ARIMA demand forecasting by aggregation

Transcription

1 ARIMA deand forecasng by aggregaon Bahan Rosa Tabar To ce hs verson: Bahan Rosa Tabar. ARIMA deand forecasng by aggregaon. Oher [cond-a.oher]. Unversé Scences e Technologes - Bordeaux I, 3. Englsh. <NNT : 3BOR58>. <el- 9864> HAL Id: el-9864 hps://el.archves-ouveres.fr/el-9864 Subed on 8 Apr 4 HAL s a ul-dscplnary open access archve for he depos and dssenaon of scenfc research docuens, wheher hey are publshed or no. The docuens ay coe fro eachng and research nsuons n France or abroad, or fro publc or prvae research ceners. L archve ouvere plurdscplnare HAL, es desnée au dépô e à la dffuson de docuens scenfques de nveau recherche, publés ou non, éanan des éablsseens d ensegneen e de recherche franças ou érangers, des laboraores publcs ou prvés.

2 THÈSE PRÉSENTÉE À L UNIVERSITÉ BORDEAUX ÉCOLE DOCTORALE DES SCIENCES PHYSIQUES ET DE L INGENIEUR Par Bahan ROSTAMI-TABAR POUR OBTENIR LE GRADE DE DOCTEUR SPÉCIALITÉ: PRODUCTIQUE ARIMA DEMAND FORECASTING BY AGGREGATION Souenue le: Décebre 3 Devan la cosson d exaen forée de: Alexandre DOLGUI Professeur, Ecole des Mnes de San-Eenne Rapporeur Yannc FREIN Professeur, INP Grenoble Rapporeur Konsannos NIKOLOPOULOS Professeur, Bangor Unversy, Royaue-Un Exanaeur Jean-Paul BOURRIERES Professeur, Unversé Bordeaux Exanaeur Ars SYNTETOS Professeur, Cardff Unversy, Royaue-Un Invé Mohaed Zed BABAI Ensegnan-Chercheur, Kedge Busness school Co-Dreceur de hése Yves DUCQ Professeur, Unversé Bordeaux Dreceur de hése

3 Absrac Deand forecasng perforance s subjec o he uncerany underlyng he e seres an organsaon s dealng wh. There are any approaches ha ay be used o reduce deand uncerany and consequenly prove he forecasng (and nvenory conrol) perforance. An nuvely appealng such approach ha s nown o be effecve s deand aggregaon. One approach s o aggregae deand n lower-frequency e buces. Such an approach s ofen referred o, n he acadec leraure, as eporal aggregaon. Anoher approach dscussed n he leraure s ha assocaed wh cross-seconal aggregaon, whch nvolves aggregang dfferen e seres o oban hgher level forecass. Ths research dscusses wheher s approprae o use he orgnal (no aggregaed) daa o generae a forecas or one should raher aggregae daa frs and hen generae a forecas. Ths Ph.D. hess reveals he condons under whch each approach leads o a superor perforance as judged based on forecas accuracy. Throughou hs wor, s assued ha he underlyng srucure of he deand e seres follows an AuoRegressve Inegraed Movng Average (ARIMA) process. In he frs par of our research, he effec of eporal aggregaon on deand forecasng s analysed. I s assued ha he non-aggregae deand follows an auoregressve ovng average process of order one, ARMA(,). Addonally, he assocaed specal cases of a frs-order auoregressve process, AR() and a ovng average process of order one, MA() are also consdered, and a Sngle Exponenal Soohng (SES) procedure s used o forecas deand. These deand processes are ofen encounered n pracce and SES s one of he sandard esaors used n ndusry. Theorecal Mean Squared Error expressons are derved for he aggregae and he non-aggregae deand n order o conras he relevan forecasng perforances. The heorecal analyss s valdaed by an exensve nuercal nvesgaon and experenaon wh an eprcal daase. The resuls ndcae ha perforance proveens acheved hrough he aggregaon approach are a The use of he words our and we hroughou he hess s purely convenonal. The wor presened n hs Ph.D. hess s he resul of research conduced by he auhor alone, albe wh suppor fro an acadec nsuon and a supervsory ea.

4 funcon of he aggregaon level, he soohng consan value used for SES and he process paraeers. In he second par of our research, he effec of cross-seconal aggregaon on deand forecasng s evaluaed. More specfcally, he relave effecveness of op-down (TD) and boo-up (BU) approaches are copared for forecasng he aggregae and sub-aggregae deands. I s assued ha ha he sub-aggregae deand follows eher a ARMA(,) or a non-saonary Inegraed Movng Average process of order one, IMA(,) and a SES procedure s used o exrapolae fuure requreens. Such deand processes are ofen encounered n pracce and, as dscussed above, SES s one of he sandard esaors used n ndusry (n addon o beng he opal esaor for an IMA() process). Theorecal Mean Squared Errors are derved for he BU and TD approach n order o conras he relevan forecasng perforances. The heorecal analyss s suppored by an exensve nuercal nvesgaon a boh he aggregae and sub-aggregae levels n addon o eprcally valdang our fndngs on a real daase fro a European supersore. The resuls show ha he superory of each approach s a funcon of he seres auocorrelaon, he cross-correlaon beween seres and he coparson level. Fnally, for boh pars of he research, valuable nsghs are offered o praconers and an agenda for furher research n hs area s provded. Keywords: deand forecasng; eporal aggregaon; cross-seconal aggregaon; saonary processes; nonsaonary processes; sngle exponenal soohng

5 v Acnowledgeens I a graeful o os ercful and ALMIGHTY ALLAH who gave e he healh and opporuny o coplee hs hess. I a delghed o place on record y deepes sense of graude and profound oblgaon o y supervsor Prof Yves Ducq for hs endless suppor, gudance and encourageen durng y hree years as a PhD suden. I apprecae all hs conrbuons of e and deas. I a deeply ndebed o y supervsor, Dr. Mohaed-Zed Baba for hs advce and paence ha sulaed and nourshed y nellecual aury. Hs never endng energy and enhusas for research ade hs PhD nsprng, challengng and very neresng. I would le o express y deep and sncere graude o Prof. Ars Syneos. Hs ndeph nowledge and parcularly hs advce have been nvaluable n ensurng I was on he rgh rac hroughou he Ph.D. sudes. I a graeful o he Capus France for he scholarshp hey provded for hs Ph.D sudy. I a also oblged o BEM (Bordeaux Manageen School) for her suppor ha enabled e o aend and presen y acadec papers a nernaonal conferences. My sncere hans o y broher, ssers, brohers-n-law, nephews and neces for her love and encourageen and for belevng n e. Words fal o express y apprecaon for y beloved wfe Rojn whose eernal love, undersang, care bu os of all confdence n e oo hs docoral sudy owards copleon. Las bu no leas, y deepes graude o y parens Ibrah and Afrooz, whose lessons on hardworng and dlgence enabled e o coplee y PhD hess.

6 v To y oher who passed away n Augus 8 To y Faher To Rojn: My bes Frend

7 v Conens Ls of Appendces v Ls of Fgures v Ls of Tables x Chaper Inroducon and Proble Saeen Defnons Busness Conex Research Bacground Research Overvew Mehodology Thess Srucure Chaper Sae of he ar Inroducon Teporal Aggregaon Teporal aggregaon denfcaon process Deand forecasng by eporal aggregaon Cross-seconal aggregaon Cross-seconal aggregaon denfcaon process Deand forecasng by cross-seconal aggregaon Dscusson on he leraure revew Chaper 3 Teporal aggregaon Theorecal Analyss Noaon and assupons

8 v 3.. MSE dervaon a dsaggregae level MSE Before Aggregaon, MSE BA MSE afer Aggregaon, MSE AA MSE dervaon a aggregae level MSE Before Aggregaon, MSE BA MSE afer Aggregaon, MSE AA Coparave analyss Ipac of he paraeers sensvy analyss Coparson a dsaggregae level Coparson a aggregae level Theorecal Coparson Coparson a dsaggregae level Coparson a aggregae level Opal aggregaon level Coparson a dsaggregae level Coparson a aaggregae level Sulaon nvesgaon Sulaon desgn Sulaon Resul Eprcal analyss Eprcal Daase and Experen Deals Eprcal Resuls Coparson a dsaggregae level Coparson a aggregae level Concluson Chaper 4 Cross-Seconal Aggregaon Theorecal analyss Noaon and assupons Varance of forecas error a aggregae level Inegraed ovng average process order one, ARIMA(,,)

9 v 4... Auoregressve ovng average process order one, ARIMA(,,) Varance of forecas error a subaggregae level Auoregressve ovng average process order one, ARIMA(,,) Theorecal Coparson Coparson a aggregae level Inegraed ovng average process order one ARIMA(,,) Auoregressve ovng average process order one ARIMA(,,) Coparson a subaggregae level Auoregressve ovng average process order one ARIMA(,,) Sulaon sudy Sulaon desgn Sulaon resuls Coparson a he Aggregae Level Coparson a Subaggregae Level Eprcal analyss Eprcal daase and experen deals Eprcal resuls Concluson Chaper 5 Conclusons and Fuure Research Conrbuons of he Thess Teporal aggregaon Cross seconal aggregaon Manageral plcaons Teporal aggregaon Cross seconal aggregaon Laons and fuure research References

10 x Ls of Appendces Appendx A: The relaonshp of auocovarance beween nonaggregae and aggregae deand Appendx B: Covarance beween he dsaggregae deand and aggregae forecas for ARIMA(,,) process Appendx C: Varance of he aggregae forecas for he ARIMA(,,) process Appendx D: Covarance of he aggregae deand and non-aggregae forecas for ARIMA(,,) process Appendx E: Coeffcen of varaon before and afer aggregaon for ARIMA(,,) Appendx F: Proof of heore Appendx G: Selecon procedure for he ARIMA(,,) process... Appendx H: Proof of heore -3, ARIMA(,,)... Appendx I: Selecon procedure for he ARIMA(,,) process... 4 Appendx J: Proof of heore 3-3 and heore Appendx K: Selecon procedure for he ARIMA(,,) process- Coparson a he aggregae level... 6 Appendx L: Proof of heore Appendx M: Proof of heore 7-3 for ARIMA(,,) Coparson a he aggregae level 9 Appendx N: Proof of heore 9-3 for an ARIMA(,,) - Coparson a aggregae level.. Appendx O: Proof of heore 9-3 for an ARIMA(,,) - Coparson a aggregae level.. Appendx P: Covarance beween deand,j and forecas,j... 3 Appendx Q: Proof of heore Appendx R: Proof of heore 4-, collary 4- and collary

11 x Ls of Fgures Fgure -: Non-overlappng eporal aggregaon (fro weely o onhly daa)... 8 Fgure -: Overlappng eporal aggregaon (fro weely o onhly daa)... 9 Fgure -3: Coparson a dsaggregae level... Fgure -4: Coparson a aggregae level... Fgure -5: Scheac dagra of TD (lef) and BU (rgh) approaches... Fgure -6: Research Overvew... 8 Fgure -7: Mehodology... 9 Fgure 3-: Saple auocorrelaon of ARIMA(,,) process when = -.8 and = Fgure 3-: Saple auocorrelaon of ARIMA(,,) process when =.8 and = Fgure 3-3: Auocorrelaon of ARIMA(,,) process Fgure 3-4: Saple auocorrelaon and he process shape of ARIMA(,,) process when = Fgure 3-5: Saple auocorrelaon and he process shape of ARIMA(,,) process when = Fgure 3-6: Saple auocorrelaon and he process shape of ARIMA(,,) process when = Fgure 3-7: Saple auocorrelaon and he process shape of ARIMA(,,) process when = Fgure 3-8: Ipac of,,, and on he rao of MSE:.,. Fgure 3-9: Ipac of,,, and on he rao of MSE:.3,. (op) (op) Fgure 3-: Ipac of,, and on he MSE rao for. (op) and. 5 (boo)... 8 Fgure 3-: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao... 8 Fgure 3-: Ipac of,, and on he MSE rao for. (op) and. 5 (boo) Fgure 3-3: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao Fgure 3-4: Ipac of,,, and on he rao of MSE:.,...5 (boo) Fgure 3-5: Ipac of,,, and on he rao of MSE:.3,. (op) (op) Fgure 3-6: Ipac of,, and on he MSE rao for. (op) and. 5 (boo)... 9 Fgure 3-7: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao... 9 Fgure 3-8: Ipac of,, and on he MSE rao for. (op) and. 5 (boo) Fgure 3-9: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao Fgure 3-: MSE rao for dfferen values of for an ARIMA(,,) process... 8 Fgure 3-: MSE rao for dfferen values of ARIMA(,,) process a dsaggregae level... 9

12 x Fgure 3-: MSE for dfferen values of, ARIMA(,,)process copared a aggregae level... Fgure 3-3: MSE for dfferen values of, ARIMA(,,) process coparson a aggregae level... Fgure 3-4: MSE for dfferen values of, ARIMA(,,) process coparson a aggregae level... Fgure 3-5: Ipac of,,, and on he rao of MSE copared a dsaggregae level:.,.(op),.. 5 (boo)... 6 Fgure 3-6: Ipac of,,, and on he rao of MSE copared a aggregae level:.,.(op),.. 5 (boo)... 7 Fgure 3-7: Eprcal resuls copared a dsaggregae level, ARIMA(,,) process... Fgure 3-8: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a dsaggregae level, ARIMA(,,) process... 3 Fgure 3-9: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a dsaggregae level, ARIMA(,,) process wh -< Fgure 3-3: Eprcal resuls copared a aggregae level, ARIMA(,,) process... 6 Fgure 3-3: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a aggregae level, ARIMA(,,) process Fgure 3-3: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a aggregae level, ARIMA(,,) process wh -<< Fgure 3-33: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a aggregae level, ARIMA(,,) process wh.33<< Fgure 4-: Saple auocorrelaon of ARIMA(,,) process when = Fgure 4-: Saple auocorrelaon of ARIMA(,,) process when = Fgure 4-3: Relave perforance of he TD and he BU approaches n forecasng aggregae deand under dfferen cobnaons of, and for an ARIMA(,,) deand process Fgure 4-4 : Relave perforance of he TD and he BU approaches n forecasng aggregae deand under dfferen cobnaons of, and for an ARIMA(,,) process Fgure 4-5: Relave perforance of TD and BU approaches n forecasng subaggregae es under dfferen values of,,, for an ARIMA(,,) process Fgure 4-6 : Relave perforance of TD and BU approaches n forecasng subaggregae es under dfferen values of,,, for an ARIMA(,,) process.... 6

13 x Ls of Tables Table -: he effec of he non-overlappng eporal aggregaon on process s srucure Table -: he effec of he overlappng eporal aggregaon on process s srucure Table -3: he effec of eporal aggregaon on neger ARIMA ype process s srucure Table -4: aggregae process of cross-seconal aggregaon Table 3-: Auocorrelaon of ARIMA(,,) process Table 3-: Selecon procedure for he ARIMA(,,) process, Coparson a dsaggregae level Table 3-3: Selecon procedure for he ARIMA(,,) process, Coparson a aggregae level... Table 3-4: Paraeers of he sulaon experen... 3 Table 3-5: Processes presen n he eprcal daa se, ARIMA(,,) process... 9 Table 3-6: Processes Presen n he Eprcal Daa Se, ARIMA(,,) process... 9 Table 3-7: Processes Presen n he Eprcal Daa Se, ARIMA(,,) process... Table 4-: The rao of V TD /V BU for dfferen conrol paraeers and. 5,.. 5 Table 4-: The rao of V TD /V BU for dfferen conrol paraeers and.5, 5 Table 4-3: Paraeers of he sulaon experen... 5 Table 4-4: The eprcal daa se for ARIMA(,,) Table 4-5: The eprcal daa se for ARIMA(,,) Table 4-6: The eprcal rao of V TD /V BU for an ARIMA(,,) process Table 4-7: The eprcal rao of V TD /V BU by consderng he aggregaon beween dfferen groups (nervals of values) Table 4-8: The eprcal resuls for ARIMA(,,)... 67

14 B.Rosa-Tabar, 3, Chaper 3 Chaper Inroducon and Proble Saeen Ths chaper provdes he overall acadec perspecve, he objecves of hs wor and he seps requred o conduc he research and ee he objecves. Frs, soe ey ers n he area of deand aggregaon and forecasng are defned. Then, he busness conex, he research bacground and an overvew of he research and s objecves are presened before dscussng he ehodologcal approach eployed for he purposes of hs wor. We elaborae on all hese ssues laer on n he hess n chapers - 4. The srucure of hs PhD hess s presened a he end of he chaper. To aan a unfed undersandng of conceps relaed o hs research wor, s necessary o ae a sep bac and provde he defnon of soe ey ers.. Defnons In hs secon, a bref descrpon of he ey ers and phrases used n hs research wor s provded. These are he ers ha are beng used all along hs hess and specfcally n chapers 3 and 4. Te seres Mardas e al (998) defned a e seres as a sequence of observaons over e. A e seres s an ordered sequence of observaons. Alhough, he order s usually hrough e, parcularly n ers of soe equally spaced e nervals, he orderng ay also be aen hrough oher densons, such as space (Harvey, 993). Te seres occur n a varey of felds such as agrculure, busness and econocs, engneerng, geophyscs, edcal scence, socal scence, ec. For exaple n he busness conex, annual producon levels, onhly spare pars deand, weely nvenory levels and daly sales all consue exaples of e seres. In hs hess, we focus on (weely) deand e seres. Wh regards o he

15 B.Rosa-Tabar, 3, Chaper 4 eprcal daa used for he purposes of our research an poran qualfcaon needs o be ade. Sales fgures are beng used as a proxy for deand. Deand self ay no necessarly equal he sales, n case of requess no beng sasfed due o soc ous. Tha s, deand would equal he (acheved) sales plus he los (or bacordered) sales. However, s reasonable o use hs approxaon and a necessary condon. Saonary e seres A saonary e seres s one whose properes do no depend on he e a whch he seres s observed (Mardas e al., 998). For a sochasc process o be saonary he expeced value of he e seres, he varance and he auocovarance of any lag does no depend on e (Harvey, 993). The os general class of saonary odels for forecasng a e seres s he class of Auoregressve Movng Average (ARMA) processes. Non-Saonary e seres Many appled e seres, parcularly arsng fro econoc and busness areas are non-saonary. Non-saonary e seres can occur n any ways. They could have nonconsan eans, e varyng varances and/or auocovarances, or all of hese properes occurrng sulaneously. Trend, seasonaly and cyclcal e seres are ypes of nonsaonary e seres (We, 6). One of he ypcal non-saonary class of odels s he AuoRegressve Inegraed Movng Average (ARIMA) one. A non-saonary e seres can be dvded n wo pars: ) Hoogeneous e seres ) Non-hoogenous e seres. In he forer case, he ean s e-dependen. By copung he dfferences beween consecuve observaons, a hoogeneous non-saonary e seres can be convered o a saonary one. Ths s nown as dfferencng. However, any non-saonary e seres are nonhoogenous. The non-saonary of hese seres s no due o her e-dependen expeced value, bu raher o her e-dependen varance and auocovarance. Forecasng ehods A forecasng ehod s a procedure for esang he fuure observaons. I depends largely on wha daa s avalable. If here s no daa avalable, or f he avalable daa s no relevan o he forecass, hen qualave forecasng ehods us be used. There are welldeveloped srucured approaches o obanng good forecass whou usng hsorcal daa

16 B.Rosa-Tabar, 3, Chaper 5 (Hyndan and Ahanasopoulos, 3). In conras, quanave forecasng can be appled when wo condons are sasfed:. Nuercal nforaon (daa) abou he pas s avalable,. I s reasonable o assue ha soe aspecs of he pas paerns wll connue no he fuure (.e. here are no srucural changes). There s a wde range of quanave forecasng ehods, ofen developed whn specfc dscplnes for specfc purposes. Each ehod s assocaed wh specfc properes, accuracy levels and coss (of pleenaon) all of whch us be consdered when choosng beween he. Mos quanave forecasng probles relae o eher e seres daa (colleced a regular nervals over e) or cross-seconal daa (colleced a a sngle pon n e). Quanave forecasng ehods are dvded n wo general caegores: ) e seres odel ) explanaory odels. An explanaory odel s very useful because ncorporaes nforaon abou oher varables, raher han only hsorcal values of he varable o be forecas. However, here are several reasons a forecaser gh selec a e seres odel raher han an explanaory odel. Frs, he syse ay no be undersood, and even f was undersood ay be exreely dffcul o easure he relaonshps assued o govern s behavour. Second, s necessary o now or forecas he varous predcors n order o be able o forecas he varable of neres, and hs ay be oo dffcul. Thrd, he an concern ay be only o predc wha wll happen raher han explanng precsely why soehng happens. Fnally, he e seres odel ay gve ore accurae forecass han an explanaory or xed odel (Hyndan and Ahanasopoulos, 3). Esaor selecon In order o evaluae he pac of each aggregaon approach on he forecasng perforance, an esaor needs o be seleced and used for exrapolaon purposes. In hs sudy, Sngle Exponenal Soohng (SES), also referred o as Exponenally Weghed Movng Average (EWMA) ehod, s used o esae he fuure deand. SES s a very popular forecasng ehod n ndusry as s nuvely appealng, easy o undersand and has nal copuer sorage requreens. Moreover, s opal for a non-saonary Inegraed Movng Average process of order one, ARIMA(,,). Alhough s applcaon

17 B.Rosa-Tabar, 3, Chaper 6 ples a non-saonary behavor of he deand, suffcenly low soohng consan values nroduce nor devaons fro he saonary assupon whls he ehod s also unbased. SES s esaor reles upon exponenally soohed forecass of he deands. The esae s updaed n each perod. For any e perod, he updang procedure of SES s ehod s presened below: f f d ( -) where d - s he deand n perod -, f s he forecas of perod and s he soohng consan. For any beween zero and one, he weghs aached o he observaons decrease exponenally as we go bac n e, hence he nae exponenal soohng. If s sall (.e., close o zero), he weghs are spread across he observaons o he very dsan pas. If s large (.e., close o one), ore wegh s gven o he ore recen observaons and he weghs declne sharply o zero for relavely recen observaons. A he exree case where =, SES becoes a naïve ehod,.e. he very las acual deand s he forecas for he nex e perod. In hs research wor, we rely upon he use of he SES ehod raher han a popular alernave (he ovng average (MA)) or any opal forecasng ehod (arsng under he ARIMA srucure), alhough hese forecasng ehods can be consdered n he nex seps of research. There are wo reasons ha suppor he choce of he SES ehod: ) On average, SES ends o ouperfor he MA ehod, as observed n an eprcal coparson of her perforance n he M3 forecasng copeon (as repored by Mardas and Hbon ()). In addon, SES corresponds o an nuvely appealng underlyng odel, whereas MA does no. I s also poran o noe ha under he saonary assupon, Brown (963) showed he correspondence beween SES and MA (correspondence beween he soohng consan value and he lengh of he ovng average).

18 B.Rosa-Tabar, 3, Chaper 7 ) In pracce, he decson aers ay no wan o spend oo uch e and effor exanng and defnng he characerscs of he daa-generang process pror o deernng he forecasng odel, as s requred by ARIMA. Besdes, n a producon plannng fraewor, forecass are requred on a perodc bass, soees as ofen as on a daly or even hourly bass. Typcally, forecasng s done sulaneously for several dfferen, bu relaed es n copuerzed syses wh nal huan nervenon. Therefore, s que praccal o deerne he opal ARIMA odel for each e n each updang perod. However, s useful o deerne he aoun of gan or loss by usng an opal forecasng ehod nsead of SES. Ths ssue wll be consdered n he nex seps of research. Accuracy easure An accuracy easure s a easure appled for judgng he effcency of a forecasng ehod. Forecas accuracy relaes o a coparson beween he forecas and he. acual values. Thhere are any accuracy easures dscussed n he leraure ha ay be used o repor perforance (Hyndan and Koehler, 6). However, such easures are no necessarly aheacally racable ang possble o use he for heorecal analyss. In hs research wor, he varance of forecas error or equvalenly he Mean Square Error (MSE) (for unbased esaon procedures) s ulsed as he only accuracy erc. Alhough we do wsh o conras perforances on eprcal daa, he a of hs wor s o undersand he underlyng reasons as o why one ehod perfors beer han anoher. To do so, a heorecal coparson needs o be underaen and he MSE s he only avalable erc. Addonally, he MSE s slar o he varance of he forecas errors (whch consss of he varance of he esaes produced by he forecasng ehod under concern and he varance of he acual deand) bu no que he sae snce any poenal bas of he esaes ay also be aen no accoun. Snce SES provdes unbased esaes for he processes consdered n hs wor he varance of forecas errors s equal o he MSE,.e. MSE = Var(Forecas Error). Deand Aggregaon An aggregaon process consss of dervng a low frequency represenaon of he process fro a hgh frequency forulaon; hs dervaon can be exered hrough e or hrough ndvduals.

19 B.Rosa-Tabar, 3, Chaper 8 Aggregaon across e, also called eporal aggregaon, refers o he process by whch a low frequency e seres (e.g. quarerly) s derved fro a hgh frequency e seres (e.g. onhly) (Nolopoulos e al., ). As shown n Fgure - and Fgure -, hs s acheved hrough he suaon (buceng) of every perods of he hgh frequency daa, where s he aggregaon level. There are wo dfferen ypes of eporal aggregaon: nonoverlappng and overlappng. In he forer case (Fgure -) he e seres are dvded no consecuve non-overlappng buces of e where he lengh of he e buce equals he aggregaon level. The aggregae deand s creaed by sung up he values nsde each buce. The nuber of aggregae perods s [N/], where N s he nuber of he orgnal perods, he aggregaon level and he [x] operaor reurns he neger par of x. As a consequence he nuber of perods n he aggregae deand s less han he orgnal deands. Fgure -: Non-overlappng eporal aggregaon (fro weely o onhly daa) The overlappng case (Fgure -) s slar o a ovng wndow echnque where he wndow s sze equals o he aggregaon level. A each perod, he wndow s oved one sep ahead, so he oldes observaon s dropped and he newes s ncluded. I s observed ha he nuber of overlappng aggregae perods s hgher han hose of he non-overlappng and equals o N-+. Therefore, he nforaon loss n neglgble as copared o he non-

20 B.Rosa-Tabar, 3, Chaper 9 overlappng case. Ths s an poran observaon n ers of daa avalably and for he cases where lle hsory of daa s avalable. Fgure -: Overlappng eporal aggregaon (fro weely o onhly daa) In hs research, only he case of he non-overlappng eporal aggregaon s consdered. The overlappng eporal aggregaon s an ssue lef for furher research. In he nex secon, he effec of eporal aggregaon on he srucure of e seres s revewed. Ofen, for he purpose of havng coparable forecass usng he eporal aggregaon approaches as copared o he classcal non-aggregaon approaches, f he coparson s underaen a he dsaggregae level, hen he aggregae forecass should be dsaggregaed o he orgnal level (by dvdng he on he aggregaon level). Furherore, f he coparson s conduced a he aggregae level, hen he orgnal forecass should be ulpled by he aggregaon level. Ths s llusraed n Fgure -3 and -4 n he case of weely and onhly forecass.

21 B.Rosa-Tabar, 3, Chaper Fgure -3: Coparson a dsaggregae level Fgure -4: Coparson a aggregae level

22 B.Rosa-Tabar, 3, Chaper Anoher ype of aggregaon referred o as cross-seconal (or herarchcal or coneporaneous) aggregaon occurs when he aggregaon aes place across a nuber of Soc Keepng Uns (SKU) a one specfc e perod o reduce varably (Slvesrn and Veredas, 8). Exsng approaches o cross-seconal forecasng usually nvolve eher a boo-up (BU) or a op-down (TD) approach (or a cobnaon of he wo). When forecasng a he aggregae level s of neres, he forer nvolves he aggregaon of ndvdual SKU forecass o he group level whereas he laer relaes o forecasng drecly a he group level (.e. frs aggregae requreens and hen exrapolae drecly a he aggregae level). Aggregae foreca s calculaed Sub-aggregae Level Aggregae Level Sub-aggregae es Aggregae forecas s dsaggregaed o oban sub-aggregae forecass Aggregae forecas s calculaed by sung up he subaggregae forecass Sub-aggregae forecass Sub-aggregae es Sub-aggregae forecass Deand Forecas Deand Forecas Fgure -5: Scheac dagra of TD (lef) and BU (rgh) approaches When he ephass s on forecasng a he subaggregae level, hen BU relaes o drec exrapolaon a he subaggregae level whereas TD nvolves he dsaggregaon of he forecass produced drecly a he group level. An poran ssue ha has araced he aenon of any researchers as well as praconers over he las few decades s he effecveness of such cross-seconal forecasng approaches. As llusraed by Fgure -5 hese approaches wor as follows: The TD approach consss of he followng seps: ) subaggregae deand es are aggregaed; ) he forecas of aggregae deand s produced

23 B.Rosa-Tabar, 3, Chaper by applyng SES a he aggregae level, and ) he forecas s subaggregaed bac o he orgnal level by applyng an approprae dsaggregaon ehod, f a subaggregae forecas s needed. In he BU approach: ) subaggregae deand forecass are produced drecly for he subaggregae es; ) he aggregae forecas s obaned by cobnng ndvdual forecass for each SKU,.e. poenally a separae forecasng odel s used for each e n he produc faly (Zoer e al., 5). These approaches are presened scheacally n Fgure -5. The presenaon syle follows ha adoped by Mohaadpour e al ().. Busness Conex Deand forecasng s he sarng pon for os plannng and conrol organzaonal acves. Moreover, one of he os poran challenges facng odern copanes s deand uncerany (Chen and Blue, ). The exsence of hgh varably n deand for fas ovng and slow/neren ovng es (es wh a hgh rao of zero observaons) pose consderable dffcules n ers of forecasng and soc conrol. Devaons fro he degree of varably accoodaed by he Noral dsrbuon ofen render sandard forecasng and nvenory heory napproprae (Chen e al., ; Syneos and Boylan, 5; Weerlov and Whybar, 984). There are any approaches ha ay be used o reduce deand uncerany and hus o prove he forecasng (and nvenory conrol) perforance of a copany. An nuvely appealng such approach ha s nown o be effecve s deand aggregaon (Chen e al., 7). One possbly s he Teporal Aggregaon. Anoher aggregaon approach ofen appled n pracce s he Cross-seconal Aggregaon (as dscussed n he prevous secon). Such an approach s equvalen o aggregang daa for one sngle SKU across a nuber of depos or soc locaons. Naural, praccally useful, assocaed fors of aggregaon also nvolve geographcal consoldaon of daa or aggregaon across ares. Alhough no eprcal sudes exs ha docuen he exen o whch aggregaon aes place n praccal sengs, hs s an approach ha s nown o be popular aongs praconers because of s nuve appeal. In praccal ers, he benef depends on he ype of aggregaon and of course he daa characerscs. Cross-seconal aggregaon for exaple usually leads o varance reducon. Ths s due o he fac ha flucuaons n he daa fro

24 B.Rosa-Tabar, 3, Chaper 3 one e seres ay be offse by he flucuaons presen n anoher e seres (Wdara e al., 9). Conrary o cross-seconal aggregaon, n eporal aggregaon, varance s ncreased. However, s shown ha eporal aggregaon can reduce he coeffcen of varaon of deand. In any case, he pled benef coupled wh he ease of pleenng such approaches renders he a popular choce n ndusry. Deand daa ay be broadly caegorzed as neren and fas. Aggregaon of deand n lower-frequency e buces enables he reducon of he presence of zero observaons n he forer case or, generally, reduces uncerany n he laer. Ineren deand es (such as spare pars) are nown o cause consderable dffcules n ers of forecasng and nvenory odellng. The presence of zeroes has sgnfcan plcaons because of he followng hree reasons. Frs, he dffculy n capurng underlyng e seres characerscs and fng sandard forecasng odels. Second, he dffculy n fng sandard sascal dsrbuons, such as he Noral. Thrd, devaons fro sandard nvenory odellng assupons and forulaons. These concerns collecvely render he anageen of hese es a very dffcul exercse. Teporal aggregaon s nown o be appled wdely n lary sengs (very sparse daa), he afer sales ndusry (servce pars) ec. Recen eprcal sudes n hs area (Baba e al., ; Nolopoulos e al., ) have resuled n soe very prosng fndngs ponng ou also he need for ore heorecal analyss. Alhough he area of forecasng wh eporal aggregaon n an neren deand conex s a very neresng one boh fro an acadec and praconer perspecve, n hs research only he os ofen occurrng cases of fas deand es are consdered. Analyss n an neren deand conex s an poran avenue for furher research and hs ssue s dscussed n ore deal n he las chaper of hs Ph.D. hess. In addon o he deand uncerany reducon assocaed wh he eporal aggregaon approach dscussed above, here s anoher poran ssue ha relaes o he forecas horzon ha renders aggregaon a very prosng approach. The forecas horzon deernes how far no he fuure he esae projecons us be. As a general rule, he furher no he fuure we loo, he ore clouded our vson becoes and consequenly long range forecass are less accurae han shor range forecass. Ths s also one of he areas where he eporal aggregaon ay prove he forecas accuracy, because as we loo furher no

25 B.Rosa-Tabar, 3, Chaper 4 he fuure, he long er vew becoes ore poran and he eporal aggregaon approach ay ulze hs nforaon ore effecvely han he classcal approaches. Fro an acadec perspecve he ephass o dae has been anly on he crossseconal aggregaon. Moreover, os nvenory forecasng sofware pacages suppor he aggregaon of daa alhough hs would ypcally cover cross-seconal aggregaon only. The consderaon of eporal aggregaon has been soewha negleced by sofware anufacurers and acadecs ale despe he poenal opporuny for addng ore value o real world pracces. In hs wor, he objecve s o advance he curren sae of nowledge n he area of deand forecasng eporal aggregaon (and exend he exsng heory on cross seconal aggregaon). In he above dscussons, he effec of eporal aggregaon on a sngle SKU s consdered. However, n realy here are ofen any relaed e seres ha can be organzed herarchcally and aggregaed a several dfferen levels n groups based on producs, cusoers, geography or oher feaures (Hyndan e al., ). The herarchcal level a whch forecasng s perfored depends on he funcon he forecass are fed no. Wh regards o producs (or SKUs) n parcular, forecasng a he ndvdual SKU level s requred for nvenory conrol whereas produc faly forecass ay be requred for Maser Producon Schedulng. Forecass across a group of es ordered fro he sae suppler ay be requred for he purpose of consoldang orders. Forecass across he es sold o a specfc large cusoer ay deerne ransporaon and roung decsons ec. TD and BU forecasng approaches are exreely useful owards provng he accuracy of forecass and plans when leveraged whn an S&OP (Sales and Operaons Plannng) process (Lapde, 6). The S&OP s a ul-funconal process ha nvolves anagers fro all deparens (Sales, Cusoer Servce, Supply Chan, Mareng, Manufacurng, Logsc, Procureen and Fnance), where each deparen requres dfferen levels of deand forecass (Lapde, 4). For exaple, n areng (Depe and Hanssens, ), forecasng of revenues by produc groups and brands s needed; sales deparens deal wh sales forecass by cusoer accouns and/or sales channels; supply chan anagers reques SKU level forecass, whle fnance requres forecass ha are aggregae no budgeary uns n ers of revenues and coss (Bozos and Nolopoulos, ). In order o produce he requred forecass, deand and/or forecass should be

26 B.Rosa-Tabar, 3, Chaper 5 aggregaed and/or dsaggregaed o varous levels. Ths nvolves he applcaon of boh TD and BU or a cobnaon of he (Lapde, 4, 6). 3. Research Bacground Aggregaon has been wdely dscussed n he acadec leraure snce as early as he 95s (Quenoulle, 958). I s seen as a eans o anage he deand flucuaon and reduce he degree of uncerany. I has been shown by Thel (954), Yehuda and Zv (96), and Agner and Goldfeld (974) ha deand uncerany can be effecvely reduced hrough approprae deand aggregaon and forecasng. In he leraure of supply chan plannng and deand plannng, deand aggregaon s nown as a rs-poolng approach o reduce deand flucuaon for ore effecve aeral/capacy plannng(chen and Blue, ). In he area of eporal aggregaon, here are boh heorecal and eprcal nvesgaons dscussed n he leraure. However, os of hese conrbuons ay be found n he Econocs dscplne. Aeya and Wu (97) evaluaed he effec of non-overlappng eporal aggregaon when he orgnal seres follows an auoregressve process of order p, AR(p) process. By consderng he rao of MSE of non-aggregae and aggregae predcon (3 lnear predcors were consdered) a he aggregae level, hey have shown ha he aggregae approach ouperfors he non-aggregae one. Tao (97) nvesgaed he effec of nonoverlappng eporal aggregaon on a non-saonary process of he Inegraed Movng Average IMA(d,q) for. A condonal expecaon was appled o oban one sep ahead forecass a he aggregae level based on he non-aggregae and aggregae seres. Subsequenly, he effcency of he aggregae forecass was defned as he rao of he varance of he forecas error of he non-aggregae o he aggregae seres when he aggregaon level s large. I was shown ha when d= and he aggregaon level n very hgh, hen he rao under concern equals one and he coparave benef of usng he non-aggregae forecass s ncreased wh d. Few recen peces of research have evaluaed he effec of eporal aggregaon on forecasng and soc conrol by eans of eprcal analyss. Nolopoulos e al. () eprcally analysed he effecs of eporal aggregaon on forecasng neren deand requreens and hey have proposed a ehodology ered as ADIDA (Aggregae Dsaggregae Ineren Deand Approach o forecasng). I was shown ha he ADIDA

27 B.Rosa-Tabar, 3, Chaper 6 ehodology ay ndeed offer consderable proveens n ers of forecas accuracy. In addon, Baba e al. () have exended he sudy dscussed above (Nolopoulos e al., ) by eans of consderng he nvenory plcaons of he ADIDA fraewor hrough a perodc order-up-o-level soc conrol polcy. The researchers concluded ha a sple echnque such as eporal aggregaon can be as effecve as coplex aheacal neren forecasng approaches. To he bes our nowledge, he only papers drecly relevan o our wor are hose by Aeya and Wu (97) and Tao (97) for he AR and he MA process respecvely. These wors focused on characerzng he aggregae deand seres n addon o evaluang he forecas perforance. However, he resuls presened n hese wors rean prelnary n naure whle her experenal seng ay also be crczed n ers of he esaon procedures consdered. In addon, no eprcal resuls were obaned. Therefore, he lac of condons ha ay deerne he superory of each approach n deand forecasng s obvous. I s no clear when he aggregaon approach provdes ore accurae forecass han he non-aggregaon one and vce versa. Consequenly, he ovaon behnd hs par of he research sudy was he lac of he heorecal analyss regardng he effec of eporal aggregaon on deand forecasng. In hs research, analycal evaluaon s appled o denfy he superory condons of each approach. The research sars wh he sple frs order ARMA ype process as dscussed earler n secon.. However, he analyss can be conduced for hgher order processes and hs wll be consdered n he fuure. In he area of cross-seconal aggregaon, os of he forecasng leraure has looed a he coparave perforance of he TD and he BU approaches. The fndngs wh regards o he perforance of hese approaches are xed. Soe auhors le Thel (954), Grunfeld and Grlches (96), Schwarzopf e al. (988), and Narashan e al., 985(985) argued ha TD ouperfors he BU approach. On he oher hand anoher auhors such as Orcu e al. (968), Edwards and Orcu (969), Dunn e al. (976), Dangerfeld and Morrs(988) and Gross and Sohl (99) found ha he BU approach perfors beer; and fnally soe oher auhors le Barnea and Laonsho (98), Fledner (999) and Wdara e al.(7, 8, 9) ae a conngen approach and analyse he condons under whch one approach produces ore accurae forecass han he oher.

