SOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM

Size: px

Start display at page:

Download "SOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM"

Jocelyn Lambert
5 years ago
Views:

Qattate Methods Iqres SOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM Dora MOANTA PhD Deartet of Matheatcs Uersty of Ecoocs Bcharest Roaa Ma blshed boos: Three desoal trasort

1 Qattate Methods Iqres SOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM Dora MOANTA PhD Deartet of Matheatcs Uersty of Ecoocs Bcharest Roaa Ma blshed boos: Three desoal trasort robles Debrdge Press Ne Yor 006; Teore s ractca roblee de trasort Cartea Uerstara Bcharest 006 E-al: dora.oata@yahoo.co Abstract: Ths aer resets the three-desoal trasortato roble a doble s odel hch the obecte fcto s the rato of to oste lear fctos. Ths aer obecte s to reset ho to obta ot th sle ethod. To llstrate the rocedre a ercal eale s ge. Key ords: the three desoal trasortato roble; rograg th fractoal lear obecte fcto; sle ethod Proble Descrto I a roosg o to sole the -desoal trasort roble a doble s odel - th the fractoal lear obecte fcto ad lear costrats: f ( ) a b 0 c a b c > 0 a b c T (7) () () () (4) (5) (6) 4

2 Qattate Methods Iqres Reqrg the follog secfcatos: the ber of sorces the ber of destatos the ber of eas of coeyace a the aalable qatty each sorce b the ecessary qatty each destato c the qatty th st be trasorted by eas of coeyace 0 Matr X { \ ; ; } hch satsfes costrats () () (4) (5) s called a trasortato la (feasble solto) ad la X s called ot f t satsfes (). Whe the codto (7) s satsfed the resltg forlato s called a balaced trasortato roble. Relato (7) s the ecessary ad sffcet codto for the estece of the solto: the leel of the atr of the costrat syste s shog that a o-degeerated trasortato la of roble (-7) cotas at least - o-ll cooets; The obecte s to establsh a trasortato la th total eeses. The fcto () s elct qas cocae S { X / () () (4) (5) }.e.: If S f( ) f( ) λ (0) ad 0 λ (- λ) the [f( ) f( )] < f( 0 ). For sch fcto local s ot ecessarly a global. Eery dfferetable [] elct qas cocae fcto s sedo cocae as ell. A otalty crtero for local s ge [] I ths aer s ade to geeralze the reslts ge by [][6]. Ths aer obecte s to reset ho to obta ot th the hel of the sle ethod. > 0 Solg the roble The cosderatos cocerg the three-desoal roble are ald. A tal feasble solto ca be obtaed by sg the o ethods fro the three-desoal trasort roble [4][5]. We deote I {() / > 0 X} De to (7) each odegeerate solto ll cota oste cooets. We cosder the dal arables (sle ltlers): defed sch that: for ( ) I ad ( ) ( ) (8) (9) (0) 44

3 Qattate Methods Iqres ( ) ( ) () Syste (8) (9) ca be soled deedetly. So syste (8) (9) has - eqatos th arables. We ca arbtrarly set 0 0 ad sole for the other ltlers. Hag detered e shall se these ales to detere oly. ad for the o-basc arables. Let X* (* ) be a feasble solto of the roble () (7). To establsh the otal crtero e eress f() ters of the o-basc arables a b c ( ) a b c V By eas of a slar rocedre e ca also rte : V a s b s c s here Vs s () Therefore the fcto f() becoes: f ( ) I I ( ) For f I I e hae V I I V I I V V V The artal derates of the fcto f() ealated at the ot f * V V ( V ) * are: 45

4 Qattate Methods Iqres Δ V V We ote The dal arables detered t old be easy to calclate The solto X* ca be roed f t ests at least a ale Δ Theore A solto X* (* ) s a local ot f arables. If oe of ths ales s ot oste e choose { Δ Δ 0} Δ < Δ < 0 for o-basc arables. Δ 0 for all o-basc ad e roe the ale of f() by trodcg 0 0 the set of basc arables. The arable hch leaes the bass ad the ale of the basc arable the bass ca be detered as sal. Eale Cosder the roble th ad a 4 b 40 c 6 a 8 b 9 c a 7 c The atrces of costs: / 5/8 9/ 9/0 5/4 4/0 4/0 8/5 4/0 0/5 /5 /0 /0 7/ /5 7/0 4/0 8/5 /6 A tal feasble solto obtaed as [4][5] s X 0 : for hch the ale of obecte fcto s f Otalty erfcato : e detere the qattes fro systes:

5 Qattate Methods Iqres We obta The atrces: / 0/0 -/-6-7/6-6/- -/- /0 9/6 0/0 0/0-4/- 0/0 7/9 /- 0/0 0/0-8/- 0/0 0/0 For hch V 88 V 549 V 5V Δ V V V ( 5 ) V Δ Ad atr Δ 0 ( for basc cooets). Matr Δ : The solto s ot ot becase there are cooets We roed solto: Δ 6 { Δ Δ < 0} It crtero : Ott crtero: as [4][5]: for basc cooets : z z z 0 z z z Δ < 0. θ ( ) (68) 8 θ 8 47

6 Qattate Methods Iqres Solto actalzato for basc cooets: θ z 0 8 The e solto X s: for hch the ale of obecte fcto s Rese fro otalty erfcato f Bblograhy. Aggaral S. P. Idefte qadratc fractoal rograg Cahers d Cetre d Etdes de Recherche Oeratoelle 97. Corba A. No Lear three-desoal rograg Re. Ro. Math. Pres et Al Datzg G. B. Wolfe F. The Decoosto Algorth for Lear Progras Ecooetrca ol. 9 o Magasara O. L. Psedo Coe Fctos J. Sa Cotrol r Moata D. Three desoal trasort robles Debrdge Press Ne Yor Moata D. Teore s ractca roblee de trasort Cartea Uerstara Bcharest Stac Masa I. M. A Three-Desoal Trasortato Proble th a Secal Strctred Obecte Fcto Bllet Math. Toe 8 o Dora Moata has receed a B.S. fro Faclty of Matheatcs Uersty of Bcharest. Also she has receed a degree Iforatcs ad has a Ph.D. Ecoocs - Ecooc Cyberetcs feld. She s the athor of ore tha 6 artcles Oeratos Research Ecooc Calcls ad Iforatcs ad atteded at eros secalty cogresses. Recetly blshed boos:. Moata D: Three desoal trasort robles Debrdge Press Ne Yor 006. Moata D: Teore s ractca roblee de trasort Cartea Uerstara Bcharest

DUALITY FOR MINIMUM MATRIX NORM PROBLEMS

DUALITY FOR MINIMUM MATRIX NORM PROBLEMS HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMNIN CDEMY, Seres, OF HE ROMNIN CDEMY Vole 6, Nber /2005,. 000-000 DULIY FOR MINIMUM MRI NORM PROBLEMS Vasle PRED, Crstca FULG Uverst of Bcharest, Faclt of Matheatcs