Research on Sustainable Cooperation Strategies for Cross-Regional Supply Chain Enterprises in Uncertain Environments

Abstract

Considering the fuzziness of cooperation and the sustainability of the redevelopment of cross-regional supply chain enterprises, the fuzzy participation degree and the generalized redistribution coefficient are introduced to describe the problem of cooperation and benefit distribution of cross-regional supply chain enterprises. A fuzzy average monotone game is constructed to study the strategies of cross-regional supply chain enterprises to increase the average benefit with the expansion of the alliance scale; a generalized fuzzy reduced game is discussed to ensure the partner selection decision of supply chain enterprises; the concepts of generalized fuzzy core, generalized fuzzy bargaining set, and generalized fuzzy proportional distribution are proposed; the equivalence between the generalized fuzzy core and the generalized fuzzy bargaining set of the fuzzy average monotone game of the cross-regional supply chain enterprises is proved; the nonempty generalized fuzzy core solution of the fuzzy average monotone game is characterized; and the example analysis shows the stability of the cross-regional supply chain enterprise alliance and the existence of the optimal generalized redistribution scheme. The research results not only satisfy the willingness of cross-regional supply chain enterprises to participate in cooperation with some resources from the “environmental” pillar of supply chain enterprise management sustainability but also achieve the strategy of retaining partial benefits for the redevelopment of supply chain alliances from the “economic” pillar of supply chain enterprise management sustainability, which provides a theoretical basis for the cooperation and benefit redistribution of cross-regional supply chain enterprises under uncertain environments.

1. Introduction

With the rapid development of the global economy, market competition is no longer the competition of a single enterprise but the competition between alliances, that is, the competition between supply chains. There is a symbiotic relationship between supply chain enterprises facing fierce market competition. On the one hand, supply chain enterprises distributed in different regions require cross-regional centralized development to reduce long-distance transaction costs. On the other hand, when supply chain enterprises experience agglomeration and lack of economy, these interest communities need to cooperate with a certain region for cross-regional cooperation. Changes in the external environment drive member enterprises in trans-regional supply chains to change their cooperation strategies [1,2,3,4,5]; the traditional cooperation strategy of either fully participating in a regional alliance or not participating in a regional alliance can no longer meet the “environmental” pillar of sustained cross-regional cooperation. Cross-regional supply chains have received extensive attention from governments and society; the scale of the industry continues to rise; the degree of refinement continues to increase; the competition of enterprises has evolved into the competition of the supply chain alliance; and enterprises distributed in various countries (regions) of the world as members of the supply chain constitute cross-regional supply chain systems, which participate to various degrees in the internal and external economic cycles of each country (region). The opening up of global resources makes supply chain enterprises form alliances to seek optimal resources on a global scale, making them no longer limited to specific regions. As an important carrier to cope with internal and external environments, cross-regional supply chain alliances have been attracting increasing attention in theory and practice. The supply chain alliance is formed based on win‒win cooperation. Whether the alliance has yet to be formed or is in the process of operation, the distribution of its benefit is always a conflict point, especially in cross-regional supply chain enterprises that form an alliance, and such cooperation continues to increase. The formation of the structure of the alliance is also becoming increasingly diversified, and the fuzzy characteristics of the member enterprises in the cooperation and the game process are more complex. It is necessary to build a stable cross-regional cooperation pattern and an optimal distribution strategy to ensure the development of high-quality resources. It is necessary to build stable cross-regional cooperation patterns and optimal distribution strategies to meet the demand for high-quality development.

However, in reality, competition continues to intensify, product life cycles are shrinking, and there are many uncertainties in supply chains, such as uncertainty in market demand and uncertainty in production costs, and cross-regional supply chain enterprises need to cooperate with multiple alliances to varying degrees, so, “cross regional” itself reflects the “environmental” pillar of the sustainability of supply chain enterprise alliance cooperation.

In view of the above, we ask the following questions: (1) In competitive supply chains, do multiple fuzzy cooperation game patterns develop in the formation of cross-regional supply chain enterprise alliances? Will their contribution behaviors affect attempts to improve the average return of the alliance? (2) How do cross-regional supply chain enterprises choose the best partners? (3) Under which cross-regional supply chain enterprises’ alliance cooperation game patterns can the optimal redistribution scheme be found? (4) Are different redistribution schemes of cross-regional supply chain enterprises’ alliances equivalent?

Our study considers the literature on fuzzy cooperative game strategies and benefit-sharing problems, and some scholars have investigated this fuzzy phenomenon [6,7,8,9,10]. Zhou Xiaoyang et al. addressed the equilibrium strategies of cross-regional supply chain networks under the influence of multitrade policies [11]. Chan proposed a dynamic equilibrium model of oligopolistic closed-loop supply chain networks with uncertain and time-varying demand [12]. Zhou considered retailers’ dynamic and static loss aversion behaviors in the equilibrium model of a supply chain network that includes each policymaker’s equilibrium model considering the dynamic and static loss aversion behavior of retailers [13], but their work does not address the impact of the contribution behaviors of cross-regional supply chain enterprises on increasing the average return of the alliance or what kind of fuzzy cooperative game patterns and related fuzzy characteristics are formed in the cooperation. There are also researchers who have studied and analyzed cross-regional supply chains with the objectives of minimizing total cost, minimizing total carbon dioxide emissions from transportation, total capacity utilization, inventory constraints, product quality, etc. [14,15,16,17,18,19], and the main novel contribution of the current study lies in the strategic choice of the optimal partner for supply chain enterprises in terms of maximizing the difference in alliance returns. Subsequently, a variety of fuzzy game models and related solution sets emerged, such as fuzzy Shapley value, fuzzy core, fuzzy stable set, fuzzy Weber set, etc. [20,21,22,23,24,25,26,27], and their research problem is to account for the fuzzy characteristics of cooperation, while our research problem is to not only account for the fuzzy characteristics of cooperation but also realize the partial preservation of the value of the alliance’s total return, which meets the requirements for the sustainable cooperation and redevelopment of the cross-region supply chain under an uncertainty environment, enterprises’ sustainable cooperation and redevelopment needs. At the same time, the current complex and changing competitive environment puts forward more new requirements for the existing basic theories, and the cooperation of cross-regional supply chain enterprises in uncertain environments presents more new development trends, which requires more systematic research from more perspectives.

To answer the above research questions, fill these research gaps, and consider both the “environmental” and “economic” pillars of the sustainability of supply chain enterprise alliances, we apply the characteristics of fuzzy uncertainty to cross-regional supply chains and regarding the formation process of a fuzzy cooperation game of cross-regional supply chain enterprises’ alliances as the process of mutual bargaining among supply chain enterprises, and the enterprises of each regional supply chain form a cooperative alliance via bargaining. The bargaining set is regarded as the set of game solutions formed by cooperative enterprises in a cross-regional supply chain in response to bargaining issues, and the bargaining set is used to compensate for the limitation that the core solution may be the null set and the optimal distribution scheme cannot be obtained.

In summary, the main contributions of this paper are as follows:

(1)

The bargaining set is expanded by considering both fuzzy participation and generalized redistribution system and constructing a fuzzy average monotonic cooperation game model for cross-regional supply chain enterprise alliances with the help of the concepts of generalized fuzzy bargaining set, and generalized fuzzy proportional distribution, which has seldom been researched.

(2)

A generalized fuzzy reduced game is used to ensure that each cross-regional supply chain enterprise finds the best partner, which is also different from most of the literature.

(3)

The equivalence of the generalized fuzzy bargaining set and the generalized fuzzy core solution to portray the equilibrium conditions of the fuzzy average monotonous cooperation game of cross-regional supply chain enterprises not only ensures the existence of the optimal redistribution scheme of the cross-regional supply chain enterprise alliance but also meets the needs of its sustainable development.

(4)

The arithmetic example is applied to analyze the nonemptiness of cross-regional supply chain enterprise alliance formation and its generalized fuzzy core solution of the fuzzy mean monotonic cooperation game, which provides choice strategies for solving the sustainable cooperation problem of cross-regional supply chain enterprises under an uncertain environment and draws more conclusions and management insights.

The rest of the paper is organized as follows. Section 2 presents the literature review. Section 3 constructs a fuzzy average monotonic cooperation game model for the contribution behavior of cross-regional supply chain enterprises. Section 4 investigates the partner selection strategy of cross-regional supply chain enterprises based on the generalized fuzzy reduced game. Section 5 analyzes the generalized fuzzy bargaining set benefit redistribution scheme for cross-regional supply chain enterprise alliances. Section 6 demonstrates the equivalence of the generalized fuzzy bargaining set and the generalized fuzzy core redistribution scheme for cross-regional supply chain enterprise alliances. Section 7 numerically simulates the conclusions. Finally, Section 8 summarizes the findings and suggests future research directions.

2. Literature Review

This paper is closely related to three major streams of literature on the drivers of enterprise alliance formation, alliance member partnerships, and alliance benefit distribution. This section reviews the literature related to each stream and highlights the differences between existing research and our work.

2.1. Motivations for the Formation of Cross-Regional Supply Chain Enterprise Alliances

In the market environment, it is very common for supply chain enterprises to form cooperative alliances. Based on resource theory, the founders of the resource school of thought, Wernerfelt, Grant, and Barney, claim that enterprises are collections of resources, and enterprises’ competitive advantages depend on the heterogeneity of the resources they possess [28]. Hamel points out that if each alliance member can satisfy the resource needs of the other members to complement each other’s resources, they can realize their common interests [29]. Teece points out that for enterprises in high-growth industries to ensure the promotion of their timely products, alliances need to be formed to realize complementary capabilities and the distribution of overall capabilities [30]. Shi and Wang, and Yi formulate the market of resources in JointCloud as a supply chain network competition model with multiple competing manufacturers and retailers offering resources; a market game is established to model the competition among clouds [31]. According to the viewpoint of the enterprise core competence school, the effective scope of the strategic activities of enterprises depends on its core competence, and the composition of enterprise advantage depends on whether enterprises can be placed in a state where their core competencies play a valuable role and are durable; that is, the core competence is the foundation of the competitive advantage of the enterprise. The principle of market path dependence represented by the American economist B. Arthur holds that because the consumption in high-tech markets embodies the characteristics of path dependence, in the context of the customer-oriented era, the market lead cannot be separated from the production lead, the technology lead or the standard lead; the phenomenon of incremental returns to scale exists in the high-tech market; and the lead depends on the speed of the market lead to realize the return on investment. Increasing returns to investment can be realized only by market leadership, which can guide the consumption path [32]. Zhang Xizheng tried to make a breakthrough in the path dependence problem of researching enterprise upgrading and transformation between enterprises and between consumers and enterprises and explores and verifies the path dependence characteristics of the subject’s decision-making behavior in the process of enterprise upgrading and transformation with the help of evolutionary game theory [33].

