Research Article Inherent Complexity Research on the ...downloads.hindawi.com/journals/aaa/2014/306907.pdf · and so forth also recognized the existence of the bullwhip e ect in supply

Research ArticleInherent Complexity Research on the Bullwhip Effectin Supply Chains with Two Retailers: The Impact of ThreeForecasting Methods Considering Market Share

Junhai Ma, Binshuo Bao, and Xiaogang Ma

College of Management and Economics, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Binshuo Bao; [email protected]

Received 16 April 2014; Revised 30 June 2014; Accepted 30 June 2014; Published 4 August 2014

Academic Editor: Hamid R. Karimi

Copyright © 2014 Junhai Ma et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An important phenomenon in supply chain management which is known as the bullwhip effect suggests that demand variabilityincreases as one moves up a supply chain. This paper contrasts the bullwhip effect for a two-stage supply chain consisting of onesupplier and two retailers under three forecasting methods based on the market share. We can quantify the correlation coefficientbetween the two retailers clearly, in consideration of market share. The two retailers both employ the order-up-to inventory policyfor replenishments. The bullwhip effect is measured, respectively, under the minimum mean squared error (MMSE), movingaverage (MA), and exponential smoothing (ES) forecasting methods. The effect of autoregressive coefficient, lead time, and themarket share on a bullwhip effectmeasure is investigated by using algebraic analysis and numerical simulation. And the comparisonof the bullwhip effect under three forecasting methods is conducted. The conclusion suggests that different forecasting methodsand various parameters lead to different bullwhip effects. Hence, the corresponding forecasting method should be chosen by themanagers under different parameters in practice.

1. Introduction

In the research of modern logistics and supply chain man-agement, a significant phenomenon which is called thebullwhip effect has attracted the attention of researchersand practitioners alike. As moving backward from a down-stream member to an upstream member, the variance oforder quantities placed by the downstream member to itsimmediate upstream member tends to be amplified. The firstinvention of the bullwhip effect could be traced back toForrester [1, 2]. After that, several researchers such as Blinder[3], Blanchard [4], Burbidge [5], Blinder [6] and Kahn [7],and so forth also recognized the existence of the bullwhipeffect in supply chains. Sterman [8] used the Beer Game,the most popular simulation of a simple production anddistribution system developed at MIT, to certificate that thebullwhip effect is an important problem. The phenomenonwas firstly called the bullwhip effect by Lee et al. [9, 10].They indicated that the main reason for this phenomenon is

demand signal processing, nonzero lead time, order batching,supply shortages, and price fluctuation.

In the five reasons mentioned above, demand signalprocessing is the research hotspot in this field. Metters[11] tried to identify the bullwhip effect by establishing anempirical lower bound on the profitability impact of thebullwhip effect. In his research, the impact of bullwhip effectwas measured by comparing results obtained in two cases,that is, high demand variability versus low demand variabilitywith weak seasonality. It was found that the importanceof the bullwhip effect differed greatly depending on thebusiness environment. Graves [12] quantified the bullwhipeffect for the supply chain in which demand pattern followsan integrated moving average process. Chen et al. [13, 14]quantified the bullwhip effect for supply chains using movingaverage and exponential smoothing techniques for demandforecasts. In their works, it was assumed that members of thechain employ base stock policy for their inventory system.The main finding of their work was that the order variance

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 306907, 13 pageshttp://dx.doi.org/10.1155/2014/306907

2 Abstract and Applied Analysis

would increase with the increasing number of members inthe chain, lower level of information sharing, and increasinglead time. Likewise, Xu et al. [15] conducted a similar researchfor a demand process that was forecasted with a simpleexponentially weighted moving average method. Zhang [16]also investigated the impact of different forecasting methodson the bullwhip effect for a simple inventory system witha first-order autoregressive demand process. Luong [17]measured the bullwhip effect for a simple two-stage supplychain that included only one retailer and one supplier inthe environment where the retailer employed the order-up-to inventory policy for their inventory and demand forecastwas performed through the AR(1) model, and the effectof autoregressive coefficient and lead time on this measurewas investigated. Duc et al. [18] examined the impact of athird-party warehouse on the bullwhip effect. Karimi et al.[19] considered the problem of local capacity 𝐻

∞control

for a class of production networks of autonomous worksystems with time-varying delays in the capacity changes.Nepal et al. [20] presented an analysis of the bullwhip effectand net-stock amplification in a three-echelon supply chainconsidering step changes in the production rates during aproduct’s lifecycle demand. Dashkovskiy [21, 22] consideredlocal input-to-state stability of complex logistics networks.Mehrsai [23, 24] tried to investigate the possibility of com-bining this new research paradigm with existing strategiesin production logistics to improve material handling andcontrol task according to material flow criteria. Ma et al.[25] stated a comparison of bullwhip effect under variousforecasting techniques in supply chains with two retailers.Shen and Jiang [26] explained the general principle ofdynamic programming algorithm. Shen et al. [27] presentedthe dynamic programming algorithm to solve the optimalcontrol variable trajectory under a given circle.

This paper continues to examine the differences in bull-whip effect under three forecasting methods. And this paperanalyzes the impact of every parameter on the bullwhipeffect. The conclusions suggest that different forecastingmethods lead to bullwhip effectmeasureswith fundamentallydifferent properties in relation to lead time and demandautocorrelation. Therefore, the corresponding forecastingmethod should be selected by the managers under differentparameters in practice.

The structure of this research is as follows. Section 2depicts a new supply chain model with two retailers whichboth follow the AR(1) demand process and apply the order-up-to stock policy. In Section 3, the bullwhip effect measurefor MMSE, MA, and ES forecasting methods is derived. InSection 4, this paper analyzes the effects of parameters onthe bullwhip effect under three forecasting methods andcompares the impact of three forecasting methods on thebullwhip effect. Finally, Section 5 concludes the paper witha short summary.

2. A Supply Chain Model

2.1. Demand Process. In this research, a two-stage supplychain with one supplier and two retailers will be considered,

Supplier

Retailer 1

Retailer 2

Customers

AR(1)

AR(1)

qt

q1, t

q2, t D2, t

D1, t𝛼

1 − 𝛼

Dt

Figure 1: Supply chain model.

and we will quantify the bullwhip effect in the simple supplychain. The two retailers order and replenish the stock from asupplier in each period 𝑡.Themarket share of the two retailersis considered as 𝛼 and 1 − 𝛼, respectively. We assumed thatthe order-up-to inventory policy. Retailer 1 and retailer 2 bothemploy an AR(1) autoregressive model:

𝐷𝑡= 𝛿 + 𝜙𝐷

𝑡−1+ 𝜀𝑡. (1)

The supply chain model shows, in Figure 1, the following:Due to the market share of retailer 1, we consider that

retailer 1 employs an AR(1)model as follows:

𝐷1,𝑡= 𝛼𝛿1+ 𝜙1𝐷1,𝑡−1

+ 𝛼𝜀1,𝑡. (2)

In (2),𝐷1,𝑡

is the demand of period 𝑡. 𝛿1is the constant of

the autoregressive model. 𝜙1is the first-order autocorrelation

coefficient, where −1 < 𝜙1< 1. 𝜀

1,𝑡is the forecast error for

period 𝑡 and 𝜀1,𝑡

is independent and identically distributedfrom a symmetric distribution with mean 0 and variance 𝜎2

1.

