- Email: [email protected]
Delayed demand information and dampened bullwhip effect
Operations Research Letters 33 (2005) 289 – 294
Operations Research Letters www.elsevier.com/locate/dsw
Delayed demand information and dampened bullwhip effect夡 Xiaolong Zhang∗ Department of Information Systems and Decision Sciences, Information System and Science, Fairleigh Dickinson University, 285 Madison Ave, The Mansion M-MS2-04, Madison, NJ 07940, USA Received 17 November 2003; accepted 20 May 2004 Available online 27 August 2004
Abstract This paper considers an inventory setting in which the historical data used for demand forecasting is delayed. When the replenishment is controlled via an order-up-to policy, we show that such delays reduce the variability of the order history and dampen the bullwhip effect. We discuss the intuition behind this result. © 2004 Elsevier B.V. All rights reserved. Keywords: Information delay; Forecasting; Supply chain management; Bullwhip effect
1. Introduction Despite the recent advances in scanning technology for real-time data collection and processing, many businesses still experience delays in providing demand data for forecasting purposes. Large companies often operate a centralized forecasting function that provides forecasting input to other operational units. In such a decoupled environment, demand information originated from an inventory control unit, for example, cannot be immediately made available to the forecasters because of data processing time and transmission delay. Even in a decentralized setting in 夡 This research is supported in part by a summer grant from Silberman College of Business at Fairleigh Dickinson University. ∗ Corresponding author. Department of Information Systems and Decision Sciences, 285 Madison Avenue, Madison, NJ 07940 USA. Tel.: +1-973-443-8873; fax: +1-973-443-8837. E-mail address: [email protected] (X. Zhang).
0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.05.010
which an operational unit performs its own forecasting, the cycle time for developing and updating forecasts does not always coincide with the data collection and processing cycle. This lack of synchronization can also cause the demand data used for forecasting to lag behind the current demand. This paper considers the impact of such delays in demand data on the bullwhip effect for a singleitem–single-installation inventory system. When the system is controlled via an order-up-to policy, we show that forecasts based on delayed demand data reduce the variability in the order process. Hence, in a two-echelon serial supply system linked by a demand and supply relationship, delays tend to stabilize the demand for the upstream echelon and dampen the bullwhip effect. The bullwhip effect reflects the magnified demand variance in a supply system and the resulting system volatility. It has received much attention
290
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
[1,3–5,9] since Lee et al. [6,7] established the theoretical bases for its existence. The effect can significantly increase the cost of maintaining an inventory for upstream installations in a supply chain, as more safety stock needs to be maintained to protect against the increasing demand variability. We argue in this paper that order quantities are oversensitive to temporary changes in the current demand data when the minimum-mean-squared-error (MMSE) forecasting method is applied. Even though delays in demand data reduce the forecasting accuracy at the installation where they occur, they help moderate the sensitivity of the orders. The reduced bullwhip effect can be financially beneficial to an upstream installation. The rest of the paper is organized as follows. Section 2 describes the inventory system including the order-up-to control policy and the demand for the system. Section 3 presents the MMSE lead-time demand forecasts and characterizes the order history process for the two cases with or without a delay. Section 4 derives the impact of a delay in demand data on the bullwhip effect. Section 5 concludes the paper with a discussion of managerial implications.
