# Research on Supply Chain Stability Driven by Consumer’s Channel Preference Based on Complexity Theory

Authors: Yi Tian; Junhai Ma; Wandong Lou
Journal: Complexity
 (2018)

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## Abstract

At present, most of the manufacturers are increasingly laying emphasis on dual-channel selling. The level of consumer’s preference is diverse with different channels. This study discusses a supply chain where one common manufacturer uses two different channels to sell product. The bounded rationality expectation has been applied to explore the decision-making mechanism in finalizing the wholesale prices and sales commissions for the manufacturer and retailers. In addition, the dynamic features of the system are simulated by 2D bifurcation diagram, the largest Lyapunov exponent, attractor variation, and time series. The simulation results suggest that if the adjustment speeds of the wholesale prices and sales commissions change drastically, the system would fall into a chaotic state. If the consumers prefer the online channel, the wholesale prices and sales commissions may be raised and vice versa. However, too strong preference of each channel will make the system appear to have a considerably periodic fluctuation, even chaos. In the meantime, the parameter adjustment method can help make the periodic or chaotic system go back to stability. Therefore, the significance of this study is with a practical meaning to make better pricing strategy for players in dual-channel selling supply chain system.


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## Body

## 1. Introduction

The rapid development of Internet technology and social economy promoted a huge amount of manufacturers who had been transferred from traditional retailing to the combination of traditional and electronic retailing. For instance, according to the data provided by the China Academy of Information and Communications Technology (CAICT), sales of smart phone in China records nearly 390 million at year-end of 2016. Huawei took a 16.4 percent market share in China, while Apple, 9.6 percent. Smart phone manufacturers always distribute their products through both traditional and online channels. In this research, a pricing problem in supply chain with one manufacturer (Huawei) who supplies an identical product (Mate 9) to two retailers (China Resources Vanguard Co. Ltd. and JD.com) is brought to discussion. CR Vanguard is one of the world’s top five hundred enterprises, and it is China’s largest supermarket chains by a number of stores for years. JD.com is a typical Chinese e-commerce company which is one of the biggest e-malls in China with self-support storage and logistics system. CR Vanguard and JD.com are the typical representatives of traditional and online channels for smartphone sales, respectively. Different consumers will choose their own preferred buying channels. Therefore, the supply chain driven by the consumer’s channel preference is necessary to be studied.Two-echelon supply chain has been widely paid attention to and discussed in academia. Ma, Ma and Bao [1, 2] investigated the bullwhip effect in a two-echelon supply chain. This study discusses a supply chain which consists of one common manufacturer and two retailers from traditional and online sales channels, respectively. The manufacturer determines the wholesale prices and the two retailers decide their own sales prices.In the current studies [3–7], the consumer preference is frequently considered as one of the chief factors that influence the marketing. Especially, consumer’s channel preference is a hot topic in supply chain research. Khouja et al. [8] analyzed the manufacturer’s channel selection and pricing problem. They considered consumer’s preferences under different distribution strategies. Gao and Su [9] investigated a stylized model where a retailer operated the dual-channel. Customers strategically make channel choices that included both online and offline. Ke and Liu [10] discussed a dual-channel supply chain competition with channel preference under an uncertain environment. Their results indicated that the customer’s direct-channel preference had the opposite effect on the profit of supplier and retailer. Gan et al. [11] presented pricing decisions and profits of supply chain members in a closed-loop supply chain and considered the parameter of customer’s direct channel preference as a key factor.Different from the above mentioned literatures, this paper discusses the decision-making of wholesale prices and sales commissions for manufacturer and retailers based on the bounded rationality expectation by employing complexity theory to explore the effect of the adjustment speed of the above variables on the system stability. More and more researchers used bounded rational theory to analyze the economic model. Ma and Wang [12] applied two noncooperative game models to study the bounded rationality expectation. Song and Zhao [13] pointed out a decision-making problem of newsvendor system with strategic customers of bounded rationality. Ma and Si [14] established a continuous Bertrand duopoly game model with two-stage delay to investigate the influence of delay and weight on the complex dynamic characteristics of the system. Ma and Xie [15] analyzed the comparison and complexity on a dual-channel supply chain under the different channel power structures and uncertain demand. Ma and Xie [16] discussed the impact of loss sensitivity on a mobile phone supply chain system stability based on the chaos theory. In conclusion, it has received a considerable amount of attention to apply bounded rationality and complexity theory to study the management problems in the supply chain.In reality, the exploration of the supply chain driven by consumer’s channel preference based on chaos theory has a great significance. In this paper, besides the adjusted basic parameters such as the wholesale prices and sales commissions, the consumer’s channel preference is also discussed. The results indicated that the preference plays an important role in status transformation from stability to chaos. In this analysis, a supply chain model which depicts one common manufacturer using two different channels to sell its product is built in the first place. Then, the bounded rationality expectation is applied to study the complex dynamics of the system. Furthermore, the dynamic characteristics of the system are discussed by 2D bifurcation diagram, the largest Lyapunov exponent, attractor variation, and time series. The results indicate that not only the drastic adjustment of traditional parameters such as the wholesale prices and sales commissions can make the system go chaotic but also the consumer’s channel preference can also lead to system considerably periodic fluctuation, even to chaos, which is the primary finding in this article. Finally, the parameter adjustment control method is performed on the system to return the periodic or the chaotic state to stability. The conclusions of this paper can give constructive suggestions for managers of both manufacturers and retailers.The structure of this paper is as follows. In Section2, the basic model considering the influence of consumer’s channel preference is established. In this part, the optimal decision of the complex dynamic system is analyzed. Following Section 2, this study further applies numerical simulation to investigate the stability aspect and the dynamical behaviors. 2D bifurcation diagram, the largest Lyapunov exponent, attractor, and time series have been utilized to study the influence on the supply chain. In Section 4, we introduce the method to control the chaos system. Finally, the conclusion of this research is provided.


