Near zero fuzzy solution of fully fuzzy linear systems 1

Abstract

This paper discusses a new approximate solution of a class of fully fuzzy linear systems $\tilde{A} \tilde{x} = \tilde{b}$ in which the coefficient matrix $\tilde{A}$ is a positive fuzzy matrix. The original fuzzy linear systems is extended into simple crisp linear equation using the obtained approximate multiplication of positive fuzzy number and near zero fuzzy number. Two cases are analysed: (a) the unknown vector $\tilde{x}$ is a near zero fuzzy vector with positive mean value; (b) the unknown vector $\tilde{x}$ is a near zero fuzzy vector with negative mean value. Two computing models are established and respective expression of the solution to fully fuzzy linear system are derived, and the sufficient condition for the existence of strong fuzzy solution are analyzed correspondingly. Some numerical examples are given to illustrated our proposed method.

Keywords

Fuzzy numbers matrix computation fuzzy linear systems fuzzy approximate solutions

1 Introduction

The system of linear equations is an essential mathematical tool in science and technology. In many applications, some parameters of the system are represented and computed by fuzzy numbers rather than crisp numbers. On the other hand, it is the linear systems that has mature theory and easy computational property. Therefore, the investigation of theory and computation for fuzzy linear systems always plays an important role in the fuzzy mathematics and its applications. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [34], Dubois et al. [12] and Nahmias [27]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [29], Goetschell et al. [16] and Wu Congxin et al. [32, 33].

In 1998, Friedman et al. [13] discussed a class of semi fuzzy linear systems $A \tilde{x} = \tilde{b}$ by an embedding approach. Later, some more general and complicated fuzzy linear systems such as dual fuzzy linear systems, general fuzzy linear systems, complex fuzzy linear systems, dual full fuzzy linear systems and general dual fuzzy linear systems were studied by many a scholar in past two decades see [1–8 , 28]. In recent years, new theories and approaches for fuzzy numbers and fuzzy linear systems has been emerging one after another [9 , 31]. For some kinds of linear matrix equations related with fuzzy numbers, Guo al. made a some investigations [17 –22].

When studying fully fuzzy linear systems, almost all the work stalled on the multiplication of two fuzzy numbers because we have to use positive or negative approximate multiplication of LR fuzzy numbers proposed by Dubois et al. [12] in 1978. In fact, there are a lot of near-zero fuzzy numbers in practical problems. [[Ten year ago, T. Allahviranloo et al. [3, 4] dealt with fully fuzzy linear systems based on interval number representation. With the help of matrix splitting technique, they realized the calculation of non zero fuzzy solutions of fully fuzzy linear systems.]] In 2021, W.S.W. Daud et al. [10] introduced the near zero LR fuzzy numbers and investigated the near zero fully fuzzy matrix equation $\tilde{A} \tilde{X} \tilde{B} - \tilde{X} = \tilde{C}$ . Recently,, M. Ghanbari al. proposed a new effective approximate multiplication operation on LR fuzzy numbers and discussed its applications [15]. On this basis, we study new approximate solution of fully fuzzy linear systems $\tilde{A} \tilde{x} = \tilde{b}$ in which the coefficient matrix $\tilde{A}$ is a positive fuzzy matrix. Two computing models are established and respective expression of the solution to fully fuzzy linear system are derived. Meanwhile, the sufficient condition for the existence of strong fuzzy solution are analyzed correspondingly. And the structure of the paper is as follows:

In Section 2, some definitions and arithmetic operations on LR fuzzy numbers are recalled. In Section 3, near zero fuzzy numbers are introduced and the model for solving fully fuzzy linear systems are extended and their corresponding solution expression are obtained. Some illustrating numerical examples are given in Section 4. And in Section 5, the conclusion and further investigations are given.

2 Preliminaries

There are several definitions for the concept of fuzzy numbers (see [12 , 34]).

2.1 The LR fuzzy number

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

(1) u is upper semi-continuous,

(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],

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

(4) 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. A fuzzy number $\tilde{m}$ is said to be a LR fuzzy number if $\begin{matrix} μ_{\tilde{m}} (x) = {\begin{matrix} L (\frac{m - x}{α}), & x \leq m, & α > 0, \\ R (\frac{x - m}{β}), & x \geq m, & β > 0, \end{matrix} \end{matrix}$ 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:

(1) L (x) = L (- x) ,

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

(3) L (x) is non increasing on [0, ∞) .

Similar, the right shape function satisfies:

(1) R (x) = R (- x) ,

(2) R (0) =1 and R (1) =0,

(3) R (x) is non decreasing on (- ∞ , 0].

A fuzzy number $\tilde{m} = (m, α, β)_{LR}$ is positive(negative) if and only if m - α > 0(m + β < 0). Also, a fuzzy number $\tilde{m} = (m, α, β)_{LR}$ is near zero if and only if m - α < 0 while m + β > 0.

Definition 2.3. For arbitrary LR fuzzy numbers $\tilde{m} = (m, α, β)_{LR}$ and $\tilde{n} = (n, γ, δ)_{LR}$ , we have

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

(2) Subtraction $\begin{matrix} \tilde{m} - \tilde{n} = (m, α, β)_{LR} - (n, γ, δ)_{LR} \\ = (m - n, α - δ, β - γ)_{LR} . \end{matrix}$

(3) Scalar multiplication $\begin{matrix} λ \tilde{m} = λ (m, α, β)_{LR} ≅ {\begin{matrix} (λ m, λ α, λ β)_{LR}, λ \geq 0, \\ (λ m, - λ β, - λ α)_{RL}, λ < 0 . \end{matrix} \end{matrix}$

(4) multiplication

If $\tilde{m} > 0$ and $\tilde{n} > 0$ , then $\begin{matrix} \tilde{m} \times \tilde{n} ≅ (mn, m γ + n α, m δ + n β) . \end{matrix}$

If $\tilde{m} < 0$ and $\tilde{n} > 0$ , then $\begin{matrix} \tilde{m} \times \tilde{n} ≅ (mn, n α - m δ, n β - m γ) . \end{matrix}$

If $\tilde{m} < 0$ and $\tilde{n} < 0$ , then $\begin{matrix} \tilde{m} \times \tilde{n} ≅ (mn, - n β - m δ, - n α - m γ) . \end{matrix}$

Definition 2.4. 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. Denoted by $\tilde{A} = (A, A^{l}, A^{r})$ in which A is its mean value matrix and A^l, A^r are its left and right spread matrices. In a similar way, we can define a LR fuzzy vector $\tilde{b} = (b, b^{l}, b^{r})$ .

