The Bullwhip Effect caused by Information Distortion in a complexSupply Chain under Exponential Smoothing Forecast
JUNHAI MATianjin University
College of ManagementTianjin 300072
WEIHUA LAITianjin University
College of ManagementTianjin 300072
XINAOGANG MATianjin University
College of ManagementTianjin 300072
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.
KeyWords: 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 playersirrational 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]
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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 = 1 + 1D1,t1 + 1,t (1)
where D1,t is the demand during period t; 1 is a con-stant that determines the mean of the demand, 1,t isa normally distributed random error with mean 0 andvariance 21; and 1 is the first-order autocorrelationcoefficient, where 1 < 1 < 1.
For the first-order autoregressive process to be s-tationary, we must have
E(D1,t) = E(D1,t1) =1
11V ar(D1,t) = V ar(D1,t1) =
21121
(2)
Similarly, we assume retailer 2 also faces anAR(1) demand model so we have
D2,t = 2 + 2D2,t1 + 2,tE(D2,t) = E(D2,t1) =
212
V ar(D2,t) = V ar(D2,t1) =22
122
(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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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,t1 +D1,t1 (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 = DL11,t + z
L11,t (5)
where DL11,t is the forecast for the lead-time demand ofretailer 1 which depends on the forecasting methodand L1, from Zhang[10]L11,t remains constant overtime. Hence L11,t is the same to
L11,t1. z is the nor-
mal z score determined by the desired service level.Combining Eqs. (4) and (5), we have
q1,t = (DL11,tD
L11,t1)+z(
L11,t
L11,t1)+D1,t1 (6)
In this paper we consider the use of an exponen-tial smoothing forecast to estimate DL11,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 denote the fraction used inthis process, also called the smoothing factor, then ESforecast can be written as
Dt = Dt1 + (1 )Dt1 (7)
Performing recursive substitutions in the above equa-tion, we arrive at an alternative expression for the one-period-ahead forecast:
Dt =i=0
(1 )iDti1 (8)
Therefore, Dt 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
Dt+i = Dt+1, i 2 (9)
and the lead-time demand forecast DLt can be ex-pressed as
DLt =Li=1
Dt+i = LDt+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 = (DL11,t D
L11,t1) + z(
L11,t
L11,t1) +D1,t1
= D1,t + 1L1(D1,t D1,t)(11)
where 1 is the smoothing exponent of retailer1, andD1,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 = (DL22,t D
L22,t1) + z(
L22,t
L22,t1) +D2,t1
= D2,t + 2L2(D2,t D2,t)(12)
where 2 is the smoothing exponent of retailer 2, andD2,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 + 1L1(D1,t D1,t)+D2,t + 2L2(D2,t D2,t)= (1 + 1L1)D1,t 1L1D1,t+(1 + 2L2)D2,t 2L2D2,t
(13)
Taking the variance for Eq.(13), we get
V ar(qt)
= (1 + 1L1)2V ar(D1,t) + (1L1)
2V ar(D1,t)+(1 + 2L2)
2V ar(D2,t) + (1 + 2L2)2V ar(D2,t)
+(2L2)2V ar(D2,t)
21L1(1 + 1L1)Cov(D1,t, D1,t)+2(1 + 1L1)(1 + 2L2)Cov(D1,t, D2,t)
22L2(1 + 1L1)Cov(D1,t, D2,t)21L1(1 + 2L2)Cov(D1,t, D2,t)+21L12L2Cov(D1,t, D2,t)
22L2(1 + 2L2)Cov(D2,t, D2,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 + 111(11)1 (21L1 +221L
21
21 ))V ar(D1,t)
+(1 + 121(12)2 (22L2 +222L
22
22 ))V ar(D2,t)
+(2(1 + 1L1)(1 + 2L2)2221L2(1+1L1)
1(12)12
212L1(1+2L2)1(11)2
+221
22L1L2(112(11)(12))
(1(11)(12))(1(11)2)(1(12)1))H
(15)
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where H = 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)
+M3V 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
V ar(D1,t)V ar(D2,t). is the correlation coef-
ficient between retailer 1 and retailer 2, representingthe degree of market competition. The greater the ab-solute value of 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 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) = 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)
+2V 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)+M3
V ar(D1,t)V ar(D2,t)
V ar(D1,t)+V ar(D2,t)+2
V ar(D1,t)V ar(D2,t)
= M1+M22+M3
1++2
(21)
where =
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 1 which is in (-1,1).
