On Solution of Min-Max Composition Fuzzy Relational Equation

Transcription

1 U-Sl Scece Jourl Vol.4()7 O Soluto of M-Mx Coposto Fuzzy eltol Equto N.M. N* Dte of cceptce /5/7 Abstrct I ths pper, M-Mx coposto fuzzy relto equto re studed. hs study s geerlzto of the works of Ohsto d Sekgush. he codtos for the exstece of solutos re studed, the the resoluto of equtos s dscussed. Itroducto he cocept of fuzzy reltol equtos troduced by Schez [5], s geerlzto of well kow oole equtos. Let A d be two fuzzy sets of two fte spces, respectvely d fuzzy relto of the set. Cosder the followg fuzzy relto equto () Where s the M-Mx coposto. Spekg wth terology of systes theory, A d represet clss of fuzzy puts d clss of fuzzy equto (). I ths pper, we llustrte other lgorths, to solve equto (). Prelres Let I, be the rel ute tervl d we set for every rel ubers, b I,, b, b x b, b,, [] of course, we hve - b b, b b (De Morg s lws) - b c c b c, bc cb c [dstrbutvty lws] Let x, x,..., x, y, y,..., y be fte sets, F : : I the set of ll fuzzy sets of. I r,,,...,r the set of frst r turl ubers. Followg Zedeh s [,], we reeber tht F s coplete dstrbutve lttce wth the potwse opertos defed x X, I s x for every - x - x x x, x x x - x x x 4- x x x d turl orderg s ff x where x, FX Let x X y Y, I, d I, we recll the followg deftos: *College of Scece for Woe, ghdd Uversty 49

2 U-Sl Scece Jourl Defto.[] A fuzzy relto betwee two fte sets d s ppg fro the Crtes product of crsp set X, Y to the ute tervl [, ] ( eleet of F ) Defto.[6] Fuzzy relto equto s gve the for of whch the coposte of fuzzy put A d fuzzy relto equls fuzzy output. he put A d the output re fuzzy sets represeted by, b respectvely. he fuzzy relto represets the cuslty of put d output. Defto.[] Let A be the set of ll possble vectors I such tht, for ll I d let prtl orderg o A be def s follows: For y pr, A, f d oly f for ll I Defto 4.[,4] A eleet of S, s clled xl soluto of Eq.(), f for ll S,, ples It s well estblshed tht wheever the soluto set S,, t s lwys cots uque xl soluto,. Defto 5.[4,5] Vol.4()7 A eleet S, s clled l soluto of Eq.() f for ll S,, of ples S, t y cot severl l solutos. d whe Exstece of solutos We ow estblsh soe theores cocerg the exstece of solutos of the equto where deotes -x coposto of two bry opertors G d, ps fro L L where L [,] [,], L the tervl [,], the or copostos wll be prtculr cses of the G coposto troduced here. More precsely, f the opertor G s ssoctve d wth the otto G d, for, G G G,,,, Eq.() c be wrtte:,,...,, G r, b J,,,..., J (4) Furtherore, f we suppose tht x, y Gx, y s ootoe o decresg ppg o L d x x, y s ootoe o decresg ppg o L for every y L the G, G r, G r L,, J J J hs leds us to stte the ecessry codto of the followg theore, whch E G,, deotes the set of ll solutos of Eq.() whe the relto s supposed ukow. hs set s sply desgted by E G,. 49

3 U-Sl Scece Jourl heore Let G d be two fuctos fro tht, for every belogs to L, G G x, y, z G x, G y, z () L to L such x, y, z d t () x, y z, t Gx, y Gz, t () x z x, y z, y f the set G, E s o-epty, the,,...,, b G,, G, J J (5) coversely f codto (5) s fulflled d f oreover: (v) the fucto G : x, y Gx, y s cotuous o L (6) (v) the fucto x x, y s cotuous o L for ech y L (7) the E G, s oepty. Ideed, f codtos (5)-(7) re t,,, fulflled, the exsts such tht G t b, d J r, exsts such tht t r,, hece E G, Corollry If the set, E s o- b, epty the x J where I,,..., (8). Coversely f (8) holds d f () x y x, y, s cotuous o L, () x x, y s cotuous o L for ech y L, the E, I the se wy, for G d, we obt: Corollry E s o- (9) If the set, epty the b I Vol.4()7 J Coversely f (9) holds d f: () x, y x, y s cotuous o L, () x x, y s cotuous o L for ech y L, the E,. ht s: ff J r, b for ll I d ff r, for ll I. heore J b he equto. () hs solutos f d oly f I. b J Fro ow, d b wll dcte the th row of, d. Where s the colu trx (s wth coeffcets the fuzzy trx)., Proof of theore If, Eq.() s equvlet to b I,. hs equto hs solutos f d oly f (corollry ) d oly f J b,.e f b, J hs codto s lso true for.. 49

