Bullwhip Effect Measurement and Its Implications

Bullwhip Effect Measurement and Its Implications
Li Chen • Hau L. Lee
Fuqua School of Business, Duke University, Durham, NC 27708
Graduate School of Business, Stanford University, Stanford, CA 94305
li.chen@duke.edu • haulee@stanford.edu
The bullwhip effect, or demand information distortion, has been a subject of both theoretical and
empirical studies in the operations management literature. Empirical studies have shown large
magnitudes of the bullwhip effect at the individual product level, but the effect does not always
exist at the macro level. The majority of studies focusing on the macro level have used monthly
data due to its availability. In practice, however, companies often order more frequently than
monthly, such as at daily or weekly intervals. In this paper, we examine how data aggregation can
affect the observation of bullwhip effect. Specifically, we show how aggregating data over relatively
long time periods can mask the magnitude of the bullwhip effect. In addition, we show that similar
impacts occur when data is aggregated across products, and how the existence of correlated demand,
seasonality, batch order, and finite capacity all can affect the measurement of the bullwhip effect.
Finally, we discuss the cost implications associated with the measured magnitude of the bullwhip
effect.
Key words: Bullwhip Effect, Data Aggregation, MMFE Model, Seasonality, Batch Order, Finite
Capacity, Supply Chain Management.
Department of review: Operations Management.
History: Revised in January, 2010.
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1.
Introduction
The supply chain bullwhip effect, as described by Lee et al. (1997a), is “the phenomenon where
orders to the supplier tend to have larger variance than sales to the buyer (i.e., demand distortion),
and the distortion propagates upstream in an amplified form (i.e., variance amplification).” This
phenomenon has been observed in many supply chains. For example, Hammond (1994) showed
large fluctuations of weekly order quantities in Barilla’s pasta supply chain. Lee et al. (1997b)
observed excess volatility in weekly orders in both Procter & Gamble’s diaper supply chain and
Hewlett-Packard’s printer supply chain. Fransoo and Wouters (2000) reported strong bullwhip
effects in daily data from a supply chain of ready-made meals and salads. Lai (2005) found a
significant level of bullwhip effect at the distribution center of a Spanish retailer. De Kok et al.
(2005) also observed substantial bullwhip effects at Philips Electronics. Most recently, Waller et
al. (2008) studied the weekly order and sales data obtained from a major U.S. retailer and found
strong bullwhip effects in all 115 products.
There also have been accounts of the bullwhip effect in the economics literature. These observa-
tions are primarily based on monthly or quarterly data aggregated across various products or firms.
For example, high production volatility was found in the TV set industry (Holt et al. 1968), retail
industry (Blinder 1981), automobile industry (Blanchard 1983), cement industry (Ghali 1987), to-
bacco, tire, and basic metal industries (Fair 1989), and in many other industries (Allen 1997). In
these studies, researchers searched for explanations to reconcile the bullwhip effect with the classic
production-smoothing theory, which posits that the motive for keeping inventory is to smooth pro-
duction variability rather than to amplify it. In a recent study, Cachon et al. (2007) used monthly
sales and inventory data from the U.S. Census Bureau to examine the bullwhip effect for various
industries. They found that if seasonality is included in the measurement, production smoothing
exists in the retail industry and in some manufacturing industries, but not in the wholesale industry.
Given these discrepant empirical findings, can one propose a unifying framework to bring order
to the observations? Specifically, how should one interpret the empirical measurements of the
bullwhip effect? What cost implications are associated with the measured magnitude of the effect?
Answering these questions requires a consideration of how the bullwhip effect is defined in the first
place.
