灰色支付合作对策的核仁解
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摘要
鉴于实际对策问题中,灰信息是普遍存在的,但经典合作对策中未能考虑对策过程中出现的灰色不确定性,使得对策模型缺乏柔性.基于合作对策理论和灰集相关理论的思想,文章建立了一种新的包含有区间灰数的合作对策模型一灰色合作对策,研究了其核仁解.首先定义了灰色集的核函数和灰度函数,在此基础上给出其排序方法,得到适合此模型的相应定义,同时提出了灰色核仁解的概念;其次运用新的排序方法,将求核仁解的问题转化为求解非线性规划问题;最后进一步探讨了灰色合作对策核仁的存在唯一性,以及核仁与其他解之间的关系.从而解决了灰色合作对策的解的结构问题.
The grey information is universal in the actual game.But the classical cooperative game failed to consider the grey uncertainty in the process of the game,which makes the game model lacks flexibility.A new cooperative game model contains interval grey numbers-grey cooperative game is established based on the idea of cooperative game theory and the related theory of grey sets in this paper.And the nucleolus solution is studied.Firstly,the concepts of kernel function and degree of greyness function of grey sets are proposed.Based on these concepts,the ranking method is presented.Then the corresponding definitions for this model are get.The concept of grey nucleolus solution is put forward at the same time.Secondly,the problem of nucleolus solution can be converted to solving nonlinear programming problem by using the new ranking method.Finally,the existence and uniqueness of nucleolus of grey cooperative game are further discussed,as well as the relationship between the nucleolus and other solution.Thus,the problem of the solution structure of grey cooperative game is solved.
The grey information is universal in the actual game.But the classical cooperative game failed to consider the grey uncertainty in the process of the game,which makes the game model lacks flexibility.A new cooperative game model contains interval grey numbers-grey cooperative game is established based on the idea of cooperative game theory and the related theory of grey sets in this paper.And the nucleolus solution is studied.Firstly,the concepts of kernel function and degree of greyness function of grey sets are proposed.Based on these concepts,the ranking method is presented.Then the corresponding definitions for this model are get.The concept of grey nucleolus solution is put forward at the same time.Secondly,the problem of nucleolus solution can be converted to solving nonlinear programming problem by using the new ranking method.Finally,the existence and uniqueness of nucleolus of grey cooperative game are further discussed,as well as the relationship between the nucleolus and other solution.Thus,the problem of the solution structure of grey cooperative game is solved.
引文
[1]高作峰,王友,王国成.对策理论与经济管理决策.北京:中国林业出版社,2006.(Gao Z F,Wang Y,Wang G C.Game Theory and Economic Management Decision-Making.Beijing:China Forestry Publishing House,2006.)
[2]Chun Y.On the symmetric and weighted shapley values.International Journal of Game Theory,1991,20(2):183-190.
[3]Branzei R,Dimitrov D,Tijs S.Models in Cooperative Game Theory.Berlin Heidelberg:SpringerVerlag,2008.
[4]高作峰,徐东方,鄂成国,等.重复模糊合作对策的核心和稳定集.运筹与管理,2006,15(4):68-72.(Gao Z F,Xu D F,E C G,et al.The cores and the stable set for repeated fuzzy cooperative games.Operations Research and Management Science,2006,15(4):68-72.)
[5]刘思峰,郭天榜,党耀国.灰色系统理论及其应用.北京:科学出版社,1999.(Liu S F,Guo T B,Dang Y G.Grey System Theory and Its Application.Beijing:Science Press,1999.)
[6]方志耕.灰色博弈理论及其经济应用研究.博士论文.南京航空航天大学,南京,2007.(Fang Z G.Research on the grey game theory and its application in economy.Doctor's Dissertation.Nanjing University of Aeronautics and Astronautics,Nanjing,2007.)
[7]朱晓峰.缺失值填充的若干问题研究.硕士论文.广西师范大学,桂林,2007.(Zhu X F.Studies on missing data imputation.Master's Dissertation.Guangxi Normal University,Guilin,2007.)
[8]刘思峰,党耀国,方志耕.灰色系统理论及其应用.北京:科学出版社,2004.(Liu S F,Dang Y G,Fang Z G.Theory and Application of Grey System.Beijing:Science Press,2004.)
[9]邓聚龙.灰理论基础.武汉:华中科技大学出版社,2002.(Deng J L.Grey Theories Basis.Wuhan:Huazhong University of Science and Technology Press,2002.)
[10]谢科范.评灰色系统理论.系统工程,1991,9(4):49-52.(Xie K F.A comment on grey systems theory.Systems Engineering,1991,9(4):49-52.)
[11]罗党,王洁方.灰色决策理论与方法.北京:科学出版社,2012.(Luo D,Wang J F.Grey Decision Theory and Method.Beijing:Science Press,2012.)