Definition 2.5. A LR fuzzy numbers matrix $\tilde{A} = (A, A^{l}, A^{r})$ is said to be near zero if and only if both A - A^l < 0 and A + A^r > 0. Also, a LR fuzzy numbers vector $\tilde{b} = (b, b^{l}, b^{r})$ is said to be near zero fuzzy numbers vector with positive(negative) mean value if and only if both b - b^l < 0 and b + b^r > 0 and b > 0 (b < 0) hold.

2.2 Positive fully fuzzy linear systems

Definition 2.6. The linear system

$(\begin{matrix} {\tilde{a}}_{11} & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{2 n} \\ \dots \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & {\tilde{a}}_{nn} \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ ⋮ \\ {\tilde{x}}_{n} \end{matrix}) = (\begin{matrix} {\tilde{b}}_{1} \\ {\tilde{b}}_{2} \\ ⋮ \\ {\tilde{b}}_{n} \end{matrix}),$ (1) where ${\tilde{a}}_{ij}, 1 \leq i, j \leq n$ are positive LR fuzzy numbers and ${\tilde{b}}_{i}, 1 \leq i \leq n$ are arbitrary LR fuzzy numbers, is called a positive fully fuzzy linear systems(PFFLS).

Using matrix notation, we have

$\tilde{A} \tilde{x} = \tilde{b} .$ (2)

A LR fuzzy vector given by $\begin{matrix} \tilde{x} = ({\tilde{x}}_{i}) = (x_{i}, x_{i}^{l}, x_{i}^{r}), 1 \leq i \leq n \end{matrix}$ is called a solution of the fuzzy linear systems Eq.(1) if $\tilde{x}$ satisfies $\tilde{A} \tilde{x} = \tilde{b} .$

3 Near zero fuzzy solution of fully fuzzy linear systems

Based on multiplication operations of LR fuzzy numbers proposed by Dubois et al. [12], we have the following results.

Theorem 3.1. Let $\tilde{m} = (m, m^{l}, m^{r})$ to be a positive fuzzy number and $\tilde{n} = (n, n^{l}, n^{r})$ to be a near zero fuzzy number, Then

$\tilde{m} \times \tilde{n} ≅ {\begin{matrix} (mn, {mn}^{l} + {nm}^{l}, {mn}^{r} + {nm}^{r}), n > 0, \\ (mn, {mn}^{l} - {nm}^{r}, {mn}^{r} - {nm}^{l}), n < 0 . \end{matrix}$ (3)

Proof. Since $\tilde{n} = (n, n^{l}, n^{r})$ to be a near zero fuzzy number, then n - n^r < 0 and n + n^l > 0. when n > 0, let $n^{r} = n_{1}^{r} + n_{2}^{r}$ , we have $\tilde{n} = (n, n_{2}^{l}, n^{r}) + (0, n_{1}^{l}, 0)$ where $(n, n_{2}^{l}, n^{r})$ is a positive number and $(0, n_{1}^{l}, 0)$ is a LR zero fuzzy number and regarded as a negative one. Thus $\begin{matrix} \tilde{m} \times \tilde{n} \\ = \tilde{m} \times ((n, n_{2}^{l}, n^{r}) + (0, n_{1}^{l}, 0)) \\ = (m, m^{l}, m^{r}) \times (n, n_{2}^{l}, n^{r}) + (m, m^{l}, m^{r}) \times (0, n_{1}^{l}, 0) \\ = (mn, {mn}_{2}^{l} + {nm}^{l}, {mn}^{r} + {nm}^{r}) + (0, {mn}_{1}^{l}, 0) \\ = (mn, {mn}^{l} + {nm}^{l}, {mn}^{r} + {nm}^{r}) . \end{matrix}$

When n < 0, let $n^{l} = n_{1}^{l} + n_{2}^{l}$ , we have $\begin{matrix} \tilde{n} = (n, n^{l}, n_{1}^{r}) + (0, 0, n_{2}^{r}) \end{matrix}$ where $(n, n^{l}, n_{1}^{r})$ is a negative number and $(0, 0, n_{2}^{r})$ is a LR zero fuzzy number and regarded as a positive one. Thus $\begin{matrix} \tilde{m} \times \tilde{n} \\ = \tilde{m} \times ((n, n^{l}, n_{1}^{r}) + (0, 0, n_{2}^{r})) \\ = (m, m^{l}, m^{r}) \times (n, n^{l}, n_{1}^{r}) + (m, m^{l}, m^{r}) \times (0, 0, n_{2}^{r}) \\ = (mn, {mn}^{l} - {nm}^{r}, {mn}_{1}^{r} - {nm}^{l}) + (0, 0, {mn}_{2}^{r}) \\ = (mn, {mn}^{l} - {nm}^{r}, {mn}^{r} - {nm}^{l}) . \end{matrix}$

It is completed the proof.

