毕达哥拉斯犹豫模糊集的相关测度
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摘要
毕达哥拉斯犹豫模糊集,既能描述隶属度与非隶属度之和超过1、而平方和不超过1的模糊现象,又能表达决策者在隶属度和非隶属度上的犹豫不决,因此它是表达不确定现象的一个强有力工具.考虑到相关测度在统计学和管理科学中发挥着重要的作用,在模糊集、直觉模糊集以及毕达哥拉斯模糊集等相关测度基础上,研究毕达哥拉斯犹豫模糊集的相关测度.为此,定义毕达哥拉斯犹豫模糊集的信息能量、相关指标以及相关系数,证明相关系数的性质.由于决策中经常要考虑到属性权重,定义毕达哥拉斯犹豫模糊集的加权相关系数,并讨论其性质.最后,通过求出每个方案与正理想方案之间的加权相关系数,实现方案的排序择优,并通过算例表明其可行性与有效性.
Pythagorean hesitant fuzzy set, which can not only describe the fuzzy phenomenon that the sum of membership degree and nonmembership degree may be bigger than 1 but their square sum is equal to or less than 1, but also express hesitations in membership degree and nonmembership degree, can be considered as a powerful tool for expressing uncertain information in the process of decision making. Considering that correlation measures play important role in statistics and management science, in this paper, correlation measures of the Pythagorean hesitant fuzzy set is studied. On the basis of the definition of the informational energy of the Pythagorean hesitant fuzzy set, the correlation coefficients between Pythagorean hesitant fuzzy sets are defined, and their natures are discussed. Then, the weights of attributes are often taken into account, and the concept of the weighted correlation coefficient between Pythagorean hesitant fuzzy sets is also introduced, and the natures are studied. Finally, through the weighted correlation coefficients between each alternative and the positive ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be identified. An example is given to illustrate the feasibility and applicability of the proposed method.
Pythagorean hesitant fuzzy set, which can not only describe the fuzzy phenomenon that the sum of membership degree and nonmembership degree may be bigger than 1 but their square sum is equal to or less than 1, but also express hesitations in membership degree and nonmembership degree, can be considered as a powerful tool for expressing uncertain information in the process of decision making. Considering that correlation measures play important role in statistics and management science, in this paper, correlation measures of the Pythagorean hesitant fuzzy set is studied. On the basis of the definition of the informational energy of the Pythagorean hesitant fuzzy set, the correlation coefficients between Pythagorean hesitant fuzzy sets are defined, and their natures are discussed. Then, the weights of attributes are often taken into account, and the concept of the weighted correlation coefficient between Pythagorean hesitant fuzzy sets is also introduced, and the natures are studied. Finally, through the weighted correlation coefficients between each alternative and the positive ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be identified. An example is given to illustrate the feasibility and applicability of the proposed method.
引文
[1] Zadeh L A. Fuzzy sets[J]. Information and control, 1965,8(3):338-353.
[2] Atanassov K. Intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1986, 20(1):87-96.
[3] Atannasov K. Intuitionistic fuzzy sets:Theory and applications[M]. New York:Physica-Verlag, 1999:1-138.
[4] Xu Z S. Intuitionistic fuzzy aggregation operators[J].IEEE Trans on Fuzzy Systems, 2007, 15(6):1179-1187.
[5] Xu Z S, Yager R R. Some geometric aggregation operators based on intuitionistic fuzzy sets[J]. Int J of General Systems, 2006, 35(4):417-433.
[6] Gerstenkorn T, Manko J. Correlation of intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1991, 44(1):39-43.
[7] Zeng W Y, Li H X. Correlation coefficient of intuitionistic fuzzy sets[J]. Int J of Industrial Engineering, 2007, 3(5):33-40.
[8] Xu Z S, Chen J, Wu J. Clustering algorithm for intuitionistic fuzzy sets[J]. Information Sciences, 2008,178(19):3775-3790.
[9] Xu Z S. Choquet integrals of weighted intuitionistic fuzzy information[J]. Information Sciences, 2010, 180(5):726-736.
[10] Zadeh L A. The concept of a linguistic variable and its application to approximate reasoning-I[J]. Information and Control, 1975, 8(3):199-249.
[11] Yager R R. Abbasov A M. Pythagorean membership grades, complex numbers and decision making[J]. Int J of Intelligent Systems, 2013, 28(5):436-452.
[12] Yager R R. Pythagorean membership grades in multicriteria decision making[J]. IEEE Trans on Fuzzy Systems, 2014, 22(4):958-965.
