A Comparison of Bullwhip Effect under Various Forecasting Techniques in Supply Chains with Two Retailers

Abstract

We examine the impact of three forecasting methods on the bullwhip effect in a two-stage supply chain with one supplier and two retailers. A first order mixed autoregressive-moving average model (ARMA(1, 1)) performs the demand forecast and an order-up-to inventory policy characterizes the inventory decision. The bullwhip effect is measured, respectively, under the minimum mean-squared error (MMSE), moving average (MA), and exponential smoothing (ES) forecasting techniques. The effect of parameters on the bullwhip effect under three forecasting methods is analyzed and the bullwhip effect under three forecasting methods is compared. Conclusions indicate that different forecasting methods lead to different bullwhip effects caused by lead time, underlying parameters of the demand process, market competition, and the consistency of demand volatility between two retailers. Moreover, some suggestions are present to help managers to select the forecasting method that yields the lowest bullwhip effect.

1. Introduction

In a supply chain, as moving backward from a downstream member to an upstream member, the variance of order quantities of orders placed by the downstream member to its (immediate) upstream member tends to be amplified. The discovery of the phenomenon can be dated back to Forrester [1, 2] who discussed its causes and possible remediation in the context of industrial dynamics. After that, more and more researchers recognized the existence of this phenomenon in supply chains. At MIT, the well known “beer game” which was widely used in teaching inventory management was developed by Sterman [3]. The above phenomenon was first called as “bullwhip effect” by Lee et al. [4, 5]. In works of Lee et al. [4, 5], demand signal processing, nonzero lead-time, order batching, supply shortages, and price fluctuation were recognized as five important sources which might lead to the emergence of the bullwhip effect in supply chains. In order to avoid the bullwhip effect, many managerial approaches have been done, that is, lead-time reduction, information sharing, or applying different replenishment rules for the inventory system.

Among the most important issues that need to be addressed in dealing with the bullwhip effect, quantifying the bullwhip effect is a challenge. In the following papers, the supply chain might employ different demand processes and different forecasting methods. Using a first-order autoregressive (AR(1)) demand process, Chen et al. [6, 7] investigated the impact of the MA and ES forecasting methods on the bullwhip effect for a simple, two-stage supply chain with one supplier and one retailer. Likewise, Xu et al. [8] conducted a similar research for the lead-time demand that is forecasted with the ES method. Luong [9] measured the bullwhip effect for a simple two-stage supply chain that includes only one retailer and one supplier in the environment where the retailer employs the order-up-to inventory policy for their inventory and demand forecast is performed through the AR(1) model, and the effect of autoregressive coefficient and lead time on this measure was investigated. For the same two-stage supply chain, Duc et al. [10] investigated effects of the autoregressive coefficient, the moving average parameter, and the lead time on the bullwhip effect when the retailer performed through the ARMA(1, 1) model. Duc et al. [11] examined the impact of a third-party warehouse on the bullwhip effect in a three-stage supply chain with one supplier, one third-party warehouse and two retailers. Then, they found that the existence of the third-party warehouse has no influence on the bullwhip effect when the lead times of the third-party warehouse, and the two retailers are equal. Zhang [12] measured the bullwhip effect for the simple supply chain with AR(1) demand process under various forecasting methods, and considered the impact of forecasting methods on the bullwhip effect. Feng and Ma [13] compared the MMSE, MA, and ES forecasting methods for the simple supply chain with ARMA(1, 1) demand process by using the dynamic simulation.

Ferrara et al. [14] examined how performances and characteristics of innovation poles and science park affect various aspects of Italian regional economies at NUTS 2 level. In particular, they found that impacts are different by geographic location but the effects are still evident even in the aggregate model. The number of IP/SP per region seems to display a positive role in sustaining the economic growth of corresponding regions. In addition, the patenting activity and the creation of research centers foster the growth of affiliated firms, which in turn affects regional economy’s parameters. To the contrary, the distance between the IP/SP and affiliated firms reduces the growing potential of the latter. Moreover, firms within an IP/SP turn out to outperforms (largely) the regional average. Finally, they found that more recent structures tend to be more prone to both patenting activity and high-level growth.

The supply chain model usually has only one retailer in the previous researches which study on the bullwhip effect, but the new supply chain we will build has two retailers. This supply chain is more in line with the structure of market. Pindyck and Rubinfeld [15] thought that the ARMA model often fits the time series of the demand process better than the AR model because the demand process usually has characteristics of both moving average and autoregressive process. The two retailers both employ the ARMA(1, 1) demand process, and both use the MMSE, MA, and ES method to forecast the lead-time demand, respectively. Our research not only determines an exact measure of the bullwhip effect but also analyzes the impact of parameters on the bullwhip effect under various forecasting methods and compare the impact of three forecasting method on the bullwhip effect.

The rest of this paper is organized as follow. Section 2 presents a new supply chain model containing two retailers which both follow the ARMA(1, 1) demand process and employ the order-up-to stock policy. In Section 3, the bullwhip effect measure for MMSE, MA, and ES forecasting method is derived. In Section 4, we analyze the effects of parameters on the bullwhip effect under different forecasting methods and compare the impact of three forecasting methods on the bullwhip effect. At last, some concluding remarks are discussed in Section 5.

2. Supply Chain Model

2.1. Demand Process

In this research, a two-stage supply chain with one supplier and two retailers is developed. Two retailers face customer demands and place orders to the supplier, respectively. We consider that retailer 1 faces an ARMA(1,1) demand model as follows: 𝐷_1,𝑡=𝛿₁+𝜙₁𝐷_1,𝑡−1+𝜀_1,𝑡−𝜃₁𝜀_1,𝑡−1. (1)

Here, 𝜙₁ and 𝜃₁ are the autoregressive and moving average coefficients of this demand model, where −1<𝜙₁<1, −1<𝜃₁<1. 𝜀_1,𝑡 is independent and identically distributed from a symmetric distribution with mean 0 and variance 𝜎². And, we have 𝐸𝐷₁=𝛿₁1−𝜙₁,Var𝐷₁=1+𝜃²₁−2𝜙₁𝜃₁1−𝜙²₁𝜎²₁. (2)

Similarly, retailer 2 also faces an ARMA(1, 1) demand model as follows: 𝐷_2,𝑡=𝛿₂+𝜙₂𝐷_2,𝑡−1+𝜀_2,𝑡−𝜃₂𝜀_2,𝑡−1, (3) here, 𝜙₂ and 𝜃₂ have same properties with 𝜙₁ and 𝜃₁ accordingly. So, we also have 𝐸𝐷₂=𝛿₂1−𝜙₂,Var𝐷₂=1+𝜃²₂−2𝜙₂𝜃₂1−𝜙²₂𝜎²₂. (4)

2.2. Inventory Policy

Assumption of the order-up-to inventory policy is used in this supply chain system for its inventory management. Two retailers both employ the order-up-to inventory policy. Retailer 1 places an order of quantity 𝑞_1,𝑡 to the supplier at the beginning of period 𝑡. And the order quantity 𝑞_1,𝑡 can be given as 𝑞_1,𝑡=𝑆_1,𝑡−𝑆_1,𝑡−1+𝐷_1,𝑡−1, (5) where 𝑆_1,𝑡 is the order-up-to lever of retailer 1 at period 𝑡, and it can be determined by the lead-time demand as 𝑆_1,𝑡=𝐷^𝐿1_1,𝑡+𝑧𝜎^𝐿1_1,𝑡, (6) where 𝐷^𝐿1_1,𝑡 is the forecast for the lead-time demand of retailer 1 which is depend on the forecasting method and 𝐿₁, 𝜎^𝐿1_1,𝑡 is the standard deviation of lead-time demand forecast error, and 𝑧 is the normal 𝑧-score that can be determined based on a given service level. Duc et al. [10] has proofed that 𝜎^𝐿1_1,𝑡 does not depend on 𝑡 and has no influence on the bullwhip effect.

