Improved Banzhaf Value Based on Participant’s Triangular Fuzzy Number-Weighted Excess Contributions and Its Application in Manufacturing Supply Chain Coalitions

Abstract

Intense market competition has driven small- and medium-sized enterprises (SMEs) in the manufacturing sector to collaborate and form supply chain coalitions, which can improve the information flow and resource sharing and significantly enhance supply chain management efficiency. However, the distribution of cooperative benefits poses a core challenge for the long-term stability of coalitions. This paper addresses the impact of dynamic changes in complex business environments by utilizing triangular fuzzy numbers to represent the value of coalition, effectively depicting the uncertainty and ambiguity in the cooperation process. Compared to traditional models (which do not use triangular fuzzy numbers), this model is better suited to dynamic changes, offering flexible response mechanisms that ensure adaptability and fairness in the decision-making process. In addition, considering the influence of each member’s weight in the coalition, the fuzzy comprehensive evaluation method is used to determine the weights. With the goal of minimizing the dissatisfaction of enterprises in benefit distribution, a least square contribution with triangular fuzzy numbers is constructed to replace the marginal contribution of the classical Banzhaf value, and an improved Banzhaf value based on the player’s triangular fuzzy number-weighted excess contribution is proposed to arrive at a fair and reasonable benefit allocation strategy in order to enhance the long-term stability and cooperative benefits of coalition. By analyzing an example of the supply chain coalition, the effectiveness of the proposed improved Banzhaf value is verified, which satisfies the uniqueness, the individual rationality, and the group rationality. It not only promotes the level of risk management and decision making under the uncertainty conditions of complex business, but also deepens the theoretical foundation of cooperative game theory and expands its possibilities in practical applications and future development.

1. Introduction

In today’s highly competitive global market environment, small- and medium-sized enterprises (SMEs), particularly in the manufacturing sector, face numerous challenges. The tight integration of global production networks can cause entire industrial chains to halt when there is a shortage of critical components, leading to significant supply chain disruptions [1]. Effectively managing the supply chain has become critical for the survival and development of these businesses. Vendor-managed inventory (VMI), as an advanced supply chain management approach, is increasingly adopted by companies aiming to enhance efficiency and responsiveness to market fluctuations [2]. In a VMI arrangement, the collaboration among suppliers, manufacturers, and third-party logistics (3PL) companies significantly optimizes supply chain management. Through integrated VMI systems and close cooperation, improved information flow and resource sharing are achieved, effectively reducing supply chain interruptions and the bullwhip effect, thereby significantly enhancing the resilience and responsiveness of the entire supply chain [3].

In the manufacturing supply chain coalition model, suppliers automatically manage and replenish manufacturers’ inventories using real-time shared sales and inventory data. This approach effectively reduces inventory costs, prevents excess or shortages of stock, and ensures a continuous supply of production materials [4]. Meanwhile, 3PL companies enhance the transportation and distribution of materials and finished products, further reducing logistics costs and improving distribution efficiency [5]. Such cooperation not only boosts the flexibility of the supply chain, but also improves market adaptability, allowing enterprises to swiftly respond to market changes, increase customer satisfaction, and strengthen market competitiveness. However, equitable benefit distribution within the supply chain is crucial for the long-term stability of the coalition [6,7]. Unfair distribution can erode trust among coalition members, decrease cooperative efficiency, and potentially lead to coalition dissolution [8,9]. A fair and reasonable benefit distribution mechanism ensures member satisfaction with the cooperation and motivates continuous resource investment and deeper collaboration, thereby enhancing the resilience of supply chain coalitions in a dynamic market environment. The benefit distribution in manufacturing supply chain coalitions represents a typical cooperative game problem. Reasonably measuring the contribution of coalition members and formulating fair benefit distribution strategies are essential to maintaining member satisfaction and sustaining cooperation.

Scholars have conducted in-depth studies on benefit distribution models in supply chain coalitions. The Shapley value is noted for its fairness and symmetry in addressing supply chain benefit distribution problems, highlighting its superiority [10]. However, the dynamic complexity and unpredictability of supply chains may limit the practicality of the model [11]. Benedek et al. [12] investigated a benefit distribution model in a large-scale international kidney exchange program. Compared with traditional solutions such as Shapley’s value and the nucleolus value, the Banzhaf value demonstrated unique advantages in addressing the contributions and benefits of different countries, especially in complex environments that require a consideration of the heterogeneity and dynamic changes among multiple parties. Similarly, in the realm of cooperative game-theoretic schemes that effectively solve linear problems, Grigoryan, G [13] conducted alignment tests to assess the sensitivity of explanatory artificial intelligence (XAI) methods. The results revealed that these methods struggle with issues of uncertainty and consistency in feature importance assessment. A comparison of results showed that the Banzhaf power index had an advantage over methods based on the nucleolus value and Shapley value due to its unique properties in interpreting feature importance values. Therefore, the Banzhaf value has become a focal point in supply chain management research and application, owing to its flexibility in adapting to changing supply chain environments. The Banzhaf value [14] is a classical single-valued solution in cooperative game theory, initially used to assess the power index of individuals in voting games and later widely applied in the distribution of cooperative benefit across various fields such as economy, politics, and environmental policy, achieving notable results [15]. As indicated by the literature review, research on Banzhaf value primarily focuses on the following three aspects:

Firstly, significant enhancements have been made to the classical Banzhaf value. These enhancements include the characterization of the Banzhaf value [16], extensions to the Banzhaf index [17,18,19], studies on the Banzhaf power index [20,21], and explorations of the Banzhaf interaction index [22]. These studies continually emphasize the importance of considering the weights of participants who have different levels of cooperation and bargaining power in multi-player cooperative games. Additionally, Alonso-Meijide and Fiestras-Janeiro [23] introduced the concept of the graph Banzhaf value, which further axiomatizes it in communication situations [24], representing an expansion of the Banzhaf value under constrained game structures. These axiomatized improvements enhance the theoretical foundation of Banzhaf value model, making it more applicable and widespread. However, in a volatile business environment, traditional improvements may not adequately address dynamic changes in all situations, and the complex computation process could hinder practical application.

Secondly, the analytical applications of the Banzhaf value. For example, improvements to the Banzhaf–Owen value using random sampling and two-stage sampling methods enhance participants’ confidence in the computational results from the initial data acquisition and processing [25]. Gallego et al. [26] defined concept-based Banzhaf value payments in a formal context. The Banzhaf value is calculated using matrix methods and applied to biological networks [27]. In addition, the Banzhaf value has also shown its effectiveness in the risk analysis of terrorist networks, by considering the alliance structure of network nodes and edges [28]. Banzhaf values have also demonstrated good application results in e-commerce [29], medical environments [30], and banking service quality assessment [31]. These studies show that the Banzhaf value is readily applicable to various practical situations, including direct applications in voting games and decision analysis. However, in specific contexts, it may oversimplify, failing to fully capture the ambiguity caused by complex economic and social factors in the dynamic environment of supply chain management.

Finally, extensions of the Banzhaf value into the fuzzy domain. Grey game theory, which addresses decision making where information is partially unknown and cannot be completely determined, uses the grey Banzhaf value as an important tool for analyzing and solving real-world cooperative game problems [32,33,34].Fuzzy game theory addresses decision-making problems where the strategies, benefits, or types of players in a game cannot be precisely defined. The fuzzy Banzhaf value considers the uncertainty and ambiguity in games, providing a flexible method for evaluating and optimizing cooperative strategies. Liao [35] applies Banzhaf values to fuzzy transfer utility games, expanding the ability of Banzhaf values to handle uncertainty and fuzziness. Scholars have refined the Banzhaf value in fuzzy cooperative game environments from multiple perspectives [36,37,38], making it more suitable for discussing interval cooperative games in scenarios where data are available but not precise. Obviously, for situations where data are available but the accuracy is insufficient, it is more suitable to discuss interval cooperative games, and then improve the interval Banzhaf value through various mathematical methods based on different conditions to obtain a more ideal solution. Scholars have combined various mathematical methods under different conditions to improve the interval Banzhaf value, achieving more ideal solutions [39,40,41,42]. For cooperative benefit distribution strategies in fuzzy environments, interval numbers provide a simple and intuitive description of uncertainty ranges. Triangular fuzzy numbers provide a more nuanced modeling of uncertainty by introducing most likely values, enabling decision makers to more accurately assess the expected effects of different decision scenarios by means of more detailed and precise analytical capabilities, leading to the development of a more reasonable and effective distribution strategy [43,44,45]. In summary, it can be seen that the introduction of fuzzy mathematics extends the application of Banzhaf values to a wider range of practical problems, permitting a better handling of uncertainty and ambiguity, especially in the context of unstable supply chains and market demand.

