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The Bullwhip Effect caused by Information Distortion in a complex Supply Chain under Exponential Smoothing Forecast JUNHAI MA Tianjin University College of Management Tianjin 300072 CHINA [email protected] WEIHUA LAI Tianjin University College of Management Tianjin 300072 CHINA [email protected] XINAOGANG MA Tianjin University College of Management Tianjin 300072 CHINA [email protected] Abstract: The bullwhip effect is the phenomenon in which information on demand is transferred in the form of orders between the nodes of a supply chain tends to be distorted when it moves from downstream to upstream. In this paper, we measure the impact of bullwhip effect under Exponential Smoothing Forecast for a simple, two-stage supply chain which consists of one supplier and two retailers. And it is a simple replenishment system where a first- order autoregressive process describes the customer demand and an order-up-to inventory policy characterizes the replenishment decision. We get the influence of information distortion on the bullwhip effect through investigating the impacts of autoregressive coefficient, the lead-time, the smoothing parameter, market competition degree, and the consistency of demand volatility on the bullwhip effect by using algebraic analysis and numerical simulation. And, we also find the ways in which these parameters affect the bullwhip effect are different. Finally, we discuss some measures to mitigate the influence of information distortion on the bullwhip effect. Key–Words: Information distortion, Bullwhip effect, Complex Supply chain, Two retailers model, Exponential Smoothing 1 Introduction The bullwhip effect suggests that the demand variabil- ity is magnified as a customer demand signal is trans- formed through various stages of a serial supply chain. It is one of the most widely investigated phenomena in the modern supply chain management research. Infor- mation on demand flows have a direct impact on the production, scheduling, inventory control and deliv- ery plans of individual members in the supply chain. Distortion in information flow is found to be a major problem. Forrester [1] provided some of the first empirical evidence of the bullwhip effect and discusses its caus- es. After that, more and more researchers recognized the existence of this phenomenon in supply chains. Blanchard [2], Blinder [3], and Kahn [4] found evi- dence of inventory volatility which is similar to the bullwhip effect. Sterman [5] illustrated the bullwhip effect, and attributed the phenomenon to the players’ irrational behavior through an experiment on the beer game. Lee et al [6,7] identified the use of demand forecasting, supply shortages, lead times, batch order- ing, and price variations are the five main causes of the bullwhip effect. Since then, many researches on bullwhip effect with respect to the five main factors. Using different forecasting methods would get d- ifferent information on the demand, so it is impor- tant to investigate the influence of forecasting meth- ods on the bullwhip effect. From past studies we could also find forecasting methods are considered as one of the most important causes because the inven- tory system of a supply chain is directly affected by the forecasting method in the five main factors of the bullwhip effect. Several researchers have examined the effects of simple forecasting techniques such as moving average or exponential smoothing on supply chains with autoregressive demand process. Chen et al [8,9] studied the impact of exponential smoothing forecasting technique on the bullwhip effect in a sim- ple, two-stage supply chain with one supplier and one retailer. Then, Zhang [10] considered the impact of forecasting methods on the bullwhip effect for a sim- ple replenishment system in which a first-order au- toregressive process describes the customer demand and an order-up-to inventory policy characterizes the replenishment decision. And he also compared the impact of different forecasting methods such as the minimum mean-squared error (MMSE), exponential smoothing(ES) method and the moving average (MA) method on the bullwhip effect. Feng and Ma [11] did a Demand and Forecasting in Supply Chains Based on ARMA(1,1) Demand Process. Bayraktar et al [12] WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma E-ISSN: 2224-2880 110 Volume 15, 2016
Transcript

The Bullwhip Effect caused by Information Distortion in a complexSupply Chain under Exponential Smoothing Forecast

JUNHAI MATianjin University

College of ManagementTianjin 300072

[email protected]

WEIHUA LAITianjin University

College of ManagementTianjin 300072

[email protected]

XINAOGANG MATianjin University

College of ManagementTianjin 300072

[email protected]

Abstract: The bullwhip effect is the phenomenon in which information on demand is transferred in the form oforders between the nodes of a supply chain tends to be distorted when it moves from downstream to upstream. Inthis paper, we measure the impact of bullwhip effect under Exponential Smoothing Forecast for a simple, two-stagesupply chain which consists of one supplier and two retailers. And it is a simple replenishment system where a first-order autoregressive process describes the customer demand and an order-up-to inventory policy characterizes thereplenishment decision. We get the influence of information distortion on the bullwhip effect through investigatingthe impacts of autoregressive coefficient, the lead-time, the smoothing parameter, market competition degree, andthe consistency of demand volatility on the bullwhip effect by using algebraic analysis and numerical simulation.And, we also find the ways in which these parameters affect the bullwhip effect are different. Finally, we discusssome measures to mitigate the influence of information distortion on the bullwhip effect.

Key–Words: Information distortion, Bullwhip effect, Complex Supply chain, Two retailers model, ExponentialSmoothing

1 IntroductionThe bullwhip effect suggests that the demand variabil-ity is magnified as a customer demand signal is trans-formed through various stages of a serial supply chain.It is one of the most widely investigated phenomena inthe modern supply chain management research. Infor-mation on demand flows have a direct impact on theproduction, scheduling, inventory control and deliv-ery plans of individual members in the supply chain.Distortion in information flow is found to be a majorproblem.

Forrester [1] provided some of the first empiricalevidence of the bullwhip effect and discusses its caus-es. After that, more and more researchers recognizedthe existence of this phenomenon in supply chains.Blanchard [2], Blinder [3], and Kahn [4] found evi-dence of inventory volatility which is similar to thebullwhip effect. Sterman [5] illustrated the bullwhipeffect, and attributed the phenomenon to the players’irrational behavior through an experiment on the beergame. Lee et al [6,7] identified the use of demandforecasting, supply shortages, lead times, batch order-ing, and price variations are the five main causes ofthe bullwhip effect. Since then, many researches onbullwhip effect with respect to the five main factors.

Using different forecasting methods would get d-

ifferent information on the demand, so it is impor-tant to investigate the influence of forecasting meth-ods on the bullwhip effect. From past studies wecould also find forecasting methods are considered asone of the most important causes because the inven-tory system of a supply chain is directly affected bythe forecasting method in the five main factors of thebullwhip effect. Several researchers have examinedthe effects of simple forecasting techniques such asmoving average or exponential smoothing on supplychains with autoregressive demand process. Chen etal [8,9] studied the impact of exponential smoothingforecasting technique on the bullwhip effect in a sim-ple, two-stage supply chain with one supplier and oneretailer. Then, Zhang [10] considered the impact offorecasting methods on the bullwhip effect for a sim-ple replenishment system in which a first-order au-toregressive process describes the customer demandand an order-up-to inventory policy characterizes thereplenishment decision. And he also compared theimpact of different forecasting methods such as theminimum mean-squared error (MMSE), exponentialsmoothing(ES) method and the moving average (MA)method on the bullwhip effect. Feng and Ma [11] dida Demand and Forecasting in Supply Chains Basedon ARMA(1,1) Demand Process. Bayraktar et al [12]

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 110 Volume 15, 2016

and Wright and Yuan [13] showed the impact of de-mand forecasting on the bullwhip effect and suggest-ed appropriate forecast methods used in reducing thebullwhip effect through some simulation studies. Lu-ong [14] developed an exact measure of bullwhip ef-fect based on a replenishment model, which is similarto the one used by Chen et al.[8,9] for a two-stagesupply chain with one retailer and one supplier. It wasassumed that the retailer employs the base stock poli-cy for replenishment and that demands are forecastedbased on an AR(1) demand process. The behavior ofthe bullwhip effect was demonstrated for different val-ues of the autoregressive coefficient and the lead time.Nepal et al [15] presented an analysis of the bullwhipeffect and net-stock amplification in a three-echelonsupply chain. Zotteri et al [16] analyzed the empiri-cal demand data for fast moving consumer goods tomeasure the bullwhip effect. Najafi and Farahani [17]investigated the effects of various forecasting method-s, such as moving average (MA), exponential smooth-ing (ES) and linear regression (LR) on the bullwhipeffect in a four-echelon supply chain. Jaipuria, S andMahapatra, SS[18] put forward an improved demandforecasting method to reduce bullwhip effect in sup-ply chains. Shan et al. [19] did an empirical study ofthe bullwhip effect in China using data on over 1200companies listed on the Shanghai and Shenzhen stockexchanges from 2002 to 2009. Fang and Ma [20]studied Hyperchaotic dynamic of Cournot-Bertrandduopoly game with multi-product and chaos control.Junhai Ma and Aiwen Ma [21] did a research on Sup-ply Chain Coordination for substitutable products incompetitive models.

