Optimization of Benefit Distribution in Green Supply Chain for Prefabricated Buildings Based on TFN-TOPSIS-Banzhaf Cooperative Game Theory

Abstract

With the rapid development of the prefabricated building industry, the green supply chain of prefabricated buildings has become a key driver of sustainable development and efficiency improvement in the industry. However, the issue of benefit distribution arising from cooperation has become the main challenge affecting the long-term stability of the supply chain. To address this, this study proposes an improved TFN-TOPSIS-Banzhaf value model, which optimizes the benefit distribution in the green supply chain of prefabricated buildings using cooperative game theory. This approach enhances both the fairness and accuracy of the distribution. The model integrates a combination of subjective and objective weighting methods based on triangular fuzzy numbers and the M-TOPSIS method for multi-factor evaluation, resulting in the corrected weight coefficients. By combining the weighting coefficients and least squares contributions, the improved Banzhaf value based on players’ weighted least squares contributions is constructed. The effectiveness and robustness of the model are verified through a case analysis, which significantly enhances the model’s ability to handle supply chain synergies and achieves a more fair and precise benefit distribution. This research provides an effective benefit distribution tool for the prefabricated building industry, promoting the continuous development of green building practices and supply chain cooperation.

1. Introduction

As resource and environmental pressures intensify, traditional construction methods urgently need to transition towards green and sustainable development. Prefabricated construction, as a new model of green building, has become increasingly important globally in the context of promoting green carbon reduction [1]. This construction method involves prefabricating components in a factory and then rapidly assembling them at the construction site, which not only significantly enhances construction efficiency and reduces waste and pollution on-site but also improves building quality through precise control [2]. However, the efficiency and sustainability of this construction model largely depend on the design and management of its supply chain [3]. The green supply chain for prefabricated construction starts with design and planning, followed by manufacturers producing prefabricated components according to blueprints, logistics providers ensuring safe and timely transportation to the construction site, and finally contractors performing on-site assembly and installation. This process requires all participants to collaborate closely to ensure that the project meets high standards and is completed smoothly. Such collaborative work not only ensures the efficient construction of prefabricated buildings but also significantly promotes resource conservation and reduces environmental impact [4]. The advantages of the green supply chain for prefabricated construction lie in the comprehensive enhancement of resource optimization, environmental protection, and construction quality, providing a practical path for the green transformation and sustainable development of the construction industry [5].

However, as construction projects become increasingly complex and diverse, the relationships of interests among participants in the green supply chain for prefabricated construction also become more intricate. In terms of production practices, the phenomenon of benefits not being equitably distributed within the green supply chain of prefabricated construction still exists [6]. In the process of distributing benefits, it is often assumed that the contributions of all participants are equal, overlooking the actual differences among them. This assumption does not align with the reality of participants’ actual investments and outputs, with varying levels of risks and degrees of investment due to multiple factors, leading to inconsistent contributions within the supply chain. An unreasonable distribution of benefits can lead to inefficient resource allocation and even provoke conflicts of interest, affecting the long-term stability of cooperation [7]. Considering that all parties in the supply chain pursue the maximization of their benefits, neglecting the interests of others can exacerbate conflicts and damage cooperative relationships, impacting the sustainable development of the supply chain [8]. Therefore, optimizing the distribution of benefits within the supply chain is crucial, especially via models that can address industry complexity and uncertainty, not only enhancing cooperative benefits but also ensuring the overall sustainability and competitiveness of the project [9].

From the existing literature, scholars have mainly focused on in-depth studies regarding the principles of benefit distribution, methods for benefit distribution, and the application areas. Guardiola et al. (2007) [10] investigated dynamic coordination and profit distribution in supply chains under decentralized control. Xu (2024) [11] combined the Stackelberg game with the Shapley value to develop a distribution model. Teng et al. (2019) [12] applied cooperative game theory to analyze benefit distribution among designers, general contractors, owners, and BIM consultants. Xu (2013) [13] combined the Shapley value with contribution-based distribution, applying it to fourth-party logistics supply chain alliances, and made reasonable adjustments to profit distribution based on practical scenarios, providing the alliance with a distribution mechanism that considers both contribution and risk. Sun et al. (2008) [14] designed a benefit distribution model based on supply chain contracts to ensure fair and reasonable profit distribution within the supply chain alliance. Panda et al. (2015) [15] addressed supplier channel conflict issues by studying a three-tier supply chain with coordinated responsibilities, proposing a bargaining approach to resolve channel conflicts and distribute profits. Hosseini-Motlagh et al. (2022) [16] introduced an “evolutionary game theory-based supply chain coordination profit surplus distribution (PSD) mechanism” to analyze the impact of long-term decision-making evolution on channel coordination and profit distribution. The results showed that manufacturers tend to opt for coordination strategies, and their share in the profits plays a significant role in evolutionary preferences. Neda Dabaghian et al. (2022) [17] analyzed the coordination issues between manufacturers, distributors, and retailers and proposed a contract negotiation model to address channel conflicts and residual profit distribution, offering a solution to the benefit distribution problem in socially responsible three-tier supply chains. Furthermore, in terms of calculation models for benefit distribution, major theoretical methods include the Shapley value (1953) [18], the Banzhaf value (1965) [19], the Nash Bargaining Solution (1950) [20], and the Core approach (1953), among others, with many scholars enriching these methods through extensive research and application in recent years. For instance, Płatkowski (2015) [21] pointed out that the Shapley value promotes long-term cooperation within alliances, with its theoretical foundation being that the value of an alliance is equivalent to the sum of the benefits of all its members. The Banzhaf value measures a player’s ability to change the outcome of the game without harming the interests of others, reflecting each player’s critical role in decision making and thus influencing benefit distribution. Van der Laan and van den Brink (2002) [22] introduced the Banzhaf coalition structure share function and demonstrated its applicability in inter-alliance and intra-alliance benefit distribution. Zhang et al. (2013) [23] improved the Nash negotiation model based on the interplay among supply chain nodes to facilitate benefit distribution. Wang and Meng (2024) [24] combined cooperative game theory with DEA to propose an asymmetric Core–Nash bargaining DEA game method for fixed cost allocation, highlighting its stability and fairness while ensuring the uniqueness of the solution. Öner and Kuyzu (2022) [25] employed cooperative game theory to study the restricted lane coverage game, proposing a core-based cost allocation method solved using column generation and row generation techniques, assessing its core stability within the supply chain. Although these methods focus on different aspects, they all aim to solve how to equitably distribute the total benefits brought by cooperation based on each party’s contribution. The Banzhaf and Shapley values emphasize the efficiency of benefit distribution, the Nash bargaining model focuses on compromise during negotiations, and the Core approach prioritizes the stability and fairness of the alliance. These theories offer various perspectives and tools, providing theoretical foundations and practical methods for addressing the complex benefit distribution challenges in the green supply chain of prefabricated construction.

The benefit distribution in the green supply chain of prefabricated buildings is a typical cooperative game problem, which requires the rational allocation of each participant’s contributions and benefits based on their cooperation. Due to significant differences among the participants in terms of risk-bearing, resource investment, technical capabilities, and market influence, and because the synergies among different supply chain links are crucial to the success of the project, the rationality of the benefit distribution directly impacts the overall project efficiency [26]. The Shapley and Banzhaf values are suitable for such distributions because these methods allocate the total cooperative benefits based on each player’s marginal contribution, better reflecting each segment’s contribution within the green supply chain and increasing the fairness of the distribution and the satisfaction of the participants [27]. Benedek (2024) [28] studied the benefit distribution model in large-scale international kidney exchange programs and found that the Shapley value, Banzhaf value, and nucleolus value have unique advantages in handling contributions and interests of different countries, especially suited for complex environments that require the consideration of heterogeneity and dynamic changes among multiple parties. Similarly, Karczmarz (2022) [29] compared the Banzhaf value and Shapley value in attributing characteristics within tree ensemble models, finding that the Banzhaf value, while providing similar attribute attribution, offers better numerical stability and faster computation. Comparative results show that the Shapley value, due to its fairness and symmetry, exhibits significant advantages in the distribution of benefits within the supply chain, particularly suitable for scenarios where there is a large disparity in contributions among cooperators. In contrast, the Banzhaf power index emphasizes the cumulative effect of marginal contributions, possessing unique attributes in explaining feature importance, especially suitable for cooperative environments with many participants and uneven influence. Therefore, it holds an important place in the study and application of supply chain management. In the complex and dynamic cooperative game environment of the green supply chain for prefabricated construction, employing the Banzhaf value can further enhance the efficiency and fairness of benefit distribution.

Although the traditional Banzhaf value treats all players in a game equally, assuming that each player has the same opportunity to influence the outcome of cooperation, this may not be entirely applicable in practice. In reality, some players may have a greater impact on the cooperative outcome due to possessing more resources, technological advantages, or market influence [30]. For example, in the green supply chain for prefabricated construction, manufacturers may have a far greater impact on component production and quality control than transportation companies have on delivery. This assumption of symmetry could lead to less influential players receiving disproportionately high benefits, thus triggering irrationalities in the game and a lack of motivation to cooperate. Conversely, key players might lack the motivation to participate in cooperation if they do not receive rewards commensurate with their contributions, thus limiting the practicality of the traditional Banzhaf value in dynamic and complex supply chains. Moreover, while the traditional Banzhaf value quantifies the contributions of different players through marginal contributions, effectively assessing the additional benefits each party may bring when joining an alliance, this method overlooks the imbalance in actual contributions. In the complex games of supply chains, parties not only influence the direct production process but may also improve the overall efficiency of the supply chain through collaboration [31]. In the green supply chain for prefabricated construction, close collaboration between production manufacturers and assembly contractors can significantly reduce overall costs, while the influence of transporters may only be indirect, depending on earlier production planning and design. This method of calculating marginal contributions ignores these synergistic effects, potentially leading to imbalanced benefit quantification, thereby affecting the fairness, stability, and cooperative motivation in benefit distribution.

In order to more effectively overcome the limitations of the traditional Banzhaf value, this paper makes an important contribution to the existing knowledge system by introducing the concepts of member weighting coefficients and least squares contribution to extend the Banzhaf value. Firstly, by comprehensively considering the weight of each participant, the problem of symmetry constraints in the traditional Banzhaf value is effectively addressed. The contribution assessment of each participant is dynamically adjusted based on their actual influence in the supply chain alliance, thus more fairly reflecting the true contribution of different participants. Secondly, by minimizing the difference between expected and actual benefits using the least squares method, the optimal benefit vector is selected to obtain the weighted least squares contribution based on players. This weighted least squares contribution can more accurately measure the contribution of each party in the cooperation, especially in scenarios where the synergies in the supply chain are significant, avoiding the shortcoming of traditional marginal contribution calculations that overlook collaborative effects. The weighted least squares method ensures that each participant’s contribution is reasonably quantified and reflected in the distribution scheme, enhancing the stability and fairness of the model. Through these improvements, the model not only solves the limitations of the traditional Banzhaf value in complex supply chain games but also adapts to the benefit distribution problem in multi-party cooperation games. It helps achieve a more scientific and reasonable benefit distribution in the green supply chain of prefabricated buildings, promoting long-term cooperation and win–win outcomes for all parties.

