An Algorithm for Capacitated n-index Transportation Problem

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1 Iteratoal Joural of Coutatoal Scece a Matheatcs ISSN Volue 3, Nuber 3 2), Iteratoal Research Publcato House htt://wwwrhouseco A Algorth for Caactate -Ie Trasortato Proble SC Shara a 2 Abha Basal Deartet of Matheatcs, Uversty of Raastha, Jaur-3255, Ia E-al: sureshcha26@galco 2 77, Taro Kute, Surya Nagar, To Roa, Jaur-328, Ia E-al: eelachusghal22@galco Abstract I ths aer a algorth was trouce for caactate trasortato roble wth -subscrts Itroucto May worl faous researchers le Detzg, Htchcoc, Katorovch a Savoure etal have roose the theoretcal a algorth bases of the classcal twoe trasortato roble Most of the eveloets are base o lear rograg techques Heley [3] a Jugger [4] trouce the ult-e trasortato roble wth out caactes I fact, the caactes of the trasortato aths are atheatcally oele as atoal costrats to eress very ortat real ees Obvously, ths volves soe theoretcal a algorthcally colatos whch are ofte ffcult to treat geeral cotet Ths ustfy art, as ost absece of sgfcat stues relate to caactate trasortato roble wth a e greater tha two I ths aer, we focus our atteto to the caactate trasortato roble wth -e Stateet of the roble The caactate -e trasortato roble CTP) s forulate as follows: Z c

2 27 SC Shara a Abha Basal st α for all,2,3,, β for all,2,3,, χ for all,2,3,, w σ for all,2,3,, I ths roble α, β, χ,, σ, a c are gve a are such that for all,,l, we have α >, β >, χ >,, σ >, > a c > Ths roble ca be equvaletly forulate as the lear rogra [ c t : A b, ] Where, c, R, b R a A s a ++ +) ) atr A feasble soluto of CTP) s calle a rogra A rogra s calle basc f the colus of the sub atr A obtae fro A by eeg oly the colus corresog to the varables such that < < are learly eeet A basc rogra s sa to be egeerate f ra A ) ra A ) Gve a basc rogra, the -tule,, ) s calle terestg f < < We assue that the followg feasblty assuto hols α β χ σ H

3 A Algorth for Caactate -Ie Trasortato Proble 27 It results that Ra A) ) -) It s useful to reset the ata of the roble va the followg trasortato table It cossts of a array of + + +) rows a ) colus, - atoal rows a a atoal colus The etres of colu P of the frst, seco,,- atoal rows are for the ata of the quattes, c,, resectvely The atoal colus are for the ata of quattes α, β, χ,, σ resectvely Fally the etry of the array o the le corresog to α a the colu P s f a f ot Sae as for β, χ,, σ Algorth The followg algorth shares wth the sle etho a the otetal ethos a structure cosstg two hases, a fte covergece a the use of the vot rcle Phase : It fs a basc rogra or says that CTP) s ot solvable) Ste : Italzato: For all,,,), ˆ α α, ˆ β β, ˆ χ χ,, ˆ σ σ a b u b s a boolea varable equal to f has alreay bee etere a f ot yet), E {,,,), such that b } Iterato: Whle E φ o Choose a -tule,,, ) E, such that c c tae,, ) E ˆ α, ˆ β, ˆ χ, ˆ σ, ), a b e, s etere), uate ˆ α, ˆ β, ˆ χ,, ˆ σ as follows ˆ α α, f ˆ α the tae for all,, ),, ) a b for all,,, ) ˆ β β, f βˆ the tae for all,,, ),,, ) a b for all,,,-) ˆ χ χ, If χˆ the tae for all, l,, ), l,, ) a b for all,l,,-) a slarly for, ˆ σ σ,

