Solving Complex Fuzzy Linear Matrix Equations

Zhiguo Yan,Junyang An,Yirong Sun,Xiaobin Guo

doi:10.1155/2021/9996566

Zhiguo Yan, Junyang An + Show 2 more

Open Access

https://doi.org/10.1155/2021/9996566

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Abstract

In this paper, a kind of complex fuzzy linear matrix equation A X ˜ B = C ˜ , in which C ˜ is a complex fuzzy matrix and A and B are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simplified procedures for solving the original fuzzy equation are proposed and the correspondingly sufficient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to illustrate our proposed method.

Highlights

In 1998, Behra and Chakraverthy [8] investigated fuzzy linear systems Ax􏽥 􏽥b by an embedding approach of fuzzy set decomposition theorem
Introduction e uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented and computed by the fuzzy numbers. e theory and computation of linear systems related with fuzzy numbers always play an important role in the fuzzy mathematics
There has a great enormous investigation in the study of fuzzy mathematics and its applications. e definition of fuzzy numbers and their arithmetic operations were first introduced by Zadeh [1], Dubois and Prade [2], and Nahmias [3]

Summary

Preliminaries

A fuzzy number is a fuzzy set such as u􏽥: R ⟶ I [0, 1] which satisfies (1) u􏽥 is upper semicontinuous (2) u􏽥 is fuzzy convex, i.e., u􏽥(λx + (1 − λ)y) ≥ min􏼈u􏽥(x), u􏽥(y)􏼉, for all x, y ∈ R, λ ∈ [0, 1]. Let E1 be the set of all fuzzy numbers on R. A fuzzy number u􏽥 in the parametric form is a pair (u, u) of functions u(r) and u(r), 0 ≤ r ≤ 1, which satisfies the requirements:. E matrix system: a11 a12 · · · a1n x􏽥11 x􏽥12 · · · x􏽥1n b11 b12 · · · b1n. A complex fuzzy matrix, a complex fuzzy linear matrix equations (CFLMEs). Is called a solution of the fuzzy matrix equation (2) if and only if it satisfies AX􏽥 B C􏽥

Solving Complex Fuzzy Linear Matrix Systems

Numerical Examples

Conclusion