Based on this, this paper establishes an optimization model for the benefit distribution in the green supply chain of prefabricated buildings. The model comprehensively considers four major factors that affect the participants in the supply chain: risk assumption, resource input, green contributions, and effort level. Triangular fuzzy numbers are introduced to flexibly and reasonably quantify the status and role of each participant in the supply chain alliance, addressing potential fuzziness and uncertainty issues. A combination of subjective and objective weighting methods and effort coefficients are employed to systematically analyze the degree of influence, calculating the benefit distribution coefficients under each individual factor. On this basis, the M-TOPSIS method is applied to conduct a multi-factor comprehensive evaluation to obtain the corrected coefficients. Additionally, the weighted least squares contribution value replaces the marginal contribution in the Banzhaf value, constructing an improved Banzhaf value based on the player’s weighted least squares contribution and applying the corrected coefficients to the Banzhaf value model. This model achieves a scientifically reasonable distribution of benefits in the green supply chain of prefabricated buildings, maintaining long-term stability in the supply chain and advancing sustainable development.

2. Theoretical Foundation and Research Process

2.1. Research Theory and Assumptions

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Research Theory

The objective of cooperative game theory is to distribute the total benefits obtained by the coalition among the players in a fair and reasonable manner, which constitutes the solution to the cooperative game. Cooperative game theory can be represented as $(N,v)$, where the set $N=\{1,2,\dots ,n\}$ consists of $n$ players. The subset $S$ represents any non-empty subset of the set $N$, and the grand coalition $N$ contains $2^n$ sub-coalitions $S(S\subseteq N)$; each sub-coalition $S$ includes $s$ players. $v$ is the utility function of the cooperative game, which quantifies the game earnings of each player, and $v(i)$ represents the payoff that player $i$ can obtain, and it signifies the contribution of the player to the grand coalition in the cooperative game. Its fundamental assumptions include the following: When a new member joins the coalition $N$, the revenue they bring should not cause the total revenue of the coalition $v(N)$ to decrease, i.e., $v(N)>{\displaystyle {\sum }_{i\subset S}v(i)}$, indicating that $v(N)$ must be greater than the sum of revenues in any subset, which is the super additivity of the coalition. If no members participate in the coalition, their interest value $v(\varnothing )$ is zero. The benefits brought by coalition cooperation should at least equal the sum of the revenues of each sub-coalition $v(S_1\cup S_2)\ge v(S_1)+v(S_2)$, which reflects the synergistic effects of cooperation. The distribution value of the revenue that each player $i$ obtains from grand coalition $N$ is denoted by $x_i$, where $x_i$ must satisfy the principle of collective rationality, i.e., ${\displaystyle {\sum }_{i\in N}{x}_i}=v(N)$, indicating that after joining the coalition, the sum of the revenue distributions of all players equals the total revenue of the grand coalition.

Furthermore, the revenue distribution value $x_i$ that each player $i$ receives should be higher than the revenue they would achieve acting alone, i.e., $x_i\ge v(i)$, which follows the principle of individual rationality. This means that for a given game, if the distribution vector $x$ and $y$ satisfy $x_i>y_i,\forall i\in S$ and ${\displaystyle {\sum }_{i\in S}{x}_i}\le v(S)$, the distribution vector $x$ is said to dominate $S$ over $y$. A strategy meets Pareto efficiency if no other strategy can make some players better off without making other players worse off. Pareto efficiency is a key criterion in cooperative games, indicating that in resource distribution, it is impossible to improve the benefit of one party without harming the benefits of others. Even if some individuals’ benefits are not maximized, it is not possible to increase any party’s benefits without disadvantaging others. Pareto efficiency ensures optimal resource allocation, avoids waste, and provides a basis for comparing other distribution schemes. It promotes the stability of cooperation, prevents irrational behavior, and enhances the sustainability of cooperation. By requiring that any adjustment in resource distribution that improves one party’s benefit does not harm others, Pareto efficiency ensures a balance of interests among all parties and guarantees that no improvement will cause harm to other participants.

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Research Assumptions

This study employs cooperative game theory as the framework for analyzing benefit distribution in the prefabricated building green supply chain. To ensure the model’s validity and adaptability, the following key assumptions are made:

①

Super additivity: Ensures that the total benefit of any two non-overlapping coalitions is at least equal to the total benefit of each acting independently. This assumption is based on the general principles of cooperative game theory, aimed at encouraging broader cooperation.

②

Concavity: Ensures that as the number of coalition members increases, the marginal benefit increases. This reflects the fact that, in supply chains, the integration of resources and capabilities often leads to super linear gains.

③

Divisibility and Linearity: Allows for the flexible and precise distribution of benefits based on each participant’s specific contribution, ensuring distribution efficiency and satisfaction.

By establishing these assumptions, our research is not only theoretically supported but also ensures the practical applicability of the method and the reliability of the results. These assumptions provide a solid theoretical foundation for subsequent optimization analysis and empirical research.

2.2. Research Process

This study unfolds in three steps: First, we analyze the stakeholders and influencing factors in the benefit distribution of the green supply chain for prefabricated construction. Second, we construct an optimized benefit distribution model for the green supply chain of prefabricated construction based on cooperative game theory, including calculating the benefit distribution coefficients under single factors, applying the improved M-TOPSIS to calculate multi-factor benefit distribution coefficients, and using the weighted Banzhaf value model improved by the least squares method—this forms the core of this paper. Lastly, we conduct empirical analysis and result validation. A detailed flowchart of the research process is shown in Figure 1.

3. Benefit Distribution Analysis

3.1. Distribution Principles and Stakeholders

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Distribution Principles

To ensure the sustainability and stability of the prefabricated building green supply chain, the rationality of benefit distribution is crucial. A reasonable distribution considers the cooperation positions and interest pursuits of each party, effectively incentivizing long-term collaboration and ensuring the achievement of supply chain goals. On the other hand, unreasonable distribution may lead members to excessively pursue their own interests, neglecting the overall goals, reducing efficiency, and even jeopardizing the stability of the supply chain. Therefore, this study proposes the following benefit distribution principles:

①

Equitable Distribution Principle

The benefit distribution must be fair and reasonable, ensuring that each participant receives an appropriate return based on their contribution, thereby consolidating cooperation and stimulating enthusiasm to create more value within the supply chain.

②

Mutual Benefit Principle

All participants should achieve a successful return through cooperation. Members joining the supply chain must receive more benefits than they would have received independently; otherwise, their willingness to participate may decrease, threatening the stability of the cooperative relationship.

③

Proportionality of Benefit and Risk Principle

Different parties within the supply chain face different risks; therefore, the benefit distribution should be proportional to the risks each party assumes. The distribution mechanism should ensure that the benefits align with the risks, promoting fairness and long-term cooperation.

④

Proportionality of Benefit and Contribution Principle

Each participant should be comprehensively evaluated based on their efforts and abilities. Those who contribute more should receive higher returns to incentivize them to invest more resources and efforts.

⑤

Information Transparency Principle

To avoid misunderstandings caused by information asymmetry, the benefit distribution scheme should ensure the transparency and openness of information, guaranteeing that all participants cooperate on a fair and clear basis.

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Stakeholders

In the prefabricated building green supply chain, the main participants in benefit distribution include contractors, manufacturers, and logistics providers. The cooperation and coordination among these three parties are the foundation for ensuring the efficient operation of the prefabricated building green supply chain.

①

Contractor

As the final executor of the supply chain, the contractor is responsible for assembling and installing prefabricated components on-site according to the design and plan. Their main responsibilities include coordinating construction schedules, ensuring quality, managing the construction site, and ensuring the timely delivery of the project. Since prefabricated buildings use pre-manufactured components, the contractor must maintain close communication with the manufacturer to adjust procurement volumes to match production capacities, avoiding inventory issues and ensuring supply chain efficiency. Additionally, the contractor must collaborate closely with the logistics provider to coordinate transportation routes and schedules, ensuring that components arrive safely and on time at the construction site and preventing transportation issues from affecting the project timeline. Therefore, the contractor plays a key role in the supply chain, ensuring the smooth and efficient completion of the project.

②

Manufacturer

The manufacturer is responsible for providing prefabricated components in a timely manner based on demand plans, participating in the design and market demand forecasting, and adjusting production quantities and inventory based on demand changes. Due to the customized and mass production nature of the components, the manufacturer must ensure the provision of high-quality, environmentally friendly components that meet green building standards, thereby guaranteeing the efficiency and sustainability of the supply chain.

③

Logistics Provider

The logistics provider is responsible for transporting the prefabricated components from the manufacturing plant to the construction site and appropriately recycling waste generated during transportation. During transport, the logistics provider must ensure that the components arrive safely and on time, preventing damage or delays. Additionally, the logistics provider needs to optimize transportation routes and schedules to ensure that the construction project progresses as planned. Due to the complexity and potential risks of the logistics process, the logistics provider assumes significant risks and costs in the supply chain.

3.2. Selection of Factors Influencing Benefit Distribution

The distribution of benefits within the prefabricated construction industry chain involves multiple participants, with the profits being jointly created by all contractors involved. In the process of generating profits, the degree of involvement and the profits obtained by each contractor vary. Benefit distribution is influenced by multiple factors, and based on the principles of benefit distribution, expert interviews, and literature review, this study categorizes the main factors influencing the distribution of benefits in the construction supply chain alliance into four types: risk assumption, resource input, green contribution, and the level of effort.

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Risk Assumption Analysis

To deeply analyze the issue of risk assumption in the green supply chain for prefabricated construction, this study uses “prefabricated construction supply chain”, “profit”, and “risk” as keywords to conduct a systematic search in the WOS database. Through a detailed combing and analysis of the relevant literature, we organized the main risk factors that may affect the profits of participants in the green supply chain for prefabricated construction. The detailed contents and analysis results of these risk factors are shown in

Table 1. Risk assumption indicator system.

Category	Resource	Source	Details
External Environmental Risks	Political Environment Risk	Wang et al. [32], Kong et al. [33], Wang et al. [34]	Changes in policies or political instability may lead to increased operational costs and market access restrictions.
	Economic Environment Risk	Jiao & Li [35], Yang & Ren [36], Xu et al. [37]	Macroeconomic factors such as inflation or interest rate fluctuations may affect capital costs and consumer demand.
	Natural Environment Risk	Feiyu & Yisheng [38], Lin et al. [39]	Natural disasters like earthquakes, floods, or hurricanes may cause project delays or losses.
	Market Demand and Competition Risk	HE Wei et al. [40], Zhang et al. [41], Zhou et al. [42]	Uncertainty in market demand and intense competition may affect sales and profits.
	Standards and Regulations Change Risk	Tatari [43], Burkhart [44]	Changes in regulations or standards may require technical adjustments or additional compliance costs.
Construction Risks	Production and Technology Risk	Zhang et al. [45], Liu et al. [46], Zhu & Hu [47]	Technical failures or efficiency issues during production may lead to production delays or quality non-conformance.
	Transportation Risk	Zhu & Hu [47], Ma et al. [48], Jin et al. [49], Gao & Wang [50]	Logistics issues such as transportation delays or damage to goods may affect supply chain efficiency.
	Cost Overrun Risk	Zhang et al. [45], Tumminia et al. [51]	Project budgets may overrun due to rising material costs, increased labor costs, or poor management.
	Schedule Risk	Wang et al. [52], Liu et al. [53]	Project delays may occur due to poor planning, insufficient resource allocation, or external factors.
Supply Chain Cooperation Management Risks	Communication and Coordination Risk	Lin et al. [39], Zhang & Liu [54], Jiang et al. [55]	Information asymmetry or the lack of coordination among supply chain parties may lead to execution errors and inefficiency.
	Management and Capability Risk	Tatari [43], Liu et al. [46], Yan [56], Liao et al. [57]	Decision-making errors or the lack of capability in management may affect project success and organizational efficiency.
	Supply Chain Design Rationality Risk	Lin et al. [39], Jiang et al. [55], Jiang et al. [5]	Irrational or suboptimal supply chain structures may lead to low operational efficiency and increased costs.
	Contract Risk	Jiang et al. [5], Han et al. [58], Zhu et al. [59]	Unclear contract terms or poor execution may lead to legal disputes and additional costs.
	Trust Risk	Zhang et al. [41], Yan [56], Zhang et al. [60]	The lack of trust among partners may lead to cooperation barriers and increased transaction costs.
Supply Chain Cooperation Management Risks	Financial Risk	Wang et al. [32], Jiang et al. [5]	Issues with cash flow, changes in credit conditions, or market fluctuations may affect the financial stability of the organization.

