基于相对熵的区间直觉模糊多属性决策方法
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摘要
多属性决策是指从不同的角度考察被评价对象（或方案），并通过综合的评价方法做出较为合理的决策。由于客观事物的复杂性、不确定性以及人类思维的模糊性，人们常常不能直接对决策信息作出精确的判断，而是以区间数、直觉模糊数的形式来表示。区间直觉模糊数是对直觉模糊数的进一步推广，它能够更加细腻地描述和刻画属性值的模糊性，因此基于区间直觉模糊数的不确定多属性决策理论、方法及应用成为当前决策研究工作者关注的热点。本文主要对属性值为区间直觉模糊数的多属性决策问题进行研究，主要工作如下：
1针对属性值为区间直觉模糊数的多属性决策问题，我们给出决策方案与理想点的区间直觉模糊数相对熵两个定义，分析其性质；同时给出了基于相对熵的区间直觉模糊多属性决策方法，并用实例加以论证及分析。
2针对属性权重未知或不完全的多属性决策问题，给出了基于区间直觉模糊相对熵的定义；基于相对熵定义，用离差最大化方法对权重信息完全未知和权重信息不完全的权重求解建立优化模型，并给出相应的决策方法，最后用实例加以论证。
The multi-attribute decision making is to examine the evaluation of the object (or program)from different angles, and make more rational decision-making through comprehensiveevaluation method. Due to the complexity of the objective things, uncertainty and the fuzzinessof human thinking, people often can not be directly in decision-making information to makeaccurate judgments, but by the interval number and in the form of intuitionistic fuzzy numbers torepresent. Interval valued intuitionistic fuzzy number is to further promote intuitionistic fuzzynumbers. It can be more delicate to describe and characterize the fuzziness of attribute values，sobased on the intuitionistic fuzzy number of uncertain multi-attribute decision theory, methodsand applications become the focus of attention of the current decision-making research workers.In this paper, the multi-attribute decision making problems of interval intuitionistic fuzzy inproperty value are studied, the main work are as follows:
1.For the property value of interval intuitionistic fuzzy multiple attribute decision makingproblems, We give the decision-making program, the ideal point of the interval intuition fuzzynumber relative entropy of the two definitions, analyze its nature; Based on relative entropy ofinterval intuitionistic fuzzy multiple attribute decision making, an example is presented toillustrate the approach.
2.With regard to the criteria weights with incomplete certain information in multicriteriadecision making, a new method is proposed based on interval-valued intuitionistic fuzzy relativeentropy, with a maximum deviation method without weight information and incompleteinformation on weights of solving weight optimization model is established. A method of thesedecision making problems is proposed. Finally, an example is used to illustrate the feasibilityand effectiveness of the proposed approach.
1针对属性值为区间直觉模糊数的多属性决策问题，我们给出决策方案与理想点的区间直觉模糊数相对熵两个定义，分析其性质；同时给出了基于相对熵的区间直觉模糊多属性决策方法，并用实例加以论证及分析。
2针对属性权重未知或不完全的多属性决策问题，给出了基于区间直觉模糊相对熵的定义；基于相对熵定义，用离差最大化方法对权重信息完全未知和权重信息不完全的权重求解建立优化模型，并给出相应的决策方法，最后用实例加以论证。
The multi-attribute decision making is to examine the evaluation of the object (or program)from different angles, and make more rational decision-making through comprehensiveevaluation method. Due to the complexity of the objective things, uncertainty and the fuzzinessof human thinking, people often can not be directly in decision-making information to makeaccurate judgments, but by the interval number and in the form of intuitionistic fuzzy numbers torepresent. Interval valued intuitionistic fuzzy number is to further promote intuitionistic fuzzynumbers. It can be more delicate to describe and characterize the fuzziness of attribute values，sobased on the intuitionistic fuzzy number of uncertain multi-attribute decision theory, methodsand applications become the focus of attention of the current decision-making research workers.In this paper, the multi-attribute decision making problems of interval intuitionistic fuzzy inproperty value are studied, the main work are as follows:
1.For the property value of interval intuitionistic fuzzy multiple attribute decision makingproblems, We give the decision-making program, the ideal point of the interval intuition fuzzynumber relative entropy of the two definitions, analyze its nature; Based on relative entropy ofinterval intuitionistic fuzzy multiple attribute decision making, an example is presented toillustrate the approach.
