measure of bullwhip effect - a dual sourcing model

396 Int. J. Operational Research, Vol. 20, No. 4, 2014

Copyright © 2014 Inderscience Enterprises Ltd.

Measure of bullwhip effect – a dual sourcing model

Kittiwat Sirikasemsuk and Huynh Trung Luong* Industrial Systems Engineering Programme, School of Engineering and Technology, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand Fax: +6625245697 E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: The bullwhip effect in which the demand in the upstream section of a supply chain is distorted gives rise to an unfavourable phenomenon. Most of the existing research studies have investigated the bullwhip effect under a single sourcing environment. However, in this research the behaviour of the bullwhip effect under a dual sourcing environment for a first-order autoregressive [AR(1)] demand process using an analytical approach is examined. The conditions under which the bullwhip effect is reduced in the dual sourcing environment in comparison to that in the single sourcing environment are subsequently derived.

Keywords: supply chain; bullwhip effect; autoregressive model; dual sourcing model; base stock policy.

Reference to this paper should be made as follows: Sirikasemsuk, K. and Luong, H.T. (2014) ‘Measure of bullwhip effect – a dual sourcing model’, Int. J. Operational Research, Vol. 20, No. 4, pp.396–426.

Biographical notes: Kittiwat Sirikasemsuk is a doctoral candidate in Industrial and Manufacturing Engineering, School of Engineering and Technology, Asian Institute of Technology, Thailand. He is on study leave from King Mongkut’s Institute of Technology Ladkrabang, Thailand. He received his Master of Engineering degree from Chulalongkorn University, Thailand. His research interests include supply chain management and statistics modelling.

Huynh Trung Luong is an Associate Professor in Industrial and Manufacturing Engineering, School of Engineering and Technology, Asian Institute of Technology, Thailand. He received his Doctor of Engineering (DEng) from Industrial Systems Engineering, Asian Institute of Technology, Thailand in 2000. His teaching and research interests include establishment of emergency inventory policies, inventory policies for perishable products, supply chain design, measures of bullwhip effect in supply chains, availability-based and reliability-based maintenance. He has published articles in various peer reviewed international journals.

Measure of bullwhip effect 397

1 Introduction

The bullwhip effect, also known as the whiplash effect, causes the amplification of variance of demand when moving upstream in the supply chain. The works of Lee et al. (1997a, 1997b) mentioned the negative ramifications of variance amplification on businesses, examples of which are excessive inventory investment, poor customer service, and lost revenue. Even with its first recognition over half a century ago, the bullwhip effect has still gained a continuous level of attention.

The first discussion of this phenomenon was found in the work of Forrester (1958), which has attracted the attention of management and research scientists. A paper by Sterman (1989), who developed the well-known ‘Beer Game’, reiterated the occurrence of the bullwhip effect. In the works of Lee et al. (1997a, 1997b), five crucial sources that lead to the occurrence of the bullwhip effect, i.e., demand forecasts, supply shortages, lead times, batch ordering, and price variations, have been discussed.

Chen et al. (2000a, 2000b) examined the effects of the simple moving average (MA) and the simple exponentially weighted moving average (EWMA) forecasting methods for the case of the first-order autoregressive, i.e., AR(1), demand process on the bullwhip effect. In their research, it was proved that the larger the smoothing parameter, the larger the magnitude of the bullwhip effect. Based upon the empirical evidence, it can also be concluded that the bullwhip effect is unavoidable even with the employment of the MA and EWMA techniques. In fact, a number of researchers, such as Alwan et al. (2003), Zhang (2004), Luong (2007), Luong and Phien (2007), Liu and Wang (2007), Duc et al. (2008a, 2008b), Agrawal et al. (2009), Pati et al. (2010) and Cho and Lee (2011), used the minimum mean square error (MMSE) forecasting method in their research studies on the bullwhip effect. The main reason for using the MMSE forecasting technique is that this technique could avoid the occurrence of the stratification error and hence reduce the bullwhip effect. However, in the case of non-stationary demand process, Graves (1999) employed the EWMA forecasting method for an ARIMA(0,1,1) demand process and confirmed that the method provides the MMSE forecast in this situation. In addition, Zhang (2004) confirmed that if the demand shifts over time (not stable), the MA and EWMA approaches are better than the MMSE approach in terms of the amplification of demand variability. More recently, Hussain et al. (2012) reported that the EWMA forecasting method is better than the MMSE forecasting method in terms of the inventory variance for the stationary demand process.

From the existing research studies on the bullwhip effect in the case of AR(1) demand process, e.g., the works of Alwan et al. (2003), Liu and Wang (2007) and Luong (2007), it could be concluded that under the MMSE forecasting scheme the bullwhip effect does not exist if the first-order autocorrelation coefficient is negative or zero. In fact, the AR(1) incoming demand process was used by several authors as the fundamental demand process both to examine the effect of information sharing on the bullwhip effect, e.g., Lee et al. (1997a, 1997b), Xu et al. (2001), Kim et al. (2006) and Agrawal et al. (2009); and to investigate the impact of stochastic lead times on the bullwhip effect, e.g., Kim et al. (2006) and Duc et al. (2008b, 2010b). Besides, in recent years the underlying AR(1) demand process has been constantly applied by several authors, examples of whom are Pereira et al. (2009) who demonstrated the influence of adjustment capability of production and responsiveness, Pati et al. (2010) who developed the bullwhip effect measure for a closed loop supply chain, Sodhi and Tang

398 K. Sirikasemsuk and H.T. Luong

(2011) who examined the incremental bullwhip effect due to operational deviations, and Nepal et al. (2012) who analysed the bullwhip effect with consideration for a product life-cycle.

Various research studies on measuring and analysing the bullwhip effect when the incoming demand process was not the AR(1) process were also conducted, examples of which were a high order autoregressive, AR(p), demand process by Luong and Phien (2007); a mixed first-order autoregressive-moving average, ARMA(1,1), demand process by Gaalman and Disney (2006) and Duc et al. (2008a); a mixed second-order autoregressive-moving average, ARMA(2,2), demand process by Gaalman and Disney (2009); a seasonal autoregressive-moving average, SARMA, demand process by Cho and Lee (2011); and a first-order bivariate vector autoregression, VAR(1), demand process by Chaharsooghi and Sadeghi (2008) and Zhang and Burke (2011). Moreover, several authors, e.g., Bayraktar et al. (2008), Wright and Yuan (2008), Coppini et al. (2010) and Ciancimino et al. (2012) examined the bullwhip effect by generating customer demands with demand functions based on time through a simulation approach. In addition, Wang et al. (2010) also derived a decision making framework to select a forecast-updating method based on three series demand models, i.e., AR(1), MA(1), and ARMA(1,1).

In previous research studies, various chain structures were considered. However, a simple two-stage supply chain with one supplier and one retailer was widely used by researchers as the fundamental structure, such as Chen et al. (2000b), Luong (2007), Luong and Phien (2007), Liu and Wang (2007), Bayraktar et al. (2008), Duc et al. (2008a, 2008b), Wang et al. (2010), Cho and Lee (2011), Hussain et al. (2012), etc. The case of more than one retailer was also attractive to many scholars, but the focus was still on a two-stage supply chain with only one supplier. Readers are advised to refer to the works of Chen et al. (2000a), Zhao et al. (2002), Raghunathan (2003), Sucky (2009), Duc et al. (2010b) and Zhang and Burke (2011) for more details. Recently, Duc et al. (2010a) considered the case of a three-stage supply chain with one supplier, one third-party warehouse, and two retailers; and proved that the existence of the third-party warehouse does not affect the bullwhip effect but saves the inventory cost. In fact, there exist some other research studies that examined the bullwhip effect in a three-stage or more than three-stage supply chain, such as the works of Xu et al. (2001), Wright and Yuan (2008), Agrawal et al. (2009), Pereira et al. (2009), Coppini et al. (2010), Ciancimino et al. (2012) and Nepal et al. (2012); and in a closed loop supply chain, such as the works of Pati et al. (2010) and Adenso-Diaz et al. (2012). However, it is important to mention that all the aforementioned research studies were conducted under the single sourcing environment.

It is also interesting to note that Helbing and Lämmer (2005) and Ouyang and Li (2010) examined the bullwhip effect for general network structures and derived some general conclusions. Helbing and Lämmer (2005) applied the network theory to investigate the stability and dynamic behaviour of supply networks for different topologies, and found that most of the supply networks display damped oscillations. Furthermore, these oscillations tend to increase in the networks with damped oscillators. Ouyang and Li (2010), on the other hand, utilised the control theory to analyse the bullwhip effect in supply chain networks operated with linear and time-invariant inventory policies, and confirmed that a number of factors, e.g., network topologies, customer demand process, and supplier operating strategies, affect the bullwhip effect. However, the above research studies examined the bullwhip effect from the system control viewpoint which requires the existence of an unbiased and unique decision


maker. This requirement might not be satisfied in many practical supply chains in which the members of the supply chains do not belong to the same organisation.

In many real supply networks, the decision to place an order upstream is not centrally decided, but it is under the control of downstream members. In such a situation, downstream members might decide to split their orders to certain upstream members for supply risk prevention purposes. This order splitting decision may affect the bullwhip effect. At present, no research articles have analytically examined the bullwhip effect in the supply chain in the case of dual sourcing environment. It should be noted that although there were many publications related to the use of multiple suppliers and order splitting, these research studies have merely investigated the supply chain from a pure inventory perspective for the downstream members. For example, Kelle and Silver (1990) investigated the effect of order splitting on the variability of lead-time demand for a Weibull distributed lead time; Ramasesh et al. (1991) derived a mathematical model for the dual sourcing environment to minimise the expected total cost for uniform and exponentially distributed lead times with a constant demand rate; and Chiang and Benton (1994) compared the cost functions between the single and dual sourcing models in the case when a shifted exponential lead time and a normally distributed demand were considered. These works focused on the reduction of the safety stock in the dual or multiple sourcing environment for a continuous review inventory system. In another study, Sedarage et al. (1999) developed a mathematical model for a multiple-supplier inventory system in which both lead time and demand were stochastic. Later, Chiang (2001) examined the consequence of multiple deliveries in the single sourcing environment on the cycle stock reduction for a periodic review inventory system. Recently, a mixed integer non-linear mathematical model for which multiple suppliers with a discount policy were considered was presented by Kamali et al. (2011). According to Minner (2003) and Thomas and Tyworth (2006), who reviewed many articles related to multiple-supplier inventory models, the major advantage of using multiple suppliers is the decrease in the inventory cost due to the reduction in the average on-hand inventory.

