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

Authors: Junhai Ma; Jing Zhang; Liqing Zhu
Journal: Complexity
 (2018)

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## Abstract

We establish in this paper a new two-stage supply chain with one manufacturer and two retailers which have a fixed market 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 order-up-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 the MMSE, MA, and ES methods 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 MA method 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.


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## Body

## 1. 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 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 [6–10]. 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-up-to 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(p) 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, 6–12], 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(p,q) 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 order-up-to inventory policy and derived a general, stationary SARMA(p,q)×(P,Q)s 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 two-stage 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 Section2, 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.


## 2. Supply Chain Model

The two-level supply chain model shown in Figure1, 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, α and 1-α, 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.Figure 1
Two-stage supply chain model.


### 2.1. Demand Process

Consider retailer 1 facing an ARMA(1,1) demand of the form(1)d1,t=αδ+ϕd1,t-1+αεt-αθεt-1.Here,d1,t represents unit demand in the period t. ϕ is the autoregressive coefficient, and ϕ<1. θ is the first-order moving average coefficient, where -1<θ<1. εt is the random error of the customer demand, and εt is independent and identically distributed from a normal distribution with mean 0 and variance σ2. α is the market share of retailer 1.Since the demand process is stationary, there is(2)Ed1,t=Ed1,t-1=Ed1,Vard1,t=Vard1,t-1=Vard1,∀t.Hence, a stationary condition can be given as (3)Ed1=αδ1-ϕ.Vard1=α21+θ2-2ϕθ1-ϕ2Varεt.Similarly, retailer 2 also has an ARMA(1,1) demand model(4)d2,t=1-αδ+ϕd2,t-1+1-αεt-1-αθεt-1.Here, ϕ and θ have the same property with retailer 1 accordingly. 1-α is the market share of retailer 2.For retailer 2, we also assume(5)Ed2,t=Ed2,t-1=Ed2,Var⁡d2,t=Var⁡d2,t-1=Var⁡d2,∀t.So, we can have(6)Ed2=1-αδ1-ϕ.Var⁡d2=1-α21+θ2-2ϕθ1-ϕ2Var⁡εt.In our research, based on the demand models of two retailers, the total customer demanddt also faces an ARMA(1,1), as follows:(7)dt=δ+ϕdt-1+εt-θεt-1.Then, we have(8)d1,t=αdt,d2,t=1-αdt.


### 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 order-up-to level is determined to achieve a desired service level. Retailer 1 places an ordered quantity q1,t to the manufacturer at the beginning of period t to be delivered at the beginning of period t+L1, where L1 is the fixed lead time for the manufacturer to fulfill an order of retailer 1. The order quantity q1,t can be given as(9)q1,t=S1,t-S1,t-1+d1,t-1,where S1,t is the order-up-to inventory position at the beginning of period t of retailer 1 after placing the order in period t. While the base stock policy [16] is employed, the order-up-to level S1,t can be determined by the sum of forecasted lead-time demand and the safety stock as(10)S1,t=D^1,tL1+zσ^1,tL1,in which D^1,tL1 is the forecast for the lead-time demand of retailer 1 which depends on the forecasting method and lead time L1, σ^1,tL1 is the standard deviation of lead-time demand forecast error, and z is the normal z score determined based on a desired service level (the optimal order-up-to level St 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 quantityq2,t, which is placed to the manufacturer in period t to be delivered at the beginning of period t+L2, where L2 is the fixed lead time of retailer 1, can be given as(11)q2,t=S2,t-S2,t-1+d2,t-1.The order-up-to level of retailer 2 at periodt is (12)S2,t=D^2,tL1+zσ^2,tL1.Equations (11)-(12) have the same meaning as (9)-(10).


### 2.3. 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 Section3, 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.


#### 2.3.1. The MMSE Forecasting Method

In the minimum mean square error (MMSE) forecasting method, the lead-time demand forecastD^tL is given as(13)D^tL=d^t+d^t+1+⋯+d^t+L-1=∑i=0L-1d^t+i,where d^t is the forecast of the demand in period t, which can be determined as (14)d^t+i=Edt+i∣dt-1,dt-2,….Under the ARMA(1,1) model, we can have the customer demand in the period of t+i by the recursive iteration of dt based on (7):(15)dt+i=1-ϕi+11-ϕδ+ϕi+1dt-1-θ∑j=0iϕjεt+i-j-1+∑j=0iϕjεt+i-j.Then, the forecasting of the demand dt+i by the MMSE method is as follows:(16)d^t+i=1-ϕi+11-ϕδ+ϕi+1dt-1-ϕiθεt-1.According to (13) and (16), the total forecasting demand for future L-periods can be simplified as(17)D^tL=L1-ϕδ-ϕ1-ϕL1-ϕ2δ+ϕ1-ϕL1-ϕdt-1-θ1-ϕL1-ϕεt-1.


#### 2.3.2. The MA Forecasting Method

Using the moving average (MA) forecasting method, we first have theτ-period-ahead demand forecast given by (18)d^t+τ=d^t=1k∑i=1kdt-i,τ≥1,in which k is the span (number of date points) for the MA forecasting method. Then, the lead-time demand forecast is given as(19)D^tL=Lk∑i=1kdt-i.


#### 2.3.3. 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(20)d^t=λdt-1+1-λd^t-1,where λ denotes the fraction used in this process, also called the smoothing factor, and 0<λ<1.


## 2.1. Demand Process

Consider retailer 1 facing an ARMA(1,1) demand of the form(1)d1,t=αδ+ϕd1,t-1+αεt-αθεt-1.Here,d1,t represents unit demand in the period t. ϕ is the autoregressive coefficient, and ϕ<1. θ is the first-order moving average coefficient, where -1<θ<1. εt is the random error of the customer demand, and εt is independent and identically distributed from a normal distribution with mean 0 and variance σ2. α is the market share of retailer 1.Since the demand process is stationary, there is(2)Ed1,t=Ed1,t-1=Ed1,Vard1,t=Vard1,t-1=Vard1,∀t.Hence, a stationary condition can be given as (3)Ed1=αδ1-ϕ.Vard1=α21+θ2-2ϕθ1-ϕ2Varεt.Similarly, retailer 2 also has an ARMA(1,1) demand model(4)d2,t=1-αδ+ϕd2,t-1+1-αεt-1-αθεt-1.Here, ϕ and θ have the same property with retailer 1 accordingly. 1-α is the market share of retailer 2.For retailer 2, we also assume(5)Ed2,t=Ed2,t-1=Ed2,Var⁡d2,t=Var⁡d2,t-1=Var⁡d2,∀t.So, we can have(6)Ed2=1-αδ1-ϕ.Var⁡d2=1-α21+θ2-2ϕθ1-ϕ2Var⁡εt.In our research, based on the demand models of two retailers, the total customer demanddt also faces an ARMA(1,1), as follows:(7)dt=δ+ϕdt-1+εt-θεt-1.Then, we have(8)d1,t=αdt,d2,t=1-αdt.


