Inherent Complexity Research on the Bullwhip Effect in Supply Chains with Two Retailers: The Impact of Three Forecasting Methods Considering Market Share

Abstract

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

1. Introduction

In the research of modern logistics and supply chain management, a significant phenomenon which is called the bullwhip effect has attracted the attention of researchers and practitioners alike. As moving backward from a downstream member to an upstream member, the variance of order quantities placed by the downstream member to its immediate upstream member tends to be amplified. The first invention of the bullwhip effect could be traced back to Forrester [1, 2]. After that, several researchers such as Blinder [3], Blanchard [4], Burbidge [5], Blinder [6] and Kahn [7], and so forth also recognized the existence of the bullwhip effect in supply chains. Sterman [8] used the Beer Game, the most popular simulation of a simple production and distribution system developed at MIT, to certificate that the bullwhip effect is an important problem. The phenomenon was firstly called the bullwhip effect by Lee et al. [9, 10]. They indicated that the main reason for this phenomenon is demand signal processing, nonzero lead time, order batching, supply shortages, and price fluctuation.

In the five reasons mentioned above, demand signal processing is the research hotspot in this field. Metters [11] tried to identify the bullwhip effect by establishing an empirical lower bound on the profitability impact of the bullwhip effect. In his research, the impact of bullwhip effect was measured by comparing results obtained in two cases, that is, high demand variability versus low demand variability with weak seasonality. It was found that the importance of the bullwhip effect differed greatly depending on the business environment. Graves [12] quantified the bullwhip effect for the supply chain in which demand pattern follows an integrated moving average process. Chen et al. [13, 14] quantified the bullwhip effect for supply chains using moving average and exponential smoothing techniques for demand forecasts. In their works, it was assumed that members of the chain employ base stock policy for their inventory system. The main finding of their work was that the order variance would increase with the increasing number of members in the chain, lower level of information sharing, and increasing lead time. Likewise, Xu et al. [15] conducted a similar research for a demand process that was forecasted with a simple exponentially weighted moving average method. Zhang [16] also investigated the impact of different forecasting methods on the bullwhip effect for a simple inventory system with a first-order autoregressive demand process. Luong [17] measured the bullwhip effect for a simple two-stage supply chain that included only one retailer and one supplier in the environment where the retailer employed the order-up-to inventory policy for their inventory and demand forecast was performed through the AR(1) model, and the effect of autoregressive coefficient and lead time on this measure was investigated. Duc et al. [18] examined the impact of a third-party warehouse on the bullwhip effect. Karimi et al. [19] considered the problem of local capacity 𝐻_∞ control for a class of production networks of autonomous work systems with time-varying delays in the capacity changes. Nepal et al. [20] presented an analysis of the bullwhip effect and net-stock amplification in a three-echelon supply chain considering step changes in the production rates during a product’s lifecycle demand. Dashkovskiy [21, 22] considered local input-to-state stability of complex logistics networks. Mehrsai [23, 24] tried to investigate the possibility of combining this new research paradigm with existing strategies in production logistics to improve material handling and control task according to material flow criteria. Ma et al. [25] stated a comparison of bullwhip effect under various forecasting techniques in supply chains with two retailers. Shen and Jiang [26] explained the general principle of dynamic programming algorithm. Shen et al. [27] presented the dynamic programming algorithm to solve the optimal control variable trajectory under a given circle.

This paper continues to examine the differences in bullwhip effect under three forecasting methods. And this paper analyzes the impact of every parameter on the bullwhip effect. The conclusions suggest that different forecasting methods lead to bullwhip effect measures with fundamentally different properties in relation to lead time and demand autocorrelation. Therefore, the corresponding forecasting method should be selected by the managers under different parameters in practice.

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

2. A Supply Chain Model

2.1. Demand Process

In this research, a two-stage supply chain with one supplier and two retailers will be considered, and we will quantify the bullwhip effect in the simple supply chain. The two retailers order and replenish the stock from a supplier in each period 𝑡. The market share of the two retailers is considered as 𝛼 and 1−𝛼, respectively. We assumed that the order-up-to inventory policy. Retailer 1 and retailer 2 both employ an AR(1) autoregressive model: 𝐷_𝑡=𝛿+𝜙𝐷_𝑡−1+𝜀_𝑡. (1)

The supply chain model shows, in Figure 1, the following:

Figure 1 

Supply chain model.

Due to the market share of retailer 1, we consider that retailer 1 employs an AR(1) model as follows: 𝐷_1,𝑡=𝛼𝛿₁+𝜙₁𝐷_1,𝑡−1+𝛼𝜀_1,𝑡. (2)

In (2), 𝐷_1,𝑡 is the demand of period 𝑡. 𝛿₁ is the constant of the autoregressive model. 𝜙₁ is the first-order autocorrelation coefficient, where −1<𝜙₁<1. 𝜀_1,𝑡 is the forecast error for period 𝑡 and 𝜀_1,𝑡 is independent and identically distributed from a symmetric distribution with mean 0 and variance 𝜎²₁.

