European Journal of Operational Research 173 (2006) 617–636
www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Quantifying the bullwhip effect in a supply chain
with stochastic lead time
Jeon G. Kim a, Dean Chatfield b, Terry P. Harrison c, Jack C. Hayya
a,*
a
c
Department of Supply Chain and Information Systems, The Smeal College of Business Administration,
Pennsylvania State University, 303 Beam Building, University Park, PA 16802, USA
b
Department of Management Science and Information Technology, Virginia Tech, 1007 Pamplin Hall (0235),
Blacksburg, VA 24061, USA
Department of Supply Chain and Information Systems, The Smeal College of Business Administration, Pennsylvania State
University, 509-L BAB, University Park, PA 16802, USA
Received 12 August 2003; accepted 10 January 2005
Available online 31 March 2005
Abstract
In a recent paper, Dejonckheere, Disney, Lambrecht, and Towill [European Journal of Operational Research 147
(2003) 567] used control systems engineering (transfer functions, frequency response, spectral analysis) to quantify
the bullwhip effect. In the present paper, we, like Chen, Ryan, Drezner, and Simchi-Levi [Management Science 46
(2000) 436], use the statistical method. But our method extends Dejonckheere et al. and Chen et al. in that we include
stochastic lead time and provide expressions for quantifying the bullwhip effect, both with information sharing and
without information sharing. We use iid demands in a k-stage supply chain for both. By contrast, Chen et al. provide
lower bounds using autoregressive demand for information sharing and for information not sharing (with zero safety
factor for stocks). Dejonckheere et al. validate Chen et al.Õs results for a 2-stage supply chain without information sharing, using both autoregressive and iid normally distributed demands. We estimate the mean and variance of lead-time
demand (LTD) from historical LTD data, rather than from the component period demands and lead time. Nevertheless, we also calculate the variance amplification like Chen et al., but with gamma lead times. With constant lead times,
which Chen et al. used, our method yields lower variance amplification. As for the effect of information, we find that the
variance increases nearly linearly in echelon stage with information sharing but exponentially in echelon stage without
information sharing.
Ó 2005 Elsevier B.V. All rights reserved.
Keywords: Supply chain management; Stochastic lead times; Bullwhip effect; Information sharing; Demand during lead time
*
Corresponding author. Tel.: +1 8148651461; fax: +1 8148632381.
E-mail addresses: [email protected] (J.G. Kim), [email protected] (D. Chatfield), [email protected] (T.P. Harrison), [email protected] (J.C.
Hayya).
0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2005.01.043
618
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
1. Introduction
1.1. Background
We shall not review the literature on the bullwhip effect, as Dejonckheere et al. (2003) have done an
excellent job of that. Our object is to quantify the bullwhip effect with and without information sharing.
By the bullwhip effect, we mean variance amplification of demand up a supply chain. By information sharing, we mean that customer demand at the lowest node of the supply chain is immediately transmitted to all
the upstream nodes. (ÔInformation sharingÕ is what Chen et al. (2000, p. 439) call Ôcentralized demand informationÕ.) Like Dejonckheere et al. and Chen et al., we use order-up-to policies. In these papers lead time
was constant, and lead-time demand (LTD) was estimated from its component lead time (L) and period
demand (D). In our paper, L is stochastic and LTD is estimated from past observations of LTD. Let D
and L be random variables. Then the demand during the lead time is
L1
X
X t ¼ Dt þ Dtþ1 þ þ DtþL1 ¼
Dtþj :
ð1AÞ
j¼0
Eq. (1A) is the conventional definition of LTD, in that the lead time is forward, but it is not ‘‘physically
realizable’’ in trying to estimate Xt from past data at time t. In that situation, the appropriate definition is1
L
X
Xt ¼
Dtj ;
ð1BÞ
j¼1
because the definition in Eq. (1A) is not ‘‘physically realizable.’’ Nevertheless, the two definitions are mathematically equivalent in terms of a priori statistical analysis. As Eqs. (1A) and (1B) are mirror images of
each other, we can just substitute the subscript j in (1B) by +j to achieve Eq. (1A).
Now when both D and L are random, X is a random sum of random variables, and demand during lead
time is the convolution (Ord, 1972, pp. 64–66; Mood et al., 1974, p. 186; Bagchi et al., 1986, p. 180) of demand rate and lead time. The LTD can also be said to be a Ôcompound distributionÕ of demand rate and
lead time (Bagchi et al., 1984). Now suppose that D and L are statistically independent. Then
Var ðX Þ ¼ r2X ¼ lL r2D þ l2D r2L
and
EðX Þ ¼ lX ¼ lL lD ;
ðlL ; r2L Þ
ð2AÞ
ðlD ; r2D Þ
are the theoretical mean and variance of the lead time and
where
variance of the demand per unit time. But if the lead time were fixed, then
Var ðX Þ ¼ r2X ¼ Lr2D
and
EðX Þ ¼ lX ¼ LlD :
the theoretical mean and
ð2BÞ
1.2. Organization of the paper
In Section 2, we quantify the bullwhip effect with stochastic lead time and with deterministic lead time, as
a special case. We show how subtle differences in estimation could lead to dramatically different variance
amplifications. In Section 3, we compare our results for variance amplification for information sharing versus information not sharing. In that section we also present comparisons with Chen et al. and Dejonckheere
et al., using our method of estimating the order-up-to level. We provide two appendices that contain the
analytical proofs for the material presented in Section 2. The long Appendix A contains Lemmas 1 and
2 and the attendant propositions needed for their proofs. Appendix B repeats the independently and identically distributed (iid) derivations in the text, but for autoregressive demand.
1
We are grateful to an anonymous referee for pointing out that our original use of Eq. (1A) was not physically realizable and that
the logical expression for estimation is of the form in Eq. (1B).
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
619
2. Quantification of the bullwhip effect
We quantify the bullwhip effect in an environment of stochastic demand and lead times. We show that
the bullwhip effect increases in approximately linear fashion with supply chain echelon, k, when customer
demand information is shared and exponentially with echelon (or as a k-power function of mean lead time,
lL) when customer demand information is not shared.
2.1. Assumptions and notation
(1) Lead time: Let Lt(k) be the lead time of the order placed at the beginning of period t at stage k.
Assume that lead times at any stage and at the different stages are iid. In other words, Lt(k) are identical and independent for k = 1, 2, . . . , and for t = 1, 2, . . . , with mean lL and variance r2L .
(2) Demand rate: Period demands, D1, D2, . . . , Dt, . . . are stationary and iid, with mean lD and variance
r2D . (We use iid period demands, because the iid structure provides a basic building block for correlated period demands, as we show in the forthcoming analysis. Also, see, for example, Chen et al.
(2000, Theorem 2, p. 44).)
(3) Statistical independence: D and L are independent.
(4) The length of the moving average, p: Following Chen et al. (2000) and Dejonckheere et al. (2003) we
concentrate on the more practical case where p P L.
