*Article* **Structural Balance under Weight Evolution of Dynamic Signed Network**

**Zhenpeng Li 1,†, Ling Ma 2,\*,†, Simin Chi 2,† and Xu Qian 2,†**


**Abstract:** The mutual feedback mechanism between system structure and system function is the 'hot spot' of a complex network. In this paper, we propose an opinions–edges co-evolution model on a weighted signed network. By incorporating different social factors, five evolutionary scenarios were simulated to investigate the feedback effects. The scenarios included the variations of edges and signed weights and the variations of the proportions of positive and negative opinions. The level of balance achieved depends on the connection weight and the distribution of negative edges/opinions on the signed graph. This paper sheds light on the analysis of constraints and opportunities of social and cognitive processes, helping us understand the real-world opinions polarization process in depth. For example, the results serve as a confirmation of the imperfect balance theory, i.e., even if the system evolves to a stable state, the signed network still cannot achieve perfect structural balance.

**Keywords:** structural balance; feedback mechanism; opinions polarization

**MSC:** 91Cxx; 91Dxx; 91Exx

#### **1. Introduction**

Nodes in social networks represent individuals or organizations, and edges among nodes represent the interaction [1–6]. The signed network is a topology with positive and negative signs on the edges. The signed edge has abundant connotations in many real complex systems. For example, a negative edge usually means disagreement, hostility, opposition, and distrust; correspondingly, a positive edge represents agreement, friendship, support, and trust [7,8]. The investigations on the signed network can effectively improve our knowledge on signed complex systems, such as international relation [9], promoting and inhibiting neurons [1], trust prediction on social networks [10], consensus and polarization of online community [11], information diffusion [12], opinion dynamics [13,14].

The most basic theory in the field of the signed network is a structural balance theory [8], which was first put forward by Heider in 1946. This theory originated from the balanced model of the node's attitude towards things [15]. Cartright and Harry [16] extended the theory by combining the graph theory. Later, scholars made great extensions, for example, Kunegis et al. [17] suggested that the network structural balance is measured by counting the proportion of balanced triads in the whole signed network, Fachetti et al. [18] proposed an energy function definition to calculate the structural balance. Real-world signed networks rarely attain a perfectly balanced state. To quantify exactly how balanced they are, Aref et al. [19] formalized the concept of a measure of partial balance. Kirkley et al. [20] proposed two measures of structural balance based on hypothesized notions of "weak" and "strong" balance.

**Citation:** Li, Z.; Ma, L.; Chi, S.; Qian, X. Structural Balance under Weight Evolution of Dynamic Signed Network. *Mathematics* **2022**, *10*, 1441. https://doi.org/10.3390/ math10091441

Academic Editors: Wen Zhang, Xiaofeng Xu, Jun Wu and Kaijian He

Received: 13 March 2022 Accepted: 11 April 2022 Published: 25 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Indeed, since the seminal work of Heider, many fundamental concepts and significant theories have been proposed for the development of the social balance theory. The extension of the classic structure balance theory can be divided into four categories. They are the balance of nodes, the balance of triangles, the balance of complete networks, and the balance of arbitrary networks. See [21] for a comprehensive review of the variations, extensions, and calculating methods related to the classic social balance theory.

Although the structural balance theory in signed networks describes a stable signed system state, the system from imbalance to balance is a dynamic process [22,23]. For example, the formation of friend or enemy groups, the constantly changing international relation. In order to investigate the dynamics of structural balance, Wu et al. [24] set up a co-evolution model in the acyclic network and cycle network. Marvel et al. [25] proposed a continuous-time model of structural balance. He et al. [26] developed a new simulation model to study the impact of structural balance on the evolution of cooperation in signed networks. However, currently, the research of dynamic networks is mostly limited to non-signed nodes or signed networks without weights. The co-evolution models of nodes and edges based on weighted signed networks have not been deeply investigated because of their complexity.

#### **2. Related Works**

Social influence network and opinion change models, such as the French-DeGroot model [27], Friedkin–Johnsen model [28], formally entail the interpersonal influence on the formation of interpersonal agreements and polarization. Social influence and its induced homogeneous effect, however, do not fully interpret the global network structure, for example, the mounting two-polarization phenomenon in US political ecology that insulates democrats and republicans from opposing opinions about current events [29]. Social network influence theory entails the interpersonal influence on the formation of interpersonal agreements and polarization. From a complex network perspective, this influence process is the feedback effect of the network structure on network functions.

In the real world, most of the signed graphs are temporal, the nodes and edges vary over time, which makes the changing of the network structure, including the clustering and the structural balance of the network. Many researchers are interested in the dynamic processes over signed networks, i.e., the co-evolution between the signed edges and the nodes, such as the agreement and disagreement evolving over random dynamic networks [30–33]. However, most of these models lack the social mechanisms of opinion formation, evolution, and dissemination.

Real-world cases call for these two threads of research, opinion dynamics and structural balance dynamics, to be combined. Holme and Newman [34] presented a simple model of this combination without any theoretical analysis. Wang et al. [35], in the latest relevant research, proposed co-evolution models for both dynamics of opinions (people's views on a particular topic) and dynamics of social appraisals (the approval or disapproval towards each other). In their model, the system evolves as opinions and edge weights are updated over time by two rules: opinion dynamics, and appraisal dynamics. Both opinion and appraisal dynamics are governed by the evolution of the time-varying matrix of a dynamic system. The social–psychological mechanisms of the co-evolution model in [35] involve the structure balance theory and social influence network theory, i.e., the Friedkin–Johnsen model. Similar to [35], Kang and Li [36] proposed a co-evolution model of discrete-time opinion evolution vectors and appraisal signed networks. Social– psychological mechanisms of the co-evolution model in [36] involve structure balance and the social distancing theory. This paper, inspired by these works, provides a ruled-based node–edge co-evolution model. However, different from a dynamic system time-varying matrix method [35,36], our model is rule-based and similar to the agent-based modeling approach. In our model, the global structure balance index is used as judgment conditions for a simulation algorithm termination. Our model's social–psychological mechanisms involve the structure balance theory and bounded confidence model of opinion dynamics [37], since the bounded confidence model is more suitable for paired interactions in large groups for agent-based modeling.

Some empirical investigations show that the unbalanced triangles will evolve into balanced ones to make the network more stable [38], and that the global level of balance of very large online singed networks is indeed extremely balanced [18,20]. Related investigations show that the structural balance of signed networks in the real world is an increasing function of evolution time. The evidence of over-represented balanced triads is well above random expectations in the vast majority of real networks, such as the statistic observation in references [18,20,38,39].

These empirical conclusions confirm the validity of the classic Heider structure balance theory. However, we still find that the complete perfect balanced structure is rarely observed in the real-world signed networks [20]; there is almost no perfect intra-/inter-group structure balance.

Relevant reference analyses urge researchers to promote this kind of research in-depth. This paper is devoted toward investigating the dynamic structural balance of groups and the emerging macro polarization patterns of signed networks. However, the data sets on related opinions and signed edges are not easy to obtain, this is why we shifted to the model simulation approach. In order to explain real-world ubiquitous opinion polarization and limited structure balance, different influence parameter values are used to explore the influences of different social factors on structural balance and polarization.

#### **3. Our Contribution**

In this paper, based on structural balance and a co-evolution model, considering the dynamic mechanics of both opinions and relationships, we employed two evolution rules: opinion renewal and edge adjustment. In addition, we defined an influence matrix and two new neutral dyadic/triadic motifs. A new co-evolutionary mutual feedback algorithm is provided to simulate our proposed model under five evolutionary scenarios.

Our findings can explain the lack of the perfect Heider balance in many real-world systems. This work verifies that signed social networks are indeed limited-balanced, but the level of balance achieved depends on the connectivity of the graph, the percentage of positive edges, and the percentage of positive opinions, most of all, on the distributions of these negative edges/opinions on the signed graph.

Our computational analysis of balance in signed networks serves as a confirmation of the balance theory. Meanwhile, our simulation results suggest that the signed network in the real world is a dynamic equilibrium process, which cannot reach a perfect equilibrium state. The comprehensive numerical results suggest that values of balance at the micro-and macro-levels may match up to some extent, especially as the macro dynamic pattern of the signed network is closely related to its micro-structural balance.

Compared with the current investigations on opinion dynamics on static networks, the proposed co-evolution model in this paper characterizes the polarization of opinions in reality and predicts the existence of imperfect balance in the social context. Our model may also help predict the potential division of social groups and public opinion dynamics.

This paper sheds light on the analysis of constraints and opportunities of social and cognitive processes, helping us to understand the real-world opinion polarization process, in depth.

#### **4. Structural Balance in Signed Network**

The social network influence theory entails interpersonal influence on the formation of interpersonal agreements and polarization. From a complex network perspective, this influence process is the feedback effect of the network structure on network functions. In this section, based on the dynamic social influence network theory, we code the dynamic weight social influence matrix, and propose two new neutral dyadic, triadic motifs, as preparation for our co-evolution model.

#### *4.1. Binary Structural Balance*

In this investigation, we introduce an edge named neutral edge ("0"), which is a neutral status between positive and negative edges. It refers to a kind of edge without a clear position, which means that the connection between nodes is not clear or neutral. After introducing the neutral edge, we propose a binary motif in a signed network, firstly. Three types of the binary motif are shown in Figure 1. Specifically, two nodes share the same attitude; at the same time, the signed edge between the two nodes is positive; then, we say the binary motif is balanced. If two nodes share an opposite attitude, and the signed edge between *i* and *j* is positive, then the binary structure is unbalanced. Based on the above analysis, a binary group includes two parameters, one is the nodes' opinion and the other is the edge sign between the two nodes. A neutral edge ("0") means that there is no clear connection between two nodes. However, in the dynamic evolution process, the neutral edge has the opportunity to evolve into negative or positive, due to the constraints of structural balance.

**Figure 1.** (**a**) Balanced binary groups, (**b**) neutral binary groups, (**c**) unbalanced binary groups.

#### *4.2. Triad Structural Balance*

Holland and Leinhardt [40] proposed the three basic binary motifs, reciprocity, asymmetric, and non-edge or null edge. It is worth noting that the null edge plays the same role as our proposed neutral edge ("0"). Since structural balance is defined on a triadic motif, which is constructed based on three basic binary motifs; binary balance is equal to triadic balance.

Figure 2a,b denote a balanced triadic structure, Figure 2c,d denote an unbalanced triadic structure [2]. If the product of signed edges in each cycle is positive, the signed graph is balanced [16]. If each triad in a signed complete graph is balanced, then the signed graph is balanced. Here, since we consider the chance of a tried with *edge* = 0, a "neutral triadic motif" is introduced, for the case, the product of three sides in a triad is 0.

**Figure 2.** Triad structure in structural balance theory.

#### *4.3. Global Structural Balance Index*

Here, we use the global structural balance definition as formulated in Equation (1) [18], where *Jij* = +1 denotes a positive edge between *i* and *j*, *Jij* = −1, there is a negative edge between *i* and *j*, and *Jij* = 0 implies a neutral edge between *i* and *j*. In the real world, a neutral edge might be considered a *i* and *j* have no influence on each other, or the two nodes remain neutral on an issue. *si* is a continuous real value, representing node *i*'s opinion, and

the range of *si* is (−1,1); 1 > *si* > 0 means node *i* holds a supportive attitude, −1 < *si* < 0 means node *i* holds a negative attitude, and *si* = 0 means node *i* hold an neutral attitude. It is worth noting that a triad-based structural balance is consistent with the global structural balance definition. A smaller value of *hs* means a more balanced social structure.

$$h\_s = \sum\_{(i,j)} (1 - f\_{i\bar{j}} \mathbb{s}\_i \mathbb{s}\_j) / 2 \tag{1}$$

#### *4.4. Social Influence Matrix*

The social influence network is a formal theory that forms attitudes and perspectives. It describes the influence processes of individuals on group attitudes in interpersonal networks. It allows the analysis of how the network structures of groups affect the formation of individual attitudes and group structures [41]. Based on this theory, we propose a social influence matrix, which provides a sociological perspective for the process of signed edge transformation. The type of edge will evolve according to the weight of the social influence matrix.

Friedkin and Johnsen's [41] social influence network theory is regarded as the cornerstone reference for social influence matrix consensus or polarization, as the important and regular group opinions dynamic pattern is generally observed in a group discussion and barging process. Friedkin and Johnsen's social influence network theory emphasizes that the interpersonal influence social structure, i.e., the social influence matrix is the underlying precondition for group consensus or opinion convergence [42]. In that model, the initial social influence structure of a group of actors is assumed to be fixed during the entire process of opinion formation. However, with the evolution of timestamps, considering both stubborn and susceptible effects, the interpersonal influence structure can be regarded as a dynamic recursive process. For this reason, the interpersonal influence structure in their model is also dynamic. Based on the dynamic social influence network theory, here we code the dynamic weight social influence matrix in Equation (2). In Equation (2), we set different influence weights to map to signed edges −1, +1, 0. To meet the needs of model simulation calculation, we set different weight ranges corresponding to different social influence intensities.

Within the framework of the social influence network theory, we set up three types of social influence, namely positive, negative, and neutral influence. Positive influence means that two individuals/nodes have homogeneous influence and a positive edge. Negative influence means that two individuals/nodes have opposite influence and a negative edge. Neutral influence means that two individuals/nodes do not influence each other. In this study, based on the adjacent matrix of the signed network, a social weighted influence matrix is randomly assigned to edges. With the assumption of strength difference—of mutual influence among individuals—we set three kinds of weights, *wij*, as shown in Figure 3 and Equation (2).

$$J\_{ij}(t) = \begin{cases} -1, & \text{if} \quad 0 < w\_{ij} \le 1 \\ 0, & \text{if} \quad 1 < w\_{ij} < 3 \\ +1, & \text{if} \quad 3 \le w\_{ij} \end{cases} \tag{2}$$

**Figure 3.** Mapping of weights and edges.

#### **5. Co-Evolution Model of Opinions and Edges**

The coupling effect of opinion propagation and network topology dynamics on networks leads to complex system behaviors. In order to study the propagation behavior and influencing factors in dynamic complex networks and the dynamic evolution process of the systems, we set up a co-evolution model of opinions and edges. In a social group, people are supposed to be motivated to keep a 'balanced edge' with others, when two people have the same attitude; if they have negative edges, they tend to change their edges or attitudes in order to maintain a balanced structure. The following is the evolution process introduced in detail.

#### *5.1. Evolutionary Rule of Opinions*

The Deffuant–Weisbuch (DW) model [37] is the most widely used continuous opinion dynamic model. The rule of this model is that two nodes change their opinions if the degree of disagreement is less than the threshold , i.e., |*si*(*t*) − *sj*(*t*)| ≤ , (*i* = *j*), the opinion is carried out as Equations (3) and (4). In this paper, we set the convergence parameter *μ* ∈ [0, 0.5], which indicates the strength of mutual influence among nodes.

$$s\_i(t+1) = s\_i(t) + \mu(s\_j(t) - s\_i(t))\tag{3}$$

$$s\_{\rangle}(t+1) = s\_{\rangle}(t) + \mu(s\_{i}(t) - s\_{\rangle}(t))\tag{4}$$

The evolutionary rule of opinion follows Equation (5). To approach structural balance, the evolution of opinions will be based on the type of edge. By adjusting the proportion of initial positive and negative opinions, we can observe the evolution of group opinion. If *i* and *<sup>j</sup>* satisfy *Jij si*∗*sj* ≥ 0, it means these two nodes follow the balanced binary conditions, and there is no need to change anyone's attitude. Considering the influence of the neighbor nodes, the opinion evolution will follow the DW model. On the contrary, if *Jij si*∗*sj* < 0, in the opinion evolution, the nodes will change their attitudes, and the attitudes in the next round will be opposite.

$$s\_i(t+1) = \begin{cases} DW, & \text{if } \quad \frac{f\_{ij}}{s\_i \* s\_j} \ge 0 \\\\ -s\_i(t), & \text{if } \quad \frac{f\_{ij}}{s\_i \* s\_j} < 0 \end{cases} \tag{5}$$

#### *5.2. Evolutionary Rule of Edges*

According to the different strengths of influence, we introduce three evolutionary rules of signed edges, respectively.

The rule for the evolution of the negative edge is illustrated in Figure 1, Equation (6), and Figure 4. When the edge is negative, it will change according to the opinion. If the attitudes of the two nodes are opposite, to achieve a balanced state, the negative edge will not change. If two nodes have the same attitude, the negative edge should evolve to a positive edge. According to the division of weight range, *wi*,*<sup>j</sup>* plus *w*<sup>1</sup> (the strength of *edge* = −1 change to *edge* = +1), and *w*<sup>1</sup> ∈ (0, 1), the negative edge evolves into the positive edge.

$$w\_{\vec{ij}}(t+1) = \begin{cases} w\_{\vec{ij}}(t), & \text{if} \quad J\_{\vec{ij}} = -1 \text{ and } (s\_i \ast s\_{\vec{j}}) < 0 \\\\ w\_{\vec{ij}}(t) + (w\_1 + 2), & \text{if} \quad J\_{\vec{ij}} = -1 \text{ and } (s\_i \ast s\_{\vec{j}}) \ge 0 \end{cases} \tag{6}$$

The rule for the evolutionary of a positive edge is shown in Equation (7) and Figure 4. When the attitude of two nodes is the same, and the edge is positive, there is a balanced structure between them. If the attitudes of two nodes are opposite and the edge is positive, in order to make the approach a balanced structure, *wij* minus *w*<sup>3</sup> (the strength of *edge* = +1 change to *edge* = −1), and *w*<sup>3</sup> ∈ (0, 1), the positive edge evolves into negative.

**Figure 4.** Evolutionary rules of *edge* = +1 and *edge* = −1.

The rule for the evolution of a neutral edge is formulated in Equation (8) and illustrated in Figure 5. No matter whether the attitudes of the two nodes are opposite or consistent, the neutral edge may evolve into a positive edge or a negative edge. When two nodes have the same attitude, to make the binary motif approach the balanced state, the neutral edge needs to evolve into a positive edge, which affects *wij* add *w*<sup>2</sup> (the strength of *edge* = 0 change to *edge* = +1). If two node opinions are opposite, in order to make the edge evolve into a negative edge, concerning the social influence weight *wij*, subtract *w*<sup>2</sup> (the strength of *edge* = 0 change to *edge* = −1). *w*<sup>2</sup> always falls into the range of (0, 1).

$$w\_{ij}(t+1) = \begin{cases} w\_{ij}(t) + w\_{2\prime} & \text{if} \quad l\_{ij} = 0 \text{ and } (s\_i \* s\_j) \ge 0 \\\\ w\_{ij}(t) - w\_{2\prime} & \text{if} \quad l\_{ij} = 0 \text{ and } (s\_i \* s\_j) < 0 \end{cases} \tag{8}$$

**Figure 5.** Evolutionary rules of *edge* = 0.

#### *5.3. Model Algorithm*

In this section, we propose the algorithm realization of the provided model, in detail. Firstly, to consider the influence of the hub node on the network, we used the BA scale-free network [4]. We initialized an undirected BA network with 100 nodes, 1539 edges, and a network density equal to 0.3109. Moreover, the observation is based on the average results of 1000 realizations. We find that |*hs*(*t* + 1) − *hs*(*t*)| ≤ 0.01 after 30 rounds. The parameter assignment is shown in Table 1. The parameter *value* = 0.5 is used to control whether the opinion evolves or the edge evolves. The algorithm is defined as Algorithm 1. In each round, a random number will be generated; when the random number is greater than *value*, the opinion evolution will be carried out; when the random number is less than *value*, the edge evolution will be carried out.


**Table 1.** Initialization Parameters.


**Table 1.** *Cont*.


#### **6. Simulation Results**

In this section, to explore the influence of different social factors on structural balance, we will integrate different social factors into the proposed co-evolution model, and simulate and discuss the simulation results, in detail. The parameters set in different scenarios are given in Table 2.