[12]Bai G Z.The grey payoff matrix game.The Journal of Grey System,1992,4(4):323-331.
[13]Xu J P.Zero sum two-person game with grey number payoff matrix in linear programming.The Journal of Grey System,1998,10(3):225-233.
[14]顾洁.电力市场辅助报价决策的灰色博弈模型研究.电力系统保护与控制,2010,3s(10):12-16.(Gu J.Study on bidding strategy model for power market based on grey game theory.Power System Protection and Control,2010,38(10):12-16.)
[15]王清印,王峰松,左其亭,等.灰色数学基础.武汉:华中理工大学出版社,1996.(Wang Q Y,Wang F S,Zuo Q T,et al.Grey Mathematics Basis.Wuhan:Huazhong University of Science and Technology Press,1996.)
[16]宋捷.灰色决策方法及应用研究.博士论文.南京航空航天大学,南京,2010.(Song J.Research on the methods of grey decision-making and its application.Doctor's Dissertation.Nanjing University of Aeronautics and Astronautics,Nanjing,2010.)
[17]郭菊花,高作峰.直觉模糊支付合作对策的核仁解.运筹与管理,2014,23(3):102-107.(Guo J H,Gao Z F.Nucleolus of intuitionistic fuzzy payoffs cooperative game.Operations Research and Management Science,2014,23(3):102-107.)
[18]Fang Z G,Liu S F,Lu F,et al.Study on improvement of token and arithmetic of interval grey numbers and its GM(1,1)model.Engineering Science,2005,7(2):57-61.
[19]Liu S F,Fang Z G,Xie N M.Algorithm rules of interval grey numbers based on the"Kernel"and the degree of greyness of grey numbers.Systems Engineering and Electronics,2010,32(2):313-316.
[20]罗党,刘思峰.灰色多指标风险型决策方法研究.系统工程与电子技术,2004,26(8):1057-1059.(Luo D,Liu S F.Research on grey multi-criteria risk decision-making method.Systems Engineering and Electronics,2004,26(8):1057-1059.)
[21]李德勇.灰矩阵博弈的灰线性规划模型求解.统计与决策,2013,(8):79-81.(Li D Y.Grey linear programming model approach to grey matrix game.Statistics and Decision,2013,(8):79-81.)
[22]Liu S F,Lin Y.Grey Information:Theory and Practical Applications.London:Springer,2006.
[23]Yin M S.Fifteen years of grey system theory research:A historical review and bibliometric analysis.Expert Systems with Applications,2013,40:2267-2275.
[24]毛慧慧,高作峰,赵英豪,等.凸随机合作对策的核仁.佳木斯大学学报,2010,28(4):595-597.(Mao H H,Gao Z F,Zhao Y H,et al.Nucleolus of the convex stochastic cooperative game.Journal of Jiamusi University,2010,28(4):595-597.)
[25]王坚强.模糊多准则决策方法研究综述.控制与决策,2008,23(6):601-606.(Wang J Q.Overview on fuzzy multi-criteria decision-making approach.Control and Decision,2008,23(6):601-606.)
[26]Schmeidler D.The nucleolus of a characteristic function game.SIAM Journal on Applied Mathematics,1969,17(6):1163-1170.
[27]闫书丽,刘思峰,朱建军.基于相对核和精确度的灰数排列方法.控制与决策,2014,29(2):315-319.(Yan S L,Liu S F,Zhu J Z.The ranking method of grey numbers based on relative kernel and degree of accuracy.Control and Decision,2014,29(2):315-319.)
[28]Chen C I,Huang S J.The necessary and sufficient condition for GM(1,1)grey prediction model.Applied Mathematics and Computation,2013,219:6152-6162.
[2]Chun Y.On the symmetric and weighted shapley values.International Journal of Game Theory,1991,20(2):183-190.
[3]Branzei R,Dimitrov D,Tijs S.Models in Cooperative Game Theory.Berlin Heidelberg:SpringerVerlag,2008.
[4]高作峰,徐东方,鄂成国,等.重复模糊合作对策的核心和稳定集.运筹与管理,2006,15(4):68-72.(Gao Z F,Xu D F,E C G,et al.The cores and the stable set for repeated fuzzy cooperative games.Operations Research and Management Science,2006,15(4):68-72.)
[5]刘思峰,郭天榜,党耀国.灰色系统理论及其应用.北京:科学出版社,1999.(Liu S F,Guo T B,Dang Y G.Grey System Theory and Its Application.Beijing:Science Press,1999.)
[6]方志耕.灰色博弈理论及其经济应用研究.博士论文.南京航空航天大学,南京,2007.(Fang Z G.Research on the grey game theory and its application in economy.Doctor's Dissertation.Nanjing University of Aeronautics and Astronautics,Nanjing,2007.)