According to a first-order autocorrelation property oftime series model, for any period 𝑡, we must have

𝐸 (𝐷1,𝑡) = 𝐸 (𝐷

1,𝑡−1) = 𝜇1,𝑑=

𝛼𝛿1

1 − 𝜙1

,

Var (𝐷1,𝑡) = Var (𝐷

1,𝑡−1) = 𝜎2

1,𝑑=𝛼2𝜎21

1 − 𝜙21

.

(3)

Analogously, with the market share 1 − 𝛼 retailer 2 alsoemploys an AR(1)model as follows:

𝐷2,𝑡= (1 − 𝛼) 𝛿

2+ 𝜙2𝐷2,𝑡−1

+ (1 − 𝛼) 𝜀2,𝑡. (4)

In (4),𝐷2,𝑡

is the demand of period 𝑡. 𝛿2is the constant of

the autoregressive model. 𝜙2is the first-order autocorrelation

coefficient, where −1 < 𝜙2< 1. 𝜀

2,𝑡is the forecast error for

period 𝑡 and 𝜀2,𝑡

is independent and identically distributedfrom a symmetric distribution with mean 0 and variance 𝜎2

2.

Similarly, we also have

𝐸 (𝐷2,𝑡) = 𝐸 (𝐷

2,𝑡−1) = 𝜇2,𝑑=(1 − 𝛼) 𝛿

2

1 − 𝜙2

,

Var (𝐷2,𝑡) = Var (𝐷

2,𝑡−1) = 𝜎2

2,𝑑=(1 − 𝛼)

2

𝜎22

1 − 𝜙22

.

(5)

Abstract and Applied Analysis 3

2.2. Inventory Policy. In order to meet the dynamic needsof the supply chain model, the supply chain model shownin Figure 1 employs the order-up-to inventory policy. Weassume that the two retailers both apply a fixed order leadtime for orders.The goal of the inventory policy is tomaintaininventory levels at the target inventory levels 𝑞

𝑡. At the

beginning of period 𝑡, the order of quantity 𝑞1,𝑡sent by retailer

1 can be given as follows:

𝑞1,𝑡= 𝑆1,𝑡− 𝑆1,𝑡−1

+ 𝐷1,𝑡−1

. (6)

In (6), 𝑆1,𝑡

is the order-up-to level and it can be deter-mined through lead-time demand by

𝑆1,𝑡= 𝐷𝐿1

1,𝑡+ 𝑧�̂�𝐿1

1,𝑡. (7)

In (7),𝐷𝐿11,𝑡

is the value of lead-time demand and forecastbased on historical sales data. 𝑧 is the normal 𝑧-score thatcan be determined based on the desired service level of theinventory policy. �̂�𝐿1

1,𝑡is the standard deviation of lead-time

demand forecast error.Analogously, the order of quantity sent by retailer 2 also

can be given as follows:

𝑞2,𝑡= 𝑆2,𝑡− 𝑆2,𝑡−1

+ 𝐷2,𝑡−1

. (8)

And the order-up-to level 𝑆2,𝑡

can be given as follows:

𝑆2,𝑡= 𝐷𝐿2

2,𝑡+ 𝑧�̂�𝐿2

2,𝑡. (9)

In (9),𝐷𝐿22,𝑡

is the value of lead-time demand and forecastbased on historical sales data. 𝑧 is the normal 𝑧-score thatcan be determined based on the desired service level of theinventory policy. �̂�𝐿2

2,𝑡is the standard deviation of lead-time

demand forecast error.

2.3. Forecasting Method. Seen from the inventory policyequation, the accuracy of the demand and forecasting of thefuture lead-time 𝐿 period is the most important factor thataffects the inventory level of the retailers in supply chainmodel. While each forecasting error is present, the impactsof different forecastingmethods on the bullwhip effect are notthe same. Then we will introduce three different forecastingmethods: MMES, MA, and ES forecasting methods.

2.3.1.TheMMSE ForecastingMethod. TheMMSE forecastingmethod is short forminimizing themean square error. UndertheMMSE forecasting method, the lead-time demand can beshown as follows:

𝐷𝐿

𝑡= 𝐷𝑡+ 𝐷𝑡+1+ ⋅ ⋅ ⋅ + 𝐷

𝑡+𝐿−1=

𝐿−1

∑𝑖=0

𝐷𝑡+𝑖. (10)

Whenusing theMMSEmethod to predict the future lead-time demand, the total demand expectations within the lead-time 𝐿 are

𝐷𝐿

𝑡= 𝐷𝑡+ 𝐷𝑡+1+ ⋅ ⋅ ⋅ + 𝐷

𝑡+𝐿−1=

𝐿−1

∑𝑖=0

𝐷𝑡+𝑖, (11)

where𝐷𝑡+𝑖

can be characterized as

𝐷𝑡+𝑖= 𝐸 [𝐷

𝑡+𝑖| 𝐷𝑡−1, 𝐷𝑡−2, . . .] . (12)

2.3.2. The MA Forecasting Method. The MA forecastingmethod is short for moving average. In the MA forecastingmethod, the lead-time demand can be determined as follows:

𝐷𝐿

𝑡=𝐿

𝑘

𝑘

∑𝑖=1

𝐷𝑡−𝑖. (13)

In (13),𝐾 is the span (number of date points) for the MAforecasting method.𝐷

𝑡−𝑖is the actual demand in period 𝑡 − 𝑖.

2.3.3.The ES Forecasting Method. TheES forecasting methodis short for exponential smoothing. Using the ES forecastingmethod, the lead-time demand can be considered as follows:

𝐷𝐿

𝑡= 𝛽𝐷𝑡−1+ (1 − 𝛽)𝐷

𝐿

𝑡−1. (14)

In (14), 𝛽 is the smoothing exponent.

3. The Measure of the Bullwhip Effect

In this section, we will talk about the measure of the bullwhipeffect under the MMSE, MA, and ES forecasting methods.The total demand of the two retailers is given as

𝐷𝑡= 𝐷1,𝑡+ 𝐷2,𝑡. (15)

3.1. The Measure of the Bullwhip Effect under the MMSEForecasting Method. Under the MMSE forecasting method,the total demand expectations within the lead-time 𝐿 ofretailer 1 can be determined as

𝐷𝐿1

1,𝑡=

𝐿1

1 − 𝜙1

𝛼𝛿1−𝜙1(1 − 𝜙

1

𝐿1)

(1 − 𝜙1)2𝛼𝛿1

+𝜙1(1 − 𝜙

1

𝐿1)

1 − 𝜙1

𝐷1,𝑡−1

.

(16)

We know that �̂�𝐿11,𝑡

does not depend on 𝑡, so the orderquantity of retailer 1 can be described as follows:

𝑞1,𝑡= (1 +

𝜙1(1 − 𝜙

1

𝐿1)

1 − 𝜙1

)𝐷1,𝑡−1

−𝜙1(1 − 𝜙

1

𝐿1)

1 − 𝜙1

𝐷1,𝑡−2

= (1 + 𝐴1)𝐷1,𝑡−1

− 𝐴1𝐷1,𝑡−2

,

(17)

where 𝐴1= 𝜙1(1 − 𝜙

1

𝐿1)/(1 − 𝜙

1).

