Analyzing the Bullwhip Effect in a Supply Chain with ARMA(1,1) Demand Using MMSE Forecasting

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1 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang choo of Economy and Managemen hangha Marme Unversy hangha 5 hna Ema:cxwang@shmu.edu.cn do:.456/ass.vo.ssue.5 Absrac Ths paper consders a suppy chan wh he frs order auoregressve and he frs order movng average ARMA demand. The ead me demand s esmaed usng he mnmum mean-suare error MME forecasng echnue. The expresson of he buwhp effec for a wo sage suppy chan conssng of one supper and one reaer s derved and he exsence condon of he buwhp effec as we as he mpac of demand nformaon sharng on he buwhp effec are anayzed heorecay and numercay. A as he exenson anayss for buwhp effec n a mupe sage suppy chan s performed by usng numerca exampes.. Inroducon Keywords: uppy chan Forecasng Buwhp effec In suppy chan managemen has been observed ha he demand varaon s ncreasngy arge as we proceed up he suppy chan9. Ths phenomenon s frs referred o as buwhp effec by ee e a.4. The buwhp effec can cause many probems for frms whn a suppy chan such as excessve nvenory ow capacy uzaon and poor cusomer servce ec 5. Receny many research wors of suppy chan managemen have been cenered on he buwhp effec mosy n he foowng areas: denfyng he possbe sources of he buwhp effec deveopng measures o reduce he mpac of buwhp effec and uanfyng and demonsrae he exsence of he buwhp effec. The exsence of he buwhp effec has been demonsraed n for posvey seray correaed demand when he reaer uses an opma nvenory pocy. In 4 fve man sources for he buwhp effec have been denfed as he use of demand sgna processng suppy shorage demand ead me order bachng and prce fucuaons and some measures have been deveoped n order o mgae he mpac of buwhp effec. Berry e a propose redesgn and reengneerng of he suppy chan as he mehods o conro demand ampfcaon. Van Acere e a. aso recommend some approaches o avod or aevae he buwhp effec..e. ead me reducon nformaon sharng or appyng dfferen repenshmen rues for he nvenory sysem. Nahmas 8 summares he poena remedes ha can hep o aevae buwhp effec. Meers 7 res o denfy he buwhp effec by deermnng an emprca ower bound on he profaby mpac of he buwhp effec. Graves uanfes he buwhp effec n a suppy chan negraed movng average demand process. In hen e a. 45 and Xu e a. based on smpe exponena smoohng forecas as we as movng average forecass he buwhp effec has been uanfed for order up o poces and he mpac of demand forecasng on he buwhp effec has been anayzed for a smpe wo-sage suppy chan conssng of one manufacurer and one reaer. Zhang derves he buwhp effec expressons usng he opma forecasng mehod ha mnmzes he mean-suared forecasng error he movng average forecasng mehod and exponena smoohng mehod. I s found ha dfferen forecasng mehods resu n buwhp effec measures wh dsnc properes n reaon o ead me and demand process parameers. uong6deveops he measure of buwhp effec for a smpe wo sage suppy chan and nvesgaes he effec of auoregressve coeffcen and ead me on hs measure. sney and Tow8 and eoncheere e a 67 have nroduced a conro engneerng-based mehod nsead of sasca mehod o uanfy he buwhp effec and he souon o he buwhp effec probem s aso presened. Hosoda and sney uanfy he varance 57

2 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang ampfcaon n a hree-echeon suppy chan wh he frs order auoregressve.e.ar demand process usng mnmum mean suare error forecasng. In hs paper as Hosoda and sney we aso use mean suare error forecasng o anayzng he buwhp effec n a suppy chan. The dfference beween hs research and Hosoda and sney s ha he demand paern n a suppy chan foows a frs order auoregressve and frs order movng average.e. ARMA process nsead of frs order auoregressve.e.ar process. The remander of hs paper s as foows. econ descrbes he basc parameers and assumpons n our research. econ derves he expresson for he buwhp effec n a smpe wo sage suppy chan conssng of one supper and one reaer boh wh nformaon sharng and whou nformaon sharng. econ 4 exends he research n econ o uanfyng he buwhp effec n a mupe sage suppy chan conssng of mupe er suppers and one reaer. ome concudng remars are dscussed n econ 5.. Basc parameers and assumpons Basc parameers : demand faced by reaer a me perod : order ead me for reaer : order ead me for supper : order-up-o nvenory eve for reaer a me perod R :order-up-o nvenory eve for supper a me perod : he order uany paced by reaer o supper a he begnnng of me perod :he order uany paced by supper o s upsream supper a he begnnng of me perod :demand durng order ead me for reaer supper :demand forecas durng order ead me for reaersupper :sandard devaon of order ead me demand forecas error for reaer supper emand Process Assumpons The fna demands faced by a reaer are assumed as random varabes n he form of foowng where > < < < < s demand auoregressve coeffcen and random error movng average coeffcen respecvey. The error erm s..d normay dsrbued wh mean zero and varance. I s assumed ha ov s < s. We furher assume ha s sgnfcany smaer han so he probaby of a negave demand s neggbe. Orderng Pocy Assumpon We assume ha he reaer and supper foow a smpe order-up-o nvenory pocy. The order-up-o pon for reaer or supper a he begnnng of me perod can hen be obaned as R λ where parameer λ s a safey facor chosen o mee a desred servce eve.. uanfyng he order uany varance n a smpe wo sage suppy chan 58

