λ(p) D(t) = where we assume that = 1 where is the demand on

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1 Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS IJAERS/Vol. II/ Issu III/Aril-Jun, /6- Rsarch Par A EOQ ODEL WIH PROGRESSIVE PAYE SCHEE UDER DCF APPROACH WIH PRICE AD CREDI SESIIVE DEAD Dr Sudita Sinha Addrss for Corrsondnc Burdwan Raj Collg, Univrsity of Burdwan, Wst Bngal ABSRAC In this ar, an EOQ modl is dvlod in which sulir offrs th rogrssiv trad crdit to th rtailr. A rogrssiv trad crdit is dfind as follows: If th rtailr ays th outstanding amount by, th sulir dos not charg any intrst. If th rtailr ays aftr but bfor (>), th rtailr will hav to ay intrst chargs at th rat Ic.If th account is sttld aftr, th rtailr will b chargd at th rat Ic (Ic >Ic ).In this ar th dmand of an itm dnds on th crdit riod as wll as on th ric offrd by th rtailr. Hr th rtailr s sals ar dividd in two catgoris: On cash(which is ric snsitiv) and On crdit(which is a function of customr s crdit riod and ric) h modl is dvlod undr Discountd-Cash-Flow (DCF) aroach. h rsnt valu of all futur cash-out-flow is drivd for all th thr ossibl scnarios. At th nd, a numrical xaml is givn to illustrat th rsults obtaind and snsitivity analysis of various aramtrs on th otimal solution is carrid out. KEYWORDS Invntory, Crdit-lind dmand, Discountd cash flow(dcf)aroach, wo-stag crdit olicy. IRODUCIO h trad crdit riod offrd by th sulirs to th rtailrs ncourags rtailrs to buy mor and it is a owrful romotional tool that attracts nw customrs. In th ast a lot of wor has bn don for studying th invntory systm bhavior undr various trad-crdit olicis offrd by th rtailr or vndors Haly and Higgins [8] dvlod an invntory modl to dtrmin EOQ undr conditions of rmissibl dlay in aymnts Goyal[7] rsntd th similar modl with no nalty cost du to lat aymnt Chang [5] thn dvlod an altrnativ aroach to th roblm. Chand and ward [4] analysd Goyal s [7] roblm undr th assumtion of th classical conomic ordr quantity modl obtaining diffrnt rsults. Jamal t al [] & Sarar t al [] xtndd th Goyal s [7] modl by considring th diffrnc btwn unit ric and unit cost. ng[7] suggstd that th buyr should ordr in similar quantity and ta advantags of trad crdit frquntly Chang t al [5] xtndd ng s[7] wor whn units in invntory ar subjct to a constant rat of dtrioration. Arclus t al [] comard scnarios of trad crdit and discount for non-dtriorating itms. Shah[4 ] and Agarwal t al [] xtndd Goyal s[7]modl to th cas of dtrioration. All th aformntiond invntory modls assumd that th customr would ay for th itms as soon as th itms ar rcivd from th rtailr. But in most of th businss transactions, th sulir offrs a crdit riod to th rtailr and th rtailr, in turn asss on som crdit riod to his customrs in ordr to stimulat his dmand. Such a situation whr both sulir as wll as th rtailr offrs crdit riod to thir rsctiv customrs is nown as two-stag crdit olicy. Rcntly Huang [9] rsntd an invntory modl assuming that th rtailr also offrs a crdit riod to his customr which is shortr than th crdit riod offrd by th sulir. An EOQ modl undr two stag crdit olicy whn th nd dmand is ric as wll as crdit riod snsitiv is dvlod by Jaggi t al[].but rogrssiv aymnt schm undr du aroach is not considrd in thir ar. Discountd cash flow (DCF) aroach is widly usd in businss dcisions to rflct th tim valu of mony which is nglctd in classical EOQ modl. It also rmits an xlicit rcognition of th xact timing of ach cash flow associatd with th invntory systm and considrs th tim valu of mony as wll. Chung [6] rsntd th discountd cash flow (DCF) aroach for th analysis of th otimal invntory olicy in th rsnc of th trad crdit. rii and Lwin[9], Bnsoussan t al [] gav th DCF/PV (t rsnt valu) aroach for th analysis of th basic EOQ modl Kim t al [] usd DCF aroach to various invntory systms, Sun & Guyram [6] stablishd that th PV of th total cost in th two aroachs, namly avrag cost and DCF ar vry clos (9.6% diffrnc btwn thir rcordr intrvals. In this ar dmand dnds not only on th ric but also on th imact of crdit riod which is not considrd in th abov mntiond ars. Hr an EOQ modl is dvlod undr th DCF aroach jointly otimiz th rtailr s slling ric and rlnishmnt cycl undr two-stag crdit olicy th dmand dnds on ric as wll as crdit riod at th nd, a numrical xaml is givn to illustrat th rsults obtaind and snsitivity analysis of various aramtrs on th otimal solutions is carrid out.. ASSUPIOS h following assumtions ar mad to dvlo th mathmatical modl (i) h invntory systm dals with singl itm only. (ii) Rlnishmnt rat is infinit and instantanous. (iii) Lad tim is ngligibl. (iv) h dmand is assumd to b th function of slling ric and th lngth of crdit riod offrd by th rtailr. his function has bn dividd into two arts viz (a) rgular cash-dmand which is th function of slling ric through out th cycl. (b) crdit riod offrd by th rtailr i.. during t.hnc dmand function at any tim t can b rrsntd as λ() + R(, ; t D( λ(); t λ(p) > and > whil R(, whr w assum that - whr is th dmand on