Similarly, we can get the order quantity of retailer 2:

𝑞2,𝑡= (1 +

𝜙2(1 − 𝜙

2

𝐿2)

1 − 𝜙2

)𝐷2,𝑡−1

−𝜙2(1 − 𝜙

2

𝐿2)

1 − 𝜙2

𝐷2,𝑡−2

= (1 + 𝐴2)𝐷2,𝑡−1

− 𝐴2𝐷2,𝑡−2

,

(18)

where 𝐴2= 𝜙2(1 − 𝜙

2

𝐿2)/(1 − 𝜙

2).

Total demand that two retailers face is

𝐷𝑡= 𝐷1,𝑡+ 𝐷2,𝑡. (19)


Taking the variance of the total demand, we have

Var (𝐷𝑡) = Var (𝐷

1,𝑡+ 𝐷2,𝑡)

= Var (𝐷1,𝑡) + Var (𝐷

2,𝑡) + 2Cov (𝐷

1,𝑡, 𝐷2,𝑡) .

(20)

The covariance between𝐷1,𝑡

and𝐷2,𝑡

can be described as

Cov (𝐷1,𝑡, 𝐷2,𝑡) =

𝛼

1 − 𝛼Var (𝐷

2,𝑡) . (21)

Total order quantity which two retailers face is

𝑞𝑡= (1 + 𝐴

1)𝐷1,𝑡−1

− 𝐴1𝐷1,𝑡−2

+ (1 + 𝐴2)𝐷2,𝑡−1

− 𝐴2𝐷2,𝑡−2

.

(22)

Taking the variance of the total order quantity, we have

Var (𝑞𝑡) = (1 + 2𝐴

1+ 2𝐴2

1)Var (𝐷

1,𝑡−1) + (1 + 2𝐴

2+ 2𝐴2

2)

× Var (𝐷2,𝑡−1

) − 2𝐴1(1 + 𝐴

1)Cov (𝐷

1,𝑡−1, 𝐷1,𝑡−2

)

+ 2 (1 + 𝐴1) (1 + 𝐴

2)Cov (𝐷

1,𝑡−1, 𝐷2,𝑡−1

) − 2𝐴2

× (1 + 𝐴1)Cov (𝐷

1,𝑡−1, 𝐷2,𝑡−2

) − 2𝐴1(1 + 𝐴

2)

× Cov (𝐷1,𝑡−2

, 𝐷2,𝑡−1

)+ 2𝐴1𝐴2Cov (𝐷

1,𝑡−2, 𝐷2,𝑡−2

)

− 2𝐴2(1 + 𝐴

2)Cov (𝐷

2,𝑡−1, 𝐷2,𝑡−2

) .

(23)

We can prove that

Cov (𝐷1,𝑡−1

, 𝐷1,𝑡−2

) = 𝜙1Var (𝐷

1,𝑡) ,

Cov (𝐷1,𝑡−1

, 𝐷2,𝑡−1

) =𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡−1

, 𝐷2,𝑡−2

) = 𝜙1

𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡−2

, 𝐷2,𝑡−1

) = 𝜙2

𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡−2

, 𝐷2,𝑡−2

) =𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷2,𝑡−1

, 𝐷2,𝑡−2

) = 𝜙2Var (𝐷

2,𝑡) .

(24)

Therefore, (23) can be described as

Var (𝑞𝑡) = (1 + 2𝐴

1+ 2𝐴2

1− 2𝐴1(1 + 𝐴

1) 𝜙1)Var (𝐷

1,𝑡)

+ (1 + 2𝐴2+ 2𝐴2

2− 2𝐴2(1 + 𝐴

2) 𝜙2)Var (𝐷

2,𝑡)

+ (2 + 2 (𝐴1+ 𝐴2) + 3𝐴

1𝐴2− 2𝜙1𝐴2(1 + 𝐴

1)

−2𝜙2𝐴1(1 + 𝐴

2))

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(25)

For simplicity, (25) can be written as

Var (𝑞𝑡) = 𝐺1Var (𝐷

1,𝑡) + 𝐺2Var (𝐷

2,𝑡)

+ 𝐺3

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(26)

In (26) 𝐺1is the coefficient of Var(𝐷

1,𝑡), 𝐺2is the

coefficient of Var(𝐷2,𝑡), and 𝐺

3is the coefficient of 𝛼/(1 −

𝛼)Var(𝐷2,𝑡).

Then, the BWE in MMSE forecasting method is

BWEMMSE =Var (𝑞

𝑡)

Var (𝐷𝑡)=𝛼2𝐺1+ (1 − 𝛼)

2

𝐺2+ 𝛼 (1 − 𝛼)𝐺

3

𝛼2 + (1 − 𝛼)2

+ 2𝛼 (1 − 𝛼).

(27)

3.2. The Measure of the Bullwhip Effect under the MA Fore-castingMethod. In theMA forecastingmethod, the lead-timedemand of retailer 1 can be determined as follows:

𝐷𝐿1

1,𝑡=𝐿1

𝑘

𝑘

∑𝑖=1

𝐷1,𝑡−𝑖

. (28)

Therefore, the total order quantity of the two retailers is

𝑞𝑡= (1 +

𝐿1

𝑘)𝐷1,𝑡−1

−𝐿1

𝑘𝐷1,𝑡−𝑘−1

+ (1 +𝐿2

𝑘)𝐷2,𝑡−1

−𝐿2

𝑘𝐷2,𝑡−𝑘−1

.

(29)


Var (𝑞𝑡) = (1 +

𝐿1

𝑘)2

Var (𝐷1,𝑡−1

) + (𝐿1

𝑘)2

Var (𝐷1,𝑡−𝑘−1

)

+ (1 +𝐿2

𝑘)2

Var (𝐷2,𝑡−1

) + (𝐿2

𝑘)2

Var (𝐷2,𝑡−𝑘−1

)

− 2 (1 +𝐿1

𝑘)𝐿1

𝑘Cov (𝐷

1,𝑡−1, 𝐷1,𝑡−𝑘−1

)

+ 2 (1 +𝐿1

𝑘) (1 +

𝐿2

𝑘)Cov (𝐷

1,𝑡−1, 𝐷2,𝑡−1

)

− 2 (1 +𝐿1

𝑘)𝐿2

𝑘Cov (𝐷

1,𝑡−1, 𝐷2,𝑡−𝑘−1

)

− 2𝐿1

𝑘(1 +

𝐿2

𝑘)Cov (𝐷

1,𝑡−𝑘−1, 𝐷2,𝑡−1

)

+ 2𝐿1

𝑘

𝐿2

𝑘Cov (𝐷

1,𝑡−𝑘−1, 𝐷2,𝑡−𝑘−1

)

− 2 (1 +𝐿2

𝑘)𝐿2

𝑘Cov (𝐷

2,𝑡−1, 𝐷2,𝑡−𝑘−1

) .

(30)


We can prove that

Cov (𝐷1,𝑡−1

, 𝐷1,𝑡−𝑘−1

) = 𝜙1

𝑘 Var (𝐷1,𝑡) ,

Cov (𝐷1,𝑡−1

, 𝐷2,𝑡−1

) =𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡−1

, 𝐷2,𝑡−𝑘−1

) = 𝜙1

𝑘𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡−𝑘−1

, 𝐷2,𝑡−1

) = 𝜙2

𝑘𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,


, 𝐷2,𝑡−𝑘−1

) =𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷2,𝑡−1

, 𝐷2,𝑡−𝑘−1

) = 𝜙2

𝑘 Var (𝐷2,𝑡) .

