EXPONENTIAL SMOOTHING MODELS: MEANS AND VARIANCES FOR LEAD-TIME DEMAND

Transcription

1 EXPONENTIAL SMOOTHING MODELS: MEANS AND VARIANCES FOR LEAD-TIME DEMAND Ralph D. Snyder Deparmen of Economerics and Business Saisics P.O. Box E, Monash Universiy, VIC 3800, Ausralia Anne B. Koehler Deparmen of Decision Sciences and Managemen Informaion Sysems Miami Universiy, Oxford, OH 45056, USA. Rob J. Hyndman Deparmen of Economerics and Business Saisics Monash Universiy, VIC 3800, Ausralia J. Keih Ord [Corresponding Auhor] McDonough School of Business 320 Old Norh, Georgeown Universiy, Washingon, DC 20057, USA. 4/5/03 Exponenial Smoohing_040803R.doc

2 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand ABSTRACT Exponenial smoohing is ofen used o forecas lead-ime demand for invenory conrol. In his paper, formulae are provided for calculaing means and variances of lead-ime demand for a wide variey of exponenial smoohing mehods. A feaure of many of he formulae is ha variances, as well as he means, depend on rends and seasonal effecs. Thus, hese formulae provide he opporuniy o implemen mehods ha ensure ha safey socks adus o changes in rend or changes in season. An example using weekly sales shows how safey socks can be seriously underesimaed during peak sales periods. KEYWORDS Forecasing; invenory; lead-ime demand; exponenial smoohing; forecas variance.

3 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand. INTRODUCTION Invenory conrol sofware ypically conains a forecasing module ha predics he mean and variance of lead-ime demand. These values are incorporaed ino an invenory conrol module for he deerminaion of ordering parameers such as reorder levels, order-up-o levels and reorder quaniies. These forecasing modules ofen rely upon exponenial smoohing mehods (iniially inroduced by R.G. Brown, 959), as hey are inuiively appealing, easy o updae and have minimal compuer sorage requiremens. Brown s iniial mehods, combined wih Hol s (957) local linear rend mehod and he Hol- Winers (Winers, 960) schemes for series displaying boh rend and seasonal paerns provide reasonably good coverage of likely behaviors o be me in pracice, paricularly when he damped rend mehod of Gardner and McKenzie (985) is included. Overall, exponenial smoohing mehods have a proven record for generaing sensible poin forecass (Gardner, 985; Makridakis and Hibon, 2000). For a review of recen developmens of saisical models for exponenial smoohing, see Chafield e al. (200). The basic problem in invenory conrol may be formulaed as follows. Suppose ha a replenishmen decision is o be made a he beginning of period n+. Any order placed a his ime is assumed o arrive a lead-ime laer a he sar of period n + λ. Invenory heory dicaes ha he primary focus should be on lead-ime demand, an aggregae of unknown fuure values y n + defined by Y ( ) λ = y. () n λ n+ = The problem is o make inferences abou he disribuion of lead-ime demand. Typically an appropriae form of exponenial smoohing is applied o pas demand daa y,..., y n, he resuls being used o predic he mean of he lead-ime demand disribuion. For mos of he paper, we assume ha λ is fixed, bu in secion 5 we briefly consider sochasic leadimes. Fixed lead-imes are relevan when suppliers make regular deliveries, an increasingly common siuaion in supply chain managemen. 2

