Further investigation to approximate fuzzy inverse 1

Abstract

A numerical procedure for calculating the inverse of LR fuzzy numbers matrix is designed and a sufficient condition for the existence of fuzzy inverse is derived. As a application, the paper considers the solution of fully fuzzy linear systems by the approximate fuzzy inverse. Some examples are given to illustrate the proposed method.

Keywords

LR fuzzy numbers matrix analysis fuzzy inverse matrix fully fuzzy linear systems

1 Introduction

In the past decades fuzzy linear systems has been paid more attention by a few scholars. In 1998, Friedman et al. [14] proposed a general model for solving an n × n fuzzy linear systems based on triangular fuzzy numbers by an embedding approach [25]. A lot of works have been done about how to deal with some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), full fuzzy linear systems (FFLS), dual full fuzzy linear systems (DFFLS) and general dual fuzzy linear systems (GDFLS) [1 , 21]. Recently, some new theory and method for fuzzy linear systems and fuzzy matrix appeared in the literature [2 , 24].

To make the multiplication of fuzzy numbers easy and handle the full fuzzy linear systems (FFLS), D. Dubois and H. Prade [13] introduced the LR fuzzy number in 1978. We know that triangular fuzzy numbers is just specious cases of LR fuzzy numbers. It is well known that the matrix is a important tool for treating crisp linear system equations. We cant help thinking that the fuzzy matrix should be a powerful one to solve fully fuzzy linear system. For example, a investigation on fuzzy inverse matrix will be helpful and useful for solving fully fuzzy linear system. To calculate the inverse of a fuzzy matrix, M.A. Basaran [10] proposed a new direction for approximating fuzzy inverse matrix by solving a n × n fully fuzzy linear system. Later, M. Mosleh and M. Otadi [22] made some correction and supplement to the proposed method [10].

In this paper we give a simple and practical matrix method for obtaining the inverse of LR fuzzy matrix. We also analyze the existence condition and computing approach of fuzzy inverse matrix. As a simple application, we consider the solution of fully fuzzy linear systems by using the inverse of a n × n fuzzy matrix $\tilde{A}$ .

2 Preliminaries

The LR fuzzy number

Definition 2.1. [23] A fuzzy number is a fuzzy set like u : R → I = [0, 1] which satisfies:

u is upper semicontinuous,

u is fuzzy convex, i.e., u (λx + (1 - λ) y) ≥ min {u (x), u (y)} for all x, y ∈ R, λ ∈ [0, 1],

u is normal, i.e., there exists x₀ ∈ R such that u (x₀) =1,

suppu = {x ∈ R ∣ u (x) >0} is the support of the u, and its closure cl(suppu) is compact.

Let E¹ be the set of all fuzzy numbers on R.

Definition 2.2. [13] A fuzzy number $\tilde{M}$ is said to be a LR fuzzy number if $μ_{\tilde{M}} (x) = {\begin{matrix} L (\frac{m - x}{α}), & x \leq m, & α > 0, \\ R (\frac{x - m}{β}), & x \geq m, & β > 0, \end{matrix}$ where m, α and β are called the mean value, left and right spreads of $\tilde{M}$ , respectively. The function L (·), which is called left shape function satisfies:

L (x) = L (- x),

L (0) =1 and L (1) =0,

L (x) is non increasing on [0, ∞).

The definition of a right shape function R (·) is similar to that of L (·).

Clearly, two LR fuzzy numbers $\tilde{M} = (m, α, β)_{LR}$ and $\tilde{N} = (n, γ, δ)_{LR}$ are said to be equal, if and only if m = n, α = γ and β = δ. Also, $\tilde{M} = (m, α, β)_{LR}$ is positive (negative) if and only if m - α > 0(m + β < 0).

Definition 2.3. (Generalized LR fuzzy number)

If α < 0 and β > 0, we define $\tilde{M} = (m, α, β)_{LR} = (m, 0, max {- α, β})_{LR}$ and $μ_{\tilde{M}} (x) = {\begin{matrix} 0, & x < m, \\ R (\frac{x - m}{max {- α, β}}), & x \geq m . \end{matrix}$

If α > 0 and β < 0, we define $\tilde{M} = (m, α, β)_{LR} = (m, max {α, - β}, 0)_{LR}$ and $μ_{\tilde{M}} (x) = {\begin{matrix} L (\frac{x - m}{max {α, - β}}), & x \geq m, \\ 0, & x < m . \end{matrix}$

If α < 0 and β < 0, we define $\tilde{M} = (m, - β, - α)_{LR}$ and $μ_{\tilde{M}} (x) = {\begin{matrix} L (\frac{m - x}{- β}), & x < m, \\ R (\frac{x - m}{- α}), & x \geq m . \end{matrix}$

