A measure of bullwhip effect in supply chains with a mixed autoregressive-moving average demand process

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European Journal of Operational Research 187 (2008) 243–256 www.elsevier.com/locate/ejor

O.R. Applications

A measure of bullwhip effect in supply chains with a mixed autoregressive-moving average demand process Truong Ton Hien Duc a

b

a,*

, Huynh Trung Luong b, Yeong-Dae Kim

a

Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Yuseong-gu, Daejeon 305-701, Republic of Korea Industrial Systems Engineering Program, School of Advanced Technologies, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand Received 17 January 2006; accepted 7 March 2007 Available online 6 April 2007

Abstract In this paper, we quantify the impact of the bullwhip effect – the phenomenon in which information on demand is distorted as moving up a supply chain – for a simple two-stage supply chain with one supplier and one retailer. Assuming that the retailer employs a base stock inventory policy, and that the demand forecast is performed via a mixed autoregressivemoving average model, ARMA(1, 1), we investigate the effects of the autoregressive coefficient, the moving average parameter, and the lead time on the bullwhip effect. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Supply chain; Bullwhip effect; Mixed autoregressive-moving average model; Base stock policy

1. Introduction A supply chain is the integration of business processes from end users to original suppliers who provide goods, services, or information that add values for the end users (Lambert et al., 1998). In a supply chain, as moving backward from a downstream member to an upstream member, the variance of order quantities of orders placed by the downstream member to its (immediate) upstream member tends to be amplified. This phenomenon is widely recognized and referred to as the bullwhip effect in supply chains (Lee et al., 1997a,b). Evidence of the bullwhip effect is first pointed out by Forrester (1958), who discusses its causes and possible remediation in the context of industrial dynamics. After that, several researchers such as Blinder (1982), Blanchard (1983), Burbidge (1984), Caplin (1985), Blinder (1986) and Kahn (1987) also recognize the existence of the bullwhip effect in supply chains, and Sterman (1989) explores and illustrates the bullwhip effect through an experiment on the well-known ‘‘beer game’’. In addition, Lee et al. (1997a,b) identify five main causes of the

*

Corresponding author. Tel.: +82 42 869 3160; fax: +82 42 869 3110. E-mail address: [email protected] (T.T.H. Duc).

0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.03.008

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bullwhip effect, i.e., demand forecasting, non-zero lead time, supply shortage, order batching, and price fluctuation. Impacts of forecasting methods on the bullwhip effect have been studied by several researchers. Johnson and Thompson (1975) show that the optimal policy for a periodic ordering system is myopic for both stationary and non-stationary demand processes when the order-up-to policy is used along with the minimum mean square error forecasting method. As discussed in Lee et al. (1997a,b), the bullwhip effect may be mitigated by eliminating its main causes. Among various causes of the bullwhip effect, forecasting methods are considered as one of the most important causes because the inventory system of a supply chain is directly affected by the forecasting method. Graves (1999) quantifies the bullwhip effect for a supply chain in which the demand pattern follows an integrated moving average process, and Lee et al. (2000) measure the benefit of information sharing between a retailer and a manufacturer in a two-stage supply chain in which the demand pattern follows a first-order autoregressive process and an order-up-to policy with a minimum mean square error forecasting technique is used at the retailers. Using a first-order autoregressive demand process, Chen et al. (2000a,b) investigate the impact of the simple moving average and exponential smoothing forecasts on the bullwhip effect for a simple, two-stage supply chain with one supplier and one retailer. Likewise, Xu et al. (2001) conduct a similar research for a demand process that is forecasted with a simple exponentially weighted moving average method. Zhang (2004) also investigates the impact of different forecasting methods on the bullwhip effect for a simple inventory system with a first-order autoregressive demand process. By quantifying the bullwhip effect, they show effects of forecasting methods on bullwhip effect measures. In addition, Chen et al. (2000a,b), Zhang (2004) show that increasing lead time enhances the bullwhip effect regardless of the forecasting methods employed. Recently, using a control systems engineering approach, Disney et al. (2006) quantify the bullwhip effect and the net inventory variance in a single-echelon supply chain with the order-up-to policy in cases of the identically and independently distributed, first-order autoregressive, firstorder moving average, and mixed first-order autoregressive-moving average demand processes. The approach of using a simple demand model such as the first-order autoregressive model, AR(1), to capture the behavior of the demand process may not be good for practical application. Instead, we need to examine a more realistic model for the demand process, such as a mixed first-order autoregressive-moving average model, ARMA(1, 1), to study the effect of the demand process on the bullwhip effect. In fact, an ARMA model often fits the time series of the demand process better than an autoregressive model because the demand process usually has characteristics of both moving average and autoregressive processes (Pindyck and Rubinfeld, 1998). In the current research, therefore, we will examine the bullwhip effect in a supply chain with an ARMA(1, 1) demand process. In addition, we will develop a measure of the bullwhip effect based on a replenishment model which is similar to the one used by Chen et al. (2000a,b) for a two-echelon supply chain with one retailer and one supplier. It is also assumed that the retailer employs the base stock policy for replenishment. Our research differs from the previous research in the following aspects. (1) Rather than relying on simple forecasting techniques such as moving average and exponential smoothing (Graves, 1999; Chen et al., 2000a; Chen et al., 2000b; Xu et al., 2001; Disney et al., 2006), we will employ the minimum mean square error technique to forecast the lead time demand. The simple forecasting techniques are very popular due to the ease of use, but they may lead to specification errors. In fact, Alwan et al. (2003) find that the use of the simple forecasting techniques leads to specification errors in case of AR(1) demand process, while the use of the minimum mean square error forecasting technique does not. The current research extends this finding to the case of ARMA(1, 1) demand process. In particular, it can be easily proven for cases in which the demand pattern follows ARMA(1, 1) demand process and the base stock policy is employed: (a) that if the minimum mean square forecasting technique is used, the ordering pattern of the downstream member also follows an ARMA(1, 1) process; (b) that if the moving average forecasting technique with length p and the exponential smoothing forecasting technique are used, the ordering process of the downstream member will become an ARMA(1, p) and ARMA(1, 1) process, respectively. These properties show clearly that when the moving average or exponential smoothing technique is employed, the upstream member in the supply chain incorrectly identifies the nature of the demand process, and therefore, specification errors occur. Since the mean

