Skip to main content
SpringerLink
Log in

A New Uncertainty Measure of Discrete Z-numbers

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

Today’s modern decision-making problem is designated by not being the most effective fuzziness; however, additionally partial reliability also plays a crucial role. The incomplete and unreliable information may also affect the selection maker to earn inaccurate decisions, ensuing in monetary losses and wastes of resources. Thus, it is vital to describe the reliability of the facts. To cope with it entirely, a notion of Z-number, i.e., a pair of fuzzy sets modeling a probability-qualified fuzzy statement, is the most suitable medium to access it. In this paper, we spout a new technique to measure the uncertainty of discrete Z-numbers based on Shannon entropy. In the given approach, by using characteristics of Z-number, all the potential probability distributions are estimated by the maximum entropy method. Then, a new fuzzy subset of the Z-number is formed based on the probability distributions and the membership functions of the fuzzy number. Finally, the centroid of the formulated set is determined to rank the degree of the uncertainty of Z-number. The applicability of the delivered approach is read with some numerical examples related to the decision-making process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inf. Sci. 290, 134–155 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aliev, R.A., Huseynov, O.H., Aliyev, R.R., Alizadeh, A.A.: The arithmetic of Z-numbers: theory and applications. World Sci (2015)

  3. Aliev, R.A., Huseynov, O.H., Zeinalova, L.M.: The arithmetic of continuous Z-numbers. Inf. Sci. 373, 441–460 (2016)

    Article  MATH  Google Scholar 

  4. Aliev, R.A., Pedrycz, W., Huseynov, O.H.: Functions defined on a set of Z-numbers. Inf. Sci. 423, 353–375 (2018)

    Article  MathSciNet  Google Scholar 

  5. Aliev, R.A., Pedrycz, W., Huseynov, O.H.: Hukuhara difference of Z-numbers. Inf. Sci. 466, 13–24 (2018)

    Article  MathSciNet  Google Scholar 

  6. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)

    Article  MATH  Google Scholar 

  7. Azadeh, A., Saberi, M., Atashbar, N.Z., Chang, E., Pazhoheshfar, P.: Z-AHP: a Z-number extension of fuzzy analytical hierarchy process. In: 2013 7th IEEE International Conference on Digital Ecosystems and Technologies, IEEE, pp. 141–147 (2013)

  8. Bakar, A.S.A., Gegov, A.: Multi-layer decision methodology for ranking Z-numbers. Int. J. Comput. Intell. Syst. 8(2), 395–406 (2015)

    Article  Google Scholar 

  9. Cao, X., Deng, Y.: A new geometric mean FMEA method based on information quality. IEEE Access 7(1), 95547–95554 (2019)

    Article  Google Scholar 

  10. Cao, Z., Lin, C.T.: Inherent fuzzy entropy for the improvement of EEG complexity evaluation. IEEE Trans. Fuzzy Syst. 26(2), 1032–1035 (2018)

    Article  MathSciNet  Google Scholar 

  11. Cao, Z., Lin, C.T., Lai, K.L., Ko, L.W., King, J.T., Liao, K.K., Fuh, J.L., Wang, S.J.: Extraction of SSVEPs-based inherent fuzzy entropy using a wearable headband EEG in migraine patients. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2905823

    Article  Google Scholar 

  12. Cao, Z., Ding, W., Wang, Y.K., Hussain, F.K., Adel, A.J., Lin, C.T.: Effects of repetitive SSVEPs on EEG complexity using multiscale inherent fuzzy entropy. Neurocomputing (2019). https://doi.org/10.1016/j.neucom.2018.08.091

    Article  Google Scholar 

  13. Dai, W.: Quadratic entropy of uncertain variables. Soft Comput. 22(17), 5699–5706 (2018)

    Article  MATH  Google Scholar 

  14. Deepak, D., Mathew, B., John, S.J., Garg, H.: A topological structure involving hesitant fuzzy sets. J. Intell. Fuzzy Syst. 36(6), 6401–6412 (2019)

    Article  Google Scholar 

  15. Deluca, A., Termini, S.: A definition of non-probabilistic entropy in setting of fuzzy set theory. Inf. Control 20, 301–312 (1971)