[7]朱晓峰.缺失值填充的若干问题研究.硕士论文.广西师范大学,桂林,2007.(Zhu X F.Studies on missing data imputation.Master's Dissertation.Guangxi Normal University,Guilin,2007.)
[8]刘思峰,党耀国,方志耕.灰色系统理论及其应用.北京:科学出版社,2004.(Liu S F,Dang Y G,Fang Z G.Theory and Application of Grey System.Beijing:Science Press,2004.)
[9]邓聚龙.灰理论基础.武汉:华中科技大学出版社,2002.(Deng J L.Grey Theories Basis.Wuhan:Huazhong University of Science and Technology Press,2002.)
[10]谢科范.评灰色系统理论.系统工程,1991,9(4):49-52.(Xie K F.A comment on grey systems theory.Systems Engineering,1991,9(4):49-52.)
[11]罗党,王洁方.灰色决策理论与方法.北京:科学出版社,2012.(Luo D,Wang J F.Grey Decision Theory and Method.Beijing:Science Press,2012.)
[12]Bai G Z.The grey payoff matrix game.The Journal of Grey System,1992,4(4):323-331.
[13]Xu J P.Zero sum two-person game with grey number payoff matrix in linear programming.The Journal of Grey System,1998,10(3):225-233.
[14]顾洁.电力市场辅助报价决策的灰色博弈模型研究.电力系统保护与控制,2010,3s(10):12-16.(Gu J.Study on bidding strategy model for power market based on grey game theory.Power System Protection and Control,2010,38(10):12-16.)
[15]王清印,王峰松,左其亭,等.灰色数学基础.武汉:华中理工大学出版社,1996.(Wang Q Y,Wang F S,Zuo Q T,et al.Grey Mathematics Basis.Wuhan:Huazhong University of Science and Technology Press,1996.)
[16]宋捷.灰色决策方法及应用研究.博士论文.南京航空航天大学,南京,2010.(Song J.Research on the methods of grey decision-making and its application.Doctor's Dissertation.Nanjing University of Aeronautics and Astronautics,Nanjing,2010.)
[17]郭菊花,高作峰.直觉模糊支付合作对策的核仁解.运筹与管理,2014,23(3):102-107.(Guo J H,Gao Z F.Nucleolus of intuitionistic fuzzy payoffs cooperative game.Operations Research and Management Science,2014,23(3):102-107.)
[18]Fang Z G,Liu S F,Lu F,et al.Study on improvement of token and arithmetic of interval grey numbers and its GM(1,1)model.Engineering Science,2005,7(2):57-61.
[19]Liu S F,Fang Z G,Xie N M.Algorithm rules of interval grey numbers based on the"Kernel"and the degree of greyness of grey numbers.Systems Engineering and Electronics,2010,32(2):313-316.
[20]罗党,刘思峰.灰色多指标风险型决策方法研究.系统工程与电子技术,2004,26(8):1057-1059.(Luo D,Liu S F.Research on grey multi-criteria risk decision-making method.Systems Engineering and Electronics,2004,26(8):1057-1059.)
[21]李德勇.灰矩阵博弈的灰线性规划模型求解.统计与决策,2013,(8):79-81.(Li D Y.Grey linear programming model approach to grey matrix game.Statistics and Decision,2013,(8):79-81.)
[22]Liu S F,Lin Y.Grey Information:Theory and Practical Applications.London:Springer,2006.
[23]Yin M S.Fifteen years of grey system theory research:A historical review and bibliometric analysis.Expert Systems with Applications,2013,40:2267-2275.
[24]毛慧慧,高作峰,赵英豪,等.凸随机合作对策的核仁.佳木斯大学学报,2010,28(4):595-597.(Mao H H,Gao Z F,Zhao Y H,et al.Nucleolus of the convex stochastic cooperative game.Journal of Jiamusi University,2010,28(4):595-597.)
[25]王坚强.模糊多准则决策方法研究综述.控制与决策,2008,23(6):601-606.(Wang J Q.Overview on fuzzy multi-criteria decision-making approach.Control and Decision,2008,23(6):601-606.)
[26]Schmeidler D.The nucleolus of a characteristic function game.SIAM Journal on Applied Mathematics,1969,17(6):1163-1170.
[27]闫书丽,刘思峰,朱建军.基于相对核和精确度的灰数排列方法.控制与决策,2014,29(2):315-319.(Yan S L,Liu S F,Zhu J Z.The ranking method of grey numbers based on relative kernel and degree of accuracy.Control and Decision,2014,29(2):315-319.)
[28]Chen C I,Huang S J.The necessary and sufficient condition for GM(1,1)grey prediction model.Applied Mathematics and Computation,2013,219:6152-6162.