This paper continues to study the role of forecast-ing in relation to the bullwhip effect, but under a newsupply chain which has two retailers is different fromthe supply chain model usually has only one retailer inprevious researches among those papers above. Here,we assume the two retailers both employ the AR(1)demand process and use the ES forecasting method.Our research not only determines an exact measureof the bullwhip effect, but also analyzes the impactof every parameter on the bullwhip effect. We haveresearched how every parameter affect the bullwhipeffect and the numerical analysis has been given forevery pattern followed by some parameters.

To quantify the increase in variability from the re-tailer to the supplier, we first consider a simple two-stage supply chain consists of a single supplier and t-wo retailers. In section 2 we describe the supply chainmodel. In section 3, the bullwhip effect measured un-der the ES forecasting method is derived. In section4, we analyze the impact of every parameter on thebullwhip effect and give the corresponding numeri-cal analysis. Finally, we conclude in Section 5 with a

t

tq

1,tq

2,tq

1,tD

2,tD

tD

Figure 1: The supply chain model

discussion of the managerial insights provided by ourresults to mitigate the influence of information distor-tion on the bullwhip effect.

2 Supply chain model2.1 Replenishment modelIn this research, we consider a two-stage supply chainwith one supplier and two retailers as shown in Figure1. The two retailers face customer demands and placeorders to the supplier respectively. We consider thatretailer1 faces an AR(1) demand model

D1,t = ξ1 + θ1D1,t−1 + ε1,t (1)

where D1,t is the demand during period t; ξ1 is a con-stant that determines the mean of the demand, ε1,t isa normally distributed random error with mean 0 andvariance σ21; and θ1 is the first-order autocorrelationcoefficient, where −1 < θ1 < 1.

For the first-order autoregressive process to be s-tationary, we must have

E(D1,t) = E(D1,t−1) =ξ1

1−θ1

V ar(D1,t) = V ar(D1,t−1) =ξ21

1−θ21

(2)

Similarly, we assume retailer 2 also faces anAR(1) demand model so we have

D2,t = ξ2 + θ2D2,t−1 + ε2,tE(D2,t) = E(D2,t−1) =

ξ21−θ2

V ar(D2,t) = V ar(D2,t−1) =ξ22

1−θ22

(3)

2.2 Inventory policyTo supply the demand, the two retailers are assumed tofollow a simple order-up-to policy in which the order-up-to level is determined to achieve a desired servicelevel. Here, the service level is defined as the proba-bility of meeting the demand with an on hand inven-tory during the lead time. In each period t, the retail-er1 observes his inventory level and places an order

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E-ISSN: 2224-2880 111 Volume 15, 2016

q1,t to the supplier. After the order is placed, the re-tailer 1 observes and fills customer demand for thatperiod, denoted by D1,t. Any unfilled demands arebacklogged. At the beginning of period t, the retailerplaces an order of quantity q1,t to the supplier. Theorder quantity q1,t can be given as

q1,t = S1,t − S1,t−1 +D1,t−1 (4)

where S1,t is the order-up-to lever of retailer 1 at pe-riod t , and it can be determined by the lead-time de-mand as

S1,t = D̂L11,t + zσ̂L1

1,t (5)

where D̂L11,t is the forecast for the lead-time demand of

retailer 1 which depends on the forecasting methodand L1, from Zhang[10]σ̂L1

1,t remains constant overtime. Hence σ̂L1

1,t is the same to σ̂L11,t−1. z is the nor-

mal z score determined by the desired service level.Combining Eqs. (4) and (5), we have

q1,t = (D̂L11,t−D̂

L11,t−1)+z(σ̂

L11,t−σ̂

L11,t−1)+D1,t−1 (6)

In this paper we consider the use of an exponen-tial smoothing forecast to estimate D̂L1

1,t for the de-mand process, first-order autoregressive process. Westart by considering the exponential smoothing fore-cast.

2.3 Exponential smoothing forecastThe ES method is an adaptive algorithm in which one-period-ahead forecast is adjusted with a fraction of theforecasting error. Let α denote the fraction used inthis process, also called the smoothing factor, then ESforecast can be written as

D̂t = αDt−1 + (1− α)D̂t−1 (7)

Performing recursive substitutions in the above equa-tion, we arrive at an alternative expression for the one-period-ahead forecast:

D̂t =∞∑i=0

α(1− α)iDt−i−1 (8)

Therefore, D̂t can be interpreted as the weighted av-erage of all past demand with exponentially decliningweights. The i-period-ahead demand forecast with ESmethod simply extends the one-period-ahead forecastsimilar to the moving average (MA) case where

D̂t+i = D̂t+1, i ≥ 2 (9)

and the lead-time demand forecast D̂Lt can be ex-

pressed as

D̂Lt =

L∑i=1

D̂t+i = LD̂t+1 (10)

3 The measure of the bullwhip effectunder the ES forecasting method

According to Zhang [10], we know the order quan-tity of retailer 1 at period t under the ES forecastingmethod is

q1,t = (D̂L11,t − D̂L1

1,t−1) + z(σ̂L11,t − σ̂L1

1,t−1) +D1,t−1

= D1,t + α1L1(D1,t − D̂1,t)(11)

where α1 is the smoothing exponent of retailer1, andD̂1,t is the forecast of the demand at period t for re-tailer 1. The same, the order quantity of retailer 2 atperiod t under the ES forecasting method is

q2,t = (D̂L22,t − D̂L2

2,t−1) + z(σ̂L22,t − σ̂L2

2,t−1) +D2,t−1

= D2,t + α2L2(D2,t − D̂2,t)(12)

where α2 is the smoothing exponent of retailer 2, andD̂2,t is the forecast of the demand at period t for re-tailer 2. Hence, total order quantity of two retailers atperiod t is

qt = q1,t + q2,t= D1,t + α1L1(D1,t − D̂1,t)

+D2,t + α2L2(D2,t − D̂2,t)

= (1 + α1L1)D1,t − α1L1D̂1,t

+(1 + α2L2)D2,t − α2L2D̂2,t

(13)

Taking the variance for Eq.(13), we get

V ar(qt)

= (1 + α1L1)2V ar(D1,t) + (α1L1)

2V ar(D̂1,t)+(1 + α2L2)

2V ar(D2,t) + (1 + α2L2)2V ar(D2,t)

+(α2L2)2V ar(D̂2,t)

−2α1L1(1 + α1L1)Cov(D1,t, D̂1,t)+2(1 + α1L1)(1 + α2L2)Cov(D1,t, D2,t)

−2α2L2(1 + α1L1)Cov(D1,t, D̂2,t)

−2α1L1(1 + α2L2)Cov(D̂1,t, D2,t)

+2α1L1α2L2Cov(D̂1,t, D̂2,t)

−2α2L2(1 + α2L2)Cov(D2,t, D̂2,t)(14)

Proposition 1 The variance of the total order quanti-ty at period t under the ES forecasting method can begiven as

V ar(qt)

= (1 + 1−θ11−(1−α1)θ1

(2α1L1 +2α2

1L21

2−α1))V ar(D1,t)

+(1 + 1−θ21−(1−α2)θ2

(2α2L2 +2α2

2L22

2−α2))V ar(D2,t)

+(2(1 + α1L1)(1 + α2L2)−2α2

2θ1L2(1+α1L1)1−(1−α2)θ1

−2α21θ2L1(1+α2L2)1−(1−α1)θ2

+2α2

1α22L1L2(1−θ1θ2(1−α1)(1−α2))

(1−(1−α1)(1−α2))(1−(1−α1)θ2)(1−(1−α2)θ1))H

(15)

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E-ISSN: 2224-2880 112 Volume 15, 2016

where H = ϕ√V ar(D1,t)V ar(D2,t).