It is evident that the study of Banzhaf values has yielded rich results in the theory of cooperative games, but it still presents some obvious shortcomings in adapting to the needs of modern business and industrial environments. Firstly, the ability to deal with uncertainty is limited: traditional Banzhaf value calculations typically rely on precise data inputs. However, such precision is often difficult to obtain in real business environments, particularly in dynamically changing supply chain systems characterized by fluctuating market demand, unstable supply, and production capacity constraints. Although there is a substantial body of literature on extending the Banzhaf value to the fuzzy domain to better accommodate uncertainty and ambiguity, these approaches still exhibit limitations when dealing with uncertain information in complex and dynamic business environments. Secondly, there is a lack of consideration of weighting differences. Most of the current research on Banzhaf values focuses on the marginal contribution of each member in the cooperative game, treating all participants equally, but often overlooks the potential differences in weights among coalition members. In supply chain coalitions with complex power structures or highly heterogeneous member contributions, members differ in their risk-taking ability, brand value, social status, and resource investment. Failure to adequately take these factors into account may lead to unfair benefit distribution or fail to accurately reflect the true status and influence of each party in the cooperation.

Fuzzy logic is widely utilized in multi-objective decision making within supply chain management to address issues of uncertainty and complexity. Eslamipoor et al. [46] applied fuzzy logic to manage uncertainty and multi-objective problems in supplier selection, effectively evaluating supplier indicators. However, quantifying and comparing the specific contributions of different suppliers to environmental impact poses challenges, particularly in terms of data availability and accuracy. To address this, a generalized fuzzy number sorting method has been proposed [47], which, despite its theoretical effectiveness, is computationally intensive in practical applications and may not efficiently handle large-scale data. Triangular fuzzy numbers simplify calculations through a well-defined three-parameter method while retaining sensitivity to uncertainty. This not only reduces computational complexity but also enhances the efficiency and feasibility of the sorting process. Molinari [48] introduced a new standard based on the concept of weighted probability mean, establishing a partial order relationship for the family of generalized triangular fuzzy numbers and highlighting their effectiveness in managing uncertainty in decision making. Consequently, triangular fuzzy numbers offer a more precise framework for representing and managing the uncertainty of environmental impacts, enhancing the model’s applicability under real-world conditions and enabling decision makers to make more informed choices in situations of incomplete information [49]. Triangular fuzzy numbers have a broad application spectrum, demonstrating their practicality and efficacy in various domains. In cooperative games, they are employed to assess the uncertainty of each participant’s contribution within the alliance and are applied to value allocation [50]. In addition, Akyar [51] introduced a new method based on the inner radii of triangles to arrange triangular fuzzy numbers and solve permutation and decision-making problems. In the field of decision making, Mohammadian [52] designed a multi-attribute decision-making framework for policy makers using interval-valued triangular fuzzy numbers, particularly in evaluating and selecting IoT applications in the agricultural sector. Customer satisfaction can also be evaluated through triangular fuzzy numbers, demonstrating its wide applicability in commercial applications [53]. Applying triangular fuzzy numbers to group decision making and complex decision-making problems increases the accuracy of decision making [54]. Yang et al. [55] employed triangular fuzzy information to tackle multi-attribute decision-making problems related to credit risk. These extensive applications underscore the maturity of research on triangular fuzzy numbers, with sophisticated calculation methods and strategic applications already well established.

In order to cope more effectively with the limitations of the existing methods, this paper makes important additions to the existing body of knowledge by introducing the concepts of triangular fuzzy numbers and membership weighting coefficients. Triangular fuzzy numbers are particularly suitable for dealing with factors such as demand forecast uncertainty, supply volatility, and price changes, which are common in the manufacturing supply chain. The settings of its three parameters (minimum, most probable, and maximum values) can effectively simulate these uncertainties and provide a more flexible and adaptable mathematical model for decision making, thus optimizing the distribution of cooperative benefits. Meanwhile, by introducing the member weighting coefficient, this study ensures that the specific situation and relative importance of each member are fully considered in the benefit distribution process, making the results of distribution fairer and more targeted. This not only enhances the accuracy of decision making, but also strengthens the recognition of the fairness of the distribution by the cooperating parties, thus promoting the harmony and stability within the coalition. In conclusion, these innovative methods will significantly improve the practicality and effectiveness of the Banzhaf value in modern supply chain management.

In light of the above, this paper introduces the weighting factor to adjust the weight of each member in the cooperative benefits of the coalition under the framework of cooperative game theory. It utilizes the least squares method [56,57,58] to minimize the difference between the expected and actual gains. The weighted least squares contribution based on the contribution excess of the triangular fuzzy number is obtained by building a quadratic programming model. The obtained least squares contribution value replaces the marginal contribution in the Banzhaf value in order to construct a triangular fuzzy number based on the participant’s contribution, improving the Banzhaf value, so as to obtain a more fair and efficient benefit distribution strategy for manufacturing supply chain coalition cooperation.

2. Preliminaries

2.1. Triangular Fuzzy Number Cooperation Game

In cooperative games, participants face many uncertainties, such as market changes, policy impacts, and cost fluctuations, which often make it difficult to accurately estimate the benefits generated by supply chain information sharing, and can only obtain a rough range of benefits and their membership and non-membership degrees. The concise mathematical expression of triangular fuzzy numbers can effectively describe the uncertainty and ambiguity of the participants in assessing the benefits [49]. It consists of three parameters (minimum value, most likely value, and maximum value), which not only describe the fluctuation range of expected results, but also intuitively represent the most likely state of the results, making the analysis of uncertainty both comprehensive and specific. Denote $\tilde{a}=(a_l,a_m,a_r)$ as a triangular fuzzy number, and $a_l\le a_m\le a_r$, and its affiliation function is:

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

(1)

where $a_m$ is the middle value, which is the most probable value. Clearly, when $a_l=a_m=a_r$, the triangular fuzzy number $\tilde{a}$ actually evolves into an exact number. In other words, an exact number is a special type of triangular fuzzy number, where the lower bound, upper bound, and mean are all equal. Assume that any two triangular fuzzy numbers are $\tilde{a}=(a_l,a_m,a_r)$ and $\tilde{b}=(b_l,b_m,b_r)$, the algorithm is:

(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)

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

In the construction and application of the triangular fuzzy number, the minimum value represents a conservative estimate under the most unfavorable circumstances, the most probable value reflects the most likely observation, and the maximum value represents an optimistic estimate under the most desirable circumstances. For example, when assessing the potential return of a product under different market responses, a triangular fuzzy number can be set to estimate the return based on information from market research, historical data, expert opinions, and industry reports. For the pessimistic market scenario, the lowest return under the assumption of a poor market response is assumed, and a subtraction operation is used to model the decline in returns due to poor market performance. For the most likely market scenario, the most likely value is an observation set based on the most common market response and median data, which is the most likely return forecast to be achieved. For the optimistic market scenario, in estimating the maximum return under the best-case scenario of an extremely positive market response, an additive operation is used to simulate the additional return that may result from market activity.

In this paper, the cooperative game with a triangular fuzzy number of payoffs is referred to as a triangular fuzzy number cooperative game. A triangular fuzzy number cooperative game involving $n$ players can be denoted by $\tilde{G}=(N,\tilde{v})$, where $N=\{1,2,\dots ,n\}$ is the finite set of $n$ participants in the grand coalition $N$, and $\tilde{v}$ represents the characteristic function of the participants’ triangular fuzzy numbers. In a triangular fuzzy number cooperative game, for any sub-coalition $S(S\subseteq N)$ consisting of $s$ participants, the payoff $\tilde{v}(S)$ can be expressed using a triangular fuzzy number $\tilde{v}(S)=(v_l(S),v_m(S),v_r(S))$, where $v_l(S)$ represents the minimum expected payoff under adverse conditions, $v_m(S)$ represents the most likely expected payoff, and $v_r(S)$ represents the maximum expected payoff under optimal conditions. The set of $n$-person triangular fuzzy number cooperative games $\tilde{v}$ is denoted by ${\tilde{G}}^n$.