In the supply chain management, for components with high supply risk or financial risk, besides using option contracts or seeking strategic partnerships with suppliers, the dual and multiple sourcing policies have also been recommended as a way to ensure supply (Simchi-Levi et al., 2008). However, the use of dual/multiple sourcing policies might cause an adverse impact on the upstream members in the supply network if the bullwhip effect increases in comparison to the use of single sourcing policies. The increase in the bullwhip effect leads to the rise in the safety stock in the supply network. In this research, the aforesaid issue will be tackled. The main objective is to establish the conditions under which the bullwhip effect will be reduced in the dual sourcing model in comparison to the single sourcing model. To achieve the above target, a measure of the bullwhip effect under the dual sourcing supply chain structure will be first developed and then compared with the bullwhip measure in the single sourcing model. Two supply chain models will be examined herein:

1 the dual sourcing model which is a three-stage supply chain consisting of a supplier, two distributors, and a retailer as shown in Figure 1

2 the single sourcing model which is a three-stage supply chain consisting of a supplier, a distributor, and a retailer as shown in Figure 2.

The notations used in this research and their definitions are provided in Table 1.


Figure 1 A two-distributor model (dual sourcing environment) (see online version for colours)

Figure 2 A single-distributor model (single sourcing environment)

Table 1 Notations used in the research

Notation Definition

i Index of distributors (i = 1 or 2)

Dt Demand of the retailer in time period t

qt Total order quantity placed by the retailer at the beginning of period t

qsi,t Order quantity placed by the retailer to distributor i in time period t

rt Total order quantity received by the supplier at the beginning of period t

rsi,t Order quantity placed by distributor i to the supplier in time period t

Li Order lead time between the retailer and distributor i

li Order lead time between the supplier and distributor i

St Overall order-up-to level of the retailer at the beginning of period t

Ss1,t Intermediate order-up-to level of the retailer at the beginning of period t which is used to decide on the order quantity to distributor 1 only

ˆ iLtD Demand forecast over Li periods

ˆ iLtσ Standard deviation of lead-time demand forecast error over Li periods

Z The normal Z score determined by the desired service level

Ysi,t Order-up-to level of distributor i at the beginning of period t

,ˆ ilsi tq Demand forecast over li periods for distributor i

,ˆ ilsi tσ Standard deviation of lead-time demand forecast error over li periods for

distributor i

tr′ Order quantity received by the supplier at the beginning of period t when only one distributor is used

LS Order lead time between the retailer and the distributor for the sole sourcing environment

lS Order lead time between the supplier and the distributor for the sole sourcing environment


Similar to various research studies in the past, the bullwhip measure in this research is defined as VAR(rt)/VAR(Dt) in the case of dual sourcing environment, in which rt will be calculated as the sum of the order quantities placed by distributors 1 and 2. For the single sourcing environment, the bullwhip measure is defined as ( ) / ( ).t tVAR r VAR D′

2 Dual sourcing supply chain model

It is assumed in this research that the MMSE forecasting method is employed for all members of the supply chain where the incoming demand process observed by the retailer is the AR(1) demand process. For the time series demand model, which is the demand process considered in this research, the MMSE forecasting technique that minimises the expected mean square forecast error helps predict the future demand with as little error as possible and hence decreases the bullwhip effect. In addition, a base-stock inventory policy is utilised by each member of the supply chain to help control inventory mainly because of its common use and ability to maximise the profit per unit time (Beckmann, 1961; Hadley and Whitin, 1963). It is also noted that the benefits of the base stock policy have been reported by several authors, examples of whom are Vassian (1955) and Hosoda and Disney (2005) who indicated in their works that the base stock policy could help minimise both the variance of the net inventory level during a long period and the variance of the forecast error over lead time. Another study by Johnson and Thompson (1975) showed that this policy is optimal for general autoregressive demand processes. This research incorporates the existence of order lead times, and it is assumed that all order lead times are deterministic. Furthermore, without loss of generality, it is assumed that L2 > L1.

Under the base stock policy, the inventory function at the retailer is presented in Figure 3 and can be detailed as follows.

Figure 3 The inventory function under the order splitting policy with two distributors


At the beginning of period t, the retailer will place the order qt upstream to raise the inventory position to level St from the current inventory level (St–1 – Dt–1). This order quantity qt will be split into qs1,t and qs2,t for distributors 1 and 2, respectively. The order amount qs1,t will arrive after lead time L1 while the order amount qs2,t after lead time L2.

3 Upstream demand process in the dual sourcing model

Prior to determining the demand variability amplification at the supplier stage, we will first determine the incoming demand processes at distributors, i.e., the order quantities placed by the retailer. Under the dual sourcing model, the demand processes at distributors cannot be instantaneously determined due to their dependence upon the demand process at the retailer.

To derive the demand process at distributors, we will consider a part of the supply chain structure between the retailer and two distributors. For an AR(1) demand process,

1 ,t t tD Dδ φ ε−= + + (1)

where δ is the constant of the autoregressive model, φ is the first-order autocorrelation coefficient, and εt’s (t =1, 2, ...) are independent and identically distributed random variables which have a normal probability distribution with mean 0 and variance σ2. For the first-order autoregressive process to be stationary, it is necessary that |φ| < 1. Under this condition, μd = δ/(1 – φ) and 2 2 2/ (1 )dσ σ φ= − in which μd is the mean of the autoregressive process which is used to describe the demand process at the retailer, and

2dσ is the variance of demand at the retailer.

If we denote Ss1,t to be the inventory position or intermediate order-up-to level at the beginning of period t taken into consideration only the order quantity to distributor 1, the following relationships hold:

1, 2, ,t s t s tq q q= + (2)

1 1,t t t tq S S D− −= − + (3)

1, 1, 1 1,s t s t t tq S S D− −= − + (4)

It can be seen from Figure 3 that the whole order amount qt will fully arrive after lead time L2. Therefore, St must cover the demand during lead time L2 and hence can be determined by 2 2ˆ ˆ .L L

t t tS D Zσ= + Similarly, the intermediate inventory position Ss1,t must cover the demand during lead time L1 and hence can be written as

1 11,

ˆ ˆ .L Ls t t tS D Zσ= +


With the employment of the MMSE forecasting scheme, qt and qs1,t as defined by equations (3) and (4) can be determined as shown later. However, it should be noted that qs2,t cannot be directly determined. Instead, it is determined through the relationship in equation (2) as

2, 1,s t t s tq q q= − (5)

From the above analysis, it should be noted that distributor 2 is not a ‘back-up’ for distributor 1. The order quantities to the two distributors depend upon each other. From equation (4), qs1,t depends on St–1 which is affected by the orders to both distributors in the past. From equation (5), qs2,t is clearly affected by qs1,t.

3.1 Determination of the total order quantity (qt)

According to Luong (2007), who studied the behaviour of the bullwhip effect in a two-stage supply chain, qt is determined by

( )22 1

1 211

1 1

LL

t t tq D Dφ φφ

φ φ

+

− −−−

= −− −

(6)

In addition, by using the same procedure as in the research of Alwan et al. (2003), it can be derived that qt will follow the ARMA(1,1) model as follows:

1 2 1t t t tq q a aδ φ θ− − −= + − + (7)

where 2

2 1(1 ) ,

1

L

Lφ φθ

φ +

−=

− and

2 1

1 11 .

1

L

t ta φ εφ

+

− −−

=−

It is noted that the order process expressed by equation (7) is slightly different from the work of Alwan et al. (2003). In this current research, the error component of qt comprises at–2 and at–1 because it is assumed that the order quantity is placed by the retailer at the beginning of period t, while in the work of Alwan et al. (2003) the order quantity is placed at the end of period t.

3.2 Determination of the order quantity placed to distributor 1 (qs1,t)

In this section, the mathematical expression of qs1,t is derived. The detailed results are presented in the following proposition.

Proposition 1: The explicit expression of qs1,t has the following form:

( )21 1

1, 1 211

1 1

LL

s t t t Zq D D Kφ φφ

φ φ

+

− −−−

= − +− −

(8)

where ( ) ( ) ( )

1 2

1 21 2

2

(1 ).ˆ ˆ

(1 )

L LL L

Z

L LK Z

φ φ φ φδ σ σφ

− +− −= + −

−


Also, qs1,t can be expressed in the form of the ARMA(1,1) model, i.e.,

1, 1 1, 1 1 1, 2 1, 1,s t s s t s s t s tq q a aδ φ θ− − −= + − + (9)

in which 2 1 2

11 11(1 ) ( ), 1 (1 ) ,

11

L L L

s s ZL Kφ φ φ φ φθ δ δ φφφ +

⎛ ⎞− −= = − + −⎜ ⎟⎜ ⎟−− ⎝ ⎠

and 1 1

1, 1 11 .

1

L

s t ta φ εφ

+

− −−

=−

Proof: See Appendix A.

3.3 Determination of the order quantity placed to distributor 2 (qs2,t)

In this section, the mathematical expression of qs2,t is derived. The detailed results are presented in the following proposition.

Proposition 2: The explicit expression of qs2,t can be determined by

( )1 2

2, 11

L L

s t t Zq D Kφ φ φ

φ −

−= −

− (10)

and it can be expressed in the form of the AR(1) model as follows:

2, 2 2, 1 2, 1s t s s t s tq q aδ φ − −= + + (11)

in which 1 2

2( ) (1 )

1

L L

s ZKφ φ φδ δ φφ−

= − −−

and 1 2

2, 1 1( ) .

