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Measure of bullwhip effect in supply chains with autoregressive demand process
European Journal of Operational Research 180 (2007) 1086–1097 www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Measure of bullwhip effect in supply chains with autoregressive demand process Huynh Trung Luong
*
Asian Institute of Technology, School of Advanced Technologies, Industrial Systems Engineering Program, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand Received 18 November 2004; accepted 27 February 2006 Available online 11 July 2006
Abstract In supply chain management, one of the most critical problems which require a lot of effort to deal with is how to quantify and alleviate the impact of bullwhip effect – the phenomenon in which information on demand is distorted while moving upstream. Although it is well established that demand forecast, lead time, order batching, shortage gaming and price fluctuation are the main sources that lead to the bullwhip effect, the problem of quantifying bullwhip effect still remain unsolved in many situations due to the complex nature of the problem. In this research, a measure of bullwhip effect will be developed for a simple two-stage supply chain that includes only one retailer and one supplier in the environment where the retailer employs base stock policy for their inventory and demand forecast is performed through the first-order autoregressive model, AR(1). The effect of autoregressive coefficient and lead time on this measure will then be investigated. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Supply chain; Bullwhip effect; Autoregressive model; Base stock policy
1. Introduction Considering the typical arrangement of a supply chain in which suppliers provide raw materials to manufacturers, manufacturers produce and provide finished products to wholesalers, who then provide manufacturers’ products for sale to retailers and from these retailers, products are delivered to consumers. In this arrangement, beside the physical flow of products downstream, there exists an information flow that goes upstream in which only the retailers have the exact information about consumer’s demand. The demands seen by wholesalers, manufacturers and the raw material suppliers are not the actual demand of the products due to the fact that the orders they received have been adjusted in accordance with some forecasting technique or inventory policy applied at their immediate successor downstream. It has been observed in many supply chains that the variability in the ordering patterns usually increase when moving upstream towards manufacturers and suppliers (e.g., Forrester, 1958; Blinder, 1982, 1986; Kahn, 1987). The Beer Game, developed at MIT *
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and reported in the paper of Sterman (1989), which was widely used in teaching inventory management, is also another demonstration of the above phenomenon. The above phenomenon was first referred to as ‘‘bullwhip effect’’ in the works of Lee et al. (1997a,b). Lee et al. (1997a,b) also propose five important sources that might lead to the emergence of bullwhip effect in supply chains, i.e., demand signal processing, non-zero lead time, order batching, supply shortages and price fluctuations. In order to avoid the bullwhip effect, all causes need to be eliminated. However, it is not so easy. Many managerial approaches have been proposed to help alleviate the bullwhip effect. Berry et al. (1995) propose redesign and reengineering of the supply chain as the tools to control demand amplification. Van Ackere et al. (1993) also recommend some approaches to avoid or alleviate the bullwhip effect, i.e., lead-time reduction, information sharing or applying different replenishment rules for the inventory system. The potential remedies that can help to alleviate bullwhip effect are summarized in Nahmias (1997). Among the most important issues that need to be addressed in dealing with bullwhip effect, quantifying bullwhip effect is a very challenging issue. Metters (1997) tries to identify the bullwhip effect by establishing an empirical lower bound on the profitability impact of the bullwhip effect. In his research, the impact of bullwhip effect is measured by comparing results obtained in two cases, i.e., high demand variability versus low demand variability with weak seasonality. It is found that the important of the bullwhip effect differs greatly depending on the business environment. Graves (1999) quantifies the bullwhip effect for the supply chain in which demand pattern follows an integrated moving average process. Chen et al. (2000a,b) quantify the bullwhip effect for supply chains using moving average and exponential smoothing techniques for demand forecasts. In their works, it is assumed that members of the chain employ base stock policy for their inventory system. The main finding of their work is that the order variance will increase with the number of members in the chain, lower level of information sharing and increasing lead time. Similar finding is also found in the research of Xu et al. (2001) for the case when demand forecast is done by use of exponential smoothing technique. Dejonckheere et al. (2004) also investigate the base stock policy and it is found that under this replenishment policy, information sharing will help to significantly reduce the variance amplification of ordering quantities in supply chains. The influence of the general (s, S) ordering policy on bullwhip effect in supply chain is also investigated by Kelle and Milne (1999) in which approximations to the exact quantitative models are proposed to help investigate the influence of (s, S) policy parameters and also demand parameters on the variability of the orders. Considering a very specific ordering policy, Disney and Towill (2003) also derive an expression for measuring bullwhip effect and the solution to the bullwhip problem in supply chains is also presented. It should be noted that the researches of Dejonckheere et al. (2003, 2004) and Disney and Towill (2003) are not based on analytical approach but on control systems engineering approach, and their target is to introduce some modification rules into the conventional order-up-to policy which can help generating smooth ordering patterns and hence, relieve variance amplification in supply chains. Although some researches have been conducted related to quantifying the bullwhip effect, more works need to be done in this field. In the current research, a measure of bullwhip effect based on a replenishment model which is similar to the one used by Chen et al. (2000a,b) will be developed. The effect of lead time on this measure will then be investigated. The difference between this research and past researches, e.g., Chen et al. (2000a,b), Graves (1999), and Xu et al. (2001), is that lead-time demand is estimated using a forecasting procedure that minimizes the expected mean-square forecast error instead of simple forecasting techniques such as moving average or