2. The inventory system Consider a periodic-review single-item–singleinstallation inventory system. The installation can be treated as a retailer that orders from its supplier with infinite capacity to meet the demand for a single item. We assume that the retailer is supplied with a constant lead time L = 1, 2, 3, . . . , all shortages are backordered, there is no fixed ordering cost, and the inventory carrying and shortage penalty costs are linear. The sequence of key events during a replenishment period t is as follows: outstanding order made L periods ago is first received; the current demand dt realizes; an order quantity decision qt is then made. Conventionally, the demand is assumed to occur after an ordering decision is made. Reversing this order implies that the demand for the current period dt is known and is not part of the lead-time demand. Lee et al. [8] use the same sequence of events to quantify the value of information sharing. Similar to Lee et al. [6] and Chen et al. [3,4], we model the customer demand as a first-order
autoregressive AR(1) process: dt = + dt−1 + t ,
|| < 1,
(1)
where is a constant that determines the mean of the demand, t is an i.i.d. normally distributed random error with a zero mean and a variance 2 , and is the first-order autocorrelation coefficient. The assumption of || < 1 assures that the demand process is covariance stationary and that the unconditional variance of the demand is finite. We use Ht to represent the history of demand observed up to and including period t, Ht = {dt , dt−1 , dt−2 , . . .}. When there is no delay in making the demand history available to the forecasters, the demand forecast is based on the information contained in Ht . When there is a delay of periods, = 1, 2, 3, . . . , the demand forecast can only use the information contained in the demand history Ht− . For deriving the MMSE leadtime demand forecasts in either case, it is convenient to express the AR(1) demand in the following meancentered form: dt − d = dt−1 − d + t , (2) where d = /(1 − ) is the unconditional mean of the AR(1) demand process. L Let L t be a forecast for the lead-time demand Dt = L 2 the varii=1 dt+i , V (x) = E[(x − E(x)) ] represent
L L ance of a random variable, and L t = V (Dt − t ) be the standard deviation of the associated forecasting error. Under the inventory system setup, a reasonable control policy is a myopic order-up-to level policy in which the order-up-to level yt is determined by (see Zipkin [10], Chapter 9): L y t = L t + z t ,
(3)
where z is the normal z score chosen to meet a desired level of stock-out probability in an order cycle. The balance of flow implies that the order quantity can be calculated from: qt = yt − yt−1 + dt .
(4)
Hence, our control policy dictates that the retailer orders an amount to replace the depleted stock from current demand and to compensate for anticipated
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
changes in its inventory position. Substituting into Eq. (4) the order-up-to level from Eq. (3), we have: qt = (L t
L − L t−1 ) + z(t
− L t−1 ) + dt .
We show in the following development that the standard deviation of lead-time demand forecasting error remains constant over time. Therefore, the second term on the right-hand side of the above equation drops out, and we obtain the following simple order quantity equation found in Lee, et al. [6], Chen, et al. [3,4], and Graves [5]: L qt = (L t − t−1 ) + dt .
(5)
In the following section, we summarize the MMSE lead-time demand forecasts for the two cases we consider: no delay or a delay of periods in demand information. We then characterize the resulting order quantity processes based on Eq. (5). To differentiate the two cases, we use a “∧ ” to denote all quantities
Single-period demand forecast dˆt+ = Et (dt+ ) = d + (dt − d ) eˆt+ =
−1 j =0
d˜t+ = Et− (dt+ ) = d + + (dt− − d ) e˜t+ =
+ −1 j =0
j t+−j = eˆt+ +
obtained from the MMSE method: dˆt+ : The MMSE forecast of future demand dt+ based on Ht eˆt+ = dt+ − dˆt+ : The forecast error associated with dˆt+ d˜t+ : The MMSE forecast of future demand dt+ based on Ht− e˜t+ = dt+ − d˜t+ : The forecast error associated with d˜t+ . In the following table, we summarize the standard MMSE forecasting results for an AR(1) demand process. Details for single-period forecasts can be found in the classical time-series text of Box and Jenkins [2]. The lead-time demand forecasts and the associated errors are found by simply summing the single period demand forecasts and the corresponding forecasting errors. Lead-time demand forecast L L+1 ˆ L dˆt+ = Ld + −1− t = (dt − d ) =1
