A neuro-fuzzy approach to compute the solution of a $Z$-numbers system with Trapezoidal fuzzy data

[1] S. Abbasbandy, R. Ezzati, A. Jafarian, LU decomposition method for solving fuzzy system of linear
equations, Appl. Math. Comput. 172(1) (2006) 633–643.
[2] R.A. Aliev, O.H. Huseynov, R.R. Aliyev, A.A. Alizadeh, Arithmetic of Z-Numbers: Theory and
Applications, World Scientific, 2015.
[3] R.A. Aliev, A.A. Alizadeh, O.H. Huseynov, The arithmetic of discrete Z-number, Inf. Sci. 290
(2015) 134–155.

[4] R. Aliev, O. Huseynov, L. Zeinalova, The arithmetic of continuous Z-numbers, Inf. Sci. 373 (2016)
441–460.
[5] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Appl. Math. Comput.
155 (2004) 493–502.
[6] T. Allahviranloo, S. Ezadi, Z-Advanced numbers processes, Inf. Sci. 480 (2019) 130–143.
[7] T. Allahviranloo, S. Ezadi, On the Z-Numbers. In: S. N. Shahbazova, M. Sugeno, J. Kacprzyk,
(eds) Recent Developments in Fuzzy Logic and Fuzzy Sets, Studies in Fuzziness and Soft Comput,
Springer, Cham, 391 (2020) 119–151.
[8] G.A. Anastassiou Fuzzy Mathematics: Approximation Theory, Part of the book series: Studies in
Fuzziness and Soft Computing 251, Publisher Springer Science & Business Media, 2010.
[9] A. Azadeh, R. Kokabi, Z-number DEA: A new possibilistic DEA in the context of Z-numbers, Adv.
Eng. Inform. 30 (2016) 604–617.
[10] S. Bandyopadhyay, S. Raha, P. Kumar Nayak, Matrix Game with Z-numbers, Int. J. Fuzzy Log.
Intell. 15 (2015) 60–71.
[11] R. Banerjee, S.K. Pal, A Decade of the Z-numbers, IEEE Trans Fuzzy Syst, 30 (2022) 2800–2812.
[12] S. Chakraverty, S.K. Jeswal, Applied Artificial Neural Network Methods for Engineers and Scien-
tists, World Scientific Publishing Co. Pte. Ltd., 2021.
[13] R. Cheng, J. Zhang, B. Kang, Ranking of Z-numbers based on the developed golden rule represen-
tative value, IEEE Trans. Fuzzy Syst. 30 (2022) 5196–5210.
[14] R. Chutia, Ranking of Z-numbers based on value and ambiguity at levels of decision making, Int. J.
Intell. Syst. 36 (2021) 313–331.
[15] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Appl. Math. Comput. 175
(2006) 645–674.
[16] Y. Deng, Z. Zhenfu, L. Qi, Ranking fuzzy numbers with an area method using radius of gyration,
Comput. Math. Appl. 51 (2006) 1127–1136.
[17] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New
York, 1980.
[18] S. Ezadi, T. Allahviranloo, New multi-layer method for Z-number ranking using hyperbolic tangent
function and convex combination, Intell. Autom. Soft. Comput. (2017) 1–7.
[19] S. Ezadi, T. Allahviranloo, S. Mohammadi, Two new methods for ranking of Z-numbers based on
sigmoid function and sign method, Int. J. Intell. Syst. 33 (2018) 1476–1487.
[20] S. Ezadi, T. Allahviranloo, Numerical solution of linear regression based on Z-numbers by im-
proved neural network, Intell. Autom. Soft Comput. (2017) 1–11.

