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Apart from the price fluctuation, the retailers’ service level becomes another key factor that affects the market demand. This paper depicts a modified price and demand game model based on the stochastic demand and the retailer’s service level which influences the market demand decided by customers’ preference, while the market demand is stochastic in this model. We explore how the price adjustment speed affects the stability of the supply chain system with respect to service level and stochastic demand. The dynamic behavior of the system is researched by simulation and the stability domain and the bifurcation phenomenon are shown clearly. The largest Lyapunov exponent and the chaotic attractor are also given to confirm the chaotic characteristic of the system. The simulation results indicate that relatively small price adjustment speed may maintain the system at stable state. With the price adjustment speed gradually increasing, the price system gets unstable and finally becomes chaotic. This chaotic phenomenon will perturb the product market and this phenomenon should be controlled to keep the system stay in the stable region. So the chaos control is done and the chaos can be controlled completely. The conclusion makes significant contribution to the system referring to the price fluctuation based on the service level and stochastic demand.

Bertrand duopoly model is frequently used when the competitive behaviors are discussed in proper research. Hence, the dynamic behaviors of the whole supply chain system have already been probed. However, the retail service level plays an important role in influencing the retail price which indirectly changes the dynamic behaviors of the whole supply chain system that contains one supplier and two retailers.

The two retailers are duopoly in the product market. Xiao and Yang [

C. H. Tremblay and V. J. Tremblay [

Ahmed et al. [

Esmaeilzadeh and Taleizadeh [

The structure of the paper is as follows. Firstly, the basic Bertrand duopoly model considering the service level and stochastic demand is established in Section

Since oligopoly game model has already been researched in previous literature, while the oligopoly situation has been discussed, we consider that the other retailer comes into the monopoly market. Due to the entrance of the other retailer, exclusive monopoly was broken. What is more, the marketing competition is more vehement. Retailer’s demand is influenced by the other retailer’s price. In addition to retail price, we also consider that the market demand is associated with the service level of themselves and the competitors. Meanwhile, we assume the products that two retailers sell are homogeneous. Hence, the demand function of the two retailers is shown as follows:

In the expression above, we find that the market demand is affected by both retail price and the service level. The higher service level will bring about the larger cost. The cost of service level is shown in

In (

While the product market is complex and stochastic, the final demand of the customers is uncertain. We set the demand as stochastic variable

The profit function of the manufacturer is depicted in (

The retailers’ operating goal is to maximize profits. The retailer’s profit function is shown as follows:

When retailers and manufacturer in the market share the information, the expected profits decision models of manufacturer and retailers are shown as follows, respectively:

We can deduce the marginal profits of two retailers through calculating the first-order partial derivative of (

For the sake of simplicity, we make

If there is only one retailer in the market, the retailer regards profit maximization as the goal to give the optimal decision in the circumstance of discrete game. When there are two retailers in the market, the optimal decision of retailer 1 is influenced by retailer 2. Meanwhile, retailer 2 optimal decision is also affected by retailer 1 decision. Hence, we assume the information in the market can be shared between two retailers; that is, the optimal decisions of two retailers are known to each other. When the marginal profits reduce to zero, we have the optimal decision for retailers. The retail price and the service level are the decision variables.

For retailer 1, we calculate the first-order derivative of its profit. When the marginal profit gets to zero, the optimal retail price and optimal service level will be derived in equation. The detailed process is as follows:

Analogously, the optimal decision of retailer 2 will be conducted. Equation (

When two retailers make their own optimal decision, we may further find the second derivatives of the retail prices and service level are less than zero. Therefore, the optimal decision of the system will be derived, which means the system has the equilibrium point:

Through jointing (

In (

Due to the complexity of the product market, the competition between enterprises in the market is increasingly fierce. Retailers do not know the perfect information of the market, while the market is complex and customers’ demands are uncertain. In addition, the more the transactions are, the greater the uncertainty will be, the more incomplete the information will be. Hence, the market demand cannot be forecasted accurately. The bounded rationality expectations solve this problem faultlessly. The retail price of next period will be decided according to the retail price in this period and the marginal profits, which makes up the discrete dynamic system of the product market.

In the equation above,

The system (

We may find that

In the equation above,

The characteristic polynomial of the Jacobian matrix takes the following form:

In order to guarantee that the Nash equilibrium is local stability,

Due to the fact that limitations are so complex, the process to solve inequality (

In consideration of the complexity of system (

The decision variables of the system are retail prices of two retailers. As retail prices obtain different values, the system will change to either the stable range or the unstable range. Figure

The stable range of system (

The fixed point is stable when all the eigenvalues of the Jacobian are in the unit circle. When

The characteristic root of system (

We discuss the bifurcation and chaos behaviors of the system in this section with parameter basin plots and the bifurcation diagram. We set the initial values of the system as

Figure

The parameter basin plots of system (

By observing this figure, we may find the retail price changes from the stable region to the even period cycles and next it gets to chaotic region. This phenomenon is the flip bifurcation, whereby the retailers adjust their own retail price, the system will go from the even period cycles to the chaotic region directly.