exponential smoothing. It should be noted that the moving average and exponential smoothing forecasting techniques are very popular mainly due to their ease of use. However, for pre-defined demand processes, e.g., AR(p) processes, these forecasting techniques might lead to specification error and hence, will further amplify the bullwhip effect. For a time series, which is the demand process considered in this research, the forecasting technique that minimizes the expected mean-square forecast error will help to predict future demand value with as little error as possible, and hence, reduce the bullwhip effect. It is also noted that Chen et al. (2000a,b) derived only the lower bound for bullwhip measure while the target of the current research is an explicit expression for the measure of bullwhip effect. The structure of this research is as follows. Section 2 describes the system under study and gives some detailed backgrounds on base stock policy and the procedure to determine order quantity in base stock policy. Section 3 discusses the AR(1) demand process and presents the relationship between variance of demand and variance of error term in AR(1) demand process. The condition for the demand process to be a stationary one
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is also taken into consideration. In Section 4, the expression of lead-time demand forecast with minimum expected mean squares of error is developed. Section 5 focuses on the determination of standard deviation of lead-time demand forecast error. Section 6 provides the expression of variance of order quantity and then, the expression for measuring the bullwhip effect can be determined as the ratio between variance of order quantity and variance of demand. The effects of autoregressive coefficient and lead time on the developed bullwhip measure are then considered in Section 7 and some concluding remarks are discussed in Section 8. The following set of notations will be used in this research: Dt qt St / d ld r2d L DLt bL D t ^Lt r et
demand of period t order quantity at the beginning of period t order-up-to level at the beginning of period t the first-order autocorrelation coefficient the constant of the autoregressive model mean of the autoregressive process which is used to describe the demand process variance of demand order lead time lead-time demand lead-time demand forecast standard deviation of lead-time demand forecast error forecast error for period t
2. Description of the system In this research, a measure of bullwhip effect will be developed for a two-stage supply chain that includes only one retailer and one supplier. Order lead time is taken into consideration and it is assumed that lead time is fixed and its value is a multiple of inventory review period. The retailer employs base stock policy, a simple order-up-to inventory policy, for their inventory and demand forecast is performed through AR(1) autoregressive model by use of a forecasting procedure that minimizes the mean-square forecast error. For the inventory policy considered in this research, let St be the order-up-to level at the beginning of period t, i.e., the inventory position at the beginning of period t after the order has been placed, the order quantity, qt, at the beginning of period t should satisfy the following relationship: qt ¼ S t S t1 þ Dt1 :
ð1Þ
If the base stock policy is employed, the order-up-to level St can be determined through lead-time demand by b L þ z^ St ¼ D rLt ; t
ð2Þ
in which z is the normal z-score that can be determined based on the desired service level of the inventory policy. It should be noted that the optimal order-up-to level St can be explicitly determined from inventory holding cost and shortage cost (e.g., Heyman and Sobel, 1984). However, it is usually difficult to accurately estimate these costs in practice. The approach using service level, which is defined as the probability for the on-hand inventory fulfilling lead-time demand requirement, to help determine order-up-to level is usually more practicable. It is noted that the base stock policy is a simple case of the order-up-to inventory policy. In the base stock policy, an order will be placed at the beginning of each period so as to increase the inventory level up to a predefined level. In general, the order-up-to level St defined in (2) is optimal, in term of total inventory cost, for inventory systems where there is no fixed ordering cost and both holding and shortage costs are proportional to the volume of on-hand inventory or shortage (Nahmias, 1997; Zipkin, 2000). b L and standard deviation of leadAt a pre-defined service level, the values of lead-time demand forecast D t L ^t are needed in order to help determine the order-up-to level St. The expressions time demand forecast error r b L and r ^Lt are developed in Sections 4 and 5, respectively. However, as will be shown later, for calculation of D t ^Lt does not depend on t and hence, this value does not have any influence on bullwhip effect. r
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3. Demand process In this research, it is assumed that demand can be modeled by an AR(1) model, in which Dt ¼ d þ /Dt1 þ et ;
ð3Þ
where et (t = 1, 2, . . .) are i.i.d. normally distributed random variables with mean 0 and variance r2. For the autoregressive process AR(1) to be stationary, we must have: E½Dt ¼ E½Dt1 ¼ ld
for all t:
Therefore, it can be easily derived that ld ¼
d : 1/
ð4Þ
From (3), we also have: r2d ¼ /2 r2d þ r2 : Hence, r2d ¼
r2 : 1 /2
ð5Þ
From (4) and (5), it can be seen that in order for the demand process to be stationary we should also have j/j < 1. 4. Determination of lead-time demand forecast In this research, demand forecast is conducted so as to minimize the expected mean squares of error. It is noted that 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 with minimum expected mean squares of error of Dt, then the minimum expected mean If D squares of error forecast for lead-time demand can be determined as bL ¼ D bt þ D b tþ1 þ þ D b tþL1 ¼ D t
L1 X
b tþi : D
ð7Þ
i¼0
b t can be determined by (Pindyck and Rubinfeld, 1998) For the AR(1) process, D b tþi ¼ E½Dtþi jDt1 ; Dt2 ; . . .: D b t can be derived and it has the following form: From the above expression, the explicit expression of D b tþi ¼ ld ð1 /iþ1 Þ þ /iþ1 Dt1 : D The detailed derivation of expression (8) is as follows: We have: Dtþi ¼ d þ /Dtþi1 þ etþi : If we express Dt+i1 in terms of Dt+i2: Dtþi1 ¼ d þ /Dtþi2 þ etþi1 : Then Dtþi ¼ dð1 þ /Þ þ /2 Dtþi2 þ etþi þ /etþi1 :
ð8Þ
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Applying the above procedure successively, we have: Dtþi ¼ dð1 þ / þ þ /i Þ þ /iþ1 Dt1 þ etþi þ /etþi1 þ þ /i et : Taking the expectation we have iþ1
b tþi ¼ E½Dtþi jDt1 ; Dt2 ; . . . ¼ dð1 þ / þ þ /i Þ þ /iþ1 Dt1 ¼ d 1 / þ /iþ1 Dt1 D 1/ ¼ ld ð1 /iþ1 Þ þ /iþ1 Dt1 : Proposition 1. The forecast of lead-time demand with minimum expected mean squares of error can be determined by L L b L ¼ ld L /ð1 / Þ þ /ð1 / Þ Dt1 : D ð9Þ t 1/ 1/ Proof. Replacing (8) into (7), we have: bL ¼ D t