2.With regard to the criteria weights with incomplete certain information in multicriteriadecision making, a new method is proposed based on interval-valued intuitionistic fuzzy relativeentropy, with a maximum deviation method without weight information and incompleteinformation on weights of solving weight optimization model is established. A method of thesedecision making problems is proposed. Finally, an example is used to illustrate the feasibilityand effectiveness of the proposed approach.
引文
[1] C.L.Hwang, K.Yoon, Multiple attribute decision making, Beidelberg, New York:Springer-Verlag,1981.
[2] Z.S. Xu, On consistency of the weighted geometric mean complex judgment matrix inAHP, European Journal of Operational Research,26,2000,683-687.
[3]樊治平，张全，不确定性多属性决策的一种线性规划方法，东北大学学报，19,1998,419-421.
[4]徐南荣，仲伟俊，现代决策理论与方法，南京：东南大学出版社，2001.
[5]王应明，运用离差最大化方法进行多指标决策与排序，系统工程与电子技术，20,1998,24-260.
[6] R.R.Yager, On ordered weighted averaging aggregation operators in multi-criteria decisionmaking,Trans Systerms Man Cybern,18,1988,183-190.
[7]程明熙，处理多目标决策问题的二项系数加权和法，系统工程理论与实践，3,1983,23-26.
[8]陈伟，关于TOPSIS法应用中的逆序问题及消除的方法，运筹与管理，14,2005,50-54.
[9] C.L.Hwang, Multi-attribute decision making methods and applications, New York,Springer-Verlag,1981.
[10] G. A.Harsanyi,Multiagent multiattribute approach for conflict resolution in acid rain impactmitigation,IEEE Transactions on Systems,19,1989,1142-1153.
[11] A.Zeeny. Rational choice under fuzzy preferences: the orlovsky choice function,Fuzzy Setsand Systems,70,1993,263-268.
[12] T.L.Saaty, The Analytic Hierarchy Rrocess, New York, McGraw-Hill,1980.
[13]徐玖平，多指标（属性）评价双基点优序法，系统工程，10,1992,39-44.
[14]王殿选，多目标决策的对比系数法，系统工程理论与实践，8,1988,66-67.
[15]王应明，张军奎，基于标准差和平均差的权系数确定方法及其应用，数理统计与管理，22,2003,22-26.
[16]徐泽水，达庆利，多属性决策的组合赋权方法研究，中国管理科学，10,2002,84-87.
[17]张吉军，评价指标权重不能完全确定的多指标决策方法研究，运筹与管理，10,2001,151-153.
[18]周文坤，武振业，鞠廷英，多目标群体决策的一种综合及称方法，西南交通大学学报，36,2001,100-103.
[19]华小文，谭景信，基于“垂面”距离的TOPSIS法——正交投影法，系统工程理论与实践，12,2004,114-119.
[20]卜广志，张宇文，基于灰色模糊关系的灰色模糊综合评判，系统工程理论与实践，4，2002,141-144.
[21]吕锋，崔晓辉，多目标决策灰色关联投影法及其应用，系统工程理论与实践，1,2002,103-107.
[22]卫贵武，基于投影的直觉模糊数多属性决策方法，管理学报，6,2009,1154-1156.
[23] D. F.Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets,Control and Decision,24,2009,1398-1401.
[24]徐泽水，直觉模糊偏好信息下的多属性决策途径，系统工程理论与实践，27,2007,62-71.
[25] K.Atanassov, G.Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems,31,1989,343-349.
[26]胡辉，徐泽水，基于TOPSIS的区间直觉模糊多属性决策方法，模糊系统与科学，21,2007,108-112.
