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Double Uncertainty Driving and Integrated Decision-Making Under the Mixed Probabilistic Hesitant Environment

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Abstract

Probabilistic hesitant fuzzy sets are an emerging and popular tool in the decision-making field, skilled at describing subjective information. However, it does not take into account the cognitive limitations of decision makers and the complexity of the actual environment can lead to situations such as multiple values or missing probabilistic information. Considering these issues, we first develop the uncertain probabilistic hesitant fuzzy set (UPHFS) to propose the mixed hesitant fuzzy set (MHFS). Moreover, we further give the computational rules and prove the generalization and robustness of the MHFS. Then, we design two new ratio models, namely the mixed probabilistic hesitant crossover ratio (MPHCR) model and mixed preference probabilistic hesitant crossover ratio (MPPHCR) model, and construct an integrated decision-making model based on them. The integrated decision-making model is a collection of integrated decision models for deriving unknown and mixed probabilities and computing optimal decision outcomes, which is different from the previous studies than the similar studies. Therefore, the model results could be more robust and reliable. Further, based on the proposed fuzzy environments and new decision-making models, we give the whole calculation process and integrated decision-making steps. Lastly, an illustrated example of project investment is provided to apply the above methods, processes, and steps and shows their effectiveness.

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References

  1. Zhang, R., Qiu, L., Xu, Y., Wen, X.: An expertise-based consensus reaching process with probability-hesitant fuzzy preference relations and its application in risk evaluation in food industry. Inf. Fusion 97, 101809 (2023)

    Article  Google Scholar 

  2. Kaur, G., Garg, H.: A novel algorithm for autonomous parking vehicles using adjustable probabilistic neutrosophic hesitant fuzzy set features. Expert Syst. Appl. 226, 120101 (2023)

    Article  MATH  Google Scholar 

  3. Alsattar, H.A., Qahtan, S., Zaidan, A.A., Deveci, M., Martinez, L., Pamucar, D., Pedrycz, W.: Developing deep transfer and machine learning models of chest X-ray for diagnosing COVID-19 cases using probabilistic single-valued neutrosophic hesitant fuzzy. Expert Syst. Appl. 236, 121300 (2024)

    Article  MATH  Google Scholar 

  4. Farhadinia, B., Aickelin, U., Khorshidi, H.A.: Uncertainty measures for probabilistic hesitant fuzzy sets in multiple criteria decision making. Int. J. Intell. Syst. 35(11), 1646–1679 (2020)

    Article  MATH  Google Scholar 

  5. Xu, T.T., Zhang, H., Li, B.Q.: Fuzzy entropy and hesitancy entropy in probabilistic hesitant fuzzy information and their applications. Soft. Comput. 26(18), 9101–9115 (2022)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Li, J., Wang, Z.X.: Multi-attribute decision making based on prioritized operators under probabilistic hesitant fuzzy environments. Soft. Comput. 23, 3853–3868 (2019)

    Article  MATH  Google Scholar 

  8. Gao, J., Xu, Z., Zhang, Y.: Integral aggregations of continuous probabilistic hesitant fuzzy sets. IEEE Trans. Fuzzy Syst. 30(3), 676–686 (2020)

    Article  MATH  Google Scholar 

  9. Zhou, W., Xu, Z.: Probability calculation and element optimization of probabilistic hesitant fuzzy preference relations based on expected consistency. IEEE Trans. Fuzzy Syst. 26(3), 1367–1378 (2017)

    Article  MATH  Google Scholar 

  10. Li, J., Wang, J.Q.: Multi-criteria decision-making with probabilistic hesitant fuzzy information based on expected multiplicative consistency. Neural Comput. Appl. 31, 8897–8915 (2019)

    Article  MATH  Google Scholar 

  11. Lin, M., Zhan, Q., Xu, Z.: Decision making with probabilistic hesitant fuzzy information based on multiplicative consistency. Int. J. Intell. Syst. 35(8), 1233–1261 (2020)

