## 4 Model Construction
### Least Square Contribution of Trapezoidal Fuzzy Numbers Considering Contribution Excess of Players
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:  
\[SHL_{i}^{*}(\widetilde{v})=\sum\limits_{S\subseteq\overline{N}:i\in S}\frac{(n -s)!(s-1)!}{n!}\overline{x}_{i}^{*}(S)(\alpha) \tag{23}\]  
where  
\[ \begin{array}{l}\overline{x}_{i}^{*}(S)(\alpha)=[\widetilde{v}_{L}^{C}(S)( \alpha)+(\widetilde{v}_{L}(S)(\alpha)-\sum\limits_{j=1}^{s}\widetilde{v}_{L}^ {C}(S)(\alpha))/s,\\ \widetilde{v}_{Rl}^{C}(S)(\alpha)+(\widetilde{v}_{R}(S)(\alpha)-\sum\limits_ {j=1}^{s}\widetilde{v}_{Rj}^{C}(S)(\alpha))/s]\;(i\in N)\end{array}  \tag{24}\]  
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 \(\widetilde{v}\in G^{N}\), the efficiency principle must be followed; that is, \(\sum\limits_{i\in N}\overline{x}_{i}^{*}(S)=\widetilde{v}(S)(S\subseteq N)\). Since the sum of the payment vectors of the players is determined, the following can be concluded:  
\[\sum\limits_{i\in N}SHL_{i}^{*}(\widetilde{v})=\sum\limits_{S\subset N}\frac{( n-s)!(s-1)!}{n!}\widetilde{v}(S)+\frac{1}{n}\widetilde{v}(N) \tag{25}\]  
Since the \(\sum\limits_{i\in N}SHL_{i}^{*}(\widetilde{v})\) obtained from Formula (25) is not equal to \(\widetilde{v}(N)\) and does not satisfy the efficiency, the differential quantity \(\Delta(\widetilde{v})\) is now calculated:  
\[\Delta(\widetilde{v})=\widetilde{v}(N)-\sum\limits_{i\in N}SHL_{i}^{*}( \widetilde{v})=\frac{n-1}{n}\widetilde{v}(N)-\sum\limits_{S\subset N}\frac{( n-s)!(s-1)!}{n!}\widetilde{v}(S) \tag{26}\]  
It is derivable from Formula (26) that \(\Delta(\widetilde{v})\geq 0\) when \(\widetilde{v}(N)\geq\sum\limits_{i\in N}SHL_{i}^{*}(\widetilde{v})\). Efficiency can merely be attained when \(\Delta(\widetilde{v})=0\). For the purpose of fulfilling the efficiency principle of the cooperative game, \(\Delta(\widetilde{v})\) 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). \[ \begin{array}{l}SHL_{i}^{*\prime}(\widetilde{v})=\sum\limits_{S\subseteq N :i\in S}\frac{(n-s)!(s-1)!}{n!}\overline{x}_{i}^{*}(S)+\frac{\Delta(\widetilde{ v})}{\sum\limits_{i\in N}\widetilde{v}(i)}\widetilde{v}(i)\\ =\sum\limits_{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}\overline{x}_{i}^{*}( S)+\frac{n-1}{n}\widetilde{v}(N)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N} \widetilde{v}(i)}-\sum\limits_{S\subset N:i\in S}\frac{(n-s)!(s-1)!}{n!} \widetilde{v}(S)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)} \\ =\sum\limits_{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}(\overline{x}_{i}^{*}( S)-\widetilde{v}(S)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)})+\frac{n-1}{n} \widetilde{v}(N)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)} +\frac{1}{n}\widetilde{v}(S)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N} \widetilde{v}(i)}\\ =\sum\limits_{S\subseteq N:i\in S}\frac{(n-s)!(s-1)!}{n!}(\overline{x}_{i}^{*}( S)-\widetilde{v}(S)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)})+ \widetilde{v}(N)\frac{\widetilde{v}(i)}{\sum\limits_{i\in N}\widetilde{v}(i)} \end{array}  \tag{27}\]  
The improved Shapley value with trapezoidal fuzzy numbers and \(\alpha\)-cut considering the contribution excess of the players is as follows:\[\left\{