**Table 2.** Simulation scenarios.


#### *6.1. Scenario 1, the Influence of Parameter*

In this experiment, we discuss the influence of different  on the structural balance weighted signed network. The ratio of *edge* = +1 is 0.2, the ratio of *edge* = 0 is 0.5, and the ratio of *edge* = −1 is 0.3. Meanwhile, the proportion of positive opinions is 0.7, the rest is negative. Moreover, convergence parameter *μ* = 0.3. The same values are set for *w*1, *w*<sup>2</sup> and *w*3, i.e., *w*<sup>1</sup> = *w*<sup>2</sup> = *w*<sup>3</sup> = 0.3.

As shown in Figure 6a–c, when  is large, the weighted signed network has the fast velocity to approach structural balance, the number of balanced triads and positive edges in the same round is the largest. The reason is that the larger  promotes nodes interacting with others, thus promoting consensus. As shown in Figure 6d, all nodes hold the positive opinion, as long as  > 0.1. According to the binary structural balance, because all opinions are positive, all edges are positive in the end, and the proportion of balanced triads is also higher. Based on the above analysis, we can conclude that the larger the  is, the faster the network approaches the structural balance.

**Figure 6.** *Cont*.

**Figure 6.** Effect of different : (**a**) *hs* changes with time, (**b**) *tri*+ changes with time, (**c**) *Rr*+ changes with time, (**d**) *Rs*+ changes with time.

In conclusion, to accelerate the network to approach the structural balance, we can improve the tolerance of opinion differences by strengthening trust among nodes. In psychology, trust is a kind of stable belief, which maintains the common value and the stability of society.

#### *6.2. Scenario 2, the Influence of Negative Opinions*

This experiment discusses the influence of the initial number of negative opinions on the structural balance. Other parameter settings are the same as that in Scenario 2, the initial ratio of *edge* = +1 is 0.2, the ratio of *edge* = 0 is 0.5, and the ratio of *edge* = −1 is 0.3. Convergence parameter *μ* = 0.3, and  = 0.3. The conversion strength between various edges are the same; that is, *w*<sup>1</sup> = *w*<sup>2</sup> = *w*<sup>3</sup> = 0.3. The parameter *value* equals 0.7. To make our expression more compact, in the following scenarios, we focus on the change of *hs*, since it is a key indicator for the evolution process of the signed networks. As shown in scenario 1, other indicators can be inferred from *hs*.

The simulation results are shown in Figure 7. When all opinions are positive or negative, which is in the case of *Rs*− = 0 and *Rs*− = 1, the global structural balance index changes faster, and the network approaches structural balance faster. However, when the proportion of negative opinions is about 0.5, the global structural balance index does not change.

In scenario 1, because the positive opinions are in the majority at the beginning, the network can approach the balanced state under different . However, in this scenario, we set the same , when the positive or negative opinions occupy the majority, the network can approach the balanced state faster. Because the more nodes with the same attitude, according to the binary structural balance, the negative edge and neutral edge evolve to the positive edge, so the number of positive edges and balanced triads are the most. It can be concluded from scenario 2 that when there are many nodes with the same attitude in a network, the network can quickly approach the structural balance.

**Figure 7.** Effect of negative opinions on network.

#### *6.3. Scenario 3, the Relation between Opinion Evolution and Edge Evolution*

In this experiment, we discuss the influence of *value* on the structural balance. The initial conditions are as follows: negative edges account for 0.3, neutral edges account for 0.5, positive edges account for 0.2, and positive opinions account for 0.7. The conversion strength between various edges are the same, i.e., *w*<sup>1</sup> = *w*<sup>2</sup> = *w*<sup>3</sup> = 0.3. Moreover, we set *μ* = 0.3,  = 0.4. Four experimental results are discussed as follows.

The first result of this experiment is the global structural balance index. As shown in Figure 8, the larger the *value* is, the faster the network approaches structural balance. When the *value* is terribly small, the network cannot approach a balanced state. It can be observed that the network approaches structural balance except for the case of *value* = 0.2.

**Figure 8.** Effect of the parameter *value* on the global structural balance: *hs* changes with time.

In Figure 9, we plot the impact of the control parameter *value* on the final network balanced triad and triad 0 (neutral triads). We observe that when *value* is larger, the number of balanced triads increase rapidly in Figure 9a. Moreover, neutral triads decrease rapidly in Figure 9b, When *value* = 0.2, a large number of neutral triads still exist after 30 rounds of evolution, it shows that the network has not reached structural balance. It can be concluded that in the process of approaching a balanced structure, it is more difficult to change the opinion than to change the edge.

**Figure 9.** Effect of the parameter *value* on the number of triads: (**a**) *tri*+ changes with time, (**b**) *tri*0 changes with time.

#### *6.4. Scenario 4, the Influence of Various Strengths of Conversion*

In this simulation, we explored the influence of the various strengths of conversion between various weighted edges. The strength of conversion represents the conversion level between different edges. The greater the strength is, the greater the connection strength between the nodes, and the more difficult the transformation of an edge is. In the initial network, we set the number of positive edges, negative edges, and neutral edges as equal, the convergence parameter *μ* = 0.3, the degree of disagreement threshold was  = 0.8, the proportion of negative opinions was 0.3, the edge and node evolution control threshold parameter *value* = 0.7.

The first experiment involved observing the strength of the negative edge change to the positive edge. As seen in Figure 10, the experiment considered two cases, *w*<sup>1</sup> = 0.5 and *w*<sup>1</sup> = 0.7, respectively. When *w*<sup>1</sup> = 0.7, *hs* reached a stable state quickly. This result shows that when *w*<sup>1</sup> is larger, i.e., the greater the strength of the negative edge change to the positive edge in the network, the network can approach a balanced structure more quickly. The result shows that it is difficult to change the negative edge to the positive.

The second experiment was to observe the transformation of neutral edges into negative and positive ones. As shown in Figure 11, when *w*<sup>2</sup> = 0.5 and *w*<sup>2</sup> = 0.7, the network approached a balanced state after 10 rounds. This experiment's results show that the strength of the neutral edge has a small influence on the structural balance. Because the neutral relationship implies that there is no clear relationship between two nodes, it is easier to convert the neutral edge to a positive edge or negative edge.

**Figure 10.** Effect of *w*1: *hs* changes with time.

**Figure 11.** Effect of *w*2: *hs* changes with time.

The third experiment was to observe the strength of the positive edge change to the negative edge. As shown in Figure 12, after 10 rounds, *hs* approached a stable state in the case of *w*<sup>3</sup> = 0.7; however, in the case of *w*<sup>3</sup> = 0.5, *hs* was still evolving toward the structural balance. The result shows that it is difficult to convert the positive edge into the negative edge.

In conclusion, the various strengths of conversion have an effect on the final network structural balance. Increasing the strength of the negative edge and positive edge can promote the network to approach structural balance. We can conclude that it is easy to transform a neutral edge into a positive or negative edge; the contrary is difficult.

**Figure 12.** Effect of *w*3: *hs* changes with time.

#### *6.5. Scenario 5, the Influence of Proportion of Positive and Negative Edges*

In this experiment, we investigated the effects of different ratios of three types of edges on structural balance. We set the proportion of negative opinions as 0.3, *μ* = 0.7,  = 0.8 and *value* = 0.7. The strength of conversion between various edges were the same; that is, *w*<sup>1</sup> = *w*<sup>2</sup> = *w*<sup>3</sup> = 0.3.

The influence of *edge* = +1 in the network is shown in Figure 13. It can be seen that the larger the ratio of *Rr*+, the faster *hs* drops. By comparing with Figure 2, we can see that the proportion of positive edges in the balanced triads is <sup>2</sup> <sup>3</sup> , and the proportion of the negative edges is <sup>1</sup> <sup>3</sup> . Therefore, we conclude that the larger the number of initial *Rr*+ is, the faster the network approaches structural balance.

**Figure 13.** Effects of different initial ratios of positive edges: *hs* changes with time.

Figure 14a,b show that positive edges are more likely to appear than negative and neutral edges. Therefore, the probability of a negative edge evolving into a positive edge is greater than that of a positive edge evolving into a negative edge within the framework of structural balance. The probability of structural transformation is illustrated

in Figure 15, and the probability of structural transformation in Figure 15a is greater than that of Figure 15b.

**Figure 14.** Changes in the number of various edges: (**a**) edges change in *Rr*+ = 0.5, *Rr*− = 0.3, *Rr*0 = 0.2; (**b**) edges change in *Rr*+ = 0.7, *Rr*− = 0.2, *Rr*0 = 0.1.

**Figure 15.** Changes of positive edge: (**a**) the edge between nodes B and C evolves from −1 to +1; (**b**) the edge between nodes B and C evolves from +1 to −1.

The influence of *edge* = −1 is shown in Figure 16. It can be observed that in the case of a low proportion of *Rr*− = 0.5, the network approaches structural balance faster. By analyzing Figure 2, we can see that the proportion of negative edges in the unbalanced triads is <sup>2</sup> <sup>3</sup> , and the proportion of positive edges is <sup>1</sup> <sup>3</sup> . Thus, the high proportion of negative edges will delay the network toward structural balance.

As shown in Figure 17a,b, we can see that the number of negative edges gradually decreased. The number of neutral edges remained unchanged in four rounds and then decreased, while the number of positive edges kept increasing until all edges became positive. Structural transformation is illustrated in Figure 18.

The influence of neutral edges on the final balanced state is shown in Figure 19. In Figure 19, it can be observed that after 20 rounds of evolution, a high number of neutral edges favor the network structural balance. The explanation is that the neutral edge may evolve to a positive edge and negative edge, and contribute to the increment of balanced triads.

**Figure 16.** Effects of different initial ratios of negative edges: *hs* changes with time.

**Figure 17.** Changes in the number of various edges: (**a**) edges change in *Rr*− = 0.5, *Rr*+ = 0.2, *Rr*0 = 0.3; (**b**) edges change in *Rr*− = 0.7, *Rr*+ = 0.1, *Rr*0 = 0.2.

**Figure 18.** Negative edge changes: the edge between nodes B and C evolves from −1 to +1.

**Figure 19.** Effects of different initial ratios of the neutral edge: *hs* changes with time.

By observing Figure 20a, we can see that both the number of neutral edges and negative edges decreased gradually before 10 rounds. This dynamic resulted in the gradual increase of positive edges. By observing Figure 20b, we can see that the number of neutral edges did not change in the first and second evolution rounds. However, from the third round, the negative and neutral edges began to decrease. These experimental results show that the probability of the neutral edge evolving into the positive edge was greater than that of the negative edge. The probability of structural transformation in Figure 21a is greater than that of Figure 21b. Since the network is toward a structural balance.

This scenario implies that we can increase the number of positive edges and neutral edges to make these edges become the main body, and reduce the negative edge to promote the harmonious development of the group. In a group, if there are more positive edges, the network will tend to be structurally balanced faster, while negative edges have a reverse effect.

**Figure 20.** Changes in the number of various edges: (**a**) edges change in *Rr*0 = 0.5, *Rr*− = 0.2, *Rr*+ = 0.3; (**b**) edges change in *Rr*0 = 0.7, *Rr*− = 0.1, *Rr*+ = 0.2.

(**a**) (**b**)

**Figure 21.** Changes of neutral edge: (**a**) the edge between nodes B and C evolves from 0 to +1; (**b**) the edge between nodes B and C evolves from 0 to −1.

#### **7. Discussion**

In our simulation, we also tested our proposed model on ER random graphs and WS small-world network structures. Our simulation suggests that final signed network evolution results are robust to different structures and sizes. The results only depend on the connection weight of the networks, the percentage of positive edges, and the percentage of positive opinions. We observed the same evolution trend on different signed networks.

In summary, five scenarios including variations of edges and signed weights and variations of the proportion of positive and negative opinions, were investigated to observe the final signed network stable state. Specifically, we performed extensive simulations to examine how different initial conditions affected the network evolution, i.e., the balance level of network structure and the opinion polarization pattern.

We observed that initial conditions, such as a high proportion of positive edges, positive opinions, and greater signed weights, could promote the signed network towards balance. Our simulation results suggest that the signed network in the real world is a dynamic equilibrium process, serving as confirmation of the imperfect balance theory, i.e., even if the system evolves to a stable state, the signed network still cannot achieve a perfect structural balance.

Importantly, the simulation results could explain the imperfect Heider balance of many real-world signed systems. Meanwhile, our computing model explains that the level of balance achieved depends on the connecting weight of the network, the percentage of positive edges, and the percentage of positive opinions, most of all, on the distributions of these positive edges/opinions and the weight on the signed network. The comprehensive numerical results suggest that values of balance at the micro-and macro-levels may match up to some extent; the macro dynamic pattern of the signed network is especially closely related to its microstructural balance.

Similar to the social influence network model, the proposed co-evolution model can be used to predict real-world signed network evolution on the final convergence state (structure balance level v.s. polarization pattern). However, the social influence network model (Friedkin, N.E. Johnsen, E.C., 1999) [28] and the co-evolution model of opinion and social tie dynamics (Wang, H.; Luo, F.; Gao J., 2021) [35] are based on rigorous timevarying matrix mathematical analyses. Our proposed model is a rule-based simulation, its effectiveness for large-scale signed social networks needs to be verified by an empirical data set. This is also the direction of our related work in the next step. In a follow-up to this study, we plan to conduct experiments based on real data sets to evaluate the effectiveness of the proposed social network evolution modeling. The basic idea is a rule-based model simulation and comparative analysis of a rule-based model adjustment with real-world empirical signed networks. Specifically, we will modify and estimate the key evolution parameters through the empirical process of edge-evolution and node-evolution of a real signed network *G* and then modify the rule-based model *G*∗. On the one hand, we can verify the effectiveness of our proposed model through the empirical evolution process of large-scale signed networks. On the other hand, we can refine our proposed model. Our

ultimate expectation is that the refined model can achieve acceptable prediction accuracy and could be extended to the prediction and regulation of a real signed social system.

In our model, it is assumed that each node is homogeneous, and there is no special node, such as an opinion leader; in addition, final signed network evolution results only depend on different initial conditions, and are robust to different structures and sizes. Therefore, this model is also applicable to the case of having non-random distributions of opinions and edges.

#### **8. Conclusions**

In this investigation, based on the social influence network theory, we introduced a neutral binary motif and a triadic motif, and focused on the contributions of interpersonal influenced to the formation of interpersonal agreements and polarization. Our proposed edges–opinions co-evolution model entails a cognitive process when it deals with conflicting influential opinions and the signs and strengths of a social structure. This article extends the implications of the social structure balance theory and assesses the social influence theory in groups of dyads—triads with neutral relations.

Extensive simulations were performed in five cases. The results are summarized as follows: (1) the higher the tolerance for the difference of opinions, the faster the network can approach a balanced state. (2) The more nodes with the same attitude, the faster the network can reach structural balance. (3) When the edge evolution is faster than the opinion, the network can approach the balanced state faster. (4) The 'larger' the strengths of the positive and negative edges, the faster the network can approach the balanced state, while the strength of the neutral edge has a trivial effect. (5) The higher the proportion of positive and neutral edges, the faster the network can approach balance.

In particular, we showed that the persistent fluctuation of opinions is consistent with the minimal global energy function or a local triadic-signed structural balance emergence. This work verifies that signed social networks are indeed of limited balanced and could be used to explain the ubiquitous polarization phenomenon in online social networks. These results can provide us with a better understanding of the inherent mechanisms and key properties of signed networks.

Our model, however simple, should find further extensions and applications in social structures, where conditions of consistence are meaningful. For example, for a real signed network, in addition to structure and node attributes, the scale of the network will also change, such as the increase of nodes and the increase or deletion of edges. There may be other complex social mechanisms for the formation of real signed social networks, such as social status, social power theories, and combing with sentiment and behavioral data analysis [43–45]. The co-evolution feedback model we proposed can be further extended to such a case.

Next, we hope to collect real signed social network data to verify the effectiveness of several main conclusions obtained in this paper. We also hope to use the model and relevant conclusions proposed in this paper to empirically study collective action problems, such as information diffusion, cooperative evolution, and public opinion dissemination on signed social networks.

**Author Contributions:** Methodology, X.Q.; Visualization, S.C.; Writing—original draft, L.M.; Writing—review & editing, Z.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Natural Science Foundation of China under grant No. 71661001.

**Data Availability Statement:** All data included in this manuscript are available upon request by contacting with the corresponding author.

**Acknowledgments:** The authors thank the anonymous reviewers for their helpful comments on an earlier draft of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


[CrossRef]


**Peiyue Cheng, Guitao Zhang \* and Hao Sun**

Departement of Management Science and Engineering, School of Business, Qingdao University, Qingdao 266071, China; 2019020339@qdu.edu.cn (P.C.); sunhao@qdu.edu.cn (H.S.)

**\*** Correspondence: zhangguitao@qdu.edu.cn

**Abstract:** Under the background of a circular economy, this paper examines multi-tiered closed-loop supply chain network competition under carbon emission permits and discusses how stringent carbon regulations influence the network performance. We derive the governing equilibrium conditions for carbon-capped mathematical gaming models of each player and provide the equivalent variational inequality formulations, which are then solved by modified projection and contraction algorithms. The numerical examples empower us to investigate the effects of diverse carbon emission regulations (cap-and-trade regulation, mandatory cap policy, and cap-sharing scheme) on enterprises' decisions. The results reveal that the cap-sharing scheme is effective in coordinating the relationship between system profit and carbon emission abatement, while cap-and-trade regulation loses efficiency compared with the cap-sharing scheme. The government should allocate caps scientifically and encourage enterprises to adopt green production technologies, especially allowing large enterprises to share carbon quotas. This study can also contribute to the enterprises' decision-making and revenue management under different carbon emissions reduction regulations.

**Keywords:** non-cooperative equilibrium; complex supply chain network; environmental policies; circular economy

#### **1. Introduction**

#### *1.1. Background*

As carbon emissions contribute to global warming through the greenhouse effect, the development of a circular economy has attracted the attention of many scholars [1–4]. Humans' industrial production continues to intensify, and carbon emissions are directly linked to supply chain activities, which include the production process, transportation, distribution, and end-of-life product disposal [5]. The production process is always accompanied by high emissions and environmental pollution, especially in industry. Therefore, sustainable supply chain development has become the focus of the *EU-ETS* and The Paris Agreement [6]. The European Commission announced that transportation has become the second-biggest greenhouse gas (GHG) emitter preceded only by energy, and accounts for almost a quarter of European GHG emissions. Especially, road transport has significant contributions to CO2 emissions in addition to the contributions from the maritime and aviation sectors.

In reality, the supply chain has become more complex and rapidly evolved into supply chain networks along with globalization and specialization. In a complex supply chain network, firms face risks not only from variable demand but also from their competitors [7–9]. Therefore, pollution and sustainability issues should be highlighted because of their fierce effect on both supply chain networks and societies and countries.

As the main advocator of a low-carbon society, international organizations and governments have taken some actions at the macro level. For example, The Paris Agreement, which entered into force on 4 November 2016, made arrangements for global action on

**Citation:** Cheng, P.; Zhang, G.; Sun, H. The Sustainable Supply Chain Network Competition Based on Non-Cooperative Equilibrium under Carbon Emission Permits. *Mathematics* **2022**, *10*, 1364. https:// doi.org/10.3390/math10091364

Academic Editors: Wen Zhang, Xiaofeng Xu, Jun Wu and Kaijian He

Received: 26 February 2022 Accepted: 15 April 2022 Published: 19 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

climate change after 2020. Many countries are facing unprecedented carbon emission stress and have thus set their abatement goals [10,11]. As the biggest developing country, the Chinese government promised that China's carbon emission intensity would be reduced by 60–65% in 2030, compared with that in 2005 [12]. The intensity of energy consumption would continue to decline, and the resource output efficiency would increase substantially. To achieve the promised emission reduction targets, governments have enacted several environmental policies which enforce firms to accomplish green transformation development.

On the other hand, the sustainable and circular economy also provides some opportunities for enterprises. The enterprises can build reverse channels or adopt green production technologies to undertake social responsibility. As a fast fashion brand, in 2016, Uniqlo established its R&D center called the Jeans Innovation Center (JIC) in California. Aiming at creating a more environmentally friendly production approach, JIC not only adopts ecological water washing materials but also develops laser-fading technology [10]. In the reverse flow, recycling and remanufacturing are efficient methods used to enhance resource utilization. The recycling process can be realized by two modes: original equipment manufacturers (OEMs) recycling and third-party recycling [13–15]. The latter mode is more efficient in dealing with the dramatically increased end-of-life (EOL) products and plays an important role in promoting a circular economy [16–18].

