COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERS

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1 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 COMPRISON OF DIFFERENT PPROXIMTIONS OF FUZZY NUMBERS D. Stephen Dinr n K.Jivn PG n Reserh Deprtment of Mthemtis T.B.M.L. Collee Poryr Ini. BSTRCT The notions of intervl pproximtions of fuzzy numers n trpezoil pproximtions of fuzzy numers hve een isusse. Comprisons hve een me etween the lose-intervl pproximtion vluemiuity intervl pproximtion n istint pproximtion with the orresponin risp n trpezoil fuzzy numers. numeril exmple is inlue to justify the ove mentione notions. KEYWORDS Close intervl pproximtions vlue miuity intervl pproximtions Distint pproximtions.. INTRODUCTION The me theory is wiely pplie to militry ffirs physil trinin n ommeril proution n so on. But the lssil me theory is se on the oule vlue loi theory lose siht of muh fuzzy informtion n rey informtion. me is esription of eision-mkin sitution involvin more thn one eision mker. Gme theory is enerlly onsiere to hve eun with the pulition of von Neumnn n Morenstern s The Theory of Gmes n Eonomi Behvior [] in 9. The evelopment of me theory elerte in ppers y Nsh [5] Shpley [7] [8] n Gillies []. The ehvior of plyers in me is ssume to e rtionl n influene y other rtionl plyers ehviors whih istinuishes me from the enerl eision-mkin prolem. In orer to nlyze the ehviors of plyers n onstrut metho for eh plyer to hoose his tion strtei me first efines eh iniviul s lterntive tions. The omintion of ll the plyers strteies will etermine unique outome to the me n the pyoffs to ll plyers. solution for plyer in me shoul llow tht plyer to win or stisfy his ojetives for the me. For exmple he n mximize his own pyoff n/or minimize his opponent s pyoff. Formlly the solution of me is sitution in whih eh plyer plys est response to the other plyers tul strtey hoies. This is the onept of equilirium. There re severl methos for otinin the equilirium of some speil kins of mes suh s the ominnt strtey for ominnt strtey equilirium or mixe strtey for mixe strtey equilirium. liner prormmin metho is use in mtrix mes. Most of these methos re se on the mximin priniple for seletin optiml strteies. But in 978 Butnriu [] pointe out tht one of the mjor ssumptions of lssil me theory is tht ll possile hoosin strteies re eqully possile hoies for plyer. DOI : 0.5/ijfls.05.50

2 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 The notion of fuzzy sets first ppere in the ppers written y Zeh [5]. This notion tries to show tht n ojet more or less orrespons to prtiulr teory. The eree to whih n element elons to teory is n element of the ontinuous intervl [0] rther thn the Boolen pir {0} [5]. Usin the notion of fuzzy sets the pyoff funtion in me n e fuzzifie. Furthermore the solution to the me my lso e fuzzy set. The pper is ornize s follows: The setion ives introution is introutory in nture. Setion ives the si efinitions n onepts neee for this work. In setion the omprisons of ifferent pproximtions in fuzzy me notion hve een presente. In setion the omprisons of ifferent pproximtions in two person zero-sum metho in fuzzy environment hve een isusse. In setion 5 relevnt numeril exmple to ompre the ifferent pproximtions of fuzzy numers to justify the ove isusse notions re inlue. The onluin remrks re lso e in the lst setion.. PRELIMINRIES.. Definition (Fuzzy Set [FS]) Let X enote universl set i.e. X{x}; then the hrteristi funtion whih ssins ertin vlues or memership re to the elements of this universl set within speifie rne {0} is known s the memership funtion n the set thus efine is lle fuzzy set. The memership rers orrespon to the eree to whih n element is omptile with the onept represente y the fuzzy set. If µ is the memership funtion efinin fuzzy set Ã then µ : X [0] where [0 ] evelope the intervl of rel numers from 0 to... Definition (Fuzzy Numer [FN]) onvex n normlize fuzzy set efine on R. whose memership funtion is pieewise ontinuous is lle fuzzy numer. fuzzy set is lle norml when t lest one of its elements ttins the mximum possile memership re. i.e. mx µ ( x) x R.. Definition (α-cut) x n α -ut of fuzzy set Ã is risp set α tht ontins ll the elements of the universl set X tht hve memership re in reter thn or equl to the speifie vlue of. Thus {x X ;µ ( x) α0 x } α.. Definition (Close Intervl pproximtion of Trpezoil Fuzzy Numers [CITFN]) [] Let ( ) e trpezoil fuzzy numer n its intervl pproximtion of fuzzy numer [] [ L U ] is si to e lose intervl pproximtion if L U α α inf{ / ( ) 0.5} x µ x { / ( ) 0.5} Sup x µ x