## 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 order-up-to level is determined to achieve a desired service level. Retailer 1 places an ordered quantity q1,t to the manufacturer at the beginning of period t to be delivered at the beginning of period t+L1, where L1 is the fixed lead time for the manufacturer to fulfill an order of retailer 1. The order quantity q1,t can be given as(9)q1,t=S1,t-S1,t-1+d1,t-1,where S1,t is the order-up-to inventory position at the beginning of period t of retailer 1 after placing the order in period t. While the base stock policy [16] is employed, the order-up-to level S1,t can be determined by the sum of forecasted lead-time demand and the safety stock as(10)S1,t=D^1,tL1+zσ^1,tL1,in which D^1,tL1 is the forecast for the lead-time demand of retailer 1 which depends on the forecasting method and lead time L1, σ^1,tL1 is the standard deviation of lead-time demand forecast error, and z is the normal z score determined based on a desired service level (the optimal order-up-to level St 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 quantityq2,t, which is placed to the manufacturer in period t to be delivered at the beginning of period t+L2, where L2 is the fixed lead time of retailer 1, can be given as(11)q2,t=S2,t-S2,t-1+d2,t-1.The order-up-to level of retailer 2 at periodt is (12)S2,t=D^2,tL1+zσ^2,tL1.Equations (11)-(12) have the same meaning as (9)-(10).


## 2.3. 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 Section3, 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.


### 2.3.1. The MMSE Forecasting Method

In the minimum mean square error (MMSE) forecasting method, the lead-time demand forecastD^tL is given as(13)D^tL=d^t+d^t+1+⋯+d^t+L-1=∑i=0L-1d^t+i,where d^t is the forecast of the demand in period t, which can be determined as (14)d^t+i=Edt+i∣dt-1,dt-2,….Under the ARMA(1,1) model, we can have the customer demand in the period of t+i by the recursive iteration of dt based on (7):(15)dt+i=1-ϕi+11-ϕδ+ϕi+1dt-1-θ∑j=0iϕjεt+i-j-1+∑j=0iϕjεt+i-j.Then, the forecasting of the demand dt+i by the MMSE method is as follows:(16)d^t+i=1-ϕi+11-ϕδ+ϕi+1dt-1-ϕiθεt-1.According to (13) and (16), the total forecasting demand for future L-periods can be simplified as(17)D^tL=L1-ϕδ-ϕ1-ϕL1-ϕ2δ+ϕ1-ϕL1-ϕdt-1-θ1-ϕL1-ϕεt-1.


### 2.3.2. The MA Forecasting Method

Using the moving average (MA) forecasting method, we first have theτ-period-ahead demand forecast given by (18)d^t+τ=d^t=1k∑i=1kdt-i,τ≥1,in which k is the span (number of date points) for the MA forecasting method. Then, the lead-time demand forecast is given as(19)D^tL=Lk∑i=1kdt-i.


### 2.3.3. 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(20)d^t=λdt-1+1-λd^t-1,where λ denotes the fraction used in this process, also called the smoothing factor, and 0<λ<1.


## 2.3.1. The MMSE Forecasting Method

In the minimum mean square error (MMSE) forecasting method, the lead-time demand forecastD^tL is given as(13)D^tL=d^t+d^t+1+⋯+d^t+L-1=∑i=0L-1d^t+i,where d^t is the forecast of the demand in period t, which can be determined as (14)d^t+i=Edt+i∣dt-1,dt-2,….Under the ARMA(1,1) model, we can have the customer demand in the period of t+i by the recursive iteration of dt based on (7):(15)dt+i=1-ϕi+11-ϕδ+ϕi+1dt-1-θ∑j=0iϕjεt+i-j-1+∑j=0iϕjεt+i-j.Then, the forecasting of the demand dt+i by the MMSE method is as follows:(16)d^t+i=1-ϕi+11-ϕδ+ϕi+1dt-1-ϕiθεt-1.According to (13) and (16), the total forecasting demand for future L-periods can be simplified as(17)D^tL=L1-ϕδ-ϕ1-ϕL1-ϕ2δ+ϕ1-ϕL1-ϕdt-1-θ1-ϕL1-ϕεt-1.


## 2.3.2. The MA Forecasting Method

Using the moving average (MA) forecasting method, we first have theτ-period-ahead demand forecast given by (18)d^t+τ=d^t=1k∑i=1kdt-i,τ≥1,in which k is the span (number of date points) for the MA forecasting method. Then, the lead-time demand forecast is given as(19)D^tL=Lk∑i=1kdt-i.


## 2.3.3. 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(20)d^t=λdt-1+1-λd^t-1,where λ denotes the fraction used in this process, also called the smoothing factor, and 0<λ<1.


## 3. 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 dt is distributed as the ARMA(1,1) model. Take the variance of dt; we have(21)Var⁡dt=1+θ2-2ϕθ1-ϕ2Var⁡εt.


### 3.1. Measure of the Bullwhip Effect under the MMSE Forecasting Method

As stated in Section3.2, the order of retailer 1 q1,t can be given as(22)q1,t=S1,t-S1,t-1+d1,t-1=D^1,tL1-D^1,t-1L1+zσ^1,tL1-σ^1,t-1L1+d1,t-1.According to (17), the forecasting of the lead-time demand for retailer 1 is(23)D^1,tL1=L11-ϕαδ-ϕ1-ϕL11-ϕ2αδ+ϕ1-ϕL11-ϕd1,t-1-αθ1-ϕL11-ϕεt-1.And the variance of lead-time demand forecast error, σ^1,tL1, does not depend on t.Thus, taking (23) into (22), q1,t can be determined as (24)q1,t=ϕ1-ϕL11-ϕd1,t-1-αθ1-ϕL11-ϕεt-1-ϕ1-ϕL11-ϕd1,t-2-αθ1-ϕL11-ϕεt-2+d1,t-1=ϕA1+1d1,t-1-ϕA1d1,t-2-αθA1εt-1-εt-2,where A1=(1-ϕL1)/(1-ϕ).Sinced1,t=αdt, q1,t can be rewritten as (25)q1,t=ϕA1+1αdt-1-ϕA1αdt-2-αθA1εt-1-εt-2.Likewise, we also haveσ^2,tL2=σ^2,t-1L2, and based on (24) and d2,t=(1-α)dt, the order of retailer 2 in period t can be written as (26)q2,t=ϕA2+11-αdt-1-ϕA21-αdt-2-1-αθA2εt-1-εt-2,where A2=(1-ϕL2)/(1-ϕ).In this way, the order quantity of both retailers under the MMSE forecasting method in the period oft is(27)qt=q1,t+q2,t=ϕA1+1αdt-1-ϕA1αdt-2-αθA1εt-1-εt-2+ϕA2+11-αdt-1-ϕA21-αdt-2-1-αθA2εt-1-εt-2=1+αϕA1+1-αϕA2dt-1-αϕA1+1-αϕA2dt-2-αθA1+1-αθA2εt-1+αθA1+1-αθA2εt-2.Proposition 1.
The variance of total order quantity of two retailers under the MMSE forecasting method in the period oft can be determined as(28)Varqt=1+αϕA1+1-αϕA22+αϕA1+1-αϕA22-21+αϕA1+1-αϕA2αϕA1+1-αϕA2ϕ-θ1-ϕθ1+θ2-2ϕθ+21-ϕ21+θ2-2ϕθαθA1+1-αθA22-1+αϕA1+1-αϕA2αθA1+1-αθA2+1+αϕA1+1-αϕA2αθA1+1-αθA2ϕ-θ-αϕA1+1-αϕA2αθA1+1-αθA2Vardt.Proof.
See AppendixA.As the definition, the bullwhip effect in the supply chain with two retailers by the MMSE method, denoted byBWEmmse, is derived as(29)BWEmmse=VarqtVardt=1+αϕA1+1-αϕA22+αϕA1+1-αϕA22-21+αϕA1+1-αϕA2αϕA1+1-αϕA2ϕ-θ1-ϕθ1+θ2-2ϕθ+21-ϕ21+θ2-2ϕθαθA1+1-αθA22+1+αϕA1+1-αϕA2αθA1+1-αθA2ϕ-θ-1-αϕA1+1-αϕA2αθA1+1-αθA2.Now, we have the exact measure of bullwhip effect under the MMSE forecasting method.


