Solving Fuzzy Linear Fractional Programming Problem using LU Decomposition Method
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1 s of Pre ppie Mhemis Vo. No ISSN: 9-8X (P) 9-888(oie) Pishe o 8 Ferr.reserhmhsi.org DOI: hp://.oi.org/./pm.v9 s of Sovig F Lier Frio Progrmmig Proem sig LU Deomposiio Meho S.Mrgm P. mik Deprme of Mhemis Imm Coege of Egieerig Tirhirppi- Tmi N Ii. Emi: smrg@hoo.i Deprme of Mhemis M..M. Coege of Egieerig & Tehoog Tirhirppi- Tmi N Ii. Emi: miksiv@gmi.om Reeive Jr ; epe Ferr sr. This pper proposes e pproh o sove f ier frio progrmmig proem (FLFPP). I his pper he FLFPP is overe io risp ier frio progrmmig proem (LFPP) sig rkig meho. The overe LFPP is he sove LU Deomposiio meho. isre empe shos he simpii of he propose pproh. Keors: Lier frio progrmmig LU eomposiio rigr f mer. MS Mhemis Sje Cssifiio (): 9C E. Iroio Lier frio progrmmig proem is mi se i eisio mkig proess i hih he ojeive fio is frio of o ier fios. Frio progrmmig proems re se i m fies sh s proio pig fii orpore pig heh re hospi pig e. M reserhers fo vrios ehiqes o sove ier frio progrmmig proems. Chres Cooper [] rsforme he ier frio progrmmig proem io he ier progrmmig proem ig e osri e vrie he he opimm soio is oie simpe meho. Chihoe he [] propose LU foriio meho o sove ier progrmmig proem. Jh Krhik Chrsekr [] sove f f ier ssems ih rpeoi f mer mries priioig he ok mries. Pi Kvih [] propose e meho me pre movig meho o fi he opim soio o he f ssigme proem. Rhkrish e.. [] fo he posiive soio of he f f ssem of ier eqios sig Crmer s re og ih Dogso s osierio. Ji [] propose Moifie Gss eimiio ehiqe o sove sepre oier progrmmig proem. Shrm s [] se rh o meho o fi he ieger soio of frio progrmmig proems. Srp [8] propose simpe gorihm o sove ier frio progrmmig proem. T [9] presee ierive meho se o he ojge grie 89
2 S.Mrgm P.mik projeio meho for sovig LFPP. LU Deomposiio is js omp reive meri se meho o sove ssem of ier eqios. For rge he ompio ime for LU Deomposiio is proporio o hie for Gssi Eimiio he ompio ime is proporio o. So for rge he rio of he ompio ime for Gssi eimiio o ompio ime for LU Deomposiio is. s empe for he oeffiie mri of orer ompio ime Gssi Eimiio o ke // imes he ime i o ke for LU Deomposiio. This pper is oie s foos. Seio gives he preimiries of he f mer. I Seio mhemi formio of he FLFPP is give. Seio esries he propose meho Seio epis ho he propose meho is ppie o he LFPP. I Seio Yger s rkig meho [] is give Seio gives he isre empe Seio 8 oes he pper.. Preimiries I his seio some si efiiios reig o f ses rigr f mers re give. Defiiio.. f se is e orm if here is es oe eeme X h ( ). µ sh Defiiio.. f se is e ove if for X λ ( λ ( λ) ) mi{ µ ( ) ( ) } µ µ Defiiio.. The -eve of f se is efie. [ ] { X / µ ( ) } Defiiio.. f se is f mer if i sisfies he oiios of ormi ovei. Defiiio.. Trigr f mer ~ rigr f mer ( ) ih memership fio µ give ( ) for µ ( ) for esehere Figre : Trigr f mer 9
3 Sovig F Lier Frio Progrmmig Proem sig LU Deomposiio Meho Mhemi Formio of FLFPP ~ T ~ Mimie ~ ~ T ~ β Sje o ~ ~ here is -imesio veor of eisio vries ~ ~ re veors ~ is m osri f mri ~ is m- imesio f veor ~ ~ β re srs.. LU eomposiio meho Give ssem of ier eqios ih kos. We rie his ssem s here Y [ ij ] [ j ] i Y [ ] (i) Wrie LU here L is he i oer rigr mri U is he pper rigr mri. From his eqio e fi L U. (ii) No he ssem of eqios eomes LUY. (iii) Le UY W. No e sove LW for W (iv) Usig W e sove UY W for Y. This i give he soio for he ssem Y.. ppiio of LU eomposiio meho o sove ier frio progrmmig proem Cosier he fooig Lier Frio Progrmmig Proem... Mimie.. β Sje o () Le s over his ier frio progrmmig proem io ier progrmmig proem sig Chres Cooper meho s eo.
