基于区间对偶犹豫不确定语言广义Banzhaf Choquet积分算子的多属性决策方法
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摘要
基于将对偶犹豫模糊集和语言变量相结合定义对偶犹豫模糊语言集的思路,提出了区间对偶犹豫不确定语言集的概念,研究了区间对偶犹豫不确定语言变量相关的基本理论与方法,并针对属性值为区间对偶犹豫不确定语言信息的关联多属性决策问题,提出了相应的决策方法.首先,定义了区间对偶犹豫不确定语言变量的概念、运算法则、得分函数、精确函数、海明距离以及排序方法.然后,提出了区间对偶犹豫不确定语言广义Banzhaf Choquet积分算子并证明了该算子的一些性质.为了确定属性集的模糊测度,建立了基于离差最大化方法以及Banzhaf函数的模型.进而,给出一种用于解决属性权重部分未知,属性值为区间对偶犹豫不确定语言变量的关联多属性决策方法.最后,通过算例验证了该方法的有效性.
The concept of dual hesitant fuzzy linguistic set is based on dual hesitant fuzzy set and linguistic variables. Motivated by this idea, this paper proposes the concept of interval-valued dual hesitant uncertain linguistic set used to solve more complex decision making. Firstly, this paper studies the concept,operations, score function, accuracy function, Hamming distance, and ranking method of interval-valued dual hesitant uncertain linguistic variables. Then, the interval-valued dual hesitant uncertain linguistic generalized Banzhaf Choquet integral operator is proposed. Meantime, some properties of this operator are investigated. To determine the optimal fuzzy measure on criteria set, the model based on maximization deviation method and Banzhaf function is established. Furthermore, an approach to muti-criteria decision making problem with incomplete weight information and interactive condition under interval-valued dual hesitant uncertain linguistic environment is developed. Finally, an example is shown to verify the effectiveness of the given method.
The concept of dual hesitant fuzzy linguistic set is based on dual hesitant fuzzy set and linguistic variables. Motivated by this idea, this paper proposes the concept of interval-valued dual hesitant uncertain linguistic set used to solve more complex decision making. Firstly, this paper studies the concept,operations, score function, accuracy function, Hamming distance, and ranking method of interval-valued dual hesitant uncertain linguistic variables. Then, the interval-valued dual hesitant uncertain linguistic generalized Banzhaf Choquet integral operator is proposed. Meantime, some properties of this operator are investigated. To determine the optimal fuzzy measure on criteria set, the model based on maximization deviation method and Banzhaf function is established. Furthermore, an approach to muti-criteria decision making problem with incomplete weight information and interactive condition under interval-valued dual hesitant uncertain linguistic environment is developed. Finally, an example is shown to verify the effectiveness of the given method.
引文
[1]Torra V.Hesitant fuzzy sets[J].International Journal of Intelligent Systems,2010,25:529-539.
[2]Torra V,Narukawa Y.On hesitant fuzzy sets and decision[C]//The 18th IEEE International Conference on Fuzzy Systems,Jeju Island,Korea,2009:1378-1382.
[3]Zhao N,Xu Z S,Liu F J.Uncertainty measures for hesitant fuzzy information[J].International Journal of Intelligent Systems,2015,30(7):818-836.
[4]Zhao N,Xu Z S,Ren Z L.On typical hesitant fuzzy prioritized"or"operator in multi-attribute decision making[J].International Journal of Intelligent Systems,2016,31(1):73-100.
[5]Chen N,Xu Z S,Xia M M.Interval-valued hesitant preference relations and their applications to group decision making[J].Knowledge-Based Systems,2013,37(2):528-540.
[6]Zhu B,Xu Z S,Xia M M.Dual hesitant fuzzy sets[J].Journal of Applied Mathematics,2012,26(5):410-425.
[7]Tyagi S K.Correlation coefficient of dual hesitant fuzzy sets and its applications[J].Applied Mathematical Modelling,2015,39(22):7082-7092.
[8]Zhao N,Xu Z S.Entropy measures for dual hesitant fuzzy information[C]//2015 Fifth International Conference on Communication Systems and Network Technologies,2015:1152-1156.
[9]Zhao N,Xu Z S,Liu F J.Group decision making with dual hesitant fuzzy preferenc relations[J].Cognitive Computation,2016,8(6):1119-1143.
[10]Ju Y B,Liu X Y,Yang S H.Interval-valued dual hesitant fuzzy aggregation operators and their applications to multiple attribute decision making[J].Journal of Intelligent&Fuzzy Systems,2014,27:1203-1218.
[11]Zhou W,Xu Z S.Generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites[J].European Journal of Operational Research,2016,254(2):610-621.
[12]Wu Z B,Xu J P.Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations[J].Omega,2016,65:28-40.
[13]彭勃,叶春明.基于不确定纯语言混合调和平均算子的多属性群决策方法[J].中国管理科学,2015,23(2):131-138.Peng B,Ye C M.An approach to multiple attribute group decision making based on uncertain pure linguistic hybrid harmonic averaging operator[J].Chinese Journal of Management Science,2015,23(2):131-138.
