Joural of Omzao Idusral Egeerg 5 (0) 55-63 Measurg ad Aalyzg he Bullwh Effec a Two-Produc ad Two Echelo Suly Cha Usg Corol Theory Aroach Ahmad Sadegh* a Asssa Professor, Faculy of Idusral ad Mechacal Egeerg, Qazv Brach, Islamc Azad Uversy, Qazv, Ira Receved Ocober, 03; Revsed 5 February, 0; Acceed 9 Arl, 0 Absrac Coordao s very mora suly cha maageme ad s oe of he ma facors suly cha rofably. Bullwh effec s oe of he basc obsacles o acheve coordao suly chas ad reduco of hs heomeo has a mora role suly cha harmoy. The oher sde, coss of suly cha ca be mgaed ad cusomer servce level ca be creased by reducg of bullwh effec. Because measureme of bullwh effec s very mora aalysg ad corollg of, rovdg equaos o vesgae bullwh effec behavour based o real world suly cha codos s ecessary. The revous sudes mosly cocerae o sgle roduc suly cha ad few sudes have bee doe o suly chas wh more ha oe roduc. Here we quafy ad vesgae he bullwh effec a wo-echelo suly cha wh wo roducs usg corol heory aroach. ue o he relaosh bewee demads of wo roducs our roosed suly cha, frs order vecor auo regressve model s used as demad rocess of he roducs. We also aly movg average mehod for lead-me demad forecasg wh he "order u o" releshme olcy. We derve a closed form bullwh measure ad he bullwh effec a wo-roduc suly cha s dscussed ad llusraed hrough a umercal examle. Keywords: Bullwh Effec, Order-U-To Polcy, Suly Cha, Sysem Egeerg, Vecor Auo Regressve Model.. Iroduco Today, ousourcg s so mora due o s effec o he suly cha beefs. I fac, we ecouer a umber of sulers, roducers, ad dsrbuers, whch work ogeher wh a suable harmoy o rovde more gas for suly cha. emad damg s a ma barrer o acheve coordao wh dffere sages of suly cha. May comaes have observed creasg flucuao orders whle movg u from dowsream ses (lke dsrbuers) o usream ses (lke sulers). The resul s a loss of suly cha rofably. Sce he frs sudy o bullwh effec coe by Forreser (958) he bullwh effec has bee meoed a large umber of researches. Sucky (009) dvded researches o he bullwh effec o sx geeral caegores: () aers amg a a quafcao of he bullwh effec, () works focusg o aalyzg ad defyg he causes of he bullwh effec, () sudes observg he bullwh effec some dusres or umerous examles from dvdual roducs ad comaes, (v) aers addressg mehods for reducg he bullwh effec, (v) works focusg o smulag he sysem behavour ad (v) aers focusg o exermeal valdao of he bullwh effec. Serma (989) develoed Beer game as a ece of evdece for exsece demad amlfcao suly chas. ee e al. (997) roduced fve ma causes of hs heomeo: demad forecas udag, order bachg, rce flucuao, raog, ad o-zero leadme. Udersadg hese causes of he bullwh effec ca be useful for maagers o fd suable soluos for halerg ad corollg of s cosequeces. Che e al. (000 a, b) quafed ad derved a lower boud for he bullwh effec a smle suly cha for wo cases of forecasg mehods: movg average ad exoeal smoohg. ejockheere e al. (003) roosed a corol heory aroach for measurg bullwh effec ad suggesed a ew geeral releshme rule ha ca reduce varace amlfcao sgfcaly. sey ad Towll (003) roduced a orderg olcy ha resuls amg bullwh effec. Zhag (00) cosdered hree forecasg mehods for a smle veory corol sysem ad reseed hree measures for bullwh effec based o hree forecasg mehods. Km e al. (006) vesgaed sochasc leadme ad vesgaed role of formao sharg o bullwh effec. Chadra ad Grabs (005) measured * Corresodg auhor E-mal: a.sadegh@qau.ac.r 55
