Fuzzy linear systems with the two-dimension fuzzy data 1

Abstract

As a key theoretical basis of fuzzy analysis, a new representation of the two-dimension fuzzy numbers is proposed in this paper. We also present a new method for solving the fuzzy system of linear equations with the two-dimension fuzzy data based on the new representation of the two-dimension fuzzy numbers, which may translate a fuzzy system of linear equations into two real systems of linear equations, one is an n × n real system of linear equations and the other is a (2n) × (2n) real system of linear equations, where the coefficients, the right hand term and the unknown term of the (2n) × (2n) real systems are all positive real numbers. Accordingly, we compare the classical methods with the technique on the basis of the study of the two-dimensional fuzzy linear system proposed in this paper for an one-dimensional fuzzy linear system, and the results shows that they have the same solutions. Finally, the conditions of the existence of the solutions for a fuzzy system of linear equations are discussed, and the examples are given to show the efficiency and effectiveness of the method investigated in this article.

Keywords

Fuzzy numbers two-dimension fuzzy numbers fuzzy linear system fuzzy system of linear equations

1 Introduction

The system of linear equations plays a vital role in real-life problems such as optimization, current flow, economics, finance, and engineering. In reality some problems in the process of the mathematical modeling tend to appear a system of linear equations with uncertain parameters, which can also be explained as fuzzy numbers in some sense. Thus, as a very important topic in the field of fuzzy mathematics, fuzzy system of linear equations has been investigated by various authors using different approaches. In 1998, Friedman et al. proposed a general model for solving an n × n fuzzy linear system whose coefficients matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector [22]. They use the embedding method given in [31] and replace the original n × n fuzzy linear system by a (2n) × (2n) crisp function linear system. Based on this method, Allahviranloo and Abbasbandy et al. used various numerical methods to solve fuzzy linear systems [1 –7], such as the iterative Jacobi and Gauss-Siedel method, the Adomian method, the successive over relaxation method, the LU decomposition method and the steepest decent method. In 2010, Ezzati developed a new method for solving fuzzy linear systems by using embedding method and replaced an n × n fuzzy linear system by two n × n crisp linear systems [21]. But in 2011, Allahviranloo have been showed by an interesting counter example that the so-called weak solution defined by Friedman et al. [22], is not always a fuzzy number vector [11]. Hence, a simple and practical method to obtain fuzzy symmetric solutions of fuzzy linear systems is proposed in [8]. In 2012, Allahviranloo et al. obtained the fuzzy exact solutions of fuzzy linear systems by a new metric and proposed a new approach for solving fuzzy linear systems by a new concept, namely “interval inclusion linear system" in [9, 10]. For more research papers see [12–19 , 32].

Although several methods have been proposed to solve the one-dimensional fuzzy linear systems, to the best of our knowledge, there are few known results about fuzzy linear systems based on the high-dimensional fuzzy data. In this paper, a new representation of the two-dimension fuzzy numbers is first proposed. Accordingly, we study the arithmetic operations of the two-dimension fuzzy numbers, which are the operations of transforming the operations of the two-dimension fuzzy numbers into the operations of the positive fuzzy numbers. At the same time, we also present a new method for solving the two-dimension fuzzy system of linear equations based on the new representation of the two-dimension fuzzy numbers, which may translate the fuzzy system of linear equations into two real systems of linear equations, one is an n × n real system of linear equations and the other is a (2n) × (2n) real system of linear equations, where the coefficients, the right hand term and the unknown term of the (2n) × (2n) real systems are all positive real numbers. The conditions of the existence of the solutions for the two-dimension fuzzy system of linear equations are considered and examples are given to show efficiency and effectiveness of the method.

Since the two-dimensional fuzzy number is an extension of the one-dimensional fuzzy number, we consider the degradation of the two-dimensional fuzzy linear system to the one-dimensional fuzzy linear system by using the same method on the basis of the study of the two-dimensional fuzzy linear system. A new representation of fuzzy numbers is first also proposed. We also study the arithmetic operations of fuzzy numbers, which are the operations of transforming the operations of fuzzy numbers into the operations of positive fuzzy numbers. Subsequently, we present a new method for solving the fuzzy system of linear equations based on the new representation of fuzzy numbers, which may translate the fuzzy system of linear equations into two real systems of linear equations, one is an n × n real system of linear equations and the other is a (2n) × (2n) real system of linear equations, where the coefficients, the right hand term and the unknown term of the (2n) × (2n) real systems are all positive real numbers. The conditions of the existence of the solutions for the fuzzy system of linear equations are considered and several examples are given to show efficiency and effectiveness of the method.

It is worth noting that the approach proposed by Friedman et al. is to convert the original n × n fuzzy linear system into a (2n) × (2n) crisp linear system with parameter r, the coefficient matrix of the crisp linear system is a block matrix with the positive part of the coefficient matrix of the original fuzzy linear system as the main diagonal and the negative part as the negative diagonal. Our approach is to convert the original one-dimensional or two-dimensional problem into an n × n crisp linear system and a (2n) × (2n) crisp linear system with parameter r, where the coefficient matrix of n × n crisp linear system is the coefficient matrix of the original one-dimensional or two-dimensional fuzzy linear systems and the coefficient matrix of the (2n) × (2n) crisp linear system is also a block matrix with the positive part of the coefficient matrix of the original fuzzy linear system as the main diagonal and the negative part as the negative diagonal which is completely consistent with the coefficient matrix of the (2n) × (2n) crisp linear system in the method proposed by Friedman et al. So our method can be seen as the development of the method proposed by Friedman et al. in the two-dimensional fuzzy linear system.

The remainder of the paper is organised as follows. In Section 2, we briefly introduce the necessary notions related to fuzzy numbers. In Section 3, a new representation of the two-dimension fuzzy numbers is first proposed and the arithmetic operations of two-dimension fuzzy number are developed based on the new representation of two-dimension fuzzy numbers. In Section 4, a new method for solving the two-dimension fuzzy system of linear equations is presented based on the new representation of the two-dimension fuzzy numbers, the conditions of the existence of the solutions for the two-dimension fuzzy system of linear equations are considered and the examples are given to show efficiency and effectiveness of the method. In Section 5, a new representation of one-dimensional fuzzy numbers is first proposed and the arithmetic operations of one-dimensional fuzzy number are developed based on the new representation of one-dimensional fuzzy numbers. In Section 6, a new method for solving the one-dimensional fuzzy system of linear equations is presented based on the new representation of one-dimensional fuzzy numbers. Finally, the conditions of the existence of the solutions for the one-dimensional fuzzy system of linear equations are considered and the examples are given to show efficiency and effectiveness of the method in Section 7 and the conclusions are given in Section 8.

2 Preliminaries

Given a fuzzy subset $\tilde{u}$ on the n-dimensional Euclidean space $ℝ^{n},$ the r-level set of $\tilde{u}$ is defined by $[\tilde{u}]^{r} = {x \in ℝ^{n} : \tilde{u} (x) \geq r}$ for r ∈ (0, 1] and $[\tilde{u}]^{0} = \bar{{x \in ℝ^{n} : \tilde{u} (x) > 0}}$ for r = 0 . The n-dimension fuzzy number space Eⁿ is the set of such $\tilde{u}$ satisfying the following properties:

$(1) \tilde{u}$ is normal, i.e. $\exists x \in ℝ^{n}$ with $\tilde{u} (x) = 1;$

$(2) \tilde{u}$ is convex fuzzy set (i.e. $\tilde{u} (λ x + (1 - λ) y) \geq min {\tilde{u} (x), \tilde{u} (y)}, \forall λ \in [0, 1], x, y \in ℝ^{n});$

$(3) \tilde{u}$ is upper semi-continuous on $ℝ^{n}$ ;

(4) supp $\tilde{u} = \bar{{x \in ℝ^{n} | \tilde{u} (x) > 0}}$ is compact.

A one-dimensional fuzzy number is also called a fuzzy number.

