OnSoftRoughTopologywithMulti-AttributeGroup DecisionMaking
MuhammadRiaz 1,* ,FlorentinSmarandache 2,AtiqaFirdous 1 andAtiqaFakhar 1
1 FacultyofScience,DepartmentofMathematics,UniversityofthePunjab,LahorePostcode54590,Pakistan; atiqafirdous08@gmail.com(A.F.);atiqafakhar254@gmail.com(A.F.)
2 DepartmentofMathematics&Sciences,UniversityofNewMexico,705GurleyAve,Gallup,NM87301, USA;fsmarandache@gmail.com
* Correspondence:mriaz.math@pu.edu.pk
Received:10November2018;Accepted:16December2018;Published:9January2019
Abstract: Roughsetapproachesencounteruncertaintybymeansofboundaryregionsinsteadof membershipvalues.Inthispaper,wedevelopthetopologicalstructureonsoftroughset(SR-set) byusingpairwise SR-approximations.Wedefine SR-openset, SR-closedsets, SR-closure, SR-interior, SR-neighborhood, SR-bases,producttopologyon SR-sets,continuousmapping,and compactnessinsoftroughtopologicalspace(SRTS).Thedevelopmentsofthetheoryon SR-set and SR-topologyexhibitnotonlyanimportanttheoreticalvaluebutalsorepresentsignificant applicationsof SR-sets.Weappliedanalgorithmbasedon SR-settomulti-attributegroupdecision making(MAGDM)todealwithuncertainty.
Keywords: roughset; SR-set; SR-topology; SR-continuity; SR-compactness;decisionmaking
MSC: 54A05;54A40;11B05;90B50
1.Introduction
Theproblemofimperfectknowledgehasbeenthecenterofattentionformanyyears. Inthefieldofmathematics,computerscience,andartificialintelligence,researchershaveused differentmethodstotackletheproblemofuncertainandincompletedata,includingprobabilitytheory, fuzzyset[1],androughset[2,3]andsoftsettechniques[4 6].Molodstov[6]introducedsoftsetas aneffectivetooltomanageimprecision;itincludesasetofparameterstodescribethesetproperly. Majietal.(2002–2003)[4,5]extendedsomeoperationsofsoftsetandeffectivelyusedthistechniquein adecision-makingproblem.Softsetwithdecisionmakinghavestudiedbymanyresearchers[7 11]. In2011,ShabirandNaz[12]andCagmanetal.[13]independentlyworkedonthetopologicalstructureof softset.Chen[14]presentedanewdefinitionrelatedtothereductionofsoftparameterization. Thestudyofhybridstructures,havingemergedfromthefusionofsoftsetswithothermathematical approaches,isbecominganactivetopicforresearchnowadays.AktasandCagman(2007)[15] efficientlyrelatedthethreeconceptsofsoftset,roughset,andfuzzyset.Riazetal.[16 18]established someresultsofsoftalgebra,softmetricspaces,andmeasurablesoftmappings.RiazandMasooma[19 23] introducedfuzzyparameterizedfuzzysoftset(fpfs-set), fpfs-topology,and fpfs-compactspaces,with someimportantapplicationsof fpfs-settodecision-makingproblems.Theypresented fns-mappings andfixedpointsof fns-mapping.Differentresearchershavetackledtheproblemofincompleteor uncertaininformationinthesystemindifferentways.Shangworkedonrobuststatistics,andhe investigatedtherobustnessofasystemunderdifferentcircumstancesandanalyzedtherobustness propertiesofsubgraphsunderattackincomplexnetworks[24,25].Theroughsetconceptpresentedby Pawlakpresentsasystematicapproachfortheclassificationofobjects.Itcharacterizesasetofobjects
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bytwoexactconcepts,knownasitsapproximations.Here,vaguenessisexpressedintheformof aboundaryregion,whereemptyboundaryregionimpliesthatthesetiscrisp,andanon-empty boundaryregionimpliesthatourknowledgeisinsufficienttoexplainthesetprecisely.Byusing equivalencerelations,Thivagaretal.[26]generatedthetopologyonroughsetwhichincludes approximationsandtheboundaryregion.EquivalencerelationplaysanimportantroleinPawlak’s roughsetmodel,andbyreplacingitwithasoftset,softroughset SR-setswereintroducedby Feng[27].Fengetal.[28]presentedsomepropertiesrelatedto SR-approximations.Xueetal.[29] presentedsomedecision-makingalgorithmsregardinghybridsoftmodels.ZouandXiao[30] analyzeddatainsoftsetsunderincompleteinformationsystems.Therearemainlytwostreamsof studyconnectingsoftroughsettheoryandtopologytheory.Atthesametime,accordingtothe topologicalpropertiesonthetopological SR-space,someapplicationsforimageprocessingand sometopologicaldiagramsareintroduced.Theremainderofthepaperiscomposedasfollows. InSection 2,webrieflydefinethenotionsofroughset R-setandsoftroughset SR-set.InSection 3, wepresentanoveltopologicalstructureof SR-set.Wepresentsomenewresultsof SR-settheory and SR-topology.AtopologicalstructureonsoftroughsetwasdefinedbyBakieretal.[31].Malik andRiaz[32,33]studiedactionofmodulargrouponrealquadraticfields.Softsets,neutrosophic setandroughsetswithdecisionmakingproblemshavestudiedbymanyresearchers[34 40]. Wedefine SR-topologyonsoftroughsetintheformofthepair τSR =(τSR , τSR ),where τSR isthelower SR-topologyand τSR istheupper SR-topologyonset Y.This SR-topologyis moreappropriate,asitlookslikeanaturalsoftroughtopologyonasoftroughset.InSection 4, continuity,homeomorphism,andprojectionmappingsin SR-setarediscussed.Section 5 describes thecompactnessin SR-set.InSection 6, SR approximationsareemployedtosolvemulti-attribute groupdecision-makingproblem.
2.Preliminaries
Inthissection,weillustratesomebasicnotionsrelatedto SR-theory.Firstwedefineroughset R-setandsoftroughset SR-setandthenexplainafewrelatedoperationson SR-set.