DtL − ˆ L t =
j t+−j
−1 j =0
+j t−j
associated with the case of no information delay and a “∼” to represent those derived for the case of an -period delay. 3. MMSE forecasting of lead-time demand Let Et (·) represent the conditional mean of a variable of interest given Ht . We employ the following notations to describe the forecasted quantities
291
˜ L t =
L
=1
L
=1
eˆt+ =
1 1−
L−1 j =0
1 − j +1 t+L−j
−L+1 ) (dt− − d ) d˜t+ = Ld + (1−
DtL − ˜ L t =
L
=1
e˜t+ = DtL − ˆ L t +
1−L 1−
−1 j =0
j +1 t−j
Note that the single-period forecast in either case is the conditional expectation of future demand given the appropriate information history Ht orHt− . The forecast dˆt+ or d˜t+ reaches back only to the last available demand dt or dt− , and it is the sum of the unconditional mean demand d and an adjustment term based on the deviation of dt or dt− from d . As increases, the adjustment term declines exponentially, and the MMSE forecast quickly approaches the
292
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
unconditional mean of the demand. Therefore, short-term forecasts are more sensitive to temporary fluctuations in demand than long-term forecasts. When comparing dˆt+ and d˜t+ , we can make an immediate observation on the impact of an information delay: the factor further dampens the influence of a temporary deviation in demand on the forecast. For instance, if we assume in both cases that the last available demand deviates the same amount from its unconditional mean, i.e., dt − d = dt− − d , we have (d˜t+ − d )/(dˆt+ − d ) = , indicating that the single-period forecast in the case of a -period delay fluctuates around d proportionally less than the case of no delay by a factor of . This observation also holds true for lead-time demand forecasts ˆ L ˜L t and t . In conclusion, a delay tends to stabilize the demand forecast, which in turn dampens the bullwhip effect, as we show in the next section. When comparing the forecasting errors, we see clearly that an -period delay increases additively the forecasting errors for both the single-period and leadtime demand forecasts. We can interpret this increase in two ways. First, demand history Ht− contains less information, which causes a less accurate forecast. Second, when a forecast is made based on the delayed demand historyHt− , the forecast for future period t+ is periods further into the future. The increase in forecasting error simply reflects the greater uncertainty of a more distant future. Recognizing that the error terms in the lead-time demand forecasting errors are independent, we can easily obtain their variances 2 2 (ˆ L ˜L t ) and ( t ) as:
2 (1 − )2 2 (1 − 2L ) 2(1 − L ) , × L+ − 1 − 2 1−
2 (ˆ L t ) =
2 2 ˆL (˜ L t ) + t ) = (
2
2 1 − L 1 − 2
(1 − )2 1 − 2
2 .
Both variances remain constant over time, as we claimed before for arriving at Eq. (5). The second term in the last equality quantifies the additive impact of a delay on the lead-time demand forecasting error variance. Substituting ˆ L ˜L t and t from the above table
for L t into Eq. (5), we have:
− L+1 1 − L+1 dt − dt−1 , 1− 1− − L+1 (dt− − dt−−1 ). q˜t = dt + 1−
qˆt =
(6)
We use these order quantities to derive their corresponding bullwhip effect measures in the next section, from which we establish the dampening effect of a delay.
4. Dampened bullwhip effect We use an absolute bullwhip effect measure defined as the difference between the order-quantity variance and the demand variance: M = V (qt ) − V (dt ).
(7)
Lee et al. [6] among others [3–5] quantify the bullwhip effect with a relative measure defined as a variance ratio R = V (qt )/V (dt ). M and R are equivalent measures because they are linked by M = (R − 1)V (dt ). Let Mˆ and M˜ represent the absolute bullwhip effect measures without a delay and with a delay of periods, respectively. There exists a simple relationship between these two measures that indicates the dampening impact of a delay on the bullwhip effect. We state the result in the following proposition: Proposition. Mˆ and M˜ are proportional such that: ˆ M˜ = 2 M.
(8)
Equivalently, the relative measures Rˆ and R˜ are related by R˜ − 1 = 2 (Rˆ − 1).
