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1 ISSN X ISBN M O N A S H U N I V E R S I T Y AUSTRALIA Forecasig Sales of Slow ad Fas Movig Iveories Ralph Syder Workig Paper 7/99 Jue 999 DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS

2 Forecasig Sales of Slow ad Fas Movig Iveories Ralph Syder Deparme of Ecoomerics ad Busiess Saisics Moash Uiversiy Page proofs should be se o: Associae Professor R. D. Syder PO Box E Deparme of Ecoomerics ad Busiess Saisics Moash Uiversiy Ausralia 3800 Correspodig auhor: Telephoe: Fax: ralph.syder@buseco.moash.edu.au C:\daa\research\oeg\Forecasig Sales of Slow ad Fas Movig Iveories04 06/07/00

3 Forecasig Sales of Slow ad Fas Movig Iveories Keywords demad forecasig, iveory corol, simulaio, parameric boosrappig, ime series aalysis. Absrac Tradiioal compuerised iveory corol sysems usually rely o expoeial smoohig o forecas he demad for fas movig iveories. Pracices i relaio o slow movig iveories are more varied, bu he Croso mehod is ofe used. I is a adapaio of expoeial smoohig ha ) icorporaes a Beroulli process o capure he sporadic aure of demad ad ) allows he average variabiliy o chage over ime. The Croso approach is criically appraised i his paper. Correcios are made o uderlyig heory ad modificaios are proposed o overcome cerai implemeaio difficulies. A parameric boosrap approach is oulied ha iegraes demad forecasig wih iveory corol. The approach is illusraed o real demad daa for car pars. /9

4 . Iroducio A udersadig of key feaures of demad daa is impora whe developig compuer sysems for forecasig ad iveory corol. Plos of demad for hree pars carried by a Ausralia subsidiary of a Japaese car compay are show i Figure. I is emphasised ha he daa geuiely measure demad ad o sales. The daa are of Ausralia-wide mohly demad over a hree-year period. The raw daa may be foud i Appedix. I should be emphasised he series are o mea o be represeaive of all ypes of demad series ecouered i pracice. The series were chose because hey were cosidered o be ypical of hose ha cause difficulies i coveioal iveory corol sysems Iser Figure abou here Car Par is slow movig ad uaffeced by srucural chage. Car Par is also slow movig, bu is level ad variabiliy appear o be i declie: i is possibly reachig he ed of is life cycle. Car Par 3 is relaively fas movig, bu is also i a decliig phase. The graphs for pars ad highligh he impora poi ha demad series ca coai may zero values. Alhough o illusraed here, series wih a majoriy of zero values are commo. A forecasig echique ha allows for he possibiliy of zero values, bu sill works wih fas movig iveories like Car Par 3, is mos desirable. I elimiaes he eed o make arificial disicios bewee slow ad fas movig iems, somehig ha researchers (Johso ad Boyla, 996b) have perceived as beig a criical issue i applied forecasig. Simple expoeial smoohig (Brow,959) has bee he maisay of forecasig for iveory corol (Garder, 985). A special adapaio (Croso, 97) of his mehod, icorporaig a Beroulli process, is ofe recommeded for cases wih iermie demad (Willemai e al, 994). The focus of his paper is o a improved versio of he Croso approach ad is use i iveory corol. Emphasis is placed o he eed o correcly specify he saisical models for he geeraio of approximaios o he probabiliy disribuios of lead-ime demad. A parameric boosrap mehod is proposed for deermiig appropriae values for iveory corol parameers. The proposed approach ad is more radiioal couerpars are applied o he demad daa i Figure. They are compared usig compued values of orderig parameers required for iveory corol. 3/9

5 . Curre Approaches o Forecasig. Simple Expoeial Smoohig (SES) A local level model, a special case of he sigle source of error sae-space family of models (Syder, 985; Ord, Koehler ad Syder, 997), is used as he saisical framework for simple expoeial. I is a srucural represeaio of he ARIMA(0,,) process, he laer beig he framework radiioally used for simple expoeial smoohig (Muh, 960; Box ad Jekis, 976). I he local level model, demad y durig a ypical period, is deermied by he equaio y = µ + ε. (.) The firs erm o he righ had side of his equaio is referred o as he uderlyig level. I is lagged because he demads ha flow i durig period are assumed o deped o he sae of he marke a he sar of period. The secod erm is a disurbace ha represes uaicipaed demad. All disurbaces are ormally disribued wih mea 0 ad commo variace also assumed ha ε is ucorrelaed wih all earlier uderlyig levels. σ. I is The uderlyig level poeially chages over ime i respose o uaicipaed chages i marke srucure. I is govered by he rasiio equaio µ = µ + αε (.) where he so-called smoohig parameer α deermies he rae of chage i he uderlyig level. I is possible ha α = 0 : he case of o srucural chage. The rasiio equaio is seeded wih µ 0 = µ. (.3) The seed level µ, he smoohig parameer α ad he variace σ are ukow ad mus be esimaed from a sample y, y, K, y of size. For ay rial values of hese quaiies, simple expoeial smoohig may be applied o deermie a series of oe-sep ahead predicio errors e, e, K, e. Simple expoeial smoohig ivolves m = µ µ, α, y, K, y, a quaiy ha will be referred o as he local level. This oaio reflecs he fac ha he local level, a ay poi of ime, depeds o he pas rajecory of he ime series, ogeher wih he specified values of he seed level µ ad smoohig parameer α. A sage he algorihm eails he calculaio of he oe-sep ahead predicio error e = y m (.4) ad he revisio of he local level wih he recurrece relaioship 4/9

