Study of the Bullwhip Effect under Various Forecasting Methods in Electronics Supply Chain with Dual Retailers considering Market Share

We establish in this paper a new two-stage supply chain with onemanufacturer and two retailers which have a fixedmarket share in the mature and stable market with specific reference to consumer electronics industry.This paper offers insights into how the three forecasting methods affect the bullwhip effect considering the market share under the ARMA(1, 1) demand process and the orderup-to inventory policy. We also discuss the stability of the order with the theory of entropy. In particular, we derive the expressions of bullwhip effect measure under theMMSE, MA, and ESmethods and compare them by numerical simulations. Results show that the MA is always better in contrast to the ES for reducing the bullwhip effect in our supply chain model. When moving average coefficient is lower than a certain value, the MMSE method is the best for reducing the bullwhip effect; otherwise, the MAmethod is the best. Moreover, the larger the market share of the retailer with a long lead time is, the greater the bullwhip effect is, no matter what the forecasting method is.


Introduction
There is a high-risk behavior occurring frequently in many companies' marketing activities, which is called bullwhip effect, first coined by Lee et al. [1,2].It affects operational costs and may lead to chaos in the system.Because of its impact on the operation and management strategies of all the upstream and downstream enterprises, it has become a focus in the supply chain management research.In this paper, we focus on the consumer electronics supply chain management, which is characterized by frequent channel competition.In the Chinese consumer electronics offline retail market, GOME and SUNING are two giant chain enterprises, which occupy more than 45% of the market share.Most of the consumer electronics manufacturers sell products through these two home appliance retail chain stores.The market demand forecasting and replenishment strategies of the two retail giants will undoubtedly affect order fluctuations and production plan of the manufacturers.We assume that the demand is independent because GOME is far apart from SUNING.So, we use the ARMA(1, 1) demand process and aim to investigate the effect of three forecasting methods on the bullwhip effect considering the market share.
Bullwhip effect refers to a phenomenon where the order variability tends to be amplified in a supply chain from the downstream member to its (immediate) upstream member.Forrester [3,4] discovered bullwhip effect's causes and possible remediation by industrial dynamics and became the first scholar proving this phenomenon.This was the first milestone in the academic research of bullwhip effect.Then, many other researchers further proved the existence of the bullwhip effect.A well-known classic "beer game" experimented on by Sterman [5] is the second milestone which has been utilized in business schools for decades to illustrate the bullwhip effect.The third milestone, also very important, is the statistical research stage which was started by Lee et al. [1,2].He identified five main causes of the bullwhip effect which are demand signal processing, supply shortage, nonzero lead time, price fluctuation, and order batching in supply chains.After that, more and more researchers studied 2 Complexity the bullwhip effect by statistical methods.Based on the works of Lee, different demand processes and forecasting methods which directly affected the inventory system of supply chains significantly are employed in a lot of papers.
Similar to Lee et al. [1,2], many papers are published for an AR (1) demand model in a two-stage supply chain with one manufacturer and one retailer [6][7][8][9][10].Chen et al. [6,7] studied the difference of bullwhip effects under two forecasting methods (MA and ES) in a simple, two-stage supply chain.Zhang [8] investigated the impact of three different forecasting methods (MA, ES, and MMSE) on the bullwhip effect under AR(1) demand process.Lee et al. [9] also measured the benefit of information sharing on reducing the bullwhip effect using AR (1) process and a simple order-upto inventory policy with the minimum mean square error (MMSE) forecasting technique.Likewise, Luong [10] used the same demand model as Lee et al. [9] to measure the bullwhip effect on the base stock policy for their inventory under the MMSE forecasting technique.Furthermore, Luong and Phien [11] investigated AR(2) demand process and higher order autoregressive models AR() with the MMSE method.By calculating the bullwhip effect, these papers investigated effects of different parameters on the bullwhip effect, such as autoregressive coefficients and order lead time.In addition, Duc et al. [12] measured the impact of a third-party warehouse on the bullwhip effect and inventory cost under AR (1) process in a three-stage supply chain with one manufacturer, one third-party warehouse, and two retailers considering market share.