1

L L

s t ta φ φ φ εφ− −−

=−

Proof: Based on equation (5), qs2,t can be determined by subtracting equation (8) from equation (6) as follows:

( )1 2

2, 1, 1 .1

L L

s t t s t t Zq q q D Kφ φ φ

φ −

−= − = −

−

On the other hand, qs2,t can also be computed by subtracting equation (9) from equation (7) as follows:

( ) ( ) ( )( )

2, 1,

1 1, 1 2 1 1, 21

1 1, 1 .

s t t s t

t s t t s s ts

t s t

q q q

q q a a

a a

θ θδ δ φ − − − −

− −

= −

− −= − + −

−+

It is noted that θat–2 – θs1as1,t–2 = 0. Then, if we denote 2 1s sδ δ δ= − 1 2( ) (1 )

1

L L

ZKφ φ φ δ φφ−

= − −−

and 1 2

2, 1 1 1, 1 1 2,( ) ,

1

L L

s t t s t t s ta a a qφ φ φ εφ− − − −−

= − =−

will be

described as follows:

2, 2 2, 1 2, 1.s t s s t s tq q aδ φ − −= + +


4 Measure of the bullwhip effect for the dual sourcing model

The total order quantity received by the supplier at the beginning of period t can be calculated as the sum of the order quantities placed by distributors 1 and 2:

1, 2,t s t s tr r r= + (12)

4.1 Determination of the order quantity placed by distributor 1 (rs1,t)

Using the order-up-to inventory policy, rs1,t can be determined as

1, 1, 1, 1 1, 1s t s t s t s tr Y Y q− −= − + (13)

where Ys1,t will be computed as 1 11, 1, 1,ˆ ˆ .l l

s t s t s tY q Zσ= + By following the same procedures as in the work of Duc et al. (2008a) for the

ARMA(1,1) demand process, rs1,t can be determined as

( )

( )( )

11

1 2 1 2 1 1

1 1 1 2 1 1 1 2

2 1

1

1, 1, 1 1, 2

1 1 1

12

1 1 1 1

22

32

111 1

1 (1 )

1 (1 )

1 1 .

(1 )

ll

s t s t s t

L L l L l L

t

L l l L l L l L

t

L l

t

r q qφ φφ

φ φ

φ φ φ φ φ εφ

φ φ φ φ φ εφ

φ φ φε

φ

+

− −

+ + + + +

−

+ + + + + + +

−

−

−−= −

− −

− − + − ++

−

− − − + +−

−

− −+

−

(14)

It should be noted that with the minimum expected mean squares of error forecasting technique, 1

1,ˆ ls tq can be determined by

( ) ( )

( ) ( )( )

1 11

1 1

11, 1 1, 1

11 1

1, 2 1, 1

1 1ˆ1 1 1

1 1 1 .

1 1

l ll ss t s t

l ls s

s t s t

q l q

a a

φ φφ φδφ φ φ

θ φ θφ φφ φ

−

−

− −

⎛ ⎞− −⎜ ⎟= − +⎜ ⎟− − −⎝ ⎠

⎛ ⎞−− −⎜ ⎟− + +⎜ ⎟− −⎝ ⎠

4.2 Determination of the order quantity placed by distributor 2 (rs2,t)

Under the order-up-to inventory policy, rs2,t can be determined as

2, 2, 2, 1 2, 1s t s t s t s tr Y Y q− −= − + (15)

in which Ys2,t will be calculated by 2 22, 2, 2,ˆ ˆ .l l

s t s t s tY q Zσ= +


The determination of rs2,t can be conducted similar to the case of rs1,t as in the work of Luong (2007) for the AR(1) demand process. Hence, rs2,t can be determined as

( )22 1 2 2 1 2 2

1 2 2 1 2 2

1 1 1 1 1

2, 2, 1 2, 2 12

1 1 1 1

22

111 1 (1 )

.(1 )

ll L L l L l L

s t s t s t t

L L l L l L

t

r q qφ φφ φ φ φ φ ε

φ φ φ

φ φ φ φ εφ

+ + + + + + +

− − −

+ + + + + +

−

−− − − += − +

− − −

− − +−

−

(16)

It is also noted that with the minimum expected mean squares of error forecasting method, 2

2,ˆ ls tq can be derived as

( ) ( )2 2 22 22, 2 2, 1 2, 1

1 1 1ˆ .1 1 1 1

l l ll ss t s t s tq l q a

φ φφ φδ φφ φ φ φ− −

⎛ ⎞− − −⎜ ⎟= − + +⎜ ⎟− − − −⎝ ⎠

4.3 Total order quantity received by the supplier (rt)

Proposition 3: The total order quantity received by the supplier can be determined as

( )

( ) ( )( )

11 2

2 2 1

2 1 1 1 1 2 1 2 2 1 2 2

1 1 1 2 2 1 2 2

1 1

1, 1 2, 1 1, 2

2, 2 32

1 1 1 1 1 1

22

1 1 1

11 11 1 1

1 1 1

1 (1 )

1(1 )

1(

ll l

t s t s t s t

l L l

s t t

L l l L l L l L l L l L

t

l L l L l L l L

r q q q

q

φ φφ φφ φ φ

φ φ φ φ φε

φ φ

φ φ φ φ φ φ φ εφ

φ φ φ φ φ

+ +

− − −

− −

+ + + + + + + + + + +

−

+ + + + + + +

−− −= + −

− − −

− − −− +

− −

− − + − + − +−

−

− + − − ++ 12 .

1 ) tεφ −−

(17)

Proof: From equation (12), rt can be calculated as the sum of the order quantities placed by distributors 1 and 2: rt = rs1,t + rs2,t. Consequently, when substituting equations (14) and (16) in equation (12), rt will be obtained as presented by equation (17).

4.4 Variance of total order quantity received by the supplier

Proposition 4: For the dual sourcing environment, the variance of total order quantity received by the supplier at period t is independent of t and can be determined by the following expression:

( )( )

( )

1 1 2 1 1 2 2 2

1 1 2 1 1 2 2 2

1 1 123 2

1 1 1

21 *(1 )

1

l L l L l L l L

t dl L l L l L l L

VAR rφ φ φ φ φ φ

φ σφ φ φ φ φ

+ + + + + + +

+ + + + + + +

⎧ ⎫⎡+ − + − − +⎪ ⎪⎣⎪ ⎪−= ⎨ ⎬⎪ ⎪⎤− + − − +⎪ ⎪⎦⎩ ⎭

(18)

Proof: See Appendix B.


From equation (18), the bullwhip measure, BWr(φ, l1, l2, L1, L2) or BWr, which is defined as the ratio of the variance of order quantity that the supplier experiences to the variance of demand observed by the retailer can be easily determined as follows:

( )

( )

( )

1 1 2 1 1 2 2 2

1 1 2 1 1 2 2 2

r 1 2 1 2

1 1 123

1 1 1

, , , ,21 *

(1 )

1 .

l L l L l L l L

l L l L l L l L

BW l l L Lϕ

φ φ φ φ φ φφ

φ φ φ φ φ

+ + + + + + +

+ + + + + + +

⎡= + − + − − +⎣−

⎤− + − − + ⎦

(19)

From equation (19), we can see that the bullwhip measure depends on the autoregressive coefficient of the demand process and all of the lead-time parameters, i.e., l1, l2, L1, and L2.

From the general expression of the bullwhip measure developed in equation (19), there are certain cases that warrant emphasis.

• Case 1: when l1 = l2 = l′, the bullwhip effect is not influenced by the lead time between the retailer and distributor 1 (L1) and can be determined as

( )( )( )2 2

r 2

2 1, , 1

1

l L l L

BW l Lφ φ φ

φφ

′ ′+ +− −′ = +

− (20)

• Case 2: when L2 = L1 + 1 = L′, the bullwhip effect is independent of the replenishment lead time from the supplier to distributor 1 (l1) and can be determined as

( )( )( )2 2

r 2

2 1, , 1

1

l L l L

BW l Lφ φ φ

φφ

′ ′+ +− −′ = +

− (21)

• Case 3: when l1 = l2 + 1 = l″, the bullwhip effect does not depend on the replenishment lead time between the retailer and distributor 2 (L2) and can be determined as

( )( )( )1 1

r 1

2 1, , 1

1

l L l L

BW l Lφ φ φ

φφ

′′ ′′+ +− −′′ = +

− (22)

5 Effect of autoregressive coefficient

In this section, the effect of autoregressive coefficient on the existence of the bullwhip effect will be examined. It is noted that all research works in the past dealing with the AR(1) demand process in the single sourcing environment have confirmed that if –1 < φ ≤ 0, the bullwhip effect does not exist; and if 0 < φ < 1, the bullwhip effect always exists. In the dual sourcing environment examined in this research, it is also found that the bullwhip effect does not exist if –1 < φ ≤ 0; however, when 0 < φ < 1, the bullwhip effect does not always exist. Our findings related to the effect of autoregressive coefficient are summarised in the following proposition.


Proposition 5: In the dual sourcing environment

a When –1 < φ ≤ 0, the bullwhip effect does not exist.

b When 0 < φ < 1, the bullwhip effect always exists if l1 ≤ l2 + 1 or L2 = L1 + 1. However, if l1 > l2 + 1 and L2 > L1 + 1, the bullwhip effect will not exist if

( )1 1 2 1 1 2 2 21 1 11 0l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + ≥

and

( )1 1 2 1 1 2 2 21 1 12 0.l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + ≤ .

Proof: See Appendix C.

To illustrate for the exception in part (b) of the above proposition, let’s consider an example where L1 = 1, L2 = 4, l1 = 7, and l2 = 2. We can easily check, by using equation (19), that the bullwhip effect does not exist when φ takes on a value in the range [0.8192, 0.8813], due to 1 1 2 1 1 2 2 21 1 11 0l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + ≥ and

1 1 2 1 1 2 2 21 1 12 0.l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + ≤ For more details on the behaviour of the bullwhip measure, various numerical

experiments at different values of φ and the lead times have been conducted, and the results are presented in Figure 4 for the case of l1 ≤ l2 + 1, Figure 5 for the case of L2 = L1 + 1, and Figures 6 and 7 for the case of l1 > l2 + 1 and L2 > L1 + 1 simultaneously.