3 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer Definition (Vlue of Fuzzy Numer [VFN]) [9] Let Ã e fuzzy numer with α-ut representtion ( ) then the vlue of Ã is efine s Ã.. Definition (miuity of Fuzzy Numer [FN]) [9] Let Ã e fuzzy numer with α-ut representtion ( ) then the miuity of Ã is efine s Ã.7. Definition (Vlue-miuity Intervl pproximtion of Fuzzy Numer [VI])[0] n intervl pproximtion opertor is efine s : F(R) in wy tht : Ã Ã [ ] where Then the opertor is lle vlue-miuity intervl pproximtion opertor lso the intervl ] is lle the vlue miuity intervl pproximtion of Ã..8. Definition (Distint pproximtion of Trpezoil Fuzzy Numer [DTrFN]) [] Let ( ) e istint pproximtion of ( ). where Then the memership funtion of DTrFN is µ ' x( ) ( ) ( x) ( ) x ( ) 0 if if if x < x < x otherwise

4 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer Definition (ssoite Rel Vlue of pproximtion of Intervl Fuzzy Numer) [ % ] [ t t ] is n lose intervl pproximtion of trpezoil fuzzy numer ( ) then its ssoite rel vlue is iven y If ˆ [ t t].0. Definition (ssoite Rel Vlue of Trpezoil Fuzzy Numer) If ( ) is trpezoil fuzzy numer then its ssoite rel vlue is iven y ˆ (.. Definition (Rnkin of Fuzzy Numers [RFN]) ) Let n B e ny two fuzzy numers. Then is reter thn B if n only if enote y f B. ˆ f Bˆ n.. Definition (Mximum of Fuzzy Numers) Let n B e ny two fuzzy numers. Then the mximum vlue of n B enote y Mx B Mx % B % % if B f. ( ) n is efine y ( ).. Definition (Minimum of Fuzzy Numers) Let n B e ny two fuzzy numers. Then the minimum vlue of n B enote y Min B Min % B% % if % p B%. ( ) n is efine y ( ).. Definition (rithmeti Opertions of Intervl Fuzzy Numers) Let [ ] [ ] ition: B R e ny two intervl fuzzy numers. Then [ ] ( )[ ] [ ] Sutrtion: [ ] ( )[ ] [ ] Multiplition: [ ] ( )[ ] [ ]

5 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer rithmeti Opertions of Trpezoil Fuzzy Numers Then ition: Let ( ) n B ( ) e ny two trpezoil fuzzy numers. ( ) B ) Sutrtion: ( ( ) B ( ) Multiplition: ( ) B ( )( )( ) ) (. FUZZY PPROCH TO STRTEGIC GMES strtey is omplete pln for plyer to eie how to ply the me []. Without etile knowlee of the sequene of moves in strtei me we n still nlyze the ehviors of plyers; ll tht is neee for solution to me is to inite wht the plyer woul o in the sitution when he must mke move. So it is suffiient if we list eh iniviul s strteies the omintion of whih will etermine unique outome (pyoff to eh plyer) of me. me is in strtei form if only the set of plyers I the set of strteies S n the pyoff funtions P i (s) for eh plyer I S S S... S I. The onepts of mes in strtei form n mixe strteies use in this pper s well s the nottions re s efine n esrie in []. i re iven where ( ).. Methooloy of Comprison of Different pproximtions y Usin Two- Person Zero Sum in Existin Sle Point Metho in Fuzzy Environment The fuzzy me with two plyers were the in of one plyer is equl to the loss of the other plyer is lle two person zero sum me. The resultin in or loss for the plyer in me etween plyers n B n e represente in the form of mtrix whih is lle the pyoff mtrix. In eh entry re fuzzy numers. To otin the solution of the me we n use the Minimx- Mximin priniple. Usin whih we fin the fuzzy sle point. Usin this fuzzy sle point the vlue of the me is otine n this sle point hs orresponin strtey for plyer n B. If sle point exists then the strtey follows y pure-strtey. (If fuzzy sle point oes not exist then we hve mixe strtey). The vlue of the me whih orresponin to the sle point represents the Mximum urntee in for plyer. (i.e.) Gurntee (i.e.) Mximum possile loss for plyer B i 5