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Resource Input Analysis

To comprehensively assess the contributions and benefit distribution issues among the participants in the green supply chain for prefabricated construction, besides considering risk assumption, resource input is also a key factor affecting benefit distribution. The level and manner of resource input by each participant in the project vary significantly, and these differences directly impact their value contribution throughout the entire supply chain. Therefore, using “prefabricated construction supply chain”, “benefit”, and “input” as keywords, we conducted a systematic search in the WOS database. Through the review and analysis of the related literature, we identified factors of resource input that might affect the benefits of each participant, including tangible resources such as capital, equipment, personnel, and professional technology, as well as intangible resources such as market share, brand value, and corporate innovation capabilities. The detailed data and analysis results can be seen in

Table 2. Resource input indicator system.

Criterion Layer	Indicator Layer	Source	Details
Tangible Resources	Capital Investment	Liao et al. [57], Türkten [61], Abidin et al. [62]	This includes necessary initial investments, operational funds, and contingency reserves, which are crucial for the smooth start-up and long-term operational efficiency of the project.
	Equipment Investment	Liao et al. [57], Zhou et al. [63], Chang & Zhao [64]	This involves purchasing or leasing necessary machinery and equipment. The modernity and maintenance status of the equipment directly impact production efficiency and the quality of the final product.
	Personnel Investment	Feiyu & Yisheng [38], Chang & Zhao [64], Zhu et al. [65]	The success of the project depends on human resources with necessary skills and experience, including technicians, engineers, and management teams, whose professional abilities play a decisive role in project execution and innovation.
	Professional Technical Investment	Gao & Wang [50], Liu et al. [53], Jiang et al. [55], Zhang [66]	This involves the adoption of new technologies and the improvement of existing technologies. These technological applications are key to enhancing project efficiency and market competitiveness.
Intangible Resources	Market Share	Zhang et al. [41], Ma et al. [48], Jiang et al. [55]	As a key indicator of market influence, market share shows the company’s position in the target market and the breadth of its customer base.
	Brand Value	Zhang et al. [41], Martillo Jeremías & Polo Peña [67], Abbasi et al. [68]	The market recognition, trustworthiness, and consumer loyalty of a brand are crucial for establishing consumer trust and attracting investors.
	Corporate Innovation Capability	Gao & Wang [50], Han et al. [58], Li et al. [69]	Reflects the participants’ ability in new product development, service innovation, and optimizing processes and management practices, which are key to maintaining a leading position in the industry and adapting to market changes.

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Green Contribution Analysis

The core of the green supply chain in prefabricated construction lies in its sustainability and environmental friendliness, and the green contribution analysis focuses on the investments and effectiveness of each participant in terms of environmental protection and resource conservation. During the green construction process, the contributions of the participants are primarily reflected in reducing environmental impacts, efficiently using energy, minimizing waste management, and utilizing green materials. Specifically, manufacturers drive the greening of building components by choosing environmentally friendly materials, optimizing production processes, and reducing energy consumption; logistics providers adopt energy-saving vehicles and optimize transportation routes to reduce carbon emissions; contractors effectively reduce the negative environmental impact through meticulous on-site construction management and reasonable resource allocation. For this purpose, using “prefabricated construction supply chain”, “benefit”, “green”, and “environmental protection” as keywords, we conducted a systematic search in the WOS database. Through the combing and analysis of related literature, we organized green technology assessment indicators and compared the green contributions of different participants. Specific analysis results are detailed in

Table 3. Green contribution indicator system.

Category	Green Technology Assessment Indicators	Source	Content
Green Procurement	Green Supply Selection	Yan [56], Pan et al. [70]	Prioritize suppliers that comply with environmental standards and practices to ensure the sustainability of raw materials and services and support environmental objectives in supply chain management.
Green Procurement	Environmental Material Procurement	Lin et al. [39], Zhu & Hu [47], Corsiuc & Mârza [71]	Use materials that meet environmental certifications, such as recycled or low-impact new materials, to reduce the environmental footprint of construction projects.
	Green Transportation of Components	Gao & Wang [50], Li et al. [72], Zhou [73], Zhang [74]	Implement energy-saving and emission reduction transportation strategies, such as using electric or low-emission vehicles and optimizing routes to reduce carbon emissions, making the transportation of building components more environmentally friendly.
Green Production	Green Production Technology Innovation	Oh & Kim [75], Chen [76], Yuan & Wang [77], Zhao et al. [78]	Apply automation and digital technologies to enhance production efficiency and environmental friendliness, significantly reducing waste and energy consumption.
	Green Construction Processes	Han et al. [58], Zhou & Zhou [79], Wu [80], Huo [81]	Use environmentally friendly construction techniques, utilizing recyclable or low-impact materials, and optimize the use of energy and materials during construction to reduce waste and environmental pollution.
	Resource and Energy Conservation	Wang et al. [82], Gao et al. [83], Liu et al. [84]	Implement resource and energy management strategies in project management, such as the efficient use of water and electricity and promoting renewable energy, to ensure the sustainability and economic efficiency of the construction process.
Green Recycling and Disposal	Reverse Logistics Management	Tumminia et al. [51], Qiao et al. [85], Qiao et al. [86]	Manage the process of recycling and reusing products and materials to maximize resource utilization and minimize environmental impact.
	Waste Disposal	Gao et al. [83], Nan & Jie [87], Zhao et al. [88]	Effectively sort, recycle, and safely dispose of waste to minimize environmental pollution and enhance the value of resource recovery.

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Effort Level Analysis

The analysis of effort levels aims to assess the actual efforts put forth by each participant in the green supply chain of prefabricated construction, which includes workload, technological innovation, coordination, and cooperation. During the project execution, there is a significant difference in the energy and labor input by different participants, which is often closely related to their key roles and responsibilities in the project. For example, manufacturers need to ensure the quality of prefabricated components and continuously optimize processes during production to improve efficiency; contractors are responsible for handling various emergencies on the construction site, coordinating resources from all parties to ensure the project is completed on time and to quality standards; logistics providers strive to ensure the timeliness and efficiency of transportation while reducing carbon emissions to ensure the smooth operation of the green supply chain. Therefore, it is crucial to perform a quantitative analysis of the efforts made by each participant at different stages.

4. Model Construction

4.1. Single-Factor Benefit Distribution Coefficient Based on Triangular Fuzzy Numbers

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Risk Assumption Coefficient

To ensure the fairness of the evaluation of benefit impact factors and to minimize the influence of human factors, this paper proposes a comprehensive analytical method based on triangular fuzzy numbers combined with an improved G1 weighting method and the entropy method. Traditional hierarchical analysis methods, which rely on expert ratings to determine the importance of indicators, are susceptible to subjectivity. Therefore, this study adopts a method that combines subjective and objective weighting to allocate weights to risk factors and uses triangular fuzzy numbers from fuzzy theory to clearly express the strength of relationships between factors, effectively resolving the issue of relational ambiguity in practical applications.

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Triangular Fuzzy Numbers

The Triangular Fuzzy Number (TFN) is a widely used form of fuzzy number in decision analysis and risk assessment, effectively handling uncertainty and expressing subjective bias and fuzzy information. It is capable of describing uncertain information in complex systems, providing an expected model that covers the potential fluctuation range and reflecting the decision maker’s subjective judgment and expectations about possible outcomes under different scenarios. Compared to traditional 0–1 relationships, triangular fuzzy numbers allow for a more comprehensive and precise quantitative analysis of the relationships between influencing factors, thereby enhancing decision quality. It helps decision makers consider the various possibilities influencing the benefit distribution, providing a scientific benefit distribution coefficient and promoting a more robust and fair decision-making process.

In this study, triangular fuzzy numbers are chosen for weight assignment primarily because they can effectively handle the uncertainty of factors such as risk assumption and green contribution. Compared to precise numbers, triangular fuzzy numbers better express uncertainty and fuzziness, and compared to interval numbers, they offer greater flexibility, allowing for a more accurate reflection of the uncertain range in decision making. Through the setting of three parameters—the minimum value, most likely value, and maximum value—the triangular fuzzy number demonstrates strong applicability when dealing with weight assignment. It effectively simulates uncertainty and provides a more flexible, adaptive mathematical model for decision making, optimizing the distribution of cooperative benefits.

The specific steps are as follows:

Let $\tilde{a}=(a_l,a_m,a_r)$ be a triangular fuzzy number, where $a_r$ is the upper limit, $a_l$ is the lower limit, and $a_m$ is the mean value; then, its membership function is as follows:

$${\mu }_{\tilde{a}}(x)=\left\{\begin{array}{l} 0\hspace{1em}\hspace{1em}\hspace{1em},x<a_l\\ {\displaystyle \frac{(x-a_l)}{(a_m-a_l)}},a_l\le x\le a_m\\ {\displaystyle \frac{(a_r-x)}{(a_r-a_m)}},a_m\le x\le a_r\\ 0\hspace{1em}\hspace{1em}\hspace{1em},x>a_r\end{array}\right.$$

(1)

When the values of ${\mu }_{\tilde{a}}(x)$ are close to 0 or 1, the fuzziness of the triangular fuzzy number $\tilde{a}$ is small. When the values of ${\mu }_{\tilde{a}}(x)$ are around 0.5, the fuzziness of the triangular fuzzy number $\tilde{a}$ is large. When $a_l=a_m=a_r$, $\tilde{a}$ is simplified to a real number. Since the contribution values studied in this paper are non-negative, that is, the lower limit $a_l>0$, the corresponding $\tilde{a}=(a_l,a_m,a_r)$ represents a non-negative triangular fuzzy number. Assuming any two non-negative triangular fuzzy numbers are $\tilde{a}=(a_l,a_m,a_r)$ and $\tilde{b}=(b_l,b_m,b_r)$, their operational rules are as follows:

①

Addition: $\tilde{a}+\tilde{b}=(a_l+b_l,a_m+b_m,a_r+b_r)$

②

Subtraction: $\tilde{a}-\tilde{b}=(a_l-b_r,a_m-b_m,a_r-b_l)$

③

Scalar multiplication: $\lambda \tilde{a}=\left\{\begin{array}{c}(\lambda a_l,\lambda a_m,\lambda a_r),\lambda \ge 0\\ (\lambda a_r,\lambda a_m,\lambda a_l),\lambda <0\end{array}\right.$

(ii)

Improved G1 Weighting Method Based on Triangular Fuzzy Numbers

The G1 weighting method was initially a subjective weighting method based on comparing the importance ratios of evaluation indicators [89]. To enhance the differentiation between weight values and improve the accuracy of the weights, this paper uses a scoring method for the importance of adjacent evaluation indicators to improve the G1 method. In expert scoring standards, to ensure the reliability of the scores, it is necessary to clearly understand the meanings of the scores on the scale. If the meanings of the scoring points on the scale are unclear, the reliability and validity of the measurements will be affected. Based on this, this study, referencing the Handbook of Survey Research [90], we propose using a 9-level scale to optimize the improved G1 weighting method. The specific calculation steps are as follows:

①

Expert scoring. A questionnaire survey was conducted to collect scores from several experts in related fields on the strength of the impact relationships between factors influencing benefit distribution among participants in the green supply chain for prefabricated construction. Triangular fuzzy numbers are used to fuzzify the evaluation indicators, reducing the subjective bias in expert scoring. The triangular fuzzy numbers correspond to each evaluation indicator, as shown in

Table 4. Expert scoring standards.

