A Genetic Algorithm Approach for Reliability of Bridge Network in Fuzzy System

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1 ISSN Engnd UK Jon of Infomton nd omptng Sene Vo. 6 No. 0 pp. - A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem G.S. Mhpt P. Roy Deptment of Engneeng Sene nd Hmnte Sg Inttte of Tehnoogy P.O.- Skn Sg Wet Beng Ind Reeved Agt 0 0 epted Septembe 0 Abtt. Th ppe peent Genet Agom GA ppoh fo ovng ontned ebty optmzton pobem of e fve nt bdge netwok. ondeng netnty fo ot e ebty optmzton pobem h been oved by GA tehnqe to mxmze ytem ebty. GA n effent meod fo ovng type of optmzton pobem. Th ppe efy ppe e GA tehnqe to obtn e optm oton of e ompex ytem ebty mode nde ot ontnt n fzzy envonment n whh e ytem ot nd omponent ot mght be mpee. Tng fzzy nmbe ed to epeent e fzzy ot oeffent. Stt penty meod h been ed to hnde e ot ontnt of e pobem. To ove e ebty optmzton mode we hve deveoped e GA ng MATLAB w tonment eeton poe etm mehnm met oove nd nfom mtton opeton. Fny omptton et e peented fo e ebty of bdge netwok n p nd fzzy envonment. Keywod: Genet gom Rebty optmzton Penty meod Bdge netwok Fzzy nmbe. Intodton Genet Agom GA obt evotony optmzton eh tehnqe t mm e poe of nt evoton to ove optmzton pobem. Genet gom e bed on e ombnton of pnpe of genet nd boog evoton. It one of e het optmzton tehnqe whh nde evotony ttege []. A genet gom toht gob optmzton eh meod t e nt eeton n boog evoton t mode of pobem ovng ntoded by Hond [] nd fe debed by Godbeg []. John Hond [][] e ponee of genet gom. GA wee nvented nd deveoped by hm h tdent nd oege. Seve type of genet gom hve been deveoped to ove nge o mtpe objetve optmzton pobem w ny ombnton of ne/nonne objetve o ontnt. Deng et. [8] popoed e heddng genet gom SGA t foow e mehnm dopted n moden beedng tehnoogy whee hey nm e tvted by ntefeng w e nt eeton poe nd fteng ot pb w ndebe htet ng e pnpe of etm. By mtng ftton poe SGA n fo e eh ond e mot mpotnt gene. Wng nd Ghon [9] popoed n extenon gom of e SGA by ombnng e beneft of e heddng nd enng opeto to e nkge poe popoed by Hk nd Godbeg [0]. heng [] popoed n tf ne netwok bed GA. Pe GA tehnoogy [] n be ed to eete e peed of GA. B-objetve genet gom [] n be ed fo mkepn nd ebty optmzed hedng fo wokfow ppton. Wng et. [] popoed e ook-hed genet gom to ntegenty optmze bo e mkepn nd e ebty fo wokfow ppton. Deb et. [] popoed e Non-domnted Sotng Genet Agom NSGA-II whh ft nd vey effent mt-objetve evotony gom. hng et. [7] popoed two-phe b popton genet gom to ove e pe mhne-hedng pobem. hen et. [] popoed hybd evotony gom w ntegton of GA nd extem optmzton to ove e hot tp m hedng pobem. Eyed et. [] deveoped mt-opeto oepondng o. Te.: 997; fx: E-m dde: g mhpt@yhoo.om Pbhed by Wod Adem Pe Wod Adem Unon