Proof. See the appendix.

For simplicity, Eq.(15) can be written as

V ar(qt) =M1V ar(D1,t) +M2V ar(D2,t)

+M3ϕ√V ar(D1,t)V ar(D2,t)

(16)

where M1 is the coefficient of V ar(D1,t), M2 is thecoefficient of V ar(D2,t), M3 is the coefficient of

ϕ√V ar(D1,t)V ar(D2,t). ϕ is the correlation coef-

ficient between retailer 1 and retailer 2, representingthe degree of market competition. The greater the ab-solute value of ϕ is, the fiercer the market competitiveis. We assume two retailers face the same perfectlycompetitive market, and their demands present a neg-ative correlation. So, we have −1 ≤ ϕ ≤ 0.

The total demand which two retailers face is

Dt = D1,t +D2,t (17)

The variance of Dt is determined as

V ar(Dt) = V ar(D1,t +D2,t)= V ar(D1,t) + V ar(D2,t) + 2Cov(D1,t, D2,t)

(18)We know

Cov(D1,t, D2,t) = ϕ√V ar(D1,t)V ar(D2,t) (19)

Using Eq.(19) in Eq.(18), we can get

V ar(Dt) = V ar(D1,t +D2,t)= V ar(D1,t) + V ar(D2,t)

+2ϕ√V ar(D1,t)V ar(D2,t)

(20)

The bullwhip effect (BWE) suggests that the de-mand variability is magnified as a customer demandsignal is transformed through various stages of a serialsupply chain. The measure of bullwhip effect, whichis defined as the ratio between variance of order quan-tity and variance of demand can be developed as fol-lows:

BWE =V ar(qt)

V ar(Dt)

So, from Eq.(16) and Eq.(20), the measure of the bull-whip under the ES forecasting method, can be deter-mined as

BWE = V ar(qt)V ar(Dt)

=M1V ar(D1,t)+M2V ar(D2,t)+M3ϕ

√V ar(D1,t)V ar(D2,t)

V ar(D1,t)+V ar(D2,t)+2ϕ√

V ar(D1,t)V ar(D2,t)

= M1+M2γ2+M3ϕγ1+γ+2ϕγ

(21)

where γ =

√V ar(D2,t)V ar(D1,t)

, which means the consistency

of demand volatility between two retailers.

4 Behavior of the bullwhip effectmeasure and numerical simulation

From the expression of the bullwhip effect, we knowthat it is a function with respect to model coefficients,lead-time of retailers, smoothing parameter, marketcompetition degree, and the consistency of demandvolatility between two retailers. In this section, alge-braic analysis and numerical simulation will be doneto investigate how those parameters affect the bull-whip effect.

4.1 The impact of autoregressive coefficientson the bullwhip effect

Proposition 2 The measure of the bullwhip effect B-WE processes the following properties on autoregres-sive coefficient θ1 which is in (-1,1).

(a) BWE is increasing when α1.= α2 andH > 1;

(b) BWE is decreasing when α1.= α2 and 0 <

H < 1;(c) BWE is decreasing when α1 > α2 and 0 <

H ≤ α2α1

;(d) BWE is decreasing and then increasing when

α1 > α2 and α2α1

< H < 2−α22−α1

, the minimum valueare obtained at θ∗1 = H−1

(1−α1)H−(1−α2);

(e) BWE is increasing when α1 > α2 and H ≥2−α22−α1

;(f) BWE is decreasing when α1 < α2 and 0 <

H ≤ 2−α22−α1

;(g) BWE is increasing and then decreasing when

α1 < α2 and 2−α22−α1

< H < α2α1

, the maximum valueare obtained at θ∗1 = H−1

(1−α1)H−(1−α2);

(h) BWE is increasing when α1 < α2 and H ≥α2α1

; where

H =2α2

2L2(α1α2(1− L1)− α1 − α2)ϕγ

α1(1− (1− α1)(1− α2)(2α1L1 +2α2

1L21

2−α1)

Proof. Take the first derivative of Eq.(21) with respectto θ1, we have

∂BWE∂θ1

=V − Y

1 + γ2 + 2ϕγ

where V =2α2

2L2(α1α2(1−L1)−α1−α2)ϕγ(1−(1−α1)(1−α2))(1−(1−α2)θ1)2

and Y =

α1(1−(1−α1)θ1)2

(2α1L1 +2α2

1L21

2−α1).

When α1.= α2, let ∂BWE

∂θ1≥ 0 where H > 1.

We can get (H−1) ≥ ((1−α1)H−(1−α2))θ1. We atfirst consider the case in which (1−α1)H−(1−α2) >0. In this case we have θ1 < H−1

(1−α1)H−(1−α2)= 1

1−α1.

Since we know 11−α1

= 1, so we can get in the interval

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 113 Volume 15, 2016

θ1 ∈ (−1, 1), ∂BWE∂θ1

≥ 0. Hence, BWE is a increas-ing function. Then, we consider the case in which(1 − α1)H − (1 − α2) < 0, here 0 < H < 1. Wecan get (1 − α1)H − (1 − α2) > 0. In this case wehave θ1 > H−1

(1−α1)H−(1−α2)= 1

1−α1, and 1

1−α1= 1,

however we have θ1 < 1, so in this case α1 is invalid.Then let ∂BWE

∂θ1< 0, the same to the case in which

∂BWE∂θ1

≥ 0, we can get the conclusion BWE is de-creasing when α1

.= α2 and 0 < H < 1;

Figure 2 presents the impact of θ1 on the bull-whip effect when α1

.= α2. Those findings have a

little difference from the results of other past research-es. In past researches, for example, Luong [14] foundthe bullwhip effect is decreasing and then increasingas autoregressive coefficient magnifies and the way inwhich it impacts the bullwhip effect is simple and sig-nificant. And it only has one pattern to affect the bull-whip effect. However, in our paper, the way autore-gressive coefficient impacts the bullwhip effect fol-lows two patterns, not singular, and the bullwhip ef-fect does not always decrease and then increase whenthe autoregressive coefficient increases.

When α1 > α2, it is noted that α2α1

< 2−α22−α1

<1−α21−α1

. Let ∂BWE∂θ1

≥ 0, we can get (H − 1) ≥ ((1 −α1)H − (1 − α2))θ1. Next, we consider the case inwhich (1−α1)H−(1−α2) > 0, henceH > 1−α2

1−α1, in

this case we can have θ1 < H−1(1−α1)H−(1−α2)

. And, wehave θ1 is in the internal (−1, 1); Then, we considertwo cases

(1) H−1(1−α1)H−(1−α2)

≥ 1 in that H ≥ α2α1

, andbecause α2

α1< 2−α2

2−α1, so we get when H > α2

α1, BWE

is increasing in the interval θ1 ∈ (−1, 1);(2) −1 < H−1

(1−α1)H−(1−α2)< 1 in that 2−α2

2−α1<

H < α2α1

, and because α2α1

< 2−α22−α1

, so the case is in-valid; Then, we can easily know that the case in which

H−1(1−α1)H−(1−α2)

≥ −1 is invalid.

Next we consider the case in which (1−α1)H −(1 − α2) < 0, hence 0 < H < 1−α2

1−α1, in this case

we can have α1 > H−1(1−α1)H−(1−α2)

. We also must

consider the cases in which H−1(1−α1)H−(1−α2)

≥ 1,

−1 < H−1(1−α1)H−(1−α2)

< 1 and H−1(1−α1)H−(1−α2)

≥−1 respectively, then we can get when H ≥ 2−α2

2−α1,

in the interval (-1,1), BWE is increasing and whenα2α1

< H < 2−α22−α1

, in the interval ( H−1(1−α1)H−(1−α2)

,1),BWE is increasing.