3 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March We consder a wo sage suppy chan conssng of one reaer and one supper. mar o exsng eraure he varance of order uany s used o anayze he buwhp effec n a suppy chan... Order uany varance of reaer For reaer he demand durng order ead me can be expressed as... If s he forecas wh mnmum mean-suare error of hen he mnmum mean-suare error forecas for order ead me can be deermned as... 4 For ARMA demand process can be deermned by 9... E 5 I can be obaned from ha 6 Repeaed use of 6yeds: 7 I can be obaned from 5 and 7 ha... E 8 Therefore he demand forecas durng order ead me for reaer can be expressed as 9 The sandard devaon of order ead me demand forecas error for reaer can be expressed as } { Therefore The order uany a he begnnng of me perod for reaer can be expressed as I can be easy obaned from 9 ha _ Therefore he order uany varance for reaer a he begnnng of me perod can be 59

4 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang expressed as ov ov ov ov ov 4 I can be easy shown from he properes of ARMA ha 5 ov 6 ov 7 ov 8 ov 9 I can be obaned by subsung and9no 4 ha Therefore we have foowng Proposon. Proposon If < hen ; If < hen Proof: I can be obaned from 5andha e A... we ony need o proof ha A >. when s obvous ha A >. f... when < consder A as a funcon of 6

5 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March Tang he dervave of f wh respec o we oban... < d df Therefore f s decreasng n. nce and > f > f The proof s compeed. Proposon demonsraes ha f he order uany varance of reaer s greaer han or eua o he varance of fna demand. On he oher hand f < he order uany varance of reaer s ess han he varance of fna demand. Ths ndcaes ha he buwhp effec exss a reaer ony when >. nce reaer drecy faces fna demand reaer possesses he nformaon of boh he fna demand process and fna demand hsorca daa. Reaer may rey a nformaon of fna demand o deermne hs order uany. In hs case he sgn of he dfference beween he order uany varaon of reaer and fna demand varaon s reaed o he demand auo-aggressve coeffcen and movng average coeffcen of random error... Order uany varance of supper To anayze he order uany varance of supper we consder he wo cases:demand nformaon sharng beween reaer and supper no demand nformaon sharng beween reaer and supper.... No demand nformaon sharng If he reaer does no share demand nformaon wh supper supper ony now he demand process bu does no now he hsorca demand nformaon. The demand faced by supper s he order uany paced by s downsream reaer. I can be obaned from and ha Repeaed use of yeds: The mnmum mean-suare error forecas for order uany paced by reaer can be deermned as... E 4 The forecas of he order uany durng supper s ead me can be obaned as 5 The order-up-o pon for he supper a he begnnng of me perod can be demonsraed as s z 6 mary can be shown ha s ndependen of. The order uany paced by supper a he begnnng of me perod can be expressed as 6

6 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang s s 7 Therefore he varance of order uany paced by supper a he begnnng of me perod can be expressed as ov 8 Thus we have he foowng Proposon. Proposon If f < <. Proof:Whou he oss of generay e herefore ov ov ov nce < < < < > If ony need o proof ha > ov I can be easy obaned from and ha > > > > ov < < ov nce I can be shown ha Thus If < I can be obaned fromand ha ov Therefore ony needs o proof ha > ov.e. > > 4 The above neuay s euvaen o > 5 onsderng he ef hand of 5as a uadrac funcon of he dscrmnan wh respec o can be shown as 6

7 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March -4 - nce < > he neuay 5s sasfed. The proof s compeed. Proposon demonsraes ha f he order uany varance of supper s greaer han or eua o he order uany varance of reaer. On he oher hand f < he order uany varance of supper s ess han he order uany varance of reaer. Ths ndcaes ha when demand nformaon s no shared beween supper and reaer supper ony nows fna demand process bu does no now he hsorca demand nformaon. upper ony rees on fna demand process o esmae he reaer s order uany and accordngy deermne hs own order uany. For supper he expeced vaues of random error for fna demand s zero herefore he sgn of he dfference beween he supper order uany varance and reaer order uany varance s ony reaed o fna demand auoregressve coeffcen.... emand nformaon sharng If demand nformaon sharng s acheved beween supper and reaer he order uany paced by supper can be expressed as 4: R R 6 The varance of order uany paced by supper wh demand nformaon sharng can be expressed ov 7 Thus we have he foowng Proposon. Proposon If. Proof Accordng o 8 6