There are two primary definitions used in the literature. Lee et al. (1997a) originally described
the bullwhip effect as a form of “information distortion,” and measured it by comparing the order
variance with the demand variance (where order can also be interpreted as production release in
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a manufacturing setting). This definition captures the distortion of information flow that goes
upstream (the downstream stage’s order is the demand input to the upstream stage). A second
definition, used in most empirical studies, compares the variance of order receipts (or shipments)
with the variance of sales. In some cases, the order receipt information, if not available, is inferred
from the sales and inventory data (see Blinder 1981, Miron and Zeldes 1988, Allen 1997, Cachon et
al. 2007). This definition essentially captures the distortion of material flow that goes downstream.
The bullwhip measurements based on these two definitions are usually good approximation to
each other, but they differ in concept. The information-based definition has a direct linkage to
supply chain cost because the upstream inventory/capacity decision is driven by the downstream
order information. Hence, the information-based bullwhip effect is a cost driver. In contrast, order
receipts information is the outcome of the upstream order-fulfillment decision process, and thus
is not a supply chain cost driver. Hence, the material-based bullwhip effect is the consequence of
the information-based bullwhip effect. Moreover, in the information-based definition, the bullwhip
effect is a result of one decision maker, i.e., the unit in question. This decision maker observes
demand, and then makes order decisions based on various structural and economic factors (see Lee
et al. 1997a). In the material-based definition, however, there are three decision effects involved.
First, the sales data is determined by the actual demand and the on-hand inventory, where the
latter is a result of the inventory decisions made in previous periods. Second, as in the information-
based case, 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’s previous production/stocking
decisions, where the order receipts may not exactly equal the orders (e.g., production shortfall,
transportation constraints, etc.). In view of these differences, we believe one should focus on the
information-based definition for theoretical development purposes. However, we recognize the need
to use the material-based definition as an empirical surrogate in some cases, and thus include a
discussion of the implications of such an approximation in Section 4.
There is also the question of the choice between ratio and difference for variance comparison
purposes. If a binary answer is needed, then both the ratio and difference metrics can be used.
However, if we want to compare the bullwhip effect for different products, the ratio metric, being
unit-independent, is probably a better choice.
For example, consider two products: one with
demand variance of 10 and order variance of 20, and the other with demand variance of 40 and order
variance of 80. With the ratio metric, the amplification ratio is 2.0 for both products. However,
with the difference metric, the second product has greater amplification (40 vs. 10). Furthermore,
if we try to calculate the aggregated bullwhip measure over these two products (assuming the
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demands are independent), the ratio would remain 2.0, but the difference would increase to 50
(10+40). The ratio metric thus appears to be more reasonable for comparison purposes in this
example as it is normalized around the demand variance. In this paper, therefore, we will employ
the ratio metric for analysis.
Before proceeding further, it is useful to discuss the main drivers for order variability. An
optimal order decision from a rational decision maker responds to both the uncertainty of supply
and demand and the cost structure of the situation. On the demand side, it is known in the
literature that positively-correlated demand coupled with long leadtime will amplify the bullwhip
effect, while negatively-correlated demand dampens it (Lee et al. 1997a, Chen and Lee 2009). On
the supply side, potential supply shortages will cause downstream stages to inflate orders and
thus trigger the bullwhip (Lee et al. 1997a). The underlying cost structure also drives the order
variability. For example, fixed ordering costs, such as machine setup costs and truckload costs, will
lead to large batch orders and cause the bullwhip (Lee et al. 1997a, Cachon 1999, Chen et al. 2002).
External cost shocks, such as promotional discounts, will induce forward-buying behavior, which
again causes the bullwhip effect (Blinder 1986, Lee et al. 1997a). Conversely, explicit penalty costs
for order variability will force the decision maker to smooth order quantities (Sobel 1969, Aviv 2007,
Cantor and Katok 2008). Furthermore, an internal capacity limit that truncates the order quantity
is likely to smooth the order sequence. A theoretical treatment for this case will be provided
in Section 4 below. In addition to these rational causes, there are also behaviorial factors. For
example, experimental studies have shown that the bounded rationality of human decision makers
can cause excess order variability (Sterman 1989, Steckel et al. 2004, Croson and Donohue 2006).