#### *1.2. Practical Motivation*

Our research investigates a real case of China's paper industrial supply chain network. The manufacturers are divided into two segments called eco manufacturers and noneco manufacturers according to whether they employ advanced low-carbon production technology. In addition, the collection centers include online recycling platforms and third-party intelligent recycling systems.

According to survey data gathered by the China Paper Association, there were about 2700 large-scale paper-making enterprises in 2018, and there were about 500 emission control enterprises included in carbon trading. The paper and paperboard production capacity was 104.35 million tons, and the total amount of wastepaper recycling in China was 49.64 million tons.

In 2015, China's total carbon emissions were 9084.62 × 10<sup>6</sup> tons of carbon dioxide equivalent, of which the carbon emissions of the paper industry accounted for 1.67% of China's total emissions and ranked seventh among all industries. The carbon emission of a paper enterprise comprises three components: carbon emission from fossil fuel combustion, carbon emission from production, carbon emission from the net purchase of electricity and heat. According to the statistics, carbon emissions from fossil fuel combustion account for the majority, about 81.3%, and coal occupies the vast majority of fossil fuel combustion. Therefore, the use of coal can represent the carbon emissions of the paper industry. Through the above calculation, we can obtain the carbon dioxide produced per ton of paper and show this information in the form of labels; that is, make carbon labels and stick carbon labels on the products.

According to the statistics of most paper-making enterprises, the production of one ton of paper emits about 3000 kg of carbon dioxide, which is both directly produced in the production process and indirectly generated in the relevant links. In addition, the transport of paper products from manufacturers to consumers also produces carbon dioxide, especially in cases in which heavy vehicles are used for long-distance transportation. After it is used by consumers, paper can be recycled by a third-party recycler such as the Little Yellow Dog recycling system, which is widespread in some communities in China, and several online recycling platforms that also conduct wastepaper recycling. After cleaning and related treatment, about 800 kg of recycled paper can be obtained by recycling one ton of wastepaper; thus, this process can greatly reduce the use of raw materials and carbon emissions.

Paper enterprises are also constantly innovating their production technologies. Some manufacturers, such as Shanghai Oriental Champion Paper Co., Ltd., one of the leading thin paper manufacturers in China, Shanghai. have switched from off-site coal-fired power generation to on-site gas-fired power generation, which has improved energy efficiency, reduced the carbon footprint by 60%, and reduced power costs by one fifth.

#### *1.3. Research Question and Contributions*

On the basis of the above background, this paper tries to investigate three questions:


To answer these questions, we first consider a strict carbon emission permits supply chain network in which the government sets an emission threshold. Then we extend the base model in two ways. First, we assume government adopts cap-and-trade regulation to stimulate enterprises' production enthusiasm. The government allocates enterprises free carbon caps before production begins, and firms should determine their strategy under certain caps. Second, we suppose caps flow among firms without transaction cost. This situation may occur among different enterprises within a large enterprise group and promote enterprises' collaboration.

Three CLSCN models were built based on practical cases. Using variational inequality theory, modified projection algorithms, and contraction algorithms, models can be transformed and solved. The equilibrium results under different situations are compared through numerical simulation.

The major contributions of this paper can be summarized as follows:


We organized the remaining parts as follows. We provide a comparative discussion of the previous research highly related to our research in Section 2. Section 3 provides the notations and assumptions to accurately describe the decision models. In Section 4, the variational inequality models and the algorithm used to solve the models are described. Section 5 analyzes the results of numerical experiments to obtain the enlightenment of the management. Finally, we present the conclusions and suggestions in Section 6. The qualitative properties of the corresponding variational inequality models are presented in the Appendix C.

#### **2. Literature Review**

This paper focuses on sustainable supply chains, cap-and-trade regulations, noncooperative equilibrium, and consumers' environmental awareness. To better highlight our research issue, we briefly reviewed some relevant studies on these subjects. We will also point out the difference between our study and previous ones.

#### *2.1. Sustainable Supply Chain*

Pressure from stakeholders for sustainable development is forcing top management to reconsider its supply chain management, and the pursuit of sustainability has evolved as a popular trend in supply chain management [10,19]. Motivated by international retailers (e.g., Walmart and H&M) cooperating with their suppliers to reduce carbon emissions across supply chains, [10] investigated information sharing and studied its effect on carbon emission reduction. Considering a supply chain consisting of two competing manufacturers and a retailer, [20] studied the optimal green technology investment strategy problem of upstream manufacturers. Guo et al. [21] established a fashion supply chain consisting of one manufacturer and two competing retailers and discussed how retailer competition and consumer returns affect the development of green products in the fashion industry.

Recently, remanufacturing has come into focus as an area of economic and environmental insight [22–25]. Savaskan et al. [22] were among the first to divide the CLSC recycling model into the manufacturer recycling model, vendor recycling model, and third-party recycling model. The results of their work illustrated that the vendor recycling model is the most effective approach. Taleizadeh et al. [26] analyzed the effects of the third-party recycler in a CLSC under deterministic demand. Zerang et al. [27] established a three-echelon closed-loop supply chain model, and the results showed that the manufacturer-Stackelberg case is often the most effective scenario in CLSC.

Although the above-mentioned papers investigate the closed-loop supply chain in depth, the cap-and-trade regulation has been neglected as an effective approach to reducing carbon emissions, and thus needs further discussion.

#### *2.2. Cap-and-Trade Regulations*

To stimulate enterprises to actively reduce their carbon emissions through economic incentives, the government launched carbon trading, which can also be called cap-and-trade regulations. The *European Union Emissions Trading Scheme* (*EU-ETS*) is a successful form of cap-and-trade regulations. China launched its first carbon trading pilot in 2013, which entered into force in 2019. Therefore, it is important to explore the impacts of cap-and-trade regulations on enterprises, and conducting a simulation study on global carbon emission rights trading can provide practical outcomes [28–35]. Zhang and Xu [36] provided a basis for decision making on the reasonable use or sale of carbon emission rights by manufacturers and made a comparative analysis of the effectiveness of carbon trading and carbon tax. Du et al. [37] analyzed the game between decision-makers on product pricing and output considering cap-and-trade regulations and obtained a unique Nash equilibrium based on the basic Newsboy model. Yang et al. [38] and Yang et al. [39] both explored the channel selection problems under cap-and-trade regulations. The former asserts that products' properties and consumers' channel preferences are key factors affecting manufacturers' channel selection. The latter highlights that both the level of carbon emissions reduction and the profits of manufacturers increase with the manufacturer's product promotion.

Unlike our research, the above studies do not combine cap-and-trade regulations with reverse logistics. Moreover, they ignore the fact that carbon trade volume should be a decision variable in decision making.

#### *2.3. Supply Chain Network Based on Non-Cooperative Equilibrium*

The business crosses and fierce competition among supply chain members present the supply chain as a hierarchical network structure, including various enterprises and demand markets. With the coexistence of a competition and cooperation relationship, according to the rational person assumption, the corporate goal is to maximize its profits. Non-cooperative competition among the same types of members in the network forms a Nash equilibrium. Our study is also related to the literature rooted in supply chain network equilibrium under different environmental policies. In this field, scholars have carried out several studies on decision-making problems with different network structures [5,38,39]. Nagurney et al. [40] first established the SCN equilibrium model, making a

great contribution to the promotion and application of supply chain network theory, and applied it to diverse fields [41,42]. With the implementation of environmental protection policies, He et al. [43] studied the joint effect of the mandatory cap policy and operational decision mode on profitability and emissions. The results illustrate that the cap-sharing scheme can achieve Pareto improvement for chain players' profit and obtain a win-win situation for system profit and GHG emission reduction. Tao et al. [44] studied two types of mandatory cap policies under a multi-period scenario supply chain network and found that decision-makers can adjust their strategies under global carbon emissions constraints in most cases. He et al. [45] considered a supply chain super network constrained by a mandatory cap policy and examined the joint effect of stringent carbon regulations and operational decision modes on system performance.

#### *2.4. Consumers' Environmental Awareness*

Currently, consumers are increasingly concerned about the energy crisis and global warming and are focused on environmentally friendly and green products [34,46]. In 2014, the Eurobarometer Commission survey stated that 75% of Europeans tended to buy green products at a higher price [47,48], which promotes the development of eco-friendly products. In China, a report by the AliResearch Institute found that the total number of consumers who have environmental awareness increased by 14% during 2011~2015, and reached 65 million in 2015 [49]. Consumers' green preferences change their purchase behavior and promote low-carbon development [48]. Therefore, in this paper, the consumers' environmental awareness level is introduced to depict the social environment more realistically.

#### *2.5. Research Gap*

We highlight the contribution of the aforementioned studies in Table 1. The literature review has shown that most previous studies examine the optimization of the supply chain under the given emission regulations. When a carbon cap exists, most studies consider it as a given constant that constrains manufacturers' decision making. Most previous research related to carbon-constrained operations optimization only considers one or two kinds of carbon reduction policies. There is a lack of literature comparatively analyzing the impact of cap-and-trade regulations, mandatory cap policy, and cap-sharing schemes on multiple decision-makers under CLSCN.


**Table 1.** Comparison of related research papers in low-carbon regulations.

To fill this gap, our study focuses on how different regulations influence members' profits and carbon emissions in a CLSCN and investigate the remanufacturing's impact on members' equilibrium decisions. The results present meaningful information for the government to enact better carbon regulations and enterprises to adopt better operational policies.

#### **3. Notations and Assumptions**

*3.1. Notations*

The following parameters, decision variables, endogenous variables, and functions shown in Tables 2–5 are used throughout the remainder of this paper.

**Table 2.** Parameters of the model.


**Table 3.** Decision variables of the model.



**Table 4.** Endogenous variables of the model.


**Table 5.** Functions of the model.


#### *3.2. Assumptions*

To highlight the research question of the models developed later in Section 4, some assumptions need to be presented as follows.

**Assumption 1.** *The manufacturers are divided into two types called "non-eco manufacturer" and "eco manufacturer" according to whether they adopt green production technology. Eco manufacturers undertake higher production costs than non-eco manufacturers due to their possession of better production technology to decrease carbon emissions, and two products have a certain substitution relationship [50]. This assumption comes from reality (e.g., Huawei mobiles phones and Apple mobile phones). As it can be seen in the demand function:*

$$d\_k^l = 250 - 2\rho\_k^l - 1.5\rho\_{3-k}^l + 0.5(\rho\_k^h + \rho\_{3-k}^h) + \sigma\sharp\sum\_{x=1}^2 (1 - a\_x^l); \\ d\_k^h = 230 - 2\rho\_k^h - 1.5\rho\_{3-k}^h + 0.5(\rho\_k^l + \rho\_{3-k}^l) + \sigma\sharp\sum\_{x=1}^2 (1 - a\_x^l) + \sigma\sharp\sum\_{x=1}^2 (1 - a\_x^l)$$

*the quantity of each kind of product is affected by both its own and another product's selling price, which represents the substitution relationship between them*.

**Assumption 2.** *The new product and remanufactured product are homogeneous [51,52]. However, re-manufactured products have lower production costs and unit carbon emissions than the new ones. This assumption refers to the literature [22,53,54]. Savaskan et al. used the Eastman Kodak Company example to illustrate this relationship. Used cameras are typically upgraded to the quality of new ones, and both products can perfectly substitute each other. In this paper, we address the carbon emissions in the production process; thus, different types of products emit the same carbon dioxide when used.*

**Assumption 3.** *Carbon emissions are generated during both the production and transportation processes [50]. To avoid trivial cases and to focus on the goals of this research, we only consider the total carbon emissions of each truck and do not carry out further analyses on the distance covered by vehicles*.

**Assumption 4.** *The carbon quota allocation mechanism is based on "Benchmarking," which can be more effective in pushing facilities to reduce carbon emissions [2,55]. Under cap-and-trade regulations, caps can be sold or bought to satisfy the target production [2].*

**Assumption 5.** *The consumers' environmental awareness level is reflected in the demand function [56,57].*

**Assumption 6.** *The cost functions in this paper are all continuous differentiable convex functions [40,58].*

#### **4. Model**

In this paper, we consider three scenarios of carbon reduction regulations. The mandatory cap policy requires manufacturers not to emit more than a specific quota. Otherwise, the firm will face heavy penalties that force it to comply with the policy. As for the cap sharing policy, the total carbon emissions of different firms cannot exceed the aggregate quota of these firms; the carbon quotas can be transferred freely between two types of manufacturers. The cap-and-trade policy requires a carbon trade market that charges a certain commission from the enterprises participating in the carbon transaction and seeks maximum profit.

In addition, the European Commission is willing to regulate heavy-duty vehicles' carbon emissions; this willingness is modeled in this paper. Trucks are only supposed to be used in the main logistics phases: (1) forward logistics: transferring products from manufacturers to customers; (2) reverse logistics: transferring reusable raw materials from collection centers to manufacturers. As assumed in [59], transferring EOL products from demand markets to collection centers is undertaken by smaller dimension vans that are not regulated strictly.

In the following, we construct three equilibrium models according to different policies, and the different closed-loop supply chain network structures are shown in Figure 1.

**Figure 1.** Closed-loop supply chain network structures.

#### *4.1. Demand Market Decisions*

Demand markets are the final demand points of product transaction in forward flow and are also the source of EOL products in reverse flow. In a forward transaction, consumers of each demand market decide the quantity of non-eco products and eco products that they want to buy according to the prices charged by manufacturers and transaction costs. In a reverse transaction, consumers sell the EOL products to collection centers when the recycling prices are reasonable and can compensate for the loss of consumers caused by recycling.

According to the previous functions and notations definition, for the non-eco products in demand market *k*, we have the following complementary relationships:

$$\rho\_{jk}^\* + c\_{jk}^{K\*} \begin{cases} \\ \ge \rho\_k^{h\*} \text{, if } q\_{jk}^\* > 0 \\ \ge \rho\_k^{h\*} \text{, if } q\_{jk}^\* = 0 \end{cases} \tag{1}$$

$$d\_k^h \begin{cases} \displaystyle = \sum\_{j=1}^f q\_{jk'}^\* \text{ if } \rho\_k^{h\*} > 0\\ \displaystyle \le \sum\_{j=1}^f q\_{jk'}^\* \text{ if } \rho\_k^{h\*} = 0 \end{cases} \tag{2}$$

When the transactions between non-eco manufacturers and demand markets occur, *q*∗ *jk* <sup>&</sup>gt; 0 and *<sup>ρ</sup>h*<sup>∗</sup> *<sup>k</sup>* > 0 hold simultaneously, and the demand and supply are equal. Otherwise, the transactions cannot occur.

Similarly, for the eco products, we have:

$$
\rho\_{ik}^{\*} + c\_{ik}^{K\*} \begin{cases}
= \rho\_{k'}^{l\*} \text{ if } q\_{ik}^{\*} > 0 \\
\ge \rho\_{k'}^{l\*} \text{ if } q\_{ik}^{\*} = 0
\end{cases} \tag{3}
$$

$$d\_k^l \begin{cases} \displaystyle = \sum\_{i=1}^l q\_{ik'}^\* \text{ if } \rho\_k^{l\*} > 0\\ \displaystyle \le \sum\_{i=1}^l q\_{ik'}^\* \text{ if } \rho\_k^{l\*} = 0 \end{cases} \tag{4}$$

In the reverse logistics, after the consumption process, part of the products become EOL products that have no use value and can be recycled. When these EOL products are sent to collection centers, it will bring disutility to consumers. We assume *α<sup>u</sup> <sup>k</sup>* (*Q*5) is a monotonically increasing function that depends on the collected volume vector *Q*5, which means more EOL products collection brings higher consumers' disutility. Further, the more recycling products are recycled by collection centers, the higher the buy-back prices are. If the buy-back price *ρu*<sup>∗</sup> *kh* can compensate the disutility of consumers, that is, *<sup>ρ</sup>u*<sup>∗</sup> *kh* = *<sup>α</sup><sup>u</sup> <sup>k</sup>* (*Q*<sup>∗</sup> 5 ), then recycling transactions will occur; otherwise, recycling transactions will not occur. This relationship can be described as the following complementary form:

$$\alpha\_k^{\mathfrak{u}}(\mathbf{Q}\_5^\*) \begin{cases} = \rho\_{k\mathbf{h'}}^{\mathfrak{u}\*} \text{ if } q\_{k\mathbf{h}}^{\mathfrak{u}\*} > 0\\ \ge \rho\_{k\mathbf{h'}}^{\mathfrak{u}\*} \text{ if } q\_{k\mathbf{h}}^{\mathfrak{u}\*} = 0 \end{cases} \tag{5}$$

$$\text{s.t.} \sum\_{h=1}^{H} q\_{kh}^{u} \le \varepsilon\_{j} \sum\_{j=1}^{J} q\_{jk} + \varepsilon\_{i} \sum\_{i=1}^{I} q\_{ik} \tag{6}$$

Constraint (6) indicates that the products in reverse logistics are always less than those in forward logistics.

Integrating the forward and reverse behavior of consumers, the optimality conditions of demand markets can be defined as follows: determine the optimal solution (*Q*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 3, *<sup>ρ</sup>J*∗, *<sup>ρ</sup>I*∗, *<sup>Q</sup>*<sup>∗</sup> 5, *<sup>γ</sup>*∗) ∈ <sup>Ω</sup>*k*, satisfying:

$$\begin{split} \sum\_{\begin{subarray}{c}j=1\\k=1\end{subarray}}^{} \sum\_{k=1}^{K} \left[\rho\_{jk}^{\*} + c\_{jk}^{K\*} - \rho\_{k}^{h\*} - \varepsilon\_{l}\gamma\_{k}^{\*}\right] &\times \left[q\_{jk} - q\_{jk}^{\*}\right] + \sum\_{i=1}^{I} \sum\_{k=1}^{K} \left[\rho\_{ik}^{\*} + c\_{ik}^{K\*} - \rho\_{k}^{l\*} - \varepsilon\_{l}\gamma\_{k}^{\*}\right] \times \left[q\_{ik} - q\_{ik}^{\*}\right] + \\ \sum\_{k=1}^{K} \left[\sum\_{j=1}^{I} \left[q\_{jk}^{\*} - d\_{k}^{h\*}\right] \times \left[\rho\_{k}^{h} - \rho\_{k}^{h\*}\right] + \sum\_{k=1}^{K} \left[\sum\_{j=1}^{I} q\_{jk}^{\*} - d\_{k}^{\*}\right] \times \left[\rho\_{k}^{l} - \rho\_{k}^{l\*}\right] + \\ \sum\_{h=1}^{H} \sum\_{k=1}^{K} \left[a\_{k}^{u}(Q\_{5}^{\*}) - \rho\_{k\bar{u}}^{u\*} + \gamma\_{k}^{\*}\right] \times \left[q\_{\bar{u}k}^{u} - q\_{\bar{u}k}^{u\*}\right] + \sum\_{k=1}^{K} \left[\varepsilon\_{j} \sum\_{j=1}^{I} q\_{jk}^{\*} + \varepsilon\_{i} \sum\_{i=1}^{I} q\_{ik}^{\*} - \sum\_{k=1}^{K} q\_{\bar{u}k}^{u\*}\right] \times \left[\gamma\_{k} - \gamma\_{k}^{\*}\right] \ge 0. \end{split} (7)$$

where <sup>Ω</sup>*<sup>k</sup>* <sup>=</sup> *<sup>R</sup>JK*+*IK*+*J*+*I*+*KH*+*<sup>K</sup>* <sup>+</sup> , *γ<sup>k</sup>* is the Lagrangian multiplier corresponding to Constraint (6), and *<sup>γ</sup>* = (*γk*)*K*×<sup>1</sup> ∈ *<sup>R</sup><sup>K</sup>* +.