2 G.S. Mhpt et : A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem genet gom nd ef-dptve mt-opeto genet gom fo ovng ontned optmzton pobem. Gonçve nd Reende [] popoed mt-popton bed ndom-key genet gom fo e nge ontne odng pobem. Meng nd Weng [7] popoed GA to eng wok zone ty k. GA n be ed to ove dffeent type of optmzton pobem h fzzy ombnto nd mt objetve optmzton pobem. In t two dede dffeent optmzton ppohe hve been popoed to ove ebty optmzton pobem. Thee nde dynm pogmmng meod ntege pogmmng meod bnh nd bond meod Lgnge mtpe meod toht pogmmng meod et. [0]-[]. Tody GA wdey ed n ebty engneeng to optmze ebty of ytem nde etn ontnt. Pnton nd mpbe [6] popoed GA to fnd mxmm ebty to tfy pef ot ontnt. ot nd Sm [8] debed GA to ove e ebty optmzton pobem fo ee-pe ytem meetng e ot nd weght ontnt. Bhn et. [9] popoed GA to ove ebty optmzton pobem of ee ytem w ntev ved omponent ebty. Shoo et. [6] deveoped GA to ove ebty edndny optmzton pobem of eepe/pe-ee/ompex ytem w ntev ved ebty of eh omponent. heng nd L [0] popoed meod fo tt ebty ny by ntegtng e nfom degn meod [] w tf ne netwok bed GA. Heh et. [] tzed GA to ove vo type of ebty degn pobem h ebty optmzton of ee ytem ee-pe ytem nd ompex ytem. Yeh et. [] popoed GA to ove e k-node et ebty optmzton pobem w pty ontnt of dtbted ytem. Gen nd heng [6] debed e ppton of GA to ebty optmzton pobem. Moghddm et. [] popoed GA fo edndny oton pobem to e ee-pe ytem when e edndny ttegy n be hoen fo ndvd bytem. Km et. [7] pefomed e ebty ny of wte en-p mnpto ng e oded GA nd fzzy meodoogy. Kho et. [8] popoed mt-objetve GA fo ompex ytem ebty optmzton pobem nde fzzy envonment whe mnmzng e ot of e ytem. In ppe we hve deveoped GA to ove e ebty optmzton pobem of bdge netwok n p nd fzzy envonment. Hee we hve peented optm oton of ytem ebty of bdge netwok w ytem ot ontnt n fzzy nd p envonment. We hve ed e-ve enodng to epeent e oton homoome. The nt popton of ze 0 h been geneted ndomy. Tonment eeton poe h been ed to eet e pent homoome fom e popton. Etm poe met oove opeton nd nfom mtton opeton e pped to genete e offpng t hve hghe ftne ve n e pent. When temnton ondton tfed e GA w gve e optm oton of e pobem. Fny nme expoe of e popoed GA tehnqe peented nd ome onon nd eommendton e mde.. Mode Fomton.. Notton A bdge netwok ebty mode deveoped ng GA tehnqe nde e foowng notton. Notton Defnton ~ ~ Rebty fo ot oeffent fo omponent of e ebty mode Fzzy ot oeffent fo Shpe pmete fo omponent of e ebty mode omponent of e ebty mode omponent of e ebty mode Avbe ytem ot of e ebty mode Fzzy ytem ot of e ebty mode Lowe end pont of e fzzy ot ntev fo Uppe end pont of e fzzy ot ntev fo Lowe end pont of e fzzy ytem ot ntev omponent omponent JI em fo ontbton: edto@j.og.k

3 Jon of Infomton nd omptng Sene Vo. 6 0 No. pp - Uppe end pont of e fzzy ytem ot ntev R Sytem ebty fnton of e ebty mode Sytem ot fnton of e ebty mode ~ Fzzy ytem ot fnton of e ebty mode.. p mode of bdge netwok The bdge netwok ondeed ytem of fve omponent to fnd ot e ytem ebty hown n Fge. The geb expeon fo ytem ebty R of e bdge ytem gven foow: R whee 0 fo. < Fg. : Shemt of fve nt bdge netwok The ot ontnt fo e bdge ytem gven foow: Hene memty e ebty optmzton pobem fo e bdge netwok n be fomted foow: Mxmze R whee 0< fo. bjet to.. Fzzy bdge netwok mode In e fe e ve of ot oeffent of omponent nd e vbe ytem ot e not gd. Thee ot e not known w etnty. We need to onde netnty fo ot. So ondeng netnty fo e ot oeffent nd ytem ot t moe et to onde em fzzy. In fzzy ytem e ot ontnt fo e bdge netwok gven foow: ~ ~ ~ Hee fzzy ot oeffent ~ nd fzzy ytem ot ~ e epeented by tng fzzy nmbe foow: ~ fo nd ~ JI em fo bpton: pbhng@wau.og.k