For any triangular fuzzy number cooperative game $\tilde{v}\in {\tilde{G}}^n$, the cooperative payoff allocated to participant $i$ from the grand coalition $N$ is denoted by ${\tilde{x}}_i=(x_{li},x_{mi},x_{ri}) (i=1,2,\dots ,n)$. For any sub-coalition $S(S\subseteq N)$, $\tilde{x}(S)={\displaystyle \sum _{i\in S}{\tilde{x}}_i}$ represents the sum of the triangular fuzzy number payoffs for all participants in coalition $S$. When $\tilde{v}(N)=\tilde{x}(N)$, the payoff of the grand coalition is fully distributed among the participants, satisfying the condition of collective rationality.

2.2. Triangular Fuzzy Excess Contribution

In triangular fuzzy number cooperative games, participants often generate a cooperative surplus during their interactions, inspired by the concept of participant surplus proposed by Liu et al. [58,59,60]:

$$e^C(i,x)={\displaystyle \sum _{S\subset N:i\in S}}{e}^C(S,x)={\displaystyle \sum _{S\subset N:i\in S}}[v(N)-v(N\backslash S)]-{\displaystyle \sum _{j\in S}}{x}_j={\displaystyle \sum _{S\subset N:i\in S}}{v}^C(S)-x(S) (i\in N)$$

Here, $e^C(S,x)$ represents the surplus contribution of coalition $S(S\subseteq N)$ when the payoff vector is $x$, $x_j$ refers to the contribution-based payoff vector for participant $j$ within coalition $S$, and $x(S)$ is the sum of the contribution-based payoff vectors for all participants within the coalition $S$. $e^C(i,x)$ indicates the dissatisfaction level of participant $i$ if the payoff vector is $x$. The larger $e^C(i,x)$ is, the higher the dissatisfaction of participant $i$, and the smaller $e^C(i,x)$ is, the lower the dissatisfaction of participant $i$.

On the basis of excess contribution, this paper proposes a participant excess contribution based on triangular fuzzy numbers:

$$\begin{array}{l}{e}^C(i,\tilde{x})={\displaystyle \sum _{S\subset N:i\in S}}{e}^C(S,\tilde{x})=({\displaystyle \sum _{S\subset N:i\in S}((v_l(N)-v_l(N\backslash S)-{\displaystyle \sum _{j\in S}{x}_l(j)),}}\\ {\displaystyle \sum _{S\subset N:i\in S}((v_m(N)-v_m(N\backslash S)-{\displaystyle \sum _{j\in S}{x}_m(j)),}}{\displaystyle \sum _{S\subset N:i\in S}((v_r(N)-v_r(N\backslash S)-{\displaystyle \sum _{j\in S}{x}_r(j)}}))\\ =({\displaystyle \sum _{S\subset N:i\in S}{v}_l^C(S)-x_l(S),{\displaystyle \sum _{S\subset N:i\in S}{v}_m^C(S)-x_m(S),}}{\displaystyle \sum _{S\subset N:i\in S}{v}_r^C(S)-x_r(S)) (i\in N)}\end{array}$$

In a triangular fuzzy number cooperative game within a fuzzy environment, $e^C(S,\tilde{x})$ represents the surplus contribution of coalition $S(S\subseteq N)$ when the payoff vector is $x$, and $\tilde{x}(j)=(x_l(j),x_m(j),x_r(j))$ refers to the contribution-based payoff vector for participant $j$ within the coalition $S$.

3. Model Construction

3.1. Weighted Excess Contribution Based on Triangular Fuzzy Numbers

In the real business environment, due to the different factors of resource control, market influence, input cost, technical capability, and strategic location among the coalition members, it is appropriate to assign different weights. Therefore, by adding the participant’s weighing factors, the above factors are differentially described by different values and reflected in the final allocation vector.

Traditional cooperative games tend to focus on the number of players when considering weights. The more players a coalition has, the more influential the coalition is, and the weight is positively proportional to the number of coalition members [61]. However, in real supply chain environments, especially in the context of vendor-managed inventory (VMI), the factors affecting the effectiveness of collaboration are far more complex than mere quantity. The five key factors that influence supply chain collaboration in a VMI environment include: supply chain role positioning and responsibilities, technology and innovation, risk management capabilities, market influence and economies of scale, and collaboration history and trust. These factors are fundamental to maintaining efficient supply chain operations and stability, and are critical to the success of collaboration. In order to systematically assess the impact of these factors on supply chain performance, this study used the fuzzy comprehensive evaluation method to determine the weights (The relevant symbol definitions are shown in

Table 1. Symbol definition of fuzzy comprehensive evaluation method.

Symbol	Definition
$X_1$	Role positioning and responsibilities
$X_2$	Technology and Innovation
$X_3$	Risk management capability
$X_4$	Market influence and economic scale
$X_5$	Collaboration history and trust
$a_i$	Weight vector

).

Let the set $X=\{X_1,X_2,X_3,X_4,X_5\}$ represent a set of key factors, the set $A=\{a_1,a_2,a_3,a_4,a_5\}$ represent the set of weight vectors for the key factors, the set $D$ represent the set of levels for the key factors, and specifically $D=\{none,low,relatively low,medium,relatively high,high\}=\{0,0.1,0.3,0.5,0.7,0.9\}$. Based on expert ratings, the fuzzy evaluation matrix is obtained as follows:

$$P=\left[\begin{array}{l}\begin{array}{ccc}{p}_{11}& p_{21}& p_{31}\end{array}\\ \begin{array}{ccc}{p}_{12}& p_{22}& p_{32}\end{array}\\ \begin{array}{ccc}{p}_{13}& p_{23}& p_{33}\end{array}\\ \begin{array}{ccc}{p}_{14}& p_{24}& p_{34}\end{array}\\ \begin{array}{ccc}{p}_{15}& p_{25}& p_{35}\end{array}\end{array}\right]$$

(2)

Selecting appropriate fuzzy comprehensive operators for operation, the key factor comprehensive evaluation matrix is obtained as follows:

$$C=AP=[a_1 a_2 a_3 a_4 a_5]\left[\begin{array}{l}\begin{array}{ccc}{p}_{11}& p_{21}& p_{31}\end{array}\\ \begin{array}{ccc}{p}_{12}& p_{22}& p_{32}\end{array}\\ \begin{array}{ccc}{p}_{13}& p_{23}& p_{33}\end{array}\\ \begin{array}{ccc}{p}_{14}& p_{24}& p_{34}\end{array}\\ \begin{array}{ccc}{p}_{15}& p_{25}& p_{35}\end{array}\end{array}\right]=[c_1 c_2 c_3 c_4 c_5]$$

(3)

Normalizing the fuzzy evaluation matrix $C$ to obtain $C^{\prime }$, the weight coefficients of the supply chain are:

$$W=C^{\prime }\cdot D^T$$

(4)

The weights of suppliers, manufacturers, and third-party logistics are respectively:

$$\left\{\begin{array}{l}{w}_1={\displaystyle \frac{W_1}{W_1+W_2+W_3}}\\ w_2={\displaystyle \frac{W_2}{W_1+W_2+W_3}}\\ w_3={\displaystyle \frac{W_3}{W_1+W_2+W_3}}\end{array}\right.$$

(5)

Based on the triangular fuzzy number excess contribution, the weighted excess contribution based on the triangular fuzzy number is proposed:

$$e^{Cw}(i,\tilde{x})=w(s)({\displaystyle \sum _{S\subset N:i\in S}{v}_l^C(S)-x_l^w(S),{\displaystyle \sum _{S\subset N:i\in S}{v}_m^C(S)-x_m^w(S),}}{\displaystyle \sum _{S\subset N:i\in S}{v}_r^C(S)-x_r^w(S) })$$

where $s$ is the number of participants in the coalition $S$, $w(s)$ is the weight function of the coalition $S(S\subseteq N)$, and $w(s)>0$.