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square error is to be minimized in the forecasting method used in this research to predict the future demand, the bullwhip effect is expected to be mitigated. This is because the smaller the forecast error is, the more accurate the demand forecast is, and hence, the smaller the bullwhip effect is. (2) As discussed in Pindyck and Rubinfeld (1998), the demand process seldom has characteristics of a pure autoregressive process or a pure moving average process. Note that Lee et al. (2000), Chen et al. (2000a,b), Xu et al. (2001), Zhang (2004) assume the pure autoregressive process and Graves (1999) assumes a pure moving average process. In this research, therefore, we express the demand process by using an ARMA(1, 1) model, as is done in the research of Disney et al. (2006). In the research of Disney et al. (2006), however, the minimum mean square error technique is not considered. Moreover, the application of z-transform technique in their research results in a very complicated expression of the bullwhip effect, and hence, it is impossible to investigate analytically the behavior of the bullwhip effect with respect to the lead time, the autoregressive coefficient and the moving average coefficient, which is done with a closed form expression for the bullwhip effect in this research. (3) Finally, unlike the research of Chen et al. (2000a,b), our research aims at determining an exact measure of the bullwhip effect rather than deriving only a lower bound on a bullwhip effect measure. The remaining part of this paper is organized as follows. In Section 2, we describe properties of the ARMA(1, 1) demand process and its stationary and invertibility conditions in a simple supply chain model. In Section 3, we develop analytical expressions for important parameters of the order-up-to inventory model, such as lead-time demand forecast and standard deviation of lead-time demand forecast error, for the retailer. Section 4 presents a bullwhip effect measure for the ARMA(1, 1) demand process as well as its behavior. Finally, Section 5 concludes the paper with a short summary. The following notation will be used in the remaining part of the paper. Dt qt St / et h d ld r2d L DLt ^L D t ^Lt r

demand quantity in period t ordered quantity in period t (orders are issued at the beginning of the period) order-up-to-level in period t first-order autocorrelation coefficient forecast error in period t first-order moving average parameter constant of the autoregressive model mean of the autoregressive process variance of demand quantities order lead time lead time demand, which is the total demand of periods t through t þ L  1 forecast for the lead time demand standard deviation of forecast errors for the lead-time demand

2. Supply chain model In this research, a measure of the bullwhip effect is developed for a two-stage supply chain with one retailer and one supplier. There is a fixed order lead time for orders placed by the retailer. The length of lead time is assumed to be a multiple of the inventory review interval. It is assumed that the retailer employs a base stock policy, a simple order-up-to inventory policy, and that the demand forecast is performed with an ARMA(1, 1), i.e., a mixed autoregressive-moving average model, by using the minimum mean-square error forecasting technique. Note that forecasting creates variability in the order-up-to level, which results in the bullwhip effect. At the beginning of period t, the retailer places an order of quantity qt to the supplier. The order quantity qt can be given as qt ¼ S t  S t1 þ Dt1 ;

ð1Þ

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where St is the order-up-to level in period t, i.e., the inventory position at the beginning of period t(after the order is placed). If the base stock policy is employed, the order-up-to level St can be determined by the leadtime demand as b L þ z^ St ¼ D rLt ; t

ð2Þ

in which z is the normal z-score that can be determined based on a given service level. (The service level is defined as the probability that demand is fulfilled by the on-hand inventory.) The optimal order-up-to level St can be determined from inventory holding cost and shortage cost (Heyman and Sobel, 1984). However, since it is usually difficult to estimate these costs accurately in practice, the approach of using the service level is often employed when the order-up-to level is to be determined. In general, the base stock policy is a special case of the order-up-to inventory policy. In the base stock policy, an order is placed at the beginning of each period so as to increase the inventory level up to a predetermined level. Order-up-to level St defined in (2) is optimal in terms of total inventory cost for inventory systems where there is no fixed ordering cost, and holding and shortage costs are proportional to the volume of onhand inventory and shortage, respectively (Nahmias, 1997; Zipkin, 2000). b L , the lead-time demand foreGiven a service level, the order-up-to level, St, is determined by obtaining D t L b L and r ^t , the standard deviation of lead-time demand forecast error. (Expressions for D ^Lt will be cast, and r t L ^t does not depend on t and thus, it does not developed in Section 3.) However, as shown later in this paper, r have any influence on the bullwhip effect. Since it is assumed that demand can be modeled with an ARMA(1, 1) model, we have Dt ¼ d þ /Dt1 þ et  het1 ;

ð3Þ

where et’s (t ¼ 1; 2; . . .) are normally and independently distributed with common mean 0 and variance r2 (i.i.d. random variables from the normal distribution). For the mixed first-order autoregressive-moving average process to be stationary, we must have E½Dt  ¼ E½Dt1  ¼ ld ;