Theorem 3.2. Let $\tilde{m} = (m, m^{l}, m^{r})$ to be a negative fuzzy number and $\tilde{n} = (n, n^{l}, n^{r})$ to be a near zero fuzzy number, Then

$\tilde{m} \times \tilde{n} ≅ {\begin{matrix} (mn, {nm}^{l} - {mn}^{r}, {nm}^{r} - {mn}^{l}), n > 0, \\ (mn, - {mn}^{r} - {nm}^{r}, - {mn}^{l} - {nm}^{l}), n < 0 . \end{matrix}$ (4)

Proof. Since $\tilde{n} = (n, n^{l}, n^{r})$ to be a near zero fuzzy number, then n - n^r < 0 and n + n^l > 0. when n > 0, let $n^{r} = n_{1}^{r} + n_{2}^{r}$ , we have $\tilde{n} = (n, n_{2}^{l}, n^{r}) + (0, n_{1}^{l}, 0)$ where $(n, n_{2}^{l}, n^{r})$ is a positive number and $(0, n_{1}^{l}, 0)$ is a LR zero fuzzy number and regarded as a positive(negative) one. Thus $\begin{matrix} \tilde{m} \times \tilde{n} \\ = \tilde{m} \times ((n, n_{2}^{l}, n^{r}) + (0, n_{1}^{l}, 0)) \\ = (m, m^{l}, m^{r}) \times (n, n_{2}^{l}, n^{r}) + (m, m^{l}, m^{r}) \times (0, n_{1}^{l}, 0) \\ = (mn, {nm}^{l} - {mn}^{r}, {nm}_{r}^{r} - {mn}_{2}^{l}) + (0, 0, {mn}_{1}^{r}) \\ = (mn, {nm}^{l} - {mn}^{r}, {nm}^{r} - {mn}^{l}) . \end{matrix}$

When n < 0, let $n^{r} = n_{1}^{r} + n_{2}^{r}$ , we have $\begin{matrix} \tilde{n} = (n, n^{l}, n_{1}^{r}) + (0, 0, n_{2}^{r}) \end{matrix}$ where $(n, n^{l}, n_{1}^{r})$ is a negative number and $(0, 0, n_{2}^{r})$ is a LR zero fuzzy number and regarded as a positive(negative) one. Thus $\begin{matrix} \tilde{m} \times \tilde{n} \\ = \tilde{m} \times ((n, n^{l}, n_{1}^{r}) + (0, 0, n_{2}^{r})) \\ = (m, m^{l}, m^{r}) \times (n, n^{l}, n_{1}^{r}) + (m, m^{l}, m^{r}) \times (0, 0, n_{2}^{r}) \\ = (mn, - {mn}_{1}^{r} - {nm}^{r}, - {mn}^{l} - {nm}^{l}) + (0, - {mn}_{2}^{r}, 0) \\ = (mn, - {mn}^{r} - {nm}^{r}, - {mn}^{l} - {nm}^{l}) . \end{matrix}$

It is completed the proof.

Remark 3.1. From the Theorem 3.1. and the Theorem 3.2., we know that when a positive (or negative) LR fuzzy number is multiplied by a near-zero LR fuzzy number, it is the positivity or negativity of the mean value of the near-zero fuzzy number has the decisive effect. When the median value of near-zero fuzzy number is positive, the result is consistent with the multiplication rule of a positive (or negative) LR fuzzy number and a positive LR fuzzy number. When the mean value of near-zero fuzzy number is negative, the result is consistent with the multiplication rules of a positive (or negative) LR fuzzy number and a negative LR fuzzy number.

Now we give conceptions of near zero LR fuzzy matrix and its basic operations.

Definition 3.1. For two LR near zero fuzzy numbers matrices $\tilde{A} = (A, A^{l}, A^{r})$ and $\tilde{B} = (B, B^{l}, B^{r})$ , we have

(1) Addition $\begin{matrix} \begin{matrix} \tilde{A} + \tilde{B} & = (A, A^{l}, A^{r}) + (B, B^{l}, B^{r}) \\ = (A + B, A^{l} + B^{l}, A^{r} + B^{r}) . \end{matrix} \end{matrix}$

(2) Subtraction $\begin{matrix} \begin{matrix} \tilde{A} - \tilde{B} & = (A, A^{l}, A^{r}) - (B, B^{l}, B^{r}) \\ = (A - B, A^{l} + B^{r}, A^{r} + B^{l}) . \end{matrix} \end{matrix}$ (3) Scalar multiplication $\begin{matrix} \begin{matrix} λ \tilde{A} & = λ (A, A^{l}, A^{r}) \\ ≅ {\begin{matrix} (λ A, λ A^{l}, λ A^{r}), λ \geq 0, \\ (λ A, - λ A^{r}, - λ A^{l}), λ < 0 . \end{matrix} \end{matrix} \end{matrix}$

(4) Multiplication

Let $\tilde{A} = (A, A^{l}, A^{r})$ to be a positive fuzzy matrix and $\tilde{b} = (b, b^{l}, b^{r})$ to be near zero fuzzy vector with positive (or negative) mean value, then the multiplication of $\tilde{A} \times \tilde{b}$ is defined as follows:

$\tilde{A} \times \tilde{b} ≅ {\begin{matrix} (Ab, {Ab}^{l} + A^{l} b, {Ab}^{r} + A^{r} b), b > 0, \\ (Ab, {Ab}^{l} - A^{r} b, {Ab}^{r} - A^{l} b), b < 0, \end{matrix}$ (5)

3.1 Extended models and its solution

In this subsection we suppose the coefficient matrix $\tilde{A} = (A, A^{l}, A^{r})$ is non negative, i.e., A - A^l ≥ O. We also suppose LR fuzzy vector $\tilde{b} = (b, b^{l}, b^{r})$ is arbitrary one. For simplification, we only find the cases that the unknown fuzzy vector $\tilde{x} = (x, x^{l}, x^{r})$ is near zero and is with positive (or negative) mean value.