#### *4.2. Collection Centers' Decisions*

In the reverse logistics, the collection center *h* recycles these EOL products by paying price *ρ<sup>u</sup> kh* to consumers in demand markets. After separating, detecting, and other treatments, these EOL products are transformed to various reusable materials and then are sold to manufacturers at price *ρ<sup>u</sup> hj* and *<sup>ρ</sup><sup>u</sup> hi*, respectively. Therefore, the collection center *h* needs to decide the recycling quantity of EOL products and the sold quantity of reusable materials. Collection center *h* seeks to maximize its profit that can be described as:

$$\max \left[ \sum\_{j=1}^{I} \rho\_{hj}^{u\*} q\_{hj}^{u} + \sum\_{i=1}^{I} \rho\_{hi}^{u\*} q\_{hi}^{u} - \sum\_{k=1}^{K} (\rho\_{kh}^{u\*} q\_{kh}^{u} + c\_{kh}) - c\_{h} \right] \tag{8}$$

$$\text{s.t.} \sum\_{j=1}^{I} q\_{lij}^{u} + \sum\_{i=1}^{I} q\_{hi}^{u} \le \delta \sum\_{k=1}^{K} q\_{kh}^{u} \tag{9}$$

The objective function is the difference of the revenues and costs. The revenues are *J* ∑ *j*=1 *ρu*<sup>∗</sup> *hj <sup>q</sup><sup>u</sup> hj* <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *ρu*<sup>∗</sup> *hi <sup>q</sup><sup>u</sup> hi* resulting from selling reusable materials to non-eco manufacturers and eco manufacturers at prices *ρu*<sup>∗</sup> *hj* and *<sup>ρ</sup>u*<sup>∗</sup> *hi* , respectively. The costs include the buyback price *<sup>K</sup>* ∑ *k*=1 *ρu*<sup>∗</sup> *kh <sup>q</sup><sup>u</sup> kh* paid for consumers in demand markets, the disposal cost *ch*, and the transaction cost *ckh*. Constraint (9) ensures the trade-off between manufacturing and re-manufacturing. Finally, all decision variables are non-negative.

All collection centers compete in a non-cooperative manner, and all functions related are assumed continuous and convex. The optimal conditions of all collection centers can be expressed as following variational inequality: determine the optimal solution (*Q*<sup>∗</sup> 2, *<sup>Q</sup>*<sup>∗</sup> 4, *<sup>Q</sup>*<sup>∗</sup> 5, *<sup>λ</sup>*∗) ∈ <sup>Ω</sup>*h*, satisfying:

$$\begin{aligned} \sum\_{j=1}^{I} \sum\_{h=1}^{H} \left[ \lambda\_h^\* - \rho\_{hj}^{u\*} \right] \times \left[ q\_{lij}^u - q\_{lij}^{u\*} \right] + \sum\_{i=1}^{I} \sum\_{h=1}^{H} \left[ \lambda\_h^\* - \rho\_{hi}^{u\*} \right] \times \left[ q\_{lii}^u - q\_{lii}^{u\*} \right] + \\ \sum\_{h=1}^{H} \sum\_{k=1}^{K} \left[ \frac{\partial \varepsilon\_{ih}^\*}{\partial q\_{ih}^u} + \frac{\partial \varepsilon\_t^\*}{\partial q\_{ih}^u} + \rho\_{kh}^{u\*} - \delta \lambda\_h^\* \right] \times \left[ q\_{lh}^u - q\_{lh}^{u\*} \right] + \sum\_{h=1}^{H} \left[ \delta \sum\_{k=1}^{K} q\_{hk}^{u\*} - \sum\_{j=1}^{I} q\_{hj}^{u\*} - \sum\_{i=1}^{I} q\_{hi}^{u\*} \right] \times \left[ \lambda\_h - \lambda\_h^\* \right] \ge 0. \end{aligned} \tag{10}$$
 
$$\forall (\mathbf{Q}\_2, \mathbf{Q}\_4, \mathbf{Q}\_5, \lambda) \in \Omega\_h$$

where <sup>Ω</sup>*<sup>h</sup>* <sup>=</sup> *<sup>R</sup>H J*+*H I*+*KH*+*<sup>H</sup>* <sup>+</sup> , *λ<sup>h</sup>* is the Lagrangian multiplier corresponding to constraint (9), and *<sup>λ</sup>* = (*λh*)*H*×<sup>1</sup> ∈ *<sup>R</sup><sup>H</sup>* +.

#### *4.3. The Supply Chain Network under Cap-and-Trade (CT) Policy*

Under the cap-and-trade policy, we assume that the unit carbon quota price is exogenous and remains unchanged.

#### 4.3.1. Non-Ecological Manufacturers' Decisions

We first study non-eco manufacturers' decisions. According to the previous Assumption 4, the non-eco manufacturer *j* needs to buy the quota that can be expressed as *tj* = *<sup>α</sup><sup>J</sup>* 1*βrqv <sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuqu <sup>j</sup>* + *t*1*τ<sup>t</sup> K* ∑ *k*=1 *xjkqjk* + *t*2*τ<sup>t</sup> H* ∑ *h*=1 *xhjq<sup>u</sup> hj* − *capj*, the corresponding payment is *εtj* + *ωtj*. When manufacturer *j* pursues the maximization of profit, the objective function can be represented as:

$$\max \left[ \sum\_{k=1}^{K} \left( \rho\_{j\bar{k}}^{\*} q\_{j\bar{k}} - \varepsilon\_{j\bar{k}} \right) - \sum\_{h=1}^{H} \left( \rho\_{h\bar{j}}^{u\*} q\_{h\bar{j}}^{u} + \varepsilon\_{h\bar{j}} \right) - f\_{\bar{j}} - f\_{\bar{j}}^{u} - \rho \left( t\_1 \sum\_{k=1}^{K} \mathbf{x}\_{j\bar{k}} q\_{j\bar{k}} + t\_2 \sum\_{h=1}^{H} \mathbf{x}\_{h\bar{j}} q\_{h\bar{j}}^{u} \right) - \varepsilon t\_{\bar{j}} - \omega t\_{\bar{j}} \right] \tag{11}$$

$$s.t. \ q\_j^u \le \sum\_{l=1}^H q\_{llj}^u \tag{12}$$

$$\sum\_{k=1}^{K} q\_{jk} \le \beta^r q\_j^v + \beta^u q\_j^u \tag{13}$$

$$\alpha\_1^I \beta^r q\_j^v + \alpha\_2^I \beta^u q\_j^u + t\_1 \tau\_l \sum\_{k=1}^K \mathbf{x}\_{jk} q\_{jk} + t\_2 \tau\_l \sum\_{h=1}^H \mathbf{x}\_{hj} q\_{hj}^u - \alpha a p\_j - t\_j = 0 \tag{14}$$

Constraint (14) ensures that the manufacturer *j*'s total carbon emissions are equal to the sum of *capj* and *tj*.

Based on the CT model, in this case, manufacturer *j* needs to make an additional decision regarding the carbon transaction amount *tj*. Hence, the optimum solution of the above objective function is characterized by the following variational inequality with (*qv*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>j</sup>* , *<sup>Q</sup>*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 2, *<sup>T</sup>*<sup>∗</sup> 1,*ϕ*<sup>∗</sup> 1,*ϕ*<sup>∗</sup> 2,*ϕ*<sup>∗</sup> <sup>3</sup>) ∈ <sup>Ω</sup><sup>1</sup> *<sup>J</sup>*, such that:

*J* ∑ *j*=1 *∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* − *βrϕ*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>α</sup><sup>J</sup>* 1*βrϕ*3<sup>∗</sup> *j* × *qv <sup>j</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *j* + *J* ∑ *j*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *j ∂q<sup>u</sup> j* + *ϕ*1<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>β</sup>uϕ*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>α</sup><sup>J</sup>* 2*βuϕ*3<sup>∗</sup> *j* × *qu <sup>j</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* + *J* ∑ *j*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *jk <sup>∂</sup>qjk* <sup>−</sup> *<sup>ρ</sup>*<sup>∗</sup> *jk* <sup>+</sup> *<sup>ρ</sup>t*1*xjk* <sup>+</sup> *<sup>ϕ</sup>*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*1*τtxjkϕ*3<sup>∗</sup> *j* × *qjk* − *q*<sup>∗</sup> *jk* + *J* ∑ *j*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hj ∂q<sup>u</sup> hj* + *ρu*<sup>∗</sup> *hj* <sup>+</sup> *<sup>ρ</sup>t*2*xhj* <sup>−</sup> *<sup>ϕ</sup>*1<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*2*τtxhjϕ*3<sup>∗</sup> *j* × *qu hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hj* + *J* ∑ *j*=1 *ε* + *ω* + *ϕ*3<sup>∗</sup> *j* × *tj* − *t* ∗ *j* + *J* ∑ *j*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* × *ϕ*1 *<sup>j</sup>* <sup>−</sup> *<sup>ϕ</sup>*1<sup>∗</sup> *j* + *J* ∑ *j*=1 *βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *jk* × *ϕ*2 *<sup>j</sup>* <sup>−</sup> *<sup>ϕ</sup>*2<sup>∗</sup> *j* + *J* ∑ *j*=1 *αJ* 1*βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuqu*<sup>∗</sup> *<sup>j</sup>* + *t*1*τ<sup>t</sup> K* ∑ *k*=1 *xjkq*<sup>∗</sup> *jk* + *t*2*τ<sup>t</sup> H* ∑ *h*=1 *xhjq<sup>u</sup>*<sup>∗</sup> *hj* − *capj* − *t* ∗ *j* × *ϕ*3 *<sup>j</sup>* <sup>−</sup> *<sup>ϕ</sup>*3<sup>∗</sup> *j* ≥ 0. (15) ∀(*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>T</sup>*1,*ϕ*1,*ϕ*2,*ϕ*3) ∈ <sup>Ω</sup><sup>1</sup> *J*

where Ω<sup>1</sup> *<sup>J</sup>* <sup>=</sup> *<sup>R</sup>*2*J*+*JK*+*H J*+3*<sup>J</sup>* <sup>+</sup> × *R<sup>J</sup>* . Note that *ϕ*<sup>1</sup> *<sup>j</sup>* , *<sup>ϕ</sup>*<sup>2</sup> *<sup>j</sup>* , and *<sup>ϕ</sup>*<sup>3</sup> *<sup>j</sup>* are the Lagrangian multiplier associated with Constraint (12), Constraint (13), and Constraint (14), respectively, while *ϕ*<sup>1</sup> = *ϕ*1 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* <sup>+</sup>, *ϕ*<sup>2</sup> = *ϕ*2 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* <sup>+</sup>, *ϕ*<sup>3</sup> = *ϕ*3 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* .

To explore the significance of management, we give some explanations for VI (15). From the 3rd term of VI (15), we have *ρ*<sup>∗</sup> *jk* <sup>=</sup> *<sup>∂</sup>c*<sup>∗</sup> *jk ∂qjk* + *ρt*1*xjk* + *ϕ*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*1*τtxjkϕ*3<sup>∗</sup> *<sup>j</sup>* when *<sup>q</sup>*<sup>∗</sup> *jk* > 0; in other words, the transaction price between non-ecological manufacturer *j* and demand market *k* comprises a marginal transaction cost, unit truck transportation cost, and related carbon emission factor. From the 1st term of VI (15), we have *ϕ*2<sup>∗</sup> *<sup>j</sup>* <sup>=</sup> <sup>1</sup> *βr ∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* − *α<sup>J</sup>* 1*βrϕ*3<sup>∗</sup> *j* when *qv*<sup>∗</sup> *<sup>j</sup>* > 0, and are mainly marginal production costs. Therefore, the previous stage cost transmits to the next stage by the transaction price.

A point that is necessary to show is that the corresponding Lagrangian multipliers of Constraint (14) may be negative because Constraint (14) is an equation. In addition, *t* ∗ *<sup>j</sup>* is affected by the sum of *ε* and *ω*.

#### 4.3.2. Ecological Manufacturers' Decisions

Similarly, the surplus carbon quotas of the eco manufacturer can be expressed as *ti* = *capi* − *α<sup>I</sup>* 1*βrqv <sup>i</sup>* − *<sup>α</sup><sup>I</sup>* 2*βuqu <sup>i</sup>* − *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikqik* − *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup> hi*. We, therefore, obtain the manufacturer *i*'s objective below to maximize its profit through aggregate revenue minus costs:

$$\max \left[ \sum\_{k=1}^{K} \left( \rho\_{\text{i\'i}}^{\*} q\_{\text{i\'i}} - \varepsilon\_{\text{i\'i}} \right) - \sum\_{h=1}^{H} \left( \rho\_{\text{li\'i}}^{\*} q\_{\text{li\'i}}^{\text{u}} + \varepsilon\_{\text{hi}} \right) - f\_{\text{i}} - f\_{\text{i}}^{\text{u}} - \rho \left( t\_{3} \sum\_{k=1}^{K} \mathbf{x}\_{\text{i}k} q\_{\text{iki}} + t\_{4} \sum\_{h=1}^{H} \mathbf{x}\_{\text{i\'i}} q\_{\text{hi}}^{\text{u}} \right) - \varepsilon t\_{1} + \omega t\_{1} \right] \tag{16}$$

$$\text{s.t. } q\_i^u \le \sum\_{h=1}^H q\_{hi}^u \tag{17}$$

$$\sum\_{k=1}^{K} q\_{ik} \le \beta^r q\_i^v + \beta^u q\_i^u \tag{18}$$

$$
\alpha a p\_i - a\_1^I \beta^r q\_i^v - a\_2^I \beta^u q\_i^u - t\_3 \pi\_t \sum\_{k=1}^K \pi\_{ik} q\_{ik} - t\_4 \pi\_t \sum\_{h=1}^H \pi\_{hi} q\_{hi}^u = t\_i \tag{19}$$

The last item of objective function *ωti* denotes the manufacturer *i*'s extra revenue. Constraint (19) ensures that the total carbon emissions of manufacturer *i* plus *ti* equals *capi*. In this case, manufacturer *i* needs to make an additional decision on the carbon transaction amount *ti*. Therefore, the optimum solution of the above objective function can be characterized by the following variational inequality with (*qv*<sup>∗</sup> *<sup>i</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>i</sup>* , *<sup>Q</sup>*<sup>∗</sup> 3, *<sup>Q</sup>*<sup>∗</sup> 4, *<sup>T</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>φ</sup>*<sup>∗</sup> <sup>1</sup> , *<sup>φ</sup>*<sup>∗</sup> <sup>2</sup> , *<sup>φ</sup>*<sup>∗</sup> <sup>3</sup> ) ∈ <sup>Ω</sup><sup>1</sup> *I* :

*I* ∑ *i*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *ik <sup>∂</sup>qik* <sup>−</sup> *<sup>ρ</sup>*<sup>∗</sup> *ik* <sup>+</sup> *<sup>ρ</sup>t*3*xik* <sup>+</sup> *<sup>φ</sup>*2<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*3*τtxikφ*3<sup>∗</sup> *i* × " *qik* − *q*<sup>∗</sup> *ik* # + *I* ∑ *i*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hi ∂q<sup>u</sup> hi* + *ρu*<sup>∗</sup> *hi* <sup>+</sup> *<sup>ρ</sup>t*4*xhi* <sup>−</sup> *<sup>φ</sup>*1<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*4*τtxhiφ*3<sup>∗</sup> *i* × " *qu hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hi* # + *I* ∑ *i*=1 " *ε* − *ω* + *φ*3<sup>∗</sup> *i* # × " *ti* − *t* ∗ *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* × " *φ*1 *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup>*1<sup>∗</sup> *i* # + *I* ∑ *i*=1 *βrqv*<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *ik* × " *φ*2 *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup>*2<sup>∗</sup> *i* # + *I* ∑ *i*=1 *capi* − *α<sup>I</sup>* 1*βrqv <sup>i</sup>* − *<sup>α</sup><sup>I</sup>* 2*βuqu <sup>i</sup>* − *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikqik* − *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup> hi* − *ti* × " *φ*3 *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup>*3<sup>∗</sup> *i* # ≥ 0. (20)

∀(*q<sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>T</sup>*2, *<sup>φ</sup>*1, *<sup>φ</sup>*2, *<sup>φ</sup>*3) ∈ <sup>Ω</sup><sup>1</sup> *I*

where Ω<sup>1</sup> *<sup>I</sup>* <sup>=</sup> *<sup>R</sup>*2*I*+*IK*+*H I*+2*<sup>I</sup>* <sup>+</sup> <sup>×</sup> *<sup>R</sup><sup>I</sup>* . Note that *φ*<sup>1</sup> *<sup>i</sup>* , *<sup>φ</sup>*<sup>2</sup> *<sup>i</sup>* and *<sup>φ</sup>*<sup>3</sup> *<sup>i</sup>* are the Lagrangian multipliers associated with Constraint (17), Constraint (18), and Constraint (19), respectively, while *φ*<sup>1</sup> = *φ*1 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* <sup>+</sup>, *<sup>φ</sup>*<sup>2</sup> = *φ*2 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* <sup>+</sup>, *<sup>φ</sup>*<sup>3</sup> = *φ*3 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* .

4.3.3. Carbon Trade Center's Decisions

Carbon trade centers charge a certain fee *ε* for unit carbon trade volume. Simultaneously, carbon trade centers should undertake associated cost ∑*<sup>J</sup> <sup>j</sup>*=<sup>1</sup> *<sup>c</sup><sup>t</sup> <sup>j</sup>* <sup>+</sup> <sup>∑</sup>*<sup>I</sup> <sup>i</sup>*=<sup>1</sup> *c<sup>t</sup> i* . The carbon trade center also pursues profit maximization, which can be described as:

$$\max \left[ \sum\_{j=1}^{I} \left( \varepsilon t\_j^\* - c\_j^t(t\_j^\*) \right) + \sum\_{i=1}^{I} \left( \varepsilon t\_i^\* - c\_i^t(t\_i^\*) \right) \right] \tag{21}$$

$$\text{s.t. } \sum\_{j=1}^{I} t\_j \le \sum\_{i=1}^{I} t\_i \tag{22}$$

Constraint (22) shows the balance between the demand supply of the carbon quota. The profit of the carbon trade center seeking to maximize can be transformed into the following variational inequality: determine the optimal solution ∀(*T*<sup>∗</sup> <sup>1</sup>, *<sup>T</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>ζ</sup>*<sup>∗</sup> *<sup>c</sup>* ) ∈ Ω*c*, satisfying:

$$\sum\_{j=1}^{I} \left[ \frac{\partial c\_j^{t\*}}{\partial t\_j} + \lambda\_c^\* - \varepsilon \right] \times \left[ t\_j - t\_j^\* \right] + \sum\_{i=1}^{I} \left[ \frac{\partial c\_i^{t\*}}{\partial t\_i} - \varepsilon - \lambda\_c^\* \right] \times \left[ t\_i - t\_i^\* \right] + \left[ \sum\_{j=1}^{I} t\_i^\* - \sum\_{j=1}^{I} t\_j^\* \right] \times \left[ \lambda\_c - \lambda\_c^\* \right] \ge 0. \tag{23}$$
 
$$\forall (T\_1, T\_2, \xi\_\varepsilon) \in \Omega\_\varepsilon$$

where <sup>Ω</sup>*<sup>c</sup>* = *<sup>R</sup>J*+*I*+<sup>1</sup> <sup>+</sup> . Note that *ζ<sup>c</sup>* is the Lagrangian multiplier associated with Constraint (22) and *ζ<sup>c</sup>* ∈ *R*+.

4.3.4. The Equilibrium Conditions of Closed-Loop Supply Chain Network in the CT Model

Under the cap-and-trade regulations, for the closed-loop supply chain network, the Nash equilibrium (Nash 1950) conditions of VI (7), VI (10), VI (15), VI (20), and VI (23) must hold simultaneously, and no one gains more from altering the current strategies.