## 2. Demand Model

A supply chain system consists of one manufacturer and two retailers, one of which is a traditional retailer (retailer 1) and the other is an online retailer (retailer 2), which has been restricted in this study. The manufacturer provides an identical product at a constant unit cost (c) and wholesale prices (wi) to retailer 1 and retailer 2. Then, the retailers distribute the product to consumers in a common market by traditional and online channels, respectively. The retailers decide their unit sales commissions ki,i=1,2. Retailer 1 and retailer 2 have a competitive relationship between each other as shown in Figure 1.Figure 1
The model of supply chain.Hence, the sales prices of this supply chain can be described aspi=wi+ki,i=1,2. The demand function is simplified as the following linear form:
(1)qi=di−δ+γpi+γp3−i,i=1,2.This simplified linear demand function has been used by Anderson and Bao [17], Wu et al. [18], Yang and Zhou [19], and Choi [20]. Here, di is the market base of channel i, pi is the sales price of retailer i, δ is the price sensitivity of the product, and γ denotes the substitutability of the retailers. In order to discuss the influence of consumer’s channel preference and pricing strategies on the performance of the supply chain members without loss of generality, the method proposed by Mcguire and Staelin [21] has been applied to rescale the demand functions in (1). The scaled model can conduct some analytical comparisons instead of numerical experiments reported in Xiao et al. [22]. Let μ=γ/γ+δ; the above linear function will be changed as
(2)qi′=di′−δ1−μpi′+δμ1−μp3−i′,i=1,2.The substitutability of the two retailers in terms of changes in prices has been denoted as parameterμ. Let ϕ1=d1′−δc′≡ϕ, ϕ2=d2′−δc′ and d1′≤d2′, and define η=d2′−δc′/d1′−δc′. The parameter ηη≥1 depicts the consumer’s channel preference.μ=0 represents the scenario in which the submarket is monopolized by one retailer. μ=1 represents the scenario in which there exist two completely substitutable retailers. Then, the prices can be standardized as ki=δ/ϕ1−μki′, wi=δ/ϕ1−μwi′−c′, and pi=δ/ϕ1−μwi′+ki′−c′.The demand functionq1′ can be formulated as follows:
(3)q1′=d1′−δ1−μp1′+δ1−μp=ϕ1−p1+μp2.q2′ can be obtained by using the same method:
(4)q2′=d2′−δ1−μp2′+δμ1−μp1′=ϕη−p2+μp1.So we can get the demands as below:(5)q1=q1′ϕ=1−p1+μp2,q2=q2′ϕ=η−p2+μp1.But in reality, the market requirements are often influenced by some other factors; here,εi is used to represent these disturbance factors in research.
(6)q1=q1′ϕ=1−p1+μp2+ε1,q2=q2′ϕ=η−p2+μp1+ε2.εii=1,2 follows a normal distribution such that Eεi=0, varεi=σ2. The expected profit decision models of the manufacturer and two retailers are shown as
(7)Eπr1=k11−k1+w1+μk2+w2,Eπr2=k2η−k2+w2+μk1+w1,Eπm=w11−k1+w1+μk2+w2+w1η−k2+w2+μk1+w1.In order to deduce the marginal profits of the manufacturer and two retailers, the first-order partial derivative of (7) can be calculated as follows:
(8)∂Eπr1∂k1=1−2k1−w1+k2+w2μ,∂Eπr2∂k2=−2k2−w2+η+k1+w1μ,∂Eπm∂w1=1−k1−2w1+w2μ+k2+w2μ,∂Eπm∂w2=−k2−2w2+η+w1μ+k1+w1μ.


### 2.1. Optimal Decision

In the market, the optimal decisions of the retailers and manufacturer are influenced by each other. Therefore, this study assumes that they share overlapping information. In this research, when the marginal profits reduce to zero, the optimal decision for retailers and manufacturer can be obtained. The wholesale prices and sales commissions are the decision variables. The research calculates the first-order derivative of the retailers’ and manufacturer’s profit. When the marginal profit equals to zero, the optimal wholesale prices and sales commissions can be derived. The detailed process is as follows:(9)∂Eπr1∂k1=0,∂Eπr1∂k2=0,∂Eπm∂w1=0,∂Eπm∂w2=0.The optimal solution of the system will be solved.(10)k1∗=−3+ημ−9+μ2,k2∗=−3η+μ−9+μ2,w1∗=−−3−4ημ−μ2−9+μ2−1+μ2,w2∗=−−3η−4μ−ημ2−9+μ2−1+μ2.