A pure autoregressive process has been supposed by a few academics [1,2,[6][7][8][9][10][11][12], and a pure moving average process has also been used by Graves [13].However, we can get that the demand model has both characteristics of them.Pindyck and Rubinfeld [14] assumed the mixed autoregressive-moving average (ARMA) demand process which is more suitable for the market demand than AR(1) model.Then, the ARMA process is frequently applied.Under the base stock policy, Disney et al. [15] used ARMA(1, 1) demand pattern to measure the bullwhip effect by the ES forecasting method in a single supply chain echelon.Duc et al. [16] focused on the impact of some parameters on the bullwhip effect via an ARMA(1, 1) model by the MMSE method.Likewise, Feng and Ma [17] evaluated the difference of MA, ES, and MMSE forecasting methods, with the ARMA(1, 1) model by using the dynamic simulation.The ARMA(1, 1) demand model is applied in all the above papers in a single supply chain with one manufacturer and one retailer.But, as an extension, a new supply chain with one manufacturer and two retailers who both employ the ARMA(1, 1) demand process is made, and emphasis is put on the competition between the two retailers analyzed [18,19].This model also compared the impact of parameters on the bullwhip effect under various forecasting methods.Moreover, Bandyopadhyay and Bhattacharya [20] primarily studied the various replenishment policies to derive expressions of BWE based on the generalized ARMA(, ) demand process in order to get the appropriate replenishment policy for the minimum BWE.
As a further expansion of the ARMA demand model, Gilbert [21] conducted a new Autoregressive Integrated Moving Average (ARIMA) time-series model to present the causes of the bullwhip effect and managerial insights about reducing the bullwhip effect in a multistage supply chain model.Dhahri and Chabchoub (2007) also used an ARIMA demand pattern to alleviate the bullwhip effect in two aspects of increase of the stock level and reduction of the service given back to customers.Nagaraja et al. [22] measured the bullwhip measure for a two-stage supply chain with an orderup-to inventory policy and derived a general, stationary SARMA(, ) × (, )  demand process.
Many scholars research the systematic problem combining the entropy theory.By using the tool of entropy, Han et al. [23] built a duopoly game model and investigated how the time delay influences the stability of the system.Ma and Si [24] researched a Bertrand duopoly game model with twostage delay and discussed the stability of the economic system.
This paper derives the expressions of the bullwhip effect under various forecasting methods in a two-stage mature and stable supply chain with one manufacturer and two retailers which have a fixed market share.With the development of information technology, all electronic manufacturers take strategies to shorten the length of supply chain and take new information technologies to speed up the logistics.For instance, Haier, one of the largest household electrical appliances manufacturers all over the world, has been benefiting from the application of ERP and BBP since the beginning of the 21st century.The information management system performs information synchronization and integration, improves the real-time and accuracy of the information, and speeds up the response speed of the supply chain.Therefore, the flat supply chain is common in reality.Based on this, we assume that both retailers face an ARMA(1, 1) demand process and employ the order-up-to replenishment policy.In the current research, we analyze the impact of parameters on the BWE under the MMSE, MA, and ES forecasting methods and compare the differences of the three methods on dampening the bullwhip effect.This paper differs from the previous research in the following.First, the ARMA(1, 1) demand model of two retailers with a certain market share which is not considered in previous papers is established in a mature and stable product market.Second, this paper evaluates the impact of the market share on the bullwhip effect and compares three forecasting methods in this two-stage supply chain.
The remaining part of this paper is organized as follows.In Section 2, we present the demand process with the market share in a new supply chain with one manufacturer and two retailers which both employ the order-up-to inventory policy.In Section 3, we derive the bullwhip effect measure under MMSE, MA, and ES forecasting methods.Under different forecasting methods, the behavior of the bullwhip effect is investigated and the effects of parameters on the bullwhip effect are discussed in Section 4, and then we compare the impact of the three forecasting methods on the bullwhip effect by numerical simulations.Finally, a short conclusion is stated for this paper in Section 5. Proofs of some propositions in this paper are summarized in the Appendix.

