Ranking Generalized Fuzzy Numbers using centroid of centroids

Transcription

1 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July ning Generlize Fuzzy Numers using entroi of entrois S.Suresh u Y.L.P. Thorni N.vi Shnr Dept. of pplie Mthemtis GIS GITM University Vishptnm Ini strt This pper esries rning metho for orering fuzzy numers se on re Moe Spres n Weights of generlize fuzzy numers. The re use in this metho is otine from the generlize trpezoil fuzzy numer first y splitting the generlize trpezoil fuzzy numers into three tringles n then lulting the Centrois of eh tringle folloe y the entroi of these Centrois n then fining the re of this entroi from the originl point. In this pper e lso pply moe n spres in those ses here the isrimintion is not possile. Some importnt results lie linerity of rning funtion n other properties re prove hih re useful for propose pproh. This metho is simple in evlution n n rn vrious types of fuzzy numers n lso risp numers hih re onsiere to e speil se of fuzzy numers. Keyors: ning funtion; Centroi Centroi points; Generlize trpezoil fuzzy numers. Introution ning fuzzy numers plys vitl role in eision ming. Most of the rel prolems tht exist in nturl orl re fuzzy thn proilisti or eterministi. In some ses the fuzzy numers must e rne efore n tion is ten y eision mer. Sine the ineption of fuzzy sets y Zeh [] in 65 mny uthors hve propose ifferent methos for rning fuzzy numers. Hoever ue to the omplexity of the prolem there is no metho hih gives stisftory result to ll situtions. Most of the methos propose so fr re non- isriminting ounterintuitive n some proue ifferent rnings for the sme sitution n some methos nnot rn risp numers. ning fuzzy numers s first propose y Jin [] for eision ming in fuzzy situtions y representing the ill-efine quntity s fuzzy set. Sine then vrious proeures to rn fuzzy quntities re propose y vrious reserhers. Yger [] first use horizontl oorinte of the entroi point in rning fuzzy numers. Murmi et l. [] hve use oth the horizontl n vertil oorintes of the entroi point s the rning inex. Cheng[5] propose istne inex hih is se on oth horizontl n vertil oorinte of the entroi point s he pointe out in ertin ses the horizontl oorintes plys n importnt role thn vertil oorinte of entroi point. This ours hen left n right spres of fuzzy numers re sme. Chu n Tso[6] propose n re metho to rn the fuzzy numers y lulting re eteen entroi point n the originl point. DOI :.5/ijfls.. 7

2 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July Chen &Chen [7] erive metho on rning Generlise Trpezoil Fuzzy Numers se on entroi point n stnr evition. Wng et l. [8] propose the formul for fining horizontl n vertil oorinte of entroi point. Ling et l. [] ompute to inies nmely istne inex n V inex to rn the fuzzy numer using entroi point. Shieh [] propose the orret formul for fining horizontl n vertil oorinte of entroi point. Chen &Chen [] propose the sore vlue of the fuzzy numers s the rning metho. Wng & Lee [] propose the rning inex on horizontl or vertil oorinte of the entroi point. In this pper e ompre not only the rning fuzzy numers using entroi ut lso rning fuzzy numers using re ompenstion y Fortemps n ouens [] Liou n Wng [] presente rning fuzzy numers ith integrl vlue Chen [5] presente rning fuzzy numers ith mximizing set n minimizing setn pproh for rning trpezoil fuzzy numers y sny n Hjjri [6] fuzzy ris nlysis se on rning generlize fuzzy numers ith ifferent heights n ifferent spres y Chen n Chen [7] n lso the rning propose y mit Kumr et l. [8] on rning generlize trpezoil fuzzy numers se on rn moe ivergene n spre. In this pper ne metho is propose hih is se on entroi of Centrois to rn fuzzy quntities. In trpezoil fuzzy numer first the trpezoi is split into three tringles n the entrois of these three tringles re lulte folloe y the lultion of the entroi of these entrois. Finlly rning proeure is efine hih is the re eteen the entroi of Centrois n the originl point n lso uses moe n spres in those ses here the isrimintion is not possile. In setion e riefly introue fuzzy efinitions n rithmeti opertions. Setion presents the propose ne metho. In Setion some importnt results lie linerity of rning funtion n other properties re prove hih re useful for propose pproh. In Setion 5 the propose metho hs een expline ith exmples hih esrie the vntges n the effiieny of the metho. In Setion 6 the metho emonstrtes its poer y ompring ith other methos tht exist in literture. Finlly the onlusions of the or re presente in Setion 7.. Fuzzy onepts In this setion some efinitions n rithmeti opertions on fuzzy set theory re reviee. Definition. Let U e universe set. fuzzy set of U is efine y memership funtion f : U [ ] here f ( x) is the egree of x in x U. Definition. fuzzy set of universe set U is norml if n only if sup f ( x) Definition. fuzzy set of universe set U is onvex if n only if f ( x ( ) y) min f ( x) f ( y) x y U n [ ]. Definition. fuzzy set x U of universe set U is fuzzy numer iff is norml n onvex on U. 8