4 272 SC Shara a Abha Basal If ˆ σ the tae for all, ),,, ) a b for all,,,-) Ste 2: Tae a b c u, such that a for a α wth,2,, b β wth,2,, c χ wth,2,, w u σ wth,2,, If, the ) s a tal basc rogra for the roble CTP), we eote t by Go to Phase 2 Costruct a roble CTPM ) by the roceure escrbe P ) below, a f a tal basc rogra for the roble CTP M ),as ste The, +, +, +,, + ), wth,2,,+,,2,,+,,2,,+,,,2,,+) If s otal the the roble CTP) s ot solvable Sto Iroveet of a basc rogra for CTP M ) Italzato: r, > s gve, Detere as hase 2 If +, +,, +, the ), wth,2,,,,2,,,,2,,,,,2,,, s a tal basc rogra for the roble CTP) Go to Phase 2 If s otal Phase 2), the the roble CTP) s ot solvable Sto Do r r+ a reeat ), to 3) Net, we escrbe the seco hase

5 A Algorth for Caactate -Ie Trasortato Proble 273 Phase 2: Research of a otal rogra for CTP) Whe Phase 2 starts, we ow a tal basc rogra a Detere the set I of the terestg -tule,,,) For all,,, ) I, solve the lear syste u v + w + + c + For all,,,),,, ) I tae Δ c u + v + + Γ Δ Such that } { r { r Γ Δ Such that u ) } Tae r a If the followg otalty cotos hols For all Δ Δ r For all Δ Δ r Γ Γ The the rogra Detere Δ,,,,,, [ Δ s otal Sto Δ Δ Γ, wth,, u Δ r <, Δ Γ, wth Δ r > ] Such that a secfy f Δ,,, Γ or Γ ), Costruct va the roceure escrbe P 2 ) below, a cycle μ cotag soe terestg -tule,,,) a the o terestg -tule,,,, ) corresog to Δ Tae,,,,, σ {,,,) such that,,,) s a -tule forcog the cycle σ {,,,) such that,,,) σ, wthα }, < μ },

6 274 SC Shara a Abha Basal + σ {,,,) such that,,,) σ, wth > α }, If Δ Γ, etere,,,, / α r,,,, ) 2 / α r +,,, ), ), Net, tae else Δ { 2 ), ), + α,,,, ) r+ ),,,, Γ ), etere +,,,, ) / α ), / α 2,,, ) et, tae, 2 { ) α,,,, ) r+ ) ), } U{ } U{,,,, ) σ },,,, ) σ } Do r r+ a reeat a), to e) utl the otalty coto hols The above algorth aes aeal to the followg roceures: P )- Costructo of a roble The roble CTP M ) s obtae fro roble CTP) by ag four fcttous ots wth ces +, +, +,, + such that: c, c c c M +, +, +,, + +,, +, l where M s a very large ube a there are o ltato o the caactes for the aths volvg a fcttous ot P 2 )- Deterato of cycles A cycle μ s etere by the solvg the lear syste

7 A Algorth for Caactate -Ie Trasortato Proble 275 α P P,,, ) I The o ull solutos α are calle coeffcets of the cycle μ Refereces [] Bazaraa, M S, Javs, J Ja Sheral, H D, 99, "Lear rograg a etwor flows", Joh Wley & Sos [2] Bulut, SA, 998, "Costructo a algebrc characterzatos of the laar a aal trasortato roble", JMathAal, 22, [3] Haley, KB, 963, "The ult-e trasortato roble", Oer Res,, [4] Jugger, W, 993, "O reresetatves of ult-e trasortato robles", Euroea JOer Res, 66, [5] Kravtsov, MK, 2, "A couter eale to the Hyothess of Mau uber of teger vertces of the ult-e aal trasortato olyhero", Dsret Mat, 2 ), 7-2 [6] Kravtsov, M Ka Lush, E V, 24, "O the oteger olyhero vertces of the three-e aal trasortato roble", Auto Reote Cotrol, 65 3), [7] Lu, S T, 23, "The total cost bous of the trasortato roble wth varyg ea a suly", Oega, 3, [8] Queyrae, M a Sesa, F C R, 997, "Aroato algorths for ult-e trasortato robles wth ecoosable costs", Dscrete Al Math, 76, [9] Shara, R R Ka Prasa, S, 23, "Obtag a goo ral soluto to the ucaactate trasortato roble", Euroea JOerRes, 44, [] Shara, R R K a Shara, K D, 2, A ew ual base roceure for the trasortato roble, Euroea JOerRes, 22, [] Stacus-Masa, IM, 974, "A three-esoal trasotato roble wth a secal structure obectve fucto", BullMath Soc Sc Math Rouae NS), 866), 3-4,