Ahmad Sadegh/ Measurg ad Aalyzg he Bullwh... bullwh effec whe order sze s calculaed accordg o mulle se forecass usg auoregressve models. uog (007) vesgaed effecs of auoregressve coeffce ad lead-me o bullwh effec whe mmum execed mea squares of error forecasg mehod s used. uog ad Phe (007) research was based o order of auoregressve demad aer. They showed ha hgh order of demad aer, bullwh effec could be reduced whe lead-me decreases. Maku ad Madad (007) ulzed he yauov exoe ad rovded a measure for bullwh effec. They reseed useful resuls o he behavor of he bullwh effec by vesgag he mahemacal relaoshs. Jaksc ad Rusja (008) demosraed ha cera releshme olces could be ducers of he bullwh effec. Chaharsoogh ad Sadegh (009) cosdered a woroduc suly cha ad quafed bullwh effec measure usg a sascal aroach. They cocluded ha here s o exlc exresso for he bullwh effec measure, whe sascal mehod s used for quafyg of he bullwh effec. Cosequely, bullwh effec measure could be calculaed for oly lmed cases. Wu e al. (0) ulzed age-based model ad smulao (ABMS), as oe of he scefc ad dyamc research mehods for comlex sysems o esablsh a suly cha model ad deerme s abuda bullwh effec heomeo uder swarm laform. Based o her aalyss, was roved ha he ABMS s he effecve way o sudy he bullwh effec comlex suly cha. Neal e al. (0) reseed a aalyss of he bullwh effec ad e-sock amlfcao a hree-echelo suly cha cosderg se-chages he roduco raes durg a roduc's lfe-cycle demad. Usg a smulao aroach, he aalyss was focused aroud hghly comlex ad egeered roducs, whch have relavely log roduco lfe cycles ad requre sgfca caal vesme maufacurg. Fazel Zarad ad Gamasaee (03) vesgaed reducg of he bullwh effec fuzzy evromes by meas of ye- fuzzy mehodology. I order o reduce he bullwh effec a suly cha, hey roosed a ew mehod for demad forecasg. Zoer (03) aalyzed he emrcal demad daa for fas movg cosumer goods o measure he bullwh effec. The daa cossed of he sell- from a large maufacurer o he realers ad he sell-ou from a realer o he cosumers. The fdgs showed ha he bullwh-effec could be subsaal. ue o he morace of forecasg mehod bullwh effec reduco, Jaura ad Mahaara (0) roosed a egraed aroach of scree Wavele Trasforms Aalyss ad Arfcal Neural Nework deoed as WT-ANN for demad forecasg. Ther model was esed ad valdaed by coducg a comarave sudy bewee Auoregressve Iegraed Movg Average (ARIMA) ad roosed WT-ANN model. The aalyss dcaes ha he mea square error of WT-ANN s comaravely less ha ha of he ARIMA model. e al. (0) suded he damed red forecasg mehod ad s bullwh geerag behavour whe used wh he Order-U-To (OUT) releshme olcy.usg z-rasform rasfer fucos hey deerme comlee sably crera for he damed red forecasg mehod ad showed ha hs forecasg mechasm s sable for a much larger rooro of he aramercal sace ha s geerally ackowledged he leraure. Furher, hey demosraed he damed red OUT sysem somemes wll geerae bullwh ad somemes wll o. Buchmeser e al. (0) smulaed a smle hree-sage suly cha usg seasoal ad deseasoalzed me seres of he marke demad daa order o defy, llusrae ad dscuss he macs of dffere level cosras o he bullwh effec. The resuls were reseed for dffere overall equme effecveess (OEE) ad cosraed veory olces. Alhough may researches have bee doe o he bullwh effec, more vesgaos are sll eeded o sudy, mmze s effec, ad quafy order o rovde soluos for comlex suly chas. Ths research cosders a wo-echelo suly cha cossg of oe realer ad oe suler or roducer. Ths suly cha roduces wo roducs whch demad aer s based o frs order vecor auoregressve (VAR()) model. Orderg olcy for each roduc s order u o olcy ad forecasg mehod s movg average. Accordg o hese assumos, bullwh effec s quafed a wo-roduc suly cha ad a closed form for bullwh effec calculaos s reseed usg corol heory aroach. The, s descrbed ad aalysed va a umercal examle.. Suly Cha Assumos I hs aer, a wo-sage suly cha wh wo roducs s ake o cosderao. emad aer s descrbed by me seres model, forecasg mehod s movg average ad orderg olcy s accordg o order u o olcy. ealed descro of hese roosos are exlaed he ex ars... Noaos We use several arameers o quafy bullwh effec measure as follows: cusomer demad Q order quay a he begg of erod varace of demad Q varace of order quay VAR() frs order vecor auoregressve demad of roduc x erod x, y, S.. demad of roduc y erod order-u-o level of erod auocorrelao coeffces 56