8t

and hence, a stationary condition can be given as ld ¼

d : 1/

ð4Þ

In addition, from (3), we have r2d ¼ /2 r2d þ r2 þ h2 r2  2/hr2 , which results in r2d ¼

1 þ h2  2/h 2 r: 1  /2

ð5Þ

From (4) and (5), it can be seen that in order to have the demand process of ARMA(1, 1) model be stationary, we should also have j/j < 1. Similarly, in order to have the demand process be invertible, we should have jhj < 1 (Vandaele, 1983; Granger and Newbold, 1986). 3. Lead-time demand forecast and forecast error In this paper, it is assumed that the demand forecast is carried out in such a way that minimizes the mean square error. It is noted that the lead-time demand can be expressed as DLt ¼ Dt þ Dtþ1 þ    þ DtþL1 ¼

L1 X

Dtþi :

ð6Þ

i¼0

b t is the forecast (for Dt) with the minimum mean square error, then the forecast for the lead-time demand If D with the minimum mean square error can be given as bL ¼ D bt þ D b tþ1 þ    þ D b tþL1 ¼ D t

L1 X i¼0

b tþi : D

ð7Þ

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b t can be determined as According to Pindyck and Rubinfeld (1998), for the ARMA(1, 1) process, D b tþi ¼ E½Dtþi jDt1 ; Dt2 ; . . .: D b tþi can be derived from the above expression. Since we have An exact expression of D Dtþi ¼ d þ /Dtþi1 þ etþi  hetþi1 and Dtþi1 ¼ d þ /Dtþi2 þ etþi1  hetþi2 ; we can obtain Dtþi ¼ ð/ þ 1Þd þ /2 Dtþi2 þ etþi þ ð/  hÞetþi1  /hetþi2 : By applying this procedure recursively, we have Dtþi ¼ ð/i þ    þ /2 þ / þ 1Þd þ /iþ1 Dt1 þ etþi þ ð/  hÞetþi1 þ /ð/  hÞetþi2 þ     /i het1 : Taking the expectation, we have b tþi ¼ E½Dtþi jDt1 ; Dt2 ; . . . ¼ ð/i þ    þ /2 þ / þ 1Þd þ /iþ1 Dt1  /i het1 D ¼d

1  /iþ1 þ /iþ1 Dt1  /i het1 : 1/

Therefore, from (4), we have b tþi ¼ ld ð1  /iþ1 Þ þ /iþ1 Dt1  /i het1 : D

ð8Þ

Note that the last term in (8) is a modification factor for the forecast which takes account of the residual, i.e., b tþi ¼ ld when the ARMA(1, 1) deknown forecast error of period t  1. It can be seen from (8) that limi!1 D mand process is stationary. This means that if the minimum mean square error technique is used, the forecast function can retain the so-called conditional mean reversion property of the stationary ARMA(1, 1) process; that is, the process is expected to return always back to its overall mean when it wanders away from the mean. This property is not valid when the moving average or exponential smoothing forecasting technique is used, b tþi ¼ D b t is a constant for any i ¼ 1; 2; . . .. since D Using (8), we can derive the following proposition. Proposition 1. The minimum mean square error forecast for the lead-time demand can be given as  L  L L b L ¼ ld L  /ð1  / Þ þ /ð1  / Þ Dt1  hð1  / Þ et1 : D t 1/ 1/ 1/

ð9Þ

Proof. From (7) and (8), we have bL ¼ D t

L1 X

ðld ð1  /iþ1 Þ þ /iþ1 Dt1  /i het1 Þ ¼ ld

i¼0



¼ ld L 

L



L1 L1 L1 X X X ð1  /iþ1 Þ þ Dt1 /iþ1  het1 /i i¼0

L

i¼0

i¼0

L

/ð1  / Þ /ð1  / Þ hð1  / Þ Dt1  et1 : þ 1/ 1/ 1/



The following proposition gives the variance of lead-time demand forecast error. Proposition 2. The variance of lead-time demand forecast error does not depend on t and is given as 0 !2 1 L2 Li2 2 2 X X r ð1  / Þ d @1 þ ð^ rLt Þ2 ¼ 1þ /j ð/  hÞ A: ð1 þ h2  2/hÞ i¼0 j¼0

ð10Þ

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b L Þ. We also have Proof. By definition, we have ð^ rLt Þ2 ¼ VARðDLt  D t b L ¼ ðDt  D b t Þ þ ðDtþ1  D b tþ1 Þ þ    þ ðDtþL1  D b tþL1 Þ ¼ et þ etþ1 þ    þ etþL1 ¼ DLt  D t

L1 X

etþi ;

i¼0

where etþi can be given from (8) as follows: et ¼ et ; etþi ¼ etþi þ

i1 X

/ij1 ð/  hÞetþj

for i ¼ 1; 2; ::; L  1:

j¼0

b L ¼ etþL1 þ Thus, DLt  D t Therefore, we have

PL2 i¼0

ð1 þ

0

b L Þ ¼ r2 @1 þ VARðDLt  D t

L2 X

PLi2 j¼0

1þ

i¼0

/j ð/  hÞÞetþi .

Li2 X

!2 1 /j ð/  hÞ A

j¼0

0

L2 Li2 X X j r2d ð1  /2 Þ @ ¼ 1 þ 1 þ / ð/  hÞ 2 ð1 þ h  2/hÞ i¼0 j¼0

This completes the proof.