Importance Level	Rating	Triangular Fuzzy Number Rating
Slightly Important	1	(0.0, 0.1, 0.2)
Between Slightly and Moderately Important	2	(0.1, 0.2, 0.3)
Moderately Important	3	(0.2, 0.3, 0.4)
Between Moderately and Quite Important	4	(0.3, 0.4, 0.5)
Quite Important	5	(0.4, 0.5, 0.6)
Between Quite and Clearly Important	6	(0.5, 0.6, 0.7)
Clearly Important	7	(0.6, 0.7, 0.8)
Between Clearly and Extremely Important	8	(0.7, 0.8, 0.9)
Extremely Important	9	(0.8, 0.9, 1.0)

, where higher values indicate greater importance of the indicator.

②

The ranking of the importance of evaluation indicators. Based on the importance scores given by experts to the indicators, the indicators are ranked according to the size of their scores. If the importance score $y_1^{\ast }$ of indicator $y_1$ is greater than the importance score $y_n^{\ast }$ of indicator $y_n$, then it is denoted as $y_1\succ y_n$, and the relative importance ranking is:

$$y_1\succ y_2\succ y_3\succ \cdots \succ y_n$$

(2)

③

The ratio of the importance of adjacent indicators. Based on the scores given by the nth expert on the importance of the evaluation indicators, the ratio of the importance scores of adjacent indicators is used as the ratio of their weights. The weight ratio of indicator $y_{k-1}$ to indicator $y_k$ is denoted as:

$${\gamma }_l^k={\displaystyle \frac{{\omega }_l^{k-1}}{{\omega }_l^k}},{\gamma }_m^k={\displaystyle \frac{{\omega }_m^{k-1}}{{\omega }_m^k}},{\gamma }_r^k={\displaystyle \frac{{\omega }_r^{k-1}}{{\omega }_r^k}}$$

(3)

In the formula, ${\tilde{\omega }}_k=({\omega }_l^k,{\omega }_m^k,{\omega }_r^k)$ represents the weight of indicator $y_k$ based on the scoring of the $i$-th expert.

④

The calculation of indicator weights. The weight of indicator $y_k$ is:

$${\omega }_l^k={(1+{\displaystyle \sum _{k=2}^n{\displaystyle \prod _{j=k}^n{\gamma }_l^k}})}^{-1},{\omega }_m^k={(1+{\displaystyle \sum _{k=2}^n{\displaystyle \prod _{j=k}^n{\gamma }_m^k}})}^{-1},{\omega }_r^k={(1+{\displaystyle \sum _{k=2}^n{\displaystyle \prod _{j=k}^n{\gamma }_r^k}})}^{-1}$$

(4)

Using the recursive formula, the weights of other indicators can be calculated as:

$${\omega }_l^{k-1}={\omega }_l^k{\gamma }_l^k,{\omega }_m^{k-1}={\omega }_m^k{\gamma }_m^k,{\omega }_r^{k-1}={\omega }_r^k{\gamma }_r^k$$

(5)

where $k=n,n-1,\cdots ,2$.

⑤

The calculation of weights. Assume there are $p$ participants in the project, with $q$ evaluation indicators for each participant. The weight of the impact factors on benefit distribution in the green supply chain for prefabricated construction as assessed by the $k$-th expert is ${\tilde{\omega }}_{mn}^k=({\omega }_l^k,{\omega }_m^k,{\omega }_r^k),(1<m<p,1<n<q)$, resulting in the initial triangular fuzzy direct relation matrix:

$$A={({\tilde{\omega }}_{mn}^k)}_{p\cdot q}=\left[\begin{array}{ccc}{\tilde{\omega }}_{11}^k& \cdots & {\tilde{\omega }}_{1n}^k\\ ⋮& \ddots & ⋮\\ {\tilde{\omega }}_{m1}^k& \cdots & {\tilde{\omega }}_{mn}^k\end{array}\right],(1<m<p,1<n<q)$$

(6)

The initial triangular fuzzy direct relation matrix reflects the fuzzy semantic information of each expert regarding risk factors and evaluation indicators. To better utilize this information in subsequent analyses, it is necessary to defuzzify it to transform it into more specific and computable values. The purpose of defuzzification, while retaining fuzziness, is to make the values more consistent with actual situations. This paper uses the CFCS method (Converting Fuzzy data into Crisp Scores) for data defuzzification, a method that is simple to compute and results in the minimal loss of information after defuzzification, specifically as follows:

⑥

Standardize triangular fuzzy numbers $L_{mn}^k$, $M_{mn}^k$, and $R_{mn}^k$:

$$\begin{array}{c}{L}_{mn}^k={\displaystyle \frac{({\omega }_l^k-\min {\omega }_l^k)}{{△}_{\min }^{\max }}}\\ M_{mn}^k={\displaystyle \frac{({\omega }_m^k-\min {\omega }_m^k)}{{△}_{\min }^{\max }}}\\ R_{mn}^k={\displaystyle \frac{({\omega }_r^k-\min {\omega }_r^k)}{{△}_{\min }^{\max }}}\\ {△}_{\min }^{\max }=\max {\omega }_r^k-\min {\omega }_l^k\end{array}$$

(7)

⑦

Calculate the left and right limit values of the standardized values $u_{mn}^k$ and $v_{mn}^k$:

$$\begin{array}{l}{u}_{mn}^k={\displaystyle \frac{M_{mn}^k}{(1+M_{mn}^k-L_{mn}^k)}}\\ v_{mn}^k={\displaystyle \frac{R_{mn}^k}{(1+R_{mn}^k-M_{mn}^k)}}\end{array}$$

(8)

⑧

Calculate the precise value of the triangular fuzzy evaluation by expert $k$:

$$\begin{array}{c}{\omega }_{mn}^k=\min {\omega }_l^k+s_{mn}^k\cdot {△}_{\min }^{\max }\\ s_{mn}^k={\displaystyle \frac{u_{mn}^k(1-u_{mn}^k)+{(v_{mn}^k)}^2}{1-u_{mn}^k+v_{mn}^k}}\end{array}$$

(9)

⑨

Calculate the weights of each risk factor: compute the standardized precise values evaluated by $t$ experts.

$${\omega }_{mn}={\displaystyle \frac{1}{t}}{\displaystyle \sum _{k=1}^t{\omega }_{mn}^k}$$

(10)

(iii)

Improved Entropy Method Based on Triangular Fuzzy Numbers

The entropy method is an objective weighting method that determines weights by analyzing the statistical characteristics of data and uses the entropy value of each risk factor to measure the degree of information loss [91]. The higher the entropy value, the greater the uncertainty of the factor, indicating a smaller contribution to the system; conversely, a lower entropy value indicates that the factor has a greater impact on the system, and therefore, its weight should be larger. Additionally, the entropy method measures the variability of each risk factor in the dataset through a differentiation coefficient, thus reflecting the importance of different risk factors [92]. Since the entropy method is an objective weighting method based on the statistical characteristics of data, it requires the defuzzification of the original triangular fuzzy numbers, transforming fuzzy ratings into precise values for input into the entropy method, thereby allowing for objective weight calculation. The objectivity of this method ensures the scientific and reasonable distribution of weights. The specific steps are as follows:

①

Construct a matrix. Collect expert ratings; this study uses the efficient CFCS method to defuzzify the rating data, following the same steps as in Formulae (7)–(10), transforming fuzzy ratings ${\tilde{x}}_{ij}$ into precise ratings $x_{ij}$. Based on objective weighting methods, construct the scoring matrix X:

$$X={(x_{ij})}_{p\cdot q}=\left[\begin{array}{ccc}{x}_{11}& \cdots & x_{1n}\\ ⋮& \ddots & ⋮\\ x_{m1}& \cdots & x_{mn}\end{array}\right],(1<m<p,1<n<q)$$

(11)

②

Quantify risk indicators. Quantify the risk indicators in the matrix:

$$P_{\mathrm{i}\mathrm{j}}={\displaystyle \frac{x_{ij}}{{\displaystyle \sum _{i=1}^m{x}_{ij}}}}$$

(12)

③

Calculate the entropy values $e_j$ for each risk factor. Use the entropy formula to calculate the entropy value of each risk factor:

$$e_j=-k*{\displaystyle \sum _{i=1}^n{P}_{ij}}\ln \left(P_{ij}\right)$$

(13)

where $k={\displaystyle \frac{1}{\ln n}}$, and $i$ and $j$ are the number of samples.

④

Calculate the differentiation coefficient for each risk factor. According to the entropy value, calculate the differentiation coefficient for the indicators:

$$g_j=1-e_j$$

(14)

The greater the variability of $X_{ij}$, the smaller the entropy, thus the larger $g_j$ is, and the more important the indicator.

⑤

Determine the weights. Based on the differentiation coefficients, calculate the weights for each risk factor:

$$w_j={\displaystyle \frac{g_j}{{\displaystyle \sum _{j=1}^m{g}_j}}},j=(1,2,3,\dots ,m)$$

(15)

(iv)

Combined Weights for Risk Assumption Factors

Combine the weights obtained from the improved G1 weighting method and the entropy method. When the sum of the squared deviations of the weights from the two methods is minimized, that is, when the weights from both subjective and objective methods each account for half, it is considered that the optimal combined weight is achieved. Derived from Equations (10) and (15), the formula for calculating the combined weight is as follows:

$$W_j=0.5{\omega }_j+0.5w_j$$

(16)

where ${\omega }_j$ represents the weight values obtained from the improved G1 weighting method, and $w_j$ represents the weight values obtained from the entropy method.

(2)

Resource Input Coefficient

This follows the same steps as with the risk assumption coefficient.

(3)

Green Contribution Coefficient

This follows the same steps as with the risk assumption coefficient.

(4)

Effort Level Coefficient

In the green supply chain alliance for prefabricated construction, the level of effort by members directly relates to their contribution to the alliance, which is crucial for maintaining stability and enhancing efficiency. Additionally, the distribution of profits significantly influences the enthusiasm of cooperative enterprises to participate in alliance activities and their level of investment. Therefore, incorporating the effort level into the benefit distribution optimization model is necessary. By setting a series of conditions, it is possible to more accurately measure the contributions made by companies in the cooperation to obtain benefits and, based on this, achieve a reasonable distribution of profits.