28 B.Rosa-Tabar, 3, Chaper 7 In hs PhD hess, he effecveness of he BU and he TD approaches s evaluaed. The research conduced by Wdara e al. s exended o consder a ore general saonary deand process ARIMA(,,) and a non-saonary ARIMA(,,) process. Moreover, he coparson s underaen a boh subaggregae and aggregae levels. Addonally, he superory of each approach s exaned by a real daa se. 4. Research Overvew Aggregaon enables forecasers o oban forecass a varous levels across e and ndvdual es. Dependng on he level of forecasng, we ay eher provde he forecass and hen aggregae he or we ay frs aggregae he orgnal seres o oban he aggregae deand and hen produce he aggregae forecas. In he laer case, a dsaggregaon ay be requred o oban he dsaggregae forecas. In hs research he pac of aggregaon on deand forecasng s evaluaed. To show he effec of aggregaon on deand forecasng, wo dfferen ypes of aggregaon are consdered: ) eporal aggregaon and ) cross seconal aggregaon. Our research overvew s suarzed n he Fgure -6. The aheacal analyss s copleened by a nuercal nvesgaon o valdae he heorecal resuls whch s also used n order o conduc a sensvy analyss by soe consranng assupons consdered n he analycal evaluaon. Nex, he fndngs are valdaed eprcally (by eans of sulaon on a daase provded by a European supersore) and by dong so soe very uch requred eprcal evdence n he area of deand aggregaon s offered. Fnally, poran anageral nsghs are derved and angble suggesons are offered o praconers dealng wh nvenory forecasng probles. Based on he research bacground and ovaons, sx objecves have been forulaed for hs research:. To evaluae analycally he effec of non-overlappng eporal aggregaon on forecasng when he basc seres follows a saonary ARMA ype process.. To denfy he condons under whch he eporal aggregaon approach ouperfors he non-aggregaon one and vce versa.

29 B.Rosa-Tabar, 3, Chaper 8 3. To deerne he opal aggregaon level ha axzes he benefs of he eporal aggregaon approach. 4. To exane he effecveness of he BU and he TD approaches o forecas subaggregae and aggregae deand n a saonary and a non-saonary envronen. 5. To analyse he effec of he conrol and he process paraeers on he superory of each approach n boh eporal and cross-seconal aggregaons. 6. To es he eprcal valdy and uly of he heorecal and sulaon resuls on a large se of real world daa. Research sudy Aggregaon ype Teporal Aggregaon Cross-seconal Aggregaon Deand process ARMA(,), MA(), AR() ARMA(,), IMA() Forecasng ehod SES SES Accuracy easure MSE (he Varance of forecas error) MSE (he Varance of forecas error) Coparson level Aggregae level, Dsaggregae level Aggregae level, Dsaggregae level Objecve Idenfy he superory condons of he aggregaon and he nonaggregaon approaches Evaluae he effecveness of he BU and he TD approaches Fgure -6: Research Overvew

30 B.Rosa-Tabar, 3, Chaper 9 5. Mehodology The research follows hree research ehods, naely aheacal analyss, sulaon and eprcal nvesgaon. The relaonshp beween he hree ehods s llusraed n Fgure -7. Fgure -7: Mehodology Frsly, he aheacal analyss s appled o exane he superory of he aggregaon approach and o dsclose he condons under whch hs approach provdes ore accurae resuls han he classcal approach. The Sulaon sudy s used for he followng reasons: To es and valdae he resuls of heorecal analyss. To relax he assupons consdered n he aheacal evaluaon. Fnally, he fndngs of hs PhD hess are o be esed on real eprcal daa o assess he praccal valdy and applcably of he an resuls of he sudy. Therefore, eprcal analyss would help us o es he applcably of he resuls n real suaons. 6. Thess Srucure The PhD hess s srucured as follows:

31 B.Rosa-Tabar, 3, Chaper 3 In Chaper, an overvew of deand forecasng by aggregaon s presened. Dfferen ypes of aggregaon,.e. eporal and cross-seconal aggregaon are dscussed and he effec of aggregaon on process srucure s descrbed. In Chaper 3, he effec of non-overlappng eporal aggregaon on deand forecasng s exaned when he underlyng seres follow a saonary process. For each process under consderaon, he heorecal MSE s derved a boh he dsaggregae and he aggregae level of coparson. Then, he MSE resuls are copared o denfy he condons under whch each approach ouperfors he oher. Nex a sulaon analyss s conduced o exane he resuls of he heorecal evaluaon followed by an eprcal nvesgaon. In Chaper 4, he effecs of cross-seconal aggregaon on deand forecasng s evaluaed. I s assued ha he underlyng seres follow eher a saonary or a nonsaonary process. An analycal evaluaon s frs consdered followed by sulaon o es and valdae he heorecal resuls. Addonally, soe assupons are relaxed copared o he heorecal analyss. The resuls are copleened by an eprcal analyss o valdae he fndngs on a real deand daa se. Fnally, he fndngs fro each chaper are suarzed and he conclusons of hs hess are dscussed n chaper 5. Manageral plcaons and laons of he research are descrbed, along wh opporunes for fuure research.

32 B.Rosa-Tabar, 3, Chaper 3 Chaper Sae of he ar The frs chaper suarzed he research wors conduced n hs sudy. I oulned he research hrough a suary of he research bacground and probles, expeced resuls, and desgnaed ehodology. Ths chaper as o provde an overvew of he leraure on forecasng by eporal and cross-seconal aggregaon.. Inroducon Deand forecasng s he sarng pon for os plannng and conrol organzaonal acves. In general pracce, accurae deand forecass lead o effcen operaons and hgh levels of cusoer servce, whle naccurae forecass nevably lead o neffcen, hgh cos operaons and/or poor levels of cusoer servce. In any organzaons, one of he os poran acons ha ay be aen o prove he effcency and he effecveness of he decson ang process s o prove he accuracy of he deand forecass. When developng he deand forecasng, he praconers need o deerne n whch level hey should produce he forecas. Forecasers need o properly denfy wha s he objecve of he forecasng process, n ers of e buce (.e., forecass are produced on a daly level, weely or on onhly one), and se of es he deand refers o (.e., sngle e or group of es). The choce of he approprae level of forecasng depends on he decsonang process he forecas s expeced o suppor. For nsance, forecasng a he ndvdual SKU level s requred for supply chan anageen, whle cuulave aggregae forecas ay be used for budgeng or plan desgn. In any organzaons, several anagers fro all deparens (Sales, Cusoer Servce, Supply Chan, Mareng, Manufacurng, Logsc, Procureen and Fnance) are nvolved n generang forecas, where each deparen requres dfferen levels of deand forecass (Lapde, 4).

33 B.Rosa-Tabar, 3, Chaper 3 In addon, one of he os poran facors ha nfluence he accuracy of forecass s deand varably. Deand dsperson and uncerany are aong he os poran challenges facng odern copanes (Chen and Blue, ). These ssues have been addressed n he acadec leraure for any years. The exsence of hgh dsperson n deand for fas ovng and slow/neren ovng es (es wh a hgh rao of zero observaons) pose consderable dffcules n ers of forecasng and soc conrol. Devaons fro he degree of varably accoodaed by he Noral dsrbuon ofen render sandard forecasng and nvenory heory napproprae (Chen e al., ; Syneos and Boylan, 5; Weerlov and Whybar, 984). There are any approaches ha ay be used o reduce he deand dsperson and provde he dfferen forecas level and consequenly prove he forecasng (and nvenory conrol) perforance of a copany. An nuvely appealng such sraegy ha s nown o be effecve s deand aggregaon (Chen e al., 7). One approach s o aggregae deand n lower-frequency e buces, hereby reducng he presence of poenal zero observaons (n case of neren deand) or generally reduce dsperson n case of fas ovng deand. Such an aggregaon sraegy s ofen referred o, n he acadec leraure, as Teporal Aggregaon (Nolopoulos e al., ). Anoher aggregaon sraegy dscussed n he leraure s he Cross-Seconal Aggregaon(also referred o as herarchcal), whch nvolves aggregang dfferen e seres o oban hgher level forecass(slvesrn and Veredas, 8). Exsng approaches o cross-seconal forecasng usually nvolve eher a boo-up (BU) or a op-down (TD) approach (or a cobnaon of he wo). Alhough he concep of aggregaon s very sple bu plays a very poran role n supply chan anageen(bonoo, 3). An neresng queson rased when applyng aggregaon o forecas deand s how exacly does ha affec he deand dsperson. The relevan pac reles enrely upon he ype of aggregaon cross-seconal versus eporal. Cross seconal aggregaon usually leads o varance reducon. Ths s due o he fac ha flucuaons n he daa fro one Soc Keepng Un (SKU) are offse by flucuaons n he daa fro oher SKUs (Wdara e al., 9). Conrary o cross seconal aggregaon, n eporal aggregaon, varance s ncreased. Schluer and Trede (Schluer and Trede, ) have shown ha for ceran ypes of daa generaon processes, boh he ean and varance of he daa ncrease hrough eporal aggregaon. However, s easy o show ha eporal aggregaon

34 B.Rosa-Tabar, 3, Chaper 33 can reduce he coeffcen of varaon of deand and hs ssue s furher dscussed laer n our paper. Aggregaon has been wdely dscussed n he acadec leraure snce as early as he 95s (Quenoulle, 958). In a producon plannng fraewor, any researchers have focused on he effecveness of cross-seconal aggregaon and especally on he boo-up and op-down approaches. However here are fewer sudes focusng on he effecs of eporal aggregaon. Moreover, and alhough os nvenory forecasng sofware pacages suppor aggregaon of daa, hs would ypcally cover cross-seconal aggregaon only; he consderaon of eporal aggregaon has been negleced by sofware anufacurers despe he poenal opporuny for addng ore value o her cusoers. In he followng secons, he exsng researches conduced n he area of eporal and cross-seconal aggregaon are presened.. Teporal Aggregaon In hs secon, he effec of he eporal aggregaon on he process srucure dscussed n he leraure revewed. Then, he pac of eporal aggregaon on deand forecasng dscussed n he leraure s presened... Teporal aggregaon denfcaon process An orgnal e seres odel s presened n ers of basc e un. Alhough he orgnal for of he odel can be used o produce he forecass, however n soe cases he e frequency of he observed daa ay no be he sae as he assued e un. For hese cases a eporally aggregae daa ay be used, so s necessary o now he effec of aggregaon on odel srucure of he daa processes. The orders of he low frequency odel (.e. onhly) fro hose of he hgh frequency odel (.e. weely) can be deerned by eporal aggregaon..e, If he hgh frequency odel s an ARIMA(,,), wha s he low frequency odel? Second, once he orders are nferred, he paraeers of he low frequency odel s derved fro he hgh frequency ones, raher han esang he.

35 B.Rosa-Tabar, 3, Chaper 34 The analyss of eporal aggregaon sars wh he wor of Aeya and Wu (97). I s shown ha f he orgnal varable follows a p h order auoregressve process, ARIMA(p,,), hen he non-overlappng aggregaes follow a xed auoregressve ovng average (ARIMA) odel of he (p,,q*). Tao (97) has nvesgaed he effec of nonoverlappng eporal aggregaon on a non-saonary process of he Inegraed Movng Average ARIMA (,d,q) for, where d s he negraed paraeer and q s he ovng average paraeer. I s shown ha he aggregae process s of he ARIMA (,d,q*). Brewer (973) suded he effecs of non-overlappng eporal aggregaon on ARIMA (p,,q) processes. I s shown ha aggregang such processes resuls n ARMA processes wh auoregressve order p and ovng average order r, ARMA (p,,r). The effec of he nonoverlappng eporal aggregaon on ARIMA(p,d,q) process s evaluaed by Wess (984). I s seen ha he eporally aggregae process s also follow an ARIMA(p,d,r) process. We (979) suded he aggregaon effec on unvarae ulplcave seasonal e seres odels. I s revealed ha for an ARIMA process of order,, (,,), he correspondng aggregae process s an ARIMA of order,, (,,) s*. Brewer(973) also presened a generalzaon of he resuls for ARMA odels wh exogenous varables (ARMAX odels), s shown ha he eporally aggregae ARIMAX(p, d, q)() odel s an ARIMAX(p, d, r)(a). Teles e al (999) sowed ha eporal aggregaon changes he order of a fraconally negraed ARFIMA process o an ARFIMA( p,d, ), whle leavng he value of d unchanged. Addonally, Souza and Sh (4) showed ha for AR Fraconally IMA (ARFIMA) odels eporal aggregaon resuls n bas reducon. Dros and Njan (993) consdered he effc of eporal aggregaon on he ARMA odels wh syerc GARCH errors, ARMA(p,q)-GARCH(P,Q). s revealed ha he aggregae odel follows an ARMA(p,r) wh wea GARCH(R,R). hey have also consdered he ARCH and GARCH ype odels. I s shown ha he eporal aggregaon of an ARCH(q) s an GARCH(q,q), s also seen ha he eporally aggregae GARCH(,q) s an GARCH(q,q). Sra and We (986) suded he relaonshp beween he auocovarance funcon of dsaggregae and aggregae processes. They have shown ha he auocovarnace funcon of he laer can be copued based on he auocovarance funcon of forer; n parcular he

36 B.Rosa-Tabar, 3, Chaper 35 auocovarance funcon afer aggregaon s a funcon of he aggregaon level and auocovarance funcon before aggregaon. Table - suarzed he effec of he non-overlappng eporal aggregaon on he srucure of he process.

37 B.Rosa-Tabar, 3, Chaper 36 Table -: he effec of he non-overlappng eporal aggregaon on process s srucure Non-aggregae process Aggregae process Paraeers Reference ARIMA (p,,q) ARMA (p, q*) q p q* p ARIMA (p,,q) ARMA (p, q*) ( )( p ) q* ARIMA (,,q) MA (n ) q n q* ARIMA (,d, q) IMA (d, n ) q d n q* d (Brewer, 973) (Aeya and Wu, 97) (We, 6) (Tao, 97) p( ) ( d )( ) q (Wess, 984) ARIMA (p, d, q) ARIMA (p, d, r) r ARIMA(p,d,q) (P,D,Q)s ARIMA(p,d,r) (P,D,R)s p r ( ) d( ) q (We, 979)

38 B.Rosa-Tabar, 3, Chaper 37 D s P Q D s P R * Frs, f < s, here s sll soe seasonaly n he eporally aggregae process. Second, f s a ulple of s, he seasonal cycle reans consan. Las, f s equal or larger han s, seasonaly vanshes. ARIMAX(p, d, q)() ARIMAX(p, d, r)(a) q d p r ) ( d v d d p r ) ( ) ( d v d d p a ) ( (Brewer, 973) ARFIMA( p,d,q) ARFIMA( p,d, ) - (Teles e al., 999) ARCH(q) GARCH(q,q) - (Dros and Njan, 993) GARCH(, q) GARCH(q, q) - (Dros and Njan, 993)

39 B.Rosa-Tabar, 3, Chaper 38 r p ( ) q (Dros and Njan, 993) ARIMA(p,,q)- GARCH(P,Q) ARMA(p,r) wh wea GARCH(R,R) R r r r r ax P, Q INAR() INARMA(,) - (Brannas e al., ) INMA() INMA() - (Brannas e al., )

40 B.Rosa-Tabar, 3, Chaper 39 Luz e al.(99) evaluaed he effec of overlappng eporal aggregaon where he orgnal seres follows an ARIMA process. I s found ha he eporally aggregae process of an ARIMA(p,d,q) s an ARIMA(P,d,Q). To he bes of our nowledge hs s he only research dealng wh he pac of overlappng eporal aggregaon on he srucure of he ARIMA ype process. Table -: he effec of he overlappng eporal aggregaon on process s srucure ARIMA(p,d,q) ARIMA(P,d,Q) P p and Q q+- (Luz e al., 99) Alhough any sudes consder he case of fas ovng es or connuous-valued e seres, neger e seres have receved less aenon n a eporal aggregaon conex. Brannas e al() frs suded he non-overlappng eporal aggregaon of an Ineger Auo-Regressve process of order one, INARIMA(,,), I s shown ha he aggregae seres follows an Ineger Auo-Regressve Movng Average process of order one, INARIMA (,,). Addonally, s observed ha he non-overlappng eporal aggregaon of an Ineger ovng average process of order one, INARIMA(,,) s an INARIMA(,,). Table -3: he effec of eporal aggregaon on neger ARIMA ype process s srucure Non-aggregae process Aggregae process Type Reference INARMA (p,, q) INARMA (p,, q) Overlappng (Mohaadpour and Boylan, ) INARMA(,, ) INARMA(,, ) Overlappng (Brannas e al., ) INARMA(,, ) INARMA(,, ) Non-overlappng (Brannas e al., ) INARMA(,, ) INARMA(,, ) Non-overlappng (Brannas e al., )

41 B.Rosa-Tabar, 3, Chaper 4 Brannas e al.() evaluaed he effec of overlappng eporal aggregaon for he INARIMA(,,) process, s seen ha he aggregae process also follow an INARIMA(,,) process. The effec of overlappng eporal aggregaon on INARIMA(p,,q) process s evaluaed by Mohaadpour and Boylan (). I s shown ha he overlappng eporally aggregae of an INARIMA(p,,q) process s also an INARIMA(p,,q) one. In he nex secon we provde a revew of he sudes ha apply eporal aggregaon approach n he area of deand forecasng... Deand forecasng by eporal aggregaon In he supply chan and deand plannng leraure, deand aggregaon s generally nown as a 'rs-poolng approach o reduce deand flucuaon for ore effecve aeral/capacy plannng (Chen and Blue, ). Deand uncerany ay consderably affec forecasng perforance wh furher derenal effecs n producon plannng and nvenory conrol. I has been shown by Thel (Thel, 954), Yehuda and Zv (Yehuda and Zv, 96), Agner and Goldfeld (Agner and Goldfeld, 974) ha deand uncerany can be effecvely reduced hrough approprae deand aggregaon and forecasng. Mos of he leraure ha deals wh eporal aggregaon ay be found n he Econocs dscplne. The analyss of eporal aggregaon sars wh he wor of Aeya and Wu (Aeya and Wu, 97). They assued ha he orgnal varable follows a p h order auoregressve process, AR(p). By consderng he rao of MSE of dsaggregae and aggregae predcon (3 lnear predcors were consdered) a he aggregae level, hey have shown ha he MSE of dsaggregae forecass s greaer han ha of he aggregae ones,.e. he aggregaon approach ouperfors he non-aggregaon one. Tao (Tao, 97) has nvesgaed he effec of non-overlappng eporal aggregaon on a non-saonary process of he Inegraed Movng Average IMA(d,q) for. They appled a condonal expecaon o oban one sep ahead forecass a he aggregae level based on he dsaggregae and aggregae seres. Subsequenly, he effcency of he aggregae forecass was defned as he rao of he varance of he forecas error of he dsaggregae o he aggregae seres when he aggregaon

42 B.Rosa-Tabar, 3, Chaper 4 level s large. They have shown ha when d= he rao under concern equals o and he coparave benef of usng he dsaggregae forecass s ncreasng wh d. Ahanasopoulos e al. () have looed a he effecs of non-overlappng eporal aggregaon on forecasng accuracy n he ours ndusry. They conduced an eprcal nvesgaon usng 366 onhly seres and soe forecasng ehods esed n he M3 copeon daa (Mardas and Hbon, ), naely Innovaons sae space odels for exponenal soohng (labeled ETS), he ARIMA ehodology, a coercal sofware (Forecas Pro), daped rend (Gardner and McKenze, 985), he Thea ehod and naïve. The onhly seres were aggregae o be quarerly, and he quarerly seres were furher aggregae o be yearly. Subsequenly, hey copared he accuracy of he forecass ade before and afer aggregaon. They consdered one and wo sep-ahead forecass and hree sascal easures were used o copare he resuls: Mean Absolue Percenage Error (MAPE), Mean Absolue Scaled Error (MASE) and Medan Absolue Scaled Error (MdASE). The aggregae forecass a he yearly level (wheher produced fro onhly or quarerly daa) were found o be ore accurae han he forecass produced fro he yearly daa drecly. Ths sudy provded consderable eprcal evdence n suppor of eporal aggregaon. Luna and Balln () have used a non-overlappng aggregaon approach o predc daly e seres of cash oney whdrawals n he neural forecasng copeon, NN5. Each e seres conssed of 735 daly observaons whch have been used o forecas 56 daly seps ahead for wo ses of and e seres. Daly saples were aggregae o gve weely e seres and hen an adapve fuzzy rule-based syse was appled o provde 8- sep-ahead forecass (hus aggregaon reduced he forecas horzon fro 56 o 8 seps). Two dfferen aggregaon approaches were evaluaed for hs purpose: he hsorcal op-down (TD-H) approach and he daly op-down (TD-DM) approach, where he an dfference beween he wo was he dsaggregaon procedure. In he forer case aggregae forecass were ds-aggregae based on hsorcal percenages. In he laer case, he daly esaons were correced by ulplyng he by he assocaed weely esaon and dvdng by he su of he seven daly esaed saples. The syerc MAPE (smape) and he Mean hp://

43 B.Rosa-Tabar, 3, Chaper 4 Absolue Error (MAE) were used o copare he resuls. The researchers showed ha he aggregae forecass produced by he wo approaches perfored slarly or beer han hose gven by he daly odels drecly. The reducon of a forecas horzon fro 56 o 8 seps ahead would be nuvely expeced o lead o perforance proveen. The effec of eporal aggregaon on deand forecasng for neger e seres have receved less aenon coparng o connues e seres. Wllean e al.(994) eprcally explored he effecs of eporal aggregaon on forecasng neren deand consderng he applcaon of Croson s ehod(croson, 97) ha has been specfcally developed for such deand paerns. The researchers consdered 6 eprcal daa ses of 95 daly observaons; he aggregaon level was consdered o be a wee. Resuls were repored by consderng he MAPE and he researchers showed a sgnfcan reducon n forecasng errors when weely deand aggregae daa were used nsead of daly daa. Mohaadpour and Boylan () have suded heorecally he effecs of overlappng eporal aggregaon of INARMA processes. They showed ha he aggregaon of an INARMA process over a gven horzon resuls n an INARMA process as well. The condonal ean of he aggregae process was derved as a bass for forecasng. A sulaon experen was conduced o assess he accuracy of he forecass produced usng he condonal ean of he aggregaon approach for hree INARMA processes: INARIMA(,,), INARIMA(,,) and INARIMA(,,), agans ha of he nonaggregaon approach. The sulaon resuls showed ha, n os cases, he aggregaon approach provdes forecass wh saller MSEs han non-aggregaon ones. The perforance of hese forecass was also esed by usng wo eprcal daases. The frs one was fro he Royal Ar Force (RAF, UK) and conssed of he ndvdual deand hsores of 6, SKUs over a perod of 6 years (onhly observaons). The second daa se conssed of he deand hsory of 3, SKUs fro he auoove ndusry (over a perod of 4 onhs). The oucoe of he eprcal nvesgaon confred he sulaon resuls. Nolopoulos e al. () have eprcally analysed he effecs of non-overlappng eporal aggregaon on forecasng neren deand requreens. Ther proposed approach, called Aggregae-Dsaggregae Ineren Deand Approach (ADIDA), was assessed on 5, SKUs conanng 7 years hsory (84 onhly deand observaons) for he Royal Ar Force (RAF, UK), by eans of eployng hree ehods: Naïve, Croson and

44 B.Rosa-Tabar, 3, Chaper 43 Syneos-Boylan Approxaon (SBA)(Syneos and Boylan, 5). The aggregaon level was vared fro o 4 onhs. Coparsons were perfored a he orgnal seres level (dsaggregae deand) and he resuls showed ha he proposed ADIDA ehodology ay ndeed offer consderable proveens n ers of forecas accuracy. The an conclusons of hs sudy were: () he ADIDA ay be perceved as an poran ehod self-provng echans; () an opal aggregaon level ay exs eher a he ndvdual seres level or across seres; (3) seng he aggregaon level equal o he lead e lengh plus one revew perod L+ (whch s he e buce requred for perodc soc conrol applcaons) shows very prosng resuls. Sphouras e al.(sphouras Georgos P. e al., ) exended he applcaon of he ADIDA approach o fas-ovng deand daa. The ehod s perforance was esed on,48 onhly e seres of he M3-Copeon by usng he Naïve, SES, Thea, Hol and daped forecasng ehods. The eprcal resuls confred he prevous fndngs repored by Nolopoulos e al.(nolopoulos e al., ). Fnally, Baba e al. () have also exended he sudy dscussed above (Nolopoulos e al., ) by eans of consderng he nvenory plcaons of he ADIDA fraewor hrough a perodc order-up-o-level soc conrol polcy. Three forecasng ehods, SES, Croson and SBA were used and he deand was assued o be negave bnoally dsrbued. Perforance was repored hrough he nvenory holdng and baclog volues and coss, for hree possble arges Cycle Servce Levels (CSL): 9%, 95% and 99%. For hgh CSLs, he aggregaon approach has been shown o be ore effcen bu for low CSLs was ouperfored by he classcal one when Croson s ehod was used. For SES, he aggregaon approach ouperfors he classcal approach even for low CSLs. The researchers concluded ha a sple echnque such as eporal aggregaon can be as effecve as coplex aheacal neren forecasng approaches. 3. Cross-seconal aggregaon In hs secon, he effec of cross-seconal aggregaon on he process srucure s suarzed. Then, he effecveness of cross-seconal aggregaon approaches on deand forecasng n he leraure s revewed.

45 B.Rosa-Tabar, 3, Chaper Cross-seconal aggregaon denfcaon process When dealng wh he pac of he cross-seconal aggregaon on forecasng, s necessary o nfer he characerscs of he aggregae daa fro he orgnal subaggregae daa..e, If he subaggregae seres follow an ARIMA process, s possble o nvesgae wheher he aggregae observed seres follows an ARIMA process as well. (Granger and Morrs, 976) showed ha he cross-seconal aggregaon of N uncorrelaed ARIMA(p,,q) processes s also an ARIMA(x,,y) process. As a specal case he showed ha he su of wo uncorrelaed ARMA processes, ARMA ( p, q ) and ARMA ( p, q ) s also an ARMA ( p + p, K), where K ax( p q, p q ). Anderson (975) saed ha he su of N ndependen Movng Average processes: MA( q ), MA( q ) MA( q n ),s an MA (q) process as well. I s seen by Harvey (993) ha when he subaggregae es follow and ARIMA(,,) process, he aggregae daa ay follow an ARIMA(,,), ARIMA(,,) or ARIMA(,,) process. Zaffaron (7) showed ha he su of wo ndependen srong GARCH(l,l) processes s wea GARCH(,). Table -4: aggregae process of cross-seconal aggregaon Sub-aggregae process Aggregae process Paraeer Reference ARIMA(p,,q ) ARIMA(x,,y) x N y ax p x p q (Granger and Morrs, 976) AR()+ AR() AR() If = AR() If =- (Harvey, 993) ARMA(,) oherwse MA(q ) MA(q) q ax( q, q... q ) (Anderson, 975) GARCH(,) GARCH(,) When + = + (Zaffaron, 7) n

46 B.Rosa-Tabar, 3, Chaper Deand forecasng by cross-seconal aggregaon Deand forecasng for sales and operaons anageen ofen concerns any es, perhaps hundreds of housands, sulaneously. The convenonal forecasng approach s o exrapolae he daa seres for each SKU ndvdually. However, os busnesses have naural groupngs of SKUs; ha s, he SKUs ay be aggregae o ge hgher levels of forecass across dfferen densons such as produc fales, geographcal area, cusoer ype, suppler ype ec (Chen and Boylan, 7). Such an approach enables he poenal denfcaon of e seres coponens such as rend or seasonaly ha ay be hdden or no parcularly prevalen a he ndvdual SKU level. Group approaches for exaple are nown o offer consderable benefs owards he esaon of seasonal ndces (Chen and Boylan 8). Mos of he forecasng leraure n hs area has looed a he coparave perforance of he op-down (TD) and he boo-up (BU) approach. The fndngs wh regards o he perforance of hese approaches are xed. Many researchers have provded evdence n favour of he TD approach. Gross and Sohl (99) for exaple, nuercally found ha he TD approach (n conjuncon wh an approprae dsaggregaon ehod) provded beer esaes han BU forecasng n wo ou of hree produc lnes exaned. Fledner (999) evaluaed by eans of sulaon he forecas syse perforance a he aggregae level resulng fro varyng degrees of cross correlaon beween wo subaggregae e seres. The subaggregae es were assued o follow a Movng Average process of order one, MA() and he forecasng ehods consdered were SES and he Sple Movng Average (SMA). Ths research showed he forecas perforance a he aggregae level o benef fro he TD approach. Barnea and Laonsho (98) exaned he effecveness of BU and TD on forecasng corporae perforance. They repored ha posve cross-correlaon conrbues o he superory of forecass based on aggregae daa (TD). On he oher hand, Orcu e al. (968) and Edwards and Orcu (969) argued ha nforaon loss s subsanal when aggregang and herefore he boo-up approach provdes ore accurae forecass. Dangerfeld and Morrs (99) and Gordon e al. (997)

47 B.Rosa-Tabar, 3, Chaper 46 used a subse of he M-copeon 3 daa (Mardas e al., 98) o exane he perforance of TD and BU approaches on subaggregae deand forecasng. They found ha forecass by he BU approach were ore accurae n os suaons especally when es were hghly correlaed or when one e donaed he aggregae seres. Weaherford e al. () evaluaed he perforance of BU and TD approaches o oban he requred forecass for hoel revenue anageen. The daa hey consdered was perceved as very ypcal whn he hoel ndusry. They experened wh four dfferen approaches (fully subaggregae d, aggregang by rae caegory only, aggregang by lengh of say only, and aggregang by boh rae caegory and lengh of say) o ge dealed forecass by day of arrval, lengh of say and rae caegory and lengh of say for revenue anageen. The resuls of her sudy showed ha a purely subaggregae forecas srongly ouperfored even he bes aggregae forecas. Soe auhors ae a conngen approach and analyse he condons under whch one ehod produces ore accurae forecass han he oher. Shlfer and Wolff (979) evaluaed analycally he superory of BU and TD on forecasng sales for specfc and enre are segens. They enoned ha BU s preferable for he purpose of forecasng he aggregae seres. In addon, hey found ha ncreasng he nuber of SKUs favours TD. However, when he coparson was perfored a he subaggregae level, hey found ha TD ofen resuls n larger forecas error han BU. Lüepohl (984) showed ha gh be preferable o forecas aggregae varables usng a TD approach when a e seres s generaed by a ulvarae ARMA process and he sascal properes of he subaggregae es are nown. However, f he processes used for forecasng are esaed fro a gven se of e seres daa hen he BU approach ouperfored TD. Wdara e al. (7) suded analycally he condons under whch one approach ouperfors he oher for forecasng he e level deands when he subaggregae es follow a frs-order auoregressve [AR()] process wh he sae auoregressve paraeer for all he es and when SES s used o exrapolae fuure deand requreens. They found ha he superory of each approach s a funcon of he auoregressve paraeer. Wdara e al. (8, 9) also evaluaed analycally he effecveness of TD and BU approaches a he subaggregae and aggregae level, respecvely. 3 The M Copeon s an eprcal forecas accuracy coparson exercse nroduced by Prof. Mardas.

48 B.Rosa-Tabar, 3, Chaper 47 They showed ha when all subaggregae es follow an MA() process wh dencal ovng average paraeers, here s no dfference n he relave perforance of TD and BU forecasng as long as he opal soohng consan s used n boh approaches. Subsequenly, hey conduced a sulaon analyss consderng non-dencal process paraeers for subaggregae es and concluded ha here s sgnfcan dfference beween he wo approaches. The superory of each approach was a funcon of he ovng average paraeer, he cross-correlaon and he proporon of a subaggregae coponen s conrbuon o he aggregae deand. Vswanahan e al. (8) used a sulaon sudy o nvesgae he effecveness of TD and BU approaches n esang he aggregae daa seres when he subaggregae es are neren. The sudy reveals ha low varably of he ner-deand nervals favours he BU approach (usng Croson s ehod (Croson, 97)). However, when deand szes and ner-deand nervals of he subaggregae seres are hghly varable and aggregaon encopasses any es, TD perfors bes. 4. Dscusson on he leraure revew In hs chaper, an overvew of he leraure on he deand forecasng by aggregaon approach s gven. The overvew presened by classfyng he leraure no wo pars: eporal and cross-seconal aggregaon approaches. In he frs par, he heorecal and eprcal nvesgaons n he area of eporal aggregaon are dscusses. The forer anly focused on he srucure of he aggregae e seres and he relaonshp beween he aggregae and dsaggregae process paraeers. The laer evaluaed he effec of he eporal aggregaon on deand forecasng n ers of forecas accuracy easures and soc conrol ercs. Accordng o he leraure, eporal aggregaon approach ay provde ore accurae forecass han classcal one n he fas and slow ovng envronens. However, he condons under whch one approach ay ouperfor oher one are no dscussed n he leraure. I s no clear when dsaggregae daa should be used and where s beer o use he aggregae daa o produce he forecas. To he bes of our nowledge, he only papers drecly relevan o our wor are hose by Aeya and Wu (97) and Tao (97) for he AR and MA process respecvely. In boh cases he researchers nvesgaed he forecas perforance of eporal aggregaon

49 B.Rosa-Tabar, 3, Chaper 48 sraeges under an (Auo-Regressve Inegraed Movng Average) ARIMA-ype fraewor. However, he resuls presened n hese wo papers rean prelnary n naure whle he experenal seng ay also be crczed n ers of he esaon procedures consdered. In addon, no eprcal resuls were provded. Iporan as hey are, boh papers focused on characerzng he aggregae deand seres raher han he forecas perforance. In hs research wor, he condons under whch aggregaon and non-aggregaon approaches yeld ore accurae forecass are deerned by analycal nvesgaon. hs wor consders he case of ARIMA(,,) and s specal cases ARIMA(,,) and ARIMA(,,) processes and as such soe of he heorecal resuls presened n he above dscussed research are of drec relevance o our analyss. Our wor dffers fro hese wors hough and exends he n soe very sgnfcan ways: ) opal esaors are seldo used n pracce no only due o he copuaonal requreens ha are ypcally prohbve bu also he lac of undersandng on he par of he anagers of her funconaly. In addon, here s evdence o suppor he fac ha sple forecasng ehods (such as SES ha s used n our wor) perfor a leas as good as ore coplex heorecally coheren alernaves (Mardas and Hbon, ); ) a dffculy assocaed wh aggregaon ehods s he fac ha a dsaggregaon echans s also requred snce very ofen forecass are needed a he orgnal/dsaggregae deand level. Boh papers consder a coparson a he aggregae level whch addresses only par of he forecasng proble. Consderaon of a coparson a he orgnal deand level, whch s he case consdered n hs wor, addresses anoher par of he proble and s an poran exenson of he research already beng done 4 ; ) no eprcal analyss has been underaen n boh papers n conras wh hs wor were he heorecal fndngs are eprcally valdaed; v) he analyss s copleened by eans of furher 4 An poran assupon n our analyss s ha we sar wh daa ha are as dsaggregae as our requred forecasng oupu. However, and as one of he referees correcly poned ou he degree of aggregaon of he forecasng oupu does no necessarly need o ach wh he exsng daa srucure (whch ay be ore aggregae or ore dsaggregae han he forecass drvng decson ang). The degree of aggregaon of he forecasng oupu (.e. he forecas we use o ae decsons) s acually a funcon of he decson ang proble forecasng res o suppor. On he conrary npus o he forecasng process are very ofen drven by exsng daa srucures. Alhough he wo ay ndeed ach soees, hs s no always he case.