Definition1 ([2]). Supposewehaveanobjectset V knownasuniverse,andanindiscernibilityrelation ⊆V×V whichrepresentsknowledgeaboutelementsof V .Wetake asanequivalencerelationanddenote itby (y).Thepair (V, ) iscalledtheapproximationspace.Let Y beanysubsetof V .Wecharacterizetheset Y withrespectto
(1)Theunionofallgranuleswhichareentirelyincludedintheset Y iscalledthelowerapproximationof theset Y w.r.t ,mathematicallydefinedas
(Y )= y∈V { (y) : (y) ⊆Y}
(2)Theunionofallthegranuleshavinganon-emptyintersectionwiththeset Y iscalledtheupper approximationoftheset Y w.r.t ,mathematicallydefinedas
(Y )= y∈V { (y) : (y) ∩Y = ∅}
(3)Thedifferencebetweentheupperandlowerapproximationsiscalledtheboundaryregionoftheset Y w.r.t ,mathematicallydefinedas
B (Y )= (Y ) − (Y )
If (Y )= (Y ),theset Y issaidtobedefined.If (Y ) = (Y ),i.e., BR (Y ) = ∅,theset Y is saidtobea(imprecise)roughsetw.r.t
Wedenotearoughset Y byapaircomprisingalowerapproximationandupperapproximation Y =( (Y ), (Y ))
Definition2 ([28]). Considerasoftset S =(T , A) overtheuniverse V ,where A⊆E and T isamapping definedas T : A→P (V ).Here,softapproximationspaceisthepair P =(V, S).Followingthesoft approximationspace P ,wedefinetwooperationsasfollows:
P (Y )= {v ∈V : ∃a ∈A, [v ∈T (a) ⊆Y ]},
P (Y )= {v ∈V : ∃a ∈A, [v ∈T (a) ∩Y = ∅]}
regardingeverysubset Y⊆V ,twosets P (Y ) and P (Y ),whicharecalledthesoftP-lowerapproximation andsoftP-upperapproximationof Y ,respectively.Ingeneral,wereferto P (Y ) and P (Y ) as SR-approximationsof Y w.r.t P.If P (Y )= P (Y ),then Y issaidtobesoftP-definable;otherwise, Y isasoftP-roughset.Then,BndP = P (Y ) − P (Y ) isthe SR-boundaryregion.
Wedenote SR-set(SR-set) Y byapaircomprising SR-lowerapproximationand SR-upper approximation Y =( P (Y ), P (Y ))
Example1. Let V = {s1, s2, s3, s4, s5} bethesetofperfumes,andlet A = {ζ1, ζ2, ζ3, ζ4} = E bethequalitieswhichMissAmalwantsinherperfume.Let S =(T , A) beasoftsetover V T (ζ1)= {s3, s5}, T (ζ2)= {s2, s4, s5}, T (ζ3)= {s1, s2, s5}, T (ζ4)= {s2, s3} andthesoftapproximation space P =(V, S).Thetabularformofsoftset (T , A) isgiveninTable 1
Table1. Softset (T , A) (T , A) s1 s2 s3 s4 s5 ζ1 01100 ζ2 01011 ζ3 11001 ζ4 00101
For Y = {s3, s4, s5}⊆V ,wehave P (Y )= {s3, s5} and P (Y )= {s1, s2, s3, s4, s5}. Since P (Y ) = P (Y );therefore, Y isasoftP-roughsetandisdenotedby Y = ({s3, s5}, {s1, s2, s3, s4, s5}) Definition3. Let A =( P (A), P (A)) and B =( P (B), P (B)) betwoarbitrary SR-setsand P =(V, S) besoftapproximationspace.Then, A isa SR-subsetof B if P (A) ⊆ P (B) and P (A) ⊆ P (B)
Example2. Suppose V = {α1, α2, α3, α4, α5, α6, α7, α8} and E = {ξ1, ξ2, ξ3, ξ4}.Let S =(T , E ) beasoft setover V , T (ξ1)= {α2, α8} T (ξ2)= {α2, α3, α6, α8}
T (ξ3)= {α2, α5, α7}
T (ξ4)= {α3, α4, α6} and P =(V, S) besoftapproximationspace.Consider A = {α2, α4, α5, α7}⊆V and B = {α3, α4} then P (A)= {α2, α5, α7} and P (A)= {α2, α3, α4, α5, α6, α7, α8},while P (B)= ∅ and P (B)= {α2, α3, α4, α6, α8}. Sowehavetwo SR-sets A =( P (A), P (A))=({α2, α5, α7}, {α2, α3, α4, α5, α6, α7, α8}) and B =( P (B), P (B))=(∅, {α2, α3, α4, α6, α8}).Since P (B) ⊆ P (A) and P (B) ⊆ P (B). Thus, B is SR-subsetof A. Definition4. Let A =( P (A), P (A)), B =( P (B), P (B)) betakenastwoarbitrary SR setsandlet (V, S) besoftapproximationspace.Then,theunionof A and B isdefinedas A∪B = ( P (A) ∪ P (B), P (A) ∪ P (B))
Definition5. Let A =( P (A), P (A)), B =( P (B), P (B)) betakenastwoarbitrary SR setsand (V, S) besoftapproximationspace.Then,theintersectionof A and B isdefinedas A∩B = ( P (A) ∩ P (B), P (A) ∩ P (B)).
Example3. ByusingExample 2,weobtain A∪B = ( P (A) ∪ P (B), P (A) ∪ P (B)) = ({α2, α5, α7}, {α2, α3, α4, α5, α6, α7, α8}) and A∩B = ( P (A) ∪ P (B), P (A) ∪ P (B)) = (∅, {α2, α3, α4, α6, α8})
Definition6 ([31]). Let V betheuniverseofdiscourseand P =(V, S) issoftapproximationspace;then, SR-topologyisdefinedas
τSR(Y )= {V, ∅, P (Y ), P (Y ), Bd(Y )}
where Y⊆V . τSR(Y ) satisfiesthefollowingaxioms:
(i) V and ∅ belongto τSR(X).
(ii)Unionofelementsofanysubcollectionof τSR(Y ) belongsto τSR(Y ). (iii)Intersectionofelementsoffinitesubcollectionof τSR(Y ) belongsto τSR(Y ). Thetopologydefinedby τSR(Y ) on V iscalled SR-topologyon V w.r.t Y and (V, τSR(Y ), E ) issaidto be SR-topologicalspace.Softroughsetwiththetopology τSR iscalledatopological SR-set.
Example4. Let V = {ϑ1, ϑ2, ϑ3, ϑ4, ϑ5, ϑ6} bethesetofcarsunderconsideration,andlet E = {ζ1, ζ2, ζ3, ζ4, ζ5} bethesetofallparametersand A = {ζ1, ζ2, ζ3}⊆E .Considerthesoftapproximation P =(V, S),where S =(T , A) isasoftsetover U givenby: T (ζ1)= {ϑ1, ϑ3}, T (ζ2)= {ϑ1, ϑ3, ϑ6} and T (ζ3)= {ϑ2, ϑ4} For Y = {ϑ2, ϑ3, ϑ4, ϑ6},weobtain P (Y )= {ϑ2, ϑ4}, P (Y )= {ϑ1, ϑ2, ϑ3, ϑ4, ϑ6} and Bd(Y )= {ϑ1, ϑ3, ϑ6}.Then, τSR(Y )= {V, ∅, {ϑ2, ϑ4}, {ϑ1, ϑ2, ϑ3, ϑ4, ϑ6}, {ϑ1, ϑ3, ϑ6}} isa SR-topology. Definition7. Let (V, τSR(Y ), E ) bea SR-topologicalspace.Anysubset A suchthat A∈ τSR A issaidto be SR-open,andanysubset A is SR-closedifandonlyif Ac ∈ τSR
Example5. InExample 2,wecanseethat {ϑ2, ϑ4}, {ϑ1, ϑ2, ϑ3, ϑ4, ϑ6}, {ϑ1, ϑ3, ϑ6} are SR-opensets,and theirrelativecomplements {ϑ1, ϑ3, ϑ5, ϑ6}, {ϑ5}, {ϑ2, ϑ4, ϑ5} are SR-closedsets,while V and ∅ areboth SR-openand SR-closed.