Supply Chain Model
The two-level supply chain model shown in Figure 1, consisting of one manufacturer with a distribution center and two retailers in the same market, is common in the consumer electronics industry.A hypothesis is that the supply chain is in a stable market.So, we assume that our two retailers are in a duopolistic competitive industry, who both have fixed share of the market of customer demand,  and 1 − , respectively.The two retailers both master the accurate market information and their customer demands, which are ARMA(1, 1) processes, and place orders to the manufacturer, respectively.The manufacturer obtains the orders directly from the retailers and arranges the delivery to the nearest distribution center as soon as possible, and the retailers take the products back immediately once what they ordered has arrived.The new supply chain model is presented in this section.

Demand Process.
Consider retailer 1 facing an ARMA(1, 1) demand of the form Here,  1, represents unit demand in the period . is the autoregressive coefficient, and || < 1.  is the first-order moving average coefficient, where −1 <  < 1.   is the random error of the customer demand, and   is independent and identically distributed from a normal distribution with mean 0 and variance  2 . is the market share of retailer 1.
Since the demand process is stationary, there is Hence, a stationary condition can be given as Similarly, retailer 2 also has an ARMA(1, 1) demand model Here,  and  have the same property with retailer 1 accordingly.1 −  is the market share of retailer 2.
For retailer 2, we also assume So, we can have In our research, based on the demand models of two retailers, the total customer demand   also faces an ARMA(1, 1), as follows: Then, we have 2.2.Inventory Policy.For supplying the demand, we assume the order-up-to inventory policy during the replenishment period, which is studied in the supply chain model by Chen et al. [6].This paper assumes that two retailers both employ a simple order-up-to inventory policy in which the orderup-to level is determined to achieve a desired service level.Retailer 1 places an ordered quantity  1, to the manufacturer at the beginning of period  to be delivered at the beginning of period  +  1 , where  1 is the fixed lead time for the manufacturer to fulfill an order of retailer 1.The order quantity  1, can be given as where  1, is the order-up-to inventory position at the beginning of period  of retailer 1 after placing the order in period .
While the base stock policy [16] is employed, the order-up-to level  1, can be determined by the sum of forecasted leadtime demand and the safety stock as in which D 1 1, is the forecast for the lead-time demand of retailer 1 which depends on the forecasting method and lead time  1 , σ 1 1, is the standard deviation of lead-time demand forecast error, and  is the normal  score determined based on a desired service level (the optimal order-up-to level   can be implicitly determined from inventory holding cost and shortage cost for backorders (Heyman and Sobel, 1984 [8]); however, since it is usually not easy to estimate these costs accurately in practice, the approach of using the service levels is often employed when the order-up-to level is to be determined).
Similarly, for retailer 2, the order quantity  2, , which is placed to the manufacturer in period  to be delivered at the beginning of period  +  2 , where  2 is the fixed lead time of retailer 1, can be given as The order-up-to level of retailer 2 at period  is Equations ( 11)-( 12) have the same meaning as ( 9)- (10).

Forecasting Method.
In this paper, we assume that two retailers both use the same forecasting method to forecast the lead-time demand.Commonly, there are three forecasting techniques, MA, ES, and MMSE, for demand forecast.As mentioned above, they have been used in most similar researches.Here, in our paper, according to the demand process, we choose appropriate forecasting methods.In Section 3, the bullwhip effect will be measured, respectively, under the MMSE, MA, and ES forecasting methods.Those three forecasting methods will be introduced in this section firstly.

