Measuring the Bullwhip Effect: Discrepancy and Alignment between Information and Material Flows

Transcription

1 Measuring he Bullwhip Effec: Discrepancy and Alignmen beween Informaion and Maerial Flows i Chen Wei uo Kevin Shang S.C. Johnson Graduae School of Managemen, Cornell Universiy, Ihaca, New York 14853, USA IESE Business School, Universiy of Navarra, Av. Pearson 21, Barcelona, Spain Fuqua School of Business, Duke Universiy, Durham, Norh Carolina 27708, USA li.chen@cornell.edu wluo@iese.edu khshang@duke.edu The bullwhip effec is a phenomenon commonly observed in supply chains. I describes how demand variance amplifies from a downsream sie o an upsream sie due o demand informaion disorion. Two differen bullwhip effec measures have been used in he lieraure. Theoriss analyze he bullwhip effec based on he informaion flow i.e., order and demand informaion), whereas mos empiriciss measure i according o he maerial flow daa i.e., shipmen and sales daa). I is unclear how much he discrepancy beween hese wo measures is, and, if significan, how o reconcile he discrepancy. In his paper, we illusrae and quanify he discrepancy under hree invenory sysems. For he sysem wih saionary demand and ample supply, we show ha he bullwhip effec measure based on he maerial flow daa is always greaer han ha based on he informaion flow. For he sysem wih correlaed demand and for he sysem wih supply shorages, we derive condiions under which he overesimaion or underesimaion occurs. We find ha he discrepancy is driven by four facors: socking level, lead ime, demand correlaion, and supply service level. We furher propose a mehod o eiher eliminae or reduce he discrepancy by using he sample variances of aggregaed sales daa. Our mehod works for common demand processes wih shor-range dependence, and i does no require he knowledge of he underlying base-sock levels. In addiion, we discuss managerial implicaions of measuring he variabiliies of informaion and maerial flows in a supply chain. Key words: Bullwhip effec, informaion and maerial flows, measuremen. Version: Ocober 30,

2 1. Inroducion The bullwhip effec is a well-known phenomenon of demand informaion disorion in supply chains. Namely, demand informaion ends o be more volaile as i propagaes upsream ee e al. 1997). The bullwhip effec has a direc impac on he supply chain performance. Specifically, he increased variabiliy in he demand process requires each firm o sock more in order o mainain he same service level. The oversocking leads o higher invenory-relaed coss as well as oher indirec coss, such as labor and warehousing coss. A higher demand variabiliy also makes scheduling harder, leading o an inefficien use of resources. Thus, i is crucial for supply chain managers o undersand he causes of he bullwhip effec and o develop miigaion sraegies. Since he seminal work of ee e al. 1997), wo research sreams have emerged: modeling and empirical. In he modeling sream, researchers have sudied he bullwhip effec in sysems wih various demand processes and ordering policies, whereas in he empirical sream, researchers have measured he exen of he bullwhip effec in many real indusry cases. These wo research sreams have reinforced each oher in deepening our undersanding of he bullwhip effec. The modeling research generaes insighs and forms hypoheses for he empirical research. The empirical research idenifies where and how he effec occurs, and discovers new phenomena o moivae he modeling research. There have been wo primary bullwhip effec measures used in he modeling and empirical research. ee e al. 1997) described he bullwhip effec as a form of demand informaion disorion of a single-iem supply chain i.e., diapers in he Procor & Gamble s supply chain). The amplificaion of demand variances is measured according o he demand and order informaion. The subsequen modeling research adoped his informaion-flow definiion for analysis e.g., Cachon 1999, Chen e al. 2000, Aviv 2003, Chen and ee 2012). On he oher hand, when measuring he bullwhip effec wih empirical daa, researchers ofen had o resor o maerial-flow daa eiher iem-, firm-, or indusry-level shipmens and sales) as proxies for he order and demand informaion, because he informaion-flow daa especially he demand daa) were hard o obain see p. 460, Cachon e al for a discussion). Inuiively, a he individual iem level, he maerial-flow measure should be consisen wih he informaion-flow measure if a supply firm can always fulfill he orders placed by a downsream firm. In realiy, however, backorders do occur due o invenory shorages and/or limied producion/shipping capaciy a he supply firm. Consequenly, hese wo measures may diverge, leading o possible inconsisen measuremen of he bullwhip effec. In his paper, we invesigae his measuremen discrepancy issue. Specifically, we seek o answer he following quesions. How much is he discrepancy beween hese wo measures? Wha are 2

3 he facors ha affec he exen of he discrepancy? When hese wo measures are significanly differen, how o reconcile hem based on he available sales daa? We invesigae hese issues in he following hree invenory models. We firs consider a base model wih saionary demand and amply supply from an ouside vendor. A saionary base-sock policy is opimal for his model see p. 378, Zipkin 2000). Under such a policy, he order quaniy in a period is equal o he demand realized in he previous period, and here is no informaion bullwhip effec. In his case, he informaion bullwhip measure, defined as he raio beween he variances of order and demand, is equal o one. On he oher hand, he maerial flow comprises he shipmens from an ouside vendor and he sales o he cusomer. Because of he ample supply from he ouside vendor, he variance of he shipmens from he vendor is he same as ha of demand. Ineresingly, we show ha he variance of shipmens is always greaer han ha of sales, and ha heir difference can be expressed as a simple funcion of he produc of on-hand invenory and backorders. This resul indicaes ha he maerial bullwhip measure, defined as he raio beween he variances of shipmen and sales, is always greaer han he informaion bullwhip measure, and ha he invenory buffer has a smoohing effec on he maerial flow going downsream. A numerical sudy suggess ha his discrepancy can be as high as 60%. Cachon e al. 2007) also observed ha he variance of demand is more han wice of ha of sales in cerain indusries. Thus, our resul is consisen wih heir empirical observaion. Our analysis furher shows ha hese wo measures are close when he base-sock level is high i.e., he service level is high) or when he lead ime is long. Indeed, some empirical work has examined he invenory/service level o ensure ha he wo measures are good approximaion o each oher. For example, Bray and Mendelson 2012) invesigaed how demand informaion updaing affecs he bullwhip effec. They verified ha he sales daa were a good proxy o he demand because he firms had high invenory levels. Similarly, Osadchiy e al. 2015) sudied he relaionships of sysem risks i.e., he covariance beween sales and marke reurns) beween supply chain firms. They also confirmed ha he service levels were high in heir daa. We nex exend he base model by considering an auoregressive AR1) demand process. A sae-dependen base-sock policy is opimal for his model, and he informaion bullwhip effec exiss under he opimal policy even when he replenishmen lead ime is zero ee e al. 1997). Under he zero lead ime assumpion, we show ha here exiss a hreshold when he auocorrelaion coefficien of he demand process is above below) he hreshold, he maerial bullwhip measure underesimaes overesimaes) he informaion bullwhip measure. This discrepancy resul suggess ha, when he demand is highly correlaed beween periods, sales variabiliy can be greaer han 3

