This paper studied system dynamics characteristics of closed-loop supply chain using repeated game theory and complex system theory. It established decentralized decision-making game model and centralized decision-making game model and then established and analyzed the corresponding continuity system. Drew the region local stability of Nash equilibrium and Stackelberg equilibrium, and a series of chaotic system characteristics, have an detail analysis of the Lyapunov index which is under the condition of different parameter combination. According to the limited rational expectations theory, it established repeated game model based on collection price and marginal profits. Further, this paper analyzed the influence of the parameters by numerical simulations and concluded three conclusions. First, when the collection price is to a critical value, the system will be into chaos state. Second, when the sale price of remanufacturing products is more than a critical value, the system will be in chaos state. Last, the initial value of the collection price is sensitive, small changes may cause fluctuations of market price. These conclusions guide enterprises in making the best decisions in each phase to achieve maximize profits.
The product life cycle becomes shorter with the rapid development of market economy. A lot of products have been washed out before life end. Such condition not only creates a tremendous waste of resources but also brings great harm to people. Therefore, the “resources-products-waste products-remanufacturing product” closed-loop type of economic growth mode appears. It realizes the economic development and utilization of resources and environmental protection in coordination strategy of sustainable development goals. Enterprise began to take a positive attitude to collect products from customers. The research on collecting control problem becomes inevitable.
This paper draws on and contributes to several streams of literature, each of whom we review below. A growing body of literature in operations management addresses reverse logistics management issues for remanufacturable products. References [
References [
The existing literatures studied major in a single phase of the mathematical model using game theory under the assumption that the closed-loop supply chain contains a manufacturer and a retailer and concludes equilibrium in perfectly rational state. However, market uncertainty caused enterprise not to be able to make decisions in perfectly rational. At present, many manufacturers devote themselves to the core competitiveness; they contract the collection of used products to a third party. So this paper sets up multistage game model of closed-loop supply chain which consists of a manufacturer and two competition collectors business based on limited rationality with China’s remanufacturing development and studies its complex dynamic characteristics.
First, this paper considers only remanufacturing products market. That means new products and remanufacturing product do not form a competitive market. It is more in line with the actual conditions of China. The manufacturer manufactures new products and remanufacturing products.
Second, the closed-loop supply chain includes a manufacturer and two competitive collecting corporations, the collecting corporations collect waste products from consumers and return the manufacturer. The manufacturer transfers payments to the collectors, and she sales directly to consumers. The manufacturers and two collectors are independent decision makers, and their strategic space is to choose the best collecting price. Their goal is to maximize returns in discrete time period as
Third, the number of the collection is increasing function of collecting price. The collecting capability and manufacturing capability are unlimited. In order to simplify the problem and emphasize main parameters influence of the system, all the collecting products can be manufactured, as shown in the MCTM collecting mode (Figure
MCTM collecting mode.
Nash equilibrium can be concluded by the first-order conditions of three reaction functions, which meet
In formula (
In reality, the game between node enterprises in closed-loop supply chain is continuous, enterprises’ decision-making is a long-term repeated process, and its action has long-term memory. And each node enterprise does not completely control the market information and also cannot fully expect future market changes, so based on limited rational expectations decision we adjust process with marginal gains. They can make the next-period price decision on the basis of the local estimate to his marginal profits in current period. Their price adjustment processes are
From the system (
We put
At present, only a few simple dynamic systems are analyzed with analytical method, but complex dynamic system mainly uses the numerical analysis method. This paper processes numerical simulation on system (
Change the discrete system (
Consider the continuous-time nonlinear dynamical system
Let the function
Supposing that
And the adjoint eigen vector
and satisfies the normalization
The first Lyapunov coefficient at the origin is defined by
Next, we calculate
It means
It can be got from calculating
(1) When
The result is as follows.
(1) Fix the value of
The change of
The change of
We can know from Figure
The change of
The change of
From Figures
(2) Fix the value of
The change of
The change of
The change of
The change of
(3) Fix the value of
The change of
The change of
The change of
The change of
(2) When
The result is as follows:
(1) Fix the value of
The change of
The change of
The change of
The change of
(2) Fix the value of
The change of
The change of
The change of
The change of
(3) Fix the value of
The change of
The change of
The change of
The change of
The result is as follows:
(1) Fix the value of
The change of
The change of
The change of
The change of
(2) Fix the value of
The change of
The change of
The change of
The change of
(3) Fix the value of
The change of
The change of
The change of
The change of
Premising
Using Matlab, parameters influence on the system (
The max Lyapunov index.
When initial setup and other parameters are the same and
(a)
(a)
According to the numerical results, no matter how large the collection price adjustment parameters are, the collection price traverses the whole value area over time, but
(a)
Suppose that the manufacturer and collectors have principal and subordinate relationship, the manufacturer is Stackelberg leader, and collectors are followers; then they process sequential dynamic game; the game equilibrium is Stackelberg equilibrium. In this game, the manufacturer makes the decision of sales price and collection price according to the market informa Lyapunov exponent function abouttion; then two collectors make decision according to the decision-making of the manufacture:
The result is
Formula (
With Nash equilibrium value the parameters values are
The manufacturer makes decision based on limited rational expectations. She adjusts the game process on the basis of marginal gains. If the marginal profits of
In formula (
(a)
(a)
(a)
(a)
(a)
Through numerical simulation and comparison analysis, it can be concluded that Stackelberg game equilibrium is better than Nash equilibrium from the point of view of the profits and collecting price. Three conclusions can be summarized as follows. First, when the manufacturer and the two collectors make decision by static game, the collection price and profits of the three parties are lower than the optimal of dynamic game equilibrium. Second, as the manufacturer and the two collectors independently make decisions, the three parties will try to lower the collecting price in order to obtain the maximum profits. Third, as the manufacturer and the two collectors independently make decisions, the system more easily reaches chaos, and the cycle of stable state is shorter.
Centralized control is that the manufacturers and the collectors codetermine to realize profits maximization of the supply chain system. We get formulas (
Value with above the top:
Manufacturers and collecting business make decision based on rational expectations, and the decision-making basis is marginal profits. Figure
(a)
This paper researches on that used products collecting pricing game model and complexity analysis in a closed-loop supply chain which consist of a manufacturer and two collectors. Corresponding to the discrete system, we have established and analyzed in detail the corresponding continuity system.
Through quantitative analysis of collecting price, time sequence diagram, and Lyapunov index, the paper describes game evolution rule in the closed-loop supply chain. We describe a series of chaotic system characteristics, based on which, we have a detailed analysis of the Lyapunov index which is under the condition of different parameter combination, and draw the figure under different conditions. The analysis found that, first, when the collection price is to a critical value, the system into chaos state. Second, the sale price of remanufacturing products are more than a critical value, the system into chaos state. Last, the collection price system of the initial value of which is sensitive, small changes in the initial value may cause market price fluctuations.
It also draws conclusions that business profits, supply chain system profits, collecting price in Stackelberg game model are superior to Nash game model. In Nash equilibrium, the manufacturer and the collectors all lower the collecting price to gain, respectively, maximum profits. The system is easier into chaos state than Stackelberg, and stable state period is shorter.
Through comparing centralized decision-making with decentralized decision-making, we have concluded that the collecting price and system profits of the latter are lower. This conclusion for the supply chain guide enterprises sure each phase of the best provide decision basis for the collection price, in order to achieve maximize returns.
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.