L1 L1 L1 X X X fld ð1 /iþ1 Þ þ /iþ1 Dt1 g ¼ ld ð1 /iþ1 Þ þ Dt1 /iþ1 i¼0
i¼0
i¼0
/ð1 /L Þ /ð1 /L Þ Dt1 : ¼ ld L þ 1/ 1/
5. Determination of standard deviation of lead-time demand forecast error In this section, the expression of variance of lead-time demand forecast error will be developed. Proposition 2. The variance of lead-time demand forecast error does not depend on t and determined by 2
ð^ rLt Þ ¼
L r2d ð1 þ /Þ X 2 ð1 /i Þ : 1 / i¼1
ð10Þ
Proof. By definition we have: b L Þ: ð^ rLt Þ2 ¼ VARðDLt D t We also have: 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 etþL1 ¼ DLt D t
L1 X
etþi :
i¼0
From the proof of expression (8), it can be seen that i X
b tþi ¼ etþi þ /etþi1 þ þ /i et ¼ etþi ¼ Dtþi D
etþj /ij :
j¼o
Hence, bL ¼ DLt D t
L1 X i X i¼0
etþj /ij ¼
j¼0
L1 Li1 X X i¼0
etþi /j ¼
j¼0
L1 X
etþi
i¼0
1 /Li : 1/
Therefore, VARðDLt
b LÞ ¼ D t
2 L1 X 1 /Li i¼0
1/
r2 ¼
L1 X
r2 2
ð1 /Þ
i¼0
2
ð1 /Li Þ ¼
L r2d ð1 þ /Þ X 2 ð1 /i Þ : ð1 /Þ i¼1
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^Lt is needed for the determination of the order-up-to level St only. The measure of It should be noted that r bullwhip effect that will be developed later in this research is not affected by this value. h 6. Determination of variance of order quantity Proposition 3. The variance of order quantity of period t can be determined by the following expression: VARðqt Þ ¼
ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 2 rd : 1/
ð11Þ
Proof. We have: b L þ z^ b L þ z^ bL D b L Þ þ Dt1 : qt ¼ S t S t1 þ Dt1 ¼ ð D rLt Þ ð D rLt1 Þ þ Dt1 ¼ ð D t t1 t t1 From (9), it can be easily derived that qt ¼
1 /Lþ1 /ð1 /L Þ Dt2 : Dt1 1/ 1/
So, 2 2 1 /Lþ1 /2 ð1 /L Þ 1 /Lþ1 /ð1 /L Þ COVðDt1 ; Dt2 Þ: VARðqt Þ ¼ VARðDt1 Þ þ VARðD Þ 2 t2 1/ 1/ 1/ ð1 /Þ2 It is noted that VARðDt1 Þ ¼ VARðDt2 Þ ¼ r2d ; COVðDt1 ; Dt2 Þ ¼ /r2d : Therefore, 2 2 1 /Lþ1 /2 ð1 /L Þ 2 1 /Lþ1 /ð1 /L Þ 2 /rd VARðqt Þ ¼ r2d þ rd 2 2 1/ 1/ 1/ ð1 /Þ ¼
ð1 /Lþ1 Þ2 þ /2 ð1 /L Þ2 2/2 ð1 /Lþ1 Þð1 /L Þ ð1 /Þ
2
r2d ¼
ð1 þ /Þ 2/Lþ1 ð1 þ /Þ þ 2/2Lþ2 2 rd 1/
ð1 þ /Þð1 2/ Þ þ 2/2Lþ2 2 rd : 1/ Lþ1
¼
Knowing the variance of order quantity, the bullwhip effect can be evaluated. It is noted that the bullwhip effect is the phenomenon in which variance of demand information is amplified when moving upstream in supply chains. Therefore, it is reasonable to measure bullwhip effect by the ratio of the variance of order quantity that the supplier experiences and the variance of demand. This measure has been used by some researchers, e.g., Chen et al. (2000a,b), and also be adopted in the current research. Hence, from (11), the measurement of bullwhip effect can be determined as BðL; /Þ ¼
VARðqt Þ ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 : ¼ r2d 1/
ð12Þ
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7. Effects of autoregressive coefficient and lead time 7.1. The effect of autoregressive coefficient on bullwhip effect In this section, we first investigate the effect of autoregressive coefficient / on the measure of bullwhip effect. From (12), we can see that BðL; /Þ 6 1 ()
ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 6 1 () ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 6 1 / 1/
() 2/ð1 /L Þð1 /Lþ1 Þ 6 0 () / 6 0: The above relationship means that the bullwhip effect occurs only when there exists a positive autoregressive relationship in the demand process. In the current research, the focus is on the measure of bullwhip effect when it exists and this situation occurs only when the autoregressive coefficient is positive. When the autoregressive is negative, bullwhip effect does not exist or equivalently, the variance of order quantity experienced by supplier is smaller than the variance of demand. From managerial point of view, the reduction in variance of order quantity in comparison with variance of demand will lead to the fact that production planning phase at supplier becomes easier and help to stabilize the production process. However, it should be noted that the autoregressive coefficient is a characteristic of the demand process and is assumed constant in the current research. The target of this research is, therefore, to measure the bullwhip effect and to investigate the effect of autoregressive coefficient and lead time on bullwhip measure to see whether it is needed for supplier to pay attention to bullwhip effect in case the bullwhip effect is so high. Managerial approaches that can help to alleviate bullwhip effect can then be investigated and employed. In the remaining parts of this paper, therefore, we consider only the case when 0 < / < 1. In order to detect the effect of autoregressive coefficient, it is noted that the measure of bullwhip effect B(L, /) developed above can be expressed as BðL; /Þ ¼ 1 þ
L X
fi ðL; /Þ;
ð13Þ
i¼1
in which fi(L, /) = 2/i(1 /L+1) (i = 1, 2, . . . , L). In fact, the above expression can be developed as follows: BðL; /Þ ¼
ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 1 þ / 2/Lþ1 2/Lþ2 ð1 /L Þ ¼ 1/ 1/
ð1 /Þð1 þ 2/ þ 2/2 þ þ 2/L Þ 2/Lþ2 ð1 /Þð1 þ / þ /2 þ þ /L1 Þ 1/ L X ¼ ð1 þ 2/ þ 2/2 þ þ 2/L Þ 2/Lþ2 ð1 þ / þ /2 þ þ /L1 Þ ¼ 1 þ 2/i ð1 /Lþ1 Þ:
¼
i¼1
Proposition 4. In the interval 0 < / < 1, the functions fi(L, /); (i = 1, 2, . . . , L) are increasing and then decreasing. The maximum values are obtained at 1=Lþ1 i /i ¼ : Lþiþ1 Moreover, /i is an increasing function with respect to i (i = 1, 2, . . . , L). Proof. Taking the first derivative of fi(L, /) = 2/i(1 /L+1) with respect to /, we have: ofi ðL; /Þ ¼ 2½i/i1 ðL þ i þ 1Þ/Lþi ¼ 2/i1 ½i ðL þ i þ 1Þ/Lþ1 : o/
H.T. Luong / European Journal of Operational Research 180 (2007) 1086–1097 1=Lþ1
ðL;/Þ i Solving the equation: ofio/ ¼ 0, we have: /i ¼ ðLþiþ1 Þ
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ðL;/Þ . In the interval ð0; /i Þ, ofio/ > 0 and in the inter-
ðL;/Þ val ð/i ; 1Þ, ofio/ < 0. Hence, fi(L, /) is an increasing and then decreasing function. i < 1. Therefore, /i is an increasing function with respect to i (i = 1,2, . . . , L). It is also noted that Lþiþ1
h
In the next session, the behavior of B(L, /) with respect to / will be investigated. Proposition 5. The measure of bullwhip effect B(L, /) possesses the following properties: 1=Lþ1
1 (a) B(L, /) is an increasing function in the interval / 2 ð0; /1 Þ, in which /1 ¼ ðLþ2 Þ (b) B(L, /) is a concave function in the interval / 2 ð/1 ; 1Þ.
.
Proof (a) Due to the fact that fi(L, /); (i = 1, 2, . . . , L) are increasing functions in ð0; /i Þ and that /i is an increasing function with respect to i, (i = 1, 2, . . . , L), the first statement of Proposition 5 is easily to be confirmed. (b) In order to prove that B(L, /) is a concave function in the interval / 2 ð/1 ; 1Þ, we first prove that fi(L, /); (i = 1, 2, . . . , L) are concave in / 2 ð/1 ; 1Þ. Consider the second-order derivative of fi(L, /) with respect to /, we have: o2 fi ðL; /Þ ¼ 2½iði 1Þ/i2 ðL þ i þ 1ÞðL þ iÞ/Lþi1 ¼ 2/i2 ½iði 1Þ ðL þ i þ 1ÞðL þ iÞ/Lþ1 : o/2 We need to prove that: o2 fi ðL; /Þ < 0 when / > /1 ¼ o/2
1 Lþ2
1=Lþ1 :
Fig. 1. Effect of autoregressive coefficient on bullwhip measure.