According to a first-order autocorrelation property of time series model, for any period 𝑡, we must have 𝐸𝐷_1,𝑡=𝐸𝐷_1,𝑡−1=𝜇_1,𝑑=𝛼𝛿₁1−𝜙₁,Var𝐷_1,𝑡=Var𝐷_1,𝑡−1=𝜎²_1,𝑑=𝛼²𝜎²₁1−𝜙²₁. (3)

Analogously, with the market share 1−𝛼 retailer 2 also employs an AR(1) model as follows: 𝐷_2,𝑡=(1−𝛼)𝛿₂+𝜙₂𝐷_2,𝑡−1+(1−𝛼)𝜀_2,𝑡. (4)

In (4), 𝐷_2,𝑡 is the demand of period 𝑡. 𝛿₂ is the constant of the autoregressive model. 𝜙₂ is the first-order autocorrelation coefficient, where −1<𝜙₂<1. 𝜀_2,𝑡 is the forecast error for period 𝑡 and 𝜀_2,𝑡 is independent and identically distributed from a symmetric distribution with mean 0 and variance 𝜎²₂.

Similarly, we also have 𝐸𝐷_2,𝑡=𝐸𝐷_2,𝑡−1=𝜇_2,𝑑=(1−𝛼)𝛿₂1−𝜙₂,Var𝐷_2,𝑡=Var𝐷_2,𝑡−1=𝜎²_2,𝑑=(1−𝛼)²𝜎²₂1−𝜙²₂. (5)

2.2. Inventory Policy

In order to meet the dynamic needs of the supply chain model, the supply chain model shown in Figure 1 employs the order-up-to inventory policy. We assume that the two retailers both apply a fixed order lead time for orders. The goal of the inventory policy is to maintain inventory levels at the target inventory levels 𝑞_𝑡. At the beginning of period 𝑡, the order of quantity 𝑞_1,𝑡 sent by retailer 1 can be given as follows: 𝑞_1,𝑡=𝑆_1,𝑡−𝑆_1,𝑡−1+𝐷_1,𝑡−1. (6)

In (6), 𝑆_1,𝑡 is the order-up-to level and it can be determined through lead-time demand by 𝑆_1,𝑡=𝐷^𝐿1_1,𝑡+𝑧𝜎^𝐿1_1,𝑡. (7)

In (7), 𝐷^𝐿1_1,𝑡 is the value of lead-time demand and forecast based on historical sales data. 𝑧 is the normal 𝑧-score that can be determined based on the desired service level of the inventory policy. 𝜎^𝐿1_1,𝑡 is the standard deviation of lead-time demand forecast error.

Analogously, the order of quantity sent by retailer 2 also can be given as follows: 𝑞_2,𝑡=𝑆_2,𝑡−𝑆_2,𝑡−1+𝐷_2,𝑡−1. (8)

And the order-up-to level 𝑆_2,𝑡 can be given as follows: 𝑆_2,𝑡=𝐷^𝐿2_2,𝑡+𝑧𝜎^𝐿2_2,𝑡. (9)

In (9), 𝐷^𝐿2_2,𝑡 is the value of lead-time demand and forecast based on historical sales data. 𝑧 is the normal 𝑧-score that can be determined based on the desired service level of the inventory policy. 𝜎^𝐿2_2,𝑡 is the standard deviation of lead-time demand forecast error.

2.3. Forecasting Method

Seen from the inventory policy equation, the accuracy of the demand and forecasting of the future lead-time 𝐿 period is the most important factor that affects the inventory level of the retailers in supply chain model. While each forecasting error is present, the impacts of different forecasting methods on the bullwhip effect are not the same. Then we will introduce three different forecasting methods: MMES, MA, and ES forecasting methods.

2.3.1. The MMSE Forecasting Method

The MMSE forecasting method is short for minimizing the mean square error. Under the MMSE forecasting method, the lead-time demand can be shown as follows: 𝐷^𝐿_𝑡=𝐷_𝑡+𝐷_𝑡+1+⋯+𝐷_{𝑡+𝐿−1}=^𝐿−1_𝑖=0𝐷_𝑡+𝑖. (10)

When using the MMSE method to predict the future lead-time demand, the total demand expectations within the lead-time 𝐿 are 𝐷^𝐿_𝑡=𝐷_𝑡+𝐷_𝑡+1+⋯+𝐷_{𝑡+𝐿−1}=^𝐿−1_𝑖=0𝐷_𝑡+𝑖, (11) where 𝐷_𝑡+𝑖 can be characterized as 𝐷_𝑡+𝑖=𝐸𝐷_𝑡+𝑖∣𝐷_𝑡−1,𝐷_𝑡−2,…. (12)

2.3.2. The MA Forecasting Method

The MA forecasting method is short for moving average. In the MA forecasting method, the lead-time demand can be determined as follows: 𝐷^𝐿_𝑡=𝐿𝑘^𝑘_𝑖=1𝐷_𝑡−𝑖. (13)

In (13), 𝐾 is the span (number of date points) for the MA forecasting method. 𝐷_𝑡−𝑖 is the actual demand in period 𝑡−𝑖.

2.3.3. The ES Forecasting Method

The ES forecasting method is short for exponential smoothing. Using the ES forecasting method, the lead-time demand can be considered as follows: 𝐷^𝐿_𝑡=𝛽𝐷_𝑡−1+1−𝛽𝐷^𝐿_𝑡−1. (14)

In (14), 𝛽 is the smoothing exponent.