(9)
Proof. From a standard time-series result [2], we have V (dt ) = 2 /(1 − 2 ), ∀t. If we substitute the definition of the AR(1) demand from Eq. (1) into the order quantity qˆt in Eq. (6), we have for the case of no delay: qˆt =
1 − L+1 1 − L+1 + t + L+1 dt−1 . 1− 1−
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
Recognizing that t and dt−1 are independent, we immediately obtain: V (qˆt ) =
1 − L+1 1−
2
and M˜ = V (q˜t ) − V (dt ) 2 1 − L+1 2(L+1) − 1 2 2 + . = 1 − 2 (1 − )2
2(L+1) 2 2 + , 1 − 2
293
and Mˆ = V (qˆt ) − V (dt ) 2 1 − L+1 2(L+1) − 1 2 + . = 1 − 2 (1 − )2
The conclusion in the proposition is immediate from the above expression.
We use a similar strategy to decompose order quantity q˜t into past observed demand and uncorrelated future errors. The decomposition can then be used to easily obtain the variance for the order quantity q˜t when there is a delay of periods. We can expand dt backward in time in the following manner: dt − d
= = .. .
(dt−1 − d ) + t 2 (dt−2 − d ) + t + t−1 ,
=
+1 (dt−−1 − d ) +
j =0
j t−j .
Substituting the above expression and dt− − d = (dt−−1 − d ) + t− into Eq. (6), we have: q˜t − d =+1 (dt−−1 − d ) +
j t−j +
j =0
( − L+1 ) 1−
×[t− − (1 − )(dt−−1 − d )] =
−1 j =0
j t−j +
(1 − L+1 ) × t− 1−
+ L++1 (dt−−1 − d ). Again the error terms in the above expression are independent of dt−−1 , and we have:
1 − 2 2 (1 − L+1 )2 + 1 − 2 (1 − )2 2 2(L+1) , + 1 − 2
V (q˜t )=
2
When there is a delay of periods in demand information, the proposition states that the bullwhip effect is dampened by a factor of 2 for|| < 1. This phenomenon can be explained from our observation in the last section. The forecast d˜t+ is less sensitive to the temporary changes in demand and it tends to gravitate closer to the unconditional mean of the demand d than dˆt+ . The same observation applies to the predicted lead-time demand ˜ L t , which gravitates closer to Ld . Consequently, the order quantity q˜t tends to have less fluctuation than qˆt .
5. Conclusions We show that a delay in demand information for forecasting purposes dampens the bullwhip effect. The rationale behind this finding has to do with a special property of MMSE forecasting. For an AR(1) demand, both the long-term single-period and lead-time demand forecasts converge to their unconditional means exponentially. One of the repercussions of this convergence is that a delay in demand information availability stabilizes the demand forecasts, which in turn generates a less variable order history in our inventory setting and causes a decrease in the bullwhip effect. The findings here have implications for managing supply chains. Suppose that the inventory system we consider here is inserted into a serial chain coupled by a demand–supply relationship. From a local perspective, delays increase the forecasting error for our system. From a global view, however, delays could benefit the upstream suppliers by reducing their demand variability. The cliché “no news is good news” is proven relevant in our setting—late demand news can sometimes be good news for the upstream suppliers.
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X. Zhang / Operations Research Letters 33 (2005) 289 – 294
Acknowledgements I thank Qing Liu, Rose Twomey, and two editors for their comments on earlier versions of this paper. References [1] Y. Aviv, A time-series framework for supply-chain inventory management, Oper. Res. 51 (2003) 210–227. [2] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Second Edition, Holden-Day, SanFrancisco, CA, 1976. [3] F. Chen, D. Zvi, K.R. Jennifer, S.-L. David, Quantifying the bullwhip effect in a simple supply chain, Manage. Sci. 46 (2000) 436–443. [4] F. Chen, D. Zvi, K.R. Jennifer, S.-L. David, The impact of exponential smoothing forecasts on the bullwhip effect, Nav. Res. Logistics 47 (2000) 269–286.
