An important observation in supply chain management, known as the bullwhip effect,

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1 Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David Simchi-Levi Decision Sciences Deartment, National University of Singaore, Singaore Deartment of MS & IS, California State University, Fullerton, California School of Industrial Engineering, Purdue University, West Lafayette, Indiana Deartment of IE & MS, Northwestern University, Evanston, Illinois An imortant observation in suly chain management, known as the bullwhi effect, suggests that demand variability increases as one moves u a suly chain. In this aer we quantify this effect for simle, two-stage suly chains consisting of a single retailer and a single manufacturer. Our model includes two of the factors commonly assumed to cause the bullwhi effect: demand forecasting and order lead times. We extend these results to multile-stage suly chains with and without centralized customer demand information and demonstrate that the bullwhi effect can be reduced, but not comletely eliminated, by centralizing demand information. (Bullwhi Effect; Forecasting; Information; Inventory; Lead Time; Suly Chain; Variability) 1. Introduction Management Science 2000 INFORMS Vol. 46, No. 3, March An imortant observation in suly chain management, known as the bullwhi effect, suggests that demand variability increases as one moves u a suly chain. For a detailed discussion of this henomenon, see Baganha and Cohen (1995), Kahn (1987), Lee et al. (1997a, b), and Metters (1996). Most of the revious research on the bullwhi effect has focused on demonstrating its existence, identifying its ossible causes, and roviding methods for reducing its imact. In articular, Lee et al. (1997a, b) identify five main causes of the bullwhi effect: the use of demand forecasting, suly shortages, lead times, batch ordering, and rice variations. This revious work has also led to a number of aroaches and suggestions for reducing the imact of the bullwhi effect. For instance, one of the most frequent suggestions is the centralization of demand information, that is, roviding each stage of the suly chain with comlete information on customer demand. This aer differs from revious research in several ways. First, our focus is on determining the imact of demand forecasting on the bullwhi effect. In other words, we do not assume that the retailer knows the exact form of the customer demand rocess. Instead, the retailer uses a standard forecasting technique to estimate certain arameters of the demand rocess. Second, the goal is not only to demonstrate the existence of the bullwhi effect, but also to quantify it, i.e., to quantify the increase in variability at each stage of the suly chain. To quantify the increase in variability from the retailer to the manufacturer, we first consider a simle two-stage suly chain consisting of a single retailer and a single manufacturer. In 2 we describe the suly chain model, the forecasting technique, and the inventory olicy used by the retailer. We derive a /00/4603/0436$ electronic ISSN

2 Quantifying the Bullwhi Effect in a Simle Suly Chain lower bound for the variance of the orders laced by the retailer to the manufacturer. In 3 we consider a multistage suly chain and the imact of centralized demand information on the bullwhi effect. Finally, in 4, we conclude with a discussion of our results. 2. A Simle Suly Chain Model Consider a simle suly chain in which in each eriod, t, a single retailer observes his inventory level and laces an order, q t, to a single manufacturer. After the order is laced, the retailer observes and fills customer demand for that eriod, denoted by D t. Any unfilled demands are backlogged. There is a fixed lead time between the time an order is laced by the retailer and when it is received at the retailer, such that an order laced at the end of eriod t is received at the start of eriod t L. For examle, if there is no lead time, an order laced at the end of eriod t is received at the start of eriod t 1, and thus L 1. The customer demands seen by the retailer are random variables of the form D t D t 1 t, (1) where is a nonnegative constant, is a correlation arameter with 1, and the error terms, t, are indeendent and identically distributed (i.i.d.) from a symmetric distribution with mean 0 and variance 2.It can easily be shown that E(D t ) /(1 ) and Var(D t ) 2 /(1 2 ). Demand rocesses of this form have been assumed by several authors analyzing the bullwhi effect (e.g., Lee et al. 1997b and Kahn 1987). Finally, note that if 0, Equation (1) imlies that the demands are i.i.d. with mean and variance The Inventory Policy and Forecasting Technique We assume that the retailer follows a simle orderu-to inventory olicy in which the order-u-to oint, y t, is estimated from the observed demand as ˆD t L y t Dˆ t L z ˆ L et, (2) where is an estimate of the mean lead time demand, ˆ L et is an estimate of the standard deviation of the L eriod forecast error, and z is a constant chosen to meet a desired service level. It is well known that for i.i.d. demands from a normal distribution an order-u-to olicy of this form is otimal. Note that the order-u-to oint is calculated based on the standard deviation of the L eriod forecast error, L e, whose estimator is ˆ L et, and not based on the standard deviation of the lead time demand, L, whose estimator is ˆ L t. While there is a simle relationshi between these two quantities, i.e., L e c L, for some constant c 1, it is more aroriate to calculate the inventory olicy based on the former quantity (Hax and Candea ). We assume that the retailer uses a simle moving average to estimate Dˆ L t and ˆ L et based on the demand observations from the revious eriods. That is, and Dˆ t L L ˆ L et C L, D t i, e t i 2, (3) where e t is the one-eriod forecast error, i.e., e t D t ˆD t 1, and C L, is a constant function of L, and. See Ryan (1997) for a more detailed discussion of this constant Quantifying the Bullwhi Effect Our objective is to quantify the bullwhi effect. To do this, we must determine the variance of q t relative to the variance of D t, i.e., the variance of the orders laced by the retailer to the manufacturer relative to the variance of the demand faced by the retailer. For this urose, we write q t as q t y t y t 1 D t 1. Observe that q t may be negative, in which case we assume, similarly to Kahn (1987) and Lee et al. (1997b), that this excess inventory is returned without cost. We discuss the imact of this assumtion on our results later in this section. Given the estimates of the lead time demand, ˆD L t, and the standard deviation of the L eriod forecast error, ˆ L et, we can write the order quantity q t as Management Science/Vol. 46, No. 3, March