8 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang 9 ubsung 56and 9no 7 can be obaned - 4 I can be shown from 8and 4 ha 4 Obvousy ony needs o proof ha > If > ony need o proof > f s obvousy ha he above neuay s sasfed. f < > ony need o proof ha > If / hen s aways posve. On he oher hand f < / hen <. nce < > >. 64

9 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March The proof s compeed. Proposon demonsraes ha f he order uany varance of supper s greaer han or eua o he order uany varance of reaer. On he oher hand f < he order uany varance of supper s ess han he order uany varance of reaer. Ths ndcaes ha when demand nformaon s shared beween supper and reaer supper no ony nows fna demand process bu aso nows he hsorca demand nformaon. upper may rey on a demand nformaon o deermne hs order uany. In hs case he sgn of he dfference beween he supper order uany varance and reaer order uany varance s reaed o fna demand auo-aggressve coeffcen and movng average coeffcen of random error.... Impac of demand nformaon sharng on supper s order uany varaon To anayze he mpac of demand nformaon sharng on supper s order uany we use numerca anayss. e based on8and 7we can cacuae. The cacuaon resu s shown n Tabe.ue o he space m we ony demonsrae par of he resus I can be shown from Tabe ha f hen <. On he oher hand f < hen >. We can oban he same concuson by changng he vaues of and. Ths ndcaes ha f he demand auoregressve coeffcen s posve demand nformaon sharng can reduce order uany varaon and vse versa. 4. uanfyng he order uany varance n a mupe sage suppy chan In pracce for reaer here exs mupe er suppers n a suppy chan. Thus n hs secon we consder a mupe sage suppy chan conssng of mupe er suppers and one reaer. Based on he resu n econ he order uany a h er supper can be expressed as 4 f f f f 4 4 where he random error coeffcens f 4 are funcons of and and can be derved from demand process. mary repeaed use of 4 yeds: g g... g Where he random error coeffcens g 4 are aso funcons of and and can be derved from No demand nformaon sharng When demand nformaon s no shared he forecas vaue of demand faced by he upsream supper for h er supper.e. h er supper can be expressed as Tabe. acuaon resu of

10 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang Noe:The daa n frs row and frs coumn are he vaues of and respecvey. E The forecas vaue of demand durng h er supper s ead me can be obaned as 45 Where s h er supper s ead me. mary The order uany paced by h er supper a he begnnng of me perod can be obaned as 46 Therefore he varance of order uany paced by h er supper a he begnnng of me perod can be expressed as ov emand nformaon sharng If a suppers and reaer acheve demand nformaon sharng hen he order uany a h er supper can be expressed as R R

11 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March Therefore when demand nformaon s shared he varance of order uany a h er supper can be obaned as ov 49 nce s dffcu o anayze he change behavors of order uany varance for mupe er suppers we perform numerca smuaon anayss. In hs paper we ony demonsrae he anayss resu for wo er suppers. e 4 and we cacuae he vaues of and. The cacuaon resus are shown n Tabe Tabe and Tabe 4 respecvey. We perform a grea dea of numerca anayss for mupe er suppers suaon he foowng concusons can be drawn from numerca anayss: Frs n a suppy chan whou demand nformaon sharng f... < <... <. On he oher hand f < Ths ndcaes ha n a suppy chan whou demand nformaon sharng f demand auoregressve coeffcen s posve he order uany varaon s ncreasngy arge as we proceed up from downsream supper o upsream supper n a suppy chan. If demand auoregressve coeffcen s negave he order uany varaon s ncreasngy sma as we proceed up from downsream supper o upsream supper n a suppy chan. econd n a suppy chan wh demand nformaon sharng f hen.... Ths ndcaes ha f order uany Tabe. The vaues of

12 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang Noe:The daa n frs row and frs coumn are he vaues of and respecvey. Tabe. The vaues of Noe:The daa n frs row and frs coumn are he vaues of and respecvey. Tabe 4. The vaues of