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Note that: o2 fi ðL; /Þ iði 1Þ : < 0 () /Lþ1 > 2 ðL þ i þ 1ÞðL þ iÞ o/ The above inequality holds true due to the fact that: /Lþ1 >
1 i iði 1Þ > > : L þ 2 L þ i þ 1 ðL þ i þ 1ÞðL þ iÞ
It is noted that the sum of concave functions is also a concave function. Hence, B(L, /), which is the sum of fi(L, /); (i = 1, 2, . . . , L) and a constant, is a concave function in the interval / 2 ð/1 ; 1Þ. h From Proposition 5, it can be concluded that there always exist a unique value of /max 2 ð/1 ; 1Þ such that B(L, /) reaches its maximum value at a fixed value of lead time L. Furthermore, from the results in Proposition 4, it can be easily seen that /max 2 ð/1 ; /L Þ and hence, bisection method can be applied in this interval to find /max. The behavior of B(L, /) with respect to / and maximum values of B(L, /) at various values of lead time Lare illustrated in Fig. 1 and Table 1. It is interesting to note from Table 1 that when L increases, the maximum value of B(L, /) increases and /max also increases. In addition, at low values of /, there is not much difference in B(L, /). Table 1 Maximum value of bullwhip measure with respect to lead time Lead time L
/max
Maximum B(L, /)
1 2 3 4 5
0.577 0.683 0.747 0.789 0.820
1.770 2.566 3.371 4.180 4.990
Fig. 2a. Effect of lead time on bullwhip measure (/ = 0.3).
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7.2. The effect of lead time on bullwhip effect In this section, we will investigate the effect of lead time on the measure of bullwhip effect. Taking the first derivative of B(L, /) with respect to L we have oB 2ð1 þ /Þ/Lþ1 ln / þ 4/2Lþ2 ln / ¼ : oL 1/ With the condition j/j < 1, it is easy to prove that oB > 0. Therefore, B(L, /) is an increasing function with oL respect to L. It can be concluded that the longer the lead time, the larger the bullwhip effect in the supply chain. This finding is similar to the results of other past researches (e.g., Chen et al., 2000a,b; Xu et al., 2001) and Fig. 1 also illustrates this trend. From expression (12), we can also prove that BðL; /Þ < 1þ/ . This upper bound can be found by taking the 1/ limit of B(L, /) when L approaches infinity. At a fixed value of autoregressive coefficient /, the above upper bound value is useful in evaluating the influence of bullwhip effect when lead time increases. The functional expression of bullwhip measure represented by (12) also exhibits an interesting property stated in the following proposition: Proposition 6. Consider values of / in the interval (0, 1) (a) If / 6 1/3 then B(L, /) is a concave function with respect to L. (b) If / > 1/3 then B(L, /) is a convex and then concave function with respect to L. Proof. Consider the second derivative of B(L, /) with respect to L, we have o2 B 2ðln /Þ2 /Lþ1 ð4/Lþ1 / 1Þ : ¼ 1/ oL2
Fig. 2b. Effect of lead time on bullwhip measure (/ = 0.9).
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(a) If / 6 1/3: (4/L+1 / 1) 6 0 when L = 0. Furthermore, when L increases /L+1 will decrease. There2
fore, (4/L+1 / 1) 6 0 for all values L P 0. This leads to the fact that ooLB2 6 0 8L P 0 and hence, B(L, /) is a concave function with respect to L. (b) If / > 1/3: (4/L+1 / 1) > 0 when L = 0. However, when L increases /L+1 will decrease and starting 2
at L ¼ log/ ð1þ/ Þ 1, (4/L+1 / 1) will receive negative values. This means that ooLB2 will first have 4 positive values and then receive negative values when L increases. Therefore, B(L, /) is a convex and then concave function with respect to L. h The above properties are illustrated in Figs. 2a and 2b for the cases / = 0.3 and / = 0.9. 8. Conclusions In this research, we have investigated a simple supply chain that includes one retailer and one supplier in which the retailer employs base stock inventory policy. The bullwhip measure is developed for the first-order autoregressive demand process and some interesting properties of this measure have been found. Firstly, it is interesting to note that the bullwhip effect does not exist not only when there is no correlation or there is negative correlation in the demand process but also when there exists a perfect positive correlation in the demand process (i.e., / = 1). This is due to the fact that B(L, /) 6 1 when / 6 0 or / = 1. In the range / 2 (0, 1), when / increases, the bullwhip effect will first increases, reaches maximum value, and then decreases as shown in Fig. 1. From Fig. 1, it is also noted that when the lead time increases, the value of / at which the bullwhip effect reaches its maximum value also increases and approaches one. Another interested finding of this research is that there exists an upper bound for the measure of bullwhip effect when lead time increases. The value of this upper bound depends on the autoregressive coefficient /. This upper bound value is useful in evaluating the influence of bullwhip effect and it can help management to allocate effort in order to alleviate the impact of bullwhip effect in supply chain when lead time increases. Lastly, it should be noted that the behavior of the bullwhip measure can have two patterns with respect to lead time. This also depends on the autoregressive coefficient /. It is expected that the findings of this research can serve as a good starting point for investigating the impact of bullwhip effect in more complicated supply chains where suppliers provide goods for more than one retailer. Acknowledgement The author is grateful to two distinguished reviewers for their valuable comments and suggestions for improving the quality and the presentation of this paper. References Berry, D., Naim, M.M., Towill, D.R., 1995. Business process reengineering an electronics products supply chain. In: Proceedings of IEE Science, Measurement, and Technology, pp. 395–403. Blinder, A.S., 1982. Inventories and stucky prices. American Economic Review 72, 334–349. Blinder, A.S., 1986. Can the production smoothing model of inventory behavior be safe? Quarterly Journal of Economics 101 431–454. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000a. Quantifying the bullwhip effect in a simple supply chain. Management Science 46 (3), 436–443. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000b. The impact of exponential smoothing forecasts on the bullwhip effect. Naval Research Logistics 47 (4), 269–286. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R., 2003. Measuring and avoiding the bullwhip effect: A control theoretical approach. European Journal of Operational Research 147 (3), 567–590. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R., 2004. The impact of information enrichment on the bullwhip effect in supply chains: A control engineering perspective. European Journal of Operational Research 153 (3), 727–750. Disney, S.M., Towill, D.R., 2003. On the bullwhip and inventory variance produced by an ordering policy. Omega 31, 157–167. Forrester, J., 1958. Industrial dynamics: A major breakthrough for decision makers. Harvard Business Review 36, 37–66. Graves, S.C., 1999. A single-item inventory model for a non-stationary demand process. Manufacturing and Service Operations Management 1 (1), 50–61. Heyman, D., Sobel, M., 1984. Stochastic Models in Operations Research. McGraw-Hill.
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Kahn, J., 1987. Inventories and the volatility of production. American Economic Review 77, 667–679. Kelle, P., Milne, A., 1999. The effect of (s, S) ordering policy on supply chain. International Journal of Production Economics 59, 113–122. Lee, H.L., Padmanabhan, V., Whang, S., 1997a. The bullwhip effect in supply chains. Sloan Management Review Spring 38 (3), 93–102. Lee, H.L., Padmanabhan, V., Whang, S., 1997b. Information distortion in a supply chain: The bullwhip effect. Management Science 43 (4), 546–558. Metters, R., 1997. Quantifying the bullwhip effect in supply chains. Journal of Operations Management 15, 89–100. Nahmias, S., 1997. Production and Operation Analysis, third ed. McGraw-Hill. Pindyck, R.S., Rubinfeld, D.L., 1998. Econometric Models and Economic Forecasts. Irwin McGraw-Hill. Sterman, J., 1989. Modeling managerial behavior: Misperceptions of feedback in a dynamic decision making experiment. Management Science 35 (3), 321–339. Van Ackere, A., Larsen, E.R., Morecroft, J.D.W., 1993. System thinking and business process redesign: An application to the Beer Game. European Management Journal 11 (4), 412–423. Xu, K., Dong, Y., Evers, P.T., 2001. Towards better coordination of the supply chain. Transportation Research Part E: Logistics and Transportation Review 37 (1), 35–54. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill.