The above studies examined the resource-based, capability-based, and path-dependent drivers of alliance formation. Via the analysis of resource-based alliance motivations, this paper provides a research approach for cross-regional supply chain enterprises to participate in alliance cooperation with partial resources, i.e., fuzzy cooperation. This paper provides a research method for cross-regional supply chain enterprises to form alliances via bargaining capabilities and the analysis of capability-based alliance motivations. By analyzing the path-dependent alliance motivations, this paper provides a research perspective on how cross-regional supply chain enterprise alliances can form different structures depending on different cooperative game patterns. That is to say, the above research results provide an important reference value for the study of the impact and selection of fuzzy cooperative game strategies in the formation of cross-regional supply chain enterprise alliances in this paper.

2.2. Collaborative Relationships among Members of Cross-Regional Supply Chain Enterprise Alliances

Supply chain enterprises need to look for partners to form supply chain alliances to ensure benefit maximization. Therefore, enterprises should pay attention to the selection of the best partners to realize sustainable cooperation of alliance members. In the cooperation of alliance members of supply chain enterprises, the desire of enterprises to establish alliances is affected by the expectation of alliance benefit distribution [34]. There is competition among alliance enterprises for alliance benefits, and alliance establishment is a game process [35]. Representatives of the school of competition and cooperation theory, such as Byry Nerenev, Benjamin Gumos Cassels, and Adam Brandenburg, have pointed out that cooperation does not weaken competition; at the same time, competition is not exclusive of cooperation, and cooperation promotes the efficiency of competition [36]. Benjamin Gumos Cassels studied the case of enterprise alliances in the high-tech field and obtained important conclusions. According to Moore, as times change, competitors form new rules of competition and cooperation [37], change from fighting with competitors to complementing each other’s strengths according to their respective contributions [38], cocreate technological standards, and form a coordinated dynamic system [39]. Andrew Inkpen’s view on alliance relationships shows that the competitive relationships between alliance partners are complex and that alliances are a way for organizations to exhibit both cooperation and competition [40]. Birnberg argued that the key way to maintain alliance relationships is to build trust [41]. Ring and Vandeven argued that inter-organizational trust is, in turn, based on trust between individuals, i.e., the key to successful alliances lies in the relationships between member enterprises, which need to be governed by alliance rules and regulations [42]. Xing and Yang introduced consumer green preferences and established a cooperation channel strategy selection model between manufacturers and retailers [43].

The above studies examined the cooperation strategies of cross-regional supply chain enterprise alliance members via the theory of alliance competition and cooperation, providing direction for this paper to study the cooperation strategies of cross-regional supply chain enterprises in uncertain environments via fuzzy cooperative games and further study the impact and strategies of fuzzy reduced game maximum difference in selecting the best partner for cross-regional supply chain enterprises.

2.3. Distribution of Benefits from Enterprise Alliances

As an effective organizational form of supply chain innovation, supply chain enterprise alliances are beneficial; whether the final benefit distribution is reasonable depends on the choice of the benefit distribution scheme. The optimal redistribution scheme is more conducive to the sustainable development of supply chain enterprise alliances. Alexander and Ruderman pointed out [44] that the fairness of benefit distribution may have a greater impact on members’ decisions to stay in their alliances or leave. Shapley L.S. studied the mathematical methods used to solve the multiplayer cooperative game and proposed Shapley’s value method [45], whose distribution strategy is based on the proportion of contribution and excess value generated by each partner. Meade pointed out that the key to the smooth formation and successful operation of dynamic alliances depends on the rational benefit distribution mechanism [46], which is also the key to maximizing the performance of each member. Lemaire constructed a cooperative game model to solve the problem of distributing the common benefits of alliance enterprises to ensure the normal operation of alliances and to reduce the cost of their dissolution [47]. Flam and Jourani studied behavior in a game with pass-through costs and provided a form of benefit distribution in the case that alliance enterprises can share the tasks and technologies of the alliance with minimum cost [48]. Meng proposed a fuzzy coalition structure cooperation game and its solution method, which accounts for the innovation of the fuzzy coalition cooperation game and traditional coalition structure cooperation game [49]. Gallego developed a fuzzy coalition structure cooperation game based on the fuzzy game of Choquet integral form by using the partition function and studied the fuzzy alliance case with the fuzzy coalition of Choquet integral form and the fuzzy coalition of Choquet integral form using a fuzzy coalitional structure cooperative game and Owen value [50]. Sun et al. developed a cooperative game with a fuzzy coalition structure based on Owen’s multilinear extension game and axiomatized the Owen value under the game form [51]. Subsequently, other scholars successively explored fuzzy cooperative games with exchange restrictions similar to the restrictions of coalition structure using undirected graphs to portray the cooperation restrictions among the players on the board. Murofushi and Soneda investigated coalition member interaction indexes from a fuzzy measurement point of view [52]. Shan, Liu, and Lyu used solidarity value as a distribution rule that combines fairness and solidarity in cooperative games to solve the problem of income distribution [53].

Although the above studies analyzed the coalition benefit distribution problem via game solutions such as the Shapley value and Owen value, their research questions are different. Unlike our paper, these studies did not consider retaining a portion of the total coalition proceeds for sustainable development, nor did they study the equivalence of various proceeds distribution schemes via specific fuzzy cooperative game solutions. However, the above research results provide inspiration for this paper to study the sustained cooperation of cross-regional supply chain enterprises from the perspective of the sustainable “environment” pillar.

3. A Fuzzy Average Monotonic Cooperative Game Model of the Contribution Behavior of Cross-Regional Supply Chain Enterprises

With the rise in industrial scale, the supply chain dimension is expanding, the number of cooperative enterprises is increasing, and cooperative relationships are becoming increasingly complex. The traditional cooperation model can no longer meet the needs of uncertain environments because under the traditional cooperative game model, each enterprise either fully participates in a supply chain alliance or does not participate in a supply chain alliance at all. However, from the perspective of many factors, such as the supply chain enterprise’s own capabilities, resources, and the attraction of other business interests, it may not fully cooperate with a specific supply chain alliance with all resources. Instead, it participates in multiple alliances with a portion of their resources, so there is an urgent need to transform into a fuzzy cooperation model. The correspondence between the traditional cooperation mode and fuzzy cooperation mode is characterized by transforming the cooperative alliance $\mathrm{S}\in 2^{\mathcal{N}}$ under traditional cooperation mode into the alliance ${\mathrm{e}}^{\mathrm{S}}$ under fuzzy cooperation mode:

$${{(\mathrm{e}}^{\mathrm{S}})}_{\mathcal{i}}=\left\{\begin{array}{c}1, \mathrm{when} i\in S \\ 0, \mathrm{when} i\in N\backslash S\end{array}\right.$$

Among them, in the traditional cooperation mode, the participation degree of enterprises in the cooperative alliance S is recorded as 1 when they fully cooperate and 0 when they do not cooperate at all. Using ${\mathcal{s}}_{\mathcal{i}}\in [0,1]$ in Fuzzy cooperation mode describes the degree of participation. Therefore, in the fuzzy cooperative game of cross-regional supply chain enterprises, an n-dimensional unit hyperspace $[0,1{]}^{\mathrm{N}}$ was formed, represented by ${\mathcal{F}}^{\mathcal{N}}$, so, in the fuzzy cooperative game of cross-regional supply chain enterprises, the formation of the $\mathcal{n}$ dimensional unit super three-dimensional space $[0,1{]}^{\mathcal{N}}$ is denoted by ${\mathcal{F}}^{\mathcal{N}}$. Then, the cross-regional supply chain enterprise alliance under an uncertain environment is denoted by vector $\mathcal{s}\in {\mathcal{F}}^{\mathcal{N}}$, i.e., $\mathcal{s}$ represents the cross-regional supply chain enterprise set $\mathcal{N}$ in a coalition. The first dimension of the $\mathcal{i}$ element on the dimension ${\mathcal{s}}_{\mathcal{i}}$ satisfies $[0,1]$ and is used to indicate that cross-regional supply chain enterprises $\mathcal{i}$ will cooperate to the degree of ${\mathcal{s}}_{\mathcal{i}}$ with alliance $\mathcal{s}$; that is, in the cross-regional supply chain enterprise alliance $\mathcal{s}$, ${\mathcal{s}}_{\mathcal{i}}$ indicates the alliance partner’s $\mathcal{i}$ participation degree, i.e., the degree of affiliation in the cross-regional supply chain enterprise alliance $\mathcal{s}$ in the cross-regional supply chain enterprise alliance. The carrier of the alliance is $\mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{s}\right)=\{\mathcal{i}\in \mathcal{N}|{\mathcal{s}}_{\mathcal{i}}>0\}$ If $\mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{s}\right)\ne \mathcal{N}$ the cross-regional supply chain enterprise alliance is a specific alliance; in the case of a cross-regional supply chain enterprise aggregation $\mathcal{s}$ is a specific alliance, and the set of specific alliances on the set of cross-regional supply chain enterprises is $\mathcal{N}.$ The set of specific alliances on the set of cross-regional supply chain enterprises is denoted by ${\mathcal{P}\mathcal{F}}^{\mathcal{N}}$ and the set of nonempty specific coalitions on the set of cross-regional supply chain enterprises $\mathcal{N}.$ The set of nonempty specific coalitions on the set of cross-regional supply chain enterprises is denoted by ${\mathcal{P}\mathcal{F}}^{\mathcal{N}}$. The set of nonempty specific alliances on the set of cross-regional supply chain enterprises is denoted by ${\mathcal{P}\mathcal{F}}_0^{\mathcal{N}}$.

