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A Programming Model for Consistency and Consensus in Group Decision Making with Probabilistic Hesitant Fuzzy Preference Relations

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Abstract

Individual consistency and group consensus are important topics in group decision-making process with preference relations. This study introduces a programming model to meet the individual consistency level and group consensus threshold value simultaneously. First, the distance measure between two probabilistic hesitant fuzzy preferences (PHFPRs) is proposed, and a consistency index for PHFPR is defined based on the proposed distance measure. Then, a mathematical programming model is constructed to improve its consistency when the PHFPR does not meet the consistency level. Second, considering that some values in PHFPRs provided by the decision makers may be missing in the decision-making process, a mathematical programming model is constructed to derive the missing values. Third, the proximity degree between any two decision makers is proposed, and a consensus index among the decision makers is defined based on the proposed proximity degree. Then, a mathematical programming model is constructed to obtain the expected consistency level and consensus threshold value simultaneously. Finally, an example is provided to illustrate the effectiveness of the proposed method. Comparative studies with several existing methods are also provided.

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References

  1. Zhang, X.Y., Wang, J.Q., Hu, J.H.: On novel operational laws and aggregation operators of picture 2-tuple linguistic information for MCDM problems. Int. J. Fuzzy Syst. 20, 958–969 (2018)

    Article  MathSciNet  Google Scholar 

  2. Liang, R.X., Wang, J.Q., Zhang, H.Y.: Projection-based PROMETHEE methods based on hesitant fuzzy linguistic term sets. Int. J. Fuzzy Syst. (2017). https://doi.org/10.1007/s40815-40017-40418-40817

    Article  Google Scholar 

  3. Liao, H.C., Mi, X.M., Xu, Z.S., Xu, J.P., Herrera, F.: Intuitionistic fuzzy analytic network process. IEEE Trans. Fuzzy Syst. (2018). https://doi.org/10.1109/TFUZZ.2017.2788881

    Article  Google Scholar 

  4. Sun, R.X., Hu, J.H., Zhou, J.D., Chen, X.H.: A hesitant fuzzy linguistic projection-based MABAC method for Patients’ prioritization. Int. J. Fuzzy Syst. (2017). https://doi.org/10.1007/s40815-40017-40345-40817

    Article  Google Scholar 

  5. Chen, S.Q., Wang, Y.M., Shi, H., Zhang, M.J., Lin, Y.: Alliance-based evidential reasoning approach with unknown evidence weights. Expert Syst. Appl. 78, 193–207 (2017)

    Article  Google Scholar 

  6. Rezaei, J., Wang, J., Tavasszy, L.: Linking supplier development to supplier segmentation using Best Worst Method. Expert Syst. Appl. 42, 9152–9164 (2015)

    Article  Google Scholar 

  7. Liao, H.C., Jiang, L.S., Xu, Z.S., Xu, J.P., Herrera, F.: A linear programming method for multiple criteria decision making with probabilistic linguistic information. Inf. Sci. 415, 341–355 (2017)

    Article  Google Scholar 

  8. Chu, J.F., Liu, X.W., Wang, Y.M., Chin, K.S.: A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations. Comput. Ind. Eng. 101, 227–242 (2016)

    Article  Google Scholar 

  9. Wu, Z.B., Xu, J.P.: Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations. Omega 65, 28–40 (2016)

    Article  Google Scholar 

  10. Zhao, M., Ma, X.Y., Wei, D.W.: A method considering and adjusting individual consistency and group consensus for group decision making with incomplete linguistic preference relations. Appl. Soft Comput. 54, 322–346 (2017)

    Article  Google Scholar 

  11. Xu, G.L., Wan, S.P., Wang, F., Dong, J.Y., Zeng, Y.F.: Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations. Knowl.-Based Syst. 98, 30–43 (2016)

    Article  Google Scholar 

  12. Liao, H.C., Xu, Z.S., Zeng, X.J., Merigo, J.M.: Framework of group decision making with intuitionistic fuzzy preference information. IEEE Trans. Fuzzy Syst. 23, 1211–1227 (2015)