6 I coras o he origial ucodiioal µ ad α. m = m + αe. (.5) µ, he m are fixed quaiies for specified values of I pracice, he seed level µ is ofe esimaed usig a heurisic such as he simple average of he firs hree series values. A aleraive is o fid ha value ha miimises he sum of squared errors = e. There are coge argumes for boh sraegies ad i is o a purpose of his paper o dwell o he choice bewee hem. I his paper, he opimisaio approach is adoped. Oe advaage is ha expoeial smoohig he accommodaes he impora special case of o srucural chage (where α = 0 ). I collapses o a classical simple average i his circumsace. If here is a preferece i pracice for he heurisic, he mehod preseed here ca be adaped accordigly. Forecass are isesiive o he seed value whe α is o close o zero, so boh iiialisaio sraegies yield similar resuls i his circumsace. The leas squares esimaes are desigaed by ˆµ ad $α. Esimaes of he correspodig codiioal meas are deoed by $m. The esimae of he variace of uaicipaed demad is give by he formula σˆ e = = (.6) Predicio wih simple expoeial smoohig has radiioally bee hadled usig ad-hoc model-free sraegies. More reliable aalyical approaches for derivig he disribuios of leadime demad (Johso ad Harriso, 986; Harvey ad Syder, 990; Syder, Koehler ad Ord, 999) ow exis ad may be used i heir place. A simple aleraive, ha explois he exesive compuaioal capaciies of moder compuers, is based o he followig simulaio mehod. Assumig ha he problem is o fid he disribuio of aggregae demad over a leadime 0+, + h5, i cosiss of he followig seps:. Use Moe Carlo radom umber geeraio mehods o obai values for he errors ε+, K, ε+ h from a ormal disribuio wih mea zero ad variace σ.. Use he local level model, as described by equaios (.) ad (.), o geerae a realisaio y, + K, y+ h of fuure series values. 3. Calculae lead-ime demad + h y. = + This procedure is repeaed may imes. Replicaio i of lead-ime demad is deoed by 5/9

7 + h = + y () i. Take ogeher, hese quaiies form a sample ha may be used o approximae he lead-ime demad disribuio. Seps ad of his simulaio procedure requires values for µ, α ad σ, bu hese are ukow. The correspodig leas squares esimaes are used i heir place, meaig ha he simulaio mehod becomes wha is commoly called a parameric boosrap. Like all such approaches, he effec of samplig error is igored. The coseque loss of accuracy is usually olerable because samplig error is a secod-order effec. Despie his drawback, he parameric boosrap mehod provides a much souder basis for he deermiaio of he lead-ime demad disribuio ha he ad hoc approaches commoly i use.. The Croso Mehod I was argued by Croso (97) ha simple expoeial smoohig is o appropriae for iveories wih iermie demad. I appedix B of his paper, he oulies a model of iermie demad, a mehod for esimaig key quaiies i he model ad a mehod for predicig lead ime demad for reorder level deermiaio from a sample. The model is cas as a ARIMA(0,,) process ha is assumed o apply oly a hose iermie periods whe rasacios occur. A Beroulli process govers he ime bewee such acive periods, he discree aalogue of a Poisso process. The Croso model may be see as a adapaio of he coveioal local level model o allow for he iermie aure of demad. A addiioal radom variable x is used o idicae hose periods i which rasacios ake place. This biary radom variable is govered by a Beroulli disribuio wih parameer p, he probabiliy of here beig some demad i a give period. I is used o force he local level ad disurbace variace o zero i iacive periods. The model is where y = xµ + ε (.7) µ = µ + αε (.8) The disurbaces are sill idepedely ad ormally disribued, wih a commo mea 0. However, he variace i period is augmeed by he biary variable o become xσ. The ARIMA model i he origial represeaio has bee replaced by is sae space aalogue, he local level model. This is a chage of form, raher ha subsace, o clarify he lik wih expoeial smoohig. Noe ha, ulike he uderlyig level µ, boh p ad σ are o 6/9