3. The Measure of the Bullwhip Effect

In this section, we will talk about the measure of the bullwhip effect under the MMSE, MA, and ES forecasting methods. The total demand of the two retailers is given as 𝐷_𝑡=𝐷_1,𝑡+𝐷_2,𝑡. (15)

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

Under the MMSE forecasting method, the total demand expectations within the lead-time 𝐿 of retailer 1 can be determined as 𝐷^𝐿1_1,𝑡=𝐿₁1−𝜙₁𝛼𝛿₁−𝜙₁1−𝜙₁^𝐿11−𝜙₁²𝛼𝛿₁+𝜙₁1−𝜙₁^𝐿11−𝜙₁𝐷_1,𝑡−1. (16)

We know that 𝜎^𝐿1_1,𝑡 does not depend on 𝑡, so the order quantity of retailer 1 can be described as follows: 𝑞_1,𝑡=𝜙1+¹1−𝜙₁^𝐿11−𝜙₁𝐷_1,𝑡−1−𝜙₁1−𝜙₁^𝐿11−𝜙₁𝐷_1,𝑡−2=1+𝐴₁𝐷_1,𝑡−1−𝐴₁𝐷_1,𝑡−2, (17) where 𝐴₁=𝜙₁(1−𝜙₁^𝐿1)/(1−𝜙₁).

Similarly, we can get the order quantity of retailer 2: 𝑞_2,𝑡=𝜙1+²1−𝜙₂^𝐿21−𝜙₂𝐷_2,𝑡−1−𝜙₂1−𝜙₂^𝐿21−𝜙₂𝐷_2,𝑡−2=1+𝐴₂𝐷_2,𝑡−1−𝐴₂𝐷_2,𝑡−2, (18) where 𝐴₂=𝜙₂(1−𝜙₂^𝐿2)/(1−𝜙₂).

Total demand that two retailers face is 𝐷_𝑡=𝐷_1,𝑡+𝐷_2,𝑡. (19)

Taking the variance of the total demand, we have Var𝐷_𝑡=Var𝐷_1,𝑡+𝐷_2,𝑡=Var𝐷_1,𝑡+Var𝐷_2,𝑡+2Cov𝐷_1,𝑡,𝐷_2,𝑡. (20)

The covariance between 𝐷_1,𝑡 and 𝐷_2,𝑡 can be described as Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛼𝐷1−𝛼Var_2,𝑡. (21)

Total order quantity which two retailers face is 𝑞_𝑡=1+𝐴₁𝐷_1,𝑡−1−𝐴₁𝐷_1,𝑡−2+1+𝐴₂𝐷_2,𝑡−1−𝐴₂𝐷_2,𝑡−2. (22)

Taking the variance of the total order quantity, we have Var𝑞_𝑡=1+2𝐴₁+2𝐴²₁Var𝐷_1,𝑡−1+1+2𝐴₂+2𝐴²₂×Var𝐷_2,𝑡−1−2𝐴₁1+𝐴₁Cov𝐷_1,𝑡−1,𝐷_1,𝑡−2+21+𝐴₁1+𝐴₂Cov𝐷_1,𝑡−1,𝐷_2,𝑡−1−2𝐴₂×1+𝐴₁Cov𝐷_1,𝑡−1,𝐷_2,𝑡−2−2𝐴₁1+𝐴₂×Cov𝐷_1,𝑡−2,𝐷_2,𝑡−1+2𝐴₁𝐴₂Cov𝐷_1,𝑡−2,𝐷_2,𝑡−2−2𝐴₂1+𝐴₂Cov𝐷_2,𝑡−1,𝐷_2,𝑡−2. (23)

We can prove that Cov𝐷_1,𝑡−1,𝐷_1,𝑡−2=𝜙₁Var𝐷_1,𝑡,Cov𝐷_1,𝑡−1,𝐷_2,𝑡−1=𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡−1,𝐷_2,𝑡−2=𝜙₁𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡−2,𝐷_2,𝑡−1=𝜙₂𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡−2,𝐷_2,𝑡−2=𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_2,𝑡−1,𝐷_2,𝑡−2=𝜙₂Var𝐷_2,𝑡. (24)

Therefore, (23) can be described as Var𝑞_𝑡=1+2𝐴₁+2𝐴²₁−2𝐴₁1+𝐴₁𝜙₁Var𝐷_1,𝑡+1+2𝐴₂+2𝐴²₂−2𝐴₂1+𝐴₂𝜙₂Var𝐷_2,𝑡+2+2𝐴₁+𝐴₂+3𝐴₁𝐴₂−2𝜙₁𝐴₂1+𝐴₁−2𝜙₂𝐴₁1+𝐴₂𝛼𝐷1−𝛼Var_2,𝑡. (25)

For simplicity, (25) can be written as Var𝑞_𝑡=𝐺₁Var𝐷_1,𝑡+𝐺₂Var𝐷_2,𝑡+𝐺₃𝛼𝐷1−𝛼Var_2,𝑡. (26)

In (26) 𝐺₁ is the coefficient of Var(𝐷_1,𝑡), 𝐺₂ is the coefficient of Var(𝐷_2,𝑡), and 𝐺₃ is the coefficient of 𝛼/(1−𝛼)Var(𝐷_2,𝑡).

Then, the BWE in MMSE forecasting method is BWE_MMSE=Var𝑞_𝑡Var𝐷_𝑡=𝛼²𝐺₁+(1−𝛼)²𝐺₂+𝛼(1−𝛼)𝐺₃𝛼²+(1−𝛼)²().+2𝛼1−𝛼 (27)

3.2. The Measure of the Bullwhip Effect under the MA Forecasting Method

In the MA forecasting method, the lead-time demand of retailer 1 can be determined as follows: 𝐷^𝐿1_1,𝑡=𝐿₁𝑘^𝑘_𝑖=1𝐷_{1,𝑡−𝑖}. (28)

Therefore, the total order quantity of the two retailers is 𝑞_𝑡=𝐿1+¹𝑘𝐷_1,𝑡−1−𝐿₁𝑘𝐷_{1,𝑡−𝑘−1}+𝐿1+²𝑘𝐷_2,𝑡−1−𝐿₂𝑘𝐷_{2,𝑡−𝑘−1}. (29)

