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Probabilistic Linguistic Z Number Decision-Making Method for Multiple Attribute Group Decision-Making Problems with Heterogeneous Relationships and Incomplete Probability Information

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Abstract

Probabilistic linguistic Z number (PLZN) is considered as an effective information representation model. It not only describes the decision-making information, but also demonstrates its reliability. To handle the increasing problems of complexity and uncertainty in real-life, PLZN is widely used to indicate qualitative information. In this paper, a novel decision-making method with PLZNs is proposed, focusing on multiple attribute group decision-making (MAGDM) problems with fewer alternatives and more interacted attributes in PLZN environment. Firstly, all basic theories of PLZNs are shown, where the possibility degree of PLZNs is defined. Then, an integration model based on evidential reasoning theory is constructed to aggregate numerous PLZNs, which fully considers the incomplete probability distributions in PLZNs. The mathematical programming model with the generalized Shapley function is introduced to determinate the important degrees of attributes and reflect the interactive characteristics among them. In addition, the probabilistic linguistic Z QUALIFLEX (PLZ-QUALIFLEX) method with the generalized Shapley function is proposed to rank small numbers of alternatives with respect to large numbers of attributes with heterogeneous relationships. Lastly, after demonstrating the rationalities and superiorities of the proposed method, it is applied to solve some numerical cases, in which is compared with other methods.

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References

  1. Chen, T.Y.: Interval-valued intuitionistic fuzzy QUALIFLEX method with a likelihood-based comparison approach for multiple criteria decision analysis. Inf. Sci. 261, 149–169 (2014)

    Article  MathSciNet  Google Scholar 

  2. Dempster, A.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)

    Article  MathSciNet  Google Scholar 

  3. Ding, X.F., Zhu, L.X., Lu, M.S., et al.: A novel linguistic Z-number QUALIFLEX method and its application to large group emergency decision making. Sci. Program. 2020, 1631869 (2020)

    Google Scholar 

  4. Feng, X., Liu, Q., Wei, C.: Probabilistic linguistic QUALIFLEX approach with possibility degree comparison. J. Intell. Fuzzy Syst. 36(1), 719–730 (2019)

    Article  Google Scholar 

  5. Grabisch, M.: K-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92(2), 167–189 (1997)

    Article  MathSciNet  Google Scholar 

  6. Huang, J., Xu, D.H., Liu, H.C., et al.: A new model for failure mode and effect analysis integrating linguistic Z-numbers and projection method. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2955916

    Article  Google Scholar 

  7. Jiang, S., Shi, H., Lin, W., et al.: A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management. Appl. Soft Comput. 86, 105900 (2020)

    Article  Google Scholar 

  8. Labella, A., Rodriguez, R.M., Martinez, L.: Computing with comparative linguistic expressions and symbolic translation for decision making: ELICIT information. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2940424

    Article  Google Scholar 

  9. Meng, F., Tang, J.: Interval-valued intuitionistic fuzzy multiattribute group decision making based on cross entropy measure and Choquet integral. Int. J. Intell. Syst. 28(12), 1172–1195 (2013)

    Article  Google Scholar 

  10. Pang, Q., Wang, H., Xu, Z.S.: Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 369, 128–143 (2016)

    Article  Google Scholar 

  11. Qiao, D., Shen, K., Wang, J., et al.: Multi-criteria PROMETHEE method based on possibility degree with Z-numbers under uncertain linguistic environment. J. Ambient. Intell. Humaniz. Comput. 11, 2187–2201 (2020)

    Article  Google Scholar 

  12. Ren, Z., Liao, H., Liu, Y.: Generalized Z-numbers with hesitant fuzzy linguistic information and its application to medicine selection for the patients with mild symptoms of the COVID-19. Comput. Ind. Eng. 145, 106517 (2020)

    Article  Google Scholar 

  13. Rodriguez, R.M., Martinez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20(1), 109–119 (2011)

    Article  Google Scholar 

  14. Seiti, H., Hafezalkotob, A., Martínez, L.: R-numbers, a new risk modeling associated with fuzzy numbers and its application to decision making. Inf. Sci. 483, 206–231 (2019)

    Article  MathSciNet  Google Scholar 

  15. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    Book  Google Scholar 

  16. Sugeno, M.: Theory of Fuzzy Integral and its Application (Ph.D. Dissertation), Tokyo Institute of Technology, Tokyo, Japan (1974)

  17. Tang, G., Chiclana, F., Lin, X.C., et al.: Interval type-2 fuzzy multi-attribute decision-making approaches for evaluating the service quality of Chinese commercial banks. Knowl. Based Syst. 193 (2020)