4 S.Mrgm P.mik 9 Le for No e rie he proem () s he fooig LPP. () We rie his LPP i he form of ess h or eqi osris. () No he ssem of eqios e osiere s here β.. i i. i Y β o Sje Mimie β β
5 Y Sovig F Lier Frio Progrmmig Proem sig LU Deomposiio Meho. Yger s rkig meho ~ Give ove rigr f mer C he - of he f mer is give C C The Yger s Rkig ie [] is efie L U here C C is -eve of f mer C ~.. Nmeri empe Cosier he fooig FLFPP ( ) ( ) C ~ L U ( ) ( ) ( ) ( ) ~ L U ( ).( C C ) R C ( ) ( ) Mimie ( ) ( ) ( 9) Sje o 9 8 () No e over he FLFPP io he fooig risp LFPP sig Yger s rkig meho. U The - of f mer is C L C ( ) R ( ) ( ) Proeeig simir he proem () e rie s he fooig risp LFPP Mimie Sje o ( ).( ) 9
6 S.Mrgm P.mik 9 Le The give LFPP eomes LPP s eo. Sje o Soio LU eomposiio meho: We rie he ove LPP s We rie he ssem s here. We rie here L is i oer rigr mri U is pper rigr mri. Mimie Y Y LU
7 Sovig F Lier Frio Progrmmig Proem sig LU Deomposiio Meho 9 Th is No O simpifiio e ge Ths No. We rie here. No e sove LW for W here L U L U LUY LW W UY W
8 S.Mrgm P.mik 9 O simpifiio e ge Fi he soio mri is give UYW. Th is O simpifiio e ge Ths he opim soio of he LFPP is give. Cosio I his pper e propose e pproh e LU Deomposiio Meho of mries o sove f ier frio progrmmig proem (FLFPP). The FLFPP is overe io risp LFPP sig Yger s Rkig meho he i is overe io LPP sig Chres Cooper meho. I his pproh e ge he soio ire he ime osmpio is ver ess. meri empe is give o sho he simpii of he propose pproh he soio is verifie LINGO. versio so. koegeme. The hors hk he omos referees he Chief-Eior for heir ve ommes sggesios hih ere ver hepf i improvig he preseio of his pper. Y 9
9 Sovig F Lier Frio Progrmmig Proem sig LU Deomposiio Meho REFERENCES..Chres W.W.Cooper Progrms ih ier frio fios Nv Reserh Logisis Qrer 9 (9) S.M.Chihoe.P.he LU foriio meho o sove ier progrmmig proem Ier. J. Emergig Tehoog ve Egieerig () () -8.. N.Jh Krhik E.Chrsekr Sovig f f ier ssems ih rpeoi f mer mries priioig he ok mries s of Pre ppie Mhemis 8() () -.. P.Pi K.Kvih e meho for sovig f ssigme proems s of Pre ppie Mhemis () () S. Rhkrish P. Gjivrh R. Govirj e simpe meho of sovig f f ier ssem s of Pre ppie Mhemis 8() () S.Ji Moifie gss eimiio ehiqe for sepre oier progrmmig proem Ierio Jor of Isri Mhemis () () -.. S.C.Shrm.s ieger soio of frio progrmmig proem Ge. Mh. Noes () K.Srp Lier frio fio progrmmig Operio Reserh (9) S.F.T e proere for sovig ier frio progrmmig proems Mhemi Comper Moeig 8 (8) R.R.Yger proere for orerig f sses of he i ierv Iformio Siees (98) -. 9
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