DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS. Exponential Smoothing for Inventory Control: Means and Variances of Lead-Time Demand

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1 ISSN X ISBN X AUSTRALIA DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS Exponenal Smoohng for Invenory Conrol: Means and Varances of Lead-Tme Demand Ralph D. Snyder, Anne B. Koehler, Rob J. Hyndman and J. Keh Ord Workng Paper 3/00

2 EXPONENTIAL SMOOTHING FOR INVENTORY CONTROL: MEANS AND VARIANCES OF LEAD-TIME DEMAND Ralph D. Snyder Deparmen of Economercs and Busness Sascs P.O. Box E, Monash Unversy, VIC 3800, Ausrala Emal: Anne B. Koehler Deparmen of Decson Scences and Managemen Informaon Sysems Mam Unversy, Oxford, OH 45056, USA. Rob J. Hyndman Deparmen of Economercs and Busness Sascs Monash Unversy, VIC 3800, Ausrala J. Keh Ord McDonough School of Busness 30 Old Norh Georgeown Unversy, Washngon, DC 0057, USA. 30/0/00 Exponenal Smoohng_40.doc

3 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand ABSTRACT Exponenal smoohng s ofen used o forecas lead-me demand for nvenory conrol. In hs paper, formulae are provded for calculang means and varances of lead-me demand for a wde varey of exponenal smoohng mehods. A feaure of many of he formulae s ha varances, as well as he means, depend on rends and seasonal effecs. Thus, hese formulae provde he opporuny o mplemen mehods ha ensure ha safey socks adus o changes n rend or changes n season. KEYWORDS Forecasng; nvenory conrol; lead-me demand; exponenal smoohng; forecas varance.

4 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand. INTRODUCTION Invenory conrol sofware ypcally conans a forecasng module based on exponenal smoohng. The purpose of such a module s o feed means and varances of lead-me demand o an nvenory conrol module for he deermnaon of orderng parameers such as reorder levels, order-up-o levels and reorder quanes. Typcally, exponenal smoohng s chosen because has a proven record for generang sensble pon forecass (Gardner, 985). To be more specfc, consder he ypcal suaon where a replenshmen decson s o be made a he begnnng of perod n+. Any order placed a hs me s assumed o arrve a lead-me laer a he sar of perod n + λ. Invenory heory dcaes ha he prmary focus should be on lead-me demand, an aggregae of unknown fuure values y n + defned by Y λ n ( λ ) yn+ = =. () The problem s o make nferences abou he dsrbuon of lead-me demand. Typcally an approprae form of exponenal smoohng s appled o pas demand daa y, y, he resuls beng used o predc he mean of he lead-me demand dsrbuon. n Varances of lead-me demand are also needed for he mplemenaon of nvenory sraeges ha provde a proecon agans he wors effecs of unceran cusomer demand. Unl Johnson and Harrson (986) derved a varance formula for use wh smple exponenal smoohng, raher ad-hoc formulae were he vogue n nvenory conrol sofware. Usng a smple sae space model, Johnson and Harrson ulzed he fac ha smple exponenal smoohng emerges as he seady sae form of he assocaed Kalman fler n large samples. Adopng a dfferen model, Snyder, Koehler and Ord (999) were able o oban he same formula whou recourse o he Kalman fler sraegy. The advanage of her approach s ha no resrcve large sample assumpon s needed. Johnson and Harrson (986) also obaned a varance formula for rend correced exponenal smoohng. Yar and Chafeld (990), however, have suggesed a slghly dfferen formula. They also provde a formula ha ncorporaes seasonal effecs for use wh he addve Wners (960) mehod. The purpose of hs paper s o ake a fresh look a he problem of dervng formulae for forecas varances of lead-me demand. We use he lnear verson of he sngle source of error