8 subscriped by ime. These quaiies are implicily assumed o remai uchaged over ime. The Croso mehod, as disic from his model, ivolves fidig expoeially weighed movig averages (EWMA) of hree quaiies:. he posiive series values y,. he associaed absolue errors e, 3. he elapsed imes bewee successive acive periods (periods i which rasacios occur). I he ivolves fidig he uderlyig mea demad from he raio of he firs ad hird EWMA s. The EWMA s i his mehod use he same smoohig parameer. Croso is vague abou how his quaiy should be chose. He idicaes if he series i s shor i may have o be chose arbirarily from experiece. He is also vague abou he choice of seed values for he EWMA s. He seems o place himself i he heurisic raher ha he opimisaio school. As idicaed earlier, his is a legiimae sace o ake whe cofroed wih he realiies of busiess evirome. The expoeially weighed averages i seps ad 3 of his mehod are desiged o deec ad allow for chages i he variabiliy of demad ad mea frequecy of acive periods. A mehod alog hese lies would herefore be expeced o work well for Car Par depiced i Figure. Because of is focus o iermie demads, he Croso mehod has bee he subjec of cosiderable ieres. Neverheless, a umber of problems have bee ideified. Rao (973) foud errors i he algebra. Johso ad Boyla (996a) expressed cocer abou a measure of he variabiliy of demad ha does o icorporae he effec of uceraiy i he elapsed imes bewee acive periods. Their proposed soluio based o reewal heory, however, assumes ha meas ad variaces are cosa, somehig ha diverges from he spiri of he expoeial smoohig mehods. I is o suiable, for example, for iveories like Car Par depiced i Figure. Thus, i ca be argued ha he variabiliy problem hey ideified wih he Croso mehod remais o be resolved. There are furher problems wih Croso s paper ha have o so far bee ideified i pri. Icosisecies exis bewee model ad mehod i relaio o he secod ad hird expoeially weighed averages. I order o jusify he use of hese EWMA s, i is ecessary o assume ha σ ad p chage over ime. I may be rue i pracice ha hese quaiies chage. 7/9

9 Bu hey are assumed o be cosa i his model. For cosisecy i is ecessary o replace he offedig EWMA s by he formulae σˆ e = = versio of his mehod will be called MCROST. = x ad pˆ = x. This modified = 3. New Mehods The simulaio mehod oulied for simple expoeial smoohig may be adaped o geerae a approximaio for he lead-ime demad disribuio from he Croso model. A ieraio would ivolve he followig seps:. Geerae values for he errors ε, K, ε from a ormal disribuio wih mea 0 ad + + h variace σ.. Geerae values for he idicaor variables x, + K, x+ h from a Beroulli disribuio wih probabiliy p. 3. Geerae a realisaio y, + K, y+ h of fuure series values from he modified local level model equaios (.7) ad (.8). 4. Calculae lead-ime demad wih y. + h = + There is, however, a serious logical difficuly. Nohig i he local level compoe of he Croso model ihibis he simulaio of egaive syheic daa, somehig ha is icompaible wih he realiy ha demad ca ever be egaive. Oe possible way aroud his difficuly is o apply expoeial smoohig o he logarihm of he daa. The weakess of his sraegy is ha he raw series may coai zeroes. log( 0 ) does o exis! Aoher possible fix migh be o roud all egaive values o zero. There would he be wo sources of zeros i he model: he Beroulli process ad he local level model. I would o be possible o disiguish bewee boh sources i real daa. The Croso model is o viable as a mechaism for geeraig predicio disribuios wihou a aleraio o overcome his basic flaw. 3. Log-Space Adapaio (LOG) A adapaio ha leads o a ew model, ad hece a ew approach o forecasig, is o eforce o-egaive demads usig he equaio y + = x exp( y ). (3.) 8/9

10 Series values are ow represeed by he o-egaive quaiy y +. We sill use y, bu i is ow reaed as a lae variable. I is govered by he local level model equaios (.) ad (.). I herefore coiues o ake boh posiive ad egaive values. Like he correced versio of he Croso model above, he variace of he disurbaces is give by govered by a Beroulli disribuio wih probabiliy p. xσ. The x are agai The correspodig smoohig equaios are: + log( y ) if x = y = arbirary if x = 0 ( ) (3.) e = x y m (3.3) m = m + αe (3.4) m 0 = µ (3.5) Boh µ ad α are agai chose o miimise he sum of squared errors crierio = e. The variace ad proporio of acive periods are agai esimaed wih σˆ = e = = = x ad pˆ = x. Predicio disribuios may be geeraed usig a appropriae boosrap procedure. 3. Adapive Variace Versio (AVAR) A sregh of he Croso mehod, i is origial form, is is capaciy o allow for chages o he uderlyig variabiliy i a ime series. This is achieved by permiig he mea absolue deviaio (MAD) o chage over ime. I his secio a ew model is iroduced. I differs from he Croso mehod i wo furher respecs:. Variabiliy is measured i erms of variaces isead of MADs. Variaces have more coveie algebraic properies.. A secod smoohig parameer β is used i he equaio ha defies how he variabiliy chages over ime. Croso uses he same smoohig parameer i he equaios ha describe how boh he level ad variabiliy chage over ime. I is difficul o believe ha srucural chage geerally has he same impac o levels ad variaces. The model cosiss of equaios (3.), (.) ad (.). Now he disurbaces are govered by 9/9