50 B.Rosa-Tabar, 3, Chaper 49 nuercal nvesgaons o denfy he opu aggregaon level and soohng consan values ha requre o be used. In he second par, he coparave perforance of he BU and TD approaches o forecas subaggregae and aggregae deand s revewed. Mos of he researches n hs area s based on he sulaon and he eprcal analyss. However, here are few wor focused on he effecveness of BU and TD by analycal nvesgaon. To he bes of our nowledge, he only papers drecly relevan o our wor are hose by Wdara e al.(7, 9) and Sbrana and Slvesrn (3). Wdara e al. evaluaed analycally he effecveness of he TD and BU approaches under he assupon of an AR() coparng a subaggregae level(auo-regressve process of order ) and MA() process coparng a aggregae level respecvely. Sbrana and Slvesrn denfy he condon of superory of Bu and TD copared a aggregae level when he deand process follow and ARIMA(,,) process wh non dencal paraeers. In suary wha can be concluded fro he cross-seconal leraure s boh BU and TD approaches appear o be assocaed wh superor perforance. Ths superory depends on he srucure of he seres and cross-correlaon relaed assupons. In hs wor he relave effecveness of he BU and TD approach for forecasng s evaluaed. I s recognzed ha forecass ay be equally requred a boh he aggregae and sub-aggregae level and as such coparsons are perfored a boh levels. In addon, a ore general unvarae saonary and a non-saonary deand processes a boh aggregae and subaggregae levels are suded. Moreover, he analyss s copleened by eans of an eprcal nvesgaon usng real daa.

51 B.Rosa-Tabar, 3, Chaper 3 5 Chaper 3 Teporal aggregaon In chaper, an overvew of he leraure on deand forecasng by aggregaon s provded, addonally he necessy of conducon ore research wor n he area of deand aggregaon s dscussed. In hs chaper he effec of eporal aggregaon on deand forecasng by eans of he analycal, sulaon and eprcal nvesgaon s evaluaed. The condons under whch eporal aggregaon ay prove he accuracy of he deand forecass are denfed. he effecs of eporal aggregaon on forecasng when he underlyng seres follows a frs order Auoregressve Movng Average process, ARIMA(,,) Auoregressve process of order one, ARIMA(,,) and a Movng Average process of order one, ARIMA(,,) s suded. Furherore, he forecasng ehod s he Sngle Exponenal Soohng (SES). These assupons bear a sgnfcan degree of reals. As s dscussed laer n he chaper here s evdence o suppor he fac ha deand ofen follows he saonary processes assued n hs wor (48% of he eprcal seres avalable n our research follow such processes). Moreover, SES s a very popular forecasng ehod n he ndusry (Acar and Gardner, ; Gardner, 99, 6; Taylor, 3). Alhough s applcaon ples a non-saonary behavor of he deand, suffcenly low soohng consan values nroduce nor devaons fro he saonary assupon whls he ehod s also unbased. In hs chaper he varance of he forecas error (or equvalenly, by consderng an unbased esaon procedure, he ean square error) obaned based on he aggregae deand o ha of he non-aggregae deand s analycally copared. Coparsons are perfored a boh dsaggregae and aggregae deand level. I s aheacally shown ha he rao of he Mean Squared Error (MSE) of he laer approach o ha of he forer s a funcon of he aggregaon level, he process paraeers and he exponenal soohng consan. The aheacal analyss s copleened by a nuercal nvesgaon o es and valdae he resuls. Nex, he heorecal resuls are valdaed eprcally (by eans of sulaon on a

52 B.Rosa-Tabar, 3, Chaper 3 5 daase provded by a European supersore) and by dong so soe very uch needed eprcal evdence n he area of eporal aggregaon are offered. To he bes of our nowledge, he only wors drecly relevan o our wor are hose by Aeya and Wu (97) and Tao (97) for he AR and MA process respecvely. In boh cases he researchers nvesgaed he forecas perforance of eporal aggregaon sraeges under an (Auo-Regressve Inegraed Movng Average) ARIMA-ype fraewor. However, he resuls presened n hese wo papers rean prelnary n naure whle he experenal seng ay also be crczed n ers of he esaon procedures consdered (Zoer and Kalchschd, 7). In addon, no eprcal resuls were provded. Iporan as hey are, boh wors focused on characerzng he aggregae deand seres raher han he forecas perforance as explaned n he chaper. Ths sudy aeps o fll hs gap and provdes helpful gudelnes o selec he approprae approach under such deand processes. The wor dscussed n hs chaper can be exended o analyse ore general cases such as ARIMA(p,,), ARIMA(,,q) or even ARIMA(p,,q) processes. However, he analyss and presenaon of such resuls would becoe oo coplex. Snce he an objecve of hs research s o oban soe ey anageral nsghs, he analyss s resrced o he ARIMA(,,), ARIMA (,,) and ARIMA (,,) processes only. Consderable par of hs chaper has been publshed n Rosa-Tabar e al (3a)and Rosa-Tabar e al (3c). Ths chaper s organzed as follows. In secon, heorecal analyss of eporal aggregaon for auoregressve ovng average process order one, ARIMA(,,) and s specal cases ovng average order one, ARIMA(,,) and auoregressve order one, ARIMA(,,) s evaluaed. In secon he resuls of he heorecal evaluaon obaned n sub-secon s presened. In secon 3 he sulaon nvesgaon o es and valdae he resuls of he aheacal analyss s used. Nex, a real daa se o valdae he resuls of heorecal and sulaon pars n pracce s appled n secon 4. Fnally he conclusons are gven n secon 5.

53 B.Rosa-Tabar, 3, Chaper 3 5. Theorecal Analyss In hs secon he varance of he forecas errors generaed by consderng he dsaggregae and he aggregae deand s derved. Coparsons are perfored a he orgnal dsaggregae and aggregae level. o ha end, he aggregaon approach wors as follows: frsly buces of aggregae deand are creaed based on he aggregaon level. Then SES s appled o hs aggregae daa o produce he aggregae forecass, now f he coparson s underaen a aggregae level hen he aggregae forecas s ananed, however, o copare a dsaggregae level he aggregae forecass are dsaggregae by dvdng by o produce forecass a he orgnal level. In addon oher dsaggregaon echanss could have been consdered (Nolopoulos e al., ) bu he one eployed for he purposes of hs research s vewed as realsc fro a praconer s perspecve and seen as a reasonable approach when dealng wh saonary deands. Noe ha n order o ensure ha he forecasng horzon s he sae n boh he aggregae and he dsaggregae cases, he aggregae SES forecas s updaed n each perod when he aggregae seres are rebul. The coparsons resul n he developen of heorecal rules ha ndcae under whch condons he forecasng of he aggregae deand s heorecally expeced o perfor beer han he forecasng of he dsaggregae deand. These heorecal rules are a funcon of he aggregaon level, he conrol, and he process paraeers. The cu-off values o be assgned o he paraeers are he oucoe of a nuercal analyss o be conduced based on he heorecal resuls. Havng obaned he cu-off values, we can hen specfy regons of superor perforance of he aggregaon approach over he non-aggregaon one. In hs sudy he varance of he forecas error s used as a forecas accuracy easure as s he only heorecally racable easure. The MSE s slar o he varance of he forecas errors (whch conss of he varance of he esaes produced by he forecasng ehod under concern and he varance of he acual deand) bu no que he sae snce any poenal bas of he esaes ay also be aen no accoun (Syneos., ). Snce SES

54 B.Rosa-Tabar, 3, Chaper 3 53 provdes unbased esaes 5 (due o he saonary of he e seres consdered n hs wor) he varance of forecas errors s equal o he MSE,.e. MSE=Var(Forecas Error). For each process under consderaon he rao of he MSE before aggregaon (MSE BA ) o he MSE afer aggregaon (MSE AA ) s calculaed. A rao ha s lower han one ples ha he aggregaon approach does no add any value. Conversely, f he rao s greaer han one, aggregaon approach perfors beer han he classcal one. 3.. Noaon and assupons For he reander of he research he noaons are denoed by: : Aggregaon level,.e. nuber of perods consdered o buld he bloc of aggregae deand. n: oal nuber of perods avalable n he deand hsory. : Te un n he orgnal dsaggregae e seres. =,,,n. T: Te un n he aggregae e seres. T=,,, n. d : Dsaggregae deand n perod D T : Aggregae deand n perod T : Independen rando varables for dsaggregae deand n perod, norally dsrbued wh zero ean and varance T : Independen rando varables for aggregae deand n perod T, norally dsrbued wh zero ean and varance f : Forecas of dsaggregae deand n perod, he forecas produced n - for he deand n. 5 Obvously oher forecasng ehods ay also provde unbased esaes under he saonary deand processes consdered n hs research bu hose are no consdered as her analyss s beyond he scope of hs research.

55 B.Rosa-Tabar, 3, Chaper 3 54 F T : Forecas of aggregae deand n perod T, he forecas produced n T- for he deand n T. α : Soohng consan used n Sngle Exponenal Soohng ehod before aggregaon, β : Soohng consan used n Sngle Exponenal Soohng ehod afer aggregaon, MSE BA : Theorecal Mean Squared Error (MSE) before aggregaon MSE AA : Theorecal Mean Squared Error (MSE) afer aggregaon : Covarance of lag of dsaggregae deand, Covd, d : Covarance of lag of aggregae deand, D, D Cov T T : Auoregressve paraeer before aggregaon, : Auoregressve paraeer afer aggregaon, : Movng average paraeer before aggregaon, : Movng average paraeer afer aggregaon, : Expeced value of dsaggregae deand n any e perod : Expeced value of aggregae deand n any e perod I s assued ha he dsaggregae deand seres d follows a frs order auoregressve ovng average, ARIMA(,,) or s specal cases ovng average order one, ARIMA (,,) and an auoregressve daa generaon process (DGP) order one, ARIMA(,,). In he followng he characerscs of each process under consderaon are dscussed o provde he nforaon based on he naure of he processes. An ARIMA(,,) process can be aheacally wren n perod as ( 3-):

56 B.Rosa-Tabar, 3, Chaper 3 55 d, where,. d ( 3-) When he deand follows an ARIMA(,,) process he auo-covarance and auocorrelaon funcons are(box e al., 8):, ( 3-). ( 3-3) For dfferen cobnaons of he process paraeers, he resulng underlyng srucure changes consderably. Table 3- presens he auocorrelaon srucure for dfferen process paraeers whch helps o beer undersand he process and can be useful o nerpre he resuls of he forhcong analyss. Table 3-: Auocorrelaon of ARIMA(,,) process Group Process paraeer Auocorrelaon < <, -<< Always posve, <Auocorrelaon lag<, -< <, -<< Oscllae beween posve and negave values 3 -< <, << Oscllae beween posve and negave values 4 < <, << For > Always posve, <Auocorrelaon lag< 5 < <, << For < Always negave, -.5<Auocorrelaon lag<

57 B.Rosa-Tabar, 3, Chaper 3 56 Fgure 3-: Saple auocorrelaon of ARIMA(,,) process when = -.8 and =.9. Fgure 3- and 3- presen he behavour of he ARIMA(,,) process for groups one and wo presened n Table 3-. In Fgure 3- can be seen ha he auocorrelaon s hghly posve no only for lag bu also for hgher lags and decays exponenally. In addon s observed ha he process shape s changng slowly and here s no flucuaon beween e perods.

58 B.Rosa-Tabar, 3, Chaper 3 57 In Fgure 3- he process shape s changng alos a each perod and here are ore flucuaons whch aes he seres ore rregular han rando seres. As can be noed ha he auocorrelaon decays exponenally and oscllaes beween posve and negave values and ends o becoe zero for hgher lags. Fgure 3-: Saple auocorrelaon of ARIMA(,,) process when =.8 and =-.7.

59 B.Rosa-Tabar, 3, Chaper 3 58 An ARIMA(,,) process s a specal case of a ore general ARIMA(,,) process where he auoregressve process s equal o zero.e =. Ths process can be aheacally shown as ( 3-4): d, where, ( 3-4) When he deand follows an ARIMA(,,) process, he auocovarance and auocorrelaon funcons are (We, 6):, ( 3-5). ( 3-6) Fgure 3-3: Auocorrelaon of ARIMA(,,) process

60 B.Rosa-Tabar, 3, Chaper 3 59 Fgure 3-4: Saple auocorrelaon and he process shape of ARIMA(,,) process when = -.9. Fgure 3-3 shows ha he auocorrelaons for an ARIMA(,,) process vares beween.5 and +.5 for hgh posve and hgh negave values of he ovng average paraeer, respecvely. In addon he auocorrelaon s equal o zero for lags greaer han one. In Fgure 3-4 he behavor of he ARIMA(,,) process s presened when he ovng average paraeer s relavely hgh. I s seen ha for hs value, he auocorrelaon funcon s close o +.5 and he process s changng slowly. However, he rae of changng s

61 B.Rosa-Tabar, 3, Chaper 3 6 slower n he case of ARIMA(,,) process where auocorrelaon s hgh, hs s naural as he auocorrelaon funcon for he ARIMA(,,) s uch saller han ARIMA(,,) process. When he ovng average paraeer aes posve values he process shape becoes ore rregular copare o Fgure 3-4. The auocorrelaon funcon s negave for lag and equals o zero for hgher lags as shown n Fgure 3-5. Fgure 3-5: Saple auocorrelaon and he process shape of ARIMA(,,) process when =.9.

62 B.Rosa-Tabar, 3, Chaper 3 6 Fnally he Auoregressve process order one, ARIMA(,,) can be represened as ( 3-7) whch s a specal case of he ARIMA(,,) process where =. d d. ( 3-7) 6): When deand follows an ARIMA(,,) process he followng properes exs (We, -, ( 3-8). ( 3-9) I s clear fro ( 3-9) ha when he auoregressve paraeer aes posve values, he auocorrelaon s always posve no only for lag bu also for hgher lags perods. I exhbs a sooh exponenal decay as shown n Fgure 3-6 for hgh posve values. When he auoregressve paraeer s negave, he auocorrelaon funcon s decays exponenally and oscllaes beween posve and negave values. The process shape s rregular as can be seen n Fgure 3-7

63 B.Rosa-Tabar, 3, Chaper 3 6 Fgure 3-6: Saple auocorrelaon and he process shape of ARIMA(,,) process when =.9..

64 B.Rosa-Tabar, 3, Chaper 3 63 Fgure 3-7: Saple auocorrelaon and he process shape of ARIMA(,,) process when = -.9. The perods non-overlappng aggregae deand of he dsaggregae deand seres as follows D T can be expressed as a funcon

65 B.Rosa-Tabar, 3, Chaper 3 64 D T l d (,,... ) l ( 3-) The forecasng ehod consdered n hs sudy s he Sngle Exponenal Soohng (SES); hs ehod s beng appled n very any copanes and os anagers use hs ehod n a producon plannng envronen due o s splcy (Gardner, 99). Usng SES, he forecas of deand n perod produced a he end of perod - s f d, ( 3-) I s furher assued ha he sandard devaon of he error er n ( 3-4), ( 3-7), and ( 3-) above s sgnfcanly saller han he expeced value of he deand, so when deand s generaed he probably of a negave value s neglgble. Consranng and o le beween - and n ( 3-4), ( 3-7), and ( 3-) eans ha he process s saonary and nverble. 3.. MSE dervaon a dsaggregae level In hs secon he MSE of he one-sep-ahead forecass resuled fro he dsaggregae and aggregae deand daa s derved. Ths secon s dvded no wo sub-secons. Frs, he MSE before aggregaon s calculaed based on he drec forecas resuled fro dsaggregae deand. Then, he MSE afer aggregaon s confgured, so he aggregae forecas s dsaggregae by dvdng he by aggregaon level MSE Before Aggregaon, MSE BA In order o calculae he MSE BA, he forecasng ehod, SES, s drecly appled o dsaggregae deand daa o produce one-sep-ahead forecass.

66 B.Rosa-Tabar, 3, Chaper 3 65 The analyss begns by dervng he MSE BA for he ARIMA(,,) process. As dscussed above he MSE BA s MSE BA Var Var Forecas Error d f Var d Var f Covd, f, ( 3-) Subsequenly, he hree pars of ( 3-) should o be deerned: ) varance of he deand, ) varance of he forecas, and ) he covarance beween he deand and he forecas. The evaluaon of MSE BA s begun by defnng he covarance beween he deand and he forecas as follows: Cov d, f Cov( d, d ) Cov( d, ) Cov ( d, d ) Cov( d, d ) Cov( d, d )..., d ( 3-3) Consderng ha Cov( d, d ) for all > and by subsung ( 3-) n ( 3-3), he covarance beween deand and s forecas s obaned: Cov ( d, f ).... ( 3-4) The varance of he forecas s calculaed as follows: Var f Var d f Var d Var f Cov d, f. ( 3-5)

67 B.Rosa-Tabar, 3, Chaper 3 66 By consderng he fac ha he process s saonary, s nown Var f Var and Covd f Covd, f f, for all values and by subsung ( 3-) and ( 3-4) no ( 3-5) he followng s obaned:, Var ( f ) ( 3-6) The equaons ( 3-), ( 3-4) and ( 3-6) s subsued n ( 3-), hese subsuons coupled wh he fac ha Var d reveals he MSE BA as follows: MSE..5 BA ( 3-7) As a specal case, when = he ARIMA(,,) becoes he ARIMA(,,) process whch s called MA() as well. Therefore, by subsung = n ( 3-7) he MSE BA for ARIMA(,,) process s obaned n ( 3-8): MSE BA. ( 3-8).5 Auoregressve order one, ARIMA(,,) or AR() s a specal case of he ARIMA(,,) process when =. Therefore, he MSE BA for he ARIMA(,,) process s obaned by subsung = n ( 3-7):.5 MSE ( 3-9) BA

68 B.Rosa-Tabar, 3, Chaper MSE afer Aggregaon, MSE AA In hs secon, he dervaon of he MSE of he forecass for he aggregaon approach s deerned. Dsaggregae deand s frs aggregae o yeld hgh frequency deand. Then, he aggregae forecass are provded based on he SES forecasng ehod. Fnally, one-sepahead esaes a he orgnal level are gven by he dsaggregaon of such forecass. Ths dsaggregaon s conduced by dvdng he aggregae forecas by he aggregaon level. The MSE AA s defned as MSE AA F T Var d Var d Var FT Covd, FT Var FT Covd, F. ( 3-) T By applyng SES, he aggregae forecas for perod T s defned as F T D. ( 3-) T In hs secon, he MSE AA s derved for an ARIMA(,,) deand process. When he dsaggregae seres follows an ARIMA(,,) process, he aggregae seres also follows an ARIMA(,,) process bu wh dfferen paraeer values (Saraslan, ; Tao, 97). The auocovarance funcon of an ARIMA(,,) process afer aggregaon s gven:. ( 3-) Fro Appendx A and Based on We (6) he relaonshp beween he auocovarance funcon of he dsaggregae and he aggregae deand for an ARIMA(,,) process s obaned as follows:

69 B.Rosa-Tabar, 3, Chaper ( 3-3). ( 3-4) By consderng ( 3-3), he auocorrelaon funcon afer aggregaon s gven as followng:. ( 3-5) Fro (B-4) and (C-4) n Appendx A and B respecvely, he covarance beween dsaggregae deand and aggregae forecas s gven n ( 3-6). Addonally, he varance of he aggregae forecas s gven n ( 3-7):,, d F T Cov ( 3-6). F T Var ( 3-7)

70 B.Rosa-Tabar, 3, Chaper 3 69 Now, he equaons ( 3-6) and ( 3-7) are subsued n ( 3-). Then, he equaons ( 3-3) and ( 3-) are subsued n ha resul. Fnally, he MSE of he forecas afer aggregaon s gven as follows:. AA MSE ( 3-8) As a specal case, when = he ARIMA(,,) process becoes an ARIMA(,,) process whch s also called MA(), herefore he MSE AA for he ARIMA(,,) process s obaned by subsung = n ( 3-8) :. MSE AA ( 3-9) To oban he MSE BA for he ARIMA(,,) or AR() process, = s subsued n ( 3-8), herefore, he MSE BA for he ARIMA(,,) process s:

71 B.Rosa-Tabar, 3, Chaper 3 7. AA MSE ( 3-3) 3..3 MSE dervaon a aggregae level In hs secon, he varance of he error of he cuulave -sep-ahead forecas s derved. Frsly, he MSE of he forecass resuled fro he dsaggregae deand daa, MSE BA, s calculaed. Then, he aggregae deand s used o calculae he aggregae forecass and, consequenly, he MSE AA s obaned MSE Before Aggregaon, MSE BA The analyss begns by dervng he MSE BA for he ARIMA(,,) process. The MSE of he forecass for he non-aggregaon approach s derved as follows: Frsly, one sep ahead deand forecass are obaned based on he SES ehod. Then, he resuls are ulpled by he aggregaon level. Ths resuls n cuulave -sep-ahead esaes a he aggregae level. The MSE BA s defned by:,, T T T BA f D Cov f Var D Var f D Var MSE ( 3-3) In hs secon, he MSE BA s derved for an ARIMA(,,) deand process. As s defned n ( 3-6), he varance of he dsaggregae forecas s:

72 B.Rosa-Tabar, 3, Chaper 3 7, Var ( f ) ( 3-3) Fro (D-4) n Appendx D, he covarance beween aggregae deand and dsaggregae forecas s gven as follows: Cov ( D T, f ). ( 3-33) When he dsaggregae seres follows an ARIMA(,,) process, he aggregae seres also follows an ARIMA(,,) process bu wh dfferen paraeer values (Brewer, 973; Saraslan, ). The aggregae deand s represened as follows: D, where,. D T T T T ( 3-34) The relaonshp beween he dsaggregae and he aggregae process paraeers s gven n ( 3-3). By consderng ha Var ( 3-3) he followng equaon s gven: and subsung ( 3-6),( 3-33) and ( 3-3) no D T MSE BA ( 3-35) Fnally, by subsung ( 3-) no ( 3-35), he MSE BA s obaned:

73 B.Rosa-Tabar, 3, Chaper 3 7 BA MSE ( 3-36) As a specal case, by subsung = n ( 3-36) he MSE BA for he ARIMA(,,) process s gven as follows:. MSE BA ( 3-37) The MSE BA for he ARIMA(,,) process s obaned by subsung = n ( 3-36): BA MSE ( 3-38) MSE afer Aggregaon, MSE AA In hs secon, he MSE of he cuulave sep ahead forecas s obaned fro he aggregae deand daa. In hs secon, The MSE AA s calculaed for he ARIMA(,,) process. The MSE AA s defned as:,, T T T T T T AA F D Cov F Var D Var F D Var MSE ( 3-39) Fro (C-3) and (C-4) n Appendx C, respecvely, he covarance beween he aggregae deand and s forecas s gven as follwng:

74 B.Rosa-Tabar, 3, Chaper 3 73, D T F T Cov ( 3-4). F T Var ( 3-4) Now by consderng ha D T Var and subsung ( 3-4) and ( 3-4) n ( 3-39), he MSE AA s obaned as follows:. AA MSE ( 3-4) By subsung ( 3-3) and ( 3-4) no ( 3-4), he followng equaon s gven:. AA MSE ( 3-43) Fnally, by subsung ( 3-) no ( 3-43), he MSE AA becoes:. AA MSE ( 3-44) The MSE AA for he ARIMA(,,) process s obaned by subsung = n ( 3-44):

75 B.Rosa-Tabar, 3, Chaper MSE AA ( 3-45) The MSE AA for he ARIMA(,,) process s obaned by subsung = n ( 3-44):. - AA MSE ( 3-46). Coparave analyss The effecveness of eporal aggregaon as copared o non-aggregaon ay be assessed by analyzng he rao of her varance of he forecas error or, equvalenly, her MSEs. Recall fro secon, ha a value of AA BA MSE MSE greaer han one ples ha he aggregaon approach s superor o he non-aggregaon one, whereas a value ha s lower han one ples he oppose. A rao value equal o one eans ha perforance s he sae. In secon 3.., he pac of he aggregaon level,, he soohng consan values, and, he ovng average paraeer,, and he auoregressve paraeer,, on he rao of AA BA MSE MSE s nvesgaed by varyng her values. In secon 3.., he condons under whch one approach ouperfors he oher are analycally deerned. Fnally n Subsecon 3..3 he deernaon of he opu aggregaon level s consdered. 3.. Ipac of he paraeers sensvy analyss In hs Sub-secon he effec of he paraeers,,,, and on he rao BA MSE AA MSE s analysed. Noe ha,,, are conrol paraeers ofen se by he

76 B.Rosa-Tabar, 3, Chaper 3 75 forecaser, whereas and are paraeers assocaed wh he underlyng deand generaon process (process paraeers). Therefore, s neresng o now whch values of he conrol paraeers lead o a rao hgher han one, for any gven values of he process paraeers. In real world sengs, daa could ypcally be aggregae as weely (=7) fro daly daa, yearly (=4) fro quarerly, onhly (=4) fro weely, quarerly (=3) fro onhly, se-annually (=6) fro onhly and annually (=) fro onhly daa or ay also be aggregae a soe oher level o reflec relevan busness concerns (e.g. equal o he lead e lengh). Gven he consderable nuber of conrol paraeer cobnaons, s naural ha only soe resuls ay be presened here Coparson a dsaggregae level In hs sub-secon, he pac of he paraeers on he rao s evaluaed when coparng a he dsaggregae level Auoregressve Movng Average Process Order One, ARIMA(,,) MSE BA MSE AA In hs sub-secon he effec of he paraeers,,, and on he rao of s evaluaed when he non-aggregae deand follows an ARIMA(,,) process. We aep o nuvely explan he effec of hese paraeers on he rao. The aggregaon level, beween and 4,.9. 9wh ncreens of.,.9. 9 wh ncreens of., wh ncreens of.5 and wh ncreens of.5 s consdered. Fgure 3-8 presens he pac of he paraeers on he rao of MSE MSE for =,,., and.,. 5,oreover Fgure 3-9 shows hs pac for =,,.3,., and.5,. when he non-aggregae deand seres follows an ARIMA(,,) process. Shaded areas represen a behavor n favor of he non-aggregaon approach. I s seen ha he superory of each approach s a funcon of,,, and. The analyss shows ha for a fxed value of he soohng consans, ncreasng he aggregaon level proves he accuracy of he aggregaon approach. Addonally, for a BA AA

77 B.Rosa-Tabar, 3, Chaper 3 76 fxed aggregaon level and soohng consan before aggregaon, ncreasng values decreases he perforance of he aggregaon approach. = = Fgure 3-8: Ipac of,,, and on he rao of MSE:.,. (op)..5(boo)

78 B.Rosa-Tabar, 3, Chaper 3 77 = = Fgure 3-9: Ipac of,,, and on he rao of MSE:.3,. (op).5. (boo)

79 B.Rosa-Tabar, 3, Chaper 3 78 In boh Fgure 3-8 and ( 3-9), s revealed ha for hgh posve values of he ovng average paraeer and for hgh negave values of he auoregressve paraeers, he aggregaon approach always yelds ore accurae forecass han he non-aggregaon approach. However, when aes negave values and aes posve values, he nonaggregaon approach ouperfors he aggregaon one. By referrng o Table 3- s obvous ha he laer case corresponds o he hgh posve auocorrelaon. Meanwhle, n he forer one, he auocorrelaon s no always posve and ossclaes beween posve and negave values. Therefore, for a hgh posve auocorrelaon he aggregaon approach does no wor and he non-aggregaon approach provdes ore accurae resuls. Ths s generally rue despe he varyng of he conrol paraeers. Thus, he aggregaon approach n no recoended when auocorrelaon s hghly posve and assocaed wh saller values of generally saller or equal o. The analyss shows ha even for hgh values of he aggregaon level he area n whch aggregaon does no wor reans alos unchanged. Generally, as θ ges ore negave and ges posve n he ARIMA(,,) process, he correlaon beween wo consecuve deand d ges larger. Noe ha for he ARIMA(,,) process he auocorrelaon spans all e lags (no only lag ). Therefore, for hghly posve values of and hghly negave values of, he correlaon beween he consecuve and non-consecuve perods becoes exreely posve. As a resul, when he deand seres are hgh posve correlaed no level of aggregaon can prove he accuracy of forecass. In Appendx E, s revealed ha he non-overlappng eporal aggregaon approach reduces he deand varably of he ARIMA(,,) process. Addonally, by ncreasng aggregaon level ore reducon n coeffcen of varaon can be obaned. I can be shown ha applyng non-overlappng eporal aggregaon decreases he value of he auocorrelaon funcon. Moreover, ncreasng he aggregaon level leads o ore reducon n auocorrelaon and becoes close o zero for hgh values of. As a resul, he aggregae seres becoes slar o a whe nose process and s alos rando. Therefore, for he aggregae seres, he saller value of he soohng consan, generally saller han, should be seleced.

80 B.Rosa-Tabar, 3, Chaper 3 79 When he dsaggregae process follows an ARIMA(,,), he aggregaon approach can reach he ore accurae resuls when he aggregaon level,, s hgh and he soohng consan afer aggregaon s low and saller han, generally for hgh posve auocorrelaon, aggregaon approach s no recoended. The presened resuls n hs secon show ha he selecon of conrol paraeers nfluences he superory of each approach and hs superory s a funcon of all paraeers,,, and, herefore n he secon 3.. we deerne heorecally he condons under whch each approach ouperfors anoher one when he dsaggregae deand seres follows an ARIMA(,,) process Movng average process order one, ARIMA(,,) Fgure 3- presens he pac of he conrol paraeer on he rao of MSE for =, and.,. 5, when he dsaggregae deand seres follows an BA MSE AA ARIMA(,,) process. Shaded areas represen a behavor n favor of he non-aggregaon approach. The resuls show ha for a fxed value of, by ncreasng he aggregaon level, he aggregaon approach provdes ore accurae forecass han he non-aggregaon one. On he oher hand, when consderng a fxed value of he aggregaon level, ncreasng resuls n a deeroraon of he aggregaon approach. If he seleced soohng consan value afer aggregaon,, s consderably hgher han he soohng consan used n he orgnal daa,, hen he aggregaon approach s no preferable. Alernavely, he aggregaon approach ay produce ore accurae forecass unless aes hghly negave values. In he parcular case where he soohng consan paraeers before and afer aggregaon are dencal ( ), he aggregaon approach ouperfors he non-aggregaon one n all cases, excep hose assocaed wh hgh negave values of (hgh posve auocorrelaon). Moreover, even n hose cases, when ncreasng he aggregaon level he perforance of he aggregaon approach s proved.

81 B.Rosa-Tabar, 3, Chaper 3 8 = = Fgure 3-: Ipac of,, and on he MSE rao for. (op) and. 5 (boo) The pac of he soohng paraeer and he aggregaon level s que nuve slar o he ARIMA(,,) process. In fac, s obvous ha he coeffcen of varaon (CV) of he non-overlappng eporally aggregae deand s saller han he CV of he orgnal (dsaggregae deand) and can be shown ha by ncreasng he aggregaon level he coeffcen of varaon of deand s furher reduced. Ths eans ha hgh aggregae order

82 B.Rosa-Tabar, 3, Chaper 3 8 seres are assocaed wh less dsperson han low aggregae order seres. In addon, by consderng he auocovarance funcon before and afer aggregaon for he ARIMA(,,) process, s seen ha he applcaon of he non-overlappng eporal aggregaon decreases he value of he auocorrelaon funcon. Addonally, ncreasng he aggregaon level leads o a hgher reducon n he auocorrelaon whch evenually becoes zero for hgh aggregaon level. Tha s, he aggregae seres has a endency owards a whe nose process n whch case sall values of he soohng consan lead o saller MSEs. Therefore, seng o be sall ( should be saller han ) n conjuncon wh hgh aggregaon levels provdes an advanage o he aggregaon approach. Ths s confred by he resuls presened n Fgure 3-. = = Fgure 3-: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao I should be noed ha even f he seleced s saller han, here are soe cases n whch he aggregaon approach s no preferable. Ths can be arbued o he poenal hgh posve auocorrelaon beween deand perods. For negave values of, he auocorrelaon s posve; for posve values of he auocorrelaon s negave and for he whe nose process, he auocorrelaon s zero. An aggregaon of hghly posve

83 B.Rosa-Tabar, 3, Chaper 3 8 auocorrelaed seres does no add as uch value as aggregang seres wh less posve auocorrelaon. However, for very hgh aggregaon level, he aggregaon approach ay ouperfor he non-aggregaon one even for hgh posve auocorrelaon. When he non-aggregae es follow an ARIMA(,,) process, as θ ges ore negave, he correlaon beween wo consecuve d ncreases. For an ARIMA(,,) process he only auocorrelaon s auocorrelaon lag and for oher lags, s equal o zero. The value of auocorrelaon lag s varyng beween -.5 and ( 3-47) I can be observed ha he axu posve auocorrelaon of lag for an ARIMA(,,) process s around.5 whle hs value alos equals o one for an ARIMA(,,) process. These exaples show ha he perforance superory of each approach s a funcon of all he conrol and he process paraeers. The selecon of he conrol paraeers, and, nfluence he effecveness of he aggregaon approach n conjuncon wh he consderaon of he process paraeers. In sub-secon 3... he condons under whch each approach produces ore accurae forecass for a fxed value of are denfed Auoregressve process order one, ARIMA(,,) When he non-aggregae deand seres follows an ARIMA(,,) process, he pac of he conrol paraeers,, on he rao of MSE BA MSE AA for =, and.,.5 s presened as can be seen n Fgure 3-. Slar o he cases of he ARIMA(,,) and he ARIMA(,,) processes, can be seen ha he superory of each approach s a funcon of all he conrol and he process paraeers. The resuls show ha for a fxed value of, ncreasng he aggregaon level resuls n an proveen n he accuracy of he aggregaon approach. Conversely, for a fxed aggregaon level, ncreasng resuls

84 B.Rosa-Tabar, 3, Chaper 3 83 n a deeroraon of he perforance. In addon, should be generally saller han n order for he aggregaon approach o produce ore accurae forecass. Fgure 3- shows ha for hghly posve values of he auoregressve paraeer he aggregaon approach does no wor well and he non-aggregaon approach provdes ore accurae resuls. Ths s generally rue regardless of he values eployed by he oher conrol paraeers. Therefore, he aggregaon approach s no recoended n such cases. (.e. When he soohng consan paraeers before and afer aggregaon are dencal ), he aggregaon approach ouperfors he non-aggregaon one n all cases, excep hose assocaed wh hghly posve values of. In hose exceponal cases he coparave perforance of he wo approaches s nsensve o he ncrease of he aggregaon level and even for very hgh aggregaon levels, no proveen s observed for he aggregaon approach. The pac of he soohng paraeer and he aggregaon level on he rao s slar o ha repored for he ARIMA(,,) and ARIMA(,,) processes. When s posve for an ARIMA(,,) process, he seres s 'slowly changng' or can be consdered as a posvely auocorrelaed process. In addon, when he non-aggregae deand follows an ARIMA(,,) process, he auocorrelaon spans all e lags (no only lag ). Therefore, for hghly posve values of, he correlaon beween he consecuve and non-consecuve perods becoes very hgh as can be obaned n ( 3-48)., for all. ( 3-48) For nsance, for lag,he auocorrelaon values vary beween - and +,. I can be seen ha he axu posve auocorrlaon of lag s around + for ARIMA(,,) process. Consder a case where he auocorrelaon =, say, d + =d =; clearly no level of aggregaon proves he accuracy of forecasng. For a hgh posve correlaed seres no level of aggregaon ay prove he accuracy of forecass.

85 B.Rosa-Tabar, 3, Chaper 3 84 = = Fgure 3-: Ipac of,, and on he MSE rao for. (op) and.5 (boo)

86 B.Rosa-Tabar, 3, Chaper 3 85 = = Fgure 3-3: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao Hence, when he non-aggregae deand follows an ARIMA(,,) process, he aggregaon approach ay lead o an proveen n accuracy when he aggregaon level,, s hgh and he soohng consan afer aggregaon s sall. However, for hghly posve values of he auoregressve paraeer, he aggregaon approach s no recoended (especally when s bgger han ). Wha ay be concluded a he end of hs sub-secon s ha f he deand daa s hghly posve auocorrelaed hen he non-aggregaon approach wors beer han he aggregaon one. In hose cases he non-aggregaon approach beer explos he very poran recen nforaon (.e. d ) (hough s ore prone o nose). On he conrary, when he auocorrelaon s less posve or negave, hen he recen deand nforaon s no ha crucal. Thus, a longer er vew of he deand s preferable (f one properly selecs how o use long er deand nforaon hrough and ). Moreover, he aggregaon perforance under he ARIMA(,,) s slghly dfferen han he ARIMA(,,) and he ARIMA(,,) due o he naure of hese processes. Posve auocorrelaon under an ARIMA(,,) or ARIMA(,,) process, wh a axu value equal o +, s poenally hgher han ha

87 B.Rosa-Tabar, 3, Chaper 3 86 assocaed wh an ARIMA(,,) process (wh a axu value equal o.5). I should be reeraed ha for he ARIMA(,,) process, he auocorrelaon s led only for o lag, whereas for he ARIMA(,,) and he ARIMA(,,) processes, he auocorrelaon spans over ore lags and s no led o lag. Ths renders he range of ouperforance of he non-aggregaon approach larger under he ARIMA(,,) and ARIMA(,,) processes. In sub-secon 3.. he condons under whch each approach ouperfors he oher one are heorecally deerned when coparson s underaen a he dsaggregae level Coparson a aggregae level In hs sub-secon, he effec of conrol and process paraeers on he rao of MSE BA / MSE AA s evaluaed when he coparson s underaen a he aggregae level Auoregressve Movng Average Process Order One, ARIMA(,,) Fgure 3-4 and Fgure 3-5 presen he pac of he conrol and he process paraeers on he rao of MSE MSE for =,,.,. and BA AA.,.5 when he non-aggregae deand seres follows an ARIMA(,,) process. Shaded areas represen a behavor n favor of he non-aggregaon approach. These fgures show ha he aggregaon approach provdes ore accurae resuls when he forecas horzon s long. Moreover, for shor horzons, he aggregaon approach perfors exreely well when s posve and aes negave values. Alernavely, he aggregaon approach does no perfor beer han he non-aggregaon one where aes negave values and aes hghly posve values. The ouperforance of he non-aggregaon approach can be arbued o he hgh posve auocorrelaon value as explaned above. The resuls show ha he effec of he soohng consan values before and afer aggregaon on he superory of each approach s slar o he case of coparng a he dsaggregae level. When consderng a fxed value of he aggregaon level, ncreasng resuls n a deeroraon of he aggregaon approach. For he aggregae daa, he responsveness of he sable forecasng ehod deeroraes he perforance because he dfferences beween he observaons are sall and low leads o beer forecass.