(31)

The proof of (31) can be seen in the Appendix.Then, taking (31) into (30), we have

Var (𝑞𝑡)

= (1 + 2𝐿1

𝑘+ 2(

𝐿1

𝑘)2

− 2𝜙1

𝑘

(1 +𝐿1

𝑘)𝐿1

𝑘)Var (𝐷

1,𝑡)

+ (1 + 2𝐿2

𝑘+ 2(

𝐿2

𝑘)2

− 2𝜙2

𝑘

(1 +𝐿2

𝑘)𝐿2

𝑘)Var (𝐷

2,𝑡)

+ 2 ((1 +𝐿1

𝑘) (1 +

𝐿2

𝑘) − (1 +

𝐿1

𝑘)𝐿2

𝑘𝜙1

𝑘

−𝐿1

𝑘(1 +

𝐿2

𝑘) 𝜙2

𝑘

+𝐿1𝐿2

𝑘)

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(32)


Var (𝑞𝑡) = 𝐻

1Var (𝐷

1,𝑡) + 𝐻2Var (𝐷

2,𝑡)

+ 𝐻3

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(33)

In (33) 𝐻1is the coefficient of Var(𝐷

1,𝑡), 𝐻2is the

coefficient of Var(𝐷2,𝑡), and 𝐻



Then, the BWE in MMSE forecasting method is

BWEMA =Var (𝑞

𝑡)

Var (𝐷𝑡)=𝛼2𝐻1+ (1 − 𝛼)

2

𝐻2+ 𝛼 (1 − 𝛼)𝐻

3

𝛼2 + (1 − 𝛼)2

+ 2𝛼 (1 − 𝛼).

(34)

3.3. The Measure of the Bullwhip Effect under the ES Forecast-ing Method. Using the ES forecasting method, the lead-timedemand of retailer 1 can be determined as follows:

𝐷𝐿1

1,𝑡= 𝛽𝐷1,𝑡−1

+ (1 − 𝛽)𝐷𝐿1

1,𝑡−1. (35)

According to Zhang [16], 𝐷𝑡+𝑖

= 𝐷𝑡+1

, 𝑖 ≥ 2, the totalorder quantity of two retailers at period t is

𝑞𝑡=(1 + 𝛽

1𝐿1)𝐷1,𝑡− 𝛽1𝐿1𝐷1,𝑡+ (1 + 𝛽

2𝐿2)𝐷2,𝑡− 𝛽2𝐿2𝐷2,𝑡.

(36)


Var (𝑞𝑡) = (1 + 𝛽

1𝐿1)2 Var (𝐷

1,𝑡) + (𝛽

1𝐿1)2 Var (𝐷

1,𝑡)

+ (1 + 𝛽2𝐿2)2 Var (𝐷

2,𝑡) + (𝛽

2𝐿2)2 Var (𝐷

2,𝑡)

− 2𝛽1𝐿1(1 + 𝛽

1𝐿1)Cov (𝐷

1,𝑡, 𝐷1,𝑡)

+ 2 (1 + 𝛽1𝐿1) (1 + 𝛽

2𝐿2)Cov (𝐷

1,𝑡, 𝐷2,𝑡)

− 2𝛽2𝐿2(1 + 𝛽

1𝐿1)Cov (𝐷

1,𝑡, 𝐷2,𝑡)

− 2𝛽1𝐿1(1 + 𝛽

2𝐿2)Cov (𝐷

1,𝑡, 𝐷2,𝑡)

+ 2𝛽1𝐿1𝛽2𝐿2Cov (𝐷

1,𝑡, 𝐷2,𝑡)

− 2𝛽2𝐿2(1 + 𝛽

2𝐿2)Cov (𝐷

2,𝑡, 𝐷2,𝑡) .

(37)

According to Zhang [16], we know that

Var (𝐷1,𝑡) =

1 + (1 − 𝛽1) 𝜙1

𝛽1(2 − 𝛽

1) (1 − (1 − 𝛽

1) 𝜙1)Var (𝐷

1,𝑡) ,

Var (𝐷2,𝑡) =

1 + (1 − 𝛽2) 𝜙2

𝛽2(2 − 𝛽

2) (1 − (1 − 𝛽

2) 𝜙2)Var (𝐷

2,𝑡) .

(38)

We can prove that

Cov (𝐷1,𝑡, 𝐷1,𝑡) =

𝛽1𝜙1

1 − (1 − 𝛽1) 𝜙1

Var (𝐷1,𝑡) ,

Cov (𝐷1,𝑡, 𝐷2,𝑡) =

𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡, 𝐷2,𝑡) =

𝛽2𝜙1

1 − (1 − 𝛽2) 𝜙1

𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡, 𝐷2,𝑡) =

𝛽1𝜙2

1 − (1 − 𝛽1) 𝜙2

𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷1,𝑡, 𝐷2,𝑡)

= (𝛽1𝛽2(1 − 𝜙

1𝜙2(1 − 𝛽

1) (1 − 𝛽

2)))

× ((1 − (1 − 𝛽1) (1 − 𝛽

2)) (1 − (1 − 𝛽

1) 𝜙2)

× (1 − (1 − 𝛽2) 𝜙1))−1 𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷2,𝑡, 𝐷2,𝑡) =

𝛽2𝜙2

1 − (1 − 𝛽2) 𝜙2

Var (𝐷2,𝑡) .

(39)

The proof of (39) can be seen in the Appendix.


Then, taking (39) into (37), we can get

Var (𝑞𝑡)

= (1 +1 − 𝜙1

1 − (1 − 𝛽1) 𝜙1

(2𝛽1𝐿1+2𝛽1

2

𝐿1

2

2 − 𝛽1

))Var (𝐷1,𝑡)

+ (1 +1 − 𝜙2

1 − (1 − 𝛽2) 𝜙2

(2𝛽2𝐿2+2𝛽2

2

𝐿2

2

2 − 𝛽2

))Var (𝐷2,𝑡)

+ (2 (1 + 𝛽1𝐿1) (1 + 𝛽

2𝐿2) −

2𝛽2

2

𝜙1𝐿2(1 + 𝛽

1𝐿1)

1 − (1 − 𝛽2) 𝜙1

−2𝛽1

2

𝜙2𝐿2(1 + 𝛽

2𝐿2)

1 − (1 − 𝛽1) 𝜙2

+2𝛽1

2

𝛽2

2

(1 − 𝜙1𝜙2(1 − 𝛽

1) (1 − 𝛽

2))

2𝛽1

2

𝛽2

2

(1− 𝜙1𝜙2(1− 𝛽

1)(1− 𝛽

2))(1− (1 − 𝛽

2) 𝜙1))

×𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(40)


Var (𝑞𝑡) = 𝑀

1Var (𝐷

1,𝑡) + 𝑀

2Var (𝐷

2,𝑡)

+ 𝑀3

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(41)

In (41) 𝑀1is the coefficient of Var(𝐷

1,𝑡), 𝑀2is the

coefficient of Var(𝐷2,𝑡), and 𝑀



Then, the BWE in ES forecasting method is

BWEES =Var (𝑞

𝑡)

Var (𝐷𝑡)

=𝑀1Var (𝐷

1,𝑡) + 𝑀

2+𝑀3(𝛼/ (1 − 𝛼))Var (𝐷

2,𝑡)

Var (𝐷1,𝑡) + Var (𝐷

2,𝑡) + 2 (𝛼/ (1 − 𝛼))Var (𝐷

2,𝑡)

=𝑀1(Var (𝐷

1,𝑡) /Var (𝐷

2,𝑡))+ 𝑀

2+ 𝑚3(𝛼/ (1 − 𝛼))

(Var (𝐷1,𝑡) /Var (𝐷

2,𝑡)) + 1 + 2 (𝛼/ (1 − 𝛼))

=𝑀1(𝛼/ (1 − 𝛼))

2

+𝑀2+𝑀3(𝛼/ (1 − 𝛼))

(𝛼/ (1 − 𝛼))2

+ 1 + 2 (𝛼/ (1 − 𝛼))

=𝛼2𝑀1+ (1 − 𝛼)

2

𝑀2+ 𝛼 (1 − 𝛼)𝑀

3

𝛼2 + (1 − 𝛼)2

+ 2𝛼 (1 − 𝛼).