[5] S.C. Graves, A single-item inventory model for a nonstationary demand process, Manuf. Serv. Oper. Manage. 1 (1999) 50–61. [6] H.L. Lee, P. Padmanabhan, S. Whang, Information distortion in a supply chain: the bullwhip effect, Manage. Sci. 43 (1997) 546–558. [7] H.L. Lee, P. Padmanabhan, S. Whang, Bullwhip effect in a supply chain, Sloan Manage. Rev. 38 (Spring, 1997) 93–102. [8] H.L. Lee, K.C. So, C.S. Tang, The value of information sharing in a two-level supply chain, Manage. Sci. 46 (2000) 626–643. [9] X. Zhang, The impact of forecasting methods on the bullwhip effect, International J. Prod. Econ. 88 (2004) 15–27. [10] P.H. Zipkin, Foundations of Inventory Management, McGrawHill, New York, 2000.
Operations Research Letters www.elsevier.com/locate/dsw
Delayed demand information and dampened bullwhip effect夡 Xiaolong Zhang∗ Department of Information Systems and Decision Sciences, Information System and Science, Fairleigh Dickinson University, 285 Madison Ave, The Mansion M-MS2-04, Madison, NJ 07940, USA Received 17 November 2003; accepted 20 May 2004 Available online 27 August 2004
Abstract This paper considers an inventory setting in which the historical data used for demand forecasting is delayed. When the replenishment is controlled via an order-up-to policy, we show that such delays reduce the variability of the order history and dampen the bullwhip effect. We discuss the intuition behind this result. © 2004 Elsevier B.V. All rights reserved. Keywords: Information delay; Forecasting; Supply chain management; Bullwhip effect
1. Introduction Despite the recent advances in scanning technology for real-time data collection and processing, many businesses still experience delays in providing demand data for forecasting purposes. Large companies often operate a centralized forecasting function that provides forecasting input to other operational units. In such a decoupled environment, demand information originated from an inventory control unit, for example, cannot be immediately made available to the forecasters because of data processing time and transmission delay. Even in a decentralized setting in 夡 This research is supported in part by a summer grant from Silberman College of Business at Fairleigh Dickinson University. ∗ Corresponding author. Department of Information Systems and Decision Sciences, 285 Madison Avenue, Madison, NJ 07940 USA. Tel.: +1-973-443-8873; fax: +1-973-443-8837. E-mail address: [email protected] (X. Zhang).
0167-6377/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.05.010
which an operational unit performs its own forecasting, the cycle time for developing and updating forecasts does not always coincide with the data collection and processing cycle. This lack of synchronization can also cause the demand data used for forecasting to lag behind the current demand. This paper considers the impact of such delays in demand data on the bullwhip effect for a singleitem–single-installation inventory system. When the system is controlled via an order-up-to policy, we show that forecasts based on delayed demand data reduce the variability in the order process. Hence, in a two-echelon serial supply system linked by a demand and supply relationship, delays tend to stabilize the demand for the upstream echelon and dampen the bullwhip effect. The bullwhip effect reflects the magnified demand variance in a supply system and the resulting system volatility. It has received much attention
290
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
[1,3–5,9] since Lee et al. [6,7] established the theoretical bases for its existence. The effect can significantly increase the cost of maintaining an inventory for upstream installations in a supply chain, as more safety stock needs to be maintained to protect against the increasing demand variability. We argue in this paper that order quantities are oversensitive to temporary changes in the current demand data when the minimum-mean-squared-error (MMSE) forecasting method is applied. Even though delays in demand data reduce the forecasting accuracy at the installation where they occur, they help moderate the sensitivity of the orders. The reduced bullwhip effect can be financially beneficial to an upstream installation. The rest of the paper is organized as follows. Section 2 describes the inventory system including the order-up-to control policy and the demand for the system. Section 3 presents the MMSE lead-time demand forecasts and characterizes the order history process for the two cases with or without a delay. Section 4 derives the impact of a delay in demand data on the bullwhip effect. Section 5 concludes the paper with a discussion of managerial implications.