[13] Zhang X L, Xu Z S. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets[J].Int J of Intelligent Systems, 2014, 29(12):1061-1078.
[14]彭新东,杨勇,宋娟萍,等.毕达哥拉斯模糊软集及其应用[J].计算机工程, 2015, 41(7):224-229.(Peng X D, Yang Y, Song J P, et al. Pythagorean fuzzy soft set and its application[J]. Computer Engineering, 2015,41(7):224-229.)
[15] Molodtsov D. Soft set theory—First results[J].Computers and Mathematics with Applications, 1999,37(4/5):19-31.
[16] Gou X J, Xu Z S, Ren P J. The properties of continuous Pythagorean fuzzy information[J]. Int J of Intelligent Systems, 2016, 31(5):401-424.
[17] Peng X, Yang Y. Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators[J]. Int J of Intelligent Systems, 2016, 31(5):444-487.
[18]刘卫锋,常娟,何霞.广义毕达哥拉斯模糊集成算子及其决策应用[J].控制与决策, 2016, 31(12):2280-2286.(Liu W F, Chang J, He X. Generalized Pythagorean fuzzy aggregation operators and applications in decision making[J]. Control and Decision, 2016, 31(12):2280-2286.)
[19]刘卫锋,杜迎雪,常娟.毕达哥拉斯模糊交叉影响集成算子及其决策应用[J].控制与决策, 2017, 32(6):1033-1040.(Liu W F, Du Y X, Chang J. Pythagorean fuzzy interaction aggregation operators and applications in decision making[J]. Control and Decision, 2017, 32(6):1033-1040.)
[20] Garg H. A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making[J]. Int J of Intelligent Systems, 2016, 31(9):886-920.
[21] Garg H. Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process[J].Int J of Intelligent Systems, 2017, 32(6):597-630.
[22] Wu S J, Wei G W. Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making[J]. Knowledge-Based Systems,2016, 97(3):24-39.
[23]何霞,杜迎雪,刘卫锋.毕达哥拉斯模糊幂平均算子[J].模糊系统与数学, 2016, 30(6):116-124.(He X, Du Y X, Liu W F. Pythagorean fuzzy power average operators[J]. Fuzzy Systems and Mathematics,2016, 30(6):116-124.)
[24] Peng X, Yang Y. Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making[J]. Int J of Intelligent Systems, 2016,31(10):989-1020.
[25]李德清,曾文艺,尹乾.勾股模糊集的距离测度及其在多属性决策中的应用[J].控制与决策, 2017, 32(10):1817-1823.(Li D Q, Zeng W Y, Yin Q. Distance measures of Pythagorean fuzzy sets and their applications in multiattribute decision making[J]. Control and Decision,2017, 32(10):1817-1823.)
[26] Garg H. A novel correlation coefficients between pythagorean fuzzy sets and its applications to decision-making processes[J]. Int J of Intelligent Systems, 2016, 31(12):1234-1252.
[27]张超,李德玉.勾股模糊粗糙集及其在多属性决策中的应用[J].小型微型计算机系统, 2016, 37(7):1531-1535.(Zhang C, Li D Y. Pythagorean fuzzy rough sets and its applications in multi-attribute decision making[J]. J of Chinese Computer Systems, 2016, 37(7):1531-1535.)
[28]刘卫锋,何霞.毕达哥拉斯犹豫模糊集[J].模糊系统与数学, 2016, 30(4):107-115.(Liu W F, He X. Pythagorean hesitant fuzzy set[J]. Fuzzy Systems and Mathematics, 2016, 30(4):107-115.)
[29] Torra V. Hesitant fuzzy sets[J]. Int J of Intelligent Systems, 2010, 25(6):529-539.
[2] Atanassov K. Intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1986, 20(1):87-96.
[3] Atannasov K. Intuitionistic fuzzy sets:Theory and applications[M]. New York:Physica-Verlag, 1999:1-138.
[4] Xu Z S. Intuitionistic fuzzy aggregation operators[J].IEEE Trans on Fuzzy Systems, 2007, 15(6):1179-1187.
[5] Xu Z S, Yager R R. Some geometric aggregation operators based on intuitionistic fuzzy sets[J]. Int J of General Systems, 2006, 35(4):417-433.
[6] Gerstenkorn T, Manko J. Correlation of intuitionistic fuzzy sets[J]. Fuzzy Sets and Systems, 1991, 44(1):39-43.
[7] Zeng W Y, Li H X. Correlation coefficient of intuitionistic fuzzy sets[J]. Int J of Industrial Engineering, 2007, 3(5):33-40.