Definition 2.4. For arbitrary LR fuzzy numbers $\tilde{M} = (m, α, β)_{LR}$ and $\tilde{N} = (n, γ, δ)_{LR}$ , we have

Addition $\begin{matrix} \tilde{M} \oplus \tilde{N} & = & (m, α, β)_{LR} \oplus (n, γ, δ)_{LR} \\ = & (m + n, α + γ, β + δ)_{LR} . \end{matrix}$

Subtraction

$\begin{matrix} \tilde{M} - \tilde{N} & = & (m, α, β)_{LR} - (n, γ, δ)_{LR} \\ = & (m - n, α - δ, β - γ)_{LR} . \end{matrix}$ (1)

If $\tilde{M} > 0$ and $\tilde{N} > 0,$ then $\begin{matrix} \tilde{M} \otimes \tilde{N} & ≅ & (m, α, β)_{LR} \otimes (n, γ, δ)_{LR} \\ ≅ & (mn, m γ + n α, m δ + n β)_{LR} . \end{matrix}$

If $\tilde{M} < 0$ and $\tilde{N} > 0,$ then $\begin{matrix} \tilde{M} \otimes \tilde{N} & ≅ & (m, α, β)_{RL} \otimes (n, γ, δ)_{LR} \\ ≅ & (mn, n α - m δ, n β - m γ)_{RL} . \end{matrix}$

If $\tilde{M} < 0$ and $\tilde{N} < 0,$ then $\begin{matrix} \tilde{M} \otimes \tilde{N} & ≅ & (m, α, β)_{LR} \otimes (n, γ, δ)_{LR} \\ ≅ & (mn, - m δ - n β, - m γ - n α)_{LR} . \end{matrix}$

Definition 2.5. A matrix $\tilde{A} = ({\tilde{a}}_{ij})$ is called a LR fuzzy matrix, if each element ${\tilde{a}}_{ij}$ of $\tilde{A}$ is a LR fuzzy number.

For example, we represent m × n LR fuzzy matrix $\tilde{A} = ({\tilde{a}}_{ij})$ , that ${\tilde{a}}_{ij} = (a_{ij}, α_{ij}, β_{ij})_{LR}$ with new notation $\tilde{A} = (A, M, N)$ , where A = (a_ij), M = (α_ij) and N = (β_ij) are three m × n crisp matrices.

Definition 2.6. Let $\tilde{A} = ({\tilde{a}}_{ij})$ and $\tilde{B} = ({\tilde{b}}_{ij})$ be two m × n and n × p fuzzy matrices, the size of the product of two fuzzy matrices is m × p and is written as follows: $\tilde{A} \otimes \tilde{B} = \tilde{C} = ({\tilde{c}}_{ij})$ where ${\tilde{c}}_{ij} = \sum_{k = 1, . . ., n}^{\oplus} {\tilde{a}}_{ik} \otimes {\tilde{b}}_{kj}$ where ⊗ is the approximated multiplication.

2.2 The fuzzy inverse matrix

Definition 2.7. [10] If the center value of a fuzzy number is 1 and the left and right spread values are δ and λ where 0 < δ, λ < 1, this fuzzy number is called one fuzzy number and denoted by $\tilde{1} = (1, δ, λ)$ . If the center value of a fuzzy number is 0 and the left and right spread values are α and β where 0 < α, β < 1, this fuzzy number is called zero fuzzy number and denoted by $\tilde{0} = (0, α, β)$ .

Definition 2.8. [10] If the diagonal elements of a fuzzy matrix are one fuzzy numbers and the off-diagonal elements are zero fuzzy numbers, then this fuzzy matrix is called fuzzy identity matrix and denoted by $\tilde{I}$ . $\tilde{I} = (= 3 pt \begin{matrix} \tilde{1} & \tilde{0} & \dots & \tilde{0} \\ \tilde{0} & \tilde{1} & \dots & \tilde{0} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \tilde{0} & \tilde{0} & \dots & \tilde{1} \end{matrix}) .$

Definition 2.9. Let $\tilde{A}$ be a LR fuzzy matrix. If there exists a LR fuzzy matrix $\tilde{X}$ such that $\tilde{A} \otimes \tilde{X} = \tilde{X} \otimes \tilde{A} = \tilde{I},$ we call the fuzzy matrix $\tilde{X}$ is the inverse of fuzzy matrix $\tilde{A}$ and is denoted by ${\tilde{A}}^{- 1} = \tilde{X}$ .

Up to rest of this paper we shall investigate the fuzzy inverse of two cases, i.e., find the positive fuzzy inverse $\tilde{X} = (X, Y, Z) \geq 0$ of nonnegative fuzzy matrix $\tilde{A} = (A, B, C)$ and the negative fuzzy inverse $\tilde{X} = (X, Y, Z) \leq 0$ of negative fuzzy matrix $\tilde{A} = (A, B, C)$ .