3.TopologicalStructureof SR-Sets
Inthissection,wedefineanewtopologicalstructureon SR-sets.Wedefine SR-openset, SR-closedsets, SR-closure, SR-interior, SR-neighborhood,and SR-bases.
Definition8. Let Y =( P (Y ), P (Y )) bea SR-subset,where P =(V, S).Let τSR and τSR betwotopologieswhichcontainonlyexactsubsetsof P (Y ) and P (Y ),respectively.Then,thepair τSR =(τSR , τSR ) iscalleda SR-topologyonthe SR-set Y andthepair (Y, τSR) isknownasasoftrough topologicalspace(SRTS).Softroughset Y withthetopology τSR =(τSR , τSR ) isknownastopological SR-set.Also,ina SR-topology, τSR =(τSR , τSR ), τSR isthelower SR-topologyand τSR istheupper SR-topologyonX.
Remark1. Since P (Y ) and P (Y ) areonlyexactlydefinedsetsin SR-approximationspace,werestrict theelementsof τ and τ tothesetofallexactordefinablesubsetsof P (Y ) and P (Y ),respectively. However,whentheyaregroupedtoformthe SR-topology τSR =(τSR , τSR ),indefinablesetscanalso be SR-open.Thepointtobenotedisthatasubsetof Y ,eitherexactorinexact,is SR-openiffitslower approximationisinthelower SR-topologyanditsupperapproximationisintheupper SR-topology.
Definition9. Let (Y, τSR) beanSRTS,where τSR =(τSR , τSR ).Let A =( P (A), P (A)) beany SR-subsetof Y =( P (Y ), P (Y )).Then, A issaidtobelower SR-openifthelowerapproximationof A belongstothelower SR-topology.Thatis, P (A) ∈ τSR .Also, A issaidtobeupper SR-openiftheupper approximationof A belongstotheupper SR-topology.Thatis, P (A) ∈ τSR A issaidtobe SR-openiff A isbothlower SR-openandupper SR-open,i.e., P (A) ∈ τSR and P (A) ∈ τSR
Theorem1. ConsideranSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ).Let T beacollectionof SR-opensubsetsof (Y, τSR).Then,Tisatopologyon Y .
Proof. Consider Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ) (i)Wehave ∅ ∈ τSR and ∅ ∈ τSR .Therefore, ∅ (∅, ∅) ∈ T.Also, P (Y ) ∈ τSR and P (Y ) ∈ τSR and,hence, Y =( P (Y ), P (Y )) ∈ T (ii)Let A =( P (A), P (A)) and B =( P (B), P (B)) beanytwoelementsof T,implying thatboth A and B are SR-opensubsetsof Y.Therefore, P (A) ∈ τSR , P (A) ∈ τSR and P (B) ∈ τSR , P (B) ∈ τSR .Beingtopologies, τSR and τSR areclosedunderfinite intersection;therefore, P (A) ∩ P (B) ∈ τSR and P (A) ∩ P (B) ∈ τSR .Hence, A∩B = ( P (A) ∩ P (B), P (A) ∩ P (B)) isan SR-opensubsetof Y,whichshowsthat A∩B∈ T Since A and B arearbitrary, T isclosedunderfiniteintersections. (iii)Let {Aµ =( P (Aµ ), P (Aµ )) | µ ∈ Ω} beanarbitraryfamilyof SR-opensubsetsof Y, andbelongstothesubcollection T Aµ =( P (Aµ ), P (Aµ )) ∈ T implies P (Aµ ) ∈ τSR and P (Aµ ) ∈ τSR forall µ ∈ Ω.Since τSR and τSR areclosedunderarbitraryunion, wehave µ∈Ω P (Aµ ) ∈ τSR and µ∈Ω P (Aµ ) ∈ τSR ,whichshowsthat µ∈Ω Aµ = µ∈Ω P (Aµ ), µ∈Ω P (Aµ ) isan SR-opensubsetof Y.Thus, T isclosedunderarbitraryunion. From(i),(ii),and(iii),thefamily T of Y formsatopologyon Y
Definition10. Inany SR-set Y =( P (Y), P (Y)),define τSR = {A⊆ P (Y) |A isP definable} and τSR = {B⊆ P (Y )/B isP definable}.Then, τSR and τSR aretopologieson P (Y ) and P (Y ),respectively,andthe SR-topology τSR =(τSR , τSR ) isknownastheDiscrete SR-topologyon Y , andthetopologicalspace (Y, τSR) isknownastheDiscrete SR-TopologicalSpaceon Y
Definition11. Inan SR-set Y =( P (Y), P (Y)),take τSR =(∅, P (Y)) and τSR =(∅, P (Y)), then τSR and τSR aretopologieson P (Y ) and P (Y ),respectively,andthe SR-topology τSR =(τSR , τSR ) on Y isknownastheindiscrete SR-topologyon Y ,and (Y, τSR) isknownasthe indiscrete SR-topologicalspaceon Y .
Definition12. InanSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ).Considera subcollection β ofsubsetsof P (Y );ifeveryelementof τSR canbeexpressedastheunionofsomeelementsof β ,then β issaidtobeabasefor τSR .Ifeverymemberof τSR canbeexpressedastheunionofsome membersof β foranothersubcollection β ofsubsetsof P (Y ),then β issaidtobeabasefor τSR .Ifthe aboveconditionsaresatisfied,thenthepair βSR =(β , β ) isknownasa SR-baseforthe SR-topology τSR on Y
Theorem2. ConsidertheSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ). βSR =(β , β ) isan SR-basefor τSR iffforany SR-openset A =( P (A), P (A)) of (Y, τSR) and (x, y) ∈A suchthat x ∈ P (A) and y ∈ P (A),thenthereexist B ∈ β and B ∈ β suchthat x ∈ B ⊆ P (A) andy ∈ B ⊆ P (A).