For retailer 2, the order quantity 𝑞_2,𝑡 also can be given as 𝑞_2,𝑡=𝑆_2,𝑡−𝑆_2,𝑡−1+𝐷_2,𝑡−1. (7)

The order-up-to level of retailer 2 at period 𝑡 is 𝑆_2,𝑡=𝐷^𝐿2_2,𝑡+𝑧𝜎^𝐿2_2,𝑡. (8) Equations (7)-(8) have the same meaning with (5)-(6).

2.3. Forecasting Technique

In this paper, we assume that two retailers both use the same forecasting technique to forecast the lead-time demand. In Section 3, the bullwhip effect will be measured, respectively, under the MMES, 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 MMSE forecasting method, the forecast for the lead-time demand can be given as 𝐷^𝐿_𝑡=𝐷_𝑡+𝐷_𝑡+1+⋯+𝐷_{𝑡+𝐿−1}=^𝐿−1_𝑖=0𝐷_𝑡+𝑖, (9) where 𝐷_𝑡 is the forecast for the demand at period 𝑡, and can be determined as 𝐷_𝑡+𝑖=𝐸𝐷_𝑡+𝑖∣𝐷_𝑡−1,𝐷_𝑡−2,…. (10)

2.3.2. The MA Forecasting Method

Using the MA forecasting method, the forecast for the lead-time demand can be given as 𝐷^𝐿_𝑡=𝐿𝑘^𝑘_𝑖=1𝐷_𝑡−𝑖, (11) where 𝑘 is the span (number of date points) for the MA forecasting method.

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, and can be written as 𝐷^𝐿_𝑡=𝛼𝐷_𝑡−1+(1−𝛼)𝐷^𝐿_𝑡−1, (12) where 𝛼  (0<𝛼<1) denotes the fraction used in the method, also called the smoothing exponent.

3. The Measure of the Bullwhip Effect under Various Forecasting Techniques

In this section, the measure of the bullwhip effect under the MMSE, MA, and ES forecasting methods will be derived, respectively.

Total demand which two retailers face is 𝐷_𝑡=𝐷_1,𝑡+𝐷_2,𝑡. (13)

The variance of 𝐷_𝑡 is determined as Var𝐷_𝑡=Var𝐷_1,𝑡+𝐷_2,𝑡=Var𝐷_1,𝑡+Var𝐷_2,𝑡+2Cov𝐷_1,𝑡,𝐷_2,𝑡. (14)

Two retailers face the same perfectly competitive market, and their demands present a negative correlation. We assume that the correlation coefficient is 𝜓 (−1≤𝜓≤0), we can get Cov𝐷_1,𝑡,𝐷_2,𝑡=𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (15)

Using (15) in (14), we can get Var𝐷_𝑡=Var𝐷_1,𝑡+𝐷_2,𝑡=Var𝐷_1,𝑡+Var𝐷_2,𝑡+2𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (16)

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

According to Duc et al. [10], for the MMSE forecasting method, the order quantity of retailer 1 can be determined as 𝑞_1,𝑡=𝜙₁1−𝜙^𝐿1₁1−𝜙₁𝐷_1,𝑡−1−𝜃₁1−𝜙^𝐿1₁1−𝜙₁𝜀_𝑡−1−𝜙₁1−𝜙^𝐿1₁1−𝜙₁𝐷_1,𝑡−2−𝜃₁1−𝜙^𝐿1₁1−𝜙₁𝜀_𝑡−2+𝐷_1,𝑡−1=𝜙₁𝐴₁+1𝐷_1,𝑡−1−𝜙₁𝐴₁𝐷_1,𝑡−2−𝜃₁𝐴₁𝜀_1,𝑡−1+𝜃₁𝐴₁𝜀_1,𝑡−2, (17) where 𝐴₁=(1−𝜙^𝐿1₁)/(1−𝜙₁).

In the same, we also have 𝑞_2,𝑡=𝜙₂𝐴₂+1𝐷_2,𝑡−1−𝜙₂𝐴₂𝐷_2,𝑡−2−𝜃₂𝐴₂𝜀_2,𝑡−1+𝜃₂𝐴₂𝜀_2,𝑡−2, (18) where 𝐴₂=(1−𝜙^𝐿2₂)/(1−𝜙₂).

Total order quantity of period 𝑡 is 𝑞_𝑡=𝜙₁𝐴₁+1𝐷_1,𝑡−1−𝜙₁𝐴₁𝐷_1,𝑡−2−𝜃₁𝐴₁𝜀_1,𝑡−1+𝜃₁𝐴₁𝜀_1,𝑡−2+𝜙₂𝐴₂+1𝐷_2,𝑡−1−𝜙₂𝐴₂𝐷_2,𝑡−2−𝜃₂𝐴₂𝜀_2,𝑡−1+𝜃₂𝐴₂𝜀_2,𝑡−2. (19)

Proposition 1. The variance of the total order quantity of period 𝑡 under the MMSE forecasting method can be given as Var𝑞_𝑡=𝜙₁𝐴₁+1²+𝜙₁𝐴₁²+2𝜃₁𝐴₁²1−𝜙₁²1+𝜃₁²−2𝜙₁𝜃₁−2𝜙₁𝐴₁𝜙₁𝐴₁+1𝜙₁−𝜃₁1−𝜙₁𝜃₁1+𝜃²₁−2𝜙₁𝜃₁+2𝜃₁𝐴₁𝜙₁𝐴₁+1𝜙₁−𝜃₁−11−𝜙₁²1+𝜃₁²−2𝜙₁𝜃₁−2𝜙₁𝜃₁𝐴²₁1−𝜙₁²1+𝜃²₁−2𝜙₁𝜃₁Var𝐷₁+𝜙₂𝐴₂+1²+𝜙₂𝐴₂²+2𝜃₂𝐴₂²1−𝜙²₂1+𝜃²₂−2𝜙₂𝜃₂−2𝜙₂𝐴₂𝜙₂𝐴₂+1𝜙₂−𝜃₂1−𝜙₂𝜃₂1+𝜃²₂−2𝜙₂𝜃₂+2𝜃₂𝐴₂𝜙₂𝐴₂+1𝜙₂−𝜃₂−11−𝜙²₂1+𝜃²₂−2𝜙₂𝜃₂−2𝜙₂𝜃₂𝐴²₂1−𝜙²₂1+𝜃²₂−2𝜙₂𝜃₂Var𝐷₂+2𝜙₁𝐴₁+1𝜙₂𝐴₂+1−𝜙₂𝐴₂𝜙₁𝐴₁+1𝜙₁−𝜙₁𝐴₁𝜙₂𝐴₂+1𝜙₂+𝜙₁𝜙₂𝐴₁𝐴₂×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (20)

Proof. See the Appendix

For simplicity, (20) can be written as Var𝑞_𝑡=𝐻₁Var𝐷_1,𝑡+𝐻₂Var𝐷_2,𝑡+𝐻₃𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡, (21) where 𝐻₁ is the coefficient of Var(𝐷_1,𝑡), 𝐻₂ is the coefficient of Var(𝐷_2,𝑡), and 𝐻₃ is the coefficient of 𝜓Var(𝐷_1,𝑡)Var(𝐷_2,𝑡) in (20).