88 B.Rosa-Tabar, 3, Chaper 3 87 = = Fgure 3-4: Ipac of,,, and on he rao of MSE:.,.(op).. 5(boo) = =

89 B.Rosa-Tabar, 3, Chaper 3 88 Fgure 3-5: Ipac of,,, and on he rao of MSE:.3,. (op).5. (boo) The resuls show ha by ncreasng he aggregaon level, he perforance of he aggregaon approach s proved. for hgher values of he aggregaon level, he aggregaon approach always ouperfors he non-aggregaon one regardless of he values of he ovng average and he auoregressve paraeers. Whereas, when he coparson s consderd a he

90 B.Rosa-Tabar, 3, Chaper 3 89 dsaggregae level, for hghly posve auocorrelaon, no level of aggregaon proves he forecas accuracy. he farher no he fuure he esaon s calculaed, he forecas errors assocaed wh he orgnal daa becoe larger copared o he eporally aggregae one. The approaches based on he eporally aggregae daa benef ore by ncreasng he forecas horzon. In hese cases a longer er vew on deand becoes val and he aggregaon approach ulzes hs nforaon uch beer han he non-aggregaon one. In he parcular case where he soohng consan paraeers before and afer aggregaon are dencal ( ), he resuls are slar o Fgure 3-4 and Fgure 3-5. These exaples show ha he perforance superory of each approach s a funcon of all he conrol and he process paraeers. The selecon of he conrol paraeers, and, nfluence he effecveness of he aggregaon approach n conjuncon wh he he process paraeers Movng average process order one, ARIMA(,,) Fgure 3-6 presens he pac of he conrol paraeers on he rao of MSE BA MSE AA for =, and.,. 5 when he non-aggregae deand seres follows an ARIMA(,,) process. Shaded areas represen a behavor n favor of he non-aggregaon approach. The resuls show ha for a fxed value of, by ncreasng he aggregaon level, he aggregaon approach provdes ore accurae forecass han he non- aggregaon one. On he oher hand, when consderng a fxed value of he aggregaon level, ncreasng resuls n a deeroraon of he aggregaon approach. If he seleced soohng consan value afer aggregaon,, s consderably hgher han he soohng consan used wh he orgnal daa,, hen he aggregaon approach s no preferred. Alernavely, he aggregaon approach yelds a ore accurae forecas. However, when aes hghly negave values he benefs of he aggregaon approach s no as uch as posve values. I s obvous fro Fgure 3-6 and Fgure 3-7 ha here s always a value of for whch he aggregaon approach ouperfors he non-aggregaon one.

91 B.Rosa-Tabar, 3, Chaper 3 9 = = Fgure 3-6: Ipac of,, and on he MSE rao for. (op) and. 5 (boo)

92 B.Rosa-Tabar, 3, Chaper 3 9 In he parcular case where he soohng consan paraeers before and afer aggregaon are dencal (.e. ), he aggregaon approach ouperfors he nonaggregaon one n all cases. = = Fgure 3-7: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao The pac of he soohng paraeer, and he aggregaon level, s que nuve and slar o he case of he ARIMA(,,) process. Therefore, seng o be sall ( should be saller han ) n conjuncon wh hgh aggregaon levels provdes an advanage o he aggregaon approach. Ths s confred by he resuls presened n Fgure 3-6 and Fgure 3-7. The weaness of he aggregaon approach for negave values of can be arbued o he poenally hgh posve auocorrelaon beween deand perods. For negave values of, he auocorrelaon s posve; for posve values of he auocorrelaon s negave and for he whe nose process, he auocorrelaon s zero. Aggregaon of a hghly posvely correlaed seres doesn' add as uch value as he aggregae seres wh less posve

93 B.Rosa-Tabar, 3, Chaper 3 9 auocorrelaon. Hence, when he non-aggregae deand follows an ARIMA(,,) process, he aggregaon approach leads o an proveen n accuracy when he aggregaon level,, s hgh and he soohng consan afer aggregaon s sall Auoregressve process order one, ARIMA(,,) Fgure 3-8 presens he pac of he conrol paraeers,, on he rao of MSE for =, and.,. 5, when he non-aggregae deand seres follows BA MSE AA an ARIMA(,,) process. s easy o see ha he superory of each approach s a funcon of all conrol and process paraeers. The resuls show ha for a fxed value of, ncreasng he aggregaon level resuls n accuracy proveens of he aggregaon approach. Conversely, for a fxed aggregaon level, ncreasng resuls n a deeroraon of he perforance. In addon, should be generally saller han n order for he aggregaon approach o produce ore accurae forecass. (.e. When he soohng consan paraeers before and afer aggregaon are dencal ), he aggregaon approach ouperfors he non-aggregaon one n all cases excep when he aggregaon level s low and assocaed wh hghly posve values of. Moreover, by ncreasng he aggregaon level he perforance of he aggregaon approach s proved and for he hgher aggregaon level, he aggregaon approach always perfors beer.

94 B.Rosa-Tabar, 3, Chaper 3 93 = = Fgure 3-8: Ipac of,, and on he MSE rao for. (op) and.5 (boo)

95 B.Rosa-Tabar, 3, Chaper 3 94 = = Fgure 3-9: Ipac of conrol paraeers for ARIMA(,,) process on he MSE rao The pac of he soohng paraeer and he aggregaon level on he rao s slar o hose repored for he ARIMA(,,) process. Fgure 3-8 shows ha even f he seleced s saller han, here are soe cases n whch he aggregaon approach s no preferred. Ths s when he auoregressve paraeer aes hgh posve values. In general, he benefs acheved by he aggregaon approach are fewer for hghly posve values of han negave values of. Hence, when he non-aggregae deand follows an ARIMA(,,) process, he aggregaon approach leads o an proveen n accuracy when he aggregaon level,, s hgh and he soohng consan afer aggregaon s sall. Wha can be concluded a he end of hs sub-secon n forecasng he aggregae level s ha f he forecas horzon s long hen he aggregaon approach s always preferred. Because n hese cases a longer er vew on deand s very poran and he aggregaon approach ulzes hs nforaon beer han he non-aggregaon one. By ncreasng he

96 B.Rosa-Tabar, 3, Chaper 3 95 forecas horzon, he forecas error assocaed wh he classcal approach ncreases as well. However, when he forecas horzon s shor, he superory of each approach depends on he aggregaon level and he auocorrelaon values. If he deand daa s hghly posve auocorrelaed hen he non-aggregaon approach wors beer han he aggregaon one. In hose cases he non-aggregaon approach beer explos he very poran recen nforaon (.e. d ). On he conrary, when he auocorrelaon s less posve or negave, he recen deand nforaon s no as crucal. Thus, a longer er vew of he deand s becoes poran. Therefore, he aggregaon approach s preferred. Moreover, he aggregaon perforance under he ARIMA(,,), he ARIMA(,,) and he ARIMA(,,) processes s slghly dfferen due o he naure of hese processes. In fac, he posve auocorrelaon n he he ARIMA(,,) and he ARIMA(,,) s hgher han ha n he ARIMA(,,) whch aes larger he range of he ouperforance of he non-aggregaon approach n he he ARIMA(,,)and he ARIMA(,,) processes. In sub-secon 3.. he condons under whch each approach ouperfors he oher one are heorecally deerned when coparson s underaen a he aggregae level. 3.. Theorecal Coparson Havng conduced a sensvy analyss n sub-secon 3.., now he condons under whch each approach ouperfors he oher one are analycally denfed Coparson a dsaggregae level In hs sub-secon he condons under whch he aggregaon and he non-aggregaon approaches perfor beer are denfed when he coparson s underaen a he dsaggregae level.

97 B.Rosa-Tabar, 3, Chaper Auoregressve Movng Average Process Order One, ARIMA(,,) In hs sub-secon he condons under whch each approach ouperfors he oher one are analycally denfed when he non-aggregae deand process s an ARIMA(,,). The rao of AA BA MSE MSE s obaned by dvdng ( 3-7) no ( 3-8) :.5 AA BA MSE MSE ( 3-49) Ths rao s a funcon of he aggregaon level, he auoregressve paraeer, ovng average paraeer, and he soohng consan paraeers before and afer aggregaon, and. Consderng ha he aggregaon level ay only ge neger values greaer han or equal o wo, he goal s o deerne he value ha enables he aggregaon approach o perfor beer. The enre range of possble values for s consdered. To show he condons under whch he aggregaon approach ouperfors he nonaggregaon one, he equaon ( 3-49) s se greaer han,.e. BA MSE AA MSE, Fro hs saeen he followng resul can be obaned: THEOREM -3: If he e seres of he non-aggregae deand follows an ARIMA(,,) process and and, hen:

98 B.Rosa-Tabar, 3, Chaper 3 97 If, he aggregaon approach provdes ore accurae forecas. If, boh approaches perfor equally. Oherwse, he non-aggregaon approach wors beer. Where defned n (F-4). PROOF: he proof of Theore -3 s gven n Appendx F. Noe ha for he presened range of and ( and ) s always posve, consequenly choosng guaranees ha he aggregaon approach always ouperfors he non-aggregaon one n hs regon. Hence, he value of reflecs a cu-off pon ha ay be used n pracce for he selecon of he soohng consan value o be used for he aggregae seres. The cu-off pon reflecs all he qualave dscusson provded n he prevous sub-secon as o when aggregaon ouperfors he non-aggregaon approach. If he e seres of he orgnal deand follows an ARIMA(,,) process and he ovng average and he auoregressve paraeers sasfy and, hen he condons under whch each approach wors beer can be obaned. These condons are suarzed n he followng selecon procedure (dscussed n Table ): Table 3-: Selecon procedure for he ARIMA(,,) process, Coparson a dsaggregae level. The procedure s begun by calculang defned n (F-3), If hen he non-aggregae approach s always superor, oherwse he values of and defned n (F-4) and (F-5) are calculaed.. If, are obaned:, he value of β and accordng o he values of β and β he followng rules If, hen he aggregaon approach wors beer. If hen boh approaches are dencal. If or hen non-aggregae sraegy wors beer. Oherwse, go o 3.

99 B.Rosa-Tabar, 3, Chaper If,, he value of s calculaed: If, hen he aggregaon approach wors beer. If, hen boh approaches are dencal. If, hen nonaggregaon approach wors beer. Where defned n (F-5). PROOF: The deals of he selecon procedure are gven n Appendx G Movng average process order one, ARIMA(,,) The rao of he MSE BA o MSE AA when he non-aggregae deand follows an ARIMA(,,) process s a funcon of he ovng average paraeer, he soohng consan before and afer aggregaon ( and ), and he aggregaon level. The cu-off pons for he value of should be deerned. Ths enables he aggregaon approach o perfor beer. The enre range of possble values for s consdered bu he soohng consan s a paraeer ha s se o s opal value by praconers, norally by nzng he MSE. Fro ( 3-7) s clear ha MSE BA s onooncally ncreasng n as he dervave of MSE BA s posve for all values of n (-, ). Hence, MSE BA can be nzed by havng he salles possble value of, whch aes sense for a saonary process. However, should be noed ha n hs heorecal analyss he ssue of nalzaon of he forecasng process s dsregarded. Ths s an poran ssue o be enoned (snce wh very low values a bad nalzaon ples naccurae esaes of he fuure deand as he forecas wll bascally be ep consan) bu one ha s no consdered as par of hs research. To show he condons under whch he aggregaon approach ouperfors he nonaggregaon approach, he rao s se greaer han one, MSE. Fro hs nequaly he followng resul can be obaned: BA MSE AA

100 B.Rosa-Tabar, 3, Chaper 3 99 THEOREM -3: If he e seres of he non-aggregae deand follows an ARIMA(,,) process, hen: If, he aggregaon approach provdes ore accurae forecass. If, boh sraeges perfor equally. Oherwse, he non-aggregaon approach wors beer. where ( ( ) ) 8 ( ( ) ) ( 3-5), and. ( 3-5) PROOF: he proof of Theore -3 s gven n Appendx H. The resuls deonsrae ha, for a gven values of and, here always exss a value of such ha he aggregaon approach ouperfors he non-aggregaon one. Hence, he value of reflecs a cu-off pon ha ay be used n pracce for he selecon of he soohng consan value o be used for he aggregae seres Auoregressve process order one, ARIMA(,,) A slar procedure s followed by seng he rao MSE BA o MSE AA greaer han for an ARIMA(,,) process. Ths s conduced o denfy he condons under whch he

101 B.Rosa-Tabar, 3, Chaper 3 aggregaon approach perfors beer. These condons are suarzed by he selecon procedure presened n Appendx I when auoregressve paraeer sasfes. As dscussed earler he soohng consan s ofen se by praconers o s opal value, so s ore neresng o dscuss he cases where such a value s consdered. To do so, he value ha nzes he MSE BA s deerned. Followng ha a value of he soohng consan afer aggregaon ha leads o ore accurae forecass s calculaed. The opal value of s gven n ( 3-5) ha can be obaned by solvng he frs dervave of ( 3-9): * 3 3. ( 3-5) 3 where s a very sall posve value. By consderng he opal value of he soohng consan before aggregaon, wo dfferen cases should be consdered. Fro MSE and ( 3-5) he followng resuls can be obaned. Case. 3 *. In hs case, 3 BA MSE AA THEOREM 3-3: If he e seres of he non-aggregae deand follows an ARIMA(,,) process, where 3 * and he opal soohng consan, 3, s used o deerne he non-aggregae deand forecas, hen he non-aggregaon approach always provdes ore accurae forecas han he aggregaon one, regardless of he soohng consan paraeer afer aggregaon, β, and he aggregaon level,. PROOF: he proof of Theore 3-3 s gven n Appendx J. * Case. 3. In hs case s a very sall posve nuber. THEOREM 4-3. If he e seres of he non-aggregae deand follows an ARIMA(,,) process, where 3 and he opal soohng consan used o deerne he nonaggregae deand forecas, *. 5, hen:

102 B.Rosa-Tabar, 3, Chaper 3 If β < β he aggregaon approach provdes ore accurae forecas. If β = β boh sraeges perfor equally. Oherwse, he non-aggregaon approach wors beer. Where 4. ( 3-53) (,, and are gven n Appendx I) PROOF: The proof of Theore 4-3 s gven n Appendx J. Slar o he case of he ARIMA(,,) process, he above resuls provde a cu-off pon ha ay be used n pracce for he selecon of he soohng consan n order o oban an ouperforance of he aggregaon approach when ARIMA(,,) processes are consdered. Obvously, as he cu-off pon ncreases for hgh aggregaon levels, s clear ha hs ples a consderable range of he soohng consan of he aggregae seres where here s a benef of usng he aggregaon approach. Hence, hese resuls provde a coprehensve way of anagng he process of forecasng of ARIMA(,,) processes when he auoregressve paraeer s nown and when he nenon s o opze he soohng consan for he non-aggregae seres Coparson a aggregae level In hs sub-secon he superory condons of each approach are denfed when he coparson s underaen a he aggregae level Auoregressve Movng Average Process Order One, ARIMA(,,) The rao of he MSE BA o MSE AA when he non-aggregae deand follows an ARIMA(,,) process s a funcon of he ovng average paraeer,, he auoregressve

103 B.Rosa-Tabar, 3, Chaper 3 paraeer,, he soohng consan before and afer aggregaon ( and ), and he aggregaon level,. The objecve s o deerne he value ha enables he aggregaon approach o perfor beer. AA BA MSE MSE ( 3-54) To show he condons under whch he aggregaon approach ouperfors he non- aggregaon approach, he rao s se o greaer han one, BA MSE AA MSE. Fro hs saeen he followng resuls can be obaned: If he e seres of he basc deand follows an ARIMA(,,) process and he ovng average and he auoregressve paraeers sasfy and, he condons under whch each approach wors beer are obaned. These condons are suarzed as follows: Table 3-3: Selecon procedure for he ARIMA(,,) process, Coparson a aggregae level. The procedure s begun by calculang defned n (K-), If hen he nonaggregaon approach s always superor, oherwse he values of and defned n (K- 3) and (K-4) are calculaed.. If,, he value of β and accordng o he values of β and β he followng rules

104 B.Rosa-Tabar, 3, Chaper 3 3 are obaned: If, hen he aggregaon approach wors beer. If hen boh approaches are dencal. If or hen non-aggregaon approach wors beer. Oherwse, go o 3., he value of s calculaed: 3. If, If, hen he aggregaon approach wors beer. If, hen boh approaches are dencal. If, hen non-aggregaon approach wors beer. PROOF: The deals of he selecon procedure are gven n Appendx K. THEOREM 5-3: If he e seres of he non-aggregae deand follows an ARIMA(,,) process and and, hen: If, he aggregaon sraegy provde ore accurae forecas. If, boh sraeges perfor equally. Oherwse, he non-aggregaon sraegy wors beer. where s defned as: - (- ) - +,,, and are defned n Appendx K. PROOF: he proof of Theore 5-3 s gven n Appendx L.

105 B.Rosa-Tabar, 3, Chaper 3 4 Theore 5-3 show ha when he auoregressve and he ovng average paraeers sasfes and, hen for a gven value of he soohng consan,, and he aggregaon level,, here s always a value of for whch he aggregaon approach provdes ore accurae forecass Movng average process order one, ARIMA(,,) The rao of he MSE BA o MSE AA when he non-aggregae deand follows an ARIMA(,,) process s a funcon of he ovng average paraeer,, he soohng consan before and afer aggregaon ( and ), and he aggregaon level,. The superory condons can be obaned by followng he sae procedure as Appendx K where he auoregressve paraeer s equal o zero. MSE BA MSE AA ( 3-55) By seng he equaon ( 3-55) o greaer han one, he followng resuls can be obaned: THEOREM 6-3: If he e seres of he non-aggregae deand follows an ARIMA(,,) process, hen for a gven values of and : If, he aggregaon approach provdes ore accurae forecass. If, boh sraeges perfor equally. Oherwse, he non-aggregaon approach wors beer. where

106 B.Rosa-Tabar, 3, Chaper 3 5 Proof: Theore 6-3 can be obaned by subsung = n Appendx L. Theore 6-3 says ha here s always a value of for whch he aggregaon approach ouperfors he non-aggregaon one. THEOREM 7-3 If he non-aggregae e seres follows an ARIMA(,,) process and he soohng consan under he aggregaon approach s saller or equal o he nonaggregaon one(), hen aggregaon approach always ouperfors he non-aggregaon one(.e. MSE BA > MSE AA ). Ths s rue regardless of he aggregaon level, and he process paraeer. In addon, when he soohng consans under he boh approaches are se sall (, <.), hen boh aggregaon and non-aggregaon approaches perfor equally. PROOF: he proof of Theore 7-3 s gven n Appendx M Auoregressve process order one, ARIMA(,,) The superory condons of each approach when he non-aggregae deand follows an ARIMA(,,) process can be obaned by seng he followng equaon greaer han one. AA BA MSE MSE ( 3-56) Slar o he case of coparson a dsaggregae level, by consderng he opal value of he soohng consan before aggregaon, wo dfferen cases are consdered.

107 B.Rosa-Tabar, 3, Chaper 3 6 THEOREM 8-3 If he e seres of he non-aggregae deand follows an ARIMA(,,) process when 3, hen here s always a value of n order o he aggregaon approach ouperfors he non-aggregaon one: If β < β he aggregaon approach provdes ore accurae forecas. If β = β boh sraeges perfor equally. Oherwse, he non-aggregae approach wors beer. Where - (- ) - + Proof: These condons can be acheved by subsung = and 3 n he Appendx K. If he auoregressve paraeer sasfes 3, hen here s always a value of for whch he aggregaon approach wors beer han he non-aggregaon one. THEOREM 9-3 If he e seres of he non-aggregae deand follows an ARIMA(,,) process when 3 and he soohng consan under he aggregaon approach s saller han non-aggregaon one(<), hen aggregaon approach always ouperfors he non-aggregaon one. Ths s always rue regardless of he aggregaon level,. In addon, when he soohng consans under he boh approaches are se o sall values (,<.), hen he dfference n he perforance of he aggregaon and non-aggregaon approaches s nsgnfcan. PROOF: he proof of Theore 9-3 s gven n Appendx N. If he e seres of he non-aggregae deand follows an ARIMA(,,) process when 3, hen he rao of MSE BA /MSE AA ay be saller, greaer han or equal o one

108 B.Rosa-Tabar, 3, Chaper 3 7 dependng on he values of he soohng consans( and ), aggregaon level, and he auoregressve paraeers. The condons under whch each approach wors beer can be obaned by subsung = and 3 n he procedure dscussed n sub-secon 3... where he case of he ARIMA(,,) s consdered Opal aggregaon level The objecve of hs secon s o denfy he opal aggregaon levels ha axze he rao or equvalenly nze he MSE AA for each deand process under consderaon. To do so, he rao of MSE BA o MSE AA for he whole range of he conrol paraeers s evaluaed Coparson a dsaggregae level In hs par he aggregaon level ha leads o ore error reducon s deerned when he coparson s underaen a dsaggregae level Auoregressve Movng Average Process Order One, ARIMA(,,) A nuercal nvesgaon o deerne he opal aggregaon level s conduced snce fro ( 3-49) s clear ha he calculaon of he frs dervave s nfeasble. wo exaples are presened: ) he whole range of where =.9, =.3, and =.; ) he whole range of where =-.5, =.3, and =.. In he laer case for soe values of and =-.5 ( Fgure 3-b) he rao s saller han one and consequenly aggregaon does no wor. Thus, n hese cases s no necessary o dscuss he opal aggregaon level. The resuls show ha by ncreasng he aggregaon level, he perforance of he aggregaon approach proves. Addonally, a hgher aggregaon level resuls n hgher values of he rao and consequenly ore benefs for he aggregaon approach.

109 B.Rosa-Tabar, 3, Chaper 3 8 a) =.3,=.,=.9 b) =.,=.,=-.4 Fgure 3-: MSE rao for dfferen values of for an ARIMA(,,) process Movng average process order one, ARIMA(,,) In order o oban he opal aggregaon level when he non-aggregae deand seres follows an ARIMA(,,), he followng heore s consdered. THEOREM -3: If he non-aggregae deand seres follows an ARIMA(,,) process, hen he opal aggregaon level s he hghes level n any consdered range. Supposng ha aggregaon s o be esed n a range [ u, u ], where u and u are he lower and upper bound, respecvely. In addon, hey are posve neger nubers. The opal aggregaon level s always u. PROOF: A calculaon of he frs dervave of MSE AA wh respec o shows ha MSE AA s a decreasng funcon of. Ths can be shown by a nuercal analyss for as well. Ths eans ha he rao MSE BA /MSE AA s an ncreasng funcon of. Therefore, a hgher value of he aggregaon level resuls n a hgher value of he rao MSE BA / MSE AA.

110 B.Rosa-Tabar, 3, Chaper Auoregressve process order one, ARIMA(,,) A nuercal nvesgaon s conduced o oban he opal aggregaon level where he subaggregae process follows an ARIMA(,,) as he calculaon of he frs dervave s nfeasble. Two exaples are consdered: ) he whole range of where =.5 and =.; ) he case dscussed n 5.. wh an opal value of. Fgure 3a shows ha he value of he aggregaon level ha axzes he MSE rao changes when varyng he conrol paraeer values. For negave and lower posve values of,.e. 3, he forecas accuracy of he aggregaon approach ncreases wh he aggregaon level whle for hgher posve values of,.e. 3, hs s no rue. Le us analyse he wo dfferen cases n whch he opal soohng consan values are consdered for MSE BA. a) ARIMA(,,) process where b) Case, ARIMA(,,) process where.33 Fgure 3-: MSE rao for dfferen values of ARIMA(,,) process a dsaggregae level

111 B.Rosa-Tabar, 3, Chaper 3 Case. 3. In hs case he opal soohng consan paraeer 3 * s used and s seen n sub-secon 5.. ha he MSE rao s always lower han. Case. 3. In hs case a very sall soohng consan value, *. 5, s used. The MSE rao for dfferen aggregaon levels s shown n Fgure 3-b for 3 and a nuercal exaple of and values where <. Ths fgure shows ha he aggregaon approach s assocaed wh ore accurae resuls for hgher aggregaon levels Coparson a aaggregae level In hs par he opal aggregaon level ha axzes he rao of MSE BA / MSE AA s denfed when he coparson s underaen a he aggregae level Auoregressve Movng Average Process Order One, ARIMA(,,) Slar o he case of coparson a dsaggregae level, here wo exaples are presened o evaluae he pac of he aggregaon level on he rao as he dervaon of ( 3-54) o deerne he opal s no feasble: ) he whole range of where =.7, =., and =.; ) he whole range of where =-.4, =., and =.. As s shown n Fgure 3-a and b, he hgher rao of MSE BA / MSE AA s assocaed wh hgher aggregaon level.

112 B.Rosa-Tabar, 3, Chaper 3 a) =.,=.,=-.4 b) =.,=.,=.7 Fgure 3-: MSE for dfferen values of, ARIMA(,,)process copared a aggregae level Movng average process order one, ARIMA(,,) By consderng ( 3-55) he values of he MSE BA / MSE AA by varyng he aggregaon level can be deerned. Fgure 3-3a and b show he pac of he aggregaon level on he rao for he whole range of when =.,=. and =.,=.5. I s shown ha hgher aggregaon level s assocaed wh hgher rao.

113 B.Rosa-Tabar, 3, Chaper 3 a) =.,=. b) =.,=.5 Fgure 3-3: MSE for dfferen values of, ARIMA(,,) process coparson a aggregae level Auoregressve process order one, ARIMA(,,) Fgure 3-3a and b presen he pac of aggregaon level on he rao for he whole range of when =.3,=. and =.,=.5. The resuls show ha a hgher value of he rao s acheved by hgher aggregaon level. a) (.3,. ) b) (.,. 5 ) Fgure 3-4: MSE for dfferen values of, ARIMA(,,) process coparson a aggregae level 3. Sulaon nvesgaon In hs sub-secon a sulaon experen based on he heorecally generaed daa s consdered. In hs par of he wor, sulaon analyss s used o es and valdae he heorecal resuls dscussed n secon.

114 B.Rosa-Tabar, 3, Chaper Sulaon desgn Dfferen auoregressve ovng average, ARMA ype processes are o es he aheacal fndngs. an ARIMA(,,) process, an ARIMA(,,) process and a xed ARIMA(,,) process are consdered. These processes are analysed n secon. The dsaggregae deands are generaed randoly n each perod subjec o he paraeers descrbed n Table 3-4. The value of s se que saller han o avod he generaon of negave sub-aggregae values. To generae he deands n each perod ha follow ARIMA(,,), ARIMA(,,) and ARIMA(,,), he error ers are frs generaed randoly. The sulaon experen s desgned and run n Malab 7... For each paraeer cobnaon descrbed n Table 3-4 a deand seres of observaons s generaed and replcaons are nroduced. Table 3-4: Paraeers of he sulaon experen, N Replcaons N Te Perods 4.: : : +.9 The generaed seres s dvded no wo pars. The frs par (whn saple) consss of 45 e perods and s used n order o nalze he SES esaes. The second par consss of 55 e perods and s used for he evaluaon of he perforance (ou-ofsaple). The values of he soohng consans before and afer aggregaon (, ) s vared fro.5 o.95 wh a sep ncrease of.5. For non-aggregaon approach, he SES s appled drecly o ge 55 one-sep ahead forecass and hen he varance of he forecas error. s calculaed. In order o oban he forecass generaed by he aggregaon approach, frs he non-overlappng buces of aggregae daa are creaed based on a specfed aggregaon level and hen SES ehod s appled o hese aggregae daa o ge he aggregae forecas. he procedure s explaned for he aggregaon level equals o wo, for hgher aggregaon level he sae procedure s followed. The calculaon s begun fro he 45 nd observaon n he nal (whn saple) par, he observaons are sued bacwards n buces of wo (), resulng n an aggregae seres conssng of 5 aggregae observaons. The average of he aggregae seres s obaned and s used as he SES s forecas for he frs buceed perod. SES s hen

115 B.Rosa-Tabar, 3, Chaper 3 4 appled all he way up o producng a forecas for buce 6 whch gves a forecas for perods 45 and 45. Then he buces of perods fro perod 45 bacwards are creaed. Thus, anoher 5 buces are creaed and he very frs observaon (perod n he orgnal daa) s no used anyore. The average of hese buces s calculaed, s used as he SES s forecas for he frs buce, hen he SES ehod s appled unl he pon ha a forecas for buce 7 (perods 45 and 453) s produced. In he nex perod, he buces are creaed bacwards fro perod 45 endng up wh 6 buces and connue le hs unl oban he forecass for 55 perods ahead. Now, f he forecas a he dsaggregae level s needed he aggregae forecass s dvded by he aggregaon level o ge he dsaggregae forecas resuled fro he aggregae daa. Fnally, he value of he varance of he forecas error before aggregaon s dvded by he varance of he forecas error afer aggregaon, o oban he rao of MSE BA o MSE AA. verfcaon s he process o ae sure ha no prograng error has been ade(klejnen and Groenendaal, 99). Ths can be esed by calculang neredae resuls anually and coparng he wh he resuls obaned by he progra. Ths s called racng(klejnen and Groenendaal, 99). Eyeballng or readng hrough he code and loong for bugs s anoher way of verfcaon(klejnen and Groenendaal, 99). The followng seps are conduced o verfy he sulaon odel: The MATLAB codes are read hrough o ae sure ha he correc logc and funcons have been used. The neredae and also he fnal resuls are copared for a led nuber of replcaons (e.g. replcaons) wh MS Excel Sulaon Resul The sulaon resuls are presened n hs sub-secon. As dscussed n chaper, he objecve of he sulaon analyss I eporal aggregaon s o es and valdae he resuls of aheacal evaluaon. In secon he condons under whch aggregaon and nonaggregaon approaches ay ouperfor each oher are dscussed. In he followng he resuls

116 B.Rosa-Tabar, 3, Chaper 3 5 of sulaon analyss are presened for hese condons o copare he wh aheacal analyss. Alhough a sulaon nvesgaon s conduced for all scenaros dscussed n he heorecal par bu he resuls of he ore general case, ARMA(,,) process are only presened, whch has he characerscs of boh ARIMA(,,) and ARIMA(,,) process. Fgure 3-5 presens he pac of he paraeers on he rao of MSE BA MSE AA when coparng a dsaggregae level for =,,., and.,. 5. Shaded areas represen a behavor n favor of he non-aggregaon approach. The sulaon resuls shows ha for posve values of and negave values of, non-aggregaon approach produce ore accurae resuls copared o aggregaon approach, however he aggregaon approach can provde ore accuracy forecass when s negave and s posve. In addon, s seen ha ncreasng he aggregaon level prove he forecasng accuracy when he aggregaon approach ouperfors he non-aggregaon one. However, by ncreasng he soohng consan afer aggregaon he perforance of he aggregaon approach deeroraes. In Fgure 3-6 he resuls of sulaon analyss for coparson a he aggregae level are presened for he sae paraeers used n he prevous case. Shaded areas represen a behavor n favor of he non-aggregaon approach. As can be observed n Fgure 3-6, here s less benefs for he aggregaon approach when aes negave values and has posve values, and s seen ha for lower aggregaon level values, he non-aggregaon approach ouperfors he aggregaon one. However, for hgher values of he aggregaon level, he aggregaon approach ouperfors he non-aggregaon one regardless of he values of he auoregressve and he ovng average paraeers. In addon, by ncreasng he aggregaon level he accuracy of he aggregaon approach proves.

117 B.Rosa-Tabar, 3, Chaper 3 6 = = Fgure 3-5: Ipac of,,, and on he rao of MSE copared a dsaggregae level:.,. (op),.. 5(boo)

118 B.Rosa-Tabar, 3, Chaper 3 7 = = Fgure 3-6: Ipac of,,, and on he rao of MSE copared a aggregae level:.,.(op),.. 5(boo) The sulaon resuls presened n boh Fgure 3-5 and Fgure 3-6 generally confr he resuls of he heorecal analyss when he underlyng seres follow an ARIMA(,,) deand process a boh dsaggregae and aggregae level of coparson.

119 B.Rosa-Tabar, 3, Chaper Eprcal analyss In hs secon he eprcal valdy of he an heorecal fndngs of hs research are assessed. In he followng sub-secon he deals of he eprcal daa avalable for he purposes of he nvesgaon along wh he experenal srucure eployed n hs wor are provded. In sub-secon 3.4. he acual eprcal resuls are presened Eprcal Daase and Experen Deals The deand daase avalable for he purposes of hs research consss of weely sales daa over a perod of wo years for,798 SKUs fro a European grocery sore. The Forecas pacage n R s used o denfy he underlyng ARIMA deand process for each seres and esae he relevan paraeers. I s found ha ore han 48% of he seres ay be represened by he processes consdered n our research. In parcular, 3.6% of he seres (544 seres) s found o be ARIMA(,,),.96% (33 seres) o be ARIMA(,,) and 5.6%(9 seres) o be ARIMA(,,), (Oher popular processes denfed are: ARIMA(,,) (6.3%) and ARIMA(,,) (3.7%). Ths analyss provdes soe eprcal jusfcaon on he frequency of saonary, and n parcular ARIMA(,,) and ARIMA(,,) processes n real world pracces. In Table 3-5 and 3-6 and 3-7 he characerscs of he SKUs relevan o hs sudy are suarzed by ndcang he esaed paraeers for ARIMA(,,), ARIMA(,,) and ARIMA(,,) processes. To faclae a clear presenaon, he esaed paraeers are grouped n nervals and he correspondng nuber of SKUs s gven for each such nerval. The average and value per nerval s also presened for he processes respecvely. Ths caegorzaon allows coparng he eprcal resuls wh he heorecal fndngs. I should be noed ha he paraeer values are all bu one negave and he paraeer values are all bu one posve for he ARIMA(,,) and he ARIMA(,,) processes respecvely. For he ARIMA(,,) process, he paraeer values are posve or negave and all paraeers are posve, bu whole paraeers lead o a posve auocorrelaon. As such, he daa do no cover he enre heorecally feasble range of he paraeers. Soe sudes (Erp e al., 99; Lee e al., 997b; Lee e al., ) ha have consdered eprcal

120 B.Rosa-Tabar, 3, Chaper 3 9 ARIMA(,,) processes, have repored ha s coon o have posve correlaon/hgh value of auoregressve paraeers n he consuer produc ndusres whch s also he case n he daase used n our research. Replcaon of our fndngs n bgger daases s ceranly an avenue for furher research. Table 3-5: Processes presen n he eprcal daa se, ARIMA(,,) process θ nervals nervals Average of Average Average of θ lagauocorrelaon No. of SKUs [.,.5[ [.6,[ [.5,.9[ [.6,[ [-.,-.5[ [.,.5[ Toal nuber of SKUs: 9 Table 3-6: Processes Presen n he Eprcal Daa Se, ARIMA(,,) process θ nervals Average of θ No. of SKUs [-.8,-.7[ -.75 [-.7,-.6[ [-.6,-.5[ [-.5,-.4[ [-.4,-.3[ [-.3,-.[ [-.,-.[ [,[.83 Toal nuber of SKUs: 33

121 B.Rosa-Tabar, 3, Chaper 3 Table 3-7: Processes Presen n he Eprcal Daa Se, ARIMA(,,) process b) ARIMA(,,) ø nervals Average of ø No. of SKUs [-.,[ -.4 [.,.[.98 [.,.3[ [.3,.4[ [.4,.5[ [.5,.6[.55 [.6,.7[ [.7,.8[ [.8,.9[ Toal nuber of SKUs: 544 The daa seres s dvded no wo pars. The frs par (whn saple) consss of 6 e perods and s used n order o nalze he SES esaes. The second par consss of he reanng 4 e perods and s used for he evaluaon of he perforance (ou-ofsaple). The values of he soohng consans are vared fro.5 o.95 wh a sep ncrease of.5. In he classcal (non-aggregae) approach, frs he 4 one-sep ahead forecass are calculaed for each seres and hen he varance of he forecas error s we calculaed. o oban he forecass va he aggregaon approach, frsly he non-overlappng buces of aggregae daa are creaed based on a specfed aggregaon level and hen he SES ehod s appled o hese aggregae daa. Aggregaon level = : Sarng fro he 6 nd weely observaon n he nal (whn saple) par, he observaons are sued bacwards n buces of wo (), resulng n a bweely seres conssng of 3 aggregae observaons. The average of aggregae seres s

122 B.Rosa-Tabar, 3, Chaper 3 obaned and s used as he SES s forecas for he frs buce perod. SES s hen appled all he way up o producng a forecas for buce 3 whch s hen dvded by (he aggregaon level, =) and gves a forecas for perods 63 and 64. The forecas for perod 64 s dropped and hose of 63 s recorded (hey are equal anyway). Then he buces of perods fro perod 63 bacwards are creaed. Therefore, anoher 3 buces are creaed and he very frs observaon (perod n he orgnal daa) s no used anyore. The average of hese buces are calculaed (hey are dfferen fro hose creaed before), ha average s used as he SES s forecas for he frs buce, he forecasng process s connues usng SES unl he pon ha a forecas for buce 3 (perods 64 and 65) s obaned. The forecas for perod 64 s ep and so on. In he nex perod, he buces are creaed bacwards fro perod 64 endng up wh 3 buces and connue le hs unl oban he forecass for 4 perods ahead. Aggregaon level = : Slarly, he sae procedure s followed wh e buces of up o 4 perods. A hs pon here are aggregae bweely observaons ( 4=48), hus 4 weely observaons a he sar of he orgnal seres rean unused. Fnally, he value of he varance of he forecas error before aggregaon s dvded by he varance of he forecas error afer aggregaon o oban he rao of MSE BA o MSE AA Eprcal Resuls In hs secon he resuls of eprcal nvesgaon copared a boh dsaggregae and aggregae level for all processes under consderaon are presened Coparson a dsaggregae level In he frs par, he valdy of aheacal resuls are evaluaed by real daa se when he non-aggregae deand follow an ARIMA(,,) process and he coparson s conduced n he dsaggregae level. The eprcal resuls show ha when he opal soohng consan values and are used, hen for all values of aggregaon level he nonaggregaon approach ouperfors he aggregaon one. Ths s n agreeen wh our fndngs as he real daa se presened n Table 3-5 aes posve auocorrelaon, no only for lag bu

123 B.Rosa-Tabar, 3, Chaper 3 spans over longer lags. Accordng o he heorecal fndngs when he auocorrelaon s posve he non-aggregaon approach perfors beer and no level of aggregaon prove he perforance of he aggregaon approach. As s shown n Fgure 3-7, for all values of he MSE BA s lower han MSE AA. I should be noed ha he resuls s presened based on he RMSE(roo ean square error) whch s slar o MSE. The MSE reducon can be as hgh as 8% for he aggregaon approach. Fgure 3-7: Eprcal resuls copared a dsaggregae level, ARIMA(,,) process In sub-secons 3... and 3 he condons under whch he aggregae forecass ay perfor beer han he non-aggregae are analycally exaned by he rao of MSE BA o MSE AA. The cu-off pons of he soohng consan of he aggregae seres ha should be used (.e. any value of ha s lower han he cu-off pon ples an ouperforance of he aggregaon approach) have also been deerned for boh he ARIMA(,,) and ARIMA(,,) process. In he followng fgures he resuls of he eprcal analyss for hese processes are presened. Addonally, he degree o whch hey valdae he heorecal fndngs s nvesgaed.

