MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS

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1 Yugoslav Joual of Opeaos Reseach Volume (), Numbe, -7 DOI:.98/YJORI MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS Sahdul ISLAM Depame of Mahemacs, Guskaa Mahavdyalaya, Guskaa, Budwa Receved: May 9 / Acceped: Novembe Absac: I hs pape, we have dscussed cosaed posyomal Mul-Objecve Geomec Pogammg Poblem. Hee we shall descbe he fuzzy opmzao echque (hough Geomec Pogammg echque) I ode o solve he above mulobjecve poblem. The soluo pocedue of he fuzzy echque s llusaed by a umecal eample ad eal lfe applcaos. Keywods: Posyomal, geomec pogammg, MOGPP, ma-m opeao, gavel bo poblem. AMS Subjec Classfcao: 9C9, 9C7.. INTRODUCTION GP mehod s a effecve mehod used o solve a o-lea pogammg poblem. I has cea advaages ove he ohe opmzao mehods. Hee, he advaage s ha s usually much smple o wok wh he dual ha he pmal oe. Solvg a o-lea pogammg poblem by GP mehod wh degee of dffculy (DD) plays a sgfca ole. (I s defed as DD oal umbe of ems objecve fuco ad cosas oal umbe of decso vaables ). Sce lae 96 s, Geomec Pogammg (GP) has bee kow ad used vaous felds (lke OR, Egeeg sceces ec.). Duff, Peeso ad Zee [4] ad Zee [] dscussed he basc heoes o GP wh egeeg applcao he books. Aohe famous book o GP ad s applcao appeaed 976 []. Thee ae may efeeces o applcaos ad mehods of GP he suvey pape by Ecke [5]. They descbed GP wh posve o zeo degee of dffculy. Today, mos of he eal-wold decso-makg poblems ecoomc, evomeal, socal, ad echcal aeas ae mul-dmesoal ad mul-objecves oes. Mul-objecve opmzao poblems dffe fom sgle-objecve opmzao

2 4 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos poblem. I s sgfca o ealze ha mulple objecves ae ofe o-commesuable ad coflc wh each ohe opmzao poblems. Howeve, s possble fo hm/he o sae he desably of achevg a aspao level a mpecse eval aoud. A objecve wh eac age value s emed as a fuzzy goal. So, a mulobjecve model wh fuzzy objecves s moe ealsc ha deemsc of. Zadeh [] fs gave he cocep of fuzzy se heoy. Lae o, Bellma ad Zadeh [] used he fuzzy se heoy o he decso-makg poblem. Taaka [7] oduced he objecve as fuzzy goal ove he α-cu of a fuzzy cosa se ad Zmmema [] gave he cocep o solve mul-objecve lea-pogammg poblem. Fuzzy mahemacal pogammg has bee appled o seveal felds. Geomec pogammg s a specal mehod used o solve a class of olea pogammg poblems; maly we use hs poblem o solve opmal desg poblems whee we mmze cos ad /o wegh, mamze volume ad/ o effcecy ec. Geeally, a egeeg desg ad maageme scece poblem has mul-objecve fucos. I hs case s o suable o use ay sgle objecve pogammg o fd a opmal compomse soluo. We ca use fuzzy pogammg o deeme such a soluo. Bswal [], Vema [9] developed fuzzy geomec pogammg echque o solve Mul-Objecve Geomec Pogammg (MOGP) poblem. Hee we have dscussed aohe fuzzy geomec pogammg echque o solve MOGPP.. MULTI-OBJECTIVE OPTIMIZATION I ece yeas hee has bee a cease eseach o mul-objecve opmzao mehods. Decsos wh mul-objecves ae que successful goveme, mlay ad ohe ogazaos. Reseaches fom a wde vaey of dscples such as mahemacs, maageme scece, ecoomcs, egeeg ad ohes have cobued o he soluo mehods fo mul-objecve opmzao poblems. The suao s fomulaed as a mul-objecve opmzao poblem whch he goal s o mmze (o mamze) o a sgle objecve fuco bu seveal objecve fucos smulaeously. The pupose of mul-objecve poblems he mahemacal pogammg famewok s o opmze he dffee objecve poblems, (say k umbe) smulaeously subjec o a se of sysem cosas. Fo eample, Mmze f ) [ f ( ), f ( ),..., f ] T k ( ) ( (.) subjec o g ( ) b j,,,m j j >. Hee ow we shall descbe he fuzzy opmzao echque (hough GP) o solve he above mul-objecve poblem... Mul-Objecve Geomec Pogammg Poblem (MOGPP) usg Fuzzy Techque A mul-objecve geomec pogammg poblem ca be saed as:

