On consistency and priority weights for interval probabilistic linguistic preference relations

Abstract

When expressing preferences with different probability weights for different linguistic terms, only partial assessment information is usually to be provided. Then the probability information can be normalized to the interval probability, hence, using interval probabilistic linguistic term sets (IPLTs) is more appropriate. Considering this situation, interval probabilistic linguistic preference relation (IPLPR) is proposed. To measure the consistency of IPLPR, the consistency definition of IPLPR is put forward. For the consistent IPLPR, from which an expected consistent PLPR can be obtained, we can obtain interval weights as the final priorities by using the pairs of linear programming models. We also create the probabilistic linguistic geometric consistency index (PLGCI) of PLPRs to judge whether the IPLPR is satisfactorily consistent. For an unsatisfied consistency IPLPR, the adjusting algorithm is proposed. Probability information is firstly considered to be adjusted. If it is not possible to achieve satisfactory consistency through the adjustment of probability information, then the linguistic terms will be adjusted. In addition to examples of different situations, such as the consistency, satisfactory consistency and consistency improvement, the application example is also given to show the practicability of the proposed methods.

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References

  1. Aguaron, J., & Moreno-Jimenez, J. M. (2003). The geometric consistency index: Approximated thresholds. European Journal of Operational Research,147, 137–145.

    Article  Google Scholar 

  2. Bai, C. Z., Zhang, R., Shen, S., et al. (2018). Interval-valued probabilistic linguistic term sets in multi-criteria group decision making. International Journal of Intelligent Systems,33, 1301–1321.

    Article  Google Scholar 

  3. Dong, Y. C., Wu, Y. Z., Zhang, H. J., & Zhang, G. Q. (2015). Multi-granular unbalanced linguistic distribution assessments with interval symbolic proportions. Knowledge-Based Systems,82, 139–151.

    Article  Google Scholar 

  4. Dong, Y. C., Xu, Y. F., & Li, H. Y. (2008). On consistency measures of linguistic preference relations. European Journal of Operational Research,189, 430–444.

    MathSciNet  Article  Google Scholar 

  5. Feng, X. Q., Zhang, L., & Wei, C. P. (2018). The consistency measures and priority weights of hesitant fuzzy linguistic preference relations. Applied Soft Computing,65, 79–90.

    Article  Google Scholar 

  6. Gao, J., Xu, Z. S., Ren, P. J., et al. (2019). An emergency decision making method based on the multiplicative consistency of probabilistic linguistic preference relations. International Journal of Machine Learning and Cybernetics,10, 1613–1629.

    Article  Google Scholar 

  7. Herrera, F., & Martinez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems,8, 746–752.

    Article  Google Scholar 

  8. Jin, C., Wang, H., & Xu, Z. S. (2019). Uncertain probabilistic linguistic term sets in group decision making. International Journal of Fuzzy Systems,21, 1241–1258.

    MathSciNet  Article  Google Scholar 

  9. Liao, H. C., Mi, X. M., & Xu, Z. S. (2020). A survey of decision-making methods with probabilistic linguistic information: Bibliometrics, preliminaries, methodologies, applications and future directions. Fuzzy Optimization and Decision Making,19(1), 81–134.

    MathSciNet  Article  Google Scholar 

  10. Lin, M. W., Chen, Z. Y., Liao, H. C., et al. (2019). ELECTRE II method to deal with probabilistic linguistic term sets and its application to edge computing. Nonlinear Dynamics,96, 2125–2143.

    Article  Google Scholar 

  11. Lin, M. W., Xu, Z. S., Zhai, Y. L., et al. (2018). Multi-attribute group decision-making under probabilistic uncertain linguistic environment. Journal of the Operational Research Society,69, 157–170.

    Article  Google Scholar 

  12. Liu, H. B., & Rodriguez, R. M. (2014). A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making. Information Sciences,258, 220–238.

    MathSciNet  Article  Google Scholar 

  13. Liu, H. C., You, J. X., Lu, C., et al. (2014). Application of interval 2-tuple linguistic MULTIMOORA method for health-care waste treatment technology evaluation and selection. Waste Management,34, 2355–2364.

    Article  Google Scholar 

  14. Pang, Q., Wang, H., & Xu, Z. S. (2016). Probabilistic linguistic term sets in multi-attribute group decision making. Information Sciences,369, 128–143.

    Article  Google Scholar 

  15. Rodriguez, R. M., Martinez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic term sets for decision making. IEEE Transactions on Fuzzy Systems,20, 109–119.

    Article  Google Scholar 

  16. Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems,12, 117–131.

    MathSciNet  Article  Google Scholar 

  17. Wang, Y. M., & Elhag, T. (2006). On the normalization of interval and fuzzy weights. Fuzzy Sets and Systems,157, 2456–2471.

    MathSciNet  Article  Google Scholar 

  18. Wang, H., & Xu, Z. S. (2015). Some consistency measures of extended hesitant fuzzy linguistic preference relations. Information Sciences,297, 316–331.

    MathSciNet  Article  Google Scholar 

  19. Wu, X. L., & Liao, H. C. (2018). An approach to quality function deployment based on probabilistic linguistic term sets and ORESTE method for multi-expert multi-criteria decision making. Information Fusion,43, 13–26.

    Article  Google Scholar 

  20. Wu, Z. B., & Xu, J. P. (2016). Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information. IEEE Transactions on Cybernetics,46, 694.

    Article  Google Scholar 

  21. Xu, Z. S. (2004). EOWA and EOWG operators for aggregating linguistic labels based on linguistic preference relations. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,12, 791–810.

    MathSciNet  Article  Google Scholar 

  22. Xu, Z. S., & Wang, H. (2017). On the syntax and semantics of virtual linguistic terms for information fusion in decision making. Information Fusion,34, 43–48.

    Article  Google Scholar 

  23. Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences,8, 199–249.

    MathSciNet  Article  Google Scholar 

  24. Zhang, X. F., Xu, Z. S., & Ren, P. J. (2019). A novel hybrid correlation measure for probabilistic linguistic term sets and crisp numbers and its application in customer relationship management. International Journal of Information Technology & Decision Making,18, 673–694.

    Article  Google Scholar 

  25. Zhang, Y. X., Xu, Z. S., Wang, H., et al. (2016). Consistency-based risk assessment with probabilistic linguistic preference relation. Applied Soft Computing,49, 817–833.

    Article  Google Scholar 

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Acknowledgements

The work was partly supported by the National Natural Science Foundation of China (No. 71971190) and University Social Sciences Project of Jiangsu Province (No. 2016SJD630014).

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Correspondence to Xiangqian Feng.

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Feng, X., Pang, X. & Zhang, L. On consistency and priority weights for interval probabilistic linguistic preference relations. Fuzzy Optim Decis Making (2020). https://doi.org/10.1007/s10700-020-09328-7

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Keywords

  • Probabilistic linguistic term sets (PLTs)
  • Probabilistic linguistic preference relation (PLPR)
  • Probabilistic Linguistic Geometric Consistency Index (PLGCI)
  • Consistency measures
  • Interval weights