124 B.Rosa-Tabar, 3, Chaper 3 3 In Fgure 3-8, he cu-off pon s presened for a fxed values of and when he nonaggregae deand of he SKUs follows an ARIMA(,,) process. Please recall ha he cuoff pon s he value below whch any value ples ha he aggregaon approach ouperfors he non-aggregaon one. Noe ha he resuls for.5 are only presened snce hs range s vewed as realsc for he saonary processes consdered n hs wor. Fgure 3-8: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a dsaggregae level, ARIMA(,,) process The eprcal resuls show ha for a low aggregaon level =, he cu-off pon s relavely low snce =. for a relavely hgh value equal o.5. In ha case, he MSE reducon when =.5 s equal o 8.89% and he MSE rao decreases for hgher values of. Obvously, he cu-off value consderably ncreases when he aggregaon level ncreases. For exaple, when we consder he aggregaon level =, he cu-off pon ay go up o =.8 for value equal o.5. In ha case he MSE reducon when =.5 s equal o.3%. Ths shows he consderable regon where he aggregaon approach ouperfors he non-aggregaon one for hgh aggregaon levels. Hence, ncreasng he aggregaon level

125 B.Rosa-Tabar, 3, Chaper 3 4 proves he perforance of he aggregaon approach and he bes resuls can be acheved for sall values of and hgh aggregaon levels. These eprcal resuls generally confr he heorecal fndngs. Fgure 3-9: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a dsaggregae level, ARIMA(,,) process wh - <.33. Fgure 3-9 shows he cu-off pon for fxed values of and when he SKUs have a non-aggregae deand ha follows an ARIMA(,,) process wh -<.33. The eprcal resuls show ha for a low aggregaon level =, low values should be seleced n order o have an ouperforance of he aggregaon approach. For exaple when an aggregaon level = s used, he cu-off pon =.33 for an value equal o.5 and he MSE reducon when =.5 s equal o.45%. The cu-off pons consderably ncrease when he aggregaon level ncreases. Fgure 3-9 shows also ha for an value equal o.5 and when he aggregaon level =, he cu-off pon s alos equal o, whch eans ha he aggregaon approach always ouperfors he non-aggregaon one n ha case. Tha

126 B.Rosa-Tabar, 3, Chaper 3 5 resuls also n a MSE reducon equal o 5.% ha decreases for hgher values of. However, should be noed ha for he SKUs where.33 < <, he eprcal resuls show ha when he opal value of s used for all values of and, he non-aggregaon approach ouperfors he aggregaon one. The eprcal analyss confrs overall he resuls of he heorecal evaluaon boh for all processes under consderaon. Wha can be concluded here s ha here s a consderable range of he values of he soohng consan of he aggregae seres ha ples a benef of usng he aggregaon approach. Ths benef can also be subsanal for hgh aggregaon levels and low soohng consans. Noe ha such analyss can be ulzed as an ndcaor on when he aggregaon approach should be used and whch paraeers lead o he ouperforance of hs approach Coparson a aggregae level In hs par he valdy of he fndngs n forecasng he aggregae deand s esed by real daa ses. In sub-secons 3... and 3... he superory condons of he aggregaon and non-aggregaon approaches are denfy when a cuulave sep ahead forecas s requred. I s shown ha for posve auocorrelaon assocaed wh low aggregaon level, non-aggregaon approach wors beer bu by ncreasng he aggregaon level he perforance s proved even for hgh posve auocrrelaon. Fgure 3-3 shows he resuls of boh aggregaon and non-aggregaon approaches for dfferen values of aggregaon level when he opal soohng consans before and afer aggregaon s used and he non-aggregae deand seres follow an ARIMA(,,) process. The resuls show ha for he aggregaon level up o sx, he MSE BA s saller han MSE AA. However, as aes hgher values han sx, he laer becoes saller. Therefore, he eprcal resuls show ha when he non-aggregae deand follow an ARIMA(,,) process and auocorrelaon s posve (refer o Table 3-5) hen for lower values of (6) he nonaggregaon approach wors beer. However for >6, aggregaon approach ouperfors he non-aggregaon one.

127 B.Rosa-Tabar, 3, Chaper 3 6 Fgure 3-3: Eprcal resuls copared a aggregae level, ARIMA(,,) process Fgure 3-3: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a aggregae level, ARIMA(,,) process.

128 B.Rosa-Tabar, 3, Chaper 3 7 Fgure 3-3: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a aggregae level, ARIMA(,,) process wh -<<.33. Fgure 3-33: Cu-off pons of plyng an ouperforance of he aggregaon approach for dfferen values of and copared a aggregae level, ARIMA(,,) process wh.33<<.

129 B.Rosa-Tabar, 3, Chaper 3 8 Fgure 3-3 presens he resuls of eprcal analyss copared a aggregae level when he non-aggregae deand follows an ARIMA(,,) process. I s shown ha here s always a value of soohng consan afer aggregaon for whch he aggregaon approach ouperfors non-aggregaon one. The cu-off pon s deerned for fxed values of and when he non-aggregae deand of he SKUs follows an ARIMA(,,) process. The resuls show ha for hgh values of aggregaon level, he aggregaon approach ouperfors he non-aggregaon one for a wde range of values. Fgure 3-3 shows he cu-off pon for fxed values of and when he nonaggregae SKUs follow an ARIMA(,,) process wh -<.33. As s dscussed above, for hese auoregressve values he auocorrelaon s no hghly posve. For hese range of here s always a value of for whch he aggregaon approach ouperfors he nonaggregaon one. The resuls show ha he cu-off pons consderably ncrease when he aggregaon level ncreases. Fgure 3-3 shows also ha for an value greaer han and equal o. and when he aggregaon level =, he cu-off pon s alos equal o, whch eans ha for hese values aggregaon approach always perfors beer. The MSE reducon assocaed wh hese values and he soohng consan afer aggregaon =. can be as hgh as 5%. In Fgure 3-33, he forecas resuls of SKUs wh.33 < < are presened. The eprcal resuls show ha when he opal value of and s used, hen for lower value of, he MSE BA s saller han he MSE AA. However, as he aggregaon level ncreases he laer becoes saller han he forer one and consequenly he non-aggregaon approach ouperfors he aggregaon one. The resuls show ha for he aggregaon level saller han or equal o seven he non-aggregaon approach perfors beer, bu for he values of greaer han seven, he aggregaon approach ouperfors he non-aggregaon one. These resuls confr he resuls of analycal evaluaon presened n sub-secons and

130 B.Rosa-Tabar, 3, Chaper Concluson In hs chaper he pac of eporal aggregaon on deand forecasng has been evaluaed by heorecal, sulaon and eprcal nvesgaon. The evaluaons were based on he consderaon of he Mean Squared Error (MSE) before and afer aggregaon (MSE BA / MSE AA ) and coparsons were underaen a boh dsaggregae and aggregae deand level. I s assued ha he deand follow an ARIMA(,,), ARIMA(,,) and ARIMA(,,) process and a Sngle Exponenal Soohng s used as a forecasng ehod. The condons under whch he aggregaon approach ouperfors he non-aggregaon are denfed. The resuls show ha when he auocorrelaon s hghly posve n he orgnal seres he non-aggregaon approach ay ouperfor he aggregaon one. In general here are fewer benefs for he aggregaon approach wh hgh posve auocorrelaon han he seres wh low posve or negave auocorrelaon. Ths s an nuve fndng snce when he auocorrelaon s hghly posve, a any e he os recen deand nforaon s val. Therefore, n ha case he dsaggregae process wors beer as fully explos such recen nforaon. However, on he conrary, for low posve auocorrelaon or negave auocorrelaon, he recen deand nforaon s no ha crucal hen a ore long er vew on deand s preferable. As dscussed above can be obaned by selecng hgh aggregaon levels and low soohng consans. Ths s also an poran eprcal nsgh snce anagers ay now wha o expec (n ers of any poenal gans) based on he auocorrelaon levels presen n her seres. When he deand process follow eher an ARIMA(,,) or an ARIMA(,,) process assocaed wh hgh posve auocorrelaon, and he coparson s underaen a dsaggregae level, he resuls show ha non level of aggregaon prove he accuracy so he non-aggregaon approach always ouperfors he aggregaon one. However, when coparng s underaen a aggregae level, for low aggregaon level he non-aggregaon approach ay ouperfors he aggregaon one, bu for hgher values of aggregaon level, he aggregaon approach always provde ore accurae forecass. I s also found ha he perforance of he aggregaon approach proves as he soohng consan value eployed a he aggregae seres reduces and he aggregaon level ncreases. Ths s rue for boh coparson a dsaggregae and aggregae level.

131 B.Rosa-Tabar, 3, Chaper 4 3 Chaper 4 Cross-Seconal Aggregaon In chaper 3, he effec of he non-overlappng eporal aggregaon on deand forecasng s analysed. In hs chaper he effecveness of cross-seconal approach on deand forecasng s evaluaed. An poran decson nvolved n he forecasng process s he deernaon of he degree of aggregaon ha forecass should refer o wh respec o he nuber of producs nvolved. The herarchcal level a whch forecasng s perfored depends on he funcon he forecass are fed no. There are several exaples wh regards o producs (or Soc Keepng Uns - SKUs) n parcular: ) forecasng a he ndvdual SKU level s requred for nvenory conrol, ) produc faly forecass ay be requred for Maser Producon Schedulng, ) forecass across a group of es ordered fro he sae suppler ay be requred for he purpose of consoldang orders, and ) forecass across he es sold o a specfc large cusoer ay deerne ransporaon and roung decsons ec. One nuvely appealng approach o oban hgher level forecass s by crossseconal (also referred o as herarchcal) aggregaon, whch nvolves aggregang dfferen es (.e. aggregang he requreens for dfferen es usually n one specfc perod) o reduce varably. Exsng approaches o he cross-seconal forecasng usually nvolve eher a boo-up (BU) or a op-down (TD) approach (or a cobnaon of he wo). When forecasng a he aggregae level s of neres, he forer nvolves he aggregaon of ndvdual SKU forecass o he group level whereas he laer relaes o forecasng drecly a he group level (.e. frs aggregae requreens and hen exrapolae drecly a he aggregae level). When he ephass s on forecasng a he subaggregae level, hen he BU approach relaes o drec exrapolaon a he subaggregae level whereas he TD nvolves he dsaggregaon of he forecass produced drecly a he group level. An poran ssue ha has araced he aenon of any researchers as well as praconers over he las few decades s he effecveness of such cross-seconal forecasng approaches.

132 B.Rosa-Tabar, 3, Chaper 4 3 TD and BU forecasng approaches are exreely useful owards provng he accuracy of forecass and plans when leveraged whn an S&OP (Sales and Operaons Plannng) process (Lapde, 6). The S&OP s a ul-funconal process ha nvolves anagers fro all deparens (Sales, Cusoer Servce, Supply Chan, Mareng, Manufacurng, Logsc, Procureen and Fnance), where each deparen requres dfferen levels of deand forecass (Lapde, 4). For exaple, n areng, forecasng of revenues by produc groups and brands s needed, sales deparens deal wh sales forecass by cusoer accouns and/or sales channels. Supply chan anagers reques SKU level forecass, whle fnance deparen requres forecass ha are aggregae no budgeary uns n ers of revenues and coss (Bozos and Nolopoulos, ). In hs chaper, he relave effecveness of he BU and TD approach for forecasng s evaluaed. I s recognzed ha forecass ay be equally requred a boh he aggregae and subaggregae level, and as such coparsons are perfored a boh levels. he effecveness of he wo approaches s analycally suded when he underlyng seres follows eher a saonary frs order Auoregressve Movng Average process ARIMA(,,) or a nonsaonary Inegraed Movng Average process of order one, ARIMA(,,), and he forecasng ehod s he Sngle Exponenal Soohng (SES) whch s he opal esaor for he ARIMA(,,) process (Box e al., 8). The assupons bear a sgnfcan degree of reals. An ARMA(Auoregressve Movng Average) odel ofen fs deand daa beer han an auoregressve or ovng average odel alone, snce ypcally deand conan srucurally boh ovng average and auoregressve characerscs (Duc e al., 8). The ARMA process have been found o f deand for long lfecycle goods such as fuel, food producs, achne ools, ec (Chopra and Mendl, ; Nahas, 993). I has also been shown ha he ARMA deand processes occur naurally n ul-sage supply chans (Gaur e al., 5; Zhang, 4). There s also consderable evdence o sugges ha nvenory deand s non-saonary and hus relevan processes should be assued for represenng her underlyng srucure. Tunc e al. () saed ha non-saonary sochasc deands are very coon n all ndusral sengs assocaed wh seasonal paerns, rends, busness cycles, and led-lfe es such as he hgh-ech ndusry (Chen e al., 8; Graves and Wlles,, 8) and grocery dsrbuon (Erp e al., 99; Lee e al., 997a; Marel e al., 995). There s also soe

133 B.Rosa-Tabar, 3, Chaper 4 3 evdence ha deand ay follow an ARIMA(,,) process n parcular (whch s he process consdered n hs sudy). Ths process has ofen been found o be useful n nvenory conrol probles and econoercs (Box e al., 8). In addon, Mahajan and Desa () saed ha realers ofen face a non-saonary deand ha follows an ARIMA(,,) process. Moreover, SES s a very popular forecasng ehod n ndusry (Acar and Gardner, ; Gardner, 99, 6; Taylor, 3). In ers of he praccal relevance of hs research we refer o a se of SKUs where a large proporon of he follow an ARIMA(,,) or an ARIMA (,,) processes. Ths s no an unypcal scenaro as deonsraed by analyss of eprcal daases ncludng our own eprcal nvesgaon. The aheacal analyss s copleened by a nuercal experen o evaluae n deal he condons under whch one approach ouperfors he oher. Such an experen also allows he nroducon of non-dencal process paraeers of he subaggregae seres and he coparson a he subaggregae level when he subaggregae es follow an ARIMA(,,) process. In addon, an eprcal nvesgaon s also conduced o assess he valdy of he resuls on real daa fro a European supersore. Consderable par of hs chaper s presened n Rosa-Tabar e al (3d) and Rosa-Tabar e al(3b). The reander of hs chaper s srucured as follows. In secon he assupons and noaons used n hs sudy are descrbed, In addon an analycal evaluaon of he varance of he forecas error relaed o boh he BU and TD approaches s conduced. In secon he analycal resuls are presened. Nex a sulaon sudy s conduced followng he sulaon resuls n secon 3. Fnally, an eprcal nvesgaon s conduced n Secon 4 and he chaper concludes n Secon 5.. Theorecal analyss In hs secon he varance of he forecas error assocaed wh he TD and BU approaches s derved. Coparsons ay be perfored a boh he aggregae and he subaggregae level alhough n hs heorecal analyss for he ARIMA(,,) process, he

134 B.Rosa-Tabar, 3, Chaper 4 33 coparsons are perfored only a he forer level snce resuls regardng he laer are nracable. However, n he sulaon sudy followng he heorecal analyss, varous assupons are relaxed and he resuls for he ARIMA(,,) and he ARIMA(,,) processes are presened. The coparson s underaen a boh subaggregae and aggregae level. When forecasng a he aggregae level s of neres, he forer nvolves he aggregaon of ndvdual SKU forecass o he group level whereas he laer relaes o forecasng drecly a he group level (.e. frs aggregae requreens and hen exrapolae drecly a he aggregae level). When he ephass s on forecasng a he subaggregae level, hen he BU relaes o drec exrapolaon a he subaggregae level whereas he TD nvolves he dsaggregaon of he forecass produced drecly a he group level. 4.. Noaon and assupons For he reander of he paper, he followng noaons are used: d, : Subaggregae deand n perod,j : Correlaon beween he error er of subaggregae e and j (cross-correlaon) D : Aggregae deand n perod, : Independen rando varable for subaggregae deand n perod, norally dsrbued wh zero ean and varance : Independen rando varable for aggregae deand n perod, norally dsrbued wh zero ean and varance f, : Forecas of subaggregae deand n perod, he forecas produced n - for he deand n. F : Forecas of aggregae deand n perod, he forecas produce n - for he deand n. α : Soohng consan used n he Sngle Exponenal Soohng ehod for each subaggregae deand n he BU approach, TD : Soohng consan used n he Sngle Exponenal Soohng ehod for aggregae deand n TD approach, TD

135 B.Rosa-Tabar, 3, Chaper 4 34 p : he relave wegh of subaggregae e 's conrbuon o he aggregae faly, N where p. V BU : Varance of Forecas Error of he BU approach V TD : Varance of Forecas Error of he TD approach : Movng average paraeer of subaggregae deand, : Auorgressve paraeer of subaggregae deand, : Movng average paraeer of aggregae deand, : Expeced value of sub-aggregae deand n any e perod : Expeced value of aggregae deand n any e perod I s assued ha all he subaggregae deand seres d, follow eher a frs order auoregressve ovng average, ARIMA(,,), or a frs order Inegraed Movng Average process, ARIMA(,,). Ths can be aheacally wren n perod by () and () respecvely: d d,, N ( 4-),,,,, d d.,,, ( 4-) Fro () s clear ha he deand n he nex perod s he deand n he curren perod plus an error er. By expandng ( 4-) we have: d ( 4-3),,,,, where. I should be noed ha only under hs condon on α, SES s opal as provdes he nu ean square forecass for he ARIMA(,,) process.

136 B.Rosa-Tabar, 3, Chaper 4 35 Here he soohng consan values are consdered as a conrol paraeer deerned by forecasers ha vares beween zero and one. Obvously, snce, under hs condon (only for ARIMA(,,) process) only aes he values beween zero and one and does no cover he whole range of -. However, he heorecal analyss s sll vald for he whole range of -. In addon, n he sulaon analyss hs assupon o cover he whole range of - are relaxed when he value of he soohng consan s fxed. Fgure 4-: Saple auocorrelaon of ARIMA(,,) process when = -.9.

137 B.Rosa-Tabar, 3, Chaper 4 36 When he underlyng process follows an ARIMA(,,) process, as oves fro + oward - he resulng underlyng srucure changes consderably. When.5< <-, he auocorrelaon s hghly posve and spans all e lags (no only lag ). For exaple, for =-.9 he auocorrelaon s very close o + wh sooh exponenal decay by ncreasng he lags (see Fgure 4-). As we ove up owards + he auocorrelaon reduces bu sll reans posve and for hgh posve values of becoes close o zero eanng ha he seres are rando (see Fgure 4-). Fgure 4-: Saple auocorrelaon of ARIMA(,,) process when =.9.

138 B.Rosa-Tabar, 3, Chaper 4 37 However, he behavour of he ARIMA(,,) process s dfferen wh hose of he ARIMA(,,) process by changng he paraeers. For dfferen cobnaons of he process paraeers, he resulng underlyng srucure changes consderably. When he deand follows an ARIMA(,,) process he auo-covarance funcon s (Box e al., 8):,,, ( 4-4) When he deand follows an ARIMA(,,) process he auo-covarance and auocorrelaon funcons are(box e al., 8). When he deand follows an ARIMA(,,) process he auo-covarance and auocorrelaon funcons are(box e al., 8). When he deand follows an ARIMA(,,) process he auo-covarance and auocorrelaon funcons are(box e al., 8):, ( 3-). ( 3-3) For dfferen cobnaons of he process paraeers, he resulng underlyng srucure changes consderably. Table 3- presens he auocorrelaon srucure for dfferen

139 B.Rosa-Tabar, 3, Chaper 4 38 process paraeers whch helps o beer undersand he process and can be useful o nerpre he resuls of he forhcong analyss For dfferen cobnaons of he process paraeers, he resulng underlyng srucure changes consderably. Table 3- presens he auocorrelaon srucure for dfferen process paraeers whch helps o beer undersand he process and can be useful o nerpre he resuls of he forhcong analyss. I s assued ha all he subaggregae deand process paraeers are dencal ( 3 N ). Ths assupon s consdered only for he purpose of he heorecal analyss and, as above, s also relaxed n he sulaon par of hs wor. The concerned assupon ples ha he aggregae deand also follows he sae process as subaggregae es. If hen he su of he subaggregae es s no necessarly he 3 N sae process (Lüepohl, 984). The aggregae deand n perod, D can be expressed as he su of he deands of he subaggregae es,.e. N,. D d The forecasng ehod consdered n hs sudy s he Sngle Exponenal Soohng (SES). Ths ehod s beng appled n any copanes. Due o s splcy, I has been specfcally appled n an nvenory producon plannng envronen (Gardner, 99). Usng SES, he forecas of subaggregae deand n perod produced a he end of perod - s f,, d,. ( 4-5) The forecas of subaggregae e n perod for he ARIMA(,,) process can be expressed as a funcon of he error ers as follows: f ( 4-6),,,,

140 B.Rosa-Tabar, 3, Chaper 4 39 I s furher assued ha he sandard devaon of he error er n ( 4-) and ( 3-7) above s sgnfcanly saller han he expeced value of he deand. Thus, when deand s generaed, he probably of a negave value s neglgble. 4.. Varance of forecas error a aggregae level The varance of forecas error correspondng o he TD (V TD ) and he BU(V BU ) approaches for boh a non-saonary ARIMA(,,) and a saonary ARIMA(,,) processes a he aggregae level are calculaed Inegraed ovng average process order one, ARIMA(,,) The analyss s begun by dervng he V BU, whch s defned as follows: V BU Var D N N N N f, Var d, f, Var d, f, ( 4-7) By subsung ( 4-3) and ( 4-6) n ( 4-7) he followng s gven: V BU Var N, ( 4-8) Snce Var and Cov j,, j j,,,, he varance of he BU approach s: VBU N N N ( 4-9) j, j j Now he varance of he forecas error for he TD approach s derved. As dscussed above, s shown ha when he subaggregae es follow an ARIMA (,,) process, he

141 B.Rosa-Tabar, 3, Chaper 4 4 aggregae faly deand also follows an ARIMA (,,) process (Lüepohl, 984).The faly aggregae process s defned as follows: D D ( 4-) where =-. Consderng 3 N resuls n he sae hea also n he aggregae deand so,. Now by consderng TD and, s obvous ha he opal soohng consan for he aggregae deand s soohng consan for he subaggregae process. TD, whch s equal o he opal The aggregae deand and s forecas can be expressed as a funcon of he error ers as followng: D ( 4-) TD TD TD Knowng ha N Var,, he followng s obaned N N N Var, Cov,, j, j ( 4-) The aggregae forecas s F ( 4-3) TD TD TD The varance of he TD forecas error s defned as: V TD Var D F ( 4-4)

142 B.Rosa-Tabar, 3, Chaper 4 4 By subsung ( 4-) and ( 4-3) no ( 4-4), he varance of TD approach s: V TD Var ( 4-5) By subsung ( 4-) no ( 4-5) we have: VTD N N N ( 4-6) j, j j 4... Auoregressve ovng average process order one, ARIMA(,,) In hs par, he varance of forecas error of he BU approach a he aggregae level s calculaed when he subaggregae es follow a saonary ARIMA(,,) process. The V BU can be obaned as follows: V BU Var N N N N D f Var d f Var,,, N N d, f Covd, f,, d j, f j, j, ( 4-7) Subsequenly, he wo pars of ( 4-7) should o be deerned: ) he varance of forecas error for subaggregae e whch s calculaed n ( 4-33), ) he covarance of he forecas error beween subaggregae and j. The covarance of he forecas error beween subaggregae and j n perod, Cov d, f,, d j, f j, s as follows: Cov d f, d f Cov( d, d ) Covd, f Cov f, d Cov f f,, j, j,, j,, j,, j,,, j, ( 4-8)

143 B.Rosa-Tabar, 3, Chaper 4 4 Now by subsung (P-), (P-), (P-3) and (P-5) n Appendx P no ( 4-8), he followng s obaned: j j j j j j j j j j j j j j j f d f d Cov,,,,, ( 4-9) Fnally by subsung ( 4-33) and ( 4-9) no ( 4-7), he varance of he forecas error of he BU approach a aggregae level s: N N j j j j j j j j j j j j j N V BU ( 4-) by subsung ( 3-), and defned n (P-) n Appendx P no ( 4-) and assung ha N, N and N, V BU s splfed as : N N j j j N BU V ( 4-) Now, he dervaon of he varance of forecas error for he TD approach a he aggregae level s preceded. All subaggregae es are aggregae o produce one-sep-ahead esaes a he op level based on SES. The V TD s defned as

144 B.Rosa-Tabar, 3, Chaper 4 43 TD F D Cov F Var D Var F D Var V, ( 4-) Assung = = = N =, and = = = N =, he aggregae faly deand also follows an ARIMA(,,) process wh he followng characerscs. he aggregae seres can be defned as. ˆ ˆ, N D d D ( 4-3) where N, ˆ and ˆ ˆ N N j j j N Var, ˆ ˆ ), ( ˆ D D Cov ( 4-4) The evaluaon of V TD s begun by defnng he varance of deand n ( 4-4). The covarance beween he aggregae deand and s forecas s:,... ), ( ), ( ), ( ), ( ), (, TD TD TD TD TD TD TD D D Cov D D Cov D D Cov D D Cov D F Cov F D Cov ( 4-5) Then by subsung ( 4-4) no ( 3-3) he followng s gven:. ˆ ), ( D F Cov ( 4-6) Fnally, he varance of forecass can be calculaed as:

145 B.Rosa-Tabar, 3, Chaper 4 44 Var F Var TDF TD F Var D Var F Cov D, F. TD TD TD TD ( 4-7) By consderng he fac ha he process s saonary, s clear ha Var F Var F and CovD, F CovD, F for all and by subsung ( 4-4) and ( 3-4) no ( 3-5) and hen by subsung N ˆ N N j j j, he followng s obaned: TD Var ) ( 4-8) ( F TD TD TD TD TD, Fnally by consderng Var ( D ) and subsung ( 3-4) and ( 3-6) no ( 4-), he varance of forecas error for TD approach s obaned: ˆ V TD N N N TD TD j j ( 4-9) j TD TD 4..3 Varance of forecas error a subaggregae level In hs sub-secon, he varance of he forecas error conssen wh he TD (V TD ) and he BU approach (V BU ) for he saonary ARIMA(,,) process a he subaggregae level s calculaed. I should be noed ha he resuls regardng he non-saonary ARIMA(,,) copared a subaggregae level are heorecally nracable. However, n he sulaon sudy followng he heorecal analyss varous assupons are relaxed and he resuls for he ARIMA(,,) process a boh levels of coparson are presened.

146 B.Rosa-Tabar, 3, Chaper Auoregressve ovng average process order one, ARIMA(,,) In hs par, he varance of forecas error of BU approach a subaggregae level s calculaed, so V BU s defned as: V BU N Var d, f, ( 4-3) Slar o ( 3-4) and ( 3-6), he varance of forecas and he covarance beween he subaggregae deand and s forecas s: Cov ( d,, f, ). ( 4-3) Var f,. ( 4-3) Now he varance of forecas error by consderng can be obaned as follows: Var d f, Var d,,, and ( 4-3) and ( 4-3),,,, ( 4-33) Fnally, by subsung ( 4-33) no ( 4-3) and consderng hs assupon ha = = = N, = = = N and = = = N he followng s obaned: V BU N ( 4-34)

147 B.Rosa-Tabar, 3, Chaper 4 46 Now he varance of he forecas error of he TD approach s derved when he coparson s underaen a subaggregae level. The varance of forecas error for he TD approach, V TD s defnes as follows: V TD N Var, N d p F Var d p Var F p Covd F,, ( 4-35) The covarance beween subaggregae es and aggregae forecas n perod s: Cov d,, F Cov d,, TD TD D,,,..., N ( 4-36) N By subsung D d, no ( 4-36) and assung ha = = = N =, and = = = N =, he value of Cov d, F Cov, s derved hrough recursve subsuons. Recall ha,,,, Cov,, j, j j, Cov,,,, for all and Cov, for all,, j, TD N Covd,, F j j,,,..., N TD j ( 4-37) Now, by subsung ( 3-), ( 3-6) and ( 4-37) no ( 4-35) he followng s gven: V TD N TD TD where ( 4-38)

148 B.Rosa-Tabar, 3, Chaper 4 47 N N TD TD p TD TD p N N j j j N N j j j. Theorecal Coparson In hs secon, he condons under whch each approach ouperfors he oher one are analycally denfed. The rao of he varance of forecas error correspondng o he TD approach (V TD ) o he varance of he forecas error assocaed wh he BU approach (V BU ) s calculaed. A rao ha s lower han one, ples a benef n favour of he TD approach. Conversely, f he rao s greaer han one, hen he BU approach perfors beer (and f he rao s equal o one, boh sraeges perfor he sae). 4.. Coparson a aggregae level In hs sub-secon, for each process under consderaon he rao of V TD o V BU s derved. The coparson s underaen a he aggregae level Inegraed ovng average process order one ARIMA(,,) Proposon. If all he subaggregae deand es follow an ARIMA(,,) process wh dencal ovng average paraeers ( ) and he opal soohng 3 N consan value s used o forecas boh he subaggregae and aggregae deand, hen he perforance of he TD and BU approaches for forecasng aggregae deand s dencal (V TD = V BU ). Proof: The effecveness of he TD and he BU approaches can be copared by evaluang he rao of he correspondng varances of forecas error (.e. by dvdng ( 4-9) and ( 4-6)):

149 B.Rosa-Tabar, 3, Chaper 4 48 V V TD BU N N N j N N N j, j j, j j ( 4-39) 4... Auoregressve ovng average process order one ARIMA(,,) The rao of he V TD o V BU when he subaggregae deand es follow an ARIMA(,,) process s obaned by dvdng ( 4-) o ( 4-9) : V V TD BU TD TD TD TD ( 4-4) Ths rao s a funcon of he ovng average paraeer ( ), he auoregressve paraeer ( ), and he soohng consans ( and TD ). Fro ( 4-) and ( 4-9) s obvous ha he opal values of and TD are equal. Hence, boh V BU and VTD nzed by havng he equal value of and TD. can be Proposon : If he e seres of he all sub-aggregae deand follows an ARIMA(,,) process when N and N, boh he TD and he BU sraeges perfor equally as long as he soohng consans used for forecasng he subaggregae deands and he aggregae deand are se opal. PROOF: By subsung ( ha V. TD V BU TD ) n ( 4-4), s easy o deonsrae These fndngs are n agreeen wh he resuls repored by Wdara e al. (9) whch heorecally shows ha here s no sgnfcan dfference beween he TD and BU approaches on forecasng aggregae deand when all subaggregae es follow an MA() process wh dencal process paraeers.

150 B.Rosa-Tabar, 3, Chaper Coparson a subaggregae level In hs par, he varance of he forecas error provded by he BU and he TD approaches are copared a he subaggregae level when he subaggregae deands follow an ARIMA(,,) process. As explaned above he coparson a he subaggregae level for he ARIMA(,,) s no raceable Auoregressve ovng average process order one ARIMA(,,) The rao of V TD o V BU coparng a he subaggregae level s gven by dvdng ( 4-38) no ( 4-34). I should be noed ha s dffcul o analyse he paraeers wh any subaggregae es, herefore he followng analyss s resrced o a faly wh wo SKUs o oban he eanngful nsghs. In addon, s assued ha, herefore he followng s gven: V V TD BU R ( 4-4) where R p p TD TD TD TD TD TD THEOREM 4-: If he e seres of all subaggregae deand follows an ARIMA(,,) process when.5 and, hen he BU ouperfors he TD approach regardless of he cross-correlaon, he relave wegh of each subaggregae e p, and he soohng consan values. PROOF: Proof n Appendx Q.

151 B.Rosa-Tabar, 3, Chaper 4 5 THEOREM 4-: If he e seres of all sub-aggregae deand follows an ARIMA(,,) process when ). 5 and and ) he soohng consans used for forecasng he subaggregae deands under he BU and TD approach are se sall(,. ), hen he axu dfference beween he BU and he TD o TD forecas he subaggregae forecass s %,.99V TD /V BU.. PROOF: Proof n Appendx R. COROLLARY 4. when he soohng consans are se equal o.5,.5 and.3 n Theore above, hen he rao of V TD /V BU aes he values presened n Table 4-. Table 4-: The rao of V TD /V BU for dfferen conrol paraeers and. 5, = TD =.5 = TD =.5 = TD =.3.95V TD /V BU..85V TD /V BU.3.7V TD /V BU.6 COROLLARY 4. If he e seres of all sub-aggregae deand follows an ARIMA(,,) process when ).5 and ) he soohng consans are se equal o.,.5,.5 and.3 n Theore 5 above, hen he rao of V TD /V BU aes he values presened ntable 4-. Table 4-: The rao of V TD /V BU for dfferen conrol paraeers and.5, = TD =. = TD =.5 = TD =.5 = TD =.3.99V TD /V BU.99.95V TD /V BU V TD /V BU V TD /V BU 6.4 The resuls of Theore show ha when s negave and aes hgh posve values hen he BU approach always provdes ore accurae forecass han he TD one regardless of he values of he soohng consan, he correlaon beween subaggregae

152 B.Rosa-Tabar, 3, Chaper 4 5 es, and he proporonal weghs. Whle, for he oher process paraeer cobnaons, he superory s a funcon of he conrol and he process paraeers. When he deand follows an ARIMA(,,) process, s dscussed ha for he negave values of and he posve values of, he auocorrelaon s hghly posve, herefore when he auocorrelaon s hghly posve he BU ouperfors he TD approach. When he auocorrelaon s posve, successve values of d are posvely correlaed and he process wll end o be sooher han he rando seres. When he aggregae forecass are dsaggregae, he perforance of he TD approach s deeroraed by he dsaggregaon process. However, he BU s no affeced by ha. Therefore, n hese cases he BU approach ouperfors he TD one. 3. Sulaon sudy In hs secon, a sulaon sudy s perfored o evaluae he relave perforance of he TD over he BU approach under ore realsc assupons. In parcular he followng scenaro for boh he ARIMA(,,) and he ARIMA(,,) processes are consdered. A sulaon nvesgaon s conduced o dscuss he effecveness of he BU and he TD approaches copared a he subaggregae and he aggregae level for non-dencal ( N, N ) process paraeers. In boh approaches, he search procedure s perfored n he whole range of - and Sulaon desgn The presenaon of he resuls and he analyss of he paraeers on he rao of V TD / V BU becoes coplex when any SKUs n he sulaon experens are consdered. Therefore, he sulaon analyss s resrced o a faly of wo SKUs o oban he eanngful nsghs. Ths s n concordance wh os of he earler papers usng sulaon approaches as hey have also resrced he nuber of es o wo (Dangerfeld and Morrs, 99; Fledner, 999; Wdara e al., 8, 9). The paraeer values for our sulaon experen are presened n Table.