The fuzzy cooperation game formed in the cooperation of cross-regional supply chain enterprises is considered a mapping $\mathcal{v}:{\mathcal{F}}^{\mathcal{N}}\to \mathcal{R}$ that is satisfied as follows:

$$\mathcal{v}\left({\mathcal{e}}^{\varnothing }\right)=0(\mathrm{Emptiness})$$

(1a)

$$\mathcal{v}\left(\mathcal{s}\vee \mathcal{t}\right)\ge \mathcal{v}\left(\mathcal{s}\right)+\mathcal{v}(\mathcal{t})(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y})$$

(1b)

The mapping, also known as the characteristic function $\mathcal{v}$, describes the possible actions in the cooperation of cross-regional supply chain enterprises and assigns a real value to each alliance to portray the value of benefits that cross-regional supply chain enterprises can obtain in the cooperation. The mapping gives the constraints of the cooperation structure of cross-regional supply chain enterprises’ alliances under the fuzzy game only at the top design level; however, many cooperation game models may be formed in the actual cross-regional supply chain enterprises’ alliance formation, and the superadditivity expresses the main idea that the various alliances of cross-regional supply chain enterprises gain more benefits via cooperation than when each enterprise acts individually.

To realize the sustainable redevelopment of cross-regional supply chains, that is to consider the sustained “economic” pillar in supply chain enterprise management, a portion of the total benefit of the supply chain enterprise alliance can be retained for remanufacturing or recooperation, that is, by introducing both participation and adjustment coefficients, supply chain enterprises can cooperate with multiple alliances at the same time with a certain degree of participation, while also meeting the application demand of retaining a portion of the benefit value for redevelopment. The distribution coefficient is also known as adjustment coefficients $\mathcal{r}$ and ${\mathcal{c}}_{\mathcal{i}}$ Introducing the benefit distribution model of supply chain enterprise alliances, this paper expands various traditional game solutions and methodology for supply chain alliance cooperation. It proposes a generalized distribution of supply chain enterprise alliances under fuzzy cooperative games. Now, with the help of the adjustment coefficients $\mathcal{r}$ and ${\mathcal{c}}_{\mathcal{i}}$ extending the traditional distribution to a generalized distribution, $\mathcal{r}$ and ${\mathcal{c}}_{\mathcal{i}}$ satisfy $0<\mathcal{r}\le 1$, $0<{\mathcal{c}}_{\mathcal{i}}\le 1$, $\le \mathcal{i}\le \mathcal{n}$, and $\mathcal{r}\ge {\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{\mathcal{c}}_{\mathcal{i}}$. The generalized fuzzy distribution vector of the fuzzy cooperation game of cross-regional supply chain enterprises $\mathcal{x}=({\mathcal{x}}_1,{\mathcal{x}}_2,\cdots ,{\mathcal{x}}_{\mathrm{n}})$ is denoted by

$${\sum }_{\mathcal{i}\in \mathcal{N}}{\mathcal{x}}_{\mathcal{i}}=\mathcal{r}·\mathcal{v}\left({\mathcal{e}}^{\mathcal{N}}\right)$$

(2a)

$${\mathcal{x}}_{\mathcal{i}}\ge {\mathcal{c}}_{\mathcal{i}}·\mathcal{v}\left(\left\{{\mathcal{e}}^{\mathcal{i}}\right\}\right)$$

(2b)

Among them, Equation (2a) depicts that under the fuzzy cooperation game of cross-regional supply chain enterprises, the total benefits of the supply chain enterprise alliance can be partially retained rather than fully distributed at one time with the help of coefficient $\mathcal{r}$. (2a) depicts that under the fuzzy cooperation game of cross-regional supply chain enterprises, the total benefit of the supply chain enterprise alliance can be partially retained rather than one-time distributed with the help of coefficients adjusted according to the actual situation; Equation (2b) shows that each cooperative enterprise of the cross-regional supply chain will not obtain less benefit than the benefit it would obtain if it does not participate in the cooperation, $\mathcal{r}\ge {\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{\mathcal{c}}_{\mathcal{i}}$. This is the necessary condition to ensure that (2a) and (2b) are consistent.

For fuzzy games in cross-regional supply chains $\mathcal{v}\in {\mathcal{F}\mathrm{G}}^{\mathcal{N}}$, the generalized fuzzy distribution set of cooperative alliances of enterprises under $\mathrm{I}(\mathcal{v},\mathcal{r},\mathcal{c})$ is the following set:

$$\mathrm{I}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)=\left\{\mathcal{x}\in {\mathcal{R}}^{\mathcal{N}}\left|\begin{array}{c}\sum _{\mathcal{i}\in \mathcal{N}}{\mathcal{x}}_{\mathcal{i}}=r·v({\mathrm{e}}^{\mathcal{N}})\\ {\mathcal{x}}_{\mathcal{i}}\ge {\mathcal{c}}_{\mathcal{i}}·v\left({\mathrm{e}}^{\mathcal{i}}\right), \forall i\in N\end{array}\right.\right\}$$

(3)

Definition 1. 

In the fuzzy cooperative game for cross-regional supply chain enterprises $\mathcal{v}\in {\mathcal{F}\mathrm{G}}^{\mathcal{N}}$, the generalized fuzzy core of the $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$ distribution is

$$\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)=\{\mathcal{x}\in \mathrm{I}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)|{\sum }_{\mathcal{i}\in \mathcal{N}}{\mathcal{s}}_{\mathcal{i}}{\cdot \mathcal{x}}_{\mathcal{i}}\ge \underset{\mathcal{j}\in \mathcal{car}(\mathcal{s})}{\mathcal{max}}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\right)\forall \mathcal{s}\in {\mathcal{F}}^{\mathcal{N}}\}$$

(4)

In the generalized fuzzy core distribution for cross-regional supply chain enterprise alliances, $\mathcal{x}\in C\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$, indicates that the total value of benefits of the cooperative alliance under the fuzzy game is partially reserved and not all distributed at once, $e^{\mathcal{N}}$. The cross-regional supply chain enterprise alliance will be partially retained and not fully distributed at one time, and based on this generalized fuzzy distribution scheme, the gains obtained by the cross-regional supply chain enterprise alliance $\mathcal{s}$ are not less than $\underset{\mathcal{j}\in \mathcal{car}(\mathcal{s})}{\mathcal{max}}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\right)$ for the partners $\mathcal{i}.$ The benefits for the partners are based on their participation level.

The application of an average monotonic game covers many fields. By introducing participation parameters, the traditional average monotonic game is extended to a fuzzy average monotonic game, which is more suitable for cross-regional supply chain management scenarios where increasing average returns can be obtained in uncertain environments, considering both the sustainable “environmental” pillar in supply chain enterprise management and the “economic” pillar.

Definition 2. 

If a fuzzy cooperative game is formed between cross-regional supply chain enterprises $\mathcal{v}\in {\mathcal{F}\mathrm{G}}^{\mathcal{N}}$ that is directed at $\forall \mathcal{s}\le \mathcal{t}\le e^{\mathcal{N}}$, it is said to be a fuzzy cooperation game if and only if $\exists \alpha \in {\mathcal{R}}_{+}^n\backslash \{0\}$ and if it satisfies the following condition $\mathcal{v}$ as a fuzzy average monotonic game:

$$\mathcal{v}\left(\mathcal{s}\right)\ge 0,\forall \mathcal{s}\le e^{\mathcal{N}} \mathrm{and}$$

(5a)

$$\alpha (\mathcal{t})\cdot \mathcal{v}(\mathcal{s})\le \alpha (\mathcal{s})\cdot \mathcal{v}(\mathcal{t})$$

(5b)

Among them, $(\mathcal{s})={\sum }_{i\in \mathcal{N}}{\mathcal{s}}_i\cdot {\alpha }_i={\sum }_{i\in \mathcal{c}\mathcal{a}\mathcal{r}(s)}{\mathcal{s}}_i\cdot {\alpha }_i· \alpha (\mathcal{t})={\sum }_{i\in \mathcal{N}}{\mathcal{t}}_i\cdot {\alpha }_i={\sum }_{i\in \mathcal{c}\mathcal{a}\mathcal{r}(t)}{\mathcal{t}}_i\cdot {\alpha }_i$. Then, there are $0\le \alpha (\mathcal{s})\le \alpha (\mathcal{t})$ and $\alpha (e^{\mathcal{N}})\ge 0$. If ${\alpha }_i\ge 0,\forall i\in \mathcal{N}$, Equation (5b) can be written as

$$\frac{v(\mathcal{s})}{\alpha (\mathcal{s})}\le \frac{v\left(\mathcal{t}\right)}{\alpha \left(\mathcal{t}\right)},\forall e^{\varnothing }\ne \mathcal{s}\le \mathcal{t}\le e^{\mathcal{N}}$$

(5c)

Among them, $\alpha =({\alpha }_i{)}_{i\in \mathcal{N}}$ represents the relative contribution of each cooperative enterprise in the cross-regional supply chain, but from the perspective of the game itself, $\alpha$ is extrinsic. Equation (5c) describes the idea that the average contribution value of the cross-regional supply chain alliance enterprises increases with the alliance size. That is, if there exists a positive nonzero contribution value vector $\alpha \in {\mathcal{R}}_{+}^{\mathcal{n}}\backslash \{0\}$, then the characteristic function of the fuzzy average monotonic game of the cross-regional supply chain represents the average return value with respect to the vector of enterprises’ contribution values $\alpha$ of the incremental average gain value. Thus, for the fuzzy average monotonic game formed by the cooperation of cross-regional supply chain enterprises, it is crucial to determine the relative contribution value of the cooperating enterprises.