!2 1 A:

h

^Lt is needed only for computation of the order-up-to level, St, and the measure of bullwhip effect Note that r ^Lt . (to be developed later in this research) is not affected by r 4. Behavior of the bullwhip effect measure As stated earlier, the bullwhip effect is a phenomenon in which the variance of demand information is amplified when moving upstream in a supply chain. Hence, it is reasonable to measure the bullwhip effect by the ratio of the variance of order quantity experienced by the supplier to the actual variance of the demand quantity. This measure has been used in previous studies such as those of Chen et al. (2000a,b), and it is adopted in this study as well. In order to develop an expression for a bullwhip measure, we first examine the variance of the order quantity. Proposition 3. The variance of the order quantity of period t can be given as VARðqt Þ ¼

r2d

! 2ð/  hÞð1  /L Þð1  /Lþ1  /hð1  /L1 ÞÞ 1þ : ð1  /Þð1 þ h2  2/hÞ

ð11Þ

Proof. See the appendix. From (11), the measure of the bullwhip effect, denoted as BðL; /; hÞ here, can be determined as BðL; /; hÞ ¼

VARðqt Þ 2ð/  hÞð1  /L Þð1  /Lþ1  /hð1  /L1 ÞÞ : ¼1þ 2 rd ð1  /Þð1 þ h2  2/hÞ

ð12Þ

Using the expression of the above bullwhip measure, we can show a necessary condition for the existence of the bullwhip effect. h Proposition 4. The bullwhip effect exists, i.e., BðL; /; hÞ > 1, only if / > h. Proof. If we let f ðL; /; hÞ ¼ 1  /Lþ1  /hð1  /L1 Þ, we have of ðL;/;hÞ ¼ /ð1  /L1 Þ. oh of ðL;/;hÞ L1 When 1 < / 6 0, we have oh ¼ /ð1  / Þ P 0. Hence, f ðL; /; hÞ is a non-decreasing function of h. Furthermore, f ðL; /; hÞjh¼1 ¼ 1  /Lþ1 þ /ð1  /L1 Þ ¼ ð1 þ /Þð1  /L Þ > 0:

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Hence, f ðL; /; hÞ > 0 for all h 2 (1, 1). When 0 < / < 1, we have of ðL;/;hÞ ¼ /ð1  /L1 Þ < 0. Hence, f ðL; /; hÞ is a decreasing function of h. oh Lþ1 Furthermore, f ðL; /; hÞjh¼1 ¼ 1  /  /ð1  /L1 Þ ¼ ð1  /Þð1 þ /L Þ > 0. Hence, f ðL; /; hÞ > 0 for all h 2 ( 1, 1). Therefore, f ðL; /; hÞ > 0 for all / 2 ð1; 1Þ and h 2 ð1; 1Þ. In addition, 1 þ h2  2/h ¼ ð1  /2 Þ þ ð/  hÞ2 > 0 , 1  / > 0, and 1  /L > 0. Therefore, BðL; /; hÞ > 1 only if / > h. h From the above proposition, we can see that the bullwhip effect occurs only when the autoregressive coefficient of the demand process is larger than the moving average coefficient. Therefore, in the remaining part of this paper, we consider only the case, / > h. Proposition 5. The bullwhip effect, measured by BðL; /; hÞ, has the following properties. (a) If / > 0, the bullwhip effect increases as L increases. (b) If h < / < 0 and L is an odd number, the larger L is, the smaller the bullwhip effect is. (c) If h < / < 0 and L is an even number, the larger L is, the larger the bullwhip effect is. Proof. Let gðL; /; hÞ  ð1  /L Þf1  /Lþ1  /hð1  /L1 Þg. Also, let DgðLÞ  gðL þ 1; /; hÞ  gðL; /; hÞ. Then, gðL; /; hÞ ¼ /2L ð/  hÞ  /L ð1 þ /Þð1  hÞ þ 1  h/, and DgðLÞ ¼ /L ð/2  1Þf/L ð/  hÞ  1 þ hg: Consider the case, 0 < / < 1. Let hðL; /; hÞ  /L ð/  hÞ  1 þ h. It can be easily seen that hðL; /; hÞ is a decreasing function with respect to L since / > h. In addition, if L = 0, hðL; /; hÞ has a negative value, i.e., hð0; /; hÞ ¼ /  1 < 0. Therefore, hðL; /; hÞ < 0. Then, from the condition 0 < / < 1, we also have /L > 0; /2  1 < 0, and hence, DgðLÞ > 0. Therefore, gðL; /; hÞ is an increasing function with respect to L and hence, the bullwhip effect, i.e., BðL; /; hÞ increases as L increases. Next, consider the case in which h < / < 0 and L is an odd number. In this case, it can be seen that as L increases, /2L decreases while /L increases. Therefore, gðL; /; hÞ is a decreasing function with respect to oddnumbered L and hence, the bullwhip effect becomes smaller as L becomes larger. In the case in which h < / < 0 and L is an even number, let Dg2 ðLÞ  gðL þ 2; /; hÞ  gðL; /; hÞ. Then, we have Dg2 ðLÞ ¼ /L ð/2  1Þð/L ð/  hÞð/2 þ 1Þ  ð1 þ /Þð1  hÞÞ. Here, we denote kðL; /; hÞ ¼ /L ð/ hÞð/2 þ 1Þ  ð1 þ /Þð1  hÞ. Then, it can be seen that kðL; /; hÞ is also a decreasing function with respect to even-numbered L and kð0; /; hÞ ¼ ð/  1Þð/2 þ / þ 1  /hÞ ¼ ð/  1Þðð/  hÞð/ þ 1Þþ ð1 þ hÞÞ < 0. Therefore, kðL; /; hÞ < 0, and hence, Dg2 ðLÞ > 0 since /L > 0 and /2  1 < 0. It can then be concluded that gðL; /; hÞ is an increasing function with respect to even-numbered L, and hence, the bullwhip effect becomes larger as L becomes larger if L is an even number. h In addition, we can derive a bound, Blim, on the bullwhip effect as 2