2. The inventory system Consider a periodic-review single-item–singleinstallation inventory system. The installation can be treated as a retailer that orders from its supplier with infinite capacity to meet the demand for a single item. We assume that the retailer is supplied with a constant lead time L = 1, 2, 3, . . . , all shortages are backordered, there is no fixed ordering cost, and the inventory carrying and shortage penalty costs are linear. The sequence of key events during a replenishment period t is as follows: outstanding order made L periods ago is first received; the current demand dt realizes; an order quantity decision qt is then made. Conventionally, the demand is assumed to occur after an ordering decision is made. Reversing this order implies that the demand for the current period dt is known and is not part of the lead-time demand. Lee et al. [8] use the same sequence of events to quantify the value of information sharing. Similar to Lee et al. [6] and Chen et al. [3,4], we model the customer demand as a first-order
autoregressive AR(1) process: dt = + dt−1 + t ,
|| < 1,
(1)
where is a constant that determines the mean of the demand, t is an i.i.d. normally distributed random error with a zero mean and a variance 2 , and is the first-order autocorrelation coefficient. The assumption of || < 1 assures that the demand process is covariance stationary and that the unconditional variance of the demand is finite. We use Ht to represent the history of demand observed up to and including period t, Ht = {dt , dt−1 , dt−2 , . . .}. When there is no delay in making the demand history available to the forecasters, the demand forecast is based on the information contained in Ht . When there is a delay of periods, = 1, 2, 3, . . . , the demand forecast can only use the information contained in the demand history Ht− . For deriving the MMSE leadtime demand forecasts in either case, it is convenient to express the AR(1) demand in the following meancentered form: dt − d = dt−1 − d + t , (2) where d = /(1 − ) is the unconditional mean of the AR(1) demand process. L Let L t be a forecast for the lead-time demand Dt = L 2 the varii=1 dt+i , V (x) = E[(x − E(x)) ] represent
L L ance of a random variable, and L t = V (Dt − t ) be the standard deviation of the associated forecasting error. Under the inventory system setup, a reasonable control policy is a myopic order-up-to level policy in which the order-up-to level yt is determined by (see Zipkin [10], Chapter 9): L y t = L t + z t ,
(3)
where z is the normal z score chosen to meet a desired level of stock-out probability in an order cycle. The balance of flow implies that the order quantity can be calculated from: qt = yt − yt−1 + dt .
(4)
Hence, our control policy dictates that the retailer orders an amount to replace the depleted stock from current demand and to compensate for anticipated
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
changes in its inventory position. Substituting into Eq. (4) the order-up-to level from Eq. (3), we have: qt = (L t
L − L t−1 ) + z(t
− L t−1 ) + dt .
We show in the following development that the standard deviation of lead-time demand forecasting error remains constant over time. Therefore, the second term on the right-hand side of the above equation drops out, and we obtain the following simple order quantity equation found in Lee, et al. [6], Chen, et al. [3,4], and Graves [5]: L qt = (L t − t−1 ) + dt .