### 2.2. The Complex Dynamics of the System

The real market is a changeable and complex system. The manufacturer and retailers cannot know the complete market information, and predict other decision-makers’ information accurately. Hence, the bounded rationality expectations have been universally adopted by market decision-makers. The decision oft+1 period is as follows:
(11)k1t+1=k1t+g1k1t∂Eπr1t∂k1t,k2t+1=k2t+g2k2t∂Eπr2t∂k2t,w1t+1=w1t+g3w1t∂Eπmt∂w1t,w2t+1=w2t+g4w2t∂Eπmt∂w2t.In the paper, the price adjustment speeds consist of two wholesale prices and sales commissions. Here,gi>0 i=1,2,3,4; g1 and g2 are the adjustment speed of the sales commissions with retailer 1 and retailer 2, respectively. g3 and g4 represent the adjustment speed of the wholesale prices with the manufacturer. Based on the above equation, the four-dimensional discrete dynamic system can be acquired and expressed in (12). The wholesale prices and the sales commissions in the next period depend on the estimation of these period marginal profits. Base on that, the adjustment of bounded rationality is completed. If the entity’s current marginal profit is positive, the price will increase during the next period for higher profits. If the entity’s current marginal profit is negative, the price should be reduced during the next period. The dynamic adjustment processes are as follows:
(12)k1t+1=k1t+g1k1t1−2k1−w1+k2+w2μ,k2t+1=k2t+g2k2t−2k2−w2+η+k1+w1μ,w1t+1=w1t+g3w1t1−k1−2w1+w2μ+k2+w2μ,w2t+1=w2t+g4w2t−k2−2w2+η+w1μ+k1+w1μ.After a long period of operation, whenkit+1=kit and wit+1=wit, the system will become steady. Under the equilibrium state, all parties in the game cannot increase profits for themselves by changing the decision variables. Therefore, when the decision variables take the values of the equilibrium state, every player in the game can obtain the maximized profit. There are sixteen equilibriums of system (12):
(13)E10,0,0,0,E20,0,0,η2,E30,0,−1+ημ2−1+μ2,−η+μ2−1+μ2,E40,0,12,0,E50,η3,−1+ημ2−1+μ2,−2η+3θ+ημ26−1+μ2,E60,η3,0,η3,E70,η2,0,0,E80,−2η+μ−4+μ2,−2+ημ−4+μ2,0,E9∗−3+ημ−9+μ2,−3η+μ−9+μ2,−−3−4ημ−μ2−9+μ2−1+μ2,−−3η−4μ−ημ2−9+μ2−1+μ2,E1013,0,−2+3ημ+μ26−1+μ2,−η+μ2−1+μ2,E11−2+ημ−4+μ2,0,0,−2η+μ−4+μ2,E1212,0,0,0,E1313,0,13,0,E14−3+2ημ2−3+μ2,−2η+μ2−3+μ2,0,−2η+μ2−3+μ2,E15−2+ημ−4+μ2,−2η+μ−4+μ2,0,0,E16−2+ημ2−3+μ2,−3η+2μ2−3+μ2,−2+ημ2−3+μ2,0.For these above equilibriums, it can be found thatE9∗ is the only point which is completely nonzero. The others are boundary points. E9∗ is the unique Nash equilibrium point. The Jacobian matrix of system (12) can be calculated as follows:
(14)JE=1−2g1k1g1k1μ−g1k1g1k1μg2k2μ1−2g2k2g2k2μ−g2k2−g3w1g3w1μ1−2g3w12g3w1μg4w2μ−g4w22g4w2μ1−2g4w2.The Jacobian matrix and Jury criterion have been applied to discuss the local stability of the Nash equilibrium in this paper.The characteristic polynomial of the Jacobian matrix is as follows:(15)Fλ=λ4+ζ3λ3+ζ2λ2+ζ1λ+ζ0.In order to guarantee the local stability of the Nash equilibrium,Fλ must satisfy the following conditions:
(16)1+ζ3+ζ2+ζ1+ζ0>0,1−ζ3+ζ2−ζ1+ζ0>0,ζ0<1,β0<β3,σ0<σ2.The details ofζ1,ζ2,ζ3,ζ4,β0,β3,σ0, and σ2 are as below:
(17)β3=1−ζ02,β2=ζ3−ζ1ζ0,β1=ζ3−ζ2ζ0,β0=ζ1−ζ3ζ0,σ2=β32−β02,σ1=β3β2−β1β0,σ0=β3β1−β2β0ζ0=1−2g1k1−2g2k2+4g1k1g2k2−2g3w1+3g1k1g3w1+4g2k2g3w1−6g1k1g2k2g3w1−2g4w2+4g1k1g4w2+3g2k2g4w2−6g1k1g2k2g4w2+4g3w1g4w2−6g1k1g3w1g4w2−6g2k2g3w1g4w2+9g1k1g2k2g3w1g4w2−g1k1g2k2μ2−g2k2g3w1μ2+2g1k1g2k2g3w1μ2−g1k1g4w2μ2+2g1k1g2k2g4w2μ2−4g3w1g4w2μ2+6g1k1g3w1g4w2μ2+6g2k2g3w1g4w2μ2−10g1k1g2k2g3w1g4w2μ2+g1k1g2k2g3w1g4w2μ2,ζ1=−4+6g1k1+6g2k2−8g1k1g2k2+6g3w1−6g1k1g3w1−8g2k2g3w1+6g1k1g2k2g3w1+6g4w2−8g1k1g4w2−6g2k2g4w2+6g1k1g2k2g4w2−8g3w1g4w2+6g1k1g3w1g4w2+6g2k2g3w1g4w2+2g1k1g2k2μ2+2g2k2g3w1μ2−2g1k1g2k2g3w1μ2+2g1k1g4w2μ2−2g1k1g2k2g4w2μ2+8g3w1g4w2μ2−6g1k1g3w1g4w2μ2−6g2k2g3w1g4w2μ2,ζ2=6−6g1k1−6g2k2+4g1k1g2k2−6g3w1+3g1k1g3w1+4g2k2g3w1−6g4w2+4g1k1g4w2+3g2k2g4w2+4g3w1g4w2−g1k1g2k2μ2−g2k2g3w1μ2−g1k1g4w2μ2−4g3w1g4w2μ2,ζ3=−4+2g1k1+2g2k2+2g3w1+2g4w2.Because the conditions are very complex, the process to solve (12) is a tedious task. If the Nash equilibrium satisfies (12), it can be considered as locally stable. In the next section, we will utilize numerical simulation to further analyze the dynamic characteristic on the system.


## 2.1. Optimal Decision

In the market, the optimal decisions of the retailers and manufacturer are influenced by each other. Therefore, this study assumes that they share overlapping information. In this research, when the marginal profits reduce to zero, the optimal decision for retailers and manufacturer can be obtained. The wholesale prices and sales commissions are the decision variables. The research calculates the first-order derivative of the retailers’ and manufacturer’s profit. When the marginal profit equals to zero, the optimal wholesale prices and sales commissions can be derived. The detailed process is as follows:(9)∂Eπr1∂k1=0,∂Eπr1∂k2=0,∂Eπm∂w1=0,∂Eπm∂w2=0.The optimal solution of the system will be solved.(10)k1∗=−3+ημ−9+μ2,k2∗=−3η+μ−9+μ2,w1∗=−−3−4ημ−μ2−9+μ2−1+μ2,w2∗=−−3η−4μ−ημ2−9+μ2−1+μ2.


