Improved Shapley Value with Trapezoidal Fuzzy Numbers and Its Application to the E-Commerce Logistics of the Forest Products

Abstract

With the rapid development of e-commerce, the traditional trade mode of forest products has undergone significant changes. Logistics as a key factor to support e-commerce trade is particularly important to build a reasonable e-commerce logistics model of forest products. In the logistics service industry, the issue of cooperative profit allocation of logistics alliance has been crucial and prevalent. However, logistics alliances often face the problem of incomplete information, such as the ambiguity of transportation cost and driving distance, which makes it difficult to effectively apply many classical cooperative game solutions. Therefore, this paper introduces an improved Shapley value for cooperative games in fuzzy situations regarding unclear profit allocation in e-commerce logistics alliance of forest products. This value maximizes the satisfaction of the players by minimizing the contribution excess, according to which the trapezoidal fuzzy number least square contribution is calculated. Based on this, we replace the marginal contribution of the classical Shapley value with this least square contribution, thus creating the improved Shapley value with trapezoidal fuzzy numbers. Through the verification of actual cases, this method not only has theoretical value, but also provides effective guidance for the actual profit allocation of e-commerce logistics alliance of forest products, which helps to promote the stability and sustainable development of alliance.

1. Introduction

In the contemporary epoch, with the rapid development of e-commerce, the number of e-commerce logistics enterprises of forest products continues to grow, and the market competition is also increasingly fierce. In order to effectively deal with such a competitive situation and effectively avoid market risks, e-commerce logistics enterprises of forest products must pay more attention to improve their competitiveness [1,2,3]. The establishment of an e-commerce logistics alliance [4] and the implementation of joint transportation have transmuted into an efficacious strategy. Within an e-commerce logistics alliance of forest products, the profitability of the alliance is affected by a number of factors, including transportation vehicles, freight volume, fuel prices, transportation distance, and customer satisfaction. A fair and reasonable distribution of alliance profits is essential to ensure the long-term stable development of alliances.

In academic research, the matters of profit allocation [5,6] and cost sharing [7,8] within e-commerce logistics alliances of forest products have been encompassed within the domain of cooperative game theory. Theoretically, numerous renowned solutions of cooperative games, such as the Shapley value [9] and the Banzhaf value [10], can be utilized to redress these problems. However, these classical solutions are mainly applicable to exact contexts and have limited effectiveness in dealing with problems in fuzzy contexts. In addition, the classical Shapley value is mainly based on the marginal contribution of the players and ignores the satisfaction of the players in making profit allocation; thus, it is not comprehensive and effective in making profit allocation.

Forest products e-commerce enterprises usually use container transportation in the export process. However, the mismatch between the type of container vehicles and the freight volume and the incompatibility of business models have resulted in a high rate of empty containers and a low utilization rate. This in turn affects the profits of e-commerce logistics enterprises of forest products, leading to a generally low level. Therefore, many e-commerce logistics enterprises of forest products tend to form logistics alliances [11,12,13] for common transportation to reduce transportation costs and increase profits. Nevertheless, in order to realize the long-term and stable development of the alliance, it is of vital importance to distribute alliance profit fairly and equitably. In order to realize the long-term stable development of alliances and solve the problem of unfair profit allocation of alliances, this paper proposes a method of improved Shapley value with trapezoidal fuzzy numbers [14]. The method not only satisfies the important properties of classical Shapley value, such as existence, efficiency, additivity, and symmetry, but it also provides a more reasonable profit allocation scheme with full consideration of player satisfaction.

This paper makes at least the following three contributions to the field of cooperative game:

(1)

This paper takes into account the dissatisfaction of the players in the game when utilizing the Shapley value for profit allocation. The least square contribution is used to replace the marginal contribution of the classical Shapley value, which makes the profit allocation more reasonable;

(2)

In the calculation of the efficiency of the improved Shapley value, instead of simply distributing the difference between the sum of the initial distribution of profits of the players in the innings and the profits of the grand alliance equally to each player, this paper takes into account the profits made by the individual players when they strike out alone and makes a more reasonable secondary distribution accordingly;

(3)

In this paper, trapezoidal fuzzy numbers are introduced to further optimize the improved Shapley value so that it has wider applicability in dealing with fuzzy numbers, breaking through the limitation that the classical Shapley value has with fuzzy numbers.

The rest of the paper is organized as follows: Section 2 reviews the related literature; Section 3 introduces the preliminary contents closely related to the study of this paper, including trapezoidal fuzzy numbers and the contribution excess of the players; Section 4 proposes the least square contribution of trapezoidal fuzzy numbers with trapezoidal fuzzy numbers that take into account the contribution excess of the players and further improves the Shapley value with it, also proving the nature of the improved Shapley value; Section 5 verifies the applicability and superiority of the improved Shapley value with trapezoidal fuzzy numbers proposed in this paper through a real case of profit allocation in e-commerce logistics alliance of forest products; Section 6 summarizes the main conclusions and discusses the possible directions of future research. Figure 1 shows the research flowchart of this study.

2. Literature Review

In cooperative games, there are some classical solutions for profit allocation, such as Shapley value [9] and Banzhaf value [10]. However, these solutions are usually applied when the profit information of the players and the coalition is precisely and completely known. In fuzzy environments, these classical cooperative game solutions are often limited to effectively solve practical problems, and there is still a gap in the research methodology. In addition, the classical Shapley value mainly focuses on the marginal contribution of the players and ignores the dissatisfaction of the players. To address this issue, this paper aims to determine the least square contribution when the satisfaction of the player is maximized by introducing the least squares method. We plan to replace the marginal contribution in the classical Shapley value with this least square contribution and improve the Shapley value based on it. Finally, this paper will also use trapezoidal fuzzy numbers to extend the application of the improved Shapley value to ensure that the improved Shapley value can be applied in fuzzy environments and that the profit allocation results are both fair and reasonable and consistent with the axiomatic system of the classical Shapley value.

For logistics alliances, the cooperative profit allocation scheme is crucial. A fair and reasonable distribution scheme can enhance the stability of the alliance; on the contrary, if the logistics enterprises are unable to obtain satisfactory profits from the cooperation, it may lead to the disintegration of the alliance [15]. In view of the fuzzy nature of the cost and profit of cooperation of some logistics enterprises, this paper studies in depth the solution of the cooperation game in a fuzzy environment. In order to explore the profit allocation solution of e-commerce logistics alliance of forest products under the fuzzy environment, this paper reviews the following literature topics: (1) cooperative game, (2) fuzzy cooperative game, (3) contribution excess, (4) least square method, and (5) e-commerce logistics of forest products. Among them, the cooperative game and fuzzy cooperative game literature form the basis of this paper’s research. And the improved Shapley value with trapezoidal fuzzy numbers proposed in this paper is inspired by the contribution of the literature on excess and on the least square method. Finally, this paper reviews the literature on e-commerce logistics of forest products to provide a reference scheme of cooperative profit allocation for e-commerce logistics alliances of forest products. For the literature on cooperative games and fuzzy cooperative games, this paper summarizes the classical cooperative game solutions and their properties; the literature on contribution excess provides theoretical support for this paper’s definition of the contribution excess of the players; the literature on the least square method elucidates a way to minimize the dissatisfaction of the players; and the literature on e-commerce logistics of forest products allows us to summarize some of the unique characteristics of e-commerce logistics of forest products.

2.1. Cooperative Game

Cooperative game theory is an important area in game theory, dedicated to solving the problem of how multiple participants can fairly distribute profits in cooperation. The importance of the theory lies in the fact that it provides a solid theoretical foundation and practical solutions for dealing with complex cooperative decisions. In cooperative games, the main theories include Shapley value [9], Banzhaf value [10], core [16], and nucleolus [17], etc. Shapley value provides a fair distribution method based on contribution by quantifying the marginal contribution of each participant to the cooperative outcome; Banzhaf value focuses on evaluating the influence of each participant on decision making in the cooperation, reflecting its contribution to the final outcome; core emphasizes the achievement of stability in the cooperative game by ensuring that no subset of participants can derive higher profits; and nucleolus seeks a fair profit allocation scheme by minimizing the level of dissatisfaction of participants.

In recent years, research on cooperative game theory has been expanding into emerging areas such as logistics alliances and supply chain management. In logistics alliances, these models are widely used to solve the problem of profit allocation. In the case of on-site logistics alliance, the participants of the cooperative alliance formed a stable cooperative structure under the guidance of Shapley value profit allocation and realized the desired expected returns [18]. In the case of cooperation between retailers and third-party logistics providers, the Shapley value distribution model, improved according to the actual operation, can align well with the profits of the participating entities and positively correlate with their contributions, ensuring a fair and reasonable distribution of profits [19]. In supply chains, cooperative game models help to optimize resource allocation and achieve profit sharing, thus improving the efficiency and stability of the entire supply chain [20]. Nevertheless, existing studies still face challenges such as computational complexity and model adaptability, and new methods and techniques are urgently needed to enhance the practical application of cooperative game models.