(5)
In the following section, we summarize the MMSE lead-time demand forecasts for the two cases we consider: no delay or a delay of periods in demand information. We then characterize the resulting order quantity processes based on Eq. (5). To differentiate the two cases, we use a “∧ ” to denote all quantities
Single-period demand forecast dˆt+ = Et (dt+ ) = d + (dt − d ) eˆt+ =
−1 j =0
d˜t+ = Et− (dt+ ) = d + + (dt− − d ) e˜t+ =
+ −1 j =0
j t+−j = eˆt+ +
obtained from the MMSE method: dˆt+ : The MMSE forecast of future demand dt+ based on Ht eˆt+ = dt+ − dˆt+ : The forecast error associated with dˆt+ d˜t+ : The MMSE forecast of future demand dt+ based on Ht− e˜t+ = dt+ − d˜t+ : The forecast error associated with d˜t+ . In the following table, we summarize the standard MMSE forecasting results for an AR(1) demand process. Details for single-period forecasts can be found in the classical time-series text of Box and Jenkins [2]. The lead-time demand forecasts and the associated errors are found by simply summing the single period demand forecasts and the corresponding forecasting errors. Lead-time demand forecast L L+1 ˆ L dˆt+ = Ld + −1− t = (dt − d ) =1
DtL − ˆ L t =
j t+−j
−1 j =0
+j t−j
associated with the case of no information delay and a “∼” to represent those derived for the case of an -period delay. 3. MMSE forecasting of lead-time demand Let Et (·) represent the conditional mean of a variable of interest given Ht . We employ the following notations to describe the forecasted quantities
291
˜ L t =
L
=1
L
=1
eˆt+ =
1 1−
L−1 j =0
1 − j +1 t+L−j
−L+1 ) (dt− − d ) d˜t+ = Ld + (1−
DtL − ˜ L t =
L
=1
e˜t+ = DtL − ˆ L t +
1−L 1−
−1 j =0
j +1 t−j
Note that the single-period forecast in either case is the conditional expectation of future demand given the appropriate information history Ht orHt− . The forecast dˆt+ or d˜t+ reaches back only to the last available demand dt or dt− , and it is the sum of the unconditional mean demand d and an adjustment term based on the deviation of dt or dt− from d . As increases, the adjustment term declines exponentially, and the MMSE forecast quickly approaches the
292
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
unconditional mean of the demand. Therefore, short-term forecasts are more sensitive to temporary fluctuations in demand than long-term forecasts. When comparing dˆt+ and d˜t+ , we can make an immediate observation on the impact of an information delay: the factor further dampens the influence of a temporary deviation in demand on the forecast. For instance, if we assume in both cases that the last available demand deviates the same amount from its unconditional mean, i.e., dt − d = dt− − d , we have (d˜t+ − d )/(dˆt+ − d ) = , indicating that the single-period forecast in the case of a -period delay fluctuates around d proportionally less than the case of no delay by a factor of . This observation also holds true for lead-time demand forecasts ˆ L ˜L t and t . In conclusion, a delay tends to stabilize the demand forecast, which in turn dampens the bullwhip effect, as we show in the next section. When comparing the forecasting errors, we see clearly that an -period delay increases additively the forecasting errors for both the single-period and leadtime demand forecasts. We can interpret this increase in two ways. First, demand history Ht− contains less information, which causes a less accurate forecast. Second, when a forecast is made based on the delayed demand historyHt− , the forecast for future period t+ is periods further into the future. The increase in forecasting error simply reflects the greater uncertainty of a more distant future. Recognizing that the error terms in the lead-time demand forecasting errors are independent, we can easily obtain their variances 2 2 (ˆ L ˜L t ) and ( t ) as:
2 (1 − )2 2 (1 − 2L ) 2(1 − L ) , × L+ − 1 − 2 1−
2 (ˆ L t ) =
2 2 ˆL (˜ L t ) + t ) = (
2
2 1 − L 1 − 2
(1 − )2 1 − 2
2 .
Both variances remain constant over time, as we claimed before for arriving at Eq. (5). The second term in the last equality quantifies the additive impact of a delay on the lead-time demand forecasting error variance. Substituting ˆ L ˜L t and t from the above table
for L t into Eq. (5), we have:
− L+1 1 − L+1 dt − dt−1 , 1− 1− − L+1 (dt− − dt−−1 ). q˜t = dt + 1−
qˆt =
(6)
We use these order quantities to derive their corresponding bullwhip effect measures in the next section, from which we establish the dampening effect of a delay.
4. Dampened bullwhip effect We use an absolute bullwhip effect measure defined as the difference between the order-quantity variance and the demand variance: M = V (qt ) − V (dt ).
