# Analysis of the Impact of Different Forecasting Techniques on the Inventory Bullwhip Effect in Two Parallel Supply Chains with a Competition Effect

Authors: Xigang Yuan; Xiaoqing Zhang; Dalin Zhang
Journal: Journal of Engineering
 (2020)

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## Abstract

This paper studies the impact of different forecasting techniques on the inventory bullwhip effect in two parallel supply chains with the competition effect, which is in contrast to the situation of a single product in a serial supply chain. In particular, this paper constructs two parallel supply chains, each of which includes one manufacturer and one retailer. Moreover, the market demand is impacted by the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the market share, and the demand shock. We then assumed that the retailer can forecast market demand by using different forecasting techniques (i.e., the moving average technique (MA), the exponential smoothing technique (ES), and the minimum mean square error technique (MMSE)). We constructed the quantity model of the bullwhip effect and the inventory bullwhip effect. Finally, we analyzed the impact of different forecasting techniques and market share on the inventory bullwhip effect. We analyzed the conditions under which the retailers should choose different types of forecasting techniques on the basis of the inventory bullwhip effect. The results show that the MMSE forecasting technique can reduce the lead-time demand forecast error to the largest extent, and the inventory bullwhip effect can obtain the lowest level using the MMSE method: retailer-1 can reduce the inventory bullwhip effect by using the MA technique, when the self-price sensitivity coefficient, the price autoregressive coefficient, and the probabilities associated with customers choosing retailer-1’s product are very low.


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## Body

## 1. Introduction

Demand amplification is the primary obstacle to achieving coordination and maintaining harmony at different stages of supply chains and is termed as the bullwhip effect (BWE). The bullwhip is the phenomenon of information distortion as ordering information percolates upstream, which means that a downstream demand fluctuation will lead to a larger fluctuation in the variance of upstream ordering [1, 2]. The application of the simple two-level supply chain modeling assumption is a widespread method for studying the bullwhip effect (e.g., [1, 3–6]). However, with the development of the economy and society, in many industries (e.g., car manufacturing, electronic enterprises, and mobile phone manufacturers), the traditional model of a simple two-level supply chain, which is used to address the issue of the bullwhip effect, is evolving into a new model that reflects a chain-to-chain competition effect which is caused by the substitutability of homogeneous products. This competition effect means that the firm is not only affected by others within the same supply chain, but is also affected by firms from other supply chains.In this paper, the inventory bullwhip effect is used as a measure of supply chain performance, and we examined the industry with respect to two parallel supply chains that show a competition effect at the level of their demand stream. Large numbers of industries satisfy the standard of our model. For example, in the mobile phone industry, which includes Huawei and Oppo, each company can form a chain relationship with its downstream retailer, who shares the same demand market. It is well-known that the selling price of mobile phones is an important factor that affects the customer demand. For example, the higher the selling price of a Huawei mobile phone, the lower the market demand will be, and vice versa. However, due to the substitutability between Huawei and Oppo, if the Oppo mobile phone’s selling price is too high, many customers will turn to Huawei, and Huawei’s market share will increase. On the other hand, if Huawei mobile phone’s selling price is too high, many customers will turn to Oppo, and Oppo’s market share will increase. In other words, customers make purchasing decisions based not only on Oppo’s selling price but also on Huawei’s selling price.The competition effect will impact the performance of a simple two-level supply chain. However, it is difficult to determine the competition effect, particularly in terms of whether or not firms will benefit from the inventory bullwhip effect. Only a limited number of papers discuss the problem of the inventory bullwhip effect from the perspective of the two-level supply chain. For example, Ma et al. [7] derived the analytical expressions of the bullwhip effect on product orders and inventory using the minimum mean squared error (MMSE), moving average (MA), and exponential smoothing (ES) forecasting techniques in a simple two-level supply chain with one manufacturer and one retailer. However, they did not discuss the impact of different forecasting techniques on the inventory bullwhip effect in two parallel supply chain systems. On the contrary, we extend the work of Ma et al. [7], who considered a two-level supply chain in which the demand was price sensitive, by quantifying the bullwhip effect on inventory (i.e., the increase in inventory variability). We elaborate upon that work by considering two parallel supply chains, in which each sells a single product and each includes one manufacturer and one retailer, such that the competition effect exists in the two parallel supply chains, and the demand that both retailers experience is price sensitive. The price follows dynamics with an autoregressive AR(1) pricing process, and the competition effect and the market share follow the demand process. Based on the analysis, we derive the mathematical modeling of the inventory bullwhip effect by using MA, ES, and MMSE forecasting techniques. Moreover, we investigated the impact of the competition effect and the market share on the inventory bullwhip effect under different forecasting techniques. Finally, we explored the condition under which the retailer chooses the best forecasting technique to minimize the inventory bullwhip effect.The remainder of this paper is organized as follows: Section2 reviews the relevant literature. Section 3 presents the parameters and modeling assumption and the problem. Section 4 deduces the bullwhip effect (BWE) and the inventory bullwhip effect (IBWE). Section 5 discusses the impact of different forecasting techniques on the IBWE and explores the condition under which the retailer chooses the best forecasting technique to minimize the IBWE. The final section presents the conclusion which outlines the limitations of this paper and the directions for future research.


## 2. Literature Review

In this section, we review the existing literature from the following three perspectives: (1) bullwhip effect in a two-level supply chain; (2) bullwhip effect in a multilevel supply chain; (3) bullwhip effect in a two-level supply chain network.


### 2.1. Bullwhip Effect in a Two-Level Supply Chain

The bullwhip effect is a widespread phenomenon in the supply chain, which means that a downstream demand fluctuation will lead to a larger fluctuation in the variance of upstream ordering [1, 2]. This phenomenon can lead to a vast amount of problems, such as superfluous inventory, erroneous product forecasts, and insufficient or excessive capacities [2]. Therefore, in recent years, it has attracted the attention of numerous administrators and researchers. Many studies quantify the bullwhip effect and determine measures by which it can be reduced in a two-level supply chain. Lee et al. [1], and Lee et al. [8] discussed the bullwhip effect in a two-level supply chain which consisted of one manufacturer and one retailer and followed the dynamics of a first-order autoregressive demand (AR(1)) process by using the moving average forecasting technique. The result has shown that the moving average forecasting technique could reduce the bullwhip effect. Alwan et al. [9], Hausman and Miyaoka [10], and Liu and Wang [11] researched the bullwhip effect in a two-level supply chain under the ARMA(1,1) demand process by using the exponential smoothing method. They compared the different influence between these two kinds of forecasting techniques. In addition, the ARMA(p, q) demand process was discussed by Gaalman and Disney [12] and Gaalman et al. [13] with the moving average forecasting technique. These aforementioned studies discussed how the bullwhip effect can be quantified by using different autoregressive demand processes in a two-level supply chain which consisted of one supplier and one retailer or one manufacturer and one retailer. Moreover, they discussed the impact of different forecasting techniques on the bullwhip effect in the two-level supply chain.In addition, many scholars discussed the bullwhip effect in a two-level supply chain using the control approach. Holt et al. [14] developed the HMMS control model in a two-level supply chain and pointed out that this model can effectively balance the relationship between ordering from the retailer and ordering from the supplier. Blinder et al. [15] proposed the reduction of the bullwhip effect by using the (S,S) ordering strategy. Baganha and Cohen [16] designed a particular inventory control strategy and pointed out that it could reduce the fluctuation of demand information. In addition, other scholars studied the bullwhip effect using the discrete control theory, H∝ control theory, control-based forecasting technique, and O-S feedback control method in a two-level supply chain (e.g., [17–22]). Udenio et al. [23] analyzed the bullwhip effect using the system dynamics (SD) model. In addition, many scholars study the bullwhip effect in the two-level supply chain using different kinds of forecasting techniques. Chen et al. [4] proposed the causal of the bullwhip effect using the exponential smoothing method and compared the result with the increase in variability due to the use of the moving average forecasting technique. Kim and Ryan [24] analyzed the impact of bullwhip on inventory using the exponential smoothing forecasting technique. Michna et al. [25] quantified the bullwhip effect (which measures how the variance of replenishment orders is amplified as the orders move up the supply chain) when both random demands and random lead times were estimated using the moving average forecasting method in the two-level supply chain. They found that maxima and minima in the bullwhip measure as a function of the demand autocorrelation. Erica et al. [26] studied a two-echelon single-product supply chain with final demand distributed according to a known AR(1) process but with unknown parameters. The results have shown that the bullwhip effect was affected by unknown parameters and was influenced by the frequency with which parameter estimates were updated.


### 2.2. Bullwhip Effect in a Multilevel Supply Chain

With economic and societal development, more and more enterprises are involved in the supply chain, and the two-level supply chain structure evolves into a multilevel supply chain structure. Many scholars attempted to analyze the bullwhip effect in a multilevel supply chain using different forecasting techniques. Holland et al. [27] pointed out that price fluctuations could cause the bullwhip effect by using the exponential smoothing forecasting technique. Zhang [28, 29] analyzed the bullwhip effect using the minimize mean square error forecasting technique and compared the results with the moving average technique and exponential smoothing techniques, respectively. Hosoda and Disney [30] studied the impact of the bullwhip effect on inventory using the minimize mean square error forecasting technique and found that it could reduce the impact of bullwhip effect on inventory. Lu et al. [31] simulated the forming process of the bullwhip in a multilevel supply chain. Cachon et al. [32] collected the microeconomic industry-level US data and observed that manufacturing industries did not have substantially greater demand volatility than retail industries. Liu and Wang [33] formulated the mathematical expression of the bullwhip effect in a multilevel supply chain using the mean square error forecasting technique and compared with the moving average forecasting technique.Liu and Wang [33] analyzed the bullwhip of a multilevel supply chain by using the exponential smoothing forecasting technique and the minimize mean square error method, respectively. Kim and Springer [34] studied demand volatility by employing a system dynamic model. In a follow-up study, Springer and Kim [35] also analyzed the inventory volatility using a system dynamic model in a multilevel supply chain. They pointed out that the inventory volatility was caused by the demand volatility. Coppini et al. [36] simulated the four-stage beer game supply chain model, and they showed that inventory oscillations provided more information on supply chain performance than the bullwhip effect on product orders. Some other forecasting techniques have also been adopted by Ma et al. [37]. Costantino et al. [38] studied the impact of demand sharing on the order strategy using the simulation method. They found that the demand sharing helped to reduce the order volatility. Robert and Haim [39] estimated both a strong bullwhip and robust smoothing and found that firms smooth both production variability and production uncertainty. Hossein et al. [40] quantified the bullwhip effect, order rate variance ratio (OVR), and inventory variance ratio (IV) in a three-stage supply chain with multiple retailers. Moreover, they surveyed the influence of the correlation coefficient on the bullwhip effect. Marieh and Mohsen [41] investigated the measure of the bullwhip effect in three different supply chains. Moreover, they pointed out that these three different supply chains were subsequently analyzed to determine which supply chain helps to reduce the bullwhip effect more. Zhu et al. [42] investigated the factors that impacted the bullwhip effect in the oil and gas supply chain using case study evidence from six companies in North America, which cover refining and marketing, exploration and production, integrated oil and gas, and drilling. The findings indicated that the existing theories of the bullwhip effect had limitations in explaining the phenomenon in the oil and gas industry.


### 2.3. Bullwhip Effect in a Two-Level Supply Chain Network

The simple two-level supply chain modeling assumption has been widely used to study the bullwhip effect. On the contrary, due to the complexity of the supply chain network’s structure, the simple two-level supply chain modeling assumption is outdated. Recently, some scholars discussed the bullwhip effect in a two-level supply chain network [43–45]. However, most of these studies assumed that customer demand follows the AR(1) autoregressive process, and they mainly discuss supply chain coordination and how to reduce the bullwhip effect. Zhang and Yuan [44] proposed that the-old-for-new policy could reduce the bullwhip effect and could increase profitability in a closed-loop supply chain network when the firm could forecast the customer’s demand by using the moving average forecasting technique. Yuan and Zhu [45] provided three quantitative models of the bullwhip effect in a two-level supply chain network consisting of a single manufacturer and two retailers. Moreover, they compared the different impact of forecasting techniques on the bullwhip effect. Zhang and Zhang [46] addressed a fuzzy robust control (FRC) approach to mitigate the bullwhip effect in the uncertain closed-loop supply chain with lead times. A new FRC approach was proposed to mitigate the bullwhip effect and realize the robust stability of the uncertain closed-loop supply chain with lead times. A simulation example verified the mitigation effect of the bullwhip effect under the proposed FRC approach.Following the work by Yuan and Zhu [45], we extended the research of Ma et al. [7] and developed a theoretical framework involving two parallel supply chains in which the competition effect exists due to product substitutability, and the two supply chains consist of one manufacturer and one retailer. We discuss the impact of different forecasting techniques on the inventory bullwhip effect and explore the condition under which the inventory bullwhip effect is amplified (or reduced) in relation to a situation without competition and market share.Though many scholars study the bullwhip effect, extensive problems still need to be resolved, such as the issue related to how the bullwhip effect can be quantified and reduced in a complex supply chain network. From the above analysis, the previous studies have many limitations: (1) the objective is mainly concentrated on the two-level or multilevel supply chain, and few scholars considered two-level or multilevel supply chain networks that consist of two manufacturers and two retailers; (2) many scholars assumed that they trade one kind of product between different firms, and fewer studies considered trades in two different kinds of products which are substitutable between different firms; (3) most studies discussed the impact of lead time, the moving average period, and other factors on the bullwhip effect, and fewer researchers studied the impact of different kinds of forecasting techniques on the inventory bullwhip effect.The major contribution of this paper is as follows: (1) the research object of this paper mainly involves two parallel supply chains consisting of two manufacturers and two retailers; (2) we assumed that demand is price sensitive, and that the price follows a first-order autoregressive pricing process. We assumed that two retailers (R1 and R2) share the market, and thus, we introduced the self-price sensitivity coefficient, the cross-price sensitivity coefficient, and the market share into the demand model. Moreover, we developed a quantitative model of the inventory bullwhip effect in two parallel supply chains; (3) we analyzed the impact of different forecasting techniques on the inventory bullwhip effect.


## 2.1. Bullwhip Effect in a Two-Level Supply Chain

The bullwhip effect is a widespread phenomenon in the supply chain, which means that a downstream demand fluctuation will lead to a larger fluctuation in the variance of upstream ordering [1, 2]. This phenomenon can lead to a vast amount of problems, such as superfluous inventory, erroneous product forecasts, and insufficient or excessive capacities [2]. Therefore, in recent years, it has attracted the attention of numerous administrators and researchers. Many studies quantify the bullwhip effect and determine measures by which it can be reduced in a two-level supply chain. Lee et al. [1], and Lee et al. [8] discussed the bullwhip effect in a two-level supply chain which consisted of one manufacturer and one retailer and followed the dynamics of a first-order autoregressive demand (AR(1)) process by using the moving average forecasting technique. The result has shown that the moving average forecasting technique could reduce the bullwhip effect. Alwan et al. [9], Hausman and Miyaoka [10], and Liu and Wang [11] researched the bullwhip effect in a two-level supply chain under the ARMA(1,1) demand process by using the exponential smoothing method. They compared the different influence between these two kinds of forecasting techniques. In addition, the ARMA(p, q) demand process was discussed by Gaalman and Disney [12] and Gaalman et al. [13] with the moving average forecasting technique. These aforementioned studies discussed how the bullwhip effect can be quantified by using different autoregressive demand processes in a two-level supply chain which consisted of one supplier and one retailer or one manufacturer and one retailer. Moreover, they discussed the impact of different forecasting techniques on the bullwhip effect in the two-level supply chain.In addition, many scholars discussed the bullwhip effect in a two-level supply chain using the control approach. Holt et al. [14] developed the HMMS control model in a two-level supply chain and pointed out that this model can effectively balance the relationship between ordering from the retailer and ordering from the supplier. Blinder et al. [15] proposed the reduction of the bullwhip effect by using the (S,S) ordering strategy. Baganha and Cohen [16] designed a particular inventory control strategy and pointed out that it could reduce the fluctuation of demand information. In addition, other scholars studied the bullwhip effect using the discrete control theory, H∝ control theory, control-based forecasting technique, and O-S feedback control method in a two-level supply chain (e.g., [17–22]). Udenio et al. [23] analyzed the bullwhip effect using the system dynamics (SD) model. In addition, many scholars study the bullwhip effect in the two-level supply chain using different kinds of forecasting techniques. Chen et al. [4] proposed the causal of the bullwhip effect using the exponential smoothing method and compared the result with the increase in variability due to the use of the moving average forecasting technique. Kim and Ryan [24] analyzed the impact of bullwhip on inventory using the exponential smoothing forecasting technique. Michna et al. [25] quantified the bullwhip effect (which measures how the variance of replenishment orders is amplified as the orders move up the supply chain) when both random demands and random lead times were estimated using the moving average forecasting method in the two-level supply chain. They found that maxima and minima in the bullwhip measure as a function of the demand autocorrelation. Erica et al. [26] studied a two-echelon single-product supply chain with final demand distributed according to a known AR(1) process but with unknown parameters. The results have shown that the bullwhip effect was affected by unknown parameters and was influenced by the frequency with which parameter estimates were updated.


## 2.2. Bullwhip Effect in a Multilevel Supply Chain

With economic and societal development, more and more enterprises are involved in the supply chain, and the two-level supply chain structure evolves into a multilevel supply chain structure. Many scholars attempted to analyze the bullwhip effect in a multilevel supply chain using different forecasting techniques. Holland et al. [27] pointed out that price fluctuations could cause the bullwhip effect by using the exponential smoothing forecasting technique. Zhang [28, 29] analyzed the bullwhip effect using the minimize mean square error forecasting technique and compared the results with the moving average technique and exponential smoothing techniques, respectively. Hosoda and Disney [30] studied the impact of the bullwhip effect on inventory using the minimize mean square error forecasting technique and found that it could reduce the impact of bullwhip effect on inventory. Lu et al. [31] simulated the forming process of the bullwhip in a multilevel supply chain. Cachon et al. [32] collected the microeconomic industry-level US data and observed that manufacturing industries did not have substantially greater demand volatility than retail industries. Liu and Wang [33] formulated the mathematical expression of the bullwhip effect in a multilevel supply chain using the mean square error forecasting technique and compared with the moving average forecasting technique.Liu and Wang [33] analyzed the bullwhip of a multilevel supply chain by using the exponential smoothing forecasting technique and the minimize mean square error method, respectively. Kim and Springer [34] studied demand volatility by employing a system dynamic model. In a follow-up study, Springer and Kim [35] also analyzed the inventory volatility using a system dynamic model in a multilevel supply chain. They pointed out that the inventory volatility was caused by the demand volatility. Coppini et al. [36] simulated the four-stage beer game supply chain model, and they showed that inventory oscillations provided more information on supply chain performance than the bullwhip effect on product orders. Some other forecasting techniques have also been adopted by Ma et al. [37]. Costantino et al. [38] studied the impact of demand sharing on the order strategy using the simulation method. They found that the demand sharing helped to reduce the order volatility. Robert and Haim [39] estimated both a strong bullwhip and robust smoothing and found that firms smooth both production variability and production uncertainty. Hossein et al. [40] quantified the bullwhip effect, order rate variance ratio (OVR), and inventory variance ratio (IV) in a three-stage supply chain with multiple retailers. Moreover, they surveyed the influence of the correlation coefficient on the bullwhip effect. Marieh and Mohsen [41] investigated the measure of the bullwhip effect in three different supply chains. Moreover, they pointed out that these three different supply chains were subsequently analyzed to determine which supply chain helps to reduce the bullwhip effect more. Zhu et al. [42] investigated the factors that impacted the bullwhip effect in the oil and gas supply chain using case study evidence from six companies in North America, which cover refining and marketing, exploration and production, integrated oil and gas, and drilling. The findings indicated that the existing theories of the bullwhip effect had limitations in explaining the phenomenon in the oil and gas industry.