6 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 The vlue of the fuzzy me is use to represent V moreover if sle point exists then the me is si to e stritly etermine. In enerl for ny iven fuzzy me the mximin vlue is enote y V n minimx vlue is enote y V for ny fuzzy me the followin inequlities is V V V if n only if However if fuzzy sle point exists. V V V If the vlue of the me V 0 tht suh fuzzy me is si to e fir me. Steps to fin the fuzzy sle point Step: Fin the minimum vrile of eh row of the pyoff mtrix n mrk them. Step: Fin the Mximum vrile of eh olumn of the pyoff mtrix n mrk them. Step: If in this proess then there exists point in the pyoff mtrix with the oth ove mrks. Then this point orrespons to the sle point. Usin this vlue of sle point is foun. Usin whih orresponin pure strteies for plyers n B re foun... Close-Intervl pproximtion of Trpezoil Fuzzy Numers (CITrFN) Let ( ) B ( ) C ( ) n D ( ) e trpezoil fuzzy numers. Then the lose-intervl pproximtions of them iven y C D () C D (B) C D (C) n C D (D) Now solve the fuzzy me whose pyoff mtrix is iven y Plyer Plyer B Min ( or) or or or Mx ( ) ( ) If Minimx Mximin ( ) Therefore the sle point is exists. Then we n fin the vlue of the fuzzy me ( V % ). n then we n fin the Strtey of plyer. i.e. S n Strtey of plyer B. i.e. S B

7 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05.. Vlue-miuity Intervl pproximtion of Trpezoil Fuzzy Numers (VI) Let ( ) ; B ( ) ; C ( ) n D ( ) re the trpezoil fuzzy numers. Then the vlue of the fuzzy numer is i.e. ( ) Vl( ) ; n similrly for ( ) ( ) Vl( B) ; Vl( C) ; ( ) Vl( D) n the miuity of fuzzy numer is i.e. ( ) m( ) ; n similrly for m( B) m( D) ( ) ( ) ; m( C) ( ) ; Therefore the Vlue-miuity intervl pproximtions of trpezoil fuzzy numer is enote y (). V [ Vl( ) m( ) Vl( ) m( ) ] V ( ) Similrly [ Vl( B/ ) m( B/ ) Vl( B/ ) m( B/ )] V ( B/ ) [ V ( C) Vl( C) m( C) Vl( C) m( C) ] [ V ( D) Vl( D) m( D) Vl( D) m( D) ]. Now solve the ove fuzzy me whose pyoff mtrix is iven y 7

8 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 8 Plyer B Min Plyer ( ) ( ) or or Mx ( ) or ( ) or If Minimx Mximin the sle point is exists. Then we n fin the vlue of the fuzzy me ( ) V %. n then we n fin Strtey of plyer. i.e. S n Strtey of plyer B. i.e. B S... Distint pproximtions of Trpezoil Fuzzy Numers (DTFN) Let ( ) e istint pproximtion of trpezoil fuzzy numer ( ) where. Distint pproximtions of ( ) D Similrly for ) ( B then its istint pproximtions ( ) B D ) ( C then its istint pproximtions ( ) D C n ) ( D then its istint pproximtions ( ) D D Now solve the ove fuzzy me whose pyoff mtrix is iven y Plyer B Min Plyer ( ) ( ) or or Mx ( ) or ( ) or If Minimx Mximin Therefore the sle point is exists. Then we n fin the vlue of the fuzzy me ( ) V %. n then we n fin Strtey of plyer i.e. S n Strtey of plyer B i.e. B S

9 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05. MIXED STRTEGY OF TWO PERSON ZERO-SUM METHOD (WITHOUT SDDLE POINT) TO COMPRE DIFFERENT PPROXIMTIONS METHODS IN FUZZY ENVIRONMENT Consier fuzzy me whose pyoff mtrix is of orer x n whih oes not hve sle point. Consier the pyoff mtrix iven y Plyer Mx Plyer B Min q q P P If there is no sle point for the ove pyoff mtrix then We know tht mixe strtey is followe. Moreover the vlue of the me is V P ( ) ( ) ( ) ( ) We know tht P P P P lso q n q q q q ( ) ( ).. Exmple: Gme with n without Sle Point Metho in Crisp Consier the me Plyer Plyer B Now solve the ove fuzzy me whose pyoff mtrix is iven y 9

