6082OSalazarJSorianoandHSerrano2thesecondonecomputingcrwhichistherightpartofthecentroidMelgarejoetal14presentedanalternativeversionofKMalgorithmreexpressingtheexpressionsforclandcrThealte ID: 326246
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AppliedMathematicalSciences,Vol.6,2012,no.122,6081-6086CentroidofanIntervalType-2FuzzySetRe-FormulationoftheProblemOmarSalazarMorales,JoseJairoSorianoMendezandJoseHumbertoSerranoDeviaUniversidadDistritalFranciscoJosedeCaldas,Bogota,Colombiaosalazarm@correo.udistrital.edu.co,jairosoriano@udistrital.edu.cojhserrano@udistrital.edu.coAbstractIntheoryofintervaltype-2(IT2)fuzzysets,type-reductionisoneofthemostdicultparts.Karnik-Mendel(KM)algorithmshavebeenthemostpopularmethodforcomputingtype-reductionofIT2fuzzysets.ThesealgorithmspresenttwoindependentproceduresforcomputingthegeneralizedcentroidofanIT2fuzzyset:the rstoneforcomputingitsleftpart(denotedbycl),andthesecondoneforcomputingitsrightpart(denotedbycr).Wepresentadiscussionwhereweshowthatthecalculationofclandcristhesameproblemandnottwodi erentproblems,andtheycanbecalculatedwithageneralexpression.MathematicsSubjectClassi cation:03E72,03B52,94D05Keywords:Intervaltype-2fuzzyset,Centroid,Karnik-Mendelalgorithms1IntroductionTheKarnik-Mendel(KM)algorithmhasbeenthemethodforcomputingtype-reductionofintervaltype-2(IT2)fuzzysets[3].Thisalgorithmhasbeenstudiedtheoreticallyandexperimentallyinordertoimproveitsperformanceonapplications.Itgivesanexactwaytogetthegeneralizedcentroid,whichisaclosedinterval,ofanIT2fuzzyset.TheconvergenceoftheKMalgorithmwasprovedbyMendelandLiu[7].AnenhancedversionofthisalgorithmisknownasEnhancedKarnik-Mendel(EKM)algorithm[5].EKMalgorithmspeedsupthesearchofthecentroidbyusinganoptimalinitialisationandabetterwaytocomputetheinvolvedarithmeticexpressions.Bothversionsalwayspresenttwoparts(eveninrecentpapers[5]):1.the rstonecomputingcl,whichistheleftpartofthecentroidand, 6082O.Salazar,J.SorianoandH.Serrano2.thesecondonecomputingcr,whichistherightpartofthecentroid.Melgarejoetal.[1,4]presentedanalternativeversionofKMalgorithmre-expressingtheexpressionsforclandcr.ThealternativeversionproposedbyMelgarejo,calledRAUL(RecursiveAlgorithmwithUniqueLoop)[1],usesinsideofanuniqueloopre-expressedexpressionsforclandcr,andit ndsaminimumforclandamaximumforcr.SuchpapersstillpresenttwopartsforcomputingthecentroidbyusingKMalgorithms:oneforclandoneforcr.Twodi erentpartsforcomputingthegeneralizedcentroidofanIT2fuzzysethavedirectimplicationsonengineeringapplications,suchasin[2]whereclandcrwerecalculatedbyhardware.Theaimofthispaperistoshowthatthecalculationofclandcrisreallythesameproblemandnottwodi erentproblemswiththedeductionofageneralexpressionthatinvolvesthecalcula-tionofboth.Expressionsofclandcrarenotindependent,oneexpressioncanbededucedfromtheotherexpression.Thispaperisorganizedasfollows:Sect.2presentsashortdescriptionofcalculationofthegeneralizedcentroidofanIT2fuzzyset.Sect.3presentstheproofthatthecalculationofclandcrisreallythesameproblem.Sect.4givesare-formulationofproblemofthecentroid.Sect.5givessome