11 he assumpio ~NID( 0, ) ε σ, he ime depede variaces evolvig accordig o he equaio: ( e ) σ = σ + βx σ. (3.6) Agai i is assumed ha he x are govered by a Beroulli disribuio wih cosa probabiliy p. The iiial codiios are µ 0 = µ ad he variace. σ 0 = σ where σ is a sarig value for The smoohig equaios (3.)-(3.5) i he previous mehod are ow modified by he equaio: ( ) s = s + β e s (3.7) where s = σ µ, σ, α, β, y, K, y. Furhermore, his recurrece relaioship is seeded wih s = σ. (3.8) 0 The ukows i his mehod of smoohig iclude he crierio where N = x. Also, = = esimaes resul from his sraegy. N = = µ, σ, α, β. These are chose o miimise s e s (3.9) pˆ = x. I is show i Appedix ha maximum likelihood Noe ha i he special case where β = 0 ad he variaces are cosa, he crierio (3.9) collapses o a coveioal sum of squared errors. I oher cases where β > 0, i appears ha he addiioal erms i (3.9) are required o allow for heeroscedasiciy. Dividig each squared error by a variace erm sadardises i. The effec is o place less weigh o hose errors associaed wih higher variaces. The geomeric mea erm is a measure of average variabiliy. Muliplyig by he geomeric mea de-sadardises he sadardised sum of squared errors. Predicio ca agai be carried ou wih a appropriae adapaio of he earlier parameric boosrap approaches. 4. Iegraio wih Iveory Corol Theory Mos iveory corol models used i pracice (Brow, 957) are buil o saioary demad disribuio assumpios. Ye he forecasig models uderlyig he expoeial smoohig mehods ivolve o-saioary sochasic processes. Thus, i ypical compuer implemeaios 0/9

12 of he heory, forecass from o-saioary models are fed io iveory formulae based o saioary demad disribuio assumpios. The use of icosise models like his is dicaed by he fac ha he heory of o-saioary iveory corol is iherely more complex ha is saioary couerpar ad is herefore perceived, righly or wrogly, as more difficul o impleme i pracice (Hax ad Cadea, 984, pp 39-40). Saisfacory mehods for iveory corol, based o he same assumpios as expoeial smoohig, are ye o be devised. I is o ieded o propose a soluio here o his difficul problem. We shall isead follow curre pracice ad show how o adap he radiioal approach o iveory corol o he case where a lead-ime demad disribuio has bee approximaed by a simulaed sample. Thus he workig hypohesis is ha he srucure of he demad process remais uchaged i all fuure periods, eve hough we have allowed for srucural chage i he pas while geeraig he required forecass. Use of a hypohesis like his is o ideal, bu i is ecessary uil a workable o-saioary iveory heory has bee developed. Brow s approach o iveory corol was adaped i Syder (984) o hadle demads geeraed by gamma probabiliy disribuio. The mehods described here are similar excep he gamma disribuio is replaced by he simulaed demad daa from he above forecasig procedures. 4. Order-Up-To Level Sysem: Zero Lead Time Case Whe here is o delivery lag, ad hece o eed o accou for ousadig repleishme orders, he order-up-o level (OUL) represes he ideal level for sock. Assumig ha repleishme orders are oly placed periodically, he aim is o order eough o esure ha sock rises o his ideal level. A he begiig of each period he order-up-o level he represes he amou of sock available o mee a ucerai demad durig he followig review period. Shorages occur if demad durig a review period exceeds he OUL. Thus he choice of he OUL is criical o he successful operaio of he sysem. The disicive feaure of he cusomer service level approach is ha a performace arge is se i erms of wha may be ermed he fill-rae. This is he proporio of demad, o average, ha is saisfied wihou backloggig. The aim is choose he OUL so ha he fill-rae equals a level specified by maageme (eg 95 perce). Le y N, K, y deoe he simulaed demads for he ex period. If S represes he bg b g + + ukow OUL, he fill-rae saisic may be defied as a esemble average /9

