Abstract
Prof. L.A. Zadeh introduced the concept of a Z-number for description of real-world information. A Z-number is an ordered pair \( Z = (A,B) \) of fuzzy numbers \( A \) and \( B \) used to describe a value of a random variable \( X \). \( A \) is an imprecise estimation of a value of \( X \) and \( B \) is an imprecise estimation of reliability of \( A \). A series of important works on computations with Z-numbers and applications were published. However, no study exists on properties of operations of Z-numbers. Such theoretical study is necessary to formulate the basics of the theory of Z-numbers. In this paper we prove that Z-numbers exhibit fundamental properties under multiplicative arithmetic operations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inf. Sci. 290, 134–155 (2015)
Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of continuous Z-numbers. Inf. Sci. 373, 441–460 (2016)
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)
Kang, B., Wei, D., Li, Y., Deng, Y.: A method of converting Z-number to classical fuzzy number. J. Inf. Comput. Sci. 9, 703–709 (2012)
Piegat, A., Plucinski, M.: Computing with words with the use of inverse RDM models of membership functions. Appl. Math. Comput. Sci. 25(3), 675–688 (2015)
Piegat, A., Plucinski, M.: Fuzzy number addition with the application of horizontal membership functions. Sci. World J. 2015, 16p. (2015). Article ID 367214
Piegat, A., Landowski, M.: Is the conventional interval-arithmetic correct? J. Theor. Appl. Comput. Sci. 6(2), 27–44 (2012)
Piegat, A., Landowski, M.: Multidimensional approach to interval uncertainty calculations. In: Atanassov, K.T., et al. (eds.) New Trends in Fuzzy Sets, Intuitionistic: Fuzzy Sets, Generalized Nets and Related Topics, Volume II: Applications, pp. 137–151. IBS PAN -SRI PAS, Warsaw (2013)
Piegat, A., Landowski, M.: Two interpretations of multidimensional RDM interval arithmetic - multiplication and division. Int. J. Fuzzy Syst. 15, 488–496 (2013)
Piegat, A., Plucinski, M.: Some advantages of the RDM-arithmetic of intervally-precisiated values. Int. J. Comput. Intell. Syst. 8(6), 1192–1209 (2015)
Williamson, R.C., Downs, T.: Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds. Int. J. Approx. Reason. 4(2), 89–158 (1990)
Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181, 2923–2932 (2011)
Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23(2), 421–427 (1968)
Zadeh, L.A.: Methods and systems for applications with Z-numbers, United States Patent, Patent No. US 8,311,973 B1, Date of Patent: 13 November 2012
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Aliev, R.A., Alizadeh, A.V. (2019). Algebraic Properties of \( Z \)-Numbers Under Multiplicative Arithmetic Operations. In: Aliev, R., Kacprzyk, J., Pedrycz, W., Jamshidi, M., Sadikoglu, F. (eds) 13th International Conference on Theory and Application of Fuzzy Systems and Soft Computing — ICAFS-2018. ICAFS 2018. Advances in Intelligent Systems and Computing, vol 896. Springer, Cham. https://doi.org/10.1007/978-3-030-04164-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-04164-9_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04163-2
Online ISBN: 978-3-030-04164-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)