Chapter 3Modeling and Measuring the Bullwhip Effect

Li Chen and Hau L. Lee

Abstract The bullwhip effect is a phenomenon commonly observed in industry. Itdescribes how the distortion of demand information in a supply chain amplifies de-mand variance as it moves from consumption point up the supply chain to layers ofsuppliers. The bullwhip effect has been a subject of intensive research activities. Re-searchers have tried to address questions such as: What causes the bullwhip effect?How would different types of demand signal processing in forecasting and replen-ishment decisions affect the bullwhip effect? Can we explain the magnitude of thebullwhip effect in terms of the characteristics of the product and the supply chain?What is the magnitude of the bullwhip effect in practice, how does it differ acrossindustries and products, and how prevalent is the phenomenon? In this chapter, wereview both theoretical and empirical research done to address these questions, aswell as research done to identify important approaches and specifications that arenecessary to correctly measure and evaluate the true extent of the bullwhip effect.

3.1 Introduction

Demand variability and uncertainty is a driver of supply chain inventory. Manag-ing supply chains can be a challenge when demand variability and uncertainty ishigh. For a company in a supply chain consisting of multiple stages, each of whichis run by a separate organization (or company), the variability of demand faced bythis company can be much higher than the variability faced by downstream stages(where “downstream stages” refers to the stages closer to the final consumption of

Li ChenSamuel Curtis Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853,e-mail: li.chen@cornell.edu

Hau L. LeeGraduate School of Business, Stanford University, Stanford, CA 94305,e-mail: haulee@stanford.edu

1

2 Chen and Lee

the product). The bullwhip effect refers to the phenomenon where demand vari-ability amplifies as one moves upstream in a supply chain (Lee et al, 1997a, orLPW). LPW described this as a form of demand information distortion. Lee et al(1997b) further discussed the managerial and practical aspects of the bullwhip ef-fect, giving more industry examples. The bullwhip effect phenomenon is closely re-lated to studies in systems dynamics (Forrester, 1961; Sterman, 1989; Senge, 1990).Sterman (1989) observed a systematic pattern of demand variation amplificationin the Beer Game, and attributed it to behavioral causes (i.e., misperceptions offeedback). Macroeconomists have also studied the phenomenon (Holt et al, 1968;Blinder, 1981; Blanchard, 1983).

According to LPW, the bullwhip effect has been observed extensively in manyindustries. However, they provided only anecdotal references. LPW developed sim-ple mathematical models to explain how the bullwhip effect could arise, and iden-tified four causes: demand signal processing, order batching, price variations, andthe rationing game. Demand signal processing refers to a company using forecastupdates, and such updates would automatically lead to larger order fluctuations thandemand. Order batching refers to companies not ordering in every single time unit,and, as a result, order variance would naturally be larger than demand variance.Price variations result in companies making more than “normal” order quantitieswhen prices are lower than normal, and vice versa, leading to higher order fluc-tuations. Rational game refers to companies anticipating supply shortages, and toensure adequate supply, exaggerate their needs through placing larger order quanti-ties than otherwise. Hence, these models showed that the bullwhip effect could bea result of “rational” decision making under limited information and myopic objec-tives. Croson and Donohue (2006), using clever experimental setups, demonstratedthat there could also be additional behavioral causes, namely, the under-reaction tolags and coordination stock. Under-reaction to lags refers to players ignoring inven-tory in the pipeline when they made ordering decisions. Coordination stock refersto players increasing their orders because they wanted to have higher safety stocks,resulting from experiences of past delays in shipments from suppliers.

The main contention of LPW is that one needs to understand the causes of thebullwhip effect in order to devise counter-strategies. Hence, each of the causes re-quires a set of strategies for companies to use. These counter-strategies tend to fallinto two broad types. First, one needs to recognize the existence of the bullwhipeffect so that one is not “fooled” by the distorted demand information. Companieswould therefore be smarter in making capacity and inventory decisions accordingly.Second, companies need to work, often collaboratively, to reduce the magnitude ofthe bullwhip effect. One of the most commonly cited counter-strategies is informa-tion sharing, a topic covered in other chapters of this book. In the ideal situation,by information sharing, a company might not be misled by the distorted demandinformation, and, as a result, might not “bullwhip” its upstream supplier.

Since the work of LPW, two streams of research have emerged: modelingand empirical. In the modeling stream, researchers have expanded the LPW workthrough more complex modeling of the demand process to show how the bullwhip

3 Modeling and Measuring the Bullwhip Effect 3

effect could arise. In the empirical stream, instead of anecdotal evidence, researchershave tried to measure the extent of the bullwhip effect in real industry cases.

These two streams of research have reinforced each other in our deepened un-derstanding of the bullwhip effect. Modeling research generates insights and formsthe bases of hypotheses in empirical research. Empirical research serves to confirmor refute some of the results derived in modeling research, but it can also suggestpotential additional causes of the bullwhip effect or additional phenomena that canlead to new modeling research. Hence, the two streams together have provided ahealthy path for both streams of research. In this chapter, we review both streams ofwork.

Fig. 3.1 Illustration of infor-mation and material flows at asupply chain stage

In studying the bullwhip effect, we note that there have been two primary def-initions of bullwhip effect measurement used in the literature. It is worthwhile forus to clarify these two definitions as they affect how one interprets the results in theliterature. LPW originally described the bullwhip effect as a form of “demand infor-mation distortion.” The amplification of demand variance is based on the measure ofdemand variance faced by each stage in the supply chain. Hence, consider one stageof the chain facing its own demand variance. This stage in turn makes its orderingdecision (where order can also be interpreted as production release in a manufac-turing setting). The orders then become the demand faced by the upstream stage.The existence of the bullwhip effect implies that the order variance is larger thanthe original demand variance. This definition captures the distortion of informationflow that goes upstream (see Fig. 3.1). A second definition, used in many empiricalstudies, compares the variance of order receipts (or shipments) with the variance ofsales. Sales represent the material outflow from the current stage under consider-ation, while order receipts (or shipments) represent the material outflow from theupstream stage, which become the material inflow to the current stage. In somecases, the order receipt information, if not available, is inferred from the sales andinventory data (see Blinder, 1981; Cachon et al, 2007). This definition essentiallycaptures the distortion of material flow that goes downstream (see Fig. 3.1).

