加权犹豫模糊集的群决策方法
|
摘要
对于犹豫模糊元中的不同隶属度值赋予不同的权重,由此构造出一种应用范围更广、更符合实际需要的犹豫模糊集合—–加权犹豫模糊集合.针对加权犹豫模糊集中的加权犹豫模糊元,定义了加权犹豫模糊集合和加权犹豫模糊元的并、交、余、数乘和幂等运算及其运算法则,并讨论它们的运算性质;同时,给出加权犹豫模糊元的得分函数和离散函数,进而给出一种比较加权犹豫模糊元的排序法则.在此基础上,提出两类集成算子:加权犹豫模糊元的加权算术平均算子和加权犹豫模糊元的加权几何平均算子,并针对专家权重(已知和未知)的两种情形,将加权犹豫模糊集合应用于群决策,给出两种基于加权犹豫模糊集合的群决策方法.最后,通过一个应用实例表明所提出的群决策方法的有效性和实用性.
In this paper, we introduce the concept of weighted hesitant fuzzy set, in which different weights are designed to these possible membership values, and the weights indicate that the decision maker has different confidence in giving every possible assessment of the membership degree. Then we define some basic operations such as union, intersection,complement, multiplication and power operation of weighted hesitant fuzzy elements and weighted hesitant fuzzy sets,discuss their operation properties, and propose the score function and variance function of the weighted hesitant fuzzy element to compare two weighted hesitant fuzzy elements. Furthermore, we present two aggregation operators such as the weighted hesitant fuzzy element weighted averaging(WHFWA) operator and the weighted hesitant fuzzy element weighted geometric(WHFWG) operator to aggregate weighted hesitant fuzzy information, and build the mathematical model of group decision making based on the expert weights(known and unknown). Finally, a numerical example is given to illustrate the effectiveness and feasibility of the proposed method.
In this paper, we introduce the concept of weighted hesitant fuzzy set, in which different weights are designed to these possible membership values, and the weights indicate that the decision maker has different confidence in giving every possible assessment of the membership degree. Then we define some basic operations such as union, intersection,complement, multiplication and power operation of weighted hesitant fuzzy elements and weighted hesitant fuzzy sets,discuss their operation properties, and propose the score function and variance function of the weighted hesitant fuzzy element to compare two weighted hesitant fuzzy elements. Furthermore, we present two aggregation operators such as the weighted hesitant fuzzy element weighted averaging(WHFWA) operator and the weighted hesitant fuzzy element weighted geometric(WHFWG) operator to aggregate weighted hesitant fuzzy information, and build the mathematical model of group decision making based on the expert weights(known and unknown). Finally, a numerical example is given to illustrate the effectiveness and feasibility of the proposed method.
引文
[1] Zadeh L A. Fuzzy sets[J]. Informmation and Control,1965, 8(3):338-353.
[2] Atanassov K. Intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1986, 20(1):87-96.
[3] Dubois D, Prade H. Fuzzy sets and systems:Theory and applications[M]. New York:Academic Press, 1980:40-60.
[4] Torra V. Hesitant fuzzy sets[J]. Int J of Intelligent Systems, 2010, 25(6):529-539.
[5] Rodríguez R M, Martínez L, Torra V, et al. Hesitant fuzzy sets:State of the art and future directions[J]. Int J of Intelligent Systems, 2014, 29(4):495-524.
[6] Xia M M, Xu Z S. Hesitant fuzzy information aggregationin decision making[J]. Int J of Approximate Reasoning,2011, 52(3):395-407.
[7] Xia M M, Xu Z S, Chen N. Some hesitant fuzzy aggregation operators with their application in group decision making[J]. Group Decision Negotiation, 2013,22(2):259-279.
[8] Zhu B, Xu Z S, Xia M M. Hesitant fuzzy geomeric Bonferroni means[J]. Information Sciences, 2012,182(11):72-85.
[9] Wei G W. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making[J].Knowledge-Based Systems, 2012, 31(1):176-182.
[10] Zhang Z M. Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making[J]. Information Sciences, 2013, 234(1):150-181.
[11]廖虎昌.复杂模糊多属性决策理论与方法[M].北京:科学出版社, 2016:80-120.(Liao H C. Theory and method for complicated fuzzy multiple attribute decision[M]. Beijing:Science Press,2016:80-120.)
[12]陈树伟,蔡丽娜.区间值犹豫模糊集[J].模糊系统与数学, 2013, 27(6):38-44.(Chen S W, Cai L N. Interval-valued hesitant fuzzy sets[J].Fuzzy Systems and Mathematics, 2013, 27(6):38-44.)
[13] Qian G, Wang H, Feng X. Generalized hesitant fuzzy sets and their application in decision support system[J].Knowledge-Based Systems, 2013, 37(3):357-365.