13 Inernaona Journa on Advances n Informaon cences and ervce cences Voume Number March varaon ampfcaon s exss n a suppy chan wh demand nformaon sharng. Thrd for h er supper f hen. On he oher hand f < hen <. Ths ndcaes ha f can reduce order uany varaon and vce vsa. 5. oncusons demand nformaon sharng Ths paper consders a suppy chan wh he frs order auoregressve and he frs order movng average ARMA demand. The ead me demand s esmaed usng he mnmum mean-suare error MME forecasng echnue. The expresson of he buwhp effec for a wo sage suppy chan conssng of one supper and one reaer s derved and he exsence condon of he buwhp effec as we as he mpac of demand nformaon sharng on he buwhp effec are anayzed heorecay and numercay. A as he exenson anayss for buwhp effec n a mupe sage suppy chan s performed usng numerca exampes. The foowng concusons can be drawn from above anayss. In a suppy chan whou demand nformaon sharng he buwhp effec does no necessary exs. If boh demand auoregressve coeffcen and he sum of demand auoregressve coeffcen and random error movng average coeffcen are posve he order uany varaon s ncreasngy arge as we proceed up from reaer o upsream supper n a suppy chan. On he oher hand f boh demand auoregressve coeffcen and he sum of demand auoregressve coeffcen and random error movng average coeffcen are negave he order uany varaon s ncreasngy sma as we proceed up from reaer o upsream supper n a suppy chan. In a suppy chan wh demand nformaon sharng he buwhp effec s no necessary aevaed. If boh demand auoregressve coeffcen and he sum of demand auoregressve coeffcen and random error movng average coeffcen are posve he buwhp effec s aways exss. In a suppy chan wh demand nformaon sharng f demand auoregressve coeffcen s posve he order uany varaon for each sage can be reduced by demand nformaon sharng. On he oher hand f demand auoregressve coeffcen s negave he order uany varaon for each sage w be enarged by demand nformaon sharng. 69

14 Anayzng he Buwhp Effec n a uppy han wh ARMA emand Usng MME Forecasng huanxu Wang 6. Acnowedgemen Ths paper s suppored by Naona Naura cence Foundaon of hna Gran No and cence and Technoogy Program of hangha Marme Unversy Gran No References Berry. Nam M. Tow.R. Busness process reengneerng an eecroncs producs suppy chan. In: Proceedngs of IEE cence Measuremen and Technoogy Bnder A.. Invenores and sucy prces. Amercan Economc Revew Bnder A.. an he producon smoohng mode of nvenory behavor be safe? uarery Journa of Economcs henf.rezner Z. Ryan J.K. ev.. uanfyng he Buwhp Effec n a mpe uppy han:the Impac of Forecasng eadmes and Informaon Managemen cence hen F. rezner Z. Ryan J.K. ev... The Impac of Exponena moohng Forecass on he Buwhp Effec Nava Research ogscs eoncheerej. sney.m. ambrech M.R. Tow.R. Measurng and avodng he buwhp effec: A conro heoreca approach. European Journa of Operaona Research eoncheerej. sney.m. ambrech M.R. Tow.R. The Impac of Informaon Enrchmen on he Buwhp Effec n uppy hans: A onro Engneerng Perspecve European Journa of Operaona Research sney.m. Tow.R. On he buwhp and nvenory varance produced by an orderng pocy. Omega Forreser J. Indusra dynamcs: A maor breahrough for decson maers Harvard Busness Revew Graves.. A snge-em nvenory mode for a non-saonary demand process. Manufacurng and ervce Operaons Managemen Hosoda T. sney.m. varance ampfcaon n a hree-echeon suppy chan wh mnmum mean suare error forecasng Omega Kahn J. Invenores and he voay of produconj Amercan Economc Revew ee H.. Padmanabhan V. Whang. The Buwhp Effec n uppy hans oan Managemen Revew prng ee H.. Padmanabhan V. Whang. soron n A uppy han:the Buwhp Effec Managemen cence ee H.. Ku.. Tang.. The Vaue of Informaon harng n a Two-eve uppy han. Managemen cence uongh.t. Measure of Buwhp Effec n uppy hans wh Auoregressve emand Process European Journa of Operaona Research Meers R. 997 uanfyng he Buwhp Effec n uppy hans. Journa of Operaons Managemen Nahmas. 997 Producon and Operaon Anayss hrd edon McGraw-H Pndyc R.. Rubnfed.. Economerc Modes and Economc Forecass. Irwn McGraw-H 998. Van Acere A. arsen E.R. Morecrof J..W. ysem hnng and busness process redesgn: An appcaon o he Beer Game. European Managemen Journa Xu K. ong Y. Evers P.T. Towards Beer oordnaon of uppy han Transporaon ResearchPar E Zhang X. The Impac of Forecasng Mehods on he Buwhp Effec Inernaona Journa of Producon Economcs

The Impact of Demand Correlation on Bullwhip Effect in a Two-stage Supply Chain with Two Retailers

The Impact of Demand Correlation on Bullwhip Effect in a Two-stage Supply Chain with Two Retailers The Impac of Demand Correlaon on Bullwhp Effec n a Two-age Supply Chan wh Two Realer Janhua J Huafeng Je Zhang and Cucu eng Ana College of Economc and anagemen Shangha Jao Tong Unvery Shangha 0005 Chna