(a) BWE is increasing when 1.= 2 andH > 1;
(b) BWE is decreasing when 1.= 2 and 0 2 and 0 2 and 21 < H 2 and H
2221 ;
(f) BWE is decreasing when 1 < 2 and 0 0. In this case we have 1 < H1(11)H(12) =
111 .
Since we know 111 = 1, so we can get in the interval
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1 (1, 1), BWE1 0. Hence, BWE is a increas-ing function. Then, we consider the case in which(1 1)H (1 2) < 0, here 0 < H < 1. Wecan get (1 1)H (1 2) > 0. In this case wehave 1 > H1(11)H(12) =
111 , and
111 = 1,
however we have 1 < 1, so in this case 1 is invalid.Then let BWE1 < 0, the same to the case in whichBWE1
0, we can get the conclusion BWE is de-creasing when 1
.= 2 and 0 < H < 1;
Figure 2 presents the impact of 1 on the bull-whip effect when 1
.= 2. 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 1 > 2, it is noted that 21 1211 , inthis case we can have 1 < H1(11)H(12) . And, wehave 1 is in the internal (1, 1); Then, we considertwo cases
(1) H1(11)H(12) 1 in that H 21
, andbecause 21
21
, BWEis increasing in the interval 1 (1, 1);
(2) 1 < H1(11)H(12) < 1 in that2221 H1(11)H(12) . We also must
consider the cases in which H1(11)H(12) 1,1 < H1(11)H(12) < 1 and
H1(11)H(12)
1 respectively, then we can get when H 2221 ,in the interval (-1,1), BWE is increasing and when21
< H < 2221 , in the interval (H1
(11)H(12) ,1),BWE is increasing.
Then we let BWE1 < 0, we can get (H 1)
2=0.3,
2=0.6,=0.6,=1
(a)L1=3,L2=3(b)L1=1,L2=8(c)L1=2,L2=8
Figure 3: The impact of 1 on the bullwhip effect
1 0.5 0 0.5 10
2
4
6
8
10
12
1
BW
E
1=0.4
BWE is decreasing.So, in conclusion we can get BWE is decreasing
when 1 > 2 and 0 < H 21 ; BWE is decreasingand increasing when 1 > 2 and 21 < H 2 and H 2221 .When 1 < 2, the same to the case in which
1 > 2, we can also get BWE is decreasing when2 > 1 and 0 < H 2221 ; BWE is increasing anddecreasing when 2 > 1 and 2221 < H 1 and H 21 .where H = 2
22L2(12(1L1)12)
1(1(11)(12)(21L1+22
1L21
21).
Figure 3-4 present the impact of 1 on the bull-whip effect when 1 > 2 and 1 < 2 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 2 which is in (1, 1)
(a) BWE is increasing when 2.= 2 andH > 1;
(b) BWE is decreasing when 2.= 2 and 0 2 and 0 2 and 2122 < H 2 and H 12
;(f) BWE is decreasing when 1 < 2 and 0 2 and 1 < 2
respectively. 2 have the same significance to the bull-whip effect as 1.
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 41 221.(b) BWE is decreasing and then increas-
ing when B > 41 221, the minimumvalue is obtained at L1 =
B41+221421
, where
B = S(21(1 + 2L2)2212(1+2L2)(1(11)2)
21221L2
(1(12)1))
+ S221
22L2(112(11)(12))
(1(11)(12)(1(12)1)(1(11)2) .
where S = (1(11)1)(21)11 .
Proof. Take the first derivative of Eq. (21) with re-spect to L1, we have
BWEL1
=(21(1+2L2)
2212(1+2L2)
(1(11)2)
21221L2
(1(12)1))
1+2+2
+(21+
421L121
)11
(1(11)1)1+2+2
+
22122L2(112(11)(12))
(1(11)(12)(1(12)1)(1(11)2))
1+2+2.
Let BWEL1 0, we can get L1 B41+221
421,
here L1 0. Hence, we consider the case in whichB41+221
421 0, we can have B 41 221. So,
we can get when B 41 221, BWEL1 0 in thatBWE is increasing in the interval(0,+). Next, weconsider B41+2
21
421> 0, we can have B > 41
221. In conclusion, we can get BWE is increasing inthe interval (B41+2
21
421,+).