From (16) and (21), the measure of the bullwhip effect under the MMSE forecasting method, denoted as BWE_MMSE here, can be determined as BWE_MMSE=𝐻₁+𝐻₂𝛾²+𝐻₃𝜓𝛾1+𝛾²,+2𝜓𝛾 (22) where 𝛾=(Var(𝐷_2,𝑡))/(Var(𝐷_1,𝑡)), which means the consistency of demand volatility between two retailers.

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

According to (5)-(6) and (11), we can get the order of retailer 1 𝑞_1,𝑡=𝑆_1,𝑡−𝑆_1,𝑡−1+𝐷_1,𝑡−1=𝐿₁𝑘^𝑘_𝑖=1𝐷_𝑡−𝑖−^𝑘_𝑖=1𝐷_{𝑡−1−𝑖}+𝑧𝜎^𝐿1_1,𝑡−𝜎^𝐿1_1,𝑡−1+𝐷_1,𝑡−1=𝐿1+¹𝑘𝐷_1,𝑡−1−𝐿₁𝑘𝐷_{1,𝑡−𝑘−1}+𝑧𝜎^𝐿1_1,𝑡−𝜎^𝐿1_1,𝑡−1. (23)

We know that 𝜎^𝐿1_1,𝑡 does not depend on 𝑡, so (23) equals 𝑞_1,𝑡=𝐿1+¹𝑘𝐷_1,𝑡−1−𝐿₁𝑘𝐷_{1,𝑡−𝑘−1}, (24) where 𝑘 is the span of the MA forecasting method for retailer 1. Retailer 2 has the same span 𝑘, so the order quantity of retailer 2 is 𝑞_2,𝑡=𝐿1+²𝑘𝐷_2,𝑡−1−𝐿₂𝑘𝐷_{2,𝑡−𝑘−1}. (25)

The total order quantity of retailers in period 𝑡 is 𝑞_𝑡=𝑞_1,𝑡+𝑞_2,𝑡=𝐿1+¹𝑘𝐷_1,𝑡−1−𝐿₁𝑘𝐷_{1,𝑡−𝑘−1}+𝐿1+²𝑘𝐷_2,𝑡−1−𝐿₂𝑘𝐷_{2,𝑡−𝑘−1}. (26)

Proposition 2. The variance of the total order quantity at period 𝑡 under the MA forecasting method can be given as Var𝑞_𝑡=𝐿1+¹𝑘²+𝐿₁𝑘²𝐿−2¹𝑘𝐿1+¹𝑘×𝜙₁−𝜃₁1−𝜙₁𝜃₁1+𝜃²₁−2𝜙₁𝜃₁𝜙₁^𝑘−1Var𝐷_1,𝑡+𝐿1+²𝑘²+𝐿₂𝑘²𝐿−2²𝑘𝐿1+²𝑘×𝜙₂−𝜃₂1−𝜙₂𝜃₂1+𝜃²₂−2𝜙₂𝜃₂𝜙₂^𝑘−1×Var𝐷_2,𝑡+2𝐿1+¹𝑘𝐿1+²𝑘𝐿−2²𝑘𝐿1+¹𝑘𝜙^𝑘₁𝐿−2¹𝑘𝐿1+²𝑘𝜙^𝑘₂𝐿+2¹𝐿₂𝑘²×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (27)

Proof. See the Appendix.

For simplicity, (27) can be written as Var𝑞_𝑡=𝐽₁Var𝐷_1,𝑡+𝐽₂Var𝐷_2,𝑡+𝐽₃𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡, (28)

where 𝐽₁ is the coefficient of Var(𝐷₁), 𝐽₂ is the coefficient of Var(𝐷_2,𝑡), and 𝐽₃ is the coefficient of 𝜓Var(𝐷_1,𝑡)Var(𝐷_2,𝑡) in (27).

From (16) and (28), the measure of the bullwhip under the MA forecasting method, denoted as BWE_MA here, can be determined as BWE_MA=𝐽₁+𝐽₂𝛾²+𝐽₃𝜓𝛾1+𝛾²,+2𝜓𝛾 (29)

the same, where 𝛾=(Var(𝐷_2,𝑡))/(Var(𝐷_1,𝑡)).

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

According to Zhang [12], we know that the order quantity of retailer 1 at period 𝑡 under the ES forecasting method is 𝑞_1,𝑡=𝐷_1,𝑡+𝛼₁𝐿₁𝐷_1,𝑡−𝐷_1,𝑡, (30) where 𝛼₁ is the smoothing exponent of retailer 1 and 𝐷_1,𝑡 is the forecast of the demand at period 𝑡 for retailer 1.

The same, the order quantity of retailer 2 at period 𝑡 under the ES forecasting method is 𝑞_2,𝑡=𝐷_2,𝑡+𝛼₂𝐿₂𝐷_2,𝑡−𝐷_2,𝑡, (31) where 𝛼₂ is the smoothing exponent of retailer 2 and 𝐷_2,𝑡 is the forecast of the demand at period 𝑡 for retailer 2.

Hence, total order quantity of two retailers at period 𝑡 is 𝑞_𝑡=1+𝛼₁𝐿₁𝐷_1,𝑡−𝛼₁𝐿₁𝐷_1,𝑡+1+𝛼₂𝐿₂𝐷_2,𝑡−𝛼₂𝐿₂𝐷_2,𝑡. (32)

Proposition 3. The variance of the total order quantity at period 𝑡 under the ES forecasting method can be given as Var𝑞_𝑡=1+𝛼₁𝐿₁²+𝛼₁𝐿₁²1+1−𝛼₁𝜙₁𝛼₁2−𝛼₁1−1−𝛼₁𝜙₁−2𝛼²₁𝐿₁1+𝛼₁𝐿₁𝜙₁−𝜃₁1−𝜙₁𝜃₁1−𝜙₁1−𝛼₁1+𝜃²₁−2𝜙₁𝜃₁×Var𝐷_1,𝑡+1+𝛼₂𝐿₂²+𝛼₂𝐿₂²1+1−𝛼₂𝜙₂𝛼₂2−𝛼₂1−1−𝛼₂𝜙₂−2𝛼²₂𝐿₂1+𝛼₂𝐿₂𝜙₂−𝜃₂1−𝜙₂𝜃₂1−𝜙₂1−𝛼₂1+𝜃²₂−2𝜙₂𝜃₂×Var𝐷_2,𝑡+21+𝛼₁𝐿₁1+𝛼₂𝐿₂+2𝛼²₂𝜙₁𝐿₂1+𝛼₁𝐿₁1−𝜙₁1−𝛼₂−2𝛼²₁𝐿₁1+𝛼₂𝐿₂𝜙₂1−𝜙₂1−𝛼₁+2𝛼₁𝛼₂𝐿₁𝐿₂𝜙₂1−𝛼₂1−1−𝛼₁1−𝛼₂1−𝜙₂1−𝛼₂×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (33)

Proof. See the Appendix.