We well known that Eⁿ is closed under addition and scalar multiplication [20] . For $\tilde{u}, \tilde{v} \in E^{n}, k \in ℝ,$ the addition $\tilde{u} + \tilde{v}$ and scalar multiplication $k \tilde{u}$ of the n-dimensional fuzzy numbers $\tilde{u}$ and $\tilde{v}$ can be defined via level sets as follows: $[\tilde{u} + \tilde{v}]^{r} = [\tilde{u}]^{r} + [\tilde{v}]^{r}, [k \tilde{u}]^{r} = k [\tilde{u}]^{r} .$ If there exists a n-dimensional fuzzy number $\tilde{w}$ such that $\tilde{u} = \tilde{v} + \tilde{w},$ then $\tilde{w}$ is called H-difference of the n-dimensional fuzzy numbers $\tilde{u}$ and $\tilde{v},$ denoted by $\tilde{u} - \tilde{v} .$

Theorem 2.1. ([26, 27]) If $\tilde{u} \in E^{n},$ then

$(1) [\tilde{u}]^{r}$ is a nonempty compact convex subset of $ℝ^{n}$ for any r ∈ [0, 1] .

$(2) [\tilde{u}]^{r_{1}} \subseteq [\tilde{u}]^{r_{2}},$ whenever 0 ≤ r₂ ≤ r₁ ≤ 1 .

(3) if r_n > 0 and r_n converging to r ∈ (0, 1] is nondecreasing, then $\cap_{n = 1}^{\infty} [u]^{r_{n}} = [u]^{r} .$

Conversely, suppose that for any r ∈ (0, 1] , an $A^{r} \subseteq ℝ^{n}$ exists that satisfies (1)-(3) above, then a unique u ∈ Eⁿ exists such that $[\tilde{u}]^{r} = A^{r}, r \in [0, 1], [\tilde{u}]^{0} = \bar{\cup_{r \in (0, 1]} [\tilde{u}]^{r}} \subseteq A^{0} .$

There is also a representation theorem for the one-dimensional fuzzy numbers.

Lemma 2.2. ([23]) Let $\tilde{u} \in E^{1}$ and $[\tilde{u}]^{r} = [u_{r}^{-}, u_{r}^{+}] .$ Then the following conditions are satisfied:

$(1) u_{r}^{-}$ is a bounded left continuous non-decreasing function on (0, 1] ;

$(2) u_{r}^{+}$ is a bounded left continuous non-increasing function on (0, 1] ;

$(3) u_{r}^{-}$ and $u_{r}^{+}$ is right at r = 0 ;

$(4) u_{r}^{-} \leq u_{r}^{+} .$

Conversely, if a pair of function a (r) and b (r) satisfy condition (1)-(4), then there exists a unique $\tilde{u} \in E^{1}$ such that $[\tilde{u}]^{r} = [a (r), b (r)]$ for each r ∈ [0, 1] .

If $\tilde{u}$ is a n-dimension fuzzy number such that $[\tilde{u}]^{0} = {0},$ then $\tilde{u}$ is called a trivial n-dimension zero fuzzy number. If $\tilde{u}$ is a n-dimension fuzzy number such that $0 \in [\tilde{u}]^{1}$ and $[\tilde{u}]^{0} \neq {0},$ then $\tilde{u}$ is called a nontrivial n-dimension zero fuzzy number. If $\tilde{u}$ is a fuzzy number and $\forall x \in [\tilde{u}]^{0}$ such that x ≥ 0, then $\tilde{u}$ is called a positive fuzzy number.

3 A new representation of the two-dimensional fuzzy numbers

Theorem 3.1. Let $\tilde{o}$ be a nontrivial two-dimension zero fuzzy number. Then $\tilde{o} = ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ}),$ where $\tilde{R} (φ) = ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, φ)])$ is a fuzzy number for all φ ∈ [0, 2π] .

Proof. If $\tilde{o}$ is a nontrivial two-dimension zero fuzzy number, then by the decompose theorem of the two-dimension fuzzy number, we have $\tilde{o} = ⋃_{r \in [0, 1]} (r^{*} \cap [\tilde{o}]^{r}),$ where r^* is a fuzzy subset on $ℝ^{2}$ whose membership function is a constant function r . Let ρ = ρ (r, φ) , φ ∈ [0, 2π] be the polar coordinates equation of boundary curve of $[\tilde{o}]^{r} .$ Then $[\tilde{o}]^{r} = ⋃_{φ \in [0, 2 π]} ([0, ρ (r, φ)] e^{i φ}),$ where ρ (r, φ) is the polar diameter of intersection point of the boundary curve which encircled $[\tilde{o}]_{r}$ and the ray from round dot making angle φ with the polar axis. Therefore

$\begin{matrix} \tilde{o} & = & ⋃_{r \in [0, 1]} (r^{*} \cap (⋃_{φ \in [0, 2 π]} ([0, ρ (r, φ)] e^{i φ}))) \\ = & ⋃_{r \in [0, 1]} ⋃_{φ \in [0, 2 π]} (r^{*} \cap ([0, ρ (r, φ)] e^{i φ})) \\ = & ⋃_{φ \in [0, 2 π]} ⋃_{r \in [0, 1]} ((r^{*} \cap [0, ρ (r, φ)]) e^{i φ}) \\ = & ⋃_{φ \in [0, 2 π]} ((⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, φ)])) e^{i φ}) . \end{matrix}$ Let $\tilde{R} (φ) = ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, φ)]),$ then $\tilde{o} = ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ})$ and $\tilde{R} (φ) = ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, φ)])$ is a fuzzy number for all φ ∈ [0, 2π] . □

A nontrivial two-dimension zero fuzzy number is shown in Fig. 1, where the shadow part in the figure represents $\tilde{R} (\frac{3 π}{2}) e^{i \frac{3 π}{2}} .$ The r-cut set $[\tilde{o}]_{r}$ of the nontrivial two-dimension zero fuzzy number is shown in Fig. 2, where the bold line represents $[0, ρ (r, \frac{π}{6})] e^{i \frac{π}{6}} .$

Fig.1

A nontrival zero fuzzy number.

Fig.2

r-cut set of nontrival zero fuzzy number.

Theorem 3.2. If $\tilde{u}$ is a two-dimension fuzzy number, then $\tilde{u} = \hat{u} + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ}),$ where $\hat{u}$ is a fuzzy point on $ℝ^{2}$ which its support is u = x₀ + iy₀, its membership is 1 and $x_{0} = \frac{\int \int_{[\tilde{u}]^{1}} xdxdy}{\int \int_{[\tilde{u}]^{1}} dxdy}, y_{0} = \frac{\int \int_{[\tilde{u}]^{1}} ydxdy}{\int \int_{[\tilde{u}]^{1}} dxdy},$ $⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ})$ is the new representation of nontrivial two-dimension zero fuzzy number $\tilde{u} - \hat{u} .$

Proof. Let $\tilde{u}$ be a two-dimension fuzzy number. Assume $\hat{u}$ is a fuzzy point on $ℝ^{2}$ which its support is u = x₀ + iy₀, its membership is 1 and $x_{0} = \frac{\int \int_{[\tilde{u}]^{1}} xdxdy}{\int \int_{[\tilde{u}]^{1}} dxdy}, y_{0} = \frac{\int \int_{[\tilde{u}]^{1}} ydxdy}{\int \int_{[\tilde{u}]^{1}} dxdy} .$ Then $\tilde{u} - \hat{u}$ is a nontrivial two-dimension zero fuzzy number. According to Theorem 3.1, we know that $\tilde{u} - \hat{u} = ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ}) .$ Hence $\tilde{u} = \hat{u} + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ}) .$ □

The two-dimension fuzzy number $\tilde{u} = \hat{u} + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ})$ can also be written as $\tilde{u} = \hat{u} + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ + \frac{k π}{2}) e^{i (φ + \frac{k π}{2})}), k \in ℤ^{+} .$