4 he demand variabiliy due o backlogging and he non-saionary base-sock policy). In his case, he high sales variabiliy may affec a firm s uilizaion of resources such as ruck flees), as well as is financial flow variabiliy if he cusomer paymens ake place upon produc delivery. We furher show ha his insigh holds for he posiive lead ime case. In he hird invenory model, we relax he ample supply assumpion. Specifically, we assume ha he orders can be backlogged a he vendor sie due o supply shorages. Boh he downsream firm and he vendor implemen a saionary base-sock policy. As a resul, here is no informaion bullwhip effec in he supply chain. We show ha he maerial bullwhip measure a he downsream firm, however, can overesimae underesimae) he informaion bullwhip measure when he basesock level a he vendor sie is eiher sufficienly high or sufficienly low wihin an inermediae range). This resul suggess ha he vendor service level can affec he discrepancy beween he wo measures. I also shows ha backlogging a he vendor sie has an inheren smoohing effec on he shipmen variabiliy o he downsream firm, as also alluded by Cachon e al. 2007). Such smoohing effec vanishes when he vendor has eiher sufficienly high or sufficienly low service levels. I is worh noing ha our analysis is based on he exac characerizaion of he maerial flow in a wo-sage invenory sysem, which has no been explored before in he bullwhip lieraure see a discussion in 2). When he discrepancy beween he informaion and maerial bullwhip measures is significan, we need o find a way o align hese wo measures. The fundamenal problem here is o esimae demand variance based on he available sales daa in he presence of backlogging). The sales in a period consis of he curren-period demand plus backorders if any) from he previous period minus he new backorders if any) of he curren period. Moreover, he aggregaed sales over muliple periods consis of he aggregaed demand plus backorders if any) from he period before aggregaion minus he backorders if any) of he final period of aggregaion. We show ha, if he wo backorder erms in he aggregaed sales are nearly) independen, such as he independen demand case or when sales are aggregaed over a long horizon and demands far apar in ime are nearly) independen, hen he demand variance can be recovered by a simple expression of he sample variances of aggregaed sales. We noe ha his esimaion mehod works for common demand processes wih shor-range dependence, and i does no require he knowledge of he underlying base-sock levels. Thus, i can be applied even if sysems are operaed under non-opimal policies. In summary, he conribuion of our sudy is hree-fold. Firs, we quanify he exen of discrepancy beween he informaion-flow and maerial-flow bullwhip measures, and idenify four facors ha affec he discrepancy: socking level, lead ime, demand correlaion, and supply service level. 4

5 One can examine he condiions of hese facors o deermine wheher or no here is a significan discrepancy beween he wo measures. Second, when he discrepancy is significan, we propose a mehod o align he wo measures based on he sales daa only. Our mehod works for demand processes wih shor-range dependence, such as he AR1) process and oher common ime-series processes. I can also be exended o he ime aggregaion and produc aggregaion frameworks of Chen and ee 2012). Finally, our comparison of demand variabiliy and shipmen/sales variabiliy also reveals ineresing managerial insighs. We show ha he maerial flow can be very differen from he informaion flow. Managers, who make operaional decisions and financial arrangemens according o he demand informaion e.g., invenory socking, labor and ransporaion scheduling, shipmen conracs, paymen erms, ec.), may also wan o ake ino accoun he maerial flow variabiliy, as he laer has a direc impac on he logisics-relaed coss and he firm s financial flow. The res of he paper is organized as follows. 2 provides a lieraure review. 3 conains he analysis of he maerial flow for a single-sage sysem wih i.i.d. demand. 4 exends he analysis o he sysem wih an AR1) demand process. 5 exends he analysis o he sysem wih supply shorages. 6 provides a discrepancy reducion mehod o align he informaion and maerial bullwhip measures. 7 concludes he paper. All proofs are presened in he Appendix. 2. ieraure Review There is an exensive lieraure on he bullwhip effec. The exising analyical models in he lieraure mosly focused on he informaion flow disorion beween orders and demand. In a singlesage model, ee e al. 1997) idenified four causes for order variabiliy amplificaion in supply chains. Cachon 1999) sudied he effec of bach orders on order variabiliies. Chen e al. 2000a, b) showed ha cerain demand forecas mehods such as moving average and exponenial smoohing) can increase order variabiliy. Aviv 2001, 2002, 2003, 2007) sudied he informaion sharing and collaboraive forecasing in supply chains and discussed he differen implicaions beween order uncerainy and variabiliy. Chen and ee 2009) proposed a generalized order-up-o policy for sudying he informaion sharing and order variabiliy conrol in supply chains. Chen and ee 2012) sudied he impac of bach size, seasonaliy, and produc aggregaion on he bullwhip effec. They poined ou ha he bullwhip effec may be masked when considering hese effecs. Our paper differs from his lieraure in ha we sudy he maerial flow variabiliy, insead of informaion flow variabiliy. 5

