基于结构元理论的模糊联盟合作对策

仅考虑局中人参与率模糊的合作对策，称为模糊联盟合作对策。将该模型中的模糊参与率用模糊结构元表示，得到基于结构元理论的具有模糊数Choquet积分表达形式的支付函数和Shapley值的理论框架，继而定义结构元线性生成的模糊支付函数和Shapley值表达式。通过算例与区间数的方法进行对比，结果表明：基于结构元理论的模糊联盟合作对策，模型中的模糊数均由结构元线性生成，模糊数之间的四则运算转化成简单的函数表达式之间的四则运算，避免了模糊数之间运算的遍历性，运算简便。运算结果包括区间及区间上各点的隶属度，结果更加精确。

Cooperative games under fuzzy coalitions based on fuzzy structured element

Fuzzy participation rate and fuzzy payoffs are both important factors in fuzzy cooperative games. Cooperative games under fuzzy coalitions only consider the fuzzy participation rate. The fuzzy participation rate is expressed by fuzzy structured element. Then theoretical framework for payment function and shapley value with fuzzy choquet integral expression based on structured element theory  is given. The linear expression of fuzzy payment function and shapley value with structured element are defined. The model is compared with interval numbers method through an example. The results show that the cooperative games under fuzzy coalitions based on fuzzy structured element is of simple operation and more precise, because of its simple operation among functions and complete expression of fuzzy numbers.