3.2. Weighted Least Squares Contribution Based on Triangular Fuzzy Numbers

The dissatisfaction of the players is measured by the squared distance between the payoffs they receive on the distribution vector ${\tilde{x}}_i$ and their contributions. Drawing on the least squares concept, the optimal payoff vector is selected by minimizing the variance of the excess contributions of all participants in the coalition. It also considers the weights of the players to minimize the overall dissatisfaction of the coalition. A quadratic programming model is established based on the players’ least squares contributions.

Model 1:

$$\min {\displaystyle \sum _{i\in N}(e^{Cw}(i,\tilde{x})-{\overline{e}}^{Cw}(i,\tilde{x})}{)}^2$$

$$\mathrm{s}.\mathrm{t}.{\displaystyle \sum _{i\in N}{\tilde{x}}_i^w}=\tilde{v}(N)$$

where ${\displaystyle \sum _{i\in N}}{e}^C(i,\tilde{x})$ represents the sum of the weighted excess contributions of participants when the final benefit vector is $\tilde{x}$. The expression for this sum is:

$$\begin{array}{l}{\displaystyle \sum _{i\in N}{e}^{Cw}(i,\tilde{x})}={\displaystyle \sum _{i\in N}{\displaystyle \sum _{S\subseteq N,i\in S}w(s)({\tilde{v}}^C(S)-{\tilde{x}}^w(S))}}={\displaystyle \sum _{S\subseteq N,S\ne \varnothing }sw(s)({\tilde{v}}^C(S)-{\tilde{x}}^w(S))}\\ ={\displaystyle \sum _{S\subseteq N,S\ne \varnothing }sw(s){\tilde{v}}^C(S)}-{\displaystyle \sum _{S\subseteq N,S\ne \varnothing }sw(s)\tilde{x}(S)}={\displaystyle \sum _{S\subseteq N,S\ne \varnothing }sw(s){\tilde{v}}^C(S)}-{\displaystyle \sum _{s=1}^{n}sw(s)C_{n-1}^{s-1}\tilde{v}(N)}\end{array}$$

Meanwhile, ${\overline{e}}^{Cw}(i,\tilde{x})$ represents the average value of all participants weighted excess contributions. Its expression is given by:

$${\overline{e}}^{Cw}(i,\tilde{x})={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i\in N}}{e}^C(i,\tilde{x})$$

(6)

Clearly, for any triangular fuzzy number cooperative game, as long as the benefit vector $\tilde{x}$ is valid, then ${\displaystyle \sum _{i\in N}}{e}^C(i,\tilde{x})$ is a definite triangular fuzzy number value. Consequently, ${\overline{e}}^{Cw}(i,\tilde{x})$ is also a definite triangular fuzzy number value.

Let ${\overline{e}}^C(i,\tilde{x})=\tilde{k}$ (where $\tilde{k}$ is any triangular fuzzy number), then:

$${\displaystyle \sum _{i\in N}(e^{Cw}(i,\tilde{x})}-\tilde{k}{)}^2={\displaystyle \sum _{i\in N}{e}^{Cw}{(i,\tilde{x})}^2-2\tilde{k}{\displaystyle \sum _{i\in N}(e^{Cw}(i,\tilde{x})}}+n{\tilde{k}}^2$$

(7)

Clearly, the value of $\tilde{k}$ does not affect the solution of Model 1; thus, $\tilde{k}$ can take any value. Therefore, we set $\tilde{k}=(0,0,0)$, transforming Model 1 into Model 2.

Model 2:

$$\min {\displaystyle \sum _{i\in N}{e}^{Cw}}{(i,\tilde{x})}^2$$

$$\mathrm{s}.\mathrm{t}.{\displaystyle \sum _{i\in N}{\tilde{x}}_i^w}=\tilde{v}(N)$$

Next, we proceed to solve Model 2. The Lagrangian function for Model 2 can be expressed as:

$$\begin{array}{l}L({\tilde{x}}^w,\lambda ,\gamma ,\mu )\\ ={\displaystyle \sum _{i\in N}{\displaystyle \sum _{S\subseteq N,i\in S}w(s)({\tilde{v}}^C}}(S)-{\tilde{x}}^w(S){)}^2+\lambda ({\displaystyle \sum _{i\in N}{x}_{li}^w}-v_l(N))\\ +\gamma ({\displaystyle \sum _{i\in N}{x}_{mi}^w-v_m(N))+\mu }({\displaystyle \sum _{i\in N}{x}_{ri}^w-v_r(N))}\end{array}$$

(8)

To solve the function $L({\tilde{x}}^w,\lambda ,\gamma ,\mu )$, we take partial derivatives with respect to the variables $x_{li}^w$, $x_{mi}^w$, $x_{ri}^w$, $\lambda$, $\gamma$, $\mu$ respectively:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial L}{\partial x_{li}^w}}=2{\displaystyle \sum _{s=1}^{n-1}s}w(s)C_{n-2}^{s-1}{x}_{li}^w+2{\displaystyle \sum _{s=2}^{n}sw(s)C_{n-2}^{s-2}}{v}_l(N)-2{\displaystyle \sum _{S\subset N}s{v}_l^C}(S)+\lambda \\ {\displaystyle \frac{\partial L}{\partial x_{mi}^w}}=2{\displaystyle \sum _{s=1}^{n-1}s}w(s)C_{n-2}^{s-1}{x}_{mi}^w+2{\displaystyle \sum _{s=2}^{n}sw(s)C_{n-2}^{s-2}}{v}_m(N)-2{\displaystyle \sum _{S\subset N}s{v}_m^C}(S)+\gamma \\ {\displaystyle \frac{\partial L}{\partial x_{ri}^w}}=2{\displaystyle \sum _{s=1}^{n-1}s}w(s)C_{n-2}^{s-1}{x}_{ri}^w+2{\displaystyle \sum _{s=2}^{n}sw(s)C_{n-2}^{s-2}}{v}_r(N)-2{\displaystyle \sum _{S\subset N}s{v}_r^C}(S)+\mu \\ {\displaystyle \frac{\partial L}{\partial \lambda }}={\displaystyle \sum _{i\in N}{x}_{li}^w}-v_l(N)\\ {\displaystyle \frac{\partial L}{\partial \gamma }}={\displaystyle \sum _{i\in N}{x}_{mi}^w}-v_m(N)\\ {\displaystyle \frac{\partial L}{\partial \lambda }}={\displaystyle \sum _{i\in N}{x}_{ri}^w}-v_r(N)\end{array}\right.$$

(9)

Setting the partial derivatives equal to zero, we obtain the values of the weighted least squares contributions based on triangular fuzzy numbers as:

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

(10)

The specific expression for the triangular fuzzy numbers is:

$$\left\{\begin{array}{l}{x}_{li}^{w*}={\displaystyle \frac{v_l(N)}{n}}+{\displaystyle \frac{{\displaystyle \sum _{S\subset N:i\in S}sw(s)v_l^C(S)}-\frac{1}{n}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{S\subset N:j\in S}sw(s)v_l^C(S)}}}{{\displaystyle \sum _{s=1}^{n-1}sw(s)C_{n-2}^{s-1}}}}\\ {\tilde{x}}_{mi}^{w*}={\displaystyle \frac{v_m(N)}{n}}+{\displaystyle \frac{{\displaystyle \sum _{S\subset N:i\in S}sw(s)v_m^C(S)}-\frac{1}{n}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{S\subset N:j\in S}sw(s)v_m^C(S)}}}{{\displaystyle \sum _{s=1}^{n-1}sw(s)C_{n-2}^{s-1}}}}\\ x_{ri}^{w*}={\displaystyle \frac{v_r(N)}{n}}+{\displaystyle \frac{{\displaystyle \sum _{S\subset N:i\in S}sw(s)v_r^C(S)}-\frac{1}{n}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{S\subset N:j\in S}sw(s)v_r^C(S)}}}{{\displaystyle \sum _{s=1}^{n-1}sw(s)C_{n-2}^{s-1}}}}\end{array}\right.$$

(11)

3.3. Improved Banzhaf Value Based on Triangular Fuzzy Numbers

For any cooperative game $(N,v)$, the expression for the classical Banzhaf value [62] is:

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

(12)

Here, the number of coalitions $S\subseteq N\backslash i$ is $2^{n-1}$, and $v(S\cup i)-v(S)$ represents the marginal contribution of participant $i$ to the grand coalition $N$. In a triangular fuzzy number cooperative game $\tilde{v}\in {\tilde{G}}^n$, combining the properties of the Banzhaf value with the least squares idea, the weighted least squares contribution based on triangular fuzzy numbers is used to replace the marginal contributions in the Banzhaf value. The proposed pre-allocation function of the triangular fuzzy number-improved Banzhaf value based on participants’ contributions is as follows.

