ANALYSIS OF ORDER-UP-TO-LEVEL INVENTORY SYSTEMS WITH COMPOUND POISSON DEMAND

8 th Intrnatonal Confrnc of Modlng and Smulaton - MOSIM - May -2, 2 - Hammamt - Tunsa Evaluaton and optmzaton of nnovatv producton systms of goods and srvcs ANALYSIS OF ORDER-UP-TO-LEVEL INVENTORY SYSTEMS WITH COMPOUND POISSON DEMAND M. Z. BABAI BEM-Bordaux Managmnt School 68 cours d la Lbératon 3345 Talnc Cdx - Franc mohamd-zd.baba@bm.du Z. JEMAI, Y. DALLERY Ecol Cntral Pars Grand Vo ds Vgns 9229 Châtnay Malabry - Franc jma@lg.cp.fr, dallry@lg.cp.fr ABSTRACT: W analys a sngl chlon sngl tm nvntory systm whr th dmand and th lad tm ar stochastc. Dmand s modlld as a compound Posson procss and th stock s controlld accordng to a contnuous tm ordr-up-to lvl polcy. W propos a nw mthod for dtrmnng th optmal ordr-up-to lvl for a cost orntd nvntory systms whr unflld dmands ar backordrd. Th condtons undr whch th systm bhavs lk a Mak-To-Ordr sttng ar also dscussd. By mans of a numrcal nvstgaton, w show that th proposd mthod provds vry good rsults. It s also shown to outprform anothr approxmat soluton provdd n th ltratur. Our work allows nsghts to b gand on stock control rlatd ssus for both fast and slow movng Stock Kpng Unts. KEYWORDS: stock control, compound Posson, ordr-up-to-lvl, slow movng tms. INTRODUCTION Invntory control polcs hav bn largly dscussd n th acadmc ltratur snc th 95s. A consdrabl amount of rsarch has bn conductd to propos xact and approxmat mthods to comput th optmal paramtrs of ths polcs (Portus, 985; Slvr t al, 998; Strjbosch, 26). Th Normal dstrbuton s oftn consdrd to modl th dmand snc t s attractv from a thortcal prspctv and t s known to provd a good mprcal ft to obsrvd dmand data. It should b notd that th normalty assumpton mak mor sns n th contxt of fast movng tms wth dmands occurrng rgularly, whch ar not hghly varabl. Obvously, f th dmand s not as such, for xampl as n th cas of slow movng tms or tms wth ntrmttnt dmand whch can b ncountrd n many contxts n practc, th normalty assumpton s judgd to b as far from approprat to b consdrd. In practc, slow movng tms and tms charactrzd by ntrmttnt dmand may b spar parts (ngnrng spars, srvc parts kpt at th wholsalng/rtalng lvl, tc.) or any Stock Kpng Unt (SKU) wthn th rang of products offrd by all compans at any lvl of a gvn supply chan. Such knd of tms s charactrzd by occasonal dmand arrvals ntrsprsd by tm ntrvals durng whch no dmand occurs. As such, dmand s bult, for modllng purposs, from consttunt lmnts (dmand arrvals and dmand szs) that rqur th consdraton of mor spcfc compound dmand dstrbutons. Undr ths contxt, two dmand gnraton procsss hav domnatd th ltratur. If tm s tratd as a dscrt (ntgr) varabl, dmand may b gnratd basd on a Brnoull procss, rsultng n a gomtrc dstrbuton of th ntr-dmand ntrvals (Boylan, 997; Tuntr t al, 2). Whn tm s tratd as a contnuous varabl, th Posson dmand gnraton procss rsults n ngatv xponntally dstrbutd ntr-arrval ntrvals. Thr s sound thory n support of xponntal dstrbuton for rprsntng th tm ntrval btwn succssv dmands n th contxt of slow movng tms (Fny and Shrbrook, 966; Archbald and Slvr, 978). Thr s also mprcal vdnc n support of ths dstrbuton (Janssn, 998; Eavs, 22). Wth Posson arrvals of dmands and an arbtrary dstrbuton of dmand szs, th rsultng dstrbuton of total dmand s compound Posson. It should b notd that from a modllng prspctv, th compound Posson procss s attractv snc t can also modl th dmand n th contxt of fast movng tms by consdrng vry low dmand tm ntrvals. Such a procss wll b consdrd to rprsnt th dmand n ths rsarch work. Furthrmor, for stock control purposs, contnuous tm polcs hav bn dscussd n th acadmc ltratur and hav bn rportd to offr tangbl bnfts to stocksts dalng wth both fast and slow movng tms (Strjbosch t al., 2; Portus, 985; Porras and Dkkr, 28). Th contnuous Ordr-Up-To-Lvl polcy, oftn calld as th bas stock polcy, s a vry appalng polcy from both a practcal and thortcal prspctv snc t s smpl, clos to optmal and rflcts to a grat xtnt ral world practcs (Larsn and Thorstnson, 28). Howvr, dspt th vry many contrbutons n ths ara and th abundanc of th nvstgatons amng at th dtrmnaton of th optmal paramtrs of ths polcs, thr ar stll many unrsolvd ssus spcally

MOSIM - May -2, 2 - Hammamt - Tunsa undr som spcfc dmand procsss such as th compound Posson procss. In fact, () most nvstgatons focus on srvc rathr than cost-orntd systms; () thr s lttl consdraton of stochastc lad-tms whn cost orntd systms ar analysd; () thr ar no smpl mthods that can b mplmntd asly by practtonrs and nabl th calculaton of th optmal ordr-up-to-lvls, spcfcally n th contxt of slow movng tms. In ths papr, w dvlop a smpl mthod than can b usd to calculat th optmal ordr-up-to-lvl n a sngl chlon, sngl tm nvntory systm undr a compound Posson dmand procss and stochastc lad tm. Th soluton of th optmal ordr-up-to-lvl s drvd for a cost orntd nvntory systms whr unflld dmands ar backordrd. Som proprts of th optmal ordrup-to-lvl ar provdd. Th qualty of th soluton s numrcally assssd n th cas of both fast and slow movng tms (.. tms wth hgh and low dmand ntrvals, rspctvly). Th bnchmark mthod usd n th contxt of fast movng tms s dscrbd n th Appndx. Th rmandr of th papr s organsd as follows. Scton 2 dscrbs th nvntory systm and prsnts th notatons usd n th papr. Th xpctd total cost and th xprsson of th optmal ordr-up-to-lvl ar drvd n Scton 3. Scton 4 prsnts th algorthm that w propos to dtrmn th optmal ordr-up-to-lvl. In Scton 5, w prsnt th assumptons and th rsults of th numrcal nvstgaton. W nd n Scton 6 wth conclusons and drctons for furthr rsarch. 2 SYSTEM DESCRIPTION AND NOTATION W consdr a sngl chlon sngl tm nvntory systm whr th dmand and th lad tm ar stochastc. Dmand s modlld as a compound Posson procss,.. th ntr-dmand arrvals ar xponntally dstrbutd and th dmand sz follows an arbtrary dstrbuton. Th stock s controlld accordng to a contnuous tm ordr-up-to-lvl polcy. Each rplnshmnt ordr s assocatd wth a stochastc lad-tm charactrsd by an arbtrary probablty dstrbuton and unflld dmands ar backordrd. A pnalty cost s ncurrd f th dmand s backordrd and a holdng cost s ncurrd f an tm s n stock, both