4 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Many invenory managemen sysems require he variance of lead-ime demand in order o implemen an invenory sraegy, bu he basic exponenial smoohing procedures originally provided only poin forecass and raher ad-hoc formulae were he vogue in invenory conrol sofware. Then Johnson and Harrison (986) derived a variance formula for use wih simple exponenial smoohing. Using a simple sae space model, Johnson and Harrison uilized he fac ha simple exponenial smoohing emerges as he seady sae form of he associaed Kalman filer in large samples. Adoping a differen model, Snyder, Koehler and Ord (999) were able o obain he same formula wihou recourse o he Kalman filer sraegy. The advanage of heir approach is ha no resricive large sample assumpion is needed. Johnson and Harrison (986) also obained a variance formula for lead-ime demand when rend-correced exponenial smoohing is employed. Yar and Chafield (990), however, have suggesed a slighly differen formula. They also provide a formula ha incorporaes seasonal effecs for use wih he addiive Winers (960) mehod. Harvey and Snyder (990) obain similar variance formulae for level, rend and seasonal cases using a srucural ime series framework. They rely on coninuous ime models so ha he links wih exponenial smoohing are more obscure. Mos of he work discussed so far makes he (someimes implici) assumpion ha he variance of he demand per uni ime [DPUT] process is consan. Ye, as Brown (959, p. 94) observed you will be very likely o find ha he sandard deviaion of demand is nearly proporional o he oal annual usage, or o he average monhly usage. Indeed, some auhors in he invenory lieraure have buil upon his idea, noably Miller (986) and Loveoy (990). However, hese auhors assume zero lead-imes. Heah and Jackson (994) generae forecass for individual fuure ime periods, bu do no examine lead-ime demand. Thus, a sysemaic framework for he developmen of forecas variances for leadime demand has no been available. The purpose of his paper is o ake a fresh look a he problem. We use he linear version of he single source of error model from Ord, Koehler and Snyder (997) o unify he derivaions. We also provide useful exensions o accommodae errors ha depend on rend and seasonal effecs. This aspec of he resuls is paricularly imporan since he variance 3

5 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand ypically increases during peak sales periods so ha safey socks could be seriously underesimaed a precisely hose imes ha are poenially mos profiable.. Srucure of he paper The model and is special cases are inroduced in Secion 2. Associaed formulae for means and variances of lead-ime demand are presened in Secion 3. The general principles used in heir derivaion are presened in he Appendix. Some numerical examples, and he resuls from applying hese formulae o real demand daa, are explored in Secion 4. Issues associaed wih sochasic lead-imes are examined in Secion 5 and conclusions and direcions for furher research are discussed in secion 6. Throughou he paper, we adop a convenion concerning he sum operaor Σ. In hose cases where he upper limi is less han he lower limi, he sum should be equaed o zero. 2. MODELS FOR EXPONENTIAL SMOOTHING Fuure values of a ime series are unknown and mus be reaed as random variables. Their behavior mus be linked o a saisical model in order o derive predicion disribuions. A model should have he poenial o include unobserved componens such as levels, growh raes and seasonal effecs, because various forms of exponenial smoohing are based on hese conceps. Common cases of exponenial smoohing and heir models are shown in Table. The column marked Code uses nomenclaure from Hyndman e al (200). Here N designaes None, A designaes Addiive and D designaes Damped. All codes involve wo leers. The firs leer is used o describe he rend. The second leer describes he seasonal componen. The various componens are l for local level, b for local growh rae, s for local seasonal effec and e for a random variable designaing he unpredicable componen. The α, β, γ are so-called smoohing parameers. The φ, anoher parameer, is a damping facor. The purpose of he care symbol is oulined laer. Each model in Table conains a measuremen equaion ha specifies how a series value is buil from unobserved componens. I conains ransiion equaions ha describe how he 4

6 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand unobserved componens change over ime in response o he effecs of srucural change. I involves a random variable represening he unpredicable componen. Case Code Model Smoohing Mehod Descripion NN y = + e l αe = + ˆ = ˆ + α ( y yˆ ) 2 AN y = + b + e = + b + αe b = b + αβ e 3 AD y = + b + e = + b + αe b = φb + αβ e yˆ = ˆ yˆ = ˆ + bˆ ˆ = ˆ + bˆ + y yˆ α ( ) ( ) bˆ = bˆ + αβ y yˆ yˆ = ˆ + bˆ ˆ = ˆ + bˆ + y yˆ α ( ) ( ) bˆ = φbˆ + αβ y yˆ Simple exponenial smoohing (Brown, 959) Trendcorreced exponenial smoohing (Hol, 957) Damped rend (Gardner and McKenzie, 985) 4 y = s m + e s s γ e = + sˆ = sˆ + γ ( y yˆ ) m yˆ = sˆ m m 5 AA y = + b + s m + yˆ ˆ ˆ ˆ = + b + s m = + b + αe ˆ ˆ ˆ ˆ = + b + α y y b = b + αβ e bˆ ˆ ( ˆ = b + αβ y y ) s = s + γ e sˆ = sˆ + γ y yˆ ( ) m ( ) 6 DA y = + b + c m + yˆ ˆ ˆ ˆ = + b + s m = + b + αe ˆ ˆ ˆ ˆ = + b + α y y b = φb + αβ e bˆ ˆ ( ˆ = φb + αβ y y ) s = s + γ e sˆ = sˆ + γ y yˆ ( ) m ( ) Elemenary seasonal case Winers addiive mehod (Winers, 960) Damped rend wih seasonal effecs Table. Models for Common Linear Forms of Exponenial Smoohing. 5

