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

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1 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 Keyword: Abrac: Demand Correlaon Bullwhp Effec Supply Chan Suppler Realer. In a wo-age upply chan wh wo realer f hey have correlaed cuomer demand forecang baed on her repecve hory order mgh caue gnfcan foreca naccuracy. Curren foreca mehod only ue upply chan member own hory demand nformaon. However when here are mul-realer havng correlaed demand he common forecang mehod gnore he foreca error caued by realer neracon. Then a queon come up ha wha he relaon beween h foreca error and he bullwhp effec. The preen paper ude relaon of mul-ermnal demand correlaon and bullwhp effec n a wo-age upply chan wh wo realer. Under cenralzed or decenralzed nformaon ( he mpac of realer demand correlaon on realer /uppler bullwhp effec uded; ( he conra of uppler and realer bullwhp effec and he conra of uppler / realer bullwhp effec under dfferen nformaon harng condon are uded. The ude how ha mul-ermnal demand correlaon a caue of upply chan bullwhp effec. BACKGROUND Today modern upply chan face more dverfed demand of cuomer and more nene horzonal compeon among he pare n he ame level of a upply chan. Epecally n a upply chan producng a homogeneou produc demand of he pare n he ame level undoubedly ge affeced by her neracon. However h correlaon no condered n common forecang mehod uch a movng average exponenal moohng or emprcal forecang. For example n one communy here are ofen more han one upermarke or convenence ore facng he ame group of cuomer and provdng produc ame n prce qualy or ervce. I obvou ha demand of hee ermnal hould be hghly correlaed. When he manager of uch real ermnal make order baed on one of he ced foreca mehod f he or he gnore h correlaon he foreca naccuracy would caue a evere nvenory backlog or ock-ou. Wha he relaonhp beween real ermnal foreca naccuracy caued by her demand correlaon and he upply chan bullwhp effec? Or more pecfcally wha characer of demand correlaon are relaed o he bullwhp effec? Under wha crcumance (uch a cenralzed nformaon or decenralzed nformaon may ermnal demand correlaon caue bullwhp effec? Alhough ubanal reearch ha been done on bullwhp effec n vercal upply chan no much reearch ha been performed on bullwhp effec n upply chan havng horzonal compeon. In he preen paper we focu on he relaon beween demand correlaon and bullwhp effec. ITERATURE REVIEW ee e al. (997 prove he exence of bullwhp effec and decrbe wh AR ( demand proce. aer ee e al. (000 prove ha bullwhp effec can be reduced by upply chan nformaon harng. Chen e al. (000a quanfy he bullwhp effec n a wo-echelon upply chan wh a ngle manufacurer and a ngle realer. They examne he mpac of forecang (movng average forecang and exponenal forecang and order lead me on he bullwhp effec and conclude ha bullwhp effec would ex f order lead me no zero and ha he bullwhp effec would become 304 J J. H. Zhang J. and eng C.. The Impac of Demand Correlaon on Bullwhp Effec n a Two-age Supply Chan wh Two Realer. DOI: 0.50/ In Proceedng of he Inernaonal Conference on Operaon Reearch and Enerpre Syem (ICORES-05 page ISBN: Copyrgh c 05 SCITEPRESS (Scence and Technology Publcaon da.