## 2.3. Bullwhip Effect in a Two-Level Supply Chain Network

The simple two-level supply chain modeling assumption has been widely used to study the bullwhip effect. On the contrary, due to the complexity of the supply chain network’s structure, the simple two-level supply chain modeling assumption is outdated. Recently, some scholars discussed the bullwhip effect in a two-level supply chain network [43–45]. However, most of these studies assumed that customer demand follows the AR(1) autoregressive process, and they mainly discuss supply chain coordination and how to reduce the bullwhip effect. Zhang and Yuan [44] proposed that the-old-for-new policy could reduce the bullwhip effect and could increase profitability in a closed-loop supply chain network when the firm could forecast the customer’s demand by using the moving average forecasting technique. Yuan and Zhu [45] provided three quantitative models of the bullwhip effect in a two-level supply chain network consisting of a single manufacturer and two retailers. Moreover, they compared the different impact of forecasting techniques on the bullwhip effect. Zhang and Zhang [46] addressed a fuzzy robust control (FRC) approach to mitigate the bullwhip effect in the uncertain closed-loop supply chain with lead times. A new FRC approach was proposed to mitigate the bullwhip effect and realize the robust stability of the uncertain closed-loop supply chain with lead times. A simulation example verified the mitigation effect of the bullwhip effect under the proposed FRC approach.Following the work by Yuan and Zhu [45], we extended the research of Ma et al. [7] and developed a theoretical framework involving two parallel supply chains in which the competition effect exists due to product substitutability, and the two supply chains consist of one manufacturer and one retailer. We discuss the impact of different forecasting techniques on the inventory bullwhip effect and explore the condition under which the inventory bullwhip effect is amplified (or reduced) in relation to a situation without competition and market share.Though many scholars study the bullwhip effect, extensive problems still need to be resolved, such as the issue related to how the bullwhip effect can be quantified and reduced in a complex supply chain network. From the above analysis, the previous studies have many limitations: (1) the objective is mainly concentrated on the two-level or multilevel supply chain, and few scholars considered two-level or multilevel supply chain networks that consist of two manufacturers and two retailers; (2) many scholars assumed that they trade one kind of product between different firms, and fewer studies considered trades in two different kinds of products which are substitutable between different firms; (3) most studies discussed the impact of lead time, the moving average period, and other factors on the bullwhip effect, and fewer researchers studied the impact of different kinds of forecasting techniques on the inventory bullwhip effect.The major contribution of this paper is as follows: (1) the research object of this paper mainly involves two parallel supply chains consisting of two manufacturers and two retailers; (2) we assumed that demand is price sensitive, and that the price follows a first-order autoregressive pricing process. We assumed that two retailers (R1 and R2) share the market, and thus, we introduced the self-price sensitivity coefficient, the cross-price sensitivity coefficient, and the market share into the demand model. Moreover, we developed a quantitative model of the inventory bullwhip effect in two parallel supply chains; (3) we analyzed the impact of different forecasting techniques on the inventory bullwhip effect.


## 3. Parameters and Modeling Assumption and Problem Description

### 3.1. Description of Parameters

In this section, we describe and explain the relevant variables, as shown in Table1.Table 1
Relevant parameters and explanations.
ParametersExplanationsa1Retailer-one (R1) potential market demanda2Retailer-two (R2) potential market demandb11R1 self-price sensitivity coefficientb12R1 cross-price sensitivity coefficientb21R2 self-price sensitivity coefficientb22R2 cross-price sensitivity coefficientψThe probabilities that customers will choose R1’s product1−ψThe probabilities for customers choosing R2’s productρ1R1 price autoregressive coefficientρ2R2 price autoregressive coefficientμ1,pThe expectation ofpt1μ2,pThe expectation ofpt2δ1,p2The variance ofpt1δ2,p2The variance ofpt2μ1,dThe expectation ofd1,tμ2,dThe expectation ofd2,tδ1,d2The variance ofd1,tδ2,d2The variance ofd2,tq1,tR1 order quantityq2,tR2 order quantityl1R1 lead timel2R2 lead timeξ^1,tl1The estimate of the standard deviation of thel1 period forecast error of R1ξ^2,tl2The estimate of the standard deviation of thel2 period forecast error of R2d^1,tl1R1 mean lead-time demandd^2,tl2R2 mean lead-time demandI1,tR1 inventory levelI2,tR2 inventory levelαSmoothing constantkThe moving average period


### 3.2. Modeling Assumption

In this section, in order to make the mathematical expression of the bullwhip effect and the inventory bullwhip effect more meaningful in a practical way, we made the following assumption:(1)
There are only two supply chains in the market, each of which consists of one manufacturer and one retailer, represented asMi and Ri (i=1,2), respectively, and they only trade one kind of product in each of the supply chains;(2)
Two retailers (R1 and R2) sell two different kinds of products which are substitutable;The probabilities that customers would choose R1’s product and R2’s product are expressed asψ and 1−ψ, respectively;(3)
The demand that is experienced by R1 is impacted by R1’s selling price and R2’s selling price. The demand that is faced by R2 is impacted by R2’s selling price and R1’s selling price [7, 37];(4)
We assumed that demand is price sensitive and that the price follows the dynamics of the AR(1) autoregressive pricing process, and in the price models, we considered the probability that customers would choose both retailers’ products [47];(5)
We only developed the mathematical expression of the bullwhip effect and the inventory bullwhip effect for two retailers. Thus, we assumed that retailers and manufacturers do not share end-demand information. The retailer forecasts customer demand using customer demand information, while the manufacturer forecasts retailer demand using the retailer’s order information;(6)
The lead timel1 and l2 are fixed, and the order is received at the start of period t+l1+1 and t+l2+1, respectively. Both retailers meet the customer demand d1,t and d2,t and backlog any excessive demand, respectively. At the same time, it was assumed that the demand of both retailers are two independent variables.


### 3.3. Problem Description

In this section, there are only two supply chains in the market, each of which consists of one manufacturer and one retailer, denoted byMi and Ri (i=1,2), respectively. Two retailers are responsible for meeting the needs of customers. Two manufacturers are responsible for supplying the product to two retailers. The demand information flows downstream to upstream, and the inventory information flows upstream to downstream. The theoretical model of the two parallel supply chains can be seen in Figure 1. Two retailers share the same market, and the differences between the two retailers’ market prices can have an impact on the customer’s purchasing behavior. We assumed that the probability of customers choosing both R1’s product and R2’s product can be expressed as ψ and 1−ψ, respectively. At the end of period t, two retailers place an order, i.e., q1,t and q2,t, to two manufacturers, respectively.Figure 1
The theoretical model of the two parallel supply chains.Two retailers face the same demand process, such that the demand that is experienced by the two retailers’ is impacted by R1’s selling price and R2’s selling price. The following basic linear demand function model of the two retailers can be obtained:(1)d1,t=a1−b11p1,t+b12p2,t+ε1,t,d2,t=a2−b21p2,t+b22p1,t+ε2,t,where a1 and a2 represent the potential market demand which are nonnegative constants, b11 and b21 denote the self-price sensitivity coefficients which are nonnegative constants, b12 and b22 are the cross-price sensitivity coefficients, and ε1,t and ε2,t are the independent and identically normally distributed demand shocks with a zero mean and variance of σ12 and σ22. It is important to emphasize that b12 and b22 are nonnegative because the two types of products are substitutable. From equation (1), we can see that the potential market demand, the self-price sensitivity coefficient, the cross-price sensitivity coefficients, and the demand shock are considered in the demand model. Thus, this makes the demand function more significant in practical terms.We assumed that the retailer trades on a perfectly competitive market and exerts no control over the market clearing price. Let market pricep1,t and p2,t in equation (1) be an AR(1) pricing process. The probability that customers will choose both retailers’ products is added to the AR(1) pricing process (the same pricing process can be seen in [7]):(2)p1,t=ψμ1+ρ1p1,t−1+ψη1,t,p2,t=1−ψμ2+ρ2p2,t−1+1−ψη2,t,where μ1 and μ2 are the nonnegative constants, ρ1 and ρ2 are the price autoregressive coefficients which satisfy 0<ρ1<1and0<ρ2<1, and η1,t and η2,t are the independent and identically normally distributed price shocks with a zero mean and variance of δ12 and δ22. From equations (1) and (2), we can obtain d1,t=a1′−b11′p1,t−1+b12′p2,t−1+ε1,t−1′, where a1′=a1, b11′=b11ρ1, and b12′=b12ρ2.ε1,t−1′=ε1,t−1′−b11ψu1−b11ψη1,t+b121−ψu2+b121−ψη2,t. We can see that the demand model is not an AR(1) process. On the contrary, the AR(1) demand process is used by most scholars (e.g., [1, 3]), and it is difficult to explain its managerial insights. Thus, we assumed that the price-sensitive demand function and the price are an AR(1) process. Thus, the self-price sensitivity coefficient, the cross-price sensitivity coefficients, and the demand shock are inputted into the demand model. The covariance of the price shock can be expressed as(3)Covη1,t,η2,t=δ122,t=t′,0,t≠t′.Based on the property of the AR(1) process, in any periodt, the expectation and the variance of pt1 and pt2 can be expressed as μ1,p=Ep1,t=ψμ1/1−ρ1; μ2,p=Ep2,t=1−ψμ2/1−ρ2 and as δ1,p2=Varp1,t=ψ2μ12/1−ρ12;δ2,p2=Varp2,t=1−ψ2μ22/1−ρ22. Moreover, the expectation and the variance of d1,t and d2,t can be expressed asμ1,d=Ed1,t=a1−b11μ1,p+b12μ2,p;μ2,d=Ed2,t=a2−b21μ2,p+b22μ1,p; δ1,d2=Vard1,t=σ12+b112δ1,p2+b122δ2,p2−2b11b12ψ1−ψδ122/1−ρ1ρ2; δ2,d2=Vard2,t=σ22+b212δ2,p2+b222δ1,p2−2b21b22ψ1−ψδ122/1−ρ1ρ2.


## 3.1. Description of Parameters

In this section, we describe and explain the relevant variables, as shown in Table1.Table 1
Relevant parameters and explanations.
ParametersExplanationsa1Retailer-one (R1) potential market demanda2Retailer-two (R2) potential market demandb11R1 self-price sensitivity coefficientb12R1 cross-price sensitivity coefficientb21R2 self-price sensitivity coefficientb22R2 cross-price sensitivity coefficientψThe probabilities that customers will choose R1’s product1−ψThe probabilities for customers choosing R2’s productρ1R1 price autoregressive coefficientρ2R2 price autoregressive coefficientμ1,pThe expectation ofpt1μ2,pThe expectation ofpt2δ1,p2The variance ofpt1δ2,p2The variance ofpt2μ1,dThe expectation ofd1,tμ2,dThe expectation ofd2,tδ1,d2The variance ofd1,tδ2,d2The variance ofd2,tq1,tR1 order quantityq2,tR2 order quantityl1R1 lead timel2R2 lead timeξ^1,tl1The estimate of the standard deviation of thel1 period forecast error of R1ξ^2,tl2The estimate of the standard deviation of thel2 period forecast error of R2d^1,tl1R1 mean lead-time demandd^2,tl2R2 mean lead-time demandI1,tR1 inventory levelI2,tR2 inventory levelαSmoothing constantkThe moving average period


## 3.2. Modeling Assumption

In this section, in order to make the mathematical expression of the bullwhip effect and the inventory bullwhip effect more meaningful in a practical way, we made the following assumption:(1)
There are only two supply chains in the market, each of which consists of one manufacturer and one retailer, represented asMi and Ri (i=1,2), respectively, and they only trade one kind of product in each of the supply chains;(2)
Two retailers (R1 and R2) sell two different kinds of products which are substitutable;The probabilities that customers would choose R1’s product and R2’s product are expressed asψ and 1−ψ, respectively;(3)
The demand that is experienced by R1 is impacted by R1’s selling price and R2’s selling price. The demand that is faced by R2 is impacted by R2’s selling price and R1’s selling price [7, 37];(4)
We assumed that demand is price sensitive and that the price follows the dynamics of the AR(1) autoregressive pricing process, and in the price models, we considered the probability that customers would choose both retailers’ products [47];(5)
We only developed the mathematical expression of the bullwhip effect and the inventory bullwhip effect for two retailers. Thus, we assumed that retailers and manufacturers do not share end-demand information. The retailer forecasts customer demand using customer demand information, while the manufacturer forecasts retailer demand using the retailer’s order information;(6)
The lead timel1 and l2 are fixed, and the order is received at the start of period t+l1+1 and t+l2+1, respectively. Both retailers meet the customer demand d1,t and d2,t and backlog any excessive demand, respectively. At the same time, it was assumed that the demand of both retailers are two independent variables.


## 3.3. Problem Description

In this section, there are only two supply chains in the market, each of which consists of one manufacturer and one retailer, denoted byMi and Ri (i=1,2), respectively. Two retailers are responsible for meeting the needs of customers. Two manufacturers are responsible for supplying the product to two retailers. The demand information flows downstream to upstream, and the inventory information flows upstream to downstream. The theoretical model of the two parallel supply chains can be seen in Figure 1. Two retailers share the same market, and the differences between the two retailers’ market prices can have an impact on the customer’s purchasing behavior. We assumed that the probability of customers choosing both R1’s product and R2’s product can be expressed as ψ and 1−ψ, respectively. At the end of period t, two retailers place an order, i.e., q1,t and q2,t, to two manufacturers, respectively.Figure 1
The theoretical model of the two parallel supply chains.Two retailers face the same demand process, such that the demand that is experienced by the two retailers’ is impacted by R1’s selling price and R2’s selling price. The following basic linear demand function model of the two retailers can be obtained:(1)d1,t=a1−b11p1,t+b12p2,t+ε1,t,d2,t=a2−b21p2,t+b22p1,t+ε2,t,where a1 and a2 represent the potential market demand which are nonnegative constants, b11 and b21 denote the self-price sensitivity coefficients which are nonnegative constants, b12 and b22 are the cross-price sensitivity coefficients, and ε1,t and ε2,t are the independent and identically normally distributed demand shocks with a zero mean and variance of σ12 and σ22. It is important to emphasize that b12 and b22 are nonnegative because the two types of products are substitutable. From equation (1), we can see that the potential market demand, the self-price sensitivity coefficient, the cross-price sensitivity coefficients, and the demand shock are considered in the demand model. Thus, this makes the demand function more significant in practical terms.We assumed that the retailer trades on a perfectly competitive market and exerts no control over the market clearing price. Let market pricep1,t and p2,t in equation (1) be an AR(1) pricing process. The probability that customers will choose both retailers’ products is added to the AR(1) pricing process (the same pricing process can be seen in [7]):(2)p1,t=ψμ1+ρ1p1,t−1+ψη1,t,p2,t=1−ψμ2+ρ2p2,t−1+1−ψη2,t,where μ1 and μ2 are the nonnegative constants, ρ1 and ρ2 are the price autoregressive coefficients which satisfy 0<ρ1<1and0<ρ2<1, and η1,t and η2,t are the independent and identically normally distributed price shocks with a zero mean and variance of δ12 and δ22. From equations (1) and (2), we can obtain d1,t=a1′−b11′p1,t−1+b12′p2,t−1+ε1,t−1′, where a1′=a1, b11′=b11ρ1, and b12′=b12ρ2.ε1,t−1′=ε1,t−1′−b11ψu1−b11ψη1,t+b121−ψu2+b121−ψη2,t. We can see that the demand model is not an AR(1) process. On the contrary, the AR(1) demand process is used by most scholars (e.g., [1, 3]), and it is difficult to explain its managerial insights. Thus, we assumed that the price-sensitive demand function and the price are an AR(1) process. Thus, the self-price sensitivity coefficient, the cross-price sensitivity coefficients, and the demand shock are inputted into the demand model. The covariance of the price shock can be expressed as(3)Covη1,t,η2,t=δ122,t=t′,0,t≠t′.Based on the property of the AR(1) process, in any periodt, the expectation and the variance of pt1 and pt2 can be expressed as μ1,p=Ep1,t=ψμ1/1−ρ1; μ2,p=Ep2,t=1−ψμ2/1−ρ2 and as δ1,p2=Varp1,t=ψ2μ12/1−ρ12;δ2,p2=Varp2,t=1−ψ2μ22/1−ρ22. Moreover, the expectation and the variance of d1,t and d2,t can be expressed asμ1,d=Ed1,t=a1−b11μ1,p+b12μ2,p;μ2,d=Ed2,t=a2−b21μ2,p+b22μ1,p; δ1,d2=Vard1,t=σ12+b112δ1,p2+b122δ2,p2−2b11b12ψ1−ψδ122/1−ρ1ρ2; δ2,d2=Vard2,t=σ22+b212δ2,p2+b222δ1,p2−2b21b22ψ1−ψδ122/1−ρ1ρ2.