153 B.Rosa-Tabar, 3, Chaper 4 5 Table 4-3: Paraeers of he sulaon experen, TD j N Replcaons N Te Perods 4 9.: : : : +.9 The subaggregae deands n each perod are generaed randoly subjec o he paraeers descrbed n Table 4-3. The value of s se que saller han o avod he generaon of negave subaggregae values. Experens have also been conduced wh oher values of and bu hey are no repored here as hey lead o he sae nsghs. To generae he deands n each perod, he error ers, and, wh a crosscorrelaon coeffcen of are frs generaed randoly hen he equaons ( 4-) and ( 3-7) are used o generae he correlaed subaggregae deands. The generaed deand s nalzed a he value of he ean plus an error er. The sulaon experen s desgned and run n Malab 7... For each paraeer cobnaon descrbed n Table 4 deand seres of observaons s generaed and replcaons are nroduced. The generaed deand s spl for each seres a boh he subaggregae and aggregae level, no hree pars. The frs par (whn saple) consss of e perods and s used n order o nalse he esaes. The second par conanng 5 perods s used o deerne he opal soohng consan (.e. he soohng consan used n he esaon procedure ha nses he ean square error - MSE). The search procedure o fnd he soohng consan ha nses he MSE s perfored n he whole range [,], wh a sep ncrease equal o.. A grd search o nse he s conduced, however we don use a connuous opsaon as hs s no he an focus of our wor and he sensvy o he value s no ha hgh. Noe ha for he BU approach, he soohng consans are opzed for each e ndvdually. Fnally, n order o evaluae he perforance of he wo forecasng approaches, he value of he varance of he forecas error for he las 55 perods of he sulaon (ou-of-saple) s calculaed. I should be noed ha he nalzaon daa of each seres have been used o calculae he proporon p whch s used o dsaggregae he aggregae forecas.

154 B.Rosa-Tabar, 3, Chaper 4 53 The relave benef of one forecasng approach over he oher s easured by V TD /V BU. As prevously dscussed, a rao lower han one ples ha he TD approach ouperfors he BU one whereas a rao greaer han one ples he oppose Sulaon resuls In hs sub-secon, he resuls of sulaon sudy are presened when he coparson s underaen a boh subaggrega and aggregae level Coparson a he Aggregae Level Frs, he relave perforance of he BU and TD approaches a he aggregae level s analysed when he subaggregae process paraeers are no necessarly dencal. For each experen, he rao of he varance of forecas error s calculaed as D F Var D f Var., The sulaon resuls show ha when he process paraeers are dencal here s no dfference beween he BU and he TD approach for boh he ARIMA(,,) and he ARIMA(,,) processes. Whereas, when he process paraeers are no dencal, whch s ore realsc, he resuls are dfferen. Fgure 4-3 presens he relave perforance of he BU and he TD approaches a he aggregae level forecasng when he subaggregae deand es follow an ARIMA(,,) process wh dfferen values of he ovng average paraeer (.e. ). I s seen ha as he cross-correlaon coeffcen changes fro -.9 oward +.9 he rao of V TD /V BU s beng reduced. The rao s hgher han or equal o one, when he crosscorrelaon s negave, when equals zero, and when aes low posve values. However, he rao s saller han one only f he cross-correlaon s (hghly) posve. The dealed resuls show ha when he ovng average paraeers, and, ae negave values (Hgh posve auocorrelaon), he perforance of he BU and he TD approaches s always dencal regardless of he values of he cross-correlaon.

155 B.Rosa-Tabar, 3, Chaper 4 54 When he cross-correlaon s posve he superory of each approach depends on he value and he sgn of he ovng average paraeers, and. The TD approach ouperfors he BU one only when he cross-correlaon s (hghly) posve and he ovng average paraeers ae hgh values and have oppose sgns,.e. eher < and > or > and <. Noe ha as he cross-correlaon decreases he superory of he TD approach decreases oo. For less posve cross-correlaon he rao of V TD /V BU becoes equal or greaer han one whch eans ha BU s preferable. In hese cases TD ouperfors BU wh a forecas error varance reducon ha can go up o 5% when he cross-correlaon s very hgh. By decreasng he cross-correlaon o.5, he axu benef of he TD approach decreases o 5% and ends oward zero when he cross-correlaon ends owards zero as well. However, under a negave cross-correlaon, he BU ouperfors he TD approach. When he and values are posve, he rao s alos equal o one for hgh posve cross-correlaon and greaer han one for less posve and negave cross-correlaon. In he laer case he rao of V TD / V BU s ncreased as aes low values and s hgh and vce versa. Fgure 4-4 presens he effec of he BU and he TD approaches on he deand forecasng n he aggregae level (op) when he subaggregae es follow an ARIMA(,,) process wh dfferen values of he ovng average and he auoregressve paraeer (.e., ). The resuls show ha as he cross-correlaon coeffcen oves fro -.9 oward +.9 he rao of V TD /V BU s reduced as well. The rao s always hgher han or equal o one when he cross-correlaon s negave, when equal zero, and when aes low posve values. Thus, for hese cases he BU approach provdes ore accurae forecass. The rao ay becoe saller han one only f he cross-correlaon s hghly posve. In hs case, he superory s a funcon of he ovng average and he auoregressve paraeers. Therefore, he TD approach ay ouperfor he BU approach when he crosscorrelaon s hghly posve.

156 B.Rosa-Tabar, 3, Chaper 4 55 a) =.9 b) =.5 c) =. d) )=-.9 e) =-.5 f) =-. Fgure 4-3: Relave perforance of he TD and he BU approaches n forecasng aggregae deand under dfferen cobnaons of, and for an ARIMA(,,) deand process.

157 B.Rosa-Tabar, 3, Chaper 4 56 a) =-.8; =.9; &3 b) =.9; =-.9; &5 c) =.4; =.9; 4&5 d) =-.9; =.7;3&4 e) =-.9; =-.75; &5 f) =-.9; =-.75; &4 Fgure 4-4 : Relave perforance of he TD and he BU approaches n forecasng aggregae deand under dfferen cobnaons of, and for an ARIMA(,,) process.

158 B.Rosa-Tabar, 3, Chaper 4 57 The resuls show ha when wo subaggregae es ae hgh posve auocorrelaon, he rao s alos equal o one regardless of he values of he cross-correlaon. For exaple, when =.9, =-.8 and =-.8, =-.3 ( case n able ) and =.4, =-.,and =.9, =.3, ( case and 4 n able ). However, when wo subaggregae es ae oppose auocorrelaon values, one wh hgh posve and he oher wh negave auocorrelaon, he rao ay becoe saller han one and consequenly he TD ouperfors he BU approach. For nsance when =.4, =.9 and =.55, =.5 ( case 4 and 5 n able ), he forecas error varance reducon can go up o 9% when he cross-correlaon s very hgh. Ths s also rue when =.8, =-.9 and =., =.6( case 4 and n able ) for hs case he varance of he forecas error reducon ay go up o 3%. In boh saonary and non-saonary cases, when boh subaggregae es ae hgh posve auocorrelaon, he BU and he TD approaches perfor equally. One possble explanaon s for a hgh posve auocorrelaon values, he opal value of he soohng consan s se a he hghes value n he gven range whch s equal o.99 for boh TD and BU approaches. When he soohng consan for he BU and he TD approaches s equal and he sae procedure of forecasng s used, he BU and he TD approaches perfor equally. When he cross-correlaon coeffcen s negave, he BU approach perfors beer. Perforance dfferences are furher nflaed when he auocorrelaon values have oppose sgns n whch case he varance reducon acheved by he BU approach can be as hgh as 4% for he saonary ARIMA(,,) and 5% for he non-saonary ARIMA(,,) for hghly negave cross-correlaon. For negave cross-correlaon, he par of seres oves n he oppose drecon (.e. f one ncreases he oher decreases), herefore he subaggregae deand seres have dfferen paerns of evoluon. A cobnaon of dfferen paerns of varaon and an oppose auocorrelaon values leads o a large forecas error for he TD approach and consequenly large values of V TD / V BU for hgh negave cross-correlaon. In hese cases s beer o forecas subaggregae requreens separaely and hen aggregae he o ge he aggregae forecas. When he wo ovng average paraeers ae oppose sgns under boh processes, hs eans ha one seres has posve auocorrelaon whle he oher has a low auocorrelaon (seres wh rando flucuaons). In addon, when he cross-correlaon s

159 B.Rosa-Tabar, 3, Chaper 4 58 posve here s a endency for he par of seres o ove ogeher n he sae drecon, so he deand seres have he sae paern. When usng he TD, all subaggregae are sued up seres o ge an aggregae one, so he flucuaons fro one seres ay be cancelled ou by ohers resulng n a less rando seres ha have a lower forecas error. Therefore TD perfors beer ha BU when he seres have he sae paern assocaed wh dfferen auocorrelaon. In suary, when he subaggregae es follow an ARIMA(,,) process and he goal s o forecas a he aggregae deand level, hen he followng resuls are acheved: ) he superory of TD and BU approaches s affeced by cross-correlaon and auocorrelaon, ) f es have dfferen paerns of flucuaon(negave cross-correlaon), he rao of V TD /V BU s saller han or close o one for lower auocorrelaon values, herefore he BU approach s preferred. ) f he es follow he sae paerns of flucuaon (hgh posve cross-correlaon) and hey have dfferen auocorrelaon paerns, one has a very hgh auocorrelaon whle he oher has a lower auocorrelaon values, he TD approach ay ouperfors he BU on, v) f he auocorrelaon of all es s hghly posve, he perforance of BU and TD s always dencal, and v) when he auocorrelaon for all es s low, BU generally donaes TD, alhough for hghly posve cross-correlaon he dfference s very low. The fndngs are soehow n agreeen wh soe of he earler sudes n hs area by Barnea and Laonsho (98) and Fledner (999) (alhough we do noe ha our resuls are no drecly coparable o hese sudes as we analyse a non-saonary case). The analyss of Barnea and Laonsho (98) based on eprcal analyss showed ha posve crosscorrelaon conrbues o he superory of forecass based on aggregae daa (TD), whch s also he case n our sudy. Fledner (999) used a sulaon sudy o copare he perforance of TD and BU n forecasng aggregae seres where he wo subaggregae es follow an MA() process. He found ha TD donaed BU regardless of he values of he cross-correlaon coeffcen. They have no repored he values of and used n her sudy, so our nerpreaon s ha hs wor consdered only he opposng sgns for and. Should hs be he case hen hese fndngs are n agreeen wh ours.

160 B.Rosa-Tabar, 3, Chaper Coparson a Subaggregae Level In hs sub-secon he relave perforance of he TD and he BU approaches n forecasng subaggregae deand s evaluaed when he ovng average paraeers are no necessarly dencal. The sulaon srucure n ers of whn and ou-of-saple arrangeens s as dscussed n he prevous sub-secon. Under he BU approach, he 55 one sep-ahead forecass are generaed for each e ndvdually usng he opal soohng consan. Under he TD approach, he su of all subaggregae deand s calculaed o oban he aggregae seres, hen he aggregae forecas s provded and fnally s ulpled by he proporonal conrbuory wegh of each subaggregae e o oban he subaggregae forecas. For each experen, he rao of he varance of forecas error s calculaed as: d p F Var d, f Var., *, Fgure 4-5 shows he rao of he varance of forecas error of he TD over he BU approach a he subaggregae level for dfferen values of,, when he subaggregae es follow an ARIMA(,,) process wh non-dencal ovng average paraeers ( ). The resuls show ha when he subaggregae es follow an ARIMA(,,) process, he BU approach always ouperfors he TD n forecasng he subaggregae es regardless of he and he process paraeers. In Fgure 4-5 s shown ha by ovng fro a cross-correlaon of -.9 oward +.9 he rao of V TD /V BU always reans greaer han regardless of he cross-correlaon coeffcen and he ovng average paraeers. When he cross-correlaon and he ovng average paraeers,,, are hghly posve,.e..99,.99 and.99, he rao of V TD /V BU becoes close o one. Fgure 4-5a shows also ha he BU approach ouperforshe TD one by a axu of abou 8% for hghly negave cross-correlaon. Addonally, he rae of superory of BU becoes very hgh when and are no hghly posve (see Fgure 4-5b, c, d).

161 B.Rosa-Tabar, 3, Chaper 4 6 a)=.9 b) =. c) =. d) =-.9 e) =-.5 f) =-. Fgure 4-5: Relave perforance of TD and BU approaches n forecasng subaggregae es under dfferen values of,,, for an ARIMA(,,) process.

162 B.Rosa-Tabar, 3, Chaper 4 6 a) =.9; =-.7;&4 b) =.9; =-.9; &3 c) =.9; =-.85; & d) =.9; =.3;&4 e) =.9; =.4; 4&5 f) =-.95; =.85; 3&5 Fgure 4-6 : Relave perforance of TD and BU approaches n forecasng subaggregae es under dfferen values of,,, for an ARIMA(,,) process.

163 B.Rosa-Tabar, 3, Chaper 4 6 In Fgure 4-6 he rao of he varance of forecas error of he TD over he BU approach s presened a he subaggregae level when he subaggregae es follow an ARIMA(,,) process wh non-dencal ovng average and auoregressve paraeers ( and ). The resuls show ha he rao of V TD /V BU s greaer han or very close o one regardless of he values of he cross-correlaon. When a leas one of he subaggregae es aes hgh posve auocorrelaon (case and 4 n Table ) he rao s greaer han one and consequenly he BU approach ouperfors he TD one. Addonally, by ovng fro hgh negave o hgh posve cross-correlaon, he rao s generally reduced. However, when none of he subaggregae es n he faly ae hgh posve auocorrelaon, he dfference beween he BU and he TD approaches s nsgnfcan. The superory of he BU a he subaggregae level can be arbued o he poenally hgh posve auocorrelaon beween deand perods. Ths aes uch ore dffcul o apporon he resulng aggregae forecas, F, o each e n he faly based on he hsorcal deand proporon, p. As a resul, he perforance of he TD approach s affeced adversely. The perforance of he BU approach, however, s no affeced as forecass he deand for each e ndvdually. By coparng he resuls presened n Fgure 4-5 and Fgure 4-6, s seen ha he rao of V TD /V BU for he non-saonary process s uch bgger han hose of he saonary process. The dfference of he rao under he non-saonary ARIMA(,,) and he saonary ARIMA(,,) process can be arbued o he naure of hese processes. When he subaggregae es follow an ARIMA(,,) process, he auocorrelaon s always hghly posve and spans all lags(no only lag one) excep for very hgh posve values of he ovng average paraeers, however for an ARIMA(,,) process he value of auocorrelaon s lower and no always posve. The fndngs are n accordance wh hose prevously repored n he acadec leraure. Wdara e al.(7) argued ha when he subaggregae e seres follows an AR() process and he value of he auocorrelaon s hgh, here s a sharp worsenng n he relave perforance of he TD approach. Gordon e al.(997) and Dangerfeld and Morrs (99) used he eprcal daa fro he M-copeon daabase and ndcaed ha he BU donaed he TD approach when forecasng he subaggregae e seres. Weaherford e

164 B.Rosa-Tabar, 3, Chaper 4 63 al.() showed ha a purely subaggregae forecas (BU) srongly ouperfored even he bes aggregae forecas (TD) a he subaggregae level. These resuls generally confr he fndngs alhough us be noed ha (as we enoned n he prevous sub-secon) here s no a drec coparson beween hese sudes and ours due o he consderaon of a non-saonary ARIMA(,,) e seres process. Conrasng our resuls wh hose repored by Wdara e al.(7, 9) on saonary MA() and AR() processes, s revealed ha he raes of superory of he BU approach when he process s non-saonary s uch hgher han he saonary case. When he deand follows a saonary AR() process, he axu rao of V TD /V BU s around 6 and s obaned wh seres wh hgh posve auocorrelaon, whle hs rao for he IMA(,) process s hgher han Eprcal analyss In hs secon, he eprcal valdy of he resuls are assess. Frs, he deals of he eprcal daa avalable for he purposes of our nvesgaon along wh he experenal srucure eployed n our wor are provded. Then, he resuls of eprcal n nvesgaon s presened Eprcal daase and experen deals The deand daase avalable for he purposes of hs research consss of 3 weely sales observaons (.e. spans a perod of wo years) for,798 SKUs fro a European grocery sore. The Forecas pacage n R s used o denfy he underlyng ARIMA deand process for each seres and esae he relevan paraeers. I s found ha ore han 3% of he seres (44 seres) ay be represened by he ARIMA(,,) and ore han 5% of he seres (9 seres) represened by ARIMA(,,). I should be noed ha for ore han 8% of SKUs (73 SKU) he auocorrelaon s relavely hgh posve. As such, he daa does no cover he enre heorecally feasble range of he paraeers. he characerscs of he SKUs relevan o hs sudy are suarzed by ndcang he esaed paraeers for he ARIMA(,,) and ARIMA(,,) process n Table 4-4 and 4-5, respecvely.

165 B.Rosa-Tabar, 3, Chaper 4 64 Table 4-4: The eprcal daa se for ARIMA(,,) Group θ nervals Average of θ No. of SKUs [.,.3[.97 4 [.3,.4[ [.4,.5[ [.5,.6[ [.6,.7[ [.7,.8[ [.8,.9[ [.9,] Toal nuber of SKUs: 44 Table 4-5: The eprcal daa se for ARIMA(,,) θ nervals nervals Average of θ Average of Average lagauocorrelaon No. of SKUs [.,.5[ [.6,[ [.5,.9[ [.6,[ [-.,- [.,.5[ [ 9 Toal nuber of SKUs: 9 To faclae a clear presenaon, he esaed paraeers are grouped n nervals and he correspondng nuber of SKUs s gven for each such nerval. The average value per nerval s also presened for he process. Ths caegorsaon allows us o copare he eprcal resuls wh he heorecal fndngs. I should be reared ha he paraeer values are all posve, excep for wo SKUs, and os of he ae hghly posve values. As such, he daa do no cover he enre heorecally feasble range of he paraeers. The daa seres s dvded no hree pars. The frs par (whn saple) consss of e perods and s used n order o nalze he SES esaes. The second par consss of 7 e perods whch are used o deerne he opal soohng consan (opsaon par); he values of he soohng consan are vared fro zero o one wh a sep ncrease of.. The reanng 56 e perods are used o evaluae he perforance of each approach (ou-ofsaple). In TD approach he aggregae forecas s dsaggregae by usng he proporon of each e n he faly, whch s calculaed based on he hsorcal deand n he nal par.

166 B.Rosa-Tabar, 3, Chaper Eprcal resuls The eprcal resuls presened n Table 4-6 are shown for he sae nervals. I can be seen ha when he soohng consan values are opsed for boh he BU and he TD approaches, he varance rao s greaer han one regardless of wheher he coparsons are underaen a he aggregae or subaggregae level. Ths eans ha he BU approach provdes ore accurae boh aggregae and subaggregae forecass han he TD when deand follows an ARIMA(,,) process and SES s he forecasng ehod. However, when he soohng consans used for he BU and he TD approaches are equal, he rao of V TD /V BU equals one n he case of aggregae deand forecasng. Table 4-6: The eprcal rao of V TD /V BU for an ARIMA(,,) process Coparson Level Group Subaggreg θ nervals Aggregae ae [.,.3[.3.73 [.3,.4[ [.4,.5[ [.5,.6[ [.6,.7[ [.7,.8[ [.8,.9[ [.9,] Average As dscussed above he ovng average paraeer, for os SKUs consdered n hs research, s hghly posve. More han 85% of he SKUs have a ovng average paraeer greaer han.6 (see Table 4-4). In addon, he subaggregae cross-correlaon coeffcens beween SKUs vary beween -.5 and +; however os of he are posve. The average of varance of forecas error reducon ay be as hgh as % when he coparson s perfored a he aggregae level, whle 5% varance error reducon ay be acheved for he coparson a he subaggregae level. By referrng o he dealed resuls of he sulaon sudy we see ha for hs range of ovng average paraeer values, <<, he BU approach perfors beer han he TD a boh coparson levels. In Table 4-6 s seen ha when coparsons are underaen a he aggregae level he rao s close o one for all ;; hs s confred by he sulaon resuls where he ovng

167 B.Rosa-Tabar, 3, Chaper 4 66 average paraeers are posve and he cross-correlaon s no hghly negave (please refer o sub-secon 4.3..). Table 4-7: The eprcal rao of V TD /V BU by consderng he aggregaon beween dfferen groups (nervals of values) Level Coparson Group,,3 Aggregae Subaggregae Wh regards o he subaggregae level coparsons, he resuls show ha he rao s greaer han one and s ncreasng by ovng fro hgher values of oward lower values. In addon for hghly posve values of and hghly posve cross-correlaon he rao becoes close o one. In Table 4-6 he resuls are presened assung ha SKUs fall whn a parcular nerval of values. In Table 4, he aggregaon of es across dfferen possble (ranges of) values s consdered and he pac of he paraeers on he superory of each approach s evaluaed. To do so a caegory conanng groups, and 3 ha ncludes 9 SKUs s creaed. Ths s regarded as a caegory wh he lowes values of. By ovng fro hs caegory o groups 4, 5 and 6 he value of ncreases. These groups wh group 8 ha represen he hghes value of are aggregae. The rao of V TD /V BU s presened n Table 4-7. The resuls ndcae ha when he ovng average paraeers are dfferen (Group,,3 wh 8) hen he rao s hgh, addonally as he values ncrease (endng owards he values covered by group 8) he rao decreases. Ths ples ha when he groups of SKUs wh low and hgh values are aggregae, hen here s a greaer benef of usng he BU approach n ers of accuracy. Ths s exacly wha s observed n he sulaon resuls for SKUs (one assocaed wh a sall and one wh a hgh value. These eprcal resuls generally confr

168 B.Rosa-Tabar, 3, Chaper 4 67 he fndngs of he heorecal and he sulaon sudy when he subaggregae es follow an ARIMA(,,) process. Table 4-8: The eprcal resuls for ARIMA(,,) Coparson Level θ nervals nervals Average of θ Average of Subaggregae level Aggregae level [.,.5[ [.6,[ [.5,.9[ [.6,[ [-.,- [.,.5[ [. Average:.3. The eprcal resuls for he ARIMA(,,) process are presened n Table 4-8. I s shown ha when he soohng consan values are opsed for boh he BU and he TD approaches, he varance rao s greaer han one regardless of wheher he coparson s underaen a he aggregae or subaggregae level. In addon, n he aggregae deand forecasng, he rao of V TD /V BU s close o one. As s explaned above, for he ovng average and auoregressve paraeers values presened n Table 3-5, he auocorrelaon s posve. For posve auocorrelaon he dfference beween BU and TD approaches copared a subaggregae level s nsgnfcan. These resuls generally confr he analycal and he sulaon resuls presened n Sub-secons and Secon 3 for he ARIMA(,,) process. 5. Concluson In hs chaper, he effecveness of he boo-up and op-down approaches s analycally evaluaed o forecas he aggregae and he subaggregae deand when he subaggregae seres follow eher a frs order negraed ovng average ARIMA(,,) or an auroregressve ovng average process order one, ARIMA(,,). Forecasng s assued o be relyng upon a Sngle Exponenal Soohng (SES) procedure and he analycal resuls were copleened by a sulaon experen a boh he aggregae and subaggregae level as well as experenaon wh an eprcal daase fro a European supersore. Soe eprcal peces of wor dscussed n secon confr such a saeen and provde suppor

169 B.Rosa-Tabar, 3, Chaper 4 68 for he frequency wh whch ARIMA(,,) and/or ARIMA(,,) processes are encounered n real world applcaons. In addon, SES s a os coonly eployed forecasng procedure n ndusry and s applcaon ples a non-saonary behavour (SES s opal for an ARIMA(,,) process). In suary, he proble seng consdered s a very realsc one. Analycal, sulaon and eprcal developens are based on he consderaon of he varance of forecas error for TD and BU approaches and coparsons are underaen a boh subaggregae and aggregae level. The condons under whch one approach ouperfors he oher are denfed. I s found ha when he subaggregae es follow an ARIMA(,,) process, hen BU ouperfors TD o provde he subaggregae forecass. However, o forecas he aggregae deand, he superory of BU and TD approaches depends on he auocorrelaon and crosscorrelaon values. For he less posve and negave cross-correlaon values, BU perfors beer ha or equally o TD. Addonally, when he cross-correlaon aes hgh posve values, TD ay ouperfor BU. TD wors beer f he cross-correlaon s hghly posve assocaed wh cobnaon of hgh auocorrelaon vs. low auocorrelaon subaggregae es. In addon, s shown ha for all dencal ovng average process paraeer he perforance of BU and TD s equal n forecasng aggregae deand. Ths s rue as well when he soohng consan used for all he subaggregae es and he aggregae level s se o be dencal (= TD ). I s shown ha, when all subaggregae es follow an ARIMA(,,) process wh dencal ovng average and auoregressve paraeers, hen he BU and TD approaches perfors equally o forecas aggregae deand. However, when he process paraeers are no dencal, he resuls are dfferen and depend on he auocorrelaon and cross-correlaon values. The sulaon resuls show ha for negave cross-correlaon, BU approach provdes ore accurae resuls han TD. However, by ncreasng he cross-correlaon values, he perforance of BU decreases and hose of TD ncreases. TD approach ay provde ore accurae forecass ha BU for hgh posve cross-correlaon. TD s always preferable for hgh posve cross-correlaon assocaed wh hgh vs. low auocorrelaon values. When he coparson s underaen a subaggregae level, f here s a leas one subaggregae e n he faly wh hgh posve auocorrelaon, hen BU ouperfors TD.

170 B.Rosa-Tabar, 3, Chaper 4 69 However, when hre s no subaggregae e n he faly wh hgh posve auocorrelaon, he dfference beween BU and TD s nsgnfcan. The resuls of coparson a subaggregae and aggregae level for ARIMA(,,) and ARIMA(,,) processes are slghly dfferen. Ths could be arbued o he naure of he subaggregae process. For an ARIMA(,,) process, he auocorrelaon s always posve, oreover for os ovng average paraeers s hghly posve. However, for an ARIMA(,,) process, he auocorrelaon spans beween - and +, addonally s hghly posve only for a sall range of process paraeers.

171 B.Rosa-Tabar, 3, Chaper 5 7 Chaper 5 Conclusons and Fuure Research In hs chaper, he an conrbuons and conclusons of hs PhD hess are gven n a concse for. Addonally, he laons of he wor are denfed and furher research avenues are suggesed. Ths chaper s dvded no four secons. Frs, he an conrbuons fro hs PhD hess are presened. Second, he an conclusons resuled for each aggregaon approach consdered are suarzed. Thrd, he anageral nsghs arsng fro hs research are dscussed. Fnally, he laons and soe areas of fuure research are consdered. The overall goal of hs research projec s o analyse he pac of aggregaon on deand forecasng. In oher words, hs research dscusses wheher s approprae o use dsaggregae daa o generae a forecas or wheher one should aggregae daa frs and hen provde a forecas. In order o address he above ssues and ee he objecves dscussed n chaper, he followng quesons have been answered:. Under whch condons are he forecass resuled fro he eporally aggregae daa preferred over hose resuled fro he dsaggregae daa?. Is here any opal aggregaon level for whch he aggregaon approach leads o he nu varance of he forecas error? 3. Under whch condons does he BU ouperfors he TD and vce versa? 4. Wha s he pac of he conrol and he process paraeers on he superory of each approach n boh eporal and cross-seconal aggregaon? In hs PhD research, all of he above quesons have been answered and he conrbuons of hs hess are suarzed n he followng secon.

172 B.Rosa-Tabar, 3, Chaper 5 7. Conrbuons of he Thess 5.. Teporal aggregaon The conrbuons of hs PhD research concernng eporal aggregaon are as follows: The superory condons of he aggregaon and he non-aggregaon approaches are denfed. The cu-off pon values are deerned for gven values of he aggregaon level and he soohng consan assocaed wh he orgnal deand seres. Ths resuls n soe heorecal rules showng he perforance of each approach a he dsaggregae and aggregae level of coparson. The perforance of he aggregaon approach s generally found o prove as he aggregaon level ncreases. The rae of proveen hough, s lower for he ARIMA(,,) and he ARIMA(,,) processes copared o he ARIMA(,,) process. In all processes, he opal aggregaon level s he hghes one n any gven aggregaon level range. The perforance of he aggregaon approach proves as he soohng consan value eployed a he aggregae seres reduces. Our analycal resuls show ha as he level of aggregaon ncreases, he auo-correlaon of he seres reduces necessang he eployen of low soohng consan values. In general, s found ha for hgh levels of posve auocorrelaon n he orgnal seres, he aggregaon approach ay be ouperfored by he non-aggregaon one: o when coparng a he dsaggregae level and where he auocorrelaon s exreely posve, (.e. hgh posve values of n he ARIMA(,,) process or hgh negave values of and hgh posve values of n he ARIMA(,,) process), no level of aggregaon proves he forecas accuracy. Consequenly, he non-aggregaon approach always provdes ore accurae forecass. Ths s an nuve fndng snce a any e perod he os recen

173 B.Rosa-Tabar, 3, Chaper 5 7 deand nforaon s precous. In such a case he dsaggregae approach wors beer as fully explos such recen nforaon. o However, when coparson s underaen a he aggregae level, even for exree posve values of he auocorrelaon, he aggregaon approach ay ouperfor he non-aggregaon one dependng on he aggregaon level. For lower values of he aggregaon level, he non-aggregaon approach wors beer. Neverheless, by ncreasng he aggregaon level, he aggregaon approach ouperfors he non-aggregaon one. Ths s because he coparson s underaen a he aggregae level where he cuulave sep ahead forecas s requred. As he aggregaon level and consequenly he forecas horzon ncreases, he forecas accuracy resulng fro he non-aggregaon approach deeroraes and yelds o a superory n favour of he aggregaon approach. For low posve or negave auocorrelaon values, he aggregaon approach s preferred regardless of he coparson level. When he auocorrelaon s negave or less posve hen he recen deand nforaon s no ha crucal, and hen a ore long er vew on deand s preferred. Ths can be acheved as dscussed above by selecng hgh aggregaon levels and he low soohng consans. Followng fro he above dscusson, our analyss suggess ha here are shades of aggregaon (a one exree no daa aggregaon) and shades of responsveness of he forecas paraeers (, ). Our fndngs sugges ha he donan soluons are eher pure whe (dsaggregae daa and responsve paraeers) or pure blac (aggregae daa and sable forecasng algorhs wh low ). Ths s, up o a ceran exen, an expeced oucoe gven he hypoheszed saonary bu: ) s no obvous and o he bes of our nowledge has never been shown before; ) sheds lgh o he general rade-off beween sable forecas paraeers (low soohng consan values) ha fler nose raher effecvely bu fal o reac o changes n deand qucly and responsve forecas paraeers (relavely hgher soohng consan values) ha however are nose sensve.

174 B.Rosa-Tabar, 3, Chaper Cross seconal aggregaon The an conrbuons regardng cross-seconal aggregaon can be suarzed as follows: When he process paraeers for all subaggregae es are dencal, here s no sgnfcan dfference beween TD and BU approaches n forecasng he aggregae level as long as he opal soohng consan s used for boh approaches. Moreover, he TD and BU approaches perfor equally when he soohng consans used for all he subaggregae es and he aggregae deand are se dencal. When he subaggregae es are hghly auo-correlaed, he BU and TD approaches perfor equally regardless of he cross-correlaon values. TD perfors beer han BU n provdng aggregae forecass when he crosscorrelaons beween subaggregae es are (hghly), he auocorrelaon of one e s posve whereas he oher one s negave. BU ay ouperfor TD when consderng aggregae forecass when he subaggregae es follow dfferen paerns of flucuaon (negave cross-correlaon). The TD appears no o be very accurae when he subaggregae es conss of dfferen paerns. BU ouperfors TD n provdng subaggregae forecass, when he auocorrelaon of a leas one e n he faly s posve and he soohng consan s se o s opal value for boh approaches, regardless of he cross-correlaon, he dsaggregaon weghs, and he values of he process paraeers. The degree of superory of he BU approach for he non-saonary case s uch hgher copared o he saonary one when coparng a subaggregae level. I s found ha for he negave or he less posve auocorrelaon, boh BU and TD approaches perfor alos equally n forecasng subaggregae deand when he opal soohng consans are used. The perforance of BU s generally proved as he cross-correlaon decreases, ovng fro posve oward negave values. Whereas, he perforance of TD deeroraes as he cross-correlaon decreases. For hghly negave cross-correlaon

175 B.Rosa-Tabar, 3, Chaper 5 74 values BU s always preferred. Ths s generally rue for he coparson a he aggregae and he subaggregae levels. The benefs acheved by BU and TD approaches for he non-saonary deand process s hgher han hose assocaed wh he saonary processes n ers of he forecas accuracy.. Manageral plcaons 5.. Teporal aggregaon Our dscussons wh praconers have revealed a sconcepon ha eporal aggregaon reduces varably, soehng ha s clearly no he case. Alhough s rue ha he non-overlappng eporal aggregaon approach reduces he coeffcen of varaon leadng o lower uncerany. Praconers have also expressed concerns wh regards o he nuvely appealng loss of nforaon assocaed wh eporal aggregaon. However, hs concern s condoned o shor deand hsores. Should long deand seres be avalable he loss of nforaon resulng fro aggregaon s ouweghed by he benefs of uncerany reducon. When applyng eporal aggregaon, praconers should always op for he hghes possble aggregaon level. However, s poran o noe ha consderaon of hgh aggregaon levels s subjec o daa avalably. Alhough, hs progressvely becoes less of an ssue n odern busness sengs. Clearly, aggregaon ay no consue a vable opon when shor deand hsores are avalable. Treendous recen developens n ers of copung sorage capacy faclae he accuulaon of very lenghy seres. Alhough, we have coe across suaons/copanes where only a few years daa s sored. In such cases aggregaon ay no be furher consdered. Long hsorcal daa seres do no only allow for he ore accurae esaon of he seres coponens bu also per he applcaon of eporal aggregaon approaches.

176 B.Rosa-Tabar, 3, Chaper 5 75 The perforance of aggregaon proves as he soohng consan value eployed a he aggregae seres reduces. Ths s an poran fndng fro a praconer s perspecve snce anagers ay se such values convenenly low o axze he benefs derved fro he aggregaon approach. The soohng consan value afer aggregaon should be generally se saller han he soohng consan before aggregaon and specfc rules and cu-off pons have been offered for ang such decsons. For hgh levels of posve auocorrelaon n he orgnal seres, he non-aggregaon approach ouperfors he aggregaon one n dsaggregae level forecasng. Ths s an nuve fndng snce a any e he os recen deand nforaon s so precous n ha case ha he dsaggregae approach wors beer as fully explos such recen nforaon. However, on he conrary, for he low posve or negave auocorrelaon when he recen deand nforaon s no ha crucal hen a ore long er vew on deand s preferable, whch can be obaned as dscussed above by selecng hgh aggregaon levels and low soohng consans. Ths s also an poran eprcal nsgh snce anagers ay now wha o expec (n ers of any poenal gans) based on he auocorrelaon levels presen n her seres. When a long range forecas s requred, he forecaser should apply he aggregaon approach o provde he forecas. Ths s because a ore long er vew on deand s preferable and he aggregaon approach ulzes beer hs nforaon. As a general rule, he farher no he fuure we loo, he ore clouded our vson becoes and he non-aggregaon approach wll be less accurae han aggregaon one. 5.. Cross seconal aggregaon In pracce, here are any seres ha are herarchcally organzed and can be aggregaed a several dfferen levels based on producs, geography or soe oher feaures. TD and BU forecasng approaches are exreely useful owards provng he accuracy of forecass on dfferen levels. For nsance, n S&OP (Sales and Operaons Plannng) process, each deparen requres dfferen levels of deand forecass ha can be acheved by applyng TD and BU approaches.

177 B.Rosa-Tabar, 3, Chaper 5 76 When he praconers requre dfferen herarchcal level of forecass, choosng beween BU or TD approaches depends on he auocorrelaon, he cross-correlaon and he coparson level. When he praconers requre deand forecass a he SKU level, he auocorrelaon values should be consdered. If here s a leas one seres n he faly wh hgh posve auocorrelaon hen would always be preferable o use he BU approach. In addon, he BU approach perfors beer when he seres are assocaed wh dfferen paerns of flucuaon (negave cross-correlaon). However, when he auocorrelaon s less posve or negave, here s no dfference beween usng BU and TD. When he aggregae deand forecas s requred, he values of cross-correlaon and auocorrelaon should be calculaed. If he subaggregae es follow he sae paerns of flucuaon (hgh posve crosscorrelaon) assocaed wh dfferen auocorrelaon values (hgh vs. low), hen TD would be appled. However, when he ndvdual es are assocaed wh dfferen paerns of evoluon BU s preferable. Addonally, f he auocorrelaon values are negave for all subaggregae es, hen he BU approach should be used. If he auocorrelaon s posve for all subaggregae es, hen boh BU and TD perfor equally. In addon, f one uses he sae value of soohng consans for boh BU and TD, hen boh approaches perfor equally as well. 3. Laons and fuure research In hs secon, suggesons for fuure research are dscussed fro heorecal, sulaon and eprcal perspecves. Throughou hs research soe assupons are consdered ha can be relaxed n fuure sudes.