(i)

In constructing the benefit distribution model for the green supply chain alliance in prefabricated construction, to reflect the relative contribution of different companies’ efforts to overall benefit creation, value creation coefficients are introduced, represented by ${\alpha }_a$, ${\alpha }_b$, and ${\alpha }_c$. Additionally, considering other uncontrollable factors that may affect the benefits of enterprises, these factors are collectively referred to as $\beta$. This setup helps to more comprehensively understand and analyze the benefit distribution situation in the alliance.

(ii)

In the green supply chain alliance for prefabricated construction, to reasonably distribute benefits among all parties, benefit distribution coefficients $\varsigma$, $\xi$, and $\zeta$ are introduced, corresponding to the benefit proportions of different companies in the alliance. These coefficients are set to follow the principle that their sum is 1, i.e., $\varsigma +\xi +\zeta =1$. Moreover, the benefit distribution coefficients are positively correlated with the level of effort in the supply chain, meaning that more effort by a company results in a larger share of benefits in the alliance. This design ensures the fairness of benefit distribution and motivates companies to increase their effort levels.

(iii)

Costs are typically divided into fixed costs and variable costs. Fixed costs are those that do not change with production volume or business volume and can usually be determined by a company’s initial investment. For companies in the green supply chain alliance for prefabricated construction, their cost functions are represented as $C(a)$, $C(b)$, and $C(c)$, corresponding to the costs of different companies. Assume that the variable costs faced by participating nodes in the supply chain reflect the level of effort they expend to optimize overall profits. Specifically, each company’s effort level can be quantified by its effort cost coefficients, represented by ${\gamma }_a$, ${\gamma }_b$, and ${\gamma }_c$. Therefore, variable costs reflect the additional costs incurred by companies to enhance their performance in the alliance due to increased efforts.

Based on the above content, a model for the total economic income and cost functions of the enterprises participating in the supply chain collaboration can be formed:

$$V=\alpha f(x)+\beta ={\alpha }_a{f}_a+{\alpha }_b{f}_b+{\alpha }_c{f}_c-C(a)-C(b)-C(c)+\beta$$

(17)

$$\left\{\begin{array}{l}C(a)=C_a+{\gamma }_a{f}_a^2\\ C(b)=C_b+{\gamma }_b{f}_b^2\\ C(c)=C_c+{\gamma }_c{f}_c^2\end{array}\right.$$

where $f_a$, $f_b$, and $f_c$ quantify the participation intensity (i.e., their level of effort) of companies A, B, and C in the alliance.

The expression for the net profit gained from cooperation among the green supply chain enterprises is:

$$\left\{\begin{array}{l}{e}_a=\varsigma v-C(a)\\ e_b=\xi v-C(b)\\ e_c=\zeta v-C(c)\end{array}\right.$$

(18)

where $e_a$, $e_b$ and $e_c$ represent the economic income obtained by companies a, b, and c within the supply chain, respectively.

Each cooperating enterprise, in order to maximize its own benefits, exhibits an effort level that is directly proportional to the economic benefits ultimately distributed. Based on this principle, specific expressions for the effort level of each company within the alliance can be derived. These expressions reflect the extent of effort each company puts forth to enhance its benefits within the alliance. By analyzing these expressions, further understanding of the behavioral patterns of enterprises within the alliance and how they might adjust their strategies to maximize benefits can be gained:

$$\left\{\begin{array}{l}{f}_a={\displaystyle \frac{\varsigma {\alpha }_a}{2{\gamma }_a}}\\ f_b={\displaystyle \frac{\xi {\alpha }_b}{2{\gamma }_b}}\\ f_c={\displaystyle \frac{\zeta {\alpha }_c}{2{\gamma }_c}}\end{array}\right.$$

Following the principle of $\varsigma +\xi +\zeta =1$, the economic return distribution coefficients for each cooperating enterprise from the perspective of effort levels can be calculated as:

$$\varsigma ={\displaystyle \frac{\frac{{\alpha }_a^2}{2{\gamma }_a}}{\frac{{\alpha }_a^2}{2{\gamma }_a}+\frac{{\alpha }_b^2}{2{\gamma }_b}+\frac{{\alpha }_c^2}{2{\gamma }_c}}},\xi ={\displaystyle \frac{\frac{{\alpha }_b^2}{2{\gamma }_b}}{\frac{{\alpha }_a^2}{2{\gamma }_a}+\frac{{\alpha }_b^2}{2{\gamma }_b}+\frac{{\alpha }_c^2}{2{\gamma }_c}}},\zeta ={\displaystyle \frac{\frac{{\alpha }_c^2}{2{\gamma }_c}}{\frac{{\alpha }_a^2}{2{\gamma }_a}+\frac{{\alpha }_b^2}{2{\gamma }_b}+\frac{{\alpha }_c^2}{2{\gamma }_c}}}$$

(19)

4.2. Improved M-TOPSIS Multi-Factor Benefit Distribution Coefficient

The previous sections constructed the weight coefficients for risk assumption, resource input, green contribution, and effort level as the factors affecting the benefit distribution in the construction supply chain alliance, providing a basis for individually determining the benefit distribution schemes for each party. However, for different enterprises within the green supply chain alliance for prefabricated construction, the outcomes of different influencing factors often vary. Therefore, this paper introduces the TOPSIS method to conduct a comprehensive evaluation of the outcomes of the four factors. This method calculates the distance between the ideal solution and the negative ideal solution, comprehensively measures the differences between the factors, and is efficient and widely applicable. It has strong adaptability, enabling a more accurate reflection of the actual benefit status of the participants in the prefabricated building green supply chain alliance.

The TOPSIS method was proposed by Hwang and Yoon in 1981 [93] and is a multi-criteria decision analysis method suitable for comprehensive evaluation with multiple variables and objectives. The method calculates the Euclidean distance between the decision object and the positive and negative ideal solutions to determine the relative closeness of each alternative, thus ranking them in order of preference. The advantages of TOPSIS lie in its simple calculation and reasonable results, allowing for the quantitative evaluation of multiple indicators. It has been widely applied in various decision-making fields. Moreover, TOPSIS has strong applicability and does not impose strict requirements on the indicators of the influencing factors, making it particularly suitable for addressing the characteristics of benefit influencing factors in the prefabricated building green supply chain alliance. By conducting a comprehensive evaluation of the contributions of each participant on different dimensions, a more scientific and comprehensive benefit distribution coefficient for the prefabricated building green supply chain can be generated. This approach takes into account the weighted importance of each factor and provides an accurate and flexible benefit distribution scheme for each participant, helping to achieve a more ideal benefit distribution coefficient.

However, the TOPSIS method may encounter a reversal issue, where sudden changes in the data for individual indicators may significantly affect the final result, sometimes even leading to a reversal in the ranking of alternatives. To address this issue, the M-TOPSIS method optimizes the traditional TOPSIS. This method enhances the reliability of the final benefit distribution coefficient by improving the relative closeness calculation rule after calculating the Euclidean distance of each scheme to the positive and negative ideal solutions.

• The specific steps are as follows:

Assuming there are $p$ factors and $q$ players in prefabricated construction, the initial decision matrix can be obtained from the single-factor benefit distribution coefficients:

$$A={(a_{mn})}_{p\cdot q}=\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right],(1<m<p,1<n<q)$$

(20)

Next, the initial decision matrix is dimensionless processed to obtain the standardized decision matrix:

$$b_{mn}={\displaystyle \frac{a_{mn}}{\sqrt{{\displaystyle \sum _{i=1}^n{a}_{mn}^2}}}},(1<m<p,1<n<q)$$

$$B={(b_{mn})}_{p\cdot q}=\left[\begin{array}{ccc}{b}_{11}& \cdots & b_{1n}\\ ⋮& \ddots & ⋮\\ b_{m1}& \cdots & b_{mn}\end{array}\right],(1<m<p,1<n<q)$$

(21)

Calculate the ideal and negative ideal solutions:

$$\begin{array}{l}{C}_m^{+}=\max \left\{b_m\right\},(1<m<p)\\ C_m^{-}=\min \left\{b_m\right\},(1<m<p)\end{array}$$

(22)

The Euclidean distances from each influencing factor to the ideal and negative ideal solutions are:

$$\begin{array}{l}{d}_m^{+}=\sqrt{{\displaystyle \sum _{n=1}^q{(C_m^{+}-b_{mn})}^2}},(1<m<q)\\ d_m^{-}=\sqrt{{\displaystyle \sum _{n=1}^q{(C_m^{-}-b_{mn})}^2}},(1<m<q)\end{array}$$

(23)

Calculate the relative closeness $C_m$:

$$C_m={\displaystyle \frac{d_m^{-}}{d_m^{+}+d_m^{-}}},(1<m<q)$$

(24)

where $C_m\in [0,1]$, and the higher the closeness value, the closer it is to the ideal point, indicating a higher priority of that influencing factor. The size of the relative closeness determines the merits of the influencing factors and provides a reference for decision making.

The improved M-TOPSIS method constructs a $(d_m^{+},d_m^{-})$ coordinate system, defining point E that represents the “optimal ideal solution” within the plane. The distances $R{S}_m$ between points within the plane EFGH, which represents the four influencing factors, and point E are used as the basis for ranking (as shown in Figure 2). The influencing factors are prioritized in ascending order of these distances.

$$R{S}_m=\sqrt{{[d_m^{+}-\min (d_m^{+})]}^2+{[\max (d_m^{-})-d_m^{-}]}^2}$$

(25)

If two influencing factors have equal $R{S}_m$ values, then their $R{S_m}^{\prime }=d_m^{+}-\min (d_m^{+})$ values are compared, and the factor with the smaller $R{S_m}^{\prime }$ value is considered superior. Figure 2 clearly illustrates the calculation and ranking principles of the M-TOPSIS method.

The $C_m$ values are normalized, resulting in a correction coefficient:

$$W_m={\displaystyle \frac{C_m}{{\displaystyle {\sum }_{m=1}^n{C}_m}}}$$

(26)

4.3. Improved Least Squares Contribution with Weighted Banzhaf Value Method

By utilizing the improved M-TOPSIS model, we calculated the composite benefit distribution coefficients under multiple factors within the green supply chain for prefabricated construction. In addressing the distribution of benefits among multiple participants, cooperative game theory methods are typically employed to ensure a fair and reasonable allocation. Cooperative game theory provides a framework for mutual negotiation and mutually beneficial outcomes, and the Banzhaf value, as a classic concept in cooperative games, provides a key basis for measuring the contributions of each party. It objectively measures the critical contributions made by each participant in all possible coalitions and uses mathematical methods to provide scientific and reasonable solutions to complex problems of cost apportionment, profit and loss distribution, and benefit distribution. Therefore, introducing the Banzhaf value in determining risk-sharing schemes allows for a more objective assessment of each party’s contributions, thereby justifying their proportions of risk assumption.