## 4. Ordering Process of the Two Retailers

In this section, we assume that the two retailers have utilized the most common inventory strategy, i.e., the order-up-to strategy. R1 and R2 can determine customer demandd1,t−1 and d2,t−1 at the end of period t−1. The two retailers then place an order of quantity q1,t and q2,t to two manufacturers at the beginning of period t. We proposed that R1 and R2’s lead time l1 and l2 is fixed. Both retailers can receive the product at the beginning of the period t+l1 and t+l2, respectively. Thus, q1,t and q2,t can be calculated relative to the demand d1,t and d2,t as follows:(4)q1,t=S1,t−S1,t−1+d1,t−1,q2,t=S2,t−S2,t−1+d2,t−1,where S1,t and S2,t represent the highest inventory levels for the two retailers, and these levels are set to satisfy the goal of the inventory strategy. They are estimated from the observed demand as follows:(5)S1,t=d^1,tl1+z1ξ^1,tl1,S2,t=d^2,tl2+z2ξ^2tl2,where d^1,tl1 and d^2,tl2 are the two estimates of the mean lead-time demands using different forecasting techniques (i.e., moving average technique (MA), exponential smoothing technique (ES), and minimum mean square error technique (MMSE)), z1 and z2 are the two constants to satisfy a desired service level, and ξ^1,tl1 and ξ^2,tl2 are the two estimates of the standard deviation of the l1 and l2 period forecast errors of R1 and R2, respectively. Thus, the order quantity q1,t and q2,t for both retailers can be calculated relative to the estimates of the lead-time demands d^1,tl1 and d^2,tl2 as(6)q1,t=d^1,tl1−d^1,t−1l1+d1,t−1+z1ξ^1,tl1−ξ^1,t−1l1,q2,t=d^2,tl2−d^2,t−1l2+d2,t−1+z2ξ^2,tl2−ξ^2,t−1l2.


## 5. The Inventory Bullwhip Effect under Different Forecasting Techniques

Ma et al. [7] research discussed the impact of the bullwhip effect on product orders and inventory in two-level supply chains which included one manufacturer and one retailer. In contrast, in this section, we mainly discuss the influence of different forecasting techniques on the inventory bullwhip effect in two parallel supply chains which include two manufacturers and two retailers. First, we developed a quantitative model of the inventory bullwhip effect (IBE) under different forecasting techniques. We analyzed the conditions under which the retailers should choose different forecasting techniques on the basis of the inventory bullwhip effect.


### 5.1. The Inventory Bullwhip Effect Using the MA Techniques

In equation (6), we can determine the mean lead-time demands, i.e., d^1,tl1 and d^2,tl2, using the MA technique, respectively, where k is the moving average period:(7)d^1,tl1=l1∑i=1p1d1,t−ik,d^2,tl2=l2∑i=1p2d2,t−ik.Then,q1,t and q2,t in equation (6) can be formulated as equation (8):(8)q1,t=1+l1kd1,t−1−l1kd1,t−k−1+z1ξ^1,tl1−ξ^1,t−1l1,q2,t==1+l2kd2,t−1−l2kd2,t−k−1+z2ξ^2,tl2−ξ^2,t−1l2.Thus, the variance of the order quantity for R1 and R2 can be derived as follows:(9)Varq1,t=1+Λ12+Λ12σ12+b112Varp1,t+b122Varp2,t−2b11b12δ122Π−21+Λ1Λ1b112Varp1,t+b122Varp2,t−b11b12ρ1k+ρ2kδ122Π,Varq2,t=1+Λ22+Λ22σ22+b212Varp2,t+b222Varp1,t−2b21b22δ122Π−21+Λ2Λ2b212ρ2kVarp2,t+b222ρ1kVarp1,t−b21b22ρ2k+ρ1kδ122Π,where Λ1=l1/k,Λ2=l2/k,andΠ=ψ1−ψ/1−ρ1ρ2.Theorem 1.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the MA forecasting technique, the quantitative model of the bullwhip effect in two supply chains is expressed as follows:(10)BEretailer−1MA=Varq1,tVard1,t=1+2Λ12+2Λ1σ12+b112δ1,p2+b122δ2,p2−2b11b12δ122Δ−21+Λ1Λ1b112δ1,p2+b122δ2,p2−b11b12ρ1k+ρ2kδ122Δδ1,d2,BEretailer−2MA=Varq2,tVard2,t=1+2Λ22+2Λ2σ22+b212δ2,p2+b222δ1,p2−2b21b22δ122Δ−21+Λ2Λ2b212ρ2kδ2,p2+b222ρ1kδ1,p2−b21b22ρ2k+ρ1kδ122Δδ2,d2,where Λ1=l1/k,Λ2=l2/k,andΔ=ψ1−ψ/1−ρ1ρ2.In order to analyze the impact of the bullwhip effect on the inventory level, we calculated the variance of the inventory level for the two retailers at periodt as I1,t and I2,t, corresponding to the bullwhip effect. I1,t and I2,t can be expressed as follows:(11)I1,t=I1,t−1+q1,t−l1−d1,t−1,I2,t=I2,t−1+q2,t−l2−d2,t−1.From equation (11), the following can be obtained:(12)q1,t−l1=I1,t−I1,t−1+d1,t−1,q2,t−l2=I2,t−I2,t−1+d2,t−1.Vassian [48] proposed that q1,t−l1 and q2,t−l2 can be expressed as follows:(13)q1,t−l1=d^1,tl1−∑i=1l1−1q1,t−i−I1,t,q2,t−l2=d^2,tl2−∑i=1l2−1q2,t−i−I2,t.We substituted equation (13) into equation (12), and we can thus obtain equation (14) as follows:(14)I1,t=ξ^1,tl1=d^1,t−l1l1−d1,t−l1l1,I2,t=ξ^2,tl2=d^2,t−l2l2−d2,t−l2l2.Lemma 1.
Using the MA forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(15)ξ^1,tl1=l1δ1,d2+2b112δ1,p2Λ1l1−Ω1−2b11b12δ122ΔΛ1l1−Ω1−2b12b11δ122ΔΛ2l2−Ω2+2b122δ2,p2Λ2l2−Ω2+l1k2kδ1,d2+2b112δ1,p2Λ1k−Θ1+2b122δ2,p2Λ2k−Θ2−2b11b12Λ1k−Θ1+Λ2k−Θ2Δδ122−2l1kb11b21ρ1Ω1Θ1δ1,p2−b11b22ρ1Ω1Θ1δ122Δ−b12b21ρ2Ω2Θ2Δδ122+b12b22ρ2Ω2Θ2δ2,p2,where Λ1=ρ1/1−ρ1, Λ2=ρ2/1−ρ2, Ω1=1−ρ1l1/1−ρ1, Ω2=1−ρ2l2/1−ρ2, Θ1=1−ρ1k/1−ρ1, Θ2=1−ρ2k/1−ρ2, and Δ=ψ1−ψ/1−ρ1ρ2.Proof.
Using the MA forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(16)ξ^1,tl1=Vard^1,tl1−d1,tl1=Vard^1,tl1+Vard1,tl1−2Covd^1,tl1,d1,tl1,where(17)Vard1,tl1=Var∑i=0l1−1d1,t+i=∑i=0l1−1Vard1,t+i+2∑i=0l1−2∑j=1l1−1−iCovd1,t+i,d1,t+i+j=∑i=0l1−1Vard1,t+i+2∑i=0l1−2∑j=1l1−1−iCovd1,t+i,d1,t+i+j=l1δ1,d2+2∑i=0l1−2∑j=1l1−1−iCova1−b11p1,t+i+b12p2,t+i+ε1,t+i,a1−b11p1,t+i+j+b12p2,t+i+j+ε1,t+i+j=l1δ1,d2+2b112δ1,p2ρ11−ρ1l1−1−ρ1l11−ρ1−2b11b12δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1−2b12b11δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l2−1−ρ2l21−ρ2+2b122δ2,p2ρ21−ρ2l2−1−ρ2l21−ρ2,Vard∧1,tl1=l1k2Var∑i=1kd1,t−i=l1k2Vard1,t−1+d1,t−2+⋯+d1,t−k=l1k2kδ1,d2+2Covd1,t−1,d1,t−2+2Covd1,t−1,d1,t−3+⋯+2Covd1,t−k−1,d1,t−k=l1k2kδ1,d2+2b112δ1,p2ρ11−ρ1k−1−ρ1k1−ρ1+2b122δ2,p2ρ21−ρ2k−1−ρ2k1−ρ2−2b11b12ρ11−ρ1k−1−ρ1k1−ρ1+ρ21−ρ2k−1−ρ2k1−ρ2ψ1−ψ1−ρ1ρ2δ122,Covd1,tl1,d∧1,tl1=Cov∑i=0l1−1d1,t+i,l1k∑j=1kd1,t−j=l1kCov∑i=0l1−1d1,t+i,l1k∑j=1kd1,t−j=l1k∑i=0l1−1∑j=1kCovd1,t+i,d1,t−j=l1k∑i=0l1−1∑j=1kCova1−b11p1,t+i+b12p2,t+i+ε1,t+i,a1−b11p1,t−j+b12p2,t−j+ε1,t−j=l1k∑i=0l1−1∑j=1kCov−b11p1,t+i+b12p2,t+i,−b11p1,t−j+b12p2,t−j=l1k∑i=0l1−1∑j=1kb112Covp1,t+i,p1,t−j−b11b12Covp1,t+i,p2,t−j−b12b11Covp2,t+i,p1,t−j+b122Covp2,t+i,p2,t−j=l1kb112ρ11−ρ1l11−ρ1kδ1,p21−ρ12−b11b12δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l11−ρ1k1−ρ12−b12b11δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l21−ρ2k1−ρ22+b122ρ21−ρ2l21−ρ2kδ2,p21−ρ22.Lemma 2.
Using the MA forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(18)ξ^2,tl2=l2δ2,d2+2b212δ2,p2Λ2l2−Ω2−2b21b22δ122ΔΛ2l2−Ω2−2b22b21δ122ΔΛ1l1−Ω1+2b222δ1,p2Λ1l1−Ω1+l2k2kδ2,d2+2b212δ2,p2Λ2k−Θ2+2b222δ1,p2Λ1k−Θ1−2b21b22Λ1k−Θ1+Λ2k−Θ2Δδ122−2l2kb212ρ2Ω2Θ2δ2,p2−b21b22δ122Δρ2Ω2Θ2−b22b21δ122Δρ1Ω1Θ1+b222ρ1δ1,p2Ω1Θ1,where Λ1=ρ1/1−ρ1, Λ2=ρ2/1−ρ2, Ω1=1−ρ1l1/1−ρ1, Ω2=1−ρ2l2/1−ρ2, Θ1=1−ρ1k/1−ρ1, Θ2=1−ρ2k/1−ρ2, and Δ=ψ1−ψ/1−ρ1ρ2.Proof.
Using the MA forecasting technique, the estimate of the standard deviation of thel2 period forecast error of retailer-2 ξ^2,tl2 is a constant and can be expressed as follows:(19)ξ^2,tl2=Vard^2,tl2−d2,tl2=Vard^2,tl2+Vard2,tl2−2Covd^2,tl2,d2,tl2,where(20)Vard2,tl2=Var∑i=0l2−1d2,t+i=∑i=0l2−1Vard2,t+i+2∑i=0l2−2∑j=1l2−1−iCovd2,t+i,d2,t+i+j=∑i=0l2−1Vard2,t+i+2∑i=0l2−2∑j=1l2−1−iCovd2,t+i,d2,t+i+j=∑i=0l2−1Vard2,t+i+2∑i=0l2−2∑j=1l2−1−iCovd2,t+i,d2,t+i+j=l2δ2,d2+2∑i=0l2−2∑j=1l2−1−iCova2−b21p2,t+i+b22p1,t+i+ε2,t+i,a2−b21p2,t+i+j+b22p1,t+i+j+ε2,t+i+j=l2δ2,d2+2b212δ2,p2ρ21−ρ2l2−1−ρ2l21−ρ2−2b21b22δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l2−1−ρ2l21−ρ2−2b22b21δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+2b222δ1,p2ρ11−ρ1l1−1−ρ1l11−ρ1,Vard^2,tl2=l2k2Var∑i=1kd2,t−i=l2k2Vard2,t−1+d2,t−2+⋯+d2,t−k=l2k2kδ2,d2+2Covd2,t−1,d2,t−2+2Covd2,t−1,d2,t−3+⋯+2Covd2,t−k−1,d2,t−k=l2k2kδ2,d2+2b212δ2,p2ρ21−ρ2k−1−ρ2k1−ρ2+2b222δ1,p2ρ11−ρ1k−1−ρ1k1−ρ1−2b21b22ρ11−ρ1k−1−ρ1k1−ρ1+ρ21−ρ2k−1−ρ2k1−ρ2ψ1−ψ1−ρ1ρ2δ122,Covd2,tl2,d^2,tl2=Cov∑i=0l2−1d2,t+i,l2k∑j=1kd2,t−j=l2kCov∑i=0l2−1d2,t+i,l2k∑j=1kd2,t−j=l2k∑i=0l2−1∑j=1kCovd2,t+i,d2,t−j=l2k∑i=0l2−1∑j=1kCova2−b21p2,t+i+b22p1,t+i+ε2,t+i,a2−b21p2,t−j+b22p1,t−j+ε2,t−j=l2k∑i=0l2−1∑j=1kCov−b21p2,t+i+b22p1,t+i,−b21p2,t−j+b22p1,t−j=l2k∑i=0l2−1∑j=1kb212Covp2,t+i,p2,t−j−b21b22Covp2,t+i,p1,t−j−b22b21Covp1,t+i,p2,t−j+b222Covp1,t+i,p1,t−j=l2kb212ρ21−ρ2l21−ρ2kδ2,p21−ρ22−b21b22δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l21−ρ2k1−ρ22−b22b21δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l11−ρ1k1−ρ12+b222ρ11−ρ1l11−ρ1kδ1,p21−ρ12.Theorem 2.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy as well as the MA forecasting technique, the quantitative model of the inventory bullwhip effect in two supply chains is expressed as follows:(21)IBEretailer−1MA=VarI1,tVard1,t=l1δ1,d2+2b112δ1,p2Λ1l1−Ω1−2b11b12δ122ΔΛ1l1−Ω1−2b12b11δ122ΔΛ2l2−Ω2+2b122δ2,p2Λ2l2−Ω2+l1/k2kδ1,d2+2b112δ1,p2Λ1k−Θ1+2b122δ2,p2Λ2k−Θ2−2b11b12Λ1k−Θ1+Λ2k−Θ2Δδ122−2l1/kb11b21ρ1Ω1Θ1δ1,p2−b11b22ρ1Ω1Θ1δ122Δ−b12b21ρ2Ω2Θ2Δδ122+b12b22ρ2Ω2Θ2δ2,p2δ1,d2,IBEretailer−2MA=VarI2,tVard2,t=l2δ2,d2+2b212δ2,p2Λ2l2−Ω2−2b21b22δ122ΔΛ2l2−Ω2−2b22b21δ122ΔΛ1l1−Ω1+2b222δ1,p2Λ1l1−Ω1+l2/k2kδ2,d2+2b212δ2,p2Λ2k−Θ2+2b222δ1,p2Λ1k−Θ1−2b21b22Λ1k−Θ1+Λ2k−Θ2Δδ122−2l2/kb212ρ2Ω2Θ2δ2,p2−b21b22δ122Δρ2Ω2Θ2−b22b21δ122Δρ1Ω1Θ1+b222ρ1δ1,p2Ω1Θ1δ2,d2,where Λ1=ρ1/1−ρ1, Λ2=ρ2/1−ρ2, Ω1=1−ρ1l1/1−ρ1, Ω2=1−ρ2l2/1−ρ2, Θ1=1−ρ1k/1−ρ1, Θ2=1−ρ2k/1−ρ2, and Δ=ψ1−ψ/1−ρ1ρ2.