The MMSE Forecasting Method.
In the minimum mean square error (MMSE) forecasting method, the lead-time demand forecast D  is given as where d is the forecast of the demand in period , which can be determined as Under the ARMA(1, 1) model, we can have the customer demand in the period of  +  by the recursive iteration of   based on (7): Then, the forecasting of the demand  + by the MMSE method is as follows: According to ( 13) and ( 16), the total forecasting demand for future -periods can be simplified as 2.3.2.The MA Forecasting Method.Using the moving average (MA) forecasting method, we first have the -period-ahead demand forecast given by in which  is the span (number of date points) for the MA forecasting method.Then, the lead-time demand forecast is given as

The ES Forecasting Method.
The ES forecasting method is an adaptive algorithm in which one-period-ahead forecast is adjusted with a fraction of the forecasting error.The demand forecast with ES can be written as where  denotes the fraction used in this process, also called the smoothing factor, and 0 <  < 1.

Measure of Bullwhip Effect considering Market Share
In this section, we derive the measure of the bullwhip effect in a supply chain with one manufacturer and two retailers which both have stable market share, under the MMSE, MA, and ES forecasting methods mentioned above, respectively.From an early start, the bullwhip effect is a phenomenon in which the variance of demand information is amplified when moving upstream in a supply chain.Thus, it is reasonable to measure the bullwhip effect by the ratio of the variance of order quantities experienced by the manufacturer to the actual variance of demand quantities.This means has been used in previous researches such as those of Chen et al. [6,7] and Duc et al. [12], and it is adopted in our research as well.
In our model, according to ( 7)-( 8), the total demand of retailers   is distributed as the ARMA(1, 1) model.Take the variance of   ; we have

Measure of the Bullwhip Effect under the MMSE Forecasting Method.
As stated in Section 3.2, the order of retailer 1  1, can be given as According to (17), the forecasting of the lead-time demand for retailer 1 is And the variance of lead-time demand forecast error, σ 1 1, , does not depend on .
Thus, taking ( 23) into ( 22),  1, can be determined as where Since  1, =   ,  1, can be rewritten as Likewise, we also have σ 2 2, = σ 2 2,−1 , and based on (24) and  2, = (1 − )  , the order of retailer 2 in period  can be written as where In this way, the order quantity of both retailers under the MMSE forecasting method in the period of  is (27) Proposition 1.The variance of total order quantity of two retailers under the MMSE forecasting method in the period of  can be determined as Proof.See Appendix A.
As the definition, the bullwhip effect in the supply chain with two retailers by the MMSE method, denoted by BWE mmse , is derived as Now, we have the exact measure of bullwhip effect under the MMSE forecasting method.

Measure of the Bullwhip Effect under the MA Forecasting
Method.Under the MA forecasting method, according to ( 9)-( 10) and ( 19), the order of retailer 1 in period  can be given as According to previous literature, σ 1 1, has no influence on the bullwhip effect.Similar to the minimum mean square error forecasting method, we have According to the above calculation process, we can get the order quantity of retailer 2 in period  similarly as follows: Take the sum of orders for two retailers; the result can be written as Proposition 2. The variance of total order quantity of two retailers under the MA forecasting method in the period of  can be determined as ] Proof.See Appendix B.
Take the variance of total order Var (  ) divided by Var (  ); the bullwhip effect measure under the MA forecasting method, called BWE MA , is given as (35)

Behavior of the Bullwhip Effect under the ES Forecasting
Method.In this section, we use the exponential smoothing forecasting technique to perform demand forecast.According to Zhang [8], we know that the forecasting demand of retailer 1 in period  is in which  1 is the smoothing exponent of retailer 1.Therefore, d1, can be interpreted as the weighted average of all past demands of retailer 1 with exponentially declining weights.Then, -period-ahead forecasting demand of retailer 1 under ES method simply extends the one-period-ahead forecast similar to the MA case where d1,+ = d1, . ≥ 1.
Then, the forecast for the lead-time demand of retailer 1 can be expressed as Using ( 20) and (38), we can have Thus, in the condition of order-up-to policy, the order quantity of retailer 1,  1, , is Here, there is also σ 1 1, = σ 1 1,−1 and  1, =   .Retailer 2 conducts a similar behavior to retailer 1, so we can get Take the sum of ( 40) and (41); the total order quantity is (42) Proposition 3. The variance of total order quantity with two retailers under the ES forecasting method in the period of  can be given as Proof.See Appendix C.
According to the most widely used method, the expression of bullwhip effect under the ES forecasting technique is as follows, named BWE ES , which seems relatively complicated: ))] . (44)