Model 3:

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

where:

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

(13)

In the process of calculating the least squares contributions based on participant contributions, a sub-coalition $S(S\subseteq N)$ is considered as a grand coalition and a sub-coalition $S^{\prime }$ within a coalition $S$, which ${\tilde{x}}_i^{w*}(S)$ can be obtained from Equation (6).

For any triangular fuzzy number cooperative game, since the sum of the benefit vectors distributed to participants is fixed, all must adhere to the principle of collective rationality: ${\displaystyle \sum _{i\in N}{\tilde{x}}_i^{w*}(S)}=\tilde{v}(S)(S\subseteq N)$, then:

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

(14)

In this regard, to verify that this single-valued solution satisfies collective rationality, calculate the excess difference $\Delta (\tilde{v})$:

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

(15)

Clearly, if $\Delta (\tilde{v})\ge 0$, then $\tilde{v}(N)\ge {\displaystyle \sum _{i\in N}{B}_i^{w*}(\tilde{v})}$. However, the overall rationality (efficiency) is only satisfied when $\Delta (\tilde{v})=0$, meaning that the total benefits produced by cooperation are fully distributed to all participants with no surplus, achieving optimal distribution. Therefore, if there is a cooperative surplus ${\displaystyle \frac{n-1}{n}}\tilde{v}(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\tilde{v}(S)}$, it indicates the need for an improvement to the pre-allocation function of Model 3.

Model 4:

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

Thus, the improved and effective weighted Banzhaf value based on triangular fuzzy numbers is given by:

$$\left\{\begin{array}{l}{B_{li}^{w*}}^{\prime }(\tilde{v})={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:i\in S}}{\tilde{x}}_{li}^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}{\tilde{v}}_l(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{{\tilde{v}}_l(S)}{s}}\\ {B_{mi}^{w*}}^{\prime }(\tilde{v})={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:i\in S}}{\tilde{x}}_{mi}^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}{\tilde{v}}_m(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{{\tilde{v}}_m(S)}{s}}\\ {B_{ri}^{w*}}^{\prime }(\tilde{v})={\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subseteq N:i\in S}}{\tilde{x}}_{ri}^{w*}(S)+{\displaystyle \frac{n-1}{n^2}}{\tilde{v}}_r(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{{\tilde{v}}_r(S)}{s}}\end{array}\right.$$

(16)

Next, we verify that the proposed effective weighted improved Banzhaf value ${B_i^{w*}}^{\prime }(\tilde{v})$ satisfies the four axioms to ensure that the value is uniquely determined. For any two triangular fuzzy number cooperative games $\tilde{v}\in {\tilde{G}}^n$ and $\tilde{u}\in {\tilde{G}}^n$:

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

From Equation (12), the sum of all participants’ benefits equals:

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

Axiom (II) (Additivity): ${\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(\tilde{v}+\tilde{u})}={\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(\tilde{v})}+{\displaystyle \sum _{i\in N}{B_i^{w*}}^{\prime }(\tilde{u})}$.

For two $n$-person triangular fuzzy number cooperative games, the results of cooperative benefit distribution are independent of each other:

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

Axiom (III) (Symmetry): ${B_i^{w*}}^{\prime }(\tilde{v})={B_j^{w*}}^{\prime }(\tilde{v})(i,j\in N)$.

If participants $i$ and $j$$(i\ne j)$ are symmetrical, and $w(i)=w(j)$, meaning participants $i$ and $j$ have the same status in the cooperative game, then changes in the order of participants within the coalition do not affect the results of benefit distribution, as can be seen from Equation (12):

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

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

Since it is assumed that participants $i$ and $j$ have the same status in the cooperative game, obviously, ${B_i^{w*}}^{\prime }(\tilde{v})={B_j^{w*}}^{\prime }(\tilde{v})$$(i,j\in N)$.

Axiom (IV) (Quasi-null player): ${B_i^{w*}}^{\prime }(\tilde{v})={\displaystyle \frac{n-1}{n^2}}\tilde{v}(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{\tilde{v}(S)}{s}}$ (when participant $i\in N$ makes no contribution to the coalition).

If whether a participant $i\in N$ joins all possible sub-coalitions $S(S\subseteq N)$ does not change the coalition’s profit, i.e., $\tilde{v}(S\cup i)=\tilde{v}(S)(S\subseteq N\backslash i)$, then the effective weighted improved Banzhaf value will be assigned to the participant $i$ with a value of ${B_i^{w*}}^{\prime }(\tilde{v})={\displaystyle \frac{n-1}{n^2}}\tilde{v}(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{\tilde{v}(S)}{s}}$, instead of the Banzhaf value assigned to 0. When $\Delta (\tilde{v})=0$, ${B_i^{w*}}^{\prime }(\tilde{v})={\displaystyle \frac{n-1}{n^2}}\tilde{v}(N)-{\displaystyle \frac{1}{2^{n-1}}}{\displaystyle \sum _{S\subset N}\frac{\tilde{v}(S)}{s}}=0$. This is because the effective weighted improved Banzhaf value considers the principles of effectiveness and fairness in cooperative games and makes a reasonable redistribution of the excess.

Therefore, the effective weighted improved Banzhaf value ${B_i^{w*}}^{\prime }(\tilde{v})$ is the only value that satisfies the aforementioned four axioms, ensuring that the distributed values possess unique validity, this is the most significant difference between the improved Banzhaf value proposed in this paper and the improved Banzhaf value.

4. Model Application

4.1. Application Background and Variable Settings

Maintaining a smooth and stable supply chain is critical in the manufacturing industry, especially in the environment of frequent fluctuations in market demand and intense competition. In traditional supply chain models, the practice of a single supplier setting up independent inventories around the manufacturing plant improves the response time. However, when the supply volume does not reach an economic scale, logistics costs can be significantly higher than the direct shipment to the manufacturer. Consequently, suppliers generally do not support this approach unless they can share expensive logistics and warehousing costs through large-scale supply. For manufacturers, fluctuations in market demand create uncertainty in inventory and production schedules, necessitating a high degree of flexibility and responsiveness in the supply chain. In this case, the vendor-managed inventory (VMI) model provides a solution to improve overall efficiency and responsiveness through more effective inventory management by suppliers. However, successful implementation of VMI requires manufacturers to ensure robust information communication and data sharing with suppliers to accurately forecast demand and make timely adjustments to production schedules. The introduction of third-party logistics (3PL) companies is an effective strategy when both sides of the VMI implementation seek to further improve the operation of the supply chain and reduce logistics costs. By centralizing the logistics needs of suppliers and manufacturers, 3PL companies are able to achieve economies of scale, thereby reducing transportation and warehousing costs [63]. The 3PL companies can also provide flexible and cost-effective logistics solutions, such as shared warehousing facilities and optimized transportation networks, to further compress costs and improve supply chain efficiency. In the VMI environment, a cooperative coalition of suppliers, manufacturers, and 3PL companies is formed, as shown in Figure 1. Such cooperative coalitions not only save costs and time, but also significantly reduce supply chain risks and improve market adaptability. Through close cooperation and data sharing, the coalitions are able to respond quickly to changes in the market and improve the level of satisfaction of customer needs, thereby enhancing the competitiveness and market position of enterprises.