### 5.2. The Inventory Bullwhip Effect Using the ES Method

In equation (6), we can obtain the mean lead-time demands d^1,tl1 and d^2,tl2 using the exponential smoothing forecasting technique, where α (0<α<1) is a smoothing constant for the two retailers:(22)d^1,tl1=l1αd1,t−1+1−αd^1,t−1,d^2,tl2=l2αd2,t−1+1−αd^2,t−1.Then,q1,t and q2,t in equation (6) can be written as the following equation:(23)q1,t=1+αl1d1,t−1−αl1d^1,t−1+z1ξ^1,tl1−ξ^1,t−1l1,q2,t=1+αl2d2,t−1−αl2d^2,t−1+z2ξ^2,tl2−ξ^2,t−1l2.Theorem 3.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy as well as the ES forecasting technique, the quantitative model of the bullwhip effect in two supply chains is expressed as follows:(24)BEretailer−1ES=Varq1,tVard1,t=1+αl12δ1,d2+α2l12α/2−αδ1,d2+α2l1221−α/2−α−2αl11+αl1ψαb112ρ1δ1,p2/1−ρ121−1−αρ1+1−ψαb122ρ2δ2,p2/1−ρ221−1−αρ2δ1,d2,BEretailer−2ES=Varq2,tVard2,t=1+αl22δ2,d2+α2l22α/2−αδ2,d2+α2l2221−α/2−α−2αl21+αl2ψαb212ρ2δ2,p2/1−ρ221−1−αρ2+1−ψαb222ρ1δ1,p2/1−ρ121−1−αρ1δ2,d2.Similar to Section5.1, in order to analyze the impact of the bullwhip effect on the inventory level, we should calculate the variance of the inventory level for the two retailers at period t as I1,t and I2,t corresponding to the bullwhip effect. I1,t and I2,t can be expressed as follows:(25)I1,t=I1,t−1+q1,t−l1−d1,t−1,I2,t=I2,t−1+q2,t−l2−d2,t−1.From equation (25), we can determine that(26)q1,t−l1=I1,t−I1,t−1+d1,t−1,q2,t−l2=I2,t−I2,t−1+d2,t−1.Vassian [48] proposed that q1,t−l1 and q2,t−l2 can be expressed as follows:(27)q1,t−l1=d^1,tl1−∑i=1l1−1q1,t−i−I1,t,q2,t−l2=d^2,tl2−∑i=1l2−1q2,t−i−I2,t.We substituted equation (27) into equation (26), and we can obtain equation (28) as follows:(28)I1,t=ξ^1,tl1=d^1,t−l1l1−d1,t−l1l1,I2,t=ξ^2,tl2=d^2,t−l2l2−d2,t−l2l2.Lemma 3.
Using the ES forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(29)ξ^1,tl1=l1+l12α2−αδ1,d2+2ψb112ρ1l1−1−ρ1l11−ρ12+l121−αα2−α1−1−αρ1−l1α1−ρ1l11−ρ11−1−αρ1δ1,p2+21−ψb122ρ2l2−1−ρ2l21−ρ22+l121−αα2−α1−1−αρ2−l1α1−ρ2l11−ρ21−1−αρ2δ2,p2.Proof.
Using the ES forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(30)ξ^1,tl1=Vard1,tl1+Vard^1,tl1−2Covd1,tl1,d^1,tl1,where(31)Vard1,tl1=Varl1a1−b11p1,t+b12p2,t+ε1,t=l1δ1,d2+2ψb112ρ1l1−1−ρ1l1δ1,p21−ρ12+21−ψb122ρ2l2−1−ρ2l2δ2,p21−ρ22,Vard^1,tl1=l12Vard^1,t=l12α2−αVard1,t+21−α2−αCovd1,t−1,d^1,t=l12α2−αδ1,d2+2ψ1−ααb112ρ12−α1−1−αρ1δ1,p2+21−ψ1−ααb122ρ22−α1−1−αρ2δ2,p2,Covd1,tl1,d^1,tl1=Cov∑i=0l1−1d1,t+i,l1d^1,t=l1∑i=0l1−1Covd1,t+i,d^1,t=l1ψαb112∑i=0l1−1∑j=1∞1−αj−1ρ1i+jδ1,p2+1−ψl1αb122∑i=0l1−1∑j=1∞1−αj−1ρ2i+jδ2,p2=ψl1αb112ρ11−ρ1l11−ρ11−1−αρ1δ1,p2+1−ψl1αb122ρ21−ρ2l11−ρ21−1−αρ2δ2,p2.Lemma 4.
Using the ES forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(32)ξ^2,tl2=l2+l22α2−αδ2,d2+2ψb212ρ2l2−1−ρ2l21−ρ22+l221−αα2−α1−1−αρ2−l2α1−ρ2l21−ρ21−1−αρ2δ2,p2+21−ψb222ρ1l2−1−ρ1l21−ρ12+l221−αα2−α1−1−αρ1−l2α1−ρ1l11−ρ11−1−αρ1δ1,p2.Proof.
Using the ES forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(33)ξ^2,tl2=Vard2,tl2+Vard^2,tl2−2Covd2,tl2,d^2,tl2,where(34)Vard2,tl2=Varl2a2−b21p2,t+b22p1,t+ε2,t=l2δ2,d2+21−ψb212ρ2l2−1−ρ2l2δ2,p21−ρ22+2ψb222ρ1l1−1−ρ1l1δ1,p21−ρ12,Vard^2,tl2=l22Vard^2,t=l22α2−αVard2,t+21−α2−αCovd2,t−1,d^2,t=l22α2−αδ2,d2+21−ψ1−ααb212ρ22−α1−1−αρ2δ2,p2+2ψ1−ααb222ρ12−α1−1−αρ1δ1,p2,Covd2,tl2,d^2,tl2=Cov∑i=0l2−1d2,t+i,l2d^2,t=l2∑i=0l2−1Covd2,t+i,d^2,t=l21−ψαb212∑i=0l2−1∑j=1∞1−αj−1ρ2i+jδ2,p2+ψl2αb222∑i=0l2−1∑j=1∞1−αj−1ρ1i+jδ1,p2=1−ψl2αb212ρ21−ρ2l21−ρ21−1−αρ2δ2,p2+ψl2αb222ρ11−ρ1l21−ρ11−1−αρ1δ1,p2.Theorem 4.
For two parallel supply chains in which interactions exist with respect to customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the ES forecasting method, the quantitative model of the inventory bullwhip effect in two supply chains is expressed as follows:(35)IBEretailer−1ES=VarI1,tVard1,t=l1+l12α/2−αδ1,d2+2ψb112ρ1l1−1−ρ1l1/1−ρ12+l121−αα/2−α1−1−αρ1−l1α1−ρ1l1/1−ρ11−1−αρ1δ1,p2+21−ψb122ρ2l2−1−ρ2l2/1−ρ22+l121−αα/2−α1−1−αρ2−l1α1−ρ2l1/1−ρ21−1−αρ2δ2,p2δ1,d2,IBEretailer−2ES=VarI2,tVard2,t=l2+l22α/2−αδ2,d2+2ψb212ρ2l2−1−ρ2l1/1−ρ22+l221−αα/2−α1−1−αρ2−l2α1−ρ2l2/1−ρ21−1−αρ2δ2,p2+21−ψb222ρ1l2−1−ρ1l2/1−ρ12+l221−αα/2−α1−1−αρ1−l2α1−ρ1l1/1−ρ11−1−αρ1δ1,p2δ2,d2.


### 5.3. The Inventory Bullwhip Effect Using the MMSE Method

Box and Jenkins [49] have pointed out that the demand forecast value d^t+i is the conditional expectation in historical demand information for the period t+ii=0,1,2,…. d^t+i=Edt+idt−1,dt−2,…, and particularly for the AR(1) process, d^t+i=Edt+idt−1. In this paper, the price follows the AR(1) process, such that p^t+i=Ept+ipt−1. Thus, the demand forecast values for R1 and R2 can be expressed as d^1,t+i=a1−b11p^1,t+i+b12p^2,t+i+ε1,t+i and d^2,t+i=a2−b21p^2,t+i+b22p^1,t+i+ε2,t+i in the period t+i. Thus, the price forecast values for the period t+i can be expressed as follows:(36)p^1,t+i=Ep1,t+ip1,t−1=1−ρ1i+1ρ1μ1ψ+ρ1i+1p1,t−1,p^2,t+i=Ep2,t+ip2,t−1=1−ρ2i+1ρ2μ21−ψ+ρ2i+1p2,t−1.Thus, the demand forecast values for the periodt+i can be expressed as follows:(37)d^1,t+i=a1−b11p^1,t+i+b12p^2,t+i=a1−b111−ρ1i+1μ1ψ1−ρ1−b121−ρ2i+1μ21−ψ1−ρ2−b11ρ1i+1p1,t−1−b12ρ2i+1p2,t−1,d^2,t+i=a2−b21p^2,t+i+b22p^1,t+i=a2−b211−ρ2i+1μ21−ψ1−ρ2−b221−ρ1i+1μ1ψ1−ρ1−b21ρ2i+1p2,t−1−b22ρ1,i+1p1,t−1.Then,q1,t and q2,t in equation (6) can be formulated as equation (37):(38)q1,t=−b11ρ1ρ1ψ1−ρ1l1p1,t−1−p1,t−21−ρ1+b12ρ2ρ21−ρ2l1p2,t−1−p2,t−21−ρ2+d1,t−1,q2,t=−b21ρ2ρ21−ψ1−ρ2l2p2,t−1−p2,t−21−ρ2+b22ρ1ρ11−ρ1l2p1,t−1−p1,t−21−ρ1+d2,t−1.Theorem 5.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the MMSE forecasting technique, the quantitative model of the bullwhip effect in two supply chains is expressed as follows:(39)BEretailer-1MMSE=Varq1,tVard1,t=1+2b112ψρ131−ρ1l11−ρ1l1+1/1+ρ11−ρ12δ12+2b122ρ231−ρ2l11−ρ2l1+1/1+ρ21−ρ22δ22−2b11b12ρ11−ρ2/1−ρ1ρ2+ρ1ρ21−ρ1/1−ρ1ρ2ρ21−ρ2l1/1−ρ22ρ11−ρ1l1/1−ρ12+ρ21−ρ1/1−ρ1ρ2+ρ2ρ11−ρ2/1−ρ1ρ2ρ11−ρ1l1/1−ρ12ρ21−ρ2l1/1−ρ22δ122δ1,d2,BEretailer-2MMSE=Varq2,tVard2,t=1+2b2121−ψρ231−ρ2l21−ρ2l2+1/1+ρ21−ρ22δ22+2b222ρ131−ρ1l21−ρ1l2+1/1+ρ11−ρ12δ12−2b21b22ρ21−ρ1/1−ρ1ρ2+ρ2ρ11−ρ2/1−ρ1ρ2ρ11−ρ1l2/1−ρ12ρ21−ρ2l2/1−ρ22+ρ11−ρ2/1−ρ1ρ2+ρ1ρ21−ρ1/1−ρ1ρ2ρ21−ρ2l2/1−ρ22ρ11−ρ1l2/1−ρ12δ122δ2,d2.Similar to Section5.1, in order to analyze the impact of the bullwhip effect on the inventory level, we calculated the variance of the inventory level for the two retailers at the period t as I1,t and I2,t corresponding to the bullwhip effect. I1,t and I2,t can be expressed as follows:(40)I1,t=I1,t−1+q1,t−l1−d1,t−1,I2,t=I2,t−1+q2,t−l2−d2,t−1.From equation (40), we can determine(41)q1,t−l1=I1,t−I1,t−1+d1,t−1,q2,t−l2=I2,t−I2,t−1+d2,t−1.Vassian [48] proposed that q1,t−l1 and q2,t−l2 can be expressed as follows:(42)q1,t−l1=d^1,tl1−∑i=1l1−1q1,t−i−I1,t,q2,t−l2=d^2,tl2−∑i=1l2−1q2,t−i−I2,t.We substituted equation (42) into equation (41), and we can obtain equation (43) as follows:(43)I1,t=ξ^1,tl1=d^1,t−l1l1−d1,t−l1l1,I2,t=ξ^2,tl2=d^2,t−l2l2−d2,t−l2l2.Lemma 5.
Using the MMSE forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(44)ξ^1,tl1=l1σ12+l2σ22+b112ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12+b1221−ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22.Proof.
Proof(45)ξ^1,tl1=Vard^1,tl1−d1,tl1=Var∑i=0l1−1d1,t−d^1,t=Var∑i=0l1−1−b11ψ∑j=0iρ1i−jη1,t+j−b121−ψ∑j=0tρ2i−jη2,t+j+ε1,t+i+ε2,t+i=Var∑i=0l1−1ε1,t+i+Var∑i=0l2−1ε2,t+i+b112ψ2Var∑i=0l1−1∑j=0iρ1i−jη1,t+j+b1221−ψ2Var∑i=0l2−1∑j=0iρ2i−jη2,t+j=l1σ12+l2σ22+b112ψ2Var∑i=0l1−1η1,t+j∑j=0l1−1−iρ1j+b1221−ψ2Var∑i=0l2−1η2,t+j∑j=0l2−1−iρ2j=l1σ12+l2σ22+b112ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12+b1221−ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22.Lemma 6.
Using the MMSE forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(46)ξ^2,tl2=l2σ22+l1σ12+b212ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22+b2221−ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12.Proof.
Proof(47)ξ^2,tl2=Vard^2,tl2−d2,tl2=Var∑i=0l2−1d2,t−d^2,t=Var∑i=0l2−1d2,t−d^2,t=Var∑i=0l2−1−b21ψ∑j=0iρ2i−jη2,t+j−b221−ψ∑j=0tρ1i−jη1,t+j+ε1,t+i+ε2,t+i=Var∑i=0l1−1ε1,t+i+Var∑i=0l2−1ε2,t+i+b212ψ2Var∑i=0l2−1∑j=0iρ2i−jη2,t+j+b2221−ψ2Var∑i=0l1−1∑j=0iρ1i−jη1,t+j=l1σ12+l2σ22+b212ψ2Var∑i=0l2−1η2,t+j∑j=0l2−1−iρ2j+b2221−ψ2Var∑i=0l1−1η1,t+j∑j=0l1−1−iρ1j=l1σ12+l2σ22+b212ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22+b2221−ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12.Theorem 6.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the MMSE forecasting technique, the quantitative model of the inventory bullwhip effect in two supply chains is expressed as follows:(48)IBEretailer−1MMSE=VarI1,tVard1,t=l1σ12+l2σ22+b112ψ2/1−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−2/1−ρ12δ12+b1221−ψ2/1−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−2/1−ρ22δ22δ1,d2,IBEretailer−2MMSE=VarI2,tVard2,t=l2σ22+l1σ12+b212ψ2/1−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−2/1−ρ22δ22+b2221−ψ2/1−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−2/1−ρ12δ12δ2,d2.


## 5.1. The Inventory Bullwhip Effect Using the MA Techniques

In equation (6), we can determine the mean lead-time demands, i.e., d^1,tl1 and d^2,tl2, using the MA technique, respectively, where k is the moving average period:(7)d^1,tl1=l1∑i=1p1d1,t−ik,d^2,tl2=l2∑i=1p2d2,t−ik.Then,q1,t and q2,t in equation (6) can be formulated as equation (8):(8)q1,t=1+l1kd1,t−1−l1kd1,t−k−1+z1ξ^1,tl1−ξ^1,t−1l1,q2,t==1+l2kd2,t−1−l2kd2,t−k−1+z2ξ^2,tl2−ξ^2,t−1l2.Thus, the variance of the order quantity for R1 and R2 can be derived as follows:(9)Varq1,t=1+Λ12+Λ12σ12+b112Varp1,t+b122Varp2,t−2b11b12δ122Π−21+Λ1Λ1b112Varp1,t+b122Varp2,t−b11b12ρ1k+ρ2kδ122Π,Varq2,t=1+Λ22+Λ22σ22+b212Varp2,t+b222Varp1,t−2b21b22δ122Π−21+Λ2Λ2b212ρ2kVarp2,t+b222ρ1kVarp1,t−b21b22ρ2k+ρ1kδ122Π,where Λ1=l1/k,Λ2=l2/k,andΠ=ψ1−ψ/1−ρ1ρ2.Theorem 1.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the MA forecasting technique, the quantitative model of the bullwhip effect in two supply chains is expressed as follows:(10)BEretailer−1MA=Varq1,tVard1,t=1+2Λ12+2Λ1σ12+b112δ1,p2+b122δ2,p2−2b11b12δ122Δ−21+Λ1Λ1b112δ1,p2+b122δ2,p2−b11b12ρ1k+ρ2kδ122Δδ1,d2,BEretailer−2MA=Varq2,tVard2,t=1+2Λ22+2Λ2σ22+b212δ2,p2+b222δ1,p2−2b21b22δ122Δ−21+Λ2Λ2b212ρ2kδ2,p2+b222ρ1kδ1,p2−b21b22ρ2k+ρ1kδ122Δδ2,d2,where Λ1=l1/k,Λ2=l2/k,andΔ=ψ1−ψ/1−ρ1ρ2.In order to analyze the impact of the bullwhip effect on the inventory level, we calculated the variance of the inventory level for the two retailers at periodt as I1,t and I2,t, corresponding to the bullwhip effect. I1,t and I2,t can be expressed as follows:(11)I1,t=I1,t−1+q1,t−l1−d1,t−1,I2,t=I2,t−1+q2,t−l2−d2,t−1.From equation (11), the following can be obtained:(12)q1,t−l1=I1,t−I1,t−1+d1,t−1,q2,t−l2=I2,t−I2,t−1+d2,t−1.Vassian [48] proposed that q1,t−l1 and q2,t−l2 can be expressed as follows:(13)q1,t−l1=d^1,tl1−∑i=1l1−1q1,t−i−I1,t,q2,t−l2=d^2,tl2−∑i=1l2−1q2,t−i−I2,t.We substituted equation (13) into equation (12), and we can thus obtain equation (14) as follows:(14)I1,t=ξ^1,tl1=d^1,t−l1l1−d1,t−l1l1,I2,t=ξ^2,tl2=d^2,t−l2l2−d2,t−l2l2.Lemma 1.
Using the MA forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(15)ξ^1,tl1=l1δ1,d2+2b112δ1,p2Λ1l1−Ω1−2b11b12δ122ΔΛ1l1−Ω1−2b12b11δ122ΔΛ2l2−Ω2+2b122δ2,p2Λ2l2−Ω2+l1k2kδ1,d2+2b112δ1,p2Λ1k−Θ1+2b122δ2,p2Λ2k−Θ2−2b11b12Λ1k−Θ1+Λ2k−Θ2Δδ122−2l1kb11b21ρ1Ω1Θ1δ1,p2−b11b22ρ1Ω1Θ1δ122Δ−b12b21ρ2Ω2Θ2Δδ122+b12b22ρ2Ω2Θ2δ2,p2,where Λ1=ρ1/1−ρ1, Λ2=ρ2/1−ρ2, Ω1=1−ρ1l1/1−ρ1, Ω2=1−ρ2l2/1−ρ2, Θ1=1−ρ1k/1−ρ1, Θ2=1−ρ2k/1−ρ2, and Δ=ψ1−ψ/1−ρ1ρ2.Proof.
Using the MA forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(16)ξ^1,tl1=Vard^1,tl1−d1,tl1=Vard^1,tl1+Vard1,tl1−2Covd^1,tl1,d1,tl1,where(17)Vard1,tl1=Var∑i=0l1−1d1,t+i=∑i=0l1−1Vard1,t+i+2∑i=0l1−2∑j=1l1−1−iCovd1,t+i,d1,t+i+j=∑i=0l1−1Vard1,t+i+2∑i=0l1−2∑j=1l1−1−iCovd1,t+i,d1,t+i+j=l1δ1,d2+2∑i=0l1−2∑j=1l1−1−iCova1−b11p1,t+i+b12p2,t+i+ε1,t+i,a1−b11p1,t+i+j+b12p2,t+i+j+ε1,t+i+j=l1δ1,d2+2b112δ1,p2ρ11−ρ1l1−1−ρ1l11−ρ1−2b11b12δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1−2b12b11δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l2−1−ρ2l21−ρ2+2b122δ2,p2ρ21−ρ2l2−1−ρ2l21−ρ2,Vard∧1,tl1=l1k2Var∑i=1kd1,t−i=l1k2Vard1,t−1+d1,t−2+⋯+d1,t−k=l1k2kδ1,d2+2Covd1,t−1,d1,t−2+2Covd1,t−1,d1,t−3+⋯+2Covd1,t−k−1,d1,t−k=l1k2kδ1,d2+2b112δ1,p2ρ11−ρ1k−1−ρ1k1−ρ1+2b122δ2,p2ρ21−ρ2k−1−ρ2k1−ρ2−2b11b12ρ11−ρ1k−1−ρ1k1−ρ1+ρ21−ρ2k−1−ρ2k1−ρ2ψ1−ψ1−ρ1ρ2δ122,Covd1,tl1,d∧1,tl1=Cov∑i=0l1−1d1,t+i,l1k∑j=1kd1,t−j=l1kCov∑i=0l1−1d1,t+i,l1k∑j=1kd1,t−j=l1k∑i=0l1−1∑j=1kCovd1,t+i,d1,t−j=l1k∑i=0l1−1∑j=1kCova1−b11p1,t+i+b12p2,t+i+ε1,t+i,a1−b11p1,t−j+b12p2,t−j+ε1,t−j=l1k∑i=0l1−1∑j=1kCov−b11p1,t+i+b12p2,t+i,−b11p1,t−j+b12p2,t−j=l1k∑i=0l1−1∑j=1kb112Covp1,t+i,p1,t−j−b11b12Covp1,t+i,p2,t−j−b12b11Covp2,t+i,p1,t−j+b122Covp2,t+i,p2,t−j=l1kb112ρ11−ρ1l11−ρ1kδ1,p21−ρ12−b11b12δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l11−ρ1k1−ρ12−b12b11δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l21−ρ2k1−ρ22+b122ρ21−ρ2l21−ρ2kδ2,p21−ρ22.Lemma 2.
Using the MA forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(18)ξ^2,tl2=l2δ2,d2+2b212δ2,p2Λ2l2−Ω2−2b21b22δ122ΔΛ2l2−Ω2−2b22b21δ122ΔΛ1l1−Ω1+2b222δ1,p2Λ1l1−Ω1+l2k2kδ2,d2+2b212δ2,p2Λ2k−Θ2+2b222δ1,p2Λ1k−Θ1−2b21b22Λ1k−Θ1+Λ2k−Θ2Δδ122−2l2kb212ρ2Ω2Θ2δ2,p2−b21b22δ122Δρ2Ω2Θ2−b22b21δ122Δρ1Ω1Θ1+b222ρ1δ1,p2Ω1Θ1,where Λ1=ρ1/1−ρ1, Λ2=ρ2/1−ρ2, Ω1=1−ρ1l1/1−ρ1, Ω2=1−ρ2l2/1−ρ2, Θ1=1−ρ1k/1−ρ1, Θ2=1−ρ2k/1−ρ2, and Δ=ψ1−ψ/1−ρ1ρ2.Proof.
Using the MA forecasting technique, the estimate of the standard deviation of thel2 period forecast error of retailer-2 ξ^2,tl2 is a constant and can be expressed as follows:(19)ξ^2,tl2=Vard^2,tl2−d2,tl2=Vard^2,tl2+Vard2,tl2−2Covd^2,tl2,d2,tl2,where(20)Vard2,tl2=Var∑i=0l2−1d2,t+i=∑i=0l2−1Vard2,t+i+2∑i=0l2−2∑j=1l2−1−iCovd2,t+i,d2,t+i+j=∑i=0l2−1Vard2,t+i+2∑i=0l2−2∑j=1l2−1−iCovd2,t+i,d2,t+i+j=∑i=0l2−1Vard2,t+i+2∑i=0l2−2∑j=1l2−1−iCovd2,t+i,d2,t+i+j=l2δ2,d2+2∑i=0l2−2∑j=1l2−1−iCova2−b21p2,t+i+b22p1,t+i+ε2,t+i,a2−b21p2,t+i+j+b22p1,t+i+j+ε2,t+i+j=l2δ2,d2+2b212δ2,p2ρ21−ρ2l2−1−ρ2l21−ρ2−2b21b22δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l2−1−ρ2l21−ρ2−2b22b21δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+2b222δ1,p2ρ11−ρ1l1−1−ρ1l11−ρ1,Vard^2,tl2=l2k2Var∑i=1kd2,t−i=l2k2Vard2,t−1+d2,t−2+⋯+d2,t−k=l2k2kδ2,d2+2Covd2,t−1,d2,t−2+2Covd2,t−1,d2,t−3+⋯+2Covd2,t−k−1,d2,t−k=l2k2kδ2,d2+2b212δ2,p2ρ21−ρ2k−1−ρ2k1−ρ2+2b222δ1,p2ρ11−ρ1k−1−ρ1k1−ρ1−2b21b22ρ11−ρ1k−1−ρ1k1−ρ1+ρ21−ρ2k−1−ρ2k1−ρ2ψ1−ψ1−ρ1ρ2δ122,Covd2,tl2,d^2,tl2=Cov∑i=0l2−1d2,t+i,l2k∑j=1kd2,t−j=l2kCov∑i=0l2−1d2,t+i,l2k∑j=1kd2,t−j=l2k∑i=0l2−1∑j=1kCovd2,t+i,d2,t−j=l2k∑i=0l2−1∑j=1kCova2−b21p2,t+i+b22p1,t+i+ε2,t+i,a2−b21p2,t−j+b22p1,t−j+ε2,t−j=l2k∑i=0l2−1∑j=1kCov−b21p2,t+i+b22p1,t+i,−b21p2,t−j+b22p1,t−j=l2k∑i=0l2−1∑j=1kb212Covp2,t+i,p2,t−j−b21b22Covp2,t+i,p1,t−j−b22b21Covp1,t+i,p2,t−j+b222Covp1,t+i,p1,t−j=l2kb212ρ21−ρ2l21−ρ2kδ2,p21−ρ22−b21b22δ122ψ1−ψ1−ρ1ρ2ρ21−ρ2l21−ρ2k1−ρ22−b22b21δ122ψ1−ψ1−ρ1ρ2ρ11−ρ1l11−ρ1k1−ρ12+b222ρ11−ρ1l11−ρ1kδ1,p21−ρ12.Theorem 2.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy as well as the MA forecasting technique, the quantitative model of the inventory bullwhip effect in two supply chains is expressed as follows:(21)IBEretailer−1MA=VarI1,tVard1,t=l1δ1,d2+2b112δ1,p2Λ1l1−Ω1−2b11b12δ122ΔΛ1l1−Ω1−2b12b11δ122ΔΛ2l2−Ω2+2b122δ2,p2Λ2l2−Ω2+l1/k2kδ1,d2+2b112δ1,p2Λ1k−Θ1+2b122δ2,p2Λ2k−Θ2−2b11b12Λ1k−Θ1+Λ2k−Θ2Δδ122−2l1/kb11b21ρ1Ω1Θ1δ1,p2−b11b22ρ1Ω1Θ1δ122Δ−b12b21ρ2Ω2Θ2Δδ122+b12b22ρ2Ω2Θ2δ2,p2δ1,d2,IBEretailer−2MA=VarI2,tVard2,t=l2δ2,d2+2b212δ2,p2Λ2l2−Ω2−2b21b22δ122ΔΛ2l2−Ω2−2b22b21δ122ΔΛ1l1−Ω1+2b222δ1,p2Λ1l1−Ω1+l2/k2kδ2,d2+2b212δ2,p2Λ2k−Θ2+2b222δ1,p2Λ1k−Θ1−2b21b22Λ1k−Θ1+Λ2k−Θ2Δδ122−2l2/kb212ρ2Ω2Θ2δ2,p2−b21b22δ122Δρ2Ω2Θ2−b22b21δ122Δρ1Ω1Θ1+b222ρ1δ1,p2Ω1Θ1δ2,d2,where Λ1=ρ1/1−ρ1, Λ2=ρ2/1−ρ2, Ω1=1−ρ1l1/1−ρ1, Ω2=1−ρ2l2/1−ρ2, Θ1=1−ρ1k/1−ρ1, Θ2=1−ρ2k/1−ρ2, and Δ=ψ1−ψ/1−ρ1ρ2.


