#### Abstract

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.

#### 1. Introduction

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 [1, 2] defined as the supply chain to reverse supply chain which formed a complete closed-loop system (closed-loop supply chain, hereinafter referred to as CLSC.) Reference [3] studied the pricing methods under three collecting channels which manufacturer, retailer, and third-party collector collected products, respectively. Reference [4] studied manufacturers should be responsible for the management of that collecting based on Xerox corporation and Kodak corporation. Reference [5] studied three kinds of collecting channels of manufacturers collecting, retailer collecting, and the third-party collecting in closed-loop supply chain. They found that the colleting distance was closer from consumers and the collecting efforts were more effective. References [6–8] described reverse logistics management mode of the third party. In this collecting mode, manufacturers entrust a third party to perform extended producer responsibility and manage used products.

References [9, 10] studied pricing analysis of reverse supply chain using the game theory method under retailers collecting mode and concluded the optimal strategy of single stage. Reference [11] established the demand model of new products and remanufacturing products from social environmental protection consciousness and consumers’ utility function to new products and remanufacturing products and drew the conclusions. The first is the optimal pricing strategy in different collecting channels in decentralized decision-making. The second is the influence of social environmental protection consciousness. Reference [12] research shows that the channel of manufacturers collecting directly is more favorable to consumers and social environment. In the autoindustry, independent third parties are handling used-product collection activities for the original equipment manufacturers (OEMs). The “big three” automanufacturers in the United States started to invest in joint research and remanufacturing partnerships with dismantling centers to benefit from their scale economies and experience. Third parties such as GENCO Distribution System are also preferred by some consumer goods manufacturers for their experience in used-product collection. In references [13–15] Analyzed the complexity of model for a class of delay complex dynamics, and some valuable conclusions are obtained.

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.

#### 2. Model Assumptions and Notation

##### 2.1. Assumption

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 1).

##### 2.2. Notation

denotes the retail price of the product, a constant, and the unit cost of remanufacturing a returned product into a new one, a constant. is the collection price of the manufacturer transfer payment to collectors in the period, a decision variable of the manufacturer. is the collection price of collectors in the period. is a decision variable of the collectors; that is, , . is the collecting cost of collectors including logistics cost and so forth, a constant. is the collecting numbers for the waste product in the market as a function of collecting price, , , , and subscript , will take values two collectors, where is the waste products number of return voluntarily when the collectors’ collecting price is zero; it denotes consumers environmental awareness; denotes consumers sensitive degree to collecting price; is competition coefficient in the two collectors, which satisfies . The collecting function of the two collectors is, respectively, as follows:

is the profits function of the manufacturer; its marginal profits are concluded by the first-order conditions, which are . can be stated as

is the profits function of the two collectors; marginal profits are concluded by the first-order conditions, which are . can be stated as

is the profits function of the closed-loop supply chain, which can be stated as

is a pricing strategy, which can be obtained by (2) and (3). Furthermore they are satisfied with , , and . It can be proved that is concave function for .

#### 3. Decentralized Control Decision-Making Model

##### 3.1. Nash Equilibrium

###### 3.1.1. Model and Analysis

*(**1) Model*. The manufacturer and collectors are equal in closed-loop supply chain; that is, the two sides make decision at the same time: manufacturer’s decision is to choose the collection price to maximize returns, and collectors’ decision is to choose to maximize their returns.

Nash equilibrium can be concluded by the first-order conditions of three reaction functions, which meet , simultaneously. The Nash equilibrium is the optimal decision of a game in all kinds of may guess, thus make it to gain maximum benefit. The Nash equilibrium is where , , and are

In formula (6), , , , can be expressed as

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 where , , , and , , are used to present the price in the th period and in the th period. Where are positive parameters, , which denotes adjustment speed, respectively, for the manufacturer and two collectors.

From the system (8) we can be conclude that the optimal collecting price of the manufacturer is related to the collection price adjustment coefficient, the two collectors’ collecting price, sale price, and consumers’ environmental protection awareness. Similarly, the optimal collecting price of collectors is related to collecting price adjustment parameters, collecting price of the manufacturer, collecting function, consumers’s environmental protection awareness, and collecting costs.

*(2) Model Analysis*. In closed-loop supply chain, any enterprise decision-making is all according to the maximum profits, the equilibrium less than zero is not practical significance, such as manufacturers to make power mainly because of benefits, collecting business is also benefit with collecting. So only research system (8) is the equilibrium; system (8) of the eight fixed points, respectively, is , , , , , , , . The analysis shows that by , , , , , and , is no real significance. Therefore, we discuss Nash equilibrium which is ; is the reaction function intersection of the manufacturer and collectors, which means marginal profits of both sides are zero, but this does not mean that the result of the game will tend to equilibrium. Instead, a party rational behavior change may cause game process to occur very complex phenomenon. has local stability, this stability region by , , decided. The stability of will be studied. The first we calculate Jacobi matrix of system (8), means

We put into formula (9), according to the Jury stability criterion, the sufficient and necessary conditions of asymptotic stability are that the zeros of characteristic polynomial are in a circular area. Moreover, the characteristic polynomial will satisfy the conditions as follows:

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 (8) by Matlab to expresses the dynamic characteristics. Because remanufacturing is still initial stage in china, consumers environmental protection awareness is lower, and manufacturing cost (including the fixed and variable cost) is higher; for autoparts collecting, the parameters of system (8) can be defined as , , , , , , , , , and to study the local stability of equilibrium point. Get the parameter value into and formula (10) to conclude the value for (17.385, 0.491, 0.489).