3 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July Definition5. rel fuzzy numer is esrie s ny fuzzy suset of the rel line ith memership funtion f ( x) possessing the folloing properties: () f ( x) () f ( x) () f ( x) () f ( x) (5) f ( x) (6) f ( x) is ontinuous mpping from to the lose intervl [ ]. < for ll x ( ] is stritly inresing on [ ] for ll x [ ] is stritly eresing on [ ] for ll x ( ] here re rel numers Definition.6. The memership funtion of the rel fuzzy numer is given y L f x x f ( x) () here < f x otherise is onstnt re rel numers n f L : [ ] [ ] f : re to [ ] [ ] stritly monotoni n ontinuous funtions from to the lose intervl[ ]. It is ustomry to rite fuzzy numer s ( ). If then ( ) fuzzy numer otherise is si to e generlize or non-norml fuzzy numer is normlize If the memership funtion f ( x) is pieeise liner then is si to e trpezoil fuzzy numer. The memership funtion of trpezoil ( x ) x < x f ( x) ( x ) x < otherise fuzzy numer is given y: ()

4 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July If then ( ) is normlize trpezoil fuzzy numer n is generlize or non norml trpezoil fuzzy numer if < <. The imge of ( ). is given y ( ; ).. s prtiulr se if the trpezoil fuzzy numer reues to tringulr fuzzy numer given y ( ). The vlue of orrespons ith the moe or ore n [ ] ith the support. If then ( ). is normlize tringulr fuzzy numer is generlize or non norml tringulr fuzzy numer if < <. Definition 7 If ( ) n ( ) re to generlize trpezoil fuzzy numers then (i) ( min( )) Θ ;min (ii) ( ( )) (iii) ; ); > ( (iv) ( ;); <. Propose rning Metho The Centroi of trpezoi is onsiere s the lning point of the trpezoi (Fig.). Divie the trpezoi into three tringles. These three tringles re PC QCDn PQC respetively. Let the Centrois of the three tringles e G G & Grespetively. The entroi of Centrois G G & Gis ten s the point of referene to efine the rning of generlize trpezoil fuzzy numers. The reson for seleting this point s point of referene is tht eh Centroi point is lning point of eh iniviul tringle n the entroi of these Centroi points is muh more lning point for generlize trpezoil fuzzy numer. Therefore this point oul e etter referene point thn the Centroi point of the trpezoi.