Proof. ConsidertheSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ).Let β and β befamiliesofsubsetsof P (Y ) and P (Y ),respectively,suchthat βSR =(β , β ) isa SR-basefor τSR.Also,considerany SR-subset A =( P (A), P (A)) andlet (x, y) ∈A be anarbitrarypointsuchthat x ∈ P (A) and y ∈ P (A).Now, x ∈ P (A), P (A) ∈ τSR and β isabasefor τSR ,whichimpliesthat P (A) canbewrittenastheunionofelementsof β .Hence, ∃Bµ ∈ β suchthat x ∈ Bµ and Bµ ⊆ P (A).Choosesucha Bµ as B .Therefore, x ∈ B ⊆ P (A)
Similarly,bythesameargument,thereexists B ∈ β suchthat y ∈ B ⊆ P (A) Conversely,supposethat β and β arefamiliesofsubsetsof P (Y ) and P (Y ),respectively, suchthatforany SR-openset A =( P (A), P (A)) of (Y, τSR), (x, y) ∈A,where x ∈ P (A) and y ∈ P (A);then,thereexist B ∈ β and B ∈ β suchthat x ∈ B ⊆ P (A) and y ∈ B ⊆ P (A).Now,wehavetoprovethat βSR =(β , β ) isan SR-basefor τSR.Let C =( P (C), P (C)) beany SR-opensubsetoftheSRTS (Y, τSR).Byourassumption,foreach x ∈ P (C),wehave Bx ∈ β suchthat x ∈ Bx ⊆ P (C).Thus, P (C)= x∈ P (C) Bx .Thisimpliesthat P (C) can beexpressedastheunionofsomeelementsof β .Since C =( P (C), P (C)) istakenarbitrarily, β isalowerbasefor SR-topology τSR. Similarly,bythesameargument, P (C) canbeexpressedastheunionofsomemembersof β ;therefore, β isanupperbaseforthe SR-topology τSR.Hence, βSR =(β , β ) isan SR-base for τSR
Definition13. InanSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ) Thecollection S =(S , S ) ofsubsetsof Y ,where S and S areacollectionofsubsetsof P (Y ) and P (Y ) S issaidtobean SR-subbaseforthetopology τSR iffthefollowingconditionsaresatisfied: (i) S ⊂ τSR and S ⊂ τSR (ii)Finiteintersectionofelementsof S givesabasefor τSR andfiniteintersectionofelementsof S givesa basefor τSR
Definition14. InanSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ). Let A =( P (A), P (A)) beany SR-subsetof Y .Then,thelowerclosureof A istheclosureof P (A) in ( P (Y ), τSR ) andisdefinedastheintersectionofallclosedsupersetsof P (A),anditisdenoted by ClSR( P (A)).Also,theupperclosureof P (A) in ( P (Y ), τSR ) istheintersectionofallclosed supersetsof P (A) andisdenotedby ClSR P (A).Then,the SR-closureof A =( P (A), P (A)) is definedasClSR(A)=(ClSR( P (A)), ClSR( P (A)))
Definition15. InanSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ) Let A =( P (A), P (A)) beany SR-subsetof Y .Then,thelowerinteriorof A istheinteriorof P (A) in ( P (Y ), τSR ) andisdefinedasunionofall SR-opensubsetsof ( P (Y ), τSR ) containedin P (A),anditisdenotedby IntSR( P (A)).Also,theupperinteriorof P (A) in ( P (Y ), τSR ) istheunionofall SR-opensubsetsof ( P (Y ), τSR ) containedin P (A) andisdenotedby IntSR( P (A)).Then,the SR-interiorof A =( P (A), P (A)) andisdefinedas IntSR(A)= (IntSR( P (A)), IntSR( P (A)))
Definition16. An SR-subset A of (Y, τSR) issaidtobedensein Y if ClSR(A)= Y ,i.e.,an SR-subset A =( P (A), P (A)) isdensein Y ifClSR( P (A))= P (Y ) andClSR( P (A))= P (Y ).
Theorem3. An SR-subset A =( P (A), P (A)) ofSRTS (Y, τSR) isdensein Y iffforeverynon-empty SR-openset B =( P (B), P (B)) of (Y, τSR), P (A) ∩ P (B) = ∅ and P (A) ∩ P (B) = ∅
Proof. Suppose A =( P (A), P (A)) isdensein Y.Then, ClSR(A)=(ClSR( P (A)), ClSR( P (A)))=( P (Y ), P (Y ))= Y.Therefore, ClSR( P (A))= P (Y ) and ClSR( P (A))= P (Y ).Now, B =( P (B), P (B)) beanynon-empty SR-opensubsetof (Y, τSR).Then, A∩B = ( P (A) ∩ P (B), P (A) ∩ P (B)) Suppose P (A) ∩ P (B)= ∅.Then, P (A) ⊆ ( P (Y ) \ P (B)),whichimplies ClSR( P (A)) ⊆ ( P (Y ) \ P (B)),since P (B) ∈ τSR and,therefore, ( P (Y ) \ P (B)) is closed.However, ( P (Y ) \ P (B)) isapropersubsetof P (Y ),whichcontradicts ClSR P (A)= P (Y ).Hence, P (A) ∩ P (B) = ∅.Similarly, P (A) ∩ P (B) = ∅. Conversely,suppose A =( P (A), P (A)) isa SR-subsetof Y suchthatforeverynon-empty SR-openset B =( P (B), P (B)) of (Y, τSR), P (A) ∩ P (B) = ∅ and P (A) ∩ P (B) = ∅. Let y ∈ P (Y ),since P (A) ∩ P (B) = ∅;so,either y ∈ P (A) oritisalimitpointof P (A).Thatis, y ∈ ClSR( P (A)).Therefore, P (Y ) ⊆ ClSR( P (A)) ⊆ P (Y ),whichimplies ClSR( P (A))= P (Y ).Byasimilarargument,wecanprovethat ClSR( P (A))= P (Y ). Hence ClSR(A)= (ClSR( P (A)), ClSR( P (A))) = ( P (Y ), P (Y )) = Y.So, A isdense in Y
Definition17. InanSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ).Iffor γ ∈Y thereexistanopenset V1 of P (Y ) suchthat γ ∈V1 ⊆N ,where N ⊆ P (Y ),thenthesubset N iscalled τSR neighborhood.Similarly,iffor γ ∈Y thereexistanopenset V2 of P (Y ) suchthat γ ∈V2 ⊆N ,where N ⊆ P (Y ),thenthesubset N iscalled τSR neighborhood.If,atthesametime, N ⊆ P (Y ) and N ⊆ P (Y ),then NSR =(N , N ) issaidtobea τSR neighborhoodof γ ∈Y
Proposition1. ConsideranSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ). Let A =( P (A), P (A)) bean SR-subsetof SR-set Y satisfying P (A) ⊆ P (Y ) ⊆ P (Y ).Then, A is SR-openiffitisaneighborhoodofeachofitspoints.
Proof. Supposethat A =( P (A), P (A)) asanopensubsetof SR-set Y =( P (Y ), P (Y )) Then,forevery µ ∈ P (A), µ ∈ P (A) ⊂ P (A),andforevery ν ∈ P (A), ν ∈ P (A) ⊂ P (A).Hence, P (A) and P (A) satisfytheneighborhooddefinitionandareneighborhoodsof eachpoint,and,hence, A =( P (A), P (A)) isaneighborhoodofeachofitspoints.
Conversely,suppose A =( P (A), P (A)) isaneighborhoodofeachofitspoints.Giventhe assumption P (A) ⊆ P (Y ) ⊆ P (Y ),if A = ∅,thenitis SR-open.For µ ∈A,thenthere existsan SR-openset V =(Vµ , Vµ ) in Y suchthat µ ∈Vµ ⊂ P (A) and µ ∈Vµ ⊂ P (A) Thisimplies P (A)= {Vµ / µ ∈ P (A)} and P (A)= {Vµ / µ ∈ P (A)}.Hence, P (A) and P (A) are SR-open,whichimplies A isopen.
4.Continuityin SR-Sets
Inthissection,wediscussthecontinuityoffunctionsin SR-topologicalspaces,thecontinuous imageofan SR-closedset.The SR-homeomorphismisthepartoftheconversation.