(a) When $\tilde{x}$ is a near zero fuzzy vector with positive mean value

Theorem 3.3. Suppose $\tilde{A} = (A, A^{l}, A^{r}), \tilde{x} = (x, x^{l}, x^{r}), \tilde{b} = (b, b^{l}, b^{r})$ . When $\tilde{x}$ is a near zero fuzzy vector with positive mean value, the fuzzy linear system Eq.(2) can be extended into a crisp linear system as follows:

$(\begin{matrix} A & O & O \\ A^{l} & A & O \\ A^{r} & O & A \end{matrix}) (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) .$ (6)

Proof. Let $\tilde{A} = (A, A^{l}, A^{r}), \tilde{x} = (x, x^{l}, x^{r}), \tilde{b} = (b, b^{l}, b^{r})$ . When $\tilde{A} > 0$ and x > 0, by the approximate multiplication Eq.(4), we extend the $\tilde{A} \tilde{x} = \tilde{b}$ into the following $\begin{matrix} \begin{matrix} \tilde{A} \tilde{x} & = (A, A^{l}, A^{r}) (x, x^{l}, x^{r}) \\ = (Ax, {Ax}^{l} + A^{l} x, {Ax}^{r} + A^{r} x) \\ = \tilde{b} = (b, b^{l}, b^{r}) . \end{matrix} \end{matrix}$

It means

${\begin{matrix} Ax = b, \\ {Ax}^{l} + A^{l} x = b^{l}, \\ {Ax}^{r} + A^{r} x = b^{r} . \end{matrix}$ (7)

By the matrix multiplication, the Eqs.(7) can be written as the Eq.(6). It is completed the proof.

When matrix A is a invertible one, the coefficient matrix of model Eq.(5) exists inverse matrix. Thus we can easily obtain the solution of the model Eq.(6) is

$\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) & = {(\begin{matrix} A & O & O \\ A^{l} & A & O \\ A^{r} & O & A \end{matrix})}^{- 1} (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{- 1} & O & O \\ - A^{- 1} A^{l} A^{- 1} & A^{- 1} & O \\ - A^{- 1} A^{r} A^{- 1} & O & A^{- 1} \end{matrix}) (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{- 1} b \\ A^{- 1} (b^{l} - A^{l} A^{- 1} b) \\ A^{- 1} (b^{r} - A^{r} A^{- 1} b) \end{matrix}) . \end{matrix}$ (8)

When matrix A is not a invertible one, the coefficient matrix of model Eq.(5) does not exist inverse matrix. Thus we obtain the minimal solution of the model Eq.(6) is

$\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) & = {(\begin{matrix} A & O & O \\ A^{l} & A & O \\ A^{r} & O & A \end{matrix})}^{†} (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} & O & O \\ - A^{†} A^{l} A^{†} & A^{†} & O \\ - A^{†} A^{r} A^{†} & O & A^{†} \end{matrix}) (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} b \\ A^{†} (b^{l} - A^{l} A^{†} b) \\ A^{†} (b^{r} - A^{r} A^{†} b) \end{matrix}) . \end{matrix}$ (9)

(b) When $\tilde{x}$ is a near zero fuzzy vector with negative mean value

Theorem 3.4. Suppose $\tilde{A} = (A, A^{l}, A^{r}), \tilde{x} = (x, x^{l}, x^{r}), \tilde{b} = (b, b^{l}, b^{r})$ . When $\tilde{x}$ is a near zero fuzzy vector with negative mean value, the fuzzy linear system Eq.(2) can be extended into a crisp linear system as follows:

$(\begin{matrix} A & O & O \\ - A^{r} & A & O \\ - A^{l} & O & A \end{matrix}) (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}),$ (10)

Proof. Let $\tilde{A} = (A, A^{l}, A^{r}), \tilde{x} = (x, x^{l}, x^{r}), \tilde{b} = (b, b^{l}, b^{r})$ . When $\tilde{A} > 0$ and x < 0, by the approximate multiplication Eq.(5), we extend the $\tilde{A} \tilde{x} = \tilde{b}$ into the following $\begin{matrix} \begin{matrix} \tilde{A} \tilde{x} & = (A, A^{l}, A^{r}) (x, x^{l}, x^{r}) \\ = (Ax, {Ax}^{l} - A^{r} x, {Ax}^{r} - A^{l} x) \\ = \tilde{b} = (b, b^{l}, b^{r}) . \end{matrix} \end{matrix}$

It means

${\begin{matrix} Ax = b, \\ {Ax}^{l} - A^{r} x = b^{l}, \\ {Ax}^{r} - A^{l} x = b^{r} . \end{matrix}$ (11)

Using matrix multiplication, the Eqs.(11) can be written as the Eq.(10). It is completed the proof.

When matrix A is a invertible one, the coefficient matrix of model Eq.(5) exists inverse matrix. Thus we can easily obtain the solution of the model Eq.(10) is

$\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) & = {(\begin{matrix} A & O & O \\ - A^{r} & A & O \\ - A^{l} & O & A \end{matrix})}^{- 1} (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{- 1} & O & O \\ A^{- 1} A^{r} A^{- 1} & A^{- 1} & O \\ A^{- 1} A^{l} A^{- 1} & O & A^{- 1} \end{matrix}) (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{- 1} b \\ A^{- 1} (b^{l} + A^{r} A^{- 1} b) \\ A^{- 1} (b^{r} + A^{l} A^{- 1} b) \end{matrix}) . \end{matrix}$ (12)

When matrix A is not a invertible one, the coefficient matrix of model Eq.(5) does not exist inverse matrix. Thus we obtain the minimal solution of the model Eq.(10) is

$\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) & = {(\begin{matrix} A & O & O \\ - A^{r} & A & O \\ - A^{l} & O & A \end{matrix})}^{†} (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} & O & O \\ A^{†} A^{r} A^{†} & A^{†} & O \\ A^{†} A^{l} A^{†} & O & A^{†} \end{matrix}) (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} b \\ A^{†} (b^{l} + A^{r} A^{†} b) \\ A^{†} (b^{r} + A^{l} A^{†} b) \end{matrix}) . \end{matrix}$ (13)