3 Calculation of fuzzy inverse matrix

Theorem 3.1. Let $\tilde{A}$ be a non negative n × n LR fuzzy matrix and be denoted by $\tilde{A} = (A, B, C)$ , where A is the center value and the B and C are the left and right spread value of $\tilde{A}$ , respectively. Matrices M and N are satisfy with the condition 0 < m_ij, n_ij < 1. Then the non negative approximate fuzzy inverse matrix of $\tilde{A}$ is as follows:

$\begin{matrix} {\tilde{A}}^{- 1} & = & \tilde{X} = (X, Y, Z) = (A^{- 1}, A^{- 1} (M - B ∣ X ∣), \\ A^{- 1} (N - C ∣ X ∣)) . \end{matrix}$ (2)

Proof. Suppose that $\tilde{I}$ is a n × n fuzzy identity matrix and is denoted by $\tilde{I} = (I, M, N)$ . According to the Definition 2.6 and the operations of LR fuzzy numbers, we have $\tilde{A} \otimes \tilde{X} = (A, B, C) \otimes (X, Y, Z) = \tilde{I}$ i.e. $\begin{matrix} (A, B, C) \otimes (X, Y, Z) \\ = (AX, AY + BX, AZ + CX) = (I, M, N) . \end{matrix}$

It is equivalent to ${\begin{matrix} AX = I, \\ AY + BX = M, \\ AZ + CX = N . \end{matrix}$

Since spread parameters are not allowed to be negative due to the definition of LR type fuzzy numbers, absolute values of the spreads should be taken. Thus the equations decided the left and right spread values should be $AY + B ∣ X ∣ = M, AZ + C ∣ X ∣ = N,$ in which ∣X∣ is the absolute values matrix of X.

It means ${\begin{matrix} AX = I, \\ AY + B ∣ X ∣ = M, \\ AZ + C ∣ X ∣ = N . \end{matrix}$

By calculation, we obtain ${\begin{matrix} X = A^{- 1}, \\ Y = A^{- 1} (M - B ∣ A^{- 1} ∣), \\ Z = A^{- 1} (N - C ∣ A^{- 1} ∣) . \end{matrix}$

So, the conclusion (3.1) of Theorem 3.1 $\begin{matrix} {\tilde{A}}^{- 1} & = & (A^{- 1}, A^{- 1} (M - B ∣ X ∣), \\ A^{- 1} (N - C ∣ X ∣)) \end{matrix}$ is obvious.

Now we consider a specious case. When $\tilde{A}$ is a non negative symmetric fuzzy matrix and is denoted by $\tilde{A} = (A, B)$ , what would be its approximate fuzzy inverse.

In fact, from $\begin{matrix} \tilde{A} \otimes \tilde{X} = (A, B) \otimes (X, Y) = \tilde{I} \\ (A, B) \otimes (X, Y) = (AX, AY + BX) = (I, M) . \end{matrix}$ i.e. ${\begin{matrix} AX = I, \\ AY + B ∣ X ∣ = M . \end{matrix}$

By calculation, we conclude ${\begin{matrix} X = A^{- 1}, \\ Y = A^{- 1} (M - B ∣ A^{- 1} ∣) . \end{matrix}$

Thus ${\tilde{A}}^{- 1} = \tilde{X} = (X, Y) = (A^{- 1}, A^{- 1} (M - B ∣ A^{- 1} ∣)) .$ (3.2)

It seems that we obtained the fuzzy inverse of fuzzy matrix ${\tilde{A}}^{- 1}$ as follows. $\begin{matrix} {\tilde{A}}^{- 1} & = & (X, Y, Z) = (A^{- 1}, ∣ A^{- 1} ∣ (M - B ∣ X ∣), \\ ∣ A ∣^{- 1} (N - C ∣ X ∣)) \end{matrix}$ where A^-1, ∣A^-1 ∣ (M - B ∣ X ∣) and ∣A ∣ ^-1 (N - C ∣ X ∣) are n × n crisp matrices. But the solution matrix may still not be an appropriate LR fuzzy numbers matrix except for Y ≥ 0, Z ≥ 0. So we give the definition of LR fuzzy inverse to the Equation (2.8) as follows:

Definition 3.1. Let $\tilde{X} = (X, Y, Z)$ . If (X, Y, Z) is an exact solution of Equation (3.2), such that Y ≥ 0, Z ≥ 0, we call $\tilde{X} = (X, Y, Z)$ is a strong LR fuzzy inverse of fuzzy matrix $\tilde{A}$ . Otherwise, the $\tilde{X} = (X, Y, Z)$ is said to a weak LR fuzzy inverse of fuzzy matrix $\tilde{A}$ given by ${\tilde{A}}^{- 1} = \tilde{X} = \tilde{x_{ij}}$ where