## 2.2. The Complex Dynamics of the System

The real market is a changeable and complex system. The manufacturer and retailers cannot know the complete market information, and predict other decision-makers’ information accurately. Hence, the bounded rationality expectations have been universally adopted by market decision-makers. The decision oft+1 period is as follows:
(11)k1t+1=k1t+g1k1t∂Eπr1t∂k1t,k2t+1=k2t+g2k2t∂Eπr2t∂k2t,w1t+1=w1t+g3w1t∂Eπmt∂w1t,w2t+1=w2t+g4w2t∂Eπmt∂w2t.In the paper, the price adjustment speeds consist of two wholesale prices and sales commissions. Here,gi>0 i=1,2,3,4; g1 and g2 are the adjustment speed of the sales commissions with retailer 1 and retailer 2, respectively. g3 and g4 represent the adjustment speed of the wholesale prices with the manufacturer. Based on the above equation, the four-dimensional discrete dynamic system can be acquired and expressed in (12). The wholesale prices and the sales commissions in the next period depend on the estimation of these period marginal profits. Base on that, the adjustment of bounded rationality is completed. If the entity’s current marginal profit is positive, the price will increase during the next period for higher profits. If the entity’s current marginal profit is negative, the price should be reduced during the next period. The dynamic adjustment processes are as follows:
(12)k1t+1=k1t+g1k1t1−2k1−w1+k2+w2μ,k2t+1=k2t+g2k2t−2k2−w2+η+k1+w1μ,w1t+1=w1t+g3w1t1−k1−2w1+w2μ+k2+w2μ,w2t+1=w2t+g4w2t−k2−2w2+η+w1μ+k1+w1μ.After a long period of operation, whenkit+1=kit and wit+1=wit, the system will become steady. Under the equilibrium state, all parties in the game cannot increase profits for themselves by changing the decision variables. Therefore, when the decision variables take the values of the equilibrium state, every player in the game can obtain the maximized profit. There are sixteen equilibriums of system (12):
(13)E10,0,0,0,E20,0,0,η2,E30,0,−1+ημ2−1+μ2,−η+μ2−1+μ2,E40,0,12,0,E50,η3,−1+ημ2−1+μ2,−2η+3θ+ημ26−1+μ2,E60,η3,0,η3,E70,η2,0,0,E80,−2η+μ−4+μ2,−2+ημ−4+μ2,0,E9∗−3+ημ−9+μ2,−3η+μ−9+μ2,−−3−4ημ−μ2−9+μ2−1+μ2,−−3η−4μ−ημ2−9+μ2−1+μ2,E1013,0,−2+3ημ+μ26−1+μ2,−η+μ2−1+μ2,E11−2+ημ−4+μ2,0,0,−2η+μ−4+μ2,E1212,0,0,0,E1313,0,13,0,E14−3+2ημ2−3+μ2,−2η+μ2−3+μ2,0,−2η+μ2−3+μ2,E15−2+ημ−4+μ2,−2η+μ−4+μ2,0,0,E16−2+ημ2−3+μ2,−3η+2μ2−3+μ2,−2+ημ2−3+μ2,0.For these above equilibriums, it can be found thatE9∗ is the only point which is completely nonzero. The others are boundary points. E9∗ is the unique Nash equilibrium point. The Jacobian matrix of system (12) can be calculated as follows:
(14)JE=1−2g1k1g1k1μ−g1k1g1k1μg2k2μ1−2g2k2g2k2μ−g2k2−g3w1g3w1μ1−2g3w12g3w1μg4w2μ−g4w22g4w2μ1−2g4w2.The Jacobian matrix and Jury criterion have been applied to discuss the local stability of the Nash equilibrium in this paper.The characteristic polynomial of the Jacobian matrix is as follows:(15)Fλ=λ4+ζ3λ3+ζ2λ2+ζ1λ+ζ0.In order to guarantee the local stability of the Nash equilibrium,Fλ must satisfy the following conditions:
(16)1+ζ3+ζ2+ζ1+ζ0>0,1−ζ3+ζ2−ζ1+ζ0>0,ζ0<1,β0<β3,σ0<σ2.The details ofζ1,ζ2,ζ3,ζ4,β0,β3,σ0, and σ2 are as below:
(17)β3=1−ζ02,β2=ζ3−ζ1ζ0,β1=ζ3−ζ2ζ0,β0=ζ1−ζ3ζ0,σ2=β32−β02,σ1=β3β2−β1β0,σ0=β3β1−β2β0ζ0=1−2g1k1−2g2k2+4g1k1g2k2−2g3w1+3g1k1g3w1+4g2k2g3w1−6g1k1g2k2g3w1−2g4w2+4g1k1g4w2+3g2k2g4w2−6g1k1g2k2g4w2+4g3w1g4w2−6g1k1g3w1g4w2−6g2k2g3w1g4w2+9g1k1g2k2g3w1g4w2−g1k1g2k2μ2−g2k2g3w1μ2+2g1k1g2k2g3w1μ2−g1k1g4w2μ2+2g1k1g2k2g4w2μ2−4g3w1g4w2μ2+6g1k1g3w1g4w2μ2+6g2k2g3w1g4w2μ2−10g1k1g2k2g3w1g4w2μ2+g1k1g2k2g3w1g4w2μ2,ζ1=−4+6g1k1+6g2k2−8g1k1g2k2+6g3w1−6g1k1g3w1−8g2k2g3w1+6g1k1g2k2g3w1+6g4w2−8g1k1g4w2−6g2k2g4w2+6g1k1g2k2g4w2−8g3w1g4w2+6g1k1g3w1g4w2+6g2k2g3w1g4w2+2g1k1g2k2μ2+2g2k2g3w1μ2−2g1k1g2k2g3w1μ2+2g1k1g4w2μ2−2g1k1g2k2g4w2μ2+8g3w1g4w2μ2−6g1k1g3w1g4w2μ2−6g2k2g3w1g4w2μ2,ζ2=6−6g1k1−6g2k2+4g1k1g2k2−6g3w1+3g1k1g3w1+4g2k2g3w1−6g4w2+4g1k1g4w2+3g2k2g4w2+4g3w1g4w2−g1k1g2k2μ2−g2k2g3w1μ2−g1k1g4w2μ2−4g3w1g4w2μ2,ζ3=−4+2g1k1+2g2k2+2g3w1+2g4w2.Because the conditions are very complex, the process to solve (12) is a tedious task. If the Nash equilibrium satisfies (12), it can be considered as locally stable. In the next section, we will utilize numerical simulation to further analyze the dynamic characteristic on the system.


## 3. Dynamic Characteristics of the System

Recently, the numerical simulation method has been widely applied to represent complex dynamic system in literatures. In this study,gii=1,2 is the sales commissions’ adjustment speed of retailer i. gii=3,4 represents the adjustment speed of the wholesale prices (w1,w2) of the manufacturer. The values of the system parameters are defined as k1=0.1, k2=0.1, w1=0.31, w2=0.4, g1=0.4, g2=0.13, g3=0.12, g4=0.15, μ=0.5, and η=1.1. The stable region, bifurcation, and chaos will be studied as follows.


### 3.1. The Stable Region

The decision variables of the system are wholesale prices and sales commissions. The system will change to be either stable or unstable, if the wholesale prices and sales commissions obtain different values. The stability region of the deterministic system is shown as the blue and green part in Figures2 and 3, respectively. The white areas are the unstable range. If the applied values come from the blue and green region of Figures 2 and 3, after the iteration of period by period, the system will tend to reach the equilibrium state. If the values landed in the white areas, the system will exhibit unstable phenomena.Figure 2
The 2D stable region ofg1 and g2.Figure 3
The 2D stable region ofg3 and g4.In Figure2, when g1 increases, the tendency of g2 should be reduced. In the stable region, it has the opposite effect on g1 and g2, so does Figure 3. Through Figures 2 and 3, we found that the stable range of g1 and g2 is almost twice as much as g3 and g4, when the value of g1 is larger than others’ price adjustment parameters. Period doubling and chaos should be avoided. The supply chain system should be kept in the stable state as long as possible.