Taking the variance of the total order quantity, we have Var𝑞_𝑡=𝐿1+¹𝑘²Var𝐷_1,𝑡−1+𝐿₁𝑘²Var𝐷_{1,𝑡−𝑘−1}+𝐿1+²𝑘²Var𝐷_2,𝑡−1+𝐿₂𝑘²Var𝐷_{2,𝑡−𝑘−1}𝐿−21+¹𝑘𝐿₁𝑘Cov𝐷_1,𝑡−1,𝐷_{1,𝑡−𝑘−1}𝐿+21+¹𝑘𝐿1+²𝑘Cov𝐷_1,𝑡−1,𝐷_2,𝑡−1𝐿−21+¹𝑘𝐿₂𝑘Cov𝐷_1,𝑡−1,𝐷_{2,𝑡−𝑘−1}𝐿−2¹𝑘𝐿1+²𝑘Cov𝐷_{1,𝑡−𝑘−1},𝐷_2,𝑡−1𝐿+2¹𝑘𝐿₂𝑘Cov𝐷_{1,𝑡−𝑘−1},𝐷_{2,𝑡−𝑘−1}𝐿−21+²𝑘𝐿₂𝑘Cov𝐷_2,𝑡−1,𝐷_{2,𝑡−𝑘−1}. (30)

We can prove that Cov𝐷_1,𝑡−1,𝐷_{1,𝑡−𝑘−1}=𝜙₁^𝑘Var𝐷_1,𝑡,Cov𝐷_1,𝑡−1,𝐷_2,𝑡−1=𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡−1,𝐷_{2,𝑡−𝑘−1}=𝜙₁^𝑘𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_{1,𝑡−𝑘−1},𝐷_2,𝑡−1=𝜙₂^𝑘𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_{1,𝑡−𝑘−1},𝐷_{2,𝑡−𝑘−1}=𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_2,𝑡−1,𝐷_{2,𝑡−𝑘−1}=𝜙₂^𝑘Var𝐷_2,𝑡. (31)

The proof of (31) can be seen in the Appendix.

Then, taking (31) into (30), we have Var𝑞_𝑡=𝐿1+2¹𝑘𝐿+2¹𝑘²−2𝜙₁^𝑘𝐿1+¹𝑘𝐿₁𝑘Var𝐷_1,𝑡+𝐿1+2²𝑘𝐿+2²𝑘²−2𝜙₂^𝑘𝐿1+²𝑘𝐿₂𝑘Var𝐷_2,𝑡𝐿+21+¹𝑘𝐿1+²𝑘−𝐿1+¹𝑘𝐿₂𝑘𝜙₁^𝑘−𝐿₁𝑘𝐿1+²𝑘𝜙₂^𝑘+𝐿₁𝐿₂𝑘𝛼𝐷1−𝛼Var_2,𝑡. (32)

For simplicity, (32) can be written as Var𝑞_𝑡=𝐻₁Var𝐷_1,𝑡+𝐻₂Var𝐷_2,𝑡+𝐻₃𝛼𝐷1−𝛼Var_2,𝑡. (33)

In (33) 𝐻₁ is the coefficient of Var(𝐷_1,𝑡), 𝐻₂ is the coefficient of Var(𝐷_2,𝑡), and 𝐻₃ is the coefficient of 𝛼/(1−𝛼)Var(𝐷_2,𝑡).

Then, the BWE in MMSE forecasting method is BWE_MA=Var𝑞_𝑡Var𝐷_𝑡=𝛼²𝐻₁+(1−𝛼)²𝐻₂+𝛼(1−𝛼)𝐻₃𝛼²+(1−𝛼)²().+2𝛼1−𝛼 (34)

3.3. The Measure of the Bullwhip Effect under the ES Forecasting Method

Using the ES forecasting method, the lead-time demand of retailer 1 can be determined as follows: 𝐷^𝐿1_1,𝑡=𝛽𝐷_1,𝑡−1+1−𝛽𝐷^𝐿1_1,𝑡−1. (35)

According to Zhang [16], 𝐷_𝑡+𝑖=𝐷_𝑡+1, 𝑖≥2, the total order quantity of two retailers at period 𝑡 is 𝑞_𝑡=1+𝛽₁𝐿₁𝐷_1,𝑡−𝛽₁𝐿₁𝐷_1,𝑡+1+𝛽₂𝐿₂𝐷_2,𝑡−𝛽₂𝐿₂𝐷_2,𝑡. (36)

Taking the variance of the total order quantity, we have Var𝑞_𝑡=1+𝛽₁𝐿₁²Var𝐷_1,𝑡+𝛽₁𝐿₁²Var𝐷_1,𝑡+1+𝛽₂𝐿₂²Var𝐷_2,𝑡+𝛽₂𝐿₂²Var𝐷_2,𝑡−2𝛽₁𝐿₁1+𝛽₁𝐿₁Cov𝐷_1,𝑡,𝐷_1,𝑡+21+𝛽₁𝐿₁1+𝛽₂𝐿₂Cov𝐷_1,𝑡,𝐷_2,𝑡−2𝛽₂𝐿₂1+𝛽₁𝐿₁Cov𝐷_1,𝑡,𝐷_2,𝑡−2𝛽₁𝐿₁1+𝛽₂𝐿₂Cov𝐷_1,𝑡,𝐷_2,𝑡+2𝛽₁𝐿₁𝛽₂𝐿₂Cov𝐷_1,𝑡,𝐷_2,𝑡−2𝛽₂𝐿₂1+𝛽₂𝐿₂Cov𝐷_2,𝑡,𝐷_2,𝑡. (37)