[14] Wei G W, Zhao X F, Lin R. Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making[J]. Knowledge-Based Systems, 2012, 31(2):176-182.
[15] 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(5):528-540.
[16]吴婉莹,何迎东,郭甦,等.直觉对偶犹豫模糊集的集结算子及其应用[J].武汉理工大学学报:信息与管理工程版, 2014, 36(2):225-228.(Wu W Y, He Y D, Guo S, et al. Intuitionistic dual hesitant fuzzy set operators and its application[J]. J of Wuhan University of Technology:Information&Management Engineering, 2014, 36(2):225-228.)
[17] Rodríguez R M, Martínez L, Herrera F. Hesitant fuzzy linguistic term sets for decision making[J]. IEEE Trans on Fuzzy Systems, 2012, 20(2):109-119.
[18] Rodríguez R M, Martínez L, Herrera F. A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets[J]. Information Sciences, 2013, 241(3):28-42.
[19] Wei C P, Zhao N, Tang X J. Comparisons of hesitant fuzzy linguistic term sets[J]. IEEE Trans on Fuzzy Systems,2014, 22(5):575-585.
[20]刘小弟,朱建军,刘思峰.犹豫模糊信息下的双向投影决策方法[J].系统工程理论与实践, 2014, 34(10):2637-2644.(Liu X D, Zhu J J, Liu S F. Bidirectional projection method with hesitant fuzzy information[J]. Systems Engineering—Theory&Practice, 2014, 34(10):2637-2644.)
[21]王坚强,吴佳亭.基于优序关系的犹豫模糊语言多准则决策方法[J].控制与决策, 2015, 30(5):887-891.(Wang J Q, Wu J T. Method for multi-criteria decision-making with hesitant fuzzy linguistic based on outranking relation[J]. Control and Decision, 2015, 30(5):887-891.)
[22] Xu Z S, Xia M M. Distance and similarity measures for hesitant fuzzy sets[J]. Information Sciences, 2011,181(11):2128-2138.
[23] Xu Z S, Xia M M. On distance and correlation measures of hesitant fuzzy information[J]. Int J of Intelligent Systems,2014, 26(5):410-425.
[24] Peng D H, Gao C Y, Gao Z F. Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making[J].Applied Mathematical Modelling, 2013, 37(8):5837-5850.
[25] Zhu B, Xu Z S. Consistency measures for hesitant fuzzy linguistic preference relations[J]. IEEE Trans on Fuzzy Systems, 2014, 22(2):35-45.
[26] Li D Q, Zeng W Y, Li J H. New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making[J]. Engineering Applications of Artificial Intelligence, 2015, 40(5):11-16.
[27] Li D Q, Zeng W Y, Zhao Y B. Note on distance measure of hesitant fuzzy sets[J]. Information Sciences, 2015,321(5):103-115.
[28] Liao H C, Xu Z S, Xia M M. Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making[J]. Int J of Information Technology&Decision Making, 2014, 13(4):47-76.
[29]岳超源.决策理论与方法[M].北京:科学出版社,2003:30-60.(Yue C Y. Decision making theory and methods[M].Beijing:Science Press, 2003:30-60.)
[30]郭海鹏,黄胜,王超.基于改进德尔菲法的舰船总体方案群决策方法[J].上海交通大学学报, 2014, 48(4):515-519.(Guo H P, Huang S, Wang C. Group decision-making method of warship overall scheme based on improved Delphi[J]. J of Shanghai Jiaotong University, 2014, 48(4):515-519.)
[31]侯远杭,胡玉龙,王文全,等.船型方案优选的多目标群决策方法[J].上海交通大学学报, 2012, 46(3):385-389.(Hou Y H, Hu Y L, Wang W Q, et al. Ship type selection based on group with multi-objects[J]. J of Shanghai Jiaotong University, 2012, 46(3):385-389.)
[2] Atanassov K. Intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1986, 20(1):87-96.
[3] Dubois D, Prade H. Fuzzy sets and systems:Theory and applications[M]. New York:Academic Press, 1980:40-60.
[4] Torra V. Hesitant fuzzy sets[J]. Int J of Intelligent Systems, 2010, 25(6):529-539.
[5] Rodríguez R M, Martínez L, Torra V, et al. Hesitant fuzzy sets:State of the art and future directions[J]. Int J of Intelligent Systems, 2014, 29(4):495-524.
[6] Xia M M, Xu Z S. Hesitant fuzzy information aggregationin decision making[J]. Int J of Approximate Reasoning,2011, 52(3):395-407.
[7] Xia M M, Xu Z S, Chen N. Some hesitant fuzzy aggregation operators with their application in group decision making[J]. Group Decision Negotiation, 2013,22(2):259-279.
[8] Zhu B, Xu Z S, Xia M M. Hesitant fuzzy geomeric Bonferroni means[J]. Information Sciences, 2012,182(11):72-85.