Blim ¼ lim BðL; /; hÞ ¼ 1 þ L!1

2ð/  hÞð1  /hÞ ð1 þ /Þð1  hÞ ¼ : 2 ð1  /Þð1 þ h  2/hÞ ð1  /Þð1 þ h2  2/hÞ

From Proposition 5(a), it can be seen that BðL; /; hÞ < Blim when / > 0. Similarly, from Propositions 5(b) and 5(c), when h < / < 0, we have Bð1; /; hÞ > Bð3; /; hÞ > Bð5; /; hÞ >    > Blim and Bð2; /; hÞ < Bð4; /; hÞ < Bð6; /; hÞ <    < Blim : This shows that the bullwhip effect does not always increase as lead time increases, and if h < / < 0, the bullwhip effect is maximized when L = 1 and minimized when L = 2. A series of numerical experiments was conducted for different values of / and h, and the results are presented in Tables 1–6. From these tables, it can be seen that for given values of L and h, the bullwhip effect measure reaches a maximum value at a certain / value, which is denoted as /max here. Given the value of

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Table 1 Values of the bullwhip effect measure, BðL; /; hÞ, in the case of L=1 h

/

0.90 0.70 0.50 0.30 0.10 0.10 0.30 0.50 0.70 0.90 hmax

/max

0.85

0.65

0.45

0.25

0.05

1.0991 – – – – – – – – – –

1.4512 1.0996 – – – – – – – – –

1.7178 1.4637 1.0997 – – – – – – – –

1.8961 1.7401 1.4688 1.0997 – – – – – – –

1.9859 1.9132 1.7481 1.4705 1.0998 – – – – – –

0.15 1.9869 1.9775 1.9077 1.7456 1.4700 1.0997 – – – – 0.8387

0.35 1.8991 1.9307 1.9323 1.8775 1.7313 1.4668 1.0997 – – – 0.5867

0.55

0.75

0.95

– – 0.2852

1.4569 1.4995 1.5469 1.5966 1.6412 1.6613 1.6152 1.4375 1.0994 – 0.0886

1.1025 1.1141 1.1285 1.1468 1.1706 1.2021 1.2438 1.2925 1.3047 1.0975 0.6378

0.55

0.75

1.7224 1.7716 1.8138 1.8350 1.8096 1.6975 1.4589 1.0996

0.05 0.17 0.29 0.40 0.52 0.63 0.74 0.83 0.92 0.98

Results of the cases in which BðL; /; hÞ < 1 are not given here (denoted as ‘‘.’’).

Table 2 Values of BðL; /; hÞ in the case of L = 2 h

/ 0.85

0.90 0.70 0.50 0.30 0.10 0.10 0.30 0.50 0.70 0.90 hmax

1.0107 – – – – – – – – – 0.9359

/max 0.65 1.0846 1.0316 – – – – – – – – 0.8919

0.45 1.2494 1.2029 1.0526 – – – – – – – 0.9035

0.25 1.5265 1.4718 1.3223 1.0736 – – – – – – 0.9549

0.05

0.15

1.8947 1.8379 1.6939 1.4411 1.0945 – – – – – 0.9976

0.35

2.2904 2.2488 2.1323 1.9077 1.5581 1.1155 – – – – 0.9744

2.6071 2.5984 2.5360 2.3843 2.1024 1.6710 1.1364 – – – 0.8415

2.6959 2.7264 2.7313 2.6847 2.5443 2.2538 1.7744 1.1572 – – 0.5686

2.3649 2.4174 2.4697 2.5138 2.5308 2.4797 2.2843 1.8477 1.1777 – 0.1250

0.95 1.3800 1.4013 1.4277 1.4607 1.5029 1.5574 1.6258 1.6954 1.6665 1.1948 0.5867

0.51 0.53 0.56 0.60 0.65 0.72 0.78 0.86 0.92 0.98

Table 3 Values of of BðL; /; hÞ in the case of L = 3 h

0.90 0.70 0.50 0.30 0.10 0.10 0.30 0.50 0.70 0.90 hmax

/

/max

0.85

0.65

0.45

0.25

0.05

1.0828 – – – – – – – – – –

1.2919 1.0744 – – – – – – – – –

1.4307 1.3097 1.0733 – – – – – – – –

1.6098 1.5337 1.3571 1.0800 – – – – – – –

1.8992 1.8416 1.6966 1.4426 1.0948 – – – – – –

0.15 2.3394 2.2922 2.1680 1.9332 1.5717 1.1178 – – – – 0.9961

0.35 2.9031 2.8741 2.7813 2.5861 2.2464 1.7474 1.1494 – – – 0.9429

0.55 3.4055 3.4120 3.3780 3.2701 3.0360 2.6119 1.9670 1.1897 – – 0.7527

0.75 3.3640 3.4113 3.4503 3.4661 3.4279 3.2749 2.9029 2.2014 1.2386 – 0.3165

0.95 1.8061 1.8356 1.8716 1.9162 1.9722 2.0422 2.1246 2.1910 2.0755 1.2913 0.5317