nalcomments.Finally,Sect.6givestheconclusion.2CentroidofanIntervalType-2FuzzySetGivenanIT2fuzzyset~(formoredetailssee[6])whichisde nedonanuniversalsetXR,withmembershipfunction~A(x),x2X,itsgeneralizedcentroidc(~)isaclosedinterval,i.e.,c(~)=[cl;cr],whereclandcrarerespectivelytheminimumandmaximumofallcentroidsoftheembeddedtype-1fuzzysetsinthefootprintofuncertainty(FOU)of~.KarnikandMendel[3]demonstratedthatclandcrcanbecomputedfromtheloweranduppermembershipfunctionsof~asfollows:cl= LX=1x ~A(x)+NX=L+1x ~A(x)!, LX=1 ~A(x)+NX=L+1 ~A(x)!(1)cr= RX=1x ~A(x)+NX=R+1x ~A(x)!, RX=1 ~A(x)+NX=R+1 ~A(x)!(2)where ~Aand ~Aarerespectivelytheupperandlowermembershipfunctionsof~(Fig.1(a)).L2Nistheswitchpointthatmarksthechangefrom ~Ato ~A(Fig.1(b)),R2Nistheswitchpointthatmarksthechangefrom ~Ato ~A(Fig.1(c))andN2Nisthenumberofdiscretepointsonwhichthex-domainof~hasbeendiscretized.Itistruethatin(1)and(2)x1x2xN,inwhichx1denotesthesmallestsampledvalueofxandxNdenotesthelargestsampledvalueofx[5]. Centroidofanintervaltype-2fuzzyset6083 (a) (b) (c)Figure1:(a)IntervalType-2fuzzyset.(b)clanditsinterpretation.(c)cranditsinterpretation.3DeductionofaGeneralExpressionLetusrewritetheexpression(2).Ifwelet=N+1 ithenwewillhave:if1iRthen1N+1 R,andhenceN R+1N;ifR+1iNthenR+1N+1 N,andhence1N R;therefore(2)canbewrittenas(bypropertiesofsums)cr=0@NXj=N R+1yj ~A(yj)+N RXj=1yj ~A(yj)1A,0@NXj=N R+1 ~A(yj)+N RXj=1 ~A(yj)1A=0@N RXj=1yj ~A(yj)+NXj=N R+1yj ~A(yj)1A,0@N RXj=1 ~A(yj)+NXj=N R+1 ~A(yj)1A=0@L0Xj=1yj ~A(yj)+NXj=L0+1yj ~A(yj)1A,0@L0Xj=1 ~A(yj)+NXj=L0+1 ~A(yj)1A(3)whereyj=xN+1 j;1jN;(4)andL=N R.Equations(1)and(3)havethesameform.WecanobtainonefromtheotheronlywiththesubstitutionofxibyyjandLbyL(orviceversa).Equations(1)and(3)di erinLandL(switchpoints)andthatthevalues 6084O.Salazar,J.SorianoandH.Serranoofxareindexedinreverseorderas(4)establishes.Equation(4)meansthaty1=xN,y2=xN 1,...,yN=x1,asweshowinFig.2.Itisjustapermutation(abijectivefunction)oftheNvaluesofx.Equation(4)canbethoughtasanindexationoftheNvaluesofxinreverseorder.Then,theproblemforcomputingclandcrcanbereducedtocalculatethemwiththesameprocedure.Itisjustnecessaryreversetheorderinwhichthevaluesofxareindexed,andifwearecomputingclthenwewillneedto ndaminimum,andifwearecomputingcrthenwewillneedto ndamaximum.y1y2yj 1yjyj+1yN 1yN x1x2xN jxN+1 jxN+2 jxN 1xN Figure2:Permutationyj=xN+1 j(1N)thatinvertstheorderinwhichthevaluesofxareindexedIfwestartform(1)byusingasimilarargumentthenwewillobtainananalogousexpressionto(2),i.e.,therewillbeanexpressioncl=0@R0Xj=1zj ~A(zj)+NXj=R0+1zj ~A(zj)1A,0@R0Xj=1 ~A(zj)+NXj=R0+1 ~A(zj)1A(5)whichisanalogousto(2),wherezj=xN+1 j,1N,andR=N L.4De nitionofaGeneralExpressionWede neageneralexpression1(6)forcomputingacentroid(clorcr)becauseofthedualitybetween(1)and(3).Itisjustnecessarytoreplaceappropriatevaluesinorderto ndclorcrasweshowinTable1.c= MX=1w ~A(w)+NX=M+1w ~A(w)!, MX=1 ~A(w)+NX=M+1 ~A(w)!(6)Thesubstitutionofandwiin(6)byLandxirespectivelygivestheex-pression(1);andthesubstitutionofandwiin(6)byL(=N R)andyi(=xN+1 i)respectivelygivestheexpression(3)(whichisthesameequation(2)asweshowedabove). 