2 TheImpacofDemandCorrelaononBullwhpEffecnaTwo-ageSupplyChanwhTwoRealer more evere wh larger order lead me. aer hey exend he concluon no a mul-age upply chan and reveal ha nformaon harng reduce bu no elmnae he bullwhp effec. uong (007 ue a forecang procedure ha mnmze he expeced mean-quare foreca error o emae he lead me demand and conclude ha he varance of order wll ncreae wh ncreang order lead me. In a laer paper uong and Phen (007 udy he bullwhp effec baed on a AR( demand proce and exend no a AR(p demand proce. They fnd ou ha n dfferen range of auoregreve coeffcen he relaon beween lead me and bullwhp effec become complcaed ha he bullwhp effec doe no alway ex and doe no alway ncreae when lead-me ncreae. e al. (006 reearch he mpac of dfference demand proce on he bullwhp effec and negrae a general ARIA (pdq demand proce no he model o analyze he valdy of he producon-moohng model. They fnd ou he an-bullwhp effec and he o-called lead-me paradox and hey alo udy he value of nformaon harng n upply chan. 3 ODE DESCRIPTION Fgure : One Suppler and Two Realer Srucure. In he above upply chan wh one uppler and wo parallel realer here ex demand correlaon beween he wo uppler. Here he concep of correlaon : ( A any perod realer demand nformaon deermned no only by own hory demand bu realer hory demand. ( A any perod he random error par of realer demand nformaon correlaed wh ha of realer. However he random error par of realer demand nformaon a perod ndependen wh ha of realer a a dfferen perod. Th aumpon n form of Cov( j Cov( j ( Cov( 0 j Generally a he end of perod he wo realer place order O (= o he uppler baed on her curren repecve nvenory poon. The uppler wll hp he produc once receve he order. Conderng he ranporaon delay we aume ha he hpmen wll arrve a he realer a he end of perod (+ and here conan mean he ame order lead me of he wo realer. 4 DEAND FORECAST AND ORDER-UP-TO POICY A menoned n he leraure revew forecang mehod ued n mo of he prevou reearch on bullwhp effec nclude he ovng Average (Ahe Exponenal Smoohng (ES and he opmal forecang mehod (or nmum ean Square Error foreca SE foreca (zhang 004 Heyman and Sobel 003 Johnon and Thompon 975 Chen e al In pracce he A he mo common forecang mehod. The advanage of h mehod ha eay o ue and ha good enough o deermne he curren change of rend when accuracy no rcly requeed. The man dadvanage ha he movng average are laggng ndcaor becaue he mehod agn he ame wegh raher han greaer wegh o he more recen hory daa whle n pracce he more recen changng rend more mporan. The ES relavely more uable n hor-o-medum erm forecang for ha more enve o recen changng rend. However no ha eay o ue becaue can be complex o chooe a proper moohng facor. The opmal foreca mehod he SE foreca whch uable n hor-o-medum erm foreca enve o recen changng rend hgh n forecang accuracy and he mo complex o ue n comparon wh oher mehod. We aume ha he wo realer ue he SE foreca mehod o emae he lead me demand. A he end of perod hory demand equence of real H D D... D D. Through he SE foreca we can ge foreca of demand n nex perod (here he lead me F D D... D D where condonal expec D ED ( D D... D0. We aume ha he wo realer follow order-up-o nvenory polcy. Ther repecve order-up-o pon are deermned by lead me demand foreca a he end of perod. Then we have y ˆ ˆ ˆ D D... D Z ˆ where ˆ an emae of he andard varance of realer 305

3 ICORES05-InernaonalConferenceonOperaonReearchandEnerpreSyem foreca error durng lead me and Z a conan meaurng realer ervce level. 5 ODE NOTATION We aume ha demand of he wo realer are correlaed whch a -dmenon AR( proce. d a d d d ( a d d are..d. followng a drbuon wh mean 0 and afe Var( Cov( j (3 Cov( j Cov( j 0 I obvou ha expreon ( become wo ndependen AR( procee when 0. For he aonary of AR proce we hould chooe proper o make he roo of ( x( x locae n he un crcle. e denoe repecvely he mean of he wo realer demand we have ( a a ( ( ( a a ( ( (4 To enure he pove value of μ and μ he followng condon hould be afed: a 0 a 0( a 0 ( a 0 (5 To mplfy expreon ( we make z d and ( can be ranferred a z z z (6 z z z Denoe z y Z Y z y A D d Z d ( Var (7 We ge he marx form of ( a below where he characerc roo of A or marx E-A nverble. Z AZ 0 (8 5. Bullwhp Effec of he Two Realer and he Suppler wh Cenralzed Demand Informaon Cenralzed demand nformaon mean ha realer hare hory demand equence H wh each oher o each realer can foreca and make order decon baed on boh realer hory demand. We ubue Z AZ for Z AZ n expreon (6 and connue h eraon o he end: Z AZ A Z A... AZA A... A (9 From E( 0 we can have EZ ( AZ. Becaue for any ARA proce SE foreca of demand of perod + equal condonal expecaon he SE foreca of Z Z Z are Z AZ Z A Z... Z AZ (0 Then he lead me demand foreca D ( Z ( AA... A Z ( EA ( AA Z The lead me demand foreca error ( D Z ( Z Z ( A A... E ( A... E... ( AE Varance of lead me demand foreca error Var ( D D Var ( Z Z ( A A... E ( A A... E ( A... E ( A... E... ( AE ( AE + ' ' ( ( (3 o Denoe O o a he marx form of realer order quany and we have 306

