Determining Reorder Point in the Presence of Stochastic Lead Time. and Box-Jenkins Time Series Demand

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1 Deermiig Reorder Poi i he Presece of Sochsic Led Time d Box-Jekis Time Series Demd Kl Nmi ), Jim Che ) ) Wiso-Slem Se Uiversiy, School of Busiess d Ecoomics (mik@wssu.edu) ) Norfolk Se Uiversiy, School of Busiess (che@su.edu) Absrc This er discusses he deermiio of reorder oi d sfey sock whe he eriod demds re o ideede, bu exhibis serilly correled demd rocess which c be rereseed s uoregressive-movig verge ARMA (.d.q)-model for boh deermiisic d discree sochsic led ime. A Excel bsed mehodology for fidig he reorder oi d he sfey sock level is lso reseed he ed of he er.. Iroducio Geerlly whe he deermiio of he oiml olicy of iveory model wih sochsic demd icludes he clculio of he reorder oi d he order size, oe hs o del wih me re of demd, sdrd deviio, sfey fcor, d led-ime. Mosly, he clculio of he re-order oi is bsed o he ssumio h he me re demd is deermiisic s fucio of ime. The deermiisic ssumio o he me re of demd is by fr remoe from he reliy. Therefore, i will be more rorie o use rorie robbiliy disribuios o rerese he led-ime d he uis demded o ccou for he icresig uceriy i he mrke evirome. Bsed o he ssumio h here is o correlio bewee wo eriod demds, Hdley d Whie [6] hve develoed wo yes of bckorder iveory models, roxime d exc oes, for boh Poisso d Norml disribued led-ime demds. I my rcicl siuios, however, he eriod demds re o ideede, bu exhibi serilly correled rocess (see, e.g. A, Fooolo, d Wg []; Chrles, Mrmorse, d Zi [3]). Tkig io cosiderio () he re of demd, () he legh of led-ime, (3) he vribiliy of demd d led-ime, d (4) he degree of cceble sock-ou risk, Ee d Mri [5] suggesed h he correc d cosise rocedure o se reorder ois wih ssumed d is o: () clerly disiguish bewee demd vriio d he vriio i forecs error, () show how o clcule he vrice of forecs error over led ime wihou ssumig forecs errors re ormlly disribued, (3) use he vrice of forecs error over led ime o se he sfey sock, d (4) show he clculio of sfey sock c be simlified if ormliy of cumulive forecs error is usified s i he cse whe he rocess geerig demd is from he Box d Jekis ime series d. I his reserch er, we look io he lysis h icorores he clculios of reorder oi d sfey sock whe he uis demded re geered by serilly correled rocess d c be rereseed by Box- Jekis ARMA ime series model []. The disribuio of forecs errors from he clculio rocess i Box- Jekis ARMA lysis will be used s he mesureme of he ccurcy wih which he reorder oi d sfey sock re deermied. I he firs r of his reserch er, he deermiio of he model s reorder oi is bsed o he ssumio h he rocureme led-ime is rdom vrible geered by ARMA rocess wih cos led-ime. I he secod r, we would ivesige he cse of sochsic led ime.. Two Differe Tyes of Led-Time Demd Disribuios I order o comue he reorder oi wih sfey sock h will mee secific service level, oe hs o kow he robbiliy desiy of he led ime demd. Two kids of oi robbiliy desiy fucios c be used o rerese he led ime demd.