[27]王坚强，信息不完全确定的多准则区间直觉模糊决策方法，控制与决策，21,2006,1253-1256.
[28] Z.S.Xu, Methods for multiple attribute decision making with intuitionistic fuzzyinformation, Int J of Uncertainty, Fuzziness and Knowledge Based Systems,15,2007,285-297.
[29]徐泽水，区间直觉模糊信息的集成方法及其在决策中的应用，控制与决策，22,2007,215-219.
[30]魏翠屏，夏梅梅，张玉忠，基于区间直觉模糊集的多准则决策方法，控制与决策，24,2009,1230-1234.
[31]卫贵武，对方案有偏好的区间直觉模糊多属性决策方法，系统工程与电子技术，31,2009,116-120.
[32]徐泽水，陈剑，一种基于区间直觉判断矩阵的群决策方法，系统工程理论与实践，27,2007,126-133.
[33] G.W.Wei, Some induced geometric aggregation operators with intuitionistic fuzzyinformation and their application to group decision making, Applied Soft Computing,10,2010,423-431.
[34] Z.S.Xu, R.Yager, Intuionistic and their measures of similarity for the evaluation ofagreement within a group, Fuzzy Optimization Decision Making,8,2009,123-139.
[35] A. D.Luca,S.Termini, A definition of nonprobabilistic entropy in the setting of fuzzy setstheory,Information and Control,20,1972,301-312.
[36] P.Burillo, H.Bustince,Entropy on intuitionistic fuzzy sets and on interval-valued fuzzysets,Fuzzy Sets and Systems,118,2001,305-316.
[37] E.Szmidt,J.Kacprzyk. Entropy for intuitionistic fuzzy sets,Fuzzy sets and Systems,118,2001,467-477.
[38]周宇峰，魏法杰，基于相对熵的多属性决策组合赋权方法，运筹与管理，15,2006,48-53.
[39]戴厚平，基于信息熵的区间直觉模糊多属性决策方法，重庆文理学院学报，28,2009,1-4.
[40]冯向前，钱钢，基于熵权的区间数多属性决策方法，计算机工程与应用，46,2010,236-238.
[41]王晓，陈华友，基于相对熵的多粒度语言信息的多属性群决策方法，运筹与管理，19,2010,95-100.
[42] W.L.Hung,M.S.Yang, Fuzzy entropy on intuitionistic fuzzy sets, International Journal ofIntelligent Systems,21,2006,443-451.
[43] K.Atanassov, Intuitionistic fuzzy sets,Fuzzy Sets and Systems,20,1986,87-96.
[44] Z.S.Xu,On consistency of the weighted geometric mean complex judgment matrix in AHP,European Journal of Operational Research,26,2000,683-687.
[45] K. Atanassov, G.Gargov, Interval valued intuitionistic fuzzy sets,Fuzzy Sets and System s.31,1989,343-349.
[46] Z.S.Xu, Chen J, On geometric aggregation over interval-valued intuitionistic fuzzyinformation, The International Conference on Natural Computation (ICNC’07) and the4thInternational Conference on Fuzzy Systems and Knowledge Discovery (FSKD’07),Haikou,china,2,2007,466-471.
[47]徐泽水，区间直觉模糊信息的集成方法及其在决策中的应用，控制与决策，22,2007,215-219.
[48] S.M.Chen, J.M.Tan, Handling multicriteria fuzzy decision-making problems based on vagueset theory,Fuzzy Sets and System,67,1994,163-172.
[49] D.H.Hong, C.H.Choi,Multicriteria fuzzy decision-making problems based on vague settheory,Fuzzy Sets and Systems,114,2000,103-113.
[50] Z.S.Xu, Intuitionistic fuzzy aggregation operators,IEEE Transactions on FuzzySystems,15,2007,1179-1187.
[51]胡辉，徐泽水，基于TOPSIS的区间直觉模糊多属性决策方法，模糊系统与数学，21,2007,108-112.