Joural of Omzao Idusral Egeerg 5 (0) 55-63 x, forecas error of roduc x erod y, forecas error of roduc y erod ˆ ˆ forecas order lead-me lead me demad forecas sadard devao of lead me demad umber of erods movg average forecas.. Bullwh effec measure Accordg o he revous researches o he bullwh effec measure (lke Che e al. (000 a, b) Km e al. (006) ad Sucky (009)) bullwh rao ca be calculaed by equao (): Q BE () whch Q ) s varace of realer orders ad ( () s varace of he cusomer demad. Therefore, o rovde bullwh effec measures we mus reare equaos for wo meoed erms..3. The VAR() demad rocess wh wo roducs We cosdered ha here are wo roducs our suly cha wh deeded demad. Therefore we use a frs order vecor auoregressve (VAR()) model for rereseg of hs relaosh bewee demad of roducs. The VAR() me seres model s gve based o We (990): x, y, xx yx x, x, xy yy y, y, Here we ca see ha demad for roduc x a me, s gve by he sum of hree comoes. The frs comoe.e. s a auoregressve erm of oe xx x, erod wh self. The secod erm.e. xy y s a, auoregressve erm wh revous realsao of roduc y. The fal erm.e. s deedely ad decally x, y, () x, dsrbued (whe ose) radom rocess. The demad rocess of he secod roduc, roduc y, s smly a mrror mage of he demad rocess for roduc x. { x, y}, ca be regarded as he forecas error ad we assume ha s has a zero mea ad s u varace. We assume, from ow o, ha he error erms are ucorrelaed as hs smlfes he mahemacs cosderably. I order for he VAR() demad rocess o be saoary; he followg crera mus be held: xx yy xx yy xy yx (3) Chaharsoogh ad Sadegh (009) showed ha a VAR() demad rocess, varace of demad for each roduc he saoary codos s gve by equao (): j j j 3 j j j () j j j j j j where, for roduc x, =x ad j=y; for roduc y, =y ad j=x. Therefore we ca use equao () as deomaor of equao () bullwh effec measureme... The OUT olcy wh movg average forecasg We cosder a sgle echelo of a suly cha whch uses he Order-U-To (OUT) olcy o geerae releshme orders o maa veory levels. The forecasg mehod s also movg average. The goal of order u o olcy s o brg he acual veory owards he desred veory. The order quay ha realer laces o he suler s gve by: Q S S (5) Usg base sock olcy, order u o level, a he begg of erod ca be deermed by equao (6): S ˆ z. ˆ (6) I equao (6), z s sadard ormal score ad ca be deermed by ormal able based o he desred servce level requred from he veory sysem. Relacg equao (6) equao (5) cocludes equao (7) ha s order quay erod : Q ˆ ˆ z( ˆ ˆ (7) ) based o movg average forecasg mehod, ˆ ca be calculaed by equao (8): j ˆ j 0 (8) As each of roducs s ordered deedely, equao (8) ca be aled o boh of roducs wh a suable chage her arameers. Fgure hghlghs a block dagram of he wo roduc suly cha we have jus descrbed. Fg.. Block dagram of he wo-roduc suly cha 57