${\tilde{x}}_{ij} = {\begin{matrix} (x_{ij}, y_{ij}, z_{ij})_{LR}, & y_{ij} > 0, & z_{ij} > 0, \\ (x_{ij}, 0, max {- y_{ij}, n_{ij}})_{LR}, & y_{ij} < 0, & z_{ij} > 0, \\ (x_{ij}, max {y_{ij}, - z_{ij}}, 0,)_{LR}, & y_{ij} > 0, & z_{ij} < 0, \\ (x_{ij}, - z_{ij}, - y_{ij})_{LR}, & y_{ij} < 0, & z_{ij} < 0 . \end{matrix} i, j = 1, \dots, n .$ (3)

Theorem 3.2. Let $\tilde{A} = (A, B, C)$ and $\tilde{I} = (I, M, N)$ be two nonnegative LR fuzzy matrices, respectively. If A^-1 is a nonnegative matrix and MA ≥ B and NA ≥ C. Then fuzzy matrix $\tilde{A}$ has a strong LR fuzzy inverse. Moreover, If the condition A ≥ MA - B holds, then the fuzzy matrix $\tilde{A}$ has a nonnegative strong LR fuzzy inverse.

Proof. Since A^-1 is a nonnegative matrix. Thus the fact X = A^-1 ≥ 0 is apparent.

On the other hand, because MA ≥ B and NA ≥ C, so with Y = A^-1 (M - BA^-1) and Z = A^-1 (N - CA^-1), we have Y ≥ 0 and Z ≥ 0. Thus $\tilde{X} = (X, Y, Z)$ is a strong LR fuzzy inverse of the fuzzy matrix $\tilde{A}$ by Definition 3.1. Since X - Y = A^-1 - A^-1 (M - BA^-1) = A^-1 (I - M + BA^-1), the nonnegative property of fuzzy inverse $\tilde{X}$ can be obtained from the condition A ≥ MA - B.

In similar way, we can consider the negative fuzzy inverse matrix $\tilde{X} = {\tilde{A}}^{- 1}$ of a negative fuzzy matrix $\tilde{A}$ .

Theorem 3.3. Let $\tilde{A}$ be a negative n × n LR fuzzy matrix and be denoted by $\tilde{A} = (A, B, C)$ , where A ≤ 0 is the center value and the B and C are the left and right spread value of $\tilde{A}$ , respectively. Matrices M and N are satisfy with the condition 0 < m_ij, n_ij < 1. Then the negative approximate fuzzy inverse matrix of $\tilde{A}$ is as follows:

$\begin{matrix} {\tilde{A}}^{- 1} & = & \tilde{X} = (X, Y, Z) = (A^{- 1}, - A^{- 1} (N - B ∣ X ∣), \\ - A^{- 1} (M - C ∣ X ∣)) . \end{matrix}$ (4)

Proof. Suppose that $\tilde{I}$ is a n × n fuzzy identity matrix and denoted by $\tilde{I} = (I, M, N)$ . According to the Definition 2.6 and the operations of LR fuzzy numbers, we have $\tilde{A} \otimes \tilde{X} = (A, B, C) \otimes (X, Y, Z) = \tilde{I}$ i.e. $\begin{matrix} (A, B, C) \otimes (X, Y, Z) \\ = (AX, - AZ - CX, - AY - BX) \\ = (I, M, N) . \end{matrix}$

It is equivalent to

${\begin{matrix} AX = I, \\ - AY - BX = N, \\ - AZ - CX = M . \end{matrix}$

$- AY + B ∣ X ∣ = N, - AZ + C ∣ X ∣ = M,$ in which ∣X∣ is the absolute values matrix of X.

It means ${\begin{matrix} AX = I, \\ - AY + B ∣ X ∣ = N, \\ - AZ + C ∣ X ∣ = M . \end{matrix}$ (3.5)

By calculation, we obtain ${\begin{matrix} X = A^{- 1}, \\ Y = - A^{- 1} (N - B ∣ X ∣), \\ Z = - A^{- 1} (M - C ∣ X ∣) . \end{matrix}$

So, the conclusion (3.4) of Theorem 3.1 $\begin{matrix} {\tilde{A}}^{- 1} & = & (A^{- 1}, - A^{- 1} (N - B ∣ X ∣), \\ - A^{- 1} (M - C ∣ X ∣)) \end{matrix}$ is obvious.

Theorem 3.4. Let $\tilde{A} = (A, B, C)$ be a negative LR fuzzy matrices with A ≤ 0. If A^-1 is a negative matrix and N≥ B ∣ X ∣ and M≥ C ∣ X ∣. Then fuzzy matrix $\tilde{A}$ has a strong LR fuzzy inverse. Moreover, If the condition (- C ∣ X ∣ + M) ≤ I holds, then the fuzzy matrix $\tilde{A}$ has a negative strong LR fuzzy inverse.