7 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand All he models in Table are special cases of wha is bes called a single source of error sae space model, inroduced by Snyder (985). The unobserved componens are sacked o give a vecor x. I is assumed ha all componens combine linearly o give he series value, so he measuremen equaion is specified as y = h x + e (2) where h is a fixed vecor of coefficiens. The lag on x is used o reflec he assumpion ha he condiions a ime - deermine wha happens during he period. The evoluion of he unobserved componens is governed by he firs-order ransiion relaionship x = Fx + ge (3) where F is a fixed marix and g is a fixed vecor ha reflecs he impac of srucural change. Example : For he AN model in Table, h = (,), x = (, b ), F =, and g = ( α, αβ ). 0 The firs componen of (2) is he underlying mean level, or one-sep-ahead forecas, and we may designae i by m = h x. The second componen represens he unpredicable error or disurbance erm. I is possible ha he disurbance is compleely independen of he mean level, bu i is also possible ha is variance increases wih his level. For example, whenever sales variaion is naurally hough of in erms of percenage changes, raher han absolue changes, he sandard deviaion is likely o depend on he mean. Boh possibiliies are capured by he assumpion ha he disurbance is governed by he relaionship e = m ε for r = 0, (4) r where he{ ε } are independen and idenically disribued wih zero mean and variance σ 2, wrien as IID(0, σ 2 ).. The measuremen equaion may now be wrien as y = m + ε when r = 0 or y = m ( + ε ) when r =. In he laer case, heε is a uni-less quaniy, convenienly hough of as a relaive error. I means ha he unpredicable componen poenially depends on he oher componens of a ime series, somehing ha can be very imporan in pracice. The elemens h, F, g poenially depend on a vecor of parameers designaed by ω. 6

8 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand I is assumed ha he same model governs boh pas and fuure values of a ime series. Pas values are known, in which case i is possible o make a pass hrough he daa, applying a compaible form of exponenial smoohing in each period. Suppose, a he beginning of ypical period, pas applicaions of exponenial smoohing have yielded he esimaed value xˆ for he sae vecor x. Afer observing y a he end of period, i is possible o calculae he error eˆ = y h xˆ. The error can be subsiued ino he ransiion equaion o give xˆ Fxˆ g ( y h xˆ ) = + for he esimaed value of he sae vecor x. Given he progressive naure of his algorihm, i is clear ha his esimae depends on he parameers, he saring values of he sae variables and he observaions hrough ime, which we wrie xˆ = x y,..., y, x, ω. Inducion may be used o confirm ha x ˆ is a fixed value. as 0 A special case of he above model, bes ermed a composie model, is now considered. The sae vecor x is pariioned ino random sub-vecors designaed by x, and x 2,. The measuremen equaion has he form y = h x + h x + e (5), 2 2, where h and h 2 are sub-vecors of h. The sub-vecors of he sae vecor are governed by ransiion equaions ( ) x = F x + g e k = (6) k, k k, k,2 where F, F 2 are ransiion marices and g, g 2 are sub-vecors of g. The special feaure of his composie model is ha he ransiion equaion for x, does no conain x 2, and vice versa. I is shown in he Appendix ha he resuls for a composie model can be buil direcly from hose of is consiuen models. All he models in Table are special cases of he single source of error model or he composie model. The links wih hese general models are provided in Table 2. Here 0 k refers o a k-vecor of zeros and I k refers o a k k ideniy marix. Noe ha alhough he seasonal cases are governed by mh-order recurrence relaionships, hey are convered o equivalen firs-order relaionships. Also noe ha ω is a vecor formed from some or all of he parameers α, β, γ, φ. 7