Figure 4 Behaviour of BWr with respect to φ in the case of l1 ≤ l2 + 1


Figure 5 Behaviour of BWr with respect to φ in the case of L2 = L1 + 1

Figure 6 Behaviour of BWr in the case of l1 > l2 + 1 and L2 > L1 + 1 when there exists no positive value of φ such that the bullwhip effect does not exist


Figure 7 Behaviour of BWr in the case of l1 > l2 + 1 and L2 > L1 + 1 when there is a range of positive values of φ such that the bullwhip effect does not exist

In the case when l1 ≤ l2 + 1 or L2 = L1 + 1, it can be seen from Figures 4 and 5 that the bullwhip effect always exists when 0 < φ < 1. Furthermore, it can be observed that when φ increases from 0, the bullwhip effect first increases and then reaches a maximum. Afterward, the bullwhip effect decreases with respect to φ and reaches one when φ approaches 1.

In the case when l1 > l2 + 1 and L2 > L1 + 1, there are two patterns for the bullwhip effect as shown in Figures 6 and 7. First, if the two inequalities

1 1 2 1 1 2 2 21 1 1(1 ) 0l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + ≥ and 1 1 2 1 1 21 12( l L l L l Lφ φ φ φ φ+ + + + +− + − − 2 2 1) 0l Lφ + ++ ≤ cannot occur simultaneously within the range 0 < φ < 1, the behaviour of

the bullwhip measure will be the same as in the case of l1 ≤ l2 + 1 or L2 = L1 + 1 as shown in Figure 6. Otherwise, there will be a range of positive values of φ that cause the bullwhip measure to be less than one as shown in Figure 7, that is, the bullwhip effect does not exist (see examples in Table 2). It should be noted that the ranges presented in Table 2 are difficult to determine analytically, instead, they are derived from a numerical method (i.e., a trial and error method).

Table 2 Examples of the range of values of φ at which the bullwhip effect does not exist

L1 L2 l1 l2 Range

1 4 7 2 [0.8191725, 0.8812714]

2 12 6 3 [0.8540635, 0.8820383]

5 10 8 4 [0.9670248, 0.9800853]


6 Comparison between the bullwhip effects measured at the supplier under the single sourcing model and the dual sourcing model

For comparison purposes, the single sourcing three-stage supply chain which consists of a supplier, a distributor, and a retailer as shown in Figure 2 is considered in this section. Under the same conditions, i.e., the MMSE forecasting method and the base stock policy are employed and the incoming demand process is the AR(1) process, tr′ can be derived, according to the works of Alwan et al. (2003) and Duc et al. (2008a), as follows:

( )

( )( ) ( ) ( ) ( )( )

1

1 2 1

2 32 2

11 11 1 1

1 1 1 1 1 1,

(1 ) (1 )

SS S S

S S S S S S

ll l L

t t t t

L l l L L l

t t

r q qφ φφ φ ε

φ φ φ

φ φ φ φ φ φ φ φε ε

φ φ

+ +

− − −

+

− −

−− −′ = − +− − −

− − + − − − −− +

− −

(23)

Then, the bullwhip effect measured at the supplier, denoted by r ( , , ),S SBW l Lφ′ can be determined as

( )( )( )2

r 2

2 1, , 1

1

S S S Sl L l Lr

S Sd

BW l Lφ φ φσ

φφσ

+ +′

′

− −= = +

− (24)

It should be noted that past research studies have confirmed that the bullwhip effect measured by equation (24) will not exist when the autocorrelation coefficient is non-positive. In this research, the same conclusion is achieved (see Proposition 5). As such, in this section the comparison is made merely for the case of positive autocorrelation coefficient. The detailed comparison results are presented in Table 3. From Table 3, it can be concluded the bullwhip effect under the dual sourcing environment can be either lower than, higher than, or equal to the bullwhip effect under the single sourcing environment. Table 3 A comparison of the bullwhip effects under the single and dual sourcing environments

Comparison of bullwhip effect

For φ > 0 Single sourcing environment Dual sourcing environment (L2 > L1)

BWsingle = BWdual LS = L1, lS = l1 l1 = l2 + 1 LS = L2, lS = l2 L2 = L1 + 1, or l1 = l2

BWsingle < BWdual LS = L1, lS = l1 l1 < l2 + 1 LS = L2, lS = l2 L2 > L1 + 1 and l1 < l2

BWsingle > BWdual LS = L1, lS = l1 l1 > l2 + 1(*) LS = L2, lS = l2 L2 > L1 + 1 and l1 > l2(*)

Note: (*) – There exists an exception in case l1 > l2 + 1 that if both 1 1 2 1 1 2 2 21 1 11 l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + and

1 1 2 1 1 2 2 21 1 12 l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + are negative, then it might happen that BWsingle ≤ BWdual or BWsingle > BWdual.

The proof of the findings in Table 3 is presented in Appendix D.


In terms of the bullwhip effect, the dual sourcing model will be better than the single sourcing model only when the bullwhip measure of the dual sourcing model is less than that of the single sourcing model with either distributor 1 or distributor 2. From the conclusions derived from Table 3, the dual sourcing model should be used when it concurrently satisfies the following conditions:

1 l1 > l2 + 1

2 L2 > L1 + 1

3 1 1 2 1 1 2 2 21 1 11 l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + and 1 1 2 1 1 2 2 21 1 12 l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + are not both negative.

In addition, when condition (3) is not satisfied, i.e., if 1 1 2 1 1 2 2 21 1 11 l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + and 1 1 2 1 1 2 2 21 1 12 l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − +

are both negative, the occurrence of BWdual > BWsingle is possible, depending on the value of φ. In this situation, numerical calculation must be performed to check whether BWdual < BWsingle. For example, let’s consider a case in which L1 = 2, L2 = 6, l1 = 9, and l2 = 1, and if φ > 0.792 then both 1 1 2 1 1 2 2 21 1 11 l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + and

1 1 2 1 1 2 2 21 1 12 l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + are negative, and thus the bullwhip effect under the dual sourcing environment might be either less than or greater than the bullwhip effect under the single sourcing environment, depending on the value of autoregressive coefficient as illustrated in Figure 8.

Figure 8 BWdual versus BWsingle when l1 > l2 + 1 and L2 > L1 + 1


7 Conclusions

In this research the impact of dual sourcing decision on the bullwhip effect in a three-stage supply chain with one supplier, two distributors, and one retailer with the AR(1) demand process is examined. The measure of the bullwhip effect is analytically developed for the situation when the base stock inventory policy is employed by members in the supply network and the demand updating is conducted through the MMSE forecasting technique. Similar to past research studies conducted for the single sourcing environment, our research still confirms that the bullwhip effect will not exist when the autocorrelation coefficient is non-positive. However, when the demand follows a positively correlated process, it is found that the bullwhip effect does not always exist. This pattern is different in the single sourcing environment in which the bullwhip effect always exists if the demand is positively correlated. This finding shows that the use of dual sourcing model might help reduce the bullwhip effect in supply networks. The conditions under which the bullwhip effect will be reduced in the dual sourcing environment in comparison to the single sourcing environment are also derived. It is anticipated that the findings of this research can serve as a good starting point for analysing the bullwhip effect in more complicated supply chain structures in which multiple sourcing decisions are considered.

References Adenso-Diaz, B., Moreno, P., Gutierrwz, E. and Lozano, S. (2012) ‘An analysis of the main factors

affecting the bullwhip effect in reverse supply chains’, International Journal of Production Economics, Vol. 135, No. 2, pp.917–928.

Agrawal, S., Sengupta, R.N. and Shanker, K. (2009) ‘Impact of information sharing and lead time on bullwhip effect and on-hand inventory’, European Journal of Operational Research, Vol. 192, No. 2, pp.576–593.

Alwan, L.C., Liu, J.J. and Yao, D. (2003) ‘Stochastic characterisation of upstream demand processes in a supply chain’, IIE Transactions, Vol. 35, No. 3, pp.207–219.

Bayraktar, E., Koh, S.C.L., Gunasekaranc, A., Sari, K. and Tatoglue, E. (2008) ‘The role of forecasting on bullwhip effect for E-SCM applications’, International Journal of Production Economics, Vol. 113, No. 1, pp.193–204.

Beckmann, M. (1961) ‘An inventory model for arbitrary interval and quantity distributions of demands’, Management Science, Vol. 8, No. 1, pp.35–57.

Chaharsooghi S.K. and Sadeghi A. (2008) ‘On the bullwhip effect measure in supply chains with VAR (1) demand process’, International Journal of Industrial Engineering & Production Research, Vol. 19, No. 4, pp.9–19.

Chen, F., Drezner, Z., Ryan, J.K. and Simchi-Levi, D. (2000a) ‘Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead time, and information’, Management Science, Vol. 46, No. 3, pp.436–443.

Chen, F., Ryan, J.K. and Simchi-Levi, D. (2000b) ‘The impact of exponential smoothing forecasts on the bullwhip effect’, Naval Research Logistics, Vol. 47, No. 4, pp.269–286.

Chiang, C. (2001) ‘Order splitting under periodic review inventory systems’, International Journal of Production Economics, Vol. 70, No. 1, pp.67–76.


Chiang, C. and Benton, W.C. (1994) ‘Sole sourcing versus dual sourcing under stochastic demands and lead times’, Naval Research Logistics, Vol. 41, No. 5, pp.609–624.

Cho, D.W., and Lee, Y.H. (2011) ‘Bullwhip effect measure in a seasonal supply chain’, Journal of Intelligent Manufacturing (forthcoming) [online] http://www.springer.com.

Ciancimino, E., Cannella, S., Bruccoleri, M. and Framinan, J.M. (2012) ‘On the bullwhip avoidance phase: the synchronised supply chain’, European Journal of Operational Research, Vol. 221, No. 1, pp.49–63.

Coppini, M., Rossignoli, C., Rossi, T. and Strozzi, F. (2010) ‘Bullwhip effect and inventory oscillations analysis using the beer game model’, International Journal of Production Research, Vol. 48, No. 13, pp.3943–3956.