178 B.Rosa-Tabar, 3, Chaper 5 77 In Chaper 3, he effec of eporal aggregaon on deand forecasng was dscussed. In hs research he case of non-overlappng eporal aggregaon s consdered when he dsaggregae daa follow a saonary deand processes and when he Sngle Exponenal Soohng forecasng ehod s used. Gven he curren under-consderaon of eporal aggregaon n nvenory forecasng sofware soluons and gven s value as a prosng uncerany reducon e seres ransforaon approach ha hs PhD has revealed, research no any of he followng areas would appear o be ered: Expanson of he analycal evaluaon dscussed n hs wor on hgher order saonary processes and ore poranly on non-saonary processes s a very poran ssue boh fro an acadec and praconer perspecve. In hs sudy, he Auoregressve Movng Average, ARMA ype processes were assued for he deand processes. Ths s a relevan assupon for fas ovng es. The analycal and eprcal consderaon of Ineger ARMA (INARMA) processes offers a grea opporuny for advanceens n he area of aggregaon. Such processes bear a consderable relevance o neren deands where he benefs of aggregaon ay be even hgher due o he reducon of zero observaons. In hs wor, he effec of non-overlappng eporal aggregaon on deand forecasng s analysed. Anoher poran exenson can be he consderaon of he overlappng eporal aggregaon. In hs research, Sngle Exponenal Soohng s appled as a forecasng ehod; one naural exenson s he consderaon of oher popular forecasng ehods. Ths sudy s focused on forecasng and no nvenory conrol. The exenson of he wor descrbed n hs research o cover nvenory/plcaon ercs would allow a lnage beween forecasng and soc conrol. Research on ore exensve daases (as well as analyss of eprcal forecasng perforance on easures oher han he MSE) should allow a beer undersandng of he dffcules and benefs assocaed wh aggregaon.

179 B.Rosa-Tabar, 3, Chaper 5 78 In hs research, he effecveness of BU and TD approaches s evaluaed o forecas he subaggregae and aggregae level. The case of saonary and non-saonary deand processes n conjuncon wh he SES forecas ehod s consdered. Naurally, here are any oher avenues for furher research and he followng possbles should be very poran n ers of advancng he curren sae of nowledge n he area of cross-seconal aggregaon. In hs research deand s assued o be srucured based on ARIMA ype processes. The evaluaon of he BU and TD approaches when he subaggregae es follow an Ineger ARMA (INARMA) processes s an neresng subjec for fuure wor. The nerface beween (and he poenal of cobnng) eporal and cross-seconal aggregaon has receved nal aenon boh n acadea and ndusry and s an ssue ha we wll explore n he nex seps of our research. Expanson of he wor dscussed n hs research for oher popular forecasng ehods such as opal forecas ehod, rend exponenal soohng and daped rend exponenal soohng odels s an poran ssue. Exendng he analyss n hs research o consder n levels herarchcal srucures would be an neresng developen. Fnally, consderaon of ore exensve eprcal daases ha cover he whole range of he process paraeers should allow a beer undersandng of he benefs of each approach.

180 B.Rosa-Tabar, 3, References 79 References Acar, Y., Gardner, E.S.,. Forecasng ehod selecon n a global supply chan. Inernaonal Journal of Forecasng 8, Agner, D.J., Goldfeld, S.M., 974. Esaon and Predcon fro Aggregae Daa when Aggregaes are Measured More Accuraely han Ther Coponens. Econoerca 4, Aeya, T., Wu, R.Y., 97. The effec of aggregaon on predcon n he auoregressve odel. Journal of he Aercan Sascal Assocaon 67, Anderson, O.D., 975. On a lea assocaed wh Box, Jenns and Granger. Journal of Econoercs 3, Ahanasopoulos, G., Hyndan, R.J., Song, H., Wu, D.C.,. The ours forecasng copeon. Inernaonal Journal of Forecasng 7, Baba, M.Z., Al, M.M., Nolopoulos, K.,. Ipac of eporal aggregaon on soc conrol perforance of neren deand esaors: Eprcal analyss. OMEGA 4, Barnea, A., Laonsho, J., 98. An Analyss of The Usefulness of Dsaggregaed Accounng Daa For Forecass of Corporae Perforance. Decson Scences, 7-6. Barezzagh, E., Vergan, R., Zoer, G., 999. Measurng he pac of asyerc deand dsrbuons on nvenores. Inernaonal Journal of Producon Econocs 6 6, Bonoo, C., 3. Forecasng fro he Cener of he Supply Chan. Journal of Busness Forecasng, 3-9. Box, G., Jenns, G.M., Rensel, G., 8. Te Seres Analyss: Forecasng & Conrol. John Wley and sons. Bozos, K., Nolopoulos, K.,. Forecasng he value effec of seasoned equy offerng announceens. European Journal of Operaonal Research 4, Brannas, K., Hellsro, J., Nordsro, J.,. A new approach o odellng and forecasng onhly gues nghs n hoels. Inernaonal Journal of Forecasng 8, 9-3. Brewer, K.R.W., 973. Soe consequences of eporal aggregaon and syseac saplng for ARMA and ARMAX odels. Journal of Econoercs,

181 B.Rosa-Tabar, 3, References 8 Chen, A., Blue, J.,. Perforance analyss of deand plannng approaches for aggregang, forecasng and dsaggregang nerrelaed deands. Inernaonal Journal of Producon Econocs 8, Chen, A., Hsu, C.H., Blue, J., 7. Deand plannng approaches o aggregang and forecasng nerrelaed deands for safey soc and bacup capacy plannng. Inernaonal Journal of Producon Research 45, Chen, F., Ryan, J.K., Sch-Lev, D.,. The pac of exponenal soohng forecass on he bullwhp effec. Naval Research Logscs (NRL) 47, Chen, H., Boylan, J.E., 7. Use of Indvdual and Group Seasonal Indces n Subaggregae Deand Forecasng. The Journal of he Operaonal Research Socey 58, Chen, H., Boylan, J.E., 8. Eprcal evdence on ndvdual, group and shrnage seasonal ndces. Inernaonal Journal of Forecasng 4, Chen, C.-F., Chen, Y.-J., Peng, J.-T., 8. Deand forecas of seconducor producs based on echnology dffuson, Sulaon Conference, 8. WSC 8. Wner, pp Chopra, S., Mendl, P.,. Supply Chan Manageen. Prence-Hall, New Jersey. Croson, J.D., 97. Forecasng and Soc Conrol for Ineren Deands. Operaonal Research Quarerly (97-977) 3, Dangerfeld, B., Morrs, J., 988. An eprcal evaluaon of op-down and boo-up forecasng sraeges., Proceedngs of he 988 Meeng of Wesern Decson Scences Insue, pp Dangerfeld, B.J., Morrs, J.S., 99. Top-down or boo-up: Aggregae versus dsaggregae exrapolaons. Inernaonal Journal of Forecasng 8, Depe, M.G., Hanssens, D.M.,. Te-seres odels n areng:: Pas, presen and fuure. Inernaonal Journal of Research n Mareng 7, Dros, F.C., Njan, T.E., 993. Teporal aggregaon of GARCH processes. Econoerca 6, Duc, T.T.H., Luong, H.T., K, Y.-D., 8. A easure of bullwhp effec n supply chans wh a xed auoregressve-ovng average deand process. European Journal of Operaonal Research 87, Dunn, D.M., Wllas, W.H., Dechane, T.L., 976. Aggregae versus subaggregae odels n local area forecasng. Journal of he Aercan Sascal Assocaon 7, Erp, N., Hausan, W.H., Nahas, S., 99. Opal Cenralzed Orderng Polces n Mul-Echelon Invenory Syses wh Correlaed Deands. Manageen Scence 36,

182 B.Rosa-Tabar, 3, References 8 Fledner, G., 999. An Invesgaon of Aggregae Varable Te Seres Forecas Sraeges wh Specfc Subaggregae Te Seres Sascal Correlaon. Copuers and Operaons Research 6, Gardner, E.S., Jr., 99. Evaluang Forecas Perforance n an Invenory Conrol Syse. Manageen Scence 36, Gardner, E.S., Jr., 6. Exponenal soohng: The sae of he ar-par II. Inernaonal Journal of Forecasng, Gardner, E.S., Jr., McKenze, E., 985. Forecasng Trends n Te Seres. Manageen Scence 3, Gaur, V., Glon, A., Seshadr, S., 5. Inforaon Sharng n a Supply Chan Under ARMA Deand. Manageen Scence 5, Gordon, T., Dangerfeld, B., Morrs, J., 997. Top-down or boo-up: Whch s he bes approach o forecasng? The Journal of Busness Forecasng 6, 3-6. Granger, C.W.J., Morrs, M.J., 976. Te seres odelng and nerpreaon. Journal of he Royal Sascal Socey. Seres A, Graves, S.C., Wlles, S.P.,. Opzng Sraegc Safey Soc Placeen n Supply Chans. Manufacurng & Servce Operaons Manageen, Graves, S.C., Wlles, S.P., 8. Sraegc Invenory Placeen n Supply Chans: Nonsaonary Deand. Manufacurng & Servce Operaons Manageen, Gross, C.W., Sohl, J.E., 99. Dsaggregaon ehods o expede produc lne forecasng. Journal of Forecasng 9, Grunfeld, Y., Grlches, Z., 96. Is aggregaon necessarly bad? Revew of Econocs and Sascs 4, -3. Harvey, A.C., 993. Te Seres Models. MIT Press. Hyndan, R.J., Ahed, R.A., Ahanasopoulos, G., Shang, H.L.,. Opal cobnaon forecass for herarchcal e seres. Copuaonal Sascs & Daa Analyss 55, Hyndan, R.J., Ahanasopoulos, G., 3. Forecasng: prncples and pracce. OTexs, Ausrala. Hyndan, R.J., Koehler, A.B., 6. Anoher loo a easures of forecas accuracy. Inernaonal Journal of Forecasng, Klejnen, J.P.C., Groenendaal, W., 99. Sulaon: a sascal perspecve. John Wley & Sons, Inc, ew Yor, NY, USA.

183 B.Rosa-Tabar, 3, References 8 Lapde, L., 4. Sales and Operaons Plannng Par I: The Process. The Journal of Busness Forecasng, 7-9. Lapde, L., 6. Top-Down & Boo-Up Forecasng In S&OP. The Journal of Busness Forecasng 5, 4-6. Lee, H.L., P., P., Whang, S., 997a. Bullwhp effec n a supply chan. Sloan Manageen Revew 38, 93-. Lee, H.L., Padanabhan, V., Whang, S., 997b. Inforaon dsoron n a supply chan: The bullwhp effec. Manageen Scence 43, Lee, H.L., So, K.C., Tang, C.S.,. The Value of Inforaon Sharng n a Two-Level Supply Chan. Manageen Scence 46, Luz, K.H., Pedro, A.M., Pedro, L.V.P., 99. The Effec of Overlappng Aggregaon on Te Seres Models: An Applcaon o he Uneployen Rae n Brazl. Brazlan Revew of Econoercs, 3-4. Luna, I., Balln, R.,. Top-down sraeges based on adapve fuzzy rule-based syses for daly e seres forecasng. Inernaonal Journal of Forecasng 7, Lüepohl, H., 984. Forecasng Coneporaneously Aggregaed Vecor ARMA Processes. Journal of Busness & Econoc Sascs, -4. Mahajan, S., Desa, O.,. Value of Inforaon n a Seral Supply Chan under a Nonsaonary Deand Process, Worng Paper Seres. Indan Insue of Manageen (IIMB), Bangalore Mardas, S., Andersen, A., Carbone, R., Fldes, R., Hbon, M., Lewandows, R., Newon, J., Parzen, E., Wnler, R., 98. The accuracy of exrapolaon (e seres) ehods: Resuls of a forecasng copeon. Journal of Forecasng, -53. Mardas, S., Hbon, M.,. The M3-Copeon: resuls, conclusons and plcaons. Inernaonal Journal of Forecasng 6, Marel, A., Daby, M., Bocor, F., 995. Mulple es procureen under sochasc nonsaonary deands. European Journal of Operaonal Research 87, Mohaadpour, M., Boylan, J., Syneos, A.,. The Applcaon of Produc-Group Seasonal Indexes o Indvdual Producs. Foresgh: The Inernaonal Journal of Appled Forecasng, 8-4. Mohaadpour, M., Boylan, J.E.,. Forecas horzon aggregaon n neger auoregressve ovng average (INARMA) odels. OMEGA 4, Nahas, S., 993. Producon and Operaons Analyss. Irwn Publshers, Boson, MA. Narashan, S.L., McLeavey, D.W., Bllngon, P., Bllngon, P.J., 985. Producon Plannng and Invenory Conrol. Allyn and Bacon Inc., 67., Boson:.

184 B.Rosa-Tabar, 3, References 83 Nolopoulos, K., Syneos, A.A., Boylan, J., Peropoulos, F., Assaopoulos, V.,. An aggregae-dsaggregae neren deand approach (ADIDA) o forecasng : an eprcal proposon and analyss. Journal of he Operaonal Research Socey 6, Orcu, G.H., Was, H.W., Edwards, J.B., 968. Daa aggregaon and nforaon loss. The Aercan Econoc Revew 58. Quenoulle, M., 958. Dscree auoregressve schees wh varyng e-nervals. Mera, -7. Rosa-Tabar, B., Baba, M.Z., Syneos, A., Ducq, Y., 3a. Deand forecasng by eporal aggregaon. Naval Research Logscs (NRL) 6, Rosa-Tabar, B., Baba, M.Z., Syneos, A., Ducq, Y., 3b. Forecasng aggregae ARMA(,) Deands: heorecal analyss of op-down versus boo-up, he Inernaonal Conference on Indusral Engneerng and Syses Manageen, Raba, Moroco. Rosa-Tabar, B., Baba, M.Z., Syneos, A., Ducq, Y., 3c. The Ipac of Teporal Aggregaon on Deand Forecasng of ARMA(, ) Process: Theorecal Analyss, n: Bahadze, N., Chernyshov, K., Dolgu, A., Loosy, V. (Eds.), Manufacurng Modellng, Manageen, and Conrol. Elsever, San Peersburg Sae Unversy and San Peersburg ITMO Unversy, S. Peersburg, Russa, pp Rosa-Tabar, B., Baba, M.Z., Syneos, A., Ducq, Y., 3d. Non-saonary Deand Forecasng by Cross-Seconal Aggregaon, n: Under Revew, I. (Ed.). Saraslan, N.,. The Effec of Teporal Aggregaon on Unvarae Te Seres Analyss, Deparen of Sascs. Mddle Eas Techncal Unversy, p. 5. Sbrana, G., Slvesrn, A., 3. Forecasng aggregae deand: Analycal coparson of op-down and boo-up approaches n a ulvarae exponenal soohng fraewor. Inernaonal Journal of Producon Econocs. Schluer, C., Trede, M.,. Esang connuous-e ncoe odels, Worng Paper 8/. Cenre for Quanave Econocs, Unversy of Münser,Gerany. Schwarzopf, A.B., Tersne, R.J., Morrs, J.S., 988. Top-down versus boo-up forecasng sraeges. Inernaonal Journal of Producon Research 6, Shlfer, E., Wolff, R.W., 979. Aggregaon and Proraon n Forecasng. Manageen Scence (pre-986) 5, 594. Slvesrn, A., Veredas, D., 8. Teporal aggregaon of unvarae and ulvarae e seres odels: A survey, Te d dscussone (worng papers). Ban of Ialy, Econoc Research Deparen. Souza, L.R., Sh, J., 4. Effecs of eporal aggregaon on esaes and forecass of fraconally negraed processes: a Mone-Carlo sudy. Inernaonal Journal of Forecasng,

185 B.Rosa-Tabar, 3, References 84 Sphouras Georgos P., Foos Peropoulos, Vasslos Assaopoulos, Mohaed Zed Baba, Nolopoulos, K.,. Iprovng he Perforance of Popular Supply Chan Forecasng Technques Supply Chan Foru: An Inernaonal Journal, 6-5. Spyros G. Mardas, Seven C. Wheelwrgh, Hyndan, R.J., 998. Forecasng: Mehods and Applcaons. John Wley & Sons. Sra, D.O., We, W.W.S., 986. Teporal aggregaon n he ARIMA process. Journal of Te Seres Analyss 7, Syneos, A.A., Boylan, J.E., 5. The accuracy of neren deand esaes. Inernaonal Journal of Forecasng, Syneos., A.A.,. Forecasng of Ineren Deand. Busness School,Bucnghashre Chlerns Unversy College,Brunel Unversy, London,UK. Taylor, J.W., 3. Exponenal soohng wh a daped ulplcave rend. Inernaonal Journal of Forecasng 9, Teles, P., We, W.W.S., Crao, N., 999. The use of aggregae seres n esng for long eory.. Bullen of he Inernaonal Sascal Insue 3, Thel, H., 954. Lnear Aggregaon of Econoc Relaons. Norh-Holland Publshng Copany, Aserda, Neherlands. Tao, G.C., 97. Asypoc behavour of eporal aggregaes of e seres. Boera 59, Tunc, H., Klc, O.A., Tar, S.A., Esoglu, B.,. The cos of usng saonary nvenory polces when deand s non-saonary. Oega 39, Vswanahan, S., Wdara, H., Pplan, R., 8. Forecasng aggregae e seres wh neren subaggregae coponens: op-down versus boo-up forecasng. IMA Journal of Manageen Maheacs 9, Weaherford, L.R., Kes, S.E., Sco, D.A.,. Forecasng for hoel revenue anageen: Tesng aggregaon agans dsaggregaon. The Cornell Hoel and Resauran Adnsraon Quarerly 4, We, W.W.S., 979. Soe Consequences of Teporal Aggregaon n Seasonal Te Seres Models, In Zellner A. (ed.) Seasonal Analyss of Econoc Te Seres. US Deparen of Coerce, Bureau of he Census, Washngon, DC. We, W.W.S., 6. Te Seres Analyss: Unvarae and Mulvarae Mehods. Addson- Wesley Publshng Copany, Boson, USA. Wess, A.A., 984. Syseac saplng and eporal aggregaon n e seres odels. Journal of Econoercs 6, 7-8.

186 B.Rosa-Tabar, 3, References 85 Weerlov, U., Whybar, D.C., 984. Lo-szng under uncerany n a rollng schedule envronen. Inernaonal Journal of Producon Research, Wdara, H., Vswanahan, S., Pplan, R., 7. On he effecveness of op-down sraegy for forecasng auoregressve deands. Naval Research Logscs (NRL) 54, Wdara, H., Vswanahan, S., Pplan, R., 8. Forecasng e-level deands: an analycal evaluaon of op down versus boo up forecasng n a producon-plannng fraewor. IMA Journal of Manageen Maheacs 9, 7-8. Wdara, H., Vswanahan, S., Pplan, R., 9. Forecasng aggregae deand: An analycal evaluaon of op-down versus boo-up forecasng n a producon plannng fraewor. Inernaonal Journal of Producon Econocs 8, Wllean, T.R., Sar, C.N., Shocor, J.H., DeSauels, P.A., 994. Forecasng neren deand n anufacurng: a coparave evaluaon of Croson's ehod. Inernaonal Journal of Forecasng, Yehuda, G., Zv, G., 96. Is aggregaon necessarly bad? Revew of Econocs and Sascs 4, -3. Zaffaron, P., 7. Coneporaneous aggregaon of GARCH processes. Journal of Te Seres Analyss 8, Zhang, X., 4. Techncal Noe: Evoluon of ARMA Deand n Supply Chans. Manufacurng & Servce Operaons Manageen 6, Zoer, G., Kalchschd, M., 7. A odel for selecng he approprae level of aggregaon n forecasng processes. Inernaonal Journal of Producon Econocs 8, Zoer, G., Kalchschd, M., Canao, F., 5. The pac of aggregaon level on forecasng perforance. Inernaonal Journal of Producon Econocs 93-94,

187 B.Rosa-Tabar, 3, Résué 86 Appendces Appendx A: The relaonshp of auocovarance beween nonaggregae and aggregae deand I has been shown ha he auocovarance funcon on nonaggregae and aggregae seres are relaed as follows: B B B (A-) Ths for can be ransfored o a arx for as follow:.... A (A-) where

188 B.Rosa-Tabar, 3, Appendces 87 C C C C C C A and zeros of vecor n n : é B B polynoal n he B of whch s he coffcen C of vecor C : Because of hs fac ha for all, n he arx A n (A-), he frs - colun correspondng o,.., can be deleed by addng he o colun correspondng o,, respecvely. Therefore, he followng s gven:..... A (A-3) Where A s a odfed arx A, afer deleon and addng requred coluns.(refer o We (6)). dfferen values of he aggregaon level s used o deerne he general relaonshps beween he auocovarance of non-aggregaon and aggregaon approach for

189 B.Rosa-Tabar, 3, Appendces 88 he deand process under consderaon n hs sudy ncludng : ARIMA(,,), ARIMA(,,) and ARIMA(,,). The calculaon s begun by subsung =3. By subsung =3 no (A-), he followng s gven: B B B B B B (A-4) Now by consderng (A-) for he ARIMA(,,) process and subsung =3 n ha (A-5) s obaned: 3.. A (A-5) Where A Then he arx A can be calculaed by addng and reovng correspondng colun: A Therefore, he followng s gven:

190 B.Rosa-Tabar, 3, Appendces (A-6) By subsung ( 3-) no (A-6), (A-7) s obaned: (A-7) By subsung =4 no (A-) he followng s gven: B B B B B B B (A-8) Now by consderng (A-) for he ARIMA(,,) process and subsung =4 n ha (A-9) s obaned:

191 B.Rosa-Tabar, 3, Appendces A (A-9) Where A Then calculae he arx A 4 s calculaed as follwos: A So by subsung A 4 no (A-9), he followng s obaned :

192 B.Rosa-Tabar, 3, Appendces (A-) by subsung ( 3-) no (A-), he followng equaons are obaned: (A-) By followng he sae procedure, he relaonshp beween he auocovarance funcon of non-aggregaon and aggregaon process when =5 : (A-)

193 B.Rosa-Tabar, 3, Appendces 9 By connung he calculaons, he general fors can be represened as follows: 3 (A-3) 3 (A-4) And fnally we have for > we have: (A-5) Now by consderng ( 3-) he followng s gven: 3 (A-6) Fro (A-5) he followng rao can be obaned: 3 (A-7) Now by coparng (A-6) wh (A-7) he relaonshp beween he auoregressve paraeer before and afer aggregaon s gven:

194 B.Rosa-Tabar, 3, Appendces 93 (A-8) When he deand process follows an ARIMA(,,) process, he relaonshp beween he auocovarance funcon of aggregaon and non-aggregaon deand can be obaned as follows:. (A-9) By followng he sae procedure he relaonshp beween he auocovarance funcon of he aggregaon and he non-aggregaon deand of an ARIMA(,,) process can be obaned as followng:, (A-), (A-), (A-) and for all >, we have: ) ( ) ( ) ( (A-3)

195 B.Rosa-Tabar, 3, Appendces 94 Appendx B: Covarance beween he dsaggregae deand and aggregae forecas for ARIMA(,,) process The covarance beween he dsaggregae deand and he forecas of aggregae deand can be calculaed as follows:... ), ( ), ( ), ( ), (, T T T T T D d Cov D d Cov D d Cov D d Cov F d Cov (B-) By subsung ( 3-) no (B-) we have: T d d d d Cov d d d d Cov d d d d Cov F d Cov )..., ( )..., (..., (, (B-) By subsung ( 3-) no (B-) and soe splfcaons, we have , d F T Cov (B-3) By dong soe sple calculaon we ge, d F T Cov (B-4)

196 B.Rosa-Tabar, 3, Appendces 95 Appendx C: Varance of he aggregae forecas for he ARIMA(,,) process The varance of he aggregae forecas can also be deerned le ( 3-6) bu wh dfferen paraeers. In order o oban he value of he varance of he forecas error, we need o calculae he covarance beween he aggregae deand and s forecas, so we begn by dervng he covarance beween he aggregae forecas and he deand n perod T: Cov D, F Cov( D, D ) Cov( D, T T Cov ( D T, D T T ) T Cov( D, D ) T T T Cov( D T D, D T T ) )... (C-) The varance of forecas afer aggregaon can be derves as: Var ( F T ) Var DT FT Var DT Var FT Cov D, F T T (C-) By subsung ( 3-) no (C-) we ge Cov D T, F T (C-3) Then, By usng he fac ha Var F Var and fac ha D T (C-3) no (C-), we have, CovD, F CovD, F for all T F T Var for all (he properes of saonary process) and by subsung T T T T Var F T (C-4)

197 B.Rosa-Tabar, 3, Appendces 96 Appendx D: Covarance of he aggregae deand and non-aggregae forecas for ARIMA(,,) process The covarance beween he aggregae deand and he subaggregae forecas s defned as follows:,... ), ( ), ( ), (,...,, ), (,... ), (, T T d d Cov d d Cov d d Cov d d Cov d d Cov d d Cov d d Cov d d d d Cov d D Cov f D Cov (D-) By subsung ( 3-) no (D-) we ge, ,, T f d Cov f D Cov (D-) Now by subsung ( 3-) no (D-) we ge:,,...,, T f d Cov f d Cov f D Cov (D-3)

198 B.Rosa-Tabar, 3, Appendces 97 By subsung ( 3-4) no (D-) we ge Cov D T, f (D-4) Appendx E: Coeffcen of varaon before and afer aggregaon for ARIMA(,,) When he non-aggregae process follows an ARIMA(,,) process, we show ha applyng he non-overlappng eporal aggregaon reduces he coeffcen of varaon (CV). CV s an poran easure n an nvenory conex (Barezzagh e al., 999). We show below ha he CV decreases as he aggregaon levels ncreases as well. The coeffcen of varaon s defned as he rao of he sandard devaon of deand o he ean of deand, he rao of he coeffcen of varaon afer aggregaon o ha before aggregaon s: CV CV AA BA. (E-) When he non-aggregae process follows ARIMA(,,), by subsung ( 3-3) and ( 3-) no (E-) we ge and CV CV AA BA. (E-) Consderng ha and we can show ha C- s saller han and by ncreasng, he rao of CV CV decreases. AA BA

199 B.Rosa-Tabar, 3, Appendces 98 Appendx F: Proof of heore -3 By consderng ( 3-49) and BA MSE AA MSE, he quadrac funcon gven by (F-) should be negave 4 (F-) where (F-) Moreover, by nvesgang he sgn of (F-) we can oban he condons under whch BA MSE AA MSE s saller, equal and greaer han one. Now, we verfy f he quadrac funcon (F-) has real roos. To do so, we defne he dscrnan of (F-) as follows

200 B.Rosa-Tabar, 3, Appendces 99, 8 4 (F-3) Now by usng he fac ha,, and, he values of he can be obaned. If eans (F-) has no real roos and f eans (F-) has wo real roos. I can shown ha n (F-3) s always posve, herefore (F-) has wo dfferen roos called and, where, 4 (F-4)

201 B.Rosa-Tabar, 3, Appendces. 4 (F-5) I can be shown ha f, s always saller han zero and and f, s greaer han one and. I s now ha he sgn of he (F-) beween he wo roos and s oppose o he sgn of A, where A defned n (F-6) s he sgn of he coeffcen of, Oherwse s ha he sae as he sgn of A. A (F-6) Now by consderng, and A ha s posve for and negave for, he sgn of (F-) s deerned. So we have If, s always saller han zero. If hen (F-) s negave n he nerval [, ] and s posve ousde hs nerval. If, s greaer han one and we can show ha hus (F-) s posve n he nerval [, ] and s negave ousde hs nerval.

202 B.Rosa-Tabar, 3, Appendces Fro he above expressons we can see ha when, (F-) s negave, oherwse when, s posve and when, (F-) s equal o zero. Equvalenly If, he rao of MSE BA MSE AA s greaer han one and consequenly he aggregaon approach ouperfors he non-aggregaon one. If, he rao of MSE BA MSE AA s equal o one and boh approaches perfor equally. If, he rao of MSE BA MSE AA s saller han one and he non-aggregaon approach ouperfors he aggregaon approach. Appendx G: Selecon procedure for he ARIMA(,,) process Usng he fac ha,, and he value of he dscrnan and he roos and can be defned by (F-3), (F-4) and (F-5) respecvely. If here are no real roos for (F-), herefore he sgn of (F-) s equvalen o he sgn of A defned n (F-6). We can show ha when, A s always posve, consequenly (F-) s posve whch eans ha MSE MSE s saller han one. BA AA If, (F-) has wo dfferen roos and. By nvesgang he sgn of, and A, we can deerne he sgn of (F-) and consequenly he perforance superory of each sraegy. If and hen (F-) s negave n he nerval [, ] and s posve ousde hs nerval. If, can be shown ha. (F-) s posve n he nerval [, ] and s negave ousde hs nerval. If, can be shown ha hen (F-) s negave n he nerval [, ] and s posve ousde hs nerval.

203 B.Rosa-Tabar, 3, Appendces By consderng he above expressons, we have If or, hen If, MSE. BA MSE AA If, MSE. BA MSE AA If, MSE. BA MSE AA Oherwse, f hen If, MSE. BA MSE AA If, MSE. BA MSE AA If and, MSE. BA MSE AA Appendx H: Proof of heore -3, ARIMA(,,) By consderng MSE (H-) should be negave BA MSE AA and soe splfcaons, he quadrac funcon gven by ( ( ) ) (H-) where. (H-) Moreover, by nvesgang he sgn of (H-) we can oban he condons under whch MSE BA MSE AA s saller, equal and greaer han one. Now, we verfy f he quadrac funcon (H-) has real roos. To do so, we defne he dscrnan of (H-) as follows ( ( ) ) 8, (H-3)

204 B.Rosa-Tabar, 3, Appendces 3 Now we use he fac ha, and o oban he values of. If eans ha (H-) has no real roos and f eans (H-) has wo real roos. We can show ha n (H-3) s always posve, herefore (H-) has wo dfferen roos denoed by and, where s defned n (F-4) and ( ( ) ) ( ( ) ) 8. (H-4) I can be shown ha f, s always saller han zero and or and, s greaer han one and or. I s now ha he sgn of he (H-) beween he wo roos and he sgn of A, where A s he sgn of he coeffcen of s oppose o, Oherwse s ha he sae as he sgn of A. Now by consderng, and A ha s posve for and negave for, we deerne he sgn of (H-). So we have If, s always saller han zero. If hen (H-) s negave n he nerval [, ] and s posve ousde hs nerval. If, s greaer han one and we can show ha hus (H-) s posve n he nerval [, ] and s negave ousde hs nerval. Fro he above expressons we can see ha when, (H-) s negave, oherwse when, s posve and when, (H-) s equal o zero. Equvalenly If, he rao of MSE BA MSE AA s greaer han one and consequenly he aggregaon approach ouperfors non-aggregaon approach. If, he rao of MSE BA MSE AA s equal o one and boh approaches perfor equally. If, he rao of MSE BA MSE AA s saller han one and he non-aggregaon approach ouperfors he aggregaon one.

205 B.Rosa-Tabar, 3, Appendces 4 Appendx I: Selecon procedure for he ARIMA(,,) process Consderng BA MSE AA MSE s equvalen o havng he quadrac funcon (I-) negave, whch subsequenly s equvalen o 4 (I-) For he quadrac funcon gven by (I-), he value of he dscrnan and he roos and can be defned as follows:, 8 4 (I-), 4 (I-3), 4 (I-4) Where, (I-5), (I-6)

206 B.Rosa-Tabar, 3, Appendces 5 3. (I-7) We defne he coeffcen of n (D-) as follows. A (I-8) f hen he non-aggregaon approach s always provdes ore accurae forecass, oherwse If hen he aggregaon approach wors beer. If hen boh approaches are dencal. If and/or hen he non-aggregaon approach wors beer. Appendx J: Proof of heore 3-3 and heore 4-3 Case. Usng he fac ha 3, and by consderng he opal soohng * consan, 3 used o calculae MSE BA, we can show ha he dscrnan defned n (I-) s negave, so here s no real roo for (I-). Consequenly, he sgn of (I-) s he sae as he sgn of A defned n (I-), we can show ha he sgn of A s always posve, herefore (I-) s always posve and BA MSE MSE s saller han one. Hence, he nonaggregaon approach always wors beer for he whole range of β and for any value of he aggregaon level,. Case. 3. Usng he fac ha 3, and by consderng he sall value of he soohng consan before aggregaon, *. 5, s sraghforward o show ha he dscrnan defned n (I-) s posve, so (I-) has wo dfferen roos denoed by and defned n(i-3) and (I-4) respecvely. We can show ha he value of β s eher less han zero or greaer han one. Now by consderng he roos, and he sgn of A, where A s defned n (I-8), we can deerne he sgn of (I-) and consequenly show he superory of each approach. AA

207 B.Rosa-Tabar, 3, Appendces 6 If and, hen (I-) s negave n he nerval [, ] and s posve ousde hs nerval. If, we can show ha and (I-) s posve n he nerval [, ] and s negave ousde hs nerval. Now fro he above expressons we can ge he followng resuls: If β < β, hen MSE. BA MSE AA If β = β, hen MSE. BA MSE AA Oherwse, MSE. BA MSE AA Appendx K: Selecon procedure for he ARIMA(,,) process- Coparson a he aggregae level By consderng MSE s equvalen o havng he quadrac funcon (K-) BA MSE AA negave, whch subsequenly s equvalen o (K-) where

208 B.Rosa-Tabar, 3, Appendces 7 For he quadrac funcon gven by (K-), he value of he dscrnan and he roos and can be defned as follows: (K-) - (- ) (- ) - + (K-3) - (- ) - + (K-4) he coeffcen of n (C-) s defned as follows: A. (K-5) If he dscrnan, here are no real roos for (K-), herefore he sgn of (K-) s equvalen o he sgn of A. We can show ha when, A s always negave, consequenly (K-) s negave whch eans ha MSE MSE s saller han one. BA AA However, If, (K-) has wo dfferen roos and. By nvesgang he sgn of, and A, we can deerne he sgn of (K-) and consequenly he perforance superory of each approach.

209 B.Rosa-Tabar, 3, Appendces 8 If and hen (K-) s posve n he nerval [, ] and s negave ousde hs nerval. If, we can show ha. (K-) s negave n he nerval [, ] and s posve ousde hs nerval. If, we can show ha hen (K-) s posve n he nerval [, ] and s negave ousde hs nerval. By consderng he above expressons, he superory condons of each approach can be obaned by followng he selecon procedure :. The procedure d begun by calculang defned n (K-), If hen he nonaggregaon approach s always superor, oherwse he values of and defned n (K- 3) and (K-4) are calculaed.. If or, he value of s calculaed, If hen he aggregaon approach wors beer. If hen boh approaches are dencal. If hen he non-aggregaon approach wors beer. Oherwse: 3. The value of s calculaed. accordng o he values of and, he followng are obaned: If hen he aggregaon approach wors beer. If hen boh approaches are dencal. If and hen he non-aggregaon approach wors beer. Appendx L: Proof of heore 5-3 Ths s a specal case of Appendx K where defned a (K-) s always posve, n hs case when and he values of s eher saller han zero or

210 B.Rosa-Tabar, 3, Appendces 9 greaer han one( or ), Therefore we follow he sae procedure as Appendx K and fnally we ge: If, he rao of MSE BA MSE AA s greaer han one and consequenly he aggregaon approach ouperfors he non-aggregaon one. If, he rao of MSE BA MSE AA s equal o one and boh approaches perfor equally. If, he rao of MSE BA MSE AA s saller han one and he non-aggregaon approach ouperfors he aggregaon one. Appendx M: Proof of heore 7-3 for ARIMA(,,) Coparson a he aggregae level In order o show ha he aggregaon approach s always ouperfors non-aggregaon one, we us show ha he nu value (lower bound) of he rao MSE BA MSE AA s always greaer han one, herefore o calculae he nu value of MSE BA MSE AA, and should be equal o he salles possble values of =- and =. By subsung hese values n he MSE MSE, we ge BA AA MSE BA MSE AA (M-) By consderng greaer han one. and < s obvous ha MSE MSE s always BA AA When he soohng consan values are very sall, s claed ha he rao s equal o one. Therefore, we us show ha MSE MSE BA ) he followng s gven: l, AA, by consderng (F-

211 B.Rosa-Tabar, 3, Appendces l, (M-) Appendx N: Proof of heore 9-3 for an ARIMA(,,) - Coparson a aggregae level In heore 9-3 s claed ha he aggregaon approach s always wors beer ha he non-aggregaon approach. we us show ha he nu value (lower bound) of he rao MSE BA MSE AA s always greaer han one, so o calculae he nu value of MSE wh ARIMA(,,) when 3, should be equal o =.33 and BA MSE AA =. By subsung hese values n he MSE MSE, we ge: BA AA MSE MSE BA AA (N-) By consderng han one, herefore, BA AA and < we can show ha boh pars of (N-) are greaer MSE MSE s always greaer han one. In addon, when he soohng consan s very sall, he rao s equal o one, we show ha MSE MSE BA now by consderng (N-) we have: l, AA, l, (N-) Appendx O: Proof of heore 9-3 for an ARIMA(,,) - Coparson a aggregae level By consderng MSE s equvalen o havng he quadrac funcon (O-) BA MSE AA negave, whch subsequenly s equvalen o:

212 B.Rosa-Tabar, 3, Appendces (O-) where For he quadrac funcon gven by (K-), he value of he dscrnan and he roos and can be defned as follows: 8 + ) - (- - (K-) + - ) (- - (K-3) + - ) (- - (K-4) We defne he coeffcen of n (C-) as follows

213 B.Rosa-Tabar, 3, Appendces A. (K-5) If he dscrnan, here are no real roos for (K-), herefore he sgn of (K-) s equvalen o he sgn of A. We can show ha when, A s always negave, consequenly (K-) s negave whch eans ha MSE MSE s saller han one. BA AA If, (K-) has wo dfferen roos and. By nvesgang he sgn of, and A, we can deerne he sgn of (K-) and consequenly he perforance superory of each approach. If and hen (K-) s posve n he nerval [, ] and s negave ousde hs nerval. If, we can show ha. (K-) s negave n he nerval [, ] and s posve ousde hs nerval. If, we can show ha hen (K-) s posve n he nerval [, ] and s negave ousde hs nerval. By consderng he above expressons, we ge he followng selecon procedure.. The procedure s begun by calculang defned n (K-), If hen he nonaggregaon approach s always superor, oherwse he values of and defned n (K- 3) and (K-4) are calculaed.. If or, he value of s calculaed, If hen he aggregaon approach wors beer. If hen boh sraeges are dencal. If hen he non-aggregaon approach wors beer. Oherwse: 3. The value of s calculaed. Accordng o he values of and we have If hen he aggregaon approach wors beer.