9 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Case x h F g x = l h = F = 2 x = [ l, b ] h = [ ] F = 0 3 x = [ l, b ] h = [ ] F = 0 φ 4 x = [ s,..., s m + ] h = [ 0 m ] 0 m F = Im 0m 5 x = [ l, b ] h = [ ] F = x2 = [ s,..., s m + ] 0 h 2 = [ 0 m ] 0 m F2 = Im 0m 6 x = [ l, b ] h = [ ] F = " #! 0 φ h 2 = [ 0 m ] $ % x2 = [ s,..., s m + ] 0 m F2 = &( )' I 0 g = α = [ ] g α αβ = [ ] g α αβ g = [ γ 0 m ] m m [ ] g = α αβ [ ] g γ 2 = 0 m [ ] g = α αβ [ ] g = γ Table 2. Conformiy of Special Cases o he General Model or Composie Model. 2 0 m In he homoscedasic cases, only he mean poenially depends on rend and seasonal effecs. However, in he heeroscedasic cases, boh he mean and he variance of he random error componen depend on rend and seasonal effecs. Thus, predicion variances reflec rend and seasonal effecs in he heeroscedasic case, a feaure ha is poenially quie useful in pracice. An inriguing insigh from Table 2 is ha each smoohing mehod applies for boh a homoscedasic and a heeroscedasic model. Now, each homoscedasic case is equivalen o an ARIMA process (Box, Jenkins and Reinsel, 994). However, no heeroscedasic case is equivalen o an ARIMA process. Thus, exponenial smoohing applies for a wider class of models han he ARIMA class (Ord, Koehler and Snyder, 997). Many oher cases are conceivable when addiion operaors are replaced in he measuremen equaion by muliplicaions. Examples of such cases are presened in Hyndman, Koehler, 8

10 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Snyder and Grose (2002). A variey of models underlying he muliplicaive version of Winers muliplicaive mehod have been inroduced in Koehler, Snyder and Ord (200). The complexiy of hese nonlinear possibiliies precludes he derivaion of resuls using he mehodology of his paper. 3. MEANS AND VARIANCES OF LEAD TIME DEMAND For he purposes of he presen discussion, we assume ha mehods similar o hose described in Ord, Koehler and Snyder (997) have been applied o pas demand daa o esimae he parameers of an appropriae model. The problem is now o find he mean and variance of he lead-ime demand disribuion. Our analysis is buil, in par, on predicion variance resuls from Hyndman, Koehler, Ord and Snyder (200) for convenional predicion disribuions. As noed earlier, we assume he lead-ime λ o be fixed; his assumpion is relaxed for a special case in secion 5. I is shown in he Appendix ha lead-ime demand can be resolved ino a linear funcion of he uncorrelaed level and error componens: where Y ( ) λ λ = µ + + C e +. (7) n n n = = µ λ = (8) n h + F xn is he mean of he -sep predicion disribuion. I is furher esablished ha he coefficiens of he errors in (7) are given by C where λ = + c for =,..., λ (9) i= i =. (0) i ci h F g Paricular cases of he formulae for he means µ n + and he coefficiens C are shown in i Table 3. Noe ha φ = φ ; φ i= 0 ( 2) i = i ; i= φ p = ; d, m + m m = if is a muliple of m and d, m = 0 oherwise. The resuls for Case 5 and Case 6 are consruced by adding he 9

11 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand corresponding resuls for consiuen basic models, an approach ha is also raionalized in he Appendix. Case µ n + c ˆ α n + ( λ ) 2 ˆ ˆ n + b n α ( + β ) ( ) 3 lˆ φ bˆ + α ( βφ ) n n 4 s ˆn+ pm d, m 5 ˆ + bˆ + ˆ n n s n + pm C α ( λ )( λ + ) + λ α + αβ 2 ( 2) + λ α + λ αβφλ αβφλ + ( ) ( ) γ ( + ) + d, α β γ m λ + γ d i= i, m ( λ )( λ + ) λ + ( λ ) α + αβ + γ 2 i= d i, m 6 ˆ + φ bˆ + sˆ n n n+ pm ( + ) + d, α βφ γ m ( ) ( ) λ ( 2) λ λ i, m i= + λ α + λ αβφ αβφ + γ d Table 3. Key Resuls for Basic models. From (7), he condiional variance is given by λ Y x C 2 2 ( n ( λ ) n ω ) σ var, =. () in he homoscedasic case. All he informaion needed o evaluae he grand mean and he grand variance is available in Table 3. In he heeroscedasic case he grand variance is 2 where θn+ E ( mn+ xn, ω ) = λ var Y λ x, ω = σ C θ + (2) 2 2 ( ( ) ) n n n = =. I is esablished, in he Appendix, ha he heeroscedasic formulae may be compued using he recurrence relaionship where he c are also given in Table n+ n+ c i n+ i i= θ = µ + θ σ (3) 0