5 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand model from Ord, Koehler and Snyder (997) o unfy he dervaons. We also provde useful exensons o accommodae errors ha depend on rend and seasonal effecs. The model and s specal cases are nroduced n Secon. Assocaed formulae for means and varances of lead-me demand are presened n Secon 3. General prncples used n her dervaon are presened n he Appendx. Throughou he paper, we adop a convenon concernng he sum operaor Σ. In hose cases where he lower lm s less han he upper lm, he sum should be equaed o zero.. MODELS FOR EXPONENTIAL SMOOTHING Fuure values of a me seres are unknown and mus be reaed as random varables. Ther behavor mus be lnked o a sascal model n order o derve predcon dsrbuons. A model should have he poenal o nclude unobserved componens such as levels, growh raes and seasonal effecs, because varous forms of exponenal smoohng are based on hese conceps. Common cases of exponenal smoohng and her models are shown n Table. The column marked Code uses nomenclaure from Hyndman e al (00). Here N desgnaes None, A desgnaes Addve and D desgnaes Damped. All codes nvolve wo leers. The frs leer s used o descrbe he rend. The second leer descrbes he seasonal componen. The varous componens are for local level, b for local growh rae, s for local seasonal effec and e for a random varable desgnang he rregular componen. The α, β, γ are so-called smoohng parameers. The φ, anoher parameer, s a dampng facor. The purpose of he care symbol s oulned laer. Case Code Model Smoohng Mehod Descrpon NN y = + e αe = + ˆ = ˆ + α ( y yˆ ) AN y = + b + e = + b + αe b = b + αβ e yˆ = ˆ yˆ = ˆ + bˆ α ˆ = ˆ + bˆ + y yˆ bˆ = bˆ + αβ y yˆ Smple exponenal smoohng (Brown, 959) Trend-correced exponenal smoohng (Hol, 957) 3

6 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand 3 AD y = + b + e = + b + αe b = φb + αβe yˆ = ˆ + bˆ α ˆ = ˆ + bˆ + y yˆ bˆ = φbˆ + αβ y yˆ Damped rend (Gardner and McKenze, 985) 4 y = s m + e s s γ e = + sˆ = sˆ + γ ( y yˆ ) m yˆ = sˆ m m Elemenary seasonal case 5 AA y = + b + s m + e = + b + αe b = b + αβ e s = s + γ e 6 DA y = + b + c m + e = + b + αe b = φb + αβ e s = s + γ e yˆ = ˆ + bˆ + sˆ m α ˆ = ˆ + bˆ + y yˆ bˆ = bˆ + αβ y yˆ sˆ = sˆ + γ y yˆ m yˆ = ˆ + bˆ + sˆ m α ˆ = ˆ + bˆ + y yˆ bˆ = φbˆ + αβ y yˆ sˆ = sˆ + γ y yˆ m Table. Models for Common Lnear Forms of Exponenal Smoohng. Wners addve mehod (Wners, 960) Damped rend wh seasonal effecs Each model n Table conans a measuremen equaon ha specfes how a seres value s bul from unobserved componens. I conans ranson equaons ha descrbe how he unobserved componens change over me n response o he effecs of srucural change. I nvolves a random varable represenng he rregular componen. All he models n Table are specal cases of wha s bes called a sngle source of error sae space model. The unobserved componens are sacked o gve a vecor x. I s assumed ha all componens combne lnearly o gve he seres value, so he measuremen equaon s specfed as y = h x + e () where h s a fxed vecor of coeffcens. The lag on x s used o reflec he assumpon ha he condons a me - deermne wha happens durng he perod. The evoluon of he unobserved componens s governed by he frs-order ranson relaonshp x = Fx + ge () 4