For simplicity, (33) can be written as Var𝑞_𝑡=𝑅₁Var𝐷_1,𝑡+𝑅₂Var𝐷_2,𝑡+𝑅₃𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡, (34)

where 𝑅₁ is the coefficient of Var(𝐷₁), 𝑅₂ is the coefficient of Var(𝐷_2,𝑡), and 𝑅₃ is the coefficient of 𝜓Var(𝐷_1,𝑡)Var(𝐷_2,𝑡) in (33).

From (16) and (34), the measure of the bullwhip under the ES forecasting method, denoted as BWE_ES here, can be determined as BWE_ES=𝑅₁+𝑅₂𝛾²+𝑅₃𝜓𝛾1+𝛾²,+2𝜓𝛾 (35)

the same, where 𝛾=(Var(𝐷_2,𝑡))/(Var(𝐷_1,𝑡)).

4. Analysis and Comparison for Various Forecasting Methods

In this section, impacts of demand process, inventory policy, forecasting methods, and other parameters on the bullwhip effect will be analyzed and illustrated by using numerical experiments. Results of these numerical experiments highlight the economic and managerial application of our model and findings.

4.1. The Effect of Parameters on the Bullwhip Effect under the MMSE

Figures 1–3 simulate the expression of the bullwhip effect under the MMSE forecasting method to illustrate the impact of parameters on the bullwhip effect. Luong [9] has indicated that the bullwhip effect occurs only when the autoregressive coefficient is positive in the demand process under MMSE forecasting method. So, we vary the autoregressive coefficient of retailer 1 𝜙₁ from 0 to 1 and set 𝜃₂=0.5, 𝐿₂=3, 𝜙₁=0.8, 𝜙₂=0.8, 𝜓=−0.5, and 𝛾=1 in Figure 1 to reveal the impact of 𝜙₁ on the bullwhip effect. We observe 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 those values are different for the different 𝐿₁. For 𝐿₁, when 𝜙₁ is smaller than a certain value, the smaller it is the larger the bullwhip effect is; when 𝜙₁ is larger than that certain value and smaller than another certain value, then the larger it is the larger the bullwhip effect is; and when 𝜙₁ is larger than another certain value, the smaller it is the larger the bullwhip effect is.

Figure 1 

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

Figure 2 

Impact of 𝜃1 on bullwhip effect for different 𝐿1 under the MMSE.

Figure 3 

Impact of 𝛾 on bullwhip effect for different 𝜓 under the MMSE.

Figure 2 shows the impact of 𝜃₁ on the bullwhip effect when we set 𝜃₂=0.5, 𝐿₂=3, 𝜙₁=0.8, 𝜙₂=0.8, 𝜓=−0.5, and 𝛾=1. The bullwhip effect increases slowly firstly, and then decreases slowly with the increase of 𝜃₁.

Figure 3 investigates how parameters 𝛾 and 𝜓 affect the bullwhip effect when we set 𝜃₁=0.5, 𝜃₂=0.5, 𝐿₁=3, 𝐿₂=3, 𝜙₁=0.8, and 𝜙₂=0.8. The bullwhip effect decreases firstly, and it reaches the minimum value when 𝛾=1, then increases with the increase of 𝛾. The phenomenon illustrates that the bullwhip effect is smaller when the volatility of demand between two retailers is more consistent. Specially, the bullwhip effect attains a minimum when two retailers have the same volatility of demand. For the correlative coefficient of demand between two retailers, the larger it is the larger the bullwhip effect is. This phenomenon indicates that the intense competition between two retailers can reduce the bullwhip effect.

4.2. The Effect of Parameters on the Bullwhip Effect under the MA

Figures 4–7 simulate the expression of the bullwhip effect under the MA forecasting method to illustrate the impact of parameters on the bullwhip effect. We vary 𝜙₁ from −1 to 1, shift 𝐿₁, and fix values for all other parameters in Figure 4. It shows that the bullwhip effect decreases slowly firstly when −1≤𝜙₁≤0, then increases slowly and decreases rapidly when 0<𝜙₁≤1. We also observe that the greater the lead-time of retailer 1 is the greater the bullwhip effect when 𝜙₁ is less than a certain number, and the greater the lead-time of retailer 1 is the less the bullwhip effect when 𝜙₁ is greater than that certain number.

Figure 4 

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

Figure 5 

Impact of 𝜃1 on bullwhip effect for different 𝐿1 under the MA.

Figure 6 

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

Figure 7 

Impact of 𝛾 on bullwhip effect for different 𝜓 under the MA.

Figure 5 shows the impact of 𝜃₁ on the bullwhip effect when we set 𝜃₂=0.5, 𝐿₂=3, 𝜙₁=0.5, 𝜙₂=0.5, 𝜓=−0.5, 𝛾=1. The bullwhip effect increases slowly all the time with the increase of 𝜃₁. And, in this situation, we shift the lead-time of retailer 1, and then the bullwhip effect becomes greater with the increase of 𝐿₁.

We shift the span for the MA forecasting method to examine how the parameter affects the bullwhip effect in Figure 6. We can observe that 𝑘 is an important factor to affect the bullwhip effect, the greater 𝑘 can lead to the less bullwhip effect.

Figure 7 investigates how parameters 𝛾 and 𝜓 affect the bullwhip effect when we set 𝜃₁=0.5, 𝜃₂=0.5, 𝐿₁=2, 𝐿₂=2, 𝜙₁=0.5, 𝜙₂=0.5, and 𝑘=3. The bullwhip effect increases firstly, and it reaches the maximum number when 𝛾=1, then decreases with the increase of 𝛾. The phenomenon illustrates that the bullwhip effect is stronger when the volatility of demand between two retailers is more consistent. Specially, the bullwhip effect attains a maximum when two retailers have the same volatility of demand. For the correlative coefficient of demand between two retailers, the larger it is the stronger the bullwhip effect is. The conclusion is different from the conclusion under the MMSE forecasting method. In proportion, this phenomenon indicates that the intense competition between two retailers can increase the bullwhip effect.