13 N + N i b g + (4.) i= i= bi g f = y S y where he superscrip + desigaes he posiive par of he associaed umber. A implici equaio solver, such as he goal seeker i Microsof Excel, ca he be used o fid he value for S ha achieves a pre-specified value of he fill-rae f. 4. Order-Up-To Level Sysem: Posiive Lead Time Case This mehodology ca be exeded o cases of a o-zero delivery lag h. The OUL ow represes he ideal level for he sock saus: he sock ad quaiy o order less he backlog. A order, which is placed a ime, is sufficie i size o raise he sock saus o he OUL. This order is delivered a ime + h ad affecs he sock level i he review period + h+. The excess demad is he differece bewee he closig ad opeig backlogs i period + h+. The fill-rae may herefore be defied as he esemble saisic: f = N + h+ + + h + () i () i y S y S i= = + = + N i= y () i + h+ + h Agai S may be chose usig a appropriae solver o achieve a pre-specified value of he fill- aif rae f. Noe ha he opeig backlog erm y S = + + (4.) i (4.) is ofe quie small. If his erm is deleed we obai he sample aalogue of Brow s (959) parial expecaio approach for deermiig he OUL. Nowadays, wih he compuerisaio of iveory sysems, socks are reviewed more frequely. I is he more likely o have sigifica opeig backlogs followig a repleishme delivery. I is safer o o use he approximaio. Noe also, whe h = 0, his erm is udefied. Formula (4.) he applies. 4.3 Reorder Level Sysems Deliveries may be cosraied o be muliples of a fixed quaiy Q. The size of his quaiy may be dicaed by packig ad rasporaio cosideraios. I may also be jusified i erms of Wilso s classical ecoomic order quaiy heory (Syder, 973) whe here is a fixed acquisiio cos associaed wih each order. Eiher way, a order may ow have he capaciy o mee demad over may review periods. To esure log ru balace, i is o ormally possible o coiue he pracice associaed wih he above OUL sysems, of placig orders a he sar of each review period. Orders are deferred uil hose reviews where he sock saus has dropped below a criical value called he reorder level (ROL). Deoig i by R, he ROL is relaed o he OUL by he equaio S= R+ Q. /9

14 Maers are made more complicaed i his ype of sysem by he fac ha he sock saus followig each review is o loger cosa. I is show i Hadley ad Whii (963), ha if he demad disribuio is saioary, he sock saus immediaely followig each review ca be modelled as a doubly sochasic Markov Chai. From his hey esablish ha is movemes are ulimaely govered by a uiform seady sae disribuio. The mass of his disribuio is Q over he domai 0RS, 5. To simulae he average performace of he sysem, N values u, K, u N are geeraed from a uiform disribuio over he ui ierval 00, 5. Correspodig values of he sock saus are he give by R+ uq i, so ha he fill-rae is ow give by f = N i= + h+ = i y R uq i y i R uq i b g b g = + N i= + y bi g + h+ + h +. (4.3) Assumig ha Q is kow ad ha maageme has specified a arge value for he fill-rae f, a implici fucio solver ca be used o fid he correspodig value of he reorder level R. This formula for he fill-rae is sricly oly applicable whe a saioary sochasic process geeraes demads. Because i relies o he seady sae disribuio of he sock saus, (4.3) is a measure of he log-erm performace of he sysem. Whe he o-saioary process uderlyig expoeial smoohig geeraes demads, a seady sae does o exis. Give ha here is o reasoable aleraive i his siuaio, however, he use of his formula is recommeded uil he maer is properly resolved. 5. Examples The forecasig mehods ad heir performace are illusraed here by applyig hem o he demad daa i Appedix. This is he daa depiced i Figure. As he Ausralia sores of he compay are repleished by deliveries from Japa, he delivery lead-ime is assumed o be 3- mohs. The review period is assumed o be oe moh i legh because he demad daa is collaed o a mohly basis. I realiy, he review period is much less ha his. However, demad daa collaed over he shorer review period was uavailable. I is also assumed ha a OUL sysem is employed o corol socks. I realiy, a reorder level corol sysem is used. This deviaio from realiy is adoped o esure ha differeces i exraeous facors, such as he size of Q, do o cofoud he coclusios. The sample size beig 36, he sar of moh 37 correspods o he curre review. Ay order placed a his poi of ime is assumed o arrive hree mohs laer a he begiig of moh 40. 3/9

15 The primary aim a he sar of moh 37, herefore, is o corol iveories i moh 40. A arge fill-rae of 95 perce is employed. Five mehods are compared: GAM The gamma probabiliy disribuio approach (Syder, 984) for obaiig order-up-o levels. Beig based o a saioary demad process, he associaed mea ad variace are esimaed by a simple average ad he classical variace formula. This case is icluded for bechmarkig purposes. SES This applies simple expoeial smoohig, as described i secio., o he daa. I igores he possibiliy ha he daa may perai o a slow movig iveory. MCROST The Croso mehod implemeed wih he modificaios specified i secio.. LOG The adapaio of MCROST described i secio 3.. A log-rasform is applied o o-zero demads. AVAR The adapive variace approach deailed i secio 3.. The adapive variace recurrece relaioship (3.7) is seeded wih a value obaied from he classical sample variace formula applied o he firs -mohs of daa. The mehod proved o be usable for opimised values of he seed variace. Lead-ime demad disribuios, for mehods -4, are derived usig suiably adaped parameric boosrap procedures. These are based o 0,000 replicaios. Each of Tables -3 summarises he resuls for a car par. Each colum correspods o oe of above mehods. I is impora o oe ha some of he resuls i he fial wo colums are o comparable wih hose i earlier colums because hey refer o saisics calculaed i log-space raher ha raw-space. The fial row coais he mos impora resuls: he OULs ha achieve he 95 perce fill-rae arge. These OULs are all expressed i erms of he raw-space ad are herefore comparable. The performace of a mehod ca be gauged by he size of he associaed OUL. Ideally, he OUL should be as low as possible. The rows before he las oe coai auxiliary iformaio. The firs row provides he simple average of he eire series. Rows ad 3 coai levels for he sar ad ed of he samplig period. The ex wo rows lis he esimaes of he level ad variace equaio smoohig parameers. The hree subseque rows coai variace esimaes. The ex row has he esimae of he acive periods proporio. The secod las row is provided for hose mehods ha do o impose o-egaiviy codiios o demads. 4/9