    Article  Google Scholar 

  13. Zhang, X.Y., Wang, J.Q.: Discussing incomplete 2-tuple fuzzy linguistic preference relations inmulti-granular linguistic MCGDM with unknown weight information. Soft. Comput. (2017). https://doi.org/10.1007/s00500-00017-02915-x

    Article  Google Scholar 

  14. Zhang, L.Y., Li, T., Xu, X.H.: Consensus model for multiple criteria group decision making under intuitionistic fuzzy environment. Knowl.-Based Syst. 57, 127–135 (2014)

    Article  Google Scholar 

  15. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)

    MATH  Google Scholar 

  16. Zhou, W., Xu, Z.S.: Probability calculation and element optimization of probabilistic hesitant fuzzy preference relations based on expected consistency. IEEE Trans. Fuzzy Syst. (2017). https://doi.org/10.1109/TFUZZ.2017.2723349

    Article  Google Scholar 

  17. Zhu, B.: Decision Method for Research and Application Based on Preference Relation. Southeast University, Nanjing (2014)

    Google Scholar 

  18. Li, J., Wang, J.Q.: An extended QUALIFLEX method under probability hesitant fuzzy environment for selecting green suppliers. Int. J. Fuzzy Syst 19, 1866–1879 (2017)

    Article  MathSciNet  Google Scholar 

  19. Li, J., Wang, J.Q.: Multi-criteria outranking methods with hesitant probabilistic fuzzy sets. Cogn. Comput. 9, 611–625 (2017)

    Article  Google Scholar 

  20. Zhang, S., Xu, Z.S., He, Y.: Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf. Fusion 38, 1–11 (2017)

    Article  Google Scholar 

  21. Peng, H.G., Zhang, H.Y., Wang, J.Q.: Cloud decision support model for selecting hotels on TripAdvisor.com with probabilistic linguistic information. Int. J. Hosp. Manag. 68, 124–138 (2018)

    Article  Google Scholar 

  22. Yu, S.M., Wang, J., Wang, J.Q., Li, L.: A multi-criteria decision-making model for hotel selection with linguistic distribution assessments. Appl. Soft Comput. 67, 741–755 (2018)

    Article  Google Scholar 

  23. Li, J., Wang, Z.X.: Multi-attribute decision-making based on prioritized operators under probabilistic hesitant fuzzy environments. Soft Comput. (2018). https://doi.org/10.1007/s00500-00018-03047-00507

    Article  Google Scholar 

  24. Hao, Z.N., Xu, Z.S., Zhao, H., Su, Z.: Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl.-Based Syst. 127, 16–28 (2017)

    Article  Google Scholar 

  25. Zhai, Y.L., Xu, Z.S., Liao, H.C.: Measures of probabilistic interval-valued intuitionistic hesitant fuzzy sets and the application in reducing excessive medical examinations. IEEE Trans. Fuzzy Syst. (2017). https://doi.org/10.1109/TFUZZ.2017.2740201

    Article  Google Scholar 

  26. Wang, Z.X., Li, J.: Correlation coefficients of probabilistic hesitant fuzzy elements and their applications to evaluation of the alternatives. Symmetry 9, 259–377 (2017)

    Article  Google Scholar 

  27. Zhou, W., Xu, Z.S.: Expected hesitant VaR for tail decision making under probabilistic hesitant fuzzy environment. Appl. Soft Comput. 60, 297–311 (2017)

    Article  Google Scholar 

  28. Gao, J., Xu, Z.S., Liao, H.C.: A dynamic reference point method for emergency response under hesitant probabilistic fuzzy environment. Int. J. Fuzzy Syst. 19, 1261–1278 (2017)

    Article  Google Scholar 

  29. Ding, J., Xu, Z.S., Zhao, N.: An interactive approach to probabilistic hesitant fuzzy multi-attribute group decision making with incomplete weight information. J. Intell. Fuzzy Syst. 32, 2523–2536 (2017)