4 G.S. Mhpt et : A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem JI em fo ontbton: edto@j.og.k 6 Appyng - t opeton on e fzzy nmbe we get e p ntev of ot foow: ] [ fo nd ] [ Hene e ebty optmzton pobem fo e bdge ytem n fzzy envonment n be fomted foow: Mxmze R bjet to ] [ ] [ whee 0 < fo nd. 0 The bove ontnt n be wtten foow: nd Theefoe e bdge netwok ebty mode tnfom nto e foowng fom : R Mxmze bjet to whee 0 < fo nd. 0. GA Bed ontnt Hndng Appoh Hee we hve ded GA bed ontnt hndng ppoh of e optmzton pobem. The penty meod h been ed to hnde e ontnt of e optmzton pobem. We hve ed tt penty [6] to tnfom e ontned optmzton pobem to n nontned optmzton pobem. Let onde e foowng ontned optmzton pobem: Mxmze x f bjet to m x h... 0 whee m e nmbe of ontnt. Aodng to e tt penty fnton meod pobem n be tnfomed to n nontned mxmzton pobem nd e objetve fnton n be wtten foow: Mxmze x P x f x F m σ whee. ] mx[0 x h x P In e bove eqton Px e penty fnton whh epeent e qe of ontnt voton nd σ e penty oeffent o penty pmete fo ontnt.

5 Jon of Infomton nd omptng Sene Vo. 6 0 No. pp - 7 f x f x S F x m f x σ P x f x S whee S e et of febe oton. If no ontnt voton o.e. e ve of x febe oton Px w be zeo fo ontnt. If ontnt voton o fo ome ontnt Px w be potve fo t voted ontnt. Hee e penty pmete σ do not depend on e ent geneton nmbe nd ontnt penty pped to nfebe oton.. GA Impementton A GA h been deveoped to ove e ebty optmzton pobem. The tep of e popoed GA e gven foow: Agom Step : Intze e pmete of e genet gom foow: popton ze0 oove te0.9 mtton te0.0 mxmm nmbe of geneton00. Step : Intze dffeent pmete of e optmzton pobem. Step : Set g whee g gnfe e ent geneton nmbe. Step : Intze e homoome of e popton Pg [Pg epeent e popton of g geneton] tkng e owe nd ppe bond of e vbe 0 nd epetvey. Step : Evte e ftne ve of eh homoome n e popton Pg ondeng e ftne fnton. Step 6: Seet pent homoome fom Pg by tonment eeton poe. Step 7: Ate Pg by ppyng etm poe met oove nd nfom mtton opeton on e eeted pent homoome. Step 8: Evte e ftne ve of eh homoome offpng n e popton Pg. Step 9: Fnd e homoome w bet ftne ve fom Pg. Step 0: If e temnton ondton tfed go to tep. Step : Pe Pg n Pg. Step : Inee g by nty. Step : Go to tep. Step : Pnt e homoome of Pg w e bet ftne fnton ve whh e oton of e optmzton pobem. Step : End. The fowht of e GA hown n Fge. JI em fo bpton: pbhng@wau.og.k

6 8 G.S. Mhpt et : A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem Fg. : Fowht of e popoed GA fo optmze ytem ebty.. Deveoped GA fo ebty optmzton of p mode of bdge ytem The GA deveoped fo p mode of bdge netwok ebty optmzton pobem debed foow: homoom-enodng of oton. In GA eh oton epeented by homoome. A homoome n enoded oton. In pobem e eng of homoome.e. e nmbe of gene n homoome ee e fve omponent n e mode. Re ve enodng h been ed to enode e oton homoome nd eh omponent gene of e homoome epeent e ebty of e omponent of e mode. The homoome epeentton of oton gven beow n exmpe: [ ] Int popton A et of pobe oton.e. homoome n geneton ed popton. The GA tted w n nt popton. The nt popton h been detemned ndomy w popton ze of 0. Ftne fnton The ftne fnton defned ombnton of e objetve fnton nd tt penty fnton. The ftne fnton ed to detemne e ftne ve of eh homoome n geneton. The ftne ve ndte whh homoome w be eeted pent to be nvoved n genet opeton. The ftne fnton of e bdge ytem ebty optmzton pobem gven beow: Mxmze R σ. mx[0 ].e. JI em fo ontbton: edto@j.og.k