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
=0.5;=2;1=0.5;
2=0.5
(a) L2=1;1=0.7;
2=0.8
(b) L2=6;1=0.3;
2=0.6
Figure 8: The impact of 1 on the bullwhip effect
ficient, the lead-time is easier to be controlled to lowerbullwhip effect.
Then let BWEL1 < 0, and we can have L1 0 in that B > 41 221. we can get
when L1 is in the interval (0,B41+221
421), BWEL1 < 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 42 222.(b) BWE is decreasing and then increasing when
C > 42 222, the minimum value is obtainedat L2 =
C42+222422
; where C = W (22(1 +
1L1)2221(1+2L1)(1(12)1)
22212L1(1(11)2)) +
W222
21L1(121(11)(12))
(1(11)(12)(1(11)2)(1(12)1) .
where W = (1(12)2)(22)12 .
Proof. Take the first derivative of Eq. (21) with re-spect to L2, we have
BWEL2
=(23(1+1L1)
2221(1+1L1)
(1(12)1)
22212L1
(1(11)2))
1+2+2
+(22+
422L222
)12
(1(12)2)1+2+2
+
22221L1(121(12)(11))
(1(12)(11)(1(11)2)(1(12)1))
1+2+2
Figure 9 presents the impact of L2 on the bull-whip effect. And those findings are similar withproposition4.
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0 5 10 15 200
10
20
30
40
50
60
70
80
L2
BW
E=0.5;=2;
1=0.5;
2=0.5
(a) L1=1;1=0.7;
2=0.8
(b) L1=6;1=0.3;
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 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 which is in[-1,0).
(a) BWE is increasing when(M3 2M2)3 + (M3 2M1) > 0.
(b) BWE is a fixed value when(M3 2M2)3 + (M3 2M1)
.= 0.
(c) BWE is decreasing when(M3 2M2)3 + (M3 2M1) < 0.
Proof. Take the first derivative of Eq.(21) with respectto , we have
BWE
=
(M3 2M2)3 + (M3 2M1)(1 + 2 + 2)2
We know that > 0,so (1 + 2 + 2)2 > 0.It is easily to known:(1) (M3 2M2)3 + (M3 2M1) > 0,BWE >
0, hence BWE is increasing;(2) (M3 2M2)3 + (M3 2M1)
.= 0,BWE
.=
0, hence BWE is a fixed value;(3) (M3 2M2)3 + (M3 2M1) < 0,BWE 0,
(M1 M2)2 2(2M2 M3)(M3 2M1) 0.(h) BWE is decreasing when (2M2 M3) > 0,
(M1 M2)2 2(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 2(2M2 M3)(M3 2M1) > 0, (M3 2M1) > 0and (M2 M1) > 0, the minimum value is obtainedat = (M1M2)+
(M1M2)22(2M2M3)(M32M1)
(2M2M3)and the maximum value is obtained at =(M1M2)
(M1M2)22(2M2M3)(M32M1)
(2M2M3) .(j) BWE is increasing and then de-
creasing when (2M2 M3) > 0, (M1 M2)
2 2(2M2 M3)(M3 2M1) > 0,(M3 2M1) < 0, the maximum value is obtained at =
(M1M2)
(M1M2)22(2M2M3)(M32M1)(2M2M3) .
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0 2 4 6 8 102
3
4
5
6
7
8
9
10
BW
E
(a)L1=5,L2=4,1=0.6,
2=0.1,=0.5,
1=0.5,
2=0.5
(b)L1=5,L2=3,1=0.4,2=0.6,=0.3,1=0.4,2=0.7(c)L1=2,L2=2,
1=0.5,
2=0.6,=0.2,
1=0.3,
2=0.5
Figure 11: The impact of on the bullwhip effect
(k) BWE is decreasing and then increasing when(2M2 M3) < 0, (M1 M2)2 2(2M2 M3)(M3 2M1) > 0, (M2 M1) > 0 and(M3 2M1) > 0, the minimum value is obtained at =
(M1M2)+
(M1M2)22(2M2M3)(M32M1)(2M2M3) .