0.65 0.66 0.67 0.70 0.73 0.77 0.82 0.87 0.93 0.98

L, /max increases as h increases. In addition, for a given value of h, /max also increases as the lead time increases. Fig. 1 shows changes in the bullwhip effect as the value for / changes. As derived in Proposition 5 and can be seen clearly from numerical experiments, the bullwhip effect increases as L increases for a given value of / > 0. However, if / < 0 and L is an even number, the bullwhip

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251

Table 4 Values of BðL; /; hÞ in the case of L = 4 h

/ 0.85

0.90 0.70 0.50 0.30 0.10 0.10 0.30 0.50 0.70 0.90 hmax

1.0193 – – – – – – – – – 0.9477

/max 0.65 1.1441 1.0460 – – – – – – – – 0.9457

0.45 1.3432 1.2595 1.0639 – – – – – – – 0.9781

0.25 1.5884 1.5179 1.3482 1.0784 – – – – – – 0.9971

0.05

0.15

1.8989 1.8414 1.6964 1.4425 1.0948 – – – – – 0.9999

0.35

2.3468 2.2988 2.1734 1.9370 1.5737 1.1182 – – – – 0.9994

3.0126 2.9757 2.8713 2.6597 2.2985 1.7748 1.1540 – – – 0.9798

0.55 3.8486 3.8376 3.7768 3.6278 3.3328 2.8247 2.0787 1.2078 – – 0.8612

0.75 4.2922 4.3286 4.3479 4.3289 4.2301 3.9733 3.4325 2.4927 1.2854 – 0.4745

0.95 2.3577 2.3939 2.4376 2.4910 2.5566 2.6357 2.7211 2.7637 2.5231 1.3868 0.4741

0.73 0.73 0.74 0.76 0.78 0.80 0.84 0.89 0.94 0.98

Table 5 Values of BðL; /; hÞ in the case of L = 5 h

0.90 0.70 0.50 0.30 0.10 0.10 0.30 0.50 0.70 0.90 hmax

/

/max

0.85

0.65

0.45

0.25

0.05

1.0717 – – – – – – – – – –

1.2346 1.0642 – – – – – – – – –

1.3814 1.2816 1.0681 – – – – – – – –

1.5937 1.5218 1.3504 1.0788 – – – – – – –

1.8989 1.8414 1.6964 1.4425 1.0948 – – – – – –

0.15

0.35

2.3479 2.2998 2.1742 1.9376 1.5740 1.1182 – – – – 0.9999

0.55

3.0516 3.0118 2.9033 2.6858 2.3169 1.7845 1.1556

4.1082 4.0864 4.0091 3.8353 3.5041 2.9465 2.1419 1.2178

– – – 0.9929

– – 0.9229

0.35

0.55

0.75 5.0890 5.1133 5.1123 5.0596 4.9045 4.5544 3.8667 2.7257 1.3211 – 0.5996

0.95 3.0141 3.0556 3.1054 3.1653 3.2370 3.3195 3.3984 3.3997 3.0014 1.4810 0.4152

0.78 0.78 0.79 0.80 0.81 0.83 0.86 0.90 0.94 0.98

Table 6 Values of BðL; /; hÞ in the case of L = 6 h

/ 0.85

0.90 0.70 0.50 0.30 0.10 0.10 0.30 0.50 0.70 0.90 hmax

1.0260 – – – – – – – – – 0.9583

/max 0.65 1.1734 1.0523 – – – – – – – – 0.9752

0.45 1.3639 1.2716 1.0662 – – – – – – – 0.9955

0.25 1.5924 1.5209 1.3499 1.0787 – – – – – – 0.9998

0.05 1.8989 1.8414 1.6964 1.4425 1.0948 – – – – – 0.9999

0.15 2.3481 2.3000 2.1743 1.9377 1.5741 1.1182 – – – – 0.9999

3.0654 3.0246 2.9146 2.6950 2.3234 1.7879 1.1562 – – – 0.9975

4.2558 4.2277 4.1408 3.9527 3.6008 3.0150 2.1771 1.2234 – – 0.9574

0.75 5.7432 5.7562 5.3770 5.6547 5.4512 5.0226 4.2132 2.9088 1.3483 – 0.6965

0.95 3.7571 3.8030 3.8572 3.9215 3.9962 4.0774 4.1417 4.0868 3.5038 1.5735 0.3560

0.81 0.81 0.82 0.82 0.83 0.85 0.87 0.91 0.94 0.98

effect measure becomes larger as L becomes larger. On the other hand, the bullwhip effect becomes smaller as L becomes larger if / < 0 and L is an odd number. Moreover, when / < 0 the maximum and minimum values of the bullwhip effect measure are obtained at L = 1 and L = 2, respectively. This can be seen from Fig. 2 for the case / > 0 and from Fig. 3 for the case / < 0.

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L=1 L=2 L=3 L=4 L=5 L=6

5

4

3

2

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Phi

Fig. 1. Effect of the autoregressive coefficient on BðL; /; hÞ in the case of h ¼ 0:1.

B 4

L=1 L=2 L=3 L=4 L=5 L=6

3.5

3

2.5

2

1.5

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4 0.5 Theta

Fig. 2. Effect of the autoregressive coefficient on BðL; /; hÞ in the case of / ¼ 0:5.