7 Jon of Infomton nd omptng Sene Vo. 6 0 No. pp - 9 Mxmze σ.mx[0 ] Seeton poede The eeton poe eet homoome odng to e ftne ve fo genet opeton. homoome e eeted fom e popton pent. To fom new offpng homoome e eeted odng to e Dwn' eoy bot evoton "Svv of e fttet" bee e moe tbe ey e e moe hne to epode. Seeton pefomed to emphze ftte ndvd n e popton n hope t e offpng hve hghe ftne ve. Hee tonment eeton poe of ze ee h been ed fo homoome eeton. Etm Etm meod ope e bet homoome o e few bet homoome of e ent geneton to e next geneton wot oove nd mtton. The et of e ndvd e ped ogh e oove nd mtton opeton. The bet homoome my be ot n ny geneton by oove nd mtton opeton when new popton eted. To oveome tton etm meod h been pped. Hee e ve of ete ont h been tken one. Ete ont ndte e nmbe of ndvd t w be gnteed to vve to e next geneton wot oove nd mtton opeton. Genet oove opeto Afte eeton poe e oove opeto pped to e etng pent homoome. The mn ppoe of oove to exhnge genet nfomton between ndomy eeted pent homoome. oove eombne e genet mte of two o moe pent homoome to pode new ndvd offpng fo e next geneton. We hve ed e met oove opeto [7] whh defne g g ne ombnton of two homoome. Sppoe nd e two pent homoome t e g g ndomy eeted fo oove n g geneton. If nd e to be oed e etng offpng e g nd g whh ne ombnton of e pent.e. g g g β β g g g β β whee β ndom ve t e between 0 nd. Hee e met oove opeto h been pped bed on e oove pobbty of Genet mtton opeto Mtton opeton pefomed to ntode ndom vton nto e gene of homoome. Mtton pped to pevent fng of oton n popton nto o optmm of e oved pobem. So opeton pefomed to mpove e gob optm oton. Mtton hep to egn e nfomton ot n ee geneton. Mtton opeton pped to nge homoome w vey ow te. Mtton n hnge nge o e gene of ndomy eeted homoome. In pobem we hve ed e nfom mtton opeto w mtton te of It eet fton of e veto ente of n ndvd fo mtton whee eh enty h e me pobbty e mtton te of beng mtted. Temntng te The temnton ondton ed to top e GA. The GA temnted when ny one of e foowng ondton tfed: e nmbe of geneton h been ehed to e mxmm nmbe of geneton. e pefed tme h been eped. e vege hnge n e ftne fnton ve ove pefed geneton e n vey m pe-gned potve nmbe. Hee we hve ed 00 geneton e mxmm nmbe of geneton fo e GA 00 geneton 0 e pefed geneton nd 0 vey m potve nmbe. JI em fo bpton: pbhng@wau.og.k

8 G.S. Mhpt et : A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem JI em fo ontbton: edto@j.og.k 0 Pmete of e GA Thee e fo pmete ed n e GA t ffet e oton obtned fom e GA. Thee pmete e popton ze p oove te/pobbty mtton te/pobbty m nd mxmm nmbe of geneton m_gen. Hee we hve tken e ve of GA pmete foow: p0 0.9 m0.0 nd m_gen00... Deveoped GA fo bdge netwok ebty n fzzy envonment The GA deveoped fo fzzy bdge netwok ebty optmzton pobem debed hee. homoom-enodng of oton. In fzzy envonment when we onde vbe e homoome ze w be 6. When not vbe.e. we gve e ve of n e optmzton pobem en e homoome ze. Int popton The nt popton h been geneted ndomy w popton ze of 0 tkng e owe nd ppe bond of e vbe 0.7 nd 0.99 epetvey. Ftne fnton The ftne fnton of e fzzy bdge netwok ebty optmzton mode gven beow. ] mx[0 ] mx[0 Mxmze σ σ 6 The eeton poede etm mehnm oove opeto mtton opeto temntng te nd pmete of e GA deveoped fo fzzy bdge ytem ebty optmzton pobem e me w e GA deveoped fo p mode of bdge ytem.. Nme Peentton.. Soton of bdge netwok ebty n p mode The popoed GA h been mpemented ng MATLAB. Let e ot oeffent hpe pmete nd e vbe ytem ot ed fo e gven foow: ; nd 8. Hee we hve tken e ve of penty pmete σ 999. Bee of e toht nte of GA 0 t hve been pefomed fo 00 geneton e mxmm nmbe of geneton eh nd e bet oton fom mong e 0 t h been ondeed e fn optm oton. In e Fge pot of e geneton nmbe ve e bet ftne ve n oe geneton peented. The fge how e bet ftne ve n e pefed geneton fo e t fo whh we get e bet oton. Fom e gph we n ee t e geneton nmbe nee e bet ftne ve n oe geneton onvege to e optm oton.