3.2 Fuzzy solution of PFFLS

However, the solution vector $\tilde{x}$ may still not be an appropriate LR fuzzy numbers vector except for x^l ≥ 0, x^r ≥ 0. Now we give the definition of LR fuzzy solution to the Eq.(2) as follows:

Definition 3.2. Let $\tilde{x} = (x, x^{l}, x^{r})$ . If (x, x^l, x^r) is the minimal solution of the Eq.(6) or (10), such that x^l ≥ 0, x^r ≥ 0 and x - x^l < 0, x + x^r > 0, we call $\tilde{x} = (x, x^{l}, x^{r})$ is a strong near zero LR fuzzy minimal solution of fuzzy linear Eq.(2). Otherwise, the $\tilde{x} = (x, x^{l}, x^{r})$ is said to a weak near zero LR fuzzy minimal solution of fuzzy linear system Eq.(2) given by $\begin{matrix} \tilde{x} = \tilde{x_{i}}, \end{matrix}$ where

$\begin{matrix} {\tilde{x}}_{i} & = {\begin{matrix} (x_{i}, x_{i}^{l}, x_{i}^{r}), & x_{i}^{l} > 0, & x_{i}^{r} > 0, \\ (x_{i}, 0, max {- x_{i}^{l}, x_{i}^{r}}), & x_{i}^{l} < 0, & x_{i}^{r} > 0, \\ (x_{i}, max {x_{i}^{l}, - x_{i}^{r}}, 0,), & x_{i}^{l} > 0, & x_{i}^{r} < 0, \\ (x_{i}, - x_{i}^{l}, - x_{i}^{r}), & x_{i}^{l} < 0, & x_{i}^{r} < 0 . \end{matrix} \\ i = 1, \dots, n . \end{matrix}$ (14)

The following Theorem shows that when the positive fully fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ exist a strong near zero LR fuzzy solution.

Theorem 3.5. Let (x, x^l, x^r) to be the minimal solution of the Eq.(6) or (10), then

(a) If A^† (b^l - A^lA^†b) ≥0 and A^† (b^r - A^rA^†b) ≥0, the positive fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong LR fuzzy solution. If A^†b > 0 and A^†b ≥ A^† (b^l - A^lA^†b), the positive fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong non negative LR fuzzy solution. Moreover, if the conditions A^†b > 0 and A^†b ≤ A^† (b^l - A^lA^†b) hold, then the positive fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong near zero LR fuzzy solution with positive mean value as

$\begin{matrix} \tilde{x} & = (x, x^{l}, x^{r}) \\ = (A^{†} b, A^{†} (b^{l} - A^{l} A^{†} b), A^{†} (b^{r} - A^{r} A^{†} b)) . \end{matrix}$ (15)

(b) If A^† (b^l + A^rA^†b) ≥0 and A^† (b^r + A^lA^†b) ≥0, the positive fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong LR fuzzy solution. If A^†b < 0 and A^†b + A^† (b^r + A^lA^†b) <0, the positive fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong non positive LR fuzzy solution. Moreover, if the conditions A^†b < 0 and A^†b + A^† (b^r + A^lA^†b) >0 hold, then the positive fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong near zero LR fuzzy solution with negative mean value as

$\begin{matrix} \tilde{x} & = (x, x^{l}, x^{r}) \\ = (A^{†} b, A^{†} (b^{l} + A^{r} A^{†} b), A^{†} (b^{r} + A^{l} A^{†} b)) . \end{matrix}$ (16)

Proof. The proof of (a) and (b) are straight forward. We only consider the proof of (a) here.

Since $\tilde{A}$ is a positive fuzzy matrix, its mean matrix A must be positive. From the Eq.(8) $\begin{matrix} \begin{matrix} \tilde{x} & = (x, x^{l}, x^{r}) \\ = (A^{†} b, A^{†} (b^{l} - A^{l} A^{†} b), A^{†} (b^{r} - A^{r} A^{†} b)), \end{matrix} \end{matrix}$ we know the fuzzy vector $\tilde{x}$ is an appropriate LR fuzzy numbers vector except for x^l ≥ 0, x^r ≥ 0 if and only if A^† (b^l - A^lA^†b) ≥0 and A^† (b^r - A^rA^†b) ≥0.

When A^†b > 0 and A^†b ≥ A^† (b^l - A^lA^†b), it means that x > 0 and x - x^l > 0. Thus we know that the fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ has a strong non negative LR fuzzy solution.

On the other hand, the conditions A^†b > 0 and A^†b ≤ A^† (b^l - A^lA^†b) are equivalent to that the conditions x > 0 and both x - x^l < 0 and x + x^r > 0 hold. So the fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ exists a strong near zero fuzzy solution by the Definition 3.2. It is completed the proof.

Remark 3.2. When the coefficient matrix of the fully fuzzy linear system is a negative LR fuzzy matrix, we can also consider its near zero fuzzy solution. In Theorem 3.2, we have put up multiplication rules for a negative fuzzy number and a near zero fuzzy number. Therefore, with the help of matrix operation, we can give the multiplication of a negative LR fuzzy matrix and a near-zero fuzzy vector.

Theorem 3.6. Let $\tilde{A} = (A, A^{l}, A^{r})$ to be a negative fuzzy numbers matrix and $\tilde{x} = (x, x^{l}, x^{r})$ to be a near zero fuzzy numbers vector, Then

$\tilde{A} \times \tilde{x} ≅ {\begin{matrix} (Ax, A^{l} x - {Ax}^{r}, A^{r} x - {Ax}^{l}), x > 0, \\ (Ax, - {Ax}^{r} - A^{r} x, - {Ax}^{l} - A^{l} x), x < 0 . \end{matrix}$ (17)

Thus, we can convert the fully fuzzy linear system Eq.(2) to the following crisp linear systems with more high orders when the coefficient matrix of the fully fuzzy linear system is a negative LR fuzzy matrix.