(5) Sequence of events: at the beginning of each period, t, the inventory manager observes the inventory
level and places an order, Qt. After the order is placed, customer demand, Dt, is observed.
Let lX(k) and r2X ðkÞ be the mean and variance of lead-time demand at stage k. Then lX(k) = lX and
¼ r2X , because Lt(k) is iid for all k and t. Now define
r2X ðkÞ
Qkt = order quantity at stage k,
S kt = the order-up-to level at stage k,
st(k) = sample standard deviation of lead-time demand at stage k,
Xt(k) = lead-time demand (LTD, also called Ôdemand during lead timeÕ) at stage k,
zk = standardized normal variable for setting safety stocks at stage k.
To further investigate Var(Qt) when the lead time is stochastic, denote the mth moment of X about
l by
ðmÞ
m
lX ¼ EðX lX Þ ;
ð3Þ
with the superscript m not to be confused with k, the stage number in the supply chain.
2.2. Subtle differences
We show that subtle differences in the way estimates are made when updating the order-up-to-level could
lead to marked differences in the magnitude of the bullwhip effect. To illustrate, let k = 1, z = 0, L be deterministic, and D iid. Calculating moving averages of demand rate and multiplying that average by L to estimate mean demand during lead time leads to (Chen et al., 2000, p. 438)
Var ðQÞ
2L 2L2
¼1þ
þ 2 ;
Var ðDÞ
p
p
ð4Þ
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J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
whereas using our method of calculating moving averages of the demand during lead time (not the demand
rate) leads to
Var ðQÞ
2 2L
¼1þ þ 2 ;
Var ðDÞ
p p
ð5Þ
a drop in variance amplification of the order of L. However, both estimation methods are legitimate, with
Chen et al.Õs perhaps the more common, because it may be easier to keep two sets of data files (D and L),
rather than three (D, L, and X). Note that the difference between the two methods is that in the Chen et al.
construction ðX t ¼ L:Dt Þ the lead time, L, is explicitly multiplicative (understandably so, because L is
PL1 Pp
Dtiþj
constant), whereas in our model X t ¼ j¼0 p i¼1
, L is implicitly additive. Thus, the difference between Eqs. (4) and (5).
Chen et al. (2000, pp. 437, 440) use the following construction:
S t ¼ LDt þ z sLet ;
ð6Þ
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
Pp
2
ðe
Þ
ti
i¼1
sLet ¼ cL;q
p
is the standard deviation of the forecast error over the lead time, with et ¼ Dt Dt the one-period-ahead
forecast error; cL,q a constant function of lead time, L; q the autoregressive correlation; and p the number
of observations in the moving average. Chen et al. (2000, Theorem 2, p. 438) show that the estimate in Eq.
(6) leads to Eq. (4). But in the present paper, we use the alternative estimate of the order-up-to level,
S t ¼ X t þ z st ðX Þ;
ð7Þ
where
Pp
i¼1
Xt ¼
p
X ti
PL1 Pp
¼
j¼0
i¼1 Dtiþj
p
¼
L1
X
Dtþj ;
j¼0
and
p
1 X
2
ðX t X t Þ ;
p 1 t¼1
"
#
p X
p X
L1
L1
X
X
1
2
¼
ðDtiþj Dtþj Þ þ
ðDtiþj Dtþj ÞðDtiþm Dtþm Þ :
p 1 i¼1 j¼0
i¼1 j6¼m
s2t ðX Þ ¼
Let k = 1, z = 0, L deterministic and D iid. This is a special case of Lemma 1 that we present later. Like in
Chen et al. (2000, p. 438), we begin with
Qt ¼ X t X t1 þ Dt1 ;
but we take moving averages of demand during lead time, X, rather than of L multiplied by the moving
average of period demand. Thus,
1
1
Qt ¼ ½X t1 X tp1 þ Dt1 ¼ Dt1 þ Dt þ þ DtþL2 Dtp1 Dtp DtpþL2 þ Dt1 :
p
p
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
621
Hence,
2L Var ðDÞ
2
þ Var ðDÞ þ Cov ½ðDt1 þ Dt þ DtþL3 DtþL2 Þ; Dt1 2
p
p
2 2L
¼ Var ðDÞ 1 þ þ 2
for p P L
p p
2
for p < L:
¼ Var ðDÞ 1 þ
p
Var ðQt Þ ¼
ð8Þ
2.3. k-Stage supply chain
We consider k echelons in the supply chain: retailer, wholesaler, distributor, factory, and tiers of suppliers. We shall put k as a superscript or in parenthesis to denote kth stage.
2.3.1. When information on customer demand is shared
In a paper on Ôinformation enrichmentÕ, Mason-Jones and Towill (1997, p. 138) say that ‘‘. . . market
information notoriously suffers from delay and distortion as it moves through the supply chain,’’ with
two main remedies being the reduction of uncertainty and the reduction of lead time. Here we deal with
the first remedy: reduction of variance amplification, where the information on customer demand is shared
by all stages in the supply chain, so that each stage can utilize this demand information for setting up its
order-up-to level and its order quantity. Therefore, when the information on customer demand is shared,
the order-up-to level at stage k is
S kt ¼ X t ðkÞ þ zk st ðkÞ;
ð9Þ
which is the same as Eqs. (2A) and (2B) in Chen et al. (2000, p. 572) and Dejonckheere et al. (2003, p. 437).
We also a use p-period moving average (MA(p)) to estimate means and variances of lead-time demand, Xt,
based on observations from the previous p periods.
Let stage zero be the customer in the supply chain. Since the demand at stage k is the orders from stage
(k 1), the order quantity at stage k is
k
k
k1
k1
1
1
0
Qkt ¼ S kt S kt1 þ Qk1
t1 ¼ S t S t1 þ S t1 S t2 þ þ S tkþ1 S tk þ Qtk :
ð10Þ
Since stage zero is the customer, let Q0tk ¼ Dtk . For convenience and without loss of generality, assume
that zi = z, for all i. Then the order quantity at stage k is
Qkt ¼
k
k
k
X
1X
1X
X t1kþi ðiÞ X t1pkþi ðiÞ þ Dtk þ z
ðstkþi ðiÞ stk1þi ðiÞÞ:
p i¼1
p i¼1
i¼1
ð11Þ
Hence, the variance of the order quantity at stage k is
Var ðQkt Þ ¼
k
k
1 X
2 X
Var
ðX
ðiÞÞ
þ
Cov ðX t1kþi ðiÞ; X t1kþj ðjÞÞ
t1kþi
p2 i¼1
p2 i<j
þ
k
k
1 X
2 X
Var
ðX
ðiÞÞ
þ
Cov ðX t1pkþi ðiÞ; X t1pkþj ðjÞÞ þ Var ðDtk Þ
t1pkþi
p2 i¼1
p2 i<j
þ z2
k
X
i¼1
Var ðstkþi ðiÞ stk1þi ðiÞÞ þ 2z2
k
X
i<j
Cov ½stkþi ðiÞ stk1þi ðiÞ; stkþj ðjÞ stk1þj ðjÞ
622
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
!