9 Bet Ftne Ve Jon of Infomton nd omptng Sene Vo. 6 0 No. pp Geneton Nmbe Fg. : Ftne ve onvegene Afte 0 geneton of e t fo whh we get e bet oton we get e foowng oton e optm oton dentfed by e GA: The mxmm ytem ebty of bdge netwok R The mxmm ebty of e fve omponent e foow: And e ytem ot of bdge netwok Hee e ot ontnt h been tfed effenty fo e hoen ve of e penty pmete nd we get e mxmm ytem ebty tfyng e ot ontnt. The hoen ve of e penty pmete fo o ebty optmzton pobem effenty hnde e ot ontnt of e pobem... Soton of fzzy bdge netwok The GA h been mpemented w MATLAB. Let e fzzy ot oeffent ~ of fve omponent ~ of e bdge netwok nd e fzzy ytem ot of e bdge mode e gven foow: ~.8 ~ ~ 789. ~.67 ~ 6. ~ nd Appyng t opeton on e bove fzzy nmbe we get e p ntev of ot foow: [.8. ] [ [7 9.. ] [ ] [ 6.. ] nd [ ] In optmzton pobem two penty pmete σ nd σ e ed fo e two ontnt nd we hve tken e ve of σ nd σ 0 9. Twenty t hve been pefomed fo e popoed GA nd e bet oton fom mong e 0 t h been tken e fn optm oton. ] JI em fo bpton: pbhng@wau.og.k

10 G.S. Mhpt et : A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem Afte geneton of e t fo whh we get e bet oton we get e foowng oton e optm oton dentfed by e popoed GA: nd R Now n e eond pt of beton we hve peented oton ppoh of e d mode fo dffeent ve of. In Tbe we hve hown e ve of ytem ebty R nd ebty of e fve omponent fo dffeent ve of. We hve o mentoned e geneton nmbe fte whh we get e ebty of e fve omponent nd ytem ebty e optm oton dentfed by e GA. Tbe : Optmm ebty fo dffeent ve of R Geneton Nmbe The ontnt hve been tfed effenty fo e hoen ve of penty pmete nd we get e mot pomng optm oton fo t ve of penty pmete. 6. onon In ppe e bdge netwok ebty optmzton pobem n p nd fzzy envonment h been oved by GA ppoh. Hee e ot of e mode hve been ondeed fzzy bee of netnty. Fo hndng ontnt tt penty meod h been tzed to ove e ontned optmzton pobem. We hve deveoped GA to ove e ebty optmzton pobem w tonment eeton poe met oove nfom mtton nd etm of ze one. We hve fnd ot e ytem ebty nd ebty of dffeent omponent of e mode ng e popoed GA. O nme et how t e GA deveoped fo ebty optmzton pobem gve vey pomng et e optm oton fo e hoen ve of penty pmete. The demontton how t e GA ppoh n be vey powef too fo ovng ebty optmzton pobem of bdge netwok. 7. Aknowedgement The o wod ke to knowedge e on of Sentf nd Indt Reeh SIR New Deh Ind fo povdng fnn ppot to y ot eeh wok. 8. Refeene [].R. Reeve. Moden Het Tehnqe fo ombnto Pobem. John Wey & Son 99. [] J.H. Hond. Adptton n Nt nd Atf Sytem. Unvety of Mhgn Pe 97. [] D.E. Godbeg. Genet Agom n Seh Optmzton nd Mhne Lenng. Addon Weey 989. [] J.H. Hond. Genet gom nd e optm oton of t. SIAM J. ompt. 97 : [] J.F. Gonçve nd M.G.. Reende. A pe mt-popton bed ndom-key genet gom fo ontne odng pobem. ompt. Ope. Re. 0 9: [6] M. Gen nd R. heng. Genet Agom nd Engneeng Degn. John Wey & Son 997. [7] Q. Meng nd J. Weng. A Genet gom ppoh to eng wok zone ty k. Sfety S : [8] L. Deng M. Ghon nd S. Sho. Sheddng genet gom fo ebty ny of tt ytem. J. Stt. Sfety. 00 7: -. JI em fo ontbton: edto@j.og.k