In a supply chain coalition, the main partners are a supplier, a manufacturer, and a 3PL company named Participant 1, Participant 2, and Participant 3. The annual revenue of each of the three firms when operating separately is: $\tilde{v}(1)=(7.4,10.5,14.1)$, $\tilde{v}(2)=(15.2,26.3,35.4)$, $\tilde{v}(3)=(31.7,52.7,78.1)$. Under the VMI system, participants vary in their capacity to bear risk, brand value, social standing, and resource inputs, contributing differently to the coalition’s success. For instance, if a 3PL company provider substantially reduces transportation costs through innovative logistics solutions, such value creation should be appropriately acknowledged in profit distribution. Similarly, if a supplier reduces inventory costs and mitigates risks associated with inventory fluctuations through advanced management techniques, these benefits should also be reflected in the profit distribution strategy. Additionally, if a manufacturer boosts the overall supply chain’s output quality and delivery speed by optimizing production processes and enhancing efficiency, these improvements should be suitably considered in profit allocation. The benefit distribution model should be developed by quantitatively analyzing the specific contributions of each company, taking into account their direct and indirect impacts on supply chain efficiency, cost savings, and customer satisfaction. Given the varied roles and responsibilities these parties hold within the supply chain, their contributions should be appropriately weighted in the distribution of profits.

The weighting of key factors needs to be considered holistically. Suppliers are frontline players in driving technological innovation. In a fast-changing market, suppliers not only need to ensure the quality of materials and stability of supply, but also through continuous technological improvements to optimize production costs and enhance product performance. Although suppliers have less direct influence in the market than manufacturers, their role in maintaining production efficiency and product innovation should not be underestimated. Manufacturers act as the core firms in the coalition. The market positioning of manufacturers determines their control over brand and product quality, and also directly affects the economic efficiency of the entire supply chain. Manufacturers score high on this factor because of their extensive market influence and economic scale. 3PL companies play a key role in optimizing logistics and distribution processes, significantly reducing costs and ensuring efficient supply chain operations through advanced logistics solutions and risk management strategies. The application of the fuzzy comprehensive evaluation method ensures fairness and transparency of benefit distribution in this process, quantifies the contribution of key factors, effectively identifies and strengthens the core competitiveness of each member, and optimizes resource allocation and strategic decision making. Therefore, the factor weights are considered as: $A=\{0.10,0.20,0.35,0.25,0.10\}$. The evaluation matrix $P$ is based on business performance reports, technological innovation records, supply chain efficiency analysis, risk management reports, cooperation history assessment, market research, customer satisfaction surveys, and assessments by industry experts, ensuring that the performance of each member of the manufacturing supply chain coalition in the comprehensive evaluation is objectively and comprehensively quantified. The evaluation matrix of suppliers, manufacturers, and 3PLs can be obtained as:

$$P=\left[\begin{array}{l}\begin{array}{ccc}0.4& 0.5& 0.3\end{array}\\ \begin{array}{ccc}0.3& 0.7& 0.5\end{array}\\ \begin{array}{ccc}0.3& 0.4& 0.9\end{array}\\ \begin{array}{ccc}0.3& 0.4& 0.9\end{array}\\ \begin{array}{ccc}0.5& 0.4& 0.5\end{array}\end{array}\right]$$

The formula $w_1\approx 0.2,w_2\approx 0.3,w_3\approx 0.5$ can be obtained through Equations (2)–(5). Obviously, the assignment of weights directly affects the output of the model because they determine the relative importance of each key factor in the final score. For example, if the weight of market demand changes is higher, then the overall risk score of the supply chain will be correspondingly higher in the case of high demand volatility, reflecting the sensitivity of the impact of market instability on the supply chain. Conversely, if technological innovation is weighted more heavily, then in industries with rapid technological advances, the supply chain may exhibit greater adaptability and competitiveness, resulting in a lower risk score in the model. In this way, the weights not only reflect the importance of the factors, but also determine the focus of supply chain management decisions.

By implementing a coalition, significant cost reductions and increased profits are achieved. First, centralized inventory management enables suppliers to more precisely adjust and optimize inventory levels, thereby reducing both surplus and shortages, and ensuring that stock levels consistently align with actual demand. This not only reduces unnecessary inventory costs but also shortens the time it takes for products to reach the market, enabling faster delivery to consumers. Secondly, the 3PL providers reduce transportation costs through optimized routing and freight consolidation. They utilize advanced logistics technologies and networks to minimize empty runs and idling, thus enhancing transportation efficiency. Further logistics cost reductions are achieved by improving load rates and reducing unnecessary return trips, which not only saves costs but also contributes to the coalition’s environmental sustainability by reducing carbon emissions. Moreover, the VMI coalition minimizes operational costs by avoiding rush orders and reducing the frequency of emergency purchases. Rush orders, often associated with high logistics expenses and administrative chaos, can be effectively circumvented through proactive forecasting and demand balancing. Simultaneously, enhanced information sharing and data transparency between suppliers and manufacturers increase the responsiveness of the entire supply chain. This allows each segment to adapt more swiftly to market changes and customer demands, thereby enhancing customer satisfaction and competitive edge. This holistic approach to supply chain optimization ensures maximal utilization of funds and resources, thereby promoting the economic benefits and competitiveness of the entire coalition.

The values of the benefits from the cooperation of the business coalition are: $\tilde{v}(12)=(65.1,105.4,140.9)$, $\tilde{v}(13)=(77.9,108.9,145.6)$, $\tilde{v}(23)=(82.6,112.4,154.3)$, and $\tilde{v}(123)=(155.2,210.7,287.3)$. Therefore, the marginal contribution of all participants to the grand coalition $N$ is:${\tilde{v}}^C(1)=(72.6,98.3,133.0)$, ${\tilde{v}}^C(2)=(77.3,101.8,141.7)$, and ${\tilde{v}}^C(3)=(90.1,105.4,146.4)$. Then, the marginal contribution of all sub-coalitions $S$ to the grand coalition $N$ is: ${\tilde{v}}^C(12)=(123.5,158.0,209.2)$, ${\tilde{v}}^C(13)=(140.0,184.4,251.9)$, ${\tilde{v}}^C(23)=(147.8,200.2,273.2)$, and ${\tilde{v}}^C(123)=(155.2,210.7,287.3)$.

4.2. Solution Procedure

The total benefit of the grand coalition $N$ is $\tilde{v}(123)=(155.2,210.7,287.3)$, which is greater than the sum of the individual benefits of each participant on their own of $\tilde{v}(1)+\tilde{v}(2)+\tilde{v}(3)=(54.3,89.6,127.6)$, demonstrating super-additivity. Additionally, the sum of the benefits of any sub-coalition $\tilde{v}(S)$ plus the profit $\tilde{v}(i)$ of its members when they operate individually is less than the total revenue of the coalition. For example, $\tilde{v}(12)+\tilde{v}(1)+\tilde{v}(2)=(87.7,142.2,190.4)$ is significantly less than $\tilde{v}(123)=(155.2,210.7,287.3)$. Therefore, the supplier, manufacturer, and third-party logistics all have the incentive to join the grand coalition to gain more benefits. However, the distribution of these benefits within the coalition becomes a crucial issue. In order to minimize the overall dissatisfaction of the participants, this paper utilizes the Banzhaf value based on the weighted least squares contributions of participants for benefit distribution. The weighted least squares contributions of each participant ${\tilde{x}}_i^{w*}(S)$ under different coalition scenarios, expressed as triangular fuzzy numbers, are shown in

Table 2. Weighted least squares contributions in different coalition scenarios (in millions of dollars, USD).

Participant	Coalition	${\tilde{\mathit{x}}}_{\mathit{i}}^{\mathit{w}*}(\mathit{S})$
1	$\left\{1\right\}$	$(72.6,98.3,133.0)$
	$\{1,2\}$	$(53.5,77.3,100.3)$
	$\{1,3\}$	$(61.3,88.7,119.2)$
	$\{1,2,3\}$	$(41.9,54.8,72.6)$
2	$\left\{2\right\}$	$(89.1,101.8,141.7)$
	$\{1,2\}$	$(70.0,80.7,108.9)$
	$\{2,3\}$	$(67.5,98.3,134.3)$
	$\{1,2,3\}$	$(48.9,67.6,90.8)$
3	$\left\{3\right\}$	$(67.0,105.4,146.4)$
	$\{1,3\}$	$(78.7,95.7,132.7)$
	$\{2,3\}$	$(80.3,101.9,139.0)$
	$\{1,2,3\}$	$(64.5,88.3,124.0)$

.