4.3. The Effect of Parameters on the Bullwhip Effect under the ES

Figures 8–11 simulate the expression of the bullwhip effect under ES forecasting method to illustrate the impact of parameters on the bullwhip effect. For the ES forecasting method, we add the analysis for smoothing exponents. In Figure 8, we set 𝜃₁=𝜃₂=0.5, 𝐿₂=3, 𝜙₂=0.5, 𝜓=−0.5, 𝛾=1, 𝛼₁=𝛼₂=0.1, and vary 𝜙₁ from −1 to 1 for different 𝐿₁. The bullwhip effect increases firstly to the maximum value, and then decreases with the increase of 𝜙₁. We observe that the greater 𝐿₁ is, the less the bullwhip effect when we fix other parameters, this result is different from cases under MMSE or MA forecasting method.

Figure 8 

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

Figure 9 

Impact of 𝜃1 on bullwhip effect for different 𝐿1 under the ES.

Figure 10 

Impact of 𝜙1 on the bullwhip effect for different 𝛼1 under the ES.

Figure 11 

Impact of 𝛾 on bullwhip effect for different 𝜓 under the ES.

Figure 9 shows the impact of 𝜃₁ on the bullwhip effect when we set 𝜃₂=0.5, 𝐿₂=3, 𝜙₁=𝜙₂=0.5, 𝜓=−0.5, 𝛾=1, and 𝛼₁=𝛼₂=0.1. The bullwhip effect increases slowly all the time with the increase of 𝜃₁. And in this situation, we shift the lead-time of retailer 1, and then the bullwhip effect becomes great with the increase of 𝐿₁. This result is the same as the situation under the MA forecasting method.

We shift the smoothing exponent of retailer 1 to indicate its impact on the bullwhip effect in Figure 10, and we observe the bullwhip effect as it grows to be greater with the increase of 𝛼₁.

Figure 11 investigates how parameters 𝛾 and 𝜓 affect the bullwhip affect under the ES forecasting method when we set 𝜃₁=𝜃₂=0.5, 𝐿₁=𝐿₂=3, 𝜙₁=𝜙₂=0.5, 𝛼₁=𝛼₂=0.1. The bullwhip effect increases firstly, and it reaches the maximum number when 𝛾=1, then decreases with the increase of 𝛾. The phenomenon illustrates that the bullwhip effect is stronger when the volatility of demand between two retailers is more consistent. Specially, the bullwhip effect attains a maximum when two retailers have the same volatility of demand. For the correlative coefficient of demand between two retailers, the larger it is, the stronger the bullwhip effect is. The conclusion is the same as the conclusion under the MMSE forecasting method.

4.4. The Comparison of Three Forecasting Methods

To compare bullwhip effects under three forecasting methods, we have to select appropriate 𝑘,𝛼₁, and 𝛼₂. According to Zhang [12], the MA and ES forecasting methods have the same average date age ((𝑘+1)/2), when we set 𝛼₁=𝛼₂=2/(𝑘+1) Figures 12–17 depict the comparison of the bullwhip effect among the MMSE, MA, and ES forecasting methods with different date ages, respectively. Then those figures illustrate which one is the best forecasting method for decreasing the bullwhip effect in different conditions.

Figure 12 

Comparison of three forecasting methods by varying 𝜓 (𝑘=4).

Figure 13 

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

Figure 14 

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

Figure 15 

Comparison of three forecasting methods by varying 𝛾 (𝑘=4).

Figure 16 

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

Figure 17 

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

From Figures 12–14, we observe that BWE_ES will always be higher than BWE_MA whatever 𝜓 is as long as 𝛼₁=𝛼₂=2/(𝑘+1). BWE_MA and BWE_ES get lower gradually with the increase of 𝑘 or with the decrease of 𝛼₁ and 𝛼₂. We set 𝑘=4, and 𝛼₁=𝛼₂=0.4 and fix other parameters in Figure 12, and BWE_MMSE is lower than BWE_MA and BWE_ES whatever 𝜓 is. It means that the MMSE forecasting method is the best to forecast lead-time demand in this situation. In Figure 13, let 𝑘=9, 𝛼₁=𝛼₂=0.2, BWE_MMSE is lower than BWE_MA when 𝜓 is smaller than a certain value, BWE_MMSE is higher than BWE_MA when 𝜓 is larger than that certain value, and BWE_MMSE is lower than BWE_ES whatever 𝜓 is. The conclusion means that the MMES method is the best when the market has fierce competition, and the MA method is the best when the market competition is not hot. In Figure 14, let 𝑘=19, 𝛼₁=𝛼₂=0.1, BWE_MMSE is lower than BWE_MA when 𝜓 is smaller than a certain value, BWE_MMSE is higher than BWE_MA when 𝜓 is larger than that certain value. And BWE_MMSE is higher than BWE_ES when 𝜓 is larger than another certain value. It reveals that the MMES method is the best when the market has fierce competition, and the MA method is the best when the market competition is not so hot, and the MMSE method is the worst when the market has little competition.

From Figures 15–17, we observe that BWE_MA is always higher than BWE_ES whatever 𝛾 is as long as 𝛼₁=𝛼₂=2/(𝑘+1). BWE_MA and BWE_ES get lower gradually with the increase of 𝑘 or with the decrease of 𝛼₁ and 𝛼₂. We set 𝑘=4, 𝛼₁=𝛼₂=0.4 and fix other parameters in Figure 15, and BWE_MMSE is lower than BWE_MA and BWE_ES whatever 𝛾 is. It means that the MMSE forecasting method is the best to forecast lead-time demand whatever the consistency of demand volatility between two retailers is.

In Figure 16, let 𝑘=9,𝛼₁=𝛼₂=0.2, BWE_MMSE is lower than BWE_MA when 𝛾 is smaller than a certain value, BWE_MMSE is higher than BWE_MA when 𝛾 is larger than that certain value, and BWE_MMSE is lower than BWE_ES whatever 𝛾 is. The conclusion means that the MMES method is the best when the volatility of demand between two retailers is more consistent, and the MA, method is the best when the consistency of demand volatility between two retailers is weaker. In Figure 17, let 𝑘=19, 𝛼₁=𝛼₂=0.1, BWE_MMSE is lower than BWE_ES when 𝛾 is smaller than a certain value, BWE_MMSE is higher than BWE_ES when 𝛾 is larger than that certain value, and BWE_MMSE is higher than BWE_MA whatever 𝛾 is. This phenomenon reveals that the MA method is the best whatever the consistency of demand volatility between two retailers is, and the MMSE method is the worst when the consistency of demand volatility between two retailers is weaker.

5. Conclusions

In this research, a new supply chain model containing two retailers which both followed the ARMA(1, 1) demand process and employed the order-up-to stock policy was established. The bullwhip effect was measured under the MMSE, MA, and ES forecasting methods respectively. For three bullwhip effect expressions, we investigated the effect of lead-times, autoregressive coefficients, moving average parameters, and the correlation coefficients between two retailers on the bullwhip effect.