Theorem 3.7. Suppose $\tilde{A} = (A, A^{l}, A^{r}), \tilde{x} = (x, x^{l}, x^{r}), \tilde{b} = (b, b^{l}, b^{r})$ . When $\tilde{x}$ is a near zero fuzzy vector with positive mean value, the fuzzy linear system Eq.(2) can be extended into a crisp linear system as follows:

$(\begin{matrix} A & O & O \\ A^{r} & - A & O \\ A^{l} & O & - A \end{matrix}) (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} b \\ b^{r} \\ b^{l} \end{matrix}) .$ (18)

When $\tilde{x}$ is a near zero fuzzy vector with negative mean value, the fuzzy linear system Eq.(2) can be extended into a crisp linear system as follows:

$(\begin{matrix} A & O & O \\ - A^{l} & - A & O \\ - A^{r} & O & - A \end{matrix}) (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} b \\ b^{r} \\ b^{l} \end{matrix}) .$ (19)

By the same way with solving the model Eq.(6) and Eq.(10), we can obtain the minimal solution of the model Eq.(18) and Eq.(19) are

$\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) & = {(\begin{matrix} A & O & O \\ A^{r} & - A & O \\ A^{l} & O & - A \end{matrix})}^{†} (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} & O & O \\ A^{†} A^{r} A^{†} & - A^{†} & O \\ A^{†} A^{l} A^{†} & O & - A^{†} \end{matrix}) (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} b \\ - A^{†} (b^{l} - A^{r} A^{†} b) \\ - A^{†} (b^{r} - A^{l} A^{†} b) \end{matrix}) \end{matrix}$ (20) and

$\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) & = {(\begin{matrix} A & O & O \\ - A^{l} & - A & O \\ - A^{r} & O & - A \end{matrix})}^{†} (\begin{matrix} b \\ b^{l} \\ b^{r} \end{matrix}) \\ = (\begin{matrix} A^{†} & O & O \\ - A^{†} A^{l} A^{†} & - A^{†} & O \\ - A^{†} A^{rl} A^{†} & O & - A^{†} \end{matrix}) (\begin{matrix} b \\ b^{r} \\ b^{l} \end{matrix}) \\ = (\begin{matrix} A^{†} b \\ - A^{†} (b^{l} + A^{l} A^{†} b) \\ - A^{†} (b^{r} + A^{r} A^{†} b) \end{matrix}) . \end{matrix}$ (21) In the similar way as Theorem 3.5, we can also analyze the sufficient condition for the existence of strong near zero fuzzy solution of the negative fully fuzzy linear system $\tilde{A} \tilde{x} = \tilde{b}$ . Here we omit its analysis now that the proof is easy and simple.

4 Numerical examples

Example 4.1. Consider the following positive fully fuzzy linear system

$\begin{matrix} (\begin{matrix} (1, 0.5, 0.3) & (2, 0.4, 0.5) \\ (3, 0.25, 0.4) & (2, 0.3, 0.25) \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) = (\begin{matrix} (1, 2.38, 1.95) \\ (3, 4.90, 3.00) \end{matrix}) . \end{matrix}$

In above equation, $\begin{matrix} A = (\begin{matrix} 1 & 2 \\ 3 & 2 \end{matrix}), A^{l} = (\begin{matrix} 0.5 & 0.4 \\ 0.25 & 0.3 \end{matrix}), A^{r} = (\begin{matrix} 0.3 & 0.5 \\ 0.4 & 0.25 \end{matrix}) \end{matrix}$ and $\begin{matrix} b = (\begin{matrix} 1 \\ 3 \end{matrix}), b^{l} = (\begin{matrix} 2.38 \\ 4.90 \end{matrix}), b^{r} = (\begin{matrix} 1.95 \\ 3.00 \end{matrix}) . \end{matrix}$

Suppose the fuzzy linear system has a near zero fuzzy vector with positive mean value. From calculation formula Eq.(5), we obtain its solution is $\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} A^{- 1} b \\ A^{- 1} (b^{l} - A^{l} A^{- 1} b) \\ A^{- 1} (b^{r} - A^{r} A^{- 1} b) \end{matrix}) = (\begin{matrix} 1 \\ 0 \\ 1.2000 \\ 0.8000 \\ 0.5000 \\ 0.4000 \end{matrix}) . \end{matrix}$

It means the LR fuzzy solution of the original fuzzy linear system is $\begin{matrix} (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) = (\begin{matrix} (1, 0.2000, 0.5000) \\ (0, 0.8000, 0.4000) \end{matrix}), \end{matrix}$ which admits a strong near zero LR fuzzy solution by Definition 3.2.

Suppose the fuzzy linear system has a near zero fuzzy vector with negative mean value. By the formula Eq.(10), we obtain its solution is $\begin{matrix} (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) = (\begin{matrix} (1, 0.7126, 0.4490) \\ (0, 0.1713, 1.2641) \end{matrix}), \end{matrix}$ it is a strong LR fuzzy solution of the original fuzzy linear system. But, $\begin{matrix} x - x^{l} = (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) - (\begin{matrix} x_{1}^{r} \\ x_{2}^{r} \end{matrix}) = (\begin{matrix} 0.2874 \\ - 0.1713 \end{matrix}), \end{matrix}$ i.e., $\tilde{x}$ is a not near zero fuzzy vector.

From the above analysis, we assert that the original fuzzy linear system only exists a near zero fuzzy solution with positive mean value.