On the other hand, van Donselaar et al. (2007) found in an empirical study that store managers
sometimes smoothed system-generated orders by moving orders from peak to non-peak days.
Measuring the bullwhip effect in terms of aggregate data prompts several questions. How does
time aggregation affect the measurement? What about product or location aggregation? A number
of studies in the economics literature have addressed these issues. Christiano and Eichenbaum
(1987) showed that time aggregation causes bias under a macro-level equilibrium model. Caplin
(1985) studied the product aggregation effect under a static (s, S) inventory policy with independent
and identically-distributed (i.i.d.) demands. Caballero and Engel (1991) extended the product
aggregation result to continuous demands. Empirical tests of the aggregation effect were reported
by Mosser (1991) and Seitz (1993). In this paper, we provide a systematic analysis for both time and
product aggregations (location aggregation is omitted as it is mathematically equivalent to product
aggregation). Our time aggregation analysis differs from that of Christiano and Eichenbaum (1987)
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in that we start with an operational-level inventory model rather than a macro-level equilibrium
model.
Our product aggregation analysis generalizes that of Caplin (1985) to systems with a
state-dependent (s, S) inventory policy.
Table 1: Bullwhip Ratio at an European Retail Store
Product
Pack Size
Weekly
Biweekly
Four-week
Apple Sauce
12
1.60
1.76
1.17
Mineral Water
6
1.87
1.51
1.32
Peanut Butter
12
1.50
1.25
1.08
Stroopwafels
15
1.31
1.23
1.17
Sugar
10
3.04
2.68
2.03
Tea Bags
15
2.09
2.46
1.70
Aggregate
2.35
2.41
1.99
To motivate our results, consider the following empirical example. Table 1 presents the bullwhip
ratios for six consumer products from an European retail store (data source from Rob Broekmeulen
of Eindhoven University of Technology). The ratios are calculated from the sales and delivery data
during a one-year span, aggregated weekly, bi-weekly, and every four weeks. The bullwhip ratios
under product aggregation are also given in the table. Note that we used store delivery as a
surrogate for store orders in the calculation (see Section 4 for a discussion of the implications of
this approximation). From the table, we can see that the bullwhip ratio tends to decrease as the
aggregation period increases. In particular, for the peanut butter product, the bullwhip ratio at
the four-week level is almost 1.0. In this paper we rigorously analyze both this time aggregation
effect and the product aggregation effect.
Specifically, we consider the following three settings in our analysis: correlated and seasonal
demand, batch order, and finite capacity. We believe these three settings, separate or in combi-
nation, capture the essence of most real-world scenarios. We show that when the demand process
follows the general martingale model of forecast evolution (MMFE) (Hausman 1969, Graves et al.
1986, Heath and Jackson 1994), the bullwhip ratio converges to one as aggregation period increases.
Under the specific autoregressive moving-average ARMA(1, 1) model, we show that if the bullwhip
ratio is greater than one at the decision-period level, then the ratio decreases monotonically under
time aggregation. For product aggregation, the bullwhip ratio depends on the characteristics of
the aggregated demand process. When demands are independent cross products, the aggregated
bullwhip ratio is simply a weighted average of the individual bullwhip ratios. Furthermore, we
show that when an additive seasonality is included in the demand process, the bullwhip ratio will
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approach to one if the variability of seasonality dominates the random demand shock.
For a system subject to batch ordering, we first show that the existence of a minimum batch
order size increases the order variability. We then show that the bullwhip ratio attributed to
batch ordering converges to one as the aggregation period increases. We further show that the
same convergence result holds for product aggregation if demands are positively-correlated across
products. This result provides a theoretical explanation for the aggregation simulation results
reported by Yan et al. (2009), who observed that under the batch-order policy and positive spatially-
correlated demands, the aggregate bullwhip ratio decreases with the number of retailers.
