )(j1 , j2 ) = ID1 (i1 j1 ) _ ID2 (i2 j2 ) (c) FD (i1 , i2 )(j1 , j2 ) = FD1 (i1 j1 ) ^ FD2 (i2 j2 )
(iv) ∀i1 j1 [ E 1 , k [ V 2
(a) TD (i1 , k )(j1 , k ) = TD1 (i1 j1 ) _ TC 2 (k ) (b) ID (i1 , k )(j1 , k ) = ID1 (i1 j1 ) _ IC 2 (k ) (c) FD (i1 , k )(j1 , k ) = FD1 (i1 j1 ) ^ FC 2 (k )
4ProposedBellman–Fordalgorithmfor neutrosophicSPP
Ourproposedalgorithmicapproachisthemodificationofclassical Bellman–FordalgorithmforneutrosophicSPP.Inthisalgorithm, wehaveincorporatedtheuncertaintiesinBellman–Fordalgorithm usingneutrosophicsetasanedgeweight.Wehaveshownthe pseudocodeofourproposedalgorithmforneutrosophicSPPin Algorithm1(seeFig. 1).The flowchartofourproposedalgorithm isgiveninFig. 2.Theproposedalgorithm findsallpossible shortestpathsbetweenthesourcenodeandallothernodesinthe neutrosophicgraph G.Ourproposedalgorithmneedsthatthe neutrosophicgraphdoesnotconsistofanyneutrosophiccyclesof negativeneutrosophiclength.However,ifthegraphcontainsany neutrosophiccycle,thenourproposedalgorithmisableto findit. Thesourceisdenoted,respectively,by source
5Numericalexamples
AnumericalexampleofneutrosophicSPPisusedtodescribeour proposedBellman–Fordalgorithm.Forthispurpose,weusean exampleneutrosophicgraph,showninFig. 3,with fivevertices andeightedges.OurmodifiedBellman–Fordalgorithmdetectsthe shortestpathbetweenthestartingvertexandallothernodesinthe neutrosophicgraphwithneutrosophicsetsasedgeweights.For thisneutrosophicSPP,weconsiderthestartingvertexisvertex s Thefollowingeightneutrosophicnumber,showninTable 1,are usedasedgeweightsoftheneutrosophicgraph.Those neutrosophicsetsarenumberfromonetoeight.Forthe neutrosophicgraph,showninFig. 3,weassignthoseneutrosophic setstotheedgesoftheneutrosophicgraphrandomly.
Thestepsofouralgorithmaregivenbelow:
Step1:Letthestartingvertexbe s.Itischangedpermanentnodeand theshortestpathdistance,i.e.dist_score[u]between(s)and(s)is0. Theaccessibleadjacentverticesbetweenstartingvertex s are v1 , v2 and v3 .Thescorevalues(distance)oftheeachadjacentedgeof startingvertex(s)arecalculatedbyusing(4).Amongallthe3 nodes,theshortestone(s) (v1 )istakenoutwithscorevalue0.43. Step2:Now,ourproposedalgorithmmovesthenode v1 andthe findingtheshortestpathstartedfromthevertex(v1 ).Thelowest score(shortestdistance)from(s) (v1 )toitsadjacentis
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Fig.1 Algorithm1:neutrosophicBellman–Fordalgorithm
Fig.2 FlowchartofneutrosophicBellman–Fordalgorithm
determined.Anyoneisminimumthanthepath(s) (v2 )by comparingallthescorevalues.
Step3:Thevertex v2 ischangedtopermanentandallthesearching ofshortestpathbeginswithvertex(v1 )andvertex(s).We findthe scoreoftheeachadjacentofthepath(s) (v2 ).Theshortest scoreamongalltheunvisitedpathis(s) (v3 ).
Step4:Similarity,wedeterminethepathwithlowestscore(i.e.the shortestpath)betweenthesourcevertexandeveryothervertex(t ), theneutrosophicshortestpathis(s) (v2 ) t
Fig.3 Neutrosophicnetworkwithneutrosophicnumbersasarclengthsfor example1
Table1 Arclengthsoftheneutrosophicgraph,representedas neutrosophicnumber IndexSVNs 1 k(4 6,5 5,8 6)l 2 k(4 7,6 9,8 5)l 3 k(6 2,7 6,8 2)l 4 k(6 2,8 9,9 1)l 5 k(4 4,5 9,7 2)l 6 k(6 6,8 8,10)l 7 k(6 3,7 5,8 9)l 8 k(6 2,7 6,8 2)l
6Conclusion
Graphtheoryhasmanyreal-lifeapplicationstotheproblemsin operationsresearch,computernetwork,economics,systems analysis,urbantrafficplanningandtransportation.Inreal-life scenarios,however,uncertaintymayexistinalmosteverygraph theoreticproblem.Neutrosophicsetisapopularandusefultoolto workinuncertainenvironment.Thispaperpresentssomenew operationofSVNGmodel.Wedescribethedefinitionofregular
CAAITrans.Intell.Technol.,2020,Vol.5,Iss.4,pp.308–313 312ThisisanopenaccessarticlepublishedbytheIET,ChineseAssociationforArtificialIntelligenceand ChongqingUniversityofTechnologyundertheCreativeCommonsAttribution-NonCommercial-NoDerivsLicense(http:// creativecommons.org/licenses/by-nc-nd/3.0/)
SVNG,completeSVNGandstrongSVNG.Inthismanuscript,we considerneutrosophicnumbertodescribetheedgeweightsofa neutrosophicgraphforneutrosophicSPP.Analgorithmis introducedtosolvethisproblem.Theuncertaintiesare incorporatedinBellman–FordalgorithmforSPPusing neutrosophicnumberasarclength.Weuseonenumericalexample toillustratetheeffectivenessoftheproposedalgorithm.The SVNGcanbeutilisedtomodelthesocialnetwork,image processing,telecommunication,expertsystemsandcomputer networks.
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