According to Zhang [16], we know that Var𝐷_1,𝑡=1+1−𝛽₁𝜙₁𝛽₁2−𝛽₁1−1−𝛽₁𝜙₁Var𝐷_1,𝑡,Var𝐷_2,𝑡=1+1−𝛽₂𝜙₂𝛽₂2−𝛽₂1−1−𝛽₂𝜙₂Var𝐷_2,𝑡. (38)

We can prove that Cov𝐷_1,𝑡,𝐷_1,𝑡=𝛽₁𝜙₁1−1−𝛽₁𝜙₁Var𝐷_1,𝑡,Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛽₂𝜙₁1−1−𝛽₂𝜙₁𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛽₁𝜙₂1−1−𝛽₁𝜙₂𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛽₁𝛽₂1−𝜙₁𝜙₂1−𝛽₁1−𝛽₂×1−1−𝛽₁1−𝛽₂1−1−𝛽₁𝜙₂×1−1−𝛽₂𝜙₁⁻¹𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_2,𝑡,𝐷_2,𝑡=𝛽₂𝜙₂1−1−𝛽₂𝜙₂Var𝐷_2,𝑡. (39)

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

Then, taking (39) into (37), we can get Var𝑞_𝑡=1+1−𝜙₁1−1−𝛽₁𝜙₁2𝛽₁𝐿₁+2𝛽₁²𝐿₁²2−𝛽₁Var𝐷_1,𝑡+1+1−𝜙₂1−1−𝛽₂𝜙₂2𝛽₂𝐿₂+2𝛽₂²𝐿₂²2−𝛽₂Var𝐷_2,𝑡+21+𝛽₁𝐿₁1+𝛽₂𝐿₂−2𝛽₂²𝜙₁𝐿₂1+𝛽₁𝐿₁1−1−𝛽₂𝜙₁−2𝛽₁²𝜙₂𝐿₂1+𝛽₂𝐿₂1−1−𝛽₁𝜙₂+2𝛽₁²𝛽₂²1−𝜙₁𝜙₂1−𝛽₁1−𝛽₂2𝛽₁²𝛽₂²1−𝜙₁𝜙₂1−𝛽₁1−𝛽₂1−1−𝛽₂𝜙₁×𝛼𝐷1−𝛼Var_2,𝑡. (40)

For simplicity, (40) can be written as Var𝑞_𝑡=𝑀₁Var𝐷_1,𝑡+𝑀₂Var𝐷_2,𝑡+𝑀₃𝛼𝐷1−𝛼Var_2,𝑡. (41)

In (41) 𝑀₁ is the coefficient of Var(𝐷_1,𝑡), 𝑀₂ is the coefficient of Var(𝐷_2,𝑡), and 𝑀₃ is the coefficient of 𝛼/(1−𝛼)Var(𝐷_2,𝑡).

Then, the BWE in ES forecasting method is BWE_ES=Var𝑞_𝑡Var𝐷_𝑡=𝑀₁Var𝐷_1,𝑡+𝑀₂+𝑀₃(𝛼/(1−𝛼))Var𝐷_2,𝑡Var𝐷_1,𝑡+Var𝐷_2,𝑡+2(𝛼/(1−𝛼))Var𝐷_2,𝑡=𝑀₁Var𝐷_1,𝑡/Var𝐷_2,𝑡+𝑀₂+𝑚₃(𝛼/(1−𝛼))Var𝐷_1,𝑡/Var𝐷_2,𝑡((=𝑀+1+2𝛼/1−𝛼))₁(𝛼/(1−𝛼))²+𝑀₂+𝑀₃(𝛼/(1−𝛼))(𝛼/(1−𝛼))²((=𝛼+1+2𝛼/1−𝛼))₂𝑀₁+(1−𝛼)²𝑀₂+𝛼(1−𝛼)𝑀₃𝛼²+(1−𝛼)²().+2𝛼1−𝛼 (42)

4. Behavior of the Bullwhip Effect Measure and Numerical Simulation

With the expression of the bullwhip effect under three forecasting methods, we can discuss the bullwhip effect by algebraic analysis and numerical simulation.

4.1. The Analysis of Parameters under the MMSE Forecasting Method

Figures 2–4 simulate the expression of the bullwhip effect under the MMSE forecasting method to illustrate the impact of parameters on the bullwhip effect. Duc et al. [28] indicated that the bullwhip effect occurs only when a positive autoregressive relationship exists in the demand process under MMSE forecasting method. So, we just consider the circumstance that 𝜙₁ varies from 0 to 1. Figure 2 indicates that the bullwhip effect increases slowly with the increase of 𝜙₁, and the bullwhip effect begins to decrease rapidly when it reaches the maximum value. The bullwhip effect does not occur only when 𝜙₁ increases to a certain value, and the values are different with different 𝛼. For 𝛼, when 𝜙₁ is smaller than a certain value, the bullwhip effect becomes smaller with 𝛼 becoming larger, when 𝜙₁ is larger than that certain value and smaller than another certain value, the bullwhip effect becomes larger with 𝛼becoming larger, when 𝜙₁ is larger than another certain value, the bullwhip effect becomes smaller with 𝛼becoming larger.

Figure 2 

Impact of 𝜙1 on bullwhip effect for different 𝛼 under the MMSE.

