勾股模糊偏好关系及其在群体决策中的应用
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摘要
以区间模糊偏好关系(IVFPR)和直觉模糊偏好关系(IFPR)的理论框架为依据,将勾股模糊数(PFN)引入偏好关系中,定义勾股模糊偏好关系(PFPR)和加性一致性PFPR.然后,提出标准化勾股模糊权重向量(PFWV)的概念,并给出构造加性一致性PFPR的转换公式.为获取任意给定的PFPR的权重向量,建立以给定的PFPR与构造的加性一致性PFPR偏差最小为目标的优化模型.针对多个勾股模糊偏好关系的集结,利用能够有效处理极端值并满足关于序关系单调的勾股模糊加权二次(PFWQ)算子作为集结工具.进一步,联合PFWQ算子和目标优化模型提出一种群体决策方法.最后,通过案例分析表明所提出方法的实用性和可行性.
Based on the theoretical framework of interval-valued fuzzy preference relation(IVFPR) and intuitionistic fuzzy preference relation(IFPR), the Pythagorean fuzzy number(PFN) is introduced into the preference relation, and the concepts of Pythagorean fuzzy preference relation(PFPR) and additive consistent PFPR are proposed. Then, the definition of normalized Pythagorean fuzzy weight vector(PFWV) is proposed, and a conversion formula is provided to convert this PFWV into the additive consistent PFPR. For any given PFPR, a goal programming model is developed to obtain its Pythagorean fuzzy weights by minimizing its deviation from the constructed additive consistent PFPR. Because the Pythagorean fuzzy weighted quadratic(PFWQ) operator can effectively deal with extremes and satisfy the monotonicity with respect to the order relation, it is used for the aggregation of multiple PFPRs. A group decision making approach is proposed by using the PFWQ operator and the goal programming model, and a practice example is given to illustrate the practicability and feasibility of the proposed approach.
Based on the theoretical framework of interval-valued fuzzy preference relation(IVFPR) and intuitionistic fuzzy preference relation(IFPR), the Pythagorean fuzzy number(PFN) is introduced into the preference relation, and the concepts of Pythagorean fuzzy preference relation(PFPR) and additive consistent PFPR are proposed. Then, the definition of normalized Pythagorean fuzzy weight vector(PFWV) is proposed, and a conversion formula is provided to convert this PFWV into the additive consistent PFPR. For any given PFPR, a goal programming model is developed to obtain its Pythagorean fuzzy weights by minimizing its deviation from the constructed additive consistent PFPR. Because the Pythagorean fuzzy weighted quadratic(PFWQ) operator can effectively deal with extremes and satisfy the monotonicity with respect to the order relation, it is used for the aggregation of multiple PFPRs. A group decision making approach is proposed by using the PFWQ operator and the goal programming model, and a practice example is given to illustrate the practicability and feasibility of the proposed approach.
引文
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[18]Ren P J,Xu Z S,Gou X J.Pythagorean fuzzy TODIMapproach to multi-criteria decision making[J].Applied Soft Computing,2016,42:246-259.
[19]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]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.
[21]Li D Q,Zeng W Y.Distance measures of pythagorean fuzzy sets and their applications in multiattribute decision making[J].Control and Decision,2017,32(10):1817-1823.
[22]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.
[23]Wei G W.Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making[J].J of Intelligent&Fuzzy Systems,2017,33(4):2119-2132.
[24]Wang Z J,Li K W.Goal programming approaches to deriving interval weights based on interval fuzzy preference relations[J].Information Sciences,2012,193(193):180-198.
[25]Wang Z J.Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations[J].Applied Mathematical Modelling,2013,37(9):6377-6388.
[26]Zadeh L A.Outline of a new approach to the analysis of complex systems and decision processes[J].IEEE Trans on Systems,Man,&Cybernetics,1973,3(1):28-44.
[27]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.
[2]Bustince H,Pagola M,Mesiar R,et al.Grouping overlap,and generalized bientropic functions for fuzzy modeling of pairwise comparisons[J].IEEE Trans on Fuzzy Systems,2012,20(3):405-415.
[3]Zadeh L A.Fuzzy sets[J].Information&Control,19658(3):338-353.
[4]Orlovsky S A.Decision-making with a fuzzy preference relation[J].Fuzzy Sets&Systems,1978,1(3):155-167.
[5]Xu Z S.On compatibility of interval fuzzy preference relations[J].Fuzzy Optimization&Decision Making2004,3(3):217-225.
[6]Xu Z S.Intuitionistic fuzzy information aggregation Theory and applications[M].Beijing:Science Press2008:1-2.
[7]Atanassov K T.Intuitionistic fuzzy sets[J].Fuzzy Sets&Systems,1986,20(1):87-96.
[8]Xu Z S.Intuitionistic preference relations and their application in group decision making[J].Information Sciences,2007,177(11):2363-2379.
[9]Yager R R,Abbasov A M.Pythagorean membership grades,complex numbers,and decision making[J].In J of Intelligent Systems,2013,28(5):436-452.
[10]Yager R R.Pythagorean membership grades in multicriteria decision making[J].IEEE Trans on Fuzzy Systems,2014,22(4):958-965.
[11]Beliakov G,Bustince H,Goswami D P,et al.On averaging operators for Atanassov’s intuitionistic fuzzy sets[J]Information Sciences,2011,181(6):1116-1124.
[12]Deschrijver G,Kerre E E.On the relationship between some extensions of fuzzy set theory[J].Fuzzy Sets&Systems,2003,133(2):227-235.
[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,2015,29(12):1061-1078.
[14]Peng X D,Yang Y.Some results for pythagorean fuzzy sets[J].Int J of Intelligent Systems,2015,30(11):1133-1160.
[15]Zhang X L.A novel approach based on similarity measure for pythagorean fuzzy multiple criteria group decision making[J].Int J of Intelligent Systems,2015,31(6):593-611.
[16]Yang Y,Ding H,Chen Z S,et al.A note on extension of TOPSIS to multiple criteria decision making with pythagorean fuzzy sets[J].Int J of Intelligent Systems,2015,31(1):68-72.
[17]Ma Z M,Xu Z S.Symmetric pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems[J].Int J of Intelligent Systems,2016,31(12):1198-1219.
[18]Ren P J,Xu Z S,Gou X J.Pythagorean fuzzy TODIMapproach to multi-criteria decision making[J].Applied Soft Computing,2016,42:246-259.
[19]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]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.
[21]Li D Q,Zeng W Y.Distance measures of pythagorean fuzzy sets and their applications in multiattribute decision making[J].Control and Decision,2017,32(10):1817-1823.
[22]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.
[23]Wei G W.Pythagorean fuzzy interaction aggregation operators and their application to multiple attribute decision making[J].J of Intelligent&Fuzzy Systems,2017,33(4):2119-2132.
[24]Wang Z J,Li K W.Goal programming approaches to deriving interval weights based on interval fuzzy preference relations[J].Information Sciences,2012,193(193):180-198.
[25]Wang Z J.Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations[J].Applied Mathematical Modelling,2013,37(9):6377-6388.
[26]Zadeh L A.Outline of a new approach to the analysis of complex systems and decision processes[J].IEEE Trans on Systems,Man,&Cybernetics,1973,3(1):28-44.
[27]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.