2 .. ( z,..., z ),..., If we use L Z + + Z Z + s rereseio of he demd durig he led ime of J eriods, he E [ L ] μ () L s Vr [ ] σ γ s where γ k Cov[ Z Z k ] is he uocovrice lg K...,..., ( z+,..., z+ z, z,..., z ) ( z,..., z+ z, z,..., z ),..., + is he codiiol robbiliy disribuio for he eriod demd durig he led ime. If we defie he led ime demd s he fuure vlue s L ( ) Z + + Z Z + (4) give h we hve observed he s vlues Z, Z,..., Z which occurred rior o he eriod ime, he we will hve he execed vlue d vrice ivolved i he reorder oi clculio s E[ L ( ),,..., ] ˆ () ˆ ()... ˆ Z Z Z Z + Z + Z ( ) (5) where Zˆ ( ) is he forecs vlue, d where g + k i, i i, k i i k > i Cov [ L ( ), L ( + k)] ψ iψ k + i σ g, + k 0 Vr [ L ( )] σ { g + g } (6) σ, d (7), is he uocorrelio of he forecs errors bewee he eriod J, d J + K. If he demd is believed o be esseilly rereseed by he Box d Jekis ime series rocess, he he demd forecs error for eriod + bsed o d hrough eriod is e ( ) Z Zˆ ( ) + (8) where Zˆ ( ) is he miimum me squred error forecs seleced from oe of he Box d Jekis models for eriods from he origi. We c see he differece bewee he firs d secod momes of he wo kids of disribuios. Ee d Mri [5] hve show h usig he firs d secod momes from he firs kid of robbiliy disribuio led o he icosise resul of he reorder oi vlue. As oied ou by Chor, Gilles d Mqbool [4], mgers hve bee uder icresig ressure o reduce iveories o sremlie he suly chi. Their gol is o reduce iveories wihou hurig he level of service rovided o cusomers. Sfey sock, however, is fucio of he cycle service level, he demd uceriy, he releishme led ime, d he led ime uceriy. I his er, we demosre how o comue he reorder oi d he sfey sock level whe he led ime demd c be roriely rereseed by Box-Jekis Time Series Model. 3. Deermiisic Led Time Of J Periods The lysis i his secio roceeds s suggesed by Ee d Mri [5], i.e., firs secify he service level (α ) give -eriod led ime. The choose reorder oi R such h he robbiliy of sockig ou durig he led ime does o exceed α. Th is, selec he smlles R such h (9) Pr[ Z > R] + i () (3)

3 The firs se i selecig R is o forecs he demd durig he led ime. Th mes h he ime, forecss re required for eriods for,,..., he use he forecs error robbiliy durig he led ime o selec he reorder oi, i.e., Pr[ Z Zˆ > R Zˆ (0) ] + i i i Pr[ e ( ) > ] where R Zˆ i, he sfey sock. If we le U ( ) e ( ), he ol forecs error durig he -eriod led ime immediely followig eriod, he ccordig o he defiiio of forecs error defied i he Box d Jekis ime series rocess, i follows h ( ) is ormlly disribued wih U d where E[ U ( )] 0 () k Vr[ U ( )] Vr[ e ( )] + Cov[ e, e ( k)] (3) Cov[ e ( h), e ( h k h + i b+ i + i 0 b)] σ ψ ψ σ g ( h, h b) (4) is he covrice bewee he -origi forecss led imes h d h + b, he vlue of is obied from esime of he rocess residul sdrd deviio usig ime series d. ψ, ψ,... re clled he error lerig coefficies clculed direcly by equig coefficies of B from he followig equio. ( ϕ B ϕ B...)( + ψ B + ψ B +...) ( θ B θ B...) (5) whereϕ is he rmeer of he uoregressive erm Z, d θ is he rmeer of he movig verge erm. Usig he defiiio of he orml iverse fucio, he reorder oi level for give α service level is R Zˆ NORMINV ( α,0, Ω ( )) (6) Ω ( ) where Ω ( ) Vr( U ( )) (7) Aoher roch o deermie reorder oi is requirig he ercege of orders filled o ime be greer h or equl o β which is defied s he fill re of he sysem β EB ( ) / Q (8) where Q is he order quiy, d EB ( ) is he execed shorge er cycle. Is formul is σ () where k EB ( ) Ω ( )[ Normdis( k,0,,0) k ( k Normdis( k,0,,))] (9) R Zˆ Ω (0)