In addition, for given values of / 6¼ 0 and L, the function BðL; /; hÞ reaches a maximum value at qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi L 2L Lþ1 hmax ¼ /  1 þ /  2/ if either L is an even number, or L is an odd number and / > 0. This can be shown as follows. L Þðh2 2/L hþ2/Lþ1 1Þ We have oBðL;/;hÞ ¼ ð1þ/Þð1/ð1þh . Since 1 þ / > 0; 1  /L > 0 and 1 þ h2  2/hq >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0; oBðL;/;hÞ ¼ 0 isffi 2 oh oh 2/hÞ2

equivalent to h2  2/L h þ 2/Lþ1  1 ¼ 0. This quadratic equation has two roots, h1 ¼ /L  1 þ /2L  2/Lþ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and h2 ¼ /L þ 1 þ /2L  2/Lþ1 . Due to the fact that j / j< 1, we can easily derive qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ /2L  2/Lþ1 >j /  /L j and hence, we have either qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ /2L  2/Lþ1 > /L  /; which results in h1 < /; or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ /2L  2/Lþ1 > /  /L ; which results in h2 > /:

Since the value h2 is not valid here because of the condition for the existence of the bullwhip effect, which is h < /, it is ignored here.

T.T.H. Duc et al. / European Journal of Operational Research 187 (2008) 243–256 B 1.8

253

L=1 L=2 L=3 L=4 L=5 L=6

1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 -1

-0.9

-0.8

-0.7

-0.6

-0.5 Theta

Fig. 3. Effect of the autoregressive coefficient on BðL; /; hÞ in the case of / ¼ 0:5.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ /2L  2/Lþ1 is the value at which BðL; /; h) is maximized, i.e., hmax as defined qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi above. However, h1 is valid only if h1 ¼ /L  1 þ /2L  2/Lþ1 > 1, or equivalently, if /L ð1 þ /Þ > 0. Therefore, h1 is valid, if either L is even, or L is odd and / > 0. This can be seen in numerical results presented in Tables 1–6 for large values of /. For small values of /, hmax is very close to 1. (Numerical results for these cases are not given in this paper as tabular forms.) Note that h1 ¼ /L 

5. Conclusions In this research, we investigated the effect of the lead time, the autoregressive coefficient, and the moving average parameter on a bullwhip effect measure in a simple two-stage supply chain with one supplier and one retailer in which the demand pattern follows a mixed autoregressive-moving average process, ARMA(1, 1), and the retailer employs the base stock policy for replenishments. We find the following properties of the bullwhip effect. (1) The bullwhip effect does not always exist, but its existence depends on the values of autoregressive and moving average coefficients of the ARMA model. In fact, the bullwhip effect occurs only when the autoregressive coefficient of the demand process is larger than the moving average parameter. For given values of the lead time L and the moving average parameter h, the bullwhip effect measure reaches the maximum value at /max 2 ðh; 1Þ, and /max increases as either L or h increases. (2) The bullwhip effect does not always increase when the lead time L increases. In fact, if the autoregressive coefficient / is positive, the longer the lead time is, the higher the bullwhip effect in a supply chain becomes, and there exists an upper bound for the bullwhip effect measure. However, if h < / < 0 and L is an odd number, the bullwhip effect becomes smaller as L becomes larger; if h < / < 0 and L is an even number, the bullwhip effect becomes larger as L becomes larger. In this case ðh < / < 0Þ, the bullwhip effect is maximized when L = 1 and minimized whenL = 2. (3) There may exist a value hmax at which the bullwhip effect measure is maximized for given values of L and /. Quantifying the bullwhip effect and investigating its behavior are helpful in allocation of efforts for mitigating the influence of the bullwhip effect in supply chains. Findings of this research are expected to pave a way for the development of more realistic models that can capture the complexity of real supply chains, e.g., multiple-stage supply chains with multiple retailers and suppliers. Also, research on the bullwhip effect in a supply

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chain with higher-order demand processes may be needed, although it would be very difficult to investigate bullwhip behaviors analytically. Acknowledgement The authors are grateful to the reviewers for their valuable comments and suggestions that help improve the quality of this paper. Appendix. Proof for Proposition 3 We have qt ¼ ¼

/ð1  /L Þ /ð1  /L Þ hð1  /L Þ hð1  /L Þ Dt1  Dt2 þ Dt1  et1 þ et2 1/ 1/ 1/ 1/ 1  /Lþ1 /ð1  /L Þ hð1  /L Þ hð1  /L Þ Dt2  et1 þ et2 : Dt1  1/ 1/ 1/ 1/

Taking the variance, we have  2  2  2 1  /Lþ1 /ð1  /L Þ hð1  /L Þ VARðqt Þ ¼ VARðDt1 Þ þ VARðDt2 Þ þ VARðet1 Þ 1/ 1/ 1/  2    hð1  /L Þ 1  /Lþ1 /ð1  /L Þ þ VARðet2 Þ  2 COVðDt1 ; Dt2 Þ 1/ 1/ 1/       1  /Lþ1 hð1  /L Þ 1  /Lþ1 hð1  /L Þ COVðDt1 ; et1 Þ þ 2 COVðDt1 ; et2 Þ 2 1/ 1/ 1/ 1/    /ð1  /L Þ hð1  /L Þ 2 COVðDt2 ; et2 Þ; 1/ 1/ where VARðDt1 Þ ¼ VARðDt2 Þ ¼ r2d , COVðDt1 ; et1 Þ ¼ COVðDt2 ; et2 Þ ¼ r2 ; and COVðDt1 ; et2 Þ ¼ E½et2 Dt1  ¼ E½et2 ðd þ /Dt2 þ et1  het2 Þ ¼ ð/  hÞr2 : The following claim gives an expression of COV(Dt ; Dt1 ), covariance of demands at periods t and t  1. Claim. The covariance of demands in periods t and t  1 can be given as COVðDt ; Dt1 Þ ¼ r2d

