Optimal Ordering Policy in a Two-Level Supply Chain with Budget Constraint

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1 Optmal Ordrng Polcy n a Two-Lvl Supply Chan wth Budgt Constrant Rasoul aj Alrza aj Babak aj ABSTRACT Ths papr consdrs a two- lvl supply chan whch consst of a vndor and svral rtalrs. Unsatsfd dmands n rtalrs ar lost. W assum that rtalrs fac Posson dmand. Th vndor polcy s that all rtalrs hav a common srvc lvl, that s, th prcntags of lost dmands at all rtalrs ar th sam. Furthr, thr s a budgt lmt on th xpctd total nvntory nvstmnt of th supply chan. t s assumd that th fxd ordrng cost for all stock ponts s nglgbl and ach rtalr appls a nw ordrng polcy calld on-for-on prod ordrng (, T polcy. Applcaton of ths polcy amounts to a dtrmnstc dmand for th vndor and ths n turn lmnats th safty stock and ts assocatd costs for hm. Thrfor, h facs a dtrmnstc dmand and adopts a dtrmnstc on-for-on polcy n rspons to dmand of ach rtalr. For ths systm w formulat th total cost of th systm pr unt tm and obtan th optmal ordr cycl for ach rtalr whch mnmzs ths total cost and satsfs th budgt constrant. Kywords: Opraton Rsarch, On-for-on prod Polcy, nvntory Control, Budgt Constrant.. NTRODUCTON n ths papr w ar concrnd wth th managmnt of a two lvl supply chan consstng of a vndor and N rtalrs. Th rtalrs rplnsh thr nvntory from th vndor whch n turn ordrs from an outsd warhous wth unlmtd stock. Th fxd ordrng cost of all stock ponts (vndor and rtalrs ar assumd to b zro or nglgbl. Th rtalrs fac Posson dmand and unsatsfd dmands n th rtalrs ar lost. Andrsson and Mlchors [] also studd th sam two chlon nvntory systm. Thy assumd All nstallatons us (S-, S polcy and proposd an approxmat mthod to valuat nvntory costs. n ths papr w assum that ach rtalr uss a nw ordrng polcy calld on-for-on prod ordrng (, T polcy [8] n whch an ordr of sz on s placd at vry fxd cycl tm T. Th Mult-chlon nvntory systm has attractd th attnton of svral rsarchrs. Axsätr [2] consdrs a two-chlon nvntory systm n whch unsatsfd dmands ar backordrd and ach chlon appls an (S-, S polcy. Thn, h [3] consdrs a two chlon systm n whch ach chlon appls formulatd (r, Q polcy. Forsbrg [4] for th cas of compound Posson dmand valuats costs of a two-lvl nvntory systm. Latr, h [5] studs an xact valuaton of (r, Q polcs for twolvl nvntory systms wth Posson dmand. Thn Forsbrg [6] for th cas of a gnral dstrbuton of dmand ntr-arrval tms valuats (r, Q polcs for two-lvl nvntory systms. Nahmas and Smth [2] consdr a two-chlon nvntory systm wth a dstrbuton cntr and a numbr of rtalrs. All nstallatons apply ordr up to S polcy. Th dmand n rtalrs has ngatv bnomal dstrbuton and a fracton of unsatsfd dmand n th rtalrs s lost. Sfbargh and Akbar [4] study a mult-chlon nvntory systm wth dntcal rtalrs controlld by contnuous rvw polcy (r, Q. Thy assum that th dmands of rtalrs ar ndpndnt Posson and stock-outs n rtalrs ar lost sals. Thy dvlop an approxmat cost functon to fnd optmal rordrs for gvn batch szs n all nstallatons. For a rparabl tm, Shrbrook [5] studs a two chlon nvntory systms and appls on for on polcy n bas-dpot supply systm. Gravs [7] consdrs a mult-chlon nvntory modl wth on-foron polcy. prsnts an xact modl for fndng th stady stat dstrbuton of th nt nvntory lvl. Monzadh [] consdrd an nvntory systm wth on supplr and M dntcal rtalrs n whch ach rtalr facs ndpndnt Posson dmand and appls contnuous rvw ( R, Q -polcy. Excss dmand s