Example 4.2. Consider the following positive fully fuzzy linear system

$\begin{matrix} (\begin{matrix} (1, 0.25, 0.3) & (2, 0.4, 0.5) \\ (2, 0.4, 0.2) & (4, 0.2, 0.25) \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) \\ = (\begin{matrix} (- 3.75, 3.8650, 4.7025) \\ (- 7.00, 5.4975, 9.7200) \end{matrix}) . \end{matrix}$

In equation, $\begin{matrix} A = (\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}), A^{l} = (\begin{matrix} 0.25 & 0.4 \\ 0.4 & 0.2 \end{matrix})), A^{r} = (\begin{matrix} 0.3 & 0.5 \\ 0.2 & 0.25 \end{matrix}) \end{matrix}$

and $\begin{matrix} b = (\begin{matrix} - 3.75 \\ - 7.00 \end{matrix}), b^{l} = (\begin{matrix} 3.8650 \\ 5.4915 \end{matrix}), b^{r} = (\begin{matrix} 4.7025 \\ 9.7200 \end{matrix}), \end{matrix}$ the mean matrix A of the coefficient matrix of the original fuzzy linear system is singular.

Suppose the fuzzy linear system has a near zero fuzzy vector with positive mean value. From calculation formula Eq.(6), we obtain its approximate solution is $\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} A^{†} b \\ A^{- 1} (b^{l} - A^{l} A^{†} b) \\ A^{- 1} (b^{r} - A^{r} A^{†} b) \end{matrix}) = (\begin{matrix} - 0.700 \\ - 1.400 \\ 26.865 \\ - 9.713 \\ - 66.875 \\ 37.020 \end{matrix}) . \end{matrix}$

It means the near zero LR fuzzy solution of the original fuzzy linear system is $\begin{matrix} (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) = (\begin{matrix} (- 0.70, 26.87, 0.000) \\ (- 1.40, 0.000, 37.02) \end{matrix}), \end{matrix}$ which is not near zero LR fuzzy numbers vector by Definition 3.2.

Suppose the fuzzy linear system has a near zero fuzzy vector with negative mean value. By the formula Eq.(10), we obtain its solution is $\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} A^{†} b \\ A^{- 1} (b^{l} + A^{r} A^{†} b) \\ A^{- 1} (b^{r} + A^{l} A^{†} b) \end{matrix}) = (\begin{matrix} - 1.5 \\ - 1 \\ 0.800 \\ 0.750 \\ 2.250 \\ 1.400 \end{matrix}) . \end{matrix}$

it means the LR fuzzy solution of the original fuzzy linear system is $\begin{matrix} (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}) = (\begin{matrix} (- 1.5, 0.80, 2.25) \\ (- 1.0, 0.75, 1.40) \end{matrix}) . \end{matrix}$

Since $\begin{matrix} x - x^{l} = (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) + (\begin{matrix} x_{1}^{l} \\ x_{2}^{l} \end{matrix}) = (\begin{matrix} - 2.30 \\ - 1.75 \end{matrix}) < 0, \end{matrix}$ $\begin{matrix} x + x^{r} = (\begin{matrix} x_{1} \\ x_{2} \end{matrix}) + (\begin{matrix} x_{1}^{r} \\ x_{2}^{r} \end{matrix}) = (\begin{matrix} 0.750 \\ 0.400 \end{matrix}) > 0, \end{matrix}$ we assert that the original fuzzy linear system exists a near zero fuzzy solution with negative mean value.

Example 4.3. Consider the following negative fully fuzzy linear system

$\begin{matrix} (\begin{matrix} (- 1.0, 0.50, 0.60) & (- 2, 1.0, 1.2) & (- 1, 0.4, 0.5) \\ (- 1.5, 0.40, 1.00) & (- 1, 0.8, 0.8) & (- 2, 0.3, 0.6) \\ (- 0.8, 0.25, 0.25) & (- 3, 0.5, 1.5) & (- 1.0.5, 0.8) \end{matrix}) (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}) \\ = (\begin{matrix} (- 3.00, 3.65, 5.30) \\ (- 3.25, 4.25, 6.00) \\ (- 4.30, 4.15, 6.68) \end{matrix}) . \end{matrix}$

In equation, $\begin{matrix} \begin{matrix} A = (\begin{matrix} - 1 & - 2 & - 1 \\ - 1.5 & - 1 & - 2 \\ - 0.8 & - 3 & - 1 \end{matrix}), A^{l} = (\begin{matrix} 0.5 & 1 & 0.4 \\ 0.4 & 0.8 & 0.3 \\ 0.25 & 0.5 & 0.5 \end{matrix}), \\ A^{r} = (\begin{matrix} 0.6 & 1.2 & 0.5 \\ 1.0 & 0.8 & 0.6 \\ 0.25 & 1.5 & 0.8 \end{matrix}) \end{matrix} \end{matrix}$ and $\begin{matrix} b = (\begin{matrix} - 3.00 \\ - 3.25 \\ - 4.30 \end{matrix}), b^{l} = (\begin{matrix} 3.650 \\ 4.250 \\ 4.155 \end{matrix}), b^{r} = (\begin{matrix} 5.30 \\ 6.00 \\ 6.68 \end{matrix}), \end{matrix}$ the mean matrix A of the coefficient matrix of the original fuzzy linear system is non singular.

Suppose the fuzzy linear system has a near zero fuzzy vector with positive mean value. From calculation formula Eq.(20), we obtain its approximate solution is $\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} A^{†} b \\ A^{- 1} (b^{l} - A^{l} A^{†} b) \\ A^{- 1} (b^{r} - A^{r} A^{†} b) \end{matrix}) = (\begin{matrix} 1.00 \\ 0.75 \\ 0.50 \\ 1.20 \\ 0.80 \\ 0.75 \\ 0.60 \\ 0.50 \\ 0.80 \end{matrix}) . \end{matrix}$