pr unt of tm. For th rmandr of th papr, w dnot by: L: man lad-tm λ: man dmand arrval rat : dmand sz random varabl μ : man of dmand sz σ : standard dvaton of dmand sz : th sum of ndpndnt and dntcally dstrbutd random varabls F : c.d.f of th random varabl I (t) : nvntory lvl at tm t S: ordr-up-to-lvl h: nvntory holdng cost pr unt pr unt of tm b: nvntory backordrng cost pr unt pr unt of tm ( ) = max(,) 3 SYSTEM MODELLING AND ANALYSIS Th nvntory systm consdrd n our work can b modlld as a quung systm. Each dmand arrval gnrats an ordr to th supply systm to rplnsh th stock. A quu s formd by th outstandng ordrs n th systm whr th arrval procss s gvn by th dmand arrvals (or quvalntly th ordrng procss) and th procssng tm corrsponds to th rplnshmnt ladtm. Th stock s rplnshd aftr L tm unts from th dmand arrval. At any tm t, th nvntory lvl I(t) s lnkd to th numbr of outstandng ordrs n th systm by N(t) = S - I(t). It s asy to not that th numbr of outstandng ordrs n th systm at any tm t s th numbr of customrs n an M/G/ quu (Klnrock, 975). In fact, th dmand arrval procss (or quvalntly th ordrng procss) s a Posson procss wth rat and th srvc tm (th rplnshmnt lad-tm) follows a Gnral dstrbuton wth man L. Thr s no capacty constrant n th rplnshmnt systm whch s quvalnt to an nfnty numbr of srvrs n th quung systm. For mor dtals about th modllng of nvntory systms by usng th quung thory, th radr s rfrrd to (Buzacott and Shanthkumar, 993; Lbropoulos and Dallry, 23). Hnc, analysng th nvntory lvl I(t) s quvalnt to analysng th numbr of customrs N(t) n an M/G/ quu. A wll-known and usful rsult from quung thory (Zpkn, 2) s that th statonary probablts for th M/G/ quu ar gvn by (). ( λl) p = for all ()! Thus, n th statonary rgm, wth probablty p thr ar ordrs n th systm (.. N(t) = ) and thr ar I(t)= S - tms n stock. For xampl, for = thr s no ordr n th systm,.. th systm s n stat S wth a stat probablty p =, for = thr s only on ordr n th systm,.. th systm s n stat S- wth a stat probablty p = λl, tc. 3. Drvaton of th Optmal Ordr-Up-To-Lvl Th xpctd holdng and backordrng nvntory n th systm ar gvn by: S p ( S ) p and = p ( rspctvly.

MOSIM - May -2, 2 - Hammamt - Tunsa Thus, th xpctd total nvntory cost s gvn by quaton (2). [ C ] = hsp [ h( S ) b( ] p E ( (2) Equaton (2) can also b wrttn as follows: S [ ] (3) E C( = hsp h ( S x) F ( x) dx b ( x F ( x) dx p S Takng th drvatv of th xpctd total cost wth rspct to S gvs: [ C ( S )] de ds = hp = hp [ hf ( S ) b( F ( S ))] = [( h b) F ( S ) b] p = Th objctv s to fnd th optmal ordr-up-to-lvl that mnmss th xpctd total cost. Not that snc F ( s ncrasng n S for all, ths drvatv s clarly ncrasng n S. Thrfor, E[C(] s convx n S. Hnc, th optmal S can smply b dtrmnd by sttng th drvatv to zro. Proposton. Th optmal ordr-up-to-lvl S s th soluton of (4). ( λl) b F S = ( ) (4)! h b Proof. Lt S * dnots th optmal ordr-up-to-lvl. Th valu of S * s th soluton of th quaton: de [ C( ] = ds de [ C( ] = [ ( ) ( ) ] ds hp h b F S b p = b h F ( p = p p ( h b) ( h b) b F ( p = p h b ( λl) b F S = ( )! h b Thus, th optmal valu S * s th