Then we let ∂BWE∂θ1

< 0, we can get (H − 1) <

((1−α1)H−(1−α2))θ1, the same to the case in which∂BWE∂θ1

≥ 0, we can get when 0 < H < α2α1

, in theinterval θ1 ∈ (−1, 1), BWE is decreasing and whenα2α1< H < 2−α2

2−α1, in the interval(-1, H−1

(1−α1)H−(1−α2)),

−1 −0.5 0 0.5 10.5

1

1.5

2

2.5

3

3.5

4

θ1

BW

E

α1=α

2=0.2,θ

2=0.5,φ=−0.7,γ=0.8

(a)L1=4,L2=3(b)L1=1,L2=5

Figure 2: The impact of θ1 on the bullwhip effect

−1 −0.5 0 0.5 10

2

4

6

8

10

12

θ1

BW

E

α1=0.6>α

2=0.3,θ

2=0.6,φ=−0.6,γ=1

(a)L1=3,L2=3(b)L1=1,L2=8(c)L1=2,L2=8

Figure 3: The impact of θ1 on the bullwhip effect

−1 −0.5 0 0.5 10

2

4

6

8

10

12

θ1

BW

E

α1=0.4<α

2=0.5,θ

1=0.5,φ=−0.6,γ=1

(a)L1=5,L2=3(b)L1=1,L2=3(c)L1=5,L2=6

Figure 4: The impact of θ1 on the bullwhip effect

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E-ISSN: 2224-2880 114 Volume 15, 2016

BWE is decreasing.So, in conclusion we can get BWE is decreasing

when α1 > α2 and 0 < H ≤ α2α1

; BWE is decreasingand increasing when α1 > α2 and α2

α1< H < 2−α2

2−α1;

BWE is increasing when α1 > α2 and H ≥ 2−α22−α1

.When α1 < α2, the same to the case in which

α1 > α2, we can also get BWE is decreasing whenα2 > α1 and 0 < H ≤ 2−α2

2−α1; BWE is increasing and

decreasing when α2 > α1 and 2−α22−α1

< H < α2α1

, themaximum value is obtained at θ∗1 = H−1

(1−α1)H−(1−α2);

BWE is increasing when α2 > α1 and H ≥ α2α1

.

where H =2α2

2L2(α1α2(1−L1)−α1−α2)ϕγ

α1(1−(1−α1)(1−α2)(2α1L1+2α2

1L21

2−α1).

Figure 3-4 present the impact of θ1 on the bull-whip effect when α1 > α2 and α1 < α2 respective-ly. The way in which the autoregressive coefficientsaffect the bullwhip effect is obvious. So, we can con-clude it is workable to adjust the bullwhip effect bycontrolling the autoregressive coefficients. ⊓⊔

Proposition 3 The measure of the bullwhip effect B-WE processes the following properties on autoregres-sive coefficient θ2 which is in (−1, 1)

(a) BWE is increasing when α2.= α2 andH > 1;

(b) BWE is decreasing when α2.= α2 and 0 <

H < 1;(c) BWE is decreasing when α1 > α2 and 0 <

H ≤ 2−α12−α2

;(d) BWE is increasing and then decreasing when

α1 > α2 and 2−α12−α2

< H < α1α2

, the maximum valueare obtained at θ∗2 = H−1

(1−α2)H−(1−α1);

(e) BWE is increasing when α1 > α2 and H ≥α1α2

;(f) BWE is decreasing when α1 < α2 and 0 <

H ≤ α1α2

;(g) BWE is decreasing and then increasing when

α1 < α2 and α1α2

< H < 2−α12−α2

, the minimum valueare obtained at θ∗2 = H−1

(1−α1)H−(1−α2);

(h) BWE is increasing when α1 < α2 and H ≥2−α12−α2

; where H =2α2

1L1(α1α2(1−L2)−α1−α2)ϕγ

α2(1−(1−α1)(1−α2)(2α2L2+2α2

2L22

2−α2).

Proof. Take the first derivative of Eq.(21) with re-spect to θ2, we have

∂BWE

∂θ2=

P − Z

1 + γ2 + 2ϕγ

where P =2α2

1L1(α1α2(1−L2)−α1−α2)ϕγ(1−(1−α1)(1−α2))(1−(1−α1)θ2)2

, Z =

α2(1−(1−α2)θ2)2

(2α2L2 +2α2

2L22

2−α2)

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

θ2

BW

E

α1=α

2=0.2,θ

1=0.5,φ=−0.7,γ=0.8

(a)L1=5,L2=3(b)L1=2,L2=5

Figure 5: The impact of θ2 on the bullwhip effect

−1 −0.5 0 0.5 10

2

4

6

8

10

12

θ2

BW

E

α1=0.6>α

2=0.3,;φ=−0.6,γ=1

(a)L1=3,L2=5,θ

1=0.2

(b)L1=3,L2=2,θ1=0.2

(c)L1=2,L2=8,θ1=0.6

Figure 6: The impact of θ2 on the bullwhip effect

−1 −0.5 0 0.5 10

2

4

6

8

10

12

θ2

BW

E

α1=0.4<α

2=0.5,θ

1=0.4,φ=−0.6,γ=1

(a)L1=1,L2=2(b)L1=7,L2=3(c)L1=5,L2=1

Figure 7: The impact of θ2 on the bullwhip effect

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 115 Volume 15, 2016

The proof of the bullwhip effect BWE with re-spect to θ2 is the same to the case in which the bull-whip effect BWE with respect to θ1. In addition, ac-cording to the symmetry between θ1 and θ2. we canget the measure of the bullwhip effect BWE propertieson autoregressive coefficient θ2 in Proposition 3. ⊓⊔

Figure 5-7 present the impact of θ2 on the bull-whip effect when α1

.= α2, α1 > α2 and α1 < α2

respectively. θ2 have the same significance to the bull-whip effect as θ1.

4.2 The impact of lead-time on the bullwhipeffect

Proposition 4 The measure of the bullwhip effect B-WE processes the following properties on the lead-time L1 which is in (0,+∞).

(a) BWE is increasing when B ≤ 4α1 − 2α21.

(b) BWE is decreasing and then increas-ing when B > 4α1 − 2α2

1, the minimum

value is obtained at L∗1 =

B−4α1+2α21

4α21

, where

B = ϕγS(2α1(1 + α2L2)−2α2

1θ2(1+α2L2)(1−(1−α1)θ2)

− 2α1α22θ1L2

(1−(1−α2)θ1))

+ ϕγS2α2

1α22L2(1−θ1θ2(1−α1)(1−α2))

(1−(1−α1)(1−α2)(1−(1−α2)θ1)(1−(1−α1)θ2).

where S = (1−(1−α1)θ1)(2−α1)1−θ1

.

Proof. Take the first derivative of Eq. (21) with re-spect to L1, we have

∂BWE∂L1

=(2α1(1+α2L2)−

2α21θ2(1+α2L2)

(1−(1−α1)θ2)−

2α1α22θ1L2

(1−(1−α2)θ1))ϕγ

1+γ2+2ϕγ

+(2α1+

4α21L1

2−α1)

1−θ1(1−(1−α1)α1)

1+γ2+2ϕγ

+

2α21α

22L2(1−θ1θ2(1−α1)(1−α2))

(1−(1−α1)(1−α2)(1−(1−α2)θ1)(1−(1−α1)θ2))ϕγ

1+γ2+2ϕγ.

Let ∂BWE∂L1

≥ 0, we can get L1 ≥ B−4α1+2α21

4α21

,here L1 ≥ 0. Hence, we consider the case in whichB−4α1+2α2

1

4α21

≤ 0, we can have B ≤ 4α1 − 2α21. So,

we can get when B ≤ 4α1 − 2α21, ∂BWE

∂L1≤ 0 in that

BWE is increasing in the interval(0,+∞). Next, weconsider B−4α1+2α2

1

4α21

> 0, we can have B > 4α1 −2α2

1. In conclusion, we can get BWE is increasing inthe interval (B−4α1+2α2

1

4α21

,+∞).Figure 8 presents the impact of L1 on the bull-

whip effect. Here, we know that the lead-time havetwo different ways to influence the bullwhip effect. Insome cases, Reducing the lead-time has no meaningto the lower of bullwhip effect, so managers shouldcontrol the bullwhip effect according to the actual sit-uation. And, in comparison of the autoregressive coef-

0 5 10 15 200

5

10

15

20

25

30

35

L1

BW

E

φ=−0.5;γ=2;α1=0.5;α

2=0.5

(a) L2=1;θ1=0.7;θ

2=0.8

(b) L2=6;θ1=0.3;θ

2=0.6

Figure 8: The impact of Ł1 on the bullwhip effect

ficient, the lead-time is easier to be controlled to lowerbullwhip effect.