Proof. Since A^-1 is a negative matrix. Thus the fact X = A^-1 ≤ 0 is apparent.

On the other hand, because N≥ B ∣ X ∣ and M≥ C ∣ X ∣, so with Y = - A^-1 (N - B ∣ X ∣) and Z = - A^-1 (M - C ∣ X ∣), we have Y ≥ 0 and Z ≥ 0. Thus $\tilde{X} = (X, Y, Z)$ is a strong LR fuzzy inverse of the fuzzy matrix $\tilde{A}$ by Definition 3.1. Since X + Z = A^-1 - A^-1 (C ∣ X ∣ - M) = A^-1 (I + C ∣ X ∣ - M), the negative property of fuzzy inverse $\tilde{X}$ can be obtained from the condition (M - C ∣ X ∣) ≤ I.

4 Numerical examples

Example 4.1. Consider the following negative fuzzy matrix $\tilde{A}$ which consists of symmetric triangular fuzzy numbers as follows: $\begin{matrix} \tilde{A} & = & (\begin{matrix} (- 6, 3) & (- 4, 2) \\ (- 4, 1) & (- 3, 1) \end{matrix}) \\ = & (\begin{matrix} (\begin{matrix} - 6 & - 4 \\ - 4 & - 3 \end{matrix}), (\begin{matrix} 3 & 2 \\ 1 & 1 \end{matrix}) \end{matrix}) . \end{matrix}$

Let $A = (\begin{matrix} - 6 & - 4 \\ - 4 & - 3 \end{matrix}), B = (\begin{matrix} 3 & 2 \\ 1 & 1 \end{matrix}), M = (\begin{matrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{matrix})$ .

According to formula (3.4), we have $\begin{matrix} X & = & A^{- 1} = (\begin{matrix} - 1.5 & 2 \\ 2 & - 3 \end{matrix}), \\ Y & = & - A^{- 1} (M - B ∣ X ∣) = (\begin{matrix} - 6 & - 8.25 \\ 7 & 9.5 \end{matrix}) . \end{matrix}$

Since the spread values of fuzzy numbers are not permitted be negative, we may take their absolute values as follows: $\begin{matrix} {\tilde{A}}^{- 1} & = & (X, ∣ Y ∣) \\ = & (\begin{matrix} (\begin{matrix} - 1.5 & - 2 \\ 2 & - 3 \end{matrix}), (\begin{matrix} 6 & 8.25 \\ 7 & 9.5 \end{matrix}) \end{matrix}) \\ = & (\begin{matrix} (- 1.5, 6) & (2, 8.25) \\ (2, 7) & (- 3, 9.5) \end{matrix}) . \end{matrix}$

Example 4.2. Consider the following fuzzy matrix $\begin{matrix} \tilde{A} & = & (\begin{matrix} (4, 2, 1) & (5, 1, 2) & (3, 1, 1) \\ (5, 1, 1) & (6, 1, 1) & (4, 2, 2) \\ (3, 1, 1) & (4, 2, 1) & (3, 1, 1) \end{matrix}) \\ = & (\begin{matrix} (\begin{matrix} 4 & 5 & 3 \\ 5 & 6 & 4 \\ 3 & 4 & 3 \end{matrix}), (\begin{matrix} 2 & 1 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{matrix}), (\begin{matrix} 1 & 2 & 1 \\ 1 & 1 & 2 \\ 1 & 1 & 1 \end{matrix}) \end{matrix}) . \end{matrix}$

Let $\begin{matrix} M & = & (\begin{matrix} 0.5000 & 0.6000 & 0.4000 \\ 0.7000 & 0.5000 & 0.8000 \\ 0.7000 & 0.8000 & 0.6000 \end{matrix}), \\ N & = & (\begin{matrix} 0.8000 & 0.4000 & 0.7000 \\ 0.6000 & 0.5000 & 0.6000 \\ 0.7000 & 0.8000 & 0.6000 \end{matrix}) . \end{matrix}$

According to formula (3.1), we have $\begin{matrix} X = A^{- 1} = (\begin{matrix} - 2.0000 & 3.0000 & - 2.0000 \\ 3.0000 & - 3.0000 & 1.0000 \\ - 2.0000 & 1.0000 & 1.0000 \end{matrix}), \\ Y = A^{- 1} (M - B ∣ X ∣) \\ = (\begin{matrix} 10.7000 & 14.7000 & 7.4000 \\ - 9.9000 & - 14.9000 & - 8.6000 \\ - 0.6000 & 2.1000 & 2.6000 \end{matrix}) \end{matrix}$ and $\begin{matrix} Z & = & A^{- 1} (M - C ∣ X ∣) \\ = & (\begin{matrix} 5.8000 & 9.1000 & 2.2000 \\ - 8.7000 & - 12.5000 & - 3.1000 \\ 3.7000 & 5.5000 & 0.8000 \end{matrix}) . \end{matrix}$