3 Quantifying the Bullwhi Effect in a Simle Suly Chain q t Dˆ t L L Dˆ t 1 z ˆ L L et ˆ e,t 1 D t 1 D t 1 z ˆ et L D t 1 D t 1 L L ˆ e,t 1 1 L/ D t 1 L/ D t 1 z ˆ L L et ˆ e,t 1. (4) It follows that Var q t 1 L/ 2 Var D t 1 2 L/ 1 L/ Cov D t 1, D t 1 L/ 2 Var D t 1 z 2 Var ˆ L L et ˆ e,t 1 2z 1 2L/ Cov D t 1, ˆ L et 1 2L 2L 2 Var D 2L 2 2L 2 2 Var D 2z 1 2L/ Cov D t 1, ˆ L et z 2 Var ˆ L L et ˆ e,t 1 2L 2 1 Var D 2 1 2L 2z 1 2L/ Cov D t 1, ˆ L et z 2 Var ˆ L L et ˆ e,t 1, where the second equation follows from Cov(D t 1, D t 1 ) [ /(1 2 )] 2 and Var(D t ) 2 /(1 2 ). To further evaluate Var(q t ) we need the following lemma. Lemma 2.1. Assume the customer demands seen by the retailer are random variables of the form given in (1) where the error terms, t, are i.i.d. from a symmetric distribution with mean 0 and variance 2. Let the estimate of the standard deviation of the L eriod forecast error be ˆ L et, as defined in (3). Then Cov D t i, ˆ L et 0 for all i 1,...,. The interested reader is referred to Ryan (1997) or Chen et al. (1998) for the roof of Lemma 2.1. We thus have the following lower bound on the increase in variability from the retailer to the manufacturer. Theorem 2.2. Under the conditions of Lemma 2.1, if the retailer uses a simle moving average forecast with demand observations, then the variance of the orders, q, laced by the retailer to the manufacturer, satisfies Var q 1. (5) Var D 1 2L 2L 2 2 The bound is tight when z Interretation of the Results Figure 1 shows the lower bound (solid lines) on the increase in variability for L 1 and z 2 for various values of and. It also resents simulation results on Var(q t )/ 2 (dotted lines), as well as the increase in variability when excess inventory is not returned (dashed lines). That is, the dashed lines reresent simulation results on Var(q t )/ 2, where q t max q t,0 max y t y t 1 D t 1,0. Notice that, given, Var(q) and Var(q ) are quite close. This imlies that the way one treats excess inventory has little imact on the increase in variability for most values of and. Although this observation alies for all ositive values of, we should note that when 3 1, Var(q) can be greater than Var(q ). Figure 2 shows the lower bound (solid lines) on the variability amlification when 0 as a function of for various values of the lead time arameter, L. It also resents simulation results on Var(q t )/ 2 (dotted lines), as well as the increase in variability when excess inventory is not returned (dashed lines). That is, the dashed lines reresent simulation results on Var(q t )/ 2. Notice that, given L, the three lines are quite close. This, again, imlies that the way one treats excess inventory has little imact on the increase in variability. Several observations regarding the increase in variability can be made from (5). First, we notice that the increase in variability from the retailer to the manufacturer is a function of three arameters, (1), the number of observations used in the moving average, (2) L, the lead time arameter, and (3), the correlation arameter. More secifically, the increase in the variability of orders from the retailer to the manufacturer is a decreasing function of, the number observations 438 Management Science/Vol. 46, No. 3, March 2000