## 5.2. The Inventory Bullwhip Effect Using the ES Method

In equation (6), we can obtain the mean lead-time demands d^1,tl1 and d^2,tl2 using the exponential smoothing forecasting technique, where α (0<α<1) is a smoothing constant for the two retailers:(22)d^1,tl1=l1αd1,t−1+1−αd^1,t−1,d^2,tl2=l2αd2,t−1+1−αd^2,t−1.Then,q1,t and q2,t in equation (6) can be written as the following equation:(23)q1,t=1+αl1d1,t−1−αl1d^1,t−1+z1ξ^1,tl1−ξ^1,t−1l1,q2,t=1+αl2d2,t−1−αl2d^2,t−1+z2ξ^2,tl2−ξ^2,t−1l2.Theorem 3.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy as well as the ES forecasting technique, the quantitative model of the bullwhip effect in two supply chains is expressed as follows:(24)BEretailer−1ES=Varq1,tVard1,t=1+αl12δ1,d2+α2l12α/2−αδ1,d2+α2l1221−α/2−α−2αl11+αl1ψαb112ρ1δ1,p2/1−ρ121−1−αρ1+1−ψαb122ρ2δ2,p2/1−ρ221−1−αρ2δ1,d2,BEretailer−2ES=Varq2,tVard2,t=1+αl22δ2,d2+α2l22α/2−αδ2,d2+α2l2221−α/2−α−2αl21+αl2ψαb212ρ2δ2,p2/1−ρ221−1−αρ2+1−ψαb222ρ1δ1,p2/1−ρ121−1−αρ1δ2,d2.Similar to Section5.1, in order to analyze the impact of the bullwhip effect on the inventory level, we should calculate the variance of the inventory level for the two retailers at period t as I1,t and I2,t corresponding to the bullwhip effect. I1,t and I2,t can be expressed as follows:(25)I1,t=I1,t−1+q1,t−l1−d1,t−1,I2,t=I2,t−1+q2,t−l2−d2,t−1.From equation (25), we can determine that(26)q1,t−l1=I1,t−I1,t−1+d1,t−1,q2,t−l2=I2,t−I2,t−1+d2,t−1.Vassian [48] proposed that q1,t−l1 and q2,t−l2 can be expressed as follows:(27)q1,t−l1=d^1,tl1−∑i=1l1−1q1,t−i−I1,t,q2,t−l2=d^2,tl2−∑i=1l2−1q2,t−i−I2,t.We substituted equation (27) into equation (26), and we can obtain equation (28) as follows:(28)I1,t=ξ^1,tl1=d^1,t−l1l1−d1,t−l1l1,I2,t=ξ^2,tl2=d^2,t−l2l2−d2,t−l2l2.Lemma 3.
Using the ES forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(29)ξ^1,tl1=l1+l12α2−αδ1,d2+2ψb112ρ1l1−1−ρ1l11−ρ12+l121−αα2−α1−1−αρ1−l1α1−ρ1l11−ρ11−1−αρ1δ1,p2+21−ψb122ρ2l2−1−ρ2l21−ρ22+l121−αα2−α1−1−αρ2−l1α1−ρ2l11−ρ21−1−αρ2δ2,p2.Proof.
Using the ES forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(30)ξ^1,tl1=Vard1,tl1+Vard^1,tl1−2Covd1,tl1,d^1,tl1,where(31)Vard1,tl1=Varl1a1−b11p1,t+b12p2,t+ε1,t=l1δ1,d2+2ψb112ρ1l1−1−ρ1l1δ1,p21−ρ12+21−ψb122ρ2l2−1−ρ2l2δ2,p21−ρ22,Vard^1,tl1=l12Vard^1,t=l12α2−αVard1,t+21−α2−αCovd1,t−1,d^1,t=l12α2−αδ1,d2+2ψ1−ααb112ρ12−α1−1−αρ1δ1,p2+21−ψ1−ααb122ρ22−α1−1−αρ2δ2,p2,Covd1,tl1,d^1,tl1=Cov∑i=0l1−1d1,t+i,l1d^1,t=l1∑i=0l1−1Covd1,t+i,d^1,t=l1ψαb112∑i=0l1−1∑j=1∞1−αj−1ρ1i+jδ1,p2+1−ψl1αb122∑i=0l1−1∑j=1∞1−αj−1ρ2i+jδ2,p2=ψl1αb112ρ11−ρ1l11−ρ11−1−αρ1δ1,p2+1−ψl1αb122ρ21−ρ2l11−ρ21−1−αρ2δ2,p2.Lemma 4.
Using the ES forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(32)ξ^2,tl2=l2+l22α2−αδ2,d2+2ψb212ρ2l2−1−ρ2l21−ρ22+l221−αα2−α1−1−αρ2−l2α1−ρ2l21−ρ21−1−αρ2δ2,p2+21−ψb222ρ1l2−1−ρ1l21−ρ12+l221−αα2−α1−1−αρ1−l2α1−ρ1l11−ρ11−1−αρ1δ1,p2.Proof.
Using the ES forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(33)ξ^2,tl2=Vard2,tl2+Vard^2,tl2−2Covd2,tl2,d^2,tl2,where(34)Vard2,tl2=Varl2a2−b21p2,t+b22p1,t+ε2,t=l2δ2,d2+21−ψb212ρ2l2−1−ρ2l2δ2,p21−ρ22+2ψb222ρ1l1−1−ρ1l1δ1,p21−ρ12,Vard^2,tl2=l22Vard^2,t=l22α2−αVard2,t+21−α2−αCovd2,t−1,d^2,t=l22α2−αδ2,d2+21−ψ1−ααb212ρ22−α1−1−αρ2δ2,p2+2ψ1−ααb222ρ12−α1−1−αρ1δ1,p2,Covd2,tl2,d^2,tl2=Cov∑i=0l2−1d2,t+i,l2d^2,t=l2∑i=0l2−1Covd2,t+i,d^2,t=l21−ψαb212∑i=0l2−1∑j=1∞1−αj−1ρ2i+jδ2,p2+ψl2αb222∑i=0l2−1∑j=1∞1−αj−1ρ1i+jδ1,p2=1−ψl2αb212ρ21−ρ2l21−ρ21−1−αρ2δ2,p2+ψl2αb222ρ11−ρ1l21−ρ11−1−αρ1δ1,p2.Theorem 4.
For two parallel supply chains in which interactions exist with respect to customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the ES forecasting method, the quantitative model of the inventory bullwhip effect in two supply chains is expressed as follows:(35)IBEretailer−1ES=VarI1,tVard1,t=l1+l12α/2−αδ1,d2+2ψb112ρ1l1−1−ρ1l1/1−ρ12+l121−αα/2−α1−1−αρ1−l1α1−ρ1l1/1−ρ11−1−αρ1δ1,p2+21−ψb122ρ2l2−1−ρ2l2/1−ρ22+l121−αα/2−α1−1−αρ2−l1α1−ρ2l1/1−ρ21−1−αρ2δ2,p2δ1,d2,IBEretailer−2ES=VarI2,tVard2,t=l2+l22α/2−αδ2,d2+2ψb212ρ2l2−1−ρ2l1/1−ρ22+l221−αα/2−α1−1−αρ2−l2α1−ρ2l2/1−ρ21−1−αρ2δ2,p2+21−ψb222ρ1l2−1−ρ1l2/1−ρ12+l221−αα/2−α1−1−αρ1−l2α1−ρ1l1/1−ρ11−1−αρ1δ1,p2δ2,d2.


## 5.3. The Inventory Bullwhip Effect Using the MMSE Method

Box and Jenkins [49] have pointed out that the demand forecast value d^t+i is the conditional expectation in historical demand information for the period t+ii=0,1,2,…. d^t+i=Edt+idt−1,dt−2,…, and particularly for the AR(1) process, d^t+i=Edt+idt−1. In this paper, the price follows the AR(1) process, such that p^t+i=Ept+ipt−1. Thus, the demand forecast values for R1 and R2 can be expressed as d^1,t+i=a1−b11p^1,t+i+b12p^2,t+i+ε1,t+i and d^2,t+i=a2−b21p^2,t+i+b22p^1,t+i+ε2,t+i in the period t+i. Thus, the price forecast values for the period t+i can be expressed as follows:(36)p^1,t+i=Ep1,t+ip1,t−1=1−ρ1i+1ρ1μ1ψ+ρ1i+1p1,t−1,p^2,t+i=Ep2,t+ip2,t−1=1−ρ2i+1ρ2μ21−ψ+ρ2i+1p2,t−1.Thus, the demand forecast values for the periodt+i can be expressed as follows:(37)d^1,t+i=a1−b11p^1,t+i+b12p^2,t+i=a1−b111−ρ1i+1μ1ψ1−ρ1−b121−ρ2i+1μ21−ψ1−ρ2−b11ρ1i+1p1,t−1−b12ρ2i+1p2,t−1,d^2,t+i=a2−b21p^2,t+i+b22p^1,t+i=a2−b211−ρ2i+1μ21−ψ1−ρ2−b221−ρ1i+1μ1ψ1−ρ1−b21ρ2i+1p2,t−1−b22ρ1,i+1p1,t−1.Then,q1,t and q2,t in equation (6) can be formulated as equation (37):(38)q1,t=−b11ρ1ρ1ψ1−ρ1l1p1,t−1−p1,t−21−ρ1+b12ρ2ρ21−ρ2l1p2,t−1−p2,t−21−ρ2+d1,t−1,q2,t=−b21ρ2ρ21−ψ1−ρ2l2p2,t−1−p2,t−21−ρ2+b22ρ1ρ11−ρ1l2p1,t−1−p1,t−21−ρ1+d2,t−1.Theorem 5.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the MMSE forecasting technique, the quantitative model of the bullwhip effect in two supply chains is expressed as follows:(39)BEretailer-1MMSE=Varq1,tVard1,t=1+2b112ψρ131−ρ1l11−ρ1l1+1/1+ρ11−ρ12δ12+2b122ρ231−ρ2l11−ρ2l1+1/1+ρ21−ρ22δ22−2b11b12ρ11−ρ2/1−ρ1ρ2+ρ1ρ21−ρ1/1−ρ1ρ2ρ21−ρ2l1/1−ρ22ρ11−ρ1l1/1−ρ12+ρ21−ρ1/1−ρ1ρ2+ρ2ρ11−ρ2/1−ρ1ρ2ρ11−ρ1l1/1−ρ12ρ21−ρ2l1/1−ρ22δ122δ1,d2,BEretailer-2MMSE=Varq2,tVard2,t=1+2b2121−ψρ231−ρ2l21−ρ2l2+1/1+ρ21−ρ22δ22+2b222ρ131−ρ1l21−ρ1l2+1/1+ρ11−ρ12δ12−2b21b22ρ21−ρ1/1−ρ1ρ2+ρ2ρ11−ρ2/1−ρ1ρ2ρ11−ρ1l2/1−ρ12ρ21−ρ2l2/1−ρ22+ρ11−ρ2/1−ρ1ρ2+ρ1ρ21−ρ1/1−ρ1ρ2ρ21−ρ2l2/1−ρ22ρ11−ρ1l2/1−ρ12δ122δ2,d2.Similar to Section5.1, in order to analyze the impact of the bullwhip effect on the inventory level, we calculated the variance of the inventory level for the two retailers at the period t as I1,t and I2,t corresponding to the bullwhip effect. I1,t and I2,t can be expressed as follows:(40)I1,t=I1,t−1+q1,t−l1−d1,t−1,I2,t=I2,t−1+q2,t−l2−d2,t−1.From equation (40), we can determine(41)q1,t−l1=I1,t−I1,t−1+d1,t−1,q2,t−l2=I2,t−I2,t−1+d2,t−1.Vassian [48] proposed that q1,t−l1 and q2,t−l2 can be expressed as follows:(42)q1,t−l1=d^1,tl1−∑i=1l1−1q1,t−i−I1,t,q2,t−l2=d^2,tl2−∑i=1l2−1q2,t−i−I2,t.We substituted equation (42) into equation (41), and we can obtain equation (43) as follows:(43)I1,t=ξ^1,tl1=d^1,t−l1l1−d1,t−l1l1,I2,t=ξ^2,tl2=d^2,t−l2l2−d2,t−l2l2.Lemma 5.
Using the MMSE forecasting technique, the estimate of the standard deviation of thel1 period forecast error of R1 ξ^1,tl1 is a constant and can be expressed as follows:(44)ξ^1,tl1=l1σ12+l2σ22+b112ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12+b1221−ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22.Proof.
Proof(45)ξ^1,tl1=Vard^1,tl1−d1,tl1=Var∑i=0l1−1d1,t−d^1,t=Var∑i=0l1−1−b11ψ∑j=0iρ1i−jη1,t+j−b121−ψ∑j=0tρ2i−jη2,t+j+ε1,t+i+ε2,t+i=Var∑i=0l1−1ε1,t+i+Var∑i=0l2−1ε2,t+i+b112ψ2Var∑i=0l1−1∑j=0iρ1i−jη1,t+j+b1221−ψ2Var∑i=0l2−1∑j=0iρ2i−jη2,t+j=l1σ12+l2σ22+b112ψ2Var∑i=0l1−1η1,t+j∑j=0l1−1−iρ1j+b1221−ψ2Var∑i=0l2−1η2,t+j∑j=0l2−1−iρ2j=l1σ12+l2σ22+b112ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12+b1221−ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22.Lemma 6.
Using the MMSE forecasting technique, the estimate of the standard deviation of thel2 period forecast error of R2 ξ^2,tl2 is a constant and can be expressed as follows:(46)ξ^2,tl2=l2σ22+l1σ12+b212ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22+b2221−ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12.Proof.
Proof(47)ξ^2,tl2=Vard^2,tl2−d2,tl2=Var∑i=0l2−1d2,t−d^2,t=Var∑i=0l2−1d2,t−d^2,t=Var∑i=0l2−1−b21ψ∑j=0iρ2i−jη2,t+j−b221−ψ∑j=0tρ1i−jη1,t+j+ε1,t+i+ε2,t+i=Var∑i=0l1−1ε1,t+i+Var∑i=0l2−1ε2,t+i+b212ψ2Var∑i=0l2−1∑j=0iρ2i−jη2,t+j+b2221−ψ2Var∑i=0l1−1∑j=0iρ1i−jη1,t+j=l1σ12+l2σ22+b212ψ2Var∑i=0l2−1η2,t+j∑j=0l2−1−iρ2j+b2221−ψ2Var∑i=0l1−1η1,t+j∑j=0l1−1−iρ1j=l1σ12+l2σ22+b212ψ21−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−21−ρ22δ22+b2221−ψ21−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−21−ρ12δ12.Theorem 6.
For two parallel supply chains in which there is a competition effect on customer demand and the retailers’ market share, when two retailers use the order-up-to inventory strategy and the MMSE forecasting technique, the quantitative model of the inventory bullwhip effect in two supply chains is expressed as follows:(48)IBEretailer−1MMSE=VarI1,tVard1,t=l1σ12+l2σ22+b112ψ2/1−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−2/1−ρ12δ12+b1221−ψ2/1−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−2/1−ρ22δ22δ1,d2,IBEretailer−2MMSE=VarI2,tVard2,t=l2σ22+l1σ12+b212ψ2/1−ρ22l2+ρ21−ρ2l2ρ2l2+1−ρ2−2/1−ρ22δ22+b2221−ψ2/1−ρ12l1+ρ11−ρ1l1ρ1l1+1−ρ1−2/1−ρ12δ12δ2,d2.