When the upstream stage can always supply perfectly the orders placed by thecurrent stage, and the current stage can always satisfy the demands that it faces, thenorders and order receipts are the same, and sales and demands also coincide. In thatcase, the measures of information-flow and material-flow bullwhip effects would be

4 Chen and Lee

identical. But once there are shortages in either the upstream or the current stages,the two measures would diverge.

The bullwhip effect measurements based on these two definitions also differ inconcept. The information-flow based definition has a direct linkage to supply chaincost because the orders issued by a stage become a driver to the upstream inven-tory/capacity decision. Hence, the information-flow bullwhip effect is a cost driver.In contrast, order receipts (or shipment) information is the outcome of the upstreamorder-fulfillment decision process, rather than an input to the decision process.Hence, the material-flow bullwhip effect is the consequence of the information-flowbullwhip effect. Moreover, in the information-flow based definition, the bullwhipeffect is a result of one decision maker, i.e., the stage in question. This decisionmaker observes demand, and then makes order decisions based on various struc-tural and economic factors. In the material-flow based definition, however, thereare three decision effects involved. First, the sales data is determined by the ac-tual demand and the on-hand inventory, where the latter is a result of the inven-tory decisions made in previous periods. Second, as in the information-flow basedcase, the unit makes order decisions, based on structural and economic conditions.Third, the actual order receipts from the supplier are the result of the supplier’sprevious production/stocking decisions, where the order receipts may not exactlyequal the orders (e.g., production shortfall, transportation constraints, etc.). In viewof these differences, we believe the information-flow based definition is more suit-able for theoretical analysis purposes. However, we recognize the need for usingthe material-flow based definition as an empirical surrogate in some cases, and thusinclude a discussion of the implications of such an approximation in Sect. 3.4.

The rest of the chapter is organized as follows. In Sect. 3.2, we review the empir-ical findings of the bullwhip effect. In Sect. 3.3, we review the literature of bullwhipeffect modeling, with an emphasis on the demand process modeling. Section 3.4discusses various issues related to the empirical measurement of the bullwhip ef-fect. We conclude the chapter by discussing some future research opportunities inSect. 3.5.

3.2 Survey of Empirical Findings

Empirical research concerning the bullwhip effect is a large literature and, ratherthan giving a comprehensive review, we highlight some significant findings. Thereare two classes of empirical research. The first one is closer to the field-based ap-proach, in which a single supply chain is the unit of analysis. The demand infor-mation in this single supply chain, often with a single class of products in focus,is analyzed to explore the existence of the bullwhip effect and measure its magni-tude. The second one uses secondary data of many companies, often aggregated, topursue statistical analysis of the bullwhip effect.

3 Modeling and Measuring the Bullwhip Effect 5

LPW started with their anecdotal observation of excessive volatility in weekly or-ders in both Procter & Gamble’s diaper supply chain and Hewlett-Packard’s printersupply chain.

We highlight a few sample studies that are based on single supply chains. In alandmark teaching case, Hammond (1994) documented how Barilla SpA, the largestpasta producer in the world, observed strong bullwhip effects. The supply chainmembers—Barilla and its customers—all processed demand signals, orders werebatched, and promotions were common. At one of the distribution centers (DC) ofits largest retail customer, the weekly orders placed by this DC to Barilla had amean of 300 quintals and a standard deviation of 227. But the weekly sell-throughat the DC (which can be viewed as shipments to the stores) had a mean of 300and standard deviation of 60. Suppose that we define the “bullwhip ratio” to beCVout/CVin, where CVout is the coefficient of variation (CV) of the outgoing orders,and CVin is the CV of the incoming orders. The bullwhip effect exists if the bullwhipratio is greater than one. The bullwhip ratio at the Barilla case was 3.75, i.e., 73%of the variation at the DC could be explained as the distortion within the supplychain, while the remaining 27% was the variation faced by the DC. Through VMI(vendor-managed inventory), Barilla was able to reduce the inventory at this DC by47%, while shortage rates dropped by 7% to almost zero.

Lai (2005) also studied a single supply chain, that of a Spanish grocery chainSebastian de la Fuente. The study was based on monthly product-level data at theDC, showing prices, markups, sales delivered to stores, supplies from suppliers,inventories and promotion. The data set contained records of 3,745 products over 29months that pass through the DC. Regression analysis by Lai (2005) demonstratesthat the bullwhip effect existed and was mainly driven by batching by the store, aswell as two behavioral causes identified by Croson and Donohue (2006).

Fransoo and Wouters (2000) studied two supply chains of convenience foods(salads and ready-made pasteurized meals) involving four companies in The Nether-lands. The supply chain consists of three stages: the producer, the regional DC andretail franchisees. Using the filtered daily sales data (from March 23 until June 5,1998), they found that the bullwhip effect was prominent across the supply chain.The bullwhip ratios found in their study are shown in Table 3.1.

Table 3.1 Summary of bullwhip ratios of different supply chain stages

Supply Chain Stage Bullwhip Ratio (Meals) Bullwhip Ratio (Salads)

Production 1.75 1.23Distribution Center 1.26 2.73Retail Franchisee 1.67 2.09

Note that both Lai (2005) and Fransoo and Wouters (2000) used sales data, and sothe bullwhip ratios that they measured were based on the material flow. Hammond’scase was based on orders placed by the DC (information flow) and the sales to the

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DC (material flow). It showed the challenges faced by empirical researchers as it isnot easy to truly measure the information distortion aspect of the bullwhip effect,i.e., the information-flow bullwhip effect. The study of Fransoo and Wouters (2000)also highlights how the bullwhip effect can vary across products. Hence, one has tobe careful when conducting empirical research with data aggregated over products.

Moving from the single supply chain approach, there also have been empiri-cal accounts of the bullwhip effect that are based on two levels of aggregation ofsecondary data. These studies usually used monthly or quarterly data aggregatedacross various products or firms. The first level of aggregation was on time units.In the sample studies of single supply chains described earlier, the time unit of thedata was based on the timing of order decisions. For example, it was a week in thecase of Barilla, since orders were generated on a weekly basis; and it was monthlyin the case of Sebastian de la Fuente, since the supply chain members order on amonthly basis. The time unit used in the convenience food supply chain study wasa day, since the supply chain ordered on a daily basis, which was necessary giventhe perishable nature of meals and salads. If companies orders on a monthly basisor a quarterly basis, then the use of monthly or quarterly data may pose no poten-tial problem. But this is rarely the case in real life. The second level of aggregationwas on products and often across firms as well, which can also be problematic onthe validity of bullwhip effect measurement. These two aggregation problems arediscussed in detail in Sect. 3.4.