### 3.2. The Bifurcation and Chaos Behavior

The parameter basin has been used to describe 2D bifurcation effect. The results are shown in Figures4 and 5. Different colors represent different cycles. Blue represents the stable region in Figure 4, and green represents the stable region in Figure 5. Orange, pink, purple, indigo, brown, yellow, and black regions represent the stable cycles of the periods from 2 to 8, respectively. The gray region is non-convergence, and the white region is divergence in both Figures 4 and 5. The system can remain stable, when the adjustment parameter gi is relatively small. As g1 and g2 increase, the system will go into the stable cycles of period 2. If g1 and g2 continue to increase, the system will go into the stable cycles of period 4, which is shown in Figure 4. However, the system continually changes to the stable cycles of periods 2, 4, 8, and so on. If g3 and g4 keep on increasing, the system will go into chaos state. This process is depicted in Figure 5. In this situation, the market will become chaotic and the market participants cannot make good decisions to reach the best profits.Figure 4
2D bifurcation diagram ofg1 and g2.Figure 5
2D bifurcation diagram ofg3 and g4.When the adjustment parameter changes in a small range from 0, the variables will be stable. However, if the decision variables adjust sharply, the variables will be unstable. Especially, in chaotic state, the system will become sensitive and unpredictable. In economic terms, when the adjustment parameter exceeds a certain limit, the intense competition and price drastic fluctuation will occur.The bifurcation diagram is an intuitive way to show the dynamic characteristics of the system. One of the parameters is fixed and the other changed within a certain range. In the following bifurcation diagrams, the blue line representsk1 and the red line k2. The green line represents w1 and the purple line w2.The largest Lyapunov exponent (LLE) is another efficient way to make clear the complex dynamic characteristics of the system, which also is corresponding to the variation of the bifurcation diagram when discussing the same parameter. The dramatic variation of LLE is accompanied by the decision variables changing. If the LLE < 0, it means the system is in a stable state. When the LLE = 0 for the first time, the system stays at a critical point and usually the bifurcation appears. If the LLE > 0, it indicates that the system runs into the unstable state. The manufacturer and retailers should be cautious to prevent system chaos.The following part gives the bifurcation diagram and LLE diagram with respect to the parametersg1, g2, g3, and g4. The bifurcation diagrams of k1, k2, w1, and w2, with respect to the adjustment speeds of the sales commissions, are shown in Figures 6 and 7. In the same condition, Figures 8 and 9 depict the variation of LLE with respect to g1 and g2.Figure 6
Bifurcation diagram ofk1, k2, w1, w2 when g1∈1.5,3.10.Figure 7
Bifurcation diagram ofk1, k2, w1, w2 when g2∈1.5,2.98.Figure 8
The largest Lyapunov exponent of the system with respect tog1∈1.5,3.10.Figure 9
The largest Lyapunov exponent of the system with respect tog2∈1.5,2.98.Wheng1∈1.5,3.10 and with the other parameters settled down, the conditions of k1, k2, w1, and w2 are shown in Figure 6. Because the dots of w1 and w2 gather together, we further enlarge critical parts in sub-windows in order to describe the bifurcation conditions of w1 and w2. The upper small picture is the blowup of the bifurcation diagram for w1 and w2, and the other one is the blowup of the bifurcation diagram for k2. When g1∈0,2.338, k1, k2, w1, and w2 are all in the stable state. When g1 grows to 2.339, k1, k2, w1, and w2 bifurcate at the first time and enter into the stable cycles of period 2. If g1 continues to increase, the system will enter into the stable cycles of period 4.Figure8 reveals the variation of LLE along with g1 increasing, which expresses the pattern of the complex dynamic behavior caused by the growth of the price adjustment speed. Figure 8 explores the LLE of the system with respect to g1∈1.5,3.10. When g1∈0,2.338, the LLE is below zero, which indicates that the system is stable. When g1 increases to 2.339, the LLE is equal to zero and the bifurcation occurs in the system. The bifurcation point in Figure 8 is the same as that in Figure 6.Figures7 and 9 also have the same bifurcation point. k1, k2, w1, and w2 bifurcate at 2.156 as g2 varies from 0 to 2.98. If g2 continues increasing, the system runs into the cycles of period 4. In order to describe the changing processes of the system from the stable state to the chaotic region more clearly, g1 and g2 start from 1.5 in Figures 6–9.The bifurcation diagrams ofk1, k2, w1, and w2, with respect to the adjustment speeds of the wholesale prices, are shown in Figures 10 and 11. In the same condition, Figures 12 and 13 depict the variation of LLE with respect to g3 and g4.Figure 10
Bifurcation diagram ofk1, k2, w1, w2 when g3∈0.7,1.70.Figure 11
Bifurcation diagram ofk1, k2, w1, w2 when g4∈0.7,1.73.Figure 12
The largest Lyapunov exponent of the system with respect tog3∈0.7,1.70.Figure 13
The largest Lyapunov exponent of the system with respect tog4∈0.7,1.73.k1, k2, w1, and w2 bifurcate at 1.096 as g3 varies from 0 to 1.70. k1, k2, w1, and w2 bifurcate at 1.107 as g4 varies from 0 to 1.73. If g3 and g4 continue increasing, the system runs into cycles of periods 4, 8, and so on. Next, the system will fall into chaos gradually. In this part, g3 and g4 start from 0.7 in Figures 10–13.Above all, Figures10–13 sufficiently indicate that if the wholesale prices of the manufacturer are adjusted drastically, the system will fall into chaos. On the other side, as shown in Figures 6–9, if the sales commissions of the retailers are adjusted radically, the bifurcation will appear but the chaos will not occur. From an economic point of view, if the adjustment parameter rapidly changes, the market will fall into the chaotic state. Therefore, the manufacturer and retailers should keep the price adjustments within a reasonable range.The substitutability of the retailers will be discussed in the following part.μ=0 represents the demands which are absolutely independent. Each retailer is a monopolist in its own area. μ=1 represents the two retailers who are totally substitutable. When μ∈0,0.89 and the other parameters are kept constant, the situations of k1, k2, w1, and w2 are shown in Figure 14. When μ grows to 0.872, the system begins the first bifurcation which is the stable cycles of period 2. If μ continues to increase, the system runs into the stable cycles of periods 4, 8, and so on. Then, the system will fall into chaos gradually. The small picture in Figure 14 is the blowup of the bifurcation region for k1, k2, w1, and w2, when μ∈0.8,0.89.Figure 14
Bifurcation diagram ofk1, k2, w1, w2 when μ∈0,0.89.Figure15 explores the LLE of the system with respect to μ∈0,0.89. When μ∈0,0.871, the LLE is less than zero, and the system is in a stable state. While μ grows to 0.872, LLE changes into zero for the first time, and the system falls into the bifurcation state, the completely same bifurcation points in Figures 15 and 14.Figure 15
The largest Lyapunov exponent of the system with respect toμ∈0.8,0.89.In Figure14, the manufacturer is more sensitive to the substitutability of the retailer, which leads to a rise in the wholesale price. However, the retailer’s profits are robust to the changes in the retailer’s substitutability. Under this condition, fierce competition may lead to the dominance of the manufacturer. Nevertheless, bifurcation and chaos will still occur in the market after the fierce competition, once a certain extent of substitutability remains among the retailers of different channels.The impacts of consumer’s channel preference on the dynamic system will be discussed in the following part. Whenη∈0,3.80, η=3.80, g1=0.4, g2=0.4, g3=0.4, and g4=0.4, the situations of k1, k2, w1, and w2 are shown in Figure 16. The lines in the bifurcation diagram indicate that other parameters are stabilized separately, when η is changing in the range of [0, 3.80]. When η grows to 2.558, the system begins the first bifurcation which is the stable cycles of period 2. If η continues to increase, the system runs into the stable cycles of periods 4, 8, and so on. After that, the system will fall into chaos gradually. Figure 17 proves the same conclusion as Figure 16 does.Figure 16
Bifurcation diagram ofk1, k2, w1, w2 when η∈0,3.80.Figure 17
The largest Lyapunov exponent of the system with respect toη∈0,3.80.The curves of the traditional retailers and online retailers are shown in Figure16. When η>1, it can be found that the channel preference of consumers will convert from one channel to another. If one channel has more consumer preference, its wholesale prices and sales commission will rise; w2>w1 and k2>k1 are displayed in Figure 16. However, if the channel preference obviously comes to one channel, it is possible that bifurcation and chaos appear in the system.In this paper, the chaos attractor has been investigated to the dynamic characteristics of the system. Strange attractor is a set of numbers of indefinite points. The larger the space of the strange attractor is, the more complex the structure will be. No matter how instable the system is, the points are always attracted along a certain trajectory.Figure18 shows the chaotic attractor of w1 and w2, when k1=0.31, k2=0.1, w1=0.31, w2=0.4, g1=0.4, g2=0.13, g3=1.65, g4=0.15, μ=0.5, and η=1.1, the system becomes complex and even moves into chaotic state. The dynamic characteristics of the system are obvious. Based on parameter studies, w1 changes in a larger range than w2 when the adjustment speeds of wholesale price g3 change more intensively than g4.Figure 18
The chaotic attractor ofw1 and w2.The time series of different decision variables are shown in Figures19 and 20. In the chaotic state, the system is more sensitive to the initial value. The phenomenon can be called as the initial value sensitivity. In Figure 19, to better show the character of the chaotic system, we assume the retailer 1 to make decisions under the condition that η=3.75 and show the time series of k1 after 100 iterations. The blue line represents the initial value with k1=0.1, and the red line represents the initial value with k1=0.101, when the dynamic system state is in a chaotic state. We also set the other parameters’ values as follows: k2=0.1, w1=0.31, w2=0.4, g1=0.4, g2=0.4, g3=0.4, g4=0.4, and μ=0.5. As shown in Figure 19, although the initial value of sales commissions only changes from 0.1 to 0.101, after 100 iterations, a large gap between the two tracks will occur.Figure 19
Time series ofk1 in a chaotic state.Figure 20
Time series ofk1, k2, w1, w2 in a stable state.Figure20 shows the time series of k1, k2, w1, and w2 in a stable state. We set the initial values of the wholesale prices and sales commissions as k1=0.3, k2=0.6, w1=1, and w2=0.5 in Figure 20. We set the initial value of k1 and w2 from a low value, while we set the initial value of k2 and w1 from a high value. We set the same initial values of k1, k2, w1, and w2 with those above. k1, k2, w1, and w2 would gradually flatten after 12 iterations shown in Figure 20.In short, Figures19 and 20 have different meanings. The adjustment speeds of the wholesale prices and sales commissions depend on the price fluctuation levels. As such, the manufacturer and retailers should keep price adjustment in reasonable ranges.