!
k
k
k
X
X
X
2
2
2 Cov
X t1kþi ðiÞ;
X t1pkþi ðiÞ þ Cov
X t1kþi ðiÞ; Dtk
p
p
i¼1
i¼1
i¼1
!
!
k
k
k
X
X
X
2z
2
þ Cov
X t1kþi ðiÞ;
½stkþi ðiÞ stk1þi ðiÞ Cov
X t1pkþi ðiÞ; Dtk
p
p
i¼1
i¼1
i¼1
!
k
k
X
X
2z
Cov
X t1pkþi ðiÞ;
½stkþi ðiÞ stk1þi ðiÞ
p
i¼1
i¼1
!
k
X
½stkþi ðiÞ stk1þi ðiÞ :
ð12Þ
þ 2z Cov Dtk ;
i¼1
We simplify Eq. (12) to the following lemma, whose proof is given in Appendix A.
Lemma 1. When information on customer demand is shared, the variance of the order quantity at stage k is
VarðQkt Þ
ð3Þ
2 2kðk 1Þ
kþ1
2k 2 2kz lX
kz
2ðlL 1Þ ð3Þ
ð3Þ
2
2
½lD þ lD rD ¼ 1þ þ
lL þ
l rD þ 2 rX þ 2 p
p2
3
p
p rX prD D
p1
"
#
"
#
ð4Þ
ð4Þ
z2 p 2 lX
2 2
ðk 1Þz2 ðp2 3p þ 3Þ lX
ðp2 6p þ 3Þ 2
þ
r :
r þ
3
ð13Þ
p3 ðp 1Þ r2X
p ðp 1Þ X
2 p3 r2X p2 X
2
Corollary. When lead time is deterministic, Eq. (13) reduces to
Var ðQkt Þ ¼
ð3Þ
2 2kðk 1Þ
kþ1
2k 2 2kz LlD
2
1þ þ
L
þ
r
þ
r
D
p
p2
3
p2 X
p2 rX
n
o
2
3
ð4Þ
2
L 3ðL 1Þr4D þ lD
kz
2ðL
1Þ
z
p
2
2
ð3Þ
ð3Þ
½lD þ lD r2D þ 4 3 þ
lD 2 r2X 5
prD
p1
p
p
2
r2X
n
o
2
3
ð4Þ
4
z2 4p 2 L 3ðL 1ÞrD þ lD
2 25
2 rX
þ
p3
p
r2X
2
n
o
2
3
ð4Þ
4
ðk 1Þz2 4ðp2 3p þ 3Þ L 3ðL 1ÞrD þ lD
ðp2 6p þ 3Þ 2 5
rX :
þ
p3 ðp 1Þ
p3 ðp 1Þ
2
r2X
ð14Þ
As we see from Eqs. (13) and (14), the variance of the order quantity at stage k is amplified with the
amplification mostly linear in k, except for quadratic and cubic terms in some system parameters. Thus,
information sharing of customer demand per se does not eliminate the Bullwhip Effect. This reinforces what
Chen et al. (2000, p. 442) have said. However, if the manager at each stage does not update the order-up-to
level, i.e., if
ki
S ki
ti ¼ S t1i ;
i ¼ 0; 1; . . . ; k 1;
then the order quantity is the same as the previous demand, i.e., Qkt ¼ Dtk . Hence, in this case, there will be
no bullwhip effect, with Var(Qk) = Var(D), which conforms to Chen et al. and Dejonckheere et al.
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
623
2.3.2. Simple supply chain: One retailer and one immediate upstream agent
As a special case, let us consider a simple supply chain, with only one retailer and one immediate upstream agent: this is the case of k = 1. If in Eq. (12) we put k = 1, that equation reduces to
Var ðQt Þ ¼ Var ðDt1 Þ þ
1
1
2
Var ðX t1 Þ þ 2 Var ðX t1p Þ þ z2 Var ðst ðX Þ st1 ðX ÞÞ þ Cov ðDt1 ; X t1 Þ
p2
p
p
2
2
Cov ðDt1 ; X t1p Þ þ 2z Cov ðDt1 ; ðst ðX Þ st1 ðX ÞÞÞ 2 Cov ðX t1 ; X t1p Þ
p
p
þ
2z
2z
Cov ðX t1 ; ðst ðX Þ st1 ðX ÞÞÞ Cov ðX t1p ; ðst ðX Þ st1 ðX ÞÞÞ;
p
p
ð15Þ
and Lemma 1 becomes
2 2
2 2
z
2ðlL 1Þ ð3Þ
ð3Þ
2
½lD þ lD rD l Var ðQt Þ ¼ 1 þ rD þ 2 rX þ
p
p
prD D
p1
2z ð3Þ
z2 p 2 ð4Þ 2 4
þ 2 lX þ 2
l 2 rX :
p rX
p3 X
p
2rX
ð16Þ
Now if, as in Chen et al. (p. 437), we used the autoregressive process,
Dt ¼ l þ qDt1 þ et ;
ð17Þ
and their estimation method, then Eq. (15) would reduce to a result similar to Chen et al.Õs (p. 438):
2L 2L2
2L
p
2
þ 2 ð1 q Þ rD þ 2z 1 þ
Var ðQt Þ ¼ 1 þ
Cov ðDt1 ; ðst st1 ÞÞ þ z2 Var ðst st1 Þ:
p
p
p
ð18Þ
A difference between our Eq. (18) and Chen et al.Õs is that Chen et al. use the standard deviation of forecast
error, set, in setting safety stocks, whereas we use the standard deviation of lead-time demand, st.
With our method of estimation and with autoregressive demand, Eq. (15) would become (see
Appendix B),
"
(
)#
2 ð1 qpLþ1 Þð1 qL Þ 2
L
2qð1 qÞ þ 2q2 ð1 qL1 Þ þ qpLþ1 ð1 qL Þ2
VarðQt Þ ¼ 1 þ
þ 2
r2D
2
p ð1 q2 Þð1 qÞ
p ð1 q2 Þ
ð1 q2 Þð1 qÞ
2
þ 2z CovðDt1 ; ðst st1 ÞÞ þ CovðX t1 ; ðst st1 ÞÞ þ z2 Varðst st1 Þ:
ð19Þ
p
Setting q = 0, z = 0, Eq. (19) reduces to our previous Eq. (8).
2.3.3. When information on customer demand is not shared
Mason-Jones and Towill (1997) speak of a ‘‘seamless supply chain,’’ where everyone in the supply
chain gets the most recent market sales data. Utilization of this information, they say improves the
responsiveness of the supply chain and reduces the bullwhip effect. Güllü (1997) demonstrates that information sharing in a two-echelon allocation model results in lower order-up-levels and diminished system
costs. Lee et al. (2000) echo the same. With one supplier and multiple identical retailers, Cachon and
Fisher (2000) find that supply chain costs are 2.2% lower on average with full information, with a maximum difference of 12.1%. Mitra and Chatterjee (2004) show through numerical examples that the optimal
624
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
expected total cost with demand information known throughout a one-warehouse, two-retailer, system is
always lower than that with information withheld. Dejonckheere et al. (2004) employ control systems
engineering to show that an information-enriched supply chain reduces the bullwhip effect, especially
at the higher echelons. This is supported in a simulation study by Chatfield et al. (2004), where information sharing reduces variance amplification and protects a supply chain against ‘‘cascading failures’’, as
occurs in an electrical power grid.