Industry-based studies are anchored on data that are aggregated over products andfirms. High production volatility was found in the TV set industry (Holt et al, 1968),retail industry (Blinder, 1981; Mosser, 1991), automobiles (Blanchard, 1983; Kahn,1992), cement industry (Ghali, 1987), high tech consumables (Hanssens, 1998), pa-per products (Carlsson and Fuller, 1999), semiconductors (De Kok et al, 2005),semiconductor equipment (Terwiesch et al, 2005), and many other industries (Mironand Zeldes, 1988; Fair, 1989). In these studies, researchers searched for explana-tions to reconcile the bullwhip effect with the classic production-smoothing theorythat posits that the motive for keeping inventory is to smooth production variabilityrather than to amplify it. One of the leading explanations is that production smooth-ing was missing because seasonality had been excluded from the data (e.g., Ghali,1987).

There are also studies that are at the multi-industry or economy level (Blinder,1986; Bivin, 1996; Cachon et al, 2007). Here, there was even more extensive aggre-gation of products and firms, as data from products and firms in different industrieshave also been combined. Fine (1999) demonstrated the bullwhip effect using thismulti-industry approach, as shown in Fig. 3.2. The fluctuation of automobile pro-duction was clearly greater than that of the GDP (mimicking the downstream stageof the automobile industry), while the fluctuation of machine tools (mimicking theupstream stage of the automobile industry) was even greater.

Several recent empirical studies are worth highlighting. Cachon et al (2007) usedmonthly sales and inventory data from the U.S. Census Bureau of 1992–2004 to ex-amine the bullwhip effect in the manufacturing, wholesale and retail sectors. Hence,there was aggregation of time units, and aggregation across products. The bullwhip

3 Modeling and Measuring the Bullwhip Effect 7

Fig. 3.2 Illustration of industry volatility

effect analysis was based on the material flow concept. They found that if season-ality is included in the measurement, production smoothing indeed exists in theretail industry and in some manufacturing industries, but not in the wholesale in-dustry. With seasonally unadjusted data: 62% of manufacturers have a bullwhip ra-tio less than one, 86% of retailers had a bullwhip ratio less than one, while 84% ofwholesalers have a bullwhip ratio larger than one. Hence, there is empirical evidencethat while there was a tendency for companies to bullwhip the upstream suppliers,sometimes the desire to smooth production may be even stronger, dampening thebullwhip effect.

Bray and Mendelson (2012) reported that about two-thirds of firms bullwhip ina sample of 4,689 public U.S. companies over 1974–2008. Building on the modelof a generalized order-up-to policy proposed by Chen and Lee (2009), the authorsdecomposed the bullwhip effect by information transmission lead time. They foundthat demand signals with shorter time notice have greater impact on the bullwhipeffect.

Bray and Mendelson (2015) further investigated the bullwhip effect and produc-tion smoothing in an automotive manufacturing sample comprising 162 car modelsand found that 75% of the sample smooth production by at least 5%, despite thefact that 99% of the sample exhibit the bullwhip effect. They measured productionsmoothing with a structural econometric production scheduling model based on thegeneralized order-up-to policy. According to their structural estimation, there existboth a strong bullwhip effect (on average, production is 220% as variable as sales)and robust smoothing (on average, production would be 22% more variable absentvolatility penalty costs).

Shan et al (2014) investigated the bullwhip effect using data from over 1200 pub-lic companies in China during 2002–2009. They found that more than two-thirds ofthe companies exhibit the bullwhip effect. Their regression analysis suggests that

8 Chen and Lee

the bullwhip effect magnitude is positively associated with average on-hand inven-tory and persistence of demand shocks, and is negatively associated with degree ofdemand seasonality.

Duan et al (2015) collected a daily product-level dataset consisting of 487 indi-vidual products from a supermarket chain in China. They found that the magnitudeof the bullwhip effect at the product level is much more significant than those mea-sured at the firm or industry level, suggesting that product and time aggregation maymask the bullwhip effect measurement.

Osadchiy et al (2015) investigated the systematic risk in demand for differentindustries and firms in the U.S. economy, including retail, wholesale, and manufac-turing sectors. They used sales as a proxy for demand, and defined the systematicrisk in sales as the correlation coefficient of sales change with the contemporane-ous market return. They found that demand signal processing does not amplify thesystematic risk, however, aggregation of orders from multiple customers and aggre-gation of orders over time can result in the amplification of systematic risk upstreamalong the supply chain.

3.3 Modeling the Bullwhip Effect

There are multiple ways of modeling the bullwhip effect along a supply chain. Forexample, the bullwhip effect can be modeled as a result of judgmental errors byhuman decision makers (e.g., Sterman, 1989; Chen, 1999; Steckel et al, 2004; Cro-son and Donohue, 2006). It can also be modeled as a result of suboptimal inventorypolicies. Chen et al (2000a,b) showed that, when certain demand forecast methods,such as moving average and exponential smoothing, are used to determine a (subop-timal) inventory policy for an AR(1) demand process, the resulting order variabilityexceeds demand variability. De Kok (2012) considered a two-echelon supply chainin which the downstream demand is stationary but the retailers forecast demand us-ing the exponential smoothing method. He quantified the bullwhip effect as a func-tion of the exponential smoothing parameter. Dejonckheere et al (2003) modeled asupply chain as an engineering system and studied the bullwhip effect under certain(suboptimal) system replenishment rules. We refer the reader to Geary et al (2006)for a survey of the control engineering approach for modeling the bullwhip effect.

While the above approaches can all account for the bullwhip effect, there is acertain degree of arbitrariness in the assumed irrational human behaviors and sub-optimal inventory policies. To eliminate such arbitrariness, a normative approach isneeded, whereby the decision maker is assumed to make rational decisions in opti-mizing the system cost performance. In the remainder of this section, we shall takethis normative approach for modeling the bullwhip effect.