7 . Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand where F s a fxed marx and g s a fxed vecor ha reflecs he mpac of srucural change. I s possble o hnk of he frs componen of () as an underlyng level and o desgnae by m = h x. I s possble ha he dsurbance s ndependen of hs level. I s also possble ha s varance ncreases wh hs level. Boh possbles are capured by he assumpon ha he dsurbance s governed by he relaonshp e r m ε = for r = 0, (3) where ε s a member of a wren as y = m + ε when 0 NID 0,σ seres? The measuremen equaon may now be r = or y m ( ε ) = + when r =. In he laer case, heε s a un-less quany, convenenly hough of as a relave error. I means ha he rregular componen poenally depends on he oher componens of a me seres, somehng ha can be very mporan n pracce. The elemens h, F, g poenally depend on a vecor of parameers desgnaed by ω. I s assumed ha he same model governs boh pas and fuure values of a me seres. Pas values are known, n whch case s possble o make a pass hrough he daa, applyng a compable form of exponenal smoohng n each perod. Suppose, a he begnnng of ypcal perod, pas applcaons of exponenal smoohng have yelded he value xˆ for he sae vecor x. Afer observng y a he end of perod, s possble o calculae he error e = y h xˆ. The error can be subsued no he ranson equaon o gve ( ) xˆ ˆ ˆ = Fx + g y h x for he value of he sae vecor x. Gven he progressve naure of hs algorhm, s clear ha xˆ = x y, y, x0, ω Inducon may be used o confrm ha x ˆ s a fxed value. A specal case of he above model, bes ermed a compose model, s now consdered. The sae vecor x s paroned no random sub-vecors desgnaed by x, and x,. The measuremen equaon has he form y = h x + h x + e (4),, 5

8 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand where h and h are sub-vecors of h. The sub-vecors of he sae vecor are governed by ranson equaons x = F x + g e k = (5) k, k k, k, where F, F are ranson marces and g, g are sub-vecors of g. The specal feaure of hs compose model s ha he ranson equaon for x, does no conan x, and vce versa. I s shown n he Appendx ha he resuls for a compose model can be bul drecly from hose of s consuen models. All he models n Table are specal cases of he sngle source of error model or he compose model. The lnks wh hese general models are provded n Table. Here 0 k refers o a k-vecor of zeros and I k refers o a k k deny marx. Noe ha alhough he seasonal cases are governed by mh-order recurrence relaonshps, hey are convered o equvalen frs-order relaonshps. Also noe ha ω s a vecor formed from some or all of he parameers α, β, γ, φ. Case x h F g x = h = F = g = α = [ ] [ ] x b = [ ] [ ] x b h = F 0 h = F 0 φ = [ ] h [ 0 m ] x s s + m [ ] x = b, [ ] x = s s +, m = m h = [ ] h = [ 0 ] m F = = [ ] = = [ ] 0 I 0 m m g α αβ g α αβ = g = [ γ 0 m ] F = 0 F 0 = m! " Im 0m [ ] g = α αβ [ ] g γ = 0 m 6

9 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand 6 [ ] x = b, [ ] x = s s +, m h = [ ] h = [ 0 ] m F = 0 φ F 0 = m! " Im 0m [ ] g = α αβ [ ] g γ = 0 m Table. Conformy of Specal Cases o he General Model or Compose Model. An nrgung nsgh from Table s ha each smoohng mehod apples for boh a homoscedasc and a heeroscedasc model. Now, each homoscedasc case s equvalen o an ARIMA process (Box, Jenkns and Rensel, 994). However, no heeroscedasc case s equvalen o an ARIMA process. Thus, exponenal smoohng apples for a wder class of models han he ARIMA class (Ord, Koehler and Snyder, 997). In he homoscedasc cases, only he mean poenally depends on rend and seasonal effecs. However, n he heeroscedasc cases, boh he mean and he varance of he rregular componen depend on rend and seasonal effecs. Thus, predcon varances reflec rend and seasonal effecs n he heeroscedasc case, a feaure ha s poenally que useful n pracce. Many oher cases are concevable when addon operaors are replaced n he measuremen equaon by mulplcaons. Examples of such cases are presened n Hyndman, Koehler, Snyder and Grose (00). A varey of models underlyng he mulplcave verson of Wners mulplcave mehod have been nroduced n Koehler, Snyder and Ord (00). The complexy of hese non-lnear possbles precludes he dervaon of resuls usng he mehodology of hs paper. 3. MEANS AND VARIANCES OF LEAD TIME DEMAND I s assumed ha mehods smlar o hose descrbed n Ord, Koehler and Snyder (997) have been appled o pas demand daa o esmae he parameers of an approprae model The problem s now o fnd he momens of he lead-me demand (). Our analyss s bul, n par, on predcon varance resuls from Hyndman, Koehler, Ord and Snyder (00) for convenonal predcon dsrbuons. I s shown n he Appendx ha lead-me demand can be resolved no a lnear funcon of he uncorrelaed rregular componens: 7