When the information on customer demand is not shared, the kth stage relies on the order quantity from
the (k 1)th stage. Because the demand at the kth stage is the order Qk1
, lead-time demand at stage k (we
t
now use Y instead of X to distinguish lead-time demand when customer demand information is not shared)
is
Y t ðkÞ ¼
L1
X
Qk1
tþi :
ð20Þ
i
The order quantity at stage k is
1
1
k1
Qkt ¼ S kt S kt1 þ Qk1
t1 ¼ Y t1 ðkÞ Y t1p ðkÞ þ Qt1 þ zðst ðY ðkÞÞ st1 ðY ðkÞÞÞ:
p
p
ð21Þ
For simplicity, let st(Y(k)) = st(k). Then,
1
1
Var ðY t1 ðkÞÞ þ 2 Var ðY t1p ðkÞÞ þ z2 Var ðst ðkÞ st1 ðkÞÞ
p2
p
2
2
k1
Cov ðQk1
þ Cov ðQk1
t1 ; Y t1 ðkÞÞ t1 ; Y t1p ðkÞÞ þ 2z Cov ½Qt1 ; ðst ðkÞ st1 ðkÞÞ
p
p
2
2z
2 Cov ðY t1 ðkÞ; Y t1p ðkÞÞ þ Cov ½Y t1 ðkÞ; ðst ðkÞ st1 ðkÞÞ
p
p
2z
Cov ½Y t1p ðkÞ; ðst ðkÞ st1 ðkÞÞ:
p
Var ðQkt Þ ¼ Var ðQk1
t1 Þ þ
ð22Þ
Eq. (22) reduces to (16) for k = 1. Using Eq. (16), we obtain the following iterative approximation of the
bullwhip effect when information on customer demand is not shared.
Lemma 2. After starting with Eq. (16) for k = 1, we can approximate the variance of the order quantity at
stage k for information not sharing by the following iterative equation:
For k P 2,
2 2lL
2 2 2 2zlk1
z2 lk1
p 2 ð4Þ 2 4
ð3Þ
k
k1
L
L
Var ðQt Þ ffi 1 þ þ 2 Var ðQt Þ þ 2 lD rL þ 2
l þ
l 2 rX
p
p
p3 X
p
p
p rX X
2r2X
zlk1
2ðlL 1Þ ð3Þ
ð3Þ
½lD þ lD r2D :
þ L
lD ð23Þ
p1
prD
The proof of Lemma 2 is in Appendix A. As we see from Eq. (23), the variance of the order quantity at
stage k can be approximated as functions of the kth exponent of mean lead time when information on customer demand is not shared. This is with stochastic lead time and MA(p).
As a special case, consider z = 0, a case that Chen et al. and Dejonckheere et al. explored. Then from
Eqs. (22) and (23),
2 2lL
2
k
Var ðQt Þ ¼ 1 þ þ 2 Var ðQk1
Þ þ 2 l2D r2L :
ð24Þ
t
p
p
p
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
625
3. Numerical comparisons
3.1. Variance amplification: With and without information sharing
In Table 1, we give an example of the behavior of the variance in a k-stage supply chain with and without
information sharing. We have customer demand (k = 0) with a standard deviation of 20 and see that as a
result of MA(p = 15) forecasting and the consequent periodic updating of the order-up-to level that the
variance of the orders at the factory (k = 4) becomes inflated, rising to 52.5 with information sharing
and 179.0 without information sharing. The agreement with the simulation results in Table 1 is heartening,
except for Ôfactory,Õ where the error is nine percent, which we attribute to sampling variability.
Fig. 1 illustrates the nearly linear behavior of the variance in echelon stage, k, with information sharing
(is) and the exponential growth in k without it (nis). Merkuryev et al. (2003) produce a similar figure via
simulation.
3.2. Comparison with Chen et al. and Dejonckheere et al.
Chen et al. (2000, p. 441, and verified in Dejonckheere et al., 2003, p. 581, for k = 1) give a lower bound
(LB) for the amplification of customer demand with deterministic lead time and information sharing,
Pk
Pk
2
Var ðQk Þ
2 i¼1 Li 2ð i¼1 Li Þ
is:
P1þ
þ
:
ð25Þ
2
Var ðDÞ
p
p
Table 1
Comparison of information sharing vs. no information sharing: Standard deviation and amplitude ratio (AR)
Customer
Retailer
Wholesaler
Distributor
Factory
Std. dev.: information sharing (is)
ARis
Std. dev.: no information sharing: (nis)
ARnis
20 (20.18)
30.0 (30.21)
38.7 (37.22)
46.1 (43.34)
52.5 (47.72)
1
1.50
1.29
1.19
1.14
20 (19.77)
30.0 (30.47)
51.1 (51.85)
92.8 (92.04)
179.0 (162.32)
1
1.50
1.70
1.82
1.93
The bold numbers in parentheses are simulation results.
AR ¼ std:dev:ðoutputÞ
std:dev:ðinputÞ ; D N(50,400); is = information sharing; L gamma with mean 4 and variance 4; nis = no information sharing.
Variance
200
nis
100
is
0
0
1
2
3
4
k
Fig. 1. Variance amplification: no information sharing (nis) vs. information sharing (is).
626
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
Table 2
A comparison: Information sharing (z = 2)
Our Lemma 1 (stochastic L)
Our Lemma 1 (deterministic L)
Chen et al.Õs LB
Retailer k = 1
Wholesaler k = 2
Distributor k = 3
Factory k = 4
2.25
1.04
1.68
3.76
1.06
2.64
4.83
1.11
3.88
6.89
1.18
5.41
Table 3
A comparison: No information sharing (no safety stock case, z = 0)
Our Lemma 2 (stochastic L)
Our Lemma 2 (deterministic L)
Chen et al.Õs LB
Retailer k = 1
Wholesaler k = 2
Distributor k = 3
Factory k = 4
1.39
1.17
1.68
1.85
1.37
2.82
2.38
1.60
4.74
3.01
1.87
7.97
With no information sharing (and deterministic lead time), the LB would be
k Y
Var ðQk Þ
2Li 2L2i
P
þ 2 :
nis:
1þ
Var ðDÞ
p
p
i¼1
ð26Þ
Using L = 4, p = Tm = 15, k = 1, the LB given to variance amplification, reported by Chen et al. and
Dejonckheere et al. would be 1.68 (deterministic lead time) > 1.04 (deterministic lead time) in our Table
2. Tables 2 and 3 compare our results with Chen et al.Õs for information sharing and information not sharing. We repeat that we and Chen et al. used different updating methods, both legitimate, and the comparison is merely intended to highlight the significant effect of different estimating procedures.