6 To our knowledge, Kahn 1987) is he only paper ha sudied he maerial flow variabiliy in single-sage sysems wih zero lead ime. He showed ha he variabiliy of shipmens is higher han ha of sales under a los-sales sysem wih ime-correlaed demand and a backorder sysem wih i.i.d demand. Our conribuions are as follows. Firs, we exend hese single-sage models wih posiive lead ime and show ha he difference of he shipmen and sales variabiliies is a funcion of he produc of he sysem s on-hand invenory and backorders. This is a sriking resul ha connecs he sysem s invenory saes o he sales variabiliy, which is unknown in he lieraure. Second, we exend Kahn s basic model o he wo-sage sysem and he ime-correlaed sysems wih backlogging, and find ha he variabiliy of shipmens is no always greaer han ha of he sales. Third, we provide a mehod ha based on he hisorical sales and shipmen daa o esimae he informaion bullwhip measure. Mos prior research on he bullwhip effec in muli-sage models ofen made simplifying assumpions o decouple he sages. For example, Graves 1999) assumed a high inernal service level beween he wo sages, so ha he wo sages could be analyzed separaely. ee e al. 2000) assumed ha he backordered invenory a he upsream sage can be borrowed from an alernaive source and is required o be reurned o he source afer usage he same assumpion were also used by Gaur e al. 2005, Gilber 2005, and Chen and ee 2009). Aviv 2003) cauioned ha he decoupling assumpion may lead o a poor sysem performance and proposed o use a coordinaed invenory policy for he wo sages similar policies were also used in Aviv 2001, 2002, 2007). De Kok 2012) sudied a one-warehouse-muli-reailer sysem in which he reailers forecas demand wih exponenial smoohing mehod. He showed ha reducing he invenory order frequency helps miigae he bullwhip effec. The wo-sage model in our paper is similar o he coordinaed wolevel invenory sysem of Aviv 2003). However, our focus is o analyze he maerial flow dynamics of he sysem so as o analyically compare he maerial bullwhip measure and he informaion bullwhip measure. There is also a large empirical lieraure on he bullwhip effec. Economiss hypohesized ha a producion smoohing effec exiss i.e., producion is less volaile han sales) as invenory can serve as a buffer o cope agains he demand uncerainy. However, he majoriy of he empirical evidence finds ha producion or he amoun of shipmen in our erm) is more volaile han sales in many indusries, e.g., TV se indusry Hol e al. 1968), reail indusry Blinder 1981, Mosser 1991), auomobile indusry Blanchard 1983, Kahn 1992), cemen indusry Ghali 1987), and many ohers Miron and Zeldes 1988, Fair 1989, Allen 1997). We refer he reader o Cachon e al. 2007) and Chen and ee 2015) for comprehensive reviews of his empirical lieraure. Rong 6

7 e al. 2009) sudied he forward and reverse bullwhip effecs in boh experimenal and simulaion sudies. More recenly, Udenio e al. 2012) sudied he impac of he 2008 financial crisis on he bullwhip effec, where he liquidiy reducion during he crisis leads o invenory de-socking in firms along a supply chain. Bray and Mendelson 2015) furher invesigaed he bullwhip effec and producion smoohing in an auomoive manufacuring sample. Duan e al. 2015) invesigaed he bullwhip effec in a daily produc-level daa se, and found ha he bullwhip effec a he produc level is more significan han ha measured a he firm or indusry level. Our sudy complemens his lieraure. We show ha under cerain condiions, on-hand invenory acually plays a role in smoohing he maerial flow going downsream, which provides an alernaive explanaion for hese empirical findings. Noe ha he maerial smoohing effec is differen from he order smoohing effec observed in Donselaar e al. 2010), where he order quaniy has a hard consrain under he capaciaed sysem. In fac, hese wo smoohing effecs are in he opposie direcion. The hard capaciy consrain has a smoohing effec on he order informaion flow propagaing upsream, because he order quaniy is runcaed by he capaciy consrain Chen and ee 2012). The on-hand invenory, however, has a smoohing effec on shipmens going downsream. 3. Saionary Demand Consider a periodic-review single-sage invenory sysem. Time is divided ino periods of lengh one, and we coun he ime forwards i.e., = 0, 1, 2,...). e D denoe he cusomer demand in period. In his secion, we shall assume D is independen and idenically-disribued i.i.d.) beween periods. e µ > 0) denoe he mean of he demand in a period. e D +τ of he demand from period o + τ, i.e., D +τ denoe he sum = τ i=0 D +i, wih D = D. The sysem operaes under a saionary base-sock policy. Tha is, he sysem reviews is invenory order posiion = ousanding orders + on-hand invenory - backorders) and orders up o he base-sock level s. We assume ha he sysem replenishes from an ouside vendor wih ample supply. The replenishmen lead ime is a consan. Cusomer demand will be fulfilled immediaely if he sysem has enough on-hand invenory; unme demand is fully backlogged. e E[ ] and V[ ] denoe he expecaion and he variance of a random variable, respecively. The sequence of evens is as follows: A he beginning of a period, 1) a shipmen sen from he ouside vendor periods ago is received, and 2) an order is placed. During he period, cusomer demand occurs and is fulfilled immediaely if here is on-hand invenory. To faciliae he subsequen discussion, we separae his even ino wo: 3) demand is realized and 4) a shipmen is sen o he 7

8 cusomer a he end of he period. Figure 1: Informaion and maerial flows in a single-sage sysem. Figure 1 illusraes he informaion and maerial flows in a single-sage sysem. following variables in period : Define he O 1 ) = order quaniy placed o he ouside vendor a Even 2); M 1 ) = shipmen released o he sysem from he ouside vendor afer Even 2); O 0 ) = cusomer demand occurred in period a Even 3) = D ; M 0 ) = realized sales o he cusomer a Even 4); I) = invenory level afer Even 4); I) = on-hand invenory = I)) + ; B) = backordered quaniy = I)) = I)) +, where ) + = max{, 0} and ) = min{, 0}. Since our main goal is o sudy he discrepancy beween he informaion and maerial bullwhip measures, i is useful o define hese measures. Specifically, we define he informaion bullwhip effec as he raio beween he variances of order and demand, i.e., r O = V[O 1)] V[O 0 )] = V[O 1)]. 1) V[D ] We refer o r O as he informaion bullwhip raio. This raio indicaes how significan he demand informaion is disored by he sysem. The bullwhip effec exiss if r O > 1, meaning ha he order variabiliy is amplified. We define he maerial bullwhip effec as he raio beween he variances of shipmens and sales, i.e., r M = V[M 1)] V[M 0 )]. 2) We refer o r M as he maerial bullwhip raio. When r M > 1, he maerial flow is smoohed going downsream. In our subsequen analysis, we will focus on comparing he difference beween he wo measures r M and r O. 8