3 Modeling and Measuring the Bullwhip Effect 9

3.3.1 Overview and Model Setup

An optimal order quantity from a rational decision maker is a response to the supplyand demand uncertainties of a given situation. On the demand side, LPW showedthat positively-correlated demand coupled with long lead time would amplify theorder variability, while negatively-correlated demand would dampen it. On the sup-ply side, LPW also showed that potential supply shortages would cause downstreamstages to inflate orders and thus trigger the bullwhip effect. However, random supplydisruptions may also dampen the bullwhip effect (Rong et al, 2009) and a capacityconstraint can smooth the order quantity (Chen and Lee, 2012).

The underlying cost structure also drives the order variability. For example, fixedordering costs, such as full truckload and machine setup costs, will lead to largebatch orders and cause the bullwhip effect (LPW; Cachon, 1999; Chen and Lee,2012). External cost shocks, such as promotional discounts, will induce forward-buying behavior, which again causes the bullwhip effect (Blinder, 1986; LPW).Conversely, explicit penalty costs for order variability will force the decision makerto smooth order quantities (Sobel, 1969; Aviv, 2007; Cantor and Katok, 2012; Brayand Mendelson, 2015).

To mitigate the bullwhip effect, one can thus either encourage information shar-ing among supply chain partners to reduce the supply and demand uncertainties,or modify the supply chain cost structure to provide economic incentives for or-der smoothing. In what follows, we highlight some bullwhip effect models that areclosely related to the study of demand information sharing.

Consider a supply chain stage for a single product. Inventory is reviewed period-ically at this stage. Time is divided into periods of length one and indexed forwards(i.e., t = 0,1,2, ...). Let Dt denote the customer demand in period t and let µ (> 0)denote the mean of demand in a period. Customer demand is fulfilled immediatelyif the stage has enough on-hand inventory; unmet demand is fully backlogged. Unitholding cost h and stockout penalty cost p are assessed and charged to the stageat the end of each period. Inventory is replenished from an upstream stage with aconstant lead time L. The upstream stage is assumed to have ample supply. For easeof exposition, we assume that the manager minimizes the long-run average cost;we note that assuming a discounted cost objective function would yield the sameinsights (e.g., LPW).

It is known in the literature that the optimal policy for such an inventory systemis a state-dependent base-stock policy; a static base-stock policy can be viewed asa special case of the state-dependent policy. Let St denote the state-dependent base-stock level in period t. Under the base-stock policy, the order quantity in period t,denoted by Ot , can be written as

Ot = St − (St−1−Dt−1) = St −St−1 +Dt−1. (3.1)

In the above expression, we have implicitly assumed that the order quantity ineach period can be negative, such that the base-stock level is achievable in eachperiod. This is equivalent to assuming that excess inventory can be freely returned to

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the supplier. We shall make this assumption throughout this chapter for tractabilityreasons. We note that the chance of a negative order quantity becomes negligiblewhen the order mean is sufficiently greater than the order variance. Justifications forthis assumption can be found in LPW, Aviv (2003, 2007), and Chen and Lee (2009).

A few quick observations can be made based on expression (3.1). First, whenthe demand process Dt is independent and identically distributed (i.i.d.), the op-timal policy is a static base-stock policy, i.e., St = St−1 for any t. It follows thatOt = Dt−1, and hence var(Ot) = var(Dt−1). Therefore, there is no bullwhip effectin such a system. Second, when the demand process Dt is not i.i.d., the optimalbase-stock policy is state dependent. In this case, var(Ot) may be greater or lessthan var(Dt−1), depending on the covariance between St − St−1 and Dt−1. Belowwe consider several different demand processes, to quantify the variance ratio be-tween order and demand.

3.3.2 AR(1) Demand Process

LPW considered an autoregressive AR(1) demand process for modeling the bull-whip effect. Specifically, the demand in a period is defined as

Dt −µ = ρ(Dt−1−µ)+ εt ,

where |ρ| < 1 and εt is an i.i.d. normal random variable with N(0,σ20 ). Let d =

(1−ρ)µ . We can rewrite the above equation as follows:

Dt = d +ρDt−1 + εt . (3.2)

To ensure the chance of a negative demand is negligible, we assume σ0� d.Under the “free-return” assumption discussed above, the optimal base-stock level

can always be achieved in each period. As a result, we can determine the optimalbase-stock level St by solving the following long-run average cost minimizationproblem:

minSt

E{

h(

St −L

∑i=0

Dt+i

)++ p( L

∑i=0

Dt+i−St

)+},

where (x)+ = max(x,0). The above problem is a standard newsvendor problem, andwe know the optimal base-stock level is given by

S∗t = G−1(

ph+ p

), (3.3)

where G(·) is the cumulative distribution function of the lead time demand ∑Li=0 Dt+i.

Thus, it remains to determine to the distribution of the lead time demand.By leveraging the recursive expression (3.2), with some algebra, we can show

that, for k ≥ 0,

3 Modeling and Measuring the Bullwhip Effect 11

Dt+k = d +ρDt+k−1 + εt+k

= d +ρ(d +ρDt+k−2 + εt+k−1)+ εt+k

= · · ·

= d1−ρk+1

1−ρ+ρ

k+1Dt−1 +k

∑i=0

ρk−i

εt+i.

Because εt+i is an i.i.d. normal random variable, the lead time demand ∑Li=0 Dt+i,

conditional on Dt−1, also follows a normal distribution, with the conditional meanand variance given as follows:

E{

∑Li=0 Dt+i

∣∣Dt−1}

= dL+1

∑k=1

1−ρk

1−ρ+

ρ(1−ρL+1)1−ρ

Dt−1,

var{

∑Li=0 Dt+i

∣∣Dt−1}

=L+1

∑k=1

( k

∑i=1

ρk−i)2

σ20 .

Substituting the above result into expression (3.3), we arrive at

S∗t = dL+1

∑k=1

1−ρk

1−ρ+

ρ(1−ρL+1)1−ρ

Dt−1 + zσ0

√√√√L+1

∑k=1

( k

∑i=1

ρk−i

)2

,

where z = Φ−1 (p/(h+ p)), with Φ−1(·) being the inverse standard normal cumu-lative distribution function. Under the above optimal base-stock policy, the orderquantity in period t is given by

Ot = S∗t −S∗t−1 +Dt−1 =ρ(1−ρL+1)

1−ρ(Dt−1−Dt−2)+Dt−1.