Theorem 3.3. Let $\tilde{u} = \hat{u} + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ}) .$ Then $k \tilde{u} = {\begin{matrix} k \hat{u} + ⋃_{φ \in [0, 2 π]} ((k \tilde{R} (φ)) e^{i φ}), & k \geq 0, \\ k \hat{u} + ⋃_{φ \in [0, 2 π]} (((- k) \tilde{R} (φ)) e^{i (φ + π)}), & k < 0 . \end{matrix}$

Proof. From proof process of Theorem 3.1, we know $[\tilde{u} - \hat{u}]^{r} = ⋃_{φ \in [0, 2 π]} ([0, ρ (r, φ)] e^{i φ}) .$ Hence $[\tilde{u}]^{r} = u + ⋃_{φ \in [0, 2 π]} ([0, ρ (r, φ)] e^{i φ}) .$ If k ≥ 0, we have $[k \tilde{u}]^{r} = k [\tilde{u}]^{r} = ku + ⋃_{φ \in [0, 2 π]} ([0, k ρ (r, φ)] e^{i φ}) .$ If k < 0, we get $[k \tilde{u}]^{r} = k [\tilde{u}]^{r} = ku + ⋃_{φ \in [0, 2 π]} ([0, (- k) ρ (r, φ)] e^{i (φ + π)}) .$ That is to say, if k ≥ 0, we have $[k \tilde{u} - k \hat{u}]^{r} = ⋃_{φ \in [0, 2 π]} ([0, k ρ (r, φ)] e^{i φ}) .$ If k < 0, we get $[k \tilde{u} - k \hat{u}]^{r} = ⋃_{φ \in [0, 2 π]} ([0, (- k) ρ (r, φ)] e^{i (φ + π)}) .$ Similar to the proof of Theorem 3.1, we have $k \tilde{u} - k \hat{u} = ⋃_{φ \in [0, 2 π]} ((k \tilde{R} (φ)) e^{i φ})$ if k ≥ 0 . i.e. $k \tilde{u} = k \hat{u} + ⋃_{φ \in [0, 2 π]} ((k \tilde{R} (φ)) e^{i φ}) .$ Similar to the proof of Theorem 3.1, when k < 0, we have $\begin{matrix} k \tilde{u} - k \hat{u} & = & ⋃_{r \in [0, 1]} (r^{*} \cap (⋃_{φ \in [0, 2 π]} ([0, (- k) ρ (r, φ)] e^{i (φ + π)}))) \\ = & ⋃_{r \in [0, 1]} {⋃_{φ \in [0, 2 π]} (r^{*} \cap ([0, (- k) ρ (r, φ)] e^{i (φ + π)}))} \\ = & ⋃_{φ \in [0, 2 π]} {⋃_{r \in [0, 1]} (r^{*} \cap ([0, (- k) ρ (r, φ)] e^{i (φ + π)}))} \\ = & ⋃_{φ \in [0, 2 π]} {⋃_{r \in [0, 1]} ((r^{*} \cap ([0, (- k) ρ (r, φ)])) e^{i (φ + π)})} \\ = & ⋃_{φ \in [0, 2 π]} {(⋃_{r \in [0, 1]} (r^{*} \cap ([0, (- k) ρ (r, φ)]))) e^{i (φ + π)}} \\ = & ⋃_{φ \in [0, 2 π]} (((- k) \tilde{R} (φ)) e^{i (φ + π)}) . \end{matrix}$ It follows that $k \tilde{u} = k \hat{u} + ⋃_{φ \in [0, 2 π]} (((- k) \tilde{R} (φ)) e^{i (φ + π)}) .$ □

Theorem 3.4. Let ${\tilde{u}}_{1} = {\hat{u}}_{1} + ⋃_{φ \in [0, 2 π]} ({\tilde{R}}_{1} (φ) e^{i φ}), {\tilde{u}}_{2} = {\hat{u}}_{2} + ⋃_{φ \in [0, 2 π]} ({\tilde{R}}_{2} (φ) e^{i φ}) .$ If |ρ₁ (r, φ₁) e^{iφ
₁} + ρ₂ (r, φ₂) e^{iφ
₂}| ≤ ρ₁ (r, φ₀) + ρ₂ (r, φ₀) , then ${\tilde{u}}_{1} + {\tilde{u}}_{2} = ({\hat{u}}_{1} + {\hat{u}}_{2}) + ⋃_{φ \in [0, 2 π]} (({\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ)) e^{i φ}),$ where φ₁, φ₂ ∈ [0, 2π] , φ₀ is the polar angle of ρ₁ (r, φ₁) e^{iφ
₁} + ρ₂ (r, φ₂) e^{iφ
₂} and $[{\tilde{R}}_{1} (φ_{1})]^{r} = [0, ρ_{1} (r, φ_{1})], [{\tilde{R}}_{2} (φ_{2})]^{r} = [0, ρ_{2} (r, φ_{2})] .$

Proof. From Theorem 3.2, we know $[{\tilde{u}}_{1}]^{r} = u_{1} + ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)] e^{i φ}),$ $[{\tilde{u}}_{2}]^{r} = u_{2} + ⋃_{φ \in [0, 2 π]} ([0, ρ_{2} (r, φ)] e^{i φ}) .$ Then $[{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r} = (u_{1} + u_{2}) + ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)] e^{i φ})$ $+ ⋃_{φ \in [0, 2 π]} ([0, ρ_{2} (r, φ)] e^{i φ}) .$

It is easy to prove that

$⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)] e^{i φ}) + ⋃_{φ \in [0, 2 π]} ([0, ρ_{2} (r, φ)] e^{i φ})$ $= ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ) + ρ_{2} (r, φ)] e^{i φ}) .$ In fact, for any x ∈ ⋃ _φ∈[0,2π] ([0, ρ₁ (r, φ)] e^iφ) + ⋃ _φ∈[0,2π] ([0, ρ₂ (r, φ)] e^iφ) , there exists x₁, x₂ such that x₁ ∈ ⋃ _φ∈[0,2π] ([0, ρ₁ (r, φ)] e^iφ) , x₂ ∈ ⋃ _φ∈[0,2π] ([0, ρ₂ (r, φ)] e^iφ) satisfying x = x₁ + x₂ . It follows that there exists φ₁ and φ₂ such that $x_{1} = ρ_{1}^{'} (r, φ_{1}) e^{i φ_{1}}, x_{2} = ρ_{2}^{'} (r, φ_{2}) e^{i φ_{2}},$ where $ρ_{1}^{'} (r, φ_{1}) \in [0, ρ_{1} (r, φ_{1})], ρ_{2}^{'} (r, φ_{2}) \in [0, ρ_{2} (r, φ_{2})] .$ Write x = x₁ + x₂ = ϱ (r, φ₀) e^{iφ
₀}, then $ϱ (r, φ_{0}) \leq | ρ_{1}^{'} (r, φ_{1}) e^{i φ_{1}} + ρ_{2}^{'} (r, φ_{2}) e^{i φ_{2}} | \leq | ρ_{1} (r, φ_{1}) e^{i φ_{1}} + ρ_{2} (r, φ_{2}) e^{i φ_{2}} | \leq ρ_{1} (r, φ_{0}) + ρ_{2} (r, φ_{0}) .$ Thus x = x₁ + x₂ ∈ [0, ρ₁ (r, φ₀) + ρ₂ (r, φ₀)] e^{iφ
₀}, which implies x = x₁ + x₂ ∈ ⋃ _φ∈[0,2π] ([0, ρ₁ (r, φ) + ρ₂ (r, φ)] e^iφ) . Therefore $⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)] e^{i φ}) + ⋃_{φ \in [0, 2 π]} ([0, ρ_{2} (r, φ)] e^{i φ})$ $\subseteq ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ) + ρ_{2} (r, φ)] e^{i φ}) .$ On the other hand, it similar to the above statement completely, we can also prove that $⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)] e^{i φ}) + ⋃_{φ \in [0, 2 π]} ([0, ρ_{2} (r, φ)] e^{i φ})$ $\supseteq ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ) + ρ_{2} (r, φ)] e^{i φ}) .$ Therefore $⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)] e^{i φ}) + ⋃_{φ \in [0, 2 π]} ([0, ρ_{2} (r, φ)] e^{i φ})$ $= ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ) + ρ_{2} (r, φ)] e^{i φ}) .$ Then $[{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r} = (u_{1} + u_{2}) + ⋃_{φ \in [0, 2 π]} ([0, ρ_{1} (r, φ)$ $+ ρ_{2} (r, φ)] e^{i φ}) .$