3 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 5 Fd T,,..., ) so as o (..) ( Mmze Mmze Mmze Subjec o T a () f c T a () f c Tk ak k() k f c whee Tm amj j j g ( ) c j,..., m >, j ks j c ( > ), c ( > ), a, a ae all eal umbes fo j,,..,m;,,.., T ; j k,,,m; s,,,. T k To solve hs mul-objecve geomec pogammg poblem, we use he Zmmema s (978) soluo pocedue. Ths pocedue cosss of he followg seps: Sep-: Solve he MOGPP as a sgle objecve GP poblem usg oly oe objecve a a me ad gog he ohes. These soluos ae kow as deal soluo. Sep-: Fom he esuls of sep-, deeme he coespodg values fo evey objecve a each soluo deved. Wh he values of all objecves a each deal soluo, pay-off ma ca be fomulaed as follows: f () f ()... f k () f ( ) f( )... fk ( ) f( ) f ( )... fk ( ) k k k k f( ) f( )... fk ( ) k Hee,,..., ae he deal soluos of he objecves f( ), f( ),..., fk ( ) especvely. U ma f ( ), f ( ),..., f k ( ) ad L f ( ) fo,,..., k So { } [ L ad U be lowe ad uppe bouds of he,...,k ]. h objecve fuco f () fo j

4 6 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos Sep : Usg aspao levels of each objecve of he MOGPP (..) may be we as follows: Fd so as o sasfy f () L (,,..., k) (..) ~ Tm a j j mj subjec o g ( ) c j,..., m >. Hee objecve fucos of he poblem (..) ae cosdeed as fuzzy cosas. Ths ype of fuzzy cosas ca be quafed by elcg a coespodg membeshp fuco μ(f ( )) o f f ( ) U u () f L f () U (,,...,k) (..) o f f () L. μ (f ()) Hee u () s a scly moooc deceasg fuco wh espec o f (). Followg fgue llusaes he gaph of he membeshp fuco μ ( ( )) f u () L U f () Fgue-.: Membeshp fuco fo mmzao poblem Havg elced he membeshp fucos ( as Eq. (..)) μ ( ( )) fo,,..., k, μ (f ()) a geeal aggegao f ~ ( D D fuco ) μ ( μ f ( )), μ ( f ( )),..., μ ( f ( ))) ~ μ s oduced. ( So a fuzzy mul-objecve decso makg poblem ca be defed as k Mamze μ ~ () (..4) D k subjec o Tm amj j j g ( ) c j,..., m

5 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 7 >. If we follow he fuzzy decso o fuzzy objecve ad cosa goals of Belma ad Zadeh (97) he usg above sad membeshp fucos μ ( f( )) (,,.,k), he poblem of choosg he mamzg decso o fd he opmal soluo (.e. ). Thee ae wo ypes of fuzzy decso ad hey ae () fuzzy decsso based o mmum opeao (lke Zmmema s appoach (978)). () cove-fuzzy decsso based o addo opeao (lke Tewa e. al. (987)). The he poblem (..4) s educed o he followg poblems () (accodg o ma-m opeao) Mamze α (..5) subjec o μ ( f ( )) α fo,,..., k Tm amj j j g ( ) c j,..., m >, α. ad () (accodg o ma-addo opeao ) m Ma μd( ) Ma λjμj( f j( )) (..6) j U f() subjec o μ ( f( )) U L (,,.,k) Tm a j j j g ( ) c j,..., m μ ( f ( )), > μ ( f ( )), >. The above poblem (..6) educes o Ma V ( ) T j ' a g j m c j j ' j g j g j j λ (..7) subjec o >. Tm a j j j g ( ) c j,..., m

6 8 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos So opmal decso vaable wh opmal objecve value V ( ) ca be j ' obaed by V ( m λ j g j ) U ( ) whee s opmal decso vaable of he ' j g g j ucosaed geomec pogammg poblem (fo gve λ j j,,,k), M U() m j T λ j j a c j ' g j g j j (..8) subjec o >. Tm a j j j g ( ) c j,..., m.. Eample: Mul-Objecve Pmal Geomec Pogammg (MOPGP) Poblem Mmze subjec Z, whee ( X > ) Z o { Z (, Z ( } ( X Y ) ( X ) ad, Y ( X ) + (..) I ode o solve hs MOGP poblem, we shall fs solve he wo sub-poblems (Sub-poblem-) Mmze Z( subjec o Y (, > ad (Sub-poblem-) Mmze Z ( subjec o Y (, > Solvg he above poblems by GP echque we have Fo (Sub-poblem-), Z( 6. 75, Z X Now he pay-off ma s gve below Fo (Sub-poblem-) ( ) 87 (..) (..)

