联盟值为梯形模糊数的合作对策最小平方求解模型与方法
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摘要
针对现实经济管理决策环境与条件具有模糊性的特点,着重研究一类联盟特征(或支付)值为梯形模糊数的合作对策,提出一种求解梯形模糊数合作对策的最小平方优化方法.利用梯形模糊数距离(平方)概念和最小平方法,建立最小化局中人联盟分配和支付值差值平方和的优化模型,根据模型推导出联盟成员梯形模糊数分配值的解析公式,探讨该最小平方解的重要性质.设计一种新的理论优化模型以避免传统梯形模糊数减法导致的计算结果不确定性扩大等问题,为求解梯形模糊数合作对策提供一种新的实践工具与参考思路.
In view of the characteristics of the fuzzy nature of real economic management decision making environment and conditions, this paper studies a class of cooperative games that the values of coalitions are expressed with trapezoidal fuzzy numbers. The main purpose of this paper is to develop a kind of least square optimization method for solving trapezoidal fuzzy number cooperative games. Firstly, according to the concept of the trapezoidal fuzzy numbers' distance(square) and the least square method, an optimized mathematical model is proposed by considering that players in coalitions try to guarantee their payoffs' sums being as close to the coalitions' values as possible. Then all players' analytical formulas of trapezoidal fuzzy number payoffs are determined by the optimized mathematical model. Finally,some important properties of the least square solutions are discussed. The new model is proposed to avoid the uncertainty expansion caused by the trapezoidal fuzzy number subtraction, which can provide a new theoretical perspective and practical tool for solving the trapezoidal fuzzy number cooperative games.
In view of the characteristics of the fuzzy nature of real economic management decision making environment and conditions, this paper studies a class of cooperative games that the values of coalitions are expressed with trapezoidal fuzzy numbers. The main purpose of this paper is to develop a kind of least square optimization method for solving trapezoidal fuzzy number cooperative games. Firstly, according to the concept of the trapezoidal fuzzy numbers' distance(square) and the least square method, an optimized mathematical model is proposed by considering that players in coalitions try to guarantee their payoffs' sums being as close to the coalitions' values as possible. Then all players' analytical formulas of trapezoidal fuzzy number payoffs are determined by the optimized mathematical model. Finally,some important properties of the least square solutions are discussed. The new model is proposed to avoid the uncertainty expansion caused by the trapezoidal fuzzy number subtraction, which can provide a new theoretical perspective and practical tool for solving the trapezoidal fuzzy number cooperative games.
引文
[1]李登峰.模糊多目标多人决策与对策[M].北京:国防工业出版社,2003:196-244.(Li D F.Fuzzy multiobjection many-person decision makings and games[M].Beijing:National Defense Industry Press,2003:196-244.)
[2]Li D F.Lexicographic method for matrix games with payoffs of triangular fuzzy numbers[J].Int J of Uncertainty,Fuzziness and Knowledge-Based Systems2008,16(3):371-389.
[3]李登峰.直觉模糊集决策与对策分析方法[M].北京:国防工业出版社,2012:235-269.(Li D F.Intuitionistic fuzzy set decision and game analysis methodologies[M].Beijing:National Defense Industry Press,2012:235-269.)
[4]Yu X H,Zhang Q.An extension of coopera-tive fuzzy games[J].Fuzzy Sets and Systems,2010,161:1614-1634.
[5]杨洁,李登峰.求解梯形模糊矩阵对策的线性规划方法[J].控制与决策,2015,30(7):1219-1226.(Yang J,Li D F.Linear programming methodology for solving trapezoidal fuzzy matrix games[J].Control and Decision,2015,30(7):1219-1226.)
[6]Li D F.Linear programming approach to solve interval-valued matrix games[J].Omega,2011,39:655-666.
[7]Li D F.Models and methods of interval-valued games in economic management[M].Switzerland:Springer Int Publishing,2016:69-128.
[8]Mare?M.Fuzzy cooperative games[M].Heidelberg:Physica-Verlag,2001:39-134.
[9]谭春桥,张强.具有区间联盟值n人对策的Shapley值[J].应用数学学报,2010,33(2):193-203.(Tan C Q,Zhang Q.Shapley value for n-person games with interval coalition worth[J].Acta Mathematicae Applicatae Sinica,2010,33(2):193-203.)
[10]黄礼健,吴祈宗,张强.具有模糊联盟值的n人合作博弈的模糊Shapley值[J].北京理工大学学报,200727(8):740-744.(Huang L J,Wu Q Z,Zhang Q.Fuzzy shapley value of n-person cooperative games with fuzzy worth[J]Transaction of Beijing Institute of Technology,200727(8):740-744.)
[11]陈雯,张强.模糊合作对策的Shapley值[J].管理科学学报,2006,9(5):50-55.(Chen W,Zhang Q.Shapley value for fuzzy cooperative games[J].J of Management Science in China,2006,9(5):50-55.)
[12]李登峰,刘家财.基于最小平方距离的区间值合作对策求解模型与方法[J].中国管理科学,2016,24(7):135-142.(Li D F,Liu J C.Models and method of interval-valued cooperative games based on the least square distance[J]Chinese J of Management Science,2016,24(7):135-142.)
[2]Li D F.Lexicographic method for matrix games with payoffs of triangular fuzzy numbers[J].Int J of Uncertainty,Fuzziness and Knowledge-Based Systems2008,16(3):371-389.
[3]李登峰.直觉模糊集决策与对策分析方法[M].北京:国防工业出版社,2012:235-269.(Li D F.Intuitionistic fuzzy set decision and game analysis methodologies[M].Beijing:National Defense Industry Press,2012:235-269.)
[4]Yu X H,Zhang Q.An extension of coopera-tive fuzzy games[J].Fuzzy Sets and Systems,2010,161:1614-1634.
[5]杨洁,李登峰.求解梯形模糊矩阵对策的线性规划方法[J].控制与决策,2015,30(7):1219-1226.(Yang J,Li D F.Linear programming methodology for solving trapezoidal fuzzy matrix games[J].Control and Decision,2015,30(7):1219-1226.)
[6]Li D F.Linear programming approach to solve interval-valued matrix games[J].Omega,2011,39:655-666.
[7]Li D F.Models and methods of interval-valued games in economic management[M].Switzerland:Springer Int Publishing,2016:69-128.
[8]Mare?M.Fuzzy cooperative games[M].Heidelberg:Physica-Verlag,2001:39-134.
[9]谭春桥,张强.具有区间联盟值n人对策的Shapley值[J].应用数学学报,2010,33(2):193-203.(Tan C Q,Zhang Q.Shapley value for n-person games with interval coalition worth[J].Acta Mathematicae Applicatae Sinica,2010,33(2):193-203.)
[10]黄礼健,吴祈宗,张强.具有模糊联盟值的n人合作博弈的模糊Shapley值[J].北京理工大学学报,200727(8):740-744.(Huang L J,Wu Q Z,Zhang Q.Fuzzy shapley value of n-person cooperative games with fuzzy worth[J]Transaction of Beijing Institute of Technology,200727(8):740-744.)
[11]陈雯,张强.模糊合作对策的Shapley值[J].管理科学学报,2006,9(5):50-55.(Chen W,Zhang Q.Shapley value for fuzzy cooperative games[J].J of Management Science in China,2006,9(5):50-55.)
[12]李登峰,刘家财.基于最小平方距离的区间值合作对策求解模型与方法[J].中国管理科学,2016,24(7):135-142.(Li D F,Liu J C.Models and method of interval-valued cooperative games based on the least square distance[J]Chinese J of Management Science,2016,24(7):135-142.)