4. Summary
We quantify the bullwhip effect in a serial supply chain, using periodic forecast updating schemes. We
employ an (R, S) inventory system, where the review period R = 1, and where S is the order-up-to level,
that is, S is revised every period. The revisions would be based on the realizations of the stochastic demand
and lead time and also the lead-time demand that obtain at the different echelons of the supply chain. We
conclude, as others before us have, that the bullwhip effect is caused by human intervention and by disruptions in information flow in the supply chain. As expected, lead times, whether deterministic or stochastic,
exacerbate the bullwhip effect by inflating the variance of the demand at the upstream echelons. However,
lead-time variability contributes further to the bullwhip effect.
We illustrate how seemingly similar estimating methods could yield dramatically different variance
amplifications and we reinforce the results in Chen et al. (2000) and Dejonckheere et al. (2003) that the bullwhip effect is due in part to demand forecasting. And as reported in earlier literature, the bullwhip effect is
attenuated when information is shared. Beyond that, we find that attenuation reducing an exponential bullwhip when information is not shared to a linear effect when it is shared.
Acknowledgment
The authors acknowledge with thanks the support from the Center of Supply Chain Research, Smeal
College of Business Administration, Pennsylvania State University.
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
627
Appendix A
More detailed proofs are posted on http://www.psu.edu.personal/jch/EJORAPPENDICES.pdf
Before proving Lemma 1, we begin with Propositions 1–7, which we need.
Proposition 1. The following holds:
Var ðst ðX Þ st1 ðX ÞÞ ¼
1 ðp 2Þ ð4Þ 2 4
l
r
:
X
p3
p2 X
2r2X
ðA:1Þ
Proof. Since
Var ðst ðX Þ st1 ðX ÞÞ ¼ Var ðst ðX ÞÞ þ Var ðst1 ðX ÞÞ 2 Cov ðst ðX Þ; st1 ðX ÞÞ;
ð4Þ
where
ðp 1Þ2 lX ðp 1Þðp 3Þr4X
4p3 r2X
¼ Var ðst1 ðX ÞÞ ðKendall and Stuart, 1977, p. 296Þ;
Var ðst ðX ÞÞ ¼
ðA:2Þ
and
Cov ðst ðX Þ; st1 ðX ÞÞ ¼
1
Cov ðs2t ðX Þ; s2t1 ðX ÞÞ ðKendall and Stuart, 1977, p. 247Þ;
4r2X
ðA:3Þ
we obtain
ð4Þ
Var ðst ðX Þ st1 ðX ÞÞ ¼
ðp 1Þ2 lX ðp 1Þðp 3Þr4X
1
2 Cov ðs2t ðX Þ; s2t1 ðX ÞÞ:
2
3
2rX
2p rX
ðA:4Þ
Now consider Cov ðs2t ðX Þ; s2t1 ðX ÞÞ in (A.4) and, for convenience, let s2t ðX Þ ¼ s2t . Then,
Cov ðs2t ; s2t1 Þ ¼
ðp3 4p2 þ 6p 3Þ ð4Þ p2 6p þ 3 4
lX rX :
p3 ðp 1Þ
p3
ðA:5Þ
Finally, we prove Proposition 1 by putting Eq. (A.5) into (A.4), obtaining
Var ðst ðX Þ st1 ðX ÞÞ ¼
1 p 2 ð4Þ 2 4
l
r
:
2r2X p3 X
p2 X
Proposition 2. The following will hold:
Cov ½X t1p ; st1 ðX Þ ¼ Cov ½X t1 ; st ðX Þ ¼
1
ð3Þ
l ;
2prX X
ðA:6Þ
and
Cov ½X t1p ; st ðX Þ ¼ Cov ½X t1 ; st1 ðX Þ ¼ 0:
ðA:7Þ
Proof. Since Cov ½X t1p ; st1 ðX Þ ¼ 2r1X Cov ½X t1p ; s2t1 ðX Þ (Kendall and Stuart, 1977, 247–248), consider
Cov ½X t1p ; s2t1 ðX Þ. Because
628
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
p
X
ðp 1Þ X t1p s2t1 ðX Þ ¼ X t1p ðX t1i X t1 Þ
2
i¼1
Pp1 2
2X 3t1p 2X t1p i¼1
X t1i
¼
þ X t1p
p
p
i¼1
Pp1
Pp1
Pp1
2
4X t1p j¼1 X t1j 2X t1p i¼1 X t1i j6¼i X t1j X 3t1p
þ
p
p
p
P
Pp1
Pp2
2
2
X t1p p1
X
2X
X
2X
X
t1p
t1p
j¼1 t1j
j¼1 t1j
j<k t1j X t1k
þ
þ
;
þ
p
p
p
ðA:8Þ
p1
X
X 3t1p
X 2t1i
ð3Þ
Cov ½X t1p ; s2t1 ðX Þ ¼ 1p lX . Hence,
Cov ½X t1p ; st1 ðX Þ ¼
1
ð3Þ
l :
2prX X
ðA:9Þ
Since
ðp 1Þ X t1p s2t ðX Þ
¼ X t1p
p
X
X 2ti
2X t1p
X t1p
Pp
2
j¼1 X tj
p
2
i¼1 X ti
p
i¼1
þ
Pp
þ
X t1p
2X t1p
Pp
i¼1 X ti
Pp
j6¼i X tj
p
Pp
j6¼k X tj X tk
p
;
Cov ½X t1p ; s2t ðX Þ ¼ 0: Therefore; Cov ½X t1p ; st ðX Þ ¼ 0:
ðA:10Þ
ðA:11Þ
Similarly we can show that
Cov ½X t1 ; st ðX Þ ¼
1
ð3Þ
l
2prX X
and
Cov ½X t1 ; st1 ðX Þ ¼ 0:
Proposition 3
1
2ðlL 1Þ ð3Þ
ð3Þ
2
½lD þ lD rD :
Cov ½Dt1 ; st ðX Þ st1 ðX Þ ¼
lD 2prD
p1
ðA:12Þ
Proof. Since Cov ½Dt1 ; st ðX Þ ¼ 2r1D Cov ½Dt1 ; s2t ðX Þ, consider Cov ½Dt1 ; s2t ðX Þ.