2 Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS crdit during customr s crdit riod which is dirctly roortional to th crdit riod and invrsly roortional to ric. W assum that R( ( whr t,> an d >. (v) h sulir rovids a fixd crdit riod to sttl th account to th rtailr and rtailr, in turn, asss on a maximum crdit riod to his customrs to sttl th account. For simlicity, it is assumd that th customr s crdit riod is lss than or qual to th rtailr s crdit riod.it is also assumd that th customr would sttl his account only on th last day of th crdit riod ( ). otations:- h following notations ar usd in this ar lngth of rlnishmnt cycl(dcision variabl) q(h invntory lvl at tim t. Qordr quantity A ordring cost r ordr c unit urchas cost of th itm unit slling ric of th itm h holding cost r unit invntory charg I intrst arnd r $ r yar. rtailr s crdit riod offrd by th sulir for stting th accounts. customr s maximum crdit riod offrd by th rtailr,. whr rdiscountd rat(oortunity cos r unit tim. Ic intrst chargd r Rs in stoc r yar by th sulir whn th rtailr ays aftr but bfor. Ic intrst chargd r Rs in stoc r yar by th sulir whn th rtailr ays aftr (Ic >Ic ). (,) Rtailr's annual rofit for subcas(i) of Cas I which is a function of and. (,) Rtailr's annual rofit for subcas(ii) of Cas I which is a function of and. (,) Rtailr's annual rofit for Cas II which is a function of and. PV (,)rsnt valu of all futur cash out flows for subcas(i) of Cas I. PV (,)rsnt valu of all futur cash out flows for subcas(ii) of Cas I. PV (,)rsnt valu of all futur cash out flows for Cas II.. AHEAICAL FORULAIO Lt q( b th invntory lvl at tim t.a batch of Q units ntrs th invntory systm at th bginning of th cycl. As th tim incrass, th invntory lvl dcrass raidly du to cash as wll as crdit dmand i.. D( + ( D( u to th riod.hraftr,it dclins only du to cash dmand i.. till th nd of th riod. h diffrntial quations govrning instantanous stat of q( at som instant of tim t ar givn by dq [ + ( ] dt ( t ) () satisfying th condition of q () Q (a) dq dt ( t ) () whr q ( ) q() (a) Solving () w gt, ( q( Q t+ () Solving () w gt, q ( ( (4) h ordr quantity can b calculatd as follows Q D( dt + (5) From () w gt, q ( ( - + ( - (6) hrfor, th invntory lvl at tim t during th cycl is ( q ( ( + q( q ( ( ; ( t ) ; ( t ) (7) IJAERS/Vol. II/ Issu III/Aril-Jun, /6-

3 () Sals Rvnu () Cost of lacing ordrs () Cost of urchasing units Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS Q (8) A (9) cq c [ + ] () h [ q(dt q(dt + (4) Cost of carrying invntory ] Ic [ + ] () CASE I Subcas (i) (< ) In this cas, th rtailr dosits th accumulatd rvnu from cash sals during th riod(,) and from crdit sals during th tim riod (,) into an account that arns on intrst rat I.So in th riod [,] th total rvnu gnratd du to cash sals dt and from crdit riod sals during th tim riod [,] is dt.at, th accounts hav to b sttld, it is assumd that accounts will b sttld by rocds of sals gnratd u to and by taing a short loan at th intrst rat of Ic and Ic for th duration of (-) and (-) rsctivly for financing th unsold stoc. Consquntly, th intrst arnd r yar h intrst ayabl r yar Subcas (ii) (< ) Hr, th intrst arnd,ie, during [,] is IE I tdt+ dt I { + ) } Ic c Icc [ q(dt] + [ q(dt] Ic.c ( ) ( ) (Ic Ic )c + I [ I Buyr has to ay intrst r yar IP tdt+ c Ic q (dt dt] { + )} ( ) Hr, th rtailr has sufficint amount in his account to ay-off th total urchas cost at. hrfor, th intrst chargs,ic. CASE II ( ) Hr,th crdit riod is mor or qual to th cycl,so th rtailr arns intrst on cash sals during th riod [,] and also on crdit sals during th riod [,] but thr is no intrst ayabl. h intrst arnd r yar I IE [ tdt+ + Qdt] ] () () (4) (5) IJAERS/Vol. II/ Issu III/Aril-Jun, /6-