Then let ∂BWE∂L1

< 0, and we can have L1 <B−4α1+2α2

1

4α21

, hereL1 ≥ 0. It is easily known that whenB−4α1+2α2

1

4α21

≤ 0 in that B ≤ 4α1 − 2α21 is invalid for

L1 is in the interval (0,+∞). Next, we consider whenB−4α1+2α2

1

4α21

> 0 in that B > 4α1 − 2α21. we can get

when L1 is in the interval (0,B−4α1+2α21

4α21

), ∂BWE∂L1

< 0

in that BWE is decreasing. ⊓⊔

Proposition 5 The measure of the bullwhip effect B-WE processes the following properties on the lead-time L2 which is in (0,+∞).

(a) BWE is increasing when C ≤ 4α2 − 2α22.

(b) BWE is decreasing and then increasing whenC > 4α2 − 2α2

2, the minimum value is obtained

at L∗2 =

C−4α2+2α22

4α22

; where C = ϕγW (2α2(1 +

α1L1)−2α2

2θ1(1+α2L1)(1−(1−α2)θ1)

− 2α2α21θ2L1

(1−(1−α1)θ2)) +

ϕγW2α2

2α21L1(1−θ2θ1(1−α1)(1−α2))

(1−(1−α1)(1−α2)(1−(1−α1)θ2)(1−(1−α2)θ1).

where W = (1−(1−α2)θ2)(2−α2)1−θ2

.

Proof. Take the first derivative of Eq. (21) with re-spect to L2, we have

∂BWE∂L2

=(2α3(1+α1L1)−

2α22θ1(1+α1L1)

(1−(1−α2)θ1)−

2α2α21θ2L1

(1−(1−α1)θ2))ϕγ

1+γ2+2ϕγ

+(2α2+

4α22L2

2−α2)

1−θ2(1−(1−α2)θ2)

1+γ2+2ϕγ

+

2α22α

21L1(1−θ2θ1(1−α2)(1−α1))

(1−(1−α2)(1−α1)(1−(1−α1)θ2)(1−(1−α2)θ1))ϕγ

1+γ2+2ϕγ

Figure 9 presents the impact of L2 on the bull-whip effect. And those findings are similar withproposition4.

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 116 Volume 15, 2016

0 5 10 15 200

10

20

30

40

50

60

70

80

L2

BW

Eφ=−0.5;γ=2;α

1=0.5;α

2=0.5

(a) L1=1;θ1=0.7;θ

2=0.8

(b) L1=6;θ1=0.3;θ

2=0.6

Figure 9: The impact of L2 on the bullwhip effect

The proof of the bullwhip effect BWE with re-spect to L2 is the same to the case in which the bull-whip effect BWE with respect to L1. In addition, ac-cording to the symmetry between L1 and L2. ⊓⊔

4.3 The impact of ϕ on the bullwhip effectProposition 6 The measure of the bullwhip effect B-WE processes the following properties on the corre-lation coefficient between two retailers ϕ which is in[-1,0).

(a) BWE is increasing when(M3 − 2M2)γ

3 + (M3 − 2M1)γ > 0.(b) BWE is a fixed value when

(M3 − 2M2)γ3 + (M3 − 2M1)γ

.= 0.

(c) BWE is decreasing when(M3 − 2M2)γ

3 + (M3 − 2M1)γ < 0.

Proof. Take the first derivative of Eq.(21) with respectto ϕ, we have

∂BWE

∂ϕ=

(M3 − 2M2)γ3 + (M3 − 2M1)γ

(1 + γ2 + 2ϕγ)2

We know that γ > 0,so (1 + γ2 + 2ϕγ)2 > 0.It is easily to known:(1) (M3 − 2M2)γ

3 + (M3 − 2M1)γ > 0,∂BWE∂ϕ >

0, hence BWE is increasing;(2) (M3 − 2M2)γ

3 + (M3 − 2M1)γ.= 0,∂BWE

∂ϕ.=

0, hence BWE is a fixed value;(3) (M3 − 2M2)γ

3 + (M3 − 2M1)γ < 0,∂BWE∂ϕ <

0, hence BWE is decreasing.Figure 10 presents the impact of ϕ on the bull-

whip effect. The bullwhip effect for ϕ is increasing ordecreasing, and it will tend to be a stable value whichdecided by other parameters. So the way in which thedegree of market competition affect the bullwhip ef-fect is simple from managerial point of view.

−1 −0.8 −0.6 −0.4 −0.2 0−70

−60

−50

−40

−30

−20

−10

0

10

20

φ

BW

E

θ1=0.2,θ

2=0.5,α

1=α

2=0.5

(a) L1=3,L2=3,γ=2

(b) L1=5,L2=5,γ=1

(c) L1=2,L2=3,γ=2

Figure 10: The impact of ϕ on the bullwhip effect

4.4 The impact of γ on the bullwhip effectProposition 7 The measure of the bullwhip effect B-WE processes the following properties on the con-sistency of demand volatility between two retailers γwhich is in (0,+∞).

(a) limγ→∞

BWE =M2.

(b) BWE is decreasing when (2M2 −M3).= 0,

(M2 −M1).= 0 and (2M1 −M3) ≥ 0.

(c) BWE is increasing when (2M2 −M3).= 0,

(M2 −M1).= 0 and (2M1 −M3) < 0.

(d) BWE is increasing when (2M2 −M3).= 0,

(M2 −M1)(2M1 −M3) ≥ 0.(e) BWE is decreasing and then increasing when

(2M2 −M3).= 0, (M2 −M1)(2M1 −M3) < 0, the

minimum value is obtained at γ∗ = (2M1−M3)ϕ2(M2−M1)

.(f) BWE is increasing when (2M2 −M3) < 0,

(M1 −M2)2 − ϕ2(2M2 −M3)(M3 − 2M1) ≤ 0.

(g) BWE is decreasing when (2M2 −M3) > 0,(M1 −M2)

2 − ϕ2(2M2 −M3)(M3 − 2M1) ≤ 0.(h) BWE is decreasing when (2M2 −M3) > 0,

(M1 − M2)2 − ϕ2(2M2 − M3)(M3 − 2M1) > 0,

(M3 − 2M1) > 0 and (M2 −M1) < 0.(i) BWE is decreasing, increasing and then de-

creasing when (2M2 −M3) > 0, (M1 − M2)2 −

ϕ2(2M2 −M3)(M3 − 2M1) > 0, (M3 − 2M1) > 0and (M2 −M1) > 0, the minimum value is obtained

at γ∗ = (M1−M2)+√

(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ

and the maximum value is obtained at γ∗ =(M1−M2)−

√(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ.

(j) BWE is increasing and then de-creasing when (2M2 −M3) > 0, (M1 −M2)

2 − ϕ2(2M2 − M3)(M3 − 2M1) > 0,(M3 − 2M1) < 0, the maximum value is obtained at

γ∗ =(M1−M2)−

√(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ.

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 117 Volume 15, 2016

0 2 4 6 8 102

3

4

5

6

7

8

9

10

γ

BW

E

(a)L1=5,L2=4,θ1=0.6,θ

2=0.1,φ=−0.5,α

1=0.5,α

2=0.5

(b)L1=5,L2=3,θ1=0.4,θ2=0.6,φ=−0.3,α1=0.4,α2=0.7

(c)L1=2,L2=2,θ1=0.5,θ

2=0.6,φ=−0.2,α

1=0.3,α

2=0.5

Figure 11: The impact of γ on the bullwhip effect

(k) BWE is decreasing and then increasing when(2M2 −M3) < 0, (M1 − M2)

2 − ϕ2(2M2 −M3)(M3 − 2M1) > 0, (M2 −M1) > 0 and(M3 − 2M1) > 0, the minimum value is obtained at

γ∗ =(M1−M2)+

√(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ.