9 Under he base-sock policy, he sysem orders he previous period s demand, i.e., O 1 ) = D 1. Consequenly, we have V[O 1 )] = V[D 1 ] = V[D ], and hus r O = 1, and here is no bullwhip effec for he informaion flow. Now, le us examine he maerial flow, i.e., he shipmen M 1 ) and he sales M 0 ). Since he ouside vendor has ample supply, we have M 1 ) = O 1 ) = D 1. For M 0 ), from he flow conservaion propery, we have M 0 ) = O 0 ) + B 1) B). 3) This equaion saes ha he shipmen in period is equal o he oal order o fill, i.e., O 0 ) + B 1), minus he new backorders B) afer shipping. Thus, o characerize M 0 ), we only need o characerize B) for each. We firs consider an order cycle saring a he beginning of period. The ordering decision made in period will affec he invenory variable in period +. Tha is, I + ) = s i=0 Thus, he on-hand invenory is given by I + ) = B + ) = I + )) + = O 0 + i) = s D +, s D + ) +, and he backorder is given by D + s) +. Subsiuing B + ) ino 3), we obain ) + + M 0 + ) = D + + D s D + s). 4) Since our goal is o compare he variabiliy of M 0 ) and M 1 ), we shif he ime index in 4) for his purpose. Tha is, M 1 ) = D 1, 5) M 0 ) = D + D 1 1 s) + D s ) +. 6) Based on he above equaions, we can esablish he following resul: Proposiion 1. In a single-sage sysem wih i.i.d. demand, for any given lead ime 0, [ s V[M 1 )] V[M 0 )] = 2E [I 1)B)] = 2E D 1 + 1) D s ) ] + 0, which implies r M r O. Proposiion 1 provides an explici expression of he difference beween he shipmen variance and he sales variance. Specifically, we show ha his difference is wo imes he expecaion of he 9

10 produc of he on-hand invenory and backorders. To our knowledge, his is he firs expression in he lieraure ha makes such a connecion. This expression enables us o sudy he impac of he base-sock level s as well as he lead ime on he gap beween he maerial bullwhip raio r M and he acual informaion bullwhip raio r O. I is clear from he expression ha he shipmen variance equals he sales variance only when s = 0 or s =. Thus, he maerial bullwhip raio r M is a good approximaion o he acual informaion bullwhip raio r O only when he base-sock level is sufficienly high or low. Noice ha he magniude of he discrepancy can be fairly significan. To see his, consider a special case wih = 0. In his case, V[M 1 )] V[M 0 )] = 2E [ s D 1 ) + D s) +] = 2E [I 1)] E [B)], where he difference of he variances is simply wo imes he average on-hand invenory and he average backorders in he sysem. 70% 70% Gap percenage 50% 30% 10% Base sock level Gap percenage 50% 30% 10% Base sock level a) Uniform demand over [0, 12] b) Exponenial demand wih mean 6 Figure 2: Impac of base-sock level on he percenage of he bullwhip raio gap. Figure 2a) illusraes he percenage of he gap beween r M and r O, i.e., r M r O )/r O 100%, under a uniform demand disribuion wih = 0. 1 We observe ha he gap percenage is unimodal in he base-sock level s wih he highes value of 60% when he base-sock level s = d/2. Figure 2b) illusraes he percenage of he gap beween r M and r O under an exponenial demand disribuion. 2 We again observe ha he gap percenage is unimodal in s wih he highes value around 48%. Proposiion 2 below shows he impac of base-sock level s on he gap of he wo bullwhip measures for he general lead ime case under a normal demand disribuion. Proposiion 2. When demand D follows an i.i.d. normal disribuion Nµ, σ 2 ), he expression [ s 2E D 1 + 1) D s ) ] + is unimodal in s. The gap beween r M and r O is monoonically increasing in s when s + 1)µ, and is monoonically decreasing in s when s + 1)µ. 1 V[M 1)] V[M 0)] = s2 2d 2 d s) + )2, where he demand densiy is uniform over [0, d]. 2 V[M 1 )] V[M 0 )] = 2e λs λ 2 λs + e λs 1 ), where he demand densiy is exponenial λe λ. 10

11 When demand follows a normal disribuion, he above resul shows ha he gap beween r M and r O is monoonically decreasing increasing, respecively) in he base-sock level s if he sysem service level is greaer less, respecively) han 50%. The gap reaches is highes value when s = µ. We can furher esablish he following resul abou he impac of he replenishmen lead ime on he difference of hese wo bullwhip measures: Proposiion 3. When demand D follows an i.i.d. normal disribuion Nµ, σ 2 ), given he opimal) base-sock level s = +1)µ+zσ + 1, where z is he safey facor deermined by he sysem invenory holding and penaly cos parameers. For a fixed z, 2E[ s D 1 1) + D s ) + ] is decreasing in lead-ime. Thus, he gap beween r M and r O is also monoonically decreasing in. The above proposiion shows ha he gap beween he wo bullwhip measures is decreasing in he lead ime when he opimal base-sock level is implemened. The inuiion is ha when he lead ime becomes longer, he firm has o sock more in order o mainain he same service level. A higher invenory safey sock) level decreases he magniude of backlogging, making he demand and sales closer o each oher. The implicaion obained from he above wo proposiions is ha he maerial bullwhip raio r M is an effecive approximaion o he informaion bullwhip raio r O when he sysem mainains a relaively high service level. Indeed, some empirical work has examined he service level o ensure he wo measures are good approximaion o each oher. For example, Bray and Mendelson 2012) invesigaed how demand informaion updaing affecs he bullwhip effec. They verified ha he sales daa were a good proxy o he demand because he firms had high invenory levels. Similarly, Osadchiy e al. 2015) sudied he relaionships of sysem risks i.e., he covariance beween sales and marke reurns) beween supply chain firms. They also confirmed ha he service levels were high in heir daa. 4. Correlaed Demand In his secion, we exend he base model o he correlaed demand case. Specifically, we assume ha he demand D follows he AR1) demand model as sudied in ee e al. 1997): D = µ + ρd 1 + ε, 7) where ρ < 1 and ε is an i.i.d. random variable wih normal disribuion N0, σ 2 ). I is easy o verify ha he lead ime demand D + follows a normal disribuion wih mean and variance given 11