2.2. Fuzzy Cooperative Game

Fuzzy cooperative game is an expansion of cooperative game, which aims to solve the cooperation problem of multiple participants under uncertainty. Different from classical cooperative games, fuzzy cooperative games introduce fuzzy set theory [21,22], which allows the contribution and cooperation level of each participant to be fuzzy, so as to better reflect the uncertainty and ambiguity in practice. The core of this theory is the concepts of fuzzy Shapley value and fuzzy core, which are based on the extension of the framework of the classical cooperative game and are applicable to the problem of profit allocation in an uncertain environment. Fuzzy Shapley value provides a more flexible allocation by considering the marginal contribution of each participant under different levels of ambiguity [23], while fuzzy core is dedicated to ensuring the stability of cooperation in ambiguous environments so that no subset of participants will benefit by splitting [24].

Currently, fuzzy cooperative game theory has been widely used in many fields. In supply chain management, it effectively solves the problem of cost allocation and profit allocation due to market demand fluctuations, production cycle changes, and other uncertainties [25]. In the field of logistics, it helps logistics alliances to solve the challenges of transportation capacity allocation, resource sharing, and cost sharing due to fuzzy contribution [26]. Overall, the fuzzy cooperative game introduces the profit allocation scheme in the fuzzy environment and effectively solves the actual profit allocation problem of the players in the game.

2.3. Contribution Excess

Contribution excess is a key concept in cooperative game theory, which focuses on the difference between the player’s contribution to the alliance and the player’s payment vector. Contribution excess not only helps to assess the value of the player in the alliance as a whole but also reflects the dissatisfaction of the player in the profit allocation, thus reflecting the fairness in the profit allocation.

The concept of contribution excess has been widely used in many fields, including logistics, supply chain, and rural e-commerce. In logistics, contribution excess is used to analyze the collaborative profit allocation scheme of logistics alliance under incomplete information, which focuses on how to effectively manage and optimize contribution excess under incomplete information conditions to achieve fairer and more efficient resource allocation [27]. In the field of rural e-commerce, contribution excess and least square method are used to improve Shapley value, take into account the dissatisfaction of the players, and optimize the profit allocation model in order to better reflect the impact of contribution excess on profit allocation [28]. The above literature demonstrates the application and management strategies of contribution excess in different scenarios, which fully reflects its important role in the fair distribution of resources and profits.

2.4. Least Square Method

The least square method is an important method for solving resource allocation problems in cooperative games. It is based on the principle of minimizing variance and aims at distributing cooperative profits in a fair way. The least square method is widely used in economics, management science, engineering, and other fields.

The basic idea of the least square method is to minimize the variance of the contribution excess of the players in the game by integrating their contributions and external factors. In cooperative games, the least squares method has been applied to a variety of environments, for example, in logistics enterprise alliances, the profit allocation strategy is formulated to promote long-term development [27]; in rural e-commerce enterprise cooperation, the stability of cooperation is enhanced by optimizing the profit allocation strategy [28]; and in supply chain alliances, the profit allocation strategy is optimized to improve the efficiency of supply chain management [29]. Numerous studies have shown that the least square method has a wide range of applications in many fields, such as logistics and supply chain, which not only effectively solves the problems of logistics cost and profit allocation but also provides practical theoretical guidance for actual operation.

2.5. E-Commerce Logistics of Forest Products

E-commerce logistics of forest products is a key link between forest products e-commerce and e-commerce logistics, and its current situation demonstrates complexity [30,31] and dynamism [32]. The rapid development of e-commerce has dramatically boosted the growth of consumer demand for forest products [33], which in turn has contributed to the increased demand for e-commerce logistics of forest products [34], while at the same time, consumers’ requirements for delivery speed [35,36] and convenience [37,38] continue to increase. This growing demand prompts e-commerce platforms and logistics service providers to continuously optimize their logistics networks [39,40] and service capabilities [41,42].

In the field of e-commerce logistics of forest products, enterprises often need to deal with complex supply chain systems, variable transportation needs, and high logistics costs [43]. As a new logistics model, cooperative transportation is gradually becoming a key strategy for e-commerce logistics enterprises of forest products to reduce transportation costs and improve operational efficiency. First of all, e-commerce logistics enterprises of forest products can integrate multiple resources through cooperative transportation, thus realizing a significant reduction in logistics costs. Through cooperation with other enterprises or logistics companies, these enterprises can share transportation resources [44,45], such as vehicles and storage facilities, using Internet of Things technology, which in turn effectively reduces logistics investment. At the same time, cooperative transport can also optimize the transport route [46] and improve the loading rate [47] and other means to further reduce the transportation cost per unit of product. Secondly, cooperative transportation helps to improve logistics efficiency. Through cooperative transportation, enterprises can more flexibly deploy transportation resources to ensure that products can be timely and accurately delivered to consumers, thereby enhancing customer satisfaction [48,49,50]. In addition, cooperative transportation can also help enterprises better respond to market changes and reduce business risks.

In summary, e-commerce logistics enterprises of forest products through cooperative transportation can integrate resources, improve efficiency, reduce transportation costs, and then improve the overall operational efficiency, providing strong support for the sustainable development of enterprises.

3. Preliminaries

In this paper, we apply the theory of cooperative game to solve the problem of profit distribution in e-commerce logistics alliance of forest products. Based on the basic assumptions of cooperative games, we first assume that all participants are rational; i.e., they all pursue the maximization of their own interests. Second, we assume that all participants understand the rules of the game and recognize that other participants are also rational. More importantly, this rational knowledge is common; i.e., each participant knows that the other participants are equally aware that they know these rules. This common knowledge not only ensures the transparency of the game but also promotes fairness. In the following, we elaborate on the knowledge related to cooperative games that needs to be used in this paper.

3.1. Trapezoidal Fuzzy Number and α-Cut

3.1.1. Definition of Trapezoidal Fuzzy Numbers

To accurately reflect the complexity and uncertainty in real-world decision-making processes, fuzzy set theory has emerged as an effective approach. In this theory, trapezoidal fuzzy numbers are considered an important component. They are highly regarded for their ability to store a large amount of information, accurately characterize information features, and offer flexibility.

Definition 1.

For each trapezoidal fuzzy number $\tilde{a}=[a_L,a_{M1},a_{M2},a_R]$, its membership function is given by the following:

$$\tilde{a}(x)=\left\{\begin{array}{ll}(x-a_L)/(a_{M1}-a_L), \hfill & a_L\le x\le a_{M1}\hfill \\ 1,\hfill & a_{M1}<x\le a_{M2}\hfill \\ (a_R-x)/(a_R-a_{M2}),\hfill & a_{M2}<x\le a_R\hfill \\ 0,\hfill & x<a_L \& x>a_R\hfill \end{array}\right.$$

(1)

In this context, $[a_{M1},a_{M2}]$ represents the mean interval of the trapezoidal fuzzy number $\tilde{a}$. The most likely values are taken within the interval $[a_{M1},a_{M2}]$, where $a_L$ and $a_R$ are the lower and upper bounds, respectively. The trapezoidal fuzzy number can only take values within the interval $[a_L,a_R]$. Specifically, when $a_{M1}=a_{M2}$, the trapezoidal fuzzy number degenerates into a triangular fuzzy number. When $a_L=a_{M1}$ and $a_{M2}=a_R$, it degenerates into an interval number, and when $a_L=a_{M1}=a_{M2}=a_R$, it becomes a precise number. Thus, triangular fuzzy numbers, interval numbers, and precise numbers can all be considered special cases of trapezoidal fuzzy numbers.