ð/  hÞð1  /hÞ : 1 þ h2  2/h

Proof. Since Dt ¼ d þ /Dt1 þ et  het1 , we have Dt Dt1 ¼ dDt1 þ /D2t1 þ et Dt1  het1 Dt1 . Taking the expectation, we have 2

E½Dt Dt1  ¼ ð1  /ÞðE½Dt1 Þ þ /E½D2t1  þ E½et Dt1   hE½et1 Dt1  2

¼ ð1  /ÞðE½Dt1 Þ þ /E½D2t1   hr2 : Since d ¼ ð1  /ÞE½Dt1 , we have E½et1 Dt1  ¼ E½et1 ðd þ /Dt2 þ et1  het2 Þ ¼ E½et1 d þ E½et1 /Dt2  þ E½e2t1   E½et1 het2  ¼ E½e2t1  ¼ r2 : 2

From (5), the variance of forecast error can be given as r2 ¼ r2d 1þh1/ . 2 2/h

T.T.H. Duc et al. / European Journal of Operational Research 187 (2008) 243–256

255

By the definition of covariance, we have 2

2

COVðDt ; Dt1 Þ ¼ E½Dt Dt1   E½Dt E½Dt1  ¼ ð1  /ÞðE½Dt1 Þ þ /E½D2t1   hr2  ðE½Dt1 Þ 2

¼ /ðE½D2t1   ðE½Dt1 Þ Þ  hr2 ¼ /r2d  hr2 ¼ /r2d  hr2d ¼

r2d

1  /2 1 þ h2  2/h

  /ð1 þ h2  2/hÞ  hð1  /2 Þ ð/  hÞð1  /hÞ : ¼ r2d 2 1 þ h  2/h 1 þ h2  2/h

This completes the proof for the claim.

h

From the above expanded expression of VARðqt Þ, it can be seen that VARðqt Þ ¼ A þ B, where  2   2   1  /Lþ1 /ð1  /L Þ 1  /Lþ1 /ð1  /L Þ A¼ VARðDt1 Þ þ VARðDt2 Þ  2 COVðDt1 ; Dt2 Þ 1/ 1/ 1/ 1/   r2d Lþ1 2 2 L 2 Lþ1 L ð/  hÞð1  /hÞ ¼ ð1  / Þ þ / ð1  / Þ  2/ð1  / Þð1  / Þ 2 1 þ h2  2/h ð1  /Þ    r2d ð/  hÞð1  /hÞ Lþ1 L 2 Lþ1 L ðð1  / Þ  /ð1  / ÞÞ þ 2/ð1  / Þð1  / Þ 1  ¼ 2 1 þ h2  2/h ð1  /Þ  ! L L1 X X r2d ð/  hÞð1  /hÞ 2 i i ð1  /Þ þ 2/ð1  /Þ / ð1  /Þ / 1 ¼ 2 1 þ h2  2/h ð1  /Þ i¼0 i¼0 !   L L1 X ð/  hÞð1  /hÞ X i i 2 ¼ rd 1 þ 2/ 1  / / 1 þ h2  2/h i¼0 i¼0 !   L L1 X 2/ð1  /Þ 1 þ hð1  /Þ þ h2 X i i 2 ¼ rd 1 þ / / : 1 þ h2  2/h i¼0 i¼0  2  2    hð1  /L Þ hð1  /L Þ 1  /Lþ1 hð1  /L Þ VARðet1 Þ þ VARðet2 Þ  2 B¼ COVðDt1 ; et1 Þ 1/ 1/ 1/ 1/       1  /Lþ1 hð1  /L Þ /ð1  /L Þ hð1  /L Þ þ2 COVðDt1 ; et2 Þ  2 COVðDt2 ; et2 Þ 1/ 1/ 1/ 1/ ¼

¼

2hð1  /L Þr2 ð1  /Þ

2

2hð1  /L Þr2 ð1  /Þ

2

ðhð1  /L Þ  ð1  /Lþ1 Þ þ ð1  /Lþ1 Þð/  hÞ  /ð1  /L ÞÞ

ðð1  /L Þðh  /Þ  ð1  /Lþ1 Þð1  / þ hÞÞ

PL1

L1 L X X /i r2d ð1  /2 Þ i ¼ ðh  /Þð1  /Þ /  ð1  / þ hÞð1  /Þ /i 1 þ h2  2/h ð1  /Þ i¼0 i¼0 ! L1 L L1 X X X 2hr2d ð1  /2 Þ i i ¼ ðh  /Þ /  ð1  / þ hÞ / /i : 1 þ h2  2/h i¼0 i¼0 i¼0

2hð1  /Þ

i¼0 2

!

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Hence, the explicit expression for VARðqt Þ can be derived as follows: 0 1 L L1 2/ð1/Þð1þhð1/Þþh2 Þ P i P i 1 þ / / 2 1þh 2/h B C i¼0 i¼0   L1 C VARðqt Þ ¼ r2d B L 1 L @ P i P i P iA 2hð1/2 Þ þ 1þh ðh  /Þ /  ð1  / þ hÞ / / 2 2/h i¼0

i¼0

Lþ1

¼ r2d 1 þ

2ð/  hÞð1  /  /hð1  / 1 þ h2  2/h

¼

L1 ÞÞ X

!

i¼0

/i

i¼0

! 2ð/  hÞð1  / Þð1  /  /hð1  /L1 ÞÞ 1þ : ð1  /Þð1 þ h2  2/hÞ L

r2d

L1

Lþ1

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