(l) BWE is increasing, decreasing and then in-creasing when (2M2 −M3) < 0, (M1 − M2)

2 −ϕ2(2M2 −M3)(M3 − 2M1) > 0, (M3 − 2M1) < 0and (M2 −M1) < 0, the maximum value is obtained

at γ∗ = (M1−M2)−√

(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ

and the minimum value is obtained at γ∗ =(M1−M2)+

√(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ.

(m) BWE is increasing when (2M2 −M3) < 0,(M1 − M2)

2 − ϕ2(2M2 − M3)(M3 − 2M1) > 0,(M3 − 2M1) < 0 and (M2 −M1) > 0.

Proof. According to L’Hospital principle, we easilyknow thatlimγ→∞

BWE = limγ→∞

M1+M2γ2+M3ϕγ1+γ2+2ϕγ

=M2.

Then, Take the first derivative of Eq. 21 with respectto γ, we can have∂BWE

∂γ = (2M2−M3)ϕγ2+2(M2−M1)γ+(M3−2M1)ϕ(1+γ2+2ϕγ)2

.When (2M2 −M3)

.= 0, we can have

∂BWE∂γ = 2(M2−M1)γ+(M3−2M1)ϕ

(1+γ2+2ϕγ)2.

Let ∂BWE∂γ ≥ 0 in that

2(M2 −M1)γ + (M3 − 2M1)ϕ ≥ 0. So, wecan have

(1) (M2 −M1).= 0, 2M1 −M3 ≥ 0 when γ is

in the interval (0,+∞) ;(2) (M2 −M1) > 0, 2M1 −M3 ≥ 0 when γ is

in the interval (0,+∞);(3) (M2 −M1)(2M1 −M3) < 0 when γ is in the

interval ( (2M1−M3)ϕ2(M2−M1)

,+∞).

0 2 4 6 8 102

3

4

5

6

7

8

γ

BW

E

(a)L1=5,L2=3,θ

1=0.5,θ

2=0.6,φ=−0.2,α

1=0.4,α

2=0.7

(b)L1=2,L2=2,θ1=0.5,θ

2=0.6,φ=−0.2,α

1=0.3,α

2=0.5

Figure 12: The impact of γ on the bullwhip effect

Let ∂BWE∂γ < 0, the same we can have:

(1) (M2 −M1).= 0 , 2M1 −M3 < 0 when γ is

in the interval (0,+∞) ;(2) (M2 −M1)(2M1 −M3) < 0 when γ is in the

interval (0, (2M1−M3)ϕ2(M2−M1)

).

Then we consider (2M2 −M3) ̸= 0. Solv-ing the equation ∂BWE

∂γ = 0, and we consider thecase in which (M1 −M2)

2 − ϕ2(2M2 −M3)(M3 −2M1) < 0, the equation has no zero root. Hence,when (2M2 −M3) > 0, we have ∂BWE

∂γ < 0 in thatBWE is decreasing when γ is in the interval (0,+∞);and when (2M2 −M3) < 0, we have ∂BWE

∂γ > 0

in that BWE is increasing when γ is in the interval(0,+∞).

However, when (M1 − M2)2 − ϕ2(2M2 −

M3)(M3 − 2M1) ≥ 0, the equation has two zeroroots:γ∗1 =

(M1−M2)+√

(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ;

γ∗2 =(M1−M2)−

√(M1−M2)2−ϕ2(2M2−M3)(M3−2M1)

(2M2−M3)ϕ.

So, we can see(1) (2M2 −M3) > 0, (M3 − 2M1) > 0 and

(M2 −M1) < 0, we have γ∗1 < γ∗2 < 0; hence wecan get When γ is in the interval (0,+∞), ∂BWE

∂γ < 0

in that BWE is decreasing function with respect to γ;(2) (2M2 −M3) > 0, (M3 − 2M1) > 0 and

(M2 −M1) > 0, we have 0 < γ∗1 < γ∗2 ; hence wecan get when γ is in the interval (0, γ∗1),

∂BWE∂γ < 0

in that BWE is decreasing; in the interval (γ∗1 , γ∗2),

∂BWE∂γ > 0 in that BWE is increasing; in the interval

(γ∗2 ,+∞), ∂BWE∂γ < 0 in that BWE is decreasing.

BWE is decreasing, increasing and then decreasingfunction with respect to γ;

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 118 Volume 15, 2016

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

γ

BW

E

(a)L1=3,L2=1,θ1=0.6,θ

2=0.6,φ=−0.5,α

1=0.6,α

2=0.4

(b)L1=1,L2=3,θ1=0.95,θ

2=0.5,φ=−0.6,α

1=0.9,α

2=0.4

(c)L1=5,L2=4,θ1=0.6,θ

2=0.1,φ=−0.5,α

1=0.5,α

2=0.5

(d)L1=5,L2=3,θ1=0.4,θ

2=0.6,φ=−0.3,α

1=0.4,α

2=0.7

(e)L1=1,L2=10,θ1=0.5,θ

2=0.8,φ=−0.6,α

1=0.3,α

2=0.5

(f)L1=10,L2=3,θ1=0.5,θ

2=0.8,φ=−0.5,α

1=0.3,α2=0.8

Figure 13: The impact of γ on the bullwhip effect

(3) (2M2 −M3) > 0 and (M3 − 2M1) < 0, wehave γ∗2 > 0 and 0 > γ∗1 < 0.

The same we can know when γ is in the interval(0, γ∗2),

∂BWE∂γ > 0 in that BWE is increasing; in the

interval (γ∗2 ,+∞), ∂BWE∂γ < 0 in that BWE is de-

creasing. BWE is increasing and then decreasing withrespect to γ.

Next we can get these cases:

(1) (2M2 −M3) < 0 and (M3 − 2M1) > 0, wehave γ∗2 < 0 and γ∗1 > 0; hence we can know when γis in the interval (0, γ∗1),

∂BWE∂γ < 0 in that BWE is

decreasing; in the interval (γ∗1 ,+∞), ∂BWE∂γ > 0 in

that BWE is increasing. So we know BWE is decreas-ing and then increasing with respect to γ;

(2) (2M2 −M3) < 0, (M3 − 2M1) < 0 and(M2 −M1) < 0, we have 0 < γ∗2 < γ∗1 ; hence wecan get when γ is in the interval (0, γ∗2),

∂BWE∂γ > 0

in that BWE is increasing; in the interval (γ∗2 , γ∗1),

∂BWE∂γ < 0 in that BWE is decreasing; in the inter-

val (γ∗1 ,+∞), ∂BWE∂γ > 0 in that BWE is increasing.

So we know BWE is increasing, decreasing and thenincreasing with respect to γ;

(3) (2M2 −M3) < 0, (M3 − 2M1) < 0 and(M2 −M1) > 0, we have γ∗2 < γ∗1 < 0; hence wecan get When γ is in the interval (0,+∞), ∂BWE

∂γ > 0

in that BWE is increasing function with respect to γ.

Figure 11-13 illustrate that γ can have a signifi-cant impact on the variability and there is a very bigdifference between the variance of the orders in eachcase. The way in which γ affect the bullwhip effectis complex. And it is difficult to lower bullwhip effectthrough controlling the consistency of demand volatil-ity between two retailers which is decided by the de-mand processes for managers. ⊓⊔

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

α1

BW

E

(a)L1=1,L2=4,θ1=0,θ

2=−0.3,φ=−0.2,γ=1,α

2=0.2

(b)L1=3,L2=4,θ1=−0.5,θ

2=−0.3,φ=−0.2,γ=1,α

2=0.2

(c)L1=2,L2=4,θ1=0.5,θ

2=0.3,φ=−0.2,γ=1,α

2=0.2

(d)L1=1,L2=4,θ1=0.5,θ

2=−0.3,φ=−0.4,γ=1,α

2=0.5

Figure 14: The impact of α1 on the M3ϕγ1+γ2+2ϕγ

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α1

BW

E

L1=1,L2=2,θ2=0.4,φ=−0.6,γ=1,α

2=0.5

(a)θ

1=−0.5

(b)θ1=−0.3

(c)θ1=0.2

(d)θ1=0.5

Figure 15: The impact of α1 on the bullwhip effectwith various values of θ1.