16 Car Par From he demad series show i Figure, Car Par appears o have a sable marke over ime. The resuls obaied from he five mehods are show i Table. The followig pois ca be observed abou he resuls: The resul α ˆ = 0 for he four expoeial smoohig mehods cofirms he srucural sabiliy i he levels. The graph for AVAR cosequely shows he smoohed series i Figure as a horizoal lie. I is a simple average of he daa. This example highlighs he eed o allow for he possibiliy ha α = 0, somehig ha is o doe i coveioal implemeaios of expoeial smoohig. I AVAR β ˆ > 0. Thus alhough he uderlyig level remais uchaged, he variace does chage. The variace i log-space almos halves over he 36 periods. Oe would expec his reducio o lead o a fall i safey socks. Despie his, he OULs of he expoeial smoohig mehods are abou he same. The OUL represes ha supply required o saisfy demad over a 4-moh period. The maximum 4-moh aggregae of demad i he fial year is 7, occurrig from April o July. The OUL s are all large eough o mee successive 4-mohly demads, he excepio beig he period April o July. The resuls are all plausible i his sese. Iser Table ad Figure abou here Car Par The series for Car Par displayed i Figure shows a disic dowward red, somehig ha is assumed o be a reflecio of srucural chage i he marke place. For example, he series migh represe demad of a par for a old model of car. The resuls i Table 3 sugges ha: The esimaes of he smoohig parameer α are all sricly posiive. All he expoeial smoohig mehods deec he dowward red see Figure for he drop i he smoohed series. The esimae of he smoohig parameer β is also posiive. The drop i he variace is quie subsaial. The classical saioary demad mehod, i his case based o a gamma probabiliy disribuio, assumes ha here is o srucural chage. Large disa pas values of he ime series are weighed equally wih more rece observaios i geeraig he forecass. Thus, he OUL from his mehod is geared o hadlig demads for a marke srucure ha o loger exiss ad is, as a cosequece, oo large. The resuls for SES, MCROST ad LOG are lower. AVAR, however, yields he lowes OUL. I allows for 5/9

17 he declie i variabiliy i he daa. The larges 4-moh ru of demad i he fial year is oly 4. Praciioers would probably argue ha all OUL s are oo high. The resul from he AVAR mehod migh jus be accepable. The proporio of 0.35 egaive simulaed demads for he coveioal local level model is quie high. Ye, he effec of he egaive values o he OUL appears o have bee miimal. Iser Table abou here Car Par 3 The fial series cosiss of demad for he fas movig Car Par 3. Agai, a slowly decliig marke is assumed o reflec he effec of srucural chage. I is ieresig o oe ha: The maximum 4-mohly ru of demad i he fial year is 67. Remarkably, AVAR yields a OUL slighly above his figure. The esimae of he smoohig parameer α is abou 0. for all expoeial smoohig mehods. This idicaes ha srucural chage impacs he daa. A compariso of he OUL s from he Gamma disribuio ad expoeial smoohig approaches demosraes sizeable beefis from he use of expoeially weighed averages isead of a simple average. The esimae of he smoohig parameer β is zero. The variabiliy i he demad series appears o o chage much over ime. The esimae of he biomial probabiliy p is oe. I his circumsace, LOG has appropriaely collapsed o classical simple expoeial smoohig, albei i log-space. AVAR has collapsed o a varia of simple expoeial smoohig ha allows for chages o variabiliy as well as chages o he mea. I oher words, hese mehods provide a uified approach o forecasig demad for slow ad fas movig iveories. The disicio bewee slow ad fas-movig iveories is made auomaically hrough he esimaio of p. There is o eed o impleme separae mehods of forecasig for slow ad fas movig iems ad o employ heurisics o disiguish bewee he wo cases. I migh be expeced ha LOG ad AVAR should yield almos ideical resuls i his siuaio. Ye, he OUL is lower for AVAR. The sadard deviaios for LOG is 0.4. The esimae of he sadard deviaio i period 36 for AVAR is The differece i hese figures leads o he observed differece i he associaed OUL s. Why does such a discrepacy occur? The variaces (equally weighed versios i log space) for years 6/9