3.1.2. The α-Cut of Trapezoidal Fuzzy Numbers

For the α-cut of the trapezoidal fuzzy number $\tilde{a}=[a_L,a_{M1},a_{M2},a_R]$, it can be represented as $\tilde{a}(\alpha )=\{x\in R|\tilde{a}(x)\ge \alpha \}$, where $\alpha \in [0,1]$. For different $\alpha$, the α-cut of the trapezoidal fuzzy number $\tilde{a}$ varies. However, any α-cut of a trapezoidal fuzzy number can be represented by an interval number in which $\tilde{a}(\alpha )=[a_L(\alpha ),a_R(\alpha )]$. Specifically, when $\alpha =1$, the α-cut is $\tilde{a}(1)=[a_{M1},a_{M2}]$, and when $\alpha =0$, the α-cut is $\tilde{a}(0)=[a_L,a_R]$. Based on the α-cut formula, the trapezoidal fuzzy number $\tilde{a}=[a_L,a_{M1},a_{M2},a_R]$ can be rewritten in the following α-cut form:

$$\begin{array}{ll}\tilde{a}(\alpha )& =[a_L(\alpha ),a_R(\alpha )]=[a_L+\alpha (a_{M1}-a_L),a_R+\alpha (a_{M2}-a_R)]\\ & =[(1-\alpha )a_L+\alpha a_{M1},(1-\alpha )a_R+\alpha a_{M2}]\end{array}$$

(2)

where $a_L(\alpha )$ and $a_R(\alpha )$ represent the lower and upper bounds of $\tilde{a}(\alpha )$ for the α-cut, respectively. The trapezoidal fuzzy number and its α-cuts are illustrated in Figure 2.

Figure 3 shows an arbitrary triangular fuzzy number and its α-cut. Compared to the triangular fuzzy numbers, the trapezoidal fuzzy number function is characterized by a gradual decrease after it first rises and keeps the maximum affiliation constant over an interval. This characteristic makes the trapezoidal fuzzy number more flexible to fit the uncertainty in many realistic scenarios. Also, when the maximum affiliation interval of the trapezoidal fuzzy number is short enough, it is practically equal to the triangular fuzzy number. In view of these advantages of trapezoidal fuzzy numbers, this paper chooses trapezoidal fuzzy numbers to improve Shapley value with trapezoidal fuzzy numbers in order to deal with the problem of profit distribution in cooperative games more accurately.

3.1.3. Basic Operations of Trapezoidal Fuzzy Numbers

Let there be two arbitrary trapezoidal fuzzy numbers: $\tilde{a}=[a_L,a_{M1},a_{M2},a_R]$ and $\tilde{b}=[b_L,b_{M1},b_{M2},b_R]$. Then, the addition and scalar multiplication operations of trapezoidal fuzzy numbers are represented by Formulas (3) and (4), respectively:

$$\begin{array}{l}\tilde{a}+\tilde{b}=[a_L+b_L,a_{M1}+b_{M1},a_{M2}+b_{M2},a_R+b_R]\\ \tilde{a}-\tilde{b}=[a_L-b_R,a_{M1}-b_{M2},a_{M2}-b_{M1},a_R-b_L]\end{array}$$

(3)

$$\lambda \tilde{a}=\left\{\begin{array}{l}[\lambda a_L,\lambda a_{M1},\lambda a_{M2},\lambda a_R],\lambda \ge 0\\ [\lambda a_R,\lambda a_{M2},\lambda a_{M1},\lambda a_L],\lambda <0\end{array}\right.$$

(4)

Let $\tilde{a}(\alpha )$ and $\tilde{b}(\alpha )$ be the α-cuts of two arbitrary trapezoidal fuzzy numbers $\tilde{a}\in N(R)$ and $\tilde{b}\in N(R)$:

$$\tilde{a}(\alpha )=\tilde{b}(\alpha ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} a_L(\alpha )=b_L(\alpha ) \mathrm{and} a_R(\alpha )=b_R(\alpha )$$

(5)

$$\tilde{a}(\alpha )\le \tilde{b}(\alpha ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} a_L(\alpha )\le b_L(\alpha ) \mathrm{and} a_R(\alpha )\le b_R(\alpha )$$

(6)

$$\tilde{a}(\alpha )<\tilde{b}(\alpha ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} \tilde{a}(\alpha )\le \tilde{b}(\alpha ) \mathrm{and} \tilde{A}(\alpha )\ne \tilde{B}(\alpha )$$

(7)

Moreover, according to the binary expansion theorem, the operation between two trapezoidal fuzzy numbers $\tilde{a}(\alpha )$ and $\tilde{b}(\alpha )$ as well as the number multiplication operation can be obtained.

$$\tilde{a}(\alpha )+\tilde{b}(\alpha )=(a_L(\alpha )+b_L(\alpha ),a_R(\alpha )+b_R(\alpha ))$$

(8)

$$\tilde{a}(\alpha )-\tilde{b}(\alpha )=(a_L(\alpha )-b_R(\alpha ),a_R(\alpha )-b_L(\alpha ))$$

(9)

$$\lambda \tilde{a}(\alpha )=\left\{\begin{array}{l}(\lambda a_L(\alpha ),\lambda a_R(\alpha )),\lambda \ge 0\\ (\lambda a_R(\alpha ),\lambda a_L(\alpha )),\lambda <0\end{array}\right.$$

(10)

3.2. Trapezoidal Fuzzy Number Cooperative Game

The trapezoidal fuzzy numbers cooperative game belongs to the category of fuzzy cooperative game. Through the use of trapezoidal fuzzy numbers, various parameters and variables in complex systems can be modelled more accurately to ensure the accuracy and reliability of the decision-making process. Therefore, cooperative games with trapezoidal fuzzy numbers have a wide range of potentials in practical applications and can help analyze and solve complex problems in real life.

A trapezoidal fuzzy number cooperative game can be represented as $(N,\tilde{v})$, where $N=\{1,2,\dots ,n\}$ denotes a finite set containing $n$ players, and $N$ represents the grand alliance consisting of all players. For any sub-alliance $S(S\subseteq N)$, let $s(s=1,2,\dots ,n)$ denote the number of players in the sub-alliance $S$ and $\tilde{v}(S)$ represent the payoff from the cooperative game for the players in alliance $S$, denoted as $\tilde{v}(S)=[v_L(S),v_{M1}(S),$ $v_{M2}(S),v_R(S)]$. According to the operational rules, the α-cut of the alliance’s payoff can be expressed as follows:

$$\tilde{v}(S)(\alpha )=[v_L(S)(\alpha ),v_R(S)(\alpha )]=[(1-\alpha )v_L(S)+\alpha v_{M1}(S),(1-\alpha )v_R(S)+\alpha v_{M2}(S)]$$

(11)

3.3. Contribution Excess of Trapezoidal Fuzzy Numbers

Each player $i(i\in N)$ desires to obtain more benefits from the cooperative profit, has expectations for the final payment of the cooperation, and hopes that it can match their own role in the alliance, that is, their contribution in the alliance. However, since the cooperative profit is fixed, during the allocation process, it is necessary to find a balance point among the players to maximize the sum of the satisfaction degrees of the players in the profit allocation. Therefore, this paper introduces the definition of contribution excess of players.

Definition 2.

For each payment vector $\overline{x}\in R$ and player $i(i\in N)$,

$$e^C(i,\overline{x})={\tilde{v}}^C(i)-\overline{x}(i)(i\in N)$$

(12)

is called the contribution excess of player $i$, where ${\tilde{v}}^C(i)=\tilde{v}(N)-\tilde{v}(N\backslash i)$.

$e^C(i,\overline{x})$ denotes the difference between a player’s contribution to the grand alliance and its payment vector, and this difference represents the player’s dissatisfaction.

Definition 3.

For the α-cut of any trapezoidal fuzzy number,

$$e^C(i,\overline{x})(\alpha )={({\tilde{v}}_{Li}^C(\alpha )-{\overline{x}}_{Li}(\alpha ))}^2+{({\tilde{v}}_{Ri}^C(\alpha )-{\overline{x}}_{Ri}(\alpha ))}^2$$

(13)

is called the contribution excess of players based on the α-cut of a trapezoidal fuzzy number, where ${\tilde{v}}_{Li}^C(\alpha )={\tilde{v}}_L(N)(\alpha )-{\tilde{v}}_L(N\backslash i)(\alpha )$ and ${\tilde{v}}_{Ri}^C(\alpha )={\tilde{v}}_R(N)(\alpha )-{\tilde{v}}_R(N\backslash i)(\alpha )$.

For the convenience of description, let $e_L^C(i,\overline{x})(\alpha )={\tilde{v}}_{Li}^C(\alpha )-{\overline{x}}_{Li}(\alpha )$ and $e_R^C(i,\overline{x})(\alpha )=$${\tilde{v}}_{Ri}^C(\alpha )-{\overline{x}}_{Ri}(\alpha )$, respectively, denote the left and right excesses of the square contribution excess of player $i\in N$ in the trapezoidal fuzzy number cooperative game. Then, $e^C(i,\overline{x})(\alpha )$ can be expressed as follows:

$$e^C(i,\overline{x})(\alpha )={(e_L^C(i,\overline{x})(\alpha ))}^2+{(e_R^C(i,\overline{x})(\alpha ))}^2$$

(14)

Let $e^C(i,\overline{x})(\alpha )$ denote the contribution excess of player $i(i\in N)$ when the payment vector of the trapezoidal fuzzy number is $\overline{x}$. The greater $e^C(i,\overline{x})(\alpha )$ is, the lower the satisfaction degree of player $i(i\in N)$ in the alliance $S$; on the contrary, the smaller $e^C(i,\overline{x})(\alpha )$ is, the higher the satisfaction degree of player $i(i\in N)$ in the alliance $S$.