Figure 3 

Impact of 𝛼 on bullwhip effect for different 𝐿1 under the MMSE.

Figure 4 

Impact of 𝜙1 on bullwhip effect for different 𝐿1 under the MMSE.

Figure 3 reveals that the bullwhip effect decreases gradually to the minimum value and then it begins to increase with the increase of 𝛼. For the different 𝐿₁, the larger it is, the larger the bullwhip effect is. The result suggests when retailer 1 occupies half of the market share and they set the short lead time, the bullwhip effect value will be the lowest.

Figure 4 depicts the impact of 𝜙₁ on the bullwhip effect. We observe that the bullwhip effect increases gradually to the maximum value with the increase of 𝜙₁, and the bullwhip effect begins to decrease rapidly when it reaches the maximum value. The bullwhip effect does not occur only when 𝜙₁ increases to a certain value, and those values are different for different 𝐿₁. For different 𝐿₁, the larger it is, the larger the bullwhip effect is.

4.2. The Analysis of Parameters under the MA Forecasting Method

Figures 5–9 simulate the expression of the bullwhip effect under the MA forecasting method to illustrate the impact of parameters on the bullwhip effect.

Figure 5 

Impact of 𝜙1 on bullwhip effect for different 𝛼 under the MA.

Figure 6 

Impact of 𝜙1 on bullwhip effect for different 𝑘 under the MA.

Figure 7 

Impact of 𝛼 on bullwhip effect for different 𝐿1 under the MA.

Figure 8 

Impact of 𝛼 on bullwhip effect for different 𝑘 under the MA.

Figure 9 

Impact of 𝜙1 on bullwhip effect for different 𝐿1 under the MA.

Figure 5 shows that the bullwhip effect decreases with the increase of 𝜙₁. With different 𝛼 the bullwhip effect changes only a little. So 𝛼 hardly affects the bullwhip effect in the circumstance of different 𝜙₁.

Figure 6 indicates that the bullwhip effect decreases slowly all the time with the increase of 𝜙₁. And, in this situation, we shift the span (number of date points) 𝑘 of retailer 1, and we can find that the bullwhip effect becomes smaller with the increase of 𝑘. We can come to the conclusion that 𝑘 is a key factor to affect the bullwhip effect.

By observing Figure 7, we know when 𝛼 takes the value of zero, the bullwhip effect value is fixed no matter what value 𝐿₁ is. Then the bullwhip effect value increases to the maximum value firstly, and after that it begins to decrease with the increase of 𝛼. In this situation, 𝐿₁ influences the BWE obviously, the smaller the 𝐿₁ is, the smaller the BWE is.

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

Figure 9 which is similar to Figure 6 simulates the impact of 𝜙₁ on the bullwhip effect based on different 𝐿₁. The bullwhip effect decreases slowly all the time with the increase of 𝜙₁. And we may find that the 𝐿₁ influences the BWE obviously; the bullwhip effect becomes smaller with the decrease of 𝐿₁.

4.3. The Analysis of Parameters under the ES Forecasting Method

Figures 10–14 simulate the expression of the bullwhip effect under the ES forecasting method to illustrate the impact of parameters on the bullwhip effect.

Figure 10 

Impact of 𝜙1 on bullwhip effect for different 𝛼 under the ES.

Figure 11 

Impact of 𝜙1 on bullwhip effect for different 𝛽1 under the ES.

Figure 12 

Impact of 𝛼 on bullwhip effect for different 𝛽1 under the ES.

Figure 13 

Impact of 𝛼 on bullwhip effect for different 𝐿1 under the ES.

Figure 14 

Impact of 𝜙1 on bullwhip effect for different 𝐿1 under the ES.

Figure 10 declares the impact of 𝜙₁ under various 𝛼. The bullwhip effect decreases continuously with the increase of 𝛼. When 𝜙₁ takes the value of 0.7 approximately, the bullwhip effect values of three different 𝛼 are the same; when 𝜙₁ is less than 0.7, the smaller the 𝛼 is, the smaller the bullwhip effect value is; when 𝜙₁ is more than 0.7, the result is opposite.

As can be seen from Figure 11, the bullwhip effect decreases all the time in pace with the increase of 𝜙₁. And, in this situation, we can observe that the smaller the 𝛽₁ is, the smaller the bullwhip effect value is. This phenomenon shows that 𝛽₁ is an important factor to influence the bullwhip effect under the ES forecasting method.

Figure 12 indicates the impact of 𝛼 on bullwhip effect for different 𝛽₁ under the ES. When 𝛽₁ takes the value of 0.4, the bullwhip effect decreases slowly first, after that it increases gradually. However, when the value of 𝛽₁ is 0.6 and 0.8, the bullwhip effect keeps increasing rapidly.

Figure 13 reveals the impact of 𝛼 on bullwhip effect for different 𝐿₁ under the ES. When 𝐿₁ takes the values of 0.4 and 0.6, the bullwhip effect decreases slowly first, after that it increases gradually. And the minimum value of the bullwhip effect occurs as the value of 𝛼 is 0.5 approximately. This phenomenon indicates that the intense competition between two retailers can increase the bullwhip effect. However, when the value of 𝐿₁ is 0.8, the bullwhip effect keeps increasing rapidly. This result is analogous with the situation in Figure 12.

Similar to Figure 11, Figure 14 shows the impact of 𝜙₁ on the bullwhip effect based on different 𝐿₁ under the ES. The bullwhip effect decreases all the time in pace with the increase of 𝜙₁. For different 𝐿₁, the smaller the 𝐿₁ is, the smaller the bullwhip effect value is. This situation indicates 𝐿₁ is also an important factor to influence the bullwhip effect under the ES forecasting method.