2 backordrd n th rtalr. Th supplr has onln nformaton on th nvntory status and dmand actvts of th rtalr, starts wth m ntal batchs (of sz Q, and placs an ordr to an outsd sourc mmdatly aftr th rtalr s nvntory poston rachs R+s, ( 0 s Q. t s also assumd that outsd sourc has ampl capacty. Monzadh [] usng th xpctd valu of th rtalr s lad tm to approxmat th lad tm dmand obtand an xcllnt approxmat valu of th xpctd total systm costs. Lattr aj and Sajadfar [9] consdrd th sam systm but wth on rtalr and mplctly drvd th xact probablty dstrbuton of th rtalr s lad tm and th xact valu of th xpctd systm costs. Lagodmos and Koukoumalos [0] consdrd a two-chlon supply chan wth on warhous and svral nd stock-ponts and us prodc rvw chlon ordr-up-to polcs to control th chan. n th vnt of warhous shortags, th avalabl matral s to b ratond among th nd stock-ponts accordng to fxd fractons. Zhang t al.[6] consdrd an ntgratd vndor-managd nvntory (VM modl for a sngl vndor and multpl buyrs n a two-chlon nvntory systm and proposd th optmal nvstmnt amount and rplnshmnt dcson for all th buyrs and th vndor. n ths papr thy analyz srvc prformanc of th systm for a class of practcal lnar ratonng ruls. n ths papr w consdr a two- lvl stochastc nvntory systm consstng of on vndor and N rtalrs. Each rtalr appls a nw ordrng polcy calld on-for-on-prod (,T polcy, whch lmnats th uncrtanty n dmand for th supplrs n th supply chan. Th mportant advantags of ths polcy, whch lads to a dtrmnstc dmand for th supplrs, ar that for constant lad tm of supplrs [8]: - Th safty stocks n supplrs ar lmnatd, (cost rducton 2- nvntory control and producton plannng n supplrs ar smplfd. 3- Shortag cost n supplrs du to uncrtanty n dmand s lmnatd. 4- nformaton xchang cost for supplr du to lmnaton of uncrtanty of ts dmand s lmnatd. 5- Ths polcy s vry asy to apply. n ths papr t s assumd that th fxd ordrng cost for all stock ponts s nglgbl and ach rtalr ordrs on unt to th supplr n a fxd tm ntrval T, =,, N, Thus th vndor s dmand s dtrmnstc. W assum that rtalrs fac ndpndnt Posson dmand wth rat μ and unsatsfd dmand n rtalrs ar lost. W also assum that all rtalrs hav a common srvc lvl, that s, th fractons of lost dmands at all rtalrs ar th sam whch s a rasonabl assumpton. Furthr, w assum that thr s a budgt lmt on th xpctd total nvntory nvstmnt of th supply chan whch s a practcal assumpton. For ths systm w show that th avrag nvntory lvl for all rtalrs ar th sam and drv th total systm cost consstng of th sum of holdng and shortag costs of rtalrs and nvntory costs of th vndor. Th objctv s to dtrmn th optmal tm ntrvals btwn two conscutv ordrs for all rtalrs, T *, =,, N, whch mnmzs th total systm cost pr unt tm and satsfs th budgt constrant ASSUMPTONS - Th rtalrs fac ndpndnt Posson dmand wth rat μ =μ, =,,N. 2- Thr s a budgt lmt on th xpctd total nvntory nvstmnt of th systm. 3- Unsatsfd dmand n rtalrs wll b lost. 4- Th fractons of lost dmands at all rtalrs ar th sam 5- Th vndor ordrs to an outsd supplr wth unlmtd capacty. 6- Fxd ordrng cost s zro or nglgbl for th vndor and all rtalrs. 7- Th lad tm for rtalrs and th vndor ar constant. 