Ahmad Sadegh/ Measurg ad Aalyzg he Bullwh... Fgure shows ha our sysem s a mul-u, mulouu sysem, so we eed mulle rasfer fucos o descrbe s dyamcs. As we have assumed he whe ose rocesses ad are ucorrelaed, here s o x y eraco bewee hem, hus her dvdual combaos o he varace exressos ca smly be added ogeher. Based o sey ad Towll (003) we ca rovde equaos for quafyg varace of orders as follows. e s cosder he combao of o. The rasfer fuco s: z z z z Pz z j j z Ths has he followg me doma mulse resose: H o, H (9) (0) where he followg subsuos have bee made as he equao s raher leghy:,, j j,, ad Hw s he Heavsde se fuco (ha s, Hw=0 f w<0, oherwse). Equao (0) was obaed by akg he verse z- rasform of equao (9). The corbuo of o he varace of orders for roduc s gve by he sum of he squared mulse resose (sey ad Towll (003)). Because of he Heavsde se-fuco roeres, s useful o sl he sum o he followg wo ars:, 0 () Equao () has he followg closed form rereseed by equao ():, j, () Now we cosder he fluece of o j. The rasfer fuco s: z z z j (3) z z z j j j Takg he verse z-rasform yelds he followg me doma mulse resose: j H o, () j The corbuo of j o he varace of s gve by he sum of he squared mulse resose (sey ad Towll (003)). Tha s:, j 0 j j (5) whch has he closed form rereseed by equao (6). A more comac ad exlc form of equao could o be rovded due o he comlexy of he relaoshs. The varace of he orders for roduc s gve by 3 3 3 3 6 3 3 3 j. However, he followg,, j alerave formulao rereseed by equao (7) has bee cocluded a more summary forma: 3 3 6 (6) 58
0 j j (7) Ths closed form s vald across all lead-mes ad movg average cosa,. 6 ) ( 3 3 j j j j j j j j j (8) Equao (8) furher smlfes o he followg: ) ( j j j (9) 3. Quafyg he Bullwh Effec Cosder equao () aga ha defes bullwh effec relaosh. As meoed before, we ca use equao () as varace of marke demad, eeded deomaor of equao (). Moreover, equao (9) rereses varace of orders ad ca be used as umeraor of equao (). Therefore bullwh effec values ca be measured by dvso of her resuls. I he ex ar, we comlee Tables ad hs maer. Now we wa o rove a rooso whch exressed codos ha bullwh effec ca be removed from he roosed suly cha. Prooso: Whe aroaches fy, varace of orders aroaches varace of demad. Tha s, whe he. Thus, whe forecass become equal o he log-erm average demad, he bullwh rao s equal o oe. Proof: Whe aroaches fy, we ca derve equao (0) based o equao (9): ( )( ) ( ) lm j (0) Subsug,,, we have: 3 lm j j j j j j j j j j j j () I s clear ha equao () s equal o derved before by equao ().Therefore whe he bullwh effec aroaches oe. Tha s: Bullwh ad whe he Bullwh. So here s o bullwh effec for roduc he suly cha f realer uses as large as umber of erods () average calculaos. Joural of Omzao Idusral Egeerg 5 (0) 55-63 59
Ahmad Sadegh/ Measurg ad Aalyzg he Bullwh.... Numercal Examle xx 0., xy 0., yx 0. ad yy 0.6 Here, a examle s reseed o aalyze bullwh effec behavor a wo-roduc suly cha. Suose ha demad of wo roducs he roosed suly cha s defed by hese coeffces: Usg revous relaoshs, bullwh effec measures are reseed Tables ad. ue o he closed form of bullwh effec measure, we ca calculae bullwh raos for each value of. Table Bullwh raos for roduc x 3 5 6 7 8 9 0 3.6596 8.8789 6.6958 7.596 0.39 55.935 7.69 95.76 8.78.878.079.77 7.039 0.6383 5.0559 0.766 6.3006 33.77 0.758 9.95 3.7735.93385.809 6.79 8.735.8.537 8.079.8856 6.0.56996.3679 3.3938.6773 6.96 7.8399 9.77735.93.337 6.9588 5.57.057.8066 3.70699.7597 5.968 7.363 8.809 0.75.79 6.379.853.388 3.308 3.930.83595 5.88 6.96703 8.9 9.533 7.3739.7.90.756 3.380.095.88798 5.76079 6.795 7.76 8.7587.6305.053.73.99 3.578.85.935 5.6898 6.57 9.369.536.8778.679.70583 3.93 3.793.38.9777 5.630 0.80.757.77305.00.8663.9088 3.35878 3.853.3895.9633 Table Bullwh raos for roduc y 3 5 6 7 8 9 0.777 5.33 9.866 5.777 3.66 3.035.3767 5.986 67.983 8.757.8633 3.309 5.366 7.90658.07.83 9.98.09 9.896 35.539 3.639.5797 3.8353 5.38 7.3896 9.5306.058.907 8.06.5366.5066.0399 3.0699.065 5.598 7.0997 8.7563 0.63.7389 5.066 5.53.96909.663 3.995.60 5.56857 6.855 8.9896 9.78.383 6.353.807.3699 3.0777 3.773.6398 5.590 6.6975 7.80996 9.0706 7.3060.6885.756.68309 3.95 3.98366.787 