i.e. $\begin{matrix} (X, Y, Z) \\ = (\begin{matrix} (\begin{matrix} - 2.00 & 3.00 & - 2.0 \\ 3.00 & - 3.00 & 1.00 \\ - 2.00 & 1.00 & 1.00 \end{matrix}), (\begin{matrix} 10.70 & 14.70 & 7.40 \\ - 9.90 & - 14.90 & - 8.6 \\ - 0.60 & 2.10 & 2.60 \end{matrix}), (\begin{matrix} 5.80 & 9.10 & 2.20 \\ - 8.70 & - 12.50 & - 3.1 \\ 3.70 & 5.50 & 0.80 \end{matrix}) \end{matrix}) \\ = (\begin{matrix} (- 2, 10.70, 5.80) & (3, 14.70, 9.10) & (- 2, 7.40, 2.20) \\ (3, - 9.90, - 8.70) & (- 3, - 14.90, - 12.50) & (1, - 8.60, - 3.10) \\ (- 2, - 0.6, 3.70) & (1, 2.10, 5.50) & (1, 2.60, 0.80) \end{matrix}) . \end{matrix}$

By the Definition 3.1., we obtain the fuzzy inverse of fuzzy matrix $\tilde{A}$ as ${\tilde{A}}^{- 1} = (\begin{matrix} (- 2, 10.70, 5.80) & (3, 14.70, 9.10) & (- 2, 7.40, 2.20) \\ (3, 8.70, 9.90) & (- 3, 12.50, 14.90) & (1.0, 3.10, 8.60) \\ (- 2, 0.00, 3.70) & (1.0, 2.10, 5.50) & (1.0, 2.60, 0.80) \end{matrix}),$ it admits a weak LR fuzzy inverse matrix.

5 A simple application

As a simple application, we consider the solution of fully fuzzy linear systems $\tilde{A} \otimes \tilde{x} = \tilde{b}$ (5.1) by using the inverse of a n × n fuzzy matrix $\tilde{A}$ . To simplicity, we suppose $\tilde{A}$ is a non negative fuzzy matrix.

Theorem 5.1. If fuzzy matrix $\tilde{A}$ of fuzzy linear system (3.6) is invertible, the approximate solution of Eqs.(5.1) can be obtained by the following $\tilde{X} = {\tilde{A}}^{- 1} \otimes \tilde{b} .$ (5.2)

Proof. We denote $\tilde{A} = (A, B, C)$ , $\tilde{b} = (b, h, g)$ and $\tilde{x} = (x, y, z)$ . To fuzzy linear system (3.4), we extend it into ${\begin{matrix} Ax = b, \\ Ay + Bx = h, \\ Az + Cx = g . \end{matrix}$

By calculation, we obtain ${\begin{matrix} x = A^{- 1} b, \\ y = A^{- 1} (h - B ∣ A^{- 1} ∣ b), \\ z = A^{- 1} (g - C ∣ A^{- 1} ∣ b), \end{matrix}$ (5.3) in which ∣X∣ is the absolute values matrix X.

According to (3.1), we have ${\tilde{A}}^{- 1} = (A^{- 1}, A^{- 1} (M - B ∣ X ∣), A^{- 1} (N - C ∣ X ∣))$ . Thus

$\begin{matrix} {\tilde{A}}^{- 1} \otimes \tilde{b} \\ = (A^{- 1}, A^{- 1} (M - B ∣ X ∣), \\ A^{- 1} (N - C ∣ X ∣)) \otimes (b, h, g) \\ = (A^{- 1} b, A^{- 1} h + A^{- 1} (M - B ∣ A^{- 1} ∣) b, \\ A^{- 1} g + A^{- 1} (N - C ∣ A^{- 1} ∣) b) \\ = (A^{- 1} b, A^{- 1} (h - B ∣ A^{- 1} ∣ b + Mb), \\ A^{- 1} (g - C ∣ A^{- 1} ∣ b + Nb)), \end{matrix}$ (5) where matrices M and N are satisfy with the condition 0 < m_ij, n_ij ≤ 0.5.

When M and N are reduce to zero matrices, Mb and Nb becomes zero vectors. Therefore Equation (5.2) is $\begin{matrix} {\tilde{A}}^{- 1} \otimes \tilde{b} & = & (A^{- 1} b, A^{- 1} (h - B ∣ A^{- 1} ∣ b), \\ A^{- 1} (g - C ∣ A^{- 1} ∣ b)) . \end{matrix}$

It is just the exact solution of fuzzy linear system (5.1).

Here we give an example to show the above result.