According to Table 1 and Figure 2, as the size of the coalition increases, the least squares contribution values for enterprises joining the coalition decrease, satisfying the convexity of cooperative games, that is, as the number of coalition members increases, the contribution values decrease. Replacing the classical Banzhaf value’s marginal contributions with the obtained weighted least squares contributions ${\tilde{x}}_i^{w*}(S)$, after pre-distribution and effectiveness improvement calculations, gets the final benefit value ${B_i^{w*}}^{\prime }(\tilde{v})$ of each participant, as shown in

Table 3. Final benefits for each enterprise in the supply chain coalition (in millions of dollars, USD).

Participant	${{\mathit{B}}_{\mathit{i}}^{\mathit{w}*}}^{\prime }(\tilde{\mathit{v}})$
1	(42.8, 61.8, 81.8)
2	(54.3, 69.1, 94.5)
3	(58.1, 79.8, 111.1)

.

5. Comparative Discussion of Application Results

After forming a cooperative coalition, the final value of the benefits allocated to each participant from the grand coalition is higher than their respective individual earnings, ensuring that all participants have sufficient incentives to participate in the cooperation. For example, a supplier’s benefit value ${B_1^{w*}}^{\prime }(\tilde{v})=(42.8,61.8,81.8)$ after joining the coalition is greater than the benefit value $\tilde{v}(1)=(7.4,10.5,14.1)$ when not joining the coalition, satisfying individual rationality. The sum of the final values of the benefits allocated to any two participants in the grand coalition is greater than the sum of the values of the benefits they would have received if they had formed a separate sub-coalition. For example, the sum of the benefits of the supplier and the manufacturer when they joined the grand coalition, ${B_1^{w*}}^{\prime }(\tilde{v})+{B_2^{w*}}^{\prime }(\tilde{v})=(97.1,130.9,176.3)$, is greater than the sum of the benefits they would have received if they had formed a sub-coalition, $\tilde{v}(12)=(65.1,105.4,140.9)$, satisfying collective rationality; The total benefits of the coalition are distributed to all participants in a reasonable manner, ${\displaystyle \sum _{i\in N}{B}_i^{w*}(\tilde{v})}=\tilde{v}(123)$, satisfying overall rationality (efficiency). Therefore, the triangular fuzzy number based on participants’ excess contribution to improve the Banzhaf value proposed in this paper is a reasonable, effective, and practical means for solving the benefit distribution problem of supply chain coalition in fuzzy environments.

The cooperative profit allocation strategy proposed in this paper has been compared with other cooperative game solutions, and the results are shown in

Table 4. Results of benefit distribution for different methods (in millions of dollars, USD).

	Participant	1	2	3
Solution
Shapley value		(42.7, 58.8, 77.9)	(48.9, 68.4, 92.9)	(63.6, 83.4, 116.6)
Banzhaf value		(44.0, 61.0, 80.0)	(50.3, 70.7, 95.0)	(64.9, 85.6, 118.7)
Bm value		(42.9, 59.2, 78.3)	(49.0, 68.5, 92.9)	(63.3, 83.0, 116.1)
$B_i^{*}$		(42.8, 62.1, 82.3)	(54.3, 69.2, 94.6)	(58.0, 79.4, 110.4)
$B_i^{w*}$		(42.8, 61.8, 81.8)	(54.3, 69.1, 94.5)	(58.1, 79.8, 111.1)

and Figure 2.

The Shapley value [64] is a traditional solution in cooperative game theory. However, a potential limitation of the Shapley value is that it fails to fully take into account the participants’ excess contributions to the grand coalition, which may lead to a sense of unfairness among the members who have contributed more to the coalition, thus affecting the stability of the coalition and the willingness to continue to cooperate. Observing the data in Table 3, under the Shapley value revenue allocation scheme, the incremental revenue of the supplier and the manufacturer after joining the grand coalition are (35.3, 48.3, 63.8) and (33.7, 42.1, 57.5) million dollars (USD), respectively, reflecting the relatively balanced contributions of these two companies in the cooperation. The marginal contribution of suppliers and manufacturers to the grand coalition after joining the coalition are ${\tilde{v}}^C(1)=(72.6,98.3,133.0)$ and ${\tilde{v}}^C(2)=(77.3,101.8,141.7)$ million dollars (USD). According to the basic principle of the cooperative game, the greater the contribution to the coalition, the higher the corresponding earnings should be. However, in this case, although the manufacturer contributes more than the supplier and ultimately receives more than the supplier, its incremental revenue is lower than the supplier’s, which is a clear violation of the principle of reasonableness of the contribution-based allocation. This inconsistency reveals the limitations of the Shapley value in dealing with members with similar contributions. In contrast, this significant difference is effectively captured and reflected in the final distribution of benefits by the improved Banzhaf value in this paper, which more realistically reflects the excess contribution of suppliers relative to manufacturers in the total coalition. Under the improved Banzhaf value benefit distribution scheme proposed in this paper, the goal is to minimize participant dissatisfaction by considering the contributions of all participants and the contributions of sub-coalitions. The least squares contributions to the grand coalition from suppliers and manufacturers joining the grand coalition are ${\tilde{x}}_i^{w*}(1)=(41.9,54.8,72.6)$ and ${\tilde{x}}_i^{w*}(2)=(48.9,67.6,90.8)$, respectively. The adjusted benefit distribution more closely matches the actual contribution levels, thus providing a more rational solution. The incremental gains of suppliers and manufacturers are (35.4, 51.6, 68.2) and (39.1, 42.8, 59.2) million dollars (USD), respectively, which more realistically reflect their actual contribution levels in the coalition. This adjustment is not only in line with the principle of distribution according to work, but also provides a more effective incentive for high-contributing members to maintain and strengthen their cooperative relationships, thus enhancing the overall effectiveness and stability of the coalition. In summary, while the Shapley value ensures theoretical efficiency and fundamental fairness in the allocation, with each participant’s marginal contribution compensated accordingly, it is insufficient to deal with the non-equalization of members’ contributions within the coalition. The improved Banzhaf value proposed in this paper provides a more refined and practical approach to deal with the complexity of the distribution of gains in cooperative games by giving greater weight and compensation to excess contributions. This approach not only promotes the fairness of cooperation, but also enhances the harmony within the coalition and the sustainability of long-term cooperation.

The Banzhaf value is the final distribution value based on the marginal contribution of the coalition members across all sub-coalitions. In this case, the sum of the distribution values of each coalition member is (159.2, 217.3, 293.8) million dollars (USD), which significantly exceeds the total revenue value of the coalition of (155.2, 210.7, 287.3) million dollars (USD). In particular, it is evident in Figure 3 that the level of gains for Banzhaf values is generally higher. This over-distribution results in a blank check that cannot be honored, thus making this revenue distribution scheme difficult to achieve. The multiplicative normalization of the Banzhaf value, termed the B^m value [65], adjusts for excess benefits by proportionally reducing each member’s allocated share, resulting in a more practical and improved distribution method. However, both the Banzhaf value and the B^m value consider only the contributions of coalition members in their respective sub-coalitions, overlooking the contributions of other members within the sub-coalitions and the contributions of the sub-coalitions formed by the members to the grand coalition. This lack of comprehensive consideration of the complex dynamics within the coalition may result in members with higher contributions not receiving their due profits, thus affecting their satisfaction with the coalition and their willingness to cooperate in the future.