    Article  MATH  Google Scholar 

  12. He, Y., Xu, Z.: Multi-attribute decision making methods based on reference ideal theory with probabilistic hesitant information. Expert Syst. Appl. 118, 459–469 (2019)

    Article  MATH  Google Scholar 

  13. Divsalar, M., Ahmadi, M., Ebrahimi, E., Ishizaka, A.: A probabilistic hesitant fuzzy Choquet integral-based TODIM method for multi-attribute group decision-making. Expert Syst. Appl. 191, 116266 (2022)

    Article  MATH  Google Scholar 

  14. 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  MATH  Google Scholar 

  15. 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  MATH  Google Scholar 

  16. Song, C.Y., Zhao, H., Xu, Z.S., Hao, Z.N.: Interval-valued probabilistic hesitant fuzzy set and its application in the Arctic geopolitical risk evaluation. Int. J. Intell. Syst. 34(4), 627–651 (2019)

    Article  MATH  Google Scholar 

  17. 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. 26(3), 1651–1670 (2017)

    Article  MATH  Google Scholar 

  18. Liu, J., Huang, C., Song, J., Du, P., Chen, H.: Group decision making based on the modified probability calculation method and DEA cross-efficiency with probabilistic hesitant fuzzy preference relations. Comput. Ind. Eng. 156, 107262 (2021)

    Article  MATH  Google Scholar 

  19. Davoudabadi, R., Mousavi, S.M., Mohagheghi, V.: A new decision model based on DEA and simulation to evaluate renewable energy projects under interval-valued intuitionistic fuzzy uncertainty. Renew. Energy 164, 1588–1601 (2021)

    Article  MATH  Google Scholar 

  20. Liu, X.D., Wu, J., Zhang, S.T., Wang, Z.W., Garg, H.: Extended cumulative residual entropy for emergency group decision-making under probabilistic hesitant fuzzy environment. Int. J. Fuzzy Syst. 24, 159–179 (2022)

    Article  MATH  Google Scholar 

  21. Fu, M., Wang, L., Zheng, B., Shao, H.Y.: The optimal emergency decision-making method with incomplete probabilistic information. Sci. Rep. 11, 23400 (2021)

    Article  MATH  Google Scholar 

  22. Krishankumar, R., Ravichandran, K.S., Liu, P., Kar, S., Gandomi, A.H.: A decision framework under probabilistic hesitant fuzzy environment with probability estimation for multi-criteria decision making. Neural Comput. Appl. 33(2), 8417–8433 (2021)

    Article  MATH  Google Scholar 

  23. Alkan, N., & Kahraman, C. Continuous intuitionistic fuzzy sets (CINFUS) and their AHP&TOPSIS extension: Research proposals evaluation for grant funding. Applied Soft Computing, 110579 (2023).

  24. Lin, Y., Wang, Y.M.: Prioritization of hesitant multiplicative preference relations based on data envelopment analysis for group decision making. Neural Comput. Appl. 31, 437–447 (2019)

    Article  MATH  Google Scholar 

  25. Meng, F.Y., Xiong, B.B.: Logical efficiency decomposition for general two-stage systems in view of cross efficiency. Eur. J. Oper. Res. 294(2), 622–632 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Namvar, H., Bamdad, S.: Resilience-based efficiency measurement of process industries with type-2 fuzzy sets. Int. J. Fuzzy Syst. 23, 1122–1136 (2021)

    Article  MATH  Google Scholar 

  27. Jin, F.F., Zhang, Y., Garg, H., Liu, J.P., Chen, J.: Evaluation of small and medium-sized enterprises’ sustainable development with hesitant fuzzy linguistic group decision-making method. Appl. Intell. 52, 4940–4960 (2022)

    Article  MATH  Google Scholar 

  28. Orlovsky, S.A.: Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1(3), 155–167 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Saaty, T.L., Vargas, L.G.: Uncertainty and rank order in the analytic hierarchy process. Eur. J. Oper. Res. 32, 107–117 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhu, B., Xu, Z.S.: Regression methods for hesitant fuzzy preference relations. Technol. Econ. Dev. Econ. 19(1), 214–227 (2013)