## 3.1. The Stable Region

The decision variables of the system are wholesale prices and sales commissions. The system will change to be either stable or unstable, if the wholesale prices and sales commissions obtain different values. The stability region of the deterministic system is shown as the blue and green part in Figures2 and 3, respectively. The white areas are the unstable range. If the applied values come from the blue and green region of Figures 2 and 3, after the iteration of period by period, the system will tend to reach the equilibrium state. If the values landed in the white areas, the system will exhibit unstable phenomena.Figure 2
The 2D stable region ofg1 and g2.Figure 3
The 2D stable region ofg3 and g4.In Figure2, when g1 increases, the tendency of g2 should be reduced. In the stable region, it has the opposite effect on g1 and g2, so does Figure 3. Through Figures 2 and 3, we found that the stable range of g1 and g2 is almost twice as much as g3 and g4, when the value of g1 is larger than others’ price adjustment parameters. Period doubling and chaos should be avoided. The supply chain system should be kept in the stable state as long as possible.


## 3.2. The Bifurcation and Chaos Behavior

The parameter basin has been used to describe 2D bifurcation effect. The results are shown in Figures4 and 5. Different colors represent different cycles. Blue represents the stable region in Figure 4, and green represents the stable region in Figure 5. Orange, pink, purple, indigo, brown, yellow, and black regions represent the stable cycles of the periods from 2 to 8, respectively. The gray region is non-convergence, and the white region is divergence in both Figures 4 and 5. The system can remain stable, when the adjustment parameter gi is relatively small. As g1 and g2 increase, the system will go into the stable cycles of period 2. If g1 and g2 continue to increase, the system will go into the stable cycles of period 4, which is shown in Figure 4. However, the system continually changes to the stable cycles of periods 2, 4, 8, and so on. If g3 and g4 keep on increasing, the system will go into chaos state. This process is depicted in Figure 5. In this situation, the market will become chaotic and the market participants cannot make good decisions to reach the best profits.Figure 4
2D bifurcation diagram ofg1 and g2.Figure 5
2D bifurcation diagram ofg3 and g4.When the adjustment parameter changes in a small range from 0, the variables will be stable. However, if the decision variables adjust sharply, the variables will be unstable. Especially, in chaotic state, the system will become sensitive and unpredictable. In economic terms, when the adjustment parameter exceeds a certain limit, the intense competition and price drastic fluctuation will occur.The bifurcation diagram is an intuitive way to show the dynamic characteristics of the system. One of the parameters is fixed and the other changed within a certain range. In the following bifurcation diagrams, the blue line representsk1 and the red line k2. The green line represents w1 and the purple line w2.The largest Lyapunov exponent (LLE) is another efficient way to make clear the complex dynamic characteristics of the system, which also is corresponding to the variation of the bifurcation diagram when discussing the same parameter. The dramatic variation of LLE is accompanied by the decision variables changing. If the LLE < 0, it means the system is in a stable state. When the LLE = 0 for the first time, the system stays at a critical point and usually the bifurcation appears. If the LLE > 0, it indicates that the system runs into the unstable state. The manufacturer and retailers should be cautious to prevent system chaos.The following part gives the bifurcation diagram and LLE diagram with respect to the parametersg1, g2, g3, and g4. The bifurcation diagrams of k1, k2, w1, and w2, with respect to the adjustment speeds of the sales commissions, are shown in Figures 6 and 7. In the same condition, Figures 8 and 9 depict the variation of LLE with respect to g1 and g2.Figure 6
Bifurcation diagram ofk1, k2, w1, w2 when g1∈1.5,3.10.Figure 7
Bifurcation diagram ofk1, k2, w1, w2 when g2∈1.5,2.98.Figure 8
The largest Lyapunov exponent of the system with respect tog1∈1.5,3.10.Figure 9
The largest Lyapunov exponent of the system with respect tog2∈1.5,2.98.Wheng1∈1.5,3.10 and with the other parameters settled down, the conditions of k1, k2, w1, and w2 are shown in Figure 6. Because the dots of w1 and w2 gather together, we further enlarge critical parts in sub-windows in order to describe the bifurcation conditions of w1 and w2. The upper small picture is the blowup of the bifurcation diagram for w1 and w2, and the other one is the blowup of the bifurcation diagram for k2. When g1∈0,2.338, k1, k2, w1, and w2 are all in the stable state. When g1 grows to 2.339, k1, k2, w1, and w2 bifurcate at the first time and enter into the stable cycles of period 2. If g1 continues to increase, the system will enter into the stable cycles of period 4.Figure8 reveals the variation of LLE along with g1 increasing, which expresses the pattern of the complex dynamic behavior caused by the growth of the price adjustment speed. Figure 8 explores the LLE of the system with respect to g1∈1.5,3.10. When g1∈0,2.338, the LLE is below zero, which indicates that the system is stable. When g1 increases to 2.339, the LLE is equal to zero and the bifurcation occurs in the system. The bifurcation point in Figure 8 is the same as that in Figure 6.Figures7 and 9 also have the same bifurcation point. k1, k2, w1, and w2 bifurcate at 2.156 as g2 varies from 0 to 2.98. If g2 continues increasing, the system runs into the cycles of period 4. In order to describe the changing processes of the system from the stable state to the chaotic region more clearly, g1 and g2 start from 1.5 in Figures 6–9.The bifurcation diagrams ofk1, k2, w1, and w2, with respect to the adjustment speeds of the wholesale prices, are shown in Figures 10 and 11. In the same condition, Figures 12 and 13 depict the variation of LLE with respect to g3 and g4.Figure 10
Bifurcation diagram ofk1, k2, w1, w2 when g3∈0.7,1.70.Figure 11
Bifurcation diagram ofk1, k2, w1, w2 when g4∈0.7,1.73.Figure 12
The largest Lyapunov exponent of the system with respect tog3∈0.7,1.70.Figure 13
