Probability-hesitant fuzzy sets and the representation of preference relations

    Bin Zhu Affiliation
    ; Zeshui Xu Affiliation


Probability interpretations play an important role in understanding decision makers’ (DMs) behaviour in decision making. In this paper, we extend hesitant fuzzy sets to probability-hesitant fuzzy sets (P-HFSs) to enhance their modeling ability by taking DMs’ probabilistic preferences into consideration. Based on P-HFSs, we propose the concept of probability-hesitant fuzzy preference relation (P-HFPR) to collect the preferences. We then develop a consensus index to measure the consensus degrees of P-HFPR, and a stochastic method to improve the consensus degrees. All these results are essential for further research on P-HFSs.

Keyword : group decision making, fuzzy sets, preference relation, simulation

How to Cite
Zhu, B., & Xu, Z. (2018). Probability-hesitant fuzzy sets and the representation of preference relations. Technological and Economic Development of Economy, 24(3), 1029-1040.
Published in Issue
May 18, 2018
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Alonso, S.; Herrera-Viedma, E.; Chiclana, F.; Herrera, F. 2010. A web based consensus support system for group decision making problems and incomplete preferences, Information Sciences 180(23): 4477–4495.

Diamond, P.; Kloeden, P. E. 1994. Metric spaces of fuzzy sets: theory and applications. World Scientific Publishing Company.

Dong, Y.; Zhang, G.; Hong, W.-C.; Xu, Y. 2010. Consensus models for AHP group decision making under row geometric mean prioritization method, Decision Support Systems 49(3): 281–289.

Herrera-Viedma, E.; Alonso, S.; Chiclana, F.; Herrera, F. 2007. A consensus model for group decision making with incomplete fuzzy preference relations, Fuzzy Systems 15(5): 863–877.

Kacprzyk, J. 1997. Mu ltistage fuzzy control: a prescriptive approach. John Wiley & Sons, Inc.

Liao, H.; Xu, Z.; Xia, M. 2014. Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, International Journal of Information Technology & Decision Making 13(01): 47–76.

Orlovsky, S. 1978. Decision-making with a fuzzy preference relation, Fuzzy sets and systems 1(3): 155–167.

Torra, V. 2010. Hesitant fuzzy sets, International Journal of Intelligent Systems 25(1): 529–539.

Torra, V.; Narukawa, Y. 2009. On hesitant fuzzy sets and decision, in 2009 IEEE International Conference on Fuzzy Systems, 20–24 August 2009, 1378–1382.

Wang, J.; Wang, J.; Zhang, H.; Chen, X. 2016. Multi-criteria group decision-making approach based on 2-tuple linguistic aggregation operators with multi-hesitant fuzzy linguistic information, International Journal of Fuzzy Systems 18(1): 81–97.

Xia, M.; Xu, Z. 2011a. Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52(3): 395–407.

Xia, M.; Xu, Z. 2011b. On consensus in group decision making based on fuzzy preference relations, in Consensual Processes, 267: 263–287.

Xu, Z. 2004. Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation, International Journal of Approximate Reasoning 36(3): 261–270.

Xu, Z. S.; Xia, M. M. 2011. Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(11): 2128–2138.

Yu, D.; Zhang, W.; Huang, G. 2016. Dual hesitant fuzzy aggregation operators, Technological and Economic Development of Economy 22(2): 194–209.

Zadeh, L. A. 1965. Fuzzy sets, Information and Control 8: 338–353.

Zhu, B.; Xu, Z. 2013. Regression methods for hesitant fuzzy preference relations, Technological and Economic Development of Economy 19(Supl. 1): S214–S227.

Zhu, B.; Xu, Z. 2014. Stochastic preference analysis in numerical preference relations, European Journal of Operational Research 237(2): 628–633.

Zhu, B.; Xu, Z. 2016. Extended hesitant fuzzy sets, Technological and Economic Development of Economy 22(1): 100–121.

Zhu, B.; Xu, Z.; Xia, M. 2012. Hesitant fuzzy geometric Bonferroni means, Information Sciences 205(1): 72–85.

Zhu, B.; Xu, Z.; Xu, J. 2014. Deriving a ranking from hesitant fuzzy preference relations under group decision making, IEEE Transactions on Cybernetics 44(8): 1328–1337.