Behavior of the Bullwhip Effect and Parameter Analysis
for the MMSE.We will explore and illustrate the impact of various parameters on the bullwhip effect by using the numerical experiments.
According to Duc et al. [12], we know that the bullwhip effect occurs only if 0 <  < 1 under the MMSE forecasting method.So, next, we will study the behavior of the bullwhip effect only when 0 <  < 1.
There are three cases that we should consider.
Case 1 ( = 0 or  = 1).In this case, the supply chain only has one retailer and one manufacturer; thus, the behaviors of the bullwhip effect have been studied by Duc et al. [16].
Case 2 ( 1 =  2 = ).When the lead times of the retailer are the same, the market share  has no influence on the bullwhip effect.Therefore, the behavior of the bullwhip effect in the supply chain of our paper is the same as that in the supply chain with one retailer and one manufacturer.This result can be shown in the next expression by BWE mmse .Under the condition of  1 =  2 = , we have  1 =  2 =  = (1 −   )/(1 − ); then, the bullwhip effect can be written as Case 3 ( ̸ = 0,  ̸ = 1, and  1 ̸ =  2 ).In this case, we take a simulation for BWE mmse at the range of  > 0 in order to see the impact of different parameters on the BWE.We perform a series of numerical experiments for different values of  1 ,  2 , , , and , and the results are shown in the following.
Figure 2 depicts the impact of  on the bullwhip effect when the other parameters are fixed.The bullwhip effect measure first increases and then decreases along with the increasing of .
From Tables 1 and 2, for given values of  1 ,  2 , , and , it can be seen that the bullwhip effect reaches the maximum at a certain value of , which is here denoted as  max .Moreover, according to Tables 1-4, it is discovered that the bullwhip effect exists if and only if  > 0.3 for any values of  1 ,  2 , , and .The results show that as the value of  2 increases, the bullwhip effect BWE mmse becomes larger when the bullwhip effect BWE mmse > 1; that is, the value of  > 0.3, and  max also increases.When the  value is smaller than 0.3, the bullwhip effect has the opposite trend on the change of  2 .In addition, compared with data between Tables 1 and 2, it is shown that as the market share  increases, both the bullwhip effect and Tables 3-4 and Figure 3 have a similar trend to Tables 1 and  2 and Figure 2. We set  1 >  2 in this part of simulations.Figure 3 illustrates the behavior of BWE mmse with respect to  for fixed  1 ,  2 , , and .First, it is shown that the bullwhip effect increases to a maximum and then drops as the autoregressive coefficient increases from 0 to 1.The bullwhip effect under the MMSE forecasting method is less than 1; that is, there is no bullwhip effect in the supply chain while  < 0.3.Second, when  1 increases and  2 , , and  are at certain values, the bullwhip effect under MMSE method increases, and  max also rises for  > 0.3.Moreover, when  increases for fixed  1 ,  2 , , and  under the condition of  1 >  2 , on the contrary, the bullwhip effect increases and  max rises.
In a word, these results can be seen clearly from data of Tables 1-4.In conclusion, the market share of the retailer which has a longer lead time affects the bullwhip effect more significantly.From our research, we discover that the larger the market share of the retailer with a long lead time is, the greater the bullwhip effect is.
Next, Figure 4 shows how the market share  affects bullwhip effect in two cases.Further, the bullwhip effect is the linear correlation with .When  goes up, the bullwhip effect drops in Figure 4(a) for the condition of  1 <  2 .However, when  goes up, the bullwhip effect changes conversely in Figure 4(b) for the case of  1 >  2 .Both figures indicate that larger lead time can result in a greater bullwhip effect.When  1 >  2 , the increasing of  means more demand needs to be forecasted in longer lead time.Thus, the lead time is critical to the bullwhip effect.
In the simulation, the first-order moving average coefficient  affects the bullwhip effect in a way that, along with the increase of  from −1 to 1, the bullwhip effect first increases slowly to the maximum where it corresponds to the value  max and then decreases quickly in Figure 5. From the figure, it is observed that when  is almost larger than 0.6, the bullwhip effect does not exist.Moreover, the increasing of  2 can lead to the increasing of bullwhip effect as well as  max under the existence of the bullwhip effect.So, Figure 8 shows that we can enlarge  to reduce the bullwhip effect appropriately when other parameters are already determined.

