This paper studied some realistic recycle fields at present in China, such as some daily machinery and equipment, electronic products, and plastic products. Under such circumstance, three oligarch retailers are a common situation. In this paper, we model and analyze the game of three oligarch retailers in a closed-loop supply chain. The stability of the Nash equilibrium, the bifurcation and chaos phenomenon of the recycling price and new product quantity when retailers change its adjustment speed, the profit trends of the three retailers, and the power spectrum are studied. At last, we used the parameter adjustment and the variable state feedback control strategy for chaos control, and the chaos of the system is delayed effectively. For closed-loop supply chain research, the conclusions of the numerical simulation in this paper not only have realistic guiding significance, but also have theoretical reference value.

As an effective method for studying multistage decision problems in supply chain, game and complexity theory has been an enduring one. Zhang and Da [

Meanwhile, the game and complexity theory has been applied to different realistic fields. Ma and Bangura [

Based on others’ former research, for the realistic recycle fields at present in China, such as some daily machinery and equipment, electronic products, and plastic products, we build a closed-loop supply chain model with three oligarch retailers in this paper. According to this model, the retailers recycle the waste products, return them to the manufacturer for repair, and then sell the second-hand products after fix and new products together. After being repaired, the second-hand products are the same as the new products in performance and appearance, but they have different customer approval degree, so their prices are different.

To be more realistic, this paper builds a three oligarch retailers competing model based on price difference between new and second-hand products. The stability of the Nash equilibrium is discussed in this paper; through bifurcation diagram, the maximum Lyapunov index diagram, and attractor, the stability and chaos of the system are studied; the trends of the three oligarch retailers’ profit and the system average profit are researched. At last, we successfully controlled the chaos of the system.

At present, the research on closed-loop supply chain at home and abroad is mainly for game equilibrium with different market structures, in which role of the supply chain is to recycle the waste products and so on. Very few papers considered the new and second-hand product market together and rarely applied complexity theory to the combination market. This paper focuses on the new and second-hand combination market and considers the game and complexity of the three oligarch retailers in the closed-loop supply chain. In this model, the retailers recycle the waste products, return them to the manufacturer for repair, and pay some cost to the manufacturer for repair. After repair, the retailers sell the second-hand products and the new products together. The second-hand products have no difference in performance and appearance from the new products, but the difference in customer approval degree between them results in different prices. In this closed-loop supply chain, the manufacturer provides new products and fixes waste products; three retailers are three oligarchs; they recycle waste products and sell new and second-hand products.

Assumptions are as follows:

three retailers

the recycle quantity of each retailer is related to its own and competitors’ recycle prices;

the manufacturer can repair all the waste products, with no waste in repair process;

for the difference in customer approval degree between new and second-hand products, their price is different;

demand is bigger than supply, so the new and second-hand products can both sell out, but price decreases when supply increases.

Symbols are as follows:

manufacturer

retailer

For the three oligarch retailers, the functions of each one’s waste products recycle quantity are as follows:

The functions of the new and second-hand products’ price are as follows:

Each retailer has two decision variables: recycle price

In real economic circumstance, the three oligarch retailers are all information limited; under the bounded rational decision rule, each retailer at period

In the previous equation,

Integrate formulas (

In order to study the behavior of different retailers, we need to solve the equilibrium of discrete dynamic system (

At present in China, the recycling industry is still in its infancy; besides the customers’ environmental protection consciousness is weak, so the customers’ recycle price sensitivity would be strong. From the customer psychology, even though the new and second-hand products have no difference in their performance and appearance, the new product would have a higher price. Based on the actual investigation of Chinese retail market, we find that the situation is in this case, especially in the sales process of some high price products. This situation is more obvious; therefore, according to the actual statistical data, the values of the market parameters are assigned as follows:

Discrete dynamic system (

Next we will discuss the stability of the equilibrium; the Jacobian matrix of discrete dynamic system (

For Nash equilibrium

From the Jacobian matrix of equilibrium

From the Jacobian matrix of equilibrium

Take

Price bifurcation with the change of

Quantity bifurcation with the change of

Price bifurcation with the change of

Quantity bifurcation with the change of

From Figures

Besides, we note that, in Figures

Figure

Lyapunov index with the change of

Lyapunov index with the change of

Comparing Figure

The chaos state of the system also can be proved by its chaos attractor; when

Chaos attractor of the system.

When each retailer changes its adjustment speed of its recycle price or its new product supply quantity, the decisions of them become complicated. In this situation, how would the profit of each retailer in the system become? Would it increase when the market uncertainty increases? Or would it decrease when the market became more complicated? Does every retailer’s profit change in the same way? Here we still use the adjustment speed

Track figure of profit with

Average profit of the oligarchs with

Track figure of profit with

Average profit of the oligarchs with

From Figures

So, if retailer 1 adjusts

Then we have the study on the power spectrum of

When the system changes from stability to chaos, all the three-oligarch retailers’ decisions become complicated, and the profit of each one declines fast, so no retailer expects the chaos. In this case, the three oligarch retailers can put some adjustment measures to the market, through the coordination and cooperation between them to control the system chaos, in order to delay or eliminate the chaos in the market and assure that their profits reach the maximum.

In this paper, we will use parameter adjustment and variable state feedback control strategy, to control the three retailers’ recycle price

Note the original discrete dynamic system (

Use the same market parameters as before, and solve the controlled system (

Obviously, the value of

Take

Chaos control of

Chaos control of

From Figures

Take

Track of

Track of the

Track of

Track of the

Comparing Figures

The larger stability range and the greater flexibility also can be reflected by the maximum Lyapunov index variation figure of retailer 1’s adjustment speed

Lyapunov index variations of

Lyapunov index variations of

Comparing Figures

From Figure

Chaos attractor of the system after adjustment.

According to the analyses of the three retailers’ profit and their average profit in Section

In the reality of economic life, the value of the adjustment parameter

In this paper, the model of a closed-loop supply chain based on price difference between new and second-hand products is established, and the game and complexity theory of the three oligarch retailers is discussed. Through numerical simulation, the conclusions are as follows.

Referencing to the model built in this paper, the participants in the supply chain can adjust their own decisions effectively in different time, and they can also work together to control the chaos in the supply chain. So the paper has a strong practical guiding significance to the market stability and the improvement of the retailers’ profits. Besides, this paper studied the closed-loop supply chain with game and complexity theory and meanwhile combined the sale of the new product with the recycle and sale of the waste product, so it also has a certain reference value to the theory research of the closed-loop supply chain. The results of this paper provide the direction for further research on closed-loop supply chain with a time delay characteristics.

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

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