Intuitionistic fuzzy geometric aggregation operators based on einstein operations

Intuitionistic fuzzy information aggregation plays an important part in Atanassov's intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and further develop some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. We also establish some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and give some numerical examples to illustrate the developed aggregation operators. In addition, we compare the proposed operators with the existing intuitionistic fuzzy geometric operators and get the corresponding relations. Finally, we apply the intuitionistic fuzzy Einstein weighted geometric operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. © 2011 Wiley Periodicals, Inc.

[1]  K. Atanassov.  Remarks on the intuitionistic fuzzy sets , 1992 .

[2]  T. Saaty,et al.  Procedures for Synthesizing Ratio Judgements , 1983 .

[3]  K. Atanassov.  New operations defined over the intuitionistic fuzzy sets , 1994 .

[4]  T. Saaty,et al.  The Analytic Hierarchy Process , 1985 .

[5]  Etienne E. Kerre,et al.  Implicators based on binary aggregation operators in interval-valued fuzzy set theory , 2005, Fuzzy Sets Syst..

[6]  Zeshui Xu,et al.  Choquet integrals of weighted intuitionistic fuzzy information , 2010, Inf. Sci..

[7]  Zeshui Xu,et al.  Intuitionistic Fuzzy Aggregation Operators , 2007, IEEE Transactions on Fuzzy Systems.

[8]  Colin O. Benjamin,et al.  Planning Facilities at the University of Missouri-Rolla , 1992 .

[9]  Shyi-Ming Chen,et al.  Handling multicriteria fuzzy decision-making problems based on vague set theory , 1994 .

[10]  Z. Xu,et al.  On consistency of the weighted geometric mean complex judgement matrix in AHP , 2000, Eur. J. Oper. Res..

[11]  R. Mesiar,et al.  ”Aggregation Functions”, Cambridge University Press , 2008, 2008 6th International Symposium on Intelligent Systems and Informatics.

[12]  Ju Wang,et al.  Reasoning within intuitionistic fuzzy rough description logics , 2009, Inf. Sci..

[13]  Guiwu Wei,et al.  Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making , 2010, Appl. Soft Comput..

[14]  Ronald R. Yager,et al.  Generalized OWA Aggregation Operators , 2004, Fuzzy Optim. Decis. Mak..

[15]  Zeshui Xu,et al.  Generalized aggregation operators for intuitionistic fuzzy sets , 2010 .

[16]  Zeshui Xu,et al.  Dynamic intuitionistic fuzzy multi-attribute decision making , 2008, Int. J. Approx. Reason..

[17]  Zeshui Xu,et al.  Generalized point operators for aggregating intuitionistic fuzzy information , 2010 .

[18]  Zeshui Xu,et al.  Clustering algorithm for intuitionistic fuzzy sets , 2008, Inf. Sci..

[19]  Deng-Feng Li,et al.  Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets , 2010, Expert Syst. Appl..

[20]  Humberto Bustince,et al.  On averaging operators for Atanassov's intuitionistic fuzzy sets , 2011, Inf. Sci..

[21]  Z. S. Xu,et al.  An overview of operators for aggregating information , 2003, Int. J. Intell. Syst..

[22]  Zeshui Xu,et al.  Some geometric aggregation operators based on intuitionistic fuzzy sets , 2006, Int. J. Gen. Syst..

[23]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[24]  Ranjit Biswas,et al.  Some operations on intuitionistic fuzzy sets , 2000, Fuzzy Sets Syst..

[25]  Etienne E. Kerre,et al.  On the relationship between some extensions of fuzzy set theory , 2003, Fuzzy Sets Syst..

[26]  Zeshui Xu,et al.  Intuitionistic Fuzzy Bonferroni Means , 2011, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[27]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[28]  Krassimir T. Atanassov,et al.  Intuitionistic Fuzzy Sets - Theory and Applications , 1999, Studies in Fuzziness and Soft Computing.

[29]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

[30]  Zeshui Xu,et al.  Induced generalized intuitionistic fuzzy operators , 2011, Knowl. Based Syst..

[31]  Zeshui Xu,et al.  An overview of methods for determining OWA weights , 2005, Int. J. Intell. Syst..

[32]  Vicenç Torra,et al.  Modeling decisions - information fusion and aggregation operators , 2007 .

[33]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decision-making , 1988 .

[34]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[35]  Dug Hun Hong,et al.  Multicriteria fuzzy decision-making problems based on vague set theory , 2000, Fuzzy Sets Syst..

[36]  Gleb Beliakov,et al.  Aggregation Functions: A Guide for Practitioners , 2007, Studies in Fuzziness and Soft Computing.

[37]  Krassimir T. Atanassov,et al.  Two theorems for intuitionistic fuzzy sets , 2000, Fuzzy Sets Syst..

[38]  Zeshui Xu,et al.  Recent advances in intuitionistic fuzzy information aggregation , 2010, Fuzzy Optim. Decis. Mak..

[39]  Francisco Herrera,et al.  Fuzzy Sets and Their Extensions: Representation, Aggregation and Models , 2008 .

[40]  Ramesh Sharda,et al.  Using the analytic hierarchy process in water resources planning: Selection of flood control projects , 1991 .

[41]  Zeshui Xu,et al.  Intuitionistic Fuzzy Information Aggregation: Theory and Applications , 2013 .

[42]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..

[43]  Etienne E. Kerre,et al.  On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision , 2007, Inf. Sci..

[44]  Kweku-Muata Osei-Bryson,et al.  Eliciting and mapping qualitative preferences to numeric rankings in group decision making , 1999, Eur. J. Oper. Res..

[45]  Xiaohong Chen,et al.  Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making , 2010, Expert Syst. Appl..

[46]  Krassimir T. Atanassov,et al.  An equality between intuitionistic fuzzy sets , 1996, Fuzzy Sets Syst..

[47]  Glad Deschrijver,et al.  Arithmetic operators in interval-valued fuzzy set theory , 2007, Inf. Sci..

[48]  Etienne Kerre,et al.  A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms , 2002 .