7 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 9 Z Z Fom he pay-off ma he lowe ad uppe boud of Z ( ) ad Z ( ) be 6.75 Z ( 6.94 ad Z ( 6.75 Le μ, μ ( ) be he fuzzy membeshp fuco of he objecve fuco ( X ( ad Z( Z especvely ad hey ae defed as: X X μ ( f Z( Z( μ ( f 6.75 Z( f Z( 6.94 The followg fgue llusaed he gaph of he fuzzy membeshp fuco μ ( Z ( ) ad Z μ (.88 ( f Z ( f Z f ( 6.75 Z ( 6.75 Now he followg fgue llusaed he fuzzy membeshp fuco μ ( ) X

8 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos μ ( Z ( Accodg o ma-addo opeao, he MOGPP (..) educes o he csp poblem. e. e Mamze( μ ( + μ ( ) 6.94 Z( 6.75 Z Mamze Z( Z( Mamze subjec o +, >. ( (..4) [cosdeg equal mpoace o boh objecve fucos.e. λ λ ] Fo mamzg he above poblem, we mmze Z( Z( + subjec o So, ou ew poblem s o solve. e Z( Z ( Mmze g( Mmze g( subjec o +, >. ( Degee of Dffculy of he poblem (..5) s (4-(+)) The dual poblem of he above poblem (..5) s ) (..5)

9 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 5.69 Mamze v( w) w ( w + w subjec ) w + w o w w w w + w, w w w, w w + w, w.699 w + w >. w w w w Solvg he above equao by Newo Raphso mehod we ulmaely ge, w.6745, w.654, w.65, w. w (..6) The value of he objecve fuco of he poblem (..6) s v ( w ) Theefoe, by usg pmal-dual vaables elao-shp, he value of he objecve fuco of he poblem (..5) s g ( X ) ad he values of decsos vaables ae. 6577,. 64. Thus, he values of he objecve fucos of he MOGPP (..) ae Z ( ad Z ( Applcaos: Poblem-: Gavel-Bo poblem 8 cubc-mee of gavel s o be feed acoss a ve o a baage.a bo (wh ope op) s o be bul fo hs pupose. Afe he ee gave has bee feed, he bo s o be dscaded. The aspo cos pe oud p of baage of bo s Rs ad he cos of maeals of sdes ad boom of bo ae Rs /m ad Rs 8/m ad eds of bo Rs /m. Fd he dmeso of he bo ha s o be buldg fo hs pupose ad oal opmal cos. Le us assume he gavel bo has legh m

10 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos wdh hegh m m The aea of he ed of he gavel bo The aea of he sde of he gavel bo m The aea of he boom of he gavel bo The volume of he gavel bo m m m Cos fucos ae: 8m Taspo cos : ( Rs/ p) Rs. 8, m / p Maeal cos: Ed of bo: ( Rs / m ) m Rs. 4 Sdes of bo: ( Rs / m ) m Rs. Boom: ( Rs 8 / m ) m Rs. The oal cos (Rupees) g ( ) I s a posyomal fuco. As saed, hs poblem ca be fomulaed as a ucosaed GP poblem Mmze g( ) (..) subjec o,, > Suppose ha we ow cosde he followg vaa of he above poblem (..) (smla dscusso have doe Duff, Peeso ad Zee(967) he book). I s equed ha he sdes ad boom of he bo should be made fom scap maeal bu oly 4 m of hs scap maeal ae avalable. Ths vaao of he poblem leads us o he followg cosaed posyomal GP poblem: 8 Mmze g ( ) + 4 subjec o g( ) + 4, whee >, >, >. (..) No oly mmzg oal cos ( oal aspoao cos + maeal cos fo wo eds of he bo) of he poblem (..) bu hee s also aohe objecve fuco whch s o mmze he oal umbe of ps.

11 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 8 Hee o. of ps. So, he poblem s o deeme dmesos of he bo, T.e. o fd,, ) so as o sasfy ( 8 Mmze g( ) Mmze g( ) subjec o + 4, whee,, >. (..) I may be we as a Mul-Objecve Geomec Pogammg Poblem (MOGPP) 8 Mmze g( ) Mmze g( ) subjec o g( ) + 4 whee,, >. Hee wo sub-poblems ae, (..4) ad 8 Mmze g ( ) + 4 (Sub-poblem-) subjec o g ( ) +, 4 whee,, >. (..5) 8 Mmze g ( ) (Sub-poblem-) subjec o g ( ) + 4 whee,, >., (..6) The above sub-poblems (..5) & (..6) ae wo GP poblem wh DD -, especvely. Solvg hs MOGPP (..4) by usg fuzzy echques, we have.9,.7 ad.4 ad opmal objecve goals g ( ) ad g ( )..