### 3.2. 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 t can be given as(30)q1,t=D^1,tL1-D^1,t-1L1+zσ^1,tL1-σ^1,t-1L1+d1,t-1=L1k∑i=1kd1,t-i-L1k∑i=1kd1,t-i-1+d1,t-1+zσ^1,tL1-σ^1,t-1L1.According to previous literature,σ^1,tL1 has no influence on the bullwhip effect. Similar to the minimum mean square error forecasting method, we have(31)q1,t=1+L1kd1,t-1-L1kd1,t-k-1=α1+L1kdt-1-αL1kdt-k-1.According to the above calculation process, we can get the order quantity of retailer 2 in periodt similarly as follows:(32)q2,t=1+L2kd2,t-1-L2kd2,t-k-1=1-α1+L2kdt-1-1-αL2kdt-k-1.Take the sum of orders for two retailers; the result can be written as(33)qt=α1+L1kdt-1-αL1kdt-k-1+1-α1+L2kdt-1-1-αL2kdt-k-1=1+αL1k+1-αL2kdt-1-αL1k+1-αL2kdt-k-1.Proposition 2.
The variance of total order quantity of two retailers under the MA forecasting method in the period oft can be determined as(34)Var⁡qt=1+αL1k+1-αL2k2+αL1k+1-αL2k2-21+αL1k+1-αL2kαL1k+1-αL2kϕk-1ϕ-θ1-ϕθ1+θ2-2ϕθVar⁡dt.Proof.
See AppendixB.Take the variance of total orderVar(qt) divided by Var(dt); the bullwhip effect measure under the MA forecasting method, called BWEMA, is given as(35)BWEMA=1+αL1k+1-αL2k2+αL1k+1-αL2k2-21+αL1k+1-αL2kαL1k+1-αL2kϕk-1ϕ-θ1-ϕθ1+θ2-2ϕθ.


### 3.3. 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 t is (36)d^1,t=∑i=0∞λ11-λ1id1,t-i-1,in which λ1is the smoothing exponent of retailer 1.Therefore,d^1,t can be interpreted as the weighted average of all past demands of retailer 1 with exponentially declining weights. Then, τ-period-ahead forecasting demand of retailer 1 under ES method simply extends the one-period-ahead forecast similar to the MA case where(37)d^1,t+τ=d^1,t.τ≥1.Then, the forecast for the lead-time demand of retailer 1 can be expressed as (38)D^1,tL1=L1d^1,t.Using (20) and (38), we can have (39)D^1,tL1-D^1,t-1L1=λ1L1d1,t-1-d^1,t-1.Thus, in the condition of order-up-to policy, the order quantity of retailer 1,q1,t, is(40)q1,t=D^1,tL1-D^1,t-1L1+zσ^1,tL1-σ^1,t-1L1+d1,t-1=λ1L1d1.t-1-d^1,t-1+d1.t-1=αdt-1+λ1L1αdt-1-d^1,t-1.Here, there is alsoσ^1,tL1=σ^1,t-1L1 and d1,t=αdt.Retailer 2 conducts a similar behavior to retailer 1, so we can get(41)q2,t=λ2L2d2,t-1-d^2,t-1+d2.t-1=1-αdt-1+λ2L21-αdt-1-d^2,t-1.Take the sum of (40) and (41); the total order quantity is (42)qt=αdt-1+λ1L1αdt-1-d^1,t-1+1-αdt-1+λ2L21-αdt-1-d^2,t-1=1+αλ1L1+1-αλ2L2dt-1-λ1L1d^1,t-1-λ2L2d^2,t-1.Proposition 3.
The variance of total order quantity with two retailers under the ES forecasting method in the period oft can be given as(43)Var⁡qt=Var⁡dt1+αλ1L1+1-αλ2L22+L12λ13α22-λ11+21-λ11-1-λ1ϕϕ-θ1-ϕθ1+θ2-2ϕθ+L22λ231-α22-λ21+21-λ21-1-λ2ϕϕ-θ1-ϕθ1+θ2-2ϕθ-21+αλ1L1+1-αλ2L2ϕ-θ1-ϕθ1+θ2-2ϕθαL1λ121-1-λ1ϕ+1-αL2λ221-1-λ2ϕ+2λ12λ22L1L2α1-αϕ-θ1-ϕθ1+θ2-2ϕθ1ϕ+λ2-11-1-λ1ϕ+11-1-λ11-λ21+ϕ-θ1-ϕθ1+θ2-2ϕθ1-λ21-1-λ2ϕ-1ϕ+λ2-1.Proof.
See AppendixC.According to the most widely used method, the expression of bullwhip effect under the ES forecasting technique is as follows, namedBWEES, which seems relatively complicated:(44)BWEES=1+αλ1L1+1-αλ2L22+L12λ13α22-λ11+21-λ11-1-λ1ϕϕ-θ1-ϕθ1+θ2-2ϕθ+L22λ231-α22-λ21+21-λ21-1-λ2ϕϕ-θ1-ϕθ1+θ2-2ϕθ-21+αλ1L1+1-αλ2L2ϕ-θ1-ϕθ1+θ2-2ϕθαL1λ121-1-λ1ϕ+1-αL2λ221-1-λ2ϕ+2λ12λ22L1L2α1-αϕ-θ1-ϕθ1+θ2-2ϕθ1ϕ+λ2-11-1-λ1ϕ+11-1-λ11-λ21+ϕ-θ1-ϕθ1+θ2-2ϕθ1-λ21-1-λ2ϕ-1ϕ+λ2-1.