4 TheImpacofDemandCorrelaononBullwhpEffecnaTwo-ageSupplyChanwhTwoRealer Table : Parameer Decrpon. Parameer Decrpon d d The demand of Realer a perod d d The demand foreca of Realer a perod The random varable of demand nformaon faced by Realer repecvely a perod. Here denoe he varance of realer random varableand j j denoe he correlaon of wo realer random varable The random varable of demand nformaon faced by Suppler le The auocorrelaon coeffcen of Realer The correlaon coeffcen decrbng he correlaon beween Realer and o o The order quany of Real a perod ˆ ˆ o o The foreca order quany of Realer a perod o The order quany of Suppler a perod le o o o o ˆ The foreca order quany of Suppler a perod le oˆ ˆ ˆ o o The meaure of Bullwhp Effec of Realer and Suppler O Y Y D D D D ( Z ( Z Z A Z ( E A ( E A (4 Realer foreca order quany ˆ O A Z (5 Varance of realer order quany error e Var O Oˆ Var E A E A (6 ( (( ( B ( EA ( E A and hen we have ˆ ' ' ' Var( O O BE( B B B (7 Aume ha realer order lead me = and we have ( ( ( ˆ O B E A E A E A Var O (8 Hence we ge he Bullwhp Effec of he wo realer a he below: Var( O Oˆ ( ( Var O ( ( Var ( ˆ O Var (9 Alo he Bullwhp Effec of he uppler Var( o o Var( Var( o o o o Var( ( ( / ( ( 5. Bullwhp Effec of he Two Realer and he Suppler wh Decenralzed Demand Informaon (0 Decenralzed demand nformaon mean ha realer ake each oher a compeor and hey do no hare nformaon of hory demand equence. Baed on h aumpon each realer can foreca and make order decon baed on only own hory demand. Accordng o expreon (6 we have z z z z ( z z ( z z z z Now ubue z z z for n he equaon above and we have z ( z ( z ( 307

5 ICORES05-InernaonalConferenceonOperaonReearchandEnerpreSyem Followng he ame procedure we have z ( z ( z e v jj j j equaon ( and (3 become (3 z z z v (4 Noce ha each realer only ha own hory demand equence. From equaon ( and (3 realer can emae he auo-regreon erm n he equaon and he auo-correlaon par n he error erm whle realer canno emae he correlaon par n he error erm. Hence neher of he realer can foreca he fuure demand baed on own hory demand equence. emma Realer can ue a able and nverble ARA proce o model hory demand. z z z Var( v Var( v 4 Cov( v v Cov( v v where and he error erm afe ( E 0 ' E( ' 0 ( E( Var( v /( Baed on emma (and(3become z z z z z z where are..d. Aume ha realer order lead me = and we can ge he lead me demand foreca and foreca error a below d z Ez ( H z z (5 d d z z Hence we ge he varance of wo realer order lead me demand foreca error Var ( d d (6 Under decenralzed nformaon realer order quany o d d d z z z z ( z z ( (7 Realer foreca order quany Oˆ ( z z (8 Varance of realer order quany error o oˆ ( ˆ (9 Var( o o ( Hence we ge he Bullwhp Effec of he wo realer a below Var( o oˆ ( (30 Var Alo he Bullwhp Effec of he uppler Var( o o Var( o o o o Var( Var( ( ( 6 BUWHIP EFFECT ANAYSIS AND COPARISION (3 In h ecor we analyze he mpac of demand correlaon on realer and uppler Bullwhp Effec. To elmnae he poble nfluence of oher parameer we aume ha = = =. Th aumpon reaonable n pracce becaue n he ame local marke here are ofen wo realer mlar n boh marke hare and produc old. 6. Numercal Analy of under Cenralzed Informaon Wh condon of cenralzed nformaon = and SE forecang he wo realer face bullwhp effec a below 308