12 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 4 Poblem-: Mul-Gavel bo poblem Suppose ha o shf gavel a fe umbe (say ) of ope ecagula boes of leghs mees, wdhs mees, ad heghs mees (,,,). The boom, sdes ad he eds of he each bo cos Rs. a, Rs. b, ad Rs. c /m. I coss Rs. fo each oud p of he boes. Assumg ha he boes wll have o salvage value, fd he mmum cos of aspog d ( d ) m of gavels. As saed, hs poblem ca be fomulaed as a ucosaed modfed geomec pogammg poblem ( ) > > > ,,...,,, ) ( whee c b a d Mmze g (..7) Suppose ha we kow he followg vaa of he above poblem. I s equed ha he sdes ad boom of he boes should be made fom scap maeal bu oly w m of hese scap maeals ae avalable. Ths vaao of he poblem leads us o he followg cosaed modfed geomec pogammg poblem: ( ) ( ) > > > + +.,,...,,,, ) ( whee w o subjec c d g Mmze (..8) I pacula, he poblem s o mmze he cos fucos.e.

13 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 5 Mmze g( ) ( g( ), g( ), g( )) subjec o ( + ) w, whee >, >, > (,,) d d g( ) + c, g( ) + c d + c, g ( ) (..9) I may be we as a Mul-Objecve Geomec Pogammg Poblem (MOGPP) d Mmze g( ) + c d Mmze g( ) + c d Mmze g( ) + c subjec o ( + ) w,, >, (,,). (..) I pacula hee we assume aspog d m of gavels by he hee dffee ope ecagula boes. The fal cos of each bo s Rs. c /m ad he amou of he aspog gavels by hee ope ecagula boes ae d d m. Ipu daa of hs MOGPP (..) s gve he able-. I s a cosaed posyomal MOGP poblem. Solvg hs MOGPP (..) by he above specfed fuzzy echque we ge opmal soluos as show able-.

14 6 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos Table- Ipu daa fo he MOGPP (..) Boes () c (Rs. /m ) d (m ) w (m ) Boe s () (mee) Table- Opmal soluos of he MOGPP (..) g ( ) (mee) (mee) (Rs.) g ( ) (Rs.) g ( ) (Rs.) Cocluso Hee, we have dscussed mul objecve geomec pogammg poblem based o fuzzy pogammg echque hough geomec pogammg. We have also fomulaed he mul objecve opmzao model of gavel bo desg poblem ad solved by fuzzy pogammg echque. Geomec Pogammg echque s used o deve he opmal soluos fo dffee pefeeces o objecve fucos. The mulobjecve veoy models may also be solved by fuzzy geomec pogammg echque. REFERENCES [] Bellma, R.E., ad Zadeh, L.A., Decso-makg a fuzzy evome, Maageme Sceces, 7 (4) (97) B4-B64. [] Beghle, C.S., ad Phlps, D.T., Appled Geomec Pogammg, Wley, New Yok, 976. [] Bswal, M.P., Fuzzy pogammg echque o solve mul-objecve geomec pogammg poblems, Fuzzy Ses ad Sysems, 5(99) [4] Duff, R.J., Peeso, E.L., ad Zee, C.M., Geomec Pogammg Theoy ad Applcaos, Wley, New Yok, 967. [5] Ecke, J., Geomec pogammg: mehods, compuaos ad applcaos, SIAM Rev., () (98) 8-6. [6] Islam, S., ad Roy, T.K., Modfed geomec pogammg poblem ad s applcaos, Joual of Appled Mahemacs ad Compug, Koea, 7(-) (5) -44 [7] Taaka, H., Okuda, T., ad Asa, K., O fuzzy mahemacal pogammg, Joual of Cybeecs, (4) (974) [8] Twa, R.N., Dhama, S., ad Rao, J.R., Fuzzy goal pogammg a addve model, Fuzzy Ses ad Sysems, 4 (987) 7-4. [9] Vema, R.K., Fuzzy Geomec Pogammg wh seveal objecve fucos, Fuzzy Ses, ad Sysems, 5 (99) 5. [] Zadeh, L.A., Fuzzy Ses, Ifomao ad Cool, 8 (965) 8-5. [] Zee, C., Egeeg Desg by Geomec Pogammg, Wley, 97.

15 S., Islam / Mul-Objecve Geomec Pogammg Poblem ad s Applcaos 7 [] Zmmema, H.J., Fuzzy lea pogammg wh seveal objecve fucos, Fuzzy Ses ad Sysems, (978)