Abstract
In this paper authors would like to make a critical review of the low-dimensional method proposed in [3] by Allahviranloo and Gandavi for solving fuzzy linear system with crisp square matrix and a fuzzy right-hand side vector. The solution presented in the mentioned work is, in general incorrect. The authors prove that given method is incorrect and propose an alternative solution based on multidimensional Relative-Distance-Measure (RDM) interval arithmetic and fuzzy RDM interval arithmetic that gives correct results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aliev, R.: Operations on z-numbers with acceptable degree of specificity. Procedia Comput. Sci. 120, 9–15 (2017). 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22–23 August 2017, Budapest, Hungary
Aliev, R., Huseynov, O., Aliyev, R.: A sum of a large number of z-numbers. Procedia Comput. Sci. 120, 16–22 (2017). 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22–23 August 2017, Budapest, Hungary
Allahviranloo, T., Ghanbari, M.: On the algebraic solution of fuzzy linear systems based on interval theory. Appl. Math. Model. 36, 5360–5379 (2012)
Dymova, L.: Soft Computing in Economics and Finance. Springer, Heidelberg (2011)
Lodwick, W.A., Dubois, D.: Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst. 281, 227–251 (2015). Special Issue Celebrating the 50th Anniversary of Fuzzy Sets
Mazandarani, M., Pariz, N., Kamyad, A.V.: Granular differentiability of fuzzy-number-valued functions. IEEE Trans. Fuzzy Syst. 26(1), 310–323 (2018)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2009)
Najariyan, M., Zhao, Y.: Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives. IEEE Trans. Fuzzy Syst. PP(99), 1–15 (2017)
Pedrycz, W., Skowron, A., Kreinovich, V.: Handbook of Granular Computing. Wiley-Interscience, New York (2008)
Piegat, A., Landowski, M.: Horizontal membership function and examples of its applications. Int. J. Fuzzy Syst. 17(1), 22–30 (2015)
Piegat, A., Landowski, M.: Fuzzy arithmetic type 1 with horizontal membership functions. In: Kreinovich, V. (ed.) Uncertainty Modeling, pp. 233–250. Springer International Publishing, Cham (2017). Dedicated to Professor Boris Kovalerchuk on his Anniversary
Piegat, A., Landowski, M.: Is an interval the right result of arithmetic operations on intervals? Int. J. Appl. Math. Comput. Sci. 27(3), 575–590 (2017)
Piegat, A., Landowski, M.: Is fuzzy number the right result of arithmetic operations on fuzzy numbers? In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K.T., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology 2017, pp. 181–194. Springer International Publishing, Cham (2018)
Piegat, A., Pluciński, M.: Computing with words with the use of inverse RDM models of membership functions. Int. J. Appl. Math. Comput. Sci. 25(3), 675–688 (2015)
Piegat, A., Pluciński, M.: Fuzzy number addition with the application of horizontal membership functions. Sci. World J. 2015, 1–16 (2015)
Piegat, A., Pluciński, M.: Fuzzy number division and the multi-granularity phenomenon. Bull. Pol. Acad. Sci. Tech. Sci. 65(4), 497–511 (2017)
Sevastjanov, P., Dymova, L.: A new method for solving interval and fuzzy equations: linear case. Inf. Sci. 179(7), 925–937 (2009)
Zeinalova, M.L.: Application of RDM interval arithmetic in decision making problem under uncertainty. Procedia Comput. Sci. 120, 788–796 (2017). 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22–23 August 2017, Budapest, Hungary
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
Piegat, A., Pietrzykowski, M. (2019). Correct Solution of Fuzzy Linear System Based on Interval Theory. In: PejaÅ›, J., El Fray, I., Hyla, T., Kacprzyk, J. (eds) Advances in Soft and Hard Computing. ACS 2018. Advances in Intelligent Systems and Computing, vol 889. Springer, Cham. https://doi.org/10.1007/978-3-030-03314-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-03314-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-03313-2
Online ISBN: 978-3-030-03314-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)