[21] H. Farooq Hasan, A. Sultana, A New Approach to Solve Fuzzy Linear Equation A · X + B = C, IOSR
J. Math. (2017) 22–30.
[22] M.A. Fariborzi Araghi, M.M. Hosseinzadeh, Solution of general dual fuzzy linear systems using
ABS algorithm, Appl. Math. Sci. 6 (2012) 163–171.
[23] M.A. Fariborzi Araghi, A. Fallahzadeh, Inherited LU factorization for solving fuzzy system of linear
equations, Soft Comput. 17 (2013) 159–163.
[24] L.V. Fausett, Fundamentals of Neural Networks: Architectures, Algorithms and Applications, Pear-
son Education India, 2006.
[25] M. Friedman, M. Ma, A. Kandel,Fuzzy linear systems, Fuzzy Sets Syst. 96 (1998) 201–209.
[26] K. Gurney, An Introduction to Neural Networks, CRC press, 1997.
[27] F. Hasankhani, B. Daneshian, T. Allahviranloo, F. Modarres Khiyabani,A new method for solving
linear programming problems using Z-numbers’ ranking, Math. Sci. 17 (2023) 121–131,
[28] S.M.R. Hashemi Moosavi, M.A. Fariborzi Araghi, Sh. Ziari, An algorithm for solving a system of
linear equations with Z-numbers based on the neural network approach, J. Intell. Fuzzy. Syst. 46
(2024) 309–320.
[29] S.S. Haykin, Neural Networks and Learning Machines, New York: Prentice Hall, 2009.
[30] W. Jiang, C. Xie , M. Zhuang , Y. Shou , Y. Tang, Sensor data fusion with Z-numbers and its
application in fault diagnosis, Sensors 16 (2016) 1509.
[31] M. Joghataee, T. Allahviranloo, F.H. Lotfi, A. Ebrahimnejad, S. Abbasbandy, A. Amirteimoori, M.
Catak, Solving Fully Linear Programming Problem based on Z-numbers, Iran. J. Fuzzy Syst. 20
(2023) 157–174.
[32] B. Kang, D. Wei, Y. Li, Y. Deng, A method of converting Z-number to classical fuzzy number, J.
Inform. Comput. Sci. 9 (2012) 703–709.
[33] B. Kang , Y. Hu , Y. Deng , D. Zhou, A new methodology of multicriteria decision-making in
supplier selection based on Z-numbers, Math. Probl. Eng. 2016 (2016) 1–17.
[34] S. Kriesel, A Brief Introduction on Neural Networks, Citeseer, 2007.
[35] F.W. Lewis, S. Jagannathan, A. Yesildirak, Neural Network Control Of Robot Manipulators And
Non-Linear Systems (Series in Systems and Control), 1st Edition, CRC Press, 1999.
[36] Z. Motamedi Pour, T. Allahviranloo, M. Afshar Kermani, S. Abbasbandy, Solving a system of linear
equations based on Z-numbers to determinate the market balance value, Adv. Fuzzy Syst. (2023)
1–28.
[37] M.A. Nielsen, Neural networks and deep learning, Determination Press, San Francisco, CA, 2015.

[38] M. Otadi, M. Mosleh, S. Abbasbandy, Numerical solution of fully fuzzy linear systems by fuzzy
neural network, Soft Comput. 15 (2011) 1513–1522.
[39] J.-Q. Wang, Y.-X. Cao, H.-Y. Zhang, Multi-criteria decision-making method based on distance
measure and choquet integral for linguistic Z-numbers, Cogn Comput. (2017) 1–16.
[40] Z. Xu, S. Shang, W. Quin, W. Shu, A method for fuzzy risk analysis based on the new similarity of
trapezoidal fuzzy numbers, Expert Syst. Appl. 37 (2010) 1920–1927.
[41] R. Yager, A procedure for ordering fuzzy subsets of the unit interval, Inf. Sci. 24 (1981) 143–161.
[42] R. Yager, On Z-valuations using Zadeh’s Z-numbers, Inter. J. Intell. Syst. 27 (2012) 259–278.
[43] R. Yager, On a View of Zadeh’s Z-numbers, IPMU 3 (2012) 90–101.
[44] A.M. Yaakob , A . Gegov, Interactive topsis based group decision making methodology using Z-
numbers, Int. J. Comput. Intell. Syst. 9 (2016) 311–324.
[45] B. Yang, G. Qi, B. Xie, The pseudo-information entropy of Z-number and its applications in multi-
attribute decision-making, Inf. Sci. 655 (2024) 119886.
[46] J. Ye, Similarity measures based on the generalized distance of neutrosophic Z-number sets and
their multi-attribute decision making method, Soft Comput. 25 (2021) 13975–13985.
[47] L.A. Zadeh, Fuzzy Sets, Inf. Control 8 (1965) 338–353.
[48] L.A. Zadeh, A note on Z-numbers, Inf. Sci. 181 (2011) 2923–2932.
[49] H.J. Zimmermann, Fuzzy Set Theory and its Applications, Springer Science & Business Media,
2011.
[50] A. Ziqan, S. Ibrahim, M. Marabeh, A. Qarariyah, Fully fuzzy linear systems with trapezoidal and
hexagonal fuzzy numbers, Granul Comput. 7 (2022) 229–238.