Using the recursive expression (3.2) again, with some algebra, we can show that

var(Ot)var(Dt−1)

= 1+2ρ(1−ρL+1)(1−ρL+2)

(1−ρ). (3.4)

The above expression provides a characterization of the bullwhip effect under theAR(1) demand process. We note that the AR(1) demand process reduces to ani.i.d. process when ρ = 0. In this case, the above expression reduces to var(Ot) =var(Dt−1), which is the same as what we obtained in the i.i.d. demand case. Thus,expression (3.4) can be viewed as a generalization of the bullwhip effect result fromthe i.i.d. demand case. From the expression, we observe that, if the demand pro-cess has a positive temporal correlation, i.e., ρ > 0, we have var(Ot) > var(Dt−1).Order variability is amplified and the bullwhip effect exists in this case. On theother hand, if the demand process is negatively correlated, i.e., ρ < 0, we havevar(Ot) < var(Dt−1). Order variability is dampened instead. Moreover, when ρ > 0,

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it can be shown that the ratio in (3.4) is increasing in the replenishment lead time L,suggesting that longer lead time induces greater bullwhip effect.

3.3.3 IMA(0,1,1) Demand Process

Graves (1999) considered another simple demand process for modeling the bullwhipeffect. Specifically, demand Dt is assumed to follow an integrated moving-averageIMA(0,1,1) process, which is defined as follows:

Dt = Dt−1− (1−α)εt−1 + εt , (3.5)

where |α|< 1 and εt is an i.i.d. normal random variable with N(0,σ20 ). We note that

the demand reduces to an i.i.d. process when α = 0, and that the demand processbecomes a random walk when α = 1.

Let Ft−1 = Dt−1− (1−α)εt−1. From (3.5), we have Dt = Ft−1 + εt . Thus, Ft−1is the best linear predictor for Dt at the end of period t−1. With some algebra, wecan show that Ft−1 satisfies the following equation:

Ft−1 = αDt−1 +(1−α)Ft−2,

which has the same form as an exponential smoothing forecast.By leveraging the recursive expression (3.5), with some algebra, we can show

that, for k ≥ 0,

Dt+k = Dt+k−1− (1−α)εt+k−1 + εt+k

= Dt+k−2− (1−α)εt+k−2 +αεt+k−1 + εt+k

= · · ·

= Ft−1 +k−1

∑i=0

αεt+i + εt+k.

Because εt+i is an i.i.d. normal random variable, the lead time demand ∑Li=0 Dt+i,

conditional on Ft−1, also follows a normal distribution, with the conditional meanand variance given as follows:

E{

∑Li=0 Dt+i

∣∣Ft−1}

= (L+1)Ft−1,

var{

∑Li=0 Dt+i

∣∣Ft−1}

=L+1

∑k=1

(1−α +αk)2σ

20 .

Following an argument analogous to that of the AR(1) demand process, we obtainthe optimal base-stock level under the IMA(0,1,1) demand process as

3 Modeling and Measuring the Bullwhip Effect 13

S∗t = (L+1)Ft−1 + zσ0

√√√√L+1

∑k=1

(1−α +αk)2,

where z = Φ−1 (p/(h+ p)). Under the above optimal base-stock policy, the orderquantity in period t is given by

Ot = S∗t −S∗t−1 +Dt−1 = (L+1)(Ft−1−Ft−2)+Dt−1.

We note that the above optimal order quantity is the same as the adaptive orderingpolicy proposed by Graves (1999). Here we have shown that this ordering policy isthe outcome of an optimal state-dependent base-stock policy.

By using the recursive expression (3.5) again, with some algebra, we can showthat

var(Ot |Ft−2)var(Dt−1|Ft−2)

= (1+α +αL)2. (3.6)

From the above expression, we observe that when α > 0, var(Ot |Ft−2) >var(Dt−1|Ft−2) and the bullwhip effect exists. Moreover, when α > 0, the ratio in(3.6) is increasing in L, suggesting again that longer lead time induces greater bull-whip effect—the same insight as shown earlier under the AR(1) demand process. Itis worth commenting that in the above expression (3.6), the conditional variances oforder and demand are used. This is because the unconditional variances of order anddemand under the IMA(0,1,1) demand process are both unbounded. Therefore, wehave to resort to the conditional variance measure, which captures the order and de-mand uncertainties conditional on the most up-to-date demand forecast information.

3.3.4 General MMFE Demand Process

Chen and Lee (2009) proposed a demand model that generalizes the demand pro-cesses discussed above. Specifically, they assume that the demand process evolvesaccording to the martingale model of forecast evolution (MMFE) process (Haus-man, 1969; Graves et al, 1986; Heath and Jackson, 1994). Under the MMFE model,the demand Dt is defined as

Dt = µ +∞

∑i=0

εt−i,t , (3.7)

where εt−i,t is the demand information obtained in period t− i with respect to de-mand Dt . For all i ≥ 0, εt−i,t is an independent normal random variable with withN(0,σ2

i ). Let σ2 = ∑∞i=0 σ2

i . For ease of exposition, we shall assume σ2 < ∞ be-low; in the case of σ2 = ∞, the bullwhip effect results can be modified with theconditional variance as in Sect. 3.3.3.

14 Chen and Lee

From (3.7), the best linear predictor for demand Dt at the end of period t − i(i≥ 0) can be defined as

Ft−i,t = µ +∞

∑j=i

εt− j,t , (3.8)

with Ftt being the actual demand Dt itself. With some algebra, we can show Ft−i,t =Ft−i−1,t + εt−i,t . Hence, εt−i,t can be viewed as the forecast revision with regard todemand Dt made at the end of period t− i, and εtt (also written as εt below) is thefinal uncertainty resolved during period t after demand realization. An illustrationof the MMFE demand process is given in Table 3.2.