12 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand In common wih mos of he lieraure on invenory sysems, we have derived only he mean and variance for lead-ime demand (LTD). Safey socks are hen deermined assuming he LTD o be normally disribued. In he homoscedasic case, LTD will be normal if he errors are normal, bu he LTD is only approximaely normal in he heeroscedasic case even when a normal error process is assumed. However, a numerical sudy in Hyndman, Koehler, Ord and Snyder (200) indicaes ha here is lile error involved in approximaing hese disribuions by he normal. The same conclusion mus apply o lead-ime disribuions where aggregaion mus help o furher reduce he approximaion error. 4. EXAMPLES To gauge wheher a move o muliplicaive models from he simpler addiive models could be worhwhile in pracice, we examine he differences beween hem for weekly sales daa for a paricular produc wih a seasonal sales paern. The series ploed in Figure shows he weekly demand for a paricular cosume ewelry produc in he Unied Saes, covering he ime period 998, week 5 o 2000, week 24. This produc is one of several hundred produced by he company and many of hem show similar seasonal paerns. The pronounced increase in sales in he pre-chrismas period beween Thanksgiving (end of November) and Chrismas is obvious [corresponding o observaions and in he figure] and is widely anicipaed in he reail rade. Given ha he series possesses such pronounced seasonal peaks, case 5 of he models from Table was fied using he condiional maximum likelihood approach described in Ord, Koehler and Snyder (997). The maximum likelihood esimaes of he smoohing parameers urned ou o be α = 0.35 and β = γ = 0. These resuls indicae he presence of a consan growh rae and an invarian seasonal cycle; in oher words, a resriced version of model AA (case 5) lised in Table. The poin predicions for he demands in individual fuure weeks are ploed in Panel A of Figure 2, using 2000 week 24 as origin. The expeced peak occurs in he forecass over he pre-chrismas period. The quesion ha arises is how he sandard deviaions of lead-ime demand may be expeced o vary over ime. We are graeful o Bill Sichel for providing his daa se.

13 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand 4. Numerical comparisons Given he parameer esimaes obained, we focus aenion upon a simplified version of case AA in Table, for which here is no slope and he seasonal effecs are fixed, so ha β = γ = b 0 = 0. Furher, o make he inerpreaion more direc, we assume ha he seasonal effec is an upward shif in mean level of demand per uni ime [DPUT], such as occurs in he example over he pre-chrismas period. The effecs may be illusraed by a numerical example. The parameer seings are summarized in he following able; hree paerns for he prediced mean level are considered, labeled as cases A, B and C. The error sandard deviaion [SD} for he muliplicaive scheme is seleced so ha, in case A, boh schemes give exacly he same value for he SD of lead-ime demand. Parameer Descripion Values λ Lengh of lead-ime, 2,, 6 m Mean levels over period + o +λ (Case A) 6 periods a 200 Mean levels over period + o +λ (Case B) 3 periods a 200 and 3 periods a 600 Mean levels over period + o +λ (Case C) 6 periods a 600 α Smoohing consan 0.35 σ Sandard deviaion of ε in (3), addiive 00 scheme κ Sandard deviaion of ε in (3), muliplicaive scheme 0.25 (see ex) The resuls are summarized in Table 4. As he expeced level of demand increases, he SD of lead-ime demand increases under he muliplicaive scheme, bu remains consan under he addiive scheme. The clear implicaion is ha if he addiive scheme is used o compue safey sock when he muliplicaive scheme is appropriae, he implied SD will be oo low. In urn, he service level will be well below ha arge figure, wih consequen likely increases in los sales. Conversely, in a period of low demand per uni ime, invenories would be excessive. The key quesion, of course, is which of he wo models is 2

14 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand appropriae in pracice? The answer will be specific o he applicaion, bu in secion 4.3 we show how he quesion may be examined empirically. Lead-ime, λ Addiive Muliplicaive All cases Case A Case B Case C Table 4: Sandard deviaion of lead-ime demand for differen lead-imes, given varying levels of demand. 4.2 Changes in LTD variance over ime We firs examine how he variance of lead-ime demand (LTD) varies wih he expeced level of sales. We consider possible lead-imes λ =, 2,, and compue he variance for LTD using expressions (0) and () for he addiive and muliplicaive cases respecively. For he parameer values assigned (in paricular, α = 0.35) he summaion in (0) reduces o: λ λ λ When ploed, his funcion looks very like a quadraic. By conras, expression () will increase more rapidly when he expeced demand in high and end o flaen ou when expeced demand falls. This behavior is illusraed in Panel B of Figure 2, which shows he variances for lead-ime demand compued for he muliplicaive model, from he forecas origin of week 24, 2000, corresponding o successive lead-imes of, 2,, 52 weeks. The variance of lead-ime demand shows a marked rae of increase in response o peaking seasonal effecs. If he muliplicaive model is correc and he company uses a consan variance model, i will seriously underesimae he safey socks required during hese peak periods. We now consider a more sysemaic empirical sudy of he sandard deviaion o mean relaionship. 3