Let ρ (r, φ) = ρ₁ (r, φ) + ρ₂ (r, φ) , then $[{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r} = (u_{1} + u_{2}) + ⋃_{φ \in [0, 2 π]} ([0, ρ (r, φ)] e^{i φ}) .$ Since $[{\tilde{u}}_{1} + {\tilde{u}}_{2}]^{r} = [{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r},$ we have ${\tilde{u}}_{1} + {\tilde{u}}_{2} = ({\hat{u}}_{1} + {\hat{u}}_{2}) + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ}),$ where

$\begin{matrix} \tilde{R} (φ) & = & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, φ)]) \\ = & ⋃_{r \in [0, 1]} (r^{*} \cap ([0, ρ_{1} (r, φ)] + [0, ρ_{2} (r, φ)])) \\ = & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ_{1} (r, φ)] + r^{*} \cap [0, ρ_{2} (r, φ)]) \\ = & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ_{1} (r, φ)]) \\ + & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ_{2} (r, φ)]) \\ = & {\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ) . \end{matrix}$ Therefore ${\tilde{u}}_{1} + {\tilde{u}}_{2} = ({\hat{u}}_{1} + {\hat{u}}_{2}) + ⋃_{φ \in [0, 2 π]} (({\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ)) e^{i φ}) .$ □

In order to keep the presentation simple, we shall write the two-dimension fuzzy number $\tilde{u} = \hat{u} + ⋃_{φ \in [0, 2 π]} (\tilde{R} (φ) e^{i φ})$ as $\tilde{u} = \hat{u} + \tilde{R} (φ) e^{i φ}, φ \in [0, 2 π] .$ From Theorem 3.3 and 3.4, we know that the addition and scalar multiplication of the two-dimension fuzzy numbers are defined by the equations ${\tilde{u}}_{1} + {\tilde{u}}_{2} = {\hat{u}}_{1} + {\hat{u}}_{2} + ({\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ)) e^{i φ},$ $φ \in [0, 2 π],$ $k \tilde{u} = {\begin{matrix} k \hat{u} + k \tilde{R} (φ) e^{i φ}, φ \in [0, 2 π], & k \geq 0, \\ k \hat{u} + (- k) \tilde{R} (φ) e^{i (φ + π)}, φ \in [0, 2 π], & k < 0 \end{matrix}$ respectively. In Theorem 3.4, the conclusion does not always hold if |ρ₁ (r, φ₁) e^{iφ
₁} + ρ₂ (r, φ₂) e^{iφ
₂}| > ρ₁ (r, φ₀) + ρ₂ (r, φ₀) , it will be illustrated by the following example.

Example 3.5. Let ${\tilde{u}}_{1} = {\tilde{R}}_{1} (φ) e^{i φ}, {\tilde{u}}_{2} = {\tilde{R}}_{2} (φ) e^{i φ},$ φ ∈ [0, 2π] , where ${\tilde{R}}_{1} (φ) = {\begin{matrix} [0, 1] e^{i φ}, & φ = 0, \\ 0, & otherwise, \end{matrix} and$ ${\tilde{R}}_{2} (φ) = {\begin{matrix} [0, 1] e^{i φ}, & φ = \frac{π}{2}, \\ 0, & otherwise . \end{matrix}$ When $φ_{1} = 0, φ_{2} = \frac{π}{2},$ $| ρ_{1} (r, φ_{1}) e^{i φ_{1}} + ρ_{2} (r, φ_{2}) e^{i φ_{2}} | = | 1 + i |$ $= \sqrt{2} > 0 = ρ_{1} (r, \frac{π}{4}) + ρ_{2} (r, \frac{π}{4}),$ where $\frac{π}{4}$ is polar angle of 1 + i . However, it is easy to verify that $({\tilde{u}}_{1} + {\tilde{u}}_{2}) (φ) = {\begin{matrix} [0, 1] e^{i φ}, & φ = 0 or \frac{π}{2}, \\ 0, & otherwise, \end{matrix}$ is not a two-dimension fuzzy number.

In what follows, we always assume that both two-dimension fuzzy numbers satisfy the condition of Theorem 3.4 when the addition of the two-dimension fuzzy numbers is involved.

4 Fuzzy linear systems with the two-dimensional fuzzy data

The n × n fuzzy system of linear equations with the two-dimension fuzzy data is written as $\begin{matrix} \begin{matrix} a_{11} {\tilde{u}}_{1} + a_{12} {\tilde{u}}_{2} + \dots + a_{1 n} {\tilde{u}}_{n} = {\tilde{v}}_{1}, \\ a_{21} {\tilde{u}}_{1} + a_{22} {\tilde{u}}_{2} + \dots + a_{2 n} {\tilde{u}}_{n} = {\tilde{v}}_{2} \\ ⋮ \\ a_{n 1} {\tilde{u}}_{1} + a_{n 2} {\tilde{u}}_{2} + \dots + a_{nn} {\tilde{u}}_{n} = {\tilde{v}}_{n} . \end{matrix} \end{matrix}$ (4.1) In matrix notation we may write the above system as $A \tilde{U} = \tilde{V},$ (4.2) where A = (a_jk) _n×n is a real n × n matrix, $\tilde{V} = ({\tilde{v}}_{1}, {\tilde{v}}_{2}, \dots, {\tilde{v}}_{n})^{T}$ is a column vector of the two-dimension fuzzy numbers and $\tilde{U} = ({\tilde{u}}_{1}, {\tilde{u}}_{2}, \dots, {\tilde{u}}_{n})^{T}$ is a vector of the two-dimension fuzzy numbers unknown. The system (4.1) can be also represented as $\sum_{j = 1}^{n} a_{kj} {\tilde{u}}_{j} = {\tilde{v}}_{k}, k = 1, 2, \dots, n .$ (4.3) Let $a_{kj} = r_{kj}^{a} e^{i θ_{kj}^{a}}, {\tilde{v}}_{k} = {\hat{v}}_{k} + {\tilde{R}}_{k}^{v} (φ) e^{i φ},$ ${\tilde{u}}_{k} = {\hat{u}}_{k} + {\tilde{R}}_{k}^{u} (φ) e^{i φ}, φ \in [0, 2 π],$ where $r_{kj}^{a} = | a_{kj} |, θ_{kj}^{a} = 0 or π;$ ${\hat{v}}_{k}, {\hat{u}}_{k}$ are fuzzy points on $ℝ^{2}$ which its bearing point are v_k, u_k and height is 1, ${\tilde{R}}_{k}^{v} (φ), {\tilde{R}}_{k}^{u} (φ)$ are fuzzy numbers, i is imaginary unit. Then the following system is obtained by substituting ${\tilde{v}}_{k}$ and ${\tilde{u}}_{k}$ in the system (4.3) . $\sum_{j = 1}^{n} a_{kj} [{\hat{u}}_{j} + {\tilde{R}}_{j}^{u} (φ) e^{i φ}] = {\hat{v}}_{k} + {\tilde{R}}_{k}^{v} (φ) e^{i φ},$ $k = 1, 2, \dots, n .$ i.e. $\sum_{j = 1}^{n} [a_{kj} {\hat{u}}_{j} + r_{kj}^{a} {\tilde{R}}_{j}^{u} (φ) e^{i (φ + θ_{kj}^{a})}] = {\hat{v}}_{k} + {\tilde{R}}_{k}^{v} e^{i φ},$ $k = 1, 2, \dots, n .$