soluton of (4) whch nds th proof of Proposton. Th dstrbuton that w consdr to rprsnt th random varabl of th dmand sz wll affct th complxty of th analyss snc n ordr to dtrmn th optmal ordr-up-to-lvl S, on nds th dstrbuton of th random varabl. Thus, th analyss s lss complx f th dmand sz dstrbuton s rgnratv,.. f th sum of dmand sz dstrbutons s of th sam typ (as.g. for Normal or Gamma dmand sz dstrbutons) so p that by knowng th typ of th dstrbuton F and th frst two momnts, th analyss s straghtforward. W hav consdrd n ths work th us of both th Normal and th Gamma dstrbuton. Both ar avalabl n commrcal spradsht packags such as Excl. Howvr, although th Normal dstrbuton s vry convnnt from an analytc standpont, n practcal stuatons, th Gamma dstrbuton s prfrabl to rprsnt dmand szs of slow movng tms snc t s nonngatv wth hgh coffcnts of varaton. Morovr, thr has bn mprcal vdnc n ts support (Kwan, 994). Consquntly, for th purpos of our numrcal nvstgatons n Scton 5, w wll us th Gamma dstrbuton. It s mportant to not that n practc th ssu wth quaton (4) whn t s usd to dtrmn numrcally th optmal ordr-up-to-lvl s that S * s a soluton of an quaton that s composd of an nfnt summaton. Thus, f ths quaton s solvd by stoppng th summaton nto a crtan ordr, th soluton that s obtand s an approxmaton of S *. Hnc, n ordr to gv a good numrcal soluton of S *, a smpl mthod should b dvlopd to tackl ths ssu, whch s th objctv of th analyss n Scton 4. 3.2 Mak-To-Stock vs. Mak-To-Ordr Systm Th nvntory systm that w ar analyzng n ths papr can bhav lk a Mak-To-Stock (MT or a Mak-To- Ordr (MTO) systm dpndng on th valu of th optmal ordr-up-to-lvl. In fact, f th optmal ordr-upto-lvl S * s qual to zro, th systm bhavs lk a MTO systm, othrws, f S * s strctly postv, th systm bhavs lk a MTS systm. Proposton 2. Th systm bhavs lk a MTO systm f and only f nqualty (5) s satsfd b < L log (5) λ h b Proof. Th systm bhavs lk an MTO systm, f and only f S * s qual to. It s asy to show that S * = f and only f * F ( S ) <. p * F ( S ) p < b λ L < h b b λ < L log h b Ths nds th proof of Proposton 2.

MOSIM - May -2, 2 - Hammamt - Tunsa 4 APPROIMATIONS OF THE ORDER-UP-TO- LEVEL In ths scton, w propos a mthod and a smpl algorthm that can b usd to drv good approxmatons of th optmal ordr-up-to-lvl S *. Lt us frst ntroduc quatons (6) and (7). ( λl) b n F ( = (6)! h b n n ( λ L) b = ( λl) F ( (7)! h b! Proposton 3. Th optmal S * s such that: S U (n) S * S L (n) whr S U (n) s th soluton of (6) and S L (n) s th soluton of (7). Morovr S U (n) and S L (n) convrg to S * whn n tnds towards nfnty. Th proof of Proposton 3 s not ncludd n th papr but t may b provdd by th frst author upon rqust. Proposton 3 shows that S U (n) and S L (n) provd bounds on S * for dffrnt valus of n and that ths bounds gt tghtr as n ncrass. As a rsult, S * can b approxmatd as closly as dsrd by usng thos bounds. Th followng algorthm can b usd to comput S *. Stp : Intals n = Stp : a: solv (6) to calculat S U (n) b: solv (7) to calculat S L (n) (ths can b don usng a smpl dchotomy algorthm) Stp 3: Calculat Δ(n) = S