4. Model Construction

In this section, we present the profit allocation model used in this paper. First, we compute the least square contribution of trapezoidal fuzzy numbers considering contribution excess of players. Second, we use this least square contribution to replace the marginal contribution in the classical Shapley value to obtain the profit allocation model proposed in this paper, i.e., the improved Shapley value with trapezoidal fuzzy numbers considering the contribution excess of the players. Finally, we show that this improved Shapley value with trapezoidal fuzzy numbers also satisfies several important properties of the classical Shapley value. Figure 4 shows the technical route for improved Shapley value with trapezoidal fuzzy numbers.

In order to present the findings of this paper more clearly, the variables used in this paper are summarized in

Table 1. Variables used in this article.

Variables	Meaning	Variables	Meaning
$\tilde{v}(i)$	Characteristic function of player $i$	$\tilde{v}(S)$	Characteristic function of alliance $S$
$\tilde{v}(N)$	Characteristic function of grand alliance $N$	$\overline{x}(\alpha )$	Payment vector in α-cut
${e_L}^C(i,\overline{x})(\alpha )$	The left endpoint of contribution excess of player $i$ in α-cut	${e_R}^C(i,\overline{x})(\alpha )$	The right endpoint of contribution excess of player $i$ in α-cut
${{\overline{e}}_L}^C(i,\overline{x})(\alpha )$	The left endpoint of the average of contribution excess player $i$ in α-cut	${{\overline{e}}_R}^C(i,\overline{x})(\alpha )$	The right endpoint of the average of contribution excess player $i$ in α-cut
${\overline{x}}_{Li}(\alpha )$	The left endpoint of least square contribution of player $i$ in α-cut	${\overline{x}}_{Ri}(\alpha )$	The right endpoint of least square contribution of player $i$ in α-cut
${\tilde{v}}_L(N)(\alpha )$	The left endpoint of characteristic function of grand alliance $N$ in α-cut	${\tilde{v}}_R(N)(\alpha )$	The right endpoint of characteristic function of grand alliance $N$ in α-cut
${\tilde{v}}_{Li}^C(\alpha )$	The left endpoint of the value of player $i$’s contribution to the grand alliance in α-cut	${\tilde{v}}_{Ri}^C(\alpha )$	The right endpoint of the value of player $i$’s contribution to the grand alliance in α-cut
${\tilde{v}}_{Lj}^C(\alpha )$	The left endpoint of the value of player $j$’s contribution to the grand alliance in α-cut	${\tilde{v}}_{Rj}^C(\alpha )$	The right endpoint of the value of player $j$’s contribution to the grand alliance in α-cut
${\overline{x}}_i^{\ast }(\alpha )$	The least square contribution of player $i$ in α-cut	${\overline{x}}_i^{\ast }(S)$	The least square contribution of alliance $S$
$SH{L}_i^{\ast }(\tilde{v})$	Initial improved Shapley values	$SH{L_i^{\ast }}^{\prime }(\tilde{v})$	Final improved Shapley value
$SH{L_{Li}^{\ast }}^{\prime }\left(\tilde{v}\right)\left(\alpha \right)$	The left endpoint of final improved Shapley value in α-cut	$SH{L_{Ri}^{\ast }}^{\prime }\left(\tilde{v}\right)\left(\alpha \right)$	The right endpoint of final improved Shapley value in α-cut

as follows.

4.1. Least Square Contribution of Trapezoidal Fuzzy Numbers Considering Contribution Excess of Players

Inspired by the utilization of the least square method by Liu et al. [27] to minimize the total dissatisfaction of the players, with the aim of balancing the dissatisfaction among the players and minimizing the variance of the contribution excess of the players in the grand alliance, a quadratic programming model for selecting the optimal payment vector is established.

Problem 1.

$\mathrm{Minimize} [{\displaystyle \sum _{i\in N}{({e_L}^C(i,\overline{x})(\alpha )-{{\overline{e}}_L}^C(i,\overline{x})(\alpha ))}^2}+{({e_R}^C(i,\overline{x})(\alpha )-{{\overline{e}}_R}^C(i,\overline{x})(\alpha ))}^2],$

$$\mathrm{s}.\mathrm{t}. \left\{\begin{array}{c}{\displaystyle \sum _{i\in N}{\overline{x}}_{Li}(\alpha )}={\tilde{v}}_L(N)(\alpha )\\ {\displaystyle \sum _{i\in N}{\overline{x}}_{Ri}(\alpha )}={\tilde{v}}_R(N)(\alpha )\end{array}\right.$$

(15)

When the final payment vector of all players is $\overline{x}(\alpha )$, ${{\overline{e}}_L}^C(i,\overline{x})(\alpha )$ and ${{\overline{e}}_R}^C(i,\overline{x})(\alpha )$ denote the average value of the contribution surplus. The average value is as follows:

$$\left\{\begin{array}{l}{{\overline{e}}_L}^C(i,\overline{x})(\alpha )={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i\in N}{e_L}^C(i,\overline{x})}(\alpha )\\ {{\overline{e}}_R}^C(i,\overline{x})(\alpha )={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i\in N}{e_R}^C(i,\overline{x})}(\alpha )\end{array}\right.$$

(16)

For any cooperative game, for any payment vector that satisfies the conditions, the sum of the values of the contribution excess all players is the same. Therefore, for the trapezoidal fuzzy number cooperative game, as long as the payment vector $\overline{x}(\alpha )$ is valid, ${\displaystyle \sum _{i\in N}{e_L}^C(i,\overline{x})}(\alpha )$ and ${\displaystyle \sum _{i\in N}{e_R}^C(i,\overline{x})}(\alpha )$ are a constant under the same α-cut. Thus, ${{\overline{e}}_L}^C(i,\overline{x})(\alpha )$ and ${{\overline{e}}_R}^C(i,\overline{x})(\alpha )$ are also a constant, and any value can be used to replace ${{\overline{e}}_L}^C(i,\overline{x})(\alpha )$ and ${{\overline{e}}_R}^C(i,\overline{x})(\alpha )$, and its optimal solution remains unchanged. Let ${{\overline{e}}_L}^C(i,\overline{x})={\overline{k}}_L$, ${{\overline{e}}_R}^C(i,\overline{x})={\overline{k}}_R$ (${\overline{k}}_L$ and ${\overline{k}}_R$ are constants), and the verification is carried out as follows:

$$\begin{array}{c}{\displaystyle \sum _{i\in N}[(}{e_L}^C(i,\overline{x})(\alpha )-{\overline{k}}_L{)}^2+({e_R}^C(i,\overline{x})(\alpha )-{\overline{k}}_R{)}^2]\\ ={\displaystyle \sum [{e_L}^C(i,\overline{x}){(\alpha )}^2+{e_R}^C(i,\overline{x}){(\alpha )}^2}]-2{\displaystyle \sum _{i\in N}[{\overline{k}}_L{e_L}^C(i,\overline{x})(\alpha )+{\overline{k}}_R{e_R}^C(i,\overline{x})(\alpha )]+n({{\overline{k}}_L}^2+{{\overline{k}}_R}^2)}\end{array}$$

(17)

Problem 2.

$\mathrm{Minimize} {\displaystyle \sum _{i\in N}[{e_L}^C(i,\overline{x}){(\alpha )}^2+}{e_R}^C(i,\overline{x}){(\alpha )}^2],$

$$\mathrm{s}.\mathrm{t}. \left\{\begin{array}{l}{\displaystyle \sum _{i\in N}{\overline{x}}_{Li}(\alpha )={\tilde{v}}_L(N)}(\alpha )\\ {\displaystyle \sum _{i\in N}{\overline{x}}_{Ri}(\alpha )={\tilde{v}}_R(N)}(\alpha )\end{array}\right.$$

(18)

This matter minimizes the sum of dissatisfaction of the players by minimizing the contribution excess of all the players so that the sum of satisfaction of the players is maximized.