## 6. Simulation and Result Analysis

### 6.1. Simulation Process

From the above theorems, we obtained the quantitative model of the bullwhip effect and the inventory bullwhip effect in two parallel supply chains. This can be contrasted with the work by Ma et al. [7], which compared the bullwhip effect on product orders and on inventory by employing the three forecasting techniques in a two-level supply chain. The research used the numerical method and included one manufacturer and one retailer. However, in this section, we analyze the impact of different forecasting techniques on the bullwhip effect and on the inventory bullwhip effect in two parallel supply chains using ExtendSim Software 7.1. Thus, the simulation model of the bullwhip effect is constructed on the basis of the MA, ES, and MMSE forecasting techniques in Figures 2–4, respectively. The simulation model of the inventory bullwhip effect is constructed on the basis of the MA, ES, and MMSE forecasting techniques in Figures 5–7, respectively.Figure 2
The simulation model of the bullwhip effect based on the MA technique.Figure 3
The simulation model of the bullwhip effect based on the ES technique.Figure 4
The simulation model of the bullwhip effect based on the MMSE technique.Figure 5
The simulation model of the inventory bullwhip effect based on the MA technique.Figure 6
The simulation model of the inventory bullwhip effect based on the ES technique.Figure 7
The simulation model of the inventory bullwhip effect based on the MMSE technique.We should explain the meaning of Figure2. The selling price of the product of both retailers can be expressed as p1,t=ψμ1+ρ1p1,t−1+ψη1,t and p2,t=1−ψμ2+ρ2p2,t−1+1−ψη2,t, respectively. The demand model which R1 faces can be expressed as d1,t=a1−b11p1,t+b12p2,t+ε1,t. The R1’s order to manufacturer-1 can be expressed as q1,t=1+l1/kd1,t−1−l1/kd1,t−k−1. Finally, the bullwhip effect can be expressed as BEretailer−1MA=Varq1,t/Vard1,t. Figure 3 indicates that the selling price of the product of both retailers can be expressed as p1,t=ψμ1+ρ1p1,t−1+ψη1,t and p2,t=1−ψμ2+ρ2p2,t−1+1−ψη2,t, respectively. The demand model which R1 faces can be expressed as d1,t=a1−b11p1,t+b12p2,t+ε1,t. R1’s order to manufacturer-1 can be expressed as q1,t=1+αl1d1,t−αl1dtg1. The bullwhip effect can be calculated using the following equation: BEretailer−1ES=Varq1,t/Vard1,t. Figure 4 illustrates that the selling price of the product of both retailers can be expressed as p^1,t+i=Ep1,t+ip1,t−1=1−ρ1i+1/ρ1μ1ψ+ρ1i+1p1,t−1 and p^2,t+i=Ep2,t+ip2,t−1=1−ρ2i+1/ρ2μ21−ψ+ρ2i+1p2,t−1, respectively. The demand model which R1 faces can be expressed as d1,t+i=a1−b11p1,t+i+b12p2,t+i. R1’s order to manufacturer-1 is as follows: q1,t=−b11ρ1ρ1ψ1−ρ1l1p1,t−1−p1,t−2/1−ρ1+b12ρ2ρ21−ρ2l1p2,t−1−p2,t−2/1−ρ2+d1,t−1. The bullwhip effect can be calculated as follows: BEretailer−1MMSE=Varq1,t/Vard1,t.The demand process, the order process, and the price process shown in Figure5 are the same as those in Figure 2. The inventory level of R1 can then be determined as follows: I1,t=dt1−d1,t. The inventory bullwhip effect can be calculated as follows: IBEretailer−1MA=VarI1,t/Vard1,t. The demand process, the order process, and the price process in Figure 6 are the same as those in Figure 3. R1’s inventory level can then be calculated as follows: I1,t=dt1−dtg1. The inventory bullwhip effect can be determined using the following equation: IBEretailer−1ES=VarI1,t/Vard1,t. The demand process, the order process, and the price process in Figure 7 are the same as those in Figure 4. R1’s inventory level can then be calculated as follows: I1,t=dt1−d1,t. The inventory bullwhip effect can be formulated using the equation: IBEretailer−1MMSE=VarI1,t/Vard1,t.


### 6.2. Analysis of Results

#### 6.2.1. Comparing the Bullwhip Effect Using Different Forecasting Techniques

Based on the quantitative model and the simulation model of the bullwhip effect using different forecasting techniques, we found that the bullwhip effect for R1, using different forecasting techniques, depends on the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the price autoregressive coefficients, the market share, and the variance. However, the bullwhip effect cannot be impacted by the potential market demand. We analyzed the impact of different forecasting techniques on the bullwhip effect for R1. The relevant parameters can be applied as follows:μ1=1,μ2=1, δ1,p2=δ2,p2=1,andδ122=0.1 (Tables 2 and 3).Table 2
The value range of the relevant factors.
Factorsl1l2kb11b12ψρ1ρ2αLevel 1224550.30.30.30.3Level 244810100.60.60.60.6Level 3661215150.90.90.90.9Level 4224550.30.30.60.3Level 5224550.30.30.60.3Level 6224550.30.30.60.3Level 7224550.30.30.90.6Table 3
The simulation number.
No.l1l2kb11b12ψρ1ρ2α11 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)22 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.3)2 (0.3)2 (0.6)33 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.3)3 (0.3)3 (0.9)41 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.6)1 (0.6)1 (0.3)52 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.6)2 (0.6)2 (0.6)63 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.6)3 (0.6)3 (0.9)71 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.9)1 (0.9)1 (0.3)82 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.9)2 (0.9)2 (0.6)93 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.9)3 (0.9)3 (0.9)101 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)111 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)122 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)132 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)143 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)153 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)164 (2)4 (2)4 (4)4 (5)4 (5)4 (0.3)4 (0.3)4 (0.6)4 (0.3)175 (2)5 (2)5 (4)5 (5)5 (5)5 (0.3)5 (0.3)5 (0.6)5 (0.3)186 (2)6 (2)6 (4)6 (5)6 (5)6 (0.3)6 (0.3)6 (0.6)6 (0.3)197 (2)7 (2)7 (4)7 (5)7 (5)7 (0.3)7 (0.3)7 (0.6)7 (0.6)The following conclusion can be drawn from Table4.(1)
From No. 1–3, when the price autoregressive coefficients of the two retailers are very low and have the same value, and regardless of how much longer the lead time is, or how much larger the smoothing coefficient is, or how bigger the market share is, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.(2)
From No. 4–9, when R1’s self-price sensitivity coefficient and R1’s cross-price sensitivity coefficient are at the low level and have the same value, other variables are equal to any values, and R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.(3)
From No. 10–15, when R1’s self-price sensitivity coefficient, cross-price sensitivity coefficient, lead time, market share, smoothing coefficient, and price autoregressive coefficient are from low to high, R1’s bullwhip effect is at the highest level using the MA forecasting technique.(4)
From No. 16–19, when R1’s self-price sensitivity coefficient, cross-price sensitivity coefficient, lead time, market share, smoothing coefficient, and price autoregressive coefficient are all at the low level, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.Table 4
The simulation result.
No.BEretailer−1MABEretailer−1ESBEretailer−1MMSE13.45.12.123.415.22.333.435.252.643.515.412.753.535.632.863.565.812.973.85.93.083.96.023.0594.06.053.11014.16.13.161114.156.213.221215.76.33.271316.16.43.371417.26.53.41517.56.63.6163.686.93.41173.727.013.43183.87.133.44193.97.33.45We converted the data into figures in order to analyze the impact of the relevant factors on R1’s bullwhip effect. Moreover, we can obtain some important managerial insights from figures. Figure8 shows the impact of b11 and b12 on the bullwhip effect using different forecasting techniques. Figure 9 depicts the impact of ρ1andρ2 on the bullwhip effect. Figure 10 depicts the impact of ψ on the bullwhip effect.Figure 8
The impact ofb11 and b12 on the bullwhip effect using different forecasting techniques.Figure 9
The impact ofρ1andρ2 on the bullwhip effect using different forecasting techniques.Figure 10
The impact ofψ on the bullwhip effect using different forecasting techniques.By comparing the bullwhip effect using different forecasting techniques, as shown in Figures8–10, we can conclude the following: (1) as the MMSE forecasting techniques can reduce the lead-time demand forecast error to the largest extent, the bullwhip effect can reach the lowest level using the MMSE techniques; (2) R1 can reduce the bullwhip effect using the MA techniques when the self-price sensitivity coefficient b11 is lower than 5.5, the price autoregressive coefficient ρ1 is lower than 0.45, and the market share ψ is lower than 0.4. On the contrary, R1 can reduce the bullwhip effect using the ES techniques. Moreover, ΔBEretailer−1=BEretailer−1MA−BEretailer−1ES, when σ12=δ12,σ22=δ22, and we can determine the following:(49)ΔBEretailer=b11b121−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+b112ρ1l1−1−ρ1l11−ρ12+ρ1l121−ρ1+l21−ρ2l21−ρ2<12ψl1k2+l12α2−α.Proposition 1.
When the self-price sensitivity coefficientb11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy the following condition, R1 can control the bullwhip effect using the MA technique. Otherwise, R1 should select the ES technique.(50)b11b121−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+b112ρ1l1−1−ρ1l11−ρ12+ρ1l121−ρ1+l21−ρ2l21−ρ2<12ψl1k2+l12α2−α.It is evident that the potential market demanda1 has no impact on R1’s bullwhip effect. Thus, the potential market demand cannot influence R1’s choice of the forecasting technique. On the other hand, when the self-price sensitivity coefficient b11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy Proposition 1, R1 can control the bullwhip effect using the MA technique. Otherwise, R1 should select the ES technique.


#### 6.2.2. Comparing the Inventory Bullwhip Effect Using Different Forecasting Techniques

Based on the quantitative model and the simulation model of the inventory bullwhip effect using different forecasting techniques, we found that R1’s inventory bullwhip effect using different forecasting techniques depends on the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the price autoregressive coefficients, the market share, and the variance. However, it cannot be impacted by the potential market demand. We analyzed the impact of different forecasting techniques on R1’s inventory bullwhip effect. The relevant parameters can be applied as follows:μ1=1,μ2=1, δ1,p2=δ2,p2=1,andδ122=0.1 (Tables 5 and 6).Table 5
The value range of the relevant factors.
Factorsl1l2kb11b12ψρ1ρ2αLevel 1224550.30.30.30.3Level 244810100.60.60.60.6Level 3661215150.90.90.90.9Level 4224550.30.30.60.3Level 5224550.30.30.60.3Level 6224550.30.30.60.3Level 7224550.30.30.90.6Table 6
The simulation number.
No.l1l2kb11b12ψρ1ρ2α11 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)22 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.3)2 (0.3)2 (0.6)33 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.3)3 (0.3)3 (0.9)41 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.6)1 (0.6)1 (0.3)52 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.6)2 (0.6)2 (0.6)63 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.6)3 (0.6)3 (0.9)71 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.9)1 (0.9)1 (0.3)82 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.9)2 (0.9)2 (0.6)93 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.9)3 (0.9)3 (0.9)101 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)111 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)122 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)132 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)143 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)153 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)164 (2)4 (2)4 (4)4 (5)4 (5)4 (0.3)4 (0.3)4 (0.6)4 (0.3)175 (2)5 (2)5 (4)5 (5)5 (5)5 (0.3)5 (0.3)5 (0.6)5 (0.3)186 (2)6 (2)6 (4)6 (5)6 (5)6 (0.3)6 (0.3)6 (0.6)6 (0.3)197 (2)7 (2)7 (4)7 (5)7 (5)7 (0.3)7 (0.3)7 (0.6)7 (0.6)The following conclusion can be drawn from Table7.(1)
In No. 1–3, when the value of the two retailers’ price autoregressive coefficient is very low and at the same level, regardless of the length of the lead time, or how much larger the smoothing coefficient is, or how much larger the market share is, R1’s inventory bullwhip effect is at the lowest level using the MMSE forecasting technique.(2)
In No. 4–9, when R1’s self-price sensitivity coefficient and R1’s cross-price sensitivity coefficient are at the low level and at the same level, regardless of the length of the lead time, or how much larger the smoothing coefficient is, or how much larger the market share is, R1’s inventory bullwhip effect is at the lowest level using the MMSE forecasting technique.(3)
In No. 16–18, when the value of the two retailers’ price autoregressive coefficient is very low and at a different level, and when the smoothing coefficient is very low and the lead time is very low, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique. Moreover, R1’s inventory bullwhip effect is at the highest level using the ES forecasting technique.Table 7
The simulation result.
No.IBEretailer−1MAIBEretailer−1ESIBEretailer−1MMSE189.6130.321.9275.5130.621.7374.3130.821.6474.1131.421.4573.8131.821.1673.3131.920.9773.2132.220.7872.8132.420.5972.6132.620.21072.4132.820.11172.1132.919.91271.9133.319.81371.6133.619.61471.2133.819.41570.8133.918.91670.5134.218.71769.9134.618.51868.6134.818.51921.5135.571.7We converted the data into figures in order to analyze the impact of the relevant factors on R1’s bullwhip effect. Moreover, we can gain some important managerial insights from the figures. Figure11 depicts the impact of b11 and b12 on the inventory bullwhip effect. Figure 12 depicts the impact of ρ1andρ2 on the inventory bullwhip effect. Figure 13 depicts the impact of ψ on the inventory bullwhip effect.Figure 11
The impact ofb11 and b12 on the inventory bullwhip effect using different forecasting techniques.Figure 12
The impact ofρ1andρ2 on the inventory bullwhip effect using different forecasting techniques.Figure 13
The impact ofψ on the inventory bullwhip effect using different forecasting techniques.By comparing the inventory bullwhip effect using different forecasting techniques, as shown in Figures11–13, it is evident that the inventory bullwhip effect can reach the highest level using the ES technique (Chen et al. and Zhang et al. reached the same conclusions). Moreover, R1 can control the impact of the inventory bullwhip effect using the MA technique or the MMSE technique. We can then analyze which forecasting technique R1 should select under different conditions. Figure 11 shows that the inventory bullwhip effect can achieve the lowest level under the MMSE technique when the value of the self-price sensitivity coefficient b11 is lower than 7. Otherwise, the inventory bullwhip effect can reach the lowest level using the MA technique when b11 is higher than 7. Figure 12 shows that the inventory bullwhip effect can reach the lowest level using the MMSE technique when the price autoregressive coefficient ρ1<0.8. Otherwise, the inventory bullwhip effect can reach the lowest level using the MA technique when the price autoregressive coefficient ρ1>0.8. Figure 13 indicates that the inventory bullwhip effect can achieve the lowest level using the MA technique when the value of the market share ψ is lower than 0.6. Otherwise, the inventory bullwhip effect can achieve the lowest level using the MMSE technique when ψ is higher than 0.6. Moreover, ΔIBEretailer−1=IBEretailer−1MMSE−IBEretailer−1MA, when σ12=δ12andσ22=δ22, and we can get(51)ΔIBE=−2b11b12ρ1ρ21−ρ11−ρ1ρ2ρ21−ρ2l11−ρ22+ρ2ρ11−ρ21−ρ1ρ2ρ11−ρ1l11−ρ12+2ρ11−ρ12+2ρ11−ρ1−21+ρ11−ρ1ρ11−ρ1b11b12ρ1k+ρ2kψ1−ψ1−ρ1ρ2.Proposition 2.
When the self-price sensitivity coefficientb11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy the following condition, R1 can control the inventory bullwhip effect using the MMSE technique. Otherwise, R1 should select the MA technique.(52)2ρ11−ρ12+2ρ11−ρ1−21+ρ11−ρ1ρ11−ρ1b11b12ρ1k+ρ2kψ1−ψ1−ρ1ρ2<2b11b12ρ1ρ21−ρ11−ρ1ρ2ρ21−ρ2l11−ρ22+ρ2ρ11−ρ21−ρ1ρ2ρ11−ρ1l11−ρ12.We found that the potential market demanda1 has no impact on R1’s inventory bullwhip effect. Thus, the potential market demand cannot influence R1’s choice of the forecasting technique. On the contrary, when the self-price sensitivity coefficient b11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy Proposition 2, R1 can control the inventory bullwhip effect using the MMSE technique. Otherwise, R1 should select the MA technique.From the above analysis, we can obtain the following managerial insights:Managerial Insight 1. Figure 8 clearly illustrates that there is a positive relationship between b12 and BE. In other words, the larger the value of the cross-price sensitivity coefficient, the greater the competition between the two supply chains. As a result, the fluctuation in demand increases. Thus, in terms of the operational and management activities of the firm, the firm should seek to reduce the competitive relationship and maintain a positive cooperative relationship.Managerial Insight 2. As shown in Figure 10, there is a positive relationship between ψ and BE. The reason for this phenomenon is that the larger the probability that customers will choose both retailers’ products, the more difficult it is for the two retailers to forecast customer demand. Thus, this will lead to greater uncertainty with respect to the retailer’s order.Managerial Insight 3. As Figure 11 clearly shows, there is a positive relationship between b12 and IBE. In other words, the larger the value of the cross-price sensitivity coefficient, the greater the competition between the two supply chains, and the fluctuation in the inventory level increases. Thus, the firms should reduce the competitive relationship and maintain a positive cooperative relationship between different firms.Managerial Insight 4. As shown in Figure 13, there is a positive relationship between ψ and IBE.As shown inManagerial Insight 2, the larger the probability that customers will choose both retailers’ products, the larger the bullwhip effect of the two retailers. As a result, this will lead to more uncertainty in the retailer’s inventory. Thus, the bullwhip effect will increase for both retailers.Managerial Insight 5. As shown in the simulation result, with the increase in relevant variables (i.e., the self-price sensitivity coefficient and the cross-price sensitivity coefficient), the retailer’s bullwhip effect and the inventory bullwhip effect become larger and larger. Moreover, the fluctuation in the inventory level is higher than the fluctuation in the level of demand. This means that the retailer should control the volatility of the demand so that the volatility of the inventory level can be controlled.