Table 3.2 Illustration of the MMFE process

Forecast revision for future demands in each period

Demand · · · 0 1 2 3 4 5 6 · · ·

D0 = µ + · · · + ε0D1 = µ + · · · + ε0,1 + ε1D2 = µ + · · · + ε0,2 + ε1,2 + ε2D3 = µ + · · · + ε0,3 + ε1,3 + ε2,3 + ε3D4 = µ + · · · + ε0,4 + ε1,4 + ε2,4 + ε3,4 + ε4D5 = µ + · · · + ε0,5 + ε1,5 + ε2,5 + ε3,5 + ε4,5 + ε5D6 = µ + · · · + ε0,6 + ε1,6 + ε2,6 + ε3,6 + ε4,6 + ε5,6 + ε6

......

......

......

......

......

The demand information obtained at the end of period t with regard to all futuredemands can be summarized in a forecast revision vector εεε t = [εt ,εt,t+1,εt,t+2, ...]T ,where εεε t is assumed to be i.i.d. with a multivariate normal distribution N(0,ΣΣΣ),where the variance-covariance matrix is given by ΣΣΣ = E{εεε tεεε

Tt }.

The above MMFE demand model generalizes many commonly used demandmodels. For example, if εεε t = [εt ,0,0, ...]T for all t, then we have an i.i.d. demandprocess. If εεε t = [εt ,ρεt ,ρ

2εt , ...]T (with 0 ≤ |ρ| < 1) for all t, then we obtain theAR(1) demand process described in Sect. 3.3.2. If εεε t = [εt ,αεt ,αεt , ...]T (with0 < α ≤ 1) for all t, then we obtain the IMA(0,1,1) demand process described inSect. 3.3.3. It can also be shown that the MMFE model is general enough to coverthe ARIMA(p,d,q) model (Box et al, 1994; Gilbert, 2005; Gaur et al, 2005), thelinear state-space model (Aviv, 2003), and the advance demand information model(Gallego and Ozer, 2001). We refer the reader to Chen and Lee (2009) for details.

Under the MMFE demand model, we have, for k ≥ 0,

Dt+k = Ft−1,t+k +k

∑i=0

εt+k−i,t+k.

3 Modeling and Measuring the Bullwhip Effect 15

Because εt+k−i,t+k is an i.i.d. normal random variable, the lead time demand ∑Li=0 Dt+i,

conditional on {εεετ : τ ≤ t− 1}, also follows a normal distribution, with the condi-tional mean and variance given as follows:

E{ L

∑i=0

Dt+i

∣∣∣∣ εεετ ,τ ≤ t−1}

=L

∑i=0

Ft−1,t+i,

var{ L

∑i=0

Dt+i

∣∣∣∣ εεετ ,τ ≤ t−1}

=L+1

∑i=1

ei1

TΣΣΣei

1,

where ek is the unitary vector with the k-th element being one and ei1 = ∑

ik=1 ek.

Therefore, we obtain the optimal base-stock level under the MMFE demand processas

S∗t =L

∑i=0

Ft−1,t+i + z

√L+1

∑i=1

ei1

TΣΣΣei

1,

where z = Φ−1 (p/(h+ p)). Under the above optimal base-stock policy, the orderquantity in period t is given by

Ot = S∗t −S∗t−1 +Dt−1 =L

∑i=0

(Ft−1,t+i−Ft−2,t+i)+Dt−1.

From the expressions (3.7) and (3.8), with some algebra, we can show that

var(Ot)var(Dt−1)

= 1+(eL+2

1 )TΣΣΣeL+21 −∑

L+2i=1 (ei)TΣΣΣei

σ2 . (3.9)

The above expression provides a general, unifying formula for the bullwhip effectfor demand processes that can be represented by the MMFE model. For example,the bullwhip effect result (3.4) under the AR(1) demand process can be recoveredfrom (3.9) by setting εεε t = [εt ,ρεt ,ρ

2εt , · · · ]T .When the demand process is either AR(1) or IMA(0,1,1), we have shown that

longer lead time induces greater bullwhip effect. However, from (3.9), we cannotclaim such a result without additional assumptions on the variance-covariance ma-trix ΣΣΣ of the forecast evolution process. The general expression (3.9) indicates thatit is actually the overall forecast correlation during the lead time period that drivesthe magnitude of the bullwhip effect.

3.4 Measuring the Bullwhip Effect

As discussed in the introduction, measurement of the bullwhip effect is anotherimportant topic. In this section, we use analytical models to demonstrate variousissues that one can encounter in measuring the bullwhip effect with empirical data.

16 Chen and Lee

When one compares the variances of the order and demand, the first questionis whether a ratio or an absolute difference metric should be used. If the objectiveis to determine whether the bullwhip effect exists or not, then both the ratio anddifference metrics can be used. However, if one wants to compare the bullwhip effectfor different products, the ratio metric, being unit-independent, appears to be a betterchoice. For example, consider two products: one with demand variance of 10 andorder variance of 20, and the other with demand variance of 40 and order varianceof 80. With the ratio metric, the amplification ratio is 2 for both products. However,with the difference metric, the second product has greater amplification than the firstone (40 versus 10). Furthermore, if one tries to calculate the aggregated bullwhipeffect measure over these two products (assuming the demands are independent),the ratio would remain 2, but the difference would increase to 50 (10 + 40). Theratio metric is thus more suitable for comparison purposes.

In some empirical studies, the standard deviation ratio metric and/or the coef-ficient of variation ratio metric have been used (see the Barilla example given inSect. 3.2). We note that the standard deviation ratio and the variance ratio containessentially the same information for comparison purposes, as the former metric isjust a squared root value of the latter. The coefficient of variation ratio is equivalentto the standard deviation ratio when the order mean is the same as the demand mean.When the order mean is different from the demand mean, one needs to normalize thevariability measure based on the different mean values. For example, suppose thatonly demand data from a partial set of customers are available in a distribution net-work. Then the order mean may be greater than the demand mean. In this case, thecoefficient of variation ratio is more appropriate for measuring the bullwhip effect.

In what follows, we shall consider a single-stage model with the order mean equalto the demand mean. Thus, the variance ratio metric is sufficient for our analysispurposes.

3.4.1 Seasonality

Cachon et al (2007) found that including seasonality in the measurement of the bull-whip effect leads to a much lower bullwhip effect measurement result. The followingmodel is used by Chen and Lee (2012) to demonstrate such an effect.