Example 5.1. Consider the following fully fuzzy linear system $\tilde{A} \otimes \tilde{x} = \tilde{b}$ , i.e., $\begin{matrix} (\begin{matrix} (4, 2, 1) & (5, 1, 2) & (3, 1, 1) \\ (5, 1, 1) & (6, 1, 1) & (4, 2, 2) \\ (3, 1, 1) & (4, 2, 1) & (3, 1, 1) \end{matrix}) \otimes (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}) \\ = (\begin{matrix} (4, 2, 2) \\ (2, 1, 1) \\ (3, 1, 2) \end{matrix}) . \\ \tilde{A} = (\begin{matrix} (4, 2, 1) & (5, 1, 2) & (3, 1, 1) \\ (5, 1, 1) & (6, 1, 1) & (4, 2, 2) \\ (3, 1, 1) & (4, 2, 1) & (3, 1, 1) \end{matrix}) \\ = (\begin{matrix} (\begin{matrix} 4 & 5 & 3 \\ 5 & 6 & 4 \\ 3 & 4 & 3 \end{matrix}), (\begin{matrix} 2 & 1 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{matrix}), (\begin{matrix} 1 & 2 & 1 \\ 1 & 1 & 2 \\ 1 & 1 & 1 \end{matrix}) \end{matrix}) . \end{matrix}$

We suppose $\begin{matrix} M & = & (= 3 pt \begin{matrix} 0.2000 & 0.3000 & 0.4000 \\ 0.1000 & 0.3000 & 0.2000 \\ 0.3000 & 0.2000 & 0.3000 \end{matrix}), \\ N & = & (= 3 pt \begin{matrix} 0.4000 & 0.1000 & 0.3000 \\ 0.1000 & 0.2000 & 0.5000 \\ 0.3000 & 0.2000 & 0.3000 \end{matrix}) . \end{matrix}$

According to formula (3.1), we have $\begin{matrix} X & = & A^{- 1} = (\begin{matrix} - 2.0000 & 3.0000 & - 2.0000 \\ 3.0000 & - 3.0000 & 1.0000 \\ - 2.0000 & 1.0000 & 1.0000 \end{matrix}), \\ Y & = & A^{- 1} (M - B ∣ X ∣) \\ = & (\begin{matrix} 10.3000 & 16.3000 & 6.2000 \\ - 9.4000 & - 16.4000 & - 7.1000 \\ - 1.0000 & 2.3000 & 1.7000 \end{matrix}) \end{matrix}$ and $\begin{matrix} Z & = & A^{- 1} (N - C ∣ X ∣) \\ = & (\begin{matrix} 5.9000 & 10.0000 & 3.3000 \\ - 8.8000 & - 13.1000 & - 4.3000 \\ 3.6000 & 5.2000 & 1.2000 \end{matrix}) . \end{matrix}$ i.e. $\begin{matrix} (X, Y, Z) \\ = (\begin{matrix} (\begin{matrix} - 2.00 & 3.00 & - 2.0 \\ 3.00 & - 3.00 & 1.00 \\ - 2.00 & 1.00 & 1.00 \end{matrix}), (\begin{matrix} 10.30 & 16.30 & 6.20 \\ - 9.40 & - 16.40 & - 7.10 \\ - 1.00 & 2.30 & 1.70 \end{matrix}), (\begin{matrix} 5.90 & 10.00 & 3.30 \\ - 8.80 & - 13.10 & - 4.30 \\ 3.60 & 5.20 & 1.20 \end{matrix}) \end{matrix}) \\ = (\begin{matrix} (- 2, 10.30, 5.90) & (3, 16.30, 10) & (- 2, 6.20, 3.30) \\ (3, - 9.40, - 8.80) & (- 3, - 16.40, - 13.10) & (1, - 7.10, - 4.30) \\ (- 2, 1.0, 3.60) & (1, 2.30, 5.20) & (1, 1.70, 1.20) \end{matrix}) . \end{matrix}$

By the (3.4.), we have $\begin{matrix} x = A^{- 1} b = (\begin{matrix} - 8 \\ 9 \\ - 3 \end{matrix}), \\ y = A^{- 1} (h - B ∣ A^{- 1} ∣ b + Mb) = (\begin{matrix} 89.4 \\ - 87.7 \\ 3.7 \end{matrix}) \end{matrix}$ and $z = A^{- 1} (g - C ∣ A^{- 1} ∣ b + Nb) = (\begin{matrix} 48.50 \\ - 69.30 \\ 27.40 \end{matrix}) .$

Thus we obtain the approximate solution of fuzzy linear systems $\tilde{A} \otimes \tilde{x} = \tilde{b}$ as $\tilde{x} = (\begin{matrix} (- 8, 89.4, 48.5) \\ (9, 69.3, 87.7) \\ (- 3, 3.7, 27.4) \end{matrix}) .$

Compare with the solution of fuzzy linear systems $\tilde{A} \otimes \tilde{x} = \tilde{b}$ that is $\tilde{x} = (\begin{matrix} (- 8, 94.0, 52.0) \\ (9, 73.3, 92.0) \\ (- 3, 4.0, 28.0) \end{matrix}),$ the error is relatively minute.

