Abstract
Dual hesitant fuzzy set (DHFS) is a new generalization of fuzzy set (FS) consisting of two parts (i.e., the membership hesitancy function and the non-membership hesitancy function), which confronts several different possible values indicating the epistemic degrees whether certainty or uncertainty. It encompasses fuzzy set (FS), intuitionistic fuzzy set (IFS), and hesitant fuzzy set (HFS) so that it can handle uncertain information more flexibly in the process of decision making. In this paper, we propose some new operations on dual hesitant fuzzy sets based on Einstein t-conorm and t-norm, study their properties and relationships and then give some dual hesitant fuzzy aggregation operators, which can be considered as the generalizations of some existing ones under fuzzy, intuitionistic fuzzy and hesitant fuzzy environments. Finally, a decision making algorithm under dual hesitant fuzzy environment is given based on the proposed aggregation operators and a numerical example is used to demonstrate the effectiveness of the method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20 (1): 87–96.
Atanassov, K.T. & Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31 (3): 343–349.
Beliakov, G., Pradera, A. & Calvo, T. (2007). Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg, Berlin, New York.
Chen, T.Y. (2011). Optimistic and pessimistic decision making with dissonance reduction using interval-valued fuzzy sets. Information Sciences, 181: 479–502.
Chen, T.Y., Wang, H.P. & Lu, Y.Y. (2011). A multicriteria group decision-making approach based on interval-valued intuitionistic fuzzy sets: a comparative perspective. Expert Systems with Applications, 38: 7647–7658.
De, S.K., Biswas, R. & Roy, A.R. (2001). An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets and Systems, 117: 209–213.
Gu, X., Wang, Y. & Yang, B. (2011). A method for hesitant fuzzy multiple attribute decision making and its application to risk investment. Journal of Convergence Information Technology, 6: 282–287.
Hung, W.L. & Yang, M.S. (2004). Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recognition Letters, 25: 1603–1611.
Liao, H.C., Xu, Z.S., Zeng, X.J. & Merigó, J.M. (2015). Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets. Knowledge-Based Systems, 76: 127–138.
Miyamoto, S. (2000). Multisets and fuzzy multisets. In: Z. Q. Liu, S. Miyamoto (eds.), Soft Computing and Human-Centered Machines, pp. 9–33. Springer, Berlin.
Miyamoto, S. (2001). Fuzzy multisets and their generalizations. In: C. S. Calude et al. (eds.), Multiset Processing, Lecture Notes in Computer Science, 2235, PP. 225–235. Springer, Berlin.
Miyamoto, S. (2005). Remarks on basics of fuzzy sets and fuzzy multisets. Fuzzy Sets and Systems, 156: 427–431.
Tan, C.Q. & Chen, X.H. (2010). Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making. Expert Systems with Applications, 37: 149–157.
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25: 529–539.
Torra, V. & Narukawa, Y. (2009). On hesitant fuzzy sets and decision. The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 1378–1382.
Wang, W.Z. & Liu, X.W. (2011). Intuitionistic fuzzy geometric aggregation operators based on Einstein operations. International Journal of Intelligent Systems, 26: 1049–1075.
Wei, G.W. (2010). GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting. Knowledge-Based Systems, 23: 243–247.
Wei, G.W. (2012). Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowledge-Based Systems, 31: 176–182.
Xia, M.M. & Xu, Z.S. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52: 395–407.
Xia, M.M., Xu, Z.S. & Zhu, B. (2012). Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm. Knowledge-Based Systems, 31: 78–88.
Xu, Y.J., Wang, H.M. & Merigó, J.M. (2014). Intuitionistic Einstein fuzzy Choquet integral operators for multiple attribute decision making. Technological and Economic Development of Economy, 20: 227–253.
Xu, Z.S. & Xia, M.M. (2011a). Distance and similarity measures for hesitant fuzzy sets. Information Sciences, 181: 2128–2138.
Xu, Z.S. & Xia, M.M. (2011b). On Distance and correlation measures of hesitant fuzzy information. International Journal of Intelligent Systems, 26: 410–425.
Xu, Z.S. (2007). Intuitionistic fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 15: 1179–1187.
Xu, Z.S. (2011). Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowledge-Based Systems, 24: 749–760.
Xu, Z.S. & Da, Q.L. (2002). The ordered weighted geometric averaging operators. International Journal of Intelligent Systems, 17: 709–716.
Xu, Z.S. & Yager, R.R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems, 35: 417–433.
Yager, R.R. (1988). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics, 18: 183–190.
Yager, R.R. (2004). Generalized OWA aggregation operators. Fuzzy Optimization and Decision Making, 3: 93–107.
Ye, J. (2010). Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Applied Mathematical Modelling, 34: 3864–3870.
Ye, J. (2011). Fuzzy cross entropy of interval-valued intuitionistic fuzzy sets and its optimal decision-making method based on the weights of alternative. Expert Systems with Applications, 38: 6179–6183.
Yu, D.J., Wu, Y.Y. & Lu, T. (2012). Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowledge-Based Systems, 30: 57–66.
Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8: 338–353.
Zadeh, L.A. (1975). Fuzzy logic and approximate reasoning. Synthese, 30: 407–428.
Zadeh, L.A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1: 1–28.
Zeng, S.Z., Wang, Q.F., Merigó, J.M. & Pan, T.J. (2014). Induced intuitionistic fuzzy ordered weighted averaging–weighted average operator and its application to business decision-making. Computer Science and Information Systems, 11(2): 839–857.
Zhao, H., Xu, Z.S., Ni, M.F. & Liu, S.S. (2010). Generalized aggregation operators for intuitionistic fuzzy sets. International Journal of Intelligent Systems, 25: 1–30.
Zhao, X.F. & Wei, G.W. (2013). Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making. Knowledge-Based Systems, 37: 472–479.
Zhu, B. Xu, Z.S. & Xia, M.M. (2012). Dual hesitant fuzzy sets. Journal of Applied Mathematics, 2012, Article ID 879629, 13 pages.
Zhu, B., Xu, Z.S. & Xia, M.M. (2012). Hesitant fuzzy geometric Bonferroni means. Information Sciences, 205: 72–85.
Author information
Authors and Affiliations
Corresponding author
Additional information
Hua Zhao received the Ph.D. degree in military operation research from PLA University of Science and Technology, Nanjing, China, in 2011. She is currently an associated professor with College of Sciences, PLA University of Science and Technology, Nanjing, China. She has contributed more than 20 journal articles to professional journals, and her current research interests include clustering analysis, decision making, and fuzzy sets.
Zeshui Xu received the Ph.D. degree in management science and engineering from Southeast University, Nanjing, China, in 2003. He is a Distinguished Young Scholar of the National Natural Science Foundation of China and the Chang Jiang Scholars of the Ministry of Education of China. He is currently a professor with the Business School, Sichuan University, Chengdu, China. He has contributed more than 450 journal articles to professional journals, and his current research interests include information fusion, decision making, and fuzzy sets.
Shousheng Liu received his M.S. degree in mathematics from Beijing Normal University, Beijng, China, in 1994, and his Ph.D. degree in signal processing from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2005. He is currently a professor with College of Sciences, PLA University of Science and Technology, Nanjing, China. He has published over 50 papers to professional journals, and his current research interests include decision making, information processing, and stochastic processes.
Rights and permissions
About this article
Cite this article
Zhao, H., Xu, Z. & Liu, S. Dual hesitant fuzzy information aggregation with Einstein t-conorm and t-norm. J. Syst. Sci. Syst. Eng. 26, 240–264 (2017). https://doi.org/10.1007/s11518-015-5289-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11518-015-5289-6