5 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July P ( ) Q ( ) G G G G O ( ) ( ) C( )D ( ) Fig. Generlize Trpezoil fuzzy numer Consier generlize trpezoil fuzzy numer ( ). (Fig.). The Centrois of the three tringles reg G ng respetively. Eqution of the line G G is y n G oes not lie on the line G G. Therefore G G n G re non-olliner n they form tringle. We efine the entroi G ( x y ) of the tringle ith verties G G n G trpezoil fuzzy numer ( ). s G ( x y ) 5 s speil se for tringulr fuzzy numer ( ). i.e. Centrois is given y G ( x y 7 ) of the generlize () the entroi of The rning funtion of the generlize trpezoil fuzzy numer ( ). hih mps the set of ll fuzzy numers to set of rel numers is efine s: 5 ( ) x y (5) This is the re eteen the entroi of the Centrois x y ) s efine in Eq.() n the originl point. G ( The Moe (m) of the generlize trpezoil fuzzy numer ( ). is efine s: () m ( ) x ( ) (6)

6 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July The Spre(s) of the generlize trpezoil fuzzy numer ( ). is efine s: s ( ) x ( ) The Left spre (ls)of the generlize trpezoil fuzzy numer ( ). is efine s: ls ( ) x ( ) The ight spre (rs) of the generlize trpezoil fuzzy numer ( ). is efine s: ( ) x ( ) rs () Using the ove efinitions e no efine the rning proeure of to generlize trpezoil fuzzy numers. Let ; ) n ; ) e to generlize trpezoil fuzzy ( ( numers. The oring proeure to ompre n is s follos: Step : Fin n Cse (i) If > then > Cse (ii) If < then < Cse (iii) If omprison is not possile then go to step. Step : Fin m n m Cse (i) If > m m then > Cse (ii) If < m m then < Cse (iii) If m m omprison is not possile then go to step. Step : Fin s n s Cse (i) If > s s then < (7) (8)

7 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July Cse (ii) If < s s then > Cse (iii) If s s omprison is not possile then go to step. Step : Fin ls n ls Cse (i) If > ls ls then > Cse (ii) If < ls ls then < Cse (iii) If ls ls omprison is not possile then go to step 5. Step 5: Exmine n Cse (i) If > then > Cse (ii) If < then < Cse (iii) If then In this setion some importnt results hih re the sis for efining the rning proeure in setion re isusse n prove. Proposition. The rning funtion efine in setion y mens of Eq. () is liner funtion for normlize 5 trpezoil fuzzy numer ( ). i.e. ( ). If ( ) n ( ) re to normlize trpezoil fuzzy numers then (i) ; (ii) (iii) ( ) ( ) Proof (i): se (i) Let >

8 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) 5 5 ( ) ( ) Similrly the result n e prove for se (ii) > < n se (iii). < > Proof (ii): Let ( ) ( ) ( ) 5 5 ( ) ( ) Proof (iii): ( ) ( ) ( ) ( ) ( ) (y (i)) ( ) ( ) Θ (y (ii)). Proposition. Let ) ; ( n ) ; ( re to generlize trpezoil fuzzy numers suh tht ; m m ; s s then (i) ls ls > > (ii) ls ls < > (iii) ls ls Proof: From the ssumptions ( ) ( ) 5 5 ()

9 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July m m ( ) ( ) s s ( ) ( ) Solving () () n () e get No to prove (i): ls > ls ( ) > ( ) > ( ) No to prove (ii): ls < ls ( ) < ( ) < ( ) No to prove (iii): ls ls ( ) ( ) ( ) Corollry : ll the results of proposition. lso hol for right spre. () () Proposition. Let ( ) n ( ) re to generlize trpezoil fuzzy numers suh tht ; m m ; s s then (i) (ii) ls > ls rs > rs ls < ls rs < rs (iii) ls ls rs rs Proof: From proposition. for the ove ssumptions e hve 5

10 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July ( ) ( ) No to prove (i): ls > ls > ( ) (from proposition.) > ( ) (from proposition.) > > ( ) ( ) ( ) rs > rs Similrly (ii) n (iii) n e prove. 5. Numeril Exmples In this setion the propose metho is first expline y rning some fuzzy numers. Exmple 5. 5 Let ( 57; ) n 5 ; 8 Then ( x y ) ( 5. ) Therefore ( ).. Sine ( ) < ( ) < G ( x y ) ( 5.6.) G n ( ) 5 Exmple 5. 7 Let ( ; ) n ; 5 Then ( x y ) (.) G x Therefore ( ).. Sine ( ) > ( ) > G n ( y ) (..) n ( ) Exmple Let ( ; ) ( ; ) n ; ; 5 5 x y. x y.. G ( ) ( ) n ( ) ( ) G 6