10 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand Y λ λ = µ + + C e +. (6) n n n = = λ where µ n h + F xn = (7) s he mean of he -sep predcon dsrbuon. I s furher esablshed ha he coeffcens of he errors n (6) are gven by where C λ = + c for =,, λ. (8) =. (9) c h F g Parcular cases of he formulae for he means µ n + and he coeffcens C are shown n Table 3. Noe ha φ = φ ; 0 φ = φ ; p = ; d, m + m m = f s a mulple of m and d, m = 0 oherwse. The resuls for Case 5 and Case 6 are consruced by addng he correspondng resuls for consuen basc models, an approach ha s also raonalzed n he Appendx. Case ˆ n µ n + c C α + ( λ ) ˆ n + b n α ( + β ) ˆ ˆ α ( βφ ) n n 3 +φ bˆ 4 s ˆn+ pm d, m ( + ) + d, 5 ˆ + bˆ + ˆ n n s n + pm 6 ˆ + φ bˆ + sˆ n n n+ pm α ( λ )( λ + ) + λ α + αβ + γ m α β γ ( + ) + d, α βφ γ m + λ α + λ αβφ αβφ + γ λ d λ λ, m ( λ )( λ + ) + ( λ ) α + αβ + γ λ + ( λ ) α + ( λ ) αβφλ αβφλ + γ d λ, m d, m 8

11 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand Table 3. Key Resuls for Basc models. From (6), he condonal varance s gven by λ Y x C ( n ( λ ) n ω ) σ var, =. (0) = n he homoscedasc case. All he nformaon needed o evaluae he grand mean and he grand varance s avalable n Table 3. In he heeroscedasc case he grand varance s where θn+ E ( mn+ xn, ω ) λ var Y λ x, ω = σ C θ + () ( ) n n n = =. I s esablshed, n he Appendx, ha he heeroscedasc formulae may be compued usng he recurrence relaonshp n+ n+ c n+ θ = µ + θ σ () where he c are also gven n Table CONCLUSIONS Formulae for calculang he mean and varance of lead-me demand have been derved for many common forms of exponenal smoohng n hs paper. For he homoscedasc cases, he predcon dsrbuons are Gaussan, so he means and varances provde all he nformaon requred o make probablsc saemens abou fuure lead-me demand. In heory, he predcon dsrbuons for he heeroscedasc cases are no Gaussan. However, a numercal sudy n Hyndman, Koehler, Ord and Snyder (00) ndcaes ha here s lle error nvolved n approxmang hem by a Gaussan dsrbuon. The same concluson mus apply o lead-me dsrbuons where aggregaon mus help o furher reduce he approxmaon error. By usng he sngle source of error sae space model, we have unfed he dervaon of he formulae. In he homoscedasc cases, many of he formulae obaned n hs paper agree wh hose found n earler work (Johnson and Harrson, 986; Yar and Chafeld, 990; Snyder, 9