Let s0, ...,sT−1 denote the additive seasonality with a regular cycle of T periods,where s0 corresponds to the seasonal factor of demand D0. Without loss of general-ity, we assume that the seasonal factors are normalized, such that ∑

T−1i=0 si = 0. We

can define the variability of seasonality as

Vs =1T

T−1

∑i=0

s2i .

Let Dt (D′t ) and Ot (O′t ) denote the seasonal (deseasonalized) demand and orderquantity in period t, respectively. Thus, the demand in a period can be expressed as

3 Modeling and Measuring the Bullwhip Effect 17

Dt = D′t + s(t mod T ).

Under the “free-return” assumption, the optimal base-stock level in a period is givenby

S∗t = S′t +L

∑i=0

s(t+i mod T ),

where S′t is the state-dependent base-stock level for the deseasonalized demand pro-cess. Therefore, the seasonal order quantity in a period is given by

Ot = S∗t − (S∗t−1−Dt−1) = S′t −S′t−1 +D′t−1 + s(t+L mod T ) = O′t + s(t+L mod T ),

where the last equality follows from relationship O′t = S′t −S′t−1 +D′t−1. Hence, wecan show that

var(Ot)var(Dt−1)

=Vs +var(O′t)

Vs +var(D′t−1)= 1+

var(O′t)−var(D′t−1)Vs +var(D′t−1)

,

where D′t and O′t are the deseasonalized demand and order quantities. Thus, if thebullwhip effect exists in the the deseasonalized demand process, i.e., var(O′t)−var(D′t−1) > 0, and if the variability of seasonality dominates the deseasonalizeddemand variability, i.e., Vs� var(D′t−1), then including seasonality in the bullwhipeffect measurement will drive the ratio close to one. Chen and Lee (2012) furthershowed that the ratio may go below one when there is a capacity limit in the system.

3.4.2 Time Aggregation

Measuring the bullwhip effect based on aggregate data also prompts the question ofpotential aggregation biases in the measurement. The following model is used byChen and Lee (2012) to investigate the effect of data aggregation on the bullwhipeffect measurement.

Consider first the time aggregation effect. Define the M-period aggregation ofdemand and order as DM

t−1 = ∑M−1τ=0 Dt−1+τ and OM

t = ∑M−1τ=0 Ot+τ , respectively. By

the relationship (3.1), it follows that

OMt = St+M−1−St−1 +DM

t−1.

For most common demand models, it can be shown that limM→∞ var(DMt−1) = ∞.

For example, it can be easily verified that this condition holds for an AR(1) demandprocess. Thus, under this condition, the variance of the aggregated demand DM

t−1 willeventually dominate the finite variance of St+M−1−St−1 as M increases. Therefore,it is straightforward to show that limM→∞ var(OM

t )/var(DMt−1) = 1. That is, time

aggregation has a masking effect.Moreover, suppose that the demand follows an ARMA(1,1) process given by

18 Chen and Lee

Dt −µ = ρ(Dt−1−µ)+ εt +θεt−1,

where ρ > 0, ρ +θ > 0. Chen and Lee (2012) showed that the bullwhip effect undertime aggregation is given by

var(OMt )

var(DMt−1)

= 1+2(ρ +θ)(1−ρL+1)(1−ρL+2 +θρ−θρL+1)

(M/(1−ρM))(1−ρ2)(1+θ)2−2(ρ +θ)(1+θρ).

It is easy to verify that the above ratio decreases to one monotonically as M in-creases.

3.4.3 Product and Location Aggregation

Besides time aggregation, empirical data are also subject to product and locationaggregation. Since location aggregation is mathematically equivalent to product ag-gregation, below we will present the analysis on product aggregation from Chenand Lee (2012). Define the N-product aggregation of demand and order quantitiesas DN

t−1 = ∑Nn=1 Dt−1,n and ON

t = ∑Nn=1 Ot,n, respectively, where Dt−1,n is the de-

mand for product n and Ot,n is the order quantity for product n. Also, define theaggregated base-stock level as SN

t = ∑Nn=1 St,n.

Consider first the case in which the demands of the N products are spatially-independent but share a common additive seasonality pattern s0, ...,sT−1. Let D′t,nand O′t,n denote the deseasonalized demand and order quantity for product n, respec-tively. Thus, the demand for product n in a period can be expressed as

Dt,n = D′t,n +αn · s(t mod T ),

where the multiplicative factor αn captures the heterogenous magnitude of season-ality across the N products. Without loss of generality, assume αn ≥ 1 for all n. Alsoassume the replenishment lead time is a constant of L periods for all products. Theoptimal order quantity for product n is given by

Ot,n = O′t,n +αn · s(t+L mod T ).

We can show that the aggregate bullwhip ratio is given by

var(ONt )

var(DNt−1)

= 1+

N

∑n=1{var(O′t,n)−var(D′t−1,n)}( N

∑n=1

αn

)2

Vs +N

∑n=1

var(D′t,n)

.

The above ratio approaches to one as N goes to infinity. Therefore, if the productsunder aggregation share a common seasonal profile (for example, the Christmasseasonality in the retail industry), including seasonality in the measurement will

3 Modeling and Measuring the Bullwhip Effect 19

drive the aggregated bullwhip ratio to one and thus mask the bullwhip effect ofindividual products.

Consider another case in which the demands of the N products are spatially-independent but all belong to the AR(1) process family. Specifically, let us assumethat the demand for product n is given by:

Dt,n−µn = ρn(Dt−1,n−µn)+ εt,n,

where 0 ≤ ρn < 1 and εt,n is an i.i.d. normal random variable with N(0,σn). Thus,we have var(Dt,n) = σ2

n /(1−ρ2n ). By setting ρ = ρn in the result of (3.4), with some

algebra, we have the following:

var(ONt )

var(DNt−1)

= 1+

N

∑n=1

2ρn ·1−ρL+1

n

1−ρn· 1−ρL+2

n

1−ρ2n·σ2

n

N

∑n=1

σ2n

1−ρ2n

.

Thus, if limN→∞ ∑Nn=1 ρnσ2

n /∑Nn=1[σ

2n /(1−ρn)] = 0, the above aggregate bullwhip

ratio approaches to one as N goes to infinity. This condition can be satisfied, forexample, when ρn = 1/(n + 1) and σn = 1 for all n, which means there is an in-creasing portion of the products with an autocorrelation coefficient close to zero.Another example is when ρn = n/(n+1) and σn = 1 for all n. In this case, there isan increasing portion of products with autocorrelation coefficient approaching one.In both cases, the bullwhip ratios for individual products are all strictly greater thanone (because ρn > 0), but product aggregation can mask the severity of the bullwhipeffect of individual products.

