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1. Introduction
With increasingly serious global environmental problems, low-carbon products for environmental protection have become an important direction in sustainable development. Both the government and consumers are calling for green production and low-carbon supply. The report of the Nineteenth National Congress of the Communist Party of China points out that high-quality development is to meet the growing needs of people for a better life and to normalize green development. At present, many manufacturers have made ecodesigns, carbon emission reductions (CERs), and green development through technology and product design [
1,
2].
Scholars have conducted research on low-carbon issues. Liu et al. [
3] analyzed the impact of income inequality on carbon emission in the US and found that income inequality exacerbates carbon emissions in the US in the short term, while it promotes carbon emission reductions in the long run. Zhang et al. [
4] found that the probability of the manufacturer introducing green technology was negatively correlated with the cost of government intervention, and it was positively correlated with punishment by the government for the manufacturer’s speculative behavior. Considering both economic and energy goals, Ye et al. [
5] established a dual-objective programming model that integrated 3E goals with quota allocation issues. They found the 3E approach was better than the ancestral approach in some respects in the case of Guangdong province. Tran et al. [
6] analyzed the relationship between an emissions trading scheme and various revenue recycling options using the case of Australian households. Results showed that Australia’s real GDP contracted slightly because of emission permit prices. Lin and Jia [
7] found emission trading mechanism fines had significant impacts on the cost of enterprise emission trading mechanisms, commodity prices, energy-intensive industrial output, GDP losses, and carbon dioxide emission reduction, but they had a small impact on the intensity of carbon emissions and emission reduction costs. Liu [
8] developed four common cost-sharing models in a low-carbon supply chain, analyzed its pricing rules, and used a revenue sharing contract to coordinate the supply chain. Lou and Ma [
9] studied the effect of sales efforts and CER efforts on the complexity of the supply chain system and found that price adjustment parameters had a greater impact on stability and profits of supply chain system than sales efforts and carbon emission reductions. Hui et al. [
10] considered a low-carbon, closed-loop supply chain and examined optimal decisions and performances in different situations (competitive pricing and competitive low-carbon promotion). Yang et al. [
11] considered two competitive supply chains under the cap and trade scheme in vertical and horizontal directions, and they found that there was a higher CER rate and a lower retail price in vertical cooperation. Liu et al. [
12] investigated the impacts of fairness concerns on the production sustainability level, low-carbon promotion level, and profitability of the supply chain based on incomplete rationality behavior. Few papers studied the simultaneous effects of CER, return rates, and service input on price determination in a closed-loop supply chain.
With the rapid development of e-commerce, the coexistence of online sales channels and traditional sales methods has gradually become a common business model, and the return rates are also on the rise. Many scholars have made keen observations and conducted research on the phenomenon of returns. Crocker and Letizia [
13] found that the retailer may take concealed actions to reduce the expected sales of products; the best policy for the retailer for return goods is that the manufacturer requires the retailer to pay full wholesale payment. Zheng et al. [
14] examined how product return rates of the reverse chain influenced the determination of market price and profits of two supply chains in different competition structures. Giri and Sharma [
15] found that, under the condition of sequential optimization, the acceptable quality levels of retailers’ returns were lower than those of global optimization, and integration of supply chain members led to a decrease in the number of deliveries from manufacturers to retailers. From the perspective of consumer utility function, Ofek et al. [
16] studied the problems of multichannel consumer returns, retailer service, and pricing under the risk of return goods. Ramanathan [
17] found that, through rating data of online consumers, the performance of enterprises in dealing with returns could affect customer loyalty.
Fairness concerns are another important behavior factor that enterprises pay attention to. Studies about fairness concerns showed that it had significant influence on price strategies and efficiency of supply chains [
18,
19]. Wang et al. [
20] studied decision-making and coordination of an e-commerce supply chain with manufacturer’s fairness concerns. Li et al. [
21] found the market shares of retailers were related to the impact of manufacturer fairness concerns on retailer profits. Niu et al. [
22] analyzed the supplier’s decision of whether to open an online direct channel or not by incorporating channel power and fairness concerns. They found that supplier fairness concerns effectively reduced their incentives to open an online channel. Ma et al. [
23] studied the pricing decisions of a closed-loop supply chain considering market efforts and fairness concerns. Du et al. [
24] indicated that fairness concerns could promote and coordinate the supplier and manufacturer to invest more in sustainable development of green technology innovations. Liang and Qin [
25] developed an estimation game model by fuzzy theory with fuzzy fairness concerns.