Definition18. Let (Y, τSR) and (Z, ρSR) betopological SR-setswithtopologies τSR =(τSR , τSR ) and ρSR =(ρSR , ρSR ),respectively.Afunction ϕ1 : P (Y ) → P (Z) iscontinuousat µ ∈Y iff every ρ1-neighborhood H1 of ϕ1(µ) in P (Z) thereexistsa τ1-neighborhood G1 of µ in P (Y ) suchthat ϕ1(G1) ⊂H1 and ϕ2 : P (Y ) → P (Z) iscontinuousat µ ∈Z iffevery ρ2-neighborhood H2 of ϕ2(µ) in P (Z) thereexistsa τ2-neighborhood G2 of µ in P (Z) suchthat ϕ2(G2) ⊂H2.Then,thefunction ϕ =( ϕ1, ϕ2) : Y→Z issaidtobeacontinuousfunctionat µ ifboth ϕ1 and ϕ2 arecontinuousfunctions at µ
Example6. Assumethat V = {℘1, ℘2, ℘3, ℘4}, E = {ζ1, ζ2, ζ3, ζ4}, A = {ζ1, ζ3, ζ4}⊂E and G = {(ζ1, (℘1, ℘4), (ζ3, ℘2), (ζ4, ℘3)} isasoftset.Thus,weget P =(V, G) asasoftapproximation space.Ifwetake Y⊂V ,where Y = {℘3, ℘4},thenwehave P (Y )= {℘3}, P (Y )= {℘1, ℘3, ℘4} and BndP = {℘1, ℘4}.Thus, τSR(Y )= {V, ∅, {℘3}, {℘1, ℘3, ℘4}, {℘1, ℘4}} isan SR-topology. Let W = { 1, 2, 3, 4} and H = {(ζ1, { 1}), (ζ3, { 2, 3}), (ζ4, { 4})} beasoftset;then, wehave P =(W, H) asasoftapproximationspace.Ifwetake Z⊂W ,where Z = { 3, 4},then P (Z)= { 4}, P (Z)= { 2, 3, 4} andBndP = { 2, 3}, and ρSR = {W, ∅, { 4}, { 2, 3, 4}, { 2, 3}} isanother SR-topology.
Defineafunction ϕ =( ϕ1, ϕ2) : V→W suchthat ϕ(℘1)= ϕ2(℘1)= 2, ϕ(℘2)= ϕ2(℘2)= 1, ϕ(℘3)= ϕ1(℘3)= ϕ1(℘1)= 4 and ϕ(℘4)= ϕ2(℘4)= 3.Then, ϕ 1({ 2, 3, 4})= {℘1, ℘3, ℘4}, ϕ 1({ 2, 3})= {℘1, ℘4} and ϕ 1({ 4})= {℘3}.Thus, ϕ is SR-continuous,sincetheinverseimagefor each SR-opensetin W is SR-openin V .
Theorem4. Consider (Y, τSR) and (Z, ρSR) aretopological SR-setsand ϕ =( ϕ1, ϕ2) : Y→Z Forevery ρ SR-openset V =(V1, V2), ϕ1 1(V1) ⊆ P (Y ) ⊆ ϕ2 1(V2) ⊆ P (Y ).Then, ϕ isa continuousfunctionifandonlyiftheinverseimageofevery SR-opensetin Z under ϕ is SR-openin Y
Proof. Suppose ϕ =( ϕ1, ϕ2) : Y→Z isacontinuousfunctionand V =(V1, V2) isan SR-openset in Z.Wehavetoprovethat ϕ 1(V )= ϕ1 1(V1), ϕ2 1(V2) isan SR-opensetin Y.If ϕ1 1(V1) and ϕ2 1(V2) areempty,thentheresultisobvious.
Suppose µ ∈ ϕ1 1(V1) ⇒ µ ∈ ϕ2 1(V2),thatis, ϕ1(µ) ∈V1 and ϕ2(µ) ∈V2.Byfollowing thedefinitionofcontinuityof ϕ1,thereexistsaneighborhood N1 of µ suchthat ϕ1(N1) ⊂V1;then, µ ∈N1 = ϕ1 1( ϕ(N1)) ⊆ ϕ1 1(V1),whichimplies ϕ1 1(V1) is SR-open.Similarly, ϕ2 1(V2) is also SR-open.Hence, ϕ 1(V ) is SR-open.
Conversely,let ϕ 1(V ) be SR-openin Y forevery SR-openset V in Z.Wehavetoprovethat ϕ isacontinuousfunction.
Consider µ ∈Y asanarbitrarypoint,and ϕ1(µ) ∈V1 implies ϕ2(µ) ∈V2 (byhypothesis).Then, µ ∈ ϕ1 1(V1) and µ ∈ ϕ2 1(V2),whichmeans ϕ1( ϕ1 1(V1)) ⊂V1 and ϕ2( ϕ2 1(V2)) ⊂V2 implies that ϕ1 and ϕ2 arecontinuousat µ.Sincewetake µ asanarbitrarypoint,then ϕ1 and ϕ2 arecontinuous everywhere.Hence, ϕ iscontinuous.
Corollary1. Afunction ϕ =( ϕ1, ϕ2) : Y→Z iscontinuousifandonlyifforevery SR-closedsubset C in Z, ϕ 1(C) is SR-closedin Y
Proof. Consider ϕ =( ϕ1, ϕ2) : Y =( P (Y ), P (Y )) →Z =( P (Z), P (Z)) isacontinuous functionand C =(C1, C2) isanarbitrary SR-closedsubsetof Z.Then, P (Z) \C1 and P (Z) \C2 are SR-openin Y =( P (Y ), P (Y )) and ϕ1 1( P (Y ) \C1)= P (Y ) \ ϕ1 1(C1) and ϕ1 1( P (Y ) \C2)= P (Y ) \ ϕ1 1(C2),whichimpliesthat ϕ1 1(C1) and ϕ1 1(C2) areclosed in P (Y ) and P (Y ),respectively.Hence, ϕ 1(C) is SR-closedin Y.
Conversely,supposethatforany SR-closedsubset C =(C1, C2) in Z, ϕ 1(C) is SR-closedin Y.Let V =(V1, V2) beany SR-opensubsetof Z =( P (Z), P (Z)). Then, Z\ (V )= ( P (Z) \ (V1), P (Z) \ (V2)) is SR-closedand ϕ 1(Z\V )= ϕ1 1( P (Z) \V1), ϕ2 1( P (Z) \V2) =ϕ 1(Z) \ ϕ 1(V )= Z\ ϕ 1(V ) is SR-closedin Y, whichimplies ϕ 1(V ) is SR-openin Y.Thus, ϕ iscontinuous.
Remark2. 1.Everyrestrictionofacontinuousmappingisalsocontinuous.
Let ψ =(ψ1, ψ2) : Y =( P (Y ), P (Y )) →Z =( P (Z), P (Z)) beacontinuousfunction and A =( P (A), P (A)) bea SR-subsetof Y .Then,therestriction ψ|A = ψA : A→Z of ψ to A iscontinuous.Thisissobecauseforeach SR-opensubset W in Z, ψA 1(W)= ψ 1(W) ∩A,whichis SR-openin A
2.Consider βSR =(β , β ) asabasefora SR-topologyon Z.Then,thefunction ψ : Y→Z iscontinuousif andonlyif,foreach SR-basicopensetin Z, ψ 1(β) is SR-openin Y 3.Afunction ψ : Y→Z isopeniftheimageofevery SR-opensetin Y is SR-open. 4.Afunction ψ : Y→Y isclosediftheimageofevery SR-closedsetin Y is SR-closed.