[52]郭效芝，李健，区间值直观模糊集的相似测度和距离测度，模糊系统与数学，23,2009,124-130
[53]赵法信，基于区间值直觉模糊集的距离测度，微电子学与计算机，27,2010,188-192.
[54] T.M.Cover, J.A.Thomas, Elements of information theory, John Wiley and Sons,2006
[55] S.Kullbuck, lnformation Theory and Atatistics, New York Wlley,1959.
[2] Z.S. Xu, On consistency of the weighted geometric mean complex judgment matrix inAHP, European Journal of Operational Research,26,2000,683-687.
[3]樊治平，张全，不确定性多属性决策的一种线性规划方法，东北大学学报，19,1998,419-421.
[4]徐南荣，仲伟俊，现代决策理论与方法，南京：东南大学出版社，2001.
[5]王应明，运用离差最大化方法进行多指标决策与排序，系统工程与电子技术，20,1998,24-260.
[6] R.R.Yager, On ordered weighted averaging aggregation operators in multi-criteria decisionmaking,Trans Systerms Man Cybern,18,1988,183-190.
[7]程明熙，处理多目标决策问题的二项系数加权和法，系统工程理论与实践，3,1983,23-26.
[8]陈伟，关于TOPSIS法应用中的逆序问题及消除的方法，运筹与管理，14,2005,50-54.
[9] C.L.Hwang, Multi-attribute decision making methods and applications, New York,Springer-Verlag,1981.
[10] G. A.Harsanyi,Multiagent multiattribute approach for conflict resolution in acid rain impactmitigation,IEEE Transactions on Systems,19,1989,1142-1153.
[11] A.Zeeny. Rational choice under fuzzy preferences: the orlovsky choice function,Fuzzy Setsand Systems,70,1993,263-268.
[12] T.L.Saaty, The Analytic Hierarchy Rrocess, New York, McGraw-Hill,1980.
[13]徐玖平，多指标（属性）评价双基点优序法，系统工程，10,1992,39-44.
[14]王殿选，多目标决策的对比系数法，系统工程理论与实践，8,1988,66-67.
[15]王应明，张军奎，基于标准差和平均差的权系数确定方法及其应用，数理统计与管理，22,2003,22-26.
[16]徐泽水，达庆利，多属性决策的组合赋权方法研究，中国管理科学，10,2002,84-87.
[17]张吉军，评价指标权重不能完全确定的多指标决策方法研究，运筹与管理，10,2001,151-153.
[18]周文坤，武振业，鞠廷英，多目标群体决策的一种综合及称方法，西南交通大学学报，36,2001,100-103.
[19]华小文，谭景信，基于“垂面”距离的TOPSIS法——正交投影法，系统工程理论与实践，12,2004,114-119.
[20]卜广志，张宇文，基于灰色模糊关系的灰色模糊综合评判，系统工程理论与实践，4，2002,141-144.
[21]吕锋，崔晓辉，多目标决策灰色关联投影法及其应用，系统工程理论与实践，1,2002,103-107.
[22]卫贵武，基于投影的直觉模糊数多属性决策方法，管理学报，6,2009,1154-1156.
[23] D. F.Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets,Control and Decision,24,2009,1398-1401.
[24]徐泽水，直觉模糊偏好信息下的多属性决策途径，系统工程理论与实践，27,2007,62-71.
[25] K.Atanassov, G.Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems,31,1989,343-349.
[26]胡辉，徐泽水，基于TOPSIS的区间直觉模糊多属性决策方法，模糊系统与科学，21,2007,108-112.
[27]王坚强，信息不完全确定的多准则区间直觉模糊决策方法，控制与决策，21,2006,1253-1256.
[28] Z.S.Xu, Methods for multiple attribute decision making with intuitionistic fuzzyinformation, Int J of Uncertainty, Fuzziness and Knowledge Based Systems,15,2007,285-297.
[29]徐泽水，区间直觉模糊信息的集成方法及其在决策中的应用，控制与决策，22,2007,215-219.