Production, Manufacturing and Logistics
Measure of bullwhip effect in supply chains with autoregressive demand process Huynh Trung Luong
*
Asian Institute of Technology, School of Advanced Technologies, Industrial Systems Engineering Program, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand Received 18 November 2004; accepted 27 February 2006 Available online 11 July 2006
Abstract In supply chain management, one of the most critical problems which require a lot of effort to deal with is how to quantify and alleviate the impact of bullwhip effect – the phenomenon in which information on demand is distorted while moving upstream. Although it is well established that demand forecast, lead time, order batching, shortage gaming and price fluctuation are the main sources that lead to the bullwhip effect, the problem of quantifying bullwhip effect still remain unsolved in many situations due to the complex nature of the problem. In this research, a measure of bullwhip effect will be developed for a simple two-stage supply chain that includes only one retailer and one supplier in the environment where the retailer employs base stock policy for their inventory and demand forecast is performed through the first-order autoregressive model, AR(1). The effect of autoregressive coefficient and lead time on this measure will then be investigated. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Supply chain; Bullwhip effect; Autoregressive model; Base stock policy
1. Introduction Considering the typical arrangement of a supply chain in which suppliers provide raw materials to manufacturers, manufacturers produce and provide finished products to wholesalers, who then provide manufacturers’ products for sale to retailers and from these retailers, products are delivered to consumers. In this arrangement, beside the physical flow of products downstream, there exists an information flow that goes upstream in which only the retailers have the exact information about consumer’s demand. The demands seen by wholesalers, manufacturers and the raw material suppliers are not the actual demand of the products due to the fact that the orders they received have been adjusted in accordance with some forecasting technique or inventory policy applied at their immediate successor downstream. It has been observed in many supply chains that the variability in the ordering patterns usually increase when moving upstream towards manufacturers and suppliers (e.g., Forrester, 1958; Blinder, 1982, 1986; Kahn, 1987). The Beer Game, developed at MIT *
Tel.: +66 2 5245683; fax: +66 2 5245697. E-mail address: [email protected]
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.02.050
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and reported in the paper of Sterman (1989), which was widely used in teaching inventory management, is also another demonstration of the above phenomenon. The above phenomenon was first referred to as ‘‘bullwhip effect’’ in the works of Lee et al. (1997a,b). Lee et al. (1997a,b) also propose five important sources that might lead to the emergence of bullwhip effect in supply chains, i.e., demand signal processing, non-zero lead time, order batching, supply shortages and price fluctuations. In order to avoid the bullwhip effect, all causes need to be eliminated. However, it is not so easy. Many managerial approaches have been proposed to help alleviate the bullwhip effect. Berry et al. (1995) propose redesign and reengineering of the supply chain as the tools to control demand amplification. Van Ackere et al. (1993) also recommend some approaches to avoid or alleviate the bullwhip effect, i.e., lead-time reduction, information sharing or applying different replenishment rules for the inventory system. The potential remedies that can help to alleviate bullwhip effect are summarized in Nahmias (1997). Among the most important issues that need to be addressed in dealing with bullwhip effect, quantifying bullwhip effect is a very challenging issue. Metters (1997) tries to identify the bullwhip effect by establishing an empirical lower bound on the profitability impact of the bullwhip effect. In his research, the impact of bullwhip effect is measured by comparing results obtained in two cases, i.e., high demand variability versus low demand variability with weak seasonality. It is found that the important of the bullwhip effect differs greatly depending on the business environment. Graves (1999) quantifies the bullwhip effect for the supply chain in which demand pattern follows an integrated moving average process. Chen et al. (2000a,b) quantify the bullwhip effect for supply chains using moving average and exponential smoothing techniques for demand forecasts. In their works, it is assumed that members of the chain employ base stock policy for their inventory system. The main finding of their work is that the order variance will increase with the number of members in the chain, lower level of information sharing and increasing lead time. Similar finding is also found in the research of Xu et al. (2001) for the case when demand forecast is done by use of exponential smoothing technique. Dejonckheere et al. (2004) also investigate the base stock policy and it is found that under this replenishment policy, information sharing will help to significantly reduce the variance amplification of ordering quantities in supply chains. The influence of the general (s, S) ordering policy on bullwhip effect in supply chain is also investigated by Kelle and Milne (1999) in which approximations to the exact quantitative models are proposed to help investigate the influence of (s, S) policy parameters and also demand parameters on the variability of the orders. Considering a very specific ordering policy, Disney and Towill (2003) also derive an expression for measuring bullwhip effect and the solution to the bullwhip problem in supply chains is also presented. It should be noted that the researches of Dejonckheere et al. (2003, 2004) and Disney and Towill (2003) are not based on analytical approach but on control systems engineering approach, and their target is to introduce some modification rules into the conventional order-up-to policy which can help generating smooth ordering patterns and hence, relieve variance amplification in supply chains. Although some researches have been conducted related to quantifying the bullwhip effect, more works need to be done in this field. In the current research, a measure of bullwhip effect based on a replenishment model which is similar to the one used by Chen et al. (2000a,b) will be developed. The effect of lead time on this measure will then be investigated. The difference between this research and past researches, e.g., Chen et al. (2000a,b), Graves (1999), and Xu et al. (2001), is that lead-time demand is estimated using a forecasting procedure that minimizes the expected mean-square forecast error instead of simple forecasting techniques such as moving average or exponential smoothing. It