4 Quantifying the Bullwhi Effect in a Simle Suly Chain Figure 1 Var(q)/Var(D) for 0.5 and 0.9, L 1, z 2 used to estimate the mean and variance of demand. In articular, when is large the increase in variability is negligible. However, when is small, there can be a significant increase in variability. In other words, the smoother the demand forecasts, the smaller the increase in variability. Also note that the increase in the variability of orders from the retailer to the manufacturer is an increasing function of L, the lead time arameter. In addition, note the relationshi between L and : If the lead time arameter doubles, then we must use twice as much demand data to maintain the same variability in the order rocess. In other words, with longer lead times, the retailer must use more demand data in order to reduce the bullwhi effect. The correlation arameter,, can also have a significant imact on the increase in variability. First, if 0, i.e., if demands are i.i.d., then Var q Var D 1 2L 2L 2 2. (6) Second, if 0, i.e., demands are ositively correlated, (1 ) 1, and the larger, the smaller the increase in variability. This can also be seen in Figure 1. On the other hand, if 0, i.e., demands are negatively correlated, then we see some strange behavior. For even values of, (1 ) 1, while for odd values of, (1 ) 1. Therefore, for 0, the lower bound on the increase in variability will be larger for odd values of than for even values of. 3. The Imact of Centralized Demand Information One frequently suggested strategy for reducing the magnitude of the bullwhi effect is to centralize demand information, i.e., to make customer demand information available to every stage of the suly chain. For examle, Lee et al. (1997a. 98) suggest that one remedy is to make demand data at a downstream site available to the ustream site. In this section, we analyze the imact of centralized customer demand information on the bullwhi effect. In articular, we demonstrate that while it will certainly reduce the magnitude of the bullwhi effect, centralizing demand information will not Management Science/Vol. 46, No. 3, March

5 Quantifying the Bullwhi Effect in a Simle Suly Chain Figure 2 Var(q)/Var(D) for L 1, 2 and 3, 0, z 2 comletely eliminate the increase in variability. That is, even if each stage of the suly chain has comlete knowledge of the demands seen by the retailer, the bullwhi effect will still exist. Consider a multistage suly chain in which the first stage (i.e., the retailer) shares all demand information with each of the subsequent stages. Assume that all stages in the suly chain use a moving average forecast with observations to estimate the mean demand. Therefore, since each stage has comlete information on customer demand, each stage will use the same estimate of the mean demand er eriod, ˆD t D t i /. Assume that each stage k follows an order-u-to inventory olicy where the order-u-to oint is of the form y k L t L k Dˆ t z k ˆ k et, where ˆD t is the estimate of the mean demand er eriod, L k is the lead time between stages k and k 1, Lk ˆ et C Lk, (e t i ) 2 /, and z k is a constant. We note that the assumtion that all stages in the suly chain use the same demand data, the same inventory olicy and the same forecasting technique allows us to determine the imact of just the demand forecasting, without considering the imact of different forecasting techniques or different inventory olicies across the stages. The sequence of events in our model is similar to the one in the traditional Beer Game (Sterman 1989) or the comuterized Beer Game (Kaminsky and Simchi-Levi 1998). At the end of eriod t 1, the retailer, or Stage 1, observes customer demand, D t 1, calculates his order-u-to oint for eriod t, y t 1, and orders q t 1 so as to raise his inventory to level y t 1. The manufacturer, or Stage 2, receives the order, q t 1, along with the most recent demand information, D t 1. This information is received at the end of eriod t 1 (i.e., there is no information lead time), allowing the manufacturer to calculate his order-u-to oint, y t 2, and immediately lace an order of q t 2 to raise his inventory to level y t 2. The next stage, Stage 3, immediately receives the 440 Management Science/Vol. 46, No. 3, March 2000