    Article  Google Scholar 

  16. Deng, W., Deng, Y.: Entropic methodology for entanglement measures. Phys. A Stat. Mech. Appl. 512, 693–697 (2018)

    Article  MathSciNet  Google Scholar 

  17. Ezadi, S., Allahviranloo, T., Mohammadi, S.: Two new methods for ranking of Z-numbers based on sigmoid function and sign method. Int. J. Intell. Syst. 33(7), 1476–1487 (2018)

    Article  Google Scholar 

  18. Fu, Z., Liao, H.: Unbalanced double hierarchy linguistic term set: the TOPSIS method for multi-expert qualitative decision making involving green mine selection. Inf. Fusion 51, 271–286 (2019)

    Article  Google Scholar 

  19. Gao, S., Deng, Y.: An evidential evaluation of nuclear safeguards. Int. J. Distrib. Sens. Netw. (2019). https://doi.org/10.1177/1550147719894550

    Article  Google Scholar 

  20. Gao, X., Deng, Y.: Quantum model of mass function. Int. J. Intell. Syst. 35(2), 267–282 (2020)

    Article  Google Scholar 

  21. Garg, H.: Generalized intuitionistic fuzzy entropy-based approach for solving multi-attribute decision-making problems with unknown attribute weights. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 89(1), 129–139 (2019)

    Article  MathSciNet  Google Scholar 

  22. Garg, H., Ansha, : Arithmetic operations on generalized parabolic fuzzy numbers and its application. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 88(1), 15–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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(106), 211 (2020). https://doi.org/10.1016/j.cie.2019.106211

    Article  Google Scholar 

  24. Garg, H., Kumar, K.: Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision-making problems. IEEE Trans. Fuzzy Syst. 27(12), 2302–2311 (2019)

    Article  Google Scholar 

  25. Garg, H., Agarwal, N., Tripathi, A.: Generalized intuitionistic fuzzy entropy measure of order \(\alpha \) and degree \(\beta \) and its applications to multi-criteria decision making problem. Int. J. Fuzzy Syst. Appl. 6(1), 86–107 (2017)

    Article  Google Scholar 

  26. Herrera, F., Herrera-Viedma, E.: Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst. 115(1), 67–82 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Herrera, F., Martinez, L.: An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 8(05), 539–562 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Herrera, F., Martínez, L.: A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making. IEEE Trans. Syst. Man Cybern. Part B Cybern. 31(2), 227–234 (2001)

    Article  Google Scholar 

  29. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jiang, W., Xie, C., Luo, Y., Tang, Y.: Ranking Z-numbers with an improved ranking method for generalized fuzzy numbers. J. Intell. Fuzzy Syst. 32(3), 1931–1943 (2017)

    Article  MATH  Google Scholar 

  31. Jiang, W., Xie, C., Wei, B., Tang, Y.: Failure mode and effects analysis based on Z-numbers. Intell. Autom. Soft Comput. 1–8 (2017)

  32. Kang, B., Deng, Y.: The maximum Deng entropy. IEEE Access 7(1), 120758–120765 (2019)

    Article  Google Scholar 

  33. Kang, B., Deng, Y., Hewage, K., Sadiq, R.: A method of measuring uncertainty for Z-number. IEEE Trans. Fuzzy Syst. 27(4), 731–738 (2018)

    Article  Google Scholar 

  34. Kang, B., Zhang, P., Gao, Z., Chhipi-Shrestha, G., Hewage, K., Sadiq, R.: Environmental assessment under uncertainty using Dempster–Shafer theory and Z-numbers. J. Ambient Intell. Humaniz. Comput. 56, 1–20 (2019). https://doi.org/10.1007/s12652-019-01228-y

    Article  Google Scholar 

  35. Karnik, N.N., Mendel, J.M.: Centroid of a type-2 fuzzy set. Inf. Sci. 132(1–4), 195–220 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Krohling, R.A., Pacheco, A.G., dos Santos, G.A.: TODIM and TOPSIS with Z-numbers. Front. Inf. Technol. Electron. Eng. 20(2), 283–291 (2019)