The Lagrangian function for Problem 2 is as follows:

$$\begin{array}{c}L({\overline{x}}_{Li}(\alpha ),{\overline{x}}_{Ri}(\alpha ),\lambda ,\gamma )={\displaystyle \sum _{i\in N}[{(e_L^C(i,\overline{x})(\alpha ))}^2+{(e_R^C(i,\overline{x})(\alpha ))}^2]}\\ +\lambda ({\displaystyle \sum _{i=1}^n{\overline{x}}_{Li}(\alpha )-{\tilde{v}}_L(N)(\alpha )})+\gamma ({\displaystyle \sum _{i=1}^n{\overline{x}}_{Ri}(\alpha )-{\tilde{v}}_R(N)(\alpha )})\end{array}$$

(19)

Calculate the partial derivatives of the variables ${\overline{x}}_{Li}(\alpha )$, ${\overline{x}}_{Ri}(\alpha )$, $\lambda$, and $\gamma$, respectively, and let them be equal to zero:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial L({\overline{x}}_{Li}(\alpha ),{\overline{x}}_{Ri}(\alpha ),\lambda ,\gamma )}{\partial {\overline{x}}_{Li}}}=-2({\tilde{v}}_{Li}^C(\alpha )-{\overline{x}}_{Li}(\alpha ))+\lambda =0\\ {\displaystyle \frac{\partial L({\overline{x}}_{Li}(\alpha ),{\overline{x}}_{Ri}(\alpha ),\lambda ,\gamma )}{\partial \lambda }}={\displaystyle \sum _{i=1}^n{\overline{x}}_{Li}(\alpha )}-{\tilde{v}}_L(N)(\alpha )=0\\ {\displaystyle \frac{\partial L({\overline{x}}_{Li}(\alpha ),{\overline{x}}_{Ri}(\alpha ),\lambda ,\gamma )}{\partial {\overline{x}}_{Ri}}}=-2({\tilde{v}}_{Ri}^C(\alpha )-{\overline{x}}_{Ri}(\alpha ))+\lambda =0\\ {\displaystyle \frac{\partial L({\overline{x}}_{Li}(\alpha ),{\overline{x}}_{Ri}(\alpha ),\lambda ,\gamma )}{\partial \lambda }}={\displaystyle \sum _{i=1}^n{\overline{x}}_{Ri}(\alpha )}-{\tilde{v}}_R(N)(\alpha )=0\end{array}\right.$$

(20)

The least square contribution of the trapezoidal fuzzy numbers considering the contribution excess of the players is calculated:

$$\left\{\begin{array}{l}{\overline{x}}_{Li}(\alpha )={\tilde{v}}_{Li}^C(\alpha )+{\displaystyle \frac{{\tilde{v}}_L(N)(\alpha )-{\displaystyle \sum _{j=1}^n{\tilde{v}}_{Lj}^C(\alpha )}}{n}}\\ {\overline{x}}_{Ri}(\alpha )={\tilde{v}}_{Ri}^C(\alpha )+{\displaystyle \frac{{\tilde{v}}_R(N)(\alpha )-{\displaystyle \sum _{j=1}^n{\tilde{v}}_{Rj}^C(\alpha )}}{n}}\end{array}\right.$$

(21)

At this point, the optimal solution of Problem 2 has been attained; the least square contribution of trapezoidal fuzzy numbers considering contribution excess of players can be presented as follows:

$${\overline{x}}_i^{\ast }(\alpha )=[{\tilde{v}}_{Li}^C(\alpha )+({\tilde{v}}_L(N)(\alpha )-{\displaystyle \sum _{j=1}^n{\tilde{v}}_{Lj}^C(\alpha )})/n,{\tilde{v}}_{Ri}^C(\alpha )+({\tilde{v}}_R(N)(\alpha )-{\displaystyle \sum _{j=1}^n{\tilde{v}}_{Rj}^C(\alpha )})/n] (i\in N)$$

(22)

Among them, ${\overline{x}}_i^{\ast }(\alpha )$ is composed of two parts, namely the average of the marginal contribution and the residual contribution of player $i$ to the alliance $N$, which is in accordance with the principle of constructing the model.

4.2. Improved Shapley Value with Trapezoidal Fuzzy Numbers Considering the Contribution Excess of Players

This paper proposes to replace the marginal contribution in the classic Shapely value with the least square contribution, and the model expression improved by trapezoidal fuzzy numbers is as follows:

$$SH{L}_i^{\ast }(\tilde{v})={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}{\overline{x}}_i^{\ast }(S)}(\alpha )$$

(23)

where

$$\begin{array}{l}{\overline{x}}_i^{\ast }(S)(\alpha )=[{\tilde{v}}_{Li}^C(S)(\alpha )+({\tilde{v}}_L(S)(\alpha )-{\displaystyle \sum _{j=1}^s{\tilde{v}}_{Lj}^C(S)(\alpha )})/s,\\ {\tilde{v}}_{Ri}^C(S)(\alpha )+({\tilde{v}}_R(S)(\alpha )-{\displaystyle \sum _{j=1}^s{\tilde{v}}_{Rj}^C(S)(\alpha )})/s] (i\in N)\end{array}$$

(24)

That is to say, in the process of calculating the least square contribution considering the contribution excess of the players, the alliance $S$ is temporarily regarded as the grand alliance, and the allocation value of player $i$ in the alliance $S$ is solved.

For any trapezoidal fuzzy number cooperative game $\tilde{v}\in G^N$, the efficiency principle must be followed; that is, ${\displaystyle \sum _{i\in N}{\overline{x}}_i^{\ast }(S)=\tilde{v}(S)(S\subseteq N)}$. Since the sum of the payment vectors of the players is determined, the following can be concluded:

$${\displaystyle \sum _{i\in N}SH{L}_i^{\ast }(\tilde{v})={\displaystyle \sum _{S\subset N}\frac{(n-s)!(s-1)!}{n!}}}\tilde{v}(S)+{\displaystyle \frac{1}{n}}\tilde{v}(N)$$

(25)

Since the ${\displaystyle \sum _{i\in N}SH{L}_i^{\ast }(\tilde{v})}$ obtained from Formula (25) is not equal to $\tilde{v}(N)$ and does not satisfy the efficiency, the differential quantity $\Delta (\tilde{v})$ is now calculated:

$$\Delta (\tilde{v})=\tilde{v}(N)-{\displaystyle \sum _{i\in N}SH{L}_i^{\ast }(\tilde{v})=\frac{n-1}{n}\tilde{v}(N)-{\displaystyle \sum _{S\subset N}\frac{(n-s)!(s-1)!}{n!}}}\tilde{v}(S)$$

(26)

It is derivable from Formula (26) that $\Delta (\tilde{v})\ge 0$ when $\tilde{v}(N)\ge {\displaystyle \sum _{i\in N}SH{L}_i^{\ast }(\tilde{v})}$. Efficiency can merely be attained when $\Delta (\tilde{v})=0$. For the purpose of fulfilling the efficiency principle of the cooperative game, $\Delta (\tilde{v})$ is reallocated in accordance with the proportion of the income of the players when they operate in isolation to the total income of all the players when they operate in isolation, and Formula (23) is ameliorated to Formula (27).

$$\begin{array}{c}SH{L_i^{\ast }}^{\prime }(\tilde{v})={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}}{\overline{x}}_i^{\ast }(S)+{\displaystyle \frac{\Delta (\tilde{v})}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}\tilde{v}(i)\\ ={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}{\overline{x}}_i^{\ast }(S)+\frac{n-1}{n}\tilde{v}(N)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}-{\displaystyle \sum _{S\subset N:i\in S}\frac{(n-s)!(s-1)!}{n!}\tilde{v}(S)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}}\\ ={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}({\overline{x}}_i^{\ast }(S)-\tilde{v}(S)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}})}+{\displaystyle \frac{n-1}{n}}\tilde{v}(N){\displaystyle \frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}+{\displaystyle \frac{1}{n}}\tilde{v}(S){\displaystyle \frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}\\ ={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}({\overline{x}}_i^{\ast }(S)-\tilde{v}(S)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}})}+\tilde{v}(N){\displaystyle \frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}\end{array}$$

(27)