[8] Xu Z S, Chen J, Wu J. Clustering algorithm for intuitionistic fuzzy sets[J]. Information Sciences, 2008,178(19):3775-3790.
[9] Xu Z S. Choquet integrals of weighted intuitionistic fuzzy information[J]. Information Sciences, 2010, 180(5):726-736.
[10] Zadeh L A. The concept of a linguistic variable and its application to approximate reasoning-I[J]. Information and Control, 1975, 8(3):199-249.
[11] Yager R R. Abbasov A M. Pythagorean membership grades, complex numbers and decision making[J]. Int J of Intelligent Systems, 2013, 28(5):436-452.
[12] Yager R R. Pythagorean membership grades in multicriteria decision making[J]. IEEE Trans on Fuzzy Systems, 2014, 22(4):958-965.
[13] Zhang X L, Xu Z S. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets[J].Int J of Intelligent Systems, 2014, 29(12):1061-1078.
[14]彭新东,杨勇,宋娟萍,等.毕达哥拉斯模糊软集及其应用[J].计算机工程, 2015, 41(7):224-229.(Peng X D, Yang Y, Song J P, et al. Pythagorean fuzzy soft set and its application[J]. Computer Engineering, 2015,41(7):224-229.)
[15] Molodtsov D. Soft set theory—First results[J].Computers and Mathematics with Applications, 1999,37(4/5):19-31.
[16] Gou X J, Xu Z S, Ren P J. The properties of continuous Pythagorean fuzzy information[J]. Int J of Intelligent Systems, 2016, 31(5):401-424.
[17] Peng X, Yang Y. Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators[J]. Int J of Intelligent Systems, 2016, 31(5):444-487.
[18]刘卫锋,常娟,何霞.广义毕达哥拉斯模糊集成算子及其决策应用[J].控制与决策, 2016, 31(12):2280-2286.(Liu W F, Chang J, He X. Generalized Pythagorean fuzzy aggregation operators and applications in decision making[J]. Control and Decision, 2016, 31(12):2280-2286.)
[19]刘卫锋,杜迎雪,常娟.毕达哥拉斯模糊交叉影响集成算子及其决策应用[J].控制与决策, 2017, 32(6):1033-1040.(Liu W F, Du Y X, Chang J. Pythagorean fuzzy interaction aggregation operators and applications in decision making[J]. Control and Decision, 2017, 32(6):1033-1040.)
[20] Garg H. A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making[J]. Int J of Intelligent Systems, 2016, 31(9):886-920.
[21] Garg H. Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process[J].Int J of Intelligent Systems, 2017, 32(6):597-630.
[22] Wu S J, Wei G W. Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making[J]. Knowledge-Based Systems,2016, 97(3):24-39.
[23]何霞,杜迎雪,刘卫锋.毕达哥拉斯模糊幂平均算子[J].模糊系统与数学, 2016, 30(6):116-124.(He X, Du Y X, Liu W F. Pythagorean fuzzy power average operators[J]. Fuzzy Systems and Mathematics,2016, 30(6):116-124.)
[24] Peng X, Yang Y. Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making[J]. Int J of Intelligent Systems, 2016,31(10):989-1020.
[25]李德清,曾文艺,尹乾.勾股模糊集的距离测度及其在多属性决策中的应用[J].控制与决策, 2017, 32(10):1817-1823.(Li D Q, Zeng W Y, Yin Q. Distance measures of Pythagorean fuzzy sets and their applications in multiattribute decision making[J]. Control and Decision,2017, 32(10):1817-1823.)
[26] Garg H. A novel correlation coefficients between pythagorean fuzzy sets and its applications to decision-making processes[J]. Int J of Intelligent Systems, 2016, 31(12):1234-1252.
[27]张超,李德玉.勾股模糊粗糙集及其在多属性决策中的应用[J].小型微型计算机系统, 2016, 37(7):1531-1535.(Zhang C, Li D Y. Pythagorean fuzzy rough sets and its applications in multi-attribute decision making[J]. J of Chinese Computer Systems, 2016, 37(7):1531-1535.)
[28]刘卫锋,何霞.毕达哥拉斯犹豫模糊集[J].模糊系统与数学, 2016, 30(4):107-115.(Liu W F, He X. Pythagorean hesitant fuzzy set[J]. Fuzzy Systems and Mathematics, 2016, 30(4):107-115.)
[29] Torra V. Hesitant fuzzy sets[J]. Int J of Intelligent Systems, 2010, 25(6):529-539.
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