毕达哥拉斯三角模糊数集成算子及其决策应用
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摘要
将毕达哥拉斯模糊数与三角模糊数相结合,提出了毕达哥拉斯三角模糊数的概念,定义了其运算,研究了其运算规律,推广了直觉三角模糊数和毕达哥拉斯模糊数,并通过定义毕达哥拉斯三角模糊数的得分函数和精确函数,实现毕达哥拉斯三角模糊数的排序。然后,定义了毕达哥拉斯三角模糊数加权平均算子(PTFWA)和有序加权平均算子(PTFOWA)、毕达哥拉斯三角模糊数加权几何算子(PTFWG)和有序加权几何算子(PTFOWG)以及毕达哥拉斯三角模糊数混合平均算子(PTFHA)和混合几何算子(PTFHG),研究了它们的性质,并给出其计算公式。最后,提出了基于毕达哥拉斯三角模糊集成算子的决策方法,通过实例说明了其可行性。
The concept of Pythagorean triangular fuzzy number(PTFN) is defined, which is one of the extensions of intuitionistic triangular fuzzy number and Pythagorean fuzzy number, and the operation of PTFN are introduced and the operations laws of PTFN are discussed. The concepts of the score and accuracy functions of PTFN are developed and a ranking method of comparing the magnitude of PTFNs is introduced. Then, Pythagorean triangular fuzzy numbers weighted averaging operator(PTFWA), Pythagorean triangular fuzzy numbers ordered weighted averaging operator(PTFOWA), Pythagorean triangular fuzzy numbers weighted geometric operator(PTFWG),Pythagorean triangular fuzzy numbers ordered weighted geometric operator(PTFOWG) and Pythagorean triangular fuzzy numbers hybrid averaging operator(PTFHA), Pythagorean triangular fuzzy numbers hybrid geometric operator(PTFHG) are defined, and their natures of these operators are discussed, and the mathematical expressions of these operators are obtained by derivation. Lastly, an approach for decision making with Pythagorean triangular fuzzy aggregation operators is developed, and a practical example is provided to illustrate the practicality of the developed method.
The concept of Pythagorean triangular fuzzy number(PTFN) is defined, which is one of the extensions of intuitionistic triangular fuzzy number and Pythagorean fuzzy number, and the operation of PTFN are introduced and the operations laws of PTFN are discussed. The concepts of the score and accuracy functions of PTFN are developed and a ranking method of comparing the magnitude of PTFNs is introduced. Then, Pythagorean triangular fuzzy numbers weighted averaging operator(PTFWA), Pythagorean triangular fuzzy numbers ordered weighted averaging operator(PTFOWA), Pythagorean triangular fuzzy numbers weighted geometric operator(PTFWG),Pythagorean triangular fuzzy numbers ordered weighted geometric operator(PTFOWG) and Pythagorean triangular fuzzy numbers hybrid averaging operator(PTFHA), Pythagorean triangular fuzzy numbers hybrid geometric operator(PTFHG) are defined, and their natures of these operators are discussed, and the mathematical expressions of these operators are obtained by derivation. Lastly, an approach for decision making with Pythagorean triangular fuzzy aggregation operators is developed, and a practical example is provided to illustrate the practicality of the developed 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]Atanassov K.New operators defined over the intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1994,61(2):137~142.
[4]De S K,Biswas R,Roy A R.Some operators on the intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,2000,114(3):477~484.
[5]Xu Z S.Intuitionistic fuzzy aggregation operators[J].IEEE Transaction on Fuzzy Systems,2007,15(6):1179~1187.
[6]Xu Z S,Yager R R.Some geometric aggregation operators based on intuitionistic fuzzy sets[J].International Journal of General Systems,2006,35(4):417~433.
[7]Zhao H,et al.Generalized aggregation operators for intuitionistic fuzzy sets[J].International Journal of intelligent Systems,2010,25(1):1~30.