15 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand 4.3 A es for consan variance The complee daa se from which he series in Figure was aken comprises weekly sales figures on 345 cosume ewelry producs. We firs deleed all producs whose sales hisories sared par way during he year, leaving n = 34. We hen pariioned he daa ino wo sub-periods: Period : week 5, 998 o week 46, 998 [res of he year] Period 2: week 47, 998 o week 5, 998 [pre-chrismas peak] Denoe he means and sandard deviaions for produc i for he wo periods by M(i,) and S(i,) respecively, i =, 2,, 34 and =, 2. We hen compued he raios: MR(i) = M(i,2)/M(i,) and SR(i) = S(i,2)/S(i,). We would expec MR o be greaer han one since sales generally increase, alhough he raio will vary by produc. In fac, MR varied beween 0.95 and 8.7 wih a mean of 3.4; SR varied beween 0.69 and 7.42 wih a mean of 3.8. These averages alone sugges ha he SD increases wih he mean, bu a more sringen es is o consider he relaionship beween SR and MR. If he addiive model holds, an increase in he mean should no induce an increase in he SD. In order o es his proposiion, we evaluaed he regression for he logarihm of SR on he logarihm of MR. The es is no exac since we are compuing he mean and SD over muliple ime periods. Neverheless, i should serve as a reasonable guide. The resuls, wih n = 34, are as follows: log_sr = log_mr The slope coefficien has a -value of 8.3 [P < 0.000] and R 2 (adused) = 0.55, showing srong suppor for he hypohesis ha SR increases wih MR. 4

16 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand 900 Acual Demand Week Figure. Weekly Demand 5

17 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Panel A: Prediced Demand Lead-ime (week s) x 0 6 Panel B: Variance of Lead Time Demand Lead-ime (week s) Figure 2: Panel A shows he prediced demand for individual weeks. Panel B shows he variance of lead-ime demand when he lead-ime is as on he horizonal axis. 6

18 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand 5. EFFECT OF STOCHASTIC LEAD-TIMES We now resric aenion o model NN in Table, for simple exponenial smoohing, bu allow he lead-ime, T, o be sochasic wih mean E(T) = λ. In he ineress of space, we omi deails of he derivaions, bu simply repor he resuls. The mean leadime demand [LTD] for boh addiive and muliplicaive models, given he level a ime n, is: E[ Y ] = E [ E[ Y T ] = λ. n T n n The variance of LTD for he addiive scheme reduces o: α [ n ] = nv ( T) + σ [ λ + ( α α ) λ[2] + λ[3] ] V Y where λ [ ] = E[ T( )...( T + )], = 2,3, known as he facorial momens of he disribuion. Example 2: When he lead-ime is fixed, λ[ ] = λ( λ )...( λ + ). When he lead-ime is Poisson wih mean λ, λ[ ] = λ. For he muliplicaive scheme, he variance of LTD reduces o: T [ n ] = { n /( ασ ) }[{ + σ + 2( + ασ ) /( ασ ) }{ E( B ) } ( ασ ) λ 2 λ( + ασ )] V Y where B σα 2 T = + ( ), and E( B ) = B λ for T fixed, E B = λ ασ for he Poisson. T 2 ( ) exp[ ( ) ] The raios of he sandard deviaions for he wo models are shown in Table 5for various parameer combinaions. The raio increases subsanially only when he lead-ime is long, he coefficien of variaion for DPUT is high, and he correlaion beween demands for successive periods is high [high alpha]. However, comparison of he resuls in Table 5 wih hose for he fixed lead-ime case [no shown] shows ha he variance of lead-ime demand generally increases subsanially in he presence of uncerain lead-imes, as we would expec. 7