(42)

4. Behavior of the Bullwhip Effect Measureand Numerical Simulation

With the expression of the bullwhip effect under threeforecasting methods, we can discuss the bullwhip effect byalgebraic analysis and numerical simulation.

0 0.2 0.4 0.6 0.8 11.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2L1 = L2 = 3, 𝜙2

𝜙1

= 0.6

BWE M

MSE

𝛼 = 0.4

𝛼 = 0.6

𝛼 = 0.8

Figure 2: Impact of 𝜙1on bullwhip effect for different 𝛼 under the

MMSE.

4.1. The Analysis of Parameters under the MMSE ForecastingMethod. Figures 2–4 simulate the expression of the bullwhipeffect under the MMSE forecasting method to illustrate theimpact of parameters on the bullwhip effect. Duc et al. [28]indicated that the bullwhip effect occurs only when a positiveautoregressive relationship exists in the demand processunder MMSE forecasting method. So, we just consider thecircumstance that 𝜙

1varies from 0 to 1. Figure 2 indicates

that the bullwhip effect increases slowly with the increase of𝜙1, and the bullwhip effect begins to decrease rapidly when

it reaches the maximum value. The bullwhip effect does notoccur onlywhen𝜙

1increases to a certain value, and the values

are different with different 𝛼. For 𝛼, when 𝜙1is smaller than

a certain value, the bullwhip effect becomes smaller with 𝛼becoming larger, when 𝜙

1is larger than that certain value

and smaller than another certain value, the bullwhip effectbecomes larger with 𝛼becoming larger, when 𝜙

1is larger than

another certain value, the bullwhip effect becomes smallerwith 𝛼becoming larger.

Figure 3 reveals that the bullwhip effect decreases grad-ually to the minimum value and then it begins to increasewith the increase of 𝛼. For the different 𝐿

1, the larger it is, the

larger the bullwhip effect is. The result suggests when retailer1 occupies half of the market share and they set the short leadtime, the bullwhip effect value will be the lowest.

Figure 4 depicts the impact of 𝜙1on the bullwhip effect.

We observe that the bullwhip effect increases graduallyto the maximum value with the increase of 𝜙

1, and the

bullwhip effect begins to decrease rapidly when it reachesthe maximum value. The bullwhip effect does not occur onlywhen 𝜙

1increases to a certain value, and those values are

different for different 𝐿1. For different 𝐿

1, the larger it is, the

larger the bullwhip effect is.


2.6

2.8

3

3.2

3.4

3.6

3.8L2 = 3, 𝜙1 = 𝜙2 = 0.6

BWE M

MSE

𝛼

L1 = 2

L1 = 3

L1 = 4

0 0.2 0.4 0.6 0.8 1

Figure 3: Impact of 𝛼 on bullwhip effect for different 𝐿1under the

MMSE.

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

BWE M

MSE

L1 = 2

L1 = 3

L1 = 4

0 0.2 0.4 0.6 0.8 1𝜙1

L2 = 3, 𝛼 = 0.4, 𝜙2 = 0.6

Figure 4: Impact of 𝜙1on bullwhip effect for different 𝐿

1under the

MMSE.

4.2. The Analysis of Parameters under the MA ForecastingMethod. Figures 5–9 simulate the expression of the bullwhipeffect under the MA forecasting method to illustrate theimpact of parameters on the bullwhip effect.

Figure 5 shows that the bullwhip effect decreases with theincrease of 𝜙

1. With different 𝛼 the bullwhip effect changes

only a little. So 𝛼 hardly affects the bullwhip effect in thecircumstance of different 𝜙

1.

Figure 6 indicates that the bullwhip effect decreasesslowly all the time with the increase of 𝜙

1. And, in this

0 0.5 12

3

4

5

6

7

8

9

−1 −0.5

𝛼 = 0.4

𝛼 = 0.6

𝛼 = 0.8

𝜙1

L1 = L2 = 3, 𝜙2 = 0.6, k = 3

BWE M

AFigure 5: Impact of 𝜙

1on bullwhip effect for different 𝛼 under the

MA.

2

3

4

5

6

7

8

𝜙1

L1 = L2 = 3, 𝜙2 = 0.6, 𝛼 = 0.4

k = 3

k = 5

k = 7

BWE M

A

0 0.5 1−1 −0.5

Figure 6: Impact of 𝜙1on bullwhip effect for different 𝑘 under the

MA.

situation, we shift the span (number of date points) 𝑘 ofretailer 1, and we can find that the bullwhip effect becomessmaller with the increase of 𝑘. We can come to the conclusionthat 𝑘 is a key factor to affect the bullwhip effect.

By observing Figure 7, we know when 𝛼 takes the valueof zero, the bullwhip effect value is fixed no matter whatvalue 𝐿

1is. Then the bullwhip effect value increases to the

maximum value firstly, and after that it begins to decreasewith the increase of𝛼. In this situation,𝐿

1influences theBWE

obviously, the smaller the 𝐿1is, the smaller the BWE is.


2.5

3

3.5

4

4.5

5

5.5

6

6.5

𝛼

L1 = 2

L1 = 3

L1 = 4

0 0.2 0.4 0.6 0.8 1

L2 = 3, 𝜙1 = 𝜙2 = 0.6, k = 3

BWE M

A


MA.

2

2.5

3

3.5

4

4.5

5

5.5

𝛼

0 0.2 0.4 0.6 0.8 1

L1 = L2 = 3, 𝜙1 = 𝜙2 = 0.6

k = 3

k = 5

k = 7

BWE M

A

Figure 8: Impact of 𝛼 on bullwhip effect for different 𝑘 under theMA.

Figure 8 shows that the trend of the bullwhip effect isincreased first to the maximum and declined gradually withthe increase of 𝛼. The larger the 𝑘 is, the smaller the BWE is.

Figure 9 which is similar to Figure 6 simulates the impactof 𝜙1on the bullwhip effect based on different 𝐿

1. The

bullwhip effect decreases slowly all the time with the increaseof 𝜙1. And we may find that the 𝐿

1influences the BWE

obviously; the bullwhip effect becomes smaller with thedecrease of 𝐿

1.

3

4

5

6

7

8

9

𝜙1

BWE M

A

L1 = 2

L1 = 3L1 = 4

L2 = 3, 𝜙2 = 0.6, k = 3, 𝛼 = 0.4

0 0.5 1−1 −0.5


1under the

MA.

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

𝛼 = 0.4

𝛼 = 0.6

𝛼 = 0.8

𝜙1

BWE E

S

L1 = L2 = 3, 𝜙2 = 0.6, 𝛽1 = 𝛽2 = 0.4

0 0.5 1−1 −0.5

Figure 10: Impact of 𝜙1on bullwhip effect for different 𝛼 under the

ES.