The application of the fuzzy mean monotonic game covers many fields, and it is also adapted to the cross-region supply chain management scenario, which can yield incremental average returns $\alpha$. In the cooperation of cross-regional supply chain enterprises, it is crucial to define what kind of contribution vector is more suitable for the formation of a fuzzy mean monotonic game, and this vector may be unique. Once the contribution value vector between cross-regional supply chain enterprises is determined, the structure of the fuzzy mean monotonic game formed by their cooperation will be clearly presented. The following properties can be used as the necessary conditions to judge whether the game formed by the cooperation between cross-regional supply chain enterprises is a fuzzy mean monotonic game about a certain vector of contribution values.

Proposition 1. 

Let $\mathcal{v}$ be a fuzzy average monotonic game with cross-regional supply chain enterprises with respect to the vector of contribution values $\alpha$. A fuzzy average monotonic game with the following property holds.

(1)

If the fuzzy cooperative alliance of cross-regional supply chain enterprises $\mathcal{s}$ has a gain value greater than zero, i.e., $\left(\mathcal{s}\right)>0$. $\forall \mathcal{s}\le {\mathcal{e}}^{\mathcal{N}}$, then its contribution value vector is greater than zero, i.e., $\left(\mathcal{s}\right)>0$.

(2)

If in the fuzzy cooperative alliance of cross-regional supply chain enterprises $\mathcal{s}$, the vector of contribution values is equal to zero, i.e., $\alpha \left(\mathcal{s}\right)=0$, where $\forall \mathcal{s}\le {\mathcal{e}}^{\mathcal{N}}$, then its alliance $\mathcal{s}$ has a gain value equal to zero, i.e., $\mathcal{v}\left(\mathcal{s}\right)=0$.

(3)

If the fuzzy cooperative alliance of two enterprises in a cross-regional supply chain $\mathcal{s},\mathcal{t}$ has equal gain values, i.e., $\mathcal{v}\left(\mathcal{s}\right)=\mathcal{v}\left(\mathcal{t}\right)$ and $\mathcal{s}<\mathcal{t}\le {\mathcal{e}}^{\mathcal{N}}$, then the contribution value vectors of the two alliances are equal, i.e., $\alpha \left(\mathcal{t}\right)=\alpha (\mathcal{s})$.

(4)

In cross-regional supply chain enterprises on the contribution value vector $\alpha$, the fuzzy average monotonic game $\mathcal{v}$ satisfies monotonicity.

(5)

In cross-regional supply chain enterprises, on the contribution value vector $\alpha$, the fuzzy average monotonic game $\mathcal{v}$ satisfies superadditivity.

Therefore, with the help of the fuzzy mean monotonic game, the total average benefits of cross-regional supply chain enterprise alliances can be predicted, which is important to determine both whether enterprises participate in the supply chain and what level of contribution they choose.

4. Partner Selection Strategies for Cross-Regional Supply Chain Enterprises Based on a Generalized Fuzzy Reduced Game

The fuzzy average monotonic game formed by the cooperation of cross-regional supply chain enterprises implies that in the cooperative alliance about the vector $\alpha$, the average return value of the cooperative alliance grows with the increase of the alliance size, and if the cooperative enterprises share the return value proportionally according to weights ${\alpha }_i$, an increase the number of cooperative enterprises in the fuzzy alliance of cross-regional supply chains will bring greater returns to the cooperative enterprises.

Definition 3. 

In a fuzzy mean monotonic game for cross-regional supply chain enterprise $\mathcal{v}$, the generalized fuzzy proportional distribution of $\mathcal{p}\left(\mathcal{v};\alpha ,\mathcal{c}\right)={({\mathcal{p}}_{\mathcal{i}}\left(\mathcal{v};\alpha ,\mathcal{c}\right))}_{\mathcal{i}=\mathrm{1,2},\cdots ,\mathcal{n}}$; is

$$\mathcal{p}\left(\mathcal{v};\alpha ,\mathcal{c}\right)={\mathsf{\alpha }}_{\mathcal{i}}·\frac{\mathcal{r}·\mathcal{v}\left({\mathcal{e}}^{\mathcal{N}}\right)}{\alpha \left({\mathcal{e}}^{\mathcal{N}}\right)}$$

(6)

Among them, $\mathcal{i}=\mathrm{1,2},\cdots ,\mathcal{n}$, $0<\mathcal{r}={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{\mathcal{c}}_{\mathcal{i}}\le 1$. $\mathcal{c}=\left({\mathcal{c}}_1,{\mathcal{c}}_2\cdots \cdots {\mathcal{c}}_{\mathcal{n}}\right),0<{\mathcal{c}}_{\mathcal{i}}\le 1$.

The generalized fuzzy proportional distribution of cross-regional supply chain enterprises is not only an element in the generalized fuzzy core but also embodies the idea that members of a cross-regional supply chain enterprise alliance can obtain higher benefits in larger alliances than in any sub-alliance. It can also prove the nonemptiness of the generalized fuzzy core with the help of the generalized fuzzy proportional distribution of cross-regional supply chain enterprises.

Theorem 1. 

Let $\mathcal{v}$ be a fuzzy average monotonic game of cross-regional supply chain enterprises on the contribution value vector $\alpha$ of the fuzzy average monotonic game; then, the following statements hold:

(1) 

The generalized fuzzy proportional distribution of cross-regional supply chain enterprise alliances is within their generalized fuzzy core.

(2) 

The fuzzy average monotonic game of cross-regional supply chain enterprises $\mathcal{v}$ is perfectly equilibrated.

Proof of Theorem 1 is attached in Appendix A.

Theorem 1 lays the foundation for arguing that the generalized fuzzy core and generalized fuzzy bargaining set distribution schemes of the fuzzy average monotonic game of cross-regional supply chain enterprises are equal.

Definition 4. 

In fuzzy alliances for cross-regional supply chain enterprises $\mathcal{s}$ and $\mathcal{t}$, $\forall {\mathcal{e}}^{\mathrm{\varnothing }}\ne \mathcal{s}\le \mathcal{t}<{\mathcal{e}}^{\mathcal{N}}$, and its fuzzy game $\mathcal{v}$, the distribution vector of $\mathcal{x}\in {\mathcal{R}}^{\mathcal{N}}$. Then, for the coalition $\mathcal{t},$ the generalized fuzzy reduced game with respect to the distribution vector $\mathcal{x}$ is

$${\mathcal{v}}_{\mathcal{x}}^{\mathcal{t}}\left({\mathcal{e}}^{\mathrm{\varnothing }}\right)=0$$

(7a)

$${\mathcal{v}}_{\mathcal{x}}^{\mathcal{t}}\left(\mathcal{s},\mathcal{c}\right)={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathrm{\varnothing }\subseteq \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{q}\right)\subseteq \mathcal{N}\backslash \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right)}\{\underset{\mathcal{j}\in \mathcal{car}(\mathcal{s})\cup \mathcal{car}(\mathcal{q})}{\mathcal{max}}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\vee \mathcal{q}\right)-{\sum }_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{q})}{\mathcal{q}}_{\mathcal{j}}·{\mathcal{x}}_{\mathcal{j}}\}$$

(7b)

The main idea of the generalized fuzzy reduced game for cross-regional supply chain enterprises is embodied in the following: for cross-regional supply chain enterprises fuzzy game $\mathcal{v}$ and ${\mathcal{e}}^{\mathcal{N}}$ the distribution vectors in $\mathcal{x}$ and its subset of the nonempty fuzzy coalition of cross-regional supply chain enterprises $\mathcal{t}={\mathcal{e}}^{\mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right)}$ and $\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})\subseteq \mathcal{N}$. In this way, we will try to form “our own game” with the help of $\mathcal{t}={\mathcal{e}}^{\mathcal{N}}$.$(\mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right),{\mathcal{v}}_{\mathcal{x}}^{\mathcal{t}})$, at which point the fuzzy coalition of cross-regional supply chain enterprises $\mathcal{s}$ will try to find partners from the ${\mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right)}^{\mathcal{c}}=\mathcal{N}\backslash \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right)$ fuzzy alliance of cross-regional supply chain enterprises, such that $\mathcal{s}$ will try to cooperate with the fuzzy alliance with a certain degree of participation $\mathcal{q}.$ In the case of $\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{q})\subseteq \mathcal{N}\backslash \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})$, the benefit gained from the cooperation is $\mathcal{v}\left(\mathcal{s}\vee \mathcal{q}\right)$. After retaining a portion of the benefits for the purpose of redevelopment, it is written as $\underset{\mathcal{j}\in \mathcal{car}(\mathcal{s})\cup \mathcal{car}(\mathcal{q})}{\mathcal{max}}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\vee \mathcal{q}\right).$ At the same time, the cost must be paid to the cross-regional supply chain partner $\mathcal{x}\left(\mathcal{q}\right)=\underset{\mathcal{j}\in \mathcal{car}(\mathcal{q})}{\mathcal{max}}{\mathcal{q}}_{\mathcal{j}}{\mathcal{x}}_{\mathcal{j}}$ after which cooperation is possible. Therefore, the difference is the most attractive for the fuzzy alliance of cross-regional supply chain enterprises $\mathcal{s}$, which can be solved by ${\mathcal{v}}_{\mathcal{x}}^{\mathcal{t}}\left(\mathcal{s},\mathcal{c}\right)$ by solving for the maximum difference to find the best partner in the cross-regional supply chain.

Meanwhile, the generalized fuzzy reduced game of cross-regional supply chain enterprises also has the following two properties.