Because E½s2t ðX Þ ¼ E½s2t1 ðX Þ, then
Cov ðDt1 ; ðs2t ðX Þ s2t1 ðX ÞÞÞ ¼ E½Dt1 s2t ðX Þ E½Dt1 s2t1 ðX Þ:
Also we can express ðp 1ÞDt1 s2t ðX Þ as
ðp 1ÞDt1 s2t ðX Þo ¼
p
L1 X
L1
Xp XL1
p1
p1 X X
Dt1 i¼1 j¼0 D2tiþj þ
Dtiþj Dtiþk
p
p i¼1 j¼0 k6¼j
p
p
p
p
L1
L1 X
L1
1XXX
1XXX
Dtiþj Dtlþj Dtiþj Dtlþk :
p i¼1 l6¼i j¼0
p i¼1 l6¼i j¼0 k6¼j
ðA:13Þ
Similarly, ðp 1ÞDt1 s2t1 ðX Þ can be obtained by replacing t with t 1 of D in the summation at the RHS of
Eq. (A.13). Hence, after some manipulation, we obtain
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
629
E½Dt1 s2t ðX Þ E½Dt1 s2t1 ðX Þ ¼ E½EfDt1 s2t ðX ÞjLg E½EfDt1 s2t1 ðX ÞjLg
1 ð3Þ 2ðlL 1Þ ð3Þ
½l þ lD r2D :
¼ lD p
pðp 1Þ D
Therefore,
1
2ðlL 1Þ ð3Þ
ð3Þ
2
½lD þ lD rD :
Cov ½Dt1 ; st ðX Þ st1 ðX Þ ¼
lD 2prD
p1
Proposition 4. The following holds:
2 2
2 2
z
2ðlL 1Þ ð3Þ
ð3Þ
2
½lD þ lD rD Var ðQt Þ ¼ 1 þ rD þ 2 rX þ
l p
p
prD D
p1
2z ð3Þ
z2 p 2 ð4Þ 2 4
þ 2 lX þ 2
l 2 rX :
p rX
p3 X
p
2rX
ðA:14Þ
Proof. Consider Cov (Dt1,Xt1) and Cov (Dt1,Xt1p):
"
Cov ðDt1 ; X t1 Þ ¼ E E Dt1 Lt1
X
!#
Dt1þi jL
2
ðlD Þ lL ¼ r2D :
ðA:15Þ
i¼0
Because we assume that L 6 p,
" "
Ltp1
Cov ðDt1 ; X t1p Þ ¼ E E Dt1
X
i¼0
##
Dt1pþi L l2D lL ¼ 0:
ðA:16Þ
From Proposition 2,
Cov ðX t1p; ðst ðX Þ st1 ðX ÞÞÞ ¼
1 ð3Þ
l ;
2prX X
ðA:17Þ
and
Cov ðX t1 ; ðst ðX Þ st1 ðX ÞÞÞ ¼
1
ð3Þ
l :
2prX X
ðA:18Þ
From Proposition 3,
1
2ðlL 1Þ ð3Þ
ð3Þ
2
½lD þ lD rD :
Cov ½Dt1 ; st ðX Þ st1 ðX Þ ¼
lD 2prD
p1
Now
"
Cov ðX t1 ; X t1p Þ ¼ E E
Lt1
X
i¼0
Therefore,
Lt1p
Dt1þi
X
j¼0
!#
l2L l2D ¼ 0:
Dt1pþj L
2 2
2
2z ð3Þ
Var ðQt Þ ¼ 1 þ rD þ 2 r2X þ z2 Var ðst ðX Þ st1 ðX ÞÞ þ 2 lX
p
p
p rX
z
2ðlL 1Þ ð3Þ
ð3Þ
2
½lD þ lD rD :
þ
lD prD
p1
ðA:19Þ
ðA:20Þ
By applying Proposition 1 to (A.21), we prove Proposition 4.
h
ðA:21Þ
630
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
Proposition 5. The following holds:
k
X
i<j
k
X
kðk 1Þ
kþ1 2
lL Cov ðX t1kþi ðiÞ; X t1kþj ðjÞÞ ¼
Cov ðX t1pkþi ðiÞ; X t1pkþj ðjÞÞ:
rD ¼
2
3
i<j
ðA:22Þ
Proof. We can express X in terms of D. For convenience, we simplify the notation to Xt1k+i
PLi 1 (i) = Xt1k+i
and Xt1k+j
(j)
=
X
.
Let
L
(i)
=
L
and
L
(j)
=
L
.
Then
X
¼
t1kþi
t1k+j
t1k+i
i
t1k+j
j
m¼0 Dt1kþiþm and
PLj 1
X t1kþj ¼ n¼0
Dt1kþjþn . Hence,
X t1kþi X t1kþj ¼
Li 1
X
Dt1kþiþm Lj 1
X
ðA:23Þ
Dt1kþjþn :
n¼0
m¼0
Now consider j > i. When j = i + 1, the i can take (k 1) different values, i.e., i = 1, 2, . . . , k 1. For these
(k 1) cases, Eq. (A.23) can be expressed as
ðLi 1ÞD2 þ ½Li Lj ðLi 1ÞDa Db ;
where a and b are arbitrary integers such that a 5 b. If we continue this procedure, then when
j = i + (k 1), the i can take only one value, i.e., i = 1. For this case, Eq. (A.23) can be expressed as
ðLi ðk 1ÞÞD2 þ ½Li Lj ðLi ðk 1ÞÞDa Db :
Since E½X t1kþi ðiÞE½X t1kþj ðjÞ ¼ l2L l2D ,
k
X
i<j
kðk 1Þ
kþ1 2
lL Cov ðX t1kþi ðiÞ; X t1kþj ðjÞÞ ¼
rD :
2
3
Similarly we can show that
k
X
i<j
kðk 1Þ
kþ1 2
lL Cov ðX t1pkþi ðiÞ; X t1pkþj ðjÞÞ ¼
rD :
2
3
Proposition 6. The following holds:
Cov
k
X
k
X
X t1kþi ðiÞ;
ðstkþi ðiÞ stk1þi ðiÞÞ
i¼1
!
¼
i¼1
and
Cov
k
X
X t1kpþi ðiÞ;
i¼1
k
X
k
ð3Þ
l ;
2prX X
!
ðstkþi ðiÞ stk1þi ðiÞÞ
¼
i¼1
k
ð3Þ
l :
2prX X
Proof. For notational convenience, let sr+i(i) = sr+i. Then,
Cov
k
X
k
X
X t1kþi ðiÞ;
ðstkþi ðiÞ stk1þi ðiÞÞ
i¼1
¼
Xk
i¼1
i¼1
Cov ðX t1kþi ; stkþi st1kþi Þ þ
!