U (n) - S L (n) If Δ(n) < ε gvn, stop Othrws, do n = n and go to stp Plas not that for n =, S U (n) rducs to th approxmaton of th optmal ordr-up-to-lvl proposd by Synttos t al (29) for an nvntory systm wth ntrmttnt dmand and short lad-tms. Whn th algorthm prsntd abov s usd n practc, a smpl mthod that can b usd n ordr to dtrmn th valu of n for whch th bounds S U (n) and S L (n) convrg to S * s to dtrmn, for th gvn stoppng crtron of th algorthm ε, th smallst valu n for whch (8) s satsfd. n λ ( L) < ε (8)! In th nxt scton, w nvstgat numrcally th qualty of th dffrnt approxmatons gvn by our mthod. In fact, for dffrnt valus of th dmand ntrval and th dmand sz, w show numrcally that S * can b asly obtand by consdrng th valus of S U (n) and S L (n) wth rlatvly low valu of n. For hgh valus of λ compard to th lad tm L (.. /λ s rlatvly low compard to L), t s clar that th tm can b consdrd as a fast movng tm. In ordr to show that our mthod provds "good" approxmatons of S *, w also conduct a numrcal comparson of th approxmatons gvn by our proposd algorthm and th approxmaton obtand by usng th classcal xprsson gvn n th ltratur for fast movng tms. For mor dtals on ths classcal xprsson, th radr s rfrrd to th Appndx. 5 NUMERICAL INVESTIGATION W start th numrcal nvstgaton by dtrmnng for dffrnt valus of th dmand arrval rat λ th smallst ordr n for whch th bounds convrg to th optmal ordr-up-to-lvl wthn a gvn rror thrshold. Th smallst valu n s computd by usng (8) and wll b dnotd by n *. To do so, th numrcal valus L =, h = and b = ar consdrd. Rsults ar rportd n Tabl for thr stoppng crtra ε = -4, -3 and -2. λ..2.3....8,9 2 3 4 5 6 7 8 9 ε = -4 3 3 4 6 6 6 9 3 5 7 9 2 23 24 n* ε = -3 2 3 3 5 5 5 8 3 5 6 8 2 2 ε = -2 2 2 3 4 4 7 8 9 2 4 5 7 8 Tabl : Convrgnc ordr of th bounds for dffrnt valus of λ As shown n Tabl, for a slow movng tm (low valus of λ), th smallst ordr of convrgnc of th algorthm n * for a stoppng crtron lss than -4 s rlatvly low. Obvously, th ordr s vn smallr f th stoppng crtron s lss than -2. For xampl, for tms wth λ (.. dmand ntrvals hgh or qual to whch corrsponds to slow movng tms), our algorthm convrgs aftr at most 6 tratons f th stoppng crtra s lss than -4 and at most 4 tratons f th stoppng crtra s lss than -2. It s also clar from Tabl, that our algorthm rqurs mor tratons to convrg n th cas of fast movng tms (.. λ ). To analys th mpact of th dgr of th dmand ntrmttnc on th qualty of th calculatd soluton of th optmal ordr-up-to-lvl, w plot th curvs of th uppr and lowr bounds S U (n) and S L (n) by varyng th dmand ntrval (.. /λ and n. Rsults n ths scton ar prsntd by consdrng th numrcal valus for th dmand szs μ = and σ = 3. Th numrcal valus of th lad-tm and th unt costs ar L =, h =, b=. Rsults ar prsntd by consdrng th cass of slow and fast movng tms wth dffrnt valus of n. Fgur corrsponds to th cas of slow movng tms (.. valus of λ such that /λ ) and low valus of n. Fgur 2