    Article  Google Scholar 

  37. Liao, H., Mi, X., Xu, Z.: A survey of decision-making methods with probabilistic linguistic information: bibliometrics, preliminaries, methodologies, applications and future directions. Fuzzy Optim. Decis. Mak. (2019). https://doi.org/10.1007/s10700-019-09309-5

    Article  Google Scholar 

  38. Liao, H., Qin, R., Gao, C., Wu, X., Hafezalkotob, A., Herrera, F.: Score-HeDLiSF: a score function of hesitant fuzzy linguistic term set based on hesitant degrees and linguistic scale functions: an application to unbalanced hesitant fuzzy linguistic MULTIMOORA. Inf. Fusion 48, 39–54 (2019)

    Article  Google Scholar 

  39. Liao, H., Wu, X.: DNMA: a double normalization-based multiple aggregation method for multi-expert multi-criteria decision making. Omega (2019). https://doi.org/10.1016/j.omega.2019.04.001

    Article  Google Scholar 

  40. Li, D., Deng, Y., Gao, X.: A generalized expression for information quality of basic probability assignment. IEEE Access 7(1), 174734–174739 (2019)

    Article  Google Scholar 

  41. Li, M., Xu, H., Deng, Y.: Evidential decision tree based on belief entropy. Entropy 21(9), 897 (2019)

    Article  Google Scholar 

  42. Liu, F., Gao, X., Zhao, J., Deng, Y.: Generalized belief entropy and its application in identifying conflict evidence. IEEE Access 7(1), 126625–126633 (2019)

    Article  Google Scholar 

  43. Liu, Q., Tian, Y., Kang, B.: Derive knowledge of Z-number from the perspective of Dempster–Shafer evidence theory. Eng. Appl. Artif. Intell. 85, 754–764 (2019)

    Article  Google Scholar 

  44. Liu, W., Li, L.: Emergency decision-making combining cumulative prospect theory and group decision making. Granul. Comput. 4(1), 39–52 (2019)

    Article  Google Scholar 

  45. Liu, Y., Jiang, W.: A new distance measure of interval-valued intuitionistic fuzzy sets and its application in decision making. Soft Comput. (2019). https://doi.org/10.1007/s00500-019-04332-5

    Article  Google Scholar 

  46. Mohamad, D., Ibrahim, S.: Decision making procedure based on jaccard similarity measure with Z-numbers. Pertanika J. Sci. Technol. 25(2), 561–574 (2017)

    Google Scholar 

  47. Mo, H., Deng, Y.: Identifying node importance based on evidence theory in complex networks. Stat. Mech. Appl. Phys. A (2019). https://doi.org/10.1016/j.physa.2019.121538

    Article  Google Scholar 

  48. Pal, N.R., Bezdek, J.C.: Measuring fuzzy uncertainty. IEEE Trans. Fuzzy Syst. 2(2), 107–118 (1994)

    Article  Google Scholar 

  49. Pal, N.R., Bezdek, J.C.: Quantifying different facets of fuzzy uncertainty. In: Fundamentals of Fuzzy Sets, pp 459–480. Springer, Berlin (2000)

  50. Pal, N.R., Pal, S.K.: Higher order fuzzy entropy and hybrid entropy of a set. Inf. Sci. 61(3), 211–231 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pan, L., Deng, Y.: An association coefficient of belief function and its application in target recognition system. Int. J. Intell. Syst. 35(1), 85–104 (2010)

    Article  Google Scholar 

  52. Peng, H.G., Wang, J.Q.: Hesitant uncertain linguistic Z-numbers and their application in multi-criteria group decision-making problems. Int. J. Fuzzy Syst. 19, 1300–1316 (2017)

    Article  Google Scholar 

  53. Pourabdollah, A., Wagner, C., Aladi, J.H., Garibaldi, J.M.: Improved uncertainty capture for nonsingleton fuzzy systems. IEEE Trans. Fuzzy Syst. 24(6), 1513–1524 (2016)

    Article  Google Scholar 

  54. Qiu, D., Jiang, H., Yu, Y.: On computing generalized hukuhara differences of Z-numbers. J. Intell. Fuzzy Syst. 36(1), 1–11 (2019)