(1)

If the original fuzzy game payoff value of cross-regional supply chain enterprise $\mathcal{v}\ge 0$ then the value of its generalized fuzzy reduced game $v_x^{\mathcal{t}}\left(\mathcal{s},\mathcal{c}\right)\ge 0,\forall \mathcal{s}\le \mathcal{t}$

(2)

The value of the maximum coalition of the generalized fuzzy reduced game for cross-regional supply chain enterprises is ${\mathit{max}}_{\phi \subseteq \mathcal{car}\left(\mathcal{q}\right)\subseteq N\backslash \mathcal{car}\left(\mathcal{t}\right)}\{\underset{\mathcal{j}\in \mathcal{car}\left(\mathcal{t}\right)\cup \mathcal{car}\left(\mathcal{q}\right)}{\mathcal{max}}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}(\mathcal{t}\vee \mathcal{q})-{\sum }_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{q})}{\mathcal{q}}_{\mathcal{j}}\cdot x_{\mathcal{j}}\}$.

It is proven that there is a commutation relationship between the restricted generalized fuzzy reduced game and the original generalized fuzzy reduced game return value of cross-regional supply chain enterprises with the help of the following primer, and the transformation of the stabilized cooperation game pattern of cross-regional supply chain enterprises is realized via the commutation relationship.

Lemma 1. 

It is hereby decreed that $\mathcal{v}$ is a generalized fuzzy reduced game for cross-regional supply chain enterprises with distribution vectors $\mathcal{r}\in R^n$ or any cooperative alliance of more than two enterprises in a cross-regional supply chain, i.e., · $|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})|\ge 2$ and $\mathcal{s}=e^{\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}\in F^{\mathcal{N}}$ and $\mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})$ then, the value of their gains has the following conversion relation:

$$[v_x^{\mathcal{s}}{]}_{x|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}^{\mathcal{s}-({\mathcal{s}}_i)}(\mathcal{r},\mathcal{c})={max}_{\phi \subseteq \mathcal{c}\mathcal{a}\mathcal{r}\left({\mathcal{q}}^{\prime }\right)\subseteq \left\{\mathcal{i}\right\},\mathcal{k}\subseteq \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{r}\right)\cup \mathcal{c}\mathcal{a}\mathcal{r}\left({\mathcal{q}}^{\prime }\right)}{\mathcal{c}}_{\mathcal{k}}·{\mathcal{v}}_x^{\mathcal{s}-({\mathcal{s}}_i)}(\mathcal{r},\mathcal{c})$$

Wherein, ${\alpha }_{|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}$ denotes the vector of contribution values restricted to the set of cross-regional supply chain cooperative enterprises $\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})\subseteq \mathcal{N}$. The vector of contribution values on $\alpha$ is denoted by ${(\mathcal{s}}_{\mathcal{i}})$. The fuzzy cooperation game in the cross-regional supply chain $\mathcal{n}$ dimension unit hypercubic space is denoted by ${\mathcal{F}}^{\mathcal{N}}$ of a nonempty fuzzy coalition in the cross-regional supply chain fuzzy cooperation game, and its first $\mathcal{i}$ component of the fuzzy coalition ${\mathcal{s}}_{\mathcal{i}}>0$ and all other components are zero. For any $\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{d})\subseteq \mathcal{N}$, ${\mathcal{v}}_{\mathcal{x}|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{d})}^{\mathcal{t}}$ represents the cross-regional supply chain enterprises restricted to $\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{d})$ on the generalized fuzzy reduced game, i.e., ${\mathcal{v}}_{\mathcal{x}|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{d})}^{\mathcal{t}}(\mathcal{s},\mathcal{c})$ = ${\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathrm{\varnothing }\subseteq \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{q})\subseteq \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{d})\backslash \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}\{\underset{\mathcal{j}\in \mathcal{car}(\mathcal{s})\cup \mathcal{car}(\mathcal{q})}{\mathcal{max}}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\vee \mathcal{q}\right)-\sum _{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{q})}{\mathcal{q}}_{\mathcal{j}}·{\mathcal{x}}_{\mathcal{j}}\}$, $\forall {\mathcal{e}}^{\mathrm{\varnothing }}\ne \mathcal{s}\le \mathcal{t}$.

Proof of Lemma 1 is attached in Appendix A.

The generalized fuzzy reduced game formed by the cooperation of cross-regional supply chain enterprises is fuzzy mean monotonic if the cross-regional supply chain enterprises participate in the fuzzy coalition with a certain level of participation to obtain more generalized gains than their generalized proportional distributions, which is now proved by the following proposition.

Proposition 2. 

Let $\mathcal{v}$ be the vector of contribution values of cross-regional supply chain enterprises with respect to $\alpha$ of a fuzzy average monotonic game if there are distribution vectors $x\in R^{\mathcal{N}}$ and $i\in \mathcal{N}$, such that $q_i\cdot x_i\ge {\alpha }_i\cdot \frac{\mathcal{r}\cdot v\left(e^{\mathcal{N}}\right)}{\alpha \left(e^{\mathcal{N}}\right)}, \forall q\in {\mathcal{F}}^{\mathcal{N}}$ $e^{\phi }<q<e^i$ and ${0<\mathcal{r}=\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{\mathcal{c}}_{\mathcal{i}}\le 1$; then,

(1) 

If the cross-regional supply chain is restricted to $\mathcal{N}\backslash \{i\}$ the vector of enterprise contribution values is zero ${\alpha }_{|\mathcal{N}\backslash \{i\}}=0$, then the value of the generalized fuzzy reduced game for enterprises in the cross-regional supply chain is zero, i.e., ${\mathcal{v}}_x^{e^{\mathcal{N}}-e^i}(\mathcal{s},\mathcal{c})=0$. $\forall \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{s}\right)\subseteq \mathcal{N}\backslash \{i\}$.

(2) 

If the cross-regional supply chain is restricted to $\mathcal{N}\backslash \{i\}$, a nonzero vector of enterprises’ contribution values on ${ \alpha }_{|\mathcal{N}\backslash \{i\}}\ne 0,$ then the generalized fuzzy reduced game of cross-regional supply chain enterprises $v_x^{e^{\mathcal{N}}-e^i}(\mathcal{s},\mathcal{c})$ is a fuzzy average monotonic game on ${\alpha }_{|\mathcal{N}\backslash \{i\}}$.

Proof of Proposition 2 is attached in Appendix A.

Proposition 2 shows that a generalized fuzzy simplifying game formed by the cooperation of cross-regional supply chain enterprises is fuzzy-averaged monotonic if the distribution vectors satisfy ${ q}_i\cdot x_i\ge {\alpha }_i\cdot \frac{\mathcal{r}\cdot v\left(e^{\mathcal{N}}\right)}{\alpha \left(e^{\mathcal{N}}\right)}$; then, the generalized fuzzy simplifying game is fuzzy mean monotonic.

5. A Generalized Fuzzy Bargaining Set Gain Redistribution Scheme for Cross-Regional Supply Chain Enterprise Alliances

Definition 5. 

On the vector of contribution values of cross-regional supply chain enterprises $\alpha$ of a fuzzy average monotonic game $\mathcal{v}$ that makes its vector of pre-distribution outside the core be $x\in {\mathrm{I}}^{\ast }\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$, the fuzzy coalition of supply chain enterprises $\mathcal{s}$ with respect to the $\mathcal{x}$ generalized fuzzy supremum is

$$\mathrm{e}\left(\mathcal{s},\mathcal{x},\mathcal{c}\right)={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\right)-{\sum }_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}{\mathcal{s}}_{\mathcal{j}}{\cdot \mathcal{x}}_{\mathcal{j}}(8-1)\forall \mathcal{s}\le {\mathrm{e}}^{\mathcal{N}}$$

(8a)

The maximum fuzzy cooperative alliance with maximum generalized fuzzy excess for cross-regional supply chain enterprises $\mathcal{t}\le {\mathrm{e}}^{\mathcal{N}}$ satisfaction is as follows:

$$\begin{gathered} \mathrm{e}\left(\mathcal{t},\mathcal{x},\mathcal{c}\right)={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{t}\right)-\sum _{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}{\mathcal{t}}_{\mathcal{j}}{\cdot \mathcal{x}}_{\mathcal{j}} \\ \ge {\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{s}\right)}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}\left(\mathcal{s}\right)-\sum _{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{s}\right)}{\mathcal{s}}_{\mathcal{j}}{\cdot \mathcal{x}}_{\mathcal{j}} \\ =\mathrm{e}(\mathcal{s},\mathcal{x},\mathcal{c}),\forall \mathcal{s}\le {\mathrm{e}}^{\mathcal{N}} \end{gathered}$$

(8b)

$$\mathrm{e}\left(\mathcal{t},\mathcal{x},\mathcal{c}\right)>\mathrm{e}(\mathcal{s},\mathcal{x},\mathcal{c}),\forall \mathcal{t}<\mathcal{s}$$

(8c)

Definition 6. 

For cross-regional supply chain enterprise alliances generalized fuzzy $\mathcal{M}\mathcal{a}\mathcal{s}-\mathcal{C}\mathcal{o}\mathcal{l}\mathcal{e}\mathcal{l}\mathcal{l}$, the bargaining set is in terms of the fuzzy alliance perspective of supply chain enterprises, and in contrast to the objections of other bargaining sets, only two fuzzy alliances of cross-regional supply chains are addressed, $with \mathcal{s}$ and $\mathcal{t}$ no longer specifying specific cooperative enterprises. Therefore, the formation conditions of fuzzy objections become weaker, but the formation conditions of fuzzy counterobjections will be enhanced.