Definition19. Let (Y, τSR) and (Z, ρSR) betopological SR-sets.Afunction ϕ =( ϕ1, ϕ2) : Y→Z is knownas SR-homeomorphismif
(i) ϕ is SR-bijective.
(ii) ϕ is SR-continuous.
(iii) ϕ 1 is SR-continuous.
Twosoftroughtopologicalspaces(SRTS)aresaidtobe SR-homeomorphicifthereisa SR-homeomorphism between Y and Z
Definition20. Consider Y =( P (Y ), P (Y )) and Z =( P (Z), P (Z)) astwotopological SR-setswithtopologies τSR =(τ1, τ2) and ρSR =(ρ1, ρ2),respectively,and Y×Z =( P (Y ) × P (Z), P (Y ) × P (Z)) istheCartesianproductof Y and Z.Thetopology ξ1 on P (Y ) × P (Y ) containingagatheringofopensetsoftheform L1 ×M1,where L1 isa τ1 −SR-openand M1 isa ρ1 −SR-open,asbasis,isknownastheproducttopology.Similarly,thetopology ξ2 on P (Y ) × P (Z) is thetopologycontainingagatheringofopensetsoftheform L2 ×M2,where L2 isa τ2 −SR-openand M2 isa ρ2 −SR-open,asbasis,isknownastheproducttopology.Hence,thetopology ξ =(ξ1, ξ2) iscalledthte producttopologyon Y×Z .
Definition21. Consider Y =( P (Y ), P (Y )) and Z =( P (Z), P (Z)) astwotopological SR-sets withtopologies τSR =(τ1, τ2) and ρSR =(ρ1, ρ2),respectively.Themapping ∏µ = P (Y ) × P (Z) → P (Y ) and ∏µ = P (Y ) × P (Z) → P (Y ),definedas ∏µ ((µ, ν)) = µ, ∀ (µ, ν) ∈ P (Y ) × P (Z) and ∏µ ((µ, ν)) = µ, ∀ (µ, ν) ∈ P (Y ) × P (Z),respectively,areknownasprojection mappings.Then, ∏µ = ∏µ , ∏µ isknownastheprojectionmappingfrom Y×Z→Y .Similarly,wecan definetheprojectionmapping ∏ν = (∏ν , ∏ν ) from Y×Z→Y
Theorem5. Consider Y and Z astwotopological SR-setsand Y×Z astheproductspace.Then, theprojections ∏µ and ∏ν arecontinuousmappings.
Proof. Suppose Y =( P (Y ), P (Y )) and Z =( P (Z), P (Z)) aretwotopological SR-sets withtopologies τSR =(τ1, τ2) and ρSR =(ρ1, ρ2),respectively.Let ξ betheproducttopologyon Y×Y and L =( P (L), P (L)) bean τ −SR-openset.Then, ∏µ 1( P (L))= P (L) × P (Z), where P (L) ∈ τ1 and P (Z) ∈ ρ1 implythat P (L) × P (Z) belongstothebasisfor τ1.Also, ∏µ 1( P (L))= P (L) × P (Z),where P (L) ∈ τ2 and P (Y ) ∈ ρ2 imply P (L) × P (Y ) belongstothebasisfor τ2,whichimpliesthat ( P (L) × P (Z), P (L) × P (Z)).Thus, ∏α and ∏α arecontinuousmappings.Therefore, ∏α = ∏µ , ∏µ isacontinuousmapping.Similarly, wecanshowthat ∏ν = (∏ν , ∏ν ) isalsoacontinuousmapping.
5.Compactnessin SR-Set
Inthissection,westudythecompactnessof SR-topologicalspaces,discussimagesof SR-compactspaces,andprovesomebasicresults.
Definition22. Let Y =( P (Y ), P (Y )) bea SR-set.Foranyopencovering V1 = {Vµ /i ∈ Ω} of P (Y ),ifwegetafinitesubcovering V1 F = {Vµ /i = 1,2,..., m},then P (Y ) issaidtobethecompact lowerapproximationof Y .Similarly,foranyopencovering V2 = {Vj /j ∈ Ω} of P (Y ),ifwegetafinite subcovering V2 F = {Vj /j = 1,2,..., n},then P (Y ) issaidtobethecompactupperapproximationof Y Then,the SR-set Y =( P (Y ), P (Y )) isknownasacompact SR-set.
Definition23. Suppose A =( P (A), P (A)) isan SR subsetof Y =( P (Y ), P (Y )).If,forany opencovering W = {Wj /j ∈ Ω} of P (A),wegetafinitesubcovering VF = {Vj /j = 1,2,..., n} of P (A),then P (A) asthesubsetof P (Y ) issaidtobecompact.If,atthesametime, P (A) isalso compact,thenwecall A acompact SR-subsetof Y .
Theorem6. Thecontinuousimageofacompacttopological SR-setiscompact.
Proof. Consider Y =( P (Y ), P (Y )) asacompact SR-setandsupposethat ψ =(ψ1, ψ2) : ( P (Y ), P (Y )) → ( P (Z), P (Z)) isacontinuousmapping.Then, ψ1 : P (Y ) → P (Z) and ψ2 : P (Y ) → P (Z) individuallyarecontinuousmappings.Let C1 = {Wν /ν ∈ Ω} bean opencoveringof P (Z).Then, ψ1 1(C1)= {ψ1 1(Wν ) /ν ∈ Ω} isanopencoveringfor P (Y ) Since P (Y ) iscompact,then,bydefinitionofcompactness,ithasafinitesubcovering,andthere areindices ν1ν2,..., νm suchthat P (Y )= m i=1 ψ1 1(C1) ψ1( P (Y )) ⊆ m i=1 (C1) ⊆ ψ1( P (Y )) Therefore, {Wν µ /i = 1,2,3,..., m} isafinitesubcoveringof ψ( P (Y ))= P (Z).So, P (Z) isalso compact.Similarly,wecanshowthat P (Z) iscompactand,hence, Z =( P (Z), P (Z)) isa compact SR-set.
Corollary2. Thehomeomorphicimageofacompact SR-spaceiscompact. Remark3. Intopological SR-sets,compactnessisatopologicalproperty.
Definition24. ConsideranSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ) Let Γ = {Aµ =( P (Aµ ), P (Aµ )) : µ ∈ Λ} beacollectionof SR-subsetsof Y .Ifeveryfinite subcollectionof Γ hasanon-emptyintersection,whichmeansthatifweconsideranyfinitesubset Λ1 of Λ, weget ν∈Λ1 Aν = ∅,thenthefiniteintersectionpropertyholdsincollection Γ.
Theorem7. ConsideranSRTS (Y, τSR),where Y =( P (Y ), P (Y )) and τSR =(τSR , τSR ); Y is SR-compactiffeverycollectionof SR-closedsubsetsin Y followingthefiniteintersectionpropertyitselfhas non-emptyintersections.