4 Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS I { ) + )} Using th quations (8) to (), () and () th rtailr s annual rofit for Subcas I (,) can b xrssd as (,) Sals rvnu-cost of urchasing units-cost of lacing ordrs-cost of carrying invntory+intrst arnd r yar-intrst ayabl r yar. ( c)( + (, ) h + Ic.c ( ) h rsnt valu of all futur cash-out flows is givn by PV (,) n nr (, ) { + )} (Ic Ic ) c A ( ) (, ) r r ( + + )(, ) (8) r 4 whr ( c)( { } + + ) A (9) (, ) h + Ic.c ( ) (Ic Ic)c ( ) Using th quations (8) to (), (4) and (5) th rtailr s annual rofit for Subcas (ii) (,) can b xrssd as (,) Sals rvnu-cost of urchasing units-cost of lacing ordrs-cost of carrying invntory+intrst arnd r yar-intrst ayabl r yar ( c)( { } + + ) A (, ) h + Ic.c ( ) () h rsnt valu of all futur cash-out flows is givn by PV (,) whr (, ) n nr (, ) (, ) r r ( + + )(, ) () r 4 ( c)( h + + Ic.c { + )} ( ) A Using th quations (8) to ()and (6) th rtailr s annual rofit for Cas II (,) can b xrssd as (,) Sals rvnu-cost of urchasing units-cost of lacing ordrs-cost of carrying invntory+intrst arnd r yar-intrst ayabl r yar. (6) (7) () IJAERS/Vol. II/ Issu III/Aril-Jun, /6-

5 Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS ( c)( + (, ) Ic.C ( ). h rsnt valu of all futur cash-out flows is givn by PV (,) n (,) nr { ( - ) + )} A (, ) r r ( + + )(, ) (4) r 4 whr ( c)( + + { + } ) I ( - ) ( ) A (, ) Ic.c ( ) (5). ow,for subcas(i) of Cas I δpv δ { ( )} + ( + )( ) r ( + + ) r 4 (+ ) c(+ ) + h( + ) + + Icc ( ) + (Ic Ic)c( ) ) r (+ ) ( + + ) [ L L] r 4 + ( + ( )( ) + (6) Whr { ( )} L L h( + cc( ) + (Ic Ic)c( ) c(+ ) (8) δ PV r (+ ) (+ )L - ( + + ) δ r 4 L+ < (9) hrfor, PV (,) is maximum w.r.t.. For Subcas (ii) (7) () δpv δ r ( + + ) r 4 ( + + I + I ) I I ) + ) h c + c + h + Ic c ( ) () L whr / L / (+ ) / + L () + + I I I ) + I ) () IJAERS/Vol. II/ Issu III/Aril-Jun, /6-

6 Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS / L c+ c + h h + Ic c (+ ) ( ) () δ PV (+ ) / / { L } + (+ )L δ hrfor, PV (,) is maximum w.r.t.. For Cas (II) w hav IJAERS/Vol. II/ Issu III/Aril-Jun, /6- < (4) (- ) + (- ( - )(- ) + (- )I ) ciic( - ) I δpv r (+ ) ( + + ) + { + } δ r 4 whr L & // // r ( + + ) r 4 [ L // (+ ) // + c L ] (5) ( ) + ( )( ) + ( ) cic( ) I L { } + (7) // δ PV (+ ) // c(+ )L L+ δ hrfor,pv (,) is maximum w.r.t.. Dtrmination of Otimal tim Substituting th valu of in i (,);i,, w hav th roblm of maximizing PV i ();i,,which now bcom a function of singl variabl. o dtrmin otimal tim, w hav to solv th following mathmatical rogramming roblms for two ossibl cass viz and. Problm(P) (for Subcas (i) of Cas I) ax PV () Problm(P) (for Subcas (ii) of Cas I) ax PV () Problm(P) (for Cas II) ax PV () h otimal valus of can b calculatd using athmatica. ) (6) < (8) 4. UERICAL AALYSIS h aramtric valus ar aroximatly as A, 4,,, 8,,, c, I.,.5,Ic.8, Ic. 7, h.8 It is obsrvd from abl that for any fixd, as incrass thr is dcras in cycl tim along with th marginal dcras in unit slling ric and also in th total rofit. As incrass for any fixd thr is marginal dcras in unit ric but cycl lngth, total rofit incrass imlying that it would b conomical for th rtailr to ot mor crdit riod () and rduc his slling ric. From abl- it is obsrvd that as th discounting rat(r) incrass, otimum cycl tim, unit slling ric, PVi(,)dcrass i.. th dvlod modl is mor snsitiv to th changs in discounting rat. 5. COCLUSIO In this ar, a mathmatical modl is dvlod whn th sulir offrs rogrssiv trad crdit to