4.4. The Comparison of the Three Different Forecasting Methods

According to the analyses above, we select appropriate parameters to compare the bullwhip effect under three different forecasting methods. We set 𝐿₁=𝐿₂=3 and 𝜙₁=𝜙₂=0.6. Then we choose appropriate 𝑘, 𝛽₁, and 𝛽₂. From Zhang [16], the MMSE method minimizes the variance of the forecasting error among all the linear forecasting methods. It obviously leads to the lowest average cost among the three forecasting approaches. If we impose the constraint of equal data age (2/(𝑘+1)) for the MA and ES forecasts by setting 𝛽₁=𝛽₂=2/(𝑘+1), we can also achieve that.

From Figures 15–17 the trends of the three bullwhip effects are the same. However the values of them change obviously.

Figure 15 

Comparison of three different forecasting methods by varying 𝛼 (𝑘=3).

Figure 16 

Comparison of three different forecasting methods by varying 𝛼 (𝑘=9).

Figure 17 

Comparison of three different forecasting methods by varying 𝛼 (𝑘=19).

According to Figure 15, we install 𝑘=3 and correspondingly 𝛽₁=𝛽₂=0.5. We may find out that BWE_MMSE is the lowest of all. BWE_MMSE and BWE_ES decrease firstly to the minimum value and then increase with the increase of 𝛼. However, BWE_MA has opposite trend. When 𝛼 is smaller than a certain value, BWE_MA is lower than BWE_ES; when 𝛼 is larger than the certain value and smaller than another certain value, BWE_MA is higher than BWE_ES; and when 𝛼 is larger than another certain value, BWE_MA is lower than BWE_ES again. It means that the MMSE forecasting method is the best to forecast lead-time demand in this situation.

Figure 16 reveals that the bullwhip effects under three forecasting methods of 𝑘=9 and 𝛽₁=𝛽₂=0.2 have the same trends with the circumstance of 𝑘=3 and 𝛽₁=𝛽₂=0.5. But BWE_MMSE is no longer the lowest of all, and it becomes the highest of the three oppositely. BWE_MA and BWE_ES are the same as Figure 15. It means that when 𝛼 is larger than a certain value and smaller than another certain value, the ES forecasting method is the best. In the other situation, the MA is the most attractive one. We can come to the conclusion that we would better adopt ES forecasting method with the intense competition between two retailers.

In Figure 17, we set 𝑘=19 and 𝛽₁=𝛽₂=0.1. In this circumstance, BWE_MMSE is the highest all the time regardless different 𝛼. BWE_ES is a fixed value with the increase of 𝛼. BWE_MA is the lowest of all. This phenomenon reveals that the MA method is the best forecasting method whatever 𝛼 is as long as 𝑘=19 or 𝑘 is larger.

5. Conclusions

This paper contrasts the bullwhip effect based on three different forecasting methods for a simple inventory system with an AR(1) demand. Previous research studied the bullwhip effect under different forecasting methods. However, the previous articles did not consider the impact of market share. This paper investigated the effect of the lead time, the autoregressive coefficient, and the market share on a bullwhip effect measure in a simple two-stage supply chain with two suppliers and two retailers, and the two retailers both employ the order-up-to inventory policy for replenishments.

We are surprised to find that the shorter the lead time is, the smaller the bullwhip effect is under three different forecasting methods. If the MMSE forecasting method is used, increasing autoregressive coefficient does not always reduce the bullwhip effect when the market share or lead time is varying. But increasing autoregressive coefficient can reduce the bullwhip effect effectively using the MA or ES forecasting method. Not only the autoregressive coefficient and lead time affect the bullwhip effect, but also market share and the average date age are two key factors to influence the bullwhip effect. Under MMSE and ES forecasting method the bullwhip effect can be reduced when the intense competition between the two retailers occurs. However, the bullwhip effect is proved to be maximum value when the intense competition between the two retailers occur using MA forecasting method. Average date age is proportional to the bullwhip effect when the MA method is used, and it is inversely proportional to the bullwhip effect when the ES method is used. And the bigger the average date age is, the smaller the bullwhip effect is.

The analyses above suggest that the bullwhip effect cannot be reduced by simply increasing the autoregressive coefficient. We must consider other parameters that affect the bullwhip effect as well. The MMSE forecasting method is likely to be the best method to qualify the bullwhip effect when the average date age is lower. But with the increase of the average date age the ES forecasting method seems better. And with the increase of the average date age, the bullwhip effect under the MA forecasting method turns into the lowest.

In the end, we must point out that quantifying the bullwhip effect and investigating its behavior are helpful for reducing the influence of the bullwhip effect in supply chains. However, firstly, controller may pay more attention to the inventory cost in practice. Hence, investigations of the impact of parameters for the inventory cost would be meaningful. Secondly, if we combine the control and bifurcation theory to study the bullwhip effect, we may produce better conclusions. There are also urgent needs for us to continue to study in this area. Only when we do more and more research, can we reduce the bullwhip effect effectively.