When a system is subject to finite capacity, we show that the order variability is reduced under
the optimal modified base-stock policy (Federgruen and Zipkin 1986a,b, Aviv and Federgruen 1997,
Kapuscinski and Tayur 1998). Our analysis can also be applied to show that the order receipt
sequence (or shipment sequence), constrained by the upstream order-fulfillment capacity, is less
variable than the original order sequence. This result, combined with our seasonality analysis,
provides a theoretical explanation for the production smoothing effect observed by Cachon et al.
(2007) (see Section 5 for a detailed discussion).
We also show that the bullwhip ratio under
finite capacity converges to one as the aggregation period increases. The cost implications of the
(aggregate) bullwhip measure are discussed in Section 5.
The rest of this paper is organized as follows: Section 2 contains the aggregation analysis with
the MMFE demand process and additive seasonality. Section 3 presents the aggregation analysis
with batch ordering. In Section 4, we derive the aggregation analysis under finite capacity. Section
5 contains a discussion of the cost implications of the bullwhip measure and our concluding remarks.
All proofs are presented in the Appendix.
2.
Demand Process Analysis
Let us first consider the case where the system is subject to a state-dependent demand process.
Assume no fixed order cost and full backlog of unmet demand. It is well-known in the literature
that a state-dependent base-stock policy is optimal in this case (see Iglehardt and Karlin 1962,
Song and Zipkin 1993). This inventory policy is also a close approximation to what is being used in
practice (such as Manhattan Associates and JDA Software). The order quantity under this policy,
denoted by Ot, is given by
Ot = St − (St−1 − Dt−1) = St − St−1 + Dt−1
(1)
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where St (St−1) is the state-dependent base-stock level in period t (t − 1) and Dt−1 is the demand
in period t − 1.
In the above expression, we have implicitly assumed that the order quantity Ot can be negative.
In other words, the retailer can freely return excess inventory to the supplier. This assumption is
needed for tractability. It becomes quite innocuous when the order mean is sufficiently greater than
the order variance (such that the chance of a negative order quantity is negligible). Justifications
for this assumption can also be found in Lee et al. (1997a, 2000), Aviv (2003, 2007), and Chen and
Lee (2009).
Taking the variance on both sides of equation (1), we have
var(Ot) = var(St − St−1) + cov(St − St−1, Dt−1) + var(Dt−1).
The order variance var(Ot) is thus dependent on the covariance between St −St−1 and Dt−1. Under
i.i.d. demand, the optimal base-stock level is constant across all periods, i.e., St ≡ St−1. Hence
var(Ot) = var(Dt−1), i.e., the bullwhip ratio equals one. For more general demand processes, we
can deduce from the expression that the bullwhip ratio can be either greater than or less than one,
depending on the demand process characteristics.
To be more specific, let us consider a general demand process that evolves according to the
MMFE process. Specifically, following the notation of Chen and Lee (2009), let us assume that the
demand in a period t is given by
∞
∑
Dt = µ +
ϵt−i,t,
(2)
i=0
where ϵt−i,t is the incremental information obtained in period t − i with regard to demand Dt. Note
that the process is assumed to start from a distant past, so the summation in (2) is from ϵ−∞,t to
ϵt,t. Each ϵt−i,t is mutually independent, stationary, and normally-distributed with N(0, σ2). Let
i
∑
σ2 =
∞ σ2. For ease of exposition, let us assume σ2 < ∞, so that the demand in each period
i=0
i
is stationary, with a normal distribution N (µ, σ2); in the case of σ2 = ∞, only the conditional
variance would be meaningful, so all the subsequent results should be modified with the conditional
expectation.
The incremental information available at the end of period t with regard to all future demands
can be summarized in a vector ϵt = [ϵt, ϵt,t+1, ϵt,t+2, ...]T . Let us assume that ϵt is i.i.d. with
a multivariate normal distribution N (0, Σ), with the variance-covariance matrix given by Σ =
E{ϵtϵT }
t
. Let ei be the unit vector with the i-th element equal to one, and also define the following
∑
notation: ej =
j
e
i
k=i
k .