4.5 The impact of the smoothing parameteron the bullwhip effect

Proposition 8 The smoothing parameter is an impor-tant and complex parameter which affects the bull-whip effect.

According to Eq.(21) we can knowBWE = V ar(qt)

V ar(Dt)= M1+M2γ2

1+γ2+2ϕγ+ M3ϕγ

1+γ2+2ϕγ.

Here, we can know that M3 = (2(1 + α1L1)(1 +

α2L2)−2α2

2θ1L2(1+α1L1)1−(1−α2)θ1

− 2α21θ2L1(1+α2L2)1−(1−α1)θ2

)

+2α2

1α22L1L2(1−θ1θ2(1−α1)(1−α2))

(1−(1−α1)(1−α2))(1−(1−α1)θ2)(1−(1−α2)θ1).

It is easily know that M3ϕγ1+γ2+2ϕγ

is decreasingwhen α1 is in the interval (0,1). The maximum val-ue of M3ϕγ

1+γ2+2ϕγis obtained at α1 = 0. We can get the

maximum value is (2(1+α2L2)−2α2

2θ1L2

1−(1−α2)θ1)ϕγ < 0.

So we can have M3ϕγ1+γ2+2ϕγ

< 0.

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E-ISSN: 2224-2880 119 Volume 15, 2016

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α1

BW

EL1=1,L2=2,θ

2=0.4,φ=−0.6,γ=1,α

2=0.5

(a)θ

2=−0.4

(b)θ2=−0.1

(c)θ2=0.4

(d)θ2=0.6

Figure 16: The impact of α1 on the bullwhip effectwith various values of θ2.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α1

BW

E

L2=2,θ1=−0.5,θ

2=0.4,φ=−0.6,γ=1,α

2=0.5

(a)L1=1(b)L1=3(c)L1=5

Figure 17: The impact of α1 on the bullwhip effectwith various values of Ł1.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α1

BW

E

L2=2,θ1=−0.5,θ

2=0.4,φ=−0.6,γ=1,α

2=0.5

(a)L2=1(b)L2=3(c)L2=5

Figure 18: The impact of α1 on the bullwhip effectwith various values of Ł2.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α1

BW

E

L1=2,L2=2,θ1=−0.5,θ

2=0.4,φ=−0.6,α

2=0.5

(a)γ=1

(b)γ=2

(c)γ=3

Figure 19: The impact of α1 on the bullwhip effectwith various values of γ.

Hence, BWE = V ar(qt)V ar(Dt)

< M1+M2γ2

1+γ2+2ϕγin that

V ar(qt)

V ar(Dt)<

H + J

(1 + γ2 + 2ϕγ)(22)

where H = (1 + 1−θ11−(1−α1)θ1

(2α1L1 +2α2

1L21

2−α1)), J =

(1 + 1−θ21−(1−α2)θ2

(2α2L2 +2α2

2L22

2−α2))γ2

In addition, notice that if 0 ≤ θ1 < 1,1−θ1

1−(1−α1)θ1≤ 1, while if −1 < θ1 ≤ 0, 1−θ1

1−(1−α1)θ1≥

1. Therefore, we see that as the demand correlationθ1 increases, the increase in variability decreases. Inaddition, for positively correlated demands, the in-crease in variability will be less than for i.i.d. demand-s (θ1=0). On the other hand, for negatively correlat-ed demands, the increase in variability will be greaterthan for i.i.d.demands. As θ2 have the same signif-icance to the supply chain model as θ1, we can getconclusions which is the same as θ1.

V ar(qt)

V ar(Dt)<

H + J

(1 + γ2 + 2ϕγ)(23)

Figure 15-20 respectively present those character-istics accordingly. We all know the smoothing param-eter have a significant impact on the bullwhip effect.The variation tendency of the bullwhip effect is notobvious when the smoothing parameter changes. Butwe can get the way in which the smoothing parameteraffect the bullwhip effect with other parameters.

Proposition 9 For smoothing parameter α2, accord-ing to the symmetry between α1 and α2, we can getthat the smoothing parameter α2 also have the sameimpact on the bullwhip effect as α1 have.

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E-ISSN: 2224-2880 120 Volume 15, 2016

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α1

BW

EL1=1,L2=2,θ

1=0.5,θ

2=0.4,φ=−0.6,γ=1

(a)α

2=0.1

(b)α2=0.3

(c)α2=0.5

(d)α2=0.7

Figure 20: The impact of α1 on the bullwhip effectwith various values of α2.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α2

BW

E

L1=1,L2=2,θ2=0.4,φ=−0.6,γ=1,α

1=0.5

(a)θ

1=−0.5

(b)θ1=−0.3

(c)θ1=0.2

(d)θ1=0.5

Figure 21: The impact of α2 on the bullwhip effectwith various values of θ1.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α2

BW

E

L1=1,L2=2,θ1=0.5,φ=−0.6,γ=1,α

1=0.5

(a)θ2=−0.4

(b)θ2=−0.1

(c)θ2=0.4

(d)θ2=0.6

Figure 22: The impact of α2 on the bullwhip effectwith various values of θ2.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α2

BW

E

L2=1,θ1=−0.5,θ

2=0.6,φ=−0.6,γ=1,α

1=0.5

(a)L1=1(b)L1=3(c)L1=5

Figure 23: The impact of α2 on the bullwhip effectwith various values of Ł1.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α2

BW

E

L1=2,θ1=−0.5,θ

2=0.4,φ=−0.6,γ=1,α

1=0.5

(a)L2=1(b)L2=3(c)L2=5

Figure 24: The impact of α2 on the bullwhip effectwith various values of Ł2.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α2

BW

E

L1=2,L2=2,θ1=−0.5,θ

2=0.4,φ=−0.6,α

1=0.4

(a)γ=1(b)γ=2(c)γ=3

Figure 25: The impact of α2 on the bullwhip effectwith various values of γ.

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E-ISSN: 2224-2880 121 Volume 15, 2016

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

α2

BW

EL1=1,L2=2,θ

1=0.5,θ

2=0.4,φ=−0.6,γ=1

(a)α

1=0.1

(b)α1=0.3

(c)α1=0.5

(d)α1=0.7

Figure 26: The impact of α2 on the bullwhip effectwith various values of α1.

It is also easily to know that M3ϕγ1+γ2+2ϕγ

is de-creasing when α2 is in the interval (0,1). The max-imum value of M3ϕγ

1+γ2+2ϕγis obtained at α2 = 0.

We can get the maximum value is (2(1 + α1L1) −2α2

1θ2L1

1−(1−α1)θ2)ϕγ < 0. So we can have M3ϕγ

1+γ2+2ϕγ< 0.

So, we can see that α2 processes the follow-ing properties on L1, L2, α1, γ, θ1, θ2 as Figure21-26present respectively. And we can know that the s-moothing parameter α2 also have an important impacton lowering the bullwhip effect.

5 ConclusionsWe know information distortion is one of the maincauses of bullwhip effect. Using different forecast-ing methods would transmit different information onthe demand to upstream. In this research, we measurethe bullwhip effect through using the Exponential S-moothing Forecasts. Then, we investigated the impactof the autoregressive coefficient, the lead time, mar-ket competition degree, the consistency of demandvolatility between two retailers and the smoothing pa-rameter on a bullwhip effect measured in a simpletwo-stage supply chain with one supplier and two re-tailers in which the demand pattern follows a autore-gressive process, AR(1), and the retailer employs theorder-up-to inventory police to get the influence of in-formation distortion on the bullwhip effect. Accord-ing to the analysis of the expression of the bullwhipeffect by using algebraic analysis and numerical sim-ulation, lots of interesting properties and manageri-al insights have been found. The autoregressive co-efficient which follows more different ways to affectthe bullwhip effect in our supply chains with two re-tailers than others with only one retailer. And those

ways are decided by the magnitude relationships be-tween parameter H , the smoothing parameter α1 andα2. Hence, managers can get how autoregressive co-efficient affect the bullwhip effect through parame-ter H , the smoothing parameter α1 and α2 so as tocontrol the bullwhip effect better. The bullwhip ef-fect does not always increase when the lead time Lincreases. The lead time L which follows differentways to affect the bullwhip effect is a complex fac-tor in our supply chains. Efforts to reduce bullwhipeffect through lead-time reduction can be misleading,especially when managers have little knowledge of theunderlying demand and are unaware of the influenceswhen different forecasting methods are applied to pre-dict lead-time demand.