11 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July Therefore ( ). n ( ). Sine ( ) < ( ) < From exmples 5. n 5. e see tht the propose metho n rn fuzzy numers n their imges s it is prove tht > <. Exmple 5. Let (...5; ) n (...; ) Step : Then G ( x y ) (..) n ( x y ) (..). Sine ( ) ( ) So go to step. G. Therefore ( ) n ( ) Step : m ( ). n m ( ). Sine m ( ) m ( ) Step: s ( ). n s ( ). Sine s ( ) > s ( ) Exmple 5.5 So go to step. < Let (...5;.8 ) n (...5; ) Then G ( x y ) (..555) n ( x y ) (..) G Therefore ( ). 66 n ( ). Sine ( ) < ( ) < From exmple 5.5 it is ler tht the propose metho n rn fuzzy numers ith ifferent height n sme spres. Exmple 5.6 Let (...5; ) n (...5; ) Then G ( x y ) (..) n ( x y ) (..) G 7

12 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July. > >. Therefore ( ) 8 n ( ) Sine ( ) ( ) 6. esults n isussion In this setion the vntges of the propose metho is shon y ompring ith other existing methos in literture here the methos nnot isriminte fuzzy numers. The results re shon in Tle I n Tle II. Exmple 6. Consier to fuzzy numers ( 5 ) n ( 6) y Liou n Wng Metho []it is ler tht the to fuzzy numers re equl for ll the eision mers s ( ).5 ( ). 5 ( ).5 ( ). 5 I T n I T Whih is not even true y intuition. y using our metho e hve ( x y ) (.77.) G x.. > > G n ( y ) (..) Therefore ( ) 6788 n ( ) Sine ( ) ( ) Exmple 6. Let (...5; ) ( ; ) > Cheng [5] rne fuzzy numers ith the istne metho using the Eulien istne eteen the Centroi point n originl point. Where s Chu n Tso [6] propose rning funtion hih is the re eteen the entroi point n originl point.their entroi formule re given y ( ) ( ) ( ) ( ) ( )( ) x y ( ) 6( ) ( ) ( ) ( ) y ( ) ( ) ( ) x ( ) ( ) 6 oth these entroi formule nnot rn risp numers hih re speil se of fuzzy numers s it n e seen from the ove formule tht the enomintor in the first oorinte of their entroi formule is zero n hene entroi of risp numers re unefine for their formule. y using our metho e hve 8

13 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July G ( x y ) (..) n ( x y ) (.). < < G. Therefore ( ) n ( ) Sine ( ) ( ) From this exmple it is prove tht the propose metho n rn risp numers heres other methos file to o so. Exmple 6. Consier four fuzzy numers (...; ) (..5.8; ) (...; ) (.6.7.8; ) Whih ere rne erlier y Yger[] Fortemps n ouens[] Liou n Wng[] n Chen n Lu [5] s shon in Tle I.

14 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July Tle I. Comprison of vrious rning methos Metho \Fuzzy ning orer numer Yger [] > > Fortemps & > > ouens[] Liou & Wng[] > > Chen [5] Propose metho > > > > > > > > > > It n e seen from Tle I tht none of the methos isrimintes fuzzy numers. Yger[] n Fortemps n ouens [] methos file to isriminte the fuzzy n Wheres the methos of Liou n Wng[] n Chen n Lu [5] nnot isriminte the fuzzy numers n y using our metho e hve G ( x y ) (..) ( x y ) (.5.) G G G ( x y ) (..) ( x y ) (.7.) Therefore. ( ) 888 ( ). ( ). 7 ( ). Exmple 6. numers > > > In this e onsier seven sets of fuzzy numers ville in literture n the omprtive stuy is presente in Tle II. Set : (...6.8;.5) n (...;.7 ) Set : (...5; ) n (...5; ) Set : (...5; ) n ( ; )