12 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand Koehler and Ord, 999). A small advance was obaned n relaon o Wners addve seasonal mehod n ha he recursve varance formulae n Yar and Chafeld (990) has been replaced by a closed counerpar. Furhermore, we have obaned, for he frs me, formulae for he varance of lead-me demand for he damped rend cases. I has been argued n he paper ha he rregular componen of a demand seres can depend on rend and seasonal effecs. Thus, a maor par of our conrbuon has been he provson of lead-me demand varance formulae for heeroscedasc exensons o exponenal smoohng. Such formulae adm he possbly of smarer approaches o safey sock deermnaon. I s now possble o mplemen schemes ha alor levels of safey sock o changes n rend or changes n season. REFERENCES Box, GEP, Jenkns, GM and Rensel, GC (994) Tme Seres Analyss: Forecasng and Conrol (hrd edon), Prence-Hall, Englewood Clffs. Brown, RG (959) Sascal Forecasng for Invenory Conrol. Mc.Graw-Hll, New York. Gardner, ES Jr (985) Exponenal Smoohng: The Sae of he Ar. Journal of Forecasng. 4: -8. Gardner, ES and E McKenze (985) Forecasng Trends n Tme Seres, Managemen Scence, 3: Harvey, AC and Snyder, RD (990) Srucural Tme Seres Models n Invenory Conrol. Inernaonal Journal of Forecasng. 6: Hol, CE (957) Forecasng Trends and Seasonal by Exponenally Weghed Averages. ONR Memorandum No. 5, Carnege Insue of Technology, Psburgh, USA. Hyndman, RJ, Koehler, AB, Snyder, RD and Grose, S (00) A Sae Space Framework for Auomac Forecasng usng Exponenal Smoohng Mehods. Inernaonal Journal of Forecasng. (forhcomng). Hyndman, RJ, Koehler, AB, Ord, JK and Snyder, RD (00) Predcon Inervals for Exponenal Smoohng Sae Space Models. Workng Paper /00, Deparmen of Economercs and Busness Sascs, Monash Unversy. 0

13 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand Johnson FR and Harrson PJ (986). The varance of lead me demand. Journal of he Operaonal Research Socey, 37: Koehler AB, Snyder, RD and Ord, JK (00). Forecasng Models and Predcon Inervals for he Mulplcave Hol-Wners Mehod. Inernaonal Journal of Forecasng, 7: Ord, JK, Koehler, AB and Snyder, RD (997) Esmaon and Predcon for a Class of Dynamc Nonlnear Sascal Models. Journal of he Amercan Sascal Assocaon. 9: Snyder, RD (985) Recursve Esmaon of Dynamc Lnear Sascal Models. Journal of he Royal Sascal Socey, B. 47: Snyder, RD, Koehler, AB and Ord, JK (999) Lead-me Demand for Smple Exponenal Smoohng. Journal of he Operaonal Research Socey. 50: Wners, PR (960) Forecasng Sales by Exponenally Weghed Movng Averages. Managemen Scence. 6: Yar, M. and Chafeld, C (990) Predcon nervals for he Hol-Wners Forecasng Procedure. Inernaonal Journal of Forecasng. 6: APPENDIX General resuls governng he formulae n Table 3 are derved n hs Appendx. To ge he formulae governng Cases -4, back solve he ranson equaon () from perod n + o perod n, o gve n+ n n+ x = F x + F ge (A) Lag (A) by one perod, pre-mulply he resul by h, and use he defnons (7) and (9) o ge

14 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand m = µ + c e. (A) n+ n+ n+ Recall ha e s gven by (3) so ha E ( en+ ) = σ E( mn+ ). Then we may square (A) and ake expecaons o gve he recurrence relaonshp () for he heeroscedasc facors. Subsue (A) no () o gve y = µ + c e + e n+ n+ n+ n+ λ n = µ n+ + n+ + n+ = Y c e e he C are defned by (8). Noe ha he dervaon of he equaons: C λ = and C = C + + c λ for = λ,,.. Subsue hs no () o gve. Rearrange erms o yeld he requred resul (6) where C s expeded usng he followng Cases 5 and 6 are compose models. Each ranson equaon (5), for a compose model, has he same srucure as (). Thus, k, n+ k k, n k k n+ x = F x + F g e. (A3) Lag () by one perod and pre-mulply he resul by h k o gve m = µ + c e (A4) k, n+ k, n+ k, n+ where µ k, n h + k Fk xk, n = (A5) and c = h F g. (A6) k, k k k Subsue (A4) no mn+ = m, n+ + m, n+ o yeld he earler equaon (A) where µ = µ + µ (A7) n+, n+, n+

15 Exponenal Smoohng for Invenory Conrol: Mean and Varances of Lead-me Demand and c = c + c. (A8),, Thus, he formula C = C, + C, may be used o derve he resuls for Case 5 and Case 6 from her consuen basc cases. In he heeroscedasc cases, he approprae facors are sll derved wh he relaonshp (). 3

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