4.3. The Analysis of Parameters under the ES ForecastingMethod. Figures 10–14 simulate the expression of the bull-whip effect under the ES forecasting method to illustrate theimpact of parameters on the bullwhip effect.

Figure 10 declares the impact of 𝜙1under various 𝛼. The

bullwhip effect decreases continuously with the increase of 𝛼.When 𝜙

1takes the value of 0.7 approximately, the bullwhip

effect values of three different 𝛼 are the same; when 𝜙1is less

than 0.7, the smaller the 𝛼 is, the smaller the bullwhip effectvalue is; when 𝜙

1is more than 0.7, the result is opposite.


1

2

3

4

5

6

7

8

9

𝜙1

BWE E

S

𝛽1 = 0.4𝛽1 = 0.6

𝛽1 = 0.8

L1 = L2 = 3, 𝜙2 = 0.6, 𝛽2 = 0.4, 𝛼 = 0.4

0 0.5 1−1 −0.5

Figure 11: Impact of 𝜙1on bullwhip effect for different 𝛽

1under the

ES.

0 0.2 0.4 0.6 0.8 13

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

BWE E

S

𝛽1 = 0.4𝛽1 = 0.6

𝛽1 = 0.8

L1 = L2 = 3, 𝜙1 = 𝜙2 = 0.6, 𝛽2 = 0.4

𝛼

Figure 12: Impact of 𝛼 on bullwhip effect for different 𝛽1under the

ES.

As can be seen from Figure 11, the bullwhip effectdecreases all the time in pace with the increase of 𝜙

1. And,

in this situation, we can observe that the smaller the 𝛽1is, the

smaller the bullwhip effect value is. This phenomenon showsthat 𝛽

1is an important factor to influence the bullwhip effect

under the ES forecasting method.Figure 12 indicates the impact of 𝛼 on bullwhip effect for

different 𝛽1under the ES. When 𝛽

1takes the value of 0.4,

the bullwhip effect decreases slowly first, after that it increases

0 0.2 0.4 0.6 0.8 12

2.5

3

3.5

4

4.5

5

BWE E

S

L1 = 0.4

L1 = 0.6L1 = 0.8

𝛼

L2 = 3, 𝜙1 = 𝜙2 = 0.6, 𝛽1 = 𝛽2 = 0.4


ES.

1.5

2

2.5

3

3.5

4

4.5

5

5.5

𝜙1

BWE E

S

L1 = 2

L1 = 3L1 = 4

L2 = 3, 𝜙2 = 0.6, 𝛽1 = 𝛽2 = 0.4, 𝛼 = 0.4

0 0.5 1−1 −0.5


1under the

ES.

gradually. However, when the value of 𝛽1is 0.6 and 0.8, the

bullwhip effect keeps increasing rapidly.Figure 13 reveals the impact of 𝛼 on bullwhip effect for

different 𝐿1under the ES. When 𝐿

1takes the values of 0.4

and 0.6, the bullwhip effect decreases slowly first, after that itincreases gradually. And the minimum value of the bullwhipeffect occurs as the value of 𝛼 is 0.5 approximately. This phe-nomenon indicates that the intense competition between tworetailers can increase the bullwhip effect. However, when the


0 0.2 0.4 0.6 0.8 12.5

3

3.5

4

4.5

5

5.5L1 = L2 = 3, 𝜙1 = 𝜙2 = 0.6

BWE

𝛼

MMSEMA (k = 3)ES (𝛽1 = 𝛽2 = 0.5)

Figure 15: Comparison of three different forecasting methods byvarying 𝛼 (𝑘 = 3).

0 0.2 0.4 0.6 0.8 11.8

2

2.2

2.4

2.6

2.8

3

3.2

BWE

𝛼

MMSEMA (k = 9)

L1 = L2 = 3, 𝜙1 = 𝜙2 = 0.6

ES (𝛽1 = 𝛽2 = 0.2)


value of 𝐿1is 0.8, the bullwhip effect keeps increasing rapidly.

This result is analogous with the situation in Figure 12.Similar to Figure 11, Figure 14 shows the impact of 𝜙

1on

the bullwhip effect based on different 𝐿1under the ES. The

bullwhip effect decreases all the time in pacewith the increaseof 𝜙1. For different 𝐿

1, the smaller the 𝐿

1is, the smaller the

bullwhip effect value is. This situation indicates 𝐿1is also an

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

BWE

0 0.2 0.4 0.6 0.8 1𝛼

MMSEMA (k = 19)

L1 = L2 = 3, 𝜙1 = 𝜙2 = 0.6

ES (𝛽1 = 𝛽2 = 0.1)


important factor to influence the bullwhip effect under the ESforecasting method.

4.4. The Comparison of the Three Different Forecasting Meth-ods. According to the analyses above, we select appropriateparameters to compare the bullwhip effect under threedifferent forecasting methods. We set 𝐿

1= 𝐿2= 3 and 𝜙

1=

𝜙2= 0.6. Then we choose appropriate 𝑘, 𝛽

1, and 𝛽

2. From

Zhang [16], theMMSEmethodminimizes the variance of theforecasting error among all the linear forecasting methods. Itobviously leads to the lowest average cost among the threeforecasting approaches. If we impose the constraint of equaldata age (2/(𝑘 + 1)) for the MA and ES forecasts by setting𝛽1= 𝛽2= 2/(𝑘 + 1), we can also achieve that.

FromFigures 15–17 the trends of the three bullwhip effectsare the same. However the values of them change obviously.

According to Figure 15, we install 𝑘 = 3 and correspond-ingly 𝛽

1= 𝛽2= 0.5. We may find out that BWEMMSE is

the lowest of all. BWEMMSE and BWEES decrease firstly tothe minimum value and then increase with the increase of 𝛼.However, BWEMA has opposite trend.When𝛼 is smaller thana certain value, BWEMA is lower than BWEES; when𝛼 is largerthan the certain value and smaller than another certain value,BWEMA is higher than BWEES; and when 𝛼 is larger thananother certain value, BWEMA is lower than BWEES again.It means that the MMSE forecasting method is the best toforecast lead-time demand in this situation.

Figure 16 reveals that the bullwhip effects under threeforecastingmethods of 𝑘 = 9 and𝛽

1= 𝛽2= 0.2 have the same

trends with the circumstance of 𝑘 = 3 and 𝛽1= 𝛽2= 0.5.

But BWEMMSE is no longer the lowest of all, and it becomesthe highest of the three oppositely. BWEMA and BWEES are


the same as Figure 15. It means that when 𝛼 is larger than acertain value and smaller than another certain value, the ESforecasting method is the best. In the other situation, the MAis themost attractive one.We can come to the conclusion thatwewould better adopt ES forecastingmethodwith the intensecompetition between two retailers.

In Figure 17, we set 𝑘 = 19 and 𝛽1= 𝛽2= 0.1. In this

circumstance, BWEMMSE is the highest all the time regardlessdifferent 𝛼. BWEES is a fixed value with the increase of 𝛼.BWEMA is the lowest of all.This phenomenon reveals that theMA method is the best forecasting method whatever 𝛼 is aslong as 𝑘 = 19 or 𝑘 is larger.