The improved Shapley value with trapezoidal fuzzy numbers and α-cut considering the contribution excess of the players is as follows:

$$\left\{\begin{array}{l}SH{L_{Li}^{\ast }}^{\prime }(\tilde{v}\left)\right(\alpha )={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}({\overline{x}}_{Li}^{\ast }(S)(\alpha )-{\tilde{v}}_L(S)(\alpha )\frac{\tilde{v}(i)(\alpha )}{{\displaystyle \sum _{i\in N}\tilde{v}(i)(\alpha )}})}+{\tilde{v}}_L(N)(\alpha ){\displaystyle \frac{\tilde{v}(i)(\alpha )}{{\displaystyle \sum _{i\in N}\tilde{v}(i)(\alpha )}}}\\ SH{L_{Ri}^{\ast }}^{\prime }(\tilde{v}\left)\right(\alpha )={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}({\overline{x}}_{Ri}^{\ast }(S)(\alpha )-{\tilde{v}}_R(S)(\alpha )\frac{\tilde{v}(i)(\alpha )}{{\displaystyle \sum _{i\in N}\tilde{v}(i)(\alpha )}})}+{\tilde{v}}_R(N)(\alpha ){\displaystyle \frac{\tilde{v}(i)(\alpha )}{{\displaystyle \sum _{i\in N}\tilde{v}(i)(\alpha )}}}\end{array}\right.$$

(28)

In contrast to the classic Shapley value, the improved Shapley value proposed in this paper takes into account the contribution excess and dissatisfaction of the players and obtains the least square contribution ${\overline{x}}_i^{\ast }(S)$ when the satisfaction is maximum by the method of least squares, replaces the marginal contribution in the classical Shapley value with it, improves it with trapezoidal fuzzy numbers, and proposes the improved Shapley value with trapezoidal fuzzy numbers, which takes into account the contribution excess of the players. This improved Shapley value not only improves the accuracy of profit allocation and the satisfaction of players but also enhances its rationality, which provides a new solution to the problem of profit allocation in e-commerce logistics alliances of forest products and has important theoretical guidance significance and further practical application value.

4.3. Properties of the Improved Shapley Value with Trapezoidal Fuzzy Numbers Considering the Contribution Excess of Players

The following describes and proves some related properties of the improved Shapley value with trapezoidal fuzzy numbers based on the contribution excess of the players. The improved Shapley value with trapezoidal fuzzy numbers based on the contribution excess of the players satisfies the following four properties.

Theorem 1.

(Existence): For any trapezoidal fuzzy number cooperative game $(N,\tilde{v})$, there always exists a unique improved Shapley value with trapezoidal fuzzy numbers that considers the contribution excess of the players. Theorem 1 can be proved according to Formula (27) or Formula (28).

Theorem 2.

(Efficiency): For any trapezoidal fuzzy number cooperative game $(N,\tilde{v})$, there is ${\displaystyle \sum _{i\in N}SH{L_i^{\ast }}^{\prime }(\tilde{v})}=\tilde{v}(N)$.

Proof of Theorem 2.

For the improved Shapley value with trapezoidal fuzzy numbers considering the contribution excess of the players, the sum of the payment values of all the players is computed:

$$\begin{array}{c}{\displaystyle \sum _{i\in N}SH{L_i^{\ast }}^{\prime }(\tilde{v})}={\displaystyle \sum _{i\in N}[{\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}{\overline{x}}_i^{\ast }(S)+\frac{n-1}{n}\tilde{v}(N)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}-{\displaystyle \sum _{S\subset N:i\in S}\frac{(n-s)!(s-1)!}{n!}\tilde{v}(S)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}}}]\\ ={\displaystyle \sum _{i\in N}[{\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}({\overline{x}}_i^{\ast }(S)-\tilde{v}(S)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}})}+\tilde{v}(N)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}]}\\ ={\displaystyle \sum _{S\subseteq N}\frac{(n-s)!(s-1)!}{n!}\tilde{v}(S)}-{\displaystyle \sum _{S\subseteq N}\frac{(n-s)!(s-1)!}{n!}\tilde{v}(S)}+\tilde{v}(N)\\ =\tilde{v}(N)\end{array}$$

(29)

Therefore, ${\displaystyle \sum _{i\in N}SH{L_i^{\ast }}^{\prime }(\tilde{v})}=\tilde{v}(N)$. Theorem 2 is proven. □

Theorem 3.

(Additivity): For any trapezoidal fuzzy number cooperative game $(N,\tilde{v})$ and $(N,\tilde{w})$, there exists $SH{L_i^{\ast }}^{\prime }(\tilde{v}+\tilde{w})=SH{L_i^{\ast }}^{\prime }(\tilde{v})+SH{L_i^{\ast }}^{\prime }(\tilde{w})$. That is, regardless of which form of cooperation is employed, the profit allocation method for each form of cooperation is mutually independent and does not rely on other cooperation outcomes.

Proof of Theorem 3.

$$\begin{array}{c}SH{L_i^{\ast }}^{\prime }(\tilde{v}+\tilde{w})={\displaystyle \sum _{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}({\overline{x}}_i^{\ast }(S)-\tilde{v}(S)\frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}+{\overline{x}}_i^{\ast }(S)-\tilde{w}(S)\frac{\tilde{w}(i)}{{\displaystyle \sum _{i\in N}\tilde{w}(i)}})}\\ +\tilde{v}(N){\displaystyle \frac{\tilde{v}(i)}{{\displaystyle \sum _{i\in N}\tilde{v}(i)}}}+\tilde{w}(N){\displaystyle \frac{\tilde{w}(i)}{{\displaystyle \sum _{i\in N}\tilde{w}(i)}}}\\ =SH{L_i^{\ast }}^{\prime }(\tilde{v})+SH{L_i^{\ast }}^{\prime }(\tilde{w})\end{array}$$

(30)

□

Theorem 4.

(Symmetry): $SH{L_i^{\ast }}^{\prime }(\tilde{v})=SH{L_j^{\ast }}^{\prime }(\tilde{v}) (i,j\in N)$, where the player $i$ and the player $j$ are symmetrical. That is, the profit allocated to each individual in the cooperation does not vary with the sequence of each individual within the cooperative alliance.

From the above Theorems 1–4, it can be seen that the improved Shapley value with trapezoidal fuzzy numbers considering the contribution excess of players proposed in this paper also satisfies the important properties of classical Shapley value, such as efficiency, efficiency, additivity, and symmetry.

5. Analysis of Calculation Examples

In contrast to general logistics enterprises, e-commerce logistics enterprises of forest products undertake limited types and quantities of goods, presenting the characteristics of small quantities and large batches. A single e-commerce logistics enterprise of forest products is often unable to make its freight vehicles fully loaded. In recent years, e-commerce logistics enterprises of forest products have undertaken a large number of export orders; for such orders, they usually use container vehicles for transportation. As a single e-commerce logistics enterprise of forest products undertakes limited types and quantities of goods, which often leads to low utilization of containers, such enterprises are more inclined to form alliances in order to increase the loading rate of containers and reduce the number of containers in use, thus reducing the cost of freight.

5.1. Case Calculation and Analysis of E-Commerce Logistics Alliance of Forest Products

The actual cost and profit of e-commerce logistics enterprises of forest products in the stage of road transportation usually depend on the following parameters:

(1)

Order quote $K$ (unit: CNY) is the quote obtained by e-commerce logistics enterprises of forest products when undertaking transportation tasks; it is affected by the market, and the value fluctuates within a certain range;

(2)

Cargo weight $W_h$ (unit: tons) is the weight of the goods contracted by e-commerce logistics enterprises of forest products;

(3)

Cargo volume $V_h$ (unit: m³) is the volume of the goods contracted by e-commerce logistics enterprises of forest products;

(4)

Vehicle load rate $R$ is the ratio of the weight of the goods to the maximum load capacity of the vehicle;

(5)

Empty driving distance $D_U$ (unit: km) is the distance that e-commerce logistics enterprises of forest products travel from the shipping company to pick up empty containers and go to forest products e-commerce enterprises for loading; it is affected by factors such as weather and road conditions, and the value fluctuates within a certain range;

(6)

Loaded driving distance $D_L$ (unit: km) is the distance from the loading of a single vehicle from the forest products e-commerce enterprises to the port of loading port; it is subject to the influence of weather and road conditions and other factors, and the value floats within a certain range;

(7)

Empty-load fuel cost $O_U$ (unit: CNY/km) is the fuel cost consumed per kilometer when a single vehicle is empty; it is affected by the market and the driver’s driving habits, and the value usually fluctuates within a certain range. The empty-load fuel costs of 20 GP and 40 GP containers are, respectively, $O_{U1}$ and $O_{U2}$;

(8)

Full-load fuel cost $O_F$ (unit: CNY/km) is the fuel cost consumed per kilometer when a single vehicle is fully loaded; it is affected by the weight of the goods, the market, and the driver’s driving habits, and the value usually fluctuates within a certain range. The full-load fuel costs of 20 GP and 40 GP containers are, respectively, $O_{F1}$ and $O_{F2}$;

(9)