The way of the market competition degree be-tween two retailers in which affects the bullwhip ef-fect is simpler. The variation tendency is clearer andthe bullwhip effect is close to a current value whichdecided by other parameters with the increase of themarket competition degree. The consistency of de-mand volatility between two retailers has a very com-plicated impact on the bullwhip effect. And it alsofollows different ways to influence the bullwhip ef-fect which are decided by the sophisticated magnituderelations with other parameters. So, for managers itmaybe more difficult and cost more to control the bull-whip effect by adjusting the consistency of demandvolatility between two retailers.

We all know the smoothing parameter has a sig-nificant impact on the bullwhip effect. The variationtendency of the bullwhip effect is not obvious whenthe smoothing parameter changes. But we can getthe way in which the smoothing parameter affect thebullwhip effect with other parameters. When one ofother parameters is fixed, the bullwhip effect increas-es as the smoothing parameter increases; And whenthe smoothing parameter is fixed, the bullwhip effectdecreases or increases as one of other parameters in-creases. For example, when parameter θ1 is fixed, thebullwhip effect increases as the smoothing parameterα1 increases; And when the smoothing parameter α1

is fixed, the bullwhip effect decreases as parameter θ1increases. However, when parameter L1 is fixed, thebullwhip effect increases as the smoothing parameterα1 increases; And when the smoothing parameter α1

is fixed, the bullwhip effect increases as parameter L1

increases.From above discussions, the influence of infor-

mation distortion using different forecasting method-s on bullwhip effect is complicated. For managers,they can mitigate the influence of information distor-tion caused by forecasting methods on the bullwhipeffect through adjusting relevant parameters. Usingappropriate forecasting methods has a significant im-

WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma

E-ISSN: 2224-2880 122 Volume 15, 2016

pact on reducing information distortion. And we alsoknow there certainly are other forecasting methods,and exponential smoothing may not be the “optimal”forecasting tool for the demand processes consideredin this paper as Chen et al[22] said. But exponentialsmoothing is one of the forecasting techniques mostcommonly used in practice. In the further research,we can try other forecasting methods to do competi-tion , experimental study and DOE analysis so that theresearch can more close to life.

Many scholars introduce Complexity, Chaos andHopf Bifurcation into social systems to do some re-searches. For example, Gao and Ma [23,24] studyChaos and Hopf Bifurcation of a Finance System andStability and Hopf Bifurcations in a Business CycleModel with Delay; Yang and Ma [25] analysed thecomplexity in evolutionary game system in the realestate market.

Appendix. Proofs

Proof for Eq.(15).

According to Feng [11], we have

Cov(D1,t−1, D1,t−i−1) = θi1V ar(D1,t)Cov(D2,t−1, D2,t−i−1) = θi2V ar(D2,t)

(24)

After iteration computation for Eq.(1), we have

D1,t−1 = ((1 + θ1 + · · ·+ θi−11 )ξ1 + θi1D1,t−i−1

+(ε1,t−1 + θ1ε1,t−2 + · · ·+ θi−11 ε1,t−i))

(25)So, we can get

Cov(D1,t−1, D2,t−i−1)

= Cov((1 + θ1 + · · ·+ θi−11 )ξ1 + θi1D1,t−i−1

+(ε1,t−1 + θ1ε1,t−2 + · · ·+ θi−11 ε1,t−i), D2,t−i−1)

= θi1ϕ√V ar(D1,t)V ar(D2,t)

(26)and we also have

Cov(D2,t−1, D1,t−i−1)

= θi2ϕ√V ar(D1,t)V ar(D2,t)

(27)

Proof of Proposition 1.Total order quantity of period t under the ES fore-

casting method is

qt = q1,t + q2,t = D1,t + α1L1(D1,t − D̂1,t) +D2,t

+α2L2(D2,t − D̂2,t)

= (1 + α1L1)D1,t − α1L1D̂1,t

+(1 + α2L2)D2,t − α2L2D̂2,t

(28)

Taking the variance for Eq.(28), we get

V ar(qt) = (1 + α1L1)2V ar(D1,t)

+(α1L1)2V ar(D̂1,t) + (1 + α2L2)

2V ar(D2,t)

+(1 + α2L2)2V ar(D2,t) + (α2L2)

2V ar(D̂2,t)

−2α1L1(1 + α1L1)Cov(D1,t, D̂1,t)+2(1 + α1L1)(1 + α2L2)Cov(D1,t, D2,t)

−2α2L2(1 + α1L1)Cov(D1,t, D̂2,t)

−2α1L1(1 + α2L2)Cov(D̂1,t, D2,t)

+2α1L1α2L2Cov(D̂1,t, D̂2,t)

−2α2L2(1 + α2L2)Cov(D2,t, D̂2,t)(29)

From Eq.(8), we have

D̂1,t =∞∑i=0

α1(1− α1)iDt−i−1 (30)

From zhang [14], we have

V ar(D̂1,t) =α1(1+(1−α1)θ1)

(2−α1)(1−(1−α1)θ1)V ar(D1,t)

V ar(D̂2,t) =α2(1+(1−α2)θ2)

(2−α2)(1−(1−α2)θ2)V ar(D2,t)

(31)

We can get

Cov(D1,t, D̂1,t) =α1θ1

(1−(1−α1)θ1)V ar(D1,t)

Cov(D2,t, D̂2,t) =α2θ2

(1−(1−α2)θ2)V ar(D2,t)

Cov(D1,t, D̂2,t) =α2θ1ϕ

(1−(1−α2)θ1)k

Cov(D2,t, D̂1,t) =α1θ2ϕ

(1−(1−α1)θ2)k

(32)

where k =√V ar(D1,t)V ar(D2,t).

From Eq.(30), we can get

Cov(D̂1,t, D̂2,t)= Cov(a, b)= α1α2(c, d)G= α1α2(e, f)G

= α1α2(1−θ1θ2(1−α1)(1−α2))(1−(1−α1)(1−α2))(1−(1−α1)θ2)(1−(1−α2)θ1)

G

(33)where

a =∞∑i=0

α1(1− α1)iD1,t−i−1

b =∞∑j=0

α2(1− α2)jD1,t−j−1)

c =∞∑i=0

∞∑j=i+1

(1− α1)i(1− α2)

jθj−i1

d =∞∑i=0

i∑j=0

(1− α1)i(1− α2)

jθi−j2

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E-ISSN: 2224-2880 123 Volume 15, 2016

e =∞∑i=0

(1− α1)i(1− α2)

i θ1(1− α2)

(1− (1− α2)θ1)

f =∞∑i=0

(1− α1)iθi2

(1− (1−α2θ2

)i+1)

1− (1−α2θ2

)

G = Cov(D1,t, D2,t) = ϕ√V ar(D1,t)V ar(D2,t).

Bring Eq.(31)-(33) into Eq.(29), then take the simpli-fication, we can get

V ar(qt)

= (1 + 1−θ11−(1−α1)θ1

(2α1L1 +2α2

1L21

2−α1))V ar(D1,t)

+(1 + 1−θ21−(1−α2)θ2

(2α2L2 +2α2

2L22

2−α2))V ar(D2,t)

+(2(1 + α1L1)(1 + α2L2)−2α2

2θ1L2(1+α1L1)1−(1−α2)θ1

−2α21θ2L1(1+α2L2)1−(1−α1)θ2

+2α2

1α22L1L2(1−θ1θ2(1−α1)(1−α2))

(1−(1−α1)(1−α2))(1−(1−α1)θ2)(1−(1−α2)θ1))H

(34)where H = ϕ

√V ar(D1,t)V ar(D2,t). This com-

pletes the proof for the proposition 1.

Acknowledgements: This work was supportedby the National Nature Science Foundation of Chi-na (No. 61273231) and the Doctoral Scientific FundProject of the Ministry of Education of China (No.2013003211073).

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