The above system can be written as $\sum_{j = 1}^{n} a_{kj} u_{j} = v_{k}, k = 1, 2, \dots, n,$ (4.4) and $\sum_{j = 1}^{n} r_{kj}^{a} {\tilde{R}}_{j}^{u} (φ) e^{i (φ + θ_{kj}^{a})} = {\tilde{R}}_{k}^{v} (φ) e^{i φ}, k = 1, 2, \dots, n,$ (4.5) where the system (4.4) is a real system of linear equations which may be solved directly. Using matrix notation we get $AU = V,$ (4.6) where U = (u₁, u₂, ⋯ , u_n) ^T, V = (v₁, v₂, ⋯ , v_n) ^T . The system (4.5) is written as $\sum_{j = 1}^{n} r_{kj}^{a} {\tilde{R}}_{j}^{u} (φ - θ_{kj}^{a}) e^{i φ} = {\tilde{R}}_{k}^{v} (φ) e^{i φ}, k = 1, 2, \dots, n .$ i.e. $\sum_{j = 1}^{n} r_{kj}^{a} {\tilde{R}}_{j}^{u} (φ - θ_{kj}^{a}) = {\tilde{R}}_{k}^{v} (φ), k = 1, 2, \dots, n,$ (4.7) where $r_{kj}^{a}$ are positive real numbers, for every φ, ${\tilde{R}}_{k}^{v} (φ)$ are fuzzy numbers known and ${\tilde{R}}_{k}^{u} (φ)$ fuzzy numbers unknown.

Let $[{\tilde{R}}_{k}^{v} (φ)]^{r} = [0, ρ_{k}^{v} (r, φ)], [{\tilde{R}}_{k}^{u} (φ)]^{r} = [0, ρ_{k}^{u} (r, φ)],$ $ρ_{k}^{v} (φ) = ρ_{k}^{v} (r, φ), ρ_{k}^{u} (φ) = ρ_{k}^{u} (r, φ) .$ Then the system (4.7) is equivalent to the following system $\sum_{j = 1}^{n} r_{kj}^{a} ρ_{j}^{u} (φ - θ_{kj}^{a}) = ρ_{k}^{v} (φ), k = 1, 2, \dots, n,$ (4.8) where $ρ_{k}^{u} (φ)$ are functions unknown period 2π, $ρ_{k}^{v} (φ)$ are functions known period 2π.

The system (4.8) is also written as a real system of linear equation $\begin{matrix} {\begin{matrix} r_{11} ρ_{1}^{u} (φ) + r_{12} ρ_{2}^{u} (φ) + \dots + r_{1 n} ρ_{n}^{u} (φ) \\ + r_{1, n + 1} ρ_{1}^{u} (φ + π) + r_{1, n + 2} ρ_{2}^{u} (φ + π) + \dots + r_{1, 2 n} ρ_{n}^{u} (φ + π) = ρ_{1}^{v} (φ), \\ ⋮ \\ r_{n 1} ρ_{1}^{u} (φ) + r_{n 2} ρ_{2}^{u} (φ) + \dots + r_{nn} ρ_{n}^{u} (φ) \\ + r_{n, n + 1} ρ_{1}^{u} (φ + π) + r_{n, n + 2} ρ_{2}^{u} (φ + π) + \dots + r_{n, 2 n} ρ_{n}^{u} (φ + π) = ρ_{n}^{v} (φ), \\ r_{n + 1, 1} ρ_{1}^{u} (φ) + r_{n + 1, 2} ρ_{2}^{u} (φ) + \dots + r_{n + 1, n} ρ_{n}^{u} (φ) \\ + r_{n + 1,, n + 1} ρ_{1}^{u} (φ + π) + r_{n + 1, n + 2} ρ_{2}^{u} (φ + π) + \dots + r_{n + 1, 2 n} ρ_{n}^{u} (φ + π) = ρ_{1}^{v} (φ + π), \\ ⋮ \\ r_{2 n, 1} ρ_{1}^{u} (φ) + r_{2 n, 2} ρ_{2}^{u} (φ) + \dots + r_{2 n, n} ρ_{n}^{u} (φ) \\ + r_{2 n, n + 1} ρ_{1}^{u} (φ + π) + r_{2 n, n + 2} ρ_{2}^{u} (φ + π) + \dots + r_{2 n, 2 n} ρ_{n}^{u} (φ + π) = ρ_{n}^{v} (φ + π), \end{matrix} \end{matrix}$ where r_ij are determined as follows: $\begin{matrix} \begin{matrix} r_{ij} = a_{ij}, r_{i + n, j + n} = a_{ij} if a_{ij} \geq 0, \\ r_{i, j + n} = - a_{ij}, r_{i + n, j} = - a_{ij} if a_{ij} < 0 \end{matrix} \end{matrix}$ (4.9) and any r_ij which is not determined by Equations 4.9 is zero. Using matrix notation we get ${RU}_{ρ} = V_{ρ},$ (4.10) where R = (r_ij) _2n×2n and $U_{ρ} = (\begin{matrix} ρ_{1}^{u} (φ) \\ ⋮ \\ ρ_{n}^{u} (φ) \\ ρ_{1}^{u} (φ + π) \\ ⋮ \\ ρ_{n}^{u} (φ + π) \end{matrix}), V_{ρ} = (\begin{matrix} ρ_{1}^{v} (φ) \\ ⋮ \\ ρ_{n}^{v} (φ) \\ ρ_{1}^{v} (φ + π) \\ ⋮ \\ ρ_{n}^{v} (φ + π) \end{matrix}) .$

The above discussion leads to the following theorem.

Theorem 4.1. A vector $\tilde{U} = ({\tilde{u}}_{1}, {\tilde{u}}_{2}, \dots, {\tilde{u}}_{n})^{T}$ is a solution (a unique solution) of Equation (4.2) if only if the vectors $U = (u_{1}, u_{2}, \dots, u_{n})^{T}, U_{ρ} = (ρ_{1}^{u} (φ), \dots, ρ_{n}^{u} (φ), ρ_{1}^{u} (φ + π), \dots, ρ_{n}^{u} (φ + π))^{T}$ are solutions (two unique solutions) of Equation (4.6) and (4.10) respectively, and $[{\tilde{u}}_{k}]^{r} = u_{k} + ⋃_{φ \in [0, 2 π]} ([0, ρ_{k}^{u} (φ)] e^{i φ}),$ where $⋃_{φ \in [0, 2 π]} ([0, ρ_{k}^{u} (φ)] e^{i φ})$ satisfy the conditions (1)-(3) of Theorem 2.1 .

Example 4.2. Consider the 2 × 2 fuzzy system of linear equations with the two-dimension fuzzy data $\begin{matrix} \begin{matrix} {\tilde{u}}_{1} - {\tilde{u}}_{2} = {\tilde{v}}_{1}, \\ {\tilde{u}}_{1} + 3 {\tilde{u}}_{2} = {\tilde{v}}_{2}, \end{matrix} \end{matrix}$ (4.11) where ${\tilde{v}}_{1} = \hat{(1, 1)} + {\tilde{R}}_{1}^{v} (φ) e^{i φ},$ $[{\tilde{R}}_{1}^{v} (φ)]^{r} = [0, 1 - r], φ \in [0, 2 π],$ ${\tilde{v}}_{2} = \hat{(5, 5)} + {\tilde{R}}_{2}^{v} (φ) e^{i φ},$ $[{\tilde{R}}_{2}^{v} (φ)]^{r} = [0, 2 (1 - r)], φ \in [0, 2 π] .$