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The MMFE model described above generalizes many demand models used in the literature,
such as the i.i.d. normal demand model, the AR(1) model (Lee et al. 2000; Raghunathan 2001),
the IMA(0,1,1) model (Graves 1999, Miyaoka and Hausman 2004), the general ARIMA model
(Gilbert 2005, Gaur et al. 2005), the linear state-space model (Aviv 2003), and the advance demand
information model (Gallego and Ozer 2001).
Under the MMFE model with a replenishment leadtime of L periods, Chen and Lee (2009) have
shown that
∞
∑ (
)
L+1
T
∑ T
St =
ei+L+1
ϵ
ei Σei ,
(3)
i+1
t−i + (L + 1)µ + z
1
1
i=1
i=1
where z is the safety stock factor determined by the critical fractile. Substituting this into (1), we
obtain the bullwhip ratio as follows:
∑
var(O
L+2
t)
(eL+2)T ΣeL+2 −
(ei)T Σei
= 1 +
1
1
i=1
.
(4)
var(Dt−1)
σ2
From the expression, we can see that the bullwhip ratio depends on the variance-covariance matrix
Σ of the demand process. This result holds true under arbitrary time-lag shifts because the demand
and order processes are stationary.
Now let us consider the time aggregation effect. Define the M -period demand and order aggre-
∑
∑
gation as DM =
M −1 D
M −1 O
t−1
τ =0
t−1+τ and OM
t
=
τ =0
t+τ , respectively. By equation (1), it is easy
to show that
OM
t
= St+M−1 − St−1 + DM
t−1.
(5)
Intuitively, as M increases, the variance of the aggregated demand DM will increase and eventually
t−1
dominate that of St+M−1 − St−1. The following proposition confirms this intuition:
Proposition 1 For a stationary MMFE process, if limM→∞ var(DM ) = ∞, then
t−1
∑L+1
var(OM
2
(ei )T ΣeM+i
t ) = 1 +
i=1
1
i+1
→ 1, as M → ∞.
var(DM )
var(DM )
t−1
t−1
Proposition 1 shows that the M -period aggregated bullwhip ratio converges to one as M goes
to infinity. It is easy to verify that limM→∞ var(DM ) = ∞ holds true for common demand models
t−1
such as i.i.d., AR(1), MA(1), and ARMA(1, 1). Specifically, for the ARMA(1, 1) model, we can
establish the following monotonicity result:
Corollary 1 Under the stationary and invertible ARMA(1, 1) demand model, where |ρ| < 1,
|θ| < 1, the following holds:
var(OM
t )
2(ρ + θ)(1 − ρL+1)(1 − ρL+2 + θρ − θρL+1)
=
1 +
→ 1, as M → ∞;
var(DM )
M
t−1
(1 − ρ2)(1 + θ)2 − 2(ρ + θ)(1 + θρ)
1−ρM
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and the bullwhip ratio decreases monotonically in M when ρ > 0 and ρ + θ > 0.
When θ = 0 (or ρ = 0), the ARMA(1, 1) model reduces to the AR(1) (or MA(1)) model. So
the above result holds for the AR(1) and MA(1) models as well. This result implies that, insofar as
an empirical demand process can be approximated by the AR(1), MA(1), or ARMA(1, 1) models
(e.g., Erkip et al. 1990), the bullwhip ratio, if greater than one (i.e., when ρ > 0 and ρ + θ > 0),
will decreases monotonically to one as the aggregation period increases.
Now let us consider the product aggregation effect. Assume that there are N products with
the MMFE process vector given by ϵt,n, n = 1, ..., N . For simplicity, let us assume that the
replenishment lead times for these N products are all L periods. Define the N -product demand
∑
∑
and order aggregation as DN
=
N
D
N
O
t−1
n=1
t−1,n and ON
t
=
n=1
t,n, respectively, where Dt−1,n is
the demand for product n and Ot,n is the order quantity for product n. By equation (1), it is easy
to show that
N
∑
N
∑
ON
t
=
St,n −
St−1,n + DN
t−1.