The largest Lyapunov exponent of the system with respect tog4∈0.7,1.73.k1, k2, w1, and w2 bifurcate at 1.096 as g3 varies from 0 to 1.70. k1, k2, w1, and w2 bifurcate at 1.107 as g4 varies from 0 to 1.73. If g3 and g4 continue increasing, the system runs into cycles of periods 4, 8, and so on. Next, the system will fall into chaos gradually. In this part, g3 and g4 start from 0.7 in Figures 10–13.Above all, Figures10–13 sufficiently indicate that if the wholesale prices of the manufacturer are adjusted drastically, the system will fall into chaos. On the other side, as shown in Figures 6–9, if the sales commissions of the retailers are adjusted radically, the bifurcation will appear but the chaos will not occur. From an economic point of view, if the adjustment parameter rapidly changes, the market will fall into the chaotic state. Therefore, the manufacturer and retailers should keep the price adjustments within a reasonable range.The substitutability of the retailers will be discussed in the following part.μ=0 represents the demands which are absolutely independent. Each retailer is a monopolist in its own area. μ=1 represents the two retailers who are totally substitutable. When μ∈0,0.89 and the other parameters are kept constant, the situations of k1, k2, w1, and w2 are shown in Figure 14. When μ grows to 0.872, the system begins the first bifurcation which is the stable cycles of period 2. If μ continues to increase, the system runs into the stable cycles of periods 4, 8, and so on. Then, the system will fall into chaos gradually. The small picture in Figure 14 is the blowup of the bifurcation region for k1, k2, w1, and w2, when μ∈0.8,0.89.Figure 14
Bifurcation diagram ofk1, k2, w1, w2 when μ∈0,0.89.Figure15 explores the LLE of the system with respect to μ∈0,0.89. When μ∈0,0.871, the LLE is less than zero, and the system is in a stable state. While μ grows to 0.872, LLE changes into zero for the first time, and the system falls into the bifurcation state, the completely same bifurcation points in Figures 15 and 14.Figure 15
The largest Lyapunov exponent of the system with respect toμ∈0.8,0.89.In Figure14, the manufacturer is more sensitive to the substitutability of the retailer, which leads to a rise in the wholesale price. However, the retailer’s profits are robust to the changes in the retailer’s substitutability. Under this condition, fierce competition may lead to the dominance of the manufacturer. Nevertheless, bifurcation and chaos will still occur in the market after the fierce competition, once a certain extent of substitutability remains among the retailers of different channels.The impacts of consumer’s channel preference on the dynamic system will be discussed in the following part. Whenη∈0,3.80, η=3.80, g1=0.4, g2=0.4, g3=0.4, and g4=0.4, the situations of k1, k2, w1, and w2 are shown in Figure 16. The lines in the bifurcation diagram indicate that other parameters are stabilized separately, when η is changing in the range of [0, 3.80]. When η grows to 2.558, the system begins the first bifurcation which is the stable cycles of period 2. If η continues to increase, the system runs into the stable cycles of periods 4, 8, and so on. After that, the system will fall into chaos gradually. Figure 17 proves the same conclusion as Figure 16 does.Figure 16
Bifurcation diagram ofk1, k2, w1, w2 when η∈0,3.80.Figure 17
The largest Lyapunov exponent of the system with respect toη∈0,3.80.The curves of the traditional retailers and online retailers are shown in Figure16. When η>1, it can be found that the channel preference of consumers will convert from one channel to another. If one channel has more consumer preference, its wholesale prices and sales commission will rise; w2>w1 and k2>k1 are displayed in Figure 16. However, if the channel preference obviously comes to one channel, it is possible that bifurcation and chaos appear in the system.In this paper, the chaos attractor has been investigated to the dynamic characteristics of the system. Strange attractor is a set of numbers of indefinite points. The larger the space of the strange attractor is, the more complex the structure will be. No matter how instable the system is, the points are always attracted along a certain trajectory.Figure18 shows the chaotic attractor of w1 and w2, when k1=0.31, k2=0.1, w1=0.31, w2=0.4, g1=0.4, g2=0.13, g3=1.65, g4=0.15, μ=0.5, and η=1.1, the system becomes complex and even moves into chaotic state. The dynamic characteristics of the system are obvious. Based on parameter studies, w1 changes in a larger range than w2 when the adjustment speeds of wholesale price g3 change more intensively than g4.Figure 18
The chaotic attractor ofw1 and w2.The time series of different decision variables are shown in Figures19 and 20. In the chaotic state, the system is more sensitive to the initial value. The phenomenon can be called as the initial value sensitivity. In Figure 19, to better show the character of the chaotic system, we assume the retailer 1 to make decisions under the condition that η=3.75 and show the time series of k1 after 100 iterations. The blue line represents the initial value with k1=0.1, and the red line represents the initial value with k1=0.101, when the dynamic system state is in a chaotic state. We also set the other parameters’ values as follows: k2=0.1, w1=0.31, w2=0.4, g1=0.4, g2=0.4, g3=0.4, g4=0.4, and μ=0.5. As shown in Figure 19, although the initial value of sales commissions only changes from 0.1 to 0.101, after 100 iterations, a large gap between the two tracks will occur.Figure 19
Time series ofk1 in a chaotic state.Figure 20
Time series ofk1, k2, w1, w2 in a stable state.Figure20 shows the time series of k1, k2, w1, and w2 in a stable state. We set the initial values of the wholesale prices and sales commissions as k1=0.3, k2=0.6, w1=1, and w2=0.5 in Figure 20. We set the initial value of k1 and w2 from a low value, while we set the initial value of k2 and w1 from a high value. We set the same initial values of k1, k2, w1, and w2 with those above. k1, k2, w1, and w2 would gradually flatten after 12 iterations shown in Figure 20.In short, Figures19 and 20 have different meanings. The adjustment speeds of the wholesale prices and sales commissions depend on the price fluctuation levels. As such, the manufacturer and retailers should keep price adjustment in reasonable ranges.