Appendix

(1)  The Proof of (31) Is as Follows. After iteration computation for (2), we have 𝐷_1,𝑡−1=1+𝜙₁+⋯+𝜙₁^𝑘−1𝛼𝛿₁+𝜙₁^𝑘𝐷_{1,𝑡−𝑘−1}+𝛼𝜙₁^𝑘−1𝜀_{1,𝑡−𝑘}+𝛼𝜙₁^𝑘−2𝜀_{1,𝑡−𝑘+1}+⋯+𝛼𝜀_1,𝑡−1=1−𝜙₁^𝑖+11−𝜙₁𝛼𝛿₁+𝜙₁^𝑘𝐷_{1,𝑡−𝑘−1}+^𝑘−1_𝑗=0𝛼𝜙₁^{𝑘−1−𝑗}𝜀_{1,𝑡−𝑘+𝑗}. (A.1)

So, we can get that Cov𝐷_1,𝑡−1,𝐷_{1,𝑡−𝑘−1}=Cov1−𝜙₁^𝑖+11−𝜙₁𝛼𝛿₁+𝜙₁^𝑘𝐷_{1,𝑡−𝑘−1}+^𝑘−1_𝑗=0𝛼𝜙₁^{𝑘−1−𝑗}𝜀_{1,𝑡−𝑘+𝑗},𝐷_{1,𝑡−𝑘−1}=𝜙₁^𝑘Var𝐷_1,𝑡,Cov𝐷_1,𝑡−1,𝐷_{2,𝑡−𝑘−1}=Cov1−𝜙₁^𝑖+11−𝜙₁𝛼𝛿₁+𝜙₁^𝑘𝐷_{1,𝑡−𝑘−1}+^𝑘−1_𝑗=0𝛼𝜙₁^{𝑘−1−𝑗}𝜀_{1,𝑡−𝑘+𝑗},𝐷_{2,𝑡−𝑘−1}=𝜙₁^𝑘Cov𝐷_{1,𝑡−𝑘−1},𝐷_{2,𝑡−𝑘−1}=𝜙₁^𝑘𝛼𝐷1−𝛼Var_2,𝑡. (A.2)

Analogously, we may get the following: Cov𝐷_{1,𝑡−𝑘−1},𝐷_2,𝑡−1=𝜙₂^𝑘𝛼𝐷1−𝛼Var_2,𝑡,Cov𝐷_2,𝑡−1,𝐷_{2,𝑡−𝑘−1}=𝜙₂^𝑘Var𝐷_2,𝑡. (A.3)

(2)  The Proof of (39) Is as Follows. From (35), Cov𝐷_1,𝑡,𝐷_1,𝑡=Cov𝐷_1,𝑡,^∞_𝑖=0𝛽₁1−𝛽₁^𝑖𝐷_{1,𝑡−𝑖−1}=𝛽₁^∞_𝑖=01−𝛽₁^𝑖Cov𝐷_1,𝑡,𝐷_{1,𝑡−𝑖−1}=𝛽₁^∞_𝑖=01−𝛽₁^𝑖𝜙₁^𝑖+1Var𝐷_1,𝑡=𝛽₁𝜙₁1−1−𝛽₁𝜙₁Var𝐷_1,𝑡,Cov𝐷_1,𝑡,𝐷_2,𝑡=Cov𝐷_1,𝑡,^∞_𝑖=0𝛽₂1−𝛽₂^𝑖𝐷_{2,𝑡−𝑖−1}=𝛽₂^∞_𝑖=01−𝛽₂^𝑖Cov𝐷_1,𝑡,𝐷_{2,𝑡−𝑖−1}=𝛽₂^∞_𝑖=01−𝛽₂^𝑖𝜙₁^𝑖+1𝛼𝐷1+𝛼Var_2,𝑡=𝛽₂𝜙₁1−1−𝛽₂𝜙₁𝛼𝐷1−𝛼Var_2,𝑡. (A.4)

The same, Cov𝐷_2,𝑡,𝐷_2,𝑡=𝛽₂𝜙₂1−1−𝛽₂𝜙₂Var𝐷_2,𝑡,Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛽₁𝜙₂1−1−𝛽₁𝜙₂𝛼𝐷1−𝛼Var_2,𝑡. (A.5)

From (35), we can also get the following: Cov𝐷_1,𝑡,𝐷_2,𝑡=Cov^∞_𝑖=0𝛽₁1−𝛽₁^𝑖𝐷_{1,𝑡−𝑖−1},^∞_𝑖=0𝛽₂1−𝛽₂^𝑖𝐷_{2,𝑡−𝑖−1}=𝛽₁𝛽₂Cov^∞_𝑖=01−𝛽₁^𝑖𝐷_{1,𝑡−𝑖−1},^∞_𝑖=01−𝛽₂^𝑖𝐷_{2,𝑡−𝑖−1}=𝛽₁𝛽₂^∞∞_{𝑖=0𝑗=𝑖+1}1−𝛽₁^𝑖1−𝛽₂^𝑗𝜙₁^𝑗−𝑖,_{∞𝑖𝑖=0𝑗=0}1−𝛽₁^𝑖1−𝛽₂^𝑗𝜙₂^𝑖−𝑗Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛽₁𝛽₂^∞_𝑖=01−𝛽₁^𝑖1−𝛽₂^𝑗𝜙₁1−𝛽₂1−1−𝛽₂𝜙_1∞𝑖=0,1−𝛽₁^𝑖1−𝛽₂^𝑖𝜙₂^𝑖1−1−𝛽₂/𝜙₂^𝑖+11−1−𝛽₂/𝜙₂×Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛽₁𝛽₂1−𝜙₁𝜙₂1−𝛽₁1−𝛽₂1−1−𝛽₁1−𝛽₂1−1−𝛽₁𝜙₂1−1−𝛽₂𝜙₁×𝛼𝐷1−𝛼Var_2,𝑡 (A.6)

Conflict of Interests

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

Acknowledgments

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