**Definition 1.** *The equilibrium of the CLSCN under cap-and-trade regulation occurs when the sum of the left-hand side (L.H.S.) of (7), L.H.S. of (10), L.H.S. of (15), L.H.S. of (20), and L.H.S. of (23) is non-negative.*

**Theorem 1.** *The equilibrium conditions of the CLSCN under cap-and-trade regulations are equivalent to the solutions of VI as follows: determine the optimal solution* (*qv*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>v*<sup>∗</sup> *<sup>i</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>i</sup>* , *<sup>Q</sup>*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 2, *<sup>Q</sup>*<sup>∗</sup> 3, *Q*∗ 4, *<sup>Q</sup>*<sup>∗</sup> 5, *<sup>T</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>T</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>ρ</sup>J*∗, *<sup>ρ</sup>I*∗,*ϕ*<sup>∗</sup> 1,*ϕ*<sup>∗</sup> 2,*ϕ*<sup>∗</sup> <sup>3</sup>, *<sup>φ</sup>*<sup>∗</sup> <sup>1</sup> , *<sup>φ</sup>*<sup>∗</sup> <sup>2</sup> , *<sup>φ</sup>*<sup>∗</sup> <sup>3</sup> , *<sup>ζ</sup>c*, *<sup>λ</sup>*∗, *<sup>γ</sup>*∗) <sup>∈</sup> <sup>Ω</sup>1*, satisfying:*

*J* ∑ *j*=1 *∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* − *βrϕ*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>α</sup><sup>J</sup>* 1*βrϕ*3<sup>∗</sup> *j* × *qv <sup>j</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *j* + *J* ∑ *j*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *j ∂q<sup>u</sup> j* + *ϕ*1<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>β</sup>uϕ*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>α</sup><sup>J</sup>* 2*βuϕ*3<sup>∗</sup> *j* × *qu <sup>j</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* + *I* ∑ *i*=1 *∂ f* <sup>∗</sup> *i ∂q<sup>v</sup> i* − *βrφ*2<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 1*βrφ*3<sup>∗</sup> *i* × " *qv <sup>i</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *i ∂q<sup>u</sup> i* + *φ*1<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>β</sup>uφ*2<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuφ*3<sup>∗</sup> *i* × " *qu <sup>i</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* # + *J* ∑ *j*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *jk ∂qjk* + *cK*<sup>∗</sup> *jk* <sup>−</sup> *<sup>ρ</sup>h*<sup>∗</sup> *<sup>k</sup>* <sup>−</sup> *<sup>ε</sup>jγ*<sup>∗</sup> *<sup>k</sup>* <sup>+</sup> *<sup>ρ</sup>t*1*xjk* <sup>+</sup> *<sup>ϕ</sup>*2<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*1*τtxjkϕ*3<sup>∗</sup> *j* × *qjk* − *q*<sup>∗</sup> *jk* + *I* ∑ *i*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *ik <sup>∂</sup>qik* <sup>+</sup> *<sup>c</sup>K*<sup>∗</sup> *ik* <sup>−</sup> *<sup>ρ</sup>l*<sup>∗</sup> *<sup>k</sup>* <sup>−</sup> *<sup>ε</sup>iγ*<sup>∗</sup> *<sup>k</sup>* <sup>+</sup> *<sup>ρ</sup>t*3*xik* <sup>+</sup> *<sup>φ</sup>*2<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*3*τtxikφ*3<sup>∗</sup> *i* × " *qik* − *q*<sup>∗</sup> *ik* # + *J* ∑ *j*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hj ∂q<sup>u</sup> hj* + *ρt*2*xhj* − *ϕ*1<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*2*τtxhjϕ*3<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>∗</sup> *h* × *qu hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hj* + *I* ∑ *i*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hi ∂q<sup>u</sup> hi* + *ρt*4*xhi* − *φ*1<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*4*τtxhiφ*3<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>∗</sup> *h* × " *qu hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hi* # + *J* ∑ *j*=1 *∂ct*<sup>∗</sup> *j ∂tj* + *λ*<sup>∗</sup> *<sup>c</sup>* + *ω* + *ϕ*3<sup>∗</sup> *j* × *tj* − *t* ∗ *j* <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *∂ct*<sup>∗</sup> *i <sup>∂</sup>ti* <sup>−</sup> *<sup>λ</sup>*<sup>∗</sup> *<sup>c</sup>* − *ω* + *φ*3<sup>∗</sup> *i* × " *ti* − *t* ∗ *i* # + *K* ∑ *k*=1 *<sup>J</sup>* ∑ *j*=1 *q*∗ *jk* <sup>−</sup> *<sup>d</sup>h*<sup>∗</sup> *k* × *ρh <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup>h*<sup>∗</sup> *k* <sup>+</sup> *<sup>K</sup>* ∑ *k*=1 *I* ∑ *i*=1 *q*∗ *ik* <sup>−</sup> *<sup>d</sup>l*<sup>∗</sup> *k* × *ρl <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup>l*<sup>∗</sup> *k* + *H* ∑ *h*=1 *K* ∑ *k*=1 *αu <sup>k</sup>* (*Q*<sup>∗</sup> <sup>5</sup> ) + *<sup>γ</sup>*<sup>∗</sup> *<sup>k</sup>* <sup>+</sup> *<sup>∂</sup>c*<sup>∗</sup> *kh ∂q<sup>u</sup> kh* <sup>+</sup> *<sup>∂</sup>c*<sup>∗</sup> *h ∂q<sup>u</sup> kh* − *δλ*<sup>∗</sup> *h* × " *qu kh* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *kh* # + *J* ∑ *j*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* × *ϕ*1 *<sup>j</sup>* <sup>−</sup> *<sup>ϕ</sup>*1<sup>∗</sup> *j* + *J* ∑ *j*=1 *βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *jk* × *ϕ*2 *<sup>j</sup>* <sup>−</sup> *<sup>ϕ</sup>*2<sup>∗</sup> *j* + *J* ∑ *j*=1 *αJ* 1*βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuqu*<sup>∗</sup> *<sup>j</sup>* + *t*1*τ<sup>t</sup> K* ∑ *k*=1 *xjkq*<sup>∗</sup> *jk* + *t*2*τ<sup>t</sup> H* ∑ *h*=1 *xhjq<sup>u</sup>*<sup>∗</sup> *hj* − *capj* − *t* ∗ *j* × *ϕ*3 *<sup>j</sup>* <sup>−</sup> *<sup>ϕ</sup>*3<sup>∗</sup> *j* + *I* ∑ *i*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* × " *φ*1 *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup>*1<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *βrqv*<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *ik* × " *φ*2 *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup>*2<sup>∗</sup> *i* # + *I* ∑ *i*=1 *capi* − *α<sup>I</sup>* 1*βrqv*<sup>∗</sup> *<sup>i</sup>* − *<sup>α</sup><sup>I</sup>* 2*βuqu*<sup>∗</sup> *<sup>i</sup>* − *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikq*<sup>∗</sup> *ik* − *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup>*<sup>∗</sup> *hi* − *ti* × " *φ*3 *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup>*3<sup>∗</sup> *i* # + *<sup>I</sup>* ∑ *i*=1 *t* ∗ *<sup>i</sup>* <sup>−</sup> *<sup>J</sup>* ∑ *j*=1 *t* ∗ *j* × [*λ<sup>c</sup>* − *λ*<sup>∗</sup> *<sup>c</sup>* ] <sup>+</sup> *<sup>H</sup>* ∑ *h*=1 *δ K* ∑ *k*=1 *qu*<sup>∗</sup> *kh* <sup>−</sup> *<sup>J</sup>* ∑ *j*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>I</sup>* ∑ *i*=1 *qu*<sup>∗</sup> *hi* × " *λ<sup>h</sup>* − *λ*<sup>∗</sup> *h* # + *K* ∑ *k*=1 *εj J* ∑ *j*=1 *q*∗ *jk* + *ε<sup>i</sup> I* ∑ *i*=1 *q*∗ *ik* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *qu*<sup>∗</sup> *kh* × " *γ<sup>k</sup>* − *γ*<sup>∗</sup> *k* # ≥ 0. (24) ∀(*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>T</sup>*1, *<sup>T</sup>*2, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* ,*ϕ*1,*ϕ*2,*ϕ*3, *φ*1, *φ*2, *φ*3, *ζc*, *λ*, *γ*) ∈ Ω<sup>1</sup>

$$where \,\Omega^1 = \Omega^1\_I \times \Omega^1\_I \times \Omega\_c \times \Omega\_k \times \Omega\_{\hbar}.$$

**Proof.** Adding VI (7), VI (10), VI (15), VI (20), and VI (23) together, we can obtain VI (24). At the same time, when VI (24) holds, then VI (7), VI (10), VI (15), VI (20), and VI (23) are also satisfied, respectively.

Let *X*<sup>1</sup> ≡ (*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>T</sup>*1, *<sup>T</sup>*2, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* ,*ϕ*1,*ϕ*2,*ϕ*3, *φ*1, *φ*2, *φ*3, *ζc*, *λ*, *γ*), *F*(*X*1) ≡ (*Fx*(*X*1))22×1, the specific parts *Fx*(*X*1) (*x* = 1, ··· , 22) of *F*(*X*1) are given by the terms proceeding the multiplication signs in VI (24). Then, we can rewrite the VI (24) in standard form of VI following: determine the optimal vector *X*<sup>∗</sup> <sup>1</sup> <sup>∈</sup> <sup>Ω</sup>1, satisfying:

$$
\langle F(X\_1^\*), X\_1^\* \rangle \ge 0, \,\forall X\_1 \in \Omega^1 \tag{25}
$$

The notation ·, · denotes the inner product in *M*1—dimensional Euclidean space, where *M*<sup>1</sup> = 2*J* + 2*I* + *JK* + *H J* + *IK* + *H I* + *KH* + *J* + *I* + 2*K* + 3*J* + 3*I* + 1 + *H* + *K*.

#### *4.4. The Supply Chain Network under Mandatory Cap Policy (MC)*

In this section, we characterize how the exogenously given strict cap policy affects the supply chain members' decisions.

#### 4.4.1. Non-Ecological Manufacturers' Decisions

We describe the decision making and operational characteristics of non-eco manufacturers and provide optimal conditions. Hence, considering the transaction between manufacturer *j* and other supply chain members, we give the manufacturer *j*'s objective function as follows:

$$\max \left[ \sum\_{k=1}^{K} \left( \rho\_{j\mathbf{k}}^{\*} q\_{j\mathbf{k}} - c\_{j\mathbf{k}} \right) - \sum\_{h=1}^{H} \left( \rho\_{h\mathbf{j}}^{u\*} q\_{h\mathbf{j}}^{u} + c\_{h\mathbf{j}} \right) - f\_{\mathbf{j}} - f\_{\mathbf{j}}^{u} - \rho \left( t\_1 \sum\_{k=1}^{K} \mathbf{x}\_{j\mathbf{k}} q\_{j\mathbf{k}} + t\_2 \sum\_{h=1}^{H} \mathbf{x}\_{h\mathbf{j}} q\_{h\mathbf{j}}^{u} \right) \right] \tag{26}$$

$$\text{s.t. } q\_j^u \le \sum\_{h=1}^H q\_{hj}^u \tag{27}$$

$$\sum\_{k=1}^{K} q\_{jk} \le \beta^r q\_j^v + \beta^u q\_j^u \tag{28}$$

$$\alpha\_1^J \beta^r q\_j^v + \alpha\_2^J \beta^u q\_j^u + t\_1 \tau\_t \sum\_{k=1}^K \mathbf{x}\_{jk} q\_{jk} + t\_2 \tau\_t \sum\_{h=1}^H \mathbf{x}\_{hj} q\_{hj}^u \le c \alpha p\_j \tag{29}$$

Constraint (28) can be called the production balance constraint; Constraint (29) ensures the total carbon emissions generated by manufacturer *j* cannot exceed its quota *capj*.

According to the previous Assumption 6, the objective functions of manufacturers are continuously concave. All decision variables are non-negative. In this situation, nonecological manufacturer *j* determines the amount of raw materials and recycled EOL products, the output and transaction amount of the new product, and the remanufactured product.

All non-eco manufacturers compete in a non-cooperative fashion, and the profits of each non-eco manufacturer seeking to maximize can be transformed simultaneously into the following variational inequality: determine the optimal solution (*qv*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>j</sup>* , *<sup>Q</sup>*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 2, *<sup>μ</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>μ</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>μ</sup>*<sup>∗</sup> 3 ) ∈ Ω<sup>2</sup> *<sup>J</sup>*, satisfying:

$$\begin{aligned} &\sum\_{i=1}^{I} \left[ \frac{\partial f\_{j}^{\*}}{\partial q\_{j}^{\*}} - \beta^{\mathsf{p}} \mu\_{j}^{2\*} + a\_{1}^{\mathsf{p}} \beta^{\mathsf{p}} \mu\_{j}^{2\*} \right] \times \left[ q\_{j}^{\mathsf{p}} - q\_{j}^{\mathsf{p}\*} \right] + \sum\_{j=1}^{I} \left[ \frac{\partial f\_{j\*}^{\*\mathsf{p}}}{\partial q\_{j}^{\*}} + \mu\_{j}^{\mathsf{p}} - \beta^{\mathsf{p}} \mu\_{j}^{2\*} + a\_{2}^{\mathsf{p}} \beta^{\mathsf{p}} \mu\_{j}^{2\*} \right] \times \left[ q\_{j}^{\mathsf{p}} - q\_{j}^{\mathsf{p}\*} \right] + \\ &\sum\_{i=1}^{I} \sum\_{k=1}^{K} \left[ \frac{\partial \mathbf{x}\_{j}^{\*}}{\partial q\_{j}} - \mu\_{jk}^{\*} + \mu\_{1} \mu\_{jk} + \mu\_{j}^{2\*} + t\_{1} \pi\_{1} \pi\_{3} \mu\_{j}^{2\*} \right] \times \left[ q\_{jk} - q\_{jk}^{\*} \right] + \\ &\sum\_{i=1}^{I} \sum\_{k=1}^{I} \left[ \frac{\partial \mathbf{x}\_{i}^{\*}}{\partial q\_{i}^{\*}} + \mu\_{kj}^{\mathsf{p}} + \mu\_{2} \mu\_{kj} - \mu\_{j}^{1\*} + t\_{2} \pi\_{k} \pi\_{k} \mu\_{j}^{2\*} \right] \times \left[ q\_{kj}^{\mathsf{p}} - q\_{kj}^{\*} \right] + \\ &\sum\_{i=1}^{I} \sum\_{k=1}^{I} \left[ q\_{ik}^{\mathsf{p}} - q\_{jk}^{\*} \right] \times$$

where Ω<sup>2</sup> *<sup>J</sup>* <sup>=</sup> *<sup>R</sup>*2*J*+*JK*+*H J*+3*<sup>J</sup>* <sup>+</sup> . Note that *μ*<sup>1</sup> *<sup>j</sup>* , *<sup>μ</sup>*<sup>2</sup> *<sup>j</sup>* , and *<sup>μ</sup>*<sup>3</sup> *<sup>j</sup>* are the Lagrangian multipliers associated with Constraint (27), Constraint (28), and Constraint (29), respectively, while *μ*<sup>1</sup> = *μ*1 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* <sup>+</sup>, *μ*<sup>2</sup> = *μ*2 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* <sup>+</sup>, *μ*<sup>3</sup> = *μ*3 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* +.

Similar to the CT model, we can give the economic interpretation of VI (30). From the 3rd term of VI (30), we have *ρ*<sup>∗</sup> *jk* <sup>=</sup> *<sup>∂</sup>c*<sup>∗</sup> *jk ∂qjk* + *ρt*1*xjk* + *μ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*1*τtxjkμ*3<sup>∗</sup> *<sup>j</sup>* when *<sup>q</sup>*<sup>∗</sup> *jk* > 0; that is, the transaction price between non-ecological manufacturer *j* and demand market *k*

comprises the marginal transaction cost, unit truck transportation cost, and the factor of carbon emission. From the 1st term of VI (30), we have *μ*2<sup>∗</sup> *<sup>j</sup>* <sup>=</sup> <sup>1</sup> *βr ∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* + *α<sup>J</sup>* 1*βrμ*3<sup>∗</sup> *j* when *qv*<sup>∗</sup> *<sup>j</sup>* > 0, which is mainly affected by the production marginal cost and conversion rate. Therefore, we can determine that the costs of the previous stage are transmitted to the next stage through the product transaction.

#### 4.4.2. Ecological Manufacturers' Decisions

Similarly, we describe the manufacturer *i*'s objective function as follows:

$$\max \left[ \sum\_{k=1}^{K} \left( \rho\_{\text{ik}}^{\*} q\_{\text{ik}} - c\_{\text{ik}} \right) - \sum\_{h=1}^{H} \left( \rho\_{\text{hi}}^{\*} q\_{\text{hi}}^{\text{u}} + c\_{\text{hi}} \right) - f\_{\text{i}} - f\_{\text{i}}^{\text{u}} - \rho \left( t\_{\text{3}} \sum\_{k=1}^{K} \mathbf{x}\_{\text{ik}} q\_{\text{ik}} + t\_{\text{4}} \sum\_{h=1}^{H} \mathbf{x}\_{\text{hi}} q\_{\text{hi}}^{\text{u}} \right) \right] \tag{31}$$

$$\text{s.t. } q\_i^u \le \sum\_{h=1}^H q\_{hi}^u \tag{32}$$

$$\sum\_{k=1}^{K} q\_{ik} \le \beta^r q\_i^v + \beta^u q\_i^u \tag{33}$$

$$a\_1^I \beta^r q\_i^v + a\_2^I \beta^u q\_i^u + t\_3 \pi\_t \sum\_{k=1}^K \pi\_{ik} q\_{ik} + t\_4 \pi\_t \sum\_{h=1}^H \pi\_{hi} q\_{hi}^u \le cap\_i \tag{34}$$

In this situation, eco manufacturer *i* determines the amount of raw materials and recycled EOL products, the output and transaction amount of the new product, and the remanufactured product.

All eco manufacturers compete in a non-cooperation fashion, and the optimality conditions of all eco manufacturers can be described simultaneously as variational inequality: determine the optimal solution (*qv*<sup>∗</sup> *<sup>i</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>i</sup>* , *<sup>Q</sup>*<sup>∗</sup> 3, *<sup>Q</sup>*<sup>∗</sup> 4, *<sup>η</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>η</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>η</sup>*<sup>∗</sup> <sup>3</sup>) ∈ <sup>Ω</sup><sup>2</sup> *<sup>I</sup>* , satisfying:

*I* ∑ *i*=1 *∂ f* <sup>∗</sup> *i ∂q<sup>v</sup> i* − *βrη*2<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 1*βrη*3<sup>∗</sup> *i* × " *qv <sup>i</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *i ∂q<sup>u</sup> i* + *η*1<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>β</sup>uη*2<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuη*3<sup>∗</sup> *i* × " *qu <sup>i</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* # + *I* ∑ *i*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *ik <sup>∂</sup>qik* <sup>−</sup> *<sup>ρ</sup>*<sup>∗</sup> *ik* <sup>+</sup> *<sup>ρ</sup>t*3*xik* <sup>+</sup> *<sup>η</sup>*2<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*3*τtxikη*3<sup>∗</sup> *i* × " *qik* − *q*<sup>∗</sup> *ik* # + *I* ∑ *i*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hi ∂q<sup>u</sup> hi* + *ρ*<sup>∗</sup> *hi* <sup>+</sup> *<sup>ρ</sup>t*4*xhi* <sup>−</sup> *<sup>η</sup>*1<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*4*τtxhiη*3<sup>∗</sup> *i* × " *qu hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hi* # + *I* ∑ *i*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* × " *η*1 *<sup>i</sup>* <sup>−</sup> *<sup>η</sup>*1<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *βrqv*<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *ik* × " *η*2 *<sup>i</sup>* <sup>−</sup> *<sup>η</sup>*2<sup>∗</sup> *i* # + *I* ∑ *i*=1 *capi* − *α<sup>I</sup>* 1*βrqv*<sup>∗</sup> *<sup>i</sup>* − *<sup>α</sup><sup>I</sup>* 2*βuqu*<sup>∗</sup> *<sup>i</sup>* − *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikq*<sup>∗</sup> *ik* − *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup>*<sup>∗</sup> *hi* × " *η*3 *<sup>i</sup>* <sup>−</sup> *<sup>η</sup>*3<sup>∗</sup> *i* # ≥ 0. (35) ∀(*q<sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>η</sup>*1, *<sup>η</sup>*2, *<sup>η</sup>*3) ∈ <sup>Ω</sup><sup>2</sup> *I*

where Ω<sup>2</sup> *<sup>I</sup>* <sup>=</sup> *<sup>R</sup>*2*I*+*IK*+*H I*+3*<sup>I</sup>* <sup>+</sup> . Note that *<sup>η</sup>*<sup>1</sup> *<sup>i</sup>* , *<sup>η</sup>*<sup>2</sup> *<sup>i</sup>* , and *<sup>η</sup>*<sup>3</sup> *<sup>i</sup>* are the Lagrangian multipliers associated with Constraint (32), Constraint (33), and Constraint (34), respectively, while *η*<sup>1</sup> = *η*1 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* <sup>+</sup>, *<sup>η</sup>*<sup>2</sup> = *η*2 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* <sup>+</sup>, *<sup>η</sup>*<sup>3</sup> = *η*3 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* +.