Let ${\tilde{u}}_{k} = {\hat{u}}_{k} + {\tilde{R}}_{k}^{u} e^{i φ}, [{\tilde{R}}_{k}^{u}]^{r} = [{\tilde{R}}_{k}^{u} (φ)]^{r} = [0, ρ_{k}^{u} (r, φ)], ρ_{k}^{u} (φ) = ρ_{k}^{u} (r, φ), k = 1, 2 .$ The system (4.11) is written as $\begin{matrix} \begin{matrix} ({\hat{u}}_{1} + {\tilde{R}}_{1}^{u} e^{i φ}) - ({\hat{u}}_{2} + {\tilde{R}}_{2}^{u} e^{i φ}) = \hat{(1, 1)} + {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ ({\hat{u}}_{1} + {\tilde{R}}_{1}^{u} e^{i φ}) + 3 ({\hat{u}}_{2} + {\tilde{R}}_{2}^{u} e^{i φ}) = \hat{(5, 5)} + {\tilde{R}}_{2}^{v} (φ) e^{i φ} . \end{matrix} \end{matrix}$ The above system is equivalent to the systems $\begin{matrix} \begin{matrix} u_{1} - u_{2} = (1, 1), \\ u_{1} + 3 u_{2} = (5, 5) \end{matrix} \end{matrix}$ (4.12) and $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} e^{i φ} - {\tilde{R}}_{2}^{u} e^{i φ} = {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{u} e^{i φ} + 3 {\tilde{R}}_{2}^{u} e^{i φ} = {\tilde{R}}_{2}^{v} (φ) e^{i φ} . \end{matrix} \end{matrix}$ (4.13)

The solution of system (4.12) are $u_{1} = (2, 2), u_{2} = (1, 1) .$ (4.14)

The system (4.13) is written as $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} e^{i φ} + {\tilde{R}}_{2}^{u} e^{i (φ + π)} = {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{u} e^{i φ} + 3 {\tilde{R}}_{2}^{u} e^{i φ} = {\tilde{R}}_{2}^{v} (φ) e^{i φ}, \end{matrix} \end{matrix}$ i.e. $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} (φ) e^{i φ} + {\tilde{R}}_{2}^{u} (φ - π) e^{i φ} = {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{u} (φ) e^{i φ} + 3 {\tilde{R}}_{2}^{u} (φ) e^{i φ} = {\tilde{R}}_{2}^{v} (φ) e^{i φ} . \end{matrix} \end{matrix}$ That is $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} (φ) + {\tilde{R}}_{2}^{u} (φ - π) = {\tilde{R}}_{1}^{v} (φ), \\ {\tilde{R}}_{1}^{u} (φ) + 3 {\tilde{R}}_{2}^{u} (φ) = {\tilde{R}}_{2}^{v} (φ) . \end{matrix} \end{matrix}$ It follows that $\begin{matrix} \begin{matrix} ρ_{1}^{u} (φ) + ρ_{2}^{u} (φ - π) = ρ_{1}^{v} (φ), \\ ρ_{1}^{u} (φ) + 3 ρ_{2}^{u} (φ) = ρ_{2}^{v} (φ), \end{matrix} \end{matrix}$ (4.15) where $ρ_{i}^{u} (φ)$ are functions unknown with period 2π, $ρ_{i}^{v} (φ)$ are functions known with period 2π, and $ρ_{1}^{v} (φ) = 1 - r, φ \in [0, π],$ $ρ_{2}^{v} (φ) = 2 (1 - r), φ \in [0, π] .$ The system (4.15) is also written as a real system of linear equation ${ρ_{1}^{u} (φ) + ρ_{2}^{u} (φ + π) = ρ_{1}^{v} (φ), ρ_{1}^{u} (φ) + 3 ρ_{2}^{u} (φ) = ρ_{2}^{v} (φ), ρ_{2}^{u} (φ) + ρ_{1}^{u} (φ + π) = ρ_{1}^{v} (φ + π), ρ_{1}^{u} (φ + π) + 3 ρ_{2}^{u} (φ + π) = ρ_{2}^{v} (φ + π) .$ It is easy to obtain that $ρ_{1}^{u} (φ) = \frac{1}{2} (1 - r), ρ_{2}^{u} (φ) = \frac{1}{2} (1 - r),$ $ρ_{1}^{u} (φ + π) = \frac{1}{2} (1 - r), ρ_{2}^{u} (φ + π) = \frac{1}{2} (1 - r) .$ (4.16)

From Theorem 4.1 and Equation (4.14) , (4.16) , we obtain that $[{\tilde{u}}_{1}]^{r} = (2, 2) + ⋃_{φ \in [0, 2 π]} ([0, \frac{1}{2} (1 - r)] e^{i φ}),$ $[{\tilde{u}}_{2}]^{r} = (1, 1) + ⋃_{φ \in [0, 2 π]} ([0, \frac{1}{2} (1 - r)] e^{i φ}) .$

Example 4.3. Consider the 2 × 2 fuzzy system of linear equations with the two-dimension fuzzy data $\begin{matrix} \begin{matrix} {\tilde{u}}_{1} - {\tilde{u}}_{2} = {\tilde{v}}_{1}, \\ {\tilde{u}}_{1} + 3 {\tilde{u}}_{2} = {\tilde{v}}_{2}, \end{matrix} \end{matrix}$ (4.17) where ${\tilde{v}}_{1} = \hat{(1, 1)} + {\tilde{R}}_{1}^{v} (φ) e^{i φ},$ $[{\tilde{R}}_{1}^{v} (φ)]^{r} = [0, ρ_{1}^{v} (φ)], φ \in [0, 2 π],$ ${\tilde{v}}_{2} = \hat{(5, 5)} + {\tilde{R}}_{2}^{v} (φ) e^{i φ},$ $[{\tilde{R}}_{2}^{v} (φ)]^{r} = [0, ρ_{2}^{v} (φ)], φ \in [0, 2 π],$ $ρ_{1}^{v} (φ) = {\begin{matrix} \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4}, \end{matrix}$ $ρ_{2}^{v} (φ) = 2 {\begin{matrix} \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4} . \end{matrix}$

Let ${\tilde{u}}_{k} = {\hat{u}}_{k} + {\tilde{R}}_{k}^{u} e^{i φ}, [{\tilde{R}}_{k}^{u}]^{r} = [{\tilde{R}}_{k}^{u} (φ)]^{r} = [0, ρ_{k}^{u} (r, φ)], ρ_{k}^{u} (φ) = ρ_{k}^{u} (r, φ), k = 1, 2 .$ The system (4.17) is written as $\begin{matrix} \begin{matrix} ({\hat{u}}_{1} + {\tilde{R}}_{1}^{u} e^{i φ}) - ({\hat{u}}_{2} + {\tilde{R}}_{2}^{u} e^{i φ}) = \hat{(1, 1)} + {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ ({\hat{u}}_{1} + {\tilde{R}}_{1}^{u} e^{i φ}) + 3 ({\hat{u}}_{2} + {\tilde{R}}_{2}^{u} e^{i φ}) = \hat{(5, 5)} + {\tilde{R}}_{2}^{v} (φ) e^{i φ} . \end{matrix} \end{matrix}$ The above system is equivalent to the systems $\begin{matrix} \begin{matrix} u_{1} - u_{2} = (1, 1), \\ u_{1} + 3 u_{2} = (5, 5) \end{matrix} \end{matrix}$ (4.18) and $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} e^{i φ} - {\tilde{R}}_{2}^{u} e^{i φ} = {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{u} e^{i φ} + 3 {\tilde{R}}_{2}^{u} e^{i φ} = {\tilde{R}}_{2}^{v} (φ) e^{i φ} . \end{matrix} \end{matrix}$ (4.19)

The solution of system (4.18) are $u_{1} = (2, 2), u_{2} = (1, 1) .$ (4.20)