    Article  Google Scholar 

  30. Li, J., Wang, Z.X.: Consensus building for probabilistic hesitant fuzzy preference relations with expected additive consistency. Int. J. Fuzzy Syst. (2018). https://doi.org/10.1007/s40815-40018-40451-40811

    Article  MathSciNet  Google Scholar 

  31. Orlovsky, S.A.: Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1, 155–167 (1978)

    Article  MathSciNet  Google Scholar 

  32. Liao, H.C., Xu, Z.S.: Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process. Appl. Soft Comput. 35, 812–826 (2015)

    Article  Google Scholar 

  33. Xia, M.M., Xu, Z.S.: Managing hesitant information in GDM problems under fuzzy and multiplicative preference relations. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 21, 865–897 (2013)

    Article  MathSciNet  Google Scholar 

  34. Zhou, W., Xu, Z.S.: Group consistency and group decision making under uncertain probabilistic hesitant fuzzy preference environment. Inf. Sci. 414, 276–288 (2017)

    Article  Google Scholar 

  35. Wu, Z.B., Jin, B.M., Xu, J.P.: Local feedback strategy for consensus building with probability-hesitant fuzzy preference relations. Appl. Soft Comput. 67, 691–705 (2018)

    Article  Google Scholar 

  36. Wu, Z.B., Xu, J.P.: A consensus model for large-scale group decision making with hesitant fuzzy information and changeable clusters. Inf. Fusion 41, 217–231 (2018)

    Article  Google Scholar 

  37. Zhu, B., Xu, Z.S., Zhang, R., Hong, M.: Hesitant analytic hierarchy process. Eur. J. Oper. Res. 250, 602–614 (2016)

    Article  MathSciNet  Google Scholar 

  38. Zhu, B., Xu, Z.S.: Probability-hesitant fuzzy sets and the representation of preference relations. Technol. Econ. Dev. Econ. (2017). https://doi.org/10.3846/20294913.2016.1266529

    Article  Google Scholar 

  39. Xu, Z.S., Zhou, W.: Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim. Decis. Making 16, 481–503 (2017)

    Article  MathSciNet  Google Scholar 

  40. Crawford, G., Williams, C.: A note on the analysis of subjective judgment matrices. J. Math. Psychol. 29, 387–405 (1985)

    Article  Google Scholar 

  41. Xia, M.M., Chen, J.: Consistency and consensus improving methods for pairwise comparison matrices based on Abelian linearly ordered group. Fuzzy Sets Syst. 266, 1–32 (2015)

    Article  MathSciNet  Google Scholar 

  42. Saaty, T.L.: Multicriteria Decision Making: The Analytic Hierarchy Process. McGraw-Hill, NewYork (1980)

    Google Scholar 

  43. Cutello, V., Montero, J.: Fuzzy rationality measures. Fuzzy Sets Syst. 62, 39–54 (1994)

    Article  MathSciNet  Google Scholar 

  44. Xu, Z.S.: Algorithm for priority of fuzzy complementary judgement matrix. J. Syst. Eng. 16, 311–314 (2001)

    Google Scholar 

  45. Oncel, S.S.: Green energy engineering: opening a green way for the future. J. Clean. Prod. 142, 3095–3100 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the editors and anonymous reviewers for providing thoughtful comments for the improvement of this paper.

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Authors and Affiliations

  1. School of XingJian College of Science and Liberal Arts, Guangxi University, Nanning, 530005, Guangxi, People’s Republic of China

    Jian Li

  2. School of Mathematics and Information Science, Guangxi University, Nanning, 530004, Guangxi, People’s Republic of China

    Zhong-xing Wang

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  1. Jian Li
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  2. Zhong-xing Wang
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Correspondence to Zhong-xing Wang.

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Li, J., Wang, Zx. A Programming Model for Consistency and Consensus in Group Decision Making with Probabilistic Hesitant Fuzzy Preference Relations. Int. J. Fuzzy Syst. 20, 2399–2414 (2018). https://doi.org/10.1007/s40815-018-0501-8

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  • DOI: https://doi.org/10.1007/s40815-018-0501-8

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