15 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July Set : (.5...; ) n (...5; ) Set 5: (..5.5; ) n ( ; ) Set 6: (..6.8; ) (..5.5.; ) n C ( ; ) Set 7: (...5; ) ; n ( ) Tle II Comprison of the rning results for ifferent pprohes 7. Conlusions n future or This pper proposes metho tht rns fuzzy numers hih is simple n onrete. This metho rns trpezoil s ell s tringulr fuzzy numers n their imges. This metho lso rns risp numers hih re speil se of fuzzy numers heres some methos propose in literture nnot rn risp numers. This metho hih is simple n esier in lultion not only gives stisftory results to ell efine prolems ut lso gives orret rning orer to prolems. Comprtive exmples re use to illustrte the vntges of the propose metho. pplition of this rning proeure in vrious eision ming prolems suh s fuzzy ris nlysis n in fuzzy optimiztion lie netor nlysis is left s future or. eferenes [] Zeh L.. (65) Fuzzy sets Informtion n ontrol 8 (): 8-5. [] Jin. (76) Deision ming in the presene of fuzzy vriles IEEE Trnstions on Systems Mn n Cyernetis 6: [] Yger.. (8) On generl lss of fuzzy onnetives Fuzzy sets n systems (6) 5-. [] MurmiS. Me S. &Immur S. (8) Fuzzy eisi on nlysis on the evelopment of entrlize regionl energy ontrol: system. IFC symposium on Fuzzy informtion nolege representtion n eision nlysis.

16 Interntionl Journl of Fuzzy Logi Systems (IJFLS) Vol. No. July [5] Cheng C.H. (8) ne pproh for rning fuzzy numers y istne metho. Fuzzy sets n systems [6] Chu T.C. & Tso C.T. () ning fuzzy numers ith n re eteen the entroi point n originl point Computers n Mthemtis ith pplitions -7. [7] Chen S.J. & Chen S.M. () ne met ho for hnling multiriteri fuzzy eision ming prolems using FN- IOW opertors Cyerntis & systems -7. [8] Wng Y.H. Yng J.. XU D.L. &Chin K.S. (6) On the entrois of fuzzy numers Fuzzy sets & system [] Ling C.Wu J. & Zhng J.(6) ning inies n rules for fuzzy numers se on grvity enter point 6th Worl ongress on Intelligent ontrol n utomtion Dlin Chin. [] Shieh.S. (7) n pproh to entrois of fuzzy numers Interntionl Journl of Fuzzy systems () 5-5. [] Chen S.J. & Chen.S.M (7) Fuzzy ris nlysis se on the rning of generlise trpezoil fuzzy numer pplie intelligene 6 () -. [] Wng Y.J. & Lee H.S. (8) The revise metho of rning fuzzy numers ith n re eteen the entroi n originl points Computers n Mthemtis ith pplitions [] Fortemps P. n ouens M. (6) ning n efuzzifition methos se on re ompenstion Fuzzy Sets n Systems 8: -. [] Liou T. S. n Wng M. J. () ning fuzzy numers ith integrl vlue Fuzzy Sets n Systems 5: [5] Chen S. H. (85) ning fuzzy numers ith mximizing set n minimizing set Fuzzy Sets n Systems 7(): -. [6] sny S. n Hjjri T. () ne pproh for rning of trpezoil fuzzy numers Computers n Mthemtis ith pplitions 57(): -. [7] Chen S.M. n Chen J.H. () Fuzzy ris nlysi s se on rning generlize fuzzy numers ith ifferent heights n ifferent spres. Expert Systems ith pplitions 6 (): [8] Kumr. Singh P. Kur. n Kur P. () ning of generlize trpezoil fuzzy numers se on rn moe ivergene n spre Turish Journl of Fuzzy Systems (): -5.