Besides those important sources which are pointed out by Lee et al. [4, 5], market competition and the consistency of demand volatility between two retailers are also two important factors leading to the bullwhip effect. The results show that the forecasting methods play an important role in determining the impact of those factors on the bullwhip effect. If the MMSE or MA forecasting method is used, shortening lead-time does not always reduce the bullwhip effect when the demand autocorrelation is varying. But, shortening lead-time has the most significant impact on reducing the bullwhip effect when the ES method is used to forecast demand. Under the MA or ES method, average date age has the significant impact on the bullwhip effect, the greater it is the weaker the bullwhip is. Market competition is proportional to the bullwhip effect when the MA or ES method is used, and it is inversely proportional to the bullwhip effect when the MMSE method is used. The consistency of demand volatility between two retailers is inversely proportional to the bullwhip effect when the MA or ES method is used, and it is proportional to the bullwhip effect when the MMSE method is used.

From a managerial perspective, the findings suggest that managers cannot reduce the bullwhip effect by shortening the lead-time blindly. So, managers should learn more knowledge of the basic demand models and different forecasting methods. The MMSE method is always the best among three methods if only the MA and ES method have the shorter average date age. The MA method is always better than the ES method whatever the average date age is. But, the bullwhip effect under the MA and ES methods will be receded with the increase of average date age. Also, we know that the MA or ES method may perform better than the MMSE method because they are more flexible and adapt better to the changing structures of the demand when the demand structure is not well specified or shifts over time. Then, the MA or ES method may be better than the MMSE method.

Quantifying the bullwhip effect and investigating its behavior are helpful in the allocation of efforts for mitigating the influence of the bullwhip effect in supply chains. This paper can be continuous in several directions that are likely to enhance our understanding of how demand signals are transmitted along the supply chain system. First, more general inventory policies can be studied. The simple order-up-to inventory policy can be misleading when an obvious fixed ordering cost exists. So, the study on the bullwhip effect for the general (𝑠,𝑆) policy would have more practical significance. Second, more factors can be used for evaluating three forecasting methods. In this paper, the bullwhip effect as the only factor is the standard to estimate various forecasting methods. Managers may pay more attention to the inventory cost in practice, so researches on the impact of parameters for the inventory cost should be interesting.

Appendix

Proofs

Var𝐷_1,𝑡−1=Var𝐷_1,𝑡−2=Var𝐷₁,Var𝐷_2,𝑡−1=Var𝐷_2,𝑡−2=Var𝐷₂,Var𝜀_1,𝑡−1=Var𝜀_1,𝑡−2=1−𝜙²₁1+𝜃²₁−2𝜙₁𝜃₁Var𝐷₁,Var𝜀_2,𝑡−1=Var𝜀_2,𝑡−2=1−𝜙²₂1+𝜃²₂−2𝜙₂𝜃₂Var𝐷₂. (A.1)(A.2)(A.3)(A.4)

According to Feng and Ma [13], we have Cov𝐷_1,𝑡−1,𝐷_{1,𝑡−𝑘−1}=𝜙₁^𝑘−1𝜙₁−𝜃₁1−𝜙₁𝜃₁1+𝜃²₁−2𝜙₁𝜃₁Var𝐷_1,𝑡,Cov𝐷_2,𝑡−1,𝐷_{2,𝑡−𝑘−1}=𝜙₂^𝑘−1𝜙₂−𝜃₂1−𝜙₂𝜃₂1+𝜃²₂−2𝜙₂𝜃₂Var𝐷_2,𝑡. (A.5)(A.6)

After iteration computation, we have 𝐷_1,𝑡−1=1+𝜙₁+⋯+𝜙₁^𝑘−1𝛿₁+𝜙^𝑘₁𝐷_{1,𝑡−𝑘−1}+𝜀_1,𝑡−1+𝜙₁𝜀_1,𝑡−2+⋯+𝜙₁^𝑘−1𝜀_{1,𝑡−𝑘}−𝜃₁𝜀_1,𝑡−2+𝜙₁𝜀_1,𝑡−3+⋯+𝜙₁^𝑘−1𝜀_{1,𝑡−𝑘−1}, (A.7) so, we can get Cov𝐷_1,𝑡−1,𝐷_{2,𝑡−𝑘−1}=Cov1+𝜙₁+⋯+𝜙₁^𝑘−1𝛿₁+𝜙^𝑘₁𝐷_{1,𝑡−𝑘−1}+𝜀_1,𝑡−1+𝜙₁𝜀_1,𝑡−2+⋯+𝜙₁^𝑘−1𝜀_{1,𝑡−𝑘}−𝜃₁𝜀_1,𝑡−2+𝜙₁𝜀_1,𝑡−3+⋯+𝜙₁^𝑘−1𝜀_{1,𝑡−𝑘−1},𝐷_{2,𝑡−𝑘−1}=𝜙^𝑘₁𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡, (A.8) and we also have Cov𝐷_2,𝑡−1,𝐷_{1,𝑡−𝑘−1}=𝜙^𝑘₂𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.9)

According to Duc et al. [10], we have Cov𝐷_1,𝑡−1,𝜀_1,𝑡−1=Var𝜀_1,𝑡−1,Cov𝐷_1,𝑡−1,𝜀_1,𝑡−2=𝜙₁−𝜃₁Var𝜀_1,𝑡−1, (A.10) so, we also have Cov𝐷_2,𝑡−1,𝜀_2,𝑡−1=Var𝜀_2,𝑡−1,Cov𝐷_2,𝑡−1,𝜀_2,𝑡−2=𝜙₂−𝜃₂Var𝜀_2,𝑡−1. (A.11)(A.12)