    Article  Google Scholar 

  55. Shanon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)

    Article  MathSciNet  Google Scholar 

  56. Shen, K.W., Wang, J.Q.: Z-VIKOR method based on a new comprehensive weighted distance measure of Z-number and its application. IEEE Trans. Fuzzy Syst. 26(6), 3232–3245 (2018)

    Article  Google Scholar 

  57. Song, Y., Deng, Y.: Divergence measure of belief function and its application in data fusion. IEEE Access 7(1), 107465–107472 (2019)

    Article  Google Scholar 

  58. Song, Y., Wang, X., Yu, X., Zhang, H., Lei, L.: How to measure non-specificity of intuitionistic fuzzy sets. J. Intell. Fuzzy Syst. 29(5), 2087–2097 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010)

    MATH  Google Scholar 

  60. Wang, F., Mao, J.: Approach to multicriteria group decision making with Z-numbers based on TOPSIS and Power aggregation operators. Math. Probl. Eng. 2019:Article ID 3014,387 (2019)

  61. Wu, X., Liao, H.: A consensus-based probabilistic linguistic gained and lost dominance score method. Eur. J. Oper. Res. 272(3), 1017–1027 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  62. Xiao, F.: A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. IEEE Trans. Syst. Man Cybern. Syst. (2019). https://doi.org/10.1109/TSMC.2019.2958635

    Article  Google Scholar 

  63. Xiao, F.: A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion. Inf. Sci. 514, 462–483 (2020)

    Article  MathSciNet  Google Scholar 

  64. Xiao, F.: EFMCDM: evidential fuzzy multicriteria decision making based on belief entropy. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2936368

    Article  Google Scholar 

  65. Xiao, F.: Generalization of Dempster–Shafer theory: a complex mass function. Appl. Intell. (2019). https://doi.org/10.1007/s10489-019-01617-y

    Article  Google Scholar 

  66. Xiao, Z.Q.: Application of Z-numbers in multi-criteria decision making. In: Proceedings 2014 International Conference on Informative and Cybernetics for Computational Social Systems, pp. 91–95. IEEE, New York (2014)

  67. Yaakob, A.M., Gegov, A.: Interactive TOPSIS based group decision making methodology using Z-numbers. Int. J. Comput. Intell. Syst. 9(2), 311–324 (2016)

    Article  Google Scholar 

  68. Yager, R.R.: On the measure of fuzziness and negation part I: membership in the unit interval. Int. J. Gen. Syst. 5, 221–229 (1979)

    Article  MATH  Google Scholar 

  69. Yager, R.R.: A note on measuring fuzziness for intuitionistic and interval-valued fuzzy sets. Int. J. Gen. Syst. 44(7–8), 889–901 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  70. Yao, K.: Sine entropy of uncertain set and its applications. Appl. Soft Comput. 22, 432–442 (2014)

    Article  Google Scholar 

  71. Yao, K., Ke, H.: Entropy operator for membership function of uncertain set. Appl. Math. Comput. 242, 898–906 (2014)

    MathSciNet  MATH  Google Scholar 

  72. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

  73. Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181(14), 2923–2932 (2011)

    Article  MATH  Google Scholar 

  74. Zamri, N., Ahmad, F., Rose, A.N.M., Makhtar, M.: A fuzzy TOPSIS with Z-numbers approach for evaluation on accident at the construction site. In: International Conference on Soft Computing and Data Mining, pp. 41–50. Springer, Berlin (2016)

Download references

Acknowledgements

The work is partially supported by the National Natural Science Foundation of China (Grant Nos. 61573290, 61973332).

Author information

Authors and Affiliations

  1. Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Yangxue Li & Yong Deng

  2. School of Mathematics, Thapar Institute of Engineering & Technology, Deemed University, Patiala, Punjab, 147004, India

    Harish Garg

Authors
  1. Yangxue Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harish Garg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Deng.

Ethics declarations

Conflict of interest

All the authors declare that they do not have any conflict of interest.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Garg, H. & Deng, Y. A New Uncertainty Measure of Discrete Z-numbers. Int. J. Fuzzy Syst. 22, 760–776 (2020). https://doi.org/10.1007/s40815-020-00819-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-020-00819-8

Keywords

Search

Navigation