Loaded fuel cost $O_L$ (unit: CNY/km) is the fuel cost of the truck that varies with the weight of the goods. The calculation formulas for the loaded fuel costs of 20 GP container trucks and 40 GP container trucks are, respectively, as follows:

$$\left\{\begin{array}{l}{O}_{L1}=(O_{F1}-O_{U1})R_1+O_U\\ O_{L2}=(O_{F2}-O_{U2})R_2+O_U\end{array}\right.$$

(31)

(10)

Expressway toll fee $H$ (unit: CNY/km) is the cost per kilometer for a single vehicle during transportation on the expressway. The expressway mileage is calculated as 80% of the driving distance;

(11)

Annual departure time $n$ (unit: time) is affected by weather, road conditions, and the number of orders, and the value fluctuates within a certain range;

(12)

Vehicle insurance premium $C$ (unit: CNY) is the annual vehicle insurance premium required to be paid for a single vehicle;

(13)

Annual vehicle maintenance cost $M$ (unit: CNY) is the annual maintenance cost required to be paid for a single vehicle;

(14)

Driver’s annual salary $S$ (unit: CNY) is the salary paid by the enterprise to the driver annually;

(15)

Vehicle purchase cost $B$ (unit: CNY) is the cost required by the enterprise to purchase a single vehicle;

(16)

Vehicle depreciation cost $G$ (unit: CNY) is the depreciation period for semi-trailers and trailers, which is set at 15 years, and the depreciation cost is $G=B/15n$;

(17)

Loading and unloading cost $L$ (unit: CNY/m³) is the labor cost required for the loading and unloading of vehicles.

Based on the above parameters, the cost (unit: CNY) of a single vehicle completing an e-commerce logistics enterprise of forest products goods transportation task by road transportation can be obtained as follows:

$$Q=D_U{O}_U+D_L{O}_L+(D_U+D_L)H\times 0.8+{\displaystyle \frac{C+M+S}{n}}+G+L{V}_h$$

(32)

Currently, there are three e-commerce logistics enterprises of forest products located in A, which need to perform long-term cargo transportation for the forest products e-commerce enterprises in the e-commerce industrial parks; i.e., the goods are transported from the e-commerce industrial parks to the ports, and then, the freight forwarders will transfer the goods from the ports to the overseas warehouses located in foreign countries through shipping. It is difficult to predict the cost of transporting goods in this transportation process. For example, although the distance from the e-commerce park to the port is a given value, the actual distance may deviate due to factors such as weather and road conditions. For these variables, trapezoidal fuzzy numbers can be used to deal with the problem.

These three e-commerce logistics enterprises of forest products are represented by Enterprise A, Enterprise B, and Enterprise C. All three enterprises undertake the transportation task from the e-commerce industrial park to the port. Among them, the volume of the goods that need to be transported in the order obtained by Enterprise A is 33 m³, the weight is 8.7 tons, and the order quote is $K_A=3252$; the volume of the goods that need to be transported in the order obtained by Enterprise B is 38 m³, the weight is 9.8 tons, and the order quote is $K_B=3447$; the volume of the goods that need to be transported in the order obtained by Enterprise C is 43 m³, the weight is 10.3 tons, and the order quote is $K_C=3690$. The effective loading volumes of common 20 GP and 40 GP containers are 28 m³ and 58 m³, respectively. It can be seen that if these three enterprises operate independently, Enterprise A, Enterprise B, and Enterprise C will need to deploy one 40 GP container truck. The relevant parameters and reference values for cost accounting are shown in

Table 2. Relevant parameters and parameter values for cost accounting.

Parameters (CNY)	Reference Value of 20 GP Container Truck	Reference Value of 40 GP Container Truck
$\mathrm{Empty} \mathrm{driving} \mathrm{distance} D_U$	[200, 206, 210, 218]	[200, 206, 210, 218]
$\mathrm{Loaded} \mathrm{driving} \mathrm{distance} D_L$	[172, 178, 182, 190]	[172, 178, 182, 190]
Empty-load fuel cost $O_U$	[2.52, 2.73, 2.94, 3.15]	[3.02, 3.27, 3.52, 3.78]
Full-load fuel cost $O_F$	[3.78, 4.09, 4.41, 4.72]	[4.53, 4.91, 5.29, 5.66]
Expressway toll fee $H$	[2.05, 2.10, 2.15, 2.20]	[2.05, 2.10, 2.15, 2.20]
Annual departure time $n$	[150, 157, 163, 170]	[150, 157, 163, 170]
Vehicle insurance premium $C$	9000	9000
Annual vehicle maintenance cost $M$	3000	3000
Driver’s annual salary $S$	80,000	80,000
Vehicle purchase cost $B$	400,000	400,000
Loading and unloading cost $L$	3	3

.

In this example, three assumptions are proposed:

(1)

The vehicles used by the three e-commerce logistics enterprises of forest products are all the same type of tractors and trailers;

(2)

There is sufficient tacit understanding among the enterprises, and the alliance will not affect the quotations of the orders;

(3)

After the alliance of the enterprises, the orders will be merged, and the volume error of the goods loading after the order merger is ignored.

Currently, the conditions of the three e-commerce logistics enterprises of forest products forming alliances, respectively, are explored. When Enterprise A and Enterprise B collaborate, the total volume of the goods reaches 71 m³, and the total weight reaches 18.5 tons. In this case, one 20 GP container and one 40 GP container are required; when Enterprise A and Enterprise C join forces, the total volume is 76 m³, and the total weight is 19 tons. Similarly, one 20 GP container and one 40 GP container are needed; when Enterprise B and Enterprise C cooperate closely, the total volume of the goods is 81 m³, and the total weight is 20.1 tons. Still, one 20 GP container and one 40 GP container are necessary. When Enterprise A, Enterprise B, and Enterprise C collaborate together, the total volume of the goods is 114 m³, and the total weight is 28.8 tons. At this point, two 40 GP containers are needed.

Next, the costs and profits of three e-commerce logistics enterprises of forest products under different alliance forms are calculated, respectively. The details are presented in

Table 3. Costs and profits of e-commerce logistics enterprises of forest products under different alliance forms.

Enterprise Alliance	Quote (CNY)	Cost (CNY)	Profit (CNY)
{A}	3252	[2416, 2535, 2650, 2809]	[443, 602, 717, 836]
{B}	3447	[2444, 2565, 2681, 2842]	[605, 766, 882, 1003]
{C}	3690	[2465, 2586, 2703, 2865]	[825, 987, 1104, 1225]
{AB}	6699	[4676, 4895, 5106, 5397]	[1302, 1593, 1804, 2023]
{AC}	6942	[4697, 4917, 5129, 5421]	[1521, 1813, 2025, 2245]
{BC}	7137	[4726, 4947, 5161, 5455]	[1682, 1976, 2190, 2411]
{ABC}	10,389	[5110, 5366, 5611, 5947]	[4442, 4778, 5023, 5279]

. Moreover, Table 3 presents the profit conditions of three e-commerce logistics enterprises of forest products before and after cooperation under different alliance forms.

According to the organizational structure of sub-alliances in the e-commerce logistics alliance of forest products, we compare the changing trends of the total costs and the total profits of three e-commerce logistics enterprises of forest products when the enterprises transport separately, when two of the e-commerce logistics enterprises of forest products form an alliance for transportation, and when all three of the e-commerce logistics enterprises of forest products participate in the same alliance for transportation. The sum of cost and profit is expressed by trapezoidal fuzzy numbers, and the trend of change of the score is consistent, so it is only plotted with its lower bound value as a reference, and the trend of change is shown in Figure 5.

It can be seen that the three e-commerce logistics enterprises of forest products, namely A, B, and C, have gradually developed from individual transportation to two-by-two cooperation and ultimately formed a grand alliance for the cooperation of the three e-commerce logistics enterprises of forest products, whose cooperation profit has increased, and the cooperation profit of the grand alliance is much larger than that of the three e-commerce logistics enterprises of forest products when they transported individually. Therefore, each e-commerce logistics enterprise of forest products is inclined to cooperative transportation. However, how to fairly and reasonably distribute the cooperative profit of alliance has become an important issue to be solved at present.

5.2. Calculation of Profit Allocation in the Case of E-Commerce Logistics Alliance of Forest Products

From Table 3 and

Table 4. Profits of e-commerce logistics enterprises of forest products before and after cooperation under different alliance forms.