12 as E[D + D 1 ] = ρ1 ρ+1 +1 ) D 1 + µ 1 ρ V[D + D 1 ] = σ 2 +1 ) 1 ρ k ρ k=1 k=1 1 ρ k 1 ρ, Thus, under he free-reurn assumpion of ee e al. 1997), we can show ha he opimal base-sock level in period is given by s = ρ1 ρ+1 ) 1 ρ +1 D 1 + µ k=1 1 ρ k 1 ρ + z σ +1 ) 1 ρ k+2 2, 1 ρ where z is he safey facor deermined by he sysem invenory holding and penaly cos parameers. Under he opimal base-sock policy s, he order in period is given by O 1 ) = s s 1 D 1 ) = ρ +2 D ρ+2 1 ρ µ + 1 ρ+2 1 ρ ε 1. 8) Because he ouside vendor has ample supply, we have M 1 ) = O 1 ). We can furher generalize he sales equaion 4) under he AR1) demand as follows: where z = z σ ) + ) + M 0 + ) = D + + D s 1 D + s 9) = ρ +2 D ρ ρ µ + ρ k+1 ε +k 1 k=0 + 1 ρ k ρ + ε +k 1 z) k+1 ε +k z), 1 ρ 1 ρ k= ρ k+2 k=1 1 ρ ) 2. Because V[M 1 )] = V[O 1 )], based on he bullwhip raio measures defined in 1) and 2), o compare r M and r O, i suffices o compare V[M 0 )] and V[D ]. Consider he case of = 0, hen, M 0 ) = ρ 2 D 2 + µ1 + ρ) + ε + ρε 1 + ε 1 z) + ε z) +, 10) where z = zσ. From he definiion 7), we know ha D 2 is independen of ε and ε 1. Therefore, V[M 0 )] = V [ ε + ρε 1 + ε 1 z) + ε z) +] + V[ρ 2 D 2 ]. Also, noe ha V[D ] = V[ε + ρε 1 ] + V[ρ 2 D 2 ]. Hence, o compare V[M 0 )] o V[D ], i suffices o compare he wo erms V [ ε + ρε 1 + ε 1 z) + ε z) +] and V[ε + ρε 1 ]. We noe ha when ρ = 0, he comparison reduces o ha of he i.i.d. demand case see Proposiion 1). We can esablish he following resul: k=1 k=0 12

13 Proposiion 4. In a single-sage sysem wih AR1) demand and zero lead ime, V[D ] V[M 0 )] = 2E[ z ε ) + ] E[ε z) + ] 2ρE [ ε ε z) +]. There exiss a hreshold 0 < ρ < 1, such ha, if ρ < ρ, V[M 0 )] < V[D ], which implies r M > r O ; oherwise, if ρ ρ, V[M 0 )] V[D ], which implies r M r O. The above resul shows ha, under he AR1) demand, he demand auocorrelaion coefficien ρ has a direc impac on he gap beween he wo bullwhip measures r M and r O. When he demands are moderaely correlaed, he maerial bullwhip raio r M overesimaes he informaion bullwhip raio r O. This resul generalizes he insigh from our base model wih i.i.d. demand. However, when he demand is highly posiively correlaed, he resul reverses, i.e., he maerial bullwhip raio r M underesimaes he acual informaion bullwhip raio r O. In his case, he sales variabiliy becomes greaer han he demand variabiliy. This is due o backlogging and he non-saionary base-sock policy in he sysem. In he case of posiive lead ime > 0, we can show ha he resul coninue o hold in he neighborhoods of ρ = 0 and ρ = 1. Proposiion 5. In a single-sage sysem wih AR1) demand and lead ime > 0, here exis wo hresholds ρ ρ, such ha, if 0 ρ ρ, V[M 0 )] < V[D ], implying r M > r O ; and if ρ ρ < 1, V[M 0 )] V[D ], implying r M r O. For illusraive purposes, we presen several numerical examples in Figure 3. Specifically, we fix σ = 1 and z = % service level), and plo he gap percenage beween r M and r O for differen lead imes = 1, 2, 5. Based on he numerical examples, i appears ha he resul of Proposiion 4 coninues o hold for he posiive lead ime case. The above wo proposiions reveal imporan managerial insighs on he variabiliy of he maerial flow. They show ha when he demand is highly correlaed beween periods, managers should expec ha he variabiliy of sales is higher han he variabiliy of demand, even if an opimal policy is implemened. A high variabiliy in sales affecs he uilizaion of resources such as ruck flees), ransporaion coss, as well as he financial flow variabiliy if he cusomer paymens occur upon produc delivery. Our numerical examples sugges ha all else being equal, a longer lead ime will lead o a higher sales variabiliy if he demand is highly correlaed. 13

14 Gap percenage 10% 5% 0 5% 10% 15% 20% =1 =2 =5 25% Demand correlaion Figure 3: Impac of he demand correlaion ρ on he percenage of he gap beween he informaion and maerial bullwhip raios, i.e., r M r O )/r O 100%. 5. Supply Shorages In his secion, we relax he ample supply assumpion of he base model. Specifically, we assume ha orders can be backlogged a he vendor sie due o supply shorages. For ease of exposiion, we refer o he downsream firm as Sage 1 and he ouside vendor as Sage 2. We also index he ouside vendor s supplier as Sage 3, and he end cusomer as Sage 0. There is a posiive lead ime j for Sage j. Sage 1 and Sage 2 operae under a local base-sock policy. Under his policy, Sage j= 1, 2) views Sage j 1 s order as is local demand and orders in each period. A he beginning of each period, Sage j reviews is local invenory order posiion = ousanding orders + on-hand invenory - Sage j s backorders) and orders up o he local base-sock level s j. The sequence of evens for Sage 2 is slighly differen from ha of he single-sage sysem inroduced in 3 as Sage 2 receives and fills Sage 1 s order a he beginning of a period. Specifically, 1) a shipmen sen from he Sage 2 s supplier 2 periods ago is received, 2) an order from Sage 1 is received, 3) an order is placed wih he Sage 2 s supplier, according o he updaed invenory order posiion in Even 2), and 4) a shipmen is sen o Sage 1. All hese evens occur a he beginning of a period. The sequence of evens for Sage 1 is he same as ha of he single-sage sysem in 3. We assume ha he sages perform he ordering even sequenially from downsream o upsream, whereas he shipping evens occur sequenially from upsream o downsream in a period. Figure 4 shows he wo-sage model wih he maerial and informaion flows in he opposie direcions. 14