[14]Yang S H,Ju Y B.Dual hesitant fuzzy linguistic aggregation operators and their applications to multi-attribute decision making[J].Journal of Intelligent&Fuzzy Systems,2014,27:1935-1947.
[15]杨尚洪,鞠彦兵.基于对偶犹豫模糊语言变量的多属性决策方法[J].运筹与管理,2015,24(5):91-96.Yang S H,Ju Y B.Multi-attribute decision-making method based on dual hesitant fuzzy linguistic variables[J].Operations Research and Management Science,2015,24(5):91-96.
[16]Zhao N,Xu Z S.A group decision making method with double hesitant linguistic preference relations[J].Journal of Southeast University,2016,32:240-249.
[17]Sugeno M.Theory of fuzzy integrals and its applications[D].Tokyo:Tokyo Institute of Technology,1974.
[18]毕克新,孙金花,张铁柱,等.基于模糊积分的区域中小企业技术创新测度与评价[J].系统工程理论与实践,2005,25(2):40-46.Bi K X,Sun J H,Zhang T Z,et al.Measurement and evaluation of regional technological innovation in small and medium enterprises based on fuzzy integral[J].Systems Engineering—Theory&Practice,2005,25(2):40-46.
[19]程楠,祝彦知.中型水电站群优化排序的模糊积分加权整数规划模型[J].系统工程理论与实践,2008,28(11):122-128.Cheng N,Zhu Y Z.Optimal sequencing and scheduling of middle-sized hydroelectric stations based on fuzzy integral weighting integer programming[J].Systems Engineering—Theory&Practice,2008,28(11):122-128.
[20]Xu Z S.Choquet integrals of weighted intuitionistic fuzzy information[J].Information Sciences,2010,180:726-736.
[21]Tan C Q,Chen X H.Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making[J].Expert Systems with Applications,2010,37:149-157.
[22]Xu Z S.Additive intuitionistic fuzzy aggregation operators based on fuzzy measure[J].International Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2016,24(1):1-12.
[23]陈岩,李庭.基于Choquet积分的直觉不确定语言信息集结算子及其应用[J].控制与决策,2016,31(5):842-852.Chen Y,Li T.Intuitionistic uncertain linguistic information aggregation operators based on Choquet integral and their application[J].Control and Decision,2016,31(5):842-852.
[24]Meng F Y,Zhang Q,Zhan J Q.The interval-valued intuitionistic fuzzy geometric Choquet aggregation operator based on the generalized Banzhaf index and 2-additive measure[J].Technological&Economic Development,2015,21(2):186-215.
[25]彭张林,张强,李珠瑞,等.改进的离差最大化决策模型及其在临近空间多任务规划中的应用[J].系统工程理论与实践,2014,34(2):421-427.Peng Z L,Zhang Q,Li Z R,et al.Improved maximizing deviation decision-making model and its application in multi-mission planning of near space system[J].Systems Engineering—Theory&Practice,2014,34(2):421-427.
[26]Banzhaf J.Weighted voting does not work:A mathematical analysis[J].Rutgers Law Review,1965,19:317-343.
[27]Herrera F,Herrera-Viedma E.Linguistic decision analysis:Steps for solving decision problems under linguistic information[J].Fuzzy Sets and Systems,2000,115(1):67-82.
[28]Xu Z S.Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment[J].Information Sciences,2004,168(1-4):171-184.
[29]Marichal J L.The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making[J].Discrete Applied Mathematics,2000,107(1-3):139-164.
[30]Murofushi T,Sugeno M.An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure[J].Fuzzy Sets and Systems,1989,29(2):201-227.
[31]李珠瑞,马溪骏,彭张林.基于离差最大化的组合评价方法研究[J].中国管理科学,2013,21(1):174-179.Li Z R,Ma X J,Peng Z L.Research on the combination evaluation method based on maximizing deviations[J].Chinese Journal of Management Science,2013,21(1):174-179.
[2]Torra V,Narukawa Y.On hesitant fuzzy sets and decision[C]//The 18th IEEE International Conference on Fuzzy Systems,Jeju Island,Korea,2009:1378-1382.
[3]Zhao N,Xu Z S,Liu F J.Uncertainty measures for hesitant fuzzy information[J].International Journal of Intelligent Systems,2015,30(7):818-836.
[4]Zhao N,Xu Z S,Ren Z L.On typical hesitant fuzzy prioritized"or"operator in multi-attribute decision making[J].International Journal of Intelligent Systems,2016,31(1):73-100.
[5]Chen N,Xu Z S,Xia M M.Interval-valued hesitant preference relations and their applications to group decision making[J].Knowledge-Based Systems,2013,37(2):528-540.
[6]Zhu B,Xu Z S,Xia M M.Dual hesitant fuzzy sets[J].Journal of Applied Mathematics,2012,26(5):410-425.
[7]Tyagi S K.Correlation coefficient of dual hesitant fuzzy sets and its applications[J].Applied Mathematical Modelling,2015,39(22):7082-7092.