Proof of Proposition 1. Total order quantity of period 𝑡 under the MMSE forecasting method is 𝑞_𝑡=𝜙₁𝐴₁+1𝐷_1,𝑡−1−𝜙₁𝐴₁𝐷_1,𝑡−2−𝜃₁𝐴₁𝜀_1,𝑡−1+𝜃₁𝐴₁𝜀_1,𝑡−2+𝜙₂𝐴₂+1𝐷_2,𝑡−1−𝜙₂𝐴₂𝐷_2,𝑡−2−𝜃₂𝐴₂𝜀_2,𝑡−1+𝜃₂𝐴₂𝜀_2,𝑡−2. (A.13)
Taking the variance for (A.12), we get Var𝑞_𝑡=𝜙₁𝐴₁+1²Var𝐷_1,𝑡−1+𝜙₁𝐴₁²Var𝐷_1,𝑡−2+𝜃₁𝐴₁²Var𝜀_1,𝑡−1+𝜃₁𝐴₁²Var𝜀_1,𝑡−2+𝜙₂𝐴₂+1²Var𝐷_2,𝑡−1+𝜙₂𝐴₂²Var𝐷_2,𝑡−2+𝜃₂𝐴₂²Var𝜀_2,𝑡−1+𝜃₂𝐴₂²Var𝜀_2,𝑡−2−2𝜙₁𝐴₁𝜙₁𝐴₁+1Cov𝐷_1,𝑡−1,𝐷_1,𝑡−2−2𝜃₁𝐴₁𝜙₁𝐴₁+1Cov𝐷_1,𝑡−1,𝜀_1,𝑡−1+2𝜃₁𝐴₁𝜙₁𝐴₁+1Cov𝐷_1,𝑡−1,𝜀_1,𝑡−2+2𝜙₁𝐴₁+1𝜙₂𝐴₂+1Cov𝐷_1,𝑡−1,𝐷_2,𝑡−1−2𝜙₂𝐴₂𝜙₁𝐴₁+1Cov𝐷_1,𝑡−1,𝐷_2,𝑡−2−2𝜙₁𝜃₁𝐴²₁Cov𝐷_1,𝑡−2,𝜀_1,𝑡−2−2𝜙₁𝐴₁𝜙₂𝐴₂+1Cov𝐷_1,𝑡−2,𝐷_2,𝑡−1+2𝜙₁𝜙₂𝐴₁𝐴₂Cov𝐷_1,𝑡−2,𝐷_2,𝑡−2−2𝜙₂𝐴₂𝜙₂𝐴₂+1Cov𝐷_2,𝑡−1,𝐷_2,𝑡−2−2𝜃₂𝐴₂𝜙₂𝐴₂+1Cov𝐷_2,𝑡−1,𝜀_2,𝑡−1+2𝜃₂𝐴₂𝜙₂𝐴₂+1Cov𝐷_2,𝑡−1,𝜀_2,𝑡−2−2𝜙₂𝜃₂𝐴²₂Cov𝐷_2,𝑡−2,𝜀_2,𝑡−2. (A.14)
Bring (A.1)–(A.12) into (A.14) and take the simplification, then we can get Var𝑞_𝑡=𝜙₁𝐴₁+1²+𝜙₁𝐴₁²+2𝜃₁𝐴₁²1−𝜙₁²1+𝜃²₁−2𝜙₁𝜃₁−2𝜙₁𝐴₁𝜙₁𝐴₁+1𝜙₁−𝜃₁1−𝜙₁𝜃₁1+𝜃²₁−2𝜙₁𝜃₁+2𝜃₁𝐴₁𝜙₁𝐴₁+1𝜙₁−𝜃₁−11−𝜙₁²1+𝜃₁²−2𝜙₁𝜃₁−2𝜙₁𝜃₁𝐴²₁1−𝜙²₁1+𝜃²₁−2𝜙₁𝜃₁Var𝐷₁+𝜙₂𝐴₂+1²+𝜙₂𝐴₂²+2𝜃₂𝐴₂²1−𝜙₂²1+𝜃₂²−2𝜙₂𝜃₂−2𝜙₂𝐴₂𝜙₂𝐴₂+1𝜙₂−𝜃₂1−𝜙₂𝜃₂1+𝜃²₂−2𝜙₂𝜃₂+2𝜃₂𝐴₂𝜙₂𝐴₂+1𝜙₂−𝜃₂−11−𝜙₂²1+𝜃₂²−2𝜙₂𝜃₂−2𝜙₂𝜃₂𝐴²₂1−𝜙₂²1+𝜃₂²−2𝜙₂𝜃₂Var𝐷₂+2𝜙₁𝐴₁+1𝜙₂𝐴₂+1−𝜙₂𝐴₂𝜙₁𝐴₁+1𝜙₁−𝜙₁𝐴₁𝜙₂𝐴₂+1𝜙₂+𝜙₁𝜙₂𝐴₁𝐴₂𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.15)
This completes the Proof for Proposition 1.

Proof of Proposition 2. Total order quantity of period 𝑡 under the MA forecasting method is 𝑞_𝑡=𝑞_1,𝑡+𝑞_2,𝑡=𝐿1+¹𝑘𝐷_1,𝑡−1−𝐿₁𝑘𝐷_{1,𝑡−𝑘−1}+𝐿1+²𝑘𝐷_2,𝑡−1−𝐿₂𝑘𝐷_{2,𝑡−𝑘−1}. (A.16)
Taking the variance for (A.16), we get Var𝑞_𝑡=𝐿1+¹𝑘²Var𝐷_1,𝑡−1+𝐿₁𝑘²Var𝐷_{1,𝑡−𝑘−1}+𝐿1+²𝑘²Var𝐷_2,𝑡−1+𝐿₂𝑘²Var𝐷_{2,𝑡−𝑘−1}𝐿−2¹𝑘𝐿1+¹𝑘Cov𝐷_1,𝑡−1,𝐷_{1,𝑡−𝑘−1}𝐿+21+¹𝑘×𝐿1+²𝑘Cov𝐷_1,𝑡−1,𝐷_2,𝑡−1𝐿−2²𝑘𝐿1+¹𝑘×Cov𝐷_1,𝑡−1,𝐷_{2,𝑡−𝑘−1}𝐿−2¹𝑘𝐿1+²𝑘Cov𝐷_{1,𝑡−𝑘−1},𝐷_2,𝑡−1𝐿+2¹𝐿₂𝑘²×Cov𝐷_{1,𝑡−𝑘−1},𝐷_{2,𝑡−𝑘−1}𝐿−2²𝑘𝐿1+²𝑘Cov𝐷_2,𝑡−1,𝐷_{2,𝑡−𝑘−1}. (A.17)
Bring (A.1)-(A.2) and (A.5)–(A.9) into (A.17), then take the simplification, and we can get Var𝑞_𝑡=𝐿1+¹𝑘²+𝐿₁𝑘²𝐿−2¹𝑘𝐿1+¹𝑘×𝜙₁−𝜃₁1−𝜙₁𝜃₁1+𝜃²₁−2𝜙₁𝜃₁𝜙₁^𝑘−1Var𝐷_1,𝑡+𝐿1+²𝑘²+𝐿₂𝑘²𝐿−2²𝑘𝐿1+²𝑘×𝜙₂−𝜃₂1−𝜙₂𝜃₂1+𝜃²₂−2𝜙₂𝜃₂𝜙₂^𝑘−1Var𝐷_2,𝑡+2𝐿1+¹𝑘𝐿1+²𝑘𝐿−2²𝑘𝐿1+¹𝑘𝜙^𝑘₁𝐿−2¹𝑘𝐿1+²𝑘𝜙^𝑘₂𝐿+2¹𝐿₂𝑘²×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.18)
This completes the Proof for Proposition 2.