[9] Wei G W. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making[J].Knowledge-Based Systems, 2012, 31(1):176-182.
[10] Zhang Z M. Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making[J]. Information Sciences, 2013, 234(1):150-181.
[11]廖虎昌.复杂模糊多属性决策理论与方法[M].北京:科学出版社, 2016:80-120.(Liao H C. Theory and method for complicated fuzzy multiple attribute decision[M]. Beijing:Science Press,2016:80-120.)
[12]陈树伟,蔡丽娜.区间值犹豫模糊集[J].模糊系统与数学, 2013, 27(6):38-44.(Chen S W, Cai L N. Interval-valued hesitant fuzzy sets[J].Fuzzy Systems and Mathematics, 2013, 27(6):38-44.)
[13] Qian G, Wang H, Feng X. Generalized hesitant fuzzy sets and their application in decision support system[J].Knowledge-Based Systems, 2013, 37(3):357-365.
[14] Wei G W, Zhao X F, Lin R. Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making[J]. Knowledge-Based Systems, 2012, 31(2):176-182.
[15] 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(5):528-540.
[16]吴婉莹,何迎东,郭甦,等.直觉对偶犹豫模糊集的集结算子及其应用[J].武汉理工大学学报:信息与管理工程版, 2014, 36(2):225-228.(Wu W Y, He Y D, Guo S, et al. Intuitionistic dual hesitant fuzzy set operators and its application[J]. J of Wuhan University of Technology:Information&Management Engineering, 2014, 36(2):225-228.)
[17] Rodríguez R M, Martínez L, Herrera F. Hesitant fuzzy linguistic term sets for decision making[J]. IEEE Trans on Fuzzy Systems, 2012, 20(2):109-119.
[18] Rodríguez R M, Martínez L, Herrera F. A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets[J]. Information Sciences, 2013, 241(3):28-42.
[19] Wei C P, Zhao N, Tang X J. Comparisons of hesitant fuzzy linguistic term sets[J]. IEEE Trans on Fuzzy Systems,2014, 22(5):575-585.
[20]刘小弟,朱建军,刘思峰.犹豫模糊信息下的双向投影决策方法[J].系统工程理论与实践, 2014, 34(10):2637-2644.(Liu X D, Zhu J J, Liu S F. Bidirectional projection method with hesitant fuzzy information[J]. Systems Engineering—Theory&Practice, 2014, 34(10):2637-2644.)
[21]王坚强,吴佳亭.基于优序关系的犹豫模糊语言多准则决策方法[J].控制与决策, 2015, 30(5):887-891.(Wang J Q, Wu J T. Method for multi-criteria decision-making with hesitant fuzzy linguistic based on outranking relation[J]. Control and Decision, 2015, 30(5):887-891.)
[22] Xu Z S, Xia M M. Distance and similarity measures for hesitant fuzzy sets[J]. Information Sciences, 2011,181(11):2128-2138.
[23] Xu Z S, Xia M M. On distance and correlation measures of hesitant fuzzy information[J]. Int J of Intelligent Systems,2014, 26(5):410-425.
[24] Peng D H, Gao C Y, Gao Z F. Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making[J].Applied Mathematical Modelling, 2013, 37(8):5837-5850.
[25] Zhu B, Xu Z S. Consistency measures for hesitant fuzzy linguistic preference relations[J]. IEEE Trans on Fuzzy Systems, 2014, 22(2):35-45.
[26] Li D Q, Zeng W Y, Li J H. New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making[J]. Engineering Applications of Artificial Intelligence, 2015, 40(5):11-16.
[27] Li D Q, Zeng W Y, Zhao Y B. Note on distance measure of hesitant fuzzy sets[J]. Information Sciences, 2015,321(5):103-115.
[28] Liao H C, Xu Z S, Xia M M. Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making[J]. Int J of Information Technology&Decision Making, 2014, 13(4):47-76.
[29]岳超源.决策理论与方法[M].北京:科学出版社,2003:30-60.(Yue C Y. Decision making theory and methods[M].Beijing:Science Press, 2003:30-60.)
[30]郭海鹏,黄胜,王超.基于改进德尔菲法的舰船总体方案群决策方法[J].上海交通大学学报, 2014, 48(4):515-519.(Guo H P, Huang S, Wang C. Group decision-making method of warship overall scheme based on improved Delphi[J]. J of Shanghai Jiaotong University, 2014, 48(4):515-519.)
[31]侯远杭,胡玉龙,王文全,等.船型方案优选的多目标群决策方法[J].上海交通大学学报, 2012, 46(3):385-389.(Hou Y H, Hu Y L, Wang W Q, et al. Ship type selection based on group with multi-objects[J]. J of Shanghai Jiaotong University, 2012, 46(3):385-389.)