The equilibrium conditions of the closed-loop supply chain network in the mandatory cap model can be obtained in the same way with the CT model, so this part is presented in Appendix A.

#### *4.5. The Supply Chain Network under Cap-Sharing Scheme (CS)*

The government examines the total emissions of a typical industry in a certain period according to the national emission reduction plan. In this section, we examine a setting in which two types of manufacturers make decisions centralized, and the carbon caps are permitted to be transferred freely, which is therefore called the cap-sharing scheme. From the perspective of the whole industry, the total carbon emissions of manufacturers do not exceed the government's regulations.

Manufacturers' Decisions

In this case, manufacturer *j* and manufacturer *i* need to decide the amount of raw materials and recycled EOL products, the transaction amount, and the EOL product transaction amount, respectively. For convenience, let *Ax* <sup>=</sup> *<sup>K</sup>* ∑ *k*=1 *ρ*∗ *xkqxk* <sup>−</sup> *cxk* <sup>−</sup> *<sup>H</sup>* ∑ *h*=1 *ρu*<sup>∗</sup> *hx <sup>q</sup><sup>u</sup> hx* + *<sup>c</sup><sup>u</sup> hx* − *fx* −*f <sup>u</sup> <sup>x</sup>* (*x* = *i*, *j*), *B*<sup>1</sup> = *t*<sup>1</sup> *K* ∑ *k*=1 *xjkqjk* + *t*<sup>2</sup> *H* ∑ *h*=1 *xhjq<sup>u</sup> hj*, *B*<sup>2</sup> = *t*<sup>3</sup> *K* ∑ *k*=1 *xikqik* + *t*<sup>4</sup> *H* ∑ *h*=1 *xhiq<sup>u</sup> hi*, then we can describe the typical manufacturer objective function as follows:

$$\max \left[ A\_x - \rho B\_y \right] \tag{36}$$

$$\text{s.t. } q\_x^u \le \sum\_{h=1}^H q\_{h\mathbf{x}}^u \tag{37}$$

$$\sum\_{k=1}^{K} q\_{\mathbf{x}k} \le \beta^r q\_{\mathbf{x}}^v + \beta^u q\_{\mathbf{x}}^u \tag{38}$$

$$\begin{bmatrix} \sum\_{j=1}^{I} \left[ \boldsymbol{a}\_1^I \boldsymbol{\beta}^\tau \boldsymbol{q}\_j^v + \boldsymbol{a}\_2^I \boldsymbol{\beta}^u \boldsymbol{q}\_j^u + t\_1 \boldsymbol{\tau}\_l \sum\_{k=1}^K \boldsymbol{x}\_{jk} \boldsymbol{q}\_{jk} + t\_2 \boldsymbol{\tau}\_l \sum\_{h=1}^H \boldsymbol{x}\_{hj} \boldsymbol{q}\_{hj}^u \right] \\ \quad + \sum\_{i=1}^I \left[ \boldsymbol{a}\_1^I \boldsymbol{\beta}^\tau \boldsymbol{q}\_i^v + \boldsymbol{a}\_2^I \boldsymbol{\beta}^u \boldsymbol{q}\_i^u + t\_3 \boldsymbol{\tau}\_l \sum\_{k=1}^K \boldsymbol{x}\_{ik} \boldsymbol{q}\_{jk} + t\_4 \boldsymbol{\tau}\_l \sum\_{h=1}^H \boldsymbol{x}\_{hj} \boldsymbol{q}\_{hj}^u \right] \end{bmatrix} \leq \sum\_{j=1}^J \boldsymbol{c} \boldsymbol{a} p\_j + \sum\_{i=1}^I \boldsymbol{c} \boldsymbol{a} p\_i \tag{39}$$

When *x* = *i* and *y* = 1, Equation (36) denotes the profit of ecological manufacturer *i*; when *x* = *j* and *y* = 2, Equation (36) denotes the profit of non-ecological manufacturer *j*. Constraint (39) can be called the carbon emissions constraint. All decision variables are non-negative; in addition, all manufacturers of the same type compete in a non-cooperation fashion, and the profits of each manufacturer seeking maximization can be transformed simultaneously into the following variational inequality to determine the optimal solution (*qv*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>v*<sup>∗</sup> *<sup>i</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>i</sup>* , *<sup>Q</sup>*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 2, *<sup>Q</sup>*<sup>∗</sup> 3, *<sup>Q</sup>*<sup>∗</sup> 4, *<sup>θ</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>θ</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>θ</sup>*<sup>∗</sup> <sup>3</sup>, *<sup>θ</sup>*<sup>∗</sup> <sup>4</sup>, *<sup>θ</sup>*5∗) <sup>∈</sup> <sup>Ω</sup><sup>3</sup> *J I*, satisfying:

*J* ∑ *j*=1 *∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* − *βrθ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 1*βrθ*5<sup>∗</sup> × *qv <sup>j</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *j* + *J* ∑ *j*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *j ∂q<sup>u</sup> j* + *θ*1<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>β</sup>uθ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuθ*5<sup>∗</sup> × *qu <sup>j</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* + *I* ∑ *i*=1 *∂ f* <sup>∗</sup> *i ∂q<sup>v</sup> i* − *βrθ*4<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 1*βrθ*5<sup>∗</sup> × " *qv <sup>i</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *i ∂q<sup>u</sup> i* + *θ*3<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>β</sup>uθ*4<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuθ*5<sup>∗</sup> × " *qu <sup>i</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* # + *J* ∑ *j*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *jk <sup>∂</sup>qjk* <sup>−</sup> *<sup>ρ</sup>*<sup>∗</sup> *jk* <sup>+</sup> *<sup>ρ</sup>t*1*xjk* <sup>+</sup> *<sup>θ</sup>*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*1*τtxjkθ*5<sup>∗</sup> × *qjk* − *q*<sup>∗</sup> *jk* + *J* ∑ *j*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hj ∂q<sup>u</sup> hj* + *ρu*<sup>∗</sup> *hj* <sup>+</sup> *<sup>ρ</sup>t*2*xhj* <sup>−</sup> *<sup>θ</sup>*1<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*2*τtxhjθ*5<sup>∗</sup> × *qu hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hj* + *I* ∑ *i*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *ik <sup>∂</sup>qik* <sup>−</sup> *<sup>ρ</sup>*<sup>∗</sup> *ik* <sup>+</sup> *<sup>ρ</sup>t*3*xik* <sup>+</sup> *<sup>θ</sup>*4<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*3*τtxikθ*5<sup>∗</sup> × " *qik* − *q*<sup>∗</sup> *ik* # + *I* ∑ *i*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hi ∂q<sup>u</sup> hi* + *ρu*<sup>∗</sup> *hi* <sup>+</sup> *<sup>ρ</sup>t*4*xhi* <sup>−</sup> *<sup>θ</sup>*3<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*4*τtxhiθ*5<sup>∗</sup> × " *qu hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hi* # + *J* ∑ *j*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* × *θ*1 *<sup>j</sup>* <sup>−</sup> *<sup>θ</sup>*1<sup>∗</sup> *j* + *J* ∑ *j*=1 *βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *jk* × *θ*2 *<sup>j</sup>* <sup>−</sup> *<sup>θ</sup>*2<sup>∗</sup> *j* + *I* ∑ *i*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* × " *θ*3 *<sup>i</sup>* <sup>−</sup> *<sup>θ</sup>*3<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *βrqv*<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *ik* × " *θ*4 *<sup>i</sup>* <sup>−</sup> *<sup>θ</sup>*4<sup>∗</sup> *i* # + ⎡ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣ *J* ∑ *j*=1 *capj* + *I* ∑ *i*=1 *capi* ⎤ ⎥ ⎥ ⎥ ⎦ − ⎡ ⎢ ⎢ ⎢ ⎣ *J* ∑ *j*=1 *αJ* 1*βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuqu*<sup>∗</sup> *<sup>j</sup>* + *t*1*τ<sup>t</sup> K* ∑ *k*=1 *xjkq*<sup>∗</sup> *jk* + *t*2*τ<sup>t</sup> H* ∑ *h*=1 *xhjq<sup>u</sup>*<sup>∗</sup> *hj* + *I* ∑ *i*=1 *αI* 1*βrqv*<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuqu*<sup>∗</sup> *<sup>i</sup>* + *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikq*<sup>∗</sup> *ik* + *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup>*<sup>∗</sup> *hi* ⎤ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎦ × " *θ*<sup>5</sup> − *θ*5∗# ≥ 0. (40) ∀(*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>θ</sup>*1, *<sup>θ</sup>*2, *<sup>θ</sup>*3, *<sup>θ</sup>*4, *<sup>θ</sup>*5) ∈ <sup>Ω</sup><sup>3</sup> *J I* where Ω<sup>3</sup> *J I* <sup>=</sup> *<sup>R</sup>*2*J*+2*I*+*JK*+*H J*+*IK*+*H I*+2*J*+2*I*+<sup>1</sup> <sup>+</sup> .

Note that *θ*<sup>1</sup> *<sup>j</sup>* , *<sup>θ</sup>*<sup>2</sup> *<sup>j</sup>* , *<sup>θ</sup>*<sup>3</sup> *<sup>j</sup>* , and *<sup>θ</sup>*<sup>4</sup> *<sup>j</sup>* are the Lagrangian multipliers associated with Constraint (37), Constraint (38) for *x* = *j*, *i*, respectively, and *θ*<sup>5</sup> *<sup>j</sup>* is the Lagrangian multiplier associated with Constraint (39), *θ*<sup>1</sup> = *θ*1 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* <sup>+</sup>, *θ*<sup>2</sup> = *θ*2 *j <sup>J</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>J</sup>* <sup>+</sup>, *<sup>θ</sup>*<sup>3</sup> = *θ*3 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* +, *θ*<sup>4</sup> = *θ*4 *i <sup>I</sup>*×<sup>1</sup> <sup>∈</sup> *<sup>R</sup><sup>I</sup>* +.

The equilibrium conditions of the closed-loop supply chain network in the CS model are shown in Appendix B.

The qualitative properties of VI. (24), VI. (40), and VI. (41) are presented in Appendix C.

#### **5. Discussion**

In this section, we provide several numerical examples to verify the foregoing theoretical results and present a further comparison of the decisions and profits with three carbon reduction regulations. In reality, the cap from the government may change with the changing of emission reduction targets; similarly, consumers' low-carbon preferences will also change with social development. Therefore, we will analyze the parameters of *capj*, *capi*, *εj*, *ε<sup>i</sup>* and *σ*.

Consider a closed-loop supply chain network comprising two non-ecological manufacturers, two ecological manufacturers, two demand markets, and two collection centers; when considering the cap-and-trade regulation, there also exists a carbon trade center.

Because the design is simple and easy to implement, the decision variables and Lagrange multipliers can be obtained at the same time. Like [40], we select a modified projection and contraction algorithm with a fixed step length to solve VI (24), VI (40), and VI (A1), and design the program with MATLAB. The convergence criterion is that the absolute value of the difference between the values of the two iterations is no more than 10<sup>−</sup>8. The selection of the function form refers to [40,58], the related parameters are set as: *β<sup>r</sup>* = 0.95, *β<sup>u</sup>* = 0.9, *ρ* = 1, *t*<sup>1</sup> = *t*<sup>2</sup> = *t*<sup>3</sup> = *t*<sup>4</sup> = 2, *xjk* = *xik* = *xhj* = *xhi* = 1, *τ<sup>t</sup>* = 1, *ε<sup>j</sup>* = 0.3, *ε<sup>i</sup>* = 0.2, *δ* = 1, *αI* <sup>1</sup> <sup>=</sup> 0.6, *<sup>α</sup><sup>J</sup>* <sup>1</sup> = 0.8, *<sup>α</sup><sup>I</sup>* <sup>2</sup> <sup>=</sup> 0.3, *<sup>α</sup><sup>J</sup>* <sup>2</sup> = 0.5. Referring to related literature [38–40,50,60], the functions are set as follows: *fi* = 8.5(*βrqv i* ) <sup>2</sup> + *βr*2*q<sup>v</sup> i qv* <sup>3</sup>−*<sup>i</sup>* + <sup>2</sup>*βrqv <sup>i</sup>* , *<sup>f</sup> <sup>u</sup> <sup>i</sup>* = <sup>3</sup>(*βuqu i* ) <sup>2</sup> + 1.5*βuqu <sup>i</sup>* + 1, *fj* = 8.0(*βrqv i* ) <sup>2</sup> + *βr*2*q<sup>v</sup> i qv* <sup>3</sup>−*<sup>i</sup>* + <sup>2</sup>*βrqv <sup>i</sup>* , *<sup>f</sup> <sup>u</sup> <sup>j</sup>* = 0.5(*βuqu j* ) <sup>2</sup> + 0.5*βuqu <sup>j</sup>* + 1, *cjk* = 0.2*t*1*qjk* + 1, *cik* = 0.2*t*3*qik* + 1, *c<sup>K</sup> jk* = 0.2*qjk* + 0.1, *<sup>c</sup><sup>K</sup> ik* = 0.2*qik* + 0.1, *ckh* = 0.1*q<sup>u</sup> kh* + 0.5, *<sup>c</sup><sup>u</sup> hj* = 0.1*t*2*q<sup>u</sup> hj* + 1, *cu hi* = 0.1*t*4*q<sup>u</sup> hi* <sup>+</sup> 1, *ch* <sup>=</sup> 2.5 <sup>2</sup> ∑ *k*=1 *qu kh* 2 <sup>+</sup> <sup>2</sup> <sup>2</sup> ∑ *k*=1 *qu kh*, *<sup>α</sup><sup>u</sup> <sup>k</sup>* (*Q*5) = 0.5 <sup>2</sup> ∑ *k*=1 2 ∑ *k*=1 *qu kh* + 5, *<sup>c</sup><sup>t</sup> <sup>x</sup>* = 0.1*tx* 2 *dl <sup>k</sup>* = <sup>250</sup> − <sup>2</sup>*ρ<sup>l</sup> <sup>k</sup>* − 1.5*ρ<sup>l</sup>* <sup>3</sup>−*<sup>k</sup>* + 0.5(*ρ<sup>h</sup> <sup>k</sup>* + *<sup>ρ</sup><sup>h</sup>* <sup>3</sup>−*k*) + *σψ* <sup>2</sup> ∑ *x*=1 (1 − *α<sup>I</sup> <sup>x</sup>*), *d<sup>h</sup> <sup>k</sup>* = <sup>230</sup> − <sup>2</sup>*ρ<sup>h</sup> <sup>k</sup>* − 1.5*ρ<sup>h</sup>* <sup>3</sup>−*<sup>k</sup>* + 0.5(*ρ<sup>l</sup> <sup>k</sup>* + *<sup>ρ</sup><sup>l</sup>* <sup>3</sup>−*k*) + *σψ* <sup>2</sup> ∑ *x*=1 (1 − *α<sup>J</sup> x*).

The demand functions are associated with the price of two types of products; due to the consumer's low carbon preference, there is also a relationship between demand and the product's unit carbon emission amount. We assume the low-carbon factor *ψ*= 10. Refs. [40,50] used a similar form of demand function in their numerical examples.

It is obvious that all the functions listed above are convex and continuously differentiable. Then, the solutions of VI (24), VI (40), and VI (41) satisfy Theorem A3, Theorem A4, and Theorem A6 in Appendix C. The detailed values and formula construction basis can be seen in Appendix D.

#### *5.1. Analyzing the Effects of Cap on Optimal Decisions and Profits*

We assume that *capj* and *capi* change from 9 to 26, respectively, then group the related equilibrium results into several matrixes corresponding to three carbon reduction regulations. We select data from the matrixes including the profits of manufacturers and recyclers and calculate the carbon emissions and the total profits of the supply chain based on the relevant data. The relevant results are illustrated in Figures 2–7.

**Figure 2.** Carbon emissions of non-eco manufacturer.

**Figure 3.** Profits of non-eco manufacturer.

**Figure 4.** Carbon emissions of eco manufacturer.

**Figure 5.** Profits of eco manufacturer.

Figures 2 and 4 illustrate the carbon emissions of two types of manufacturers under three carbon emission reduction regulations, respectively. Figures 3 and 5 illustrate the profits of two types of manufacturers under three policies. Figures 6 and 7 show the network performance.

#### **Figure 7.** Total profits.

#### 5.1.1. The Effects on Non-Ecological Manufacturer

As it can be seen from Figure 2, the trends are similar under the three regulations. Overall, the larger *capj* or *capi* incurs more carbon emissions. In Figure 2a, we can see that the carbon emission under MC is only affected by *capj*, Figure 2b shows that the carbon emission under CT is greatly affected by *capj*, and Figure 2c shows that the carbon emission under CS is affected by *capj* and *capi* simultaneously. The maximum value and minimum value appear in Figure 2c.

Comparing the three subfigures in Figure 2, the results show its carbon emissions and profits under policy MC are always lower than the other two policies, which means policy MC limits the enterprise's production activity and injures its profit. However, there is a special interval: when *capi* is less than 14, policy CS is conductive to decrease carbon emissions because the lower cap cannot activate the recycling process. Policy CT and Constraint (6) are invalid. In addition, there is no cap trade between two manufacturers, manufacturers only use the allocated cap to produce, and policy MC and CT have the same effects on the equilibrium results.

When *capi* is larger than 14, policy CS and CT are valid. Under policy CS, carbon caps are transferred freely from eco manufacturers to non-eco manufacturers. When caps transaction exists, policy CT promotes the effective allocation of carbon caps and benefits social-economic development. The maximum profit appears in the region when *capi* is relatively small while *capj* is large under policy CT. This can be explained by the adequacy caps reducing its carbon trade activities.

In Figure 3, the trends are also similar under three regulations. The maximum values appear in Figure 3a,b, while the minimum value appears in Figure 3c. Figure 3a states that the profit is mainly affected by *capj* and is slightly affected by *capi*. This phenomenon is different from carbon emissions. The reason can be explained by: the increasing caps will stimulate production activities, more customers turn to buy eco-products, and eco manufacturers' profits increase. Figure 3c illustrates *capj* and *capi* have the same effects on non-eco manufacturers' profit, and the profit is only influenced by caps.

The comparison of these three subfigures shows policy CT is a cost-effective carbon reduction policy. Particularly, when *capi* is lower, the profit in Figure 3b is similar to that in Figure 3a; when *capi* is higher, the profit in Figure 3b is similar to that in Figure 3c.

In addition, it should be noted that when *capj* and *capi* are large enough, the changing carbon emission trend is not exactly with that of the profit under policy CS, because the transportation cost increases rapidly with the intense production and recycling activities.

#### 5.1.2. The Effects on Ecological Manufacturer

The analysis of eco manufacturers is similar to that of non-ecological manufacturers.

According to Figure 4, the equilibrium results of eco manufacturers are opposite to non-eco manufacturers in Figure 2. In Figure 4a, its carbon emission is only affected by *capi* in most ranges, which is similar to Figure 2a. In Figure 4b, when *capi* is at a relatively low level, it has no extra caps for sale, and policy CT is the same as MC. When *capi* gradually increases, the extra carbon quotas bring cap transactions. In Figure 4c, due to the adoption of ecological production technology, its carbon emission changes slightly.

Comparing the three subfigures of Figure 4, policy CS is the most effective method to reduce carbon emissions. Particularly, when *capi* is relatively small, carbon quotas are not transferred from eco manufacturers to non-eco manufacturers. Combined with Figure 5b, the carbon emissions and profits are identical under policy MC and CT. Policy MC is always beneficial for the eco manufacturer to obtain higher profits when *capi* is large. This phenomenon occurs because carbon quotas are adequate for it to produce more products, while policy CT and CS may force carbon caps to transfer to non-eco manufacturers.