Behavior of the Bullwhip Effect and Parameter Analysis
for the MA.Similar to Section 4.1, there are also three cases that we should consider.However, there is one difference of simulation: that we need to consider the whole range of  from −1 to 1 in this part.
Case 1 ( = 0 or  = 1).When the market share of retailer is zero or one, it means there is only one retailer in the supply chain.We do not study this case in our paper.
Case 2 ( 1 =  2 = ).In the case of  1 =  2 = , (35) can be written by simplification as The bullwhip effect has no effect with the market share  from (46).

Case 3 (𝛼 ̸
= 0,  ̸ = 1, and  1 ̸ =  2 ).When we set  ̸ = 0,  ̸ = 1, and  1 ̸ =  2 , we know the market share will affect the bullwhip effect.Next, it is studied how the parameters influence the bullwhip effect under the MA forecasting technique by simulations on the base of the expression of (35).
As shown in Figure 6, the bullwhip effect varies differently with respect to the first autoregressive coefficient under the different value of .In this figure, we fix the values of  1 = 1,  = 0.3, and  = 0.4 and run a function of  for  2 = 2, 3, and 4, with  = 4 and 5.The bullwhip effect first increases to the maximum and then decreases quickly with the increasing of  for even .When  = −1 or  = 1, the values of the bullwhip effect are both one.Likewise, for odd , the bullwhip effect first quickly decreases to a stable value and then also decreases with the continuous increase of .When  is near negative one, the bullwhip effect is most pronounced.Moreover, BWE MA is a function of  2 and increases as the lead time  2 increases.Equation (36) shows that the behavior of  1 is the same as  1 .So, the bullwhip effect increases unconditionally with the increasing of the lead time.
Figure 7(a) mainly depicts the influence of market share on the bullwhip effect.We set  1 <  2 in Figure 7(a) and  1 >  2 in Figure 7(b) when other parameters are fixed.Figure 7(a) shows that the bullwhip effect decreases as the market share  increases.And, however, Figure 7(b) shows that the bullwhip effect increases as the market share  increases.In a word, when the market share of the retailer which has a longer lead time increases, the bullwhip effect increases too.
From the two figures, we also observe that the bullwhip effect under the MA forecasting method is approximately symmetrical about  when the span  is even.
The first-order moving average coefficient  has a simple relationship with the bullwhip effect.In Figure 8, the larger Figure 6: Effect of  on the bullwhip effect under the MA method for different  2 and .
the value of  from −1 to 1, the greater the bullwhip effect.So, in the supply chain, we can reduce  to decrease the bullwhip effect.The span  is negative with the bullwhip effect, and the increasing of  can reduce BWE MA .