[30]魏翠屏，夏梅梅，张玉忠，基于区间直觉模糊集的多准则决策方法，控制与决策，24,2009,1230-1234.
[31]卫贵武，对方案有偏好的区间直觉模糊多属性决策方法，系统工程与电子技术，31,2009,116-120.
[32]徐泽水，陈剑，一种基于区间直觉判断矩阵的群决策方法，系统工程理论与实践，27,2007,126-133.
[33] G.W.Wei, Some induced geometric aggregation operators with intuitionistic fuzzyinformation and their application to group decision making, Applied Soft Computing,10,2010,423-431.
[34] Z.S.Xu, R.Yager, Intuionistic and their measures of similarity for the evaluation ofagreement within a group, Fuzzy Optimization Decision Making,8,2009,123-139.
[35] A. D.Luca,S.Termini, A definition of nonprobabilistic entropy in the setting of fuzzy setstheory,Information and Control,20,1972,301-312.
[36] P.Burillo, H.Bustince,Entropy on intuitionistic fuzzy sets and on interval-valued fuzzysets,Fuzzy Sets and Systems,118,2001,305-316.
[37] E.Szmidt,J.Kacprzyk. Entropy for intuitionistic fuzzy sets,Fuzzy sets and Systems,118,2001,467-477.
[38]周宇峰，魏法杰，基于相对熵的多属性决策组合赋权方法，运筹与管理，15,2006,48-53.
[39]戴厚平，基于信息熵的区间直觉模糊多属性决策方法，重庆文理学院学报，28,2009,1-4.
[40]冯向前，钱钢，基于熵权的区间数多属性决策方法，计算机工程与应用，46,2010,236-238.
[41]王晓，陈华友，基于相对熵的多粒度语言信息的多属性群决策方法，运筹与管理，19,2010,95-100.
[42] W.L.Hung,M.S.Yang, Fuzzy entropy on intuitionistic fuzzy sets, International Journal ofIntelligent Systems,21,2006,443-451.
[43] K.Atanassov, Intuitionistic fuzzy sets,Fuzzy Sets and Systems,20,1986,87-96.
[44] Z.S.Xu,On consistency of the weighted geometric mean complex judgment matrix in AHP,European Journal of Operational Research,26,2000,683-687.
[45] K. Atanassov, G.Gargov, Interval valued intuitionistic fuzzy sets,Fuzzy Sets and System s.31,1989,343-349.
[46] Z.S.Xu, Chen J, On geometric aggregation over interval-valued intuitionistic fuzzyinformation, The International Conference on Natural Computation (ICNC’07) and the4thInternational Conference on Fuzzy Systems and Knowledge Discovery (FSKD’07),Haikou,china,2,2007,466-471.
[47]徐泽水，区间直觉模糊信息的集成方法及其在决策中的应用，控制与决策，22,2007,215-219.
[48] S.M.Chen, J.M.Tan, Handling multicriteria fuzzy decision-making problems based on vagueset theory,Fuzzy Sets and System,67,1994,163-172.
[49] D.H.Hong, C.H.Choi,Multicriteria fuzzy decision-making problems based on vague settheory,Fuzzy Sets and Systems,114,2000,103-113.
[50] Z.S.Xu, Intuitionistic fuzzy aggregation operators,IEEE Transactions on FuzzySystems,15,2007,1179-1187.
[51]胡辉，徐泽水，基于TOPSIS的区间直觉模糊多属性决策方法，模糊系统与数学，21,2007,108-112.
[52]郭效芝，李健，区间值直观模糊集的相似测度和距离测度，模糊系统与数学，23,2009,124-130
[53]赵法信，基于区间值直觉模糊集的距离测度，微电子学与计算机，27,2010,188-192.
[54] T.M.Cover, J.A.Thomas, Elements of information theory, John Wiley and Sons,2006
[55] S.Kullbuck, lnformation Theory and Atatistics, New York Wlley,1959.