## 3.1. Measure of the Bullwhip Effect under the MMSE Forecasting Method

As stated in Section3.2, the order of retailer 1 q1,t can be given as(22)q1,t=S1,t-S1,t-1+d1,t-1=D^1,tL1-D^1,t-1L1+zσ^1,tL1-σ^1,t-1L1+d1,t-1.According to (17), the forecasting of the lead-time demand for retailer 1 is(23)D^1,tL1=L11-ϕαδ-ϕ1-ϕL11-ϕ2αδ+ϕ1-ϕL11-ϕd1,t-1-αθ1-ϕL11-ϕεt-1.And the variance of lead-time demand forecast error, σ^1,tL1, does not depend on t.Thus, taking (23) into (22), q1,t can be determined as (24)q1,t=ϕ1-ϕL11-ϕd1,t-1-αθ1-ϕL11-ϕεt-1-ϕ1-ϕL11-ϕd1,t-2-αθ1-ϕL11-ϕεt-2+d1,t-1=ϕA1+1d1,t-1-ϕA1d1,t-2-αθA1εt-1-εt-2,where A1=(1-ϕL1)/(1-ϕ).Sinced1,t=αdt, q1,t can be rewritten as (25)q1,t=ϕA1+1αdt-1-ϕA1αdt-2-αθA1εt-1-εt-2.Likewise, we also haveσ^2,tL2=σ^2,t-1L2, and based on (24) and d2,t=(1-α)dt, the order of retailer 2 in period t can be written as (26)q2,t=ϕA2+11-αdt-1-ϕA21-αdt-2-1-αθA2εt-1-εt-2,where A2=(1-ϕL2)/(1-ϕ).In this way, the order quantity of both retailers under the MMSE forecasting method in the period oft is(27)qt=q1,t+q2,t=ϕA1+1αdt-1-ϕA1αdt-2-αθA1εt-1-εt-2+ϕA2+11-αdt-1-ϕA21-αdt-2-1-αθA2εt-1-εt-2=1+αϕA1+1-αϕA2dt-1-αϕA1+1-αϕA2dt-2-αθA1+1-αθA2εt-1+αθA1+1-αθA2εt-2.Proposition 1.
The variance of total order quantity of two retailers under the MMSE forecasting method in the period oft can be determined as(28)Varqt=1+αϕA1+1-αϕA22+αϕA1+1-αϕA22-21+αϕA1+1-αϕA2αϕA1+1-αϕA2ϕ-θ1-ϕθ1+θ2-2ϕθ+21-ϕ21+θ2-2ϕθαθA1+1-αθA22-1+αϕA1+1-αϕA2αθA1+1-αθA2+1+αϕA1+1-αϕA2αθA1+1-αθA2ϕ-θ-αϕA1+1-αϕA2αθA1+1-αθA2Vardt.Proof.
See AppendixA.As the definition, the bullwhip effect in the supply chain with two retailers by the MMSE method, denoted byBWEmmse, is derived as(29)BWEmmse=VarqtVardt=1+αϕA1+1-αϕA22+αϕA1+1-αϕA22-21+αϕA1+1-αϕA2αϕA1+1-αϕA2ϕ-θ1-ϕθ1+θ2-2ϕθ+21-ϕ21+θ2-2ϕθαθA1+1-αθA22+1+αϕA1+1-αϕA2αθA1+1-αθA2ϕ-θ-1-αϕA1+1-αϕA2αθA1+1-αθA2.Now, we have the exact measure of bullwhip effect under the MMSE forecasting method.


## 3.2. 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 t can be given as(30)q1,t=D^1,tL1-D^1,t-1L1+zσ^1,tL1-σ^1,t-1L1+d1,t-1=L1k∑i=1kd1,t-i-L1k∑i=1kd1,t-i-1+d1,t-1+zσ^1,tL1-σ^1,t-1L1.According to previous literature,σ^1,tL1 has no influence on the bullwhip effect. Similar to the minimum mean square error forecasting method, we have(31)q1,t=1+L1kd1,t-1-L1kd1,t-k-1=α1+L1kdt-1-αL1kdt-k-1.According to the above calculation process, we can get the order quantity of retailer 2 in periodt similarly as follows:(32)q2,t=1+L2kd2,t-1-L2kd2,t-k-1=1-α1+L2kdt-1-1-αL2kdt-k-1.Take the sum of orders for two retailers; the result can be written as(33)qt=α1+L1kdt-1-αL1kdt-k-1+1-α1+L2kdt-1-1-αL2kdt-k-1=1+αL1k+1-αL2kdt-1-αL1k+1-αL2kdt-k-1.Proposition 2.
The variance of total order quantity of two retailers under the MA forecasting method in the period oft can be determined as(34)Var⁡qt=1+αL1k+1-αL2k2+αL1k+1-αL2k2-21+αL1k+1-αL2kαL1k+1-αL2kϕk-1ϕ-θ1-ϕθ1+θ2-2ϕθVar⁡dt.Proof.
See AppendixB.Take the variance of total orderVar(qt) divided by Var(dt); the bullwhip effect measure under the MA forecasting method, called BWEMA, is given as(35)BWEMA=1+αL1k+1-αL2k2+αL1k+1-αL2k2-21+αL1k+1-αL2kαL1k+1-αL2kϕk-1ϕ-θ1-ϕθ1+θ2-2ϕθ.


## 3.3. 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 t is (36)d^1,t=∑i=0∞λ11-λ1id1,t-i-1,in which λ1is the smoothing exponent of retailer 1.Therefore,d^1,t can be interpreted as the weighted average of all past demands of retailer 1 with exponentially declining weights. Then, τ-period-ahead forecasting demand of retailer 1 under ES method simply extends the one-period-ahead forecast similar to the MA case where(37)d^1,t+τ=d^1,t.τ≥1.Then, the forecast for the lead-time demand of retailer 1 can be expressed as (38)D^1,tL1=L1d^1,t.Using (20) and (38), we can have (39)D^1,tL1-D^1,t-1L1=λ1L1d1,t-1-d^1,t-1.Thus, in the condition of order-up-to policy, the order quantity of retailer 1,q1,t, is(40)q1,t=D^1,tL1-D^1,t-1L1+zσ^1,tL1-σ^1,t-1L1+d1,t-1=λ1L1d1.t-1-d^1,t-1+d1.t-1=αdt-1+λ1L1αdt-1-d^1,t-1.Here, there is alsoσ^1,tL1=σ^1,t-1L1 and d1,t=αdt.Retailer 2 conducts a similar behavior to retailer 1, so we can get(41)q2,t=λ2L2d2,t-1-d^2,t-1+d2.t-1=1-αdt-1+λ2L21-αdt-1-d^2,t-1.Take the sum of (40) and (41); the total order quantity is (42)qt=αdt-1+λ1L1αdt-1-d^1,t-1+1-αdt-1+λ2L21-αdt-1-d^2,t-1=1+αλ1L1+1-αλ2L2dt-1-λ1L1d^1,t-1-λ2L2d^2,t-1.Proposition 3.
The variance of total order quantity with two retailers under the ES forecasting method in the period oft can be given as(43)Var⁡qt=Var⁡dt1+αλ1L1+1-αλ2L22+L12λ13α22-λ11+21-λ11-1-λ1ϕϕ-θ1-ϕθ1+θ2-2ϕθ+L22λ231-α22-λ21+21-λ21-1-λ2ϕϕ-θ1-ϕθ1+θ2-2ϕθ-21+αλ1L1+1-αλ2L2ϕ-θ1-ϕθ1+θ2-2ϕθαL1λ121-1-λ1ϕ+1-αL2λ221-1-λ2ϕ+2λ12λ22L1L2α1-αϕ-θ1-ϕθ1+θ2-2ϕθ1ϕ+λ2-11-1-λ1ϕ+11-1-λ11-λ21+ϕ-θ1-ϕθ1+θ2-2ϕθ1-λ21-1-λ2ϕ-1ϕ+λ2-1.Proof.
See AppendixC.According to the most widely used method, the expression of bullwhip effect under the ES forecasting technique is as follows, namedBWEES, which seems relatively complicated:(44)BWEES=1+αλ1L1+1-αλ2L22+L12λ13α22-λ11+21-λ11-1-λ1ϕϕ-θ1-ϕθ1+θ2-2ϕθ+L22λ231-α22-λ21+21-λ21-1-λ2ϕϕ-θ1-ϕθ1+θ2-2ϕθ-21+αλ1L1+1-αλ2L2ϕ-θ1-ϕθ1+θ2-2ϕθαL1λ121-1-λ1ϕ+1-αL2λ221-1-λ2ϕ+2λ12λ22L1L2α1-αϕ-θ1-ϕθ1+θ2-2ϕθ1ϕ+λ2-11-1-λ1ϕ+11-1-λ11-λ21+ϕ-θ1-ϕθ1+θ2-2ϕθ1-λ21-1-λ2ϕ-1ϕ+λ2-1.