Behavior of the Bullwhip Effect and Parameter Analysis for the ES.
Based on the analytical expression of BWE ES , there are also three cases that we should consider similar to the MMSE and MA methods.
Figure 7: Effect of  on the bullwhip effect for different  under Case 1 ( = 0 or  = 1).Here, the conclusion is identical to the cases of MMSE and MA method.
Case 3 ( ̸ = 0,  ̸ = 1, and  1 ̸ =  2 or  1 ̸ =  2 ).In order to discuss the impact of market share  on the bullwhip effect, we make the specific rules as  1 ̸ =  2 and  1 ̸ =  2 .By numerical simulations, we will study the changes of BWE ES .Figure 9 shows the trend of bullwhip effect along with the varying of  from −1 to 1 under different values of  2 .BWE ES is decreased when  increases and is increased when  2 increases from 2 to 4. In the supply chain, the longer the lead time is, the more difficult the forecast to the demand is.So, the bullwhip effect is larger and larger when the lead time of retailer 2 is increasing.
Figure 10 depicts how the bullwhip effect is affected by the market share under the different conditions of the lead time.A similar trend can be observed to Figure 10 where the bullwhip effect decreases when  increases while  1 ,  2 , , ,  1 , and  2 are fixed.
Figure 10: The impact of  and  on the bullwhip effect on the condition of  1 <  2 and  1 >  2 .
Moreover, when the values of  1 ,  2 , ,  1 , and  2 are fixed such that  1 <  2 , the bullwhip effect decreases as  increases.Under the fixed values of  1 ,  2 , ,  1 , and  2 such that  1 >  2 , yet, the bullwhip effect increases as  increases.This consequence can be observed from Figure 11(a) for the case  1 <  2 and from Figure 10(b) for the case  1 >  2 .
In conclusion, when all other parameters are fixed, the bullwhip effect decreases as the market share of the retailer with a shorter lead time increases.
Figure 11(a) studies the influence of  1 on the bullwhip effect; it is indicated that BWE ES has a positive correlation with the smoothing coefficient of retailer 1.In addition, the trend of BWE ES is roughly the same for different  2 .That is, the influence of the smoothing coefficient  1 on the bullwhip effect is determinate regardless of the measurement of  1 and  2 .
Figure 11(b) indicates the behavior of the bullwhip effect with regard to  for fixed values of other parameters.The bullwhip effect is increased slowly along with the increasing of  from −1 to 0; however, it increases quickly when  increases from 0 to 1.Some other information can be discovered from Figure 11(b).The larger the value of  2 is, the faster the bullwhip effect increases as  increases.

Comparison of the Bullwhip Effect by Three Forecasting
Methods.We study the bullwhip effect using three forecasting methods in Sections of 4.1-4.3.Next, we will compare the different influences of MMSE, MA, and ES on the bullwhip effect.For the comparability of the MA and ES methods, we set a constraint on the span  of MA and the smoothing factors  1 and  2 .According to Zhang [8], we obtain that the smoothing exponents are fixed as  1 =  2 = 2/( + 1).This means the average data ages are the same for the MA and ES forecasting method.Further, we know  +  2 − 1 ̸ = 0 from the expression of the bullwhip effect (see (46)) under the ES forecasting method.
Figure 12 depicts the difference of the three forecasting methods on measuring the bullwhip effect by varying  for different  and .We observe that the bullwhip effect measure of MA is always smaller than that of ES no matter what  is as long as  1 =  2 = 2/( + 1).It can be seen that the bullwhip effect under the MMSE forecasting method does not exist when  ≤ 0.3 from the two figures.The bullwhip effect for the three forecasting methods converges to one as  approaches one.
Moreover, for fixed values of , ,  1 , and  2 , BWE MMSE is the least when  is smaller than a certain value, called  1 .After  becomes larger than  1 and smaller than another certain value, called  2 , BWE MA is the smallest and BWE MMSE is lower than BWE ES .When  is larger than  2 , BWE MMSE is the largest compared to the other two methods.In general, while  is lower than  1 , the MMSE method is the best for reducing the bullwhip effect, and while  is higher than  1 , the MA method is the best.
However, comparing the two figures, we find that as the span  increases, the bullwhip effects for the three forecasting methods overall decrease, and the values of  1 and  2 are lessened.This phenomenon reveals that when we choose a larger span , the extent of  is larger for the MA which is the best method to predict the bullwhip effect.
In Figure 13, we set  = 0.5,  = 0.3,  1 = 1,  2 = 2,  1 =  2 = 0.4, and  = 4 and then study the bullwhip effect for the three forecasting methods under the condition of various market shares .We observe that the bullwhip effect is the smallest by the MMSE forecasting method whatever  is.And the bullwhip effect under the ES method is the worst whatever  is.This conclusion is related to the value of  when we select a certain .
Figure 14 shows the curved surface of the bullwhip effect by varying  and  for the three forecasting methods.It is observed that BWE ES is always larger than BWE MA whatever  is as long as  1 =  2 = 2/( + 1) for the fixed value of .When  increases from −1 to 1, BWE MA is the smallest while  is less than a certain value, and BWE MMSE is the best while  is more than that certain value.In addition, there is also a phenomenon where BWE MMSE is all along the largest for any value of  when  approaches negative one.In summary, in order to reduce the bullwhip effect, we will select a reasonable forecasting method according to the values of various parameters.