Enterprise Alliance	Profit Before Cooperation (CNY)	Profit After Cooperation (CNY)	Super- Additivity	Whether to Form an Alliance
{AB}	[1048, 1368, 1599, 1839]	[1302, 1593, 1804, 2023]	Yes	Yes
{AC}	[1268, 1589, 1820, 2061]	[1521, 1813, 2025, 2245]	Yes	Yes
{BC}	[1430, 1752, 1986, 2228]	[1682, 1976, 2190, 2411]	Yes	Yes
{ABC}	[1873, 2354, 2702, 3064]	[4442, 4778, 5023, 5279]	Yes	Yes

, it can be concluded that when the three forest product e-commerce logistics enterprises form a grand alliance for transportation, the profit ${\tilde{v}}_{ABC}$ = [4442, 4778, 5023, 5279] is much larger than the sum of the profits ${\tilde{v}}_A+{\tilde{v}}_B+{\tilde{v}}_C$ = [1873, 2354, 2702, 3064] when the three forest product e-commerce logistics enterprises operate independently. Moreover, the profits gained by any sub-alliance plus the profits of the enterprises that are not participating in the alliance when they operate alone are all smaller than the total profit of the grand alliance, which meets the conditions for the establishment of an alliance; thus, we calculate the profit allocated to each e-commerce logistics enterprise of forest products in the common transportation.

The least square contribution of each enterprise in the e-commerce logistics alliance of forest products is calculated according to Formula (24) as shown in

Table 5. Least square contribution of each enterprise in the e-commerce logistics alliance of forest products.

Enterprise	Alliance $\mathit{S}$	${\overline{\mathit{x}}}_{\mathit{i}}^{\ast }(\mathit{S})$	Enterprise	Alliance $\mathit{S}$	${\overline{\mathit{x}}}_{\mathit{i}}^{\ast }(\mathit{S})$	Enterprise	Alliance $\mathit{S}$	${\overline{\mathit{x}}}_{\mathit{i}}^{\ast }(\mathit{S})$
A	A	[443, 602, 717, 836]	B	B	[605, 766, 882, 1003]	C	C	[825, 987, 1104, 1225]
A	AB	[570, 715, 819, 928]	B	AB	[732, 878, 985, 1095]	C	AC	[952, 1099, 1206, 1317]
A	AC	[570, 714, 819, 928]	B	BC	[731, 878, 984, 1094]	C	BC	[951, 1098, 1206, 1317]
A	ABC	[1300, 1411, 1491, 1575]	B	ABC	[1461, 1574, 1656, 1741]	C	ABC	[1680, 1794, 1876, 1963]

.

Next, the marginal contribution of the classical Shapley value is replaced by the calculated least square contribution of the trapezoidal fuzzy numbers α-cut set, and the profit of each e-commerce logistics of forest products company is calculated, as shown in

Table 6. Final profit of each enterprise in the e-commerce logistics alliance of forest products.

Enterprise	Profit (CNY)
A	[1146, 1293, 1392, 1491]
B	[1445, 1562, 1646, 1733]
C	[1851, 1923, 1985, 2055]

.

It is evident from Table 6 that the improved Shapley value with the trapezoidal fuzzy number proposed in this paper satisfies the individual rationality and overall rationality. Specifically, the profit [1146, 1293, 1392, 1491] of e-commerce logistics Enterprise A after joining the grand alliance is significantly higher than the profit [443, 602, 717, 836] before joining. And for any two e-commerce logistics enterprises of forest products, the sum of the profits obtained after joining the grand alliance exceeds the sum of the profits of the sub-alliance formed by them independently. Meanwhile, the improved Shapley value of the proposed trapezoidal fuzzy number satisfies the overall rationality, and the total profit of the alliance can be fully distributed to each e-commerce logistics enterprise of forest products without surplus. Therefore, the new improved Shapley value of trapezoidal fuzzy number proposed in this paper can effectively solve the problem of profit allocation of e-commerce logistics alliance of forest products under the fuzzy environment.

5.3. Comparison of Profit Allocation Schemes of E-Commerce Logistics Alliance of Forest Products

In the field of cooperative games, the classical profit allocation schemes include Shapley value and Banzhaf value. The improved Shapley value with trapezoidal fuzzy numbers considering contribution excess proposed in this paper is improved based on the classical Shapley value. Compared with the classical Shapley value, the improved Shapley value with trapezoidal fuzzy numbers takes into account the satisfaction of the players and can distribute the profits of e-commerce logistics alliance of forest products more fairly and reasonably.

Figure 6 shows the profits of the three e-commerce logistics enterprises of forest products under different profit allocation schemes. Since the three e-commerce logistics enterprises of forest products have the same trend of changes in profits under different profit allocation schemes, Figure 6 is plotted using only the lower bound value of profits as a reference.

Table 7. Results of profit allocation among enterprises in e-commerce logistics alliance of forest products based on classical Shapley value.

Enterprise	Profit (CNY)
A	[1300, 1410, 1490, 1575]
B	[1461, 1574, 1656, 1741]
C	[1681, 1794, 1877, 1963]

shows the profits of e-commerce logistics enterprises of forest products under the classic Shapley value profit allocation scheme, from which it can be seen that the lower limit values of the profits of e-commerce logistics Enterprises A, B, and C are CNY 1300, CNY 1461, and CNY 1681, respectively, and the lower limit values of the final profits are CNY 1146, CNY 1445 and CNY 1851, respectively, under the improved Shapley value profit allocation scheme. Under the classic Shapley value profit allocation scheme, the incremental value of the lower limit of profit after Enterprise A joins the grand alliance is CNY 857, an increase of 93.45% year-on-year; under the improved Shapley value profit allocation scheme, the incremental value of the lower limit of profit after Enterprise A joins the grand alliance is CNY 703, an increase of 58.69% year-on-year. In contrast, the incremental value of the lower limit of grand alliance after Enterprise A joins the grand alliance is CNY 2760, an increase of 64.09 percent year-on-year. Compared with the 93.45% year-on-year increase in profit of Enterprise A in the classic Shapley value, the 58.69% year-on-year increase in profit of Enterprise A in the improved Shapley value is closer to the 64.09% year-on-year increase in profit of the grand alliance after Enterprise A joins the grand alliance, which is more in line with the actual situation of profit allocation of the e-commerce logistics alliance of forest products, and this better embodies the principle of distribution according to labor and improves the e-commerce logistics enterprises’ satisfaction of forest products.

From

Table 8. Results of profit allocation among enterprises in e-commerce logistics alliance of forest products based on Banzhaf value.

Enterprise	Profit (CNY)
A	[1149, 1264, 1348, 1436]
B	[1311, 1428, 1514, 1602]
C	[1530, 1648, 1735, 1824]

, it can be seen that the final profit lower bounds of e-commerce logistics Enterprises A, B, and C under the Banzhaf value profit allocation scheme are CNY 1149, CNY 1311, and CNY 1530, respectively, with a total profit of CNY 3990. However, this cannot fully distribute the cooperative profit of CNY 4442, which fails to meet the efficiency of cooperative game; however, under the improved Shapley value profit allocation scheme, the final profit lower bounds of e-commerce logistics Enterprises A, B, and C are CNY 1146, CNY 1445 and CNY 1851, respectively, which can fully distribute the cooperative profit of CNY 4442, meeting cooperative game efficiency and better reflecting the actual profit allocation of e-commerce logistics alliance of forest products.

6. Conclusions and Future Researches

This paper addresses the fuzzy numbers problem encountered by e-commerce logistics alliance of forest products in the process of profit allocation and discusses the cooperative game when the profit value is trapezoidal fuzzy numbers. This paper considers the contribution excess of the players and the dissatisfaction of the players, adopts the least squares contribution to replace the marginal contribution of the classical Shapley value, and then proposes the improved Shapley value with trapezoidal fuzzy numbers considering the contribution excess of the players. It is thus proven that this improved Shapley value satisfies the key properties of cooperative game solution, including existence, efficiency, additivity, and symmetry, which ensures the fairness and rationality of profit allocation. The solution method proposed in this paper is concise and efficient, closely fits the reality, provides a fair and reasonable profit allocation strategy for e-commerce logistics alliance of forest products, and further enriches the theoretical system of profit allocation of e-commerce logistics alliance under a fuzzy environment.

The profit allocation strategy proposed in this paper has not yet considered the factor of coalition weights, which may lead to some bias in the final profit allocation results. However, as far as the current strategy is concerned, the satisfaction of the players has been maximized. In the near future, we will study the effect of coalition weights on contributions and profits in depth and propose new cooperative game solutions accordingly. In addition, we plan to extend the concept of contribution excess to other cooperative game models and improve it with fuzzy numbers so that the outgoing cooperative game solutions take into account the satisfaction of the players and can also be applied in a fuzzy environment.