4 U-Sl Scece Jourl heore Let be the set of solutos of fuzzy relto equto A, the : fuzzy relto ff. A If, the s gretest eleet. Exstece codto for theore : he ecessry d suffcet codto for the set s: here exsts, J such tht bk for ll p. hs exstece codto hs bee docueted by pedrycz[7] for the Mx-M coposto. esoluto to fuzzy reltol equto Let F d F we defe F, the (), -x coposto of d A s y x, y, x () Let the ebershp trces of, d deoted, r, b, respectvely, by where x, r x, y, b y for ll I d hs e tht ll the etres the trces A. J. d re rel ubers I,, whe ute tervl trces A d re gve d trx s to be detered Vol.4()7 fro Eq.() the proble s trvl. It s solved sply by perforg the M-Mx coposto-lke operto o A d s defed by Eq.(). Clerly the soluto ths cse exsts d s uque. Exple Gve d Detere the soluto of A fro Eq.() y x, y, x, y x y, x.7,.8,. 7.9,,..7,,. 4.7,.9,..7 he.7 y 5 y 6 y 7 he proble becoes fr fro trvl whe oe of two trces o the left-hd sde of Eq.() s ukow. I ths cse, the soluto s ether gurteed to exst or to be uque. Sce Eq.() s obted by coposg A d, t s suggestve to vew the proble of deterg A fro d s decoposto of wth respect to. Let us ssue tht pr of specfc trces d fro Eq.() s gve d tht we wsh to detere the set of 494

5 U-Sl Scece Jourl ll prtculr trces of the for A tht stsfy Eq.(). Let ech prtculr trx A tht stsfes Eq.() be clled ts soluto d let S, A : deote the set of ll solutos, (the soluto set). It follows edtely tht whe we tke the verse of both sdes of Eq.() we wll get: () ote tht:, d tht s x, y y, x : I, b : J d r : I, J, Now we c solve Eq.() ore sply th Eq.(). he f r b the vlues, I exst tht stsfy Eq.() d, trx exsts tht stsfes the trx equto thus S,. Whe S,, the of Eq.() s detered by: xu soluto, x r,, b where r,, b () b f f r r,, b b I We ext detere the set S, of ts l solutos of Eq.() c be detered by the followg procedure: - Detere the sets J for I xr, b :, ll J d the costruct ther product J J J Vol.4()7 deote eleets of J J J J by : J - For ech J d ech I detere the set, J : - For ech J geerte the -tuple g g : by tkg I x b f, g, f, 4- Fro ll the -tuples g geerted step () select ll the xu oes d by prwse coposto. he resultg set of - tuples s the set S, of the l soluto of Eq.(). Flly the soluto set S, s fully chrcterzed by the xu d l solutos the followg sese: It cossts exctly of the xu soluto, ll the l solutos d ll eleets of A tht re betwee d ech of the l soluto. Forlly S,, 495

6 U-Sl Scece Jourl Where the uo s tke for ll S,. We got the set of solutos of Eq.(). We ust ow tke the trspose of d ech of the l solutos tht s the xl solutos of Eq.(). Ad soluto of Eq.(). So S,, A the l Exple ; Gve..4.. Detere ll solutos of Sol ke the verse of the equto bove so () Where..4. b b b,, hus, we frst ust fd the solutos of Eq.(). So we ust detere whether S,, or ot by: Frst we detere whether.,,.4. b,,. b S, or ot, by: M M M,,. b hus sce Vol.4()7 S, [ow S, ].We detere the xu soluto Eq.() by: r,, b Mx,,. 5 r,, b Mx,, r b Mx,,. 5 Mx Mx Mx,, of,,. we c esly stsfy..4 S, , hece S, Next we pply the four steps of the procedure for deterg the set S, of ll l soluto of ths reduced trx equtto: - Eployg the xu soluto of the reduced equto, we obt J I : Mx r,, b, J I : Mx r,, b, J I : Mx r,, b, hece J J,,,,,,,,,, 496