  18. Teng, F., Liu, P.: A large group decision-making method based on a generalized Shapley probabilistic linguistic Choquet average operator and the TODIM method. Comput. Ind. Eng. 151, 106971 (2020)

    Article  Google Scholar 

  19. Mahmoodi, A.H., Sadjadi, S.J., Sadi-Nezhad, S., et al.: Linguistic Z-number Bonferroni mean and Linguistic Z-number geometric Bonferroni mean operators: their applications in portfolio selection problems. IEEE Access 8, 98742–98760 (2020)

    Article  Google Scholar 

  20. Wang, J.Q., Cao, Y.X., Zhang, H.Y.: Multi-criteria decision-making method based on distance measure and choquet integral for linguistic Z-numbers. Cogn. Comput. 9(6), 827–842 (2017)

    Article  Google Scholar 

  21. Wang, J., Wu, J., Wang, J., et al.: Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf. Sci. 288, 55–72 (2014)

    Article  MathSciNet  Google Scholar 

  22. Wang, X.K., Wang, Y.T., Wang, J.Q., et al.: A TODIM-PROMETHEE II based multi-criteria group decision making method for risk evaluation of water resource carrying capacity under probabilistic linguistic Z-number circumstances. Mathematics 8(7), 1190 (2020)

    Article  Google Scholar 

  23. Wang, X.T., Triantaphyllou, E.: Ranking irregularities when evaluating alternatives by using some ELECTRE methods. Omega 36, 45–63 (2008)

    Article  Google Scholar 

  24. Xu, Z.S.: Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Inf. Sci. 168(1–4), 171–184 (2004)

    Article  Google Scholar 

  25. Yaakob, A.M., Gegov, A.: Interactive TOPSIS based group decision making methodology using Z-numbers. International Journal of Computational Intelligence Systems 9(2), 311–324 (2016)

    Article  Google Scholar 

  26. Yang, J.B., Singh, M.G.: An evidential reasoning approach for multiple-attribute decision making with uncertainty. IEEE Trans. Syst. Man Cybern. 24(1), 1–18 (1994)

    Article  Google Scholar 

  27. Yang, J.B., Xu, D.L.: On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 32(3), 289–304 (2002)

    Article  Google Scholar 

  28. Ye, J., Xu, Z., Gou, X.: Virtual linguistic trust degree-based evidential reasoning approach and its application to emergency response assessment of railway station. Inf. Sci. 513, 341–359 (2020)

    Article  MathSciNet  Google Scholar 

  29. Zadeh, L.A.: The concept of a linguistic variable and its applications to approximate reasoning-Part I. Inf. Sci. 8, 199–249 (1975)

    Article  Google Scholar 

  30. Zadeh, L.A.: The concept of a linguistic variable and its applications to approximate reasoning pt II. Inf. Sci. 8, 301–357 (1975)

    Article  Google Scholar 

  31. Zadeh, L.A.: The concept of a linguistic variable and its applications to approximate reasoning pt III. Inf. Sci. 9, 43–80 (1975)

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 71771140, 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), the Social Science Planning Project of Shandong Province (No. 20CSDJ23), the Natural Science Foundation of Shandong Province (No. ZR2020QG002).

Author information

Authors and Affiliations

  1. School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, 250014, Shandong, China

    Fei Teng, Lili Rong & Peide Liu

  2. School of International Education, Shandong University of Finance and Economics, Jinan, 250014, Shandong, China

    Lei Wang

  3. School of Science and Technology, Shandong TV University, Jinan, 250014, Shandong, China

    Lili Rong

Authors
  1. Fei Teng
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  2. Lei Wang
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  3. Lili Rong
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  4. Peide Liu
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Corresponding author

Correspondence to Fei Teng.

Appendices

Appendix A

See Table 8.

Table 8 The concordance/discordance index in Example 2
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Appendix B

For the sake of easy to grasp this paper, these notations used in the proposed method are summarized in Table 9.

Table 9 The notations in proposed PLZ-QUALIFLEX method
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Teng, F., Wang, L., Rong, L. et al. Probabilistic Linguistic Z Number Decision-Making Method for Multiple Attribute Group Decision-Making Problems with Heterogeneous Relationships and Incomplete Probability Information. Int. J. Fuzzy Syst. 24, 552–573 (2022). https://doi.org/10.1007/s40815-021-01161-3

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  • DOI: https://doi.org/10.1007/s40815-021-01161-3

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