6 TheImpacofDemandCorrelaononBullwhpEffecnaTwo-ageSupplyChanwhTwoRealer Var( o o Var ( ( Var( o o Var ( ( When = = = ge (3 we =. Then obvou ha d ( j d d j and j or d j jj j ( j j (33 d e 0 d we ge j ( j j jj From j we ge 0 o d / d j and j have he ame gn. Fgure Fgure 3 and Fgure 4 dplay he relaon beween Bullwhp Effec and demand correlaon under cenralzed nformaon. e Noce ha o enure he ably le ( vare wh a hown n Fgure. e To enure he ably ( vare wh j a hown n Fgure =0.5 = -0-0 (hn->hck = (hn->hck Fgure 3: Suppler Bullwhp Effec under Cenralzed demand nformaon. Nex compare he realer bullwhp effec wh he uppler under cenralzed demand nformaon when = = = : Rao (34 [( ( ] / e Rao vare wh a hown n Fgure =0.5 =-0-0 (hn->hck.0 Raon Raon = -0.5 = -0-0 (hn->hck Fgure : Realer Bullwhp Effec under Cenralzed demand nformaon =-0.5 =-0-0 (hn->hck Fgure 4: Bullwhp Effec Conra under Cenralzed demand nformaon. 309

7 ICORES05-InernaonalConferenceonOperaonReearchandEnerpreSyem 6. Numercal Analy of under Decenralzed Informaon Wh condon of decenralzed nformaon = and SE forecang he wo realer face bullwhp effec a below o ( Var( o (35 Var where Vv ( Vv ( 4 Covv ( v (36 Cov( v v Vv ( Vv ( 4 Covv ( v When vare wh a hown n Fgure 5. When vare wh a hown n Fgure 6. j =0.5 =-0-00 (hn->hck =-0.5 =-0-00 (hn->hck Fgure 6: Suppler Bullwhp Effec under Decenralzed demand nformaon =0.5 =-0-0 (hn->hck =-0.5 =-0-0 (hn->hck Fgure 5: Realer Bullwhp Effec under Decenralzed demand nformaon. Under decenralzed nformaon bullwhp effec of he uppler ( ( ( (37 Under decenralzed nformaon f = = = we have =.Now we compare he realer bullwhp effec wh he uppler. Raon [( ( ]/ ( (38 [( ]/ When Raon vare wh j a hown n Fgure 7. Raon =0.5 =-0-00 (hn->hck Fgure 7: Bullwhp Effec Conra under Decenralzed demand nformaon. 30

8 TheImpacofDemandCorrelaononBullwhpEffecnaTwo-ageSupplyChanwhTwoRealer Raon.0 R =-0.5 =-0-00(hn->hck Fgure 7: Bullwhp Effec Conra under Decenralzed demand nformaon (con Bullwhp Effec Conra beween Cenralzed and Decenralzed Demand Informaon In h econ we analyze he realer /uppler bullwhp effec conra beween cenralzed and decenralzed nformaon. e R repreen he rao of realer bullwhp effec under cenralzed nformaon o ha under decenralzed nformaon ( decenralzed R ( cenralzed [( ] / [( ( ] / j j j jj ( j j j jj ( ( When (39 we ge R R and R vare wh a hown n Fgure 8. e S repreen he rao of uppler bullwhp effec under cenralzed nformaon o ha under decenralzed nformaon S ( ( ( ( ( decenralzed cenralzed R ( =-0.5 = (hn->hck Fgure 8: Realer B.E. Conra Beween Cenralzed And Decenralzed D-I (con.. When S vare wh j a hown n Fgure =0.5 = (hn->hck =-0.5 = (hn->hck Fgure 9: Suppler B.E. Conra Beween Cenralzed And Decenralzed D-I. S S CONCUSIONS AND INSIGHTS =0.5 = (hn->hck Fgure 8: Realer B.E. Conra Beween Cenralzed And Decenralzed D-I. 7. an Concluon ( Under decenralzed nformaon when >0: 3