Conclusions
In this paper, we mainly study the impact of three forecasting methods on the bullwhip effect measure considering the market share in a stable mature supply chain with two retailers.We first quantify the bullwhip effect in a two-level supply chain with two retailers considering the market share.In the ARMA(1, 1) demand model, we use the three forecasting methods to establish the expressions of the bullwhip effect.By simulations, some results are obtained.(1) The larger the market share of the retailer with long lead time is, the greater the bullwhip effect is, no matter what the forecasting method is.(2) The bullwhip effect is increased along with the increasing of  for the MA and ES method and, however, it is dampened for the MMSE method.(3) The bullwhip effect has a positive correlation with the lead time and the smoothing factors  and has a negative correlation with the span .(4) By comparison, the MA is always better than the ES for reducing the bullwhip effect in our supply chain model.When  is lower than a certain value, the MMSE method is the best for reducing the bullwhip effect; otherwise, the MA method is the best.
Our findings give some useful insights for supply chain managers in the consumer electronics industry.Firstly, the retailers should make it clear what kind of pattern the demand process follows and choose a suitable forecasting method to dampen the bullwhip effect.Secondly, the manufacturer should pay more attention to the retailer who occupies a large market share on purpose and reduce its lead time to dampen the bullwhip effect.
From the view of entropy, we see that the autocorrelation coefficient can impact the uncertainty of the order.The minimum of the entropy is around  = 0.672, and the maximum of the entropy is around  = 1.
The research presented in this paper proposes several future directions for improvement of our understanding of market share.First, we can further study the effect of unfixed market share on the bullwhip effect in an unstable supply chain.Second, we only consider the order-up-to inventory policy and three traditional demand forecasting methods.Therefore, other new forecast methods and complex inventory policies can be studied in future research.Furthermore, price is the key factor on the bullwhip effect.So, introducing the retailer's price into the demand model to analyze the complexity and stability of the supply chain system is another future direction. Complexity This completes the proof of Proposition 1.

B. A Proof of Proposition 2
The variance of the total order quantity under the MA forecasting method can be derived from (33) as follows: According to (A.2), the variance can be simplified as This completes the proof of Proposition 2.

C. A Proof of Proposition 3
The variance of the total order quantity under the ES forecasting method can be derived from (42) as follows: Var (  ) = Var ((1 +

Figure 5 :
Figure 5: Effect of  on the bullwhip effect under the MMSE method for different .

Figure 8 : 3 Figure 9 :
Figure 8: Effect of  on the bullwhip effect for different span .

Figure 11 :
Figure 11: The impact of the smoothing coefficient  1 and  on BWE ES for different  2 .

Figure 12 : 3 Figure 13 :
Figure 12: Comparison of the three forecasting methods by varying  in the case of  = 4 or  = 9.

Figure 14 :
Figure 14: Comparison of the three forecasting methods by varying  and  in the case of  = 8.

Table 1 :
Values of bullwhip effect under MMSE in the case of  1 = 1 and  = 0.4.

Table 2 :
Values of bullwhip effect under MMSE in the case of  1 = 1 and  = 0.7.

Table 3 :
Values of bullwhip effect under MMSE in the case of  2 = 1 and  = 0.4.

Table 4 :
Values of bullwhip effect under MMSE in the case of  2 = 1 and  = 0.7. max decrease, at the fixed values of ,  1 , and  2 , which meets the condition of  1 <  2 .