    MATH  Google Scholar 

  31. Xu, Z.S.: Intuitionistic preference relations and their application in group decision making. Inf. Sci. 177, 2363–2379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou, W., Xu, Z.S.: Probability calculation and element optimization of probabilistic hesitant fuzzy preference relations based on expected consistency. IEEE Trans. Fuzzy Syst. 26(3), 1367–1378 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, J.P., Zhang, J., Suo, W.L.: Risk evaluation in cross-border transport infrastructure projects: A fuzzy hybrid method considering dual interdependent effects. Inf. Sci. 488, 140–157 (2019)

    Article  MATH  Google Scholar 

  34. Wang, J.J., Ma, X.L., Xu, Z.S., Zhan, J.M.: A three-way decision approach with risk strategies in hesitant fuzzy decision information systems. Inf. Sci. 588, 293–314 (2022)

    Article  MATH  Google Scholar 

  35. Zhou, W., Xu, Z.S.: Hesitant fuzzy linguistic portfolio model with variable risk appetite and its application in the investment ratio calculation. Appl. Soft Comput. 84, 105719 (2019)

    Article  MATH  Google Scholar 

  36. Zhang, Z., Liao, H.: A stochastic cross-efficiency DEA approach based on the prospect theory and its application in winner determination in public procurement tenders. Ann. Oper. Res. (2022). https://doi.org/10.1007/s10479-022-04539-0

    Article  MATH  Google Scholar 

  37. Chu, X.L., Sun, B.Z., Li, X., Han, K.Y., Wu, J.Q., Zhang, Y., Huang, Q.C.: Neighborhood rough set-based three-way clustering considering attribute correlations: An approach to classification of potential gout groups. Inf. Sci. 535, 28–41 (2020)

    Article  MathSciNet  Google Scholar 

  38. Garg, H., Kaur, G.: Quantifying gesture information in brain hemorrhage patients using probabilistic dual hesitant fuzzy sets with unknown probability information. Comput. Ind. Eng. 140, 106211 (2020)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  40. Xu, Z.S., Xia, M.M.: On distance and correlation measures of hesitant fuzzy information. Int. J. Intell. Syst. 26(5), 410–425 (2011)

    Article  MATH  Google Scholar 

  41. Zhang, Y., Zhang, Y.X., Gong, C., Dinçer, H., Yüksel, S.: An integrated hesitant 2-tuple Pythagorean fuzzy analysis of QFD-based innovation cost and duration for renewable energy projects. Energy 248, 123561 (2022)

    Article  Google Scholar 

  42. Tutsoy, O.: Pharmacological, non-pharmacological policies and mutation: an artificial intelligence based multi-dimensional policy making algorithm for controlling the casualties of the pandemic diseases. IEEE Trans. Pattern Anal. Mach. Intell. 44(12), 9477–9488 (2021)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of China [No. 72071176], the Social Science Innovation Team Project of Yunnan Province [No. 2022CX01] and the Applied Basic Research of Science and Technology Commission of Yunnan Province (No. 202301AV070010).

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Authors and Affiliations

  1. Business School, Yunnan University of Finance and Economics, Kunming, 650221, People’s Republic of China

    Wei Zhou & Danxue Luo

  2. Yunnan Key Laboratory of Service Computing, Kunming, 650221, People’s Republic of China

    Wei Zhou & Danxue Luo

  3. Business School, Sichuan University, Chengdu, 610064, People’s Republic of China

    Zeshui Xu

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  1. Wei Zhou
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Correspondence to Danxue Luo.

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Zhou, W., Luo, D. & Xu, Z. Double Uncertainty Driving and Integrated Decision-Making Under the Mixed Probabilistic Hesitant Environment. Int. J. Fuzzy Syst. 27, 27–42 (2025). https://doi.org/10.1007/s40815-024-01755-7

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