[8]Tan C Q.Generalized intuitionistic fuzzy geometric aggregation operator and its application to multi-criteria group decision making[J].Soft Computing,2011,15:867~876.
[9]Yang W,Chen Z P.The quasi-arithmetic intuitionistic fuzzy OWA operators[J].Knowledge-Based Systems,2012,27(3):219~233.
[10]何迎东,陈华友,周礼刚.基于隶属度和非隶属交叉影响的直觉模糊集运算法则及其应用[J].模糊系统与数学,2013,27(3):134~142.
[11]何迎东等.基于交叉影响的IFWGA算子及其在多属性决策中的应用[J].数学的实践与认识,2013,43(6):55~61.
[12]何霞,刘卫锋.一种有方案偏好的直觉模糊多属性决策方法[J].运筹与管理,2013,22(1):36~40.
[13]Xu Z S,Chen J,Wu J J.Clustering algorithm for intuitionistic fuzzy sets[J].Information Sciences,2008,178(19):3775~3790.
[14]Gerstenkorn T,Mańko J.Correlation of intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,2007,44(1):39~43.
[15]Tan C Q,Chen X H.Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making[J].Expert Systems with Applications,2010,37(7):149~157.
[16]Xu Z S.Choquet integrals of weighted intuitionistic fuzzy information[J].Information Sciences,2010,180(5):726~736.
[17]Yager R R,Abbasov A M.Pythagorean membership grades,complex numbers and decision making[J].International Journal of Intelligent Systems,2013,28(5):436~452.
[18]Yager R R.Pythagorean membership grades in multicriteria decision making[J].IEEE Transaction on Fuzzy Systems,2014,22(4):958~965.
[19]Zhang X L,Xu Z S.Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets[J].International Journal of Intelligent Systems,2014,29(12):1061~1078.
[20]彭新东等.毕达哥拉斯模糊软集及其应用[J].计算机工程,2015,41(7):224~229.
[21]Molodtsov D.Soft set theory-first results[J].Computers and Mathematics with Applications,1999,37:19~31.[22 Liang W,Zhang X,Liu M.The maximizing deviation method based on interval-valued Pythagorean fuzzy weighted aggregating operator for multiple criteria group decision analysis[J].Discrete Dynamics in Nature and Society,2015:1~15.
[23]Peng X,Yang Y.Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators[J].International Journal of Intelligent Systems,2016,31(5):444~487.
[24]Gou X,Xu Z,Ren P.The properties of continuous Pythagorean fuzzy information[J].International Journal of Intelligent Systems,2016,31(5):401~424.
[25]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.
[26]Peng X D,Yang Y.Pythagorean fuzzy Choquet integral based MMBAC method for multiple attribute group decision making[J].International Journal of Intelligent Systems,2016,31(10):989~1020.
[27]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.
[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.
[29]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.
[30]Garg H.A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making[J].International Journal of Intelligent Systems,2016,31(9):886~920.
[31]Garg H.Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process[J].International Journal of Intelligent Systems,2017,32(6):597~630.
[32]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.
[33]Garg H.A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes[J].International Journal of Intelligent Systems,2016,31(12):1234~1252.
[34]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.
[35]Ma Z M,Xu Z S.Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision making problems[J].International Journal of Intelligent Systems,2016,31(12):1198~1219.
[36]Liang D C,Xu Z S,Darko A P.Projection model for fusing the information in Pythagorean fuzzy multi-criteria group decision making based on geometric Bonferroni mean[J].International Journal of Intelligent Systems,2017,32(9):966~987.
[37]Xue W T,et al.Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making[J].International Journal of Intelligent Systems,2018,33(1):93~125.
[38]Ren P J,Xu Z S,Gou X J.Pythagorean fuzzy TODIM approach to multi-criteria decision making[J].Applied Soft Computing,2016,42:246~259.
[39]Liang D C,Xu Z S.The new extension of TOPSIS method for MCDM with hesitant Pythagorean fuzzy sets[J].Applied Soft Computing,2017,60:167~179.