1Thisproblemcanalsobere-formulatedwiththede nitionofageneralexpressionbyusingthedualitybetween(2)and(5) Centroidofanintervaltype-2fuzzyset6085Table1:Summaryforcomputingacentroid(clorcr)byusing(6)c M w Observation (1iN) cl L x Ifweare ndingcl,wewillhaveto ndLsuchthat(6)isminimumbyusingxcr L0 y Ifweare ndingcr,wewillhaveto ndL0(=N R)suchthat(6)ismaximumbyusingy(=xN+1 )5RelationshipwiththeConceptofConvexCombinationZadeh[8,page345]de nedtheconceptofconvexcombinationofthreearbi-trary(type-1)fuzzysets,Band,withmembershipfunctionsA,Band,bytherelationhA;B;i=+B,whereisthecomplementof.WrittenoutintermsofmembershipfunctionsA;B;(w)=(w)A(w)+(1 (w))B(w);w2X:(7)IfweapplythisconcepttoanIT2fuzzyset~(withloweranduppertype-1fuzzysets and whosemembershipfunctionsare ~Aand ~A):A(w)= ~A(w);B(w)= ~A(w);and(w)=(1;ifiM;0;ifiM+1;withi=1;:::;N,then(7)reducesto A;A ;(w)=(w) ~A(w)+(1 (w)) ~A(w)=( ~A(w);ifiM; ~A(w);ifiM+1:(8)Withtheaidof(8),thenumeratorof(6)canbewrittenasNX=1w A;A ;(w)=MX=1w A;A ;(w)+NX=M+1w A;A ;(w)=MX=1w ~A(w)+NX=M+1w ~A(w);andsimilarlyforitsdenominator.Hence(6)canbewrittenasc= NX=1w A;A ;(w)!, NX=1 A;A ;(w)!:Melgarejoetal.[1,4]deducedandusedanalogousexpressionsinhisworkoftheRAULalgorithm,buthedidnotcitetheconceptofconvexcombination. 6086O.Salazar,J.SorianoandH.Serrano6ConclusionThispapershowedthatexpressions(1)and(2),whichweregivenbyKarnikandMendelinordertocalculateclandcr,havethesameformwithasimplesubstitutionofitsindexvariable.Then,thereisadualitybetweenthem.Wepresentedageneralexpression(6)forcomputingclandcr.Itisjustnecessarytoreplaceappropriatevaluesinorderto ndclorcrasweshowedinTable1.WealsodeducedageneralexpressionbasedontheconceptofconvexcombinationgivenbyZadeh.Thecalculationofclorcrcanbedonewithamembershipfunctionthatistheconvexcombinationofthreetype-1fuzzysets: ~A(lowermembershipfunctionof~), ~A(uppermembershipfunctionof~)and(acrispset).References[1]HectorBernal,KarinaDuran,andMiguelMelgarejo.Acomparativestudybetweentwoalgorithmsforcomputingthegeneralizedcentroidofaninter-valtype-2fuzzyset.InProceedingsoftheIEEEInternationalConferenceonFuzzySystems(FUZZ2008),pages954{959,2008.[2]LindaK.DuranandMiguelA.Melgarejo.ImplementacionhardwaredelalgoritmoKarnik-MendelmejoradobasadaenoperadoresCORDIC.Inge-nierayCompetitividad,11(2):21{39,2009.[3]NileshN.KarnikandJerryM.Mendel.Centroidofatype-2fuzzyset.InformationSciences,132:195{220,2001.[4]MiguelMelgarejo.Afastrecursivemethodtocomputethegeneralizedcentroidofanintervaltype-2fuzzyset.InAnnualMeetingoftheNorthAmericanFuzzyInformationProcessingSocietyNAFIPS2007,pages190{194,SanDiego,California,USA,June2007.[5]JerryM.Mendel.Oncentroidcalculationsfortype-2fuzzysets.Appl.Comput.Math.,10(1):88{96,2011.[6]JerryM.MendelandRobertI.BobJohn.Type-2fuzzysetsmadesimple.IEEETransactionsonFuzzySystems,10(2):117{127,2002.[7]JerryM.MendelandFeilongLiu.Super-exponentialconvergenceoftheKarnik-Mendelalgorithmsforcomputingthecentroidofanintervaltype-2fuzzyset.IEEETransactionsonFuzzySystems,15(2):309{320,April2007.[8]LoftiA.Zadeh.Fuzzysets.InformationandControl,8(3):338{353,1965.Received:June,2012