Let order $\mathcal{x}$ be the fuzzy cooperation game of cross-regional supply chain enterprises and $\mathcal{v}$ be the generalized benefit distribution vector for $\mathcal{x}.$ The generalized fuzzy objection is a pair $\left(\mathcal{y},\mathcal{s}\right),$ where $\mathcal{s}$ is the fuzzy coalition of cross-regional supply chain enterprises, and $\mathcal{y}$ is the distribution vector, which is satisfied as follows:

$${\sum }_{\mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}{\mathcal{s}}_{\mathcal{i}}{\cdot \mathcal{y}}_{\mathcal{i}}={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}(\mathcal{s})$$

(9a)

$${\mathcal{s}}_{\mathcal{i}}{\cdot \mathcal{y}}_{\mathcal{i}}\ge {\mathcal{s}}_{\mathcal{i}}{\cdot \mathcal{x}}_{\mathcal{i}}(9-2)\forall \mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})$$

(9b)

where Equation (9b) holds for at least one strictly greater than the inequality. In response to the generalized fuzzy objection $(\mathcal{y},\mathcal{s})$, supply chain enterprises $\mathcal{l}$, in response to the enterprises $\mathcal{k}$, may take corresponding resistance measures, e.g., the $\mathcal{l}$ may organize a fuzzy coalition that does not contain $\mathcal{k}$ fuzzy coalition that does not contain $\mathcal{z}$. If a generalized benefit sharing vector $\mathcal{t}$ is brought in, $\mathcal{t}\in {\mathcal{F}}^{\mathcal{N}}$, ${\mathcal{t}}_{\mathcal{k}}>0$, $\mathrm{a}\mathrm{n}\mathrm{d} {\mathcal{t}}_{\mathcal{l}}=0$ in the fuzzy coalition $\mathcal{z}.$ The generalized benefit distribution vector of each enterprise in $\mathcal{t}$ is the component of the generalized benefit distribution vector of the enterprises from $\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})$ members in the fuzzy coalition, which is simultaneously satisfied:

$${\mathcal{t}}_{\mathcal{i}}{\cdot \mathcal{z}}_{\mathcal{i}}\ge {\mathcal{t}}_{\mathcal{i}}{\cdot \mathcal{x}}_{\mathcal{i}},\forall \mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right)\backslash \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})$$

(9c)

$${\mathcal{t}}_{\mathcal{i}}{\cdot (\mathcal{z}}_{\mathcal{i}}-{\mathcal{x}}_{\mathcal{i}})\ge {\mathcal{t}}_{\mathcal{i}}{\cdot ({\mathcal{y}}_{\mathcal{i}}-\mathcal{x}}_{\mathcal{i}})(9-4)\forall \mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{t}\right)\cap \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})$$

(9d)

$${\sum }_{\mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}{\mathcal{t}}_{\mathcal{i}}{\cdot \mathcal{z}}_{\mathcal{i}}={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{j}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}{\mathcal{c}}_{\mathcal{j}}·\mathcal{v}(\mathcal{t})$$

(9e)

It is clear that the cross-regional supply chain fuzzy alliance $\mathcal{z}$ across a regional supply chain will not realize fewer broad benefits than they would if they cooperated with the alliance $\mathcal{s},$ and for those enterprises that are involved in both the alliance $\mathcal{s}$ and the fuzzy alliance $\mathcal{t},$ the generalized gains of the enterprises in the fuzzy alliance are greater than the generalized gains of the enterprises in the fuzzy alliance $\mathcal{s}$. The binary couple is the supply chain enterprises’ counterargument to the generalized benefits. $(\mathcal{t},\mathcal{z})$ is the generalized fuzzy counterobjection formed by the supply chain enterprises to refute the generalized fuzzy objection. $(\mathcal{y},\mathcal{s})$ and the generalized fuzzy counterobjection is formed.

In a fuzzy average monotonic game with cross-regional supply chain enterprises $\mathcal{v}$, the generalized benefit sharing vector $\mathcal{x}$ as a bargaining point of enterprises $\mathcal{k}$ in response to $\mathcal{l}$, any generalized fuzzy objections about $\mathcal{x},$ $\left(\mathcal{y},\mathcal{s}\right)$, the enterprise, the enterprise will be given a $\mathcal{l}$ against $\mathcal{k},$ the generalized fuzzy counterobjection $\left(\mathcal{t},\mathcal{z}\right),$ and then the fuzzy mean monotonic game of cross-regional supply chain enterprises $\mathcal{v}$ of bargaining points is called the generalized fuzzy bargaining set of cross-regional supply chain enterprises $\mathcal{M}\mathcal{a}\mathcal{s}-\mathcal{C}\mathcal{o}\mathcal{l}\mathcal{e}\mathcal{l}\mathcal{l}$, described as

$${MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)=\{\mathcal{x}\in I^{\ast }(\mathcal{v},\mathcal{r},\mathcal{c})|\mathrm{E}\mathrm{a}\mathrm{c}\mathrm{h} \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d} \mathrm{f}\mathrm{u}\mathrm{z}\mathrm{z}\mathrm{y} \mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathcal{x} \mathrm{h}\mathrm{a}\mathrm{s} \mathrm{a} \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d} \mathrm{f}\mathrm{u}\mathrm{z}\mathrm{z}\mathrm{y} \mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\}$$

(9f)

In generalized fuzzy bargaining for cross-regional supply chain enterprises $\mathcal{M}\mathcal{a}\mathcal{s}-\mathcal{C}\mathcal{o}\mathcal{l}\mathcal{e}\mathcal{l}\mathcal{l}$, the concept of bargaining set portrays that for a fuzzy average game formed by the cooperation of cross-regional supply chain enterprises, even if its generalized fuzzy core is empty, its generalized fuzzy $\mathcal{M}\mathcal{a}\mathcal{s}-\mathcal{C}\mathcal{o}\mathcal{l}\mathcal{e}\mathcal{l}\mathcal{l}$ bargaining set contains a generalized benefit distribution vector other than the generalized fuzzy core, i.e., the generalized fuzzy bargaining set of cross-regional supply chain enterprises $\mathcal{x}\left(\mathcal{v}\right).$ The nonempty nature of the generalized fuzzy bargaining set of cross-regional supply chain enterprises suggests that the nonemptiness of the generalized fuzzy $\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{e}\mathrm{t} \mathcal{M}\mathcal{a}\mathcal{s}-\mathcal{C}\mathcal{o}\mathcal{l}\mathcal{e}\mathcal{l}\mathcal{l}$ indicates the existence of its generalized optimal benefit distribution scheme [26,27].

The above maximum fuzzy cooperative coalition of cross-regional supply chain enterprises with maximum generalized fuzzy supremacy can be used to construct a generalized fuzzy reduced game on the subject of $\mathcal{x}$ generalized fuzzy objections. It can also be used to prove whether the generalized fuzzy reduced game of cross-regional supply chain enterprises satisfies the fuzzy mean monotonic property.

Proposition 3. 

Let $\mathcal{v}$ be the cross-regional supply chain enterprises on the contribution value vector $\mathsf{\alpha }$ of the fuzzy mean monotonic game, with $x$ being the distribution within the generalized predistribution set but not in the generalized fuzzy core, i.e., $x\in I^{*}(\mathcal{v},\mathcal{r},\mathcal{c})\backslash C(\mathcal{v},\mathcal{r},\mathcal{c})$, and let $\mathcal{t}$ be the maximum fuzzy coalition of cross-regional supply chain enterprises with maximum generalized fuzzy supremum, then

(1) 

If the vector of contribution values of cross-regional supply chain enterprises is zero, ${\alpha }_{|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}=0$, then the generalized fuzzy reduced game value of cross-regional supply chain enterprises is zero, ${\mathcal{v}}_{\mathcal{x}}^{\mathcal{t}}\left(\mathcal{s},\mathcal{c}\right)=0$

(2) 

If the vector of contribution values of cross-regional supply chain enterprises is nonzero, ${\alpha }_{|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}\ne 0$ then the generalized fuzzy reduced game of cross-regional supply chain enterprises ${\mathcal{v}}_x^{\mathcal{t}}\left(\mathcal{s},\mathcal{c}\right)$ is a fuzzy average monotonic game about ${\alpha }_{|\mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{t})}$.

Proof of Proposition 3 is attached in Appendix A.

Proposition 3 shows the relationship between the generalized fuzzy reduced game and the fuzzy average monotonic game for the vector of contribution values of cross-regional supply chain enterprises under various conditions.

6. Equivalence Analysis of Generalized Fuzzy Bargaining Sets and Generalized Fuzzy Core Redistribution Schemes for Cross-Regional Supply Chain Enterprise Alliances

As an increasing number of cooperative game patterns are formed during the alliance formation process of cross-regional supply chain enterprises, the optimal benefit distribution scheme does not exist under each cooperative game pattern, especially under the current fuzzy cooperation mode. The fuzzy characteristics and game process presented in the cooperation of cross-regional supply chain enterprises are becoming increasingly complicated from the generalized fuzzy bargaining sets ${\mathrm{M}\mathrm{C}}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)$. From the perspective of equivalence between the generalized fuzzy bargaining set and generalized fuzzy core $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$, the existence of a generalized fuzzy core in the fuzzy mean monotonic game of cross-regional supply chain enterprises can be better portrayed in the equilibrium of the fuzzy mean monotonic game of cross-regional supply chain enterprises to realize the stability and continuity of the alliance structure of cross-regional supply chain enterprises.

Theorem 2. 

For the fuzzy mean monotonic game of cross-regional supply chain enterprises $\mathcal{v}$ and its generalized fuzzy bargaining set $\mathcal{M}\mathcal{a}\mathcal{s}-\mathcal{C}\mathcal{o}\mathcal{l}\mathcal{e}\mathcal{l}\mathcal{l}$, ${MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)$ is equal to the generalized fuzzy core $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$. The steps for demonstrating the equivalence of its redistribution scheme are shown in Figure 1.

Step 1: Proof of generalized fuzzy bargaining set distribution ${MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)$ contains generalized fuzzy core distribution $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$, i.e., $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)\subseteq {MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)$.$\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)\subseteq {MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)$ is obvious because from their respective definitions, it can be concluded that there is no generalized fuzzy objection to any generalized fuzzy core element.

Step 2: Prove that the generalized fuzzy core distribution $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$ includes the generalized fuzzy bargaining set distribution ${MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)$, i.e., ${MC}_{\mathcal{F}}\left(\mathcal{v},\mathcal{c}\right)\subseteq \mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$.