5. Conclusions

This paper contrasts the bullwhip effect based on threedifferent forecasting methods for a simple inventory systemwith an AR(1) demand. Previous research studied the bull-whip effect under different forecasting methods. However,the previous articles did not consider the impact of marketshare. This paper investigated the effect of the lead time, theautoregressive coefficient, and themarket share on a bullwhipeffect measure in a simple two-stage supply chain with twosuppliers and two retailers, and the two retailers both employthe order-up-to inventory policy for replenishments.

We are surprised to find that the shorter the lead timeis, the smaller the bullwhip effect is under three differentforecasting methods. If the MMSE forecasting method isused, increasing autoregressive coefficient does not alwaysreduce the bullwhip effect when the market share or leadtime is varying. But increasing autoregressive coefficient canreduce the bullwhip effect effectively using the MA or ESforecasting method. Not only the autoregressive coefficientand lead time affect the bullwhip effect, but also marketshare and the average date age are two key factors to influ-ence the bullwhip effect. Under MMSE and ES forecastingmethod the bullwhip effect can be reduced when the intensecompetition between the two retailers occurs. However, thebullwhip effect is proved to be maximum value when theintense competition between the two retailers occur usingMA forecasting method. Average date age is proportional tothe bullwhip effect when the MA method is used, and it isinversely proportional to the bullwhip effect when the ESmethod is used. And the bigger the average date age is, thesmaller the bullwhip effect is.

The analyses above suggest that the bullwhip effectcannot be reduced by simply increasing the autoregressivecoefficient. We must consider other parameters that affectthe bullwhip effect as well. The MMSE forecasting methodis likely to be the best method to qualify the bullwhip effectwhen the average date age is lower. Butwith the increase of theaverage date age the ES forecastingmethod seems better. Andwith the increase of the average date age, the bullwhip effectunder the MA forecasting method turns into the lowest.

In the end, we must point out that quantifying thebullwhip effect and investigating its behavior are helpful forreducing the influence of the bullwhip effect in supply chains.However, firstly, controller may pay more attention to the

inventory cost in practice. Hence, investigations of the impactof parameters for the inventory cost would be meaningful.Secondly, if we combine the control and bifurcation theory tostudy the bullwhip effect, wemay produce better conclusions.There are also urgent needs for us to continue to study inthis area. Only when we do more and more research, can wereduce the bullwhip effect effectively.

Appendix

(1) The Proof of (31) Is as Follows.After iteration computationfor (2), we have

𝐷1,𝑡−1

= (1 + 𝜙1+ ⋅ ⋅ ⋅ + 𝜙

1

𝑘−1

) 𝛼𝛿1+ 𝜙1

𝑘

𝐷1,𝑡−𝑘−1

+ 𝛼𝜙1

𝑘−1

𝜀1,𝑡−𝑘

+ 𝛼𝜙1

𝑘−2

𝜀1,𝑡−𝑘+1

+ ⋅ ⋅ ⋅ + 𝛼𝜀1,𝑡−1

=1 − 𝜙1

𝑖+1

1 − 𝜙1

𝛼𝛿1+ 𝜙1

𝑘

𝐷1,𝑡−𝑘−1

+

𝑘−1

∑𝑗=0

𝛼𝜙1

𝑘−1−𝑗

𝜀1,𝑡−𝑘+𝑗

.

(A.1)

So, we can get that

Cov (𝐷1,𝑡−1

, 𝐷1,𝑡−𝑘−1

)

= Cov(1 − 𝜙1

𝑖+1

1 − 𝜙1

𝛼𝛿1+ 𝜙1

𝑘

𝐷1,𝑡−𝑘−1

+

𝑘−1

∑𝑗=0

𝛼𝜙1

𝑘−1−𝑗


, 𝐷1,𝑡−𝑘−1

) = 𝜙1

𝑘 Var (𝐷1,𝑡) ,

Cov (𝐷1,𝑡−1

, 𝐷2,𝑡−𝑘−1

)

= Cov(1 − 𝜙1

𝑖+1

1 − 𝜙1

𝛼𝛿1+ 𝜙1

𝑘

𝐷1,𝑡−𝑘−1

+

𝑘−1

∑𝑗=0

𝛼𝜙1

𝑘−1−𝑗


, 𝐷2,𝑡−𝑘−1

)

= 𝜙1

𝑘Cov (𝐷1,𝑡−𝑘−1

, 𝐷2,𝑡−𝑘−1

) = 𝜙1

𝑘𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(A.2)

Analogously, we may get the following:


, 𝐷2,𝑡−1

) = 𝜙2

𝑘𝛼

1 − 𝛼Var (𝐷

2,𝑡) ,

Cov (𝐷2,𝑡−1

, 𝐷2,𝑡−𝑘−1

) = 𝜙2

𝑘 Var (𝐷2,𝑡) .

(A.3)


(2) The Proof of (39) Is as Follows. From (35),

Cov (𝐷1,𝑡, 𝐷1,𝑡) = Cov(𝐷

1,𝑡,

∞

∑𝑖=0

𝛽1(1 − 𝛽

1)𝑖

𝐷1,𝑡−𝑖−1

)

= 𝛽1

∞

∑𝑖=0

(1 − 𝛽1)𝑖Cov (𝐷

1,𝑡, 𝐷1,𝑡−𝑖−1

)

= 𝛽1

∞

∑𝑖=0

(1 − 𝛽1)𝑖

𝜙𝑖+1

1Var (𝐷

1,𝑡)

=𝛽1𝜙1

1 − (1 − 𝛽1) 𝜙1

Var (𝐷1,𝑡) ,

Cov (𝐷1,𝑡, 𝐷2,𝑡) = Cov(𝐷

1,𝑡,

∞

∑𝑖=0

𝛽2(1 − 𝛽

2)𝑖

𝐷2,𝑡−𝑖−1

)

= 𝛽2

∞

∑𝑖=0

(1 − 𝛽2)𝑖Cov (𝐷

1,𝑡, 𝐷2,𝑡−𝑖−1

)

= 𝛽2

∞

∑𝑖=0

(1 − 𝛽2)𝑖

𝜙𝑖+1

1

𝛼

1 + 𝛼Var (𝐷

2,𝑡)

=𝛽2𝜙1

1 − (1 − 𝛽2) 𝜙1

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(A.4)

The same,

Cov (𝐷2,𝑡, 𝐷2,𝑡) =

𝛽2𝜙2

1 − (1 − 𝛽2) 𝜙2

Var (𝐷2,𝑡) ,

Cov (𝐷1,𝑡, 𝐷2,𝑡) =

𝛽1𝜙2

1 − (1 − 𝛽1) 𝜙2

𝛼

1 − 𝛼Var (𝐷

2,𝑡) .