4 4. Sochsic Led Time If we ssume h he led ime rdom vrible kes o he vlues (,, 3, ) wih robbiliy P. If f (.) is he desiy fucio for U he he desiy of demd durig he led ime, is give by d f ( U ) f ( U ) () F ( U ) F ( U ) To isure h he robbiliy of o more h α erce of sockig ou durig he led ime, R is seleced o be he smlles umber such h () is F( R) F ( R) (3) If we defie he sdrd deviio of he ( ) U o be Ω ( ), he he Excel formul for he service level NORMDIST (,0, Ω ( )) α (4) The formul for he fill re wih sochsic led ime is EB( ) Q β (5) where EB ( ) is defied i (9). For bckorder model, he execed ol ul iveory sysem cos cosis of he sum of orderig cos d holdig cos (see Hdley d Whii [6], d Silver d Peerso [7]). If he ul demd is W, d he cos er order is A, he he ul orderig coss re (W/Q)A. If he ul holdig coss er ui is IC, d he execed me led ime is d Z, he holdig cos c be rereseed s ( Q / + r d) IC, where Q/ i is he verge iveory, d r-d is he verge level of sfey sock. If β is he fill re, he he ler is ieresed i meeig redeermied frcio of uis (β ) off he shelf. The he ler c use he soluio of he followig model o fid he oiml olicy. Miimize TC ( Q, r) Mi( W / Q) A+ ( Q / + r d) IC (6) Subec o EB ( ) Q β (7) 5. Illusrio Of The Comuios Of The Forecs Zˆ ( ) Ad Vr[ U ( )] This secio illusred he comuios of Zˆ ( ) d Vr[ U ( )]. To obi he forecs Zˆ ( ), oe wries he model i differece form, d use he followig rules:

5 . Use he vilble d o comue he kow rdom shocks ' s, he oe-se hed forecs errors from Z Zˆ (). Noe h which is reled o he uvilble Z d will be ssiged vlue of zero.. Leve Z (,,...) uchged becuse hey lredy heed origi. 3. Relce Zˆ + (,,...) wih heir forecss Zˆ ( ) origi becuse hey hve o heed. 4. Le + (,,...) be zero becuse hey hve o ye heed. For exosiory urose, he followig ime series model d d re used i Figure o illusre he comuios usig our desiged Excel Temles. Suose h he led-ime demd c be rereseed by ARIMA(,) model s Z.6Z + 0.6Z , wih he d for Z 47, Z 48, Z 49, Z 50, d σ Figure Excel Temle for Comuig Z ' s Vlues d l ψ s Weighs 5.. Comuio of he Forecs vlues Z ( )' s () Eer he followig vilble d Z 47., Z48., Z49.9, Z50 3 i cells C4:C7. () Use he equio Z.6Z + 0.6Z o comue 49. Sice here were o d vilble o comue 47 d 48, we equed hem o zeroes, heir execed vlues. The cell formul for 49 i cell D6 is C6.6*C *C *D5 0.4*D4.58. I ws exeded o cell D7 o fid 50. Sice he forecs origi is 50, for 5, 5, 53, i cells D8,, D7 re he fuure vlues of s which hve o occurred d hus re give he vlues of zeroes. (3) Geere he forecs vlues Z50( ), Z50(),..., Z50(0) usig Z.6Z + 0.6Z The formul i C8 o comue Z ( ) is 50.6*C7 0.6*C6 + D7 0.83*D *D5. The vlues for Z50( ),..., Z50(0) c be obied by exedig he formul.