Duc, T.T.H., Luong, H.T. and Kim, Y. (2008a) ‘A measure of bullwhip effect in supply chains with a mixed autoregressive-moving average demand process’, European Journal of Operational Research, Vol. 187, No. 1, pp.243–256.

Duc, T.T.H., Luong, H.T. and Kim, Y. (2008b) ‘A measure of the bullwhip effect in supply chains with stochastic lead time’, The International Journal of Advanced Manufacturing Technology, Vol. 38, Nos. 11–12, pp.1201–1212.

Duc, T.T.H., Luong, H.T. and Kim, Y. (2010a) ‘Effect of the third-party warehouse on bullwhip effect and inventory cost in supply chains’, International Journal of Production Economics, Vol. 124, No. 2, pp.395–407.

Duc, T.T.H., Luong, H.T. and Kim, Y. (2010b) ‘Minimizing bullwhip effect in supply chain’, Proceeding of the International Conference on Industrial Engineering and Engineering Management (IEEM), Macao.

Forrester, J.W. (1958) ‘Industrial dynamics, A major breakthrough for decision makers’, Harvard Business Review, Vol. 36, No. 4, pp.37–66.

Gaalman, G. and Disney, S.M. (2006) ‘State space investigation of the bullwhip problem with ARMA(1,1) demand processes’, International Journal of Production Economics, Vol. 104, No. 2, pp.327–339.

Gaalman, G. and Disney, S.M. (2009) ‘On bullwhip in a family of order-up-to policies with ARMA(2,2) demand and arbitrary lead-times’, International Journal of Production Economics, Vol. 121, No. 2, pp.454–463.

Graves, S.C. (1999) ‘A single item inventory model for a nonstationary demand process’, Manufacturing and Service Operations Management, Vol. 1, No. 1, pp.50–61.

Hadley, G. and Whitin T. (1963) Analysis of inventory systems, Prentice-Hall, Englewood Cliffs, NJ.

Helbing, D. and Lämmer, S. (2005) ‘Supply and production networks: From the bullwhip effect to business cycles’, in D. Armbruster, A.S. Mikhailov and K. Kaneko (Eds.): Networks of Interacting Machines: Production Organization in Complex Industrial Systems and Biological Cells, pp.33–66, World Scientific, Singapore.

Hosoda, T. and Disney, S.M. (2005) ‘On variance amplification in a three-echelon supply chain with minimum mean square error forecasting’, Omega, Vol. 34, No. 4, pp.344–358.

Hussain, M., Shome, A. and Lee, D.M. (2012) ‘Impact of forecasting methods on variance ratio in order-up-to level policy’, The International Journal of Advanced Manufacturing Technology, Vol. 59, Nos. 1–4, pp.413–420.

Johnson, G.D. and Thompson, H.E. (1975) ‘Optimality of myopic inventory policies for certain dependent demand processes’, Management Science, Vol. 21, No. 11, pp.1303–1307.

Kamali, A., Fatemi Ghomia, S.M.T. and Jolai, F. (2011) ‘A multi-objective quantity discount and joint optimization model for coordination of a single-buyer multi-vendor supply chain’, Computers and Mathematics with Applications, Vol. 62, No. 8, pp.3251–3269.


Kelle, P. and Silver, E.A. (1990) ‘Safety stock reduction by order splitting’, Naval Research Logistics, Vol. 37, No. 5, pp.725–743.

Kim, J.G., Chatfield, D., Harrison, T.P. and Hayya, J.C. (2006) ‘Quantifying the bullwhip effect in a supply chain with stochastic lead time’, European Journal of Operational Research, Vol. 173, No. 2, pp.617–636.

Lee, H., Padmanabhan, V. and Whang, S. (1997a) ‘Information distortion in a supply chain: the bullwhip effect’, Management Science, Vol. 43, No. 4, pp.546–558.

Lee, H., Padmanabhan, V. and Whang, S. (1997b) ‘The bullwhip effect in supply chains’, Sloan Management Review, Vol. 38, No. 3, pp.93–102.

Liu, H. and Wang, P. (2007) ‘Bullwhip effect analysis in supply chain for demand forecasting technology’, Systems Engineering-Theory & Practice, Vol. 27, No. 7, pp.26–33.

Luong, H.T. (2007) ‘Measure of bullwhip effect in supply chain with autoregressive demand process’, European Journal of Operational Research, Vol. 180, No. 3, pp.1086–1097.

Luong, H.T. and Phien, N.H. (2007) ‘Measure of bullwhip effect in supply chain: the case of high order autoregressive demand process’, European Journal of Operational Research, Vol. 183, No. 1, pp.197–209.

Minner, S. (2003) ‘Multiple-supplier inventory models in supply chain management: a review’, International Journal of Production Economics, Vol. 81–82, No. 1, pp.265–279.

Nepal, B., Murat, A. and Chinnam, R.B. (2012) ‘The bullwhip effect in capacitated supply chains with consideration for product life-cycle aspects’, International Journal of Production Economics, Vol. 136, No. 2, pp.318–331.

Ouyang, Y. and Li, X. (2010) ‘The bullwhip effect in supply chain networks’, European Journal of Operational Research, Vol. 201, No. 3, pp.799–810.

Pati, R.K., Vrat, P. and Kumar, P. (2010) ‘Quantifying bullwhip effect in a closed loop supply chain’, OPSEARCH, Vol. 47, No. 4, pp.231–253.

Pereira, J., Takahashi, K., Ahumada, L. and Paredes, F. (2009) ‘Flexibility dimensions to control the bullwhip effect in a supply chain’, International Journal of Production Research, Vol. 47, No. 22, pp.6357–6374.

Raghunathan, S. (2003) ‘Impact of demand correlation on the value of and incentives for information sharing in a supply chain’, European Journal of Operational Research, Vol. 146, No. 3, pp.634–649.

Ramasesh, R.V., Ord, J.K., Hayya, J.C. and Pan, A. (1991) ‘Sole versus dual sourcing in stochastic lead-times (s,Q) inventory models’, Management Science, Vol. 37, No. 4, pp.428–443.

Sedarage, D., Fujiwara, O. and Luong, H.T. (1999) ‘Determining optimal order splitting and reorder levels for N-supplier inventory systems’, European Journal of Operational Research, Vol. 116, No. 2, pp.389–404.

Simchi-Levi, D., Kaminsky, P. and Simchi-Levi, E. (2008) Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies, 3rd ed., McGraw-Hill.

Sodhi, M.S. and Tang, C.S. (2011) ‘The incremental bullwhip effect of operational deviations in an arborescent supply chain with requirements planning’, European Journal of Operational Research, Vol. 215, No. 2, pp.374–382.

Sterman, J.D. (1989) ‘Modeling managerial behavior: misperceptions of feedback in a dynamic decision making experiment’, Management Science, Vol. 35, No. 3, pp.321–339.

Sucky, E. (2009) ‘The bullwhip effect in supply chains – an overestimated problem?’, International Journal of Production Economics, Vol. 118, No. 1, pp.311–322.

Thomas, D.J. and Tyworth, J.E. (2006) ‘Pooling lead-time risk by order splitting: a critical review’, Transportation Research Part E, Vol. 42, No. 4, pp.245–257.


Vassian, H.J. (1955) ‘Application of discrete variable servo theory to inventory control’, Journal of the Operations Research Society of America, Vol. 3, No. 3, pp.272–282.

Wang, J., Kuo, J., Chou, S. and Wang, S. (2010) ‘A comparison of bullwhip effect in a single-stage supply chain for autocorrelated demands when using correct, MA, and EWMA methods’, Expert Systems with Applications, Vol. 37, No. 7, pp.4726–4736.

Wright, D. and Yuan, X. (2008) ‘Mitigating the bullwhip effect by ordering policies and forecasting methods’, International Journal of Production Economics, Vol. 113, No. 2, pp.587–597.

Xu, K., Dong, Y. and Evers, P.T. (2001) ‘Towards better coordination of the supply chain’, Transportation Research Part E, Vol. 37, No. 1, pp.35–54.

Zhang, X. (2004) ‘The impact of forecasting methods on the bullwhip effect’, International Journal of Production Economics, Vol. 88, No. 1, pp.15–27.

Zhang, X. and Burke, G.J. (2011) ‘Analysis of compound bullwhip effect causes’, European Journal of Operational Research,, Vol. 210, No. 3, pp.514–526.

Zhao, X., Xie, J. and Leung, J. (2002) ‘The impact of forecasting model selection on the value of information sharing in a supply chain’, European Journal of Operational Research, Vol. 142, No. 2, pp.321–344.

Appendix A

Proof for Proposition 1

The order quantity placed by the retailer to distributor 1 at the beginning of period t, qs1,t, should satisfy the following relationship: qs1,t = Ss1,t – St–1 + Dt–1, where the order-up-to levels can be determined as 2 2

1 1 1ˆ ˆL L

t t tS D Zσ− − −= + and 1 11,

ˆ ˆ .L Ls t t tS D Zσ= + It is noted

that the overall order-up-to level, St, covers the forecast demand in time periods t, t + 1, …, t + L2 – 1. However, the intermediate order-up-to level of the retailer for distributor 1, Ss1,t, which is used to determine the order quantity to distributor 1, will encompass the forecast demand in time periods t, t + 1, …, t + L1 – 1.

According to Luong (2007), St–1 can be determined as follows:

( ) ( )2 2

2 22

1 1 1

22 12

ˆ ˆ

(1 ) 1 1 ˆ .1(1 )

L Lt t t

L LL

t t

S D Z

LD Z

σ

φ φ φφ φδ σ

φφ

− − −

− −

= +

⎛ ⎞− − − −⎜ ⎟= + +⎜ ⎟ −−⎝ ⎠

(A.1)

Similarly, we can derive that

( ) ( )1 1

1 11

1,

112

ˆ ˆ

(1 ) 1 1 ˆ .1(1 )

L Ls t t t

L LL

t t

S D Z

LD Z

σ

φ φ φφ φδ σ

φφ −

= +

⎛ ⎞− − − −⎜ ⎟= + +⎜ ⎟ −−⎝ ⎠

(A.2)

It has been proved that 1ˆ Ltσ and 2

1ˆ Ltσ − do not depend on the time, t (Luong, 2007).