The proof of Theorem 2 is attached in Appendix A.

Theorem 2 shows that the distribution vectors outside the generalized fuzzy core of cross-regional supply chain enterprises are not in the generalized fuzzy bargaining set, i.e., for the fuzzy average monotonic game of cross-regional supply chain enterprises, the generalized fuzzy core and the generalized fuzzy bargaining set of redistribution schemes are equal. In other words, if the fuzzy average monotonic game pattern is formed in the cooperation of cross-regional supply chain enterprises, the optimal generalized redistribution scheme can be found, and its alliance structure is stable, thus realizing the sustainable development of the supply chain.

7. Numerical Simulation

The most basic conditions for selecting alliance members are a rational analysis of interests and the degree of resource participation among alliance members. In addition, the cooperative game pattern formed by alliance members and the bargaining equilibrium under different patterns are indicators that decision-makers need to consider. For example, the establishment of a 5G alliance reflects the adherence to the basic principles of cooperation and benefit distribution among alliance members, as well as the application of cooperative game models. However, in addition to considering these key elements, the selection of alliance members should also consider the characteristics of the alliance itself, the factors of sustained cooperation, and the redevelopment of the alliance. Therefore, the concepts and theories proposed in this paper are more suitable for application in 5G alliances and more real-world case studies in uncertain environments.

Now, order $\mathcal{N}=\{Q_1,Q_2,Q_3,Q_4\}$ denotes those four enterprises from different regional supply chains. To realize cross-regional collaborative innovation R&D projects, if each enterprise studies and develops individually, it will surely invest a lot of costs relying on its own core resources to obtain elements, structural synergies, functional enhancement, etc., and may not be able to obtain the benefits, but the total benefits brought by the fuzzy cooperation from different scales will be much larger than those obtained by the separate actions. Therefore, these four cross-regional supply chain enterprises form a cooperative alliance and try to provide resource conditions for the redistribution of benefits and supply chain redevelopment based on cooperation. The process of each cooperative enterprise using its own resources to form an alliance for optimization and integration is the fuzzy mean monotonic game $\mathcal{v}.$ The formation process of each cooperative enterprise and alliance benefit distribution problem is the game model solution process. Among them, the value of the characteristic function (unit: million dollars) of the gain obtained by each enterprise when it studies and develops alone is zero, and the gain obtained by the two enterprises when they cooperate in R&D is zero $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)=\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)=\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)=\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)=0.$ The value of the characteristic function of each enterprise’s gain when it cooperates in R&D is zero, and the value of the characteristic function of its gain when it cooperates in R&D is zero, $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2\right\}}\right)=\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=0$ The value of the characteristic function of the benefit when two companies cooperate in R&D is zero, $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3\right\}}\right)=20$. $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)=25$. $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=30$. The values of the characteristic function of the benefit obtained when three enterprises cooperate in R&D are $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=30$, $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=40$, $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=35$, and $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=45$, respectively; the value of the eigenfunction of the gain obtained when four enterprises work together is $\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=55$. The fuzzy average monotonic game collaborative innovation cooperative alliance of cross-regional supply chain enterprises is shown in Figure 2.

It can be verified that this cross-regional supply chain enterprise fuzzy game satisfies the superadditivity property, i.e., the cooperative enterprises $Q_1,Q_2,Q_3,Q_4$ gained by forming any fuzzy cooperative alliance when cooperating with participation degree 1 is not less than that gained when they act alone, and the comparison process is shown in

Table 1. Comparison of benefits of each fuzzy cooperative alliance of cross-regional supply chain enterprises.

For Enterprise Q1	For Enterprise Q2	For Enterprise Q3	For Enterprise Q4
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)=0 \\ \le \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2\right\}}\right)=0 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)=0 \\ \le \mathcal{v}\left({\mathrm{e}}^{\left\{,{\mathrm{Q}}_2,{\mathrm{Q}}_1\right\}}\right)=0 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_3,\mathrm{Q}}_1\right\}}\right)=20 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_4,\mathrm{Q}}_1\right\}}\right)=25 \end{gathered}$$
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)=0 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}<\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3\right\}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=20 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=25 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_3,\mathrm{Q}}_2\right\}}\right)=25 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)=0 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}<\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4,{\mathrm{Q}}_2\right\}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=30 \end{gathered}$$
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)=0< \\ \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)=25 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=30 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)=0 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)=0 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4,{\mathrm{Q}}_3\right\}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0 \end{gathered}$$
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=25 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=30 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3\right\}}\right)=20 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_1,{\mathrm{Q}}_3\right\}}\right)=30 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_3,\mathrm{Q}}_1,{\mathrm{Q}}_2\right\}}\right)=30 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_4,\mathrm{Q}}_1,{\mathrm{Q}}_2\right\}}\right)=35 \end{gathered}$$
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=30 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=35 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)= \\ 25<\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)= \end{gathered}$$ 35	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)=25 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_3,\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)=40 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3\right\}}\right)=20 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_4,\mathrm{Q}}_1,{\mathrm{Q}}_3\right\}}\right)=40 \end{gathered}$$
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=40 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=0 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=45 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=30 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_3,\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=45 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=25 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_4,\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=45 \end{gathered}$$
$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=45 \\ <\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=55 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right) \\ =40<\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_2,\mathrm{Q}}_1,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=55 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_3\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right) \\ =35<\mathcal{v}\left({\mathrm{e}}^{\left\{{{\mathrm{Q}}_3,\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_4\right\}}\right)=55 \end{gathered}$$	$$\begin{gathered} \mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4\right\}}\right)+\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right) \\ =30<\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_4,{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=55 \end{gathered}$$

.

Based on the above vector of cross-regional supply chain enterprise alliance returns and relative contribution values of enterprises $\alpha =(\mathrm{0.25,0.5,0.75,1})$ all fuzzy cooperative alliances $\mathcal{s}\le \mathcal{t}\le e^{\mathcal{N}}$ can be made to satisfy $\alpha (\mathcal{t})\cdot \mathcal{v}(\mathcal{s})\le \alpha (\mathcal{s})\cdot \mathcal{v}(\mathcal{t})$; i.e., the fuzzy average monotonic game of cross-regional supply chain enterprise cooperative innovation can be constructed. If the total gain value of its game is partially retained, i.e., the distribution coefficient is adjusted according to the participation degree of each cooperative enterprise in the cross-regional supply chain, for example, to find the vector $\mathcal{c}=\{{\mathcal{c}}_1,{\mathcal{c}}_2,{\mathcal{c}}_3,{\mathcal{c}}_4\}$ that satisfies $0\le {\mathcal{c}}_{\mathcal{i}}\le \mathrm{1,1}\le \mathcal{i}\le \mathcal{n}$, and $\mathcal{i}={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{\mathcal{c}}_{\mathcal{i}}$, such that the generalized fuzzy core solution of the fuzzy average monotonic game of cross-regional supply chain enterprises $\mathrm{C}\left(\mathcal{v},\mathcal{r},\mathcal{c}\right)$ is nonempty, one can find the vector that satisfies ① $\sum _{\mathcal{i}\in \mathcal{N}}{\mathcal{x}}_{\mathcal{i}}=\mathcal{r}·\mathcal{v}\left({\mathrm{e}}^{\mathcal{N}}\right),\mathcal{r}\ge {\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{\mathcal{c}}_{\mathcal{i}}$ and ② ${\mathcal{x}}_{\mathcal{i}}\ge {\mathcal{c}}_{\mathcal{i}}·\mathcal{v}\left({\mathrm{e}}^{\mathcal{i}}\right) \forall \mathcal{i}\in \mathcal{N}$ of the generalized optimal redistribution scheme. The existing adjustment distribution vector $\left(\mathrm{0.6,0.3,0.3,0.2}\right)$ is ${\mathcal{c}}_1=0.6$, ${\mathcal{c}}_2=0.3, {\mathcal{c}}_3=0.3, {\mathcal{c}}_4=0.2, \mathcal{r}={\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{N}}{,\mathrm{a}\mathrm{n}\mathrm{d} \mathcal{c}}_{\mathcal{i}}=0.6$, where ${\mathcal{c}}_{\mathcal{i}}$ represents the cross-regional supply chain cooperative enterprises $Q_{\mathcal{i}}$ The distribution coefficient of each fuzzy alliance generalized gain of cross-regional supply chain enterprises = ${\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \mathcal{c}\mathcal{a}\mathcal{r}(\mathcal{s})}{\mathcal{c}}_{\mathcal{i}}\times$ the fuzzy alliance gains, $\mathcal{c}\mathcal{a}\mathcal{r}\left(\mathcal{s}\right)\subset \{Q_1,Q_2,Q_3,Q_4\}$, retaining $\left(1-\mathcal{r}\right)·\mathcal{v}\left({\mathrm{e}}^{\mathcal{N}}\right)=0.4\times 55=22$ million dollars of gains not distributed for supply chain enterprise alliance redevelopment. From the definition of the generalized fuzzy core solution, it can be proved that the generalized distribution under the fuzzy average monotonic game of each cooperative enterprise in the supply chain ${\mathcal{x}}_1=16.5$, ${\mathcal{x}}_2=6.6$, ${\mathcal{x}}_3=6.6$, and ${\mathcal{x}}_4=3.3$ is included in the generalized fuzzy core solution, and the test process is

$${\mathcal{x}}_1+{\mathcal{x}}_2+{\mathcal{x}}_3+{\mathcal{x}}_4=33$$

at the same time

$${\mathcal{x}}_1+{\mathcal{x}}_2+{\mathcal{x}}_3=29.7>{\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \{Q_1,Q_2,Q_3\}}{\mathcal{c}}_{\mathcal{i}}·\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3\right\}}\right)=18$$

$${\mathcal{x}}_1+{\mathcal{x}}_3+{\mathcal{x}}_4=26.4>{\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \{Q_1,Q_3,Q_4\}}{\mathcal{c}}_{\mathcal{i}}·\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_1,{\mathrm{Q}}_3,{\mathrm{Q}}_4\right\}}\right)=24$$

$${\mathcal{x}}_2+{\mathcal{x}}_1+{\mathcal{x}}_4=26.4>{\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \{Q_2,Q_1,Q_4\}}{\mathcal{c}}_{\mathcal{i}}·\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,{\mathrm{Q}}_1,{\mathrm{Q}}_4\right\}}\right)=21$$

$${\mathcal{x}}_2+{\mathcal{x}}_3+{\mathcal{x}}_4=16.5>{\mathcal{m}\mathcal{a}\mathcal{x}}_{\mathcal{i}\in \{Q_2,Q_3,Q_4\}}{\mathcal{c}}_{\mathcal{i}}·\mathcal{v}\left({\mathrm{e}}^{\left\{{\mathrm{Q}}_2,Q_3,{\mathrm{Q}}_4\right\}}\right)=13.5$$