[40]Shu M H,Cheng C H,Chang J R.Using intuitionistic fuzzy sets for fault tree analysis on printed circuit board assembly[J].Microelectronics Reliability,2006,46(12):2139~2148.
[41]Li D F,Nan J X,Zhang M J.A ranking method of triangular intuitionistic fuzzy numbers and application to decision making[J].International Journal of Computational Intelligence Systems,2010,3(5):522~530.
[42]Li D F.A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems[J].Computer and Mathematics with Applications,2010,60:1557~1570.
[43]Yue X Y,et al.A multiattribute group decision method based on triangular intuitionistic fuzzy number[J].Communications in Computer and Information Science,2011,244(6):486~493.
[44]Chen D F,Zhang L,Jiao J S.Triangular fuzzy number intuitionistic fuzzy aggregation operators and their application to group decision making[J].Lecture Notes in Computer Science,2010,6320:350~357.
[45]Robinson J,Henry A.A search for the correlation coefficient of triangular and trapezoidal intuitionistic sets for multiple attribute group decision making[J].Communications in Computer and Information Science,2012,283:333~342.
[46]万树平,董九英.基于三角直觉模糊数Choquet积分算子的多属性决策方法[J].中国管理科学,2014,22(3):121~130.
[47]Xu Z S,Da Q L.An overview of operators for aggregating information[J].International Journal of Intelligent Systems,2003,18:953~969.
[48]Xu Z S.An overview of methods for determining OWA weights[J].International Journal of Intelligent Systems,2005,20(8):843~865.
[2]Atanassov K.Intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1986,20(1):87~96.
[3]Atanassov K.New operators defined over the intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1994,61(2):137~142.
[4]De S K,Biswas R,Roy A R.Some operators on the intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,2000,114(3):477~484.
[5]Xu Z S.Intuitionistic fuzzy aggregation operators[J].IEEE Transaction on Fuzzy Systems,2007,15(6):1179~1187.
[6]Xu Z S,Yager R R.Some geometric aggregation operators based on intuitionistic fuzzy sets[J].International Journal of General Systems,2006,35(4):417~433.
[7]Zhao H,et al.Generalized aggregation operators for intuitionistic fuzzy sets[J].International Journal of intelligent Systems,2010,25(1):1~30.
[8]Tan C Q.Generalized intuitionistic fuzzy geometric aggregation operator and its application to multi-criteria group decision making[J].Soft Computing,2011,15:867~876.
[9]Yang W,Chen Z P.The quasi-arithmetic intuitionistic fuzzy OWA operators[J].Knowledge-Based Systems,2012,27(3):219~233.
[10]何迎东,陈华友,周礼刚.基于隶属度和非隶属交叉影响的直觉模糊集运算法则及其应用[J].模糊系统与数学,2013,27(3):134~142.
[11]何迎东等.基于交叉影响的IFWGA算子及其在多属性决策中的应用[J].数学的实践与认识,2013,43(6):55~61.
[12]何霞,刘卫锋.一种有方案偏好的直觉模糊多属性决策方法[J].运筹与管理,2013,22(1):36~40.
[13]Xu Z S,Chen J,Wu J J.Clustering algorithm for intuitionistic fuzzy sets[J].Information Sciences,2008,178(19):3775~3790.
[14]Gerstenkorn T,Mańko J.Correlation of intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,2007,44(1):39~43.
[15]Tan C Q,Chen X H.Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making[J].Expert Systems with Applications,2010,37(7):149~157.
[16]Xu Z S.Choquet integrals of weighted intuitionistic fuzzy information[J].Information Sciences,2010,180(5):726~736.
[17]Yager R R,Abbasov A M.Pythagorean membership grades,complex numbers and decision making[J].International Journal of Intelligent Systems,2013,28(5):436~452.
[18]Yager R R.Pythagorean membership grades in multicriteria decision making[J].IEEE Transaction on Fuzzy Systems,2014,22(4):958~965.