Xk
i6¼j
Cov ðX t1kþi ; stkþj st1kþj Þ:
ðA:24Þ
ðA:25Þ
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
631
Now consider Cov (Xt1k+i, stk+i). Since
ðp 1ÞX t1kþi s2tkþi
Pp
2X 3t1kþi 2X t1kþi m¼2 X 2tkþim
¼
þ X t1kþi
p
p
m¼2
Pp
Pp
Pp
2
4X
X tkþim 2X t1kþi m¼2 X tkþim n6¼m X tkþin
t1kþi m¼2
p
p
Pp
P
3
2
2
X t1kþi X t1kþi m¼2 X tkþim 2X t1kþi pm¼2 X tkþim
þ
þ
þ
p
p
p
Pp
2X t1kþi m<n X tkþim X tkþin
;
þ
p
p
X
X 3t1kþi
X 2tkþim
ðA:26Þ
by the way similar to Eqs. (A.8) and (A.9),
Cov ðX t1kþi ; stkþi Þ ¼
1
ð3Þ
l :
2prX X
ðA:27Þ
Next consider Cov (Xt1k+i, st1k+i). Since
ðp 1ÞX t1kþi s2t1kþi
¼ X t1kþi
p
X
X 2t1kþim
2X t1kþi
Pp
2
m¼1 X t1kþim
p
Pp
m¼1
þ
þ
X t1kþi
Pp
X t1kþi
Pp
2
m¼1 X t1kþim
p
2X t1kþi
n6¼m X t1kþim X t1kþin
p
m¼1 X t1kþim
Pp
n6¼m X t1kþin
p
ðA:28Þ
;
then similar to Eqs. (A.10) and (A.11)
Cov ðX t1kþi ; st1kþi Þ ¼ 0:
ðA:29Þ
Therefore,
k
X
Cov ðX t1kþi ; stkþi st1kþi Þ ¼
i¼1
k
ð3Þ
l :
2prX X
ðA:30Þ
Also consider
k
X
Cov ðX t1kþi ; stkþj st1kþj Þ ¼
i6¼j
k
X
Cov ðX t1kþi ; stkþj Þ i6¼j
k
X
Cov ðX t1kþi ; st1kþj Þ:
i6¼j
From the relationships in Eqs. (A.26)–(A.29), we obtain
k
X
Cov ðX t1kþi ; stkþj Þ ¼
i6¼j
k
kðk 1Þ ð3Þ X
lX ¼
Cov ðX t1kþi ; st1kþj Þ:
4prX
i6¼j
Therefore,
k
X
i6¼j
Cov ðX t1kþi ; stkþj st1kþj Þ ¼ 0:
ðA:31Þ
632
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
Hence, by Eqs. (A.30) and (A.31),
Cov
k
X
k
X
X t1kþi ðiÞ;
ðstkþi ðiÞ stk1þi ðiÞÞ
i¼1
!
i¼1
Also, in the same way, we can show that
Cov
k
X
¼
X t1kpþi ðiÞ;
i¼1
k
X
ðstkþi ðiÞ stk1þi ðiÞÞ
k
ð3Þ
l :
2prX X
!
¼
i¼1
k
ð3Þ
l :
2prX X
Proposition 7. The following holds:
k
X
Cov stkþi ðiÞ stk1þi ðiÞ; stkþj ðjÞ stk1þj ðjÞ
i6¼j
ðk 1Þ p 2 ð4Þ 2 4
ðk 1Þ p2 3p þ 3 ð4Þ p2 6p þ 3 4
l 3
r :
l 2 rX þ
¼
p3 X
p
p3 ðp 1Þ X
p ðp 1Þ X
2r2X
2r2X
ðA:32Þ
Proof. For notational convenience, let sr+i(i) = sr+i. Then,
k
X
Cov stkþi ðiÞ stk1þi ðiÞ; stkþj ðjÞ stk1þj ðjÞ
i6¼j
¼
k
X
Cov ðstkþi ; stkþj Þ i6¼j
k
X
Cov ðstkþi ; stk1þj Þ þ
i6¼j
k
X
k
X
Cov ðstk1þi ; stk1þj Þ
i6¼j
Cov ðstkþj ; stk1þi Þ:
i6¼j
Since Cov (stk+i, stk+i+m) = Cov (st, st+m) = Cov (st, stm) = Cov (stk+i, stk+im), we obtain
k
X
Cov stkþi ðiÞ stk1þi ðiÞ; stkþj ðjÞ stk1þj ðjÞ
i6¼j
¼ 2ðk 1Þ½Cov ðst ; st1 Þ Var ðst Þ þ 2½Cov ðst ; st1 Þ Cov ðst ; stk Þ:
Now consider Cov (st,stk). From Eqs. (A.3) and (A.5), we obtain
"
#
1 ðp kÞðp3 4p2 þ 6p 3Þ ð4Þ p4 ðk þ 7Þp3 þ ð7k þ 9Þp2 ð9k þ 3Þp þ 3k 4
lX rX :
Cov ðst ; stk Þ ¼ 2
2
2
4rX
p3 ðp 1Þ
p3 ðp 1Þ
ðA:33Þ
Using Eqs. (A.2), (A.3), (A.5) and (A.33), we obtain
k
X
Cov stkþi ðiÞ stk1þi ðiÞ; stkþj ðjÞ stk1þj ðjÞ
i6¼j
ðk 1Þ p 2 ð4Þ 2 4
ðk 1Þ p2 3p þ 3 ð4Þ p2 6p þ 3 4
l 3
r :
l 2 rX þ
¼
p3 X
p
p3 ðp 1Þ X
p ðp 1Þ X
2r2X
2r2X
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
633
Proof of Lemma
Pk1. By using Propositions
Pk1–7, we prove Lemma 1. We consider the terms in Eq. (12).
First, consider i¼1 Var ðX t1kþi ðiÞÞ and i¼1 Var ðX t1kpþi ðiÞÞ. We obtain
k
k
k
1 X
1 X
k 2
1 X
2
Var
ðX
ðiÞÞ
¼
r
ðiÞ
¼
r
¼
Var ðX t1kpþi ðiÞÞ:
t1kþi
p2 i¼1
p2 i¼1 X
p2 X p2 i¼1
ðA:34Þ
By Proposition 5, we obtain
k
2 X
kðk 1Þ
kþ1 2
Cov ðX t1kþi ðiÞ; X t1kþj ðjÞÞ ¼
lL rD
p2 i<j
p2
3
¼
k
2 X
Cov ðX t1pkþi ðiÞ; X t1pkþj ðjÞÞ:
2
p i<j
By definition, Var ðDtk Þ ¼ r2D .
By Proposition 1,
"
#
ð4Þ
k
X
kz2 p 2 lX
2 2
2
Var ðstkþi ðiÞ stk1þi ðiÞÞ ¼
2 rX ;
z
p3 r2X
p
2
i¼1
ðA:35Þ
ðA:36Þ
and by Proposition 7,
z2
k
X
Cov stkþi ðiÞ stk1þi ðiÞ; stkþj ðjÞ stk1þj ðjÞ
i6¼j
¼
ðk 1Þz2 p 2 ð4Þ 2 4
ðk 1Þz2 p2 3p þ 3 ð4Þ p2 6p þ 3 4
l
r
l
r
þ
:
p3 X
p2 X
p3 ðp 1Þ X
p3 ðp 1Þ X
2r2X
2r2X
From Eq. (A.20),
k
k
X
X
2
Cov
X
ðiÞ;
X t1pkþi ðiÞ
t1kþi
p2
i¼1
i¼1
ðA:37Þ
!
¼ 0:
ðA:38Þ
Now consider the covariance between X and D:
!
k
k
X
2
2X
Cov
X t1kþi ðiÞ; Dtk ¼
Cov ðX t1kþi ; Dtk Þ;
p
p i¼1
i¼1
where
X t1kþi ¼
Li 1
X
Dt1kþiþm :
m¼0
Hence, when i = 1, Cov ðX t1kþi ; Dtk Þ ¼ r2D .