15 Figure 4: Informaion and maerial flows in a wo-sage sysem. Define he following sae variables in period for Sage j = 1, 2: O j ) = order quaniy from Sage j o is upsream supplier; M j ) = shipmen released o Sage j from is upsream supplier a Even 4); I j ) = local invenory level for Sage j afer Even 4); B j ) = local backorders = I j )) = I j )) +. Similarly, we define O 0 ) and M 0 ) o represen he cusomer demand and sales occurred in period, respecively. We exend he definiion of he informaion bullwhip raio and he maerial bullwhip raio in 3 by adding he sage index j. Tha is, he maerial bullwhip raio for Sage j is and he informaion bullwhip raio for Sage j is r M j = V[M j)] V[M j 1 )], r O j = V[O j)] V[O j 1 )]. Clearly, O 0 ) = D. Under he local base-sock policy, Sage j = 1, 2 will order he previous period s local demand. Thus, O 2 ) = O 1 ) = D 1. Consequenly, we have r O 1 = ro 2 = 1, implying ha here is no bullwhip effec for he informaion flow in he wo-sage sysem. Below we examine he maerial flow, i.e., he shipmens M 1 ) and M 2 ), and he sales M 0 ) in he sysem. 5.1 Maerial Flow Dynamics We can apply he flow conservaion propery in 6) for j = 0, 1, 2, M j ) = O j ) + B j+1 1) B j+1 ), 11) where B 3 ) 0. To characerize M j ), we only need o deermine B j+1 ) in each period. Obaining expressions for he backorder variables in he wo-sage sysem is more complicaed, because Sage 1 may no be able o receive wha i orders from Sage 2. Thus, we have o consider 15

16 he impac of Sage 2 s order decision on he backorder variables. We firs consider an order cycle saring from Sage 2 in period. I is clear ha he ordering decision made a Sage 2 in period will direcly and indirecly affec he invenory variables from period o Specifically, Sage 2 s order in period will arrive Sage 1 in period + 2, which affecs he invenory availabiliy for Sage 1 in ha period as well as he resuling ne invenory level a Sage 1 in period We now specify he deailed dynamics. Since Sage 2 s supplier has ample supply, he invenory order posiion is equal o s 2 in each period. Thus, 2 I ) = s 2 O 1 + i) = s 2 D i=1 B ) = I )) + = D s 2 ) +. 12) Now, consider Sage 1 in period + 2. Sage 1 places an order o s 1, bu Sage 2 may have backorders. Thus, Sage 1 canno ge wha i orders so he invenory in-ransi posiion is given by IT P ) = s 1 B ), and he resuling invenory level in period is Therefore, I ) = IT P ) O 0 + i) i= 2 B ) = I )) + = = s 1 B ) D D B ) s 1 ) +. 13) Wih 12) and 13), we are able o characerize backorders B j ) in he sysem for all. Subsiuing 12) and 13) ino 11), we can derive he shipmen variables as follows: M ) = D M ) = D ) + D s 2 D D s 2 ) +, 14) D s 2 ) + s1 ) + ) ) + + D D s 2 s1. 15) Figure 8 in Appendix A provides an illusraion of hese shipmen variables for a wo-sage model. 16

17 Since our goal is o compare he variabiliy of M 0 ), M 1 ), and M 2 ), we shif he ime index in Equaions 14) o 15) for his purpose. Tha is, M 2 ) = D 1, 16) ) + ) + M 1 ) = D 1 + D s 2 D 1 2 s 2, 17) ) ) + + M 0 ) = D + D D s 2 s1 ) ) + + D 1 + D s 2 s1. 18) I is clear ha Equaions 16) and 17) have he same srucure as Equaions 5) and 6) in he single-sage sysem. Therefore, he resuls of Proposiions 1-3 can be direcly carried over o Sage 2, i.e., he maerial bullwhip raio r M 2 r O 2 = 1. Moreover, when s 2 =, Sage 2 has ample supply and he wo-sage sysem is effecively reduced o a single-sage sysem. In his case, he resuls of Proposiions 1-3 can also be carried over o Sage 1, i.e., he maerial bullwhip raio r M 1 r O 1 = 1. Below we will focus on sudying r M 1 = V[M 1 )]/V[M 0 )] in he case of s 2 <, i.e., he ouside vendor Sage 2) does no have ample supply and orders may be backlogged a he vendor sie. In his case, he variabiliy of he maerial flow a Sage 1 depends on he lead imes and he base-sock levels of boh Sages 1 and 2, which makes i difficul o obain an explici expression as in he single-sage sysem. We shall adop an alernaive approach based on he sample pah analysis o compare he variances of shipmen and sales for Sage Comparison of Bullwhip Measures e us firs consider he case wih s 2 s 1 and 2 = 1. For illusraive purposes, consider a special case wih 1 = 0 below. Under his sysem configuraion, he shipmen dynamics are given by M 1 ) = D 1 + D 2 s 2 ) + D 1 s 2 ) +, 19) M 0 ) = D + D 1 + D 2 s 2 ) + s 1 ) + D + D 1 s 2 ) + s 1 ) +. 20) Consider a demand sample pah. Since demand is random, here will be periods wih backorders. e us call he ime inerval ha conains consecuive backorder periods he backorder cycle, and he oher ime inervals he non-backorder inervals. We firs consider he non-backorder inerval. The shipmen M 0 ) in he non-backorder inerval may no necessarily be equal o he demand D because Sage 2 may have backorders. However, under he condiion s 2 > s 1, we can show ha when Sage 1 is in he non-backorder period, Sage 2 will naurally be in he non-backorder period 17

18 as well, because boh face he same local demand D. Thus, we have M 0 ) = D if is in he non-backorder inerval. We nex consider a backorder cycle from period o + τ a Sage 1, wih B 1 1) = 0, B 1 ) > 0, B 1 + 1) > 0,..., B 1 + τ 1) > 0, and B 1 + τ) = 0. In his cycle, he sequence of sales are given by s 1, if i = 0, M 0 + i) = M 1 + i), if 1 i τ 1, D +τ 1 + D +τ + D +τ 2 s 2 ) + s 1, if i = τ. 21) From he above expression, we can show ha τ i=0 M 1 + i + 1) 2 τ i=0 M 0 + i) 2. Thus, by he ergodic heorem, i follows ha V[M 1 )] V[M 0 )]. The above analysis can be furher exended o he general case wih 1 0. The resul is summarized in he following proposiion: Proposiion 6. In a wo-sage sysem wih i.i.d. demand and Sage 2 lead ime 2 = 1, for any given Sage 1 lead ime 1 0, if s 2 s 1, hen V[M 1 )] V[M 0 )], which implies r M 1 r O 1. The above proposiion can be viewed as a furher generalizaion of Proposiion 1. Here we show ha he insigh r M 1 r O 1 coninues o hold if he vendor sie is subjec o invenory shorages. However, o ensure he resul o hold, he vendor sage needs o have a shor replenishmen lead ime and a higher base-sock level han he downsream sage. We noe ha he above resul holds for any i.i.d. demand disribuion. The above resul can be generalized he sysem wih 2 > 1, which is summarized in he nex wo proposiions. Proposiion 7. In a wo-sage sysem wih i.i.d. demand and lead imes 1 0 and 2 1, for any s 1, here exiss a hreshold s 2 s 1 ) 0, such ha for any s 2 s 2 s 1 ), V[M 1 )] V[M 0 )], which implies r M 1 r O 1. The above resul provides furher suppor for he inuiion ha r M 1 r O 1 when he vendor sage keeps a relaively high base-sock level. We noe ha, when 2 = 1, we have s 2 s 1 ) = s 1 from he resul of Proposiion 6. Now le us consider he oher exreme case where Sage 2 keeps zero invenory, i.e., s 2 = 0. In his case, he vendor sie becomes a cross-docking faciliy e.g., Eppen and Schrage 1981). The maerial flow dynamics equaions 17) and 18) become he following: M 1 ) = D 2, M 0 ) = D + D s 1) + D 1 2 s 1 ) +. 18