[8]Zhao N,Xu Z S.Entropy measures for dual hesitant fuzzy information[C]//2015 Fifth International Conference on Communication Systems and Network Technologies,2015:1152-1156.
[9]Zhao N,Xu Z S,Liu F J.Group decision making with dual hesitant fuzzy preferenc relations[J].Cognitive Computation,2016,8(6):1119-1143.
[10]Ju Y B,Liu X Y,Yang S H.Interval-valued dual hesitant fuzzy aggregation operators and their applications to multiple attribute decision making[J].Journal of Intelligent&Fuzzy Systems,2014,27:1203-1218.
[11]Zhou W,Xu Z S.Generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites[J].European Journal of Operational Research,2016,254(2):610-621.
[12]Wu Z B,Xu J P.Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations[J].Omega,2016,65:28-40.
[13]彭勃,叶春明.基于不确定纯语言混合调和平均算子的多属性群决策方法[J].中国管理科学,2015,23(2):131-138.Peng B,Ye C M.An approach to multiple attribute group decision making based on uncertain pure linguistic hybrid harmonic averaging operator[J].Chinese Journal of Management Science,2015,23(2):131-138.
[14]Yang S H,Ju Y B.Dual hesitant fuzzy linguistic aggregation operators and their applications to multi-attribute decision making[J].Journal of Intelligent&Fuzzy Systems,2014,27:1935-1947.
[15]杨尚洪,鞠彦兵.基于对偶犹豫模糊语言变量的多属性决策方法[J].运筹与管理,2015,24(5):91-96.Yang S H,Ju Y B.Multi-attribute decision-making method based on dual hesitant fuzzy linguistic variables[J].Operations Research and Management Science,2015,24(5):91-96.
[16]Zhao N,Xu Z S.A group decision making method with double hesitant linguistic preference relations[J].Journal of Southeast University,2016,32:240-249.
[17]Sugeno M.Theory of fuzzy integrals and its applications[D].Tokyo:Tokyo Institute of Technology,1974.
[18]毕克新,孙金花,张铁柱,等.基于模糊积分的区域中小企业技术创新测度与评价[J].系统工程理论与实践,2005,25(2):40-46.Bi K X,Sun J H,Zhang T Z,et al.Measurement and evaluation of regional technological innovation in small and medium enterprises based on fuzzy integral[J].Systems Engineering—Theory&Practice,2005,25(2):40-46.
[19]程楠,祝彦知.中型水电站群优化排序的模糊积分加权整数规划模型[J].系统工程理论与实践,2008,28(11):122-128.Cheng N,Zhu Y Z.Optimal sequencing and scheduling of middle-sized hydroelectric stations based on fuzzy integral weighting integer programming[J].Systems Engineering—Theory&Practice,2008,28(11):122-128.
[20]Xu Z S.Choquet integrals of weighted intuitionistic fuzzy information[J].Information Sciences,2010,180:726-736.
[21]Tan C Q,Chen X H.Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making[J].Expert Systems with Applications,2010,37:149-157.
[22]Xu Z S.Additive intuitionistic fuzzy aggregation operators based on fuzzy measure[J].International Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2016,24(1):1-12.
[23]陈岩,李庭.基于Choquet积分的直觉不确定语言信息集结算子及其应用[J].控制与决策,2016,31(5):842-852.Chen Y,Li T.Intuitionistic uncertain linguistic information aggregation operators based on Choquet integral and their application[J].Control and Decision,2016,31(5):842-852.
[24]Meng F Y,Zhang Q,Zhan J Q.The interval-valued intuitionistic fuzzy geometric Choquet aggregation operator based on the generalized Banzhaf index and 2-additive measure[J].Technological&Economic Development,2015,21(2):186-215.
[25]彭张林,张强,李珠瑞,等.改进的离差最大化决策模型及其在临近空间多任务规划中的应用[J].系统工程理论与实践,2014,34(2):421-427.Peng Z L,Zhang Q,Li Z R,et al.Improved maximizing deviation decision-making model and its application in multi-mission planning of near space system[J].Systems Engineering—Theory&Practice,2014,34(2):421-427.
[26]Banzhaf J.Weighted voting does not work:A mathematical analysis[J].Rutgers Law Review,1965,19:317-343.
[27]Herrera F,Herrera-Viedma E.Linguistic decision analysis:Steps for solving decision problems under linguistic information[J].Fuzzy Sets and Systems,2000,115(1):67-82.
[28]Xu Z S.Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment[J].Information Sciences,2004,168(1-4):171-184.
[29]Marichal J L.The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making[J].Discrete Applied Mathematics,2000,107(1-3):139-164.
[30]Murofushi T,Sugeno M.An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure[J].Fuzzy Sets and Systems,1989,29(2):201-227.
[31]李珠瑞,马溪骏,彭张林.基于离差最大化的组合评价方法研究[J].中国管理科学,2013,21(1):174-179.Li Z R,Ma X J,Peng Z L.Research on the combination evaluation method based on maximizing deviations[J].Chinese Journal of Management Science,2013,21(1):174-179.