Thus, from equations (A.1) and (A.2), qs1,t can be determined as follows:

( )

( ) ( )( )

21

1 2

1 2

1, 1, 1 1

1

1 2

1 22

111 1

(1 ) .ˆ ˆ

(1 )

s t s t t t

LL

t t

L LL L

q S S D

D D

L LZ

φ φφφ φ


− −

+

− −

= − +

−−= −

− −

⎡ ⎤− +− −⎣ ⎦+ + −−

(A.3)

If we denote

( ) ( )( )

1 2

1 21 2

2

(1 ),ˆ ˆ

(1 )

L LL L

Z

L LK Z


⎡ ⎤− +− −⎣ ⎦= + −−

we will obtain equation (8). By using the similar procedure as in the research of Alwan et al. (2003), we can

derive that qs1,t will follow the ARMA(1,1) model. This completes the proof.

Appendix B


For convenience, let’s denote

( ) ( )1 21 2

1 1 1 2 2 1 2 2

1 1

1 1 1

2

1 11 1, 1 , , 1 ,1 1 1 1

1 ,(1 )

l ll l

l L l L l L l L

a a b b

c

φ φ φ φφ φφ φ φ φ

φ φ φ φ φφ

+ +

+ + + + + + +

− −− −= + = = + =

− − − −

− + − − +=

−

and

( )( )( ) ( )

2 1 2 1 1 21 1 1

2 2

1 1.

1 1

L l L l l L

dφ φ φ φ φ φ φ

φ φ

+ + + +− − − − += =

− −

This allows us to write equation (17) as

1, 1 1, 2 2, 1

2, 2 1 2 3

( 1) ( 1) ( )

t s t s t s t

s t t t t

r a q aq b qbq c c d dε ε ε

− − −

− − − −

= + − + +

− + − + + (B.1)

Taking the variance of equation (B.1), we have


( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( )

2

2 21, 1 1, 2

2 2 22, 1 2, 2 1

2 22 3 1, 1 1, 2

1, 1 2, 1 1, 1 2, 2

( )

( 1)

( 1)

( ) 2 ( 1) ,

2( 1)( 1) , 2 ( 1) ,

2 ( 1)

r t

s t s t

s t s t t

t t s t s t

s t s t s t s t

VAR r

a VAR q a VAR q

b VAR q b VAR q c VAR

c d VAR d VAR a a COV q q

a b COV q q b a COV q q

a b C

σ

ε

ε ε

− −

− − −

− − − −

− − − −

=

= + +

+ + + +

+ + + − +

+ + + − +

− + ( ) ( )( ) ( )( ) ( )( ) ( )

1, 2 2, 1 1, 2 2, 2

2, 1 2, 2 1, 1 2

1, 1 3 2, 1 2

2, 1 3 1, 2 3

2,

, 2 ,

2 ( 1) , 2( 1)( ) ,

2 ( 1) , 2( 1)( ) ,

2 ( 1) , 2 ,

2

s t s t s t s t

s t s t s t t

s t t s t t

s t t s t t

s t

OV q q abCOV q q

b b COV q q a c d COV q

d a COV q b c d COV q

d b COV q adCOV q

bdCOV q

ε

ε ε

ε ε

− − − −

− − − −

− − − −

− − − −

+

− + − + +

+ + − + +

+ + −

− ( )2 3, .tε− −

(B.2)

It is noted that

( )2 221 dσ σφ= −

( ) ( ) ( )( )( )

1 2 1 2

1

1 1 1 2 1 2

2 21 12 22 2

2 2

1 3 2 2 2 2 322

2

1 1 2 1 1

1 (1 )

1 2 2 2(1 )

s

L L L L

q

L L L L L L

d

φ φ φ φ φ φσ σ

φ φ

φ φ φ φ φ φ σφ

+ +

+ + + + + +

− + − − − −=

− −

− − + + + −=

−

( )( )

1 2

2

1 1 2 2

222 2

2 2

2 2 2 2 22

2

1 (1 )

2(1 )

s

L L

q

L L L L

d

φ φ φσ σ

φ φ

φ φ φ σφ

+ + + +

−=

− −

− +=

−

We also have

( )( )( )

( )

1 2

1

1 1 1 2 2 2 1 2 1 2

1, 1 1, 2

12 2

2

2 4 2 3 1 3 2 3 2 42

2

,

1 1

(1 )

1

s

s t s t

L L

q

L L L L L L L L L L

d

COV q q

φ φ φφσ σ

φ

φ φ φ φ φ φ φ φ σφ

− −

+

+ + + + + + + + + +

− −= −

−

− + + + − + − −=

−


( )( )

( ) ( ) ( )( )

( )

1 2 1 2

1 1 1 2 2 2 1 2 1 2

1, 1 2, 1

1, 2 2, 2

1 22

2 2

1 3 2 2 1 3 2 3 2 32

2

,

,

1 1

1 (1 )

1

s t s t

s t s t

L L L L

L L L L L L L L L L

d

COV q q

COV q q

φ φ φ φ φ φσ

φ φ


− −

− −

+

+ + + + + + + + + +

=

⎡ ⎤− − − ∗ −⎣ ⎦=− −

− − − + − + +=

−

( )( ) ( ) ( )

( )1 2 1 2

1 2 1 2 1 2

1, 1 2, 2

12

2 2

2 3 2 2 2 32

2

,

1 1

1 (1 )

(1 )

s t s t

L L L L

L L L L L L

d

COV q q

φ φ φ φ φ φ φσ

φ φ

φ φ φ φ σφ

− −

+

+ + + + + +

⎡ ⎤− − − ∗ −⎣ ⎦=− −

− − + +=

−

( )( ) ( ) ( )

( )1 2 1 2

1 1 1 2 2 2 1 2 1 2

1, 2 2, 1

1 2 22

2 2

2 4 2 3 2 4 2 4 3 42

2

,

1 1

1 (1 )

(1 )

s t s t

L L L L

L L L L L L L L L L

d

COV q q

φ φ φ φ φ φσ

φ φ


− −

+

+ + + + + + + + + +

⎡ ⎤− − − ∗ −⎣ ⎦=− −

− − − + − + +=

−

( )( )

1 2 1 2

2

2 3 2 3 32 2

2, 1 2, 2 22,

1s

L L L L

s t s t q dCOV q q φ φ φφσ σφ

+ + + +

− −+ −

= =−

( ) ( ) ( )1 11 1

2 221, 1 2 1, 2 3

1 1, , 11 1

L L

s t t s t t dCOV q COV q φ φε ε σ σφφ φ

+ +

− − − −− −

= = = −− −

( ) ( ) ( ) ( ) ( )1 2 1 21 1

2 221, 1 3

1 1, 1

1 1

L L L L

s t t dCOV qφ φ φ φ φ φ φ

ε σ σφφ φ

+ +

− −

− − + − += = −

− −

( ) ( ) ( )

( ) ( )

1 2

1 2

22, 1 2 2, 2 3

22

, ,1

11

L L

s t t s t t

L L

d

COV q COV qφ φ φ

ε ε σφ

φ φ φσφ

φ

− − − −

−= =

−

−= −

−

( ) ( ) ( ) ( )1 2 1 22 2

2 222, 1 3, 1

1 1

L L L L

s t t dCOV qφ φ φ φ φ φ

ε σ σφφ φ− −

− −= = −

− −


We can rewrite equation (B.2) as

( )

( )( )( ) ( )

( )( )( )

1 2

22 2 2 2 2 2 2 2 2 2

1, 1 1, 2

1, 1 2, 1

1, 1 2, 2

1, 2 2, 1

2, 1 2, 2

( 1) ( 1)

2 ( 1) ,

2 1 1 ,

2 ( 1) ,

2 ( 1) ,

2 ( 1) ,

2 (

s sr q q

s t s t

s t s t

s t s t

s t s t

s t s t

a a b b c d c d

a a COV q q

ab a b COV q q

b a COV q q

a b COV q q

b b COV q q

a

σ σ σ σ

− −

− −

− −

− −

− −

⎡ ⎤⎡ ⎤ ⎡ ⎤= + + + + + + + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

− +

⎡ ⎤+ + + +⎣ ⎦

− +

− +

− +

− +[ ] ( )( )

[ ] ( )( )

1, 1 2

1, 1 3

2, 1 2

2, 1 3

1)( ) ,

2 ( 1) ,

2 ( 1)( ) ,

2 ( 1) , .

s t t

s t t

s t t

s t t

c d ad COV q

d a COV q

b c d bd COV q

d b COV q

ε

ε

ε

ε

− −

− −

− −

− −

+ +

+ +

− + + +

+ +

(B.3)

Let’s first compute the term 2[ab + (a + 1)(b + 1)]COV(qs1,t–1, qs2,t–1) as follows:

[ ] ( )

( )( )( )

1 1 2 2 1 2

1 1 2 2

1 1 2 2 1 2

1

1, 1 2, 1

1 2 1 2 223

1 2 1 2 2

1 2 1 2 224

2 2 2

2 ( 1)( 1) ,

2 1 2(1 )

*

2 1 2(1 )

*

s t s t

l l l l l l

L L L Ld

l l l l l l

L

ab a b COV q q

φ φ φ φ φ φφ

φ φ φ φ σ

φ φ φ φ φ φφ

φ φ

− −

+ + + + + +

+ + + +

+ + + + + +

+

+ + +

= + − − − − +−

+ − −

+ + − − − − +−

− −( )2 1 2 1 23 2 3 2.L L L L Ldφ φ σ+ + + + ++ +

(B.4)

For convenience, let’s denote

( ) ( )

( )( )

1 2

1 1 2 2 1 2

1 1 2 2

2 22 2 2 2

1, 1 1, 2 1, 1 2, 2

1 2 1 2 223

1 2 1 2 2

( 1) ( 1)

2 ( 1) , 2 ( 1) ,

2 1 2(1 )