Proof of Proposition 3. Total order quantity of period 𝑡 under the MA forecasting method is 𝑞_𝑡=1+𝛼₁𝐿₁𝐷_1,𝑡−𝛼₁𝐿₁𝐷_1,𝑡+1+𝛼₂𝐿₂𝐷_2,𝑡−𝛼₂𝐿₂𝐷_2,𝑡. (A.19)
Taking the variance for (A.19), we get Var𝑞_𝑡=1+𝛼₁𝐿₁²Var𝐷_1,𝑡+𝛼₁𝐿₁²Var𝐷_1,𝑡+1+𝛼₂𝐿₂²Var𝐷_2,𝑡+𝛼₂𝐿₂²Var𝐷_2,𝑡−2𝛼₁𝐿₁1+𝛼₁𝐿₁Cov𝐷_1,𝑡,𝐷_1,𝑡+21+𝛼₁𝐿₁×1+𝛼₂𝐿₂Cov𝐷_1,𝑡,𝐷_2,𝑡−2𝛼₂𝐿₂1+𝛼₁𝐿₁×Cov𝐷_1,𝑡,𝐷_2,𝑡−2𝛼₁𝐿₁1+𝛼₂𝐿₂Cov𝐷_1,𝑡,𝐷_2,𝑡+2𝛼₁𝐿₁𝛼₂𝐿₂Cov𝐷_1,𝑡,𝐷_2,𝑡−2𝛼₂𝐿₂1+𝛼₂𝐿₂×Cov𝐷_2,𝑡,𝐷_2,𝑡. (A.20)
From Zhang [12], we have 𝐷_1,𝑡=^∞_𝑖=0𝛼₁1−𝛼₁^𝑖𝐷_{1,𝑡−𝑖−1},Var𝐷_1,𝑡=1+1−𝛼₁𝜙₁𝛼₁2−𝛼₁1−1−𝛼₁𝜙₁Var𝐷_1,𝑡𝐷,Var_2,𝑡=1+1−𝛼₂𝜙₂𝛼₂2−𝛼₂1−1−𝛼₂𝜙₂Var𝐷_2,𝑡. (A.21)(A.22)(A.23)
According to (A.21), we can get Cov𝐷_1,𝑡,𝐷_1,𝑡=Cov𝐷_1,𝑡,^∞_𝑖=0𝛼₁1−𝛼₁^𝑖𝐷_{1,𝑡−𝑖−1}=𝛼₁^∞_𝑖=01−𝛼₁^𝑖Cov𝐷_1,𝑡,𝐷_{1,𝑡−𝑖−1}, (A.24) and according to (A.5), (A.24) equals to Cov𝐷_1,𝑡,𝐷_1,𝑡=𝛼₁^∞_𝑖=01−𝛼₁^𝑖𝜙^𝑖₁𝜙₁−𝜃₁1−𝜙₁𝜃₁1+𝜃²₁−2𝜙₁𝜃₁Var𝐷_1,𝑡=𝛼₁𝜙₁−𝜃₁1−𝜙₁𝜃₁1−𝜙₁1−𝛼₁1+𝜃²₁−2𝜙₁𝜃₁Var𝐷_1,𝑡. (A.25)
The same, we also have Cov𝐷_2,𝑡,𝐷_2,𝑡=𝛼₂𝜙₂−𝜃₂1−𝜙₂𝜃₂1−𝜙₂1−𝛼₂1+𝜃²₂−2𝜙₂𝜃₂Var𝐷_2,𝑡. (A.26)
According to (A.21), we can get Cov𝐷_1,𝑡,𝐷_2,𝑡=Cov𝐷_1,𝑡,^∞_𝑖=0𝛼₂1−𝛼₂^𝑖𝐷_{2,𝑡−𝑖−1}=𝛼₂^∞_𝑖=01−𝛼₂^𝑖Cov𝐷_1,𝑡,𝐷_{2,𝑡−𝑖−1}, (A.27) and according to (A.9), (A.27) equals to Cov𝐷_1,𝑡,𝐷_2,𝑡=𝛼₂^∞_𝑖=01−𝛼₂^𝑖𝜙₁^𝑖+1𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡=𝛼₂𝜙₁𝜓1−𝜙₁1−𝛼₂Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.28)
The same, we also have Cov𝐷_2,𝑡,𝐷_1,𝑡=𝛼₁𝜙₂𝜓1−𝜙₂1−𝛼₁Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.29)
From (A.21), we also get 𝐷_1,𝑡=^∞_𝑖=0𝛼₁1−𝛼₁^𝑖𝐷_{1,𝑡−𝑖−1}. (A.30)
So, we can get Cov𝐷_1,𝑡,𝐷_2,𝑡=Cov^∞_𝑖=0𝛼₁1−𝛼₁^𝑖𝐷_{1,𝑡−𝑖−1},^∞_𝑗=0𝛼₂1−𝛼₂^𝑗𝐷_{2,𝑡−𝑗−1}=𝛼₁𝛼₂Cov^∞_𝑖=01−𝛼₁^𝑖𝐷_{1,𝑡−𝑖−1},^∞_𝑗=01−𝛼₂^𝑗𝐷_{2,𝑡−𝑗−1}=𝛼₁𝛼₂^∞∞_{𝑖=0𝑗=𝑖+1}1−𝛼₁^𝑖1−𝛼₂^𝑗𝜙₂^𝑗−𝑖+^∞_𝑖=0^𝑖−1_𝑗=01−𝛼₁^𝑖1−𝛼₂^𝑗𝜙₁^𝑖−𝑗×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡=𝛼₁𝛼₂^∞_𝑖=01−𝛼₂^𝑖+11−𝛼₁^𝑖𝜙₂1−𝜙₂1−𝛼₂+^∞_𝑖=01−𝛼₁^𝑖1−𝛼₂𝜙^𝑖₁1−(1−𝛼₂)/𝜙₁^𝑖1−1−𝛼₂/𝜙₁×𝜓Var𝐷_1,𝑡=𝜙Var₂1−𝛼₂1−1−𝛼₁1−𝛼₂1−𝜙₂1−𝛼₂×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.31)
Bing (A.1)-(A.2), (A.22)-(A.23), (A.28)-(A.29), and (A.31) into (A.20), then take the simplification, and we can get Var𝑞_𝑡=1+𝛼₁𝐿₁²+𝛼₁𝐿₁²1+1−𝛼₁𝜙₁𝛼₁2−𝛼₁1−1−𝛼₁𝜙₁−2𝛼²₁𝐿₁1+𝛼₁𝐿₁𝜙₁−𝜃₁1−𝜙₁𝜃₁1−𝜙₁1−𝛼₁1+𝜃²₁−2𝜙₁𝜃₁×Var𝐷_1,𝑡+1+𝛼₂𝐿₂²+𝛼₂𝐿₂²1+1−𝛼₂𝜙₂𝛼₂2−𝛼₂1−1−𝛼₂𝜙₂−2𝛼²₂𝐿₂1+𝛼₂𝐿₂𝜙₂−𝜃₂1−𝜙₂𝜃₂1−𝜙₂1−𝛼₂1+𝜃²₂−2𝜙₂𝜃₂×Var𝐷_2,𝑡+21+𝛼₁𝐿₁1+𝛼₂𝐿₂+2𝛼²₂𝜙₁𝐿₂1+𝛼₁𝐿₁1−𝜙₁1−𝛼₂−2𝛼²₁𝐿₁1+𝛼₂𝐿₂𝜙₂1−𝜙₂1−𝛼₁+2𝛼₁𝛼₂𝐿₁𝐿₂𝜙₂1−𝛼₂1−1−𝛼₁1−𝛼₂1−𝜙₂1−𝛼₂×𝜓Var𝐷_1,𝑡Var𝐷_2,𝑡. (A.32)
This completes the Proof for Proposition 3.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (no. 61273231) and the Doctoral Scientific Fund Project of the Ministry of Education of China (no. 20090032110031).