Proof. First,wesuppose Y is SR-compactand Γ = {Dµ =( P (Dµ ), P (Dµ )) : µ ∈ Λ} isan arbitrarycollectionof SR-closedsetssatisfyingthefiniteintersectionproperty.Wehavetoprovethat thecollection {Dµ =( P (Dµ ), P (Dµ )) : µ ∈ Λ} itselfhasnon-emptyintersection.Suppose,onthe contrary,that µ∈Λ Dµ = ∅.Bytakingthecomplement ( µ∈Λ Dµ ) = ∅ ,wehave Y = µ∈Λ Dµ ,which implies {Dµ =( P (Dµ ), P (Dµ )) : µ ∈ Λ} isanopencoverfor Y =( P (Y ), P (Y )).Byour assumption, Y =( P (Y ), P (Y )) is SR-compact,andthereareindices µ1, µ2, µ3,..., µk suchthat P (Y )= k ι=1 P (Dµ ι ) and P (Y )= k ι=1 P (Dµ ι ).Again,bytakingthecomplement,weget k ι=1 P (Dµ ι )= ∅ and k ι=1 P (Dµ ι )= ∅,thatis, k ι=1 Dµ ι = k ι=1 P (Dµ ι ), k ι=1 P (Dµ ι ) = ∅,which contradictsthefiniteintersectionproperty.So,ourassumptioniswrongand µ∈Λ Dµ = ∅. Conversely,supposethateverycollectionof SR-closedsetssatisfyingthefiniteintersection propertyhasanon-emptyintersectionitself.Wenowhavetoprovethat Y =( P (Y ), P (Y )) is SR-compact.Forthis,letusconsider {Vε =( P (Vε ), P (Vε )) : ε ∈ Υ} asanopencoverof Y,i.e., Y = ( P (Y ), P (Y )) = ε∈Υ P (Vε ), ε∈Υ P (Vε ) .Toprovethat Y is SR-compact, wehavetoshowthatthisopencoverhasafinitesubcover.Onthecontrary,supposethattheredoesnot existanyfinitesubcoverforthisopencover.Then,foranyfinitesubcover Υ1 of Υ, ∈Υ1 V = Y, i.e., ∈Υ1 P (V ), ∈Υ1 P (V ) = ( P (Y ), P (Y )).Thisimplies ∈Υ1 V = ∅.Now, {Vε =( P (Vε ), P (Vε ) ) : ε ∈ Υ} isacollectionof SR-closedsetssatisfyingthefiniteintersection property,so ε∈Υ1 Vε = ∅ i.e., ε∈Υ P (Vε ), ε∈Υ P (Vε ) = ∅.Bytakingthecomplement, weget ε∈Υ P (Vε ), ε∈Υ P (Vε ) = ( P (Y ), P (Y )),whichcontradictsoursuppositionthat {Vε =( P (Vε ), P (Vε )) : ε ∈ Υ} isanopencoverof Y.Hence, {Vε =( P (Vε ), P (Vε )) : ε ∈ Υ} hasafinitesubcover,so Y is SR-compact. Theorem8. Every SR-closedsubsetof SR-compactspaceis SR-compact. Proof. Let Y bean SR-compactspaceand D =( P (D), P (D)) bea SR-closedsubsetof Y.Let {Vε =( P (Vε ), P (Vε )) : ε ∈ Υ} beanopencoverfor D =( P (D), P (D)); thereexistan SR-openset Wε =( P (Wε ), P (Wε )) in Y =( P (Y ), P (Y )) suchthat Vε = Wε ∩D, ε ∈ Υ,i.e., P (Vε )= P (Wε ) ∩ P (D) and P (Vε )= P (Wε ) ∩ P (D) Thecollection {D , Wε : ε ∈ Υ} isanopencoverfor Y.Since Y iscompact,thereexistsafinitesubcover {D , Wε : ε ∈ Υ} of Y,thatis, Y = D k ι=1 Wε ι ,whichimplies P (Y )= P (D ) k ι=1 P (Wε ι ) and P (Y )= P (D ) k
6.Applicationof SR-SetinMulti-AttributeGroupDecisionMaking
Decision-makingperformsavitalroleinourdailylife,andthisprocessyieldsthebestalternative amongdifferentchoices.Inthissection,wepresentanapplicationofan SR-setinmulti-attribute groupdecisionmaking(MAGDM)forcosmeticbrandselection.First,wepresentAlgorithm 1 andits flowchartformulti-attributegroupdecisionmaking.
Algorithm1 Theschemeofthealgorithmisgivenas.
Step-1:Writethesoftset G =(T , A) whichdescribesthegivendata.
Step-2:Basedoninitialassessmentresultsofthegroupofanalysts S,defineasoftset.
Step-3:Obtainan SR-approximationsintheformofsoftsets Λ =(λ , S) and Λ =(λ , S)
Step-4:Definefuzzysets νΛ , νΛ and νΛ correspondingtothesoftsets Λ =(λ , S), Λ =(λ, S) and Λ =(λ , S) definedbytheformulas:
ν
m ι=1 CλDι (
)
k ),
Step-6:Findthefinaldecisionsetbyadding Λ , Λ,and Λ ,calculatedas
Step-7:Finally,thealternativehavingthemaximumdecisionvaluecanbechosenastheoptimal solution.
Figure1. GraphicalrepresentationofAlgorithm 1
Figure1:GraphicalrepresentationofAlgorithm
Example7. Thetradeofqualitycosmeticsisgrowingrapidlyamongthelower-middleclassofdeveloping countrieslikePakistanandIndia.Assumethatapopulardepartmentalstoreofthecitywantstomakea contractwithamultinationalcompanyfortheproductionofcosmetics.Themanagingcommitteeofthestore consistofthreemanagers, S = {M1, M2, M3}:theproductmanager,marketingmanager,andaccounts manager.Theteamofthesethreemanagersiselectedtochooseonebrandwhichcoversthemajorproductionof cosmetics.Theyconsidersevenbrands: V = {h1,¯h2,¯h3,¯h4,¯h5,¯h6,¯h7,¯h8},where h1:Loreal, h2:Maybelline, h3:Remmil, h4:ArtDeco,
h5:Essence, h6:ColorStudio, h7:Mac, h8:Sephora. Theydefineasetofcriteriafortheselectionofasuitablebrandfortheirstoreasfollows, E = {ρ1, ρ2, ρ3, ρ4, ρ5, ρ6, ρ7, ρ8},where
ρ1:Productquality
ρ2:Relationshipcloseness(customer–brandrelationship)
ρ3:Deliveryperformance
ρ4:Pricestability
ρ5:Plansformajorevents
ρ6:Distributionplans(in-storefurniture)
ρ7:Recoveryservicesincaseofdamages
ρ8:Shoppermarketingactivities.
Weconstructasoftset G =(T , A) whichexplainsthequalitiesofthebrandsunderconsideration. ThetabularformofthesoftsetisgiveninTable 2
Table2. Softset (T , A).