*

s sq q

s t s t s t s t

l l l l l l

L L L Ld

U a a b b

a a COV q q b b COV q q

σ σ

φ φ φ φ φ φφ

φ φ φ φ σ

− − − −

+ + + + + +

+ + + +

⎡ ⎤ ⎡ ⎤= ++ + + +⎣ ⎦ ⎣ ⎦

− + − +

+ + − − − − +−

+ − − ∗

( ) ( )( )

( )

1 1 2 2 1 2

1 2 1 2 1 2

1, 1 2, 2 1, 2 2, 1

1 2 1 2 224

2 2 2 3 2 3 2

2 ( 1) , 2 ( 1) ,

2 1 2(1 )

*

s t s t s t s t

l l l l l l

L L L L L Ld

V b a COV q q a b COV q q

φ φ φ φ φ φφ

φ φ φ φ σ

− − − −

+ + + + + +

+ + + + + +

= − + − +

+ + − − − − +−

− − + +


[ ] ( )( )

[ ] ( )( )

22 2 2

1, 1 2

1, 1 3

2, 1 2

2, 1 3

( )

2 ( 1)( ) ,

2 ( 1) ,

2 ( 1)( ) ,

2 ( 1) , .

s t t

s t t

s t t

s t t

W c d c d

a c d ad COV q

d a COV q

b c d bd COV q

d b COV q

σ

ε

ε

ε

ε

− −

− −

− −

− −

⎡ ⎤= + + +⎣ ⎦

− + + +

+ +

− + + +

+ +

From equation (B.3), 2rσ can be expressed as

2r U V Wσ = + + (B.5)

First, considering the term U, we can derive the expression of U as

1 1 1

1 2 2 2 2 2

2 1 2 1 2 1 1 1

1 1 1 1 1 1 1 1 1 1

1

3 4 2 22 33

2 3 1 2 3 4 2 2

2 3 2 3 1 2 3

2 2 2 3 2 3 4 5

2

1 1 2 2(1 )

2 4 4 2 2

2 2 2 2 4 2

2 2 2 2 2 2

2

L L L

L L L L L L

L L L L L l l l

l l l L l L l L l L

l

U φ φ φ φ φ φφ

φ φ φ φ φ φ

φ φ φ φ φ φ

φ φ φ φ φ φ

φ

+ + +

+ + + + + +

+ + + + + + + +

+ + + + + + + + + +

+

⎡= + + + + + +⎣−

+ − − − − +

+ − − − − −

+ + + + − −

− 1 1 1 1 1 1 1 1 1

1 1 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 2 1

2 1 2 1 2 1 2 1

3 2 4 2 3 2 4 2 5

2 2 4 2 3 4 2 3

2 4 2 3 2 4 2 2 4 2

3 4 2 3 2

2 4 2 2

2 4 8 4 2

2 2 2 2 2

4 2 2 2

L l L l L l L l L

l L l L l L l L l L

l L l L l L l L l L

l L l L l L l L

φ φ φ φ

φ φ φ φ φ

φ φ φ φ φ

φ φ φ φ

+ + + + + + + + +

+ + + + + + + + + +

+ + + + + + + + + +

+ + + + + + + +

− − − +

+ + + + −

− − − + −

− − − − 2 1

2 2 2 2 2 2 2 2 2 2

2 2 1 1 2 1 1 2 1 1 2

1 1 2 2 1 2 2 1 2 2 1 2

1 2 1 1 2 1

4 2 2 4

2 3 4 2 3 2 4

2 2 4 3 5 2 4

2 5 3 4 2 4

3

2

2 4 2 2 2

2 2 2 2

2 4 4 4

4 4

l L

l L l L l L l L l L

l L l L L l L L l L L

l L L l L L l L L l L L

l l L l l L

φ

φ φ φ φ φ

φ φ φ φ

φ φ φ φ

φ φ

+ +

+ + + + + + + + + +

+ + + + + + + + + + +

+ + + + + + + + + + + +

+ + + + + +

+

+ + + − −

+ − + +

− + + −

+ + 1 2 2 1 2 24 3 4 24 4 * .l l L l l Ldφ φ σ+ + + + + + ⎤− − ⎦

(B.6)

Second, the term V can be determined as

1 1 2 1 2

1 1 1 1 1 2 2 1 2 1

2 2 2 2 2 2 1 1 2 2 1 2

1 2 1 1 2 2 1 2 1 2

3 2 2 3 23

3 2 3 3 4 2 3

4 2 3 2 4 3 4

4 4 4

(1 ) 2 2 2 2(1 )

2 2 2 2 2

2 2 2 2 2

2 2 2

L L L L L

l L l L l L l L l L

l L l L l L l L L l L L

l l L l l L l l L L

V φ φ φ φ φφ

φ φ φ φ φ

φ φ φ φ φ

φ φ φ

+ + + + +

+ + + + + + + + + +

+ + + + + + + + + + + +

+ + + + + + + + + +

+ ⎡= − − + +⎣−

+ + − + +

− − + − −

− + +

1 2 1 1 2 2

2

2 4 2 42 2

3 24 2 .

(1 ) (1 )

d

l l L l l L

d d

σ

φ φσ σφ φ

+ + + + + +

⎤⎦

− +− −

(B.7)

Finally, the term W can be determined as


2 2 1 12

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 2 1 2

1 2 1 2 1 2

2 2 2 1 213

2 2 1 2 3 4 2 3

2 4 2 2 2 2 2 3 1

2 3 2 3

(1 ) 2 2 2 22 2(1 )

2 2 4 2 2 4

2 2 2 2 2

4 2 2

L L l lL

l l L l L l L l L l L

l L l L l L l L l L

l L l L l L

W φ φ φ φ φφ φφ

φ φ φ φ φ φ

φ φ φ φ φ

φ φ φ

+ + + ++

+ + + + + + + + + + +

+ + + + + + + + +

+ + + + + +

+⎡= + − + +− +⎣−

− + − − + +

− + − − +

− − + 1 2 1 2

1 2 1 2 1 2 2 1 2 1

2 1 2 1 2 1 2 1 2 2

2 2 2 2 2 2 2 2 2 2

2 2

2 3 2 2

2 2 1 2 2 2 2 2 3 1 2

3 4 2 2 2 2 2 3 1

2 3 4 2 3 2 4

2 2 2

2 2

2 2 2 2 4

2 2 2 2 2

4 2 2 4 2

2

l L l L

l L l L l L l L l L

l L l L l L l L l L

l L l L l L l L l L

l L

φ φ

φ φ φ φ φ

φ φ φ φ φ

φ φ φ φ φ

φ

+ + +

+ + + + + + + + + +

+ + + + + + + + + +

+ + + + + + + + + +

+ +

+ +

− + − − +

+ − + − +

− − + + −

+ − 2 2 1 1 2 1 1 2 1 1 2

1 1 2 1 1 2 1 1 2 2 1 2 2 1 2

2 1 2 2 1 2 1 2 1 1 2 1 1 2 1

1 2

2 2 3 3 4 2 1

2 2 2 3 2 4 3 4

2 2 2 3 3 4 2 2

2

2 4 2 4

4 4 2 4 2

4 4 4 2 4

4

l L l L L l L L l L L

l L L l L L l L L l L L l L L

l L L l L L l l L l l L l l L

l l

φ φ φ φ

φ φ φ φ φ

φ φ φ φ φ

φ

+ + + + + + + + + + +

+ + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + +

+ +

+ − −

+ − + − +

− + − + −

+ 1 1 2 2 1 2 2 1 2 2 1 2 2

1 2 2 1 2 2 1 2 1 2 1 2 1 2

3 3 4 2 1 2 2

2 3 2 4 1 24

4 2 4 4

4 2 4 .2

L l l L l l L l l L l l L

l l L l l L l l L L l l L Ld

φ φ φ φ

φ φ φ σφ

+ + + + + + + + + + + + +

+ + + + + + + + + + + + + +

+ − − +

⎤− + + − ⎦

(B.8)

Substituting equations (B.6) to (B.8) into equation (B.5), the variance of rt can be derived as

1 1 1 1

1 1 1 2 1 2 1 2 2 1

2 1 2 1 2 2 2 2 2 2

1 1 2 2 1 2 1 2

2

1 32 33

2 2 2 2 2 2 1

3 2 2 2 1 3 2 2 2

2 1 2 2

( ) ( )1 1 2 2

(1 )

2 2 2 2 2

2 2 2 2 2

4 4 4

r t

l L l L

l L l L l L l L l L

l L l L l L l L l L

l L L l L L l l

VAR r U W Vσ

φ φ φ φ φφ

φ φ φ φ φ

φ φ φ φ φ

φ φ φ

+ + + +

+ + + + + + + +

+ + + + + + + + + +

+ + + + + + +

= = + +

⎡= − − + + −⎣−

+ − + + −

+ + + − +

− − − 1 1 2 2

1 2 1 2 1 2 1 2

2 2 2 1

1 2 2

4

4 4 .

L l l L

l l L L l l L Ld

φ

φ φ σ

+ + + + +

+ + + + + + + +

−

⎤+ + ⎦

(B.9)

After rearranging, equation (B.9) can be rewritten as

( ) ( )

( )( ) ( )

( )

2 21 2 1 2

1 11 2 1 2

1 2 1 2 1 2 1 2

1 1 2 1 1

222 2 1 13

21 2 22 223

1 2 2 1 1

1 123

21 (1 ) (1 )(1 )

21

(1 )

2

21(1 )

L Ll l l l

L Ll l l lr d

L L l l l l l l

l L l L l L

φ φ φ φ φφ φ φ φφ

σ φ φ σφ φ φ φ φφ

φ φ φ φ φ

φ φ φ φ φφ

+ +

+ +

+ + + + + +

+ + + + +

⎧ ⎫⎡ ⎤+ − − − +− −⎪ ⎪⎢ ⎥⎣ ⎦−⎪ ⎪⎪ ⎪⎡= + +− − −⎨ ⎬⎢⎣−⎪ ⎪⎪ ⎪⎤− + − −⎪ ⎪⎦⎩ ⎭

+ − + − −−=

( )

( )

2 2 2

1 1 2 1 1 2 2 2

1

2

1 1 1

*

1

l L

dl L l L l L l L

φσ

φ φ φ φ φ

+ +

+ + + + + + +

⎧ ⎫⎡ +⎪ ⎪⎣⎪ ⎪⎨ ⎬⎪ ⎪⎤− + − − +⎪ ⎪⎦⎩ ⎭

This completes the proof.