5.5905 6.5089 7.5085 8.698.5989.9870.3565.9 3.539.09.88 5.576 6.38369 9.3968.573.8685.63.67776 3.573 3.68.5966.8883 5.55393 0.56.70.76.0976.7.8859 3.3339 3.839.3559.9999 Fgures, 3, show bullwh effec raos whle ad are varyg for each of roducs searaely. The resuls are comable wh he revous researches sgle roduc suly chas ad are cofrmed by hem (e.g. Che e al., 000a; Zhag, 00; ad uog, 007). I addo, he resuls are smlar o Chaharsoogh ad Sadegh (009) bu he dfferece s he umber of movg average calculaos (). ue o he aforemeoed lmaos her sascal aroach, bullwh effec measures could be rovded for oly a few values of (=,, 3,, 5). Here, usg corol heory aroach, we rese a closed form for ( Q ) equao (9) whch eables us o calculae bullwh effec measures for all of values (as he real world we may use may values of he movg average forecasg). A revew of Tables ad ad Fgures dcaes ha bullwh effec s relaed o lead-me drecly ad s releva o he umber of observaos movg average calculaos () reversely. BE(x) 0 8 6 3 5 6 7 8 9 0 Fg. (a). Bullwh effec varao wh resec o for roduc x Fgure (a) decs ha bullwh effec of he roduc x creases whe s lead-me creases. The uer curve s bullwh raos for = ad ohers are for =, =3, =, resecvely. I s clear ha creasg of has a mora role bullwh effec reduco. Fgure (b) shows bullwh effec of roduc y wh resec o s lead-me. All he above oes abou he bullwh effec of he frs roduc s vald for he roduc y. 60
Joural of Omzao Idusral Egeerg 5 (0) 55-63 BE(y) 8 6 3 5 6 7 8 9 0 BE(y) 7 5 3 9 7 5 3 3 5 6 7 8 9 0 Fg. (b). Bullwh effec varao wh resec o for roduc y Fgure 3(a) shows bullwh effec measures of he roduc x wh resec o he umber of observaos lead-me demad forecasg,. The uer curve s bullwh rao for = ad he ohers are for =3, =, =, resecvely. I s clear ha more he observao resuls, he less he bullwh effec. I addo, a shar fall ca be see whe he umber of observaos chages from o. Moreover, Fgure 3(a) shows ha sloe of he bullwh effec curve decreases dramacally whe leadme creases. I oher words f lead-me s large, he creasg from o ca reduce bullwh effec cosderably. BE(x) 6 6 Fg. 3(b). Bullwh effec varao wh resec o for roduc y Fally, Fgures (a) ad (b) show bullwh raos for wo roducs; searaely for each roduc x ad roduc y whe (y dmeso of he curve) ad (x dmeso of he curve) vary smulaeously. Also z dmeso of curve shows bullwh effec measure a roosed value of ad. I fac, whe a realer creases umber of erods hs movg average calculaos ad suler reduces lead-me of rocureme, bullwh effec decreases. So we have mmum of bullwh measure maxmum value of ad mmum value of. Whle decreases or creases, bullwh effec measure creases, as we ca see Fgures (a) ad (b), he maxmum o of he fla curve occurs maxmum measure of lead-me ad he mmum measurer of movg average arameer (). I fac coordao realer ad suler seleco of ad ca reduce bullwh effec of roducs. 6 3 5 6 7 8 9 0 Fg. 3(a). Bullwh effec varao wh resec o for roduc x Fgure 3(b) shows bullwh effec curve for he roduc y wh resec o umber of observaos leadme demad forecasg,. I s clear ha bullwh effec decreases whe umber of observaos creases. A dramac fall ca be see whe lead-me of he secod roduc creases from o. Sloe of curves bewee bullwh effecs of wo roducs dffers Fgure 3(a) ad Fgure 3(b). Fg. (a). Bullwh effec varao wh resec o ad for roduc x 6