MOSIM - May -2, 2 - Hammamt - Tunsa corrsponds to th cas of fast movng tms (.. valus of λ such that /λ ) and hgh valus of n. To mak Fg. asy to rad, snc th valus of th uppr and lowr bounds ar vry clos from n = 2, w only prsnt th curvs for n =, 2 and 5. Fgur and fgur 2 clarly show that th bounds ar vry clos and thy convrg quckly to th optmal ordr-up-to-lvl onc th algorthm rachs th smallst valu n *. For xampl, for th numrcal xampl prsntd abov, f /λ = th uppr and lowr bounds convrg to th optmal soluton (S * = 25.2) for n * = 6. If /λ =, th uppr and lowr bounds convrg to th optmal soluton (S * = 45.6) for n * = 24 although as shown n Fgur 2, t s dffcult to dstngush graphcally btwn th uppr and lowr bounds from th valu n = 7. Bounds 9 8 7 6 5 4 3 2,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 SU() SL() SU(2) SL(2) SU(5) SL(5) 8,5 9 9,5,5 Fgur : Uppr & lowr bounds of S corrspondng to hgh valus of /λ & low valus of n. / λ To analys th prformanc qualty of th approxmaton provdd n th Appndx that s oftn usd n stock control for both fast and slow movng tms, w compar th prformanc of ths approxmaton wth th optmal soluton gvn by our algorthm. In th rmandr of th papr, ths approxmaton wll b calld th "classcal approxmaton". Th classcal approxmaton s dtrmnd by usng a Gamma dstrbutd dmand wth man and varanc that w comput as dscrbd th n Appndx basd on th sam numrcal valus prsntd abov for th dmand sz and th dmand ntrval. Fgur 3 shows th curv of th optmal soluton and th classcal approxmaton for hgh valus of /λ (.. slow movng tm). Fgur 4 shows th curv of th optmal soluton and th classcal approxmaton for low valus of /λ (.. fast movng tm). S 3 25 2 5 5,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 Optmal Soluton Classcal Approxmaton 7,5 8 8,5 9 9,5,5 Fgur 3 : Optmal soluton and classcal approxmaton for hgh valus of /λ 6 4 / λ Bounds 6 4 2 8 6 4 2 SU(5) SL(5) SU(7) SL(7) SU(9) SL(9),,2,3,4,5,6,7,8,9, / λ S 2 8 6 4 2 Classcal approxmaton Optmal soluton,,2,3,4,5,6,7,8,9, Fgur 4 : Optmal soluton and classcal approxmaton for low valus of /λ / λ Fgur 2 : Uppr & lowr bounds of S corrspondng to low valus of /λ & hgh valus of n. It should also b notd that th uppr bound s gnrally closr to th optmal ordr-up-to-lvl than th lowr bound vn for low valus of n. Consquntly, f a smpl approxmaton s ndd n practc wthout usng our algorthm, th uppr bound can b consdrd as a good soluton of th optmal ordr-up-to-lvl. Fgur shows that th nvntory systm bhavs lk a Mak-To-Ordr systm for th dmand ntrval /λ.5. It s asy to chck that ths s th rght valu obtand whn th numrcal valus ar rplacd n (5). Fgur 3 shows that for hgh valus of /λ (.. slow movng tm) th qualty of th classcal approxmaton s poor. Th rlatv rror btwn th ordr-up-to-lvl whn computd by usng th classcal approxmaton and th optmal soluton computd by usng our algorthm can go up to 6%. As xpctd, ths dffrnc dcrass whn /λ dcrass. In contrast, Fg. 4 shows that for low valus of /λ (.. fast movng tm), th classcal approxmaton gvs good prformanc snc t s almost qual to th optmal soluton for all valus of /λ. Ths shows that th classcal approxmaton s a good soluton only for fast movng tms, but for slow movng ons, t s far to b a good approxmaton.