(l) BWE is increasing, decreasing and then in-creasing when (2M2 M3) < 0, (M1 M2)2 2(2M2 M3)(M3 2M1) > 0, (M3 2M1) < 0and (M2 M1) < 0, the maximum value is obtainedat = (M1M2)
(M1M2)22(2M2M3)(M32M1)
(2M2M3)and the minimum value is obtained at =(M1M2)+
(M1M2)22(2M2M3)(M32M1)
(2M2M3) .(m) BWE is increasing when (2M2 M3) < 0,
(M1 M2)2 2(2M2 M3)(M3 2M1) > 0,(M3 2M1) < 0 and (M2 M1) > 0.
Proof. According to LHospital principle, we easilyknow thatlim
BWE = lim
M1+M22+M31+2+2
=M2.
Then, Take the first derivative of Eq. 21 with respectto , we can haveBWE
=(2M2M3)2+2(M2M1)+(M32M1)
(1+2+2)2.
When (2M2 M3).= 0, we can have
BWE =
2(M2M1)+(M32M1)(1+2+2)2
.
Let BWE 0 in that2(M2 M1) + (M3 2M1) 0. So, wecan have
(1) (M2 M1).= 0, 2M1 M3 0 when is
in the interval (0,+) ;(2) (M2 M1) > 0, 2M1 M3 0 when is
in the interval (0,+);(3) (M2 M1)(2M1 M3) < 0 when is in the
interval ( (2M1M3)2(M2M1) ,+).
0 2 4 6 8 102
3
4
5
6
7
8
BW
E
(a)L1=5,L2=3,
1=0.5,
2=0.6,=0.2,
1=0.4,
2=0.7
(b)L1=2,L2=2,1=0.5,
2=0.6,=0.2,
1=0.3,
2=0.5
Figure 12: The impact of on the bullwhip effect
Let BWE < 0, the same we can have:(1) (M2 M1)
.= 0 , 2M1 M3 < 0 when is
in the interval (0,+) ;(2) (M2 M1)(2M1 M3) < 0 when is in the
interval (0, (2M1M3)2(M2M1) ).
Then we consider (2M2 M3) = 0. Solv-ing the equation BWE = 0, and we consider thecase in which (M1 M2)2 2(2M2 M3)(M3 2M1) < 0, the equation has no zero root. Hence,when (2M2 M3) > 0, we have BWE < 0 in thatBWE is decreasing when is in the interval (0,+);and when (2M2 M3) < 0, we have BWE > 0in that BWE is increasing when is in the interval(0,+).
However, when (M1 M2)2 2(2M2 M3)(M3 2M1) 0, the equation has two zeroroots:1 =
(M1M2)+
(M1M2)22(2M2M3)(M32M1)(2M2M3) ;
2 =(M1M2)
(M1M2)22(2M2M3)(M32M1)
(2M2M3) .
So, we can see(1) (2M2 M3) > 0, (M3 2M1) > 0 and
(M2 M1) < 0, we have 1 < 2 < 0; hence wecan get When is in the interval (0,+), BWE < 0in that BWE is decreasing function with respect to ;
(2) (2M2 M3) > 0, (M3 2M1) > 0 and(M2 M1) > 0, we have 0 < 1 < 2 ; hence wecan get when is in the interval (0, 1),
BWE < 0
in that BWE is decreasing; in the interval (1 , 2),
BWE > 0 in that BWE is increasing; in the interval
(2 ,+), BWE < 0 in that BWE is decreasing.BWE is decreasing, increasing and then decreasingfunction with respect to ;
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
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BW
E
(a)L1=3,L2=1,1=0.6,
2=0.6,=0.5,
1=0.6,
2=0.4
(b)L1=1,L2=3,1=0.95,
2=0.5,=0.6,
1=0.9,
2=0.4
(c)L1=5,L2=4,1=0.6,
2=0.1,=0.5,
1=0.5,
2=0.5
(d)L1=5,L2=3,1=0.4,
2=0.6,=0.3,
1=0.4,
2=0.7
(e)L1=1,L2=10,1=0.5,
2=0.8,=0.6,
1=0.3,
2=0.5
(f)L1=10,L2=3,1=0.5,
2=0.8,=0.5,
1=0.3,2=0.8
Figure 13: The impact of on the bullwhip effect
(3) (2M2 M3) > 0 and (M3 2M1) < 0, wehave 2 > 0 and 0 >
1 < 0.
