Abstract Applied Mathematics and Computation 175 (2006) 645–674 Iterative solution of fuzz

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IterativesolutionoffuzzylinearsystemsMehdiDehghan*,BehnamHashemiDepartmentofAppliedMathematics,FacultyofMathematicsandComputerSciences,AmirkabirUniversityofTechnology,No.424,HafezAve.,Tehran,Iran

AbstractLinearsystemshaveimportantapplicationstomanybranchesofscienceandengi-neering.Inmanyapplications,atleastsomeoftheparametersofthesystemarerepre-sentedbyfuzzyratherthancrispnumbers.Soitisimmenselyimportanttodevelopnumericalproceduresthatwouldappropriatelytreatgeneralfuzzylinearsystems(willbedenotedbyFLS)andsolvethem.Inthispaperfirstlyageneralfuzzylinearsystemusingtheembeddingapproach,hasbeeninvestigatedandthenseveralwell-knownnumericalalgorithmsforsolvingsystemoflinearequationssuchasRichardson,Extrap-olatedRichardson,Jacobi,JOR,Gaus

–Seidel,EGS,SOR,AOR,ESOR,SSOR,

USSOR,EMAandMSORareextendedforsolvingFLS.Theiterativemethodsarefol-lowedbyconvergencetheoremsandthepresentedalgorithmsaretestedbysolvingsomenumericalexamples.(

HackbuschnoticedthatGaussspellingislesscorrectthanGaus

[W.Hackbusch,IterativeSolutionofLargeSparseSystemsofEquations,Springer-VerlagInc.,NewYork,1994[9]].)Ó2005ElsevierInc.Allrightsreserved.

Keywords:Fuzzylinearsystems(FLS);Nonnegativematrix;Iterativemethods;Jacobi;Gaus–Seidel(GS);Successiveoverrelaxation(SOR)method;Acceleratedoverrelaxation(AOR);Sym-metricSOR(SSOR);UnsymmetricSOR(USSOR);Rateofconvergence;ExtrapolatedRichardson(ER);ExtrapolatedmodifiedAitken(EMA)

0096-3003/$-seefrontmatterÓ2005ElsevierInc.Allrightsreserved.doi:10.1016/j.amc.2005.07.033

*Correspondingauthor.

E-mailaddresses:mdehghan@aut.ac.ir(M.Dehghan),hashemi_am@aut.ac.ir(B.Hashemi).

AppliedMathematicsandComputation175(2006)645–6741.IntroductionOneofthemajorapplicationsusingfuzzynumberarithmeticistreatinglin-earsystemswhoseparametersareallorpartiallyrepresentedbyfuzzynumbers[8].AgeneralmodelforsolvingaFLSwhosecoefficientmatrixiscrispandtheright-handsidecolumnisanarbitraryfuzzynumberwasfirstproposedbyFriedmanetal.[8].Theyusedtheembeddingmethodandreplacedtheoriginalfuzzylinearsystembyacrisplinearsystemwithanonnegativecoefficientma-trixS,whichmaybesingularevenifAbenonsingular.Anotherclassofmeth-odsforsolvingfuzzylinearsystemsisiterativemethods.WithourbestknowledgeAllahviranloo[1]introducedtheJacobimethodforsolvingFLSforthefirsttime.AlsoheproposedtheGaus–SeidelandtheSORmethodsin[1,2].ThemainaimofthispaperistomodifythesemethodsandproposeotheriterativeschemesforsolvingFLShere.Alsowithsmalladditionalcom-putationaleffort,itseemsadvisabletoaddtoanyiterativeschemeitsextrap-olatedformasanadditionterm.InthispaperseveraldifferentnumericaltechniquesandtheirextrapolatedformwillbedevelopedandwillbecomparedforsolvingthistypeofFLS.Thispaperisorganizedinthefollowingway:Somebasicdefinitionsandre-sultsonfuzzynumbersandFLSaregiveninSection2.InSection3,basiccon-ceptsofaniterativemethodareintroduced.IterativemethodsareintroducedforsolvingFLSinSection4andtheproposedalgorithmsarediscussedandcomparedbysolvingseveralexamplesinSection5.Section6endsthispaperwithconclusion.

2.Preliminaries2.1.FuzzynumbersFuzzynumbersareonewaytodescribethevaguenessandlackofprecisionofdata.Thetheoryoffuzzynumbersisbasedonthetheoryoffuzzysetswhichwasintroducedin1965byZadeh[19].TheconceptofafuzzynumberwasfirstusedbyNahmiasintheUnitedStatesandbyDuboisandPradeinFranceinthelate1970s.Ourdefinitionofafuzzynumberisexplainedinthefollowing.

Definition2.1.WerepresentanarbitraryfuzzynumberbyapairoffunctionsðuðrÞ;uðrÞÞ;06r61whichsatisfythefollowingrequirements:

•u(r)isaboundedleftcontinuousnondecreasingfunctionover[0,1].•uðrÞisaboundedleftcontinuousnonincreasingfunctionover[0,1].•uðrÞ6uðrÞ;06r61.

646M.Dehghan,B.Hashemi/Appl.Math.Comput.175(2006)645–674AcrispnumberaissimplyrepresentedbyuðrÞ¼uðrÞ¼a;06r61.ThesetofallfuzzynumbersfuðrÞ;uðrÞgbecomesaconvexconethatisdenotedbyE1whichisthenembeddedisomorphicallyandisometricallyintoaBanach

space.Apopularfuzzynumberisthetrapezoidalfuzzynumberu(x0,y0,r,b)withtwodefuzzifiersx0,y0andleftfuzzinessrandrightfuzzinessb.

Itsparametricformis

uðrÞ¼x0Àrþrr;uðrÞ¼y0þbÀbr.Letv=(x0,y0,r,b)beatrapezoidalfuzzynumberandx0=y0,thenviscalledatriangularfuzzynumberandisshownbyv=(x0,r,b).

Forexample,thefuzzynumber(1+r,6À2r)isshowninFig.1,wherex0=2,y0=4,r=1andb=2.NowsupposethatyðrÞ¼ðyðrÞ;yðrÞÞisanexactfuzzysolutionand

y0ðrÞ¼ðy0ðrÞ;y0ðrÞÞistheestimatedfuzzysolution(seeFig.2).

Fig.1.Afuzzynumber.Fig.2.Thedistanceoftwofuzzynumbers.

M.Dehghan,B.Hashemi/Appl.Math.Comput.175(2006)645–674647