## 4. Chaos Control

The excessive competition will cause the chaotic phenomenon of a market, which poises negative effect on all competitive subjects in the market and impedes the economic operation. Therefore, some chaos control methods have been made use of to propel the system to return to stable equilibrium. There are a lot of chaos control methods such as continuous feedback control, self-adaptation control, and OGY. In this research, parameter adjustment control method has been used. The manager can control the chaotic behavior through introducing the control parameterυ. Huang [23] conducted the chaos control with setting the control parameter as υ=0.6. The values of the other parameters have been set in the sections above. The stable range and the bifurcation diagram will be given to account for the state of the system. Under the parameter adjustment control, the system can be expressed as follows:
(18)k1t+1=1−υk1t+g1k11‐2k1−w1+k2+w2μ+υk1t,k2t+1=1−υk2t+g2k2−2k2−w2+η+k1+w1μ+υk2t,w1t+1=1−υw1t+g3w11−k1−2w1+w2μ+k2+w2μ+υw1t,w2t+1=1−υw2t+g4w2−k2−2w2+η+w1μ+k1+w1μ+υw2t.Under the process of chaos control, three parameters, including wholesale price, consumer’s channel preference, and the substitutability of retailers, will be discussed in this part. Ifg3 equals 1.65, the system will be chaotic. When the control parameter υ varies from 0 to 0.7, the changes of the system are shown in Figure 21. Besides, if μ equals 0.892, the system will be chaotic. When the control parameter υ varies from 0 to 0.3, the changes of the system are shown in Figure 22. Finally, if we set η=3.8, g1=0.4, g2=0.4, g3=0.4, and g4=0.4, the system will be chaotic. When the control parameter υ varies from 0 to 0.4, the changes of system are shown in Figure 23.Figure 21
Bifurcation diagram whenυ varies from 0 to 0.7 and g3=1.65.Figure 22
Bifurcation diagram whenυ varies from 0 to 0.3 and μ=0.892.Figure 23
Bifurcation diagram whenυ varies from 0 to 0.4 and η=3.8.The above figures indicate that the chaos of the system has been controlled when introducing the controlling parameterυ. With υ increasing, the system changes from the chaotic region to the bifurcation region. When υ is big enough, the system will fall into the steady state. For the manufacturer and retailer, they should try their best to control the system into the stable state and maximize their profits.


## 5. Conclusion

A supply chain system consisted of one manufacturer and two retailers have been discussed in this study. One is a traditional retailer and the other an online retailer. The complex dynamics of the system is studied by numerical simulation. The dynamic characteristics are investigated through 2D bifurcation diagram, the largest Lyapunov exponent, chaotic attractor, and time series. Finally, parameter adjustment control method is used to control the chaos.In this paper, a pricing problem in the supply chain with one manufacturer who supplies an identical product to two retailers has been researched. For example, Huawei supplies Mate 9 to China Resources Vanguard Co. Ltd. and JD.com. CR Vanguard and JD.com are the typical representatives of traditional and online channels, respectively. These research results indicate that if the retailers compete with each other, the substitutability between the two sales channels will be generated. The competitions between retailers are helpful for the manufacturers. However, when the competition is fierce, the traditional and online channel can be replaced by each other, and the system will be reduced into a chaotic state. This article has demonstrated that consumer’s channel preference can affect the wholesale prices and sales commissions. If the consumers prefer any channel of the two, its wholesale prices and sales commissions may be raised. When the preference is too intense, the system will be unstable and even lead to chaotic phenomena. The parameter adjustment control method is performed on the parameters of wholesale price; the substitution of retailers and consumer’s channel preference make the system stable by adjusting the parameterυ. For all these works, the achievements of this study have both theoretical and practical significances for the manufacturer and retailers.The conclusions of this paper can be helpful to the managers; however, there still exist some limitations. On the one hand, it considered a model of two-stage and dual-channel supply chain, which might be simple in practice. On the other hand, we only did the numerical simulation but did not make an empirical analysis of influence factors as an example. Hence, in the future, research based on large amounts of real data is significant. We may do further study in multi-stage and multi-channel supply chains in a practical market.



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