For a cooperative game $(N,v)$ involving $n$ people, the classical expression for the Banzhaf value is [94]:

$$B_i(N,v)={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum }_{S\subseteq N\backslash i}(v(S\cup i)-v(S))$$

(27)

$B_i(N,v)$ represents the benefits allocated to player $i$ from grand coalition $N$, where $v(S\cup i)-v(S)$ represents the player’s marginal contribution to the grand coalition $N$. Although the Banzhaf value effectively measures a player’s critical role in cooperative games, its methodology has certain limitations. Specifically, the Banzhaf value assesses a player’s influence by calculating their marginal contributions to the coalition, a method that performs well in symmetric, balanced games. However, the original calculation of marginal contributions is somewhat simplified and does not fully consider the interactions and complex relationships among players in cooperative games, especially as game structures become more complex and players’ heterogeneity increases. Moreover, the Banzhaf value assumes that each player’s contribution is based solely on their ability to influence the outcome of cooperation, without considering that players often have different resources, capabilities, and objectives, which may lead to varying degrees of contribution in actual cooperation. Therefore, to ensure the scientific accuracy of the risk-sharing scheme, this paper modifies the Banzhaf value based on the least squares method to more accurately reflect the actual contributions of players in projects, resulting in a more equitable distribution of risk-sharing proportions and aligning risk-sharing strategies more closely with project realities and stakeholder interests.

Liu et al. [95] demonstrated through the optimized Shapley value based on least squares contributions that the traditional method of marginal contributions is insufficient to capture the true influence of players in complex cooperative games. Extending the least squares contributions to the improvement of the Banzhaf value in a fuzzy environment has shown good results in the distribution of benefits within a supply chain alliance. This study extends this method to the calculation of the Banzhaf value in a precise environment, using weighted least squares contributions instead of marginal contributions in the Banzhaf value, proposing a modified Banzhaf value based on player contributions.

Triangular fuzzy number weighted least squares contributions are measured by the squared distance between the benefits received by players on the distribution vector and their actual contributions. By minimizing the variance of the excess contributions of all players in the coalition, the determined solution for the triangular fuzzy number weighted least squares contribution is:

$${\tilde{x}}_i^{w*}={\displaystyle \frac{\tilde{v}(N)}{n}}+{\displaystyle \frac{{\displaystyle \sum _{S\subset N:i\in S}sw(s){\tilde{v}}^C(S)}-\frac{1}{n}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{S\subset N:j\in S}sw(s){\tilde{v}}^C(S)}}}{{\displaystyle \sum _{s=1}^{n-1}sw(s)C_{n-2}^{s-1}}}}$$

(28)

where $w(s)$ represents the weight of coalition $S(S\subseteq N)$, and $0\le w(s)\le 1$. ${\tilde{v}}^C(S)$ is the profit attainable by grand coalition $N$ excluding subset $S$, representing the marginal contribution of sub-coalition $S$ to grand coalition $N$.

According to the membership function of the triangular fuzzy number, when $a_l=a_m=a_r$, the upper and lower limits of the profits do not change, becoming certain, and at this point, the triangular fuzzy number $\tilde{a}$ can degenerate into an exact number. When $x_{li}^{w*}=x_{mi}^{w*}=x_{ri}^{w*}$, ${\tilde{x}}_i^{w*}$ degenerates into the exact number $x_i^{w*}$. Similarly, the total revenue of the grand coalition as a triangular fuzzy number $\tilde{v}(N)$ can degenerate into the exact number $v(N)$, and the marginal contribution of the sub-coalition as a triangular fuzzy number ${\tilde{v}}^C(S)$ can also degenerate into an exact number $v^C(S)$.

Therefore, based on the weighted least squares contributions of the players, the value is:

$$x_i^{w*}={\displaystyle \frac{v(N)}{n}}+{\displaystyle \frac{{\displaystyle \sum _{S\subset N:i\in S}sw(s)v^C(S)}-\frac{1}{n}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{S\subset N:j\in S}sw(s)v^C(S)}}}{{\displaystyle \sum _{s=1}^{n-1}sw(s)C_{n-2}^{s-1}}}}$$

(29)

where $v^C(S)=v(S\cup i)-v(S)$ represents the marginal contribution of player $i$ to grand coalition $N$, and number of coalitions $S\subseteq N\backslash i$ is $2^{n-1}$.

Integrating the properties of the Banzhaf value with the concept of least squares, using the least squares contribution values in place of the marginal contributions in the Banzhaf value, yields a preliminary distribution function based on player contributions:

$$B_i^{*}(v)={\displaystyle \frac{1}{2^{n-1}}}\sum _{S\subseteq N:i\in S}{x}_i^{w*}(S)$$

(30)

where

$$x_i^{w*}(S)={\displaystyle \frac{v(S)}{s}}+{\displaystyle \frac{{\displaystyle \sum _{S^{\prime }\subset S:i\in S^{\prime }}{s}^{\prime }w(s^{\prime })v^C(S^{\prime })-\frac{1}{s}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{S^{\prime }\subset S:j\in S^{\prime }}{s}^{\prime }w(s^{\prime })v^C(S^{\prime })}}}}{{\displaystyle \sum _{s^{\prime }=1}^{s-1}{s}^{\prime }w(s^{\prime })C_{s-2}^{s^{\prime }-1}}}}$$

(31)

In this case, treating sub-coalition $S(S\subseteq N)$ as the grand coalition, where $S^{\prime }$ is a sub-coalition within coalition $S$, $x_i^{w*}(S)$ can still be obtained from the formula. To verify whether the single-value solution of this preliminary distribution function satisfies the collective rationality principle ${\displaystyle \sum _{i\in N}{x}_i^{w*}(S)}=v(S)(S\subseteq N)$, we calculate the excess difference:

$$\Delta (v)=v(N)-{\displaystyle \sum _{i\in N}{B}_i^{w*}(v)}={\displaystyle \frac{n-1}{n}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}v(S)}$$

(32)

At this time, $\Delta (v)\ge 0$ means $v(N)>{\displaystyle \sum _{i\in N}{B}_i^{w*}(v)}$; thus, it does not satisfy collective rationality, and there exists a cooperative surplus ${\displaystyle \frac{n-1}{n}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}v(S)}$, requiring the surplus value to be evenly redistributed. The final result, based on the weighted contributions of players, yields an improved Banzhaf value:

$${B_i^{w*}}^{\prime }(N,v)={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum }_{S\subseteq N:i\in S}{x}_i^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}}$$

(33)

The Shapley value is one of the most classic single-value solutions in cooperative game theory. For any cooperative game, the Shapley value can be expressed as [18]:

$${\mathrm{Sh}}_i(v)={\displaystyle \sum _{S\subseteq N,i\in S}\frac{(n-s)!(s-1)!}{n!}}[v(S)-v(S\backslash i)]$$

(34)

The marginal contribution $v(S)-v(S\backslash i),(i\in S)$ of the Shapley value is equivalent to the marginal contribution $v(S\cup i)-v(S),(S\subseteq N\backslash i)$ of the Banzhaf value, differing only in the form of expression. The Shapley value of each participant is their average marginal contribution, with the calculation involving all possible participant combinations. Unlike the Shapley value, the Banzhaf value does not consider the probability of various coalitions forming but instead simply averages all possible marginal contributions. The least squares method selects the optimal payoff vector by minimizing the difference between expected and actual returns $\min {\displaystyle {\sum }_{i\in N}(e^{Cw}(i,x)-{\overline{e}}^{Cw}(i,x)}$, thus minimizing the overall dissatisfaction of the coalition. It not only considers the marginal contributions of players within the coalition but also ensures a more stable and balanced benefit distribution through mathematical optimization. Moreover, the least squares method emphasizes reducing errors and biases, ensuring that the benefit distribution does not excessively rely on changes in specific coalition structures, thereby increasing the model’s adaptability and robustness in complex cooperative environments.

Next, we verify that the proposed improved Banzhaf value satisfies the four axioms of cooperative game theory and confirm that this cooperative game solution is uniquely determined. For any two cooperative games $(N,v)$ and $(N,g)$, the following applies:

1.

Efficiency: ${\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(v)}=v(N)$

From Equation (33), the sum of all players’ earnings is:

$$\begin{array}{ll}{\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(v)}& ={\displaystyle \sum _{i\in N}[}{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:i\in S}}{x}_i^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}}]\\ & ={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}v(S)}+{\displaystyle \frac{1}{n}}v(N)+{\displaystyle \frac{n-1}{n}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}v(S)}\\ & =v(N)\end{array}$$

This indicates that the total sum of the payoffs distributed to all players is exactly equal to the total value of the entire coalition, satisfying the efficiency axiom.

2.

Additivity: ${\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(v+g)}={\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(v)}+{\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(g)}$

The results of cooperative games are always independent:

$$\begin{array}{l}{\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(v+g)}={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle {\displaystyle \sum }_{S\subseteq N:i\in S}}[x_i^{w*}(S)-{\displaystyle \frac{v(S)}{s}}+x_i^{w*}(S)-{\displaystyle \frac{g(S)}{s}}]+{\displaystyle \frac{n-1}{n^2}}(v(N)+g(N))\\ ={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:i\in S}}{x}_i^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}}\\ +{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:i\in S}}{x}_i^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}g(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{g(S)}{s}}\\ ={\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(v)}+{\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(g)}\end{array}$$

This indicates that regardless of how the cooperative game solution is combined, the payoffs for each player are simply the direct sum of their individual game payoffs, verifying the additivity axiom of cooperative games.

3.

Symmetry: ${B_j^{w*}}^{\prime }(v)={B_k^{w*}}^{\prime }(v),(j,k\in N)$

The distribution of earnings between player $j$ and player $k$ in the cooperative game is independent of the order of distribution. If two players have the same position and contribution in the cooperative game, they should receive the same payoff. This can be derived from Equation (29):

$$\begin{gathered} {B_j^{w*}}^{\prime }(v)={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:j\in S}}{x}_j^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}} \\ {B_k^{w*}}^{\prime }(v)={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:k\in S}}{x}_k^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}} \end{gathered}$$

Given the assumption that player $j$ and player $k$ have the same position and contribution in the cooperative game, obviously, ${B_j^{w*}}^{\prime }(v)={B_k^{w*}}^{\prime }(v),(j,k\in N)$. This shows that the benefit distribution is independent of the order in which players are assigned, reflecting the symmetry axiom.

4.

Dummy player: ${B_i^{w*}}^{\prime }(v)={\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}}$ (when player $i\in N$ contributes nothing to the coalition).

Since $v(S\cup i)=v(S)(S\subseteq N\backslash i)$, the improved Banzhaf value assigns to player $i$ a value of ${B_i^{w*}}^{\prime }(v)={\displaystyle \frac{n-1}{n^2}}v(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{v(S)}{s}}$, not zero as with the traditional Banzhaf value. When $\Delta (v)=0$, ${B_i^{w*}}^{\prime }(v)=0$. This is because the improved Banzhaf value considers principles of efficiency and fairness, providing a reasonable redistribution for $\Delta (v)$. This method more accurately reflects the actual contributions of the players, enhancing the fairness and rationality of cooperative benefit distribution.

In summary, the improved Banzhaf value satisfies three fundamental axioms of cooperative game theory: efficiency, additivity, and symmetry. Additionally, it introduces a new interpretation in the virtual participant setting that considers indirect impacts and fairness, more accurately reflecting the actual contributions of the players and enhancing the fairness and rationality of the cooperative benefit distribution. Therefore, the improved Banzhaf value is the unique solution that satisfies the four aforementioned axioms, and the resulting benefit distribution has unique validity. Thus, the improved Banzhaf value satisfies these four axioms uniquely, and the distributed ${B_i^{w*}}^{\prime }(v)$ values possess unique validity.