It means the near zero LR fuzzy solution of the original fuzzy linear system is $\begin{matrix} (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}) = (\begin{matrix} (1.00, 1.20, 0.60) \\ (0.75, 0.80, 0.50) \\ (0.50, 0.75, 0.80) \end{matrix}), \end{matrix}$ it admits is a strong near zero LR fuzzy numbers solution by Definition 3.2 since that { $\begin{matrix} x - x^{l} = (\begin{matrix} - 0.2 \\ - 0.05 \\ - 0.25 \end{matrix}) < 0, x + x^{r} = (\begin{matrix} 1.10 \\ 1.25 \\ 1.30 \end{matrix}) > 0 . \end{matrix}$ ∥Suppose the fuzzy linear system has a near zero fuzzy vector with negative mean value. By the formula Eq.(21), we obtain its solution is $\begin{matrix} (\begin{matrix} x \\ x^{l} \\ x^{r} \end{matrix}) = (\begin{matrix} A^{†} b \\ A^{- 1} (b^{l} + A^{l} A^{- 1} b) \\ A^{- 1} (b^{r} + A^{r} A^{- 1} b) \end{matrix}) = (\begin{matrix} 1.00 \\ 0.75 \\ 0.50 \\ 5.38 \\ 0.60 \\ - 1.68 \\ 5.10 \\ 1.21 \\ - 0.48 \end{matrix}) . \end{matrix}$ it means the LR fuzzy solution of the original fully fuzzy linear system is $\begin{matrix} (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}) = (\begin{matrix} (1.00, 5.38, 5.10) \\ (0.75, 0.60, 1.21) \\ (0.50, 0.48, 1.60) \end{matrix}), \end{matrix}$ which is a weak LR fuzzy solution of the original negative fully fuzzy linear system. So we assert that the original fuzzy linear system does not exist a near zero fuzzy solution with negative mean value.

5 Conclusion

In this work the near zero solution to a class of positive fully fuzzy linear systems is investigated. According to the mean value of the unknown fuzzy solution vector $\tilde{x}$ is positive or negative, two computing models are established and respective expression of the solution to fully fuzzy linear system are derived and the sufficient condition for the existence of strong fuzzy solution are derived correspondingly. Illustrating examples verified the correctness of our results. Since near zero fuzzy numbers are more extensive than pure positive fuzzy numbers or negative fuzzy numbers in operation and application, our work enriched the computation theory of fuzzy linear systems.

References

Abbasbandy

, Otadi

and Mosleh

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

Allahviranloo

and Ghanbari

, A new approach to obtain algebraicsolution 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

and Mikaeilvand

, Non zero fuzzy solutions of the fully fuzzy linear systems, Applied Mathematics and Computation 2 (2011), 271–282.

Allahviranloo

, Mikaeilvand

, Kiani

N.A.

and Shabestari

R.M.

, Signed decomposition of fully fuzzy linear systems, Applications and Applied Mathematics: An International Journal 1 (2008), 77–88.

Asady

, Abbasbandy

and Alavi

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

Babbar

, Kumar

Abhinav Bansal , Solving fully fuzzy linearsystem with arbitrary triangular fuzzy numbers, Soft Computing 17 (2013), 691–702.

Basaran

M.A.

, Calculating fuzzy inverse matrix using fuzzy linear equation system, Applied Soft Computing 12 (2012), 1810–1813.

Chen

, Guo

X.B.

Solving Complex 445 LR-Trapezoidal Fuzzy Linear System, The 17th International Conference on Computational Intelligence and Security, 2021, 23–25.

10.

Daud

W.S.W.

, Ahmad

and Malkawi

, A modification of fuzzy arithmetic operators for solving near zero fully fuzzy matrix equation, TELKOMNIKA Telecommun Comput EI Control 19(2021), 583–598.

11.

Dehghan

, Hashemi

and Ghatee

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

12.

Dubois

and Prade

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

13.

Friedman

, Ma

and Kandel

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

14.

Ghanbari

, Solutions of fuzzy LR algebraic linear systems using linear programs, Applied Mathematical Modelling 39 (2015), 5164–5173.

15.

Ghanbari

, Allahviranloo

, Nuraei

and Pedrycz

, A new effective approximate multiplication operation on LR fuzzy numbers and its application, Soft Computing 26 (2022), 4103–4113.

16.

Goetschel

and Voxman

, Elementary calculus, Fuzzy Sets and Systems 18 (1986), 31–43.

17.

Gong

Z.T.

and Guo

X.B.

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

18.

Gong

Z.T.

, Guo

X.B.

and Liu

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

19.

Guo

X.B.

and Shang

D.Q.

, Solving LR fuzzy linear matrix equation, Iranian Journal of Fuzzy Systems 16 (2019), 33–44.

20.

Guo

X.B.

, Wu

L.J.

Inconsistent LR fuzzy matrix equation, Mathematical Problems in Engineering 2020.

21.

Guo

X.B.

, Zhang

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

22.

Guo

X.B.

and Han

Y.L.

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

23.

Liu

, Guo

X.B.

Fuzzy least squares approximation using fuzzy polynomial, Mathematical Problems in Engineering 2021.

24.

, Friedman

and Kandel

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

25.

Malkawi

, Ahmad

and Ibrahim

, An algorithm for a positive solution of arbitrary fully fuzzy linear system, Computational Mathematics and modeling 26 (2015), 436–465.

26.

Mosleh

and Otadi

, A discussion on calculating fuzzy inverse matrix using fuzzy linear equation system, Applied Soft Computing 28 (2015), 511–513.

27.

Nahmias

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

28.

Nuraei

, Allahviranloo

and Ghanbari

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

29.

Puri

M.L.

and Ralescu

D.A.

, Differentials for fuzzy functions, Journal of Mathematics Analyisis and Application 91 (1983), 552–558.

30.

Wang

G.X.

, Shi

and Xie

Y.Y.

, Two-dimensional discrete fuzzy numbers and applications, Information Sciences 326(2016), 258–269.

31.

Wang

G.X.

and Li

, Approximations of fuzzy numbers by step type fuzzy numbers, Fuzzy Sets and Systems 310 (2017), 47–59.

32.

C.X.

and Ma

, Embedding problem of fuzzy number space: Part I, Fuzzy Sets and Systems 44 (1991), 33–38.

33.

C.X.

and Ma

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

34.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(1965), 338–353.