10 Plyer Mx Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 Plyer B Min If Minmx Mximin Therefore the sle point is exist. Therefore the vlue of the fuzzy me ( V ) 5. The Strtey for plyer. i.e. S 5 Strtey for plyer B. i.e. S 7. B.. Fuzzy Gme with n without Sle Point Metho Let (57 ) B (57) C (9) D (78 ) re the trpezoil fuzzy numers. Now solve the ove fuzzy me whose pyoff mtrix is iven y Plyer Plyer B Min (57) ( 57) 57 ( 9) ( 78 ) 9 78 ( ) ( ) Mx ( ) ( 9) If Minmx Mximin Therefore the sle point is exist. Therefore the vlue of the fuzzy me ( V ) 5.. The Strtey for plyer. i.e. S. 7 Strtey for plyer B. i.e. 7. S B.. Close Intervl pproximtion of Fuzzy Gme without Sle Point Metho Let (57 ) e trpezoil fuzzy numer n its lose intervl pproximtion is [ ] [.5] (y ef.). Similrly for B (57 ) then its lose intervl pproximtion is [ B ] [.5.5] C % ( 9) then the lose intervl pproximtion is [ C %] [.5 ] n D (78 ) then the lose intervl pproximtion of [ D ] [ ]. Now solve the ove fuzzy me whose pyoff mtrix is iven y Plyer Mx [.5] Plyer B Min [.5.5] [.5] [.5] [ ] [ ] [.5] [ ] 0

11 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 If Minmx Mximin Therefore the sle point is exist. Therefore the vlue of the fuzzy me V. 7. The Strtey for plyer. i.e. S [.5]. 75 Strtey for plyer B. i.e. S [ ] 7. 5 B.. Vlue-miuity Intervl pproximtion of Fuzzy Gme without Sle Point Metho Let (57 ) B (57) C (7) D (78 ) numers n its vlue miuity intervl pproximtion of is re the trpezoil fuzzy ( ) Vlue( ) ( ) Vlue( ) Vlue ( B) 5. Vlue ( C). 8 n Vlue ( D ) lso ( ) miuity( ) miuity ( ) miuity ( B). miuity ( C). 5 n miuity ( D ). 5. Then the vlue miuity intervl pproximtion of is V ( ) [ Vl( ) m( ) Vl( ) m( )] whih implies V ( ) [..] Similrly V ( B) [.] V ( C) [..] V ( D) [9]. Now solve the ove fuzzy me whose pyoff mtrix is iven y Plyer B Min [..] [.] [.] [..] [ 9] [..] Plyer [..] [ 9] Mx If Minmx Mximin

12 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 Therefore the sle point oes not exist. Hene we hve Plyer Mx Plyer B Min [.. ] [. ] P [..] [ 9] P q q We know tht Mixe strtey is followe. Moreover the vlue of the me is V ( ) ( ) [..][9] [.][..] V [[..] [9]] [[.] [..]] Therefore V % 5.9 We know tht P ( ) ( ) P P P P lso q 0. 0 ( ) ( ) q q q q n 0. 8 [ 9] [..] P 0. 9 (.7) [ 9] [.] 0. 8 [.7] Therefore the vlue of the fuzzy me ( V ) The Strtey for plyer. i.e. S ( ) 0. 5 Strtey for plyer B. i.e. ( ) 0. 5 S B.5. Distint pproximtion of Fuzzy Gme without Sle Point Metho Let (57 ) B (57) C (7) D (78 ) re the trpezoil fuzzy numers n its Distint pproximtion of Trpezoil fuzzy numer is D ( ). Usin this D ( ) (.5.58) D ( B ) ( ) D ( C ) (.57.5 ) D ( D ) ( ). Now solve the ove fuzzy me whose pyoff mtrix is iven y

13 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 Plyer B Min (.5.58 ) ( ) (.57.5 ) ( ) (.57.5 ) Plyer (.5.58 ) ( ) Mx If Minmx Mximin Therefore the sle point oes not exist. Hene we hve Plyer B Min (.5.58) ( ) P (.57.5 ) ( ) P Plyer q q Mx We know tht ( ) Mixe strtey is followe. Moreover the vlue of the me is V ( ) ( ) [(.5.58)( )] [(.5.7.5)(.7.5)] V. [( )] [(.55)] We know tht P ( ) ( ) P P P P 0. ( 0.78) lso q ( ) ( ) 0. n q q q q (0.79) Therefore the vlue of the fuzzy me ( V ) (.). S ( ) 0. S The Strtey for plyer. i.e. [ ] 5 Strtey for plyer B. i.e. [( )] 5 B

