多目标合作博弈最小二乘预核仁与核仁解

在实际生活中,常常存在许多带有不对等的联盟和不止一个关联或无关目标的复杂博弈情景.对此,本文首先构建了带有与联盟、目标相关的综合权重的多目标合作博弈,并在此基础上提出含有关联目标与无关目标的混合多目标合作博弈最小二乘预核仁与核仁解求解模型.其次,我们将经典的合作博弈最小二乘预核仁求解方法与核仁解算法推广到多目标合作博弈中,利用拉格朗日乘子法与伪逆理论得到了多目标合作博弈的最小二乘预核仁的显性表达式与最小二乘核仁解算法,并通过凸函数的性质,重新证明了该算法的有效性.最后,利用水资源的数值算例,说明并验证了文中构建的模型的正确性与有效性,并通过对比可知所构建模型的优越性.

Least squares prenucleolus and nucleolus solution of multi-objective cooperative games

In the real life, there exist many complicated game situations with the unequal coalitions and more than one relevant or irrelevant objections. First of all, this paper constructs the multi-objective cooperative games with comprehensive weights, and those weights are associated with coalitions and objections. Then the least squares prenucleolus model and the least squares nucleolus model of the multi-objective cooperative games are proposed with hybrid objections, including the irrelevant objections and relevant objections. Second, the methods of least squares prenucleolus and the algorithm of least squares nucleolus in the classical cooperative games are extended to the multi-objective cooperative games. Using Lagrange multiplier method and pseudo-inverse theory, we have the explicit expression of the least squares prenucleolus of the multi-objective cooperative games and an algorithm for least squares nucleolus of the multi-objective cooperative games, and also prove the validity of the algorithm by properties of convex function. Finally, the correctness and effectiveness of the proposed models are verified though an numerical example of water resources allocation, and the advantages of the proposed models are reflected by comparison.