#### 5.1.3. The Effects on Supply Chain Performance

In this subsection, we focus on the impact of different policies on the whole supply chain performance. Total carbon emission and profit are given in Figure 6. From the environmental perspective, total carbon emissions are equal in three scenarios. From the economic perspective, according to Figure 7, we can clearly see that the best policy is cap sharing, but the difference between each policy is small. Combined with the previous figures, when caps increase, policy CS also perform well in reducing manufacturers' carbon emissions and promoting their profits. From the view of the government, the carbon trade model sacrifices part of the efficiency of the supply chain in exchange for government control and supervision of the carbon trading market. Although policy CS is an ideal regulation, if it lowers the carbon trade cost, policy CT may have a similar performance to policy CS, which makes it easier for the government to achieve the emissions reduction target.

Policy CT is conducive to the government to control the carbon emissions of enterprises; at the same time, the government may permit cap sharing within a large enterprise when there are different levels of low carbon technology applied in production.

#### *5.2. Analyzing Effects of ε<sup>j</sup> and ε<sup>i</sup> on Optimal Decisions and Profits*

Figure 8 illustrates the impacts of parameter *ε<sup>j</sup>* and *ε<sup>i</sup>* in the interval [0, 0.3] on carbon emission amount and EOL product quantity, while Figure 9 shows the impacts on decisionmakers' profits.

From Figure 8, we can see that manufacturers' carbon emission curves remain unchanged in policy MC. In policy CT and CS, manufacturer *j*'s carbon emission curve almost decreases, then increases, and finally stays invariant, while manufacturer *i*'s curve has an opposite trend, and *ε<sup>i</sup>* = *ε<sup>j</sup>* = 0.09 is the turning point. The observed phenomenon can be explained in the following manner. From the equilibrium decision value, we can see that the use of raw materials has been in a downward trend; thus, the emissions from raw materials continue to decrease. At first, non-eco manufacturer *j*'s carbon emissions are higher due to the higher unit emission factor. For EOL products, the point at which manufacturer *i* begins to have EOL is 0, while the point at which manufacturer *j* begins to have EOL is 0.6. Therefore, the carbon emissions of *j* decreases at the beginning, and when *ε<sup>i</sup>* = *ε<sup>j</sup>* = 0.09, carbon emissions are minimum. After this point, the emission increased by the production and transportation of EOL remanufacturing is higher than the decrease

in raw materials, and the emissions increase again. For eco manufacturer *i*, the same explanation can be made.

**Figure 8.** The effects on carbon emission amount and EOL product quantity.

**Figure 9.** The effects on decision-makers' profits.

As for EOL product quantity, it is worth noting that when *ε<sup>i</sup>* = *ε<sup>j</sup>* > 0, the manufacturer *i*'s EOL product quantity increases gradually, when *ε<sup>j</sup>* and *ε<sup>i</sup>* are in the interval (0.06,0.09), the manufacturer *j*'s EOL product quantity becomes positive, and there is a turning point at 0.09 for manufacturer *i* under policy CS. When *ε<sup>i</sup>* = *ε<sup>j</sup>* > 0.24, these curves in three scenarios gradually stabilize. In this scenario, Constraint (6) does not hold, which means that the collection center is unable to recycle at a specified proportion.

According to Figure 9, manufacturer *i* always obtains more profit than manufacturer *j*. Overall, there is little difference in the total profits of the three cases; in particular, the total profits of all decision-makers in the policy CS are always higher than that in the other two cases. As for the collection center, the profit is almost the same under different regulations. For government, the recycling ratio should be set in an appropriate range. When it is too high, the enterprise will not comply with it, and it is meaningless. When the recycling ratio is set too low, it will fail to achieve the goal of resource utilization.

#### *5.3. Analysis Effects of σ on Optimal Decisions and Profits*

Figure 10 illuminates the impacts of parameter *σ* in the interval [0,1] on the product transaction amount and the carbon emissions amount, while Figure 11 shows the impacts on decision-makers' profits.

In CS policy, increasing *σ* has positive effects on the manufacturer *i*'s products transaction amount, whereas for manufacturer *j*, the situation is reversed. In the other two scenarios, the products transaction amounts stay unchanged. The carbon emission curve has a synchronous changing trend with the product transaction.

As can be seen from Figure 11, the profit of non-ecological manufacturer *j* maintains stability in policy CS and increases a little in policy CT and MC. However, the profit of ecological manufacturer *i* increases rapidly under the three regulations. The change in profits shows that this situation is more favorable to ecological manufacturers. For the total profit of the two types of manufacturers, there are almost no differences in these three cases. Similar to the analysis of previous examples, the profit of collection center *h* is mainly affected by the EOL collection amount; thus, it maintains stability at first and decreases later. Therefore, the increasement of *σ* will promote the development and impacts little on carbon emission.

**Figure 10.** The effects on products transaction amount and carbon emission amount.

**Figure 11.** The effects on decision-makers' profits.

#### *5.4. Managerial Insights*

Compared with the literature [3], this research highlights the difference between carbon emission reduction policies, and based on numerical examples and analysis, we present several managerial insights as follows. This may facilitate the government and enterprises to refer to when issuing policies and enterprises making operation decisions.


#### **6. Conclusions**

In this paper, we expand the previous research to a CLSC network based on noncooperative equilibrium. This paper provides a research framework for a series of Nash game problems of low-carbon supply chain networks with complex relationships between horizontal and vertical competitive members. The profit maximum problem with nonlinear constraints can be transformed into variational inequality, and the equilibrium results can be obtained through a modified projection and contraction algorithm.

In the forward flow, we suppose three different environmental regulations, namely mandatory cap, cap-and-trade, and cap sharing. The collection and remanufacturing of EOL products are taken into consideration in the reverse flow. The effects of policies on network performance are discussed in detail. The results show that:


This study mainly contributes to the enterprises' decision making and revenue management under three carbon emissions reduction regulations. Through numerical simulations, we verified the validity of each policy. For future research, possible extensions can be as follows:


**Author Contributions:** Conceptualization, P.C. and G.Z.; methodology and model, P.C. and G.Z.; software, P.C.; validation, P.C., G.Z. and H.S.; numerical analysis, G.Z.; investigation, P.C.; resources, P.C.; data curation, P.C.; writing—original draft preparation, P.C.; writing—review and editing, P.C.; visualization, G.Z.; supervision, G.Z.; project administration, H.S.; funding acquisition, G.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Social Science Foundation of China [grant number 19BGL091].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We greatly appreciate the associate editor and the anonymous reviewers for their insightful comments and constructive suggestions, which have greatly helped us to improve the manuscript and guide us forward to the future research.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Appendix A**

The equilibrium conditions of the closed-loop supply chain network in the MC model

Under the government's mandatory cap regulation, at equilibrium conditions of the supply chain network, the Nash equilibrium (Nash 1950) conditions of VI (7), VI (10), VI (30), and VI (35) must hold simultaneously, and no one gains more from altering strategies.

**Definition A1.** *The equilibrium of the CLSCN under mandatory cap occurs when the sum of the L.H.S. of (7), L.H.S. of (10), L.H.S. of (30), and L.H.S. of (35) is non-negative.*

**Theorem A1.** *The equilibrium conditions of the CLSCN under mandatory cap are equivalent to the solutions of the VI as follows: determine the optimal solution* (*qv*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>v*<sup>∗</sup> *<sup>i</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>i</sup>* , *<sup>Q</sup>*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 2, *<sup>Q</sup>*<sup>∗</sup> 3, *<sup>Q</sup>*<sup>∗</sup> 4, *Q*∗ 5, *<sup>ρ</sup>J*∗, *<sup>ρ</sup>I*∗, *<sup>μ</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>μ</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>μ</sup>*<sup>∗</sup> <sup>3</sup>, *<sup>η</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>η</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>η</sup>*<sup>∗</sup> <sup>3</sup>, *<sup>λ</sup>*∗, *<sup>γ</sup>*∗) <sup>∈</sup> <sup>Ω</sup>2*, satisfying:*

*J* ∑ *j*=1 *∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* − *βrμ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 1*βrμ*3<sup>∗</sup> *j* × *qv <sup>j</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *j* + *J* ∑ *j*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *j ∂q<sup>u</sup> j* + *μ*1<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>β</sup>uμ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuμ*3<sup>∗</sup> *j* × *qu <sup>j</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* + *I* ∑ *i*=1 *∂ f* <sup>∗</sup> *i ∂q<sup>v</sup> i* − *βrμ*2<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 1*βrη*3<sup>∗</sup> *i* × " *qv <sup>i</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *i ∂q<sup>u</sup> i* + *η*1<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>β</sup>uμ*2<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuη*3<sup>∗</sup> *i* × " *qu <sup>i</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* # + *J* ∑ *j*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *jk <sup>∂</sup>qjk* <sup>+</sup> *<sup>ρ</sup>t*1*xjk* <sup>+</sup> *<sup>μ</sup>*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*1*τtxjkμ*3<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>c</sup>K*<sup>∗</sup> *jk* <sup>−</sup> *<sup>ρ</sup>h*<sup>∗</sup> *<sup>k</sup>* <sup>−</sup> *<sup>ε</sup>jγ*<sup>∗</sup> *k* × *qjk* − *q*<sup>∗</sup> *jk* + *J* ∑ *j*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hj ∂q<sup>u</sup> hj* + *ρt*2*xhj* − *μ*1<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*2*τtxhjμ*3<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>∗</sup> *h* × *qu hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hj* + *I* ∑ *i*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *ik <sup>∂</sup>qik* <sup>+</sup> *<sup>ρ</sup>t*3*xik* <sup>+</sup> *<sup>η</sup>*2<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*3*τtxikη*3<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>c</sup>K*<sup>∗</sup> *ik* <sup>−</sup> *<sup>ρ</sup>l*<sup>∗</sup> *<sup>k</sup>* <sup>−</sup> *<sup>ε</sup>iγ*<sup>∗</sup> *k* × " *qik* − *q*<sup>∗</sup> *ik* # + *I* ∑ *i*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hi ∂q<sup>u</sup> hi* + *ρt*4*xhi* − *η*1<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*4*τtxhiη*3<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>∗</sup> *h* × " *qu hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hi* # + *H* ∑ *h*=1 *K* ∑ *k*=1 *∂cu*<sup>∗</sup> *kh ∂q<sup>u</sup> kh* <sup>+</sup> *<sup>∂</sup>c*<sup>∗</sup> *h ∂q<sup>u</sup> kh* − *δλ*<sup>∗</sup> *<sup>h</sup>* + *<sup>α</sup><sup>u</sup> <sup>k</sup>* (*Q*<sup>∗</sup> <sup>5</sup> ) + *<sup>γ</sup>*<sup>∗</sup> *k* × " *qu kh* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *kh* # + *K* ∑ *k*=1 *<sup>J</sup>* ∑ *j*=1 *q*∗ *jk* <sup>−</sup> *<sup>d</sup>h*<sup>∗</sup> *k* × *ρh <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup>h*<sup>∗</sup> *k* <sup>+</sup> *<sup>K</sup>* ∑ *k*=1 *I* ∑ *i*=1 *q*∗ *ik* <sup>−</sup> *<sup>d</sup>l*<sup>∗</sup> *k* × *ρl <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup>l*<sup>∗</sup> *k* + *J* ∑ *j*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* × *μ*1 *<sup>j</sup>* <sup>−</sup> *<sup>μ</sup>*1<sup>∗</sup> *j* + *J* ∑ *j*=1 *βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *jk* × *μ*2 *<sup>j</sup>* <sup>−</sup> *<sup>μ</sup>*2<sup>∗</sup> *j* + *J* ∑ *j*=1 *capj* − *<sup>α</sup><sup>J</sup>* 1*βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>α</sup><sup>J</sup>* 2*βuqu*<sup>∗</sup> *<sup>j</sup>* − *t*1*τ<sup>t</sup> K* ∑ *k*=1 *xjkq*<sup>∗</sup> *jk* − *t*2*τ<sup>t</sup> H* ∑ *h*=1 *xhjq<sup>u</sup>*<sup>∗</sup> *hj* × *μ*3 *<sup>j</sup>* <sup>−</sup> *<sup>μ</sup>*3<sup>∗</sup> *j* + *I* ∑ *i*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* × " *η*1 *<sup>i</sup>* <sup>−</sup> *<sup>η</sup>*1<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *βrqv*<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *ik* × " *η*2 *<sup>i</sup>* <sup>−</sup> *<sup>η</sup>*2<sup>∗</sup> *i* # + *I* ∑ *i*=1 *capi* − *α<sup>I</sup>* 1*βrqv*<sup>∗</sup> *<sup>i</sup>* − *<sup>α</sup><sup>I</sup>* 2*βuqu*<sup>∗</sup> *<sup>i</sup>* − *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikq*<sup>∗</sup> *ik* − *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup>*<sup>∗</sup> *hi* × " *η*3 *<sup>i</sup>* <sup>−</sup> *<sup>η</sup>*3<sup>∗</sup> *i* # + *H* ∑ *h*=1 *δ K* ∑ *k*=1 *qu*<sup>∗</sup> *kh* <sup>−</sup> *<sup>J</sup>* ∑ *j*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>I</sup>* ∑ *i*=1 *qu*<sup>∗</sup> *hi* × " *λ<sup>h</sup>* − *λ*<sup>∗</sup> *h* # <sup>+</sup> *<sup>K</sup>* ∑ *k*=1 *εj J* ∑ *j*=1 *q*∗ *jk* + *ε<sup>i</sup> I* ∑ *i*=1 *q*∗ *ik* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *qu*<sup>∗</sup> *kh* × " *γ<sup>k</sup>* − *γ*<sup>∗</sup> *k* # ≥ 0. (A1) ∀(*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* , *μ*1, *μ*2, *μ*3, *η*1, *η*2, *η*3, *λ*, *γ*) ∈ Ω<sup>2</sup>

*where* Ω<sup>2</sup> = Ω<sup>2</sup> *<sup>J</sup>* × <sup>Ω</sup><sup>2</sup> *<sup>I</sup>* × Ω*<sup>k</sup>* × Ω*h*.

**Proof.** Adding VI (7), VI (10), VI (30), and VI (35) together, we can obtain VI (A1). Meanwhile, when VI (A1) holds, then VI (7), VI (10), VI (30), and VI (35) are also satisfied, respectively.

Let *X*<sup>2</sup> ≡ (*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* , *μ*1, *μ*2, *μ*3, *η*1, *η*2, *η*3, *λ*, *γ*), *F*(*X*2) ≡ (*Fx*(*X*2))19×1, The specific parts *Fx*(*X*2) (*x* = 1, ··· , 19) of *F*(*X*2) are given by the terms proceeding the multiplication signs in VI (A1). Then, we can rewrite the VI (A1) in the standard form of VI following: determine the optimal vector *X*<sup>∗</sup> <sup>2</sup> ∈ <sup>Ω</sup>2, satisfying: *F*(*X*<sup>∗</sup> <sup>2</sup> ), *<sup>X</sup>*<sup>∗</sup> <sup>2</sup> ≥ 0, ∀*X*<sup>2</sup> ∈ <sup>Ω</sup>2.

The notation ·, · denotes the inner product in *M*2—dimensional Euclidean space, where *M*<sup>2</sup> = 2*J* + 2*I* + *JK* + *H J* + *IK* + *H I* + *KH* + 2*K* + 3*J* + 3*I* + *H* + *K*.

#### **Appendix B**

The equilibrium conditions of closed-loop supply chain network in CS model

Under the government's cap-sharing regulations, for the closed-loop supply chain network, the Nash equilibrium (Nash 1950) conditions of VI (7), VI (10), and VI (40) must hold simultaneously, and no one gains more from altering current strategies.

**Definition A2.** *The equilibrium of the CLSCN under cap-sharing regulations occurs when the sum of the L.H.S. of (7), L.H.S. of (10), and L.H.S. of (40) is non-negative.*

**Theorem A2.** *The equilibrium conditions of the CLSCN under cap-sharing regulations are equivalent to the solutions of the VI as follows, determine the optimal solution* (*qv*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>j</sup>* , *<sup>q</sup>v*<sup>∗</sup> *<sup>i</sup>* , *<sup>q</sup>u*<sup>∗</sup> *<sup>i</sup>* , *<sup>Q</sup>*<sup>∗</sup> 1, *<sup>Q</sup>*<sup>∗</sup> 2, *Q*∗ 3, *<sup>Q</sup>*<sup>∗</sup> 4, *<sup>Q</sup>*<sup>∗</sup> 5, *<sup>ρ</sup>J*∗, *<sup>ρ</sup>I*∗, *<sup>θ</sup>*<sup>∗</sup> <sup>1</sup>, *<sup>θ</sup>*<sup>∗</sup> <sup>2</sup>, *<sup>θ</sup>*<sup>∗</sup> <sup>3</sup>, *<sup>θ</sup>*<sup>∗</sup> <sup>4</sup>, *<sup>θ</sup>*5∗, *<sup>λ</sup>*∗, *<sup>γ</sup>*∗) <sup>∈</sup> <sup>Ω</sup>3, satisfying:

*J* ∑ *j*=1 *∂ f* <sup>∗</sup> *j ∂q<sup>v</sup> j* − *βrθ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 1*βrθ*5<sup>∗</sup> × *qv <sup>j</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *j* + *J* ∑ *j*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *j ∂q<sup>u</sup> j* + *θ*1<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>β</sup>uθ*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuθ*5<sup>∗</sup> × *qu <sup>j</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* + *I* ∑ *i*=1 *∂ f* <sup>∗</sup> *i ∂q<sup>v</sup> i* − *βrθ*4<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 1*βrθ*5<sup>∗</sup> × " *qv <sup>i</sup>* <sup>−</sup> *<sup>q</sup>v*<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *∂ f <sup>u</sup>*<sup>∗</sup> *i ∂q<sup>u</sup> i* + *θ*3<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>β</sup>uθ*4<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuθ*5<sup>∗</sup> × " *qu <sup>i</sup>* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* # + *J* ∑ *j*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *jk <sup>∂</sup>qjk* <sup>+</sup> *<sup>ρ</sup>t*1*xjk* <sup>+</sup> *<sup>θ</sup>*2<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*1*τtxjkθ*5<sup>∗</sup> <sup>+</sup> *<sup>c</sup>K*<sup>∗</sup> *jk* <sup>−</sup> *<sup>ρ</sup>h*<sup>∗</sup> *<sup>k</sup>* <sup>−</sup> *<sup>ε</sup>jγ*<sup>∗</sup> *k* × *qjk* − *q*<sup>∗</sup> *jk* + *J* ∑ *j*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hj ∂q<sup>u</sup> hj* + *ρt*2*xhj* − *θ*1<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*2*τtxhjθ*5<sup>∗</sup> <sup>+</sup> *<sup>λ</sup>*<sup>∗</sup> *h* × *qu hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hj* + *I* ∑ *i*=1 *K* ∑ *k*=1 *∂c*<sup>∗</sup> *ik <sup>∂</sup>qik* <sup>+</sup> *<sup>ρ</sup>t*3*xik* <sup>+</sup> *<sup>θ</sup>*4<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*3*τtxikθ*5<sup>∗</sup> <sup>+</sup> *<sup>c</sup>K*<sup>∗</sup> *ik* <sup>−</sup> *<sup>ρ</sup>l*<sup>∗</sup> *<sup>k</sup>* <sup>−</sup> *<sup>ε</sup>iγ*<sup>∗</sup> *k* × " *qik* − *q*<sup>∗</sup> *ik* # + *I* ∑ *i*=1 *H* ∑ *h*=1 *∂cu*<sup>∗</sup> *hi ∂q<sup>u</sup> hi* + *ρt*4*xhi* − *θ*3<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>t</sup>*4*τtxhiθ*5<sup>∗</sup> <sup>+</sup> *<sup>λ</sup>*<sup>∗</sup> *h* × " *qu hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *hi* # + *H* ∑ *h*=1 *K* ∑ *k*=1 *αu <sup>k</sup>* (*Q*<sup>∗</sup> <sup>5</sup> ) + *<sup>γ</sup>*<sup>∗</sup> *<sup>k</sup>* <sup>+</sup> *<sup>∂</sup>c*<sup>∗</sup> *kh ∂q<sup>u</sup> kh* <sup>+</sup> *<sup>∂</sup>c*<sup>∗</sup> *h ∂q<sup>u</sup> kh* − *δλ*<sup>∗</sup> *h* × " *qu kh* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *kh* # + *K* ∑ *k*=1 *<sup>J</sup>* ∑ *j*=1 *q*∗ *jk* <sup>−</sup> *<sup>d</sup>h*<sup>∗</sup> *k* × *ρh <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup>h*<sup>∗</sup> *k* <sup>+</sup> *<sup>K</sup>* ∑ *k*=1 *I* ∑ *i*=1 *q*∗ *ik* <sup>−</sup> *<sup>d</sup>l*<sup>∗</sup> *k* × *ρl <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup>l*<sup>∗</sup> *k* + *J* ∑ *j*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *j* × *θ*1 *<sup>j</sup>* <sup>−</sup> *<sup>θ</sup>*1<sup>∗</sup> *j* + *J* ∑ *j*=1 *βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>j</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *jk* × *θ*2 *<sup>j</sup>* <sup>−</sup> *<sup>θ</sup>*2<sup>∗</sup> *j* + *I* ∑ *i*=1 *H* ∑ *h*=1 *qu*<sup>∗</sup> *hi* <sup>−</sup> *<sup>q</sup>u*<sup>∗</sup> *i* × " *θ*3 *<sup>i</sup>* <sup>−</sup> *<sup>θ</sup>*3<sup>∗</sup> *i* # <sup>+</sup> *<sup>I</sup>* ∑ *i*=1 *βrqv*<sup>∗</sup> *<sup>i</sup>* <sup>+</sup> *<sup>β</sup>uqu*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *q*∗ *ik* × " *θ*4 *<sup>i</sup>* <sup>−</sup> *<sup>θ</sup>*4<sup>∗</sup> *i* # + ⎡ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣ *J* ∑ *j*=1 *capj* + *I* ∑ *i*=1 *capi* ⎤ ⎥ ⎥ ⎥ ⎦ − ⎡ ⎢ ⎢ ⎢ ⎣ *J* ∑ *j*=1 *αJ* 1*βrqv*<sup>∗</sup> *<sup>j</sup>* <sup>+</sup> *<sup>α</sup><sup>J</sup>* 2*βuqu*<sup>∗</sup> *<sup>j</sup>* + *t*1*τ<sup>t</sup> K* ∑ *k*=1 *xjkq*<sup>∗</sup> *jk* + *t*2*τ<sup>t</sup> H* ∑ *h*=1 *xhjq<sup>u</sup>*<sup>∗</sup> *hj* + *I* ∑ *i*=1 *αI* 1*βrqv*<sup>∗</sup> *<sup>i</sup>* + *<sup>α</sup><sup>I</sup>* 2*βuqu*<sup>∗</sup> *<sup>i</sup>* + *t*3*τ<sup>t</sup> K* ∑ *k*=1 *xikq*<sup>∗</sup> *ik* + *t*4*τ<sup>t</sup> H* ∑ *h*=1 *xhiq<sup>u</sup>*<sup>∗</sup> *hi* ⎤ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎦ × " *θ*<sup>5</sup> − *θ*5∗# + *K* ∑ *k*=1 *εj J* ∑ *j*=1 *q*∗ *jk* + *ε<sup>i</sup> I* ∑ *i*=1 *q*∗ *ik* <sup>−</sup> *<sup>K</sup>* ∑ *k*=1 *qu*<sup>∗</sup> *kh* × " *γ<sup>k</sup>* − *γ*<sup>∗</sup> *k* # <sup>+</sup> *<sup>H</sup>* ∑ *h*=1 *δ K* ∑ *k*=1 *qu*<sup>∗</sup> *kh* <sup>−</sup> *<sup>J</sup>* ∑ *j*=1 *qu*<sup>∗</sup> *hj* <sup>−</sup> *<sup>I</sup>* ∑ *i*=1 *qu*<sup>∗</sup> *hi* × " *λ<sup>h</sup>* − *λ*<sup>∗</sup> *h* # ≥ 0 (A2) ∀(*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* , *θ*1, *θ*2, *θ*3, *θ*4, *θ*5, *λ*, *γ*) ∈ Ω<sup>3</sup>

*where* Ω<sup>3</sup> = Ω<sup>3</sup> *J I* × Ω*<sup>k</sup>* × Ω*h*.

**Proof.** Adding VI (7), VI (10), and VI (40) together, we can obtain VI (A2). At the same time, when VI (A2) holds, then VI (7), VI (10), and VI (40) are also satisfied, respectively.

Let *X*<sup>3</sup> ≡ (*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2, *<sup>Q</sup>*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* , *θ*1, *θ*2, *θ*3, *θ*4, *θ*5, *λ*, *γ*), *F*(*X*3) ≡ (*Fx*(*X*3))18×1, the specific parts *F*3(*X*3) (*x* = 1, ··· , 18) of *F*(*X*3) are given by the terms proceeding the multiplication signs in VI (A2). Then, we can rewrite the VI (A2) in standard form of VI following: determine the optimal vector *X*<sup>∗</sup> <sup>3</sup> <sup>∈</sup> <sup>Ω</sup>3, *satisfying: F*(*X*<sup>∗</sup> <sup>3</sup> ), *<sup>X</sup>*<sup>∗</sup> 3 ! ≥ 0, ∀*X*<sup>3</sup> ∈ Ω3.

The notation ·, · denotes the inner product in *M*3—dimensional Euclidean space, where *M*<sup>3</sup> = 2*J* + 2*I* + *JK* + *H J* + *IK* + *H I* + *KH* + 2*K* + 2*J* + 2*I* + 1 + *H* + *K*.

#### **Appendix C**

Qualitative Properties

In this appendix, we provide the existence and uniqueness results of VI (24), VI (40), and VI (A1), and prove that the solutions of these VIs are the equilibriums of the closed-loop supply chain network under different regulations. Because the process and steps of the proof are basically the same, we only give the proof process of VI (24). Similar to [39,40], we give the following theorems, a variational inequality admits at least one solution if the entering function *F*(*X*1) is continuous and the feasible region is compact. Obviously, *F*(*X*1) is continuous, while the feasible region Ω<sup>1</sup> is not compact; thus, we impose a weak condition on Ω<sup>1</sup> to guarantee the solution existence of VI (24).

Similar with [58], let Ω =

*q<sup>v</sup> <sup>j</sup>* , *<sup>q</sup><sup>u</sup> <sup>j</sup>* , *<sup>q</sup><sup>v</sup> <sup>i</sup>* , *<sup>q</sup><sup>u</sup> <sup>i</sup>* , *<sup>Q</sup>*1, *<sup>Q</sup>*2*Q*3, *<sup>Q</sup>*4, *<sup>Q</sup>*5, *<sup>ρ</sup><sup>J</sup>* , *ρ<sup>I</sup>* , *μ*1, *μ*2, *μ*3, *η*1, *η*2, *η*3, *λ*, *γ* 0 ≤ *q<sup>v</sup> <sup>j</sup>* ≤ *<sup>r</sup>*1; 0 ≤ *<sup>q</sup><sup>u</sup> <sup>j</sup>* ≤ *<sup>r</sup>*2; 0 ≤ *<sup>q</sup><sup>v</sup> <sup>i</sup>* ≤ *<sup>r</sup>*3; 0 ≤ *<sup>q</sup><sup>u</sup> <sup>i</sup>* ≤ *r*4; 0 ≤ *Q*<sup>1</sup> ≤ *r*5; 0 ≤ *Q*<sup>2</sup> ≤ *r*6; 0 ≤ *Q*<sup>3</sup> ≤ *r*7; 0 ≤ *Q*<sup>4</sup> ≤ *r*8; 0 ≤ *Q*<sup>5</sup> ≤ *r*9; 0 ≤ *T*<sup>1</sup> ≤ *r*10; 0 ≤ *T*<sup>2</sup> ≤ *r*11; 0 ≤ *ρ<sup>J</sup>* ≤ *r*12; 0 ≤ *ρ<sup>I</sup>* ≤ *r*13; 0 ≤ *ϕ*<sup>1</sup> ≤ *r*14; 0 ≤ *ϕ*<sup>2</sup> ≤ *r*15; 0 ≤ *ϕ*<sup>3</sup> ≤ *r*16; 0 ≤ *φ*<sup>1</sup> ≤ *r*17; 0 ≤ *φ*<sup>2</sup> ≤ *r*18; 0 ≤ *φ*<sup>3</sup> ≤ *r*19; 0 ≤ *ζ<sup>c</sup>* ≤ *r*20; 0 ≤ *λ* ≤ *r*21; 0 ≤ *γ* ≤ *r*22},

> where *r* = (*r*1,*r*2,*r*3,*r*4,*r*5,*r*6,*r*7,*r*8,*r*9,*r*10,*r*11,*r*12,*r*13,*r*14,*r*15,*r*16,*r*17,*r*18,*r*19,*r*20,*r*21,*r*22) ≥ 0, and *q<sup>v</sup> <sup>j</sup>* ≤ *<sup>r</sup>*<sup>1</sup> means *<sup>q</sup><sup>v</sup> <sup>j</sup>* ≤ *r*<sup>1</sup> for all *j* = 1, ··· *J*, and other notations can be interpreted in the same manner. Obviously, Ω is a bounded, closed convex set, and Ω ∈ Ω1. From Hammond et al. [58], the following VI *F*(*X*<sup>∗</sup> <sup>1</sup> ), *<sup>X</sup>*<sup>∗</sup> 1 ! ≥ 0, ∀*X*<sup>1</sup> ∈ Ω, admits at least one solution. We have the following theorem.

> **Theorem A3.** *(Existence) Variational inequality (24) admits a solution if and only if there is an r* > 0*, such that variational inequality (41) admits at least one solution in* Ω *with q<sup>v</sup> <sup>j</sup>* < *r*1*, qu <sup>j</sup>* < *<sup>r</sup>*2*, <sup>q</sup><sup>v</sup> <sup>i</sup>* < *<sup>r</sup>*3*, <sup>q</sup><sup>u</sup> <sup>i</sup>* < *r*4*, Q*<sup>1</sup> < *r*5*, Q*<sup>2</sup> < *r*6*, Q*<sup>3</sup> < *r*7*, Q*<sup>4</sup> < *r*8*, Q*<sup>5</sup> < *r*9*, T*<sup>1</sup> ≤ *r*10*,* 0 ≤ *T*<sup>2</sup> ≤ *r*11*,ρ<sup>J</sup>* < *r*12*,ρ<sup>I</sup>* < *r*13*, ϕ*<sup>1</sup> < *r*14*, ϕ*<sup>2</sup> < *r*15*, ϕ*<sup>3</sup> < *r*16*, φ*<sup>1</sup> < *r*17*, φ*<sup>2</sup> < *r*18*, φ*<sup>3</sup> < *r*19*, ζ<sup>c</sup>* < *r*20*, λ* < *r*21*, γ* < *r*22.

**Proof.** The proof of this theorem follows from Theorem 2.

**Theorem A4.** *(Monotonicity) When the cost functions and demand functions in VI (24) are convex, then the vector function F*(*X*1) *in VI (25) is monotone.*

**Proof.** Let *X*<sup>1</sup> <sup>1</sup> ∈ <sup>Ω</sup> and *<sup>X</sup>*<sup>2</sup> <sup>1</sup> ∈ Ω, ∇*H*(*X*1) = *F*(*X*1), according to Assumption 5 in Section 2, all functions in this paper are convex, then we have *H*(*X*<sup>1</sup> <sup>1</sup>) ≥ *<sup>H</sup>*(*X*<sup>2</sup> <sup>1</sup>) + ∇*H*(*X*<sup>2</sup> 1) *T* (*X*<sup>1</sup> <sup>1</sup> − *<sup>X</sup>*<sup>2</sup> 1) and *H*(*X*<sup>2</sup> <sup>1</sup>) ≥ *<sup>H</sup>*(*X*<sup>1</sup> <sup>1</sup>) + ∇*H*(*X*<sup>1</sup> 1) *T* (*X*<sup>2</sup> <sup>1</sup> − *<sup>X</sup>*<sup>1</sup> <sup>1</sup>), adding two formulas, " ∇*H*(*X*<sup>1</sup> <sup>1</sup>) − ∇*H*(*X*<sup>2</sup> 1) #*T* (*X*<sup>1</sup> <sup>1</sup> − *<sup>X</sup>*<sup>2</sup> <sup>1</sup>) ≥ 0, that is, *F*(*X*1) − *F*(*X*2), *X*<sup>1</sup> − *X*2 ≥ 0. Thus, we conclude that VI (25) is monotone.

**Theorem A5.** *(Strict monotonicity) When one of the cost functions and demand functions in VI (24) is strictly convex, then VI (25) is strictly monotone.*

**Proof.** Let *X*<sup>1</sup> <sup>1</sup>, *<sup>X</sup>*<sup>2</sup> <sup>1</sup> ∈ <sup>Ω</sup>, and *<sup>X</sup>*<sup>1</sup> <sup>1</sup> = *<sup>X</sup>*<sup>2</sup> 1, we can know at least one element in the vector *X*<sup>1</sup> <sup>1</sup> and *<sup>X</sup>*<sup>2</sup> <sup>1</sup> is not equal. No loss generality, let us suppose *<sup>q</sup>v*<sup>1</sup> *<sup>j</sup>* = *<sup>q</sup>v*<sup>2</sup> *<sup>j</sup>* . At the same time, we also suppose the production cost function *fj* is strictly convex. Thus, we have *F*(*X*1) − *F*(*X*2), *X*<sup>1</sup> − *X*2 > 0; that is, VI (25) is strictly monotone.

**Theorem A6.** *(Uniqueness) When VI (25) is strictly monotone, VI (24) has a unique solution in* Ω.

**Proof.** The proof of uniqueness of solution follows easily from Kinderlehrer and Stampacchia [61].

**Theorem A7.** *(Lipschitz continuity) Suppose that fj, f <sup>u</sup> <sup>j</sup> , fi, <sup>f</sup> <sup>u</sup> <sup>i</sup> , cjk, <sup>c</sup><sup>u</sup> hj, cik, <sup>c</sup><sup>u</sup> hi, <sup>c</sup><sup>u</sup> kh and ch have bounded second-order derivatives. Suppose that c<sup>K</sup> jk, <sup>c</sup><sup>K</sup> ik, <sup>α</sup><sup>u</sup> <sup>k</sup>* (*Q*5)*,* −*d<sup>h</sup> <sup>k</sup> and* −*d<sup>l</sup> <sup>k</sup> have bounded firstorder derivatives. The VI (24) is Lipschitz continuous. That is F*(*X*<sup>1</sup> <sup>1</sup>) − *<sup>F</sup>*(*X*<sup>2</sup> <sup>1</sup>) ≤ *<sup>L</sup> <sup>X</sup>*<sup>1</sup> <sup>1</sup> − *<sup>X</sup>*<sup>2</sup> 1*, X*1 1*, X*<sup>2</sup> <sup>1</sup> ∈ Ω*, with L* > 0.

**Proof.** Applying the mean value theorem of integrals to vector function *F*(*X*1) can immediately demonstrate Theorem A7.

#### **Appendix D**

*αI* <sup>1</sup> <sup>=</sup> 0.6 is lower than *<sup>α</sup><sup>J</sup>* <sup>1</sup> = 0.8 and *<sup>α</sup><sup>I</sup>* <sup>2</sup> <sup>=</sup> 0.3 is lower than *<sup>α</sup><sup>J</sup>* <sup>2</sup> = 0.5, which illustrates the result of the eco manufacturers' adoption of green technology. The selection of *ti* and *xxy* refers to Allevi et al. [50]. The other parameters are decided from the operation of paper industry enterprises.

The production cost of eco manufacturer *i*: *fi* = 8.5(*βrqv i* ) <sup>2</sup> + *βr*2*q<sup>v</sup> i qv* <sup>3</sup>−*<sup>i</sup>* + <sup>2</sup>*βrqv i* , *i* = 1, 2.

The production cost depends on the amount of raw materials used by both eco manufacturers, so it reflects the competitive relationship between eco manufacturers. In the numerical examples of the SCN equilibrium model, Nagurney et al. [40] first used this production cost function form, then other researchers such as [38,39] adopted this production cost function form.

The remanufacturing cost of eco manufacturer *i*: *f <sup>u</sup> <sup>i</sup>* = <sup>3</sup>(*βuqu i* ) <sup>2</sup> + 1.5*βuqu <sup>i</sup>* + 1, *i* = 1, 2.

Similarly, the production cost function and remanufacturing cost function of non-eco manufacturer *j* can be described as:

$$f\_j^u = 0.5(\beta^u q\_j^u)^2 + 0.5\beta^u q\_j^u + 1,\; j = 1,2;\; f\_i = 8.0(\beta^r q\_i^v)^2 + \beta^{r2} q\_i^v q\_{3-i}^v + 2\beta^r q\_i^v,\; i = 1,2... $$

We need to point out that the production cost and remanufacturing cost of the eco manufacturer is higher than that of non-eco manufacturer *j*, which is consistent with the previous Assumption 1.

The transaction cost functions between manufacturers and demand markets: *cjk* = 0.2*t*1*qjk* + 1, *cik* = 0.2*t*3*qik* + 1, *c<sup>K</sup> jk* = 0.2*qjk* + 0.1, *<sup>c</sup><sup>K</sup> ik* = 0.2*qik* + 0.1, *j* = 1, 2, *i* = 1, 2, *k* = 1, 2.

The transaction cost burdened by manufacturers and consumers comprises two parts: variable cost, which is associated with product quantity, and fixed cost, which is associated with the transaction action; whereas the cost burdened by manufacturers is also associated with the truck number.

The transaction cost functions between the collection center and demand market, and between the collection center and manufacturers: *ckh* = 0.1*q<sup>u</sup> kh* + 0.5, *<sup>c</sup><sup>u</sup> hj* = 0.1*t*2*q<sup>u</sup> hj* + 1, *cu hi* = 0.1*t*4*q<sup>u</sup> hi* + 1, *i* = 1, 2, *h* = 1, 2, *k* = 1, 2, *j* = 1, 2.

Similar to [50], *c<sup>u</sup> hj* and *<sup>c</sup><sup>u</sup> hi* include the number of trucks, which shows the transport effect. The disposal cost function of the collection center: *ch* <sup>=</sup> 2.5 <sup>2</sup> ∑ *k*=1 *qu kh* 2 <sup>+</sup> <sup>2</sup> <sup>2</sup> ∑ *k*=1 *qu kh*, *h* = 1, 2, the disutility function of consumers: *α<sup>u</sup> <sup>k</sup>* (*Q*5) = 0.5 <sup>2</sup> ∑ *k*=1 2 ∑ *k*=1 *qu kh* + 5, *k* = 1, 2.

According to carbon trading data related to the paper industry and related study [60], the carbon trade cost of manufacturers: *c<sup>t</sup> <sup>x</sup>* = 0.1*tx* 2, *x* = *i*, *j*, *i* = 1, 2, *j* = 1, 2.

The demand functions:

$$d\_k^l = 250 - 2\rho\_k^l - 1.5\rho\_{3-k}^l + 0.5(\rho\_k^h + \rho\_{3-k}^h) + \sigma\psi\sum\_{x=1}^2 (1 - a\_x^l), \ k = 1, 2;$$

$$d\_k^l = 230 - 2\rho\_k^h - 1.5\rho\_{3-k}^h + 0.5(\rho\_k^l + \rho\_{3-k}^l) + \sigma\psi\sum\_{x=1}^2 (1 - a\_x^l), \ k = 1, 2.$$

The manufacturers' production functions indicate that competition exists between the same types of manufacturers, and there is no competition between different types of manufacturers. The demand functions are associated with the price of two types of products; due to the consumer's low carbon preference, there is also a relationship between demand and the product's unit carbon emission amount. We assume the low carbon factor *ψ*= 10.

It is obvious that all the functions listed above are convex and continuously differentiable. Then, the solutions of VI (24), VI (40), and VI (A1) satisfy Theorem A3, Theorem A4, and Theorem A6.

#### **References**