Some researchers studied dynamic characteristics of the supply chain. Puu [
26] briefly analyzed three oligopoly competition conditions and pointed out that strange attractors could appear in the duopoly model. Matouk et al. [
27] analyzed the stability of a discrete-time dynamical game model and discussed the Neimark–Sacker bifurcation Huang et al. [
28] investigated the influences of parameters on the stability of three dynamic game models, with the risk-averse manufacturer providing the complementary product. Wu and Ma [
29] found that the game model introduced chaos in two ways: flip fluctuation and Neimark–Sacker bifurcation. Huang and Li [
30] found the probabilistic selling supply chain system would introduce chaos from the higher system’s entropy through flip bifurcation or Neimark–Sacker bifurcation. Li and Ma [
31] analyzed system stability affected by the customer risk aversion and the customer preference for probabilistic products. The papers mentioned above provided many methods and perspectives for analyzing complex characteristics of the supply chain.
However, few papers simultaneously considered customer’s behavior for return goods, retailer’s service, and manufacturer’s behavior for CER in a closed-loop supply chain. It is a very interesting topic to study the influence of return rates, the service level of the retailer, and CER of the manufacturer on the price and stability of a low-carbon, closed-loop supply chain system.
This study can be considered as an extension of the work of Zhang et al. [
32]. They set up the demand function considering price difference and return risk; however, their research did not consider the impact of CER and service input on the demand function. This paper considers more realistic factors affecting the demand of low-carbon products (i.e., serves inputs, return rate, and CER) and focuses on complex analysis of a low-carbon, closed-loop supply chain.
Our theoretical contribution is as follows. The first contribution is to construct a low-carbon, closed-loop supply chain model, in which the demand function is affected by sales efforts, return rates, and CER. The second contribution is to study the stability and profitability of the low-carbon, closed-loop supply chain system under a dynamic game structure.
This paper is organized as follows. The dynamic game model of a low-carbon, closed-loop supply chain is developed in
Section 2. In
Section 3, the local stability of the dynamic game model is given using analytical analysis and numerical simulation. The complexity analysis of the dynamic game model is investigated in
Section 4.
Section 5 gives the global stability analysis of the dynamic game model using basins of attraction.
Section 6 discusses chaos control of the dynamic game model.
Section 7 gives conclusions.
3. The Local Stability of Dynamic System (7)
3.1. Equilibrium Points
When , the four equilibrium solutions can be obtained:
where and .
Economically, it makes no sense for the manufacturer and retailer to have zero prices, so the features of
,
, and
were not studied. Next, we only considered the stability of the Nash equilibrium point (
), and the Jacobian matrix of dynamic System (7) is given as follows:
where
The corresponding characteristic polynomial of dynamic System (7) can be written as follows:
where
= 2 +
and
= (
(
[
.
represent the trace and determinant of the Jacobian matrix (, respectively.
The eigenvalues of the Jacobian matrix at the corresponding equilibrium point determine system stability. According to the Routh–Hurwitz condition, when the nonzero eigenvalues of the equilibrium point are all less than one, dynamic System (7) will be in a stable state; when one of the nonzero eigenvalues of the equilibrium point are larger than one, dynamic System (7) will be in an unstable state. The stable conditions of dynamic System (7) must satisfy the following conditions:
Solving the inequality Equation (9), we can obtain the stable region of dynamic System (7). In the stable region, dynamic System (7) is locally stable with the initial values of prices within a certain range. Because these limitations are very complex, solving inequality Equation (9) is very complicated. Next, we give the stable region of dynamic System (7) through numerical simulation.
According to the current situation and characteristics of the low-carbon, closed-loop supply chain, we set parameter values as follows: .
According to the parameter values above, we can obtain
. The Jaobian matrix is:
The characteristic equation of Jacobian Matrix (10) is:
where
, and
.
The parameter basin is a powerful tool for numerical simulation, which can show the evolution process of dynamic System (7) into a chaotic state. Based on the stability conditions in Equation (9),
Figure 2 shows the route of dynamic System (7) to chaos. Different colors represent different periods, for example: stable (red), period-2 (pink), period-3 (yellow), period-4 (green), period-5 (black), period-7 (wine red), period-8 (blue), chaos (white), and divergence (grey). From
Figure 2, we find that dynamic System (7) goes into chaos through period bifurcation with an increasing
or
. If the price adjustment parameters are in the red region, dynamic System (7) is in a stable state. If the price adjustment parameters are in the grey region, the manufacture or the retailer will withdraw from the competitive market.
3.2. The Stable Region of Dynamic System (7) with Changing Parameters
Figure 3 shows the parameter basins of dynamic System (7) when
, and
take different values. Comparing the sizes of stability regions (red region) in
Figure 2 and
Figure 3, we found that a high level of CER decreased the stable region of dynamic System (7). An increasing customer loyalty to the direct marketing channel decreased the stable region of the manufacturer’s price adjustment and increased that of the retailer. An increasing
decreased the stable region of the manufacturer’s price adjustment and had no effect on the stable range of the retailer’s price adjustment. Thus, the manufacturer and retailer should adjust parameters according to the actual market conditions so that dynamic System (7) is in a stable state.
Figure 4 gives the effect that changing parameters had on the stability region of dynamic System (7).