3. NOTATON N Numbr of rtalrs. μ Dmand rat at rtalr, =,2,,N. π Unt cost of a lost sal at rtalr, =,2,,N. Π Π= π C unt cost of product h oldng cost rat at rtalr, =,2,,N. = h T Tm ntrval btwn any two conscutv ordrs for rtalr, =,2,,N. Avrag nvntory lvl at rtalr, =,2,,N. C R Expctd total cost pr unt tm for rtalr, =, 2,, N. C R Expctd total cost pr unt tm for all rtalrs. C s Total vndor s nvntory cost. TC Total systm cost pr unt tm, TC=C S +C R. Vndor s nvntory Cost Snc th ordrng cost at vndor s zro, to rduc hs nvntory on hand and ts assocatd holdng cost, h uss a dtrmnstc on for on polcy for satsfyng th ordrs of rtalr, =,, N. That s, at vry T = T unts of tm h placs an ordr (of sz to th outsd supplr to satsfy th ordr of rtalr, =,, N. Furthrmor, snc th lad tm to th vndor from th outsd supplr s constant, to furthr rduc hs holdng

3 cost du to satsfyng th rtalr ordrs, h placs hs ordrs to outsd supplr n such a way that th arrval tms of hs ordr concd wth th arrval tms of th rtalr s ordrs. Adoptng ths polcy, h dos not carry any nvntory for satsfyng th ordrs of rtalr, =,, N. Thrfor, th vndor s total nvntory cost s qual to zro. That s, C s = 0. Rtalrs nvntory Costs To obtan th rtalrs nvntory cost, w nd to comput th holdng cost and th shortag cost at rtalr. Usng th concpt of quung thory th sum of holdng and shortag costs for a sngl stock-pont and ts assocatd optmal soluton hav bn obtand by aj and aj [8]. Accordng to thr drvaton th total cost at rtalr s: CR( = h + π μ( ρ ( Whr s hs xpctd nvntory on hand, ( ρ s th proporton of tm that th rtalr s out of stock, π μ ( ρ s hs total lost sals cost pr unt tm, and ρ = = (, (2 Tμ n addton, thy showd that th avrag nvntory s ρ =, (3 β whr β ρ β = =. (4 Snc w assum that all th rtalrs hav an qual srvc lvl, w can wrt ρ = ρ, =,..., N (5 Now t can b shown that for ρ <, th rlaton ( β ρ β = has a unqu soluton, call t β, whch s btwn 0 and [3]. Thus, from (4 and (5 w can wrt β = β, and =, =,..., N (6 Also, from (2 and (6,w hav ρ = ( (7 Now from (, (5, and (6, th total cost for all rtalrs s C = C ( = [ h + π μ ( ρ ] R R = h + ( ρ π μ (8 or from (7 CR = h + ( π μ N N Lt (9 Π= π μ. Thrfor, (9 can b = h and N N wrttn as: CR = +Π ( or quvalntly, from (7, (0 C = +Π( ρ ( R Budgt Constrant Th total xpctd nvntory nvstmnt of th systm, M, s M = C = C Whr = s th xpctd nvntory lvl at rtalr, =,2,,N. thrfor w can wrt M = Lt γ = th lmt on total avrag nvntory nvstmnt n th supply chan (st by th managmnt. Thus, w can wrt th budgt constrant as γ (2 5. OPTMAL SOLUTON To obtan th optmal soluton, frst w nd to calculat, TC, th systm total cost functon pr unt tm. Th total cost s th sum of holdng and shortag costs of all rtalrs, C R whch w valuatd n scton 4.2 plus th vndor s nvntory cost C S. Snc th vndor s nvntory cost s zro, w can wrt th total cost of th systm as TC = C =Π+ Π( R Th frst drvatv of TC wth rspct to s TC = Π Th scond drvatv of TC can b wrttn as 2 TC 2 =Π 3

4 t s clar that th scond drvatv of TC wth rspct to s gratr than zro. Thus, th total systm cost TC s convx. Lt 0 dnot that valu of whch mnmzs TC. On can obtan 0 by lttng th frst drvatv of TC qual to zro. nc 0 s th soluton of o o o TC = +Π( = 0 or o o o + =. (4 Π Snc ρ = ρ = = ( w can fnd T o,, μt =,2,,N, whch mnmzs th total cost as To =, =,..., N o μ ( o (5 Now our problm s th mnmzaton of TC subjct to th constrant that th total xpctd nvntory nvstmnt of th systm, M, must not xcd γ. Thus, from (2 and (3 w can wrt th problm as ( Mn TC =Π+ Π( Subjct to γ or quvalntly ( Mn TC =Π+ Π( Subjct to γ Snc TC s a convx functon, th optmal soluton can b obtand asly. Frst, w solv th unconstrand problm and fnd 0 from (4 [or quvalntly fnd T 0, =,2,,N,from (5]. Thn f 0 satsfs th