## 4. Behavior and Comparison of Bullwhip Effect by Three Forecasting Methods

### 4.1. 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<ϕ<1 under the MMSE forecasting method. So, next, we will study the behavior of the bullwhip effect only when 0<ϕ<1.There are three cases that we should consider.Case 1 (α=0 or α=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 (L1=L2=L).
When the lead times of the retailer are the same, the market shareα 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 BWEmmse.
Under the condition ofL1=L2=L, we have A1=A2=A=(1-ϕL)/(1-ϕ); then, the bullwhip effect can be written as(45)BWEmmse=1+ϕA2+ϕA2-2ϕA1+ϕAϕ-θ1-ϕθ1+θ2-2ϕθ+21-ϕ21+θ2-2ϕθθA2+1+ϕAθAϕ-θ-1-ϕθA2.Case 3 (α≠0, α≠1, and L1≠L2).
In this case, we take a simulation forBWEmmse at the range of ϕ>0 in order to see the impact of different parameters on the BWE. We perform a series of numerical experiments for different values of L1, L2, ϕ, θ, and α, and the results are shown in the following.Figure2 depicts the impact of ϕ on the bullwhip effect when the other parameters are fixed. The bullwhip effect measure first increases and then decreases along with the increasing of ϕ.Figure 2
Effect ofϕ on BWEmmse in the case of L1=1, α=0.4 and 0.7.
(a)
(b)From Tables1 and 2, for given values of L1, L2, θ, and α, it can be seen that the bullwhip effect reaches the maximum at a certain value of ϕ, which is here denoted as ϕmax. Moreover, according to Tables 1–4, it is discovered that the bullwhip effect exists if and only if ϕ>0.3 for any values of L1, L2, θ, and α. The results show that as the value of L2 increases, the bullwhip effect BWEmmse becomes larger when the bullwhip effect BWEmmse>1; that is, the value of ϕ>0.3, and ϕmax also increases. When the ϕ value is smaller than 0.3, the bullwhip effect has the opposite trend on the change of L2. In addition, compared with data between Tables 1 and 2, it is shown that as the market share α increases, both the bullwhip effect and ϕmax decrease, at the fixed values of ϕ,L1, and L2, which meets the condition of L1<L2.Table 1
Values of bullwhip effect under MMSE in the case ofL1=1 and α=0.4.
L
2
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.5969
0.7805
1.0000
1.2493
1.5134
1.7637
1.9502
1.9899
1.7473
0.771
3
0.5951
0.7763
1.0000
1.2704
1.5837
1.9186
2.2191
2.3641
2.1120
0.800
4
0.5949
0.7754
1.0000
1.2789
1.6197
2.0159
2.4221
2.6982
2.4931
0.824Table 2
Values of bullwhip effect under MMSE in the case ofL1=1 and α=0.7.
L
2
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.6062
0.7912
1.0000
1.2233
1.4454
1.6417
1.7732
1.7806
1.5714
0.757
3
0.6052
0.7891
1.0000
1.2336
1.4791
1.7141
1.8958
1.9465
1.7288
0.778
4
0.6051
0.7887
1.0000
1.2378
1.4962
1.7587
1.9852
2.0880
1.8837
0.797Table 3
Values of bullwhip effect under MMSE in the case ofL2=1 and α=0.4.
L
1
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.6031
0.7877
1.0000
1.2319
1.4678
1.6816
1.8308
1.8483
1.6279
0.762
3
0.6018
0.7848
1.0000
1.2458
1.5134
1.7804
1.9995
2.0789
1.8489
0.787
4
0.6017
0.7842
1.0000
1.2514
1.5366
1.8417
2.1241
2.2789
2.0713
0.808Table 4
Values of bullwhip effect under MMSE in the case ofL2=1 and α=0.7.
L
1
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.5939
0.7770
1.0000
1.2580
1.5366
1.8058
2.0120
2.0639
1.8101
0.775
3
0.5917
0.7720
1.0000
1.2828
1.6197
1.9903
2.3350
2.5169
2.2550
0.805
4
0.5915
0.7711
1.0000
1.2928
1.6623
2.1071
2.5812
2.9266
2.7274
0.829Tables3-4 and Figure 3 have a similar trend to Tables 1 and 2 and Figure 2. We set L1>L2 in this part of simulations. Figure 3 illustrates the behavior of BWEmmse with respect to ϕ for fixed L1, L2, θ, and α. 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 ϕ<0.3. Second, when L1 increases and L2,θ, and α are at certain values, the bullwhip effect under MMSE method increases, and ϕmax also rises for ϕ>0.3. Moreover, when α increases for fixed L1,L2,θ, and ϕ under the condition of L1>L2, on the contrary, the bullwhip effect increases and ϕmax rises.Figure 3
Effect ofϕ on BWEmmse in the case of L2=1, α=0.4 and 0.7.
(a)
(b)In a word, these results can be seen clearly from data of Tables1–4. 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, Figure4 shows how the market share α affects bullwhip effect in two cases. Further, the bullwhip effect is the linear correlation with α. When α goes up, the bullwhip effect drops in Figure 4(a) for the condition of L1<L2. However, when α goes up, the bullwhip effect changes conversely in Figure 4(b) for the case of L1>L2. Both figures indicate that larger lead time can result in a greater bullwhip effect. When L1>L2, the increasing of α means more demand needs to be forecasted in longer lead time. Thus, the lead time is critical to the bullwhip effect.Figure 4
The bullwhip effect corresponding toα under the MMSE in case of L1<L2.
(a)
(b)In the simulation, the first-order moving average coefficientθ affects the bullwhip effect in a way that, along with the increase of θ from −1 to 1, the bullwhip effect first increases slowly to the maximum where it corresponds to the value θmax and then decreases quickly in Figure 5. From the figure, it is observed that when θ is almost larger than 0.6, the bullwhip effect does not exist. Moreover, the increasing of L2 can lead to the increasing of bullwhip effect as well as θmax under the existence of the bullwhip effect. So, Figure 8 shows that we can enlarge θ to reduce the bullwhip effect appropriately when other parameters are already determined.Figure 5
Effect ofθ on the bullwhip effect under the MMSE method for different α.