[19]Zhang X L,Xu Z S.Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets[J].International Journal of Intelligent Systems,2014,29(12):1061~1078.
[20]彭新东等.毕达哥拉斯模糊软集及其应用[J].计算机工程,2015,41(7):224~229.
[21]Molodtsov D.Soft set theory-first results[J].Computers and Mathematics with Applications,1999,37:19~31.[22 Liang W,Zhang X,Liu M.The maximizing deviation method based on interval-valued Pythagorean fuzzy weighted aggregating operator for multiple criteria group decision analysis[J].Discrete Dynamics in Nature and Society,2015:1~15.
[23]Peng X,Yang Y.Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators[J].International Journal of Intelligent Systems,2016,31(5):444~487.
[24]Gou X,Xu Z,Ren P.The properties of continuous Pythagorean fuzzy information[J].International Journal of Intelligent Systems,2016,31(5):401~424.
[25]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.
[26]Peng X D,Yang Y.Pythagorean fuzzy Choquet integral based MMBAC method for multiple attribute group decision making[J].International Journal of Intelligent Systems,2016,31(10):989~1020.
[27]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.
[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.
[29]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.
[30]Garg H.A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making[J].International Journal of Intelligent Systems,2016,31(9):886~920.
[31]Garg H.Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process[J].International Journal of Intelligent Systems,2017,32(6):597~630.
[32]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.
[33]Garg H.A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes[J].International Journal of Intelligent Systems,2016,31(12):1234~1252.
[34]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.
[35]Ma Z M,Xu Z S.Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision making problems[J].International Journal of Intelligent Systems,2016,31(12):1198~1219.
[36]Liang D C,Xu Z S,Darko A P.Projection model for fusing the information in Pythagorean fuzzy multi-criteria group decision making based on geometric Bonferroni mean[J].International Journal of Intelligent Systems,2017,32(9):966~987.
[37]Xue W T,et al.Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making[J].International Journal of Intelligent Systems,2018,33(1):93~125.
[38]Ren P J,Xu Z S,Gou X J.Pythagorean fuzzy TODIM approach to multi-criteria decision making[J].Applied Soft Computing,2016,42:246~259.
[39]Liang D C,Xu Z S.The new extension of TOPSIS method for MCDM with hesitant Pythagorean fuzzy sets[J].Applied Soft Computing,2017,60:167~179.
[40]Shu M H,Cheng C H,Chang J R.Using intuitionistic fuzzy sets for fault tree analysis on printed circuit board assembly[J].Microelectronics Reliability,2006,46(12):2139~2148.
[41]Li D F,Nan J X,Zhang M J.A ranking method of triangular intuitionistic fuzzy numbers and application to decision making[J].International Journal of Computational Intelligence Systems,2010,3(5):522~530.
[42]Li D F.A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems[J].Computer and Mathematics with Applications,2010,60:1557~1570.
[43]Yue X Y,et al.A multiattribute group decision method based on triangular intuitionistic fuzzy number[J].Communications in Computer and Information Science,2011,244(6):486~493.
[44]Chen D F,Zhang L,Jiao J S.Triangular fuzzy number intuitionistic fuzzy aggregation operators and their application to group decision making[J].Lecture Notes in Computer Science,2010,6320:350~357.
[45]Robinson J,Henry A.A search for the correlation coefficient of triangular and trapezoidal intuitionistic sets for multiple attribute group decision making[J].Communications in Computer and Information Science,2012,283:333~342.
[46]万树平,董九英.基于三角直觉模糊数Choquet积分算子的多属性决策方法[J].中国管理科学,2014,22(3):121~130.
[47]Xu Z S,Da Q L.An overview of operators for aggregating information[J].International Journal of Intelligent Systems,2003,18:953~969.
[48]Xu Z S.An overview of methods for determining OWA weights[J].International Journal of Intelligent Systems,2005,20(8):843~865.
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