should be noted that the moving average and exponential smoothing forecasting techniques are very popular mainly due to their ease of use. However, for pre-defined demand processes, e.g., AR(p) processes, these forecasting techniques might lead to specification error and hence, will further amplify the bullwhip effect. For a time series, which is the demand process considered in this research, the forecasting technique that minimizes the expected mean-square forecast error will help to predict future demand value with as little error as possible, and hence, reduce the bullwhip effect. It is also noted that Chen et al. (2000a,b) derived only the lower bound for bullwhip measure while the target of the current research is an explicit expression for the measure of bullwhip effect. The structure of this research is as follows. Section 2 describes the system under study and gives some detailed backgrounds on base stock policy and the procedure to determine order quantity in base stock policy. Section 3 discusses the AR(1) demand process and presents the relationship between variance of demand and variance of error term in AR(1) demand process. The condition for the demand process to be a stationary one
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is also taken into consideration. In Section 4, the expression of lead-time demand forecast with minimum expected mean squares of error is developed. Section 5 focuses on the determination of standard deviation of lead-time demand forecast error. Section 6 provides the expression of variance of order quantity and then, the expression for measuring the bullwhip effect can be determined as the ratio between variance of order quantity and variance of demand. The effects of autoregressive coefficient and lead time on the developed bullwhip measure are then considered in Section 7 and some concluding remarks are discussed in Section 8. The following set of notations will be used in this research: Dt qt St / d ld r2d L DLt bL D t ^Lt r et
demand of period t order quantity at the beginning of period t order-up-to level at the beginning of period t the first-order autocorrelation coefficient the constant of the autoregressive model mean of the autoregressive process which is used to describe the demand process variance of demand order lead time lead-time demand lead-time demand forecast standard deviation of lead-time demand forecast error forecast error for period t
2. Description of the system In this research, a measure of bullwhip effect will be developed for a two-stage supply chain that includes only one retailer and one supplier. Order lead time is taken into consideration and it is assumed that lead time is fixed and its value is a multiple of inventory review period. The retailer employs base stock policy, a simple order-up-to inventory policy, for their inventory and demand forecast is performed through AR(1) autoregressive model by use of a forecasting procedure that minimizes the mean-square forecast error. For the inventory policy considered in this research, let St be the order-up-to level at the beginning of period t, i.e., the inventory position at the beginning of period t after the order has been placed, the order quantity, qt, at the beginning of period t should satisfy the following relationship: qt ¼ S t S t1 þ Dt1 :
ð1Þ
If the base stock policy is employed, the order-up-to level St can be determined through lead-time demand by b L þ z^ St ¼ D rLt ; t
ð2Þ
in which z is the normal z-score that can be determined based on the desired service level of the inventory policy. It should be noted that the optimal order-up-to level St can be explicitly determined from inventory holding cost and shortage cost (e.g., Heyman and Sobel, 1984). However, it is usually difficult to accurately estimate these costs in practice. The approach using service level, which is defined as the probability for the on-hand inventory fulfilling lead-time demand requirement, to help determine order-up-to level is usually more practicable. It is noted that the base stock policy is a simple case of the order-up-to inventory policy. In the base stock policy, an order will be placed at the beginning of each period so as to increase the inventory level up to a predefined level. In general, the order-up-to level St defined in (2) is optimal, in term of total inventory cost, for inventory systems where there is no fixed ordering cost and both holding and shortage costs are proportional to the volume of on-hand inventory or shortage (Nahmias, 1997; Zipkin, 2000). b L and standard deviation of leadAt a pre-defined service level, the values of lead-time demand forecast D t L ^t are needed in order to help determine the order-up-to level St. The expressions time demand forecast error r b L and r ^Lt are developed in Sections 4 and 5, respectively. However, as will be shown later, for calculation of D t ^Lt does not depend on t and hence, this value does not have any influence on bullwhip effect. r
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3. Demand process In this research, it is assumed that demand can be modeled by an AR(1) model, in which Dt ¼ d þ /Dt1 þ et ;
ð3Þ
where et (t = 1, 2, . . .) are i.i.d. normally distributed random variables with mean 0 and variance r2. For the autoregressive process AR(1) to be stationary, we must have: E½Dt ¼ E½Dt1 ¼ ld
for all t:
Therefore, it can be easily derived that ld ¼
d : 1/
ð4Þ
From (3), we also have: r2d ¼ /2 r2d þ r2 : Hence, r2d ¼
r2 : 1 /2
ð5Þ
From (4) and (5), it can be seen that in order for the demand process to be stationary we should also have j/j < 1. 4. Determination of lead-time demand forecast In this research, demand forecast is conducted so as to minimize the expected mean squares of error. It is noted that 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 with minimum expected mean squares of error of Dt, then the minimum expected mean If D squares of error forecast for lead-time demand can be determined as bL ¼ D bt þ D b tþ1 þ þ D b tþL1 ¼ D t
L1 X
b tþi : D
ð7Þ
i¼0
b t can be determined by (Pindyck and Rubinfeld, 1998) For the AR(1) process, D b tþi ¼ E½Dtþi jDt1 ; Dt2 ; . . .: D b t can be derived and it has the following form: From the above expression, the explicit expression of D b tþi ¼ ld ð1 /iþ1 Þ þ /iþ1 Dt1 : D The detailed derivation of expression (8) is as follows: We have: Dtþi ¼ d þ /Dtþi1 þ etþi : If we express Dt+i1 in terms of Dt+i2: Dtþi1 ¼ d þ /Dtþi2 þ etþi1 : Then Dtþi ¼ dð1 þ /Þ þ /2 Dtþi2 þ etþi þ /etþi1 :
ð8Þ
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Applying the above procedure successively, we have: Dtþi ¼ dð1 þ / þ þ /i Þ þ /iþ1 Dt1 þ etþi þ /etþi1 þ þ /i et : Taking the expectation we have iþ1
b tþi ¼ E½Dtþi jDt1 ; Dt2 ; . . . ¼ dð1 þ / þ þ /i Þ þ /iþ1 Dt1 ¼ d 1 / þ /iþ1 Dt1 D 1/ ¼ ld ð1 /iþ1 Þ þ /iþ1 Dt1 : Proposition 1. The forecast of lead-time demand with minimum expected mean squares of error can be determined by L L b L ¼ ld L /ð1 / Þ þ /ð1 / Þ Dt1 : D ð9Þ t 1/ 1/ Proof. Replacing (8) into (7), we have: bL ¼ D t