19 The above expressions have he same srucure as he maerial flow equaions 5) and 6) in he single-sage sysem. Thus, he resul of Proposiion 1 can be direcly applied o his case, i.e., r M 1 r O 1 = 1. In fac, his resul can be generalized as follows: Proposiion 8. In a wo-sage sysem wih i.i.d. demand and lead imes 1 0 and 2 1, suppose ha he demand has a finie bound, i.e., D < d d = if D < ). Then, for any given s 1 < )d, here exiss a hreshold s 2 s 1 ) > 0, such ha for any s 2 s 2 s 1 ), V[M 1 )] V[M 0 )], which implies r1 M r1 O. The above resul shows ha he resul r1 M r1 O coninues o hold if he vendor sage keeps low invenory. The inuiion behind his resul is ha, when he vendor sage becomes closely resembling a cross-docking faciliy, he combinaion of he vendor sage and is supplier can be viewed as one ample supply source wih some exra lead ime delay. Thus, he single-sage sysem resul can be applied here. We noe ha in he above resul we need he condiion s 1 < )d o rule ou he unrealisic case in which Sage 1 provides 100% service level o he end cusomer. Combining Proposiions 7 and 8, we arrive a he conclusion ha, when Sage 2 has eiher relaively high or low invenory, he maerial bullwhip raio r1 M is greaer han or equal o he acual informaion flow bullwhip raio r1 O a Sage 1. 20% 15% Gap percenage 10% 5% 0 5% 10% 15% Sage 2 base sock level s 2 Figure 5: Impac of Sage 2 base-sock level s 2 on he percenage of he bullwhip raio gap, r1 M ro 1 )/ro 1 100%. Figure 5 provides an illusraion of he gap percenage beween r1 M and r1 O under he normal demand disribuion N20, 8) wih lead imes 2 = 1 and 1 = 0 and Sage 1 base-sock level s 1 = 30). We observe from he figure ha here exiss hree regions based on Sage 2 basesock level: When s 2 is eiher high or low, r1 M > r1 O, and when s 2 akes value in an inermediae range, he resul reverses o r1 M < r1 O. These resuls sugges ha he vendor service level can 19

20 affec he discrepancy beween he informaion and maerial bullwhip measures. I also shows ha backlogging a he vendor sie has an inheren smoohing effec on he shipmen variabiliy o he downsream firm, which helps reduce he logisics-relaed coss. Such smoohing effec vanishes when he vendor has eiher sufficienly high or sufficienly low service levels. We furher noe ha our analysis in his secion is based on he exac characerizaion of he maerial flow in a wo-sage invenory sysem, which is new o he bullwhip lieraure. 6. Discrepancy Reducion Mehod In he previous secions, we have illusraed and quanified he discrepancy beween he informaion and maerial bullwhip measures in hree invenory models. When he discrepancy is significan, we need o find a way o align hese wo measures. The fundamenal problem here is o esimae demand variance based on he available sales daa in he presence of backlogging. In his secion, we provide an esimaion mehod o achieve his under a fairly general demand process. Consider ha he demand process D follows a general, common saionary ime series which can be emporally correlaed). e N be he ime aggregaion period. Define he raio beween he variance of he aggregaed demand and he variance of he single-period demand as ] fn) = V[D+N 1, 22) V[D ] where fn) is a funcion depending on he aggregaion period and he underlying demand model parameers. For example, for he i.i.d. demand process, i is sraighforward ha fn) = N. For he AR1) demand process, i can be shown ha 1 ρ N fn) = 1 ρ ) ρ 2 ) N 1 i=1 ) 1 ρ N i 2. 23) 1 ρ e s be an arbirary sae-dependen base-sock level for period. For example, s can be a non-saionary value ha depends on he hisorical demand daa. We know ha I follows ha N i=0 ) + ) + M 0 + ) = D + + D s 1 D + s. ) + ) + M i) = D +N+ + + D s 1 D +N+ +N s +N. 24) Inuiively, he oal sales in [ +, + + N] is equal o he oal demand in he same ime period he firs erm on he righ hand side) plus he backorder level in period + 1 he 20

21 second erm) minus he backorder level in period + N + he hird erm). If he backorder erms D s 1 ) + and D +N+ +N s +N ) + are i.i.d. for any, we can hen obain a simple expression for he demand variance based on he variances of he aggregaed sales. The following heorem presens his resul. Theorem 1. Suppose ha D is a saionary ime series, and here exiss T > 0, such ha D and D +i are independen for any and i T. Then, for any N T + 1, he following holds: [ 2N+1 ] [ N ] V i=0 M 0 + i) V i=0 M 0 + i) V[D ] =, f2n + 2) fn + 1) where f ) is defined in 22). An imporan feaure abou he above heorem is ha he demand variance can be recovered from he formula wihou knowing he underlying base-sock levels. Thus, he formula holds even if he sysem is operaed under non-opimal policies. In general, he wo backorder erms may be correlaed. However, for demand processes wih shor-range dependence i.e., α-mixing wih exponenial decay rae; see Billingsley 1995), he wo backorder erms are nearly independen as hey become far apar in ime when he aggregaion period N increases. Thus, we can sill apply he formula o esimae he demand variance as long as N 0, i.e., V[D ] [ 2N+1 ] [ N ] V i=0 M 0 + i) V i=0 M 0 + i). f2n + 2) fn + 1) Many common ime-series possess he propery of shor-range dependence. Below we provide wo illusraive examples. 6.1 IID Demand For he i.i.d. demand model, i is clear ha when N, he wo backorder erms are independen. Recall ha fn) = N. Thus, he demand variance can be recovered from he variances of he aggregaed sales as follows: Corollary 1. Suppose ha D follows an i.i.d. process. In a single-sage sysem, for any given lead ime 0 and N, V[D ] = [ 2N+1 ] [ N ] V i=0 M 0 + i) V i=0 M 0 + i). N