(T , A) ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 h1 10110100 h2 10111101 h3 01010010 h4 01010000 h5 10100101 h6 01110000 h7 10110110 h8 01010010
Let Xi betheinitialassessmentresultofthemanagerteam.Werepresentthisevaluationbymeansofasoft set Λ =(λ, S) whosetabularrepresentationisgivenbyTable 3
Table3. Softset (λ, S)
D1 D2 D3 h1 101 h2 101 h3 010 h4 010 h5 001 h6 101 h7 101 h8 010
Fromthissoftset Λ =(λ, S),theprimaryevaluationresultofexpertsis
X1 = λ(D1)= {h1,¯h2,¯h6,¯h7},
X2 = λ(D2)= {h3,¯h4,¯h7,¯h8},
X3 = λ(D3)= {h1,¯h2,¯h5,¯h6}
Now,wefindthe SR-approximationsas
λ (D1)= P (X1)= {h1}, λ (D2)= P (X2)= {h3,¯h7,¯h8}, λ (D3)= P (X3)= {h2,¯h5}, and λ (D1)= P (X1)= V, λ (D2)= P (X2)= V, λ (D3)= P (X3)= V
Followingthese SR-approximations,wegettwosoftsets, Λ =(λ , S) and Λ =(λ , S),where λ (Di )= P (Xi ) and λ (Di )= P (Xi ).Tabularrepresentationofthesesoftsetsaregivenin Tables 4 and 5.
Table4. Softset Λ D1 D2 D3 h1 000 h2 101 h3 010 h4 000 h5 001 h6 000 h7 010 h8 010 Table5. Softset Λ D1 D2 D3 h1 111 h2 111 h3 111 h4 111 h5 111 h6 111 h7 111 h8 111
Now,wedefineafuzzyset νΛ (hk ), νΛ (hk ),and νΛ (hk ) asfollows: νΛ (hk )= 1 3 3 ∑ i=1 Cλ Di (hk ), νΛ (hk )= 1 3 3 ∑ i=1 CλDi (hk ), νΛ (hk )= 1 3 3 ∑ i=1 Cλ Di (hk ).
Thus,wehave νΛ (hk )= {(h1,0), (h2,2/3), (h3,1/3), (h4,0), (h5,1/3), (h6,0), (h7,1/3), (h8,1/3)}, νΛ (hk )= {(h1,2/3), (h2,2/3), (h3,1/3), (h4,1/3), (h5,1/3), (h6,2/3), (h7,2/3), (h8,1/3)}, νΛ (hk )= {(h1,0), (h2,2/3), (h3,1/3), (h4,0), (h5,1/3), (h6,0), (h7,1/3), (h8,1/3)},
Now,wefindthedecisionsetbyadding Λ , Λ,and Λ .Then,wehave
(hk )=
(hk )+
(hk )+
Since h2 isthebrandhavingthemaximumdecisionvalueinTable 6,then h2 isselectedbythethemanager teamasthemajorproductionbrandforcosmeticsinthedepartmentalstore.
Intheproposedalgorithm,weobservethattheuseof SR-methodologyfilterstheprimaryassessment resultsandpermitstheexpertstochoosetheoptimalalternativeinasuitablemanner.Particularly,the SR-upper approximationcanbeusedtoaddoptimalobjectspossiblyneglectedbytheselectorsintheprimaryassessment, whilethe SR-lowerapproximationcanbeusedtoremovetheobjectsthatareirregularlyselectedasoptimal. Hence, SR reducestheerror,tosomeextent,thatiscausedbythesubjectivenatureofexpertsduringgroup decisionmaking.
Table6. Decisionvaluetable.
DecisionValue
h1 1.000 h2 1.889 h3 1.556 h4 1.000 h5 1.556 h6 1.778 h7 1.778 h8 1.556
NowwepresentbarchartasgivenbyFigure 2 ofthedecisionvalues.
Figure2. GraphicalrepresentationofDecisionValues.
7.Applicationsofthe SR-TopologicalSpacesinImageProcessing
Since 2 isthebrandhavingmaximumdecisionvalueintable6.So, 2 isselectedbythethe managersteamasamajorproductionbrandforcosmeticsinthedepartmentalstore.
Weknowthatageometricalfigurecanbeobtainedbyitspartinformationanditstopological structureproperties.Similarly,accordingtothe SR-topologicalpropertiesofthe SR-space,wecan alsorestorethe SR-topologicaldiagramforsomeincompletediagram.
Example8. Oneofthebestexamplesofanincompleteimageisthatoffingerprints.Figure 3 showsaportionof fingerprintinformationfromaperson;however,thefingerprintinformationisincomplete.Wecanobtainthe realfingerprintimageonthebasisofthisFigure 3 byusing SR-approximationsand SR-topologicalproperties.
Intheproposedalgorithmweobservethattheuseof SR-methodologyfilterstheprimary assessmentresultsandpermittheexpertstochoosetheoptimalalternativeinasuitablemanner. Particularly, SR-upperapproximationcanbeusedtoaddtheoptimalobjectspossiblyneglected bytheselectorsinprimaryassessmentwhile SR-lowerapproximationcanbeusedtoremovethe objectsthatareirregularlyselectedasoptimal.Hence SR-reducetheerrortosomeextentcaused bysubjectivenatureofexpertsduringgroupdecisionmaking.
ApplicationsoftheTopological SR-SpacesinImageProcessing Weknowthatgeometricalfigurescanbeobtainedbyitspartinformationanditstopological structureproperties.Similarly,accordingtothe SR-topologicalpropertiesofthe SR-space,we canalsorestorethe SR-topologicaldiagramforsomeincompletediagram.
7Conclusion
Figure3. (a)Incompleteinformationofafingerprint.(b)Renewedfingerprint.
Thedevelopmentofthesetheoriescanformthetheoreticalbasisforfurtherapplicationsofthe SR-setand SR-topologyinmanyscienceandengineeringareas,suchasimageprocessing,protein structureprediction,targetrecognition,andgenestructureprediction.
8.Conclusions
Weestablishedthetopologicalstructureonthe SR-setinanewway.Wedefinevarioustopological terms,define SR-continuity,producttopologyin SR-set,andcompactnessin SR-setsbytaking an SR-setasapairofsetscorrespondingtothelowerandupperapproximations.Furthermore, wepresentanalgorithmtocopewithuncertaintiesinmulti-attributegroupdecision-makingproblems byutilizing SR-sets.Theeffectivenessofthealgorithmwasverifiedbyacasestudyforcosmetic brandselection.However,undertopologicaltransformation,somepropertiesofthe SR-setand topologicaltheory,likeconnectednessandseparationaxiomsonthe SR-set,needtobefurtherstudied. Ifwecombinethe SR-setwithothersoftcomputingmethods,suchasbipolarfuzzy,neutrosophicset, andotherhybridstructures,andusetheminimageprocessing,expertsystems,andcognitivemaps, ahighmachineIQandhybridintelligentsystemcanbedesigned,whichwillbeaproductiveattempt.
AuthorContributions: Theauthorscontributedtoeachpartofthispaperequally.Theauthorsreadandapproved thefinalmanuscript.
Acknowledgments: Theauthorsarehighlythankfultotheeditorandrefereesforthevaluablecommentsand suggestionsforimprovingthequalityofthepaper.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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