The same we can know when is in the interval(0, 2),
BWE > 0 in that BWE is increasing; in the
interval (2 ,+), BWE < 0 in that BWE is de-creasing. BWE is increasing and then decreasing withrespect to .
Next we can get these cases:
(1) (2M2 M3) < 0 and (M3 2M1) > 0, wehave 2 < 0 and
1 > 0; hence we can know when
is in the interval (0, 1),BWE
< 0 in that BWE isdecreasing; in the interval (1 ,+), BWE > 0 inthat BWE is increasing. So we know BWE is decreas-ing and then increasing with respect to ;
(2) (2M2 M3) < 0, (M3 2M1) < 0 and(M2 M1) < 0, we have 0 < 2 < 1 ; hence wecan get when is in the interval (0, 2),
BWE > 0
in that BWE is increasing; in the interval (2 , 1),
BWE < 0 in that BWE is decreasing; in the inter-
val (1 ,+), BWE > 0 in that BWE is increasing.So we know BWE is increasing, decreasing and thenincreasing with respect to ;
(3) (2M2 M3) < 0, (M3 2M1) < 0 and(M2 M1) > 0, we have 2 < 1 < 0; hence wecan get When is in the interval (0,+), BWE > 0in that BWE is increasing function with respect to .
Figure 11-13 illustrate that 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 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 15
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1
BW
E
(a)L1=1,L2=4,1=0,
2=0.3,=0.2,=1,
2=0.2
(b)L1=3,L2=4,1=0.5,
2=0.3,=0.2,=1,
2=0.2
(c)L1=2,L2=4,1=0.5,
2=0.3,=0.2,=1,
2=0.2
(d)L1=1,L2=4,1=0.5,
2=0.3,=0.4,=1,
2=0.5
Figure 14: The impact of 1 on the M31+2+2
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
18
20
1
BW
E
L1=1,L2=2,2=0.4,=0.6,=1,
2=0.5
(a)
1=0.5
(b)1=0.3
(c)1=0.2
(d)1=0.5
Figure 15: The impact of 1 on the bullwhip effectwith various values of 1.
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+M22
1+2+2+ M3
1+2+2.
Here, we can know that M3 = (2(1 + 1L1)(1 +2L2)
2221L2(1+1L1)1(12)1
2212L1(1+2L2)1(11)2 )
+221
22L1L2(112(11)(12))
(1(11)(12))(1(11)2)(1(12)1) .
It is easily know that M31+2+2
is decreasingwhen 1 is in the interval (0,1). The maximum val-ue of M3
1+2+2is obtained at 1 = 0. We can get the
maximum value is (2(1+2L2)2221L2
1(12)1 ) < 0.
So we can have M31+2+2
< 0.
WSEAS TRANSACTIONS on MATHEMATICS Junhai Ma, Weihua Lai, Xinaogang Ma
E-ISSN: 2224-2880 119 Volume 15, 2016
0 0.2 0.4 0.6 0.8 10
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1
BW
EL1=1,L2=2,
2=0.4,=0.6,=1,
2=0.5
(a)
2=0.4
(b)2=0.1
(c)2=0.4
(d)2=0.6
Figure 16: The impact of 1 on the bullwhip effectwith various values of 2.
0 0.2 0.4 0.6 0.8 10
2
4
6
8
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12
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20
1
BW
E
L2=2,1=0.5,
2=0.4,=0.6,=1,
2=0.5
(a)L1=1(b)L1=3(c)L1=5
Figure 17: The impact of 1 on the bullwhip effectwith various values of 1.
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
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20
1
BW
E
L2=2,1=0.5,
2=0.4,=0.6,=1,
2=0.5
(a)L2=1(b)L2=3(c)L2=5
Figure 18: The impact of 1 on the bullwhip effectwith various values of 2.
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
18
20
1
BW
E
L1=2,L2=2,1=0.5,
2=0.4,=0.6,
2=0.5
(a)=1(b)=2(c)=3
Figure 19: The impact of 1 on the bullwhip effectwith various values of .
Hence, BWE = V ar(qt)V ar(Dt)