## 4 Model Construction
### Least Square Contribution of Trapezoidal Fuzzy Numbers Considering Contribution Excess of Players
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Meaning** & **Variables** & **Meaning** \\ \hline \({\widetilde{v}}(i)\) & Characteristic function of player \(i\) & \({\widetilde{v}}(S)\) & Characteristic function of alliance \(S\) \\ \hline \({\widetilde{v}}(N)\) & Characteristic function of grand alliance \(N\) & \({\overline{x}}(\alpha)\) & Payment vector in \(\alpha\)-cut \\ \hline \({e_{L}}^{C}(i,\overline{x})(\alpha)\) & The left endpoint of contribution excess of player \(i\) in \(\alpha\)-cut & \({e_{R}}^{C}(i,\overline{x})(\alpha)\) & The right endpoint of contribution excess of player \(i\) in \(\alpha\)-cut \\ \hline \({\bar{e}_{L}}^{C}(i,\overline{x})(\alpha)\) & The left endpoint of the average of contribution excess player \(i\) in \(\alpha\)-cut & \({\bar{e}_{R}}^{C}(i,\overline{x})(\alpha)\) & The right endpoint of the average of contribution excess player \(i\) in \(\alpha\)-cut \\ \hline \({\overline{x}_{Li}}(\alpha)\) & The left endpoint of least square contribution of player \(i\) in \(\alpha\)-cut & \({\overline{x}_{Ri}}(\alpha)\) & The right endpoint of least square contribution of player \(i\) in \(\alpha\)-cut \\ \hline \({\widetilde{v}_{L}}(N)(\alpha)\) & The left endpoint of characteristic function of grand alliance \(N\) in \(\alpha\)-cut & \({\widetilde{v}_{R}}(N)(\alpha)\) & The right endpoint of characteristic function of grand alliance \(N\) in \(\alpha\)-cut \\ \hline \({\widetilde{v}_{Li}}^{C}(\alpha)\) & The left endpoint of the value of player \(i\)’s contribution to the grand alliance in \(\alpha\)-cut & \({\widetilde{v}_{Ri}}^{C}(\alpha)\) & The right endpoint of the value of player \(i\)’s contribution to the grand alliance in \(\alpha\)-cut \\ \hline \({\widetilde{v}_{Li}}^{C}(\alpha)\) & The left endpoint of the value of player \(j\)’s contribution to the grand alliance in \(\alpha\)-cut & \({\widetilde{v}_{Rj}}^{C}(\alpha)\) & The right endpoint of the value of player \(j\)’s contribution to the grand alliance in \(\alpha\)-cut \\ \hline \({\overline{x}_{i}}^{*}(\alpha)\) & The least square contribution of player \(i\) in \(\alpha\)-cut & \({\overline{x}_{i}}^{*}(S)\) & The least square contribution of alliance \(S\) \\ \hline \({SHL_{i}^{*}}({\widetilde{v}})\) & Initial improved Shapley values & \({SHL_{i}^{*}}^{\prime}({\widetilde{v}})\) & Final improved Shapley value \\ \hline \({SHL_{Li}^{*}}^{\prime}({\widetilde{v}})(\alpha)\) & The left endpoint of final improved Shapley value in \(\alpha\)-cut & \({SHL_{Ri}^{*}}^{\prime}({\widetilde{v}})(\alpha)\) & The right endpoint of final improved Shapley value in \(\alpha\)-cut \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables used in this article. 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, \(\sum\limits_{i\in N}e_{L}^{\rm C}(i,\overline{x})(\alpha)\) and \(\sum\limits_{i\in N}e_{R}^{\rm C}(i,\overline{x})(\alpha)\) are a constant under the same \(\alpha\)-cut. Thus, \(\overline{e}_{L}^{\rm C}(i,\overline{x})(\alpha)\) and \(\overline{g}_{R}^{\rm C}(i,\overline{x})(\alpha)\) are also a constant, and any value can be used to replace \(\overline{e}_{L}^{\rm C}(i,\overline{x})(\alpha)\) and \(\overline{g}_{R}^{\rm C}(i,\overline{x})(\alpha)\), and its optimal solution remains unchanged. Let \(\overline{e}_{L}^{\rm C}(i,\overline{x})=\overline{k}_{L},\overline{g}_{R}^{ \rm C}(i,\overline{x})=\overline{k}_{R}\) (\(\overline{k}_{L}\) and \(\overline{k}_{R}\) are constants), and the verification is carried out as follows:  
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