14 Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer COMPRISON OF DIFFERENT PPROXIMTIONS OF FUZZY NUMBERS IN TWO-PERSON ZERO SUM GME METHOD Here we hve ompre the ifferent pproximtions of fuzzy numers with its orresponin risp numer. Exmple. The risp numer vlue of the me ( V ) 5.. The vlue of the me with the i of trpezoil fuzzy numers ( V ) 5.. The vlue of the me with the i of lose intervl pproximtions of fuzzy numers ( V ).7. The vlue of the me with the i of Vlue-miuity intervl pproximtions of fuzzy numers ( V ) The vlue of the me with the i of Distint pproximtion of fuzzy numers ( V ). From the tle it is note tht the vlue-miuity intervl pproximtion is very lose to the risp numer vlue of the me. Then we n onlue tht the vlue-miuity intervl pproximtion is muh etter thn ny other methos..conclusion: In this pper omprisons etween ifferent pproximtions of fuzzy numers hve een me y the i of fuzzy me theory. The ifferent pproximtions suh s lose intervl vluemiuity n istint pproximtions hve een employe in fuzzy me re foun n it is oserve n onlue tht the vlue-miuity pproximtion of fuzzy numers is the est pproximtion thn other methos.

15 REFERENCES Interntionl Journl of Fuzzy Loi Systems (IJFLS) Vol.5 No. Otoer 05 [] Billot.. (99) Eonomi Theory of Fuzzy Equiliri: n xiomti nlysis. ew York: Spriner- Verl. [] Butnriu.D.( 978) Fuzzy Gmes: Desription of the Conept Fuzzy Sets Syst. Vol. Pp [] Eiherer.J. (99) Gme Theory for Eonomists. New York: emi. [] Gillies.D.(95) Lotions of Solutions Informl Conf. Theory of N-Person Gmes Prineton Mthemtis Prineton Nj Pp.. [5] Nsh.J. (950) Equilirium Points in N-Person Gmes Pro. Nt. emy Si. New York Ny Vol. Pp [] Nsh.J. (950) The Brinin Prolem Eonometri Vol. 8 Pp. 55 [7] Shpley.L. (95) Vlue for N-Person Gmes Kuhn n Tuker Es. in Contriutions to the Theory of Gmes. Prineton Nj: Prineton Univ.Press Pp [8] Shpley.L. (95) Open Question Informl Conf. Theory of N-Person Gmes Prineton Mthemtis Prineton Nj Pp.. [9] Stephen Dinr.D n Jivn.K. (0). Note on Intervl pproximtion of Fuzzy Numers Proeeins of the Interntionl Conferene on Mthemtil Methos n Computtion ICOMC. [0] Stephen Dinr.D n Jivn.K. (05) Solvin Trnsporttion Prolems usin Vlue- miuity intervl pproximtions of fuzzy numers. Interntionl Journl of Reent Development in Enineerin n Tehnoloy. (ISSN-7-5). Volume Issue 5. [] Stephen Dinr.D n Jivn.K. (05) Dulity in Close Intervl pproximtion of Fuzzy Numer Liner Prormmin. "Jml emi Reserh Journl (JRJ)" (ISSN No.:097-00). Proeeins of the Interntionl Conferene on Mthemtil Methos n Computtion ICOMC pp.7-8. [] Stephen Dinr.D n Jivn.K. (05) On Complement of Distint pproximtions of Fuzzy Numer. Interntionl Journl of Sientifi Reserh n Enineerin Trens. Volume Issue Mrh-05 ISSN (Online):95-5X. [] Von Neumnn.J n Morenstern.D. (9). The Theory of Gmes in Eonomi Bhvior. New York: Wiley. [] Zeh.L.. (95) Fuzzy Sets Inform. Contr. Vol. 8 Pp [5] Zeh.L.. (98) Proility Mesures n Fuzzy Events J. Mthemtis nl.pplit. Vol. No. Pp. 7. 5

Ranking Generalized Fuzzy Numbers using centroid of centroids

Ranking Generalized Fuzzy Numbers using centroid of centroids 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