6 Quantifying the Bullwhi Effect in a Simle Suly Chain order, q t 2, and the demand observation D t 1, and the rocess continues. We can now state the following lower bound on the variance of the orders laced by each stage of the suly chain relative to the variance of customer demand. Theorem 3.1. Consider a multistage suly chain where the demands seen by the retailer are random variables of the form given in (1) where the error terms, t, are i.i.d. from a symmetric distribution with mean 0 and variance 2. Each stage of the suly chain follows an order-u-to olicy of the form y k Lk t L k ˆD t z k ˆ et, where L k is the lead time between stages k and k 1. The variance of the orders laced by stage k, denoted q k, satisfies Var q k Var D 1 2 k L i 2 k 2 L i 2 1 k. This bound is tight when z i 0 for i 1,..., k. The reader is referred to Ryan (1997) or Chen et al. (1998) for a roof of Theorem 3.1. This result demonstrates that even when (i) all demand information is centralized, (ii) every stage of the suly chain uses the same forecasting technique, and (iii) every stage uses the same inventory olicy, there will still be an increase in variability at every stage of the suly chain. In other words, we have not comletely eliminated the bullwhi effect by centralizing customer demand information. Finally, it would be interesting to comare the increase in variability for the case of centralized information, as given in Theorem 3.1, with the increase in the case of decentralized information. To do this, we consider a suly chain similar to the one just analyzed, but without centralized information. In this case, the retailer does not rovide the ustream stages with any customer demand information and each stage determines its forecast demand based on the orders laced by the revious stage, not based on actual customer demands. In this case, we can state the following lower bound on the variance of the orders laced by each stage of the suly chain. Theorem 3.2. Consider a multistage suly chain where the demands seen by the retailer are random variables of the form D t t, where the error terms, t, are i.i.d. from a symmetric distribution with mean 0 and variance 2. Each stage of the suly chain follows an order-u-to olicy of the form y k t L k ˆD (k) t, where L k is the lead time between stages k and k 1, Dˆ t 1 D t i and Dˆ t k j 0 1 k 1 q t j for k 2, where q k t is the order laced by stage k in eriod t. The variance of the orders laced by stage k, denoted q k, satisfies Var q k k Var D 1 2L i 2L 2 i k. (7) 2 The reader is referred to Ryan (1997) or Chen et al. (1998) for a roof of Theorem 3.2. Note that Theorem 3.2 only alies when each stage of the suly chain uses an inventory olicy as defined in (2) with z 0. When a olicy of this form is used in ractice, an inflated value of L k is often used, with the excess inventory reresenting the safety stock. For examle, a retailer facing an order lead time of three weeks may choose to kee inventory equal to four weeks of forecast demand, with the extra week of inventory reresenting his safety stock. Policies of this form are often used in ractice. Indeed, we have recently collaborated with a major U.S. retail comany that uses a olicy of this form. See also Johnson et al. (1995). We are interested in comaring the increase in variability at each stage of the suly chain for the centralized and decentralized systems. We ll consider the case of i.i.d. demands, i.e., 0, and z i 0 for all i. For a discussion of the case where 0, see Ryan (1997) or Chen et al. (1998). If demand information is shared with each stage of the suly chain, then the increase in variability from the retailer to stage k is Var q k Var D 1 2 k L i 2 k L i 2 (8) 2 On the other hand, if demand information is not shared with each stage of the suly chain, then a Management Science/Vol. 46, No. 3, March