(A.5)

From (35), we can also get the following:

Cov (𝐷1,𝑡, 𝐷2,𝑡)

= Cov(∞

∑𝑖=0

𝛽1(1 − 𝛽

1)𝑖

𝐷1,𝑡−𝑖−1

,

∞

∑𝑖=0

𝛽2(1 − 𝛽

2)𝑖

𝐷2,𝑡−𝑖−1

)

= 𝛽1𝛽2Cov(

∞

∑𝑖=0

(1 − 𝛽1)𝑖

𝐷1,𝑡−𝑖−1

,

∞

∑𝑖=0

(1 − 𝛽2)𝑖

𝐷2,𝑡−𝑖−1

)

= 𝛽1𝛽2(

∞

∑𝑖=0

∞

∑𝑗=𝑖+1

(1 − 𝛽1)𝑖

(1 − 𝛽2)𝑗

𝜙𝑗−𝑖

1,

∞

∑𝑖=0

𝑖

∑𝑗=0

(1 − 𝛽1)𝑖

(1 − 𝛽2)𝑗

𝜙𝑖−𝑗

2)Cov (𝐷

1,𝑡, 𝐷2,𝑡)

= 𝛽1𝛽2(

∞

∑𝑖=0

(1 − 𝛽1)𝑖

(1 − 𝛽2)𝑗𝜙1(1 − 𝛽

2)

1 − (1 − 𝛽2) 𝜙1

,

∞

∑𝑖=0

(1 − 𝛽1)𝑖

(1 − 𝛽2)𝑖

𝜙2

𝑖1 − ((1 − 𝛽

2) /𝜙2)𝑖+1

1 − ((1 − 𝛽2) /𝜙2))

× Cov (𝐷1,𝑡, 𝐷2,𝑡)

=𝛽1𝛽2(1 − 𝜙

1𝜙2(1 − 𝛽

1) (1 − 𝛽

2))

(1 − (1 − 𝛽1) (1 − 𝛽

2)) (1 − (1 − 𝛽

1) 𝜙2) (1 − (1 − 𝛽

2) 𝜙1)

×𝛼

1 − 𝛼Var (𝐷

2,𝑡)

(A.6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to thank the reviewers for theircareful reading and for providing some pertinent suggestions.The research was supported by the National Natural ScienceFoundation of China (no. 61273231) and Doctoral Fund ofMinistry of Education of China (Grant no. 20130032110073),and it was supported by Tianjin University Innovation Fund.

References

[1] J. W. Forrester, “Industrial dynamicsa major breakthrough fordecision making,” Harvard Business Review, vol. 36, no. 4, pp.37–66, 1958.

[2] J. W. Forrester, Industrial Dynamics, MIT Press, Cambridge,Mass, USA, 1961.

[3] A. S. Blinder, “Inventories and sticky prices,” American Eco-nomic Review, vol. 72, pp. 334–349, 1982.

[4] O. J. Blanchard, “The production and inventory behavior ofAmerican automobile industry,” Journal of Political Economy,vol. 91, pp. 365–400, 1983.

[5] J. L. Burbidge, “Automated production control with a simulationcapability,” IFIP Working Paper WG 5(7), Copenhagen, Den-mark, 1984.

[6] A. S. Blinder, “Can the production smoothing model of inven-tory behavior be safe?” Quarterly Journal of Economics, vol. 101,pp. 431–454, 1986.

[7] J. Kahn, “Inventories and the volatility of production,”AmericanEconomics Review, vol. 77, pp. 667–679, 1987.

[8] J. Sterman, “Optimal policy for a multi-product, dynamic,nonstationary inventory problem,”Management Science, vol. 12,pp. 206–222, 1989.

[9] H. L. Lee, V. Padmanabhan, and S. Whang, “Informationdistortion in a supply chain: the bullwhip effect,” ManagementScience, vol. 43, no. 4, pp. 546–558, 1997.

[10] H. L. Lee, P. Padmanabhan, and S. Whang, “Bullwhip effect ina supply chain,” Sloan Management Review, vol. 38, pp. 93–102,1997.


[11] R. Metters, “Quantifying the bullwhip effect in supply chains,”Journal of Operations Management, vol. 15, no. 2, pp. 89–100,1997.

[12] S. C. Graves, “A single-item inventory model for a non-stationary demand process,” Manufacturing and Service Oper-ations Management, vol. 1, pp. 50–61, 1999.

[13] F. Chen, Z. Drezner, J. K. Ryan, and D. Simchi-Levi, “Quantify-ing the bullwhip effect in a simple supply chain: the impact offorecasting, lead times, and information,”Management Science,vol. 46, no. 3, pp. 436–443, 2000.

[14] F. Chen, J. K. Ryan, and D. Simchi-Levi, “The impact ofexponential smoothing forecasts on the bullwhip effect,” NavalResearch Logistics, vol. 47, no. 4, pp. 269–286, 2000.

[15] K. Xu, Y. Dong, and P. T. Evers, “Towards better coordinationof the supply chain,” Transportation Research E: Logistics andTransportation Review, vol. 37, no. 1, pp. 35–54, 2000.

[16] X. Zhang, “The impact of forecasting methods on the bullwhipeffect,” International Journal of Production Economics, vol. 88,no. 1, pp. 15–27, 2004.

[17] H. T. Luong, “Measure of bullwhip effect in supply chainswith autoregressive demand process,” European Journal ofOperational Research, vol. 180, no. 3, pp. 1086–1097, 2007.

[18] T. T. H. Duc, H. T. Luong, and Y. D. Kim, “Effect of the third-party warehouse on bullwhip effect and inventory cost in supplychains,” International Journal of Production Economics, vol. 124,no. 2, pp. 395–407, 2010.

[19] H. R. Karimi, N. A. Duffie, and S. Dashkovskiy, “Localcapacity 𝐻

∞control for production networks of autonomous

work systems with time-varying delays,” IEEE Transactions onAutomation Science and Engineering, vol. 7, no. 4, pp. 849–857,2010.

[20] B. Nepal, A. Murat, and R. Babu Chinnam, “The bullwhip effectin capacitated supply chains with consideration for product life-cycle aspects,” International Journal of Production Economics,vol. 136, no. 2, pp. 318–331, 2012.

[21] S. Dashkovskiy, H. R. Karimi, and M. Kosmykov, “A Lyapunov-Razumikhin approach for stability analysis of logistics networkswith time-delays,” International Journal of Systems Science, vol.43, no. 5, pp. 845–853, 2012.

[22] S. Dashkovskiy, H. R. Karimi, and M. Kosmykov, “Stabilityanalysis of logistics networks with time-delays,” ProductionPlanning and Control, vol. 24, no. 7, pp. 567–574, 2013.

[23] A. Mehrsai, H. Karimi, K. Thoben, and B. Scholz-Reiter,“Application of learning pallets for real-time scheduling by theuse of radial basis function network,” Neurocomputing, vol. 101,pp. 82–93, 2013.

[24] A. Mehrsai, H. Karimi, and B. Scholz-Reiter, “Toward learningautonomous pallets by using fuzzy rules, applied in a Conwipsystem,” International Journal of AdvancedManufacturing Tech-nology, vol. 64, no. 5–8, pp. 1131–1150, 2013.

[25] J. H. Ma and X. G. Ma, “A comparison of bullwhip effectunder various forecasting techniques in supply chains with tworetailers,” Abstract and Applied Analysis, vol. 2013, Article ID796384, 14 pages, 2013.

[26] W. Shen and J. H. Jiang, “Parameter matching analysis ofhydraulic hybrid excavators based on dynamic programmingalgorithm ,” Journal of Applied Mathematics, vol. 2013, ArticleID 615608, 10 pages, 2013.

[27] W. Shen, J. Jianga, X. Sub, and H. R. Karimic, “Control strategyanalysis of the hydraulic hybrid excavator,” Journal of theFranklin Institute, 2014.

[28] T. T. H. Duc, H. T. Luong, and Y. Kim, “A measure of bullwhipeffect in supply chains with a mixed autoregressive-movingaverage demand process,” The European Journal of OperationalResearch, vol. 187, no. 1, pp. 243–256, 2008.

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Research Article Inherent Complexity Research on the ...downloads.hindawi.com/journals/aaa/2014/306907.pdf · and so forth also recognized the existence of the bullwhip e ect in supply