Figure 4a shows the stable regions of dynamic System (7) with
having different values when other parameters are fixed. Stability regions of dynamic System (7) increased when
.
Figure 4b shows that the stable regions of dynamic System (7) enlarged with different values of
.
Figure 4c shows that the stable regions of dynamic System (7) shrank with different values of
.
Figure 4d,e are the changes of stable regions with the change of
and
; the scope of
decreased and that of
remained unchanged with the change of
. Similarly, the stability regions of dynamic System (7) remained unchanged with
increasing (shown in
Figure 4e). So dynamic System (7) was more sensitive to
than
. The high level of fairness concerns of the retailer shrank the stability region of the Nash equilibrium point, while the fairness concern level of the manufacturer had little effect on system stability. The retailer should be more cautious on adjusting the price strategy than the manufacturer.
From the above analysis, we made some conclusions: (1) The increase of CER shrank the stability region of dynamic System (7), which weakened the market competition. The customers’ preference for the direct marketing channel decreased the stable range of price adjustment of the direct marketing channel and enlarged the stable range of price adjustment of the traditional channel. The change of share in the profit target of the manufacturer had no effect on the stable range of the retailer’s price adjustment, and it shrank that of the manufacturer’s price adjustment. (2) The fairness concern behavior of the manufacturer had little effect on the stability region of dynamic System (7), while the fairness concern of the retailer was disadvantageous in keeping dynamic System (7) in the stability region, which indicated the retailer’s fairness concern behavior had greater impact on the stability of dynamic System (7) than that of the manufacturer’s fairness concern behavior. (3) Raising the service level decreased the stable region of dynamic System (7), while the increase of return rate increased the stability region of dynamic System (7).
6. Chaos Control
When the supply chain system was in chaos, the utilities of the manufacturer and retailer declined in an unstable state, which was harmful for all firms to obtain their business objectives. In order to reduce the risk caused by a chaotic state of the supply chain system, the decision makers should select suitable adjustment parameters to keep the supply chain system in a stable state or delay the supply chain system entering into a chaotic state. According to the above numerical simulation, if the parameter values or the price adjustment speeds were beyond the stable region, the supply chain system lost stability and even fell into chaos.
There are many methods to control the supply chain from the chaos state to the stable state such as the modified straight-line stabilization method [
38], the time-delayed feedback method [
39], and the OGY method [
40]. In this section, the state feedback control method was used to delay or eliminate the chaos in the supply chain system. The controlled system is represented as follows:
where
can be considered as the learning ability, adjustment ability, or adaptability of the manufacturer and retailer. For example, the manufacturer and retailer can adjust price or service input and CER through market research or by analyzing historical information. As what can be seen from
Figure 20, chaos in control System (11) was delayed when
, and the first bifurcation in control System (11) occurred when
. In the stable state, the system’s entropy was equal to zero; in the chaotic state, the system’s entropy approximately increased to five. Therefore, the manufacturer and retailer can delay the occurrence of price bifurcation in dynamic System (7) by choosing appropriate control parameters.
Figure 21 depicts the evolution process and entropy of controlled System (11) with the change of
when
. Controlled System (11) was in an unstable state and had large entropy when
; when
, controlled System (11) returned to the stable state. Therefore, when the dynamic system was in a chaotic state, the manufacturer and retailer can select appropriate control parameters to restore the competitive market to a stable state.
7. Conclusions
In this paper, we developed a dynamic game model of a low-carbon, closed-loop supply chain system in which the manufacturer had fair concern behaviors, fair CER, and made market share and profit maximization their objective. In addition, the retailer showed fair concern behaviors in market competition and provided service input to reduce return rate. The retailer recycled old products from customers and sold them to the manufacturer, the manufacturer remanufactured the recycled old products. Dynamic behaviors of the dynamic game model were analyzed using bifurcation, basin of attraction, chaotic attractors, and so on. The utilities of the manufacturer and retailer were described when parameters changed. The state feedback control method was used to control the chaos of the dynamic system. The following conclusions can be obtained.
(1) An increasing customer loyalty to the direct marketing channel will decrease the stable region of the manufacturer’s price adjustment and increase that of the retailer. The stable region of the system shrinks with an increasing CER level and retailer service level, which expands with return rates.
(2) The dynamic system enters into chaos through flip bifurcation with an increase in price adjustment speed. In the stable state, the utilities of the manufacturer and retailer increase with the increase of the retailer’s fair concern, while utilities decrease with the increase of the manufacturer’s fair concern. The level of CER is beneficial for utility acquisition of the manufacturer and retailer. The retailer’s service level and the balance coefficient of the manufacturer make the utility of the manufacturer decline and that of the retailer increase. In the chaotic state, the average utility of the manufacturer and retailer all decline, while that of the retailer declines even more.
(3) By selecting appropriate control parameters, the dynamic system can return to a stable state from chaos again.
The research of this paper is of great significance to the participants’ price decision-making and supply chain operation management.