### 4.2. Behavior of the Bullwhip Effect and Parameter Analysis for the MA

Similar to Section4.1, 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 ϕ from −1 to 1 in this part.Case 1 (α=0 or α=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 (L1=L2=L).
In the case ofL1=L2=L, (35) can be written by simplification as(46)BWEMA=1+Lk2+Lk2-2Lk1+Lkϕk-1ϕ-θ1-ϕθ1+θ2-2ϕθ.
The bullwhip effect has no effect with the market shareα from (46).Case 3 (α≠0, α≠1, and L1≠L2).
When we setα≠0, α≠1, and L1≠L2, 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 Figure6, the bullwhip effect varies differently with respect to the first autoregressive coefficient under the different value of k. In this figure, we fix the values of L1=1, θ=0.3, and α=0.4 and run a function of ϕ for L2=2, 3, and 4, with k=4 and 5. The bullwhip effect first increases to the maximum and then decreases quickly with the increasing of ϕ for even k. When ϕ=-1 or ϕ=1, the values of the bullwhip effect are both one. Likewise, for odd k, the bullwhip effect first quickly decreases to a stable value and then also decreases with the continuous increase of ϕ. When ϕ is near negative one, the bullwhip effect is most pronounced.Figure 6
Effect ofϕ on the bullwhip effect under the MA method for different L2 and k.Moreover,BWEMA is a function of L2 and increases as the lead time L2 increases. Equation (36) shows that the behavior of L1 is the same as L1. So, the bullwhip effect increases unconditionally with the increasing of the lead time.Figure7(a) mainly depicts the influence of market share on the bullwhip effect. We set L1<L2 in Figure 7(a) and L1>L2 in Figure 7(b) when other parameters are fixed. Figure 7(a) shows that the bullwhip effect decreases as the market share α increases. And, however, Figure 7(b) shows that the bullwhip effect increases as the market share α increases. In a word, when the market share of the retailer which has a longer lead time increases, the bullwhip effect increases too.Figure 7
Effect ofϕ on the bullwhip effect for different α under L1<L2 and L1>L2.
(a)
(b)Figure 8
Effect ofθ on the bullwhip effect for different span k.From the two figures, we also observe that the bullwhip effect under the MA forecasting method is approximately symmetrical aboutϕ when the span k is even.The first-order moving average coefficientθ has a simple relationship with the bullwhip effect. In Figure 8, the larger the value of θ from −1 to 1, the greater the bullwhip effect. So, in the supply chain, we can reduce θ to decrease the bullwhip effect. The span k is negative with the bullwhip effect, and the increasing of k can reduce BWEMA.


### 4.3. Behavior of the Bullwhip Effect and Parameter Analysis for the ES

Based on the analytical expression ofBWEES, there are also three cases that we should consider similar to the MMSE and MA methods.Case 1 (α=0 or α=1).
Here, the conclusion is identical to the cases of MMSE and MA method.Case 2 (L1=L2=L, and λ1=λ2=λ).
We set two retailers having the same smoothing exponent; then, whenL1=L2=L, the market share α has no influence on the bullwhip effect.Case 3 (α≠0, α≠1, and L1≠L2 or λ1≠λ2).
In order to discuss the impact of market shareα on the bullwhip effect, we make the specific rules as L1≠L2 and λ1≠λ2. By numerical simulations, we will study the changes of BWEES.Figure9 shows the trend of bullwhip effect along with the varying of ϕ from −1 to 1 under different values of L2. BWEES is decreased when ϕ increases and is increased when L2 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 9
The impact ofϕ on BWEES by varying L2.Figure10 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 ϕ increases while L1, L2, θ, α, λ1, and λ2 are fixed.Figure 10
The impact ofϕ and α on the bullwhip effect on the condition of L1<L2 and L1>L2.
(a)
(b)Moreover, when the values ofλ1, λ2, θ, L1, and L2 are fixed such that L1<L2, the bullwhip effect decreases as α increases. Under the fixed values of λ1, λ2, θ, L1, and L2 such that L1>L2, yet, the bullwhip effect increases as α increases. This consequence can be observed from Figure 11(a) for the case L1<L2 and from Figure 10(b) for the case L1>L2.Figure 11
The impact of the smoothing coefficientλ1 and θ on BWEES for different L2.
(a)
(b)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.Figure11(a) studies the influence of λ1 on the bullwhip effect; it is indicated that BWEES has a positive correlation with the smoothing coefficient of retailer 1. In addition, the trend of BWEES is roughly the same for different L2. That is, the influence of the smoothing coefficient λ1 on the bullwhip effect is determinate regardless of the measurement of L1 and L2.Figure11(b) indicates the behavior of the bullwhip effect with regard to θ for fixed values of other parameters. The bullwhip effect is increased slowly along with the increasing of θ from −1 to 0; however, it increases quickly when θ increases from 0 to 1. Some other information can be discovered from Figure 11(b). The larger the value of L2 is, the faster the bullwhip effect increases as θ increases.


### 4.4. Comparison of the Bullwhip Effect by Three Forecasting Methods

We study the bullwhip effect using three forecasting methods in Sections of4.1–4.3. 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 k of MA and the smoothing factors λ1 and λ2. According to Zhang [8], we obtain that the smoothing exponents are fixed as λ1=λ2=2/(k+1). This means the average data ages are the same for the MA and ES forecasting method. Further, we know ϕ+λ2-1≠0 from the expression of the bullwhip effect (see (46)) under the ES forecasting method.Figure12 depicts the difference of the three forecasting methods on measuring the bullwhip effect by varying ϕ for different k and λ. We observe that the bullwhip effect measure of MA is always smaller than that of ES no matter what ϕ is as long as λ1=λ2=2/(k+1). It can be seen that the bullwhip effect under the MMSE forecasting method does not exist when ϕ≤0.3 from the two figures. The bullwhip effect for the three forecasting methods converges to one as ϕ approaches one.Figure 12
Comparison of the three forecasting methods by varyingϕ in the case of k=4 or k=9.
(a)
(b)Moreover, for fixed values ofα, θ, L1, and L2, BWEMMSE is the least when ϕ is smaller than a certain value, called ϕ1. After ϕ becomes larger than ϕ1 and smaller than another certain value, called ϕ2, BWEMA is the smallest and BWEMMSE is lower than BWEES. When ϕ is larger than ϕ2, BWEMMSE is the largest compared to the other two methods. In general, while ϕ is lower than ϕ1, the MMSE method is the best for reducing the bullwhip effect, and while ϕ is higher than ϕ1, the MA method is the best.However, comparing the two figures, we find that as the spank increases, the bullwhip effects for the three forecasting methods overall decrease, and the values of ϕ1 and ϕ2 are lessened. This phenomenon reveals that when we choose a larger span k, the extent of ϕ is larger for the MA which is the best method to predict the bullwhip effect.In Figure13, we set ϕ=0.5, θ=0.3, L1=1, L2=2, λ1=λ2=0.4, and k=4 and then study the bullwhip effect for the three forecasting methods under the condition of various market shares α. We observe that the bullwhip effect is the smallest by the MMSE forecasting method whatever α is. And the bullwhip effect under the ES method is the worst whatever α is. This conclusion is related to the value of ϕ when we select a certain k.Figure 13
Comparison of the three forecasting methods by varyingα in the case of k=4.Figure14 shows the curved surface of the bullwhip effect by varying ϕ and θ for the three forecasting methods. It is observed that BWEES is always larger than BWEMA whatever θ is as long as λ1=λ2=2/(k+1) for the fixed value of ϕ. When θ increases from −1 to 1, BWEMA is the smallest while θ is less than a certain value, and BWEMMSE is the best while θ is more than that certain value. In addition, there is also a phenomenon where BWEMMSE is all along the largest for any value of ϕ when θ approaches negative one.Figure 14
Comparison of the three forecasting methods by varyingϕ and θ in the case of k=8.In summary, in order to reduce the bullwhip effect, we will select a reasonable forecasting method according to the values of various parameters.