Change the discrete system (8) into a continuous system and the points of Jacobi matrix for

Consider the continuous-time nonlinear dynamical system

Let the function be written as where and are bilinear and trilinear functions, respectively. In coordinates, we have

Supposing that has a pair of complex eigenvalues on the imaginary axis, they are , and these eigenvalues are the only eigenvalues with . Let be a complex eigenvector to :

And the adjoint eigen vector admits the properties

and satisfies the normalization .

The first Lyapunov coefficient at the origin is defined by

Next, we calculate , , , and ; If , calculate : having nonzero solutions.

It means

It can be got from calculating

(1) When , matrix has pure imaginary eigenvalues . Calculate , , , .

The result is as follows. where is the Lyapunov exponent function about , , and and is the real part. The following is the figure of .

(1) Fix the value of and , .(i), , and (see Figure 3).(ii) , , and (see Figure 2).

We can know from Figure 3 that when = 0.02, = 0.048, and is between zero and 0.05, is always equal to 1.375.(iii), , and (see Figure 4).(iv), , and (see Figure 5).

From Figures 4 and 5 we can observe the same change of as in Figures 2 and 3.

(2) Fix the value of and , . , , and (see Figure 6). , , and (see Figure 7). , , and (see Figure 8). , , and (see Figure 9).

(3) Fix the value of and , .(i), , and (see Figure 10).(ii), , and (see Figure 11).(iii), , and (see Figure 12).(iv), , and (see Figure 13).

(2) When , matrix has pure imaginary eigenvalues . Calculate , , , .

The result is as follows: where is the Lyapunov exponent function about , , and and is the real part. The following is the figure of .

(1) Fix the value of and , .(i) , , and (see Figure 14).(ii), , and (see Figure 15).(iii), , and (see Figure 16).(iv), , and (see Figure 17).

(2) Fix the value of and , .(i), , and (see Figure 18).(ii), , and (see Figure 19).(iii), , and (see Figure 20).(iv), , and (see Figure 21).

(3) Fix the value of and , .(i), , and (see Figure 22).(ii), , and (see Figure 23).(iii), , and (see Figure 24).(iv), , and (see Figure 25).

When , matrix has pure imaginary eigenvalues . Calculate , , , .

The result is as follows: where is the Lyapunov exponent function about , , and and is the real part. The following is the figure of .

(1) Fix the value of and .(i), , and (see Figure 26).(ii), , and (see Figure 27).(iii), , and (see Figure 28).(iv), , and (see Figure 29).

(2) Fix the value of and , .(i), , and (see Figure 30).(ii), , and (see Figure 31).(iii), , (see Figure 32).(iv), , and (see Figure 33).

(3) Fix the value of and , .(i), , and (see Figure 34).(ii), , and (see Figure 35).(iii), , and (see Figure 36).(iv), , and (see Figure 37).

Premising , the stable area in the , plane is determined by inequality group (10). Figure 38 shows the system fixed point in the area of the asymptotic stability.

###### 3.1.2. Numerical Simulation

Using Matlab, parameters influence on the system (8) can be analyzed through numerical simulation.

*(1) *,* *,* and ** Influence on the Collecting Market.* The second collector on the assumption that the manufacturer and the first collector stem variables parameters is fixed with , , her system variables parameters is at . Manufacturers’ initial collecting price , collector 1 and 2 for manufacturers of the collecting price of depreciate rate for 20, 0.35, respectively, 0.3. And, as shown in Figure 39 changes Figures 39(a), 39(b), and 39(c) shows, the corresponding Lyapunov index as shown in Figure 39(d). Figure 39 showing, when , the system is in stable state. after multi game, , and is stable at the point of (17.4, 0.4904, 0.4825). When , the system (8) occurs the first bifurcation, then after cycle 2, 4 cycle, and so forth, the system is gradually into the chaotic state. Figure 39(d) is Lyapunov index spectrum distribution which can also confirmed the phenomena. at , the max Lyapunov index is zero, the system is in critical condition. At , most of Lyapunov index greater than zero which explains system at chaos state (Figure 39 and Figure 44).

(a) , , and , , |

(b) , and , |

(c) , , and , , |

(d) , , and |