## 4 Model Construction
### Least Square Contribution of Trapezoidal Fuzzy Numbers Considering Contribution Excess of Players
\begin{array}{l}{SH{L_{L}^{*}}_{l}}^{\prime}(\widetilde{v})(\alpha)= \sum\limits_{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}(\overline{x}_{Li}^{*}(S) (\alpha)-\widetilde{v}_{L}(S)(\alpha)\frac{\widetilde{v}(i)(\alpha)}{\sum \limits_{i\in N}\widetilde{v}(i)(\alpha)})+\widetilde{v}_{L}(N)(\alpha)\frac{ \widetilde{v}(i)(\alpha)}{\sum\limits_{i\in N}\widetilde{v}(i)(\alpha)}\\ {SH{L_{R}^{*}}_{l}}^{\prime}(\widetilde{v})(\alpha)=\sum\limits_{S\subseteq N: i\in S}\frac{(n-s)!(s-1)!}{n!}(\overline{x}_{R!}^{*}(S)(\alpha)-\widetilde{v}_{R}(S)( \alpha)\frac{\widetilde{v}(i)(\alpha)}{\sum\limits_{i\in N}\widetilde{v}(i)( \alpha)})+\widetilde{v}_{R}(N)(\alpha)\frac{\widetilde{v}(i)(\alpha)}{\sum \limits_{i\in N}\widetilde{v}(i)(\alpha)}\end{array} \right. \tag{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 \(\overline{x}_{i}^{*}(S)\) 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. 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 \((N,\widetilde{v})\), 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 \((N,\widetilde{v})\), there is \(\sum\limits_{i\in N}{SH{L_{i}^{*}}_{l}}^{\prime}(\widetilde{v})=\widetilde{v} (N)\)._  
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:  
\[ \begin{array}{l}{\sum\limits_{i\in N}{SH{L_{i}^{*}}_{l}}^{\prime }(\widetilde{v})=\sum\limits_{i\in N}{[\sum\limits_{S\subseteq N:i\in S} \frac{(n-s)!(s-1)!}{n!}\overline{x}_{i}^{*}(S)+\frac{n-1}{n}\widetilde{v}(N) \frac{\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)}-\sum\limits_{ S\subset N:i\in S}\frac{(n-s)!(s-1)!}{n!}\widetilde{v}(S)\frac{\widetilde{v}(i)}{ \sum\limits_{i\in N}\widetilde{v}(i)}]}}\\ {\hskip 14.226378 pt=\sum\limits_{i\in N}{[\sum\limits_{S\subseteq N:i\in S} \frac{(n-s)!(s-1)!}{n!}(\overline{x}_{i}^{*}(S)-\widetilde{v}(S)\frac{ \widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)})+\widetilde{v}(N)\frac {\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)}]}}\\ {\hskip 14.226378 pt=\sum\limits_{S\subseteq N}\frac{(n-s)!(s-1)!}{n!} \widetilde{v}(S)-\sum\limits_{S\subseteq N}\frac{(n-s)!(s-1)!}{n!}\widetilde{v }(S)+\widetilde{v}(N)}\\ {\hskip 14.226378 pt=\widetilde{v}(N)}\end{array}  \tag{29}\]  
Therefore, \(\sum\limits_{i\in N}{SH{L_{i}^{*}}_{l}}^{\prime}(\widetilde{v})=\widetilde{v} (N)\). Theorem 2 is proven. \(\Box\)  
**Theorem 3**.