6 Conclusion

In this work we presented a matrix method for solving the inverse of a LR fuzzy matrix $\tilde{A}$ . The model was made of three crisp linear matrix equations which determined the mean value and the left and right spreads of the fuzzy inverse. The LR fuzzy inverse of the fuzzy matrix was derived from solving the model and the existence condition of strong LR fuzzy inverse was studied. As a application, the paper considered the solution of fully fuzzy linear systems by the fuzzy inverse matrix. Numerical examples showed that our method is feasible to solve fuzzy inverse.

References

Abbasbandy

, Otadi

and Mosleh

, Minimal solution of general dual fuzzy linear systems, Chaos, Solitions and Fractals 29 (2008), 638–652.

Allahviranloo

and Ghanbari

, A new approach to obtain algebraic solution of interval linear systems, Soft Computing 16 (2012), 121–133.

Allahviranloo

, Hosseinzadeh Lotfi

, Khorasani Kiasari

and Khezerloo

, On the fuzzy solution of LR fuzzy linear systems, Applied Mathematical Modelling 37 (2013), 1170–1176.

Allahviranloo

, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation 153 (2004), 493–502.

Allahviranloo

, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation 162 (2005), 189–196.

Allahviranloo

, The adomian decomposition method for fuzzy system of linear equations, Applied Mathematics and Computation 163 (2005), 553–563.

Allahviranloo

, Ghanbari

and Nuraei

, A note on Fuzzy linear systems, Fuzzy Sets and Systems 177 (2011), 87C92.

Asady

, Abbasbandy

and Alavi

, Fuzzy general linear systems, Applied Mathematics and Computation 169 (2005), 34–40.

Babbar

, Kumar

and Bansal

, Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers, Soft Computing 17 (2013), 691C702.

10.

Basaran

M.A.

, Calculating fuzzy inverse matrix using fuzzy linear equation system,C, Applied Soft Computing 12 (2012), 1810C1813.

11.

Dehghan

, Hashemi

and Ghatee

, Solution of the full fuzzy linear systems using iterative techniques, Chaos, Solitons and Fractals 34 (2007), 316–336.

12.

Dookhitram

, Kanaksabee

and Bhuruth

, Krylov subspace method for fuzzy eigenvalue problem, Journal of Intelligent and Fuzzy Systems 27 (2014), 717–727.

13.

Dubois

and Prade

, Operations on fuzzy numbers, Journal of Systems Science 9 (1978), 613–626.

14.

Friedman

, Ma

and Kandel

, Fuzzy linear systems, Fuzzy Sets and Systems 96 (1998), 201–209.

15.

Ghanbari

, Solutions of fuzzy LR algebraic linear systems using linear programs, Applied Mathematical Modelling 39 (2015), 5164C5173.

16.

Gong

Z.T.

and Guo

X.B.

, Inconsistent fuzzy matrix equations and its fuzzy least squares solutions, Applied Mathematical Modelling 35 (2011), 1456–1469.

17.

Gong

Z.T.

, Guo

X.B.

and Liu

, Approximate solution of dual fuzzy matrix equations, Information Sciences 266 (2014), 112–133.

18.

Guo

X.B.

and Zhang

, Minimal solution of complex fuzzy linear systems, Advances in Fuzzy Systems 2016, 9. Article ID5293917.

19.

Guo

X.B.

and Zhang

, Solving fuzzy matrix equation of the form

\tilde{XA} = \tilde{B}

, Journal of Intelligent and Fuzzy Systems 32 (2017), 2771–2778.

20.

Guo

X.B.

and Han

Y.L.

, Further investigation to dual fuzzy matrix equation, Journal of Intelligent and Fuzzy Systems 33 (2017), 2617–2629.

21.

, Friedman

and Kandel

, Duality in Fuzzy linear systems, Fuzzy Sets and Systems 109 (2000), 55–58.

22.

Mosleh

and Otadi

, A discussion on calculating fuzzy inverse matrix using fuzzy linear equation system, Applied Soft Computing 28 (2015), 511C513.

23.

Nahmias

, Fuzzy variables, Fuzzy Sets and Systems 1(2) (1978), 97–111.

24.

Nuraei

, Allahviranloo

and Ghanbari

, Finding an inner estimation of the solution set of a fuzzy linear system, Applied Mathematical Modelling 37 (2013), 5148–5161.

25.

C.X.

and Ma

, Embedding problem of fuzzy number space: Part III, Fuzzy Sets and Systems 46 (1992), 281–286.