19 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Lambda Level Alpha Sigma Kappa Mean SD(A) SD(M SD raio ) MEAN: mean lead-ime demand SD(A), SD(M): sandard deviaions for addiive and muliplicaive schemes respecively SD raio = SD(M)/SD(A) Table 5: Comparison of addiive and muliplicaive models, wih Poisson Lead Times We now assume he onse of a seasonal increase in sales, represened by muliplying expeced sales by (+c). The impac of he seasonal increases is shown in Table 6 for fixed lead-imes and for wo varians of Poisson lead-imes. For a fixed lead-ime, he sandard deviaion is always increased by he facor (+c). For he Poisson schemes, he increase lies in he range [, +c]. The service levels corresponding o each case, for differen values of c, are given in he able, showing he expeced drop in performance. 8

20 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Lead-ime Seasona l Fixed Poisson () Poisson (2) Facor, c SD raio SL SD raio SL SD raio SL Poisson (): lead-ime = 5, level = 25, alpha = 0.5, SD = 5 Poisson (2): lead-ime =20, level = 00, alpha = 0., SD = 5 Table 6: Comparison Of Service Levels [SL] For Given Shifs In Demand Per Uni Time When The Muliplicaive Scheme Is Correc [Targe Level =0.99]. 6. CONCLUSIONS AND DIRECTIONS FOR FURTHER RESEARCH We have derived formulae for he mean and variance of lead-ime demand for many common forms of exponenial smoohing. For he general cases, we have assumed he leadime o be fixed, as is increasingly common in managed supply chain sysems. However, in he las par of he paper we have examined he impac of sochasic lead-imes for he special case corresponding o simple exponenial smoohing. By using he single source of error sae space model, we have unified he derivaion of he formulae. In he homoscedasic cases, many of he formulae obained in his paper agree wih hose found in earlier work (Johnson and Harrison, 986; Yar and Chafield, 990; Snyder, Koehler and Ord, 999). In addiion, for he Winers addiive seasonal mehod, he recursive variance formula in Yar and Chafield (990) has been replaced by a closed-form counerpar. Furhermore, we have obained, for he firs ime, formulae for he variance of lead-ime demand for he damped rend cases. The resuls for he heeroscedasic cases are also new. 9

21 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand I has been argued in he paper ha he random error componen of a demand series can depend on rend and seasonal effecs. Thus, a maor par of our conribuion has been he provision of lead-ime demand variance formulae for heeroscedasic exensions o exponenial smoohing. Such formulae admi he possibiliy of smarer approaches o safey sock deerminaion. I is now possible o implemen schemes ha ailor levels of safey sock o changes in rend or changes in season. The numerical resuls in he paper indicae he following conclusions, some of hem familiar: The failure o recognize ha he variabiliy in demand may be proporional o he mean level (raher han consan) can lead o service levels much lower han desired during peak periods (and excess invenory during periods of low demand). Incorporaing known seasonal and rend paerns ino safey sock planning leads o improved invenory managemen. Lead-ime uncerainy can lead o considerable increases in safey socks, making careful managemen of supplier delivery schedules a valuable sraegy. The principal direcion where furher research would be useful lies in he impac of esimaion error upon safey sock planning decisions. In common wih nearly all of he lieraure, we have no allowed for he uncerainy in he esimaion of model parameers from shor series. The combined perils of esimaion error and model misspecificaion have been clearly deailed in Chafield (993) for predicion inervals, and hey apply equally o he curren problem. 20

22 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand REFERENCES Box, G.E.P., Jenkins, G.M. and Reinsel, G.C., 994. Time Series Analysis: Forecasing and Conrol (hird ediion), Prenice-Hall, Englewood Cliffs. Brown, R.G Saisical Forecasing for Invenory Conrol. McGraw-Hill, New York. Chafield, C Calculaing Inerval Forecass. Journal of Business and Economic Saisics, 2-44 (wih discussion). Chafield, C., Koehler, A.B., Ord, J.K. and Snyder, R.D A New Look A Exponenial Smoohing. Journal of he Royal Saisical Sociey, series D 50, Gardner, E.S. Jr Exponenial Smoohing: The Sae of he Ar. Journal of Forecasing 4, -28. Gardner, E.S. and McKenzie, E., 985. Forecasing Trends in Time Series. Managemen Science 3, Harvey, A.C. and Snyder, R.D., 990. Srucural Time Series Models in Invenory Conrol. Inernaional Journal of Forecasing 6, Heah, D.C. and Jackson, P.L., 994. Modeling he Evoluion of Demand Forecass wih Applicaion o Safey Sock Analysis in Producion / Disribuion Sysems. Insiue of Indusrial Engineers, Transacions 26, Hol, C.E Forecasing Trends and Seasonal by Exponenially Weighed Averages. ONR Memorandum No. 52, Carnegie Insiue of Technology, Pisburgh, USA. 2