In addition, ${\mathcal{x}}_1\ge 0$, ${\mathcal{x}}_2\ge 0$, ${\mathcal{x}}_3\ge 0,$ and ${\mathcal{x}}_4\ge 0$. Therefore, the generalized fuzzy core solution of the fuzzy average monotonic game formed by the four cooperative enterprises in the cross-regional supply chain is nonempty, indicating that the generalized optimal redistribution scheme for supply chain redevelopment can be found. The existence of a generalized fuzzy core solution is a guarantee of the stability of cross-regional supply chain enterprise alliances, and the fuzzy average monotonic game between cross-regional supply chain enterprises has an equivalent relationship between the generalized fuzzy core and the generalized fuzzy bargaining set. The Shapley value in reference [45] is used as a solution tool to solve the problem of supply chain enterprise $\mathcal{i}$ based on participation degree ${\mathcal{s}}_{\mathcal{i}}=1,\mathcal{i}=\mathrm{1,2},\mathrm{3,4}$. The equilibrium results of forming the largest alliance when participating in the alliance are shown in Figure 3. In contrast, when cross-regional supply chain enterprises form the largest cooperative alliance in a fuzzy average monotonic game pattern, the fuzzy bargaining set distribution in this paper can achieve the overall optimization of the largest alliance. The comparison of distribution schemes is shown in Figure 4.

The stability of cross-regional supply chain enterprise alliances is guaranteed via the existence of generalized fuzzy core solutions, Moreover, the generalized fuzzy core of the fuzzy mean monotonic game and the generalized fuzzy bargaining set of the cross-regional supply chain enterprises have an equivalence relationship, compared to the Shapley value as a solution tool in reference [45], When cross-regional supply chain enterprises form the largest cooperative alliance under the fuzzy average monotonic game pattern, the fuzzy bargaining set distribution in this paper can achieve the overall optimization of the largest alliance. Coordinating and cooperating via bargaining is more reasonable than the original classical Shapley value and can be better applied to practical cooperation environments. It can also motivate all members of the alliance to cooperate based on the uncertainty characteristics of the environment. At the same time, the optimal redistribution scheme of any of the generalized fuzzy game solutions can be selected according to the actual needs, and the cooperative participation degree and redistribution parameters can be adjusted at the same time according to the actual needs ${\mathcal{c}}_{\mathcal{i}}$ and redistribution parameter $\mathcal{r}.$ It can also adjust the values of the cooperative participation degree and redistribution parameters according to the actual needs. From the perspective of sustainable “environmental” pillar research, achieving cross-regional supply chain enterprises to cooperate with various alliances with different degrees of participation, and retaining some benefits for sustainable development of the supply chain from the perspective of sustainable “economic” pillar research, can better ensure the effectiveness and stable sustainable development of cooperative alliances, and provide decision-making basis for solving the problem of cross-regional supply chain enterprise cooperation and benefit redistribution in uncertain and complex environments.

8. Conclusions

8.1. Theoretical Results

In an uncertain environment, cross-regional supply chain enterprises have shown a fuzzy characteristic of cooperation and a sustained demand for re-cooperation. This paper uses the fuzzy average monotonic cooperative game and its generalized fuzzy bargaining set solution and generalized fuzzy core solution to study the sustainable cooperation strategies of cross-regional supply chain enterprises in an uncertain environment. Not only does it meet the actual needs of cross-regional supply chain enterprises to participate in cooperation with partial resources from the perspective of the “environmental” pillar of supply chain enterprise management sustainability, but it also achieves a strategy of retaining partial benefits for the redevelopment of supply chain alliances from the perspective of the “economic” pillar of supply chain enterprise management sustainability, the main theoretical results are as follows:

(1)

Considering the fuzzy characteristics of cross-regional supply chain enterprise cooperation and the contribution behavior of supply chain enterprises, we constructed a fuzzy average monotonic cooperation game model of cross-regional supply volume enterprise alliance and investigated various contribution vectors of supply volume enterprises in the alliance cooperation $\mathsf{\alpha }.$ The fuzzy average monotonic game formed by the alliance of supply volume enterprises implies that the average return value of their alliance on vector $\mathsf{\alpha }$ increases with the size of the coalition.

(2)

The construction of the generalized fuzzy reduced game of cross-regional supply chain enterprises reflects the strategy of supply chain enterprises to find the best partners in cross-regional cooperation because only the difference of the alliance game is the most attractive for cross-regional supply chain enterprise alliances, and at the same time, the generalized fuzzy reduced game and the fuzzy average monotonic game also have a certain conversion relationship.

(3)

Under the fuzzy cooperation mode of cross-regional supply chain enterprises, an increasing number of cooperative game patterns are formed, but not every cooperative game pattern has an optimal benefit distribution scheme. The generalized fuzzy bargaining set distribution scheme of the fuzzy average monotonic game of cross-regional supply chain enterprises is investigated, which accounts for the degree of cooperative participation and meets the demand of retaining a portion of the total benefit value of the game for sustainable development.

(4)

The equivalence of the generalized fuzzy core distribution scheme and the generalized fuzzy bargaining set distribution scheme of the fuzzy average monotonic game of the cross-regional supply chain enterprises is studied, and its validity is verified by the arithmetic example, in which the appropriate generalized optimal redistribution scheme can be selected according to the actual needs.

8.2. Managerial Implications

The above theoretical results provide the following important management insights, consider the “environmental” pillar of supply chain enterprise management sustainability, mechanisms for supply chain enterprises to cooperate with multiple alliances can be established, the long-term and stable partnership of supply chain enterprise alliances can be constructed. Considering the “economic” pillar of supply chain enterprise management sustainability, a benefit distribution mechanism suitable for the sustainable development of supply chain enterprise alliances can be proposed.

(1)

Establish a mechanism for supply chain enterprises to cooperate with multiple alliances. The idea of a classical game in the application of a supply chain enterprise alliance is embodied as follows: (i) Participants are fully involved in a specific alliance, i.e., participants either participate in a certain alliance or do not participate in a certain alliance, and there is no situation in which participants participate in a certain alliance with a certain degree of participation. (ii) Participants are fully aware of the benefits of various cooperation strategies and the distribution of their participation in a particular coalition before they cooperate. However, in reality, more often than not, in the face of many uncertainties, enterprises participate in multiple supply chain alliances with different participation levels or participation rates, and they are not sure or even aware of the benefits of different cooperation strategies and their respective distributions in a particular alliance before cooperation. Therefore, it is necessary to use the distributional solution of the fuzzy game of supply chain enterprise alliances to describe the uncertain phenomenon via the degree of affiliation or participation, to portray the quantitative relationship in the real problem more reasonably and to provide a powerful analytical tool to deal with the uncertain phenomenon.

(2)

Establish long-term stable partnerships of supply chain enterprise alliances. The total benefit value of the supply chain enterprise alliance should be distributed fairly and reasonably so that all participating enterprises can gain from successful cooperation. A long-term stable partnership is based on fair benefit sharing and risk sharing. Therefore, to establish a long-term and stable partnership, enterprises can cooperate with multiple supply chain alliances at the same time with different degrees of participation, and at the same time, they can partially retain the total benefit value of the supply chain alliance, which is conducive to its sustainable development.

(3)

Construct a benefit distribution mechanism suitable for the sustainable development of supply chain enterprise alliances. Most of the research carried out by the previous researchers on the cooperative game is in the sense of a traditional solution to distribute all the cooperative game benefits to the participants at one time, but to ensure the sustainable development of the supply chain alliance, it should be considered that not all the cooperative benefits will be distributed and that a part of them will be retained for sustainable development. In view of this idea, it is necessary to introduce a distribution coefficient or an adjustment coefficient for the value of the cooperative benefits and the distribution of the benefits to each participant. In view of this idea, it is necessary to introduce a distribution coefficient or adjustment coefficient for the value of cooperation gains and the value of each participant’s gain distribution to realize the partial retention of the value of game gains and thus to solve the problem of sustainable development of supply chain enterprise alliances.

8.3. Future Research

Our research provides some strategic references for sustainable cooperation among cross-regional supply chain enterprises. However, the model still has certain limitations, from the perspective of the “economic” pillar of supply chain enterprise management sustainability, we only studied the generalized fuzzy core solution and the generalized fuzzy bargaining set distribution scheme of the fuzzy average monotone game in cross-regional supply chain enterprise alliances but did not study other distribution schemes and their equivalent relationships, these limitations leave room for future research. For example, supply chain enterprises may form many fuzzy game patterns when forming cooperative alliances, with properties varying with game patterns, while the benefit distribution scheme is affected by a variety of factors in addition to cooperative participation and distribution coefficients. In the future, we will consider the generalized fuzzy kernel distribution scheme and its equivalence to coordinate the continuous cooperation of supply chain enterprises. In addition, the influence of other fuzzy cooperative game patterns on the cooperative strategy choices of cross-district supply chain enterprises is also a future research direction.