Z-number-valued function analysis is a new direction of studies on processing of information characterized by fuzziness and probabilistic uncertainty. Differentiability and integrability are among the related key problems. In this paper, we made a first step toward formulation of a definite integral of Z-number-valued function. A definition of definite integral is proposed and its relation with the concept of derivative is analyzed. In the proposed approach, the formalism of approximate limits is used.


Z-number Bimodal information Derivative Integral Approximate limit 


  1. 1.
    Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181, 2923–2932 (2011)CrossRefGoogle Scholar
  2. 2.
    Aliev, R.A., Alizadeh, A.V., Huseynov, O.H., Jabbarova, K.I.: Z-number based linear programming. Int. J. Intell. Syst. 30, 563–589 (2015)CrossRefGoogle Scholar
  3. 3.
    Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of continuous Z-numbers. Inf. Sci. 373, 441–460 (2016)CrossRefGoogle Scholar
  4. 4.
    Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inf. Sci. 290, 134–155 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aliev, R.A., Huseynov, O.H., Aliyev, R.R., Alizadeh, A.V.: The Arithmetic of Z-Numbers: Theory and Applications. World Scientific, Singapore (2015)CrossRefGoogle Scholar
  6. 6.
    Aliev, R.A., Huseynov, O.H.: Decision Theory with Imperfect Information. World Scientific, Singapore (2014)CrossRefGoogle Scholar
  7. 7.
    Aliev, R.A., Perdycz, W., Huseynov, O.H.: Hukuhara difference of Z-numbers. Inf. Sci. 466, 13–24 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Aliev, R.A., Perdycz, W., Huseynov, O.H.: Functions defined on a set of Z-numbers. Inf. Sci. 423, 353–375 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Aliev, R.A.: Uncertain Computation Based on Decision Theory. World Scientific Publishing, Singapore (2017)Google Scholar
  10. 10.
    Yager, R.: On Z-valuations using Zadeh’s Z-numbers. Int. J. Intell. Syst. 27, 259–278 (2012)CrossRefGoogle Scholar
  11. 11.
    Jafari, R., Razvarz, S., Gegov, A.: Solving differential equations with Z-numbers by utilizing fuzzy Sumudu transform. In: Arai, K., Kapoor, S., Bhatia, R. (eds.) Intelligent Systems and Applications. IntelliSys 2018. Advances in Intelligent Systems and Computing, vol. 869, pp. 1125–1138. Springer, Cham (2019)CrossRefGoogle Scholar
  12. 12.
    Lorkowski, J., Aliev, R., Kreinovich, V.: Towards decision making under interval, set-valued, fuzzy, and Z-number uncertainty: a fair price approach. In: Proceedings of the IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2014, pp. 2244–2253. IEEE (2014)Google Scholar
  13. 13.
    Aliev, R.A., Kreinovich, V.: Z-numbers and type-2 fuzzy sets: a representation result. Intell. Autom. Soft Comput. (2017). Scholar
  14. 14.
    Jiang, W., Cao, Y., Deng, X.: A novel Z-network model based on Bayesian network and Z-number. IEEE Trans. Fuzzy Syst. (2019).
  15. 15.
    Shen, K., Wang, J.: 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)CrossRefGoogle Scholar
  16. 16.
    Kang, B., Deng, Y., Hewage, K., Sadiq, R.: A method of measuring uncertainty for Z-number. IEEE Trans. Fuzzy Syst. 27, 731–738 (2018)CrossRefGoogle Scholar
  17. 17.
    Aliev, R.A., Pedrycz, W.: Fundamentals of a fuzzy-logic-based generalized theory of stability. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 39(4), 971–988 (2009)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Research Laboratory of Intelligent Control and Decision Making Systems in Industry and EconomicsAzerbaijan State Oil and Industry UniversityBakuAzerbaijan

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