The system (4.19) is written as $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} e^{i φ} + {\tilde{R}}_{2}^{u} e^{i (φ + π)} = {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{u} e^{i φ} + 3 {\tilde{R}}_{2}^{u} e^{i φ} = {\tilde{R}}_{2}^{v} (φ) e^{i φ}, \end{matrix} \end{matrix}$ i.e. $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} (φ) e^{i φ} + {\tilde{R}}_{2}^{u} (φ - π) e^{i φ} = {\tilde{R}}_{1}^{v} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{u} (φ) e^{i φ} + 3 {\tilde{R}}_{2}^{u} (φ) e^{i φ} = {\tilde{R}}_{2}^{v} (φ) e^{i φ} . \end{matrix} \end{matrix}$ That is $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{u} (φ) + {\tilde{R}}_{2}^{u} (φ - π) = {\tilde{R}}_{1}^{v} (φ), \\ {\tilde{R}}_{1}^{u} (φ) + 3 {\tilde{R}}_{2}^{u} (φ) = {\tilde{R}}_{2}^{v} (φ) . \end{matrix} \end{matrix}$ It follows that $\begin{matrix} \begin{matrix} ρ_{1}^{u} (φ) + ρ_{2}^{u} (φ - π) = ρ_{1}^{v} (φ), \\ ρ_{1}^{u} (φ) + 3 ρ_{2}^{u} (φ) = ρ_{2}^{v} (φ), \end{matrix} \end{matrix}$ (4.21) where $ρ_{i}^{u} (φ)$ are functions unknown with period 2π, $ρ_{i}^{v} (φ)$ are functions known with period 2π, and $ρ_{1}^{v} (φ) = {\begin{matrix} \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4}, \end{matrix}$ $ρ_{2}^{v} (φ) = 2 {\begin{matrix} \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4} . \end{matrix}$ The system (4.21) is also written as a real system of linear equation ${\begin{matrix} ρ_{1}^{u} (φ) + ρ_{2}^{u} (φ + π) = ρ_{1}^{v} (φ), \\ ρ_{1}^{u} (φ) + 3 ρ_{2}^{u} (φ) = ρ_{2}^{v} (φ), \\ ρ_{2}^{u} (φ) + ρ_{1}^{u} (φ + π) = ρ_{1}^{v} (φ + π), \\ ρ_{1}^{u} (φ + π) + 3 ρ_{2}^{u} (φ + π) = ρ_{2}^{v} (φ + π) . \end{matrix}$ Since $ρ_{1}^{v} (φ + π) = {\begin{matrix} \frac{1 - r}{\cos (φ + π)}, & - \frac{π}{4} \leq φ + π \leq \frac{π}{4}, \\ \frac{1 - r}{\sin (φ + π)}, & \frac{π}{4} \leq φ + π \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos (φ + π)}, & \frac{3 π}{4} \leq φ + π \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin (φ + π)}, & \frac{5 π}{4} \leq φ + π \leq \frac{7 π}{4} \end{matrix}$ $= {\begin{matrix} \frac{1 - r}{- \cos φ}, & - \frac{5 π}{4} \leq φ \leq - \frac{3 π}{4}, \\ \frac{1 - r}{- \sin φ}, & - \frac{3 π}{4} \leq φ \leq - \frac{π}{4}, \\ \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4} \end{matrix}$ $= {\begin{matrix} \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4}, \\ \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \end{matrix}$ $= {\begin{matrix} \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4}, \end{matrix}$ $ρ_{2}^{v} (φ + π) = 2 {\begin{matrix} \frac{1 - r}{\cos φ}, & - \frac{π}{4} \leq φ \leq \frac{π}{4}, \\ \frac{1 - r}{\sin φ}, & \frac{π}{4} \leq φ \leq \frac{3 π}{4}, \\ \frac{1 - r}{- \cos φ}, & \frac{3 π}{4} \leq φ \leq \frac{5 π}{4}, \\ \frac{1 - r}{- \sin φ}, & \frac{5 π}{4} \leq φ \leq \frac{7 π}{4} . \end{matrix}$ It is easy to obtain that $ρ_{1}^{u} (φ) = \frac{1}{2} ρ_{1}^{v} (φ), ρ_{2}^{u} (φ) = \frac{1}{2} ρ_{1}^{v} (φ),$ $ρ_{1}^{u} (φ + π) = \frac{1}{2} ρ_{1}^{v} (φ), ρ_{2}^{u} (φ + π) = \frac{1}{2} ρ_{1}^{v} (φ) .$ (4.22)

From Theorem 4.1 and Equation (4.20) , (4.22) , we obtain that $[{\tilde{u}}_{1}]^{r} = (2, 2) + ⋃_{φ \in [0, 2 π]} ([0, \frac{1}{2} ρ_{1}^{v} (φ)] e^{i φ}),$ $[{\tilde{u}}_{2}]^{r} = (1, 1) + ⋃_{φ \in [0, 2 π]} ([0, \frac{1}{2} ρ_{1}^{v} (φ)] e^{i φ}) .$

From Example 4.2 and 4.3, we know that the structure of R in Equation 4.10 implies that r_i,j ≥ 0, 1 ≤ i, j ≤ 2n and that $R = (\begin{matrix} C & D \\ D & C \end{matrix}),$ where C contains the positive entries of A, D the absolute values of the negative entries of A and A = C - D is the coefficient matrix of Equation 4.2 .

Example 4.4. Consider the 2 × 2 fuzzy system of linear equations with the two-dimension fuzzy data $\begin{matrix} \begin{matrix} {\tilde{u}}_{1} - {\tilde{u}}_{2} = {\tilde{v}}_{1}, \\ {\tilde{u}}_{1} + 3 {\tilde{u}}_{2} = {\tilde{v}}_{2}, \end{matrix} \end{matrix}$ (4.17) From Example 4.2 and 4.3, we have $R = (\begin{matrix} C & D \\ D & C \end{matrix}),$ where $C = (\begin{matrix} 1 & 0 \\ 1 & 3 \end{matrix}), D = (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) .$

Example 4.5. The following table shows the four foods on the diet and the amount of nutrients per 100 grams of food. For example, each of the two types of skim milk contains only one protein respectively, the first 100 grams of skim milk with 36 grams of the first protein.

Nutrition	Nutrients per 100 grams of food		The amount of daily nutrients required to lose weight
	Two skim milk	Two soy flour
Two proteins	36	50	(33,36)
Two carbohydrates	52	25	(45,30)

To ensure the amount of daily nutrients required to lose weight, we assume that we need to eat two kinds of skim milk every day for x₁ and y₁, and two kinds of soy flour every day for x₂ and y₂ . Therefore, we establish the following linear system. $\begin{matrix} \begin{matrix} 36 (x_{1}, y_{1}) + 50 (x_{2}, y_{2}) = (33, 36), \\ 52 (x_{1}, y_{1}) + 25 (x_{2}, y_{2}) = (45, 30) . \end{matrix} \end{matrix}$ It is easy to obtain that $(x_{1}, y_{1}) \approx (0.84, 0.35), (x_{2}, y_{2}) \approx (0.06, 0.47) .$ If the amount of daily nutrients is only an estimated quantity, then using a two-dimensional fuzzy number ${\tilde{v}}_{i} (i = 1, 2)$ to express the quantity is more appropriate than using a crisp two-dimensional quantity. Let ${\tilde{u}}_{1}$ is an estimated quantity of skim milk every day, ${\tilde{u}}_{2}$ is an estimated quantity of soy flour every day, ${\tilde{v}}_{1} = \hat{(33, 36)} + {\tilde{R}}_{1}^{v} (φ) e^{i φ},$ $[{\tilde{R}}_{1}^{v} (φ)]^{r} = [0, 1 - r], φ \in [0, 2 π],$ ${\tilde{v}}_{2} = \hat{(45, 30)} + {\tilde{R}}_{2}^{v} (φ) e^{i φ},$ $[{\tilde{R}}_{2}^{v} (φ)]^{r} = [0, 2 (1 - r)], φ \in [0, 2 π] .$ then we have the following fuzzy system

$\begin{matrix} \begin{matrix} 36 {\tilde{u}}_{1} + 50 {\tilde{u}}_{2} = {\tilde{v}}_{1}, \\ 52 {\tilde{u}}_{1} + 25 {\tilde{u}}_{2} = {\tilde{v}}_{2} . \end{matrix} \end{matrix}$ It is easy to obtain that $[{\tilde{u}}_{1}]^{r} \approx (0.84, 0.35) + ⋃_{φ \in [0, 2 π]} ([0, 0.015 (1 - r)] e^{i φ}),$ $[{\tilde{u}}_{2}]^{r} \approx (0.06, 0.47) + ⋃_{φ \in [0, 2 π]} ([0, 0.015 (1 - r)] e^{i φ}) .$

5 A new representation of fuzzy numbers

Theorem 5.1. Let $\tilde{o}$ be a nontrivial zero fuzzy number. Then $\tilde{o} = (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}),$ where $\tilde{R} (π) = ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, π)]),$ $\tilde{R} (0) = ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, 0)])$ are two positive fuzzy number and ρ (r, φ) ≥0, φ ∈ {0, π} .