18 o 3 are 0.08, 0.4 ad 0.3 respecively. Thus, here was a sigifica icrease i relaive variabiliy bewee years ad. The i sabilised bewee years ad 3. Thus, he variace recurrece relaioship is iiiaed wih he lower value Because he variabiliy sabilised i years ad 3, he bes esimae of he smoohig parameer β was foud o be almos zero. Therefore, he variace did o adjus much o he higher variabiliy i he secod ad hird years. I is o possible, o he basis of his paricular daa se, o coclude ha AVAR is iherely beer ha LOG. I migh be more saisfacory if he seed value of he variace recurrece relaioship could be opimised isead of beig chose by a heurisic. As idicaed previously, however, is opimisaio wih he crierio (3.9) ad similar crieria seems o be usable. This example highlighs he eed o use disic smoohig parameers i he updaig equaios for he level ad variace. Iser Table 3 abou here Coclusios ad Fial Commes Mehods of forecasig ha ca be applied o boh fas ad slow movig iveories have bee proposed i his paper. May feaures of hese mehods reflec he ifluece of Croso s approach o forecasig. Bu here are key differeces, hese beig: smoohig i log-space o avoid egaive demads; differe smoohig parameers for he level ad variace; he use of compaible models ad mehods; he use of models i a parameric boosrap approach o geerae lead-ime demads; reorder level ad order-up-o level deermiaio usig he fill-rae crierio from boosrapped demads; a cosa probabiliy i he Beroulli process goverig he occurrece of acive mohs. This las poi may seem o be a backward sep. However, whe radom walk models of his probabiliy were implemeed, maximum likelihood esimaes of he smoohig parameer associaed wih he resulig expoeially weighed average always ured ou o be zero. I is o clear why his should be so. Oe cojecure is ha very large samples are required o esure a large eough umber of ier-rasacio imes wih which o work. I pracice, such samples are rarely available. A feaure of he mehods is ha hey ca be applied o boh slow ad fas movig demad daa. 7/9

19 For fas movig iems, he esimae of he biomial probabiliy ieviably equals oe. LOG he collapses o he applicaio of simple expoeial smoohig, albei i log-space. AVAR collapses o a exesio of simple expoeial smoohig ha allows he variabiliy as well as he uderlyig level o chage i respose o srucural chage. There remai a umber of poeial difficulies wih he approaches described i his paper. Firs, he parameric boosrap approach igores he effecs of esimaio error. Thus here may be a edecy for hese mehods o uderesimae he variabiliy of lead-ime demad. Esimaio error is a secod-order effec compared wih he predicio error. Is impac, i all bu small samples, is usually fairly small. I mos circumsaces, i is probably o worhwhile o seek he refiemes ecessary o allow for esimaio error. However, i hose cases where i is, a adapaio of he mehods i Ord, Koehler ad Syder (997) is a possibiliy. Ayway, i he examples, he OUL s eded o be o he high side wihou his ype of adjusme. Secod, he heory preseed here is based o he ormal disribuio. Whe rasacios are small, he discree aure of demad ca become impora. Furhermore, a skewed disribuio may be required o properly model demad daa. The use of a discree probabiliy disribuio defied over he whole umbers, combied wih expoeial smoohig updaes of is mea, is problemaic because associaed simulaed daa always exhibis bizarre behaviour (Gruwald, Hamza ad Hydma, R.J. (997). Thus, he problem of forecasig demad for slow movig iems remais a challegig area for furher research. 8/9

20 Refereces Box, G.E.P., Jekis, G.M., 976. Time Series Aalysis: Forecasig ad Corol, Holde-Day, Sa Fracisco. Brow, R.G., 959. Saisical Forecasig for Iveory Corol, McGraw-Hill, New York, 959. Croso, J.E., 97. Forecasig ad sock corol for iermie demads, Operaioal Research Quarerly, 3, Garder, E.S., 985. Expoeial smoohig: he sae of he ar, Joural of Forecasig, 4, - 8. Gruwald, G.K., Hamza, K. ad Hydma, R.J., 997. Some properies ad geeralisaios of Bayesia ime series models, Joural of he Royal Saisical Sociey B, 59, Hadley, G., Whii, T.M., 963. Aalysis of Iveory Sysems, Preice-Hall, Eglewood Cliffs, N.J. Harvey, A.C., Syder, R.D., 990. Srucural ime series i iveory corol, Ieraioal Joural of Forecasig, 6, Hax, A.C., Cadea, D., 984. Producio ad Iveory Maageme, Preice-Hall, Eglewood Cliffs, N.J. Johso, F.R.,Boyla, J.E., 996a, Forecasig for iems wih iermie demad, Joural of he Operaioal research Sociey, 996, 3-. Johso, F.R., Boyla, J.E., 996b, Forecasig iermie demad: a comparaive evaluaio of Croso s mehod. Comme, Ieraioal Joural of Forecasig,, Johso, F.R., Harriso, P.J., 986. The variace of lead ime demad, Joural of he Operaioal Research Sociey, 37, Muh, J.K., 960. Opimal properies of expoeially weighed averages, Joural of he America Saisical Associaio, 9, Ord, J.K., Koehler, A.B., Syder, R.D., 997. Esimaio ad predicio of a class of dyamic oliear saisical models, Joural of he America Saisical Associaio, 9, Syder, R.D., 973. The classical ecoomic order quaiy formula, Operaioal Research Quarerly, 4, 5-7. Rao, A.V., 973, A comme o: forecasig ad sock corol for iermie demads, Operaioal research Quarerly, 4, Syder, R.D., 984. Iveory corol wih he gamma disribuio, Europea Joural of Operaioal Research, 7, Syder, R.D., 985. Recursive esimaio of dyamic liear models, Joural of he Royal Saisical Sociey, 47, Syder, R.D., Koehler, A.B., Ord, J. K., 999. Lead ime demad for simple expoeial 9/9