Appendix C


We first consider the case in which 0 < φ < 1. For simplicity, denote L2 = L1 + j in which j is a positive integer number. Substituting into equation (19), the measure of the bullwhip effect can be rewritten as

( ) ( ) ( )( ){( ) ( )( )}

1 1 2 1

1 1 2 1

1 11r 3

1 11

21 1 1 1(1 )

* 1 1 1 .

l L l Lj j

l L l Lj j

BW φ φ φ φ φ φφ

φ φ φ φ φ

+ + + +−

+ + + +−

= + − + − − −−

− + − − − (C.1)

Since 1 2

1

0 0

(1 ) (1 )( ) (1 )( ),j j

j k j k

k k

φ φ φ φ φ φ− −

−

= =

− = − = − +∑ ∑ we will obtain

( )

( )

2 1 1 1 2

2 1 1 1 2

21

r0

21

0

21(1 )

* 1 .

jl L j L l l k

k

jl L j L l l k

k


φ φ φ φ φ

−+ + +

=

−+ + +

=

⎧⎛ ⎞⎛ ⎞⎪⎜ ⎟= + − + − ⎜ ⎟⎨ ⎜ ⎟⎜ ⎟− ⎪ ⎝ ⎠⎝ ⎠⎩⎫⎛ ⎞⎛ ⎞ ⎪⎜ ⎟− + − ⎜ ⎟ ⎬⎜ ⎟⎜ ⎟⎪⎝ ⎠⎝ ⎠⎭

∑

∑ (C.2)

Based on equation (C.2), it can be seen that the bullwhip effect always exists if l1 ≤ l2. When l1 = l2 + 1 or L2 = L1 + 1, we have:

( ) ( )1 1 2 1 1 2 2 2 2 11 1 1 11 (1 ) 0,1l L l L l L l L l Lφ φ φ φ φ φ φ+ + + + + + + + +− + − − + = − >−

( ) ( )1 1 2 1 1 2 2 2 2 11 1 12 (1 ) 0.1l L l L l L l L l Lφ φ φ φ φ φ φ φ φ+ + + + + + + +− + − − + = − >−

Therefore, the bullwhip effect exists. For the case of l1 > l2 + 1 and L2 > L1 + 1, it should be noted that

1 1 2 1 1 2 2 2

1 1 2 1 1 2 2 2

1 1 1 2

1 1 1

1

.

l L l L l L l L

l L l L l L l L

φ φ φ φ φ φ φ

φ φ φ φ

+ + + + + + +

+ + + + + + +

− + − − + > −

+ − − +

Depending on the values of lead times, the two expressions on both sides of the above inequality can take negative (or zero) or positive value (or zero). Hence, the bullwhip effect will not exist only when

1 1 2 1 1 2 2 21 1 11 0l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + ≥

and

1 1 2 1 1 2 2 21 1 12 0.l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + ≤

Next, consider the case in which –1 < φ < 0. The combinations of the four lead times from Table C.1 should be examined.


Table C.1 All possible scenarios for the combination of lead times

Scenario Lead time

Scenario Lead time

L1 L2 l1 l2 L1 L2 l1 l2

1 Odd Odd Odd Odd 5 Odd Odd Odd Even Even Even Even Even Even Even Even Odd

2 Odd Even Odd Odd 6 Odd Even Odd Even Even Odd Even Even Even Odd Even Odd

3 Odd Even Even Even 7 Odd Even Even Odd Even Odd Odd Odd Even Odd Odd Even

4 Odd Odd Even Even 8 Odd Odd Even Odd Even Even Odd Odd Even Even Odd Even

It is proved that the bullwhip effect does not exist when the autoregressive coefficient of the demand process is negative for all scenarios. For more specific, we will prove that

1 1 2 1 1 2 2 21 1 12( ) 0l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + < and 1 1 2 1 1 2 2 21 1 1(1 ) 0.l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + > For Scenario 1, the bullwhip effect measure can be expressed as

( ){( )}

1 1 2 1 1 2 2 2

1 1 2 1 1 2 2 2

2 1 1 113

1 1 1

21(1 )

* 1 .

l L l L l L l LScenarior

l L l L l L l L


φ φ φ φ φ

+ + + + + + +

+ + + + + + +

= + − − − + − −−

+ − + − −

It is noted that for the first parenthesised component in the curly brackets, we can easily derive that

( ) ( )2 1 1 1 1 2 2 21 2 1 1 0l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− − − + + + <

and for the second parenthesised component in the same curly brackets, we can see that

( ) ( )1 1 1 2 2 1 2 21 1 11 0.l L l L l L l Lφ φ φ φ φ+ + + + + + ++ − − + − >

Thus, 1 1.ScenariorBW <

The proofs of Scenarios 2 to 4 are similar to the proof of Scenario 1. For Scenario 5, the measure of the bullwhip effect can be rearranged as

( ){( )}

1 1 2 1 1 2 2 2

1 1 2 1 1 2 2 2

2 1 1 153

1 1 1

21(1 )

* 1 .

l L l L l L l LScenarior

l L l L l L l L


φ φ φ φ φ

+ + + + + + +

+ + + + + + +

= + − − − − − +−

+ − − − +

For the first parenthesised component in the curly brackets, we can derive that

( ) ( )1 1 1 2 2 1 2 22 1 1 1 0.l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + + + − − <


The second parenthesised component in the curly brackets can be rewritten as

( ) ( ) ( ){ }1 1 1 2 2 1 2 11 1 11 1 1 .l L l L l L L Lφ φ φ φ φ+ + + − + + −− + − − −

Noted that 1 1 2 11 11 0, ,l L l Lφ φ φ+ + + +− > > and 1 1 11 1l L j jφ φ+ + −− > − making the above

expression positive. Thus, 5 1.ScenariorBW <

The proofs of Scenarios 6 to 8 are similar to the proof of Scenario 5. Last, if φ = 0, we can see that the bullwhip effect does not exist because

BWr = 1. This completes the proof.

Appendix D

Proof of conclusions in Table 3

The bullwhip effect measure for the single sourcing environment can be rewritten as

( )( )( )1 12

r 3

2 1, , 1

(1 )

S S S S S S S Sl L l L l L l L

S SBW l Lφ φ φ φ φ φ φ

φφ

+ + + + + +

′

⎡ ⎤− − + − − +⎣ ⎦= +−

(D.1)

Noted that both 12( )S S S Sl L l Lφ φ φ φ+ + +− − + and 1(1 )S S S Sl L l Lφ φ φ+ + +− − + are greater than zero when 0 < φ < 1.

It can be seen that the bullwhip effect measures under the dual sourcing environment in equation (19) and the single sourcing environment in equation (D.1) are structurally similar. Subtracting the first parenthesised component in the square brackets of equation (D.1) from the first parenthesised component in the square brackets of equation (19), we have

( ) ( )1 1 2 1 1 2 2 2

1 1 2 1 1 2 2 2

11 1 12 2

11 1 1 .

S S S S

S S S S

l L l Ll L l L l L l L

l L l Ll L l L l L l L

φ φ φ φ φ φ φ φ φ φ

φ φ φ φ φ φ

+ + ++ + + + + + +

+ + ++ + + + + + +

− + − − + − − − +

= − − + + − (D.2)

It is also noted that subtracting the second parenthesised component in the square brackets of equation (D.1) from the second parenthesised component in the square brackets of equation (19) we will also have equation (D.2).

For convenience, let’s denote r 1 2 1 2 dual( , , , , )BW l l L L BWφ = and r ( , , )S SBW l Lφ′

single .BW= The following cases will be considered:

• Case 1: LS = L1 and lS = l1 (the single distributor is the distributor 1)

Expression (D.2) can be rewritten as:

( )( )2 1 1 2 2 2 1 1 1 2 1 21 1 1 .l L l L l L l L L L l lφ φ φ φ φ φ φ φ+ + + + + + +− − + + = − − (D.3)


Noted that 1 2 0.L Lφ φ− > Thus, it can be concluded that

1 If l1 = l2 + 1, the value of expression (D.3) = 0 and hence BWsingle = BWdual.

2 If 121 +< ll , the value of expression (D.3) > 0 and hence BWsingle<BWdual.

3 If 121 +> ll , the value of expression (D.3) < 0. In this case, it should be noted

that there exists a possibility for both 1 1 2 1 1 2 2 21 1 11 l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + and 1 1 2 1 1 2 2 21 1 12 l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + to be negative (see Appendix C). If the

above situation occurs, it is possible that BWsingle ≤ BWdual or BWsingle > BWdual, depending on the value of parameter φ.

• Case 2: LS = L2 and lS = l2 (the single distributor is the distributor 2)

Expression (D.2) can be rewritten as:

( )( )1 1 2 1 1 2 2 2 1 2 1 21 1 1l L l L l L l L L L l lφ φ φ φ φ φ φ φ+ + + + + + +− − + = − − (D.4)

From expression (D.4), it can be seen that 1 If L2 = L1 + 1 or l1 = l2, BWsingle = BWdual 2 If L2 > L1 + 1 and l1 < l2, BWsingle < BWdual 3 If L2 > L1 + 1 and l1 > l2: Similar to case 1, if l1 > l2 + 1 and both

1 1 2 1 1 2 2 21 1 11 l L l L l L l Lφ φ φ φ φ+ + + + + + +− + − − + and 1 1 2 1 1 2 2 21 1 12 l L l L l L l Lφ φ φ φ φ φ+ + + + + + +− + − − + are negative, then it might happen that BWsingle ≤ BWdual or BWsingle > BWdual, depending on the value of parameter φ.

This completes the proof.

measure of bullwhip effect - a dual sourcing model

Documents