23 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Hyndman, R.J., Koehler, A.B., Snyder, R.D. and Grose, S., A Sae Space Framework for Auomaic Forecasing using Exponenial Smoohing Mehods. Inernaional Journal of Forecasing 8, Hyndman, R.J., Koehler, A.B., Ord, J.K. and Snyder, R.D., 200. Predicion Inervals for Exponenial Smoohing Sae Space Models. Working Paper /200, Deparmen of Economerics and Business Saisics, Monash Universiy. Johnson, F.R. and Harrison, P.J The variance of lead-ime demand. Journal of he Operaional Research Sociey 37, Koehler A.B., Snyder, R.D. and Ord, J.K., 200. Forecasing Models and Predicion Inervals for he Muliplicaive Hol-Winers Mehod. Inernaional Journal of Forecasing 7, Loveoy, W.S., 990. Myopic Policies for some Invenory Models wih Uncerain Demand Disribuions. Managemen Science 36, Makridakis, S. and Hibon, M The M3 Compeiion. Inernaional Journal of Forecasing 6, Miller, B Scarf s Sae Reducion Mehod, Flexibiliy, and a Dependen Demand Invenory Model. Operaions Research 34, Ord, J.K., Koehler, A.B. and Snyder, R.D., 997. Esimaion and Predicion for a Class of Dynamic Nonlinear Saisical Models. Journal of he American Saisical Associaion 92, Snyder, R.D Recursive Esimaion of Dynamic Linear Saisical Models. Journal of he Royal Saisical Sociey, B. 47,

24 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand Snyder, R.D., Koehler, A.B. and Ord, J.K., 999. Lead-ime Demand for Simple Exponenial Smoohing. Journal of he Operaional Research Sociey 50, Winers, P.R Forecasing Sales by Exponenially Weighed Moving Averages. Managemen Science 6, Yar, M. and Chafield, C., 990. Predicion inervals for he Hol-Winers Forecasing Procedure. Inernaional Journal of Forecasing 6,

25 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand APPENDIX General resuls governing he formulae in Table 3 are derived in his Appendix. To ge he formulae governing Cases -4, back solve he ransiion equaion (3) from period n + o period n, o give i n+ n n+ i i= x = F x + F ge (A) Lag (A) by one period, pre-muliply he resul by h, and use he definiions (8) and (0) o ge m = µ + c e. (A2) n+ n+ i n+ i i= Recall ha e is given by (3) so ha E ( en+ i ) = σ E( mn+ i ). Then we may square (A2) and ake expecaions o give he recurrence relaionship (3) for he heeroscedasic facors. Subsiue (A) ino (2) o give give ( ) where he λ n = µ n+ + i n+ i + n+ = i= Y c e e y = µ + c e + e n+ n+ i n+ i n+ i= C are defined by (9). Noe ha he derivaion of he. Subsiue his ino () o. Rearrange erms o yield he required resul (7) following equaions: C λ = and C = C + + c λ for = λ,...,. C is expedied using he Cases 5 and 6 are composie models. Each ransiion equaion (6), for a composie model, has he same srucure as (3). Thus, i k, n+ k k, n k k n+ i i= x = F x + F g e. (A3) Lag () by one period and pre-muliply he resul by h k o give where i m = µ + c e (A4) k, n+ k, n+ k, i n+ i i= µ k, n h + k Fk xk, n = (A5) and 24

26 Exponenial Smoohing Models: Means and Variances for Lead-Time Demand c = h F g. (A6) i k, i k k k Subsiue (A6) ino mn+ = m, n+ + m2, n+ o yield he earlier equaion (A) where µ = µ + µ (A7) n+, n+ 2, n+ and c = c + c. (A8) i, i 2, i Thus, he formula C = C, + C2, may be used o derive he resuls for Case 5 and Case 6 i i i from heir consiuen basic cases. In he heeroscedasic cases, he appropriae facors are sill derived wih he relaionship (3). 25