7 U-Sl Scece Jourl - he sets, tht we ust detere for ll J d ll I re lsted the followg tble:, = g,,,,,, - For ech,,. 5,, J, we geerte the trples g whch re lso lsted tble bove. g 4- Oe of the trples tble bove s l soluto. 5. Hece so we S, hve the set of soluto of equto () s S, A : Vol.4()7 So to detere the solutos of equto A, we ust tke the trspose of d,, tht s Now the set S, of ll soluto of the gve trx equto s ow fully cptured by the xu soluto. 5 d the l. 5 soluto So we hve: S, A : Exple Gve.., Detere ll solutos of Sol ke the verse of the equto bove so where...4,.9.9, We frst ust fd the solutos of Eq.(). So we ust detere whether S,, or ot by: Frst we detere whether S, or ot by: 497

8 U-Sl Scece Jourl., ,.4. 9 hus S, We deterer the xu soluto by: r, b x.9,.9. 9 x.9,.9. 9 x, x r,, b. 9,.9 S, S, of Eq.(), hece Next we pply the four steps of the procedure for deterg the set S, for ll l soluto of ths reduced trx equto: - We obt J I : x r,, b, J I : x r,, b, hece J J,,,, - he sets,,, tht we ust detere for ll I re d ll J lsted the followg tble:, g, ,.9 - For ech J Vol.4()7, we geerte the trples g whch re lso lsted tble bove. g 4- wo of the trples tble bove re l. 9,.9 solutos S,.9,.9 So we hve the set of soluto of equto () S, A : A : S,. So, sc procedure to detere ll solutos of the equto () - ke the verse (trspose) of both sdes of Eq.() ths results the ew equto () - If r b the the, I equto hs soluto S, d the procedure tertes, otherwse proceed to step. - Detere by procedure. 4- If s ot soluto of Eq.(), the the equto hs o soluto, S,. 5- Detere ll l solutos of the reduced equto 498

9 U-Sl Scece Jourl () by procedure : ths results S,. 6- Detere the soluto set of the reduced equto (): where the S,, uo s tke over ll. S, 7- ke the trspose of d ech of the l solutos soluto set S,, : ths results the S, whch s Procedure () Fro the vector whch, r, b where r, b, b f f r r,, b b x, Procedure () - Perute eleets of d the correspodg colus of pproprtely to rrge the decresg order. - Detere the set I : x, r J, b for ll J d the costruct ther crtes product. J J J - For ech J Vol.4()7 d ech I detere the set, J :. 4- For ech J geerte the -tuple g g : by tkg g, I x b f, otherwse 5- Fro ll the -tuples g geerted step 4 select oly the l oes, ths results S, Note : f, d re fuzzy relto o, d, respectvely. Where s fuzzy reltol equto where s the Mx-M product coposto. If s ukow Eq. we c fd the xl soluto by the Eq. where x, y y, x y x, Exple: gve.7, Detere xl soluto of Sol S.7 We c esly prove, 499

10 U-Sl Scece Jourl Note : If A d re two fuzzy sets, respectvely d fuzzy relto of the set. where Mx product coposto g y x, y x f A s ukow we c fd the soluto by equto Exple Gve Sol , We ust fd the soluto: x.4.. x x ust prove to prove S, we..4.7 Vol.4() x x.49 x efereces. Dru,,. Itellget Hybrd Systes: Fuzzy Logc, Neurl Network d Geetc Algorths elg Nucler eserch Ceter (SCK. CEN) Mol, elgu.. George J.K,, 997.Fuzzy set foudto d pplctos Utest, Clr, oyuou.. Klr, Folger A., 994. Fuzzy set ucertty d forto, Uv of New York. 4. Mlk D.S. d Joh N., 99. Fuzzy reltos o groups, fuzzy sets d systes, North Holld 4: Skr K Muder D.K., Fuzzy thetcl pproch to ptter recogto Clcutt, Id. 6. Ohsto A. d Sekguch., 994.Cobed for of fuzzy reltol equtos d ts pplcto, Proc. It. Systes M Cyboret., oby : Pedrycz W., 996 Processg reltol structures: Fuzzy reltol equtos, Fuzzy sets d systes 4. * * Ohsto d Sekguch 5