22 We can furher exend he above resul o he case when he ouside vendor does no have ample supply, i.e., he wo-sage sysem sudied in 5. In his sysem, from expression 18), we can show ha N i=0 ) ) + + M 0 + i) = D +N + D D s 2 s1 ) ) + + D +N +N 1 + D +N 1 1 +N 1 2 s 2 s1. When N 1 + 2, he second and hird erms in he above expression do no have overlapping demand erms and hus are i.i.d. everaging his observaion, we obain he following resul: Corollary 2. In a wo-sage sysem wih i.i.d. demand and lead imes 1 0 and 2 1, for any N 1 + 2, he following holds: [ 2N+1 ] [ N ] V i=0 M 0 + i) V i=0 M 0 + i) V[D ] =. N + 1 Thus, o esimae he demand variance in his sysem, one only needs a sufficienly long hisory of sales daa. There is no need o know he underlying base-sock levels a eiher sage, even hough he observed sales daa depends on hese base-sock levels. In fac, his formula can be generalized o a general J-sage serial sysem as long as N J j=1 j, where j is he lead ime a Sage j. 5% 0 Gap percenage -5% -10% -15% -20% Discrepancy reducion mehod -25% mean performance and 95% confidence inerval) Sales variance approximaion mean performance and 95% confidence inerval) -30% Sage 2 base-sock level Figure 6: Comparison of he discrepancy reducion mehod and he sales variance approximaion for he i.i.d. demand model. To illusrae how our discrepancy reducion mehod can align beween he informaion and maerial bullwhip measures, we compue he esimaed demand variance V e [D ] from simulaed daa based on Corollary 2, and compare is performance wih ha of he commonly-used sales variance approximaion mehod. The performance is measured based on he gap percenage relaive 22

23 o he acual demand variance V[D ], i.e., V e [D ] V[D ])/V[D ]. Figure 6 shows he performance comparison beween our mehod and he sales variance approximaion mehod under he normal demand disribuion N20, 8), wih lead imes 2 = 1, 1 = 0, and Sage 1 base-sock level s 1 = 30. To reduce esimaion error, we compue V e [D ] for N = 2, 3, 4 and 5 based on he simulaed daa, and hen ake an average of he four esimaed values o obain he demand variance esimaion. We repea he simulaion for 30 imes o generae 30 independen demand variance esimaes based on which we compue he mean and 95% confidence inerval of he gap percenage. I is clear from he figure ha our mehod yields a significanly beer esimae of V[D ] han does he sales variance approximaion mehod in all cases. 6.2 AR1) Demand Now suppose ha D follows an AR1) model. I can be verified ha he AR1) process is α-mixing wih exponenial decay rae Billingsley 1995). Thus, D and D +i becomes nearly independen as i increases, and we can apply he formula given in Theorem 1 o esimae he demand variance as long as N is sufficienly large. Specifically, we can choose K disincive sufficienly large N values, denoed by N k. We have, for 1 k K, V[D ] [ 2Nk ] [ +1 Nk ] V i=0 M 0 + i) V i=0 M 0 + i), f2n k + 2) fn k + 1) where f ) is given by 23). Take naural logarihm on boh sides of he above equaion, and define [ 2Nk ] [ +1 Nk ]) yn k ) = ln V i=0 M 0 + i) V i=0 M 0 + i) gn k, ρ) = ln f2n k + 2) fn k + 1)) a = lnv[d ]). We arrive a, for 1 k K, yn k ) = a + gn k, ρ) + e k, where yn k ) is he observaion poin based on he sales daa and e k is he error erm. Thus, given he K observaion poins, we can apply he leas square mehod o esimae he unknown parameers a and ρ. Tha is, a and ρ can be esimaed by solving he following opimizaion problem: min â,0 ˆρ 1 k=1 K [yn k ) â gn k, ˆρ)] 2. 25) 23

24 Afer obaining he esimae â, we can conver i o he esimaed demand variance as V e [D ] = eâ. Noe ha he above esimaion procedure is no limied o he AR1) process. One can apply he same approach o esimae he demand variance for any demand processes wih shor-range dependence. All one needs o do is firs deermining he funcion fn) based on he underlying demand process parameers accordingly, and hen ensuring ha he number of observaion poins are greaer han he number of parameers o be esimaed, such ha he leas square mehod can work properly. Gap percenage 25% 20% 15% 10% 5% 0 Discrepancy reducion mehod mean performance and 95% confidence inerval) Sales variance approximaion mean performance and 95% confidence inerval) Gap percenage 25% 20% 15% 10% 5% 0 Discrepancy reducion mehod performance mean and 95% confidence inerval) Sales variance approximaion performance mean and 95% confidence inerval) 5% 5% 10% Demand correlaion a) = 1 10% Demand correlaion b) = 5 Figure 7: Comparison of he discrepancy reducion mehod and he sales variance approximaion for he AR1) demand model. As a numerical illusraion, we esimae he demand variance V e [D ] based on he above leas square mehod 25), and hen compare is performance wih ha of he sales variance approximaion mehod. As in 6.1, we measure he performance based on he gap percenage relaive o he acual demand variance. In each simulaion run, we fix σ = 1, z = % service level), and compue he gap percenage for boh mehods under lead ime = 1 and = 5. In our leas square esimaion mehod, we choose N k = 2, 5, 8, and 11 when = 1, and N k = 6, 8, 10, and 12 when = 5. We repea he simulaion for 30 imes o generae 30 independen demand variance esimaes, based on which we compue he mean and 95% confidence inerval of he gap percenage. Figure 7 shows he performance comparison beween our esimaion mehod and he commonly used sales variance approximaion mehod. I is clear from he figure ha our mehod yields a significanly beer esimae of V[D ] han does he sales variance approximaion mehod in almos all cases, excep for wo cases ρ = 0.5 for = 1 and ρ = 0.2 for = 5) in which boh mehods yield esimaes close o he acual demand variance. 24