(7)
Lee et al. [6] among others [3–5] quantify the bullwhip effect with a relative measure defined as a variance ratio R = V (qt )/V (dt ). M and R are equivalent measures because they are linked by M = (R − 1)V (dt ). Let Mˆ and M˜ represent the absolute bullwhip effect measures without a delay and with a delay of periods, respectively. There exists a simple relationship between these two measures that indicates the dampening impact of a delay on the bullwhip effect. We state the result in the following proposition: Proposition. Mˆ and M˜ are proportional such that: ˆ M˜ = 2 M.
(8)
Equivalently, the relative measures Rˆ and R˜ are related by R˜ − 1 = 2 (Rˆ − 1).
(9)
Proof. From a standard time-series result [2], we have V (dt ) = 2 /(1 − 2 ), ∀t. If we substitute the definition of the AR(1) demand from Eq. (1) into the order quantity qˆt in Eq. (6), we have for the case of no delay: qˆt =
1 − L+1 1 − L+1 + t + L+1 dt−1 . 1− 1−
X. Zhang / Operations Research Letters 33 (2005) 289 – 294
Recognizing that t and dt−1 are independent, we immediately obtain: V (qˆt ) =
1 − L+1 1−
2
and M˜ = V (q˜t ) − V (dt ) 2 1 − L+1 2(L+1) − 1 2 2 + . = 1 − 2 (1 − )2
2(L+1) 2 2 + , 1 − 2
293
and Mˆ = V (qˆt ) − V (dt ) 2 1 − L+1 2(L+1) − 1 2 + . = 1 − 2 (1 − )2
The conclusion in the proposition is immediate from the above expression.
We use a similar strategy to decompose order quantity q˜t into past observed demand and uncorrelated future errors. The decomposition can then be used to easily obtain the variance for the order quantity q˜t when there is a delay of periods. We can expand dt backward in time in the following manner: dt − d
= = .. .
(dt−1 − d ) + t 2 (dt−2 − d ) + t + t−1 ,
=
+1 (dt−−1 − d ) +
j =0
j t−j .
Substituting the above expression and dt− − d = (dt−−1 − d ) + t− into Eq. (6), we have: q˜t − d =+1 (dt−−1 − d ) +
j t−j +
j =0
( − L+1 ) 1−
×[t− − (1 − )(dt−−1 − d )] =
−1 j =0
j t−j +
(1 − L+1 ) × t− 1−
+ L++1 (dt−−1 − d ). Again the error terms in the above expression are independent of dt−−1 , and we have:
1 − 2 2 (1 − L+1 )2 + 1 − 2 (1 − )2 2 2(L+1) , + 1 − 2
V (q˜t )=
2
When there is a delay of periods in demand information, the proposition states that the bullwhip effect is dampened by a factor of 2 for|| < 1. This phenomenon can be explained from our observation in the last section. The forecast d˜t+ is less sensitive to the temporary changes in demand and it tends to gravitate closer to the unconditional mean of the demand d than dˆt+ . The same observation applies to the predicted lead-time demand ˜ L t , which gravitates closer to Ld . Consequently, the order quantity q˜t tends to have less fluctuation than qˆt .
5. Conclusions We show that a delay in demand information for forecasting purposes dampens the bullwhip effect. The rationale behind this finding has to do with a special property of MMSE forecasting. For an AR(1) demand, both the long-term single-period and lead-time demand forecasts converge to their unconditional means exponentially. One of the repercussions of this convergence is that a delay in demand information availability stabilizes the demand forecasts, which in turn generates a less variable order history in our inventory setting and causes a decrease in the bullwhip effect. The findings here have implications for managing supply chains. Suppose that the inventory system we consider here is inserted into a serial chain coupled by a demand–supply relationship. From a local perspective, delays increase the forecasting error for our system. From a global view, however, delays could benefit the upstream suppliers by reducing their demand variability. The cliché “no news is good news” is proven relevant in our setting—late demand news can sometimes be good news for the upstream suppliers.
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X. Zhang / Operations Research Letters 33 (2005) 289 – 294
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