11 Jon of Infomton nd omptng Sene Vo. 6 0 No. pp - [9] J. Wng nd M. Ghon. Lnkge-heddng genet gom fo ebty ement of tt ytem. J. Stt. Sfety. 00 7: 9-7. [0] G.R. Hk nd D.E. Godbeg. Lnkge enng ogh pobbt expeon. J. ompt. Me. App. Meh. Eng : 9-0. [] J. heng. Hybd genet gom fo tt ebty ny. J. ompt. Stt : -. [] D. Lm Y. S. Ong Y. Jn B. Sendhoff nd B. S. Lee. Effent heh pe genet gom ng gd omptng. Fte Gene. omp. Sy. 007 : [] A. Doğn nd F. Özgüne. Bobjetve hedng gom fo exeton tme-ebty tde-off n heteogeneo omptng ytem. ompt. J. 00 8: 00-. [] X. Wng.S. Yeo R. Byy nd J. S. Optmzng e mkepn nd ebty fo wokfow ppton w eptton nd ook-hed genet gom. Fte Gene. omp. Sy. 0 78: -. [] K. Deb S. Agw A. Ptp nd T. Meyvn. A ft ett non-domnted otng genet gom fo mtobjetve optmzton: NSGA-II. Po. of e Pe Pobem Sovng fom Nte VI onfeene. 000 pp [6] L. Shoo A.K. Bhn nd D. Roy. A genet gom bed ebty edndny optmzton fo ntev ved ebte of omponent. J. App. Qnt. Me. 00 : [7] P.. hng S.H. hen nd K.L. Ln. Two-phe b popton genet gom fo pe mhne-hedng pobem. Expet Syt. App. 00 9: [8] S.N. Svnndm nd S.N. Deep. Intodton to Genet Agom. Spnge 008. [9] M. Gen nd R. heng. Genet Agom nd Engneeng Optmzton. John Wey & Son 000. [0] F.A. Tmn.L. Hwng nd W. Ko. Optmzton tehnqe fo ytem ebty w edndny - evew. IEEE T. Reb.977 R-6: 8-. [] W. Ko V.R. Pd F.A. Tmn nd.l. Hwng. Optm Rebty Degn: Fndment nd Appton. mbdge Unvety Pe 00. [] X.L. Sn K.I.M. Mknnon nd D. L. A onvexfton meod fo of gob optmzton pobem w ppton to ebty optmzton. J. Gob Optm. 00 : [] X.L. Sn nd D. L. Optmty ondton nd bnh nd bond gom fo ontned edndny optmzton n ee ytem. Optm. Eng. 00 : -6. [] R. Zho nd B. L. Stoht pogmmng mode fo gene edndny- optmzton pobem. IEEE T. Reb. 00 R: 8-9. [] Y.W. hen Y.Z. L M. Ge G.K. Yng nd.. Pn. Deveopment of hybd evotony gom fo podton hedng of hot tp m. ompt. Ope. Re. 0 9: 9-9. [6] L. Pnton nd J. mpbe. Genet gom n optmzton of ytem ebty. IEEE T. Reb. 99 : [7] N. Km J.H. Bom nd A. Km. Rebty ny of wte en-p mnpto ng genet gom nd fzzy meodoogy. ompt. Ope. Re. 0 9: 0-9. [8] D.W. ot nd A.E. Sm. Rebty optmzton of ee-pe ytem ng genet gom. IEEE T. Reb : -60. [9] A.K. Bhn L. Shoo nd D. Roy. Rebty toht optmzton fo ee ytem w ntev omponent ebty v genet gom. App. M. ompt. 00 6: [0] J. heng nd Q.S. L. Rebty ny of tte ng tf ne netwok bed genet gom. ompt. Meod App. 008 M. 97-8: [] K.T. Fng nd Y. Wng. Nmbe-Theoet Meod n Sttt. hpmn & H 99. [] Y.. Heh T.. hen nd D.L. Bke. Genet gom fo ebty degn pobem. Moeeton. Reb : [] Y.S. Yeh.. h nd R.S. hen. A genet gom fo k-node et ebty optmzton w pty ontnt of dtbted ytem. Po. of e Nton Sene on. 00 pp. 7-. [] S.M. Eyed R.A. Ske nd D.L. Em. Mt-opeto bed evotony gom fo ovng ontned optmzton pobem. ompt. Ope. Re. 0 8: [] R. Tvkko-Moghddm J. Sf nd F. Sn. Rebty optmzton of ee-pe ytem w hoe of edndny ttege ng genet gom. Reb. Eng. Syt. Sfe : 0-6. [6] A. Homf S.H.Y. L nd X. Q. ontned optmzton v genet gom. Smton. 99 6: -. JI em fo bpton: pbhng@wau.og.k

12 G.S. Mhpt et : A Genet Agom Appoh fo Rebty of Bdge Netwok n Fzzy Sytem [7] Z. Mhewz. Genet Agom Dt Stte Evoton Pogm. Spnge 996. [8] A. Kho S.P. Ydv nd S. Km. A Mt-objetve Genet Agom fo Rebty Optmzton Pobem. Int. J. Pefom. Eng. 009 : 7-. JI em fo ontbton: edto@j.og.k