rlaton (2 that s, γ thn th optmal soluton of th problm, * s * = 0 (or quvalntly T * = T 0, =,2,,N. But f 0 dos not satsfy th rlaton (4, namly f > γ thn th constrant s actv and th optmal soluton can b obtand asly from th followng rlaton * γ = (6 To obtan th optmal cycl tm, T *, from (5 and (6 w can wrt. * T =, =,..., N μγ( γ 6- CONCLUSONS n ths papr w consdrd a two- lvl stochastc nvntory systm consstng of on vndor and N rtalrs. W assumd that th fxd ordrng cost for all stock ponts s nglgbl and ach rtalr ordrs on unt to th supplr n a fxd tm ntrval T, =,, N, Thus th vndor s dmand s dtrmnstc. W assumd that rtalrs fac ndpndnt Posson dmand wth rat μ and unsatsfd dmand n rtalrs ar lost. W also assumd that th prcntags of lost dmands for all rtalrs ar th sam. Furthr, w assumd that thr s a budgt constrant on th xpctd nvstmnt for total nvntory of th systm. For ths systm w showd that th avrag nvntory lvl for all rtalrs s th sam and drvd th total systm cost pr unt tm. Thn w obtand th optmal avrag nvntory and also th tm ntrvals btwn two conscutv ordrs for all rtalrs, T *, =,, N, whch mnmzs th total systm cost pr unt tm and satsfs th budgt constrant. 7. REFRENCES [] Andrsson, J., Mlchors, Ph., A two-chlon nvntory modl wth lost sals, ntrnatonal Journal of Producton Economcs, Vol. 69, 200, pp [2] Axsatr, S., Smpl Soluton Procdurs for a Class of Two-Echlon nvntory Problm, Opratons Rsarch, Vol. 38, 990, pp [3] Axsätr, S., Exact and Approxmat Evaluaton of Batch-Ordrng Polcs for Two-Lvl nvntory Systms, Opratons.Rsarch, Vol. 4,993, pp [4] Forsbrg, F., Optmzaton of Ordr-up-to-S Polcs for Two-Lvl nvntory Systms wth Compound Posson Dmand, Europan Journal of Opratonal Rsarch, Vol. 8,995, pp [5] Forsbrg, F., Exact Evaluaton of (R,Q Polcs for Two-Lvl nvntory Systms wth Posson Dmand, Europan Journal of Opratonal Rsarch, Vol. 96, 996, pp [6] Forsbrg, F., Evaluaton of (R,Q Polcs for Two-Lvl nvntory Systms wth Gnrally Dstrbutd Customr ntr-arrval Tms, Europan Journal of Opratonal Rsarch, Vol. 99, 997, pp [7] Gravs, S., A Mult-Echlon nvntory Modl for a Rparabl tm wth On for-on

5 Rplnshmnt, Managmnt Scnc, Vol. 3, 985, pp [8] aj, R., and aj, A., "ntroducng On-for-On Prod Ordrng Polcy and ts Optmal Soluton Basd on Quung Thory", Journal of ndustral and Systms Engnrng, Vol., 2007, pp [9] aj, R., Sajadfar, S. M., Drvng th Exact Cost Functon for a Two-Lvl nvntory Systm wth nformaton Sharng, Journal of ndustral and Systms Engnrng, Vol. 2, 2008, pp [0] Lagodmos, A.G., Koukoumalos, S., Srvc Prformanc of Two-Echlon Supply Chans undr Lnar Ratonng, ntrnatonal Journal of Producton Economcs, Vol. 2, 2008, pp [] Monzadh K., A Mult-Echlon nvntory Systm wth nformaton Exchang, Managmnt Scnc, Vol. 48, No. 3; 2002, pp [2] Nahmas, S., Smth, S.A., Optmzng nvntory Lvls n a Two-Echlon Rtalr Systm wth Partal Lost Sals, Managmnt Scnc, Vol. 40,994, pp [3] Ross, S.M., ntroducton to Probablty Modls, 5 th dton, Nw York, Acadmc prss, 993. [4] Sfbarghy, M., Akbar, M.R., Cost Evaluaton of a Two-Echlon nvntory Systm wth Lost Sals and Approxmatly Posson Dmand, ntrnatonal Journal of Producton Economcs, Vol. 02, 2006, pp [5] Shrbrook, C.C., METRC: A Mult-Echlon Tchnqu for Rcovrabl tm Control, Opratons Rsarch, Vol. 6, 986, pp [6] Zhang, T., Lang, L., Yu, Y., Yu, Y., 2007, An ntgratd Vndor-Managd nvntory Modl for a Two-Echlon Systm wth Ordr Cost Rducton, ntrnatonal Journal of Producton Economcs, Vol.09, 2007, pp