## 4.1. 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<ϕ<1 under the MMSE forecasting method. So, next, we will study the behavior of the bullwhip effect only when 0<ϕ<1.There are three cases that we should consider.Case 1 (α=0 or α=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 (L1=L2=L).
When the lead times of the retailer are the same, the market shareα 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 BWEmmse.
Under the condition ofL1=L2=L, we have A1=A2=A=(1-ϕL)/(1-ϕ); then, the bullwhip effect can be written as(45)BWEmmse=1+ϕA2+ϕA2-2ϕA1+ϕAϕ-θ1-ϕθ1+θ2-2ϕθ+21-ϕ21+θ2-2ϕθθA2+1+ϕAθAϕ-θ-1-ϕθA2.Case 3 (α≠0, α≠1, and L1≠L2).
In this case, we take a simulation forBWEmmse at the range of ϕ>0 in order to see the impact of different parameters on the BWE. We perform a series of numerical experiments for different values of L1, L2, ϕ, θ, and α, and the results are shown in the following.Figure2 depicts the impact of ϕ on the bullwhip effect when the other parameters are fixed. The bullwhip effect measure first increases and then decreases along with the increasing of ϕ.Figure 2
Effect ofϕ on BWEmmse in the case of L1=1, α=0.4 and 0.7.
(a)
(b)From Tables1 and 2, for given values of L1, L2, θ, and α, it can be seen that the bullwhip effect reaches the maximum at a certain value of ϕ, which is here denoted as ϕmax. Moreover, according to Tables 1–4, it is discovered that the bullwhip effect exists if and only if ϕ>0.3 for any values of L1, L2, θ, and α. The results show that as the value of L2 increases, the bullwhip effect BWEmmse becomes larger when the bullwhip effect BWEmmse>1; that is, the value of ϕ>0.3, and ϕmax also increases. When the ϕ value is smaller than 0.3, the bullwhip effect has the opposite trend on the change of L2. In addition, compared with data between Tables 1 and 2, it is shown that as the market share α increases, both the bullwhip effect and ϕmax decrease, at the fixed values of ϕ,L1, and L2, which meets the condition of L1<L2.Table 1
Values of bullwhip effect under MMSE in the case ofL1=1 and α=0.4.
L
2
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.5969
0.7805
1.0000
1.2493
1.5134
1.7637
1.9502
1.9899
1.7473
0.771
3
0.5951
0.7763
1.0000
1.2704
1.5837
1.9186
2.2191
2.3641
2.1120
0.800
4
0.5949
0.7754
1.0000
1.2789
1.6197
2.0159
2.4221
2.6982
2.4931
0.824Table 2
Values of bullwhip effect under MMSE in the case ofL1=1 and α=0.7.
L
2
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.6062
0.7912
1.0000
1.2233
1.4454
1.6417
1.7732
1.7806
1.5714
0.757
3
0.6052
0.7891
1.0000
1.2336
1.4791
1.7141
1.8958
1.9465
1.7288
0.778
4
0.6051
0.7887
1.0000
1.2378
1.4962
1.7587
1.9852
2.0880
1.8837
0.797Table 3
Values of bullwhip effect under MMSE in the case ofL2=1 and α=0.4.
L
1
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.6031
0.7877
1.0000
1.2319
1.4678
1.6816
1.8308
1.8483
1.6279
0.762
3
0.6018
0.7848
1.0000
1.2458
1.5134
1.7804
1.9995
2.0789
1.8489
0.787
4
0.6017
0.7842
1.0000
1.2514
1.5366
1.8417
2.1241
2.2789
2.0713
0.808Table 4
Values of bullwhip effect under MMSE in the case ofL2=1 and α=0.7.
L
1
ϕ
ϕ
m
a
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
0.5939
0.7770
1.0000
1.2580
1.5366
1.8058
2.0120
2.0639
1.8101
0.775
3
0.5917
0.7720
1.0000
1.2828
1.6197
1.9903
2.3350
2.5169
2.2550
0.805
4
0.5915
0.7711
1.0000
1.2928
1.6623
2.1071
2.5812
2.9266
2.7274
0.829Tables3-4 and Figure 3 have a similar trend to Tables 1 and 2 and Figure 2. We set L1>L2 in this part of simulations. Figure 3 illustrates the behavior of BWEmmse with respect to ϕ for fixed L1, L2, θ, and α. 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 ϕ<0.3. Second, when L1 increases and L2,θ, and α are at certain values, the bullwhip effect under MMSE method increases, and ϕmax also rises for ϕ>0.3. Moreover, when α increases for fixed L1,L2,θ, and ϕ under the condition of L1>L2, on the contrary, the bullwhip effect increases and ϕmax rises.Figure 3
Effect ofϕ on BWEmmse in the case of L2=1, α=0.4 and 0.7.
(a)
(b)In a word, these results can be seen clearly from data of Tables1–4. 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, Figure4 shows how the market share α affects bullwhip effect in two cases. Further, the bullwhip effect is the linear correlation with α. When α goes up, the bullwhip effect drops in Figure 4(a) for the condition of L1<L2. However, when α goes up, the bullwhip effect changes conversely in Figure 4(b) for the case of L1>L2. Both figures indicate that larger lead time can result in a greater bullwhip effect. When L1>L2, the increasing of α means more demand needs to be forecasted in longer lead time. Thus, the lead time is critical to the bullwhip effect.Figure 4
The bullwhip effect corresponding toα under the MMSE in case of L1<L2.
(a)
(b)In the simulation, the first-order moving average coefficientθ affects the bullwhip effect in a way that, along with the increase of θ from −1 to 1, the bullwhip effect first increases slowly to the maximum where it corresponds to the value θmax and then decreases quickly in Figure 5. From the figure, it is observed that when θ is almost larger than 0.6, the bullwhip effect does not exist. Moreover, the increasing of L2 can lead to the increasing of bullwhip effect as well as θmax under the existence of the bullwhip effect. So, Figure 8 shows that we can enlarge θ to reduce the bullwhip effect appropriately when other parameters are already determined.Figure 5
Effect ofθ on the bullwhip effect under the MMSE method for different α.