MOSIM - May -2, 2 - Hammamt - Tunsa 6 CONCLUSIONS AND FUTURE RESEARCH Basd on a quung thory approach, w dvlopd n ths papr a smpl mthod for dtrmnng th optmal ordr-up-to-lvl n a sngl chlon nvntory systm undr a compound procss dmand and stochastc ladtm. W provdd xprssons of uppr and lowr bounds of th optmal ordr-up-to-lvl for a cost orntd systm whn unflld dmands ar backordrd. Ths xprssons can b consdrd as approxmatons of th optmal ordr-up-to-lvl and any dsrd lvl of accuracy can b achvd. By conductng a numrcal nvstgaton, w showd that th proposd bounds rprsnt good approxmatons of th optmal ordr-up-to-lvl for rasonabl valus of n. Th algorthm usd to comput th optmal soluton has bn shown to hav rlatvly quck convrgnc spcally for slow movng tms. Furthrmor, f a smpl approxmaton s ndd n practc wthout th nd to us th algorthm, w rcommndd th us of th uppr bound snc t s gnrally closr to th optmal ordrup-to-lvl than th lowr bound. It s mportant to not that th approxmatons of th ordr-up-to-lvl that w proposd can b asly mplmntd and dtrmnd on spradshts lk Excl whch s mportant ssu for practtonrs. Although anothr approxmaton proposd n th ltratur (classcal approxmaton) s oftn usd n stock control for both fast and slow movng tms, w showd n ths papr that ths approxmaton provds vry poor prformanc n a contxt of slow movng tms. Th rlatv rror btwn th classcal approxmaton and th optmal th ordr-up-to-lvl can go up to 6% whch s a hug dffrnc that th practtonrs should b awar of t. Obvously, as t s xpctd th classcal approxmaton gvs good prformanc whn appld to a fast movng dmand. As a concluson to ths analyss, t s thn rcommndd to not consdr ths approxmaton n futur rsarch dalng wth stock control of slow movng tms. In th othr hand, t s clar that our approach gvs bttr approxmatons of th optmal ordrup-to-lvl n both th slow and th fast movng tms contxts. Thr ar many ways to xtnd ths rsarch work. It would b ntrstng n a frst stp to xtnd ths nvstgaton by conductng an mprcal nvstgaton usng ral data to assss th bnft of applyng our approxmatons n a ral contxt and to rport th bnft n trms of cost savngs. W should say that such an nvstgaton ncsstats non only ral data on dmand hstors for slow movng SKUs but also rqurs ral cost fgurs whch could b a dffcult task n practcal sttngs. W also attract th attnton of th radr about th ssu of th dstrbuton to consdr whl modllng th dmand n th contxt of slow movng tms (.. rprsntaton of dmand szs and dmand ntrvals). Ths ssu has bn rpatdly qustond n th acadmc ltratur (Wllman t al, 994, Synttos t al, 2) so anothr ntrstng rsarch would b to assss th valdty of th consdrd dstrbutons. It s also worthwhl to hghlght n ths last scton of th papr th lmts rgardng our assumpton on th statonarty of th dmand szs and dmand ntrvals, spcally n ths contxt of slow movng tms. Thus, anothr avnu of furthr rsarch s to fnd a way to tak nto account n th analyss th non-statonarty of th dmand pattrn whch wll b of a consdrabl bnft to both th acadmc rsarchrs and th practtonrs. APPENDI An approxmaton of th optmal ordr-up-to-lvl S for fast movng SKUs (Slvr t al, 998) s gvn by: b F DL ( = (9) h b whr F DL (.) dnots th cumulatv probablty dstrbuton of th dmand durng th lad-tm. Th lad-tm dmand s wth man μ DL and standard dvaton σ DL, 2 2 whr μ DL = λlμ and σ DL = λl( μ σ ). For slow movng SKUs (9) has also bn usd n th ltratur to dtrmn th optmal ordr-up-to-lvl (Synttos t al, 26). Ths s obvously rasonabl only f approprat probablty dstrbutons for th lad-tm dmand and approprat stmats of th man and varanc of th dstrbuton ar consdrd. REFERENCES Archbald B.C. and Slvr E.A. (978). (s, Polcs undr Contnuous Rvw and Dscrt Compound Posson Dmand. Managmnt Scnc, vol. 24, 9, p. 899-99. Boylan, J.E. (997). Th cntralsaton of nvntory and th modllng of dmand. Unpublshd Ph.D. thss, Unvrsty of Warwck, UK. Buzacott, J.A. and Shanthkumar, J.G. (993), Stochastc Modls of Manufacturng Systms, Englwood Clffs, NJ: Prntc-Hall. Eavs, A.H.C. (22), Forcastng for th ordrng and stock holdng of consumabl spar parts, Unpublshd Ph.D. thss, Lancastr Unvrsty, UK. Fny, G.J. and Shrbrook, C.C. (966) th (S-, Invntory Polcy Undr Compound Posson Dmand. Managmnt Scnc, vol. 2, p. 39-4. Janssn, F.B.S.L.P. (998) Invntory managmnt systms; control and nformaton ssus. Publshd

MOSIM - May -2, 2 - Hammamt - Tunsa Ph.D. thss, Cntr for Economc Rsarch, Tlburg Unvrsty, Th Nthrlands. Klnrock, L. (975). Quung Systms, Vol. I: Thory. Wly, Nw York. Kwan, H.W. (99) On th dmand dstrbutons of slow movng tms. Unpublshd Ph.D. thss, Lancastr Unvrsty, UK. Larsn, C. and Thorstnson, A. (28). A comparson btwn th ordr and th volum fll rat for a bas-stock nvntory control systm undr a compound rnwal dmand procss. Journal of th Opratonal Rsarch Socty, p. -7. Lbropoulos, G. and Dallry, Y. (23). Comparatv modllng of mult-stag producton-nvntory control polcs wth lot szng, Intrnatonal Journal of Producton Rsarch, vol. 4, 6, p. 273 298. Porras, E.M. and Dkkr, R. (28). An nvntory control systm for spar parts at a rfnry: An mprcal comparson of dffrnt rordr pont mthods. Europan Journal of Opratonal Rsarch, vol. 84,, p. -32. Portus, E.L. (985). Numrcal comparsons of nvntory polcs for prodc rvw systms. Opratons Rsarch, vol. 33,, p. 34-52. San, B. and Kngsman, B.G. (997). Slctng th bst prodc nvntory control and dmand forcastng mthods for low dmand tms. Journal of th Opratonal Rsarch Socty, vol. 48, p. 7-73. Slvr, E.A., Pyk, D.F. and Ptrson, R. (998) Invntory managmnt and producton plannng and schdulng. Thrd dton, John Wly & Sons, Inc., Nw York. Strjbosch, L.W.G., Huts, R.M.J. and van dr Schoot, E.H.M (2). A combnd forcast-nvntory control procdur for spar parts. Journal of th Opratonal Rsarch Socty, vol. 5, p. 84-92. Strjbosch, L.W.G., Moors, J.J.A. (26). Modfd normal dmand dstrbutons n (R,-nvntory control. Europan Journal of Opratonal Rsarch, vol. 72, p. 2 22. Tuntr, R., Synttos, A.A. and Baba, M.Z. (2). Dtrmnng Ordr-Up-To Lvls undr Prodc Rvw for Compound Bnomal (Intrmttnt) dmand. Europan Journal of Opratonal Rsarch, vol. 23, p. 69-624. Synttos A.A. and Boylan J.E. (26). On th stockcontrol prformanc of ntrmttnt dmand stmators. Intrnatonal Journal of Producton Economcs, vol. 3, p. 36-47. Synttos, A.A., Baba, M.Z., Dallry, Y. and Tuntr, R. (29). Prodc control of ntrmttnt dmand tms: thory and mprcal analyss. Journal of th Opratonal Rsarch Socty, vol. 6, p. 6-68. Synttos, A.A., Baba, M.Z., Lngu, D., Altay, N. (2). Dstrbutonal assumptons for paramtrc forcastng of ntrmttnt dmand. Srvc Parts Managmnt: Dmand Forcastng and Invntory (ds: Altay, N. and Lttral, L.A.), Sprngr Vrlag. Wllams, T.M. (982). Rordr lvls for lumpy dmand. Journal of th Opratonal Rsarch Socty, vol. 33, p. 85-89. Zpkn P.H. (2). Foundatons of Invntory Managmnt. McGraw-Hll, Inc.: Boston, MA.