L1 L1 L1 X X X fld ð1 /iþ1 Þ þ /iþ1 Dt1 g ¼ ld ð1 /iþ1 Þ þ Dt1 /iþ1 i¼0
i¼0
i¼0
/ð1 /L Þ /ð1 /L Þ Dt1 : ¼ ld L þ 1/ 1/
5. Determination of standard deviation of lead-time demand forecast error In this section, the expression of variance of lead-time demand forecast error will be developed. Proposition 2. The variance of lead-time demand forecast error does not depend on t and determined by 2
ð^ rLt Þ ¼
L r2d ð1 þ /Þ X 2 ð1 /i Þ : 1 / i¼1
ð10Þ
Proof. By definition we have: b L Þ: ð^ rLt Þ2 ¼ VARðDLt D t We also have: 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 etþL1 ¼ DLt D t
L1 X
etþi :
i¼0
From the proof of expression (8), it can be seen that i X
b tþi ¼ etþi þ /etþi1 þ þ /i et ¼ etþi ¼ Dtþi D
etþj /ij :
j¼o
Hence, bL ¼ DLt D t
L1 X i X i¼0
etþj /ij ¼
j¼0
L1 Li1 X X i¼0
etþi /j ¼
j¼0
L1 X
etþi
i¼0
1 /Li : 1/
Therefore, VARðDLt
b LÞ ¼ D t
2 L1 X 1 /Li i¼0
1/
r2 ¼
L1 X
r2 2
ð1 /Þ
i¼0
2
ð1 /Li Þ ¼
L r2d ð1 þ /Þ X 2 ð1 /i Þ : ð1 /Þ i¼1
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^Lt is needed for the determination of the order-up-to level St only. The measure of It should be noted that r bullwhip effect that will be developed later in this research is not affected by this value. h 6. Determination of variance of order quantity Proposition 3. The variance of order quantity of period t can be determined by the following expression: VARðqt Þ ¼
ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 2 rd : 1/
ð11Þ
Proof. We have: b L þ z^ b L þ z^ bL D b L Þ þ Dt1 : qt ¼ S t S t1 þ Dt1 ¼ ð D rLt Þ ð D rLt1 Þ þ Dt1 ¼ ð D t t1 t t1 From (9), it can be easily derived that qt ¼
1 /Lþ1 /ð1 /L Þ Dt2 : Dt1 1/ 1/
So, 2 2 1 /Lþ1 /2 ð1 /L Þ 1 /Lþ1 /ð1 /L Þ COVðDt1 ; Dt2 Þ: VARðqt Þ ¼ VARðDt1 Þ þ VARðD Þ 2 t2 1/ 1/ 1/ ð1 /Þ2 It is noted that VARðDt1 Þ ¼ VARðDt2 Þ ¼ r2d ; COVðDt1 ; Dt2 Þ ¼ /r2d : Therefore, 2 2 1 /Lþ1 /2 ð1 /L Þ 2 1 /Lþ1 /ð1 /L Þ 2 /rd VARðqt Þ ¼ r2d þ rd 2 2 1/ 1/ 1/ ð1 /Þ ¼
ð1 /Lþ1 Þ2 þ /2 ð1 /L Þ2 2/2 ð1 /Lþ1 Þð1 /L Þ ð1 /Þ
2
r2d ¼
ð1 þ /Þ 2/Lþ1 ð1 þ /Þ þ 2/2Lþ2 2 rd 1/
ð1 þ /Þð1 2/ Þ þ 2/2Lþ2 2 rd : 1/ Lþ1
¼
Knowing the variance of order quantity, the bullwhip effect can be evaluated. It is noted that the bullwhip effect is the phenomenon in which variance of demand information is amplified when moving upstream in supply chains. Therefore, it is reasonable to measure bullwhip effect by the ratio of the variance of order quantity that the supplier experiences and the variance of demand. This measure has been used by some researchers, e.g., Chen et al. (2000a,b), and also be adopted in the current research. Hence, from (11), the measurement of bullwhip effect can be determined as BðL; /Þ ¼
VARðqt Þ ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 : ¼ r2d 1/
ð12Þ
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7. Effects of autoregressive coefficient and lead time 7.1. The effect of autoregressive coefficient on bullwhip effect In this section, we first investigate the effect of autoregressive coefficient / on the measure of bullwhip effect. From (12), we can see that BðL; /Þ 6 1 ()
ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 6 1 () ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 6 1 / 1/
() 2/ð1 /L Þð1 /Lþ1 Þ 6 0 () / 6 0: The above relationship means that the bullwhip effect occurs only when there exists a positive autoregressive relationship in the demand process. In the current research, the focus is on the measure of bullwhip effect when it exists and this situation occurs only when the autoregressive coefficient is positive. When the autoregressive is negative, bullwhip effect does not exist or equivalently, the variance of order quantity experienced by supplier is smaller than the variance of demand. From managerial point of view, the reduction in variance of order quantity in comparison with variance of demand will lead to the fact that production planning phase at supplier becomes easier and help to stabilize the production process. However, it should be noted that the autoregressive coefficient is a characteristic of the demand process and is assumed constant in the current research. The target of this research is, therefore, to measure the bullwhip effect and to investigate the effect of autoregressive coefficient and lead time on bullwhip measure to see whether it is needed for supplier to pay attention to bullwhip effect in case the bullwhip effect is so high. Managerial approaches that can help to alleviate bullwhip effect can then be investigated and employed. In the remaining parts of this paper, therefore, we consider only the case when 0 < / < 1. In order to detect the effect of autoregressive coefficient, it is noted that the measure of bullwhip effect B(L, /) developed above can be expressed as BðL; /Þ ¼ 1 þ
L X
fi ðL; /Þ;
ð13Þ
i¼1
in which fi(L, /) = 2/i(1 /L+1) (i = 1, 2, . . . , L). In fact, the above expression can be developed as follows: BðL; /Þ ¼
ð1 þ /Þð1 2/Lþ1 Þ þ 2/2Lþ2 1 þ / 2/Lþ1 2/Lþ2 ð1 /L Þ ¼ 1/ 1/
ð1 /Þð1 þ 2/ þ 2/2 þ þ 2/L Þ 2/Lþ2 ð1 /Þð1 þ / þ /2 þ þ /L1 Þ 1/ L X ¼ ð1 þ 2/ þ 2/2 þ þ 2/L Þ 2/Lþ2 ð1 þ / þ /2 þ þ /L1 Þ ¼ 1 þ 2/i ð1 /Lþ1 Þ:
¼
i¼1
Proposition 4. In the interval 0 < / < 1, the functions fi(L, /); (i = 1, 2, . . . , L) are increasing and then decreasing. The maximum values are obtained at 1=Lþ1 i /i ¼ : Lþiþ1 Moreover, /i is an increasing function with respect to i (i = 1, 2, . . . , L). Proof. Taking the first derivative of fi(L, /) = 2/i(1 /L+1) with respect to /, we have: ofi ðL; /Þ ¼ 2½i/i1 ðL þ i þ 1Þ/Lþi ¼ 2/i1 ½i ðL þ i þ 1Þ/Lþ1 : o/
H.T. Luong / European Journal of Operational Research 180 (2007) 1086–1097 1=Lþ1
ðL;/Þ i Solving the equation: ofio/ ¼ 0, we have: /i ¼ ðLþiþ1 Þ
1093
ðL;/Þ . In the interval ð0; /i Þ, ofio/ > 0 and in the inter-
ðL;/Þ val ð/i ; 1Þ, ofio/ < 0. Hence, fi(L, /) is an increasing and then decreasing function. i < 1. Therefore, /i is an increasing function with respect to i (i = 1,2, . . . , L). It is also noted that Lþiþ1
h
In the next session, the behavior of B(L, /) with respect to / will be investigated. Proposition 5. The measure of bullwhip effect B(L, /) possesses the following properties: 1=Lþ1
1 (a) B(L, /) is an increasing function in the interval / 2 ð0; /1 Þ, in which /1 ¼ ðLþ2 Þ (b) B(L, /) is a concave function in the interval / 2 ð/1 ; 1Þ.
.
Proof (a) Due to the fact that fi(L, /); (i = 1, 2, . . . , L) are increasing functions in ð0; /i Þ and that /i is an increasing function with respect to i, (i = 1, 2, . . . , L), the first statement of Proposition 5 is easily to be confirmed. (b) In order to prove that B(L, /) is a concave function in the interval / 2 ð/1 ; 1Þ, we first prove that fi(L, /); (i = 1, 2, . . . , L) are concave in / 2 ð/1 ; 1Þ. Consider the second-order derivative of fi(L, /) with respect to /, we have: o2 fi ðL; /Þ ¼ 2½iði 1Þ/i2 ðL þ i þ 1ÞðL þ iÞ/Lþi1 ¼ 2/i2 ½iði 1Þ ðL þ i þ 1ÞðL þ iÞ/Lþ1 : o/2 We need to prove that: o2 fi ðL; /Þ < 0 when / > /1 ¼ o/2
1 Lþ2
1=Lþ1 :
Fig. 1. Effect of autoregressive coefficient on bullwhip measure.
