具有联盟优先关系的模糊合作博弈的目标规划求解模型

The Goal Programming Model Solving Fuzzy Cooperative Games with Priority Coalition

用合作博弈研究实际管理问题中的分配方案时,常常存在一些不重要联盟或无效联盟,这些联盟影响公平合理的分配方案。因此,联盟的重要程度成为求解合作博弈必不可少的因素。本文考虑了联盟的重要性和局中人参与联盟的不确定性,研究了具有优先关系的模糊联盟合作博弈(简称为模糊合作博弈)。首先,借助于目标规划模型的优先因子可以表征联盟重要程度的思想,通过构建多优先级目标规划模型,得到模糊合作博弈新的解。其次,证明了构建的多优先级目标规划模型的解和模糊合作博弈的核心之间具有重要对应关系。最后,通过数值实例和比较分析,说明本文提出的多优先级目标规划模型求解模糊合作博弈问题的合理性和有效性。研究表明:(1)本文提出的多优先级目标规划模型考虑不同联盟重要程度,得到的解符合“多劳多得”原则,能够更公平合理解决实际管理中的分配问题。(2)本文的目标规划模型同时适用于求解存在联盟特征函数值缺失的合作博弈。与已有合作博弈的解进行比较分析,该模型无需估算无效联盟的特征函数缺失数据得到的分配值更为准确。从而,说明本文给出的目标规划求解模糊合作博弈解的模型,更符合许多管理学问题的实际情况。(3)通过多优先级目标规划模型最优解是否存在可判断模糊合作博弈的核心存在情况,若核心存在则该模型通过目标规划软件包可得到核心内的一个解,这样也得到了一个判断合作博弈核心是否存在的标准。(4)目标规划模型可弥补已有合作博弈解的一些不足,如核心可能为空集,Shapley值和最小二乘预核仁可能不满足个体合理性等。本文构建的多优先级目标规划模型不仅能求解联盟具有优先关系的模糊合作博弈,而且能够求解一般合作博弈的解,该目标规划模型作为合作博弈一种新的求解方法,能更有效地解决实际管理中的分配问题,具有更加广泛的应用价值。

As the allocation problem in practical management problem is studied by cooperative game theory,there exist some unimportant coalitions or invalid coalitions,which affect the fairness and rationality of the allocation.For example,the joint investment problem:if the three enterprises which are denoted as enterprise 1,enterprise 2 and enterprise 3,they partly participate in the joint investment in H company due to the consideration of risk,while enterprise 2 can provide the H company with advanced technology at the same time.If only enterprise 1 and enterprise 3 participate in the cooperation,the characteristic function value of the coalition{1,3}is far less than coalitions{1,2}and{2,3},so the coalition{1,3}will not be given firstly priority in the cooperation,which is a cooperative game with priority coalitions.However,these coalitions in cooperative games are treated equally solving by some exiting solutions such as core,Shapley value and the least square pre-nucleolus,which cannot reflect the different importance of the coalitions in the cooperation.Thus,it is necessary to find a new solution to consider the importance of coalitions.In this paper,the importance of coalitions and the uncertainty of players participating in any coalition are considered,and the fuzzy cooperative games with priority coalitions are suited.Firstly,the priority factors P,P,…,P,Pare used to represent the priority level of different coalitions,andP>>Pis stipulated,which means the importance degree of the k th coalition takes precedence over the importance degree of the k+1 coalition.A new solution of coalition cooperative games with priority coalitions is obtained by constructing the multi-priority goal programming model.Secondly,it has been proved that there is an important corresponding relationship between the core of fuzzy cooperative games and the solution of fuzzy cooperative games with priority coalitions obtained by the multi-priority goal programming model.Finally,some number examples are used to explain the validity and applicability of our proposed method by the comparison analysis with other methods.Our researching results show that:(1)The multi-priority goal programming model considers the importance of different coalitions,and the solution obtained by the model conforms the principle that players who contribute a lot to coalitions are allocated more values,which can solve the allocation problem in practical management more fairly and reasonably.(2)The multi-priority goal programming model is also applicable to fuzzy cooperative games with invalid coalitions.Comparing with the existing solutions,such as Shapley value and the least square pre-nucleolus,the goal programming model does not need to estimate the missing data of the characteristic function of the invalid coalitions,and the allocation values obtained by the model is more accurate.Thus,it is shown that the model given in this paper is more suitable for the practical situation of many management problems.(3)The multi-priority goal programming model can be used as a criterion to judge whether the core of fuzzy coalition cooperative games with priority coalitions are empty sets.If the core of fuzzy coalition cooperative games exists,then a solution can easily be implemented using common software packages,which is included in the core.(4)The multi-priority goal programming model may make up for some shortcomings of the existing solutions of fuzzy cooperative games with priority coalitions,such as the core may be an empty set,the Shapley value and the least square pre-nucleolus may not satisfy individual rationality.The multi-priority goal programming model proposed in this paper not only can be used to find the fuzzy coalition cooperative games with priority coalition,but also to the classic fuzzy coalition cooperative games.The model has more extensive application value.