3.4.4 Material Flow Data

So far, we have considered bullwhip effect measurement based on the variances oforder and demand (i.e., the information flow). As discussed in Sect. 3.2, in manyempirical studies, material flow data (such as shipments and sales) are used as prox-ies for the order and demand data to measure the bullwhip effect. Chen et al (2014)provided an analytical comparison of these two bullwhip effect measurements.

Consider a simple i.i.d. demand process. The optimal policy in this system is astatic base-stock policy S. From (3.1), it follows that Ot = Dt−1, and there is nobullwhip effect based on the information flow data. Now let us examine the materialflow in such a system. Let M1(t) denote the shipment from an upstream stage inperiod t. Since the upstream stage is assumed to have ample supply, we have

M1(t) = Ot = Dt−1. (3.10)

20 Chen and Lee

Let M0(t) denote the sales to the downstream customer in period t. It can be shownthat

M0(t) = Dt +( L

∑i=0

Dt−L−1+i−S)+−( L

∑i=0

Dt−L+i−S)+

, (3.11)

where the last two terms represent the demand backlogs in periods t− 1 and t, re-spectively. From the above expressions, the sales M0(t) can be written as a summa-tion of Dt and two additional random variable terms. Intuitively, one would expectthe variance of M0(t) to be greater than the variance of Dt . However, it can be shownthat the opposite is true. Specifically, from (3.11), the following expression can beobtained:

var(Dt)−var(M0(t)) = 2E{(

S−L

∑i=0

Dt−L−1+i

)+( L

∑i=0

Dt−L+i−S)+}

≥ 0. (3.12)

From (3.10), we know that var(M1(t)) = var(Dt). Thus, it follows that var(M1(t))≥var(M0(t)). In other words, measurement based on the material flow data wouldoverstate the underlying information-flow bullwhip effect in such a system. Kahn(1987) derived a similar insight in a model with zero lead time. The above resultgeneralizes the insight to the case with general, positive lead time. It is clear fromthe expression that the shipment variance equals the sales variance only when S = 0or S = ∞. Thus, measurement based on the material flow data may provide a goodapproximation to the underlying information-flow bullwhip effect when the base-stock level is either (sufficiently) high or low.

When demand Dt follows an i.i.d. normal distribution N(µ,σ2), it can be shownthat the expression (3.12) is unimodal in S, reaching a peak value at S = (L + 1)µ

(where the system service level is 50%). It can be further shown that the expression(3.12) is decreasing in lead time L, suggesting that measurement based on the mate-rial flow data may provide a good approximation to the information-flow bullwhipeffect when the replenishment lead time increases.

Chen et al (2014) further considered the AR(1) demand process for comparingthe bullwhip effect measurements. They showed that the autocorrelation parameterρ in the AR(1) demand model has a direct impact on the measurement biases. Whenthe demand is moderately correlated, measurement based on the material flow datamay overstate the information-flow bullwhip effect, which generalizes the insightfrom the i.i.d. demand case. However, when the demand has a strong temporal cor-relation, the result reverses, i.e., measurement based on the material flow data wouldunderstate the information-flow bullwhip effect. To correct the measurement biasesfrom the material flow data, Chen et al (2014) proposed simple debiasing methodsbased on the sample autocovariances of the sales and shipment data.

3 Modeling and Measuring the Bullwhip Effect 21

3.5 Future Research Opportunities

The vast literature of modeling and empirical research on the bullwhip effect pro-vides several important observations. First, we need to be careful to distinguish be-tween information-flow versus material-flow bullwhip effects. Second, the magni-tudes of the bullwhip effects of different products do differ. Third, data aggregationover time, product, firms and industry sectors may play a role in masking the bull-whip effect. For practical managerial insight, a single firm has to deal with bullwhipeffects for each of its products (since production and inventory are based on a prod-uct), and the demand orders arrive or are issued in accordance with the time unitsthat the partners in the supply chain use as their decision points. Hence, when prod-ucts and firms data are aggregated, the resulting aggregate measurement may not bethat meaningful, unless one is dealing with capacities that are shared by multipleproducts, or in the extreme cases, resources that are shared by multiple firms.

The second and third observations above imply that much research is still neededfor a full understanding of the bullwhip effect. Richer demand models, as well asassociated analysis of the impacts of different levels of aggregation on the measure-ment of the bullwhip effect, are needed. We have discussed some of the recent de-velopments in this direction in Sect. 3.3 and 3.4. Most of the models assume a linearsupply chain. It would be interesting to expand the models to include more complexdistribution networks. For example, in a system where multiple customers are sup-plied by a common supplier, how would the inventory allocation rule at the supplieraffect the bullwhip effect and its measurement? How does the inventory poolingeffect at the supplier interact with the demand variability amplification in such asystem? How does the demand correlation among the downstream customers affectthe bullwhip effect? When each customer has a different ordering process resultingfrom a different demand process, decision time unit, and ordering economics, howdoes this affect the bullwhip effect?

The empirical research indicated that the tendency toward demand amplificationthrough demand signal processing is sometimes counteracted by the tendency to-ward production smoothing, so that the bullwhip effect may be dampened. Chen andLee (2009) was one of the articles that modeled both such drivers, using a simpleway of smoothing by postponing some orders to a later time. To gain a deeper under-standing of the interactive effects of demand amplification and production smooth-ing, more complex production smoothing models can be constructed (e.g., Bray andMendelson, 2015).

Finally, we need empirical research that is based on the right units (time, prod-ucts, or firm) to truly measure the bullwhip effect. We also need empirical researchto explore the drivers or characteristics of the products and supply chains that giverise to different bullwhip effect magnitudes. In addition, Ozer et al (2014) have iden-tified cultural differences between China and the US on trusts and information shar-ing. Since the bullwhip effect is closely tied to trust factors between the buyer andthe supplier, it would also be of interest to conduct behavioral research on culturaldifferences in the bullwhip effect magnitudes.

22 Chen and Lee

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