5. Model Application

5.1. Application Background and Variable Settings

The L project, developed by a real estate company, is a residential project. The study focuses on a green modular construction supply chain alliance composed of Manufacturer X, Contractor Y, and Logistics Provider Z. Before the cooperation began, the three parties negotiated the details and signed the “X, Y, Z Green Modular Construction Industry Alliance Agreement”. The focus of this agreement is to resolve the profit distribution issues among the three partners, ensuring that all stakeholders find the solutions acceptable and satisfactory.

Independently, Manufacturer X, Contractor Y, and Logistics Provider Z earned revenues of USD 8.270 million, USD 23.000 million, and USD 11.890 million, respectively. After establishing the green supply chain alliance, the collaboration between Manufacturer X and Contractor Y yielded profits of USD 36.250 million; Manufacturer X and Logistics Provider Z, USD 22.770 million; and Contractor Y and Logistics Provider Z, USD 37.120 million. When Manufacturer X, Contractor Y, and Logistics Provider Z collaborate together, the total revenue amounts to USD 50.200 million. Based on these figures, expressions for the characteristic function can be constructed, as shown in

Table 5. Sub-coalitions and their characteristic values (unit: USD million).

Eigenvalue	X	Y	Z	XY	XZ	YZ	XYZ
$v(S)$	8.270	23.000	11.890	36.250	22.770	37.120	50.200
$v^C(S)$	13.080	27.430	13.950	38.310	27.200	41.930	50.200

.

The total benefit $v(\mathrm{XYZ})=50.200$ of grand coalition $N$ exceeds the sum of the benefits generated by each participant operating independently $v(\mathrm{X})+v(\mathrm{Y})+v(\mathrm{Z})=43.160$, satisfying super additivity. Additionally, the benefit of any sub-coalition $v(S)$ plus the benefit of non-participating stakeholders $v(i)$ is always less than the total benefit of the grand coalition, for example $v(\mathrm{XY})+v(\mathrm{Z})=48.140<v(123)=50.200$. The model satisfies the concavity of the cooperative game and demonstrates that Manufacturer X, Contractor Y, and Logistics Provider Z have sufficient cooperation incentives to join the grand coalition in order to gain more benefits. Furthermore, the project involves the distribution of financial benefits, which are inherently divisible. The model can linearly calculate the total benefits based on the contributions of the cooperating parties, satisfying the divisibility and linearity conditions.

Therefore, the project clearly aligns with the assumptions and premises of the model in this study. In this context, a reasonable benefit distribution mechanism becomes the key to the success of the cooperation. This model aims to maximize the benefits for all partners while promoting the stability of long-term cooperative relationships.

5.2. Solution Process

To ensure the accuracy and scientific validity of the evaluation indicators, this study invited 15 experts with extensive experience in consultancy and related fields. Based on their professional knowledge and experience, these experts used triangular fuzzy numbers to assess the importance of evaluation indicators (see Criteria Table 4 for details). Subsequently, the weights for risk assumption, resource input, and green contribution were calculated using the G1 method (derived from Equations (2)–(10)), entropy weights (derived from Equations (11)–(15)), and combined weights (derived from Equation (16)). The performance of the three companies in terms of value creation and effort cost coefficients was also analyzed to determine the weight coefficients from a single-factor perspective (derived from Equations (17)–(19)). Detailed calculation results and analysis can be found in

Table 6. Weight coefficients from a single-factor perspective.

Factor	Method	Manufacturer X	Contractor Y	Logistics Provider Z
Risk Assumption	G1 Method	0.357	0.368	0.276
	Entropy Method	0.335	0.337	0.328
	Combined Weight	0.346	0.352	0.302
Resource Input	G1 Method	0.377	0.295	0.328
	Entropy Method	0.393	0.379	0.228
	Combined Weight	0.385	0.337	0.278
Green Contribution	G1 Method	0.476	0.272	0.252
	Entropy Method	0.348	0.338	0.313
	Combined Weight	0.412	0.305	0.282
Effort Level	Value Creation	0.600	0.750	0.700
	Effort Cost	0.400	0.450	0.500
	Final Weight	0.288	0.399	0.313

.

Next, this study will use the M-TOPSIS method to comprehensively evaluate the benefit distribution coefficients under four impact factors, determining the comprehensive benefit distribution share due to each stakeholder in the alliance. This method integrates risk assumption, resource input, green contribution, and effort level to ensure the fairness and rationality of benefit distribution, reflecting the actual contributions and inputs of each party.

The initial decision matrix is:

$$\left[\begin{array}{llll}0.346\hfill & 0.288\hfill & 0.385\hfill & 0.412\hfill \\ 0.352\hfill & 0.399\hfill & 0.337\hfill & 0.305\hfill \\ 0.302\hfill & 0.313\hfill & 0.278\hfill & 0.282\hfill \end{array}\right]$$

The initial decision matrix is normalized using Equation (21) to obtain a standardized decision matrix:

$$\left[\begin{array}{llll}0.598\hfill & 0.493\hfill & 0.704\hfill & 0.661\hfill \\ 0.609\hfill & 0.685\hfill & 0.521\hfill & 0.579\hfill \\ 0.521\hfill & 0.537\hfill & 0.482\hfill & 0.477\hfill \end{array}\right]$$

Based on this, the ideal and negative ideal solutions for each indicator are determined using Equation (22):

$$\begin{array}{l}{z}^{+}=(z_1^{+},z_2^{+},z_3^{+},z_4^{+})=(0.609,0.685,0.704,0.661)\\ z^{-}=(z_1^{-},z_2^{-},z_3^{-},z_4^{-})=(0.521,0.493,0.482,0.477)\end{array}$$

Euclidean distances are then calculated using Equation (23), leading to the relative closeness measure derived from Equation (24), which results in the final adjustment coefficients determined by Equation (26). These are detailed in

Table 7. Adjusted coefficients from a multi-factor perspective.

	Manufacturer X	Contractor Y	Logistics Provider Z
Positive Euclidean Distance $d_m^{+}$	0.192	0.200	0.335
Negative Euclidean Distance $d_m^{-}$	0.298	0.237	0.044
Relative Closeness $C_m$	0.608	0.542	0.116
Adjustment Coefficient $W_m$	0.480	0.428	0.091

.

The resulting adjustment coefficients are $W=(W_1,W_2,W_3)=(0.480,0.428,0.091)$.

To minimize the overall dissatisfaction among stakeholders, this study utilizes the Banzhaf value based on players’ weighted least squares contributions, as proposed in this article, to distribute benefits. Derived from Equation (29), the weighted least squares contributions of each stakeholder under different alliance scenarios $x_i^{w*}(S)$ are shown in

Table 8. Weighted least squares contributions in different alliance scenarios (unit: USD million).

Stakeholder	Alliance	${\mathit{x}}_{\mathit{i}}^{\mathit{w}*}(\mathit{S})$
X	$\left\{X\right\}$	13.080
	$\{X,Y\}$	11.980
	$\{X,Z\}$	13.165
	$\{X,Y,Z\}$	10.992
Y	$\left\{Y\right\}$	27.430
	$\{X,Y\}$	26.330
	$\{Y,Z\}$	27.705
	$\{X,Y,Z\}$	25.585
Z	$\left\{Z\right\}$	13.950
	$\{X,Z\}$	14.035
	$\{Y,Z\}$	14.225
	$\{X,Y,Z\}$	13.624

.

Continuing from Table 8, it is evident that as the size of the alliance increases, the minimum squared contribution of firms joining the alliance decreases, satisfying the convexity of cooperative games. By substituting the obtained weighted least squares contribution $x_i^{w*}(S)$ for the classic Banzhaf value’s marginal contribution, we determine the final benefit amounts for each stakeholder ${B_i^{w*}}^{\prime }(v)$ based on Equation (33), as shown in

Table 9. Final benefits of each enterprise (unit: USD million).

Stakeholder	${{\mathit{B}}_{\mathit{i}}^{\mathit{w}*}}^{\prime }(\mathit{v})$
Manufacturer	11.362
Contractor	25.821
Logistics	13.017

.

Regarding the rationality of the final distribution results, after forming the alliance, each stakeholder’s benefits within the grand coalition exceed their individual operating benefits, ensuring all stakeholders’ motivation to cooperate. For example, Contractor Y’s benefit after joining the alliance is USD 25.821 million, higher than its independent operation benefit of USD 23.000 million. This not only shows the increase in benefits after joining the coalition but also highlights the added value achieved through cooperation, satisfying individual rationality.

The combined benefits of any two stakeholders within the grand coalition are greater than the sum of their benefits when forming a sub-coalition on their own.

For example, when Manufacturer X and Logistics Provider Z join the grand coalition, their total benefit is ${B_i^{w*}}^{\prime }(X)+{B_i^{w*}}^{\prime }(Z)=24.379$, whereas when they form an independent sub-coalition, their total benefit is only $v(XZ)=27.200$. This comparison verifies the synergy effect in the grand coalition and confirms that the maximization of benefits is achieved through comprehensive cooperation, aligning with the standard of coalition rationality.

Finally, from a global perspective, the total benefits of the alliance can be distributed to each participant ${\displaystyle {\sum }_{i\in N}{B_i^{w*}}^{\prime }(v)=v(XYZ)}$ in a reasonable way. Each participant receives their corresponding share, with no remainder, demonstrating the precision and fairness of the distribution, which satisfies overall rationality.

Therefore, the improved Banzhaf value proposed in this paper is a cooperative game solution that conforms to individual, coalition, and overall rationality.

6. Result Analysis

The distribution results of the revenue allocation strategy proposed in this paper considering single-factor and multi-factor perspectives in the green supply chain of prefabricated construction are compared with other cooperative game solutions. The specific results are shown in

Table 10. Summary of income distribution results (unit: USD million).

Typology	Method of Distribution	Manufacturer X	Contractor Y	Logistics Provider Z
single factor	Risk Assumption	11.355	25.816	13.029
	Resource Input	11.338	25.805	13.057
	Green Contribution	11.392	25.840	12.968
	Effort Level	11.326	25.798	13.077
multi-factorial	Correction Factor	11.362	25.821	13.017
other solutions	Shapley	11.138	25.678	13.383
improved solution	Banzhaf	11.370	25.910	13.615
	Bm	11.215	25.556	13.429

, which demonstrates the distribution of the benefits of each participant under different methods.

6.1. Sensitivity Analysis

The benefit values of the cooperative alliance are fixed in advance, with the primary variable parameter being the weight coefficients. Therefore, the sensitivity analysis focuses on testing the robustness of the model by changing the weight coefficients. This comparative analysis helps evaluate the effectiveness and rationality of the strategy proposed in this paper, ensuring that the proposed strategy can fairly and reasonably distribute the coalition benefits in practical applications.

