zy numbers ( À Áwith different r and also all triangular fuz-

1. Author'spersonalcopy ExpertSystemswithApplications39(2012)690–695 ContentslistsavailableatSciVerseScienceDirect ExpertSystemswithApplications journalhomepage:www.elsevier.com/locate/eswa Anapproachforrankingoffuzzynumbers R.Ezzatia,⇑,T.Allahviranloob,S.Khezerlooa,M.Khezerloob a b DepartmentofMathematics,KarajBranch,IslamicAzadUniversity,Karaj,Iran DepartmentofMathematics,ScienceandResearchBranch,IslamicAzadUniversity,Tehran,Iran article info abstract Inordertorankallfuzzynumbers,wemodifythemethodof‘‘anewapproachforrankingoftrapezoidal fuzzynumbers’’byAbbasbandyandHajjari(2009).Ourproposedmethodisusedforrankingsymmetric fuzzynumbers.Theadvantageofthismethodisillustratedbysomecomparativeexamples. Ó2011ElsevierLtd.Allrightsreserved. 2.Preliminaries Keywords: Rankingoffuzzynumbers Parametricformoffuzzynumber Magnitudeoffuzzynumber 1.Introduction Inmanyapplications,rankingoffuzzynumbersisanimportant andprerequisiteprocedurefordecisionmakers.Firstly,In1976, Jain(1976,1977)proposedamethodforrankingoffuzzynumbers, thenalargeofvarietyofmethodshavebeendevelopedtorankfuz- zynumbers.WangandKerre(2001a,2001b)classifiedtheordering methodintothreecategoriesandproposedsevenreasonableprop- ertiestoevaluatetheorderingmethod.In2007,AsadyandZendeh- nam(2007)proposedanewmethodbasedon‘‘distance minimizing’’andthenin2009,AbbasbandyandHajjari(2009)pro- posedanewmethodforrankingoftrapezoidalfuzzynumbersand showedthattheirnewmethodovercometosomedrawbacksof distanceminimizing.Butbytheirnewmethod,alltrapezoidalfuz- x0,y0,r,r) zynumbersx0þ2y0;r;rwithdifferentrarethesameorder. Therearevariousdefinitionsfortheconceptoffuzzynumbers (Dubois&Prade,1982;Gal,2000;Goetschel&Voxman,1986) Definition2.1(GoetschelandVoxman,1986).Afuzzynumberis afuzzysetlikeu:R?[0,1]satisfyingthefollowingproperties: (i)uisuppersemi-continuous, (ii)u(x)=0outsideofinterval[0,1], (iii)therearerealnumbersa,b,canddsuchthata6b6c6d and (a)u(x)ismonotonicincreasingon[a,b], (b)u(x)ismonotonicdecreasingon[c,d], (c)u(x)=1,b6x6c, andthemembershipfunctionucanbeexpressas zynumbers(ÀÁwithdifferentrandalsoalltriangularfuz- Forexample,considerthetwofuzzynumbers,A=(3,2,2)and B=(3,1,1),seeFig.1,fromChuandTsao(2002). Mag(A)=Mag(B)=3,i.e.A$BandalsoconsiderA=(À1,1,5,5) andB=(0,2,2)thenMag(A)=Mag(B)=0soA$B,seeFig.2. However,itisclearthattheresultoforderingisnotreasonable 8 > >uRðxÞ; >uLðxÞ; a6x6b; b6x6c; c6x6d; otherwise; > <1; uðxÞ¼ > : 0; andfuzzynumbersAandBdonotbelongtoanequivalenceclass. Inthispaper,wemodifytheabovementionedmethodinorder torankallfuzzynumbersandovercometoaboveunreasonable results.Thestructureofthispaperisorganizedasfollows:InSec- tion2webringsomebasicdefinitionsandresultsonfuzzynum- bers.InSection3weproposenewmethodforrankingoffuzzy numbers.Comparingtheproposedrankingmethodwithsome otherapproaches,somenumericalexamplesareprovidedin Section4.Finally,conclusionsaredrowninSection5. ⇑Correspondingauthor.Tel.:+989123618518;fax:+982614405031. E-mailaddress:ezati@kiau.ac.ir(R.Ezzati). 0957-4174/$-seefrontmatterÓ2011ElsevierLtd.Allrightsreserved. doi:10.1016/j.eswa.2011.07.060 whereuL:[a,b]?[0,1]anduR:[c,d]?[0,1]areleftandrightmem- bershipfunctionoffuzzynumberu,respectively. Definition2.2(Ma,Friedman,andKandal,1999).Anarbitrary fuzzynumberintheparametricformisrepresentedbyanordered pairoffunctionsðuðrÞ;u ðrÞÞ;06r61,whichsatisfiesthefollow- ingrequirements: 1.u(r)isaboundedleft-continuousnon-decreasingfunctionover [0,1]. 2.u ðrÞisaboundedleft-continuousnon-increasingfunctionover [0,1]. 3.uðrÞ6u ðrÞ;06r61.