## 6.1. Simulation Process

From the above theorems, we obtained the quantitative model of the bullwhip effect and the inventory bullwhip effect in two parallel supply chains. This can be contrasted with the work by Ma et al. [7], which compared the bullwhip effect on product orders and on inventory by employing the three forecasting techniques in a two-level supply chain. The research used the numerical method and included one manufacturer and one retailer. However, in this section, we analyze the impact of different forecasting techniques on the bullwhip effect and on the inventory bullwhip effect in two parallel supply chains using ExtendSim Software 7.1. Thus, the simulation model of the bullwhip effect is constructed on the basis of the MA, ES, and MMSE forecasting techniques in Figures 2–4, respectively. The simulation model of the inventory bullwhip effect is constructed on the basis of the MA, ES, and MMSE forecasting techniques in Figures 5–7, respectively.Figure 2
The simulation model of the bullwhip effect based on the MA technique.Figure 3
The simulation model of the bullwhip effect based on the ES technique.Figure 4
The simulation model of the bullwhip effect based on the MMSE technique.Figure 5
The simulation model of the inventory bullwhip effect based on the MA technique.Figure 6
The simulation model of the inventory bullwhip effect based on the ES technique.Figure 7
The simulation model of the inventory bullwhip effect based on the MMSE technique.We should explain the meaning of Figure2. The selling price of the product of both retailers can be expressed as p1,t=ψμ1+ρ1p1,t−1+ψη1,t and p2,t=1−ψμ2+ρ2p2,t−1+1−ψη2,t, respectively. The demand model which R1 faces can be expressed as d1,t=a1−b11p1,t+b12p2,t+ε1,t. The R1’s order to manufacturer-1 can be expressed as q1,t=1+l1/kd1,t−1−l1/kd1,t−k−1. Finally, the bullwhip effect can be expressed as BEretailer−1MA=Varq1,t/Vard1,t. Figure 3 indicates that the selling price of the product of both retailers can be expressed as p1,t=ψμ1+ρ1p1,t−1+ψη1,t and p2,t=1−ψμ2+ρ2p2,t−1+1−ψη2,t, respectively. The demand model which R1 faces can be expressed as d1,t=a1−b11p1,t+b12p2,t+ε1,t. R1’s order to manufacturer-1 can be expressed as q1,t=1+αl1d1,t−αl1dtg1. The bullwhip effect can be calculated using the following equation: BEretailer−1ES=Varq1,t/Vard1,t. Figure 4 illustrates that the selling price of the product of both retailers can be expressed as p^1,t+i=Ep1,t+ip1,t−1=1−ρ1i+1/ρ1μ1ψ+ρ1i+1p1,t−1 and p^2,t+i=Ep2,t+ip2,t−1=1−ρ2i+1/ρ2μ21−ψ+ρ2i+1p2,t−1, respectively. The demand model which R1 faces can be expressed as d1,t+i=a1−b11p1,t+i+b12p2,t+i. R1’s order to manufacturer-1 is as follows: q1,t=−b11ρ1ρ1ψ1−ρ1l1p1,t−1−p1,t−2/1−ρ1+b12ρ2ρ21−ρ2l1p2,t−1−p2,t−2/1−ρ2+d1,t−1. The bullwhip effect can be calculated as follows: BEretailer−1MMSE=Varq1,t/Vard1,t.The demand process, the order process, and the price process shown in Figure5 are the same as those in Figure 2. The inventory level of R1 can then be determined as follows: I1,t=dt1−d1,t. The inventory bullwhip effect can be calculated as follows: IBEretailer−1MA=VarI1,t/Vard1,t. The demand process, the order process, and the price process in Figure 6 are the same as those in Figure 3. R1’s inventory level can then be calculated as follows: I1,t=dt1−dtg1. The inventory bullwhip effect can be determined using the following equation: IBEretailer−1ES=VarI1,t/Vard1,t. The demand process, the order process, and the price process in Figure 7 are the same as those in Figure 4. R1’s inventory level can then be calculated as follows: I1,t=dt1−d1,t. The inventory bullwhip effect can be formulated using the equation: IBEretailer−1MMSE=VarI1,t/Vard1,t.


## 6.2. Analysis of Results

### 6.2.1. Comparing the Bullwhip Effect Using Different Forecasting Techniques

Based on the quantitative model and the simulation model of the bullwhip effect using different forecasting techniques, we found that the bullwhip effect for R1, using different forecasting techniques, depends on the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the price autoregressive coefficients, the market share, and the variance. However, the bullwhip effect cannot be impacted by the potential market demand. We analyzed the impact of different forecasting techniques on the bullwhip effect for R1. The relevant parameters can be applied as follows:μ1=1,μ2=1, δ1,p2=δ2,p2=1,andδ122=0.1 (Tables 2 and 3).Table 2
The value range of the relevant factors.
Factorsl1l2kb11b12ψρ1ρ2αLevel 1224550.30.30.30.3Level 244810100.60.60.60.6Level 3661215150.90.90.90.9Level 4224550.30.30.60.3Level 5224550.30.30.60.3Level 6224550.30.30.60.3Level 7224550.30.30.90.6Table 3
The simulation number.
No.l1l2kb11b12ψρ1ρ2α11 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)22 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.3)2 (0.3)2 (0.6)33 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.3)3 (0.3)3 (0.9)41 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.6)1 (0.6)1 (0.3)52 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.6)2 (0.6)2 (0.6)63 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.6)3 (0.6)3 (0.9)71 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.9)1 (0.9)1 (0.3)82 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.9)2 (0.9)2 (0.6)93 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.9)3 (0.9)3 (0.9)101 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)111 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)122 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)132 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)143 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)153 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)164 (2)4 (2)4 (4)4 (5)4 (5)4 (0.3)4 (0.3)4 (0.6)4 (0.3)175 (2)5 (2)5 (4)5 (5)5 (5)5 (0.3)5 (0.3)5 (0.6)5 (0.3)186 (2)6 (2)6 (4)6 (5)6 (5)6 (0.3)6 (0.3)6 (0.6)6 (0.3)197 (2)7 (2)7 (4)7 (5)7 (5)7 (0.3)7 (0.3)7 (0.6)7 (0.6)The following conclusion can be drawn from Table4.(1)
From No. 1–3, when the price autoregressive coefficients of the two retailers are very low and have the same value, and regardless of how much longer the lead time is, or how much larger the smoothing coefficient is, or how bigger the market share is, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.(2)
From No. 4–9, when R1’s self-price sensitivity coefficient and R1’s cross-price sensitivity coefficient are at the low level and have the same value, other variables are equal to any values, and R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.(3)
From No. 10–15, when R1’s self-price sensitivity coefficient, cross-price sensitivity coefficient, lead time, market share, smoothing coefficient, and price autoregressive coefficient are from low to high, R1’s bullwhip effect is at the highest level using the MA forecasting technique.(4)
From No. 16–19, when R1’s self-price sensitivity coefficient, cross-price sensitivity coefficient, lead time, market share, smoothing coefficient, and price autoregressive coefficient are all at the low level, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.Table 4
The simulation result.
No.BEretailer−1MABEretailer−1ESBEretailer−1MMSE13.45.12.123.415.22.333.435.252.643.515.412.753.535.632.863.565.812.973.85.93.083.96.023.0594.06.053.11014.16.13.161114.156.213.221215.76.33.271316.16.43.371417.26.53.41517.56.63.6163.686.93.41173.727.013.43183.87.133.44193.97.33.45We converted the data into figures in order to analyze the impact of the relevant factors on R1’s bullwhip effect. Moreover, we can obtain some important managerial insights from figures. Figure8 shows the impact of b11 and b12 on the bullwhip effect using different forecasting techniques. Figure 9 depicts the impact of ρ1andρ2 on the bullwhip effect. Figure 10 depicts the impact of ψ on the bullwhip effect.Figure 8
The impact ofb11 and b12 on the bullwhip effect using different forecasting techniques.Figure 9
The impact ofρ1andρ2 on the bullwhip effect using different forecasting techniques.Figure 10
The impact ofψ on the bullwhip effect using different forecasting techniques.By comparing the bullwhip effect using different forecasting techniques, as shown in Figures8–10, we can conclude the following: (1) as the MMSE forecasting techniques can reduce the lead-time demand forecast error to the largest extent, the bullwhip effect can reach the lowest level using the MMSE techniques; (2) R1 can reduce the bullwhip effect using the MA techniques when the self-price sensitivity coefficient b11 is lower than 5.5, the price autoregressive coefficient ρ1 is lower than 0.45, and the market share ψ is lower than 0.4. On the contrary, R1 can reduce the bullwhip effect using the ES techniques. Moreover, ΔBEretailer−1=BEretailer−1MA−BEretailer−1ES, when σ12=δ12,σ22=δ22, and we can determine the following:(49)ΔBEretailer=b11b121−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+b112ρ1l1−1−ρ1l11−ρ12+ρ1l121−ρ1+l21−ρ2l21−ρ2<12ψl1k2+l12α2−α.Proposition 1.
When the self-price sensitivity coefficientb11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy the following condition, R1 can control the bullwhip effect using the MA technique. Otherwise, R1 should select the ES technique.(50)b11b121−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+b112ρ1l1−1−ρ1l11−ρ12+ρ1l121−ρ1+l21−ρ2l21−ρ2<12ψl1k2+l12α2−α.It is evident that the potential market demanda1 has no impact on R1’s bullwhip effect. Thus, the potential market demand cannot influence R1’s choice of the forecasting technique. On the other hand, when the self-price sensitivity coefficient b11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy Proposition 1, R1 can control the bullwhip effect using the MA technique. Otherwise, R1 should select the ES technique.


### 6.2.2. Comparing the Inventory Bullwhip Effect Using Different Forecasting Techniques

Based on the quantitative model and the simulation model of the inventory bullwhip effect using different forecasting techniques, we found that R1’s inventory bullwhip effect using different forecasting techniques depends on the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the price autoregressive coefficients, the market share, and the variance. However, it cannot be impacted by the potential market demand. We analyzed the impact of different forecasting techniques on R1’s inventory bullwhip effect. The relevant parameters can be applied as follows:μ1=1,μ2=1, δ1,p2=δ2,p2=1,andδ122=0.1 (Tables 5 and 6).Table 5
The value range of the relevant factors.
Factorsl1l2kb11b12ψρ1ρ2αLevel 1224550.30.30.30.3Level 244810100.60.60.60.6Level 3661215150.90.90.90.9Level 4224550.30.30.60.3Level 5224550.30.30.60.3Level 6224550.30.30.60.3Level 7224550.30.30.90.6Table 6
The simulation number.
No.l1l2kb11b12ψρ1ρ2α11 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)22 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.3)2 (0.3)2 (0.6)33 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.3)3 (0.3)3 (0.9)41 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.6)1 (0.6)1 (0.3)52 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.6)2 (0.6)2 (0.6)63 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.6)3 (0.6)3 (0.9)71 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.9)1 (0.9)1 (0.3)82 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.9)2 (0.9)2 (0.6)93 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.9)3 (0.9)3 (0.9)101 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)111 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)122 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)132 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)143 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)153 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)164 (2)4 (2)4 (4)4 (5)4 (5)4 (0.3)4 (0.3)4 (0.6)4 (0.3)175 (2)5 (2)5 (4)5 (5)5 (5)5 (0.3)5 (0.3)5 (0.6)5 (0.3)186 (2)6 (2)6 (4)6 (5)6 (5)6 (0.3)6 (0.3)6 (0.6)6 (0.3)197 (2)7 (2)7 (4)7 (5)7 (5)7 (0.3)7 (0.3)7 (0.6)7 (0.6)The following conclusion can be drawn from Table7.(1)
In No. 1–3, when the value of the two retailers’ price autoregressive coefficient is very low and at the same level, regardless of the length of the lead time, or how much larger the smoothing coefficient is, or how much larger the market share is, R1’s inventory bullwhip effect is at the lowest level using the MMSE forecasting technique.(2)
In No. 4–9, when R1’s self-price sensitivity coefficient and R1’s cross-price sensitivity coefficient are at the low level and at the same level, regardless of the length of the lead time, or how much larger the smoothing coefficient is, or how much larger the market share is, R1’s inventory bullwhip effect is at the lowest level using the MMSE forecasting technique.(3)
In No. 16–18, when the value of the two retailers’ price autoregressive coefficient is very low and at a different level, and when the smoothing coefficient is very low and the lead time is very low, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique. Moreover, R1’s inventory bullwhip effect is at the highest level using the ES forecasting technique.Table 7
The simulation result.
No.IBEretailer−1MAIBEretailer−1ESIBEretailer−1MMSE189.6130.321.9275.5130.621.7374.3130.821.6474.1131.421.4573.8131.821.1673.3131.920.9773.2132.220.7872.8132.420.5972.6132.620.21072.4132.820.11172.1132.919.91271.9133.319.81371.6133.619.61471.2133.819.41570.8133.918.91670.5134.218.71769.9134.618.51868.6134.818.51921.5135.571.7We converted the data into figures in order to analyze the impact of the relevant factors on R1’s bullwhip effect. Moreover, we can gain some important managerial insights from the figures. Figure11 depicts the impact of b11 and b12 on the inventory bullwhip effect. Figure 12 depicts the impact of ρ1andρ2 on the inventory bullwhip effect. Figure 13 depicts the impact of ψ on the inventory bullwhip effect.Figure 11
The impact ofb11 and b12 on the inventory bullwhip effect using different forecasting techniques.Figure 12
The impact ofρ1andρ2 on the inventory bullwhip effect using different forecasting techniques.Figure 13
The impact ofψ on the inventory bullwhip effect using different forecasting techniques.By comparing the inventory bullwhip effect using different forecasting techniques, as shown in Figures11–13, it is evident that the inventory bullwhip effect can reach the highest level using the ES technique (Chen et al. and Zhang et al. reached the same conclusions). Moreover, R1 can control the impact of the inventory bullwhip effect using the MA technique or the MMSE technique. We can then analyze which forecasting technique R1 should select under different conditions. Figure 11 shows that the inventory bullwhip effect can achieve the lowest level under the MMSE technique when the value of the self-price sensitivity coefficient b11 is lower than 7. Otherwise, the inventory bullwhip effect can reach the lowest level using the MA technique when b11 is higher than 7. Figure 12 shows that the inventory bullwhip effect can reach the lowest level using the MMSE technique when the price autoregressive coefficient ρ1<0.8. Otherwise, the inventory bullwhip effect can reach the lowest level using the MA technique when the price autoregressive coefficient ρ1>0.8. Figure 13 indicates that the inventory bullwhip effect can achieve the lowest level using the MA technique when the value of the market share ψ is lower than 0.6. Otherwise, the inventory bullwhip effect can achieve the lowest level using the MMSE technique when ψ is higher than 0.6. Moreover, ΔIBEretailer−1=IBEretailer−1MMSE−IBEretailer−1MA, when σ12=δ12andσ22=δ22, and we can get(51)ΔIBE=−2b11b12ρ1ρ21−ρ11−ρ1ρ2ρ21−ρ2l11−ρ22+ρ2ρ11−ρ21−ρ1ρ2ρ11−ρ1l11−ρ12+2ρ11−ρ12+2ρ11−ρ1−21+ρ11−ρ1ρ11−ρ1b11b12ρ1k+ρ2kψ1−ψ1−ρ1ρ2.Proposition 2.
When the self-price sensitivity coefficientb11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy the following condition, R1 can control the inventory bullwhip effect using the MMSE technique. Otherwise, R1 should select the MA technique.(52)2ρ11−ρ12+2ρ11−ρ1−21+ρ11−ρ1ρ11−ρ1b11b12ρ1k+ρ2kψ1−ψ1−ρ1ρ2<2b11b12ρ1ρ21−ρ11−ρ1ρ2ρ21−ρ2l11−ρ22+ρ2ρ11−ρ21−ρ1ρ2ρ11−ρ1l11−ρ12.We found that the potential market demanda1 has no impact on R1’s inventory bullwhip effect. Thus, the potential market demand cannot influence R1’s choice of the forecasting technique. On the contrary, when the self-price sensitivity coefficient b11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy Proposition 2, R1 can control the inventory bullwhip effect using the MMSE technique. Otherwise, R1 should select the MA technique.From the above analysis, we can obtain the following managerial insights:Managerial Insight 1. Figure 8 clearly illustrates that there is a positive relationship between b12 and BE. In other words, the larger the value of the cross-price sensitivity coefficient, the greater the competition between the two supply chains. As a result, the fluctuation in demand increases. Thus, in terms of the operational and management activities of the firm, the firm should seek to reduce the competitive relationship and maintain a positive cooperative relationship.Managerial Insight 2. As shown in Figure 10, there is a positive relationship between ψ and BE. The reason for this phenomenon is that the larger the probability that customers will choose both retailers’ products, the more difficult it is for the two retailers to forecast customer demand. Thus, this will lead to greater uncertainty with respect to the retailer’s order.Managerial Insight 3. As Figure 11 clearly shows, there is a positive relationship between b12 and IBE. In other words, the larger the value of the cross-price sensitivity coefficient, the greater the competition between the two supply chains, and the fluctuation in the inventory level increases. Thus, the firms should reduce the competitive relationship and maintain a positive cooperative relationship between different firms.Managerial Insight 4. As shown in Figure 13, there is a positive relationship between ψ and IBE.As shown inManagerial Insight 2, the larger the probability that customers will choose both retailers’ products, the larger the bullwhip effect of the two retailers. As a result, this will lead to more uncertainty in the retailer’s inventory. Thus, the bullwhip effect will increase for both retailers.Managerial Insight 5. As shown in the simulation result, with the increase in relevant variables (i.e., the self-price sensitivity coefficient and the cross-price sensitivity coefficient), the retailer’s bullwhip effect and the inventory bullwhip effect become larger and larger. Moreover, the fluctuation in the inventory level is higher than the fluctuation in the level of demand. This means that the retailer should control the volatility of the demand so that the volatility of the inventory level can be controlled.