7 Quantifying the Bullwhi Effect in a Simle Suly Chain lower bound on the increase in variability from the retailer to stage k is given by (7). Note that for suly chains with centralized information, the increase in variability at each stage is an additive function of the lead time and the lead time squared, while for suly chains without centralized information, the lower bound on the increase in variability at each stage is multilicative. This imlies that centralizing customer demand information can significantly reduce the bullwhi effect. However, as mentioned above, centralizing customer demand information does not comletely eliminate the bullwhi effect. In addition, these results indicate that the difference between the variability in the centralized and decentralized suly chains increases as we move u the suly chain, i.e., as we move from the first stage to the second and third stages of the suly chain. A simulation study has confirmed these insights for the case in which z i 0 for all i, and 0. See Ryan (1997) or Chen et al. (1998) for the results of this simulation study. 4. Summary In this aer we have demonstrated that the henomenon known as the bullwhi effect is due, in art, to the effects of demand forecasting. In articular, we have shown that if a retailer eriodically udates the mean and variance of demand based on observed customer demand data, then the variance of the orders laced by the retailer will be greater than the variance of demand. More imortantly, we have shown that roviding each stage of the suly chain with comlete access to customer demand information can significantly reduce this increase in variability. However, we have also shown that the bullwhi effect will exist even when demand information is shared by all stages of the suly chain and all stages use the same forecasting technique and inventory olicy. To further exlain this oint, consider the first stage of the suly chain, the retailer. Notice that if the retailer knew the mean and the variance of customer demand, then the orders laced by the retailer would be exactly equal to the customer demand and there would be no increase in variability. In our model, however, even though the retailer has comlete knowledge of the observed customer demands, he must still estimate the mean and variance of demand. As a result, the manufacturer sees an increase in variability. This aer would be incomlete if we did not mention several imortant limitations of our model and results. The main drawback of our model is the assumtion that excess inventory is returned without cost. The simulation results demonstrate, however, that this assumtion has little imact on the increase in variability for all reasonable values of,, and L. Another limitation of our model is the fact that we are not using the otimal order-u-to olicy, but rather the olicy defined by Equation (2). It is imortant to oint out, however, that olicies of this form are frequently used in ractice. In addition, we are not using the otimal forecasting technique for the correlated demand rocess considered here. However, we again oint out that the moving average is one of the most commonly used forecasting techniques in ractice. Indeed, we believe that when evaluating the bullwhi effect it is most aroriate to consider inventory olicies and forecasting techniques that are used in ractice. Finally, our model clearly does not cature many of the comlexities involved in real-world suly chains. For examle, we have not considered a multistage system with multile retailers and manufacturers. Fortunately, extending our results to the multiretailer case when demand between retailers may be correlated is straightforward. See Ryan (1997). 1 1 Research suorted in art by ONR Contracts N J-1649 and N , NSF Contracts DDM , DDM and DMI References Baganha, M., M. Cohen The stabilizing effect of inventory in suly chains. Oer. Res. Forthcoming. Chen, F., Z. Drezner, J. Ryan, D. Simchi-Levi Quantifying the bullwhi effect: The imact of forecasting, lead times and information. Working Paer, Deartment of IE/MS Northwestern University, Evanston, IL. and School of Industrial Engineering, Purdue University, Lafayette, IN. Hax, A. C., D. Candea Production and Inventory Management. Prentice-Hall, Englewood Cliffs, NJ. 442 Management Science/Vol. 46, No. 3, March 2000

8 Quantifying the Bullwhi Effect in a Simle Suly Chain Johnson, M. E., H. L. Lee, T. Davis, R. Hall Exressions for item bill rates in eriodic inventory systems. Naval Res. Logist Kahn, J Inventories and the volatility of roduction. The Amer. Econom. Rev Kaminsky, P., D. Simchi-Levi A new comuterized beer game: A tool for teaching the value of integrated suly chain management. Hau Lee and Shu Ming Ng, eds. Global Suly Chain and Technology Management. POMS Series in Technology and Oerations Management Lee, H., P. Padmanabhan, S. Whang. 1997a. The bullwhi effect in suly chains. Sloan Management Rev b. Information distortion in a suly chain: The bullwhi effect. Management Sci Metters, R Quantifying the bullwhi effect in suly chains. Proc MSOM Conf Porteus, E Stochastic Inventory Theory. Handbooks in Oerations Research and Management Science, Volume Ryan, J. K Analysis of inventory models with limited demand information. Ph.D. Dissertation, Deartment of Industrial Engineering and Management Science, Northwestern University, Evanston, IL. Sterman, J. D Modeling managerial behavior: Misercetions of feedback in a dynamic decision making exeriment. Management Sci Acceted by Hau Lee; received Aril This aer was with the authors 15 months for 1 revision. Management Science/Vol. 46, No. 3, March

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