## 4.2. Behavior of the Bullwhip Effect and Parameter Analysis for the MA

Similar to Section4.1, 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 ϕ from −1 to 1 in this part.Case 1 (α=0 or α=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 (L1=L2=L).
In the case ofL1=L2=L, (35) can be written by simplification as(46)BWEMA=1+Lk2+Lk2-2Lk1+Lkϕk-1ϕ-θ1-ϕθ1+θ2-2ϕθ.
The bullwhip effect has no effect with the market shareα from (46).Case 3 (α≠0, α≠1, and L1≠L2).
When we setα≠0, α≠1, and L1≠L2, 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 Figure6, the bullwhip effect varies differently with respect to the first autoregressive coefficient under the different value of k. In this figure, we fix the values of L1=1, θ=0.3, and α=0.4 and run a function of ϕ for L2=2, 3, and 4, with k=4 and 5. The bullwhip effect first increases to the maximum and then decreases quickly with the increasing of ϕ for even k. When ϕ=-1 or ϕ=1, the values of the bullwhip effect are both one. Likewise, for odd k, the bullwhip effect first quickly decreases to a stable value and then also decreases with the continuous increase of ϕ. When ϕ is near negative one, the bullwhip effect is most pronounced.Figure 6
Effect ofϕ on the bullwhip effect under the MA method for different L2 and k.Moreover,BWEMA is a function of L2 and increases as the lead time L2 increases. Equation (36) shows that the behavior of L1 is the same as L1. So, the bullwhip effect increases unconditionally with the increasing of the lead time.Figure7(a) mainly depicts the influence of market share on the bullwhip effect. We set L1<L2 in Figure 7(a) and L1>L2 in Figure 7(b) when other parameters are fixed. Figure 7(a) shows that the bullwhip effect decreases as the market share α increases. And, however, Figure 7(b) shows that the bullwhip effect increases as the market share α increases. In a word, when the market share of the retailer which has a longer lead time increases, the bullwhip effect increases too.Figure 7
Effect ofϕ on the bullwhip effect for different α under L1<L2 and L1>L2.
(a)
(b)Figure 8
Effect ofθ on the bullwhip effect for different span k.From the two figures, we also observe that the bullwhip effect under the MA forecasting method is approximately symmetrical aboutϕ when the span k is even.The first-order moving average coefficientθ has a simple relationship with the bullwhip effect. In Figure 8, the larger the value of θ from −1 to 1, the greater the bullwhip effect. So, in the supply chain, we can reduce θ to decrease the bullwhip effect. The span k is negative with the bullwhip effect, and the increasing of k can reduce BWEMA.


## 4.3. Behavior of the Bullwhip Effect and Parameter Analysis for the ES

Based on the analytical expression ofBWEES, there are also three cases that we should consider similar to the MMSE and MA methods.Case 1 (α=0 or α=1).
Here, the conclusion is identical to the cases of MMSE and MA method.Case 2 (L1=L2=L, and λ1=λ2=λ).
We set two retailers having the same smoothing exponent; then, whenL1=L2=L, the market share α has no influence on the bullwhip effect.Case 3 (α≠0, α≠1, and L1≠L2 or λ1≠λ2).
In order to discuss the impact of market shareα on the bullwhip effect, we make the specific rules as L1≠L2 and λ1≠λ2. By numerical simulations, we will study the changes of BWEES.Figure9 shows the trend of bullwhip effect along with the varying of ϕ from −1 to 1 under different values of L2. BWEES is decreased when ϕ increases and is increased when L2 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 9
The impact ofϕ on BWEES by varying L2.Figure10 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 ϕ increases while L1, L2, θ, α, λ1, and λ2 are fixed.Figure 10
The impact ofϕ and α on the bullwhip effect on the condition of L1<L2 and L1>L2.
(a)
(b)Moreover, when the values ofλ1, λ2, θ, L1, and L2 are fixed such that L1<L2, the bullwhip effect decreases as α increases. Under the fixed values of λ1, λ2, θ, L1, and L2 such that L1>L2, yet, the bullwhip effect increases as α increases. This consequence can be observed from Figure 11(a) for the case L1<L2 and from Figure 10(b) for the case L1>L2.Figure 11
The impact of the smoothing coefficientλ1 and θ on BWEES for different L2.
(a)
(b)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.Figure11(a) studies the influence of λ1 on the bullwhip effect; it is indicated that BWEES has a positive correlation with the smoothing coefficient of retailer 1. In addition, the trend of BWEES is roughly the same for different L2. That is, the influence of the smoothing coefficient λ1 on the bullwhip effect is determinate regardless of the measurement of L1 and L2.Figure11(b) indicates the behavior of the bullwhip effect with regard to θ for fixed values of other parameters. The bullwhip effect is increased slowly along with the increasing of θ from −1 to 0; however, it increases quickly when θ increases from 0 to 1. Some other information can be discovered from Figure 11(b). The larger the value of L2 is, the faster the bullwhip effect increases as θ increases.


## 4.4. Comparison of the Bullwhip Effect by Three Forecasting Methods

We study the bullwhip effect using three forecasting methods in Sections of4.1–4.3. 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 k of MA and the smoothing factors λ1 and λ2. According to Zhang [8], we obtain that the smoothing exponents are fixed as λ1=λ2=2/(k+1). This means the average data ages are the same for the MA and ES forecasting method. Further, we know ϕ+λ2-1≠0 from the expression of the bullwhip effect (see (46)) under the ES forecasting method.Figure12 depicts the difference of the three forecasting methods on measuring the bullwhip effect by varying ϕ for different k and λ. We observe that the bullwhip effect measure of MA is always smaller than that of ES no matter what ϕ is as long as λ1=λ2=2/(k+1). It can be seen that the bullwhip effect under the MMSE forecasting method does not exist when ϕ≤0.3 from the two figures. The bullwhip effect for the three forecasting methods converges to one as ϕ approaches one.Figure 12
Comparison of the three forecasting methods by varyingϕ in the case of k=4 or k=9.
(a)
(b)Moreover, for fixed values ofα, θ, L1, and L2, BWEMMSE is the least when ϕ is smaller than a certain value, called ϕ1. After ϕ becomes larger than ϕ1 and smaller than another certain value, called ϕ2, BWEMA is the smallest and BWEMMSE is lower than BWEES. When ϕ is larger than ϕ2, BWEMMSE is the largest compared to the other two methods. In general, while ϕ is lower than ϕ1, the MMSE method is the best for reducing the bullwhip effect, and while ϕ is higher than ϕ1, the MA method is the best.However, comparing the two figures, we find that as the spank increases, the bullwhip effects for the three forecasting methods overall decrease, and the values of ϕ1 and ϕ2 are lessened. This phenomenon reveals that when we choose a larger span k, the extent of ϕ is larger for the MA which is the best method to predict the bullwhip effect.In Figure13, we set ϕ=0.5, θ=0.3, L1=1, L2=2, λ1=λ2=0.4, and k=4 and then study the bullwhip effect for the three forecasting methods under the condition of various market shares α. We observe that the bullwhip effect is the smallest by the MMSE forecasting method whatever α is. And the bullwhip effect under the ES method is the worst whatever α is. This conclusion is related to the value of ϕ when we select a certain k.Figure 13
Comparison of the three forecasting methods by varyingα in the case of k=4.Figure14 shows the curved surface of the bullwhip effect by varying ϕ and θ for the three forecasting methods. It is observed that BWEES is always larger than BWEMA whatever θ is as long as λ1=λ2=2/(k+1) for the fixed value of ϕ. When θ increases from −1 to 1, BWEMA is the smallest while θ is less than a certain value, and BWEMMSE is the best while θ is more than that certain value. In addition, there is also a phenomenon where BWEMMSE is all along the largest for any value of ϕ when θ approaches negative one.Figure 14
Comparison of the three forecasting methods by varyingϕ and θ in the case of k=8.In summary, in order to reduce the bullwhip effect, we will select a reasonable forecasting method according to the values of various parameters.


## 5. 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 θ 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 λ and has a negative correlation with the span k. (4) By comparison, the MA is always better than the ES for reducing the bullwhip effect in our supply chain model. When ϕ 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 aroundϕ=0.672, and the maximum of the entropy is around ϕ=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.



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*Source: 8539740-2018-01-08.xml*