214 B.Rosa-Tabar, 3, Appendces 3 If hen boh sraeges are dencal. If and hen he non-aggregaon approach wors beer. Appendx P: Covarance beween deand,j and forecas,j The covarance beween sub aggregae deand and j defned as followng:,,,, d d Cov j j j j j j j j j j j (P-) Slar o (P-), j d d Cov,,, can be calculaed where we subsue by j and vce versa. By consderng (A-), he covarance beween sub aggregae deand and sub aggregae forecas j s calculaed as follows:, ), ( ), ( ), ( ), ( ), (,,,,,,,,,,,,, j j j j j j j j j j j j j j j j j j j j d d Cov d d Cov d d Cov d d Cov d d Cov f d Cov (P-) Slar o (P-), he covarance beween sub aggregae deand j and sub aggregae forecas s: j j d j f Cov,,, (P-3) The covarance beween subaggregae forecas and j s as follows:

215 B.Rosa-Tabar, 3, Appendces 4,,,,,,,,,,,,,,,,,,,,, j j j j j j j j j j j j j f f Cov d f Cov f d Cov d d Cov f d f d Cov f f Cov (P-4) Snce,,,,,, j j f f Cov f f Cov and by consderng j d d Cov,,, and subsung (P-), (P-), and (P-3) no (P-4), we ge j j j j j j j j j j f f Cov,,, (P-5) Appendx Q: Proof of heore 4- I s suffcen o show ha he lower bound s greaer han or equal o one, we use hese facs ha,.5 p p, By consderng he lower bounds of, he value of R s equal o zero, herefore we rewre ( 4-4) as follows BU TD V V (Q-) Now by =.5 and = for he nerval of.5< = <, -< = we can calculae he upper bound of he rao of V TD /V BU, now by subsung hese values n (B-) we have.4.5 BU TD V V (Q-) Now we can see ha he nu value of V TD /V BU s obaned when he soohng consan becoes close o zero. Addonally, when= he rao equals o one.

216 B.Rosa-Tabar, 3, Appendces 5 Appendx R: Proof of heore 4-, collary 4- and collary 4- By usng hese facs ha - =.5, - = <,,.5 p p, he rao of V TD /V BU for dfferen values can be calculaed fro ( 4-4). By consderng he lower bounds of, R n ( 4-4) s equal o zero, herefore ( 4-4) can be rewren as V V TD BU (R-) ( To ge he lower bound of (R-) we need o se o he nu value =-.99, and should be he axu value, =.99 for he nerval of -.5, -<, now we subsue hese values n (R-), so we have V V TD BU (R-) ( Now by subsung =. no (R-) s seen ha V TD /V BU =.99. To ge he upper bound of MSE TD /MSE BU, he axu values of.5 p p and are subsued no ( 4-4), we ge V V TD BU R (R-3) ( where

217 B.Rosa-Tabar, 3, Appendces 6 R TD TD TD TD TD TD To ge he lower bound of (R-3), he auoregressve paraeers s se o he axu value =.5, and he ovng average paraeer should be he nu value, =-.99. Now, by subsung hese values n (R-3), he followng s gven: V V TD BU.97 R (R-4) ( where R 5.94 TD TD.3 TD.97.5 TD TD TD By subsung = TD =. no (R-4), he rao equals o V TD /V BU =.. Proof of Collary 5.. By subsung TD= =.5,.5,.3 n (R-) and (R-4) he resuls presened n Table can be obaned. Proof of Collary 5.. By subsung =.5, =.99 and =.99, =. no (R-) and (R-3) he lower and upper bound of MSE TD /MSE BU can be obaned for he nerval of.5< = <, < = <. Fnally, by subsung TD= =.,.5,.5,.3 no ha he resuls presened n Table 3 can be obaned.

218 B.Rosa-Tabar, 3, Résué 7 Résué Ce résué a pour objecf de fournr une vson globale, les prncpaux objecfs e les éapes nécessares de cee recherche. Nous coençons ou d'abord par défnr cerans eres clés dans le doane de l agrégaon e de la prévson de la deande afn d assurer une copréhenson cohérene des conceps lés à ce raval de recherche. Par la sue, le conexe anagéral e scenfque, l'aperçu e les objecfs de la recherche son présenés. Enfn, une fos la déarche éhodologque adopée dans ce raval es exposée, nous dscuons les résulas e les conrbuons de ce raval.. Défnons Une brève descrpon des eres e des expressons clés ulsés dans ce raval de recherche es présenée dans les secons suvanes. Il s'ag des éléens applqués ou au long de cee hèse. Les séres chronologques Mardas e al (998) défnssen une sére chronologque coe une séquence d'observaons ordonnées dans le eps. Ben que l'ordre so généraleen sur le eps, l ordre peu égaleen êre consdéré sur d aures densons, coe l'espace (Harvey, 993). Les séres chronologques se produsen dans des doanes varés els que l'agrculure, le coerce, l'éconoe, l'ngénere, la géophysque, la édecne, les scences socales, ec. A re d'llusraon, dans le conexe de l'enreprse, le nveau de producon annuel, la deande ensuelle de pèces déachées, le nveau des socs hebdoadares e des venes quodennes son oues des séres chronologques. Séres chronologques saonnares Par sére chronologque saonnare, on enend une sére don les propréés ne dépenden pas du eps duran lequel la sére es observée (Mardas e al., 998).

219 B.Rosa-Tabar, 3, Résué 8 Pour qu un processus sochasque so saonnare, l fau que l espérance ahéaque de la sére chronologque, la varance e l'auo-covarance de ou décalage d ordre soen consanes au cours du eps (Harvey, 993). La classe la plus générale des odèles saonnares pour la prévson des séres chronologques es celu des processus auorégressfs e à oyenne oble (ARMA). Séres chronologques non-saonnares La ajoré des séres chronologques exsanes, en parculer dans les seceurs éconoques e coercaux son non-saonnares. Les séres chronologques nonsaonnares peuven se produre de pluseurs façons. Elles peuven avor des oyennes non consanes, des écars e/ou auocovarances varan dans le eps, ou oues ces propréés sulanéen. Les séres chronologques concernan les endances, sasonnalés e les séres cyclques son des séres eporelles non-saonnares (We, 6). L'un des odèles ypques non-saonnares es le processus auorégressfs e à oyenne oble négrée (ARIMA). Une sére chronologque non-saonnare peu êre dvsée en deux pares: ) séres chronologques hoogènes ) séres eporelles non-hoogène. Dans le preer cas, la oyenne es dépendane du eps. En calculan les dfférences enre les observaons consécuves, une sére chronologque hoogène peu êre convere en sére saonnare: c'es la dfférencaon. Cependan, de nobreuses séres chronologques non-saonnares son non-hoogènes. La non-saonnaré de ces séres ne découle pas des oyennes dépendan du eps, as résule de la dépendance au eps de leurs varances e auocovarances. Méhodes de prévson Une éhode de prévson es une procédure pour eser les observaons fuures. Elle dépend largeen de la dsponblé des données. En cas d'ndsponblé, aureen d s les données dsponbles ne son pas pernenes pour les prévsons, les éhodes de prévson qualaves doven êre ulsées. Il exse des approches srucurées eux développées pour l'obenon de bonnes prévsons sans l'ade de données hsorques (Hyndan and Ahanasopoulos, 3). En revanche, les éhodes quanaves peuven êre applquées lorsque les condons suvanes son reples:

220 B.Rosa-Tabar, 3, Résué 9. Dsponblé des données nuérques sur le passé,. Il es rasonnable de supposer que cerans aspecs des donnés passées von se reprodure dans le fuur. Il exse un large évenal de éhodes de prévson quanaves, souven élaborées dans les dscplnes spécfques à des fns spécfques. Chaque éhode a ses propres propréés, sa précson e son coû qu doven êre consdérés au oen de leur chox. La plupar des éhodes de prévson quanaves ulsen so des séres chronologques (collecées à des nervalles régulers dans le eps) so des données ransversales (collecées à un oen précs). Les éhodes quanaves de prévson son dvsées en deux caégores: ) odèles de séres chronologques ) odèles explcafs. Un odèle explcaf es rès ule car l nègre des nforaons sur d'aures varables, pluô que seuleen les valeurs hsorques de la varable à prévor. Cependan, dverses rasons peuven pousser un prévsonnse à séleconner un odèle de sére chronologque pluô qu un odèle explcaf. Preèreen, le sysèe peu ne pas êre coprs, e êe s'l l'éa, l peu êre exrêeen dffcle de esurer les relaons qu déernen son coporeen. Deuxèeen, l es nécessare de connaîre ou de prévor les dverses varables afn d'êre en esure d'ancper sur la varable d'nérê, e cela peu êre rès dffcle. Trosèeen, la préoccupaon prncpale peu êre seuleen de prévor ce qu va se passer sans savor pourquo. En fn de cope, un odèle de séres chronologques peu donner des prévsons plus précses qu'un odèle explcaf ou xe (Hyndan and Ahanasopoulos, 3). Les odèles de séres chronologques ulsés pour la prévson ncluen des odèles ARIMA, le lssage exponenel e les odèles srucurels. Sélecon de l esaeur Afn d'évaluer l'pac de chaque approche d'agrégaon sur la perforance de la prévson, la sélecon d'un esaeur dans un bu d'exrapolaon s'avère nécessare. Dans cee éude, le lssage exponenel sple (SES) es ulsé pour eser la prévson de la deande. Il s'ag d'une éhode de prévson rès populare dans l'ndusre car elle es nuveen sédusane, facle à ere à jour e possède des exgences nales de socage nforaque des données. En oure, elle es opale pour un processus non-

221 B.Rosa-Tabar, 3, Résué saonnare à oyenne oble négrée, IMA() ou ARIMA (,,). Ben que son applcaon plque un coporeen non-saonnare de la deande, les valeurs suffsaen fables de la consane de lssage (ou coeffcen de lssage) nrodusen des écars neurs de l'hypohèse de saonnaré, ands que la éhode es auss parale. Le lssage exponenel sple s'appue sur des prévsons de la deande exponenelleen lssées. L'esaon es se à jour à chaque pérode. Pour oue pérode de eps, la procédure d'acualsaon de la éhode SES es présenée suvan l'équaon caprès: f d f () où d - es la deande à la pérode -, f es la prévson à la pérode e la consane de lssage. Le coeffcen, coprs enre e, s applque à la dernère réalsaon. Il s'ag de la consane de lssage chose à ce nveau. S es fable (par exeple, proche de zéro), plus de pods sera accordé aux observaons plus lon dans le passé. S par conre es grand (so près d un), plus de pods sera accordé aux observaons plus récenes. Dans le cas exrêe ( = ), SES deven la éhode naïve. Dans ce raval de recherche, la éhode SES es préférée à la oyenne oble (MA) e la éhode de prévson opale, ben que ces éhodes de prévson peuven êre envsagées pour les fuures recherches. Deux rasons jusfen ce chox de éhode: ) En oyenne, SES a endance à donner de elleures perforances que la éhode MA, coe on l'observe dans une coparason eprque de leur perforance dans la copéon de prévson M3 (el que rapporée par Mardas and Hbon ()). De plus, SES correspond à un odèle nuf sédusan conrareen à MA. ) En praque, les décdeurs ne veulen pas passer rop de eps e d'effors pour exaner e défnr les caracérsques du processus de données avan de déerner le odèle de prévson opal, coe l'exge ARIMA. Par alleurs, dans un cadre de planfcaon de la producon, les prévsons son enues sur une base pérodque, parfos auss souven que quodenne ou êe horare. Typqueen, la prévson es fae sulanéen

222 B.Rosa-Tabar, 3, Résué pour pluseurs arcles dfférens dans les sysèes nforaques avec un nu d'nervenon huane. Par conséquen, l es relaveen possble de déerner le odèle ARIMA opal pour chaque éléen à chaque se à jour. Or, l sera ule de déerner le onan du gan ou de pere en ulsan une éhode de prévson opale au leu de SES. Nous aborderons cee queson dans les ravaux fuurs. Indcaeurs de précson Un ndcaeur de précson es une esure applquée afn de juger l'effcacé du processus de prévson. Il exse de nobreux ndcaeurs perean de esurer la précson des prévsons. Dans le cadre de cee éude, la varance de l'erreur de prévson, auss appelée erreur quadraque oyenne (MSE) es ulsée coe un ndcaeur de précson. Le chox de séleconner le MSE pour la coparason héorque des éhodes consdérées dans cee éude es jusfé par le fa que ce derner es une esure de la précson ahéaqueen arayane. En oure, l se rapproche de la varance des erreurs de prévson (qu se copose de la varance des esaons produes par la éhode de prévson e la varance de la deande réelle), as en dffère par le bas poenel des esaons qu peu égaleen êre prs en cope. Éan donné que SES fourn des esaons non-basées des processus consdérés dans ce raval, la varance des erreurs de prévson es égale à la MSE,.e. MSE = Var (erreur de la prévson) Agrégaon de la deande Un processus d'agrégaon consse à dérver le odèle de basse fréquence à parr du odèle à haue fréquence; cee dérvaon peu êre exercée dans le eps ou par l'nerédare des ndvdus. L'agrégaon dans le eps, auss appelée agrégaon eporelle, fa en parculer référence au processus par lequel une sére de eps de basse fréquence (par exeple resrelle) es dérvée d'une sére eporelle à haue fréquence (par exeple ous les os) (Nolopoulos e al., ). Coe onré dans les Fgures e, ce résula es obenu grâce à la soe de oues les pérodes de données à haue fréquence, où es le nveau d'agrégaon.

223 B.Rosa-Tabar, 3, Résué Fgure : L'agrégaon eporelle non-cuulée d'hebdoadare à ensuelle Fgure : L'agrégaon eporelle cuulée d'hebdoadare à ensuelle

224 B.Rosa-Tabar, 3, Résué 3 Il exse deux ypes dfférens d'agrégaon eporelle: non-cuulée e cuulée. Dans le preer cas, les séres chronologques son dvsées en segens consécufs non-cuulée de eps, où la longueur de la ranche de eps es égale au nveau de l'agrégaon. La deande agrégée es ans créée en addonnan les valeurs dans chaque ranche. Le nobre de pérodes agrégées es [N/], où N es le nobre de pérodes d'orgne, le nveau d'agrégaon e [x] es la pare enère de x. En conséquence, le nobre de pérodes de la deande agrégée es nféreur à la deande d'orgne. Souven, pour avor des prévsons coparables enre une approche d agrégaon e une approche de non-agrégaon, s la coparason es effecuée au nveau désagrégé, les prévsons agrégées doven êre désagrégées au nveau nal (en les dvsan par le nveau d'agrégaon). Par alleurs, s la coparason es effecuée au nveau agrégé, dans ce cas les prévsons nales doven êre ulplées par le nveau d'agrégaon. Cec es llusré dans les Fgure 3 e 4 dans le cas de prévsons hebdoadares e ensuelles. Fgure 3: Nveau de coparason désagrégé

225 B.Rosa-Tabar, 3, Résué 4 Fgure 4: Nveau de coparason agrégé Un aure ype d agrégaon (l'agrégaon ransversale encore appelée agrégaon hérarchque ou coneporane) se fa au ravers d un ceran nobre d'unés de geson des socs (SKU) à une pérode de eps précse afn de rédure la varablé (Slvesrn and Veredas, 8). Les approches exsanes de prévson ransversale plquen généraleen so une approche ascendane (BU), so une approche descendane (TD), vore une cobnason des deux. Lorsque la prévson au nveau agrégé es en queson, cee dernère plque l'agrégaon des prévsons des unés de geson des socs ndvduelles au nveau du groupe, ands que la deuxèe concerne la prévson dreceen au nveau du groupe (.e. cec exge preèreen l agrégaon de la deande, pus exrapoler dreceen la prévson au nveau global). Lorsque l'accen es s sur la prévson au nveau désagrégé, l approche BU concerne l exrapolaon drece au nveau désagrégé alors que TD plque la désagrégaon des prévsons agrégés produes dreceen au nveau du groupe. Coe l'llusre la Fgure 5, L'approche TD se copose des éapes suvanes: ) les deandes sous-agrégas son agrégées; ) producon des prévsons de deande agrégée va

226 B.Rosa-Tabar, 3, Résué 5 la éhode SES au nveau agrégé, e ) la prévson es désagrégée pour revenr à son nveau nal en applquan une éhode de désagrégaon approprée, s une prévson désagrégée es exgée. Dans l'approche BU: ) les prévsons de la deande désagrégée son produes dreceen pour les arcles désagrégés; ) la prévson agrégée es obenue en cobnan les prévsons ndvduelles pour chaque SKU, so poenelleen un odèle de prévson séparé ulsé pour chaque éléen de la falle de produs (Zoer e al., 5). Ces approches son présenées schéaqueen dans la Fgure 5. Nous adopons ans le syle de présenaon de Mohaadpour e al. (). Nveau agrégé Prévson agrégé es calculé Deande dés agrégée es agrégée Prévsons agrégé es venlé pour obenr des prévsons désagrégées Prévsons agrégée es calculée en addonnan les prévsons désagrégées Nveau désagrégé Arcles désagrégées Prévsons désagrégées Arcles désagrégées Prévsons désagrégées Deande Prévson Deande Prévson Fgure 5: Schéa de TD (gauche) e BU approches (droe). Conexe Managéral La prévson de la deande es le pon de dépar de la plupar des acvés de la planfcaon e du conrôle des organsaons. En oure, l'un des défs les plus porans des

227 B.Rosa-Tabar, 3, Résué 6 socéés odernes es l ncerude de la deande (Chen and Blue, ). L'exsence d'une fore varablé de la deande des arcles à grande ou à fable roaon pose des dffculés consdérables en eres de prévson e de geson de soc (Chen e al., ; Syneos and Boylan, 5; Weerlov and Whybar, 984). Il exse pluseurs approches qu peuven êre ulsées pour rédure l'ncerude de la deande e par conséquence aélorer la perforance de la prévson (e la geson des socs) d'une enreprse. Une approche nuveen arayane, connue pour êre effcace, es l agrégaon de la deande (Chen e al., 7). Une possblé es l'agrégaon eporelle. Une aure approche d agrégaon souven applquée dans la praque es l'agrégaon ransversale, c es-à-dre l agrégaon des données de pluseurs SKUs. Cee approche es équvalene auss à l'agrégaon des données d un seul SKU à ravers d un ceran nobre de dépôs ou des leux d'socage. Naurelles e ules dans la praque, des fores d'agrégaon assocées plquen égaleen la consoldaon géographque des données ou le regroupeen enre les archés. Ben qu'l n'y a pas d'éude eprque qu docuene la esure dans laquelle l'agrégaon a leu dans un conexe praque, l s'ag d'une approche qu es connue pour êre effcace par les professonnels en rason de son ara nuf. En eres praques, la presaon dépend du ype d'agrégaon e ben sûr des caracérsques des données. Une agrégaon ransversale par exeple condu généraleen à la réducon de la varance. Cela es dû au fa que les flucuaons dans les données d'une sére chronologque peuven êre copensées par les flucuaons présenes dans une aure sére (Wdara e al., 9). Conrareen à l'agrégaon ransversale, dans l'agrégaon eporelle la varance augene. Cependan, l peu facleen êre onré que l'agrégaon eporelle peu rédure le coeffcen de varaon de la deande. Dans ous les cas, l'avanage plce assocé à la faclé de se en œuvre de ces approches les rend un chox populare dans l'ndusre. En praque, la deande peu êre classée coe nerene ou à fore roaon. Dans le preer cas, l'agrégaon eporelle de la deande enranera la réducon de la présence d'observaons nulles e, plus généraleen, la réducon des ncerudes dans le second cas. Les arcles à deande nerene (coe pèces de rechange) son connus pour causer des dffculés consdérables en eres de prévson e odélsaon des socs. La présence de zéros a des plcaons poranes en rason des ros rasons suvanes. Tou d'abord, la dffculé à capurer les caracérsques des séres chronologques éudées e des

228 B.Rosa-Tabar, 3, Résué 7 odèles de prévson sandards qu leurs corresponden. Deuxèeen, la dffculé de s adaper à une dsrbuon sasque sandard elle que la lo norale. Trosèeen, les écars par rappor aux hypohèses de odélsaon de soc sandard e leurs forulaons. Ceux-c renden la geson de ces éléens un exercce rès dffcle. L agrégaon eporelle es connue pour êre largeen applquée dans les leux lares (données rès rares), le seceur après-vene (pèces déachées ou de servce), ec. Des éudes eprques récenes (Baba e al., ; Nolopoulos e al., ) dans ce doane on abou à des résulas rès proeeurs en soulgnan égaleen la nécessé d'une analyse plus héorque. Ben que le doane de la prévson à l ade de l'agrégaon eporelle dans un conexe de deandes nerenes es rès néressan an d'un pon de vue acadéque e professonnel, dans cee recherche le conexe des deandes à fore roaon, qu rese le conexe le plus renconré, es celu prs en cope. L analyse dans un conexe de deandes nerenes es une voe néressane de recherches fuures e cee queson es abordée avec plus en déal dans le derner chapre de cee hèse. En plus de la réducon de l'ncerude de la deande assocée à l'approche de l'agrégaon eporelle dscuée c-dessus, l y a une queson porane dans un processus de prévson où l agrégaon eporelle peu êre ule. Il es appelé "horzon de la prévson" qu déerne la le de la prévson fuure. En règle générale, plus on regarde lon dans le fuur, plus la précson décroî. C'es auss l'un des doanes où l'agrégaon eporelle peu aélorer la précson des prévsons, parce que coe nous regardons plus lon dans l'avenr, la vson à long ere deven plus porane e la éhode d'agrégaon eporelle peu ulser cee nforaon eux que les approches classques. Donc, l'approche d'agrégaon eporelle peu auss êre rès effcace lorsque les professonnels on beson de prévsons à long ere au leu d'une prévson pour une seule pérode fuure. D'un pon de vue héorque, l'accen à ce jour a éé prncpaleen sur l'agrégaon ransversale. En oure, la plupar des logcels de prévson prend en charge l'agrégaon des données, ce sera auss couvrr seuleen l agrégaon ransversale. La consdéraon de l'agrégaon eporelle a éé quelque peu néglgée par les édeurs de logcels e les chercheurs algré la possblé d'ajouer plus de valeur en praque. Dans ce raval, l'objecf es de fare progresser l'éa acuel des connassances dans le doane de la prévson de la deande à l ade de l agrégaon eporelle.

229 B.Rosa-Tabar, 3, Résué 8 Dans les dscussons c-dessus, l'effe de l'agrégaon eporelle sur un seul SKU es consdéré. Alors qu en réalé, l y a souven de nobreuses séres chronologques qu peuven êre organsées de façon hérarchque e groupées à dfférens nveaux dans les groupes basés sur des références de produs, des clens, de la géographe ou d'aures caracérsques (Hyndan e al., ). Le nveau hérarchque auquel la prévson es effecuée dépend du beson de chaque foncon. En ce qu concerne les produs (ou références), en parculer, la prévson au nveau SKU ndvduel es nécessare pour la geson des socs, les prévsons de la falle de produs peuven êre requses pour le prograe dreceur de producon. Les prévsons à ravers d un groupe d'arcles coandés auprès du êe fournsseur peuven êre nécessares dans le bu de regrouper les coandes. Les prévsons à ravers des arcles vendus à un grand clen spécfque peuven pacer le ranspor, les décsons de rouage, ec. Une approche a pror néressane pour obenr des prévsons de nveau supéreur es l agrégaon ransversale, ce qu plque généraleen so une approche TD ou une approche BU (ou une cobnason des deux). Une queson porane qu a aré l'aenon de nobreux chercheurs e professonnels au cours de ces dernères décennes es l'effcacé de ces approches de prévson ransversales. Les approches de prévson BU e TD son exrêeen ules pour aélorer la précson des prévsons e des plans au sen d'un processus S&OP (la planfcaon des venes e des opéraons) (Lapde, 6). Le S&OP es un processus ulfonconnel qu plque les gesonnares de ous les dépareens (venes, servce clen, chaîne logsque, areng, fabrcaon, achas e fnances), où chaque dépareen a beson de dfférens nveaux des prévsons de la deande (Lapde, 4). Par exeple, dans le areng (Depe and Hanssens, ), la prévson du chffre d'affares par groupes de produs e par arques es nécessare. Les servces coercaux raen avec des prévsons de venes par les copes clens e/ou des canaux de vene. Les gesonnares de la chaîne d'approvsonneen deanden les prévsons au nveau du SKU, ands que la fnance a beson de prévsons qu son agrégées dans les unés budgéares en eres de revenus e de coûs (Bozos and Nolopoulos, ). Afn de produre les prévsons requses, la deande e/ou les prévsons devraen êre agrégés e/ou désagrégés à dfférens nveaux. Il s'ag de l'applcaon des approches TD e BU ou une cobnason des deux (Lapde, 4, 6).

230 B.Rosa-Tabar, 3, Résué 9 3. Conexe scenfque L'agrégaon a éé largeen dscuée dans la léraure acadéque depus les années 95 (Quenoulle, 958). Elle es consdérée coe un oyen de rédure les flucuaons de la deande e le degré d'ncerude. Il a éé déonré par Thel (954), Yehuda and Zv (96), e Agner and Goldfeld (974) que l'ncerude de la deande peu êre effcaceen rédue par l'agrégaon e une bonne prévson de la deande. Dans la léraure de la planfcaon de la chaîne logsque e la planfcaon de la deande, l'agrégaon de la deande es connue coe un approche de uualsaon des rsques pour rédure les flucuaons de la deande afn d avor une planfcaon des aères/capacé plus effcace (Chen and Blue, ). Dans le doane de l'agrégaon eporelle, l y a à la fos des éudes héorques e eprques dscuées dans la léraure. Cependan, la plupar de ces éudes son dans le doane de l éconoe. Les propréés du processus agrégé son fournes sur la base des données non-agrégées. De plus, l'effe de l'agrégaon eporelle sur la prévson es évalué par l ajuseen d'un odèle e de l'esaon des paraères. Aeya and Wu (97) on évalué l'effe de l agrégaon eporelle non-cuulée lorsque la sére orgnale su un processus auorégressf d'ordre p, AR (p). En consdéran le rao des MSE de la prévson non-agrégée e agrégée (3 prédceurs lnéares on éé consdérés au nveau agrégé, ls on onré que l'approche d'agrégaon perfore eux que l approche non-agrégée. Tao (97) a éudé l'effe de l agrégaon eporelle non-cuulée sur un processus non-saonnare oyenne oble négrée d ordre (p,q), l'ima (p,q). Une espérance condonnelle es applquée pour obenr une prévson à l horzon d une pérode au nveau agrégé basé sur les séres non-agrégées e agrégées. Par la sue, l'effcacé des prévsons agrégées a éé défne coe le rao de la varance de l'erreur de prévson de la sére non-agrégée à la sére agrégée lorsque le nveau d'agrégaon es grand. On onre que lorsque d = e le nveau d'agrégaon es rès grand, alors le rao en queson es égal à un e l'avanage coparaf de l'ulsaon des prévsons non-agrégées augene avec d. Peu d'éudes récenes on évalué l'effe de l'agrégaon eporelle sur la prévson e la geson des socs par des recherches eprques. Nolopoulos e al. () on eprqueen analysé les effes de l'agrégaon eporelle sur la prévson de deandes nerenes e ls on proposé la éhodologe ADIDA. Il es déonré que la éhodologe ADIDA peu en effe apporer des aéloraons consdérables en eres de précson des

231 B.Rosa-Tabar, 3, Résué 3 prévsons. Enfn, Baba e al. () on égaleen éendu l'éude décre c-dessus (Nolopoulos e al., ) afn d'exaner les plcaons de la éhodologe ADIDA sur les socs en consdéran une polque à suv pérodque appelée polque avec nveau de recoplèeen (ordre-up-o-level). Les chercheurs on conclu qu'une echnque sple coe l agrégaon eporelle peu êre auss effcace que les approches ahéaques coplexes de prévson des deandes nerenes. Au elleur de nore connassance, les seuls éudes dreceen pernenes pour nore raval son celles par Aeya and Wu (97) e Tao (97) pour les processus AR e MA respecveen. Ces ravaux on poré sur la caracérsaon de la sére de la deande agrégée en plus de l'évaluaon de la perforance des prévsons. Cependan, les résulas présenés dans ces ravaux resen prélnares alors que le conexe expérenal peu égaleen êre crqué en eres des procédures d'esaon consdérées. De plus, aucun résula eprque n a éé fourn. Par conséquen, l'absence des condons qu déernen la supéroré d une approche, en aère de prévson de la deande, es évdene. Il n'es pas clar s l'approche d'agrégaon fourn des prévsons plus précses que celle de la non-agrégaon, e vce versa. Par conséquen, la ovaon derrère cee pare de l'éude es l'absence de l'analyse héorque en ce qu concerne l'effe de l'agrégaon eporelle sur la prévson de la deande. Dans cee recherche, l'évaluaon analyque es applquée pour déerner les condons de supéroré de chaque approche. La recherche es coencée avec le processus sple ARMA d ordre un. Cependan, l'analyse peu êre effecuée pour les processus d'ordres supéreurs as les résulas devennen plus coplexes à présener donc cec es consdéré dans les recherches fuures. Dans le doane de l'agrégaon ransversale, la plupar de la léraure de la prévson s'es penchée sur les perforances coparées des approches TD e BU. Les conclusons en ce qu concerne les perforances de ces approches son élangées. Cerans aueurs coe Thel (954), Grunfeld and Grlches (96), Schwarzopf e al. (988), e Narashan e al., 985(985) on fa valor que l'approche TD perfore eux que BU, d'aure par, des aueurs coe Orcu e al. (968), Edwards and Orcu (969), Dunn e al. (976), Dangerfeld and Morrs(988) and Gross and Sohl (99) on consaé que l'approche BU es perforane e enfn quelques aures aueurs coe Barnea and Laonsho (98), Fledner (999) and Wdara e al.(7, 8, 9) adopen une

232 B.Rosa-Tabar, 3, Résué 3 approche conngene e analysen les condons dans lesquelles une éhode produ des prévsons plus précses que les aures. Dans cee hèse, l'effcacé de la BU e la TD es évaluée. Les ravaux présenés par Wdara e al. son éendues dans cee hèse en consdéran un processus de deande saonnare plus général ARIMA(,,) e un processus non-saonnare ARIMA(,,). Par alleurs, la coparason es effecuée an au nveau désagrégé e agrégé. En oure, la supéroré de chaque approche es exanée en ulsan un enseble de données réelles. 4. Aperçu de la recherche L'agrégaon es un oyen effcace pour rédure la varablé de la deande. De plus, l pere aux prévsonnses d'obenr dfférens nveaux de prévsons dans le eps e des nveaux hérarchques. Selon le nveau des prévsons, nous produsons d'abord les prévsons e les agrégeons par la sue so nous regroupons d abord les séres orgnales ndvduelles pour obenr la deande agrégée e pus de produre la prévson agrégée. Dans ce derner cas, un écanse de désagrégaon es nécessare pour obenr les prévsons désagrégées. Dans cee recherche, l'pac de l'agrégaon sur la prévson de la deande es évalué. Pour onrer l'effe de l'agrégaon sur la prévson de la deande, deux ypes d'agrégaon son consdérés: ) l'agrégaon eporelle e ) l'agrégaon ransversale. Nore aperçu de la recherche es résué dans la Fgure 6. On suppose que la sére chronologque su un processus de ype ARIMA e la éhode de prévson es SES. Dans l'agrégaon eporelle, l es supposé que la deande désagrégée su un processus saonnare auoregressf oyenne oble d ordre un, ARIMA (,,), ce qu veu dre que leurs cas parculers, oyenne oble d ordre un, ARIMA (,,) e l auorégressf d'ordre un, ARIMA (,,) son égaleen consdérés. Ensue, l es dscué s des données désagrégées ou des données agrégées doven êre ulsées pour fournr les prévsons requses. De plus, les condons dans lesquelles, une approche perfore eux que l'aure son présenées.

233 B.Rosa-Tabar, 3, Résué 3 Recherche Type d Agrégaon Agrégaon eporelle Agrégaon ransversale Processus de la deande ARMA(,), MA(), AR() ARMA(,), IMA() Méhode de prévson SES SES Mesure de précson MSE MSE Nveau de la coparason Nveau agrégé, Nveau désagrégé Nveau agrégé, Nveau désagrégé Objecf Idenfer les condons de supéroré des approches de l'agrégaon e non-agrégaon Évaluer l'effcacé des approches BU e TD Fgure 6: Vue d enseble de la recherche Dans l'agrégaon ransversale, l'effcacé des approches BU e TD, pour fournr des prévsons désagrégées e agrégées, es analysée. On suppose que la sére désagrégée su so un processus saonnare auoregressf oyenne oble d ordre un, ARIMA(,,) so un processus non-saonnare oyenne oble négrée d ordre un, IMA (,). Dans cee pare de la hèse, la varance de l'erreur de prévson es ulsée pour coparer la perforance de chaque approche. La varance de l'erreur de prévson es équvalene à la MSE en consdéran une éhode de prévson non basée. Les varances des erreurs de prévson son obenues sur la base de la deande désagrégée e agrégée. Les coparasons son effecuées au nveau de la deande désagrégée e agrégée. Les condons dans lesquelles chaque approche surpasse les aures son ahéaqueen denfées. L'analyse ahéaque es copléée

234 B.Rosa-Tabar, 3, Résué 33 par une éude nuérque pour valder les résulas héorques. De plus, l éude nuérque es applquée pour évaluer en déal les condons de supéroré de l'approche en relaxan ceranes hypohèses consdérées dans l'évaluaon analyque. Ensue, les résulas son valdés eprqueen (à l ade des sulaons sur un enseble de données réelles fournes par un hyperarché européen). Enfn, des leçons anagérales rès poranes son dérvées e des suggesons concrèes son proposées aux professonnels qu s'néressen aux problèes de prévson e de geson des soc. Dans cee recherche, l'ordre des processus de ype ARIMA es lé à un car ce ype de processus es plus observé dans la léraure pour les séres non-sasonnères, oure l'objecf prncpal qu es de rer pluseurs éclarages clés pour les anagers. Par conséquen, nous allons ler nore aenon aux processus AR(), MA() e ARMA(,). Touefos, l conven de noer que l'exenson du raval à analyser des cas plus généraux els que AR (p), MA (q), vore ARMA (p, q) es fasable, as l'analyse e la présenaon des résulas devendraen coplexe. Cee analyse sera exanée dans les ravaux fuurs. L'objecf prncpal de cee recherche es d'analyser les effes de l'agrégaon sur la prévson de la deande. Ce effe es exané par l'analyse ahéaque e l éude de sulaon. L'analyse es copléée en exanan les résulas sur un enseble de données réelles. Basé sur le conexe scenfque e anagéral de la recherche e des ovaons, sx objecfs on éé forulés pour cee recherche: 7. Evaluer analyqueen l'effe de l agrégaon eporelle non-cuulée sur la prévson lorsque la sére de base su un processus saonnare de ype ARMA. 8. Idenfer les condons dans lesquelles l'approche d'agrégaon eporelle perfore eux que celle de non- agrégaon, e vce versa. 9. Déerner le nveau d'agrégaon opale qu axse les avanages de l'approche d'agrégaon eporelle.. Exaner l'effcacé des approches BU e TD afn de prévor la deande désagrégée e agrégée dans un envronneen saonnare e non-saonnare.. Analyser l'effe des paraères du processus e de conrôle sur la supéroré de l'approche dans les agrégaons eporelles e ransversale.

235 B.Rosa-Tabar, 3, Résué 34. Teser la valdé eprque e l'ulé des résulas héorques e de sulaon sur un large enseble de données réelles. 5. Mehodologe La recherche su ros éhodes de recherche, l'analyse ahéaque, la sulaon e l éude eprque. La relaon enre les ros éhodes es llusrée dans la Fgure 7. Fgure 7: Méhodologe Preèreen, l'analyse ahéaque es applquée afn d exaner la supéroré de l'approche d'agrégaon e de dévoler les condons dans lesquelles cee approche donne des résulas plus précs par rappor à l'approche classque. La varance héorque de l'erreur de prévson assocée à chaque approche es calculée pour ous les processus de la deande à l'éude. Cec es ené afn d'denfer les condons de la supéroré de chaque approche. L'éude de sulaon es ulsée pour les rasons suvanes: Pour eser e valder les résulas de l'analyse héorque. Pour relaxer les hypohèses prses en cope dans l'évaluaon ahéaque. Enfn, les résulas de cee hèse son esés sur des données eprques réelles pour évaluer la valdé e l'applcablé praque des prncpaux résulas de l'éude. Par