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A New Method for Ranking Fuzzy Numbers: Enhancements and Comparative Analysis

This paper proposes a modified method for ranking fuzzy numbers, specifically symmetric fuzzy numbers, building on previous approaches. The new technique enhances the existing methodology established by Abbasbandy and Hajjari (2009) for trapezoidal fuzzy numbers. By introducing comparative examples, we illustrate the advantages of our method over traditional ranking techniques. We also delve into the theoretical underpinnings of fuzzy numbers and provide numerical analyses to demonstrate the efficacy of the proposed approach for decision-making applications.

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A New Method for Ranking Fuzzy Numbers: Enhancements and Comparative Analysis

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  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.

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