6 5.. Comuio of Vr[ U ( )] We hve develoed he Excel formule o comue he lerig error weighs for he ARIMA model. () Before usig our Excel formule, he ime series model hs o be exressed i he form of Equio (5), he use SUMPRODUCT(OFFSET(THI,0,-,,),OFFSET(PSI,0,0,,)) o comue ψ for,,,. d use SUMPRODUCT(OFFSET(THI,0,0,, +),OFFSET(PSI,0, -,,) o comuie ψ for +, N, where THI, THETA, d PSI re defied s he mes for he row vecors ( ϕ, ϕ,..., ϕ, ), (,θ,...,θ q ), d (,ψ,...,ψ s ), resecively. h g h, h + b) ψ iψ b+ i 0 () Use he ψ - weigh from (..) o comue ( of Equio (4). We emloyed D/Tble o geere he vlues of g ( h, h + b) usig b,,.., h- for he row, h,,, L for he colum, d SUMPRODUCT(OFFSET(PSI, 0, 0,, l), OFFSET(PSI, 0, b,, l)) s he fucio for he Tble. Noice h cusom umber form code ;;; is used i cell H5 o hide is vlue. Figure is he Tble of cumulive vlues of G. Cumulive of g i,i s re lised i he rge F6:F35. The cumulive g h,h+b s re lised log he rows i he rge G6:O34. We me Figure II- (E6:O35) s Cumulive_Tble, he cree he G-Vlue Tble o lis he cumulive vlues of g i,i s d g h,h+b s by usig he D/Tble commd wih IF(D4>D4, VLOOKUP(D4-D4, Cumulive_Tble,D4+,0),0) i cell F40, d G38 d G39 s row iu cell d colum iu cell, resecively. The vlues of g i,i s d g h,h+b s re he used o fid he vrices for differe led ime vlues. Give σ 5. 78, we eered (G4+*SUM(H4:N4))*5.78^ i cell O4 o comue Vr[ U ( )] for h. The vrices for led ime vlues rgig from o 8 c be obied by exedig he formul. Figure. Cumulive G Tble, b. G-Vlues Tble d Vrice Vr[ U ( )]

7 6. Coclusio I his er, we emloyee he Box-Jekis forecsig echique o del wih he cses whe he eriod demds re o ideede, bu exhibis serilly correled demd rocess which c be rereseed s uoregressive-movig verge ARMA (.d.q)-model for boh deermiisic d discree sochsic led ime. We lso rese Excel bsed mehodology for fidig he reorder oi d he sfey sock level. This roch is flexible d cble of hdlig he relisic scerio whe boh he demd d led ime re rdomly disribued. Develoig Excel emle does o require high-level rogrmmig kowledge d skills. I ddiio, he build-i robbiliy desiy fucios, disribuio fucios, d he D Tble commd remedously simlify he ierive comuios by elimiig he eed o look for vlues from he sisicl bles. Moreover, he udig equios of Box-Jekis ogeher wih he wh-if lysis cbiliy of Excel mke i ossible o ude he reorder oi d sfey sock eriodiclly. The user friedliess d buil-i cbiliies of Excel mkes sredshee iveory-corol licio low-cos ool d model simulor which is esy for whever modificio ecessry o beer d o eeds d eviromes of he mrke. Refereces [] A, B. G., S. B. Foooulos, d M. C. Wg, Esimig he Led-Time Demd Disribuio for Auocorreled Demd by he Perso Sysem d Norml Aroximio, Nvl Reserch Logisics, Vol. 36, No. 4, 989. [] Box, G. E. P., d G. M. Jekis, Time Series Forecsig d Corol, Holde-Dy, S Frcisco, 976. [3] Chrles, J. M., H. Mrmorsei, d W. Zi, Sfey Sock Deermiio wih Serilly Correled Demd i Periodic-Review Iveory Sysem, Jourl of he Oeriol Reserch Sociey, Vol. 46, No. 8, 995. [4] Chor, S., R. Gilles, d D. Mqbool, The Effec of Led Time Uceriy o Sfey Socks, Decisio Sciece, Vol. 35, No., Wier 004. [5] Ee, G. D. d R. K. Mri, Deermiig Sfey Sock i The Presece of Sochsic Led Time d Demd, Mgeme Sciece, Vol. 34, No., November 988. [6] Hdley, G., d T. M. Whie, Alysis of Iveory Sysems, Preice-Hll., Eglewood Cliffs, N.J, 984. [7] Silver, E. A., d R. Peerso, Decisio Sysem for Iveory Mgeme d Producio Plig. New York: Joh Wiley 985.