7 Sudita Sinha., Intrnational Journal of Advancd Enginring Rsarch and Studis E-ISS th rtailr. h modl is dvlod undr DCF aroach. wo-stag crdit olicy with th crdit lind dmand is considrd hr. A numrical xaml is rsntd to illustrat th thortical rsult which suggsts that rtailr s should ordr mor and charg fwr ric as th rtailr s crdit riod incrass. REFERECES. Aggarwal,SP,Jaggi CK,(995), Ordring olicis of dtriorating itms undr rmissibl dlay in aymnts,journal.of th Orational Rsarch socity;46; Arclus,F,Saha,H,Srinivasan, () Rtails rsons to scial sals ric discount vs trad crdit,oega,9(5), Bnsoussan,A,Crouhy, and Proth, J[98], athmatical thory of roduction and invntory lanning, orth Holland, Amstrdam,98th thrlands. 4. Chand,S and Ward J. (987), A not on conomic ordr quantity undr conditions of rmissibl dlay in aymnts, Journal of Orational Rsarch Socity,8, Chang HJ; Hung CH and Dy,CY.(4), An Invntory modl with stoc dndnt dmand and tim-valu of mony whn crdit riod is rovidd, Journal of Information and Otimization Scincs, 5(), Chung,K.J(), h invntory rlnishmnt olicy for dtriorating itms undr rmissibl dlay in aymnts Or. Rsarch,7, Goyal SK.( 985 ), Economic ordr quantity undr condition of rmissibl dlay in aymnts jjournal of th orational rsarch socity,6, Haly,CW and Higgins,RC. (97), Invntory olicy and trad crdit financing gnt Scincs,, Huang,YF.(), Otimal Rtaitr s ordring olics in th EOQ odl undr rad crdit Financing Journal of Orational Rsarch socity 54,-5.. Jaggi,CK,Kausar,A&Khanna,A,(7) Joint otimization of rtailr s unit slling ric and cycl lngth undr wo-stag crdit olicy whn th End Dmand is ric as wll as crdit riod snsitiv, Osarch, Vol 44,o, Jamal,A, Sarar, BR and Wang,S,( 997), Otimal aymnt tim undr rmissibl dlay in aymnt, Journal of th Orational Rsarch Socity,48; Kim,YH,Philiratos,GC and Chung,KH.(986), Evaluating invstmnts in invntory systms: A nt rsnt valu framwor, Enginring Economics,,9-6.. Sarar,BR,Jamal,A and Wang,S. () Otimal aymnt tim undr rmissibl dlay for roduction with dtrioration Production Planning & Control,, Shah,H, (99) A lot-siz modl for xonntially dlaying invntory whn dlay in aymnt is rmissibl, Cahirs D Etuds D Rchrch Orationnld Oration rsarch, Statistics and Alid athmatics,95, Soni H,Shah,BJ& Shah H.ita, (6) An EOQ modl for dtriorating itms for rogrssiv aymnt schm undr DCF Aroach Osarch, Vol 4,o, Sum D and Quyrann,, (), Production and invntory modl using nt rsnt valu Orations Rsarch,5(), ng, J.(), On Economic ordr quantity undr condition of rmissibl dlay in aymnts, Journal of Orational Rsarch,5, hangam,a and Uthayumar,R.() An invntory modl for dtriorating itms with inflation inducd dmand and xonntial artial bacordrs-a discountd cash flow aroach, Intrnational Journal of anagmnt Scincs and Enginring, Vol 5(), rii, RR and Lwin DE, (974), A rsnt valu formulation of th classical EOQ roblm, Dcision Scincs,5,-5 IJAERS/Vol. II/ Issu III/Aril-Jun, /6-