By observing the distribution results under single- versus multi-factor conditions in Figure 3, we can gain a deeper understanding of the roles and contributions of each party in the supply chain. An analysis of the data shows that Contractor Y has the highest distribution of benefits in terms of risk-taking (USD 25.816 million) and resource input (USD 25.805 million), which reflects its core responsibility in the project and the high level of uncertainty it faces. This high risk-taking should be reflected in the contractual and financial arrangements to ensure that the contractor is properly compensated in the face of potential project delays or cost overruns. This not only recognizes their contribution but also incentivizes them to maintain high standards in future projects. At the same time, Manufacturer X excelled in green contribution (USD 11.392 million), highlighting its core competency in sustainability and attracting an environmentally conscious client base. Government green subsidies or additional premiums from the market could be potential avenues for their returns. In addition, Logistics Provider Z has the highest return distribution in terms of effort level (USD 13.077 million), highlighting its key role in ensuring timely supply and optimizing the transportation process. Such efforts by logistics providers are key to reducing overall project time delays and costs and should be appropriately rewarded through service fees or incentives to ensure their continued service quality and efficiency. The single-factor analysis helps to better understand the specific impact of each factor on the participants’ benefit distribution.

After integrating the single-factor allocation coefficients through the M-TOPSIS methodology, the calculation of the multi-factor correction coefficients takes into account not only the individual factors but also how these factors interact with each other to collectively affect the final benefit distribution. This avoids the limitations and errors that may arise from single-factor evaluations, allowing the model to comprehensively consider the combined contributions of each participant. This integration reduces the impact of extreme values, resulting in a more balanced distribution of returns. The multi-factor integration results show greater balance relative to the single-factor results, but the overall trend does not change much, suggesting that the weights and impacts of the single factors are relatively balanced and that no single factor dominates the distribution of benefits at the extremes. The results show that, despite Contractor Y’s high score on a single factor, the Modified Factor Allocation result (USD 25.821 million) takes into account its combined risk and ensures that the distribution of benefits reflects its irreplaceable role in the alliance. Manufacturer X’s correction factor allocation result (USD 11.362 million) is slightly higher than the single-factor average of its risk-taking and resource investment (USD 11.346 million), reflecting the market’s high regard for green building materials. The correction factor allocation result (USD 13.017 million) for logistics provider Z is slightly lower than that obtained for its effort (USD 13.077 million). This is because in the overall supply chain, after considering its relative contribution in resource input and risk taking, logistics, although critical, is more substitutable compared to project management for contractors and material supply for manufacturers, and reasonable adjustments ensure a balanced and fair distribution of the overall benefits. Based on the comparison of the single-factor and multi-factor benefit distribution coefficients, the correction coefficients can well balance the impact of each single factor, reflecting a more balanced result. This shows the necessity and effectiveness of comprehensive multi-factor evaluation in earnings distribution, which can consider the impact of various factors more comprehensively and thus make a more reasonable earnings distribution decision. This multidimensional analysis and revision not only facilitates further investment by the collaborating parties in their areas of expertise but also strengthens the trust and cooperative relationship within the alliance, provides management with decision support, identifies and rewards key contributors to the project’s success more equitably, and helps them to more accurately position their resources and priorities, thus not only ensuring that the project is completed on schedule but also enhancing the sustainability and social responsibility of the entire supply chain.

The combination of single-factor analysis and multi-factor integrated evaluation aims to make the results more robust and provide decision-making support for management. In single-factor analysis, we separately consider the independent impacts of each influencing factor on the benefit distribution, which helps provide a more detailed analytical perspective. On the other hand, the multi-factor integrated evaluation consolidates these factors and comprehensively considers their interactions, providing a more balanced solution for the final benefit distribution, ensuring the comprehensiveness and systematization of the evaluation. Although the final adjustment coefficient result does not show significant numerical fluctuations compared to the single-factor analysis, the core value of this integration process lies in effectively reducing the mutual bias among the factors, making the results more scientific, rational, and able to reflect the multidimensional contributions of the parties in the supply chain.

6.2. Comparative Discussion

Next, we compare the adjustment coefficient distribution results obtained from the multi-factor integrated evaluation with classic cooperative game solutions: the Shapley value, the Banzhaf value, and the B^m value. These game theoretic solutions provide different perspectives and methods for assessing and distributing the total benefits generated by cooperation.

The Shapley value results for the benefit distribution obtained for each participant are calculated using Equation (34) and shown in Table 10. The Shapley value allocates revenue by calculating the marginal contribution of each participant, emphasizing fairness and the direct impact of each participant on the final coalition revenue. For example, Manufacturer X has a Shapley value of USD 11.138 million, which shows the marginal contribution of its membership in the grand coalition to the total revenue $v^C(X)=50.200-37.120=13.080$. However, this approach may not adequately capture the combined role of participant in multiple sub-coalitions, especially if the impact of its contribution on the coalition is not linear or direct. In contrast, the improved Banzhaf value is calculated through a least squares contribution approach, which takes into account not only the participant’s contribution to the grand coalition but also its role in all possible sub-coalitions. For example, Manufacturer X has a least squares contribution of $x_X^{w*}(XY)=11.980$ in the sub-alliance with Contractor Y, as well as a least squares contribution of $x_X^{w*}(XZ)=13.165$ in the sub-alliance with Logistics Provider Z and a least squares contribution of $x_X^{w*}(XYZ)=10.992$ in Grand Alliance XYZ. In the end, Manufacturer X’s adjustment coefficient Banzhaf value is USD 11.362 million. Through the least squares contribution, the improved Banzhaf value more comprehensively considers the contribution of Manufacturer X in all possible cooperative alliances and more realistically reflects its actual contribution level and cooperative value in the alliance. This adjustment not only follows the principle of distribution according to work but also effectively incentivizes high-contributing members to maintain and strengthen the cooperative relationship, thus enhancing the overall benefits and stability of the alliance. The marginal contribution of all participants sums to ${\displaystyle {\sum }_{i\in N}{v}^C(v)=54.46}0$, which obviously exceeds the total benefit of the alliance, due to the non-independence and overlapping of marginal contributions. The marginal contribution of each inning participant depends on the presence and actions of other inning participants, and in a multi-inning participant scenario, different cooperative combinations may lead to certain contributions being double-counted. For example, two inning players may both significantly enhance the effectiveness of a coalition in different combinations, but this enhancement is double-counted in the total benefit. The improved Banzhaf value evaluates the contribution of the inning player by considering all possible cooperation configurations, not just based on marginal contributions. The least squares contribution is calculated to optimize the distribution of gains by minimizing the variance of the excess contributions of all the players in the coalition, usually taking into account interdependencies and overlapping contributions in the cooperation. The improved Banzhaf value ensures that the total gain matches: ${\displaystyle {\sum }_{i\in N}{x}_i^{w*}(v)=50.200}=v(N)$, i.e., the minimum contribution value, and is equal to the total gain of the coalition. By comprehensively evaluating the impact of each bureau in all possible sub-coalitions, it provides a more realistic reflection of the contribution versus the value of the collaboration. This approach is more suitable for complex and highly interdependent cooperative environments, such as large supply chain collaborations. From the comparison results, it can be seen that the Banzhaf value of the correction coefficient provides higher balance and rationality compared to the Shapley value, especially when dealing with players with different contribution characteristics. The modified coefficient Banzhaf value not only takes into account the marginal contributions of the parties but also synthesizes their roles in different sub-coalitions, so that the contributions and benefits of the parties are more closely related to the dynamics and complexity of the actual cooperation.

According to the calculation of Equation (23), when using the traditional Banzhaf value method for benefit distribution, the total sum of the benefits for each participant is USD 50.895 million, exceeding the total benefit of the coalition, which is USD 50.200 million. This excess allocation issue makes the benefit distribution scheme infeasible. To solve this problem, the B^m value makes a simple adjustment to the traditional Banzhaf allocation by deducting the excess proportionally from the allocation value of each participant to obtain an improved revenue allocation value. However, this method lacks a comprehensive consideration of the complex dynamics within the alliance, which may result in the higher-contributing participants failing to receive the corresponding benefits, affecting their satisfaction and willingness to cooperate with the allocation results. The improved Banzhaf value proposed in this paper is based on the least squares contributions of the participants, and integrates the contributions of all participants and the contributions of the sub-coalitions. This approach minimizes the dissatisfaction of the participant by decreasing the contribution value as the number of alliance members increases and avoids the situation in which the participant contributes a lot to the alliance but is assigned a lower benefit. By introducing weights and considering the correction coefficients of participants’ benefit allocation under multiple factors, this method demonstrates significant hierarchy and variability in coalition contributions. This method effectively matches the actual contribution and market position of each participant, which enhances the fairness and incentives of the alliance, and thus better adapts to the complex and highly interdependent cooperative environment.

The Shapley value, Banzhaf value, and B^m value are classic and well-established solutions in cooperative game theory. They have been widely validated in long-term applications and used in various complex cooperative situations. Although the calculation approaches of these methods differ, they all emphasize fairness, the marginal contributions of participants, and the reasonable returns for each party in the cooperation. Since these methods are mature and relatively stable, they tend to yield similar results in benefit distribution. The robustness of the multi-factor evaluation result is not only reflected in the comprehensive consideration of each participant’s contribution but also in the mutual validation of different strategies, effectively avoiding the extreme results that may arise from a single game solution. Compared to traditional game solutions, the multi-factor integrated evaluation method proposed in this paper enables benefit distribution based on multiple influencing factors (risk assumption, resource input, green contribution, and effort level). Although the differences in results are small when compared to other game solutions, this slight difference indicates that our method remains consistent with other classic solutions when integrating multiple influencing factors, thereby validating the rationality and reliability of our approach.

7. Conclusions

This paper innovatively improves the Banzhaf value model by introducing the concepts of member weighting coefficients and least squares contributions, aiming to optimize the benefit distribution in the green supply chain of prefabricated buildings. By comprehensively considering multiple influencing factors such as risk assumption, resource input, green contribution, and effort intensity, a combined objective–subjective comprehensive weighting method and triangular fuzzy numbers are used to address the uncertainty and fuzziness in decision making, thereby obtaining more practical single-factor weight coefficients. The M-TOPSIS method is employed to conduct a comprehensive evaluation of the four influencing factors, derive the adjustment coefficients, and apply them to the weighted least squares contribution modified Banzhaf value. This leads to the proposal of a more suitable benefit distribution model for the prefabricated building supply chain, aiming to achieve a fairer and more reasonable benefit distribution.

The research findings from the model application demonstrate that the TFN-TOPSIS-Banzhaf method exhibits both innovation and strong applicability in theory and practice. By integrating the weighted least squares method and the Banzhaf value, this study offers a new perspective to help the industry assess and optimize stakeholder benefit distribution from multiple dimensions. Particularly in the management practice of the green supply chain for prefabricated buildings, this approach contributes to promoting a fairer and more efficient resource distribution, providing an effective tool for advancing green building practices. As a result, it supports the sustainable development of the prefabricated building supply chain and fosters the fulfillment of environmental responsibilities. Therefore, this research not only deepens the application of cooperative game theory but also provides a new theoretical and practical framework for benefit distribution in supply chain management.

However, this study also has certain limitations. For example, the application of the model is constrained by the availability and quality of data, and the selection of specific parameters may require broader industry data to validate its universality. Additionally, the assumptions of the model may not be fully applicable in certain non-standard supply chain structures, which could impact its effectiveness in different environments. Future research could improve the model in several aspects. Firstly, the dataset could be expanded to include data from more industry experts and research institutions to test and optimize the model’s adaptability. Secondly, exploring the integration of the model with other economic models or machine learning algorithms could enhance prediction accuracy and the model’s ability to automatically adjust. Finally, further research into the model’s application in multi-cultural and multi-regulatory environments may reveal new adjustment factors, better adapting the model to the needs of global supply chains.