Ahmad Sadegh/ Measurg ad Aalyzg he Bullwh... Fg. (b). Bullwh effec varao wh resec o ad for roduc y 5. Cocluso I hs aer, a wo-echelo suly cha wh wo roducs s cosdered ad bullwh effec values are measured; usg sysem egeerg ad block dagram aroach. emad of each roduc deeds o he demad of he oher oe. Ths relaosh s descrbed by VAR () model. Realer used OUT olcy hs releshmes for boh of roducs. Order of frs roduc s deede o he secod roduc. We assumed ha realer uses smle movg average mehod o forecas lead-me demad of each roduc deedely. Afer descro of he model, we derved a closed form of bullwh effec rao for boh of roducs usg block dagram. Fally, he las seco we aalyzed behavor of he bullwh effec hrough a umercal examle. We showed ha lead-me reduco, as well as, creasg umber of observao lead-me demad forecasg ca reduce bullwh effec for boh roducs, smulaeously. I oher words, f realer uses more hsorcal formao forecasg lead-me demad, bullwh effec ca be decreased. I addo, f a maufacurer decreases lead-me, he he bullwh heomeo ca be removed from he suly cha. Ths research maly focused o wo-roduc suly cha whle he basc roblem suly eworks s remaed. If we have several roducs (as we see real world), how ca we measure ad aalyze bullwh measures? Aoher drawback of our research s he assumo abou orderg olcy. If orderg of wo roducs deeds o each oher, how ca quafy bullwh raos? So coug he research he above areas s srogly recommeded. 6. Refereces Buchmese, B. Frscc,. (0). Bullwh effec sudy a cosraed suly cha. Proceda Egeerg, 69, 63-7. Chaharsoogh, S.K. Sadegh, A. (009). Measurg he bullwh effec suly chas wh vecor auoregressve demad rocess. Kuwa Joural of Scece ad Egeerg, 36, -30. 3 Chadra, C. Grabs, J. (005). Alcao of mul-ses forecasg for resrag he bullwh effec ad mrovg veory erformace uder auoregressve demad. Euroea Joural of Oeraoal Research, 66, 337-350. Chafeld,.C. Prchard, A. M. (03). Reurs ad he bullwh effec. Trasorao Research Par E: ogscs ad Trasorao Revew, 9, 59 75. 5 Che, F. rezer, Z. Rya, J.K. Smh-ev,. (000a). Quafyg he bullwh effec a smle suly cha: The mac of forecasg, lead mes ad formao. Maageme Scece, 6, 36-3. 6 Che, F. rezer, Z. Rya, J.K. Smh - ev,. (000b). The mac of exoeal smoohg forecass o he bullwh effec. Naval Research ogscs, 7, 69-86. 7 ejockheere, J. sey, S.M. ambrech, M.R. Towll,.R. (003). Measurg ad avodg he bullwh effec: A corol heorec aroach. Euroea Joural of Oeraoal Research, 7, 567-590. 8 sey, S.M. Towll.R. (003). O he bullwh ad veory varace roduced by a orderg olcy. Omega, 3, 57-67. 9 Fazel Zarad. M.R. Gamasaee. R. (03). A ye- fuzzy sysem model for reducg bullwh effecs suly chas ad s alcao seel maufacurg. Scea Iraca, 0, 879-899. 0 Forreser, J.W. (958). Idusral dyamcs - a major break hrough for decso-makg. Harvard Busess Revew. Jaura, S. Mahaara, S. (0). A mroved demad forecasg mehod o reduce bullwh effec suly chas. Exer Sysems wh Alcaos,, 395-08. Jaksc, M. Rusja, B. (008). The effec of releshme olces o he bullwh effec: A rasfer fuco aroach. Euroea Joural of Oeraoal Research, 8, 96-96. 3 Km, J.G. Chafeld,. Harrso, T.P. Hayya, J.C. (006). Quafyg he bullwh effec suly cha wh sochasc lead-me. Euroea Joural of he Oeraoal Research, 73, 67-636. ee, H.. Padmaabha, V. Whag, S. (997). The bullwh effec suly chas. Sloa Maageme Revew, 38, 93-0. 5, Q. sey, S.M. Gaalma, G. (0). Avodg he bullwh effec usg damed red forecasg ad he Order-U-To releshme olcy. Ieraoal Joural of Produco Ecoomcs, 9, 3 6. 6 uog, H.T. (007). Measure of bullwh effec suly chas wh auoregressve demad rocess. Euroea Joural of Oeraoal Research, 80, 086-097. 7 uog, H.T. Phe, N.H. (007). Measure of bullwh effec suly chas: The case of hgh order auoregressve demad rocess. Euroea Joural of Oeraoal Research, 83, 97-09. 8 Maku, A. Madad, A. (007). The bullwh effec ad yauov exoe. Aled Mahemacs ad Comuao, 89, 35-0. 9 Neal. B, Mura. A, Cham. R.B. (0). The bullwh effec caacaed suly chas wh cosderao for roduc lfe-cycle asecs. Ieraoal Joural of Produco Ecoomcs, 36, 38-33. 6
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