21 smoohig: a adjusme facor for he sadard deviaio, Joural of Operaioal Research, forhcomig. Willemai, T.R., Smar, C.N., Shockor, J.H., DeSauels, P.A., 994. Forecasig iermie demad i maufacurig: a comparaive evaluaio of Croso s mehod, Ieraioal Joural of Forecasig, 0, /9

22 Ackowledgemes A Faculy of Busiess ad Ecoomics research gra fuded he work for his paper. I would like o hak Gopal Bose for his assisace durig he course of he research projec. /9

23 Appedix (Likelihood Fucio ad Fiig Crierio) The likelihood fucio for AVAR is derived i his appedix. The eire argume is preseed i he log-space. I ivolves ormal disribuios. The seed variace is desigaed by variace i period is represeed by heeroscedasic-scalig facors v by σ. Furhermore, i is coveie o defie s σ. The = vσ. (A) where s = σ y, x, µ, α, β. Because s = σ i follows ha v 0 =. 0 The joi ormal desiy of he series, codiioal o a paricular se of acive ad iacive periods specified by x, K x ad he ukow parameers, is deoed by, (,,,,,, ) p y Ky x K x µ σ α β. Progressively codiioig o earlier series values yields ( ) ( ) (,, µ, σ, α, β,, µ, σ, α, β, µ, σ, α, β ) p y K y x = p y y x p y x (A) = where = [ K ] ad [,, ] x x,, x his equaio simplifies o y y y = K. Give ha (,,, µσ, α, β) ( ) (,, µ, σ, α, β, µ, σ, α, β ) = E y y x = m, p y K y x = p e x. (A3) The desiy of he oe-sep ahead error is Noe ha whe 0 becomes e p( e x, µσ,, α, β) = exp. (A4) x π σ v σ ( v ) x =, i reduces o p( e x µ α β) ( x µσ α β) p y,,,, =. Thus, he Equaio (A3),,,, = exp x = v σ ( πv σ ) = e. (A5) Give ha he x follow a Beroulli disribuio, he joi desiy of he observable quaiies y ad x is e x p( y, x µσ,, α, β) = exp p p x σ = v ( πv σ ) = ( ) ( x ) (A6) /9

24 Viewed as a fucio of he ukow parameers, he righ had side of (A6) is he formula for he likelihood fucio. The maximum likelihood esimae of p is pˆ = x. The maximum = likelihood esimaor of σ is σˆ e v = = x =. (A7) The corollable par of he likelihood fucio, afer subsiuig for p ad σ i (A6), is he deermia erm. Thus, i is clear ha he maximum likelihood esimaes of he parameers µ, α ad β are obaied by miimisig he quaiy x N v ˆ σ where = N = x. Usig (A7), droppig he ucorollable deomiaor (A7), ad fidig he Nh-roo, a equivale = i crierio is x x N v e v. This is he same as N ( vσ ) e vσ = = = = so ha he crierio becomes N = = s e s. I compuaioal work i is ormal pracice o use he log rasform = log N ( s ) e +. log = s 3/9

25 Appedix (Mohly Demad Daa for Three Car Pars) Par Year Moh /9

26 Mehod GAM SES MCROST LOG AVAR y 0.78 $m $m $α $β 0.5 $σ ˆv ˆv $p Pr y < S Table. Summary of Resuls for Car Par 5/9

27 Mehod GAM SES MCROST LOG AVAR y.75 $m $m $α $β 0.30 $σ ˆv ˆv $p Pr y < S Table. Summary of Resuls for Car Par 6/9

28 Mehod GAM SES MCROST LOG AVAR y 50.8 $m $m $α $β 0.00 $σ ˆv ˆv $p Pr y < S Table 3. Summary of Resuls for Car Par 3 7/9

29 Car Par Car Par Car Par Mohs Mohs Mohs 9 33 Figure. Demad Series for Car Pars C:\daa\research\oeg\Forecasig Sales of Slow ad Fas Movig Iveories04 06/07/00

30 Car Par Car Par Car Par Mohs Mohs Mohs Figure. Demad Series (solid lie) ad AVAR Smoohed Series (dashed lie) 9/9