## 6.2.1. Comparing the Bullwhip Effect Using Different Forecasting Techniques

Based on the quantitative model and the simulation model of the bullwhip effect using different forecasting techniques, we found that the bullwhip effect for R1, using different forecasting techniques, depends on the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the price autoregressive coefficients, the market share, and the variance. However, the bullwhip effect cannot be impacted by the potential market demand. We analyzed the impact of different forecasting techniques on the bullwhip effect for R1. The relevant parameters can be applied as follows:μ1=1,μ2=1, δ1,p2=δ2,p2=1,andδ122=0.1 (Tables 2 and 3).Table 2
The value range of the relevant factors.
Factorsl1l2kb11b12ψρ1ρ2αLevel 1224550.30.30.30.3Level 244810100.60.60.60.6Level 3661215150.90.90.90.9Level 4224550.30.30.60.3Level 5224550.30.30.60.3Level 6224550.30.30.60.3Level 7224550.30.30.90.6Table 3
The simulation number.
No.l1l2kb11b12ψρ1ρ2α11 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)22 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.3)2 (0.3)2 (0.6)33 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.3)3 (0.3)3 (0.9)41 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.6)1 (0.6)1 (0.3)52 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.6)2 (0.6)2 (0.6)63 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.6)3 (0.6)3 (0.9)71 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.9)1 (0.9)1 (0.3)82 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.9)2 (0.9)2 (0.6)93 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.9)3 (0.9)3 (0.9)101 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)111 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)122 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)132 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)143 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)153 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)164 (2)4 (2)4 (4)4 (5)4 (5)4 (0.3)4 (0.3)4 (0.6)4 (0.3)175 (2)5 (2)5 (4)5 (5)5 (5)5 (0.3)5 (0.3)5 (0.6)5 (0.3)186 (2)6 (2)6 (4)6 (5)6 (5)6 (0.3)6 (0.3)6 (0.6)6 (0.3)197 (2)7 (2)7 (4)7 (5)7 (5)7 (0.3)7 (0.3)7 (0.6)7 (0.6)The following conclusion can be drawn from Table4.(1)
From No. 1–3, when the price autoregressive coefficients of the two retailers are very low and have the same value, and regardless of how much longer the lead time is, or how much larger the smoothing coefficient is, or how bigger the market share is, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.(2)
From No. 4–9, when R1’s self-price sensitivity coefficient and R1’s cross-price sensitivity coefficient are at the low level and have the same value, other variables are equal to any values, and R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.(3)
From No. 10–15, when R1’s self-price sensitivity coefficient, cross-price sensitivity coefficient, lead time, market share, smoothing coefficient, and price autoregressive coefficient are from low to high, R1’s bullwhip effect is at the highest level using the MA forecasting technique.(4)
From No. 16–19, when R1’s self-price sensitivity coefficient, cross-price sensitivity coefficient, lead time, market share, smoothing coefficient, and price autoregressive coefficient are all at the low level, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique.Table 4
The simulation result.
No.BEretailer−1MABEretailer−1ESBEretailer−1MMSE13.45.12.123.415.22.333.435.252.643.515.412.753.535.632.863.565.812.973.85.93.083.96.023.0594.06.053.11014.16.13.161114.156.213.221215.76.33.271316.16.43.371417.26.53.41517.56.63.6163.686.93.41173.727.013.43183.87.133.44193.97.33.45We converted the data into figures in order to analyze the impact of the relevant factors on R1’s bullwhip effect. Moreover, we can obtain some important managerial insights from figures. Figure8 shows the impact of b11 and b12 on the bullwhip effect using different forecasting techniques. Figure 9 depicts the impact of ρ1andρ2 on the bullwhip effect. Figure 10 depicts the impact of ψ on the bullwhip effect.Figure 8
The impact ofb11 and b12 on the bullwhip effect using different forecasting techniques.Figure 9
The impact ofρ1andρ2 on the bullwhip effect using different forecasting techniques.Figure 10
The impact ofψ on the bullwhip effect using different forecasting techniques.By comparing the bullwhip effect using different forecasting techniques, as shown in Figures8–10, we can conclude the following: (1) as the MMSE forecasting techniques can reduce the lead-time demand forecast error to the largest extent, the bullwhip effect can reach the lowest level using the MMSE techniques; (2) R1 can reduce the bullwhip effect using the MA techniques when the self-price sensitivity coefficient b11 is lower than 5.5, the price autoregressive coefficient ρ1 is lower than 0.45, and the market share ψ is lower than 0.4. On the contrary, R1 can reduce the bullwhip effect using the ES techniques. Moreover, ΔBEretailer−1=BEretailer−1MA−BEretailer−1ES, when σ12=δ12,σ22=δ22, and we can determine the following:(49)ΔBEretailer=b11b121−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+b112ρ1l1−1−ρ1l11−ρ12+ρ1l121−ρ1+l21−ρ2l21−ρ2<12ψl1k2+l12α2−α.Proposition 1.
When the self-price sensitivity coefficientb11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy the following condition, R1 can control the bullwhip effect using the MA technique. Otherwise, R1 should select the ES technique.(50)b11b121−ψ1−ρ1ρ2ρ11−ρ1l1−1−ρ1l11−ρ1+b112ρ1l1−1−ρ1l11−ρ12+ρ1l121−ρ1+l21−ρ2l21−ρ2<12ψl1k2+l12α2−α.It is evident that the potential market demanda1 has no impact on R1’s bullwhip effect. Thus, the potential market demand cannot influence R1’s choice of the forecasting technique. On the other hand, when the self-price sensitivity coefficient b11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy Proposition 1, R1 can control the bullwhip effect using the MA technique. Otherwise, R1 should select the ES technique.


## 6.2.2. Comparing the Inventory Bullwhip Effect Using Different Forecasting Techniques

Based on the quantitative model and the simulation model of the inventory bullwhip effect using different forecasting techniques, we found that R1’s inventory bullwhip effect using different forecasting techniques depends on the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the price autoregressive coefficients, the market share, and the variance. However, it cannot be impacted by the potential market demand. We analyzed the impact of different forecasting techniques on R1’s inventory bullwhip effect. The relevant parameters can be applied as follows:μ1=1,μ2=1, δ1,p2=δ2,p2=1,andδ122=0.1 (Tables 5 and 6).Table 5
The value range of the relevant factors.
Factorsl1l2kb11b12ψρ1ρ2αLevel 1224550.30.30.30.3Level 244810100.60.60.60.6Level 3661215150.90.90.90.9Level 4224550.30.30.60.3Level 5224550.30.30.60.3Level 6224550.30.30.60.3Level 7224550.30.30.90.6Table 6
The simulation number.
No.l1l2kb11b12ψρ1ρ2α11 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)22 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.3)2 (0.3)2 (0.6)33 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.3)3 (0.3)3 (0.9)41 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.6)1 (0.6)1 (0.3)52 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.6)2 (0.6)2 (0.6)63 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.6)3 (0.6)3 (0.9)71 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.9)1 (0.9)1 (0.3)82 (4)2 (4)2 (8)2 (5)2 (5)2 (0.6)2 (0.9)2 (0.9)2 (0.6)93 (6)3 (6)3 (12)3 (5)3 (5)3 (0.9)3 (0.9)3 (0.9)3 (0.9)101 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)111 (2)1 (2)1 (4)1 (5)1 (5)1 (0.3)1 (0.3)1 (0.3)1 (0.3)122 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)132 (4)2 (4)2 (8)2 (10)2 (10)2 (0.6)2 (0.6)2 (0.6)2 (0.6)143 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)153 (6)3 (6)3 (12)3 (15)3 (15)3 (0.9)3 (0.9)3 (0.9)3 (0.9)164 (2)4 (2)4 (4)4 (5)4 (5)4 (0.3)4 (0.3)4 (0.6)4 (0.3)175 (2)5 (2)5 (4)5 (5)5 (5)5 (0.3)5 (0.3)5 (0.6)5 (0.3)186 (2)6 (2)6 (4)6 (5)6 (5)6 (0.3)6 (0.3)6 (0.6)6 (0.3)197 (2)7 (2)7 (4)7 (5)7 (5)7 (0.3)7 (0.3)7 (0.6)7 (0.6)The following conclusion can be drawn from Table7.(1)
In No. 1–3, when the value of the two retailers’ price autoregressive coefficient is very low and at the same level, regardless of the length of the lead time, or how much larger the smoothing coefficient is, or how much larger the market share is, R1’s inventory bullwhip effect is at the lowest level using the MMSE forecasting technique.(2)
In No. 4–9, when R1’s self-price sensitivity coefficient and R1’s cross-price sensitivity coefficient are at the low level and at the same level, regardless of the length of the lead time, or how much larger the smoothing coefficient is, or how much larger the market share is, R1’s inventory bullwhip effect is at the lowest level using the MMSE forecasting technique.(3)
In No. 16–18, when the value of the two retailers’ price autoregressive coefficient is very low and at a different level, and when the smoothing coefficient is very low and the lead time is very low, R1’s bullwhip effect is at the lowest level using the MMSE forecasting technique. Moreover, R1’s inventory bullwhip effect is at the highest level using the ES forecasting technique.Table 7
The simulation result.
No.IBEretailer−1MAIBEretailer−1ESIBEretailer−1MMSE189.6130.321.9275.5130.621.7374.3130.821.6474.1131.421.4573.8131.821.1673.3131.920.9773.2132.220.7872.8132.420.5972.6132.620.21072.4132.820.11172.1132.919.91271.9133.319.81371.6133.619.61471.2133.819.41570.8133.918.91670.5134.218.71769.9134.618.51868.6134.818.51921.5135.571.7We converted the data into figures in order to analyze the impact of the relevant factors on R1’s bullwhip effect. Moreover, we can gain some important managerial insights from the figures. Figure11 depicts the impact of b11 and b12 on the inventory bullwhip effect. Figure 12 depicts the impact of ρ1andρ2 on the inventory bullwhip effect. Figure 13 depicts the impact of ψ on the inventory bullwhip effect.Figure 11
The impact ofb11 and b12 on the inventory bullwhip effect using different forecasting techniques.Figure 12
The impact ofρ1andρ2 on the inventory bullwhip effect using different forecasting techniques.Figure 13
The impact ofψ on the inventory bullwhip effect using different forecasting techniques.By comparing the inventory bullwhip effect using different forecasting techniques, as shown in Figures11–13, it is evident that the inventory bullwhip effect can reach the highest level using the ES technique (Chen et al. and Zhang et al. reached the same conclusions). Moreover, R1 can control the impact of the inventory bullwhip effect using the MA technique or the MMSE technique. We can then analyze which forecasting technique R1 should select under different conditions. Figure 11 shows that the inventory bullwhip effect can achieve the lowest level under the MMSE technique when the value of the self-price sensitivity coefficient b11 is lower than 7. Otherwise, the inventory bullwhip effect can reach the lowest level using the MA technique when b11 is higher than 7. Figure 12 shows that the inventory bullwhip effect can reach the lowest level using the MMSE technique when the price autoregressive coefficient ρ1<0.8. Otherwise, the inventory bullwhip effect can reach the lowest level using the MA technique when the price autoregressive coefficient ρ1>0.8. Figure 13 indicates that the inventory bullwhip effect can achieve the lowest level using the MA technique when the value of the market share ψ is lower than 0.6. Otherwise, the inventory bullwhip effect can achieve the lowest level using the MMSE technique when ψ is higher than 0.6. Moreover, ΔIBEretailer−1=IBEretailer−1MMSE−IBEretailer−1MA, when σ12=δ12andσ22=δ22, and we can get(51)ΔIBE=−2b11b12ρ1ρ21−ρ11−ρ1ρ2ρ21−ρ2l11−ρ22+ρ2ρ11−ρ21−ρ1ρ2ρ11−ρ1l11−ρ12+2ρ11−ρ12+2ρ11−ρ1−21+ρ11−ρ1ρ11−ρ1b11b12ρ1k+ρ2kψ1−ψ1−ρ1ρ2.Proposition 2.
When the self-price sensitivity coefficientb11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy the following condition, R1 can control the inventory bullwhip effect using the MMSE technique. Otherwise, R1 should select the MA technique.(52)2ρ11−ρ12+2ρ11−ρ1−21+ρ11−ρ1ρ11−ρ1b11b12ρ1k+ρ2kψ1−ψ1−ρ1ρ2<2b11b12ρ1ρ21−ρ11−ρ1ρ2ρ21−ρ2l11−ρ22+ρ2ρ11−ρ21−ρ1ρ2ρ11−ρ1l11−ρ12.We found that the potential market demanda1 has no impact on R1’s inventory bullwhip effect. Thus, the potential market demand cannot influence R1’s choice of the forecasting technique. On the contrary, when the self-price sensitivity coefficient b11, the cross-price sensitivity coefficient b12, the price autoregressive coefficients ρ1 and ρ2, and the market share ψ satisfy Proposition 2, R1 can control the inventory bullwhip effect using the MMSE technique. Otherwise, R1 should select the MA technique.From the above analysis, we can obtain the following managerial insights:Managerial Insight 1. Figure 8 clearly illustrates that there is a positive relationship between b12 and BE. In other words, the larger the value of the cross-price sensitivity coefficient, the greater the competition between the two supply chains. As a result, the fluctuation in demand increases. Thus, in terms of the operational and management activities of the firm, the firm should seek to reduce the competitive relationship and maintain a positive cooperative relationship.Managerial Insight 2. As shown in Figure 10, there is a positive relationship between ψ and BE. The reason for this phenomenon is that the larger the probability that customers will choose both retailers’ products, the more difficult it is for the two retailers to forecast customer demand. Thus, this will lead to greater uncertainty with respect to the retailer’s order.Managerial Insight 3. As Figure 11 clearly shows, there is a positive relationship between b12 and IBE. In other words, the larger the value of the cross-price sensitivity coefficient, the greater the competition between the two supply chains, and the fluctuation in the inventory level increases. Thus, the firms should reduce the competitive relationship and maintain a positive cooperative relationship between different firms.Managerial Insight 4. As shown in Figure 13, there is a positive relationship between ψ and IBE.As shown inManagerial Insight 2, the larger the probability that customers will choose both retailers’ products, the larger the bullwhip effect of the two retailers. As a result, this will lead to more uncertainty in the retailer’s inventory. Thus, the bullwhip effect will increase for both retailers.Managerial Insight 5. As shown in the simulation result, with the increase in relevant variables (i.e., the self-price sensitivity coefficient and the cross-price sensitivity coefficient), the retailer’s bullwhip effect and the inventory bullwhip effect become larger and larger. Moreover, the fluctuation in the inventory level is higher than the fluctuation in the level of demand. This means that the retailer should control the volatility of the demand so that the volatility of the inventory level can be controlled.


## 7. Conclusion

In this paper, we researched the impact of different forecasting techniques and market share on the inventory bullwhip effect in two parallel supply chains which include one manufacturer and one retailer in each supply chain. We assumed that the two retailers order products from two manufacturers using the order-up-to strategy and that the two retailers predict customer demand using different forecasting techniques. First, we developed a quantitative model of the inventory bullwhip effect based on the bullwhip effect and by means of inventory. We then analyzed the impact of the self-price sensitivity coefficient, the cross-price sensitivity coefficient, the different forecasting techniques, and the market share on the inventory bullwhip effect. Finally, we analyzed the condition under which different forecasting techniques and the market share can increase or reduce the inventory bullwhip effect. We reached the following conclusions:(1)
The bullwhip effect can achieve the lowest level under the MMSE technique when the value of the self-price sensitivity coefficientb11 is lower than 7. On the contrary, the bullwhip effect can achieve the lowest level using the MA technique when b11 is higher than 7. The bullwhip effect can reach the lowest level using the MMSE technique when the price autoregressive coefficient ρ1<0.8.(2)
The bullwhip effect can achieve the lowest level using the MA technique when the value of the market shareψ is lower than 0.6. Otherwise, the bullwhip effect can achieve the lowest level using the MMSE technique when ψ is higher than 0.6. Moreover, ΔBEretailer−1=BEretailer−1MMSE−BEretailer−1MA, when σ12=δ12andσ22=δ22.(3)
The MMSE forecasting technique can reduce the lead-time demand forecast error to the largest extent, and the inventory bullwhip effect can achieve the lowest level using the MMSE technique.(4)
R1 can reduce the inventory bullwhip effect using the MA method when the self-price sensitivity coefficientb11 is lower than 5.5, the price autoregressive coefficient ρ1 is lower than 0.45, and the market share ψ is lower than 0.4.Our research suggests that the manager should strengthen cooperation with the partner. If two products are substitutable or two supply chains are competitors, they should reduce the competitive relationship and maintain positive cooperative relationship. Moreover, the firm should forecast the market demand using the mean square error forecasting method under some conditions. On the contrary, the company can do this by using other forecasting methods under some conditions. In this paper, we only researched the inventory bullwhip effect in two parallel supply chains. In the future, we will discuss the inventory bullwhip effect in a supply chain network which includes multiple suppliers, multiple manufacturers, multiple distributors, and multiple retailers. Lastly, we should point out that many suppliers or retailers build the network selling channel to sell products by means of electronic commerce. There is a competitive relationship between the network selling channel and the traditional entity selling channel. In the future, we will discuss the impact of different selling channels on the inventory bullwhip effect.



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*Source: 2987218-2020-09-11.xml*