9 ICORES05-InernaonalConferenceonOperaonReearchandEnerpreSyem monoone ncreang a abolue value of ncreae. R aon vare around and monoone decreang a ncreae no gnfcan. Th uaon ndcae ha no rongly relaed o he amplfcaon of bullwhp effec gong up he upply chan. ha lle mpac on bullwhp effec. ( Under cenralzed nformaon when >0: Bullwhp effec of realer/uppler monoone ncreang a ncreae and he amplfcaon gnfcan. When >0 Raon cenr / cenr >and Raon monoone ncreang a ncreae. I mean ha he amplfcaon of varance of order n uppler age larger han ha n realer age and h dfference ncreae wh he value of. Th uaon ndcae ha larger wll ncreae he amplfcaon of varance of order quany preadng o he upream upply chan. When <0 R aon < and Raon monoone ncreang a ncreae. I mean ha he amplfcaon of varance of order n uppler age maller han ha n realer age and h dfference decreae a ncreae. The mpac of number of age of upply chan on bullwhp effec no effeced by. j ha lle mpac on bullwhp effec. (3 When <0 and boh have lle mpac on bullwhp effec. 7. anagemen Ingh To um up wha we hould pay aenon o are a followng: ( When realer demand are pove correlaed no maer under cenralzed or decenralzed nformaon h correlaon ha gnfcan mpac on realer /uppler bullwhp effec. ( Under decenralzed nformaon boh realer and uppler bullwhp effec ncreae a he abolue value of realer demand correlaon ncreae and bullwhp effec n uppler age and realer age are almo he ame. (3 Under cenralzed nformaon when realer demand are pove correlaed boh realer and uppler bullwhp effec ncreae a realer demand correlaon ncreae and bullwhp effec level n uppler age larger han ha n realer level. I ndcae ha under cenralzed nformaon he mpac of number of upply chan age on bullwhp effec relaed wh he realer demand correlaon. (4 Under cenralzed nformaon when and only when realer demand are negave correlaed ( j 0 he uppler bullwhp effec wll be le han realer. I ndcae ha under cenralzed nformaon uppler demand foreca become more accurae a he reul of realer compeon. Hence when realer demand are correlaed bede he well-known caue of bullwhp effec (uch a lead me number of upply chan age any member n he upply chan hould conder he mpac of mul-ermnal demand correlaon on bullwhp effec when makng producon plan. Furhermore under cenralzed nformaon when realer demand are pove correlaed he bullwhp effec n uppler age hgher han ha n realer age; on he conrary under cenralzed nformaon when realer demand are negave correlaed he bullwhp effec n uppler age lower han ha n realer age. Thee concluon provde heorecal reference abou bullwhp caued by ermnal demand correlaon for enerpre o make producon plan. ACKNOWEDGENTS Th reearch uppored n par by Naonal Scence Foundaon of Chna Gran ( REFERENCES Chen F. Drezner Z. Ryan J.K. and Smch-ev D. (000a Quanfyng he bullwhp effec n a mple upply chan anagemen Scence Chen F. Ryan J.K. and Smch ev D. (000 The mpac of exponenal moohng foreca on he bullwhp effec Naval Reearch ogc Heyman D.P. and Sobel.J. (003 Sochac odel n Operaon Reearch: Sochac Opmzaon (Vol. New York: Courer Dover Publcaon. Johnon G.D. and Thompon H.E. (975 Opmaly of myopc nvenory polce for ceran dependen demand procee anagemen Scence

10 TheImpacofDemandCorrelaononBullwhpEffecnaTwo-ageSupplyChanwhTwoRealer ee H. Padmanabhan V. and Whang S. (997 Informaon Doron n a Supply Chan: The Bullwhp Effec anagemen Scence ee H. So K.C. and Tang C.S. (000 The value of nformaon harng n a wo-level upply chan anagemen Scence G. Wang S.Y. and Yu G. (006 A udy on bullwhp effec and nformaon harng n upply chan Changha: Hunan Unvery Pre. (n Chnee. uong H.T. (007 eaure of Bullwhp Effec n Supply Chan Wh Auoregreve Demand Proce European Journal of Operaonal Reearch uong H.T. and Phen N.H. (007 eaure of Bullwhp Effec n Supply Chan The cae of hgh order Auoregreve Demand Proce European Journal of Operaonal Reearch Zhang X.. (004 The mpac of forecang mehod on he bullwhp effec Inernaonal Journal of Producon Economc