Approximate Fuzzy Inverse Matrix Calculation Method using Scenario-based Inverses and Bisection

Abstract

In this paper, we introduce a numerical method to construct the inverse of a square matrix whose elements are trapezoidal or triangular fuzzy numbers (FNs). A set of fuzzy linear equations is required to be solved in order to determine the fuzzy inverse matrix. The proposed technique first iteratively searches the possible solution intervals and then narrows those too-wide estimated intervals via bisection. Using interval arithmetic in left and right matrix multiplication, we aim to approximate the identity matrix as a result of product operations. The dissimilarity of the endpoints of intervals belonging to multiplication matrices with the identity matrix is considered to be an error function to be minimized. In this way, even if the entries of a matrix are uncertain, the fuzzy inverse matrix containing all inverse matrices can be found quickly with the use of computer technology. The method is explained, and comparisons are drawn with inverse stable examples from the literature.

Keywords

• Approximate fuzzy inverse
• Bisection method
• Fully fuzzy linear system of equations
• Fuzzy matrix
• Fuzzy numerical simulation

References

1. [1] S. Chanas, M. Nowakowski, Single value simulation of fuzzy variable, Fuzzy Sets Syst., 25(1) (1988), 43-57.
2. [2] J. Rohn, Inverse interval matrix, SIAM J. Numer. Anal., 30(3) (1993), 864-870.
3. [3] S. S. Rao, L. Chen, Numerical solution of fuzzy linear equations in engineering analysis, Int. J. Numer. Methods Eng., 42(5) (1998), 829-846.
4. [4] M. Dehghan, M. Ghatee, B. Hashemi, Inverse of a fuzzy matrix of fuzzy numbers, Int. J. Comput. Math., 86(8) (2009), 1433-1452.
5. [5] M. A. Basaran, Calculating fuzzy inverse matrix using fuzzy linear equation system, Appl. Soft Comput., 12(6) (2012), 1810-1813.
6. [6] M. Mosleh, Evaluation of fully fuzzy matrix equations by fuzzy neural network, Appl. Math. Modell., 37(9) (2013), 6364-6376.
7. [7] M. Mosleh, M. Otadi, A discussion on “Calculating fuzzy inverse matrix using fuzzy linear equation system”, Appl. Soft Comput., 28 (2015), 511-513.
8. [8] J. Kaur, A. Kumar, Commentary on “Calculating fuzzy inverse matrix using fuzzy linear equation system”, Appl. Soft Comput., 58 (2017), 324-327.

9. [9] X. Guo, Y. Wei, Z. Li, Further investigation to approximate fuzzy inverse, J. Intell. Fuzzy Syst., 35(1) (2018), 1161-1168.
10. [10] C. Y. Chen, J. J. Huang, Deriving fuzzy weights of the fuzzy analytic network process via fuzzy inverse matrix, Mathematics, 7(10) (2019), 914.
11. [11] F. Babakordi, N. A. Taghi-Nezhad, Calculating fuzzy inverse matrix using linear programming problem: An improved approach, Pak. J. Stat. Oper. Res. (2021), 983-996.
12. [12] H. Farahani, M. J. Ebadi, Finding fuzzy inverse matrix using Wu’s method, J. Mahani Math. Res. Cent., 10(1) (2021), 37-52.
13. [13] H. G. Akdemir, H. G. Kocken, A new fuzzy linear regression algorithm based on the simulation of fuzzy samples and an application on popularity prediction of Covid-19 related videos, J. Stat. Manage. Syst., 25(8) (2022), 2025-2041.
14. [14] P. V. Saraev, Interval pseudo-inverse matrices and interval Greville algorithm, Reliab. Comput., 18 (2013), 147-156.
15. [15] J. Lebedinska, On another view of an inverse of an interval matrix, Soft Comput., 14(10) (2010), 1043-1046.