For the effect of weights on the benefit distribution results, the improved Banzhaf values without considering the weighting factor were obtained by setting all the weight values to 1 and bringing them into the operation, where all the participants were considered to be of equal importance and the distribution results depended only on the size of the contribution values. The results for the unweighted improved Banzhaf values are (42.8, 62.1, 82.3) (54.3, 69.2, 94.6), and (58.0, 79.4, 110.4), which shows that there is less difference in the benefits among the participants. This suggests that the unweighted approach maintains a certain degree of equilibrium despite the consideration of the triangular fuzzy number, but may not fully reflect the market influence and strategic position of each participant in actual business activities. The weighted improved Banzhaf values are (42.8, 61.8, 81.8), (54.3, 69.1, 94.5), and (58.1, 79.8, 111.1). The higher weighted participant 3 received a larger distribution, while the lower weighted participant 1 had a decrease in distribution. This makes the benefit distribution more refined and personalized, and better reflects the market position and actual contribution of the participants. The modified model is able to adjust the participant’s allocation results according to the actual situation, reflecting the actual position or influence of each participant in the coalition. Increased benefit distribution to high-weighted participants incentivizes them to invest further resources or contributions in the future, thus enhancing the overall benefits of the coalition. At the same time, the distribution result for low-weight participants is slightly lower, but the distribution result is more reasonable, which avoids the situation of “less contribution but too much distribution”, reduces the possible unfairness within the coalition, and contributes to the stability of the coalition in the long run. For example, for participants who invest fewer resources but take higher risks, the weightings could be adjusted to reflect this characteristic.

In addition, Figure 3 provides a clear view of the range and distributional characteristics of benefits affecting each participant by different methods of benefit distribution. The Shapley value demonstrates a relatively even distribution of gains, reflecting its equal treatment of each member, but lacks sensitivity to differences in member contributions. In contrast, Banzhaf values show slightly higher differences in earnings, reflecting the marginal contribution of each participant in the different sub-coalitions. However, Banzhaf values may over-recognize participants who provide contributions in multiple possible coalitions, which does not always accurately represent their true importance to the overall coalition. The less hierarchical variation of the B^m value demonstrates effective control of excess allocation to avoid the problem of the sum of allocations exceeding the total coalition return. Moreover, from a three-dimensional perspective, all three approaches show that 3PLs have a wider range of benefits. It highlights the central role of 3PLs in supply chain coalitions and their key contribution in integrating upstream and downstream and improving overall operational efficiency. In contrast, the distribution of benefits for suppliers and manufacturers varies less across the allocation methods, showing a more compact distribution of benefits. While this compactness can enhance fairness in benefit distribution to some extent, it also presents the risk of over-averaging, which may obscure the strategic importance and actual contribution of 3PLs, and consequently, fail to fully capture their true influence within the coalition. A lack of sensitivity to the varying market positions and strategic importance of each member may not adequately motivate high contributors and lacks the flexibility to adapt to market changes. Especially in a market environment with rapidly changing supply chain dynamics, such an inflexible revenue-sharing model may impede the long-term stability and development of the coalition. The improved Banzhaf value proposed in this paper introduces weight adjustments that significantly reflect the hierarchy and variability, effectively aligning with the actual contributions and market positions of each participant. This approach ensures that high contributors receive their due rewards, thus enhancing fairness and incentives in coalitions.

Therefore, the improved Banzhaf value based on least squares contributions proposed in this paper not only comprehensively considers the contributions of all participants, the contributions of sub-coalitions, and the weights of participants, but also bases itself on the least squares contributions of participants to minimize dissatisfaction. This approach prevents situations where participants contribute significantly to the coalition but receive low profit shares. Moreover, the introduction of weights effectively solves the problem that the differences in the actual contributions of participants are not adequately reflected (e.g., resource input, risk taking, market position, etc.). For high-weighted participants (e.g., Participant 3), the allocation result is increased, in line with the important role they assume in the coalition. For low-weighted participants (e.g., Participant 1), the allocation result is reduced but more in line with their contribution level, which helps to reduce allocation conflicts within the coalition and enhance the fairness of the allocation scheme. It ensures that profit distribution is not only fair but also closely reflects the actual contributions and market influence of each member. Thus, it more effectively motivates member cooperation and contribution, enhancing the long-term stability and cooperative benefits of the coalition.

6. Conclusions

By introducing triangular fuzzy numbers to represent the coalition value, this study effectively describes the uncertainty and ambiguity in the cooperation process. It avoids reliance on deterministic assumptions in traditional benefit distribution methods, and thus better meets the practical application scenarios. In addition, considering the influence of participants’ weights on the cooperative profit allocation strategy, the fuzzy comprehensive evaluation method is used to determine the weights, which better balances the fairness and incentives, and fully reflects the actual contributions of the participants. By minimizing the dissatisfaction of all participants, the weighted least squares contribution is constructed to reduce the difference between expected and actual benefits, replacing the marginal contribution of the classical Banzhaf value. To satisfy the principle of wholeness, the model is subjected to the necessary validity improvements, and an improved Banzhaf value based on participant’s triangular fuzzy number-weighted excess contributions is proposed. By combining the weights and fuzzy numbers, the model not only effectively addresses the uncertainty in the benefit distribution process, but also meets the incentive needs of high-weighted participants and reflects fairness to low-weighted participants, thus promoting the stability and sustainable development of the coalition. This improvement significantly enhances the practicality and scientific of Banzhaf value in actual coalition benefit distribution, which is especially applicable to complex and uncertain multi-party cooperation scenarios such as supply chain coalitions. Moreover, the theoretical results of this study also support the development of solutions to similar problems in other fields, such as logistics, international cooperation, and financial coalitions, which promote the construction of green supply chains and cooperation in environmental governance, and provide solid theoretical support for the realization of the goal of sustainable development.

The results of the study show that the improved Banzhaf value is not only innovative in theory but also operational in practice. The model takes into account the contributions of all participants, the contributions of sub-coalitions, and the weights of the participants, providing a fairer and more efficient benefit distribution strategy for manufacturing supply chain coalitions. In particular, the validity of the model is verified through an example analysis, which satisfies individual rationality, coalition rationality, and overall rationality (efficiency), and significantly improves the stability and cooperative benefits of the coalition. The theoretical and methodological innovations of this paper provide a new development direction for cooperative game theory, which foretells the prospect of its application in a wider range of social, economic, and industrial problems. The results of this study not only deepen the theoretical foundations of cooperative game theory but also expand its possibilities in practical applications and future development. The improved Banzhaf-valued benefit distribution model can help various economic entities deal with benefit distribution more fairly and efficiently, and at the same time provides a data-driven and theoretically-supported framework for policy makers to design decision-support systems involving economic and social policies.

The flexibility and adaptability of this model enable it to cope with dynamic changes in the supply chain, assisting supply chain managers in making more rational decisions within a complex market environment. This adaptation helps to better manage market competition while maintaining supply chain stability and improving overall efficiency. Additionally, the model provides a scientific basis for benefit distribution in multi-party collaboration within the manufacturing supply chain. It can serve as a reference for formulating policies aimed at promoting the integration of small- and medium-sized enterprises (SMEs) into the supply chain and other industrial collaboration policies, thereby fostering the development of supply chain cooperation towards long-term stability. However, the current model depends on accurate weight allocation and precise estimation of triangular fuzzy numbers, which enhances the rationality and accuracy of the allocation but also increases computational complexity. Consequently, the applicability of the model may be limited in large-scale supply chains or scenarios requiring rapid decision making. Future research could investigate how model parameters might be adjusted with limited data access, or how the model could be applied to different types of cooperative games and industry contexts. Additionally, further research into environmental impacts and social responsibility could broaden the range of applications for this model.

In conclusion, the improved Banzhaf value based on participant’s triangular fuzzy number-weighted excess contributions proposed in this paper provides a new perspective and an effective tool for understanding and optimizing the benefit distribution problem in manufacturing supply chain coalitions, ensuring that each member’s contribution is reasonably rewarded, thus motivating closer cooperation and common development. In the future, we will consider further exploring the dynamic adjustment mechanism and multi-dimensional weight design, combined with intelligent optimization algorithms (e.g., genetic algorithm, particle swarm optimization algorithm) to solve the optimal allocation scheme, especially in large-scale coalitions. In addition, the impact of member withdrawal, accession, or change in coalitions on benefit distribution and coalition stability will be further studied to promote the study of dynamic optimization of coalition structure. This will not only enhance the applicability of the model, but also provide a more comprehensive solution to the benefit distribution problem in complex coalitions and promote the long-term stability and sustainable development of coalitions.