For i = 2, 3, . . ., k, we obtain Cov (Xt1k+i, Dtk) = 0. Hence,
!
k
X
2
2
Cov
X t1kþi ðiÞ; Dtk ¼ r2D :
p
p
i¼1
ðA:39Þ
From Eq. (A.24) in Proposition 6,
!
k
k
X
X
2z
kz ð3Þ
Cov
X t1kþi ðiÞ;
½stkþi ðiÞ stk1þi ðiÞ ¼ 2 lX :
p
p
rX
i¼1
i¼1
ðA:40Þ
634
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
From Eq. (A.16) from Proposition 4,
!
k
X
2
Cov
X t1pkþi ðiÞ; Dtk ¼ 0:
p
i¼1
ðA:41Þ
From Eq. (A.25) in Proposition 6,
!
k
k
X
X
2z
kz ð3Þ
Cov
X t1pkþi ðiÞ;
½stkþi ðiÞ stk1þi ðiÞ ¼ 2 lX :
p
p
rX
i¼1
i¼1
ðA:42Þ
By Proposition 3,
k
X
2z Cov Dtk ;
½stkþi ðiÞ stk1þi ðiÞ
!
i¼1
¼
kz
2ðlL 1Þ ð3Þ
ð3Þ
½lD þ lD r2D :
lD prD
p1
Hence, from Eqs. (A.35)–(A.44), we prove Lemma 1.
ðA:43Þ
h
Proof of Lemma 2. First consider EðQkt Þ. Since Qkt ¼ S kt S kt1 þ Qk1
t1 ,
0
EðQkt Þ ¼ EðQk1
t1 Þ ¼ EðQtk Þ ¼ EðDtk Þ ¼ lD :
ðA:44Þ
Now consider Var (Yt1(k)) and Var (Yt1p(k)). By Eq. (A.44),
k1
2 2
t1
Qk1
Var ðY t1 ðkÞÞ ¼ Var sumLi¼0
t1þi ¼ lL Var ðQt1 Þ þ lD rL ¼ Var ðY t1p ðkÞÞ:
Also,
"
Cov ðQk1
t1 ; Y t1 ðkÞÞ
¼E E
Qk1
t1
Lt1
X
Qk1
t1þi Lt1
i¼0
Similar to Eq. (A.16),
" (
k1
Cov ðQk1
t1 ; Y t1p ðkÞÞ ¼ E E Qt1
L1
X
i¼0
Similar to Eq. (A.20),
" (
Cov ½ðY t1 ðkÞ; Y t1p ðkÞÞ ¼ E E
Lt1
X
i¼0
!#
2
k1
lL E½Qk1
t1 ¼ Var ðQt1 Þ:
¼ L:
)#
E½Qk1
Qk1
t1pþi L
t1 E½Y t1p ðkÞ ¼ 0:
Qk1
t1þi
Ltp
X
i¼0
ðA:45Þ
)#
Qk1
t1pþi L
2
l2L E½Qk1
t1 ¼ 0:
Therefore, Eq. (22) reduces to
2 2l
2 2 2
2
Var ðQkt Þ ¼ 1 þ þ 2L Var ðQk1
t1 Þ þ 2 lD rL þ z Var ðst ðkÞ st1 ðkÞÞ
p
p
p
2z
Cov ½Y t1 ðkÞ; ðst ðkÞ st1 ðkÞÞ
þ 2z Cov ½Qk1
t1 ; ðst ðkÞ st1 ðkÞÞ þ
p
2z
Cov ½Y t1p ðkÞ; ðst ðkÞ st1 ðkÞÞ:
p
ðA:46Þ
ðA:47Þ
ðA:48Þ
ðA:49Þ
Instead of finding an exact expression for Var ðQkt Þ, we shall approximate it. Since we can express Yt(k), the
LTD at stage k, in terms of the LTD at the first stage, Xt, as
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
Y t ðkÞ ¼
XX
X
635
ðA:50Þ
X t;
all covariance terms and Var (st(k) st1(k)) can be approximated by using relationships in Eq. (A.50). As a
result, we obtain the following iterative approximation of the bullwhip effect when information on customer demand is not shared: When k = 1, Eq. (A.49) becomes Eq. (16). For k P 2,
Var ðQkt Þ
2 2lL
2 2 2 2zlk1
z2 lk1
p 2 ð4Þ 2 4
ð3Þ
k1
L
L
ffi 1 þ þ 2 Var ðQt Þ þ 2 lD rL þ 2
l þ
l 2 rX
p
p
p3 X
p
p
p rX X
2r2X
k1
zl
2ðlL 1Þ ð3Þ
ð3Þ
½lD þ lD r2D :
þ L
lD p1
prD
ðA:51Þ
Appendix B
Let
Dt ¼ lD þ qDt1 þ et ;
ðB:1Þ
lD
and Var ðDt Þ ¼
then, EðDt Þ ¼ 1q
fined as
Xt ¼
L1
X
r2D
.
1q2
Let the lead time, L, be deterministic, with lead-time demand, Xt, de-
ðB:2Þ
Dtþi :
i¼0
m
q
2
From Eq. (15), let us consider Var (Xt1) first. Since Cov ðDt ; Dtþm Þ ¼ 1q
2 rD ,
Var ðX t1 Þ ¼
L1
X
Var ðDt1þi Þ þ 2
X
Cov ðDt1þi ; Dt1þj Þ
i<j
i¼0
L
2q
qð1 qL1 Þ
ðL
1Þ
þ
r2D ¼ Var ðX t1p Þ:
¼
1 q2 ð1 qÞð1 q2 Þ
1q
Now consider the covariance terms,
Cov ðDt1 ; X t1 Þ ¼ Cov Dt1 ;
L1
X
!
Dt1þi
i¼0
¼
1 qL
r2 :
ð1 q2 Þð1 qÞ D
ðB:3Þ
ðB:4Þ
Similarly,
Cov ðDt1 ; X t1p Þ ¼
qpLþ1 ð1 qL Þ 2
r :
ð1 q2 Þð1 qÞ D
ðB:5Þ
Next consider
Cov ðX t1 ; X t1p Þ ¼
L1 X
L1
X
i¼0
Cov ðDt1þi ; Dt1pþj Þ ¼
j¼0
qpLþ1 ð1 qL Þ
ð1 q2 Þð1
2
qÞ
2
r2D :
ðB:6Þ
From Eqs. (A.17) and (A.18),
Cov ðX t1 ; ðst ðX Þ st1 ðX ÞÞÞ ¼ Cov ðX t1p ; ðst ðX Þ st1 ðX ÞÞÞ:
By putting (B.3)–(B.7) into Eq. (15), we obtain Eq. (19).
ðB:7Þ
636
J.G. Kim et al. / European Journal of Operational Research 173 (2006) 617–636
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