(6)
n=1
n=1
From the expression, as N increases, the variance of DN
does not necessarily dominate that of
t−1
∑
∑
N
S
N
S
n=1
t,n −
n=1
t−1,n.
∑
Proposition 2 Let ΣN = E{ϵN
N
t (ϵN
t )T }, where ϵN
t
=
ϵ
n=1
t,n.
The bullwhip ratio under N -
product aggregation is given by
∑
var(ON
− L+2
t )
(eL+2)T ΣN eL+2
(ei)T ΣN ei
= 1 +
1
1
i=1
∑∞
.
var(DN )
(e
t−1
i=1
i)T ΣN ei
The product-aggregated bullwhip ratio thus depends on the variance-covariance matrix ΣN of
the aggregated demand process. If the diagonal elements in ΣN dominate the first L + 2 non-
diagonal elements, then, according to the above formula, the bullwhip ratio will be close to one.
Such a case arises, for example, when a large portion of the products has i.i.d. demand. When the
demands are independent across all products, it is straightforward to show that
N
var(ON
∑
t )
=
wn · var(Ot,n) ,
var(DN )
var(D
t−1
t−1,n)
n=1
∑
with w
N
n = var(Dt−1,n)/
var(D
n=1
t−1,n). Thus, in this case, the aggregated bullwhip ratio is
a weighted average of the individual bullwhip ratios, with the weight being the relative ratio of
demand variance of each product.
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2.1
Seasonality Analysis
Now let us examine the effect of including seasonality in the demand process. For simplicity, let us
assume that an additive seasonality is present in the demand of each period. Let s0, ..., sT −1 denote
the normalized seasonality with a regular cycle of T periods, where s0 corresponds to the seasonal
∑
factor of demand D
T −1
0. By “normalized” seasonality, we mean
s
i=0
i = 0. Let us also define the
variability of seasonality as
T −1
1 ∑
Vs =
s2
T
i .
(7)
i=0
With this additive seasonality, the demand Dt given in (2) can be rewritten as
∞
∑
Dt = s
+ µ +
ϵ
(t mod T )
t−i,t.
i=0
Extending the analysis of Chen and Lee (2009) to this additive seasonal demand process, we have
the following result:
Proposition 3 The optimal base-stock level in a period is given by
∞
∑ (
)
L
L+1
T
∑
∑ T
St =
ei+L+1
ϵ
s
+ (L + 1)µ + z
ei Σei .
i+1
t−i +
(t+i mod T )
1
1
i=1
i=0
i=1
The bullwhip ratio including seasonality is given by
∑
{
}
∑
1
T −1
var(O
E (O
L+2
t)
j=0
t+j − µ)2
(eL+2)T ΣeL+2 −
(ei)T Σei
βσ2
=
T∑
= 1 +
1
1
i=1
= 1 +
,
var(D
1
T −1
t−1)
E {(D
Vs + σ2
Vs + σ2
T
j=0
t+j−1 − µ)2}
where 1 + β is the bullwhip ratio without seasonality as given in (4), and Vs is defined in (7).
From Proposition 3, we can see that including seasonality in the measurement will make the
ratio go to one if Vs ≫ σ2, namely, if the variability of seasonality dominates the random de-
mand shock. With some algebra, we can show that the time aggregation result of Proposition 1
continues to hold under the seasonal demand process. For product aggregation, if the aggregated
seasonality dominates the aggregated random demand shock, then, by the same reasoning, the
product-aggregated bullwhip ratio (including seasonality) will approach to one.
It is worth pointing out here that the essence of our analysis in this section lies in the order-
demand relationship given in (1), (5)and (6), not in the underlying MMFE demand process. With
additional notation and technicality, our analysis can be directly extended to other state-dependent
demand models, such as the Markov-modulated demand model.
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