Figures

Bifurcation diagram of

Bifurcation diagram of

Figure

Figures

Bifurcation diagram of

Bifurcation diagram of

Analogously, the complex dynamic behavior of retail price 2 considering the price adjustment speed of retailer 2 varying from zero to 3.5 will be shown in Figure

Through the analysis of Figures

3D bifurcation diagram of

The largest Lyapunov exponent (LLE) is another way to make clear the complex dynamic characteristics of the system. And the LLE represents the extreme sensitivity to initial value of the system vividly that two of the same initial value generated in the trajectory separate according to index methods with the passage of time. As we want to judge whether the chaotic phenomenon exists in the system, we just need to observe the LLE intuitively. If the LLE is more than zero, it means that the differences of the initial value increase with exponential form after periods of iteration; that is, the system falls into chaotic region. On the contrary, if the LLE is less than zero, it means that the system is in stable state, while the system is not initially sensitive and keeps a fixed value all the time. When the LLE equals zero, the system is periodic and quasi-periodic. The expression of LLE is as follows:

We used MATLAB to simulate Figures

The largest Lyapunov exponent of system with respect to

The largest Lyapunov exponent of system with respect to

Entropy is an important quantity to characterize chaotic systems. In different types of dynamic systems, the entropy values are different. Entropy values can be used to distinguish between regular motion, chaotic motion, and random motion. In the random motion system, the entropy is unbounded; in the regular motion system, the entropy is zero; in the chaotic motion system, the entropy is greater than zero. The greater the entropy, the greater the rate of loss of information, the greater the degree of chaos in the system, or the more complex the system.

It can be seen from Figure

The entropy of system with respect to

The entropy of system with respect to

The chaotic attractor investigates the chaotic characteristics of the system and the chaotic attractors will be given in Figures

The chaotic attractor of system when

The chaotic attractor of system when

The chaotic attractor of system when

In Figure

From the parameter basin plots, bifurcation diagram, the largest Lyapunov exponent, and entropy of system, we may see that the systems are in chaotic states while

The time domain response depicts the changing process of the price data with the increase of time, which is arranged by the whole time index. There are different shapes for the time domain response. Figures

The time domain response of system when

The time domain response of system when

The time domain response of system when

In Figure

With the findings of sections above, we come to the conclusion that if the adjustment speeds of the retail prices are in chaotic region, the system will be complex, and the manager cannot make decision in time that phenomenon is not expected to appear in the product market. Therefore, we should take measure to expand the stable region and control the chaos further.

Here, we employ the widely used control method, that is, parameter adjustment control method to control the chaotic phenomenon. Ma and Li [

Figure

The stable range of system (

The complex dynamic behavior of system (

Bifurcation diagram of

Bifurcation diagram of

Bifurcation diagram of

Bifurcation diagram of

Previous researches have explored the price fluctuation with price and demand game model in the supply chain management. However the retailers’ service level has not been involved in the basic demand model. This paper depicts a modified Bertrand duopoly model based on the stochastic demand and the retailer’s service level which influences the market demand decided by customers’ preference, while the market demand is stochastic in this model. How the price adjustment speed affects the stability of the supply chain system with respect to service level and stochastic demand has been studied in this paper. The dynamic behavior of the system is researched by simulation and the stability domain and the bifurcation phenomenon are shown clearly. The largest Lyapunov exponent, the chaotic attractor, and the time series are also given to confirm the chaotic characteristic of the system.

We come to the conclusion that (1) with the price adjustment speed gradually increasing, the price system gets unstable and finally becomes chaotic; (2) a relatively small price adjustment speed may maintain the system at stable state; (3) chaos can be controlled completely with the parameter control method. This chaotic phenomenon will perturb the product market. Hence, the conclusion makes significant contribution to the system, referring to the price fluctuation based on the service level and stochastic demand.

In spite of the contribution that this paper has offered to the managers, some limitations still exist in this paper. Firstly, the supply chain only contains two retailers; in order for this study to be more close to the reality and be used in more scales of enterprises, we may study the dynamic game behavior in the three-oligarch product market. Secondly, this paper only does the numerical simulation, while there are no real data to support the conclusion of the paper. Hence, some real data should be collected to identify the conclusion.

The authors declare no conflicts of interest.

Weiya Di and Junhai Ma built the supply chain model; Junhai Ma provided economic interpretation; Weiya Di carried out numerical simulation; Hao Ren performed mathematical derivation; the authors wrote this research manuscript together. All authors have read and approved the final manuscript.

The research was supported by the National Natural Science Foundation of China (71571131).