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Note that: o2 fi ðL; /Þ iði 1Þ : < 0 () /Lþ1 > 2 ðL þ i þ 1ÞðL þ iÞ o/ The above inequality holds true due to the fact that: /Lþ1 >
1 i iði 1Þ > > : L þ 2 L þ i þ 1 ðL þ i þ 1ÞðL þ iÞ
It is noted that the sum of concave functions is also a concave function. Hence, B(L, /), which is the sum of fi(L, /); (i = 1, 2, . . . , L) and a constant, is a concave function in the interval / 2 ð/1 ; 1Þ. h From Proposition 5, it can be concluded that there always exist a unique value of /max 2 ð/1 ; 1Þ such that B(L, /) reaches its maximum value at a fixed value of lead time L. Furthermore, from the results in Proposition 4, it can be easily seen that /max 2 ð/1 ; /L Þ and hence, bisection method can be applied in this interval to find /max. The behavior of B(L, /) with respect to / and maximum values of B(L, /) at various values of lead time Lare illustrated in Fig. 1 and Table 1. It is interesting to note from Table 1 that when L increases, the maximum value of B(L, /) increases and /max also increases. In addition, at low values of /, there is not much difference in B(L, /). Table 1 Maximum value of bullwhip measure with respect to lead time Lead time L
/max
Maximum B(L, /)
1 2 3 4 5
0.577 0.683 0.747 0.789 0.820
1.770 2.566 3.371 4.180 4.990
Fig. 2a. Effect of lead time on bullwhip measure (/ = 0.3).
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7.2. The effect of lead time on bullwhip effect In this section, we will investigate the effect of lead time on the measure of bullwhip effect. Taking the first derivative of B(L, /) with respect to L we have oB 2ð1 þ /Þ/Lþ1 ln / þ 4/2Lþ2 ln / ¼ : oL 1/ With the condition j/j < 1, it is easy to prove that oB > 0. Therefore, B(L, /) is an increasing function with oL respect to L. It can be concluded that the longer the lead time, the larger the bullwhip effect in the supply chain. This finding is similar to the results of other past researches (e.g., Chen et al., 2000a,b; Xu et al., 2001) and Fig. 1 also illustrates this trend. From expression (12), we can also prove that BðL; /Þ < 1þ/ . This upper bound can be found by taking the 1/ limit of B(L, /) when L approaches infinity. At a fixed value of autoregressive coefficient /, the above upper bound value is useful in evaluating the influence of bullwhip effect when lead time increases. The functional expression of bullwhip measure represented by (12) also exhibits an interesting property stated in the following proposition: Proposition 6. Consider values of / in the interval (0, 1) (a) If / 6 1/3 then B(L, /) is a concave function with respect to L. (b) If / > 1/3 then B(L, /) is a convex and then concave function with respect to L. Proof. Consider the second derivative of B(L, /) with respect to L, we have o2 B 2ðln /Þ2 /Lþ1 ð4/Lþ1 / 1Þ : ¼ 1/ oL2
Fig. 2b. Effect of lead time on bullwhip measure (/ = 0.9).
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(a) If / 6 1/3: (4/L+1 / 1) 6 0 when L = 0. Furthermore, when L increases /L+1 will decrease. There2
fore, (4/L+1 / 1) 6 0 for all values L P 0. This leads to the fact that ooLB2 6 0 8L P 0 and hence, B(L, /) is a concave function with respect to L. (b) If / > 1/3: (4/L+1 / 1) > 0 when L = 0. However, when L increases /L+1 will decrease and starting 2
at L ¼ log/ ð1þ/ Þ 1, (4/L+1 / 1) will receive negative values. This means that ooLB2 will first have 4 positive values and then receive negative values when L increases. Therefore, B(L, /) is a convex and then concave function with respect to L. h The above properties are illustrated in Figs. 2a and 2b for the cases / = 0.3 and / = 0.9. 8. Conclusions In this research, we have investigated a simple supply chain that includes one retailer and one supplier in which the retailer employs base stock inventory policy. The bullwhip measure is developed for the first-order autoregressive demand process and some interesting properties of this measure have been found. Firstly, it is interesting to note that the bullwhip effect does not exist not only when there is no correlation or there is negative correlation in the demand process but also when there exists a perfect positive correlation in the demand process (i.e., / = 1). This is due to the fact that B(L, /) 6 1 when / 6 0 or / = 1. In the range / 2 (0, 1), when / increases, the bullwhip effect will first increases, reaches maximum value, and then decreases as shown in Fig. 1. From Fig. 1, it is also noted that when the lead time increases, the value of / at which the bullwhip effect reaches its maximum value also increases and approaches one. Another interested finding of this research is that there exists an upper bound for the measure of bullwhip effect when lead time increases. The value of this upper bound depends on the autoregressive coefficient /. This upper bound value is useful in evaluating the influence of bullwhip effect and it can help management to allocate effort in order to alleviate the impact of bullwhip effect in supply chain when lead time increases. Lastly, it should be noted that the behavior of the bullwhip measure can have two patterns with respect to lead time. This also depends on the autoregressive coefficient /. It is expected that the findings of this research can serve as a good starting point for investigating the impact of bullwhip effect in more complicated supply chains where suppliers provide goods for more than one retailer. Acknowledgement The author is grateful to two distinguished reviewers for their valuable comments and suggestions for improving the quality and the presentation of this paper. References Berry, D., Naim, M.M., Towill, D.R., 1995. Business process reengineering an electronics products supply chain. In: Proceedings of IEE Science, Measurement, and Technology, pp. 395–403. Blinder, A.S., 1982. Inventories and stucky prices. American Economic Review 72, 334–349. Blinder, A.S., 1986. Can the production smoothing model of inventory behavior be safe? Quarterly Journal of Economics 101 431–454. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000a. Quantifying the bullwhip effect in a simple supply chain. Management Science 46 (3), 436–443. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000b. The impact of exponential smoothing forecasts on the bullwhip effect. Naval Research Logistics 47 (4), 269–286. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R., 2003. Measuring and avoiding the bullwhip effect: A control theoretical approach. European Journal of Operational Research 147 (3), 567–590. Dejonckheere, J., Disney, S.M., Lambrecht, M.R., Towill, D.R., 2004. The impact of information enrichment on the bullwhip effect in supply chains: A control engineering perspective. European Journal of Operational Research 153 (3), 727–750. Disney, S.M., Towill, D.R., 2003. On the bullwhip and inventory variance produced by an ordering policy. Omega 31, 157–167. Forrester, J., 1958. Industrial dynamics: A major breakthrough for decision makers. Harvard Business Review 36, 37–66. Graves, S.C., 1999. A single-item inventory model for a non-stationary demand process. Manufacturing and Service Operations Management 1 (1), 50–61. Heyman, D., Sobel, M., 1984. Stochastic Models in Operations Research. McGraw-Hill.
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