Proof. If $\tilde{o}$ is a nontrivial zero fuzzy number, then by the decompose theorem of fuzzy number, we have $\tilde{o} = ⋃_{r \in [0, 1]} (r^{*} \cap [\tilde{o}]_{r}),$ where $[\tilde{o}]_{r} = [o_{-}^{r}, o_{+}^{r}] .$ Let $ρ (r, π) = - o_{-}^{r}, ρ (r, 0) = o_{+}^{r},$ where ρ (r, φ) ≥0, φ ∈ {0, π} , and ρ (r, φ) be a function period 2π with respect to φ . Then $[\tilde{o}]_{r} = [- ρ (r, π), ρ (r, 0)] = [- ρ (r, π), 0] \cup [0, ρ (r, 0)] .$ Therefore $\begin{matrix} \tilde{o} = ⋃_{r \in [0, 1]} (r^{*} \cap ([- ρ (r, π), 0] \cup [0, ρ (r, 0)])) \\ = ⋃_{r \in [0, 1]} ((r^{*} \cap [- ρ (r, π), 0]) \cup (r^{*} \cap [0, ρ (r, 0)])) \end{matrix}$ $\begin{matrix} = & (⋃_{r \in [0, 1]} (r^{*} \cap [- ρ (r, π), 0])) ⋃ (⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, 0)])) \\ = & (⋃_{r \in [0, 1]} [(r^{*} \cap [0, ρ (r, π)]) (- 1)]) ⋃ (⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, 0)])) \\ = & {(⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, π)])) (- 1)} ⋃ {(⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, 0)])) \cdot 1} \\ = & {(⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, π)])) e^{i π}} ⋃ {(⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, 0)])) e^{i 0}} . \end{matrix}$ We using $\tilde{R} (π), \tilde{R} (0)$ to denote ⋃_r∈[0,1] (r^* ∩ [0, ρ (r, π)]) , ⋃ _r∈[0,1] (r^* ∩ [0, ρ (r, 0)]) respectively. It is easy to see that $\tilde{R} (π), \tilde{R} (0)$ are two positive fuzzy number and $\tilde{o} = (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}) .$ □

Theorem 5.2. Let $\tilde{u}$ be a fuzzy number. Then $\tilde{u} = \hat{u} + (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}),$ where $\hat{u}$ is a fuzzy point on $ℝ$ which its support is $u = \frac{u_{1}^{-} + u_{1}^{+}}{2}$ and membership is 1. $(\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0})$ is the new representation of nontrivial zero fuzzy number $\tilde{u} - \hat{u} .$

Proof. Let $\tilde{u}$ be a fuzzy number. Assume $u = \frac{u_{1}^{-} + u_{1}^{+}}{2} .$ Then $\tilde{u} - \hat{u}$ is a nontrivial zero fuzzy number. According to Theorem 5.1, we know that $\tilde{u} - \hat{u} = (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}) .$ Hence $\tilde{u} = \hat{u} + (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}) .$ □

In the following, we give two arithmetic operations of fuzzy number based on the new representation of fuzzy numbers.

Theorem 5.3. Let $\tilde{u} = \hat{u} + (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}) \in E^{1}, k \in ℝ .$ Then $k \tilde{u} = {\begin{matrix} k \hat{u} + (k \tilde{R} (π) e^{i π}) \cup (k \tilde{R} (0) e^{i 0}) & k \geq 0, \\ k \hat{u} + ((- k) \tilde{R} (π) e^{i 0}) \cup ((- k) \tilde{R} (0) e^{i π}) & k < 0 . \end{matrix}$

Proof. From proof process of Theorem 5.1, we know $[\tilde{u} - \hat{u}]^{r} = [- ρ (r, π), 0] \cup [0, ρ (r, 0)] .$ Hence $[\tilde{u}]^{r} = u + [- ρ (r, π), 0] \cup [0, ρ (r, 0)] .$ If k ≥ 0, we have $[k \tilde{u}]^{r} = k [\tilde{u}]^{r} = ku + [- k ρ (r, π), 0] \cup [0, k ρ (r, 0)] .$ If k < 0, we get $[k \tilde{u}]^{r} = k [\tilde{u}]^{r} = ku + [0, - k ρ (r, π)] \cup [k ρ (r, 0), 0] .$ That is to say, if k ≥ 0, we have $[k \tilde{u} - k \hat{u}]^{r} = [- k ρ (r, π), 0] \cup [0, k ρ (r, 0)] .$ If k < 0, we get $[k \tilde{u} - k \hat{u}]^{r} = [0, - k ρ (r, π)] \cup [k ρ (r, 0), 0] .$ Similar to the proof of Theorem 5.1, we have $k \tilde{u} - k \hat{u} = (k \tilde{R} (π) e^{i π}) \cup (k \tilde{R} (0) e^{i 0})$ if k ≥ 0 . i.e. $k \tilde{u} = k \hat{u} + (k \tilde{R} (π) e^{i π}) \cup (k \tilde{R} (0) e^{i 0}) .$ Similar to the proof of Theorem 5.1, when k < 0, we have $\begin{matrix} k \tilde{u} - k \hat{u} = ⋃_{r \in [0, 1]} (r^{*} \cap ([0, - k ρ (r, π)] \cup [k ρ (r, 0), 0])) \\ = ⋃_{r \in [0, 1]} (r^{*} \cap ([k ρ (r, 0), 0] \cup [0, - k ρ (r, π)])) \end{matrix}$ $\begin{matrix} = & ⋃_{r \in [0, 1]} ((r^{*} \cap [k ρ (r, 0), 0]) \cup (r^{*} \cap [0, - k ρ (r, π)])) \\ = & (⋃_{r \in [0, 1]} (r^{*} \cap [k ρ (r, 0), 0])) ⋃ (⋃_{r \in [0, 1]} (r^{*} \cap [0, - k ρ (r, π)])) \\ = & (⋃_{r \in [0, 1]} [(r^{*} \cap [0, - k ρ (r, 0)]) (- 1)]) ⋃ (⋃_{r \in [0, 1]} (r^{*} \cap [0, - k ρ (r, π)])) \\ = & {(⋃_{r \in [0, 1]} (r^{*} \cap [0, - k ρ (r, 0)])) (- 1)} ⋃ {(⋃_{r \in [0, 1]} (r^{*} \cap [0, - k ρ (r, π)])) \cdot 1} \\ = & {(⋃_{r \in [0, 1]} (r^{*} \cap [0, - k ρ (r, 0)])) e^{i π}} ⋃ {(⋃_{r \in [0, 1]} (r^{*} \cap [0, - k ρ (r, π)])) e^{i 0}} . \end{matrix}$ i.e. $k \tilde{u} - k \hat{u} = ((- k) \tilde{R} (π) e^{i 0}) \cup ((- k) \tilde{R} (0) e^{i π}) .$ It follows that $k \tilde{u} = k \hat{u} + ((- k) \tilde{R} (π) e^{i 0}) \cup ((- k) \tilde{R} (0) e^{i π}) .$ □

Theorem 5.4. Let ${\tilde{u}}_{1} = {\hat{u}}_{1} + ({\tilde{R}}_{1} (π) e^{i π}) \cup ({\tilde{R}}_{1} (0) e^{i 0}), {\tilde{u}}_{2} = {\hat{u}}_{2} + ({\tilde{R}}_{2} (π) e^{i π}) \cup ({\tilde{R}}_{2} (0) e^{i 0}) .$ Then ${\tilde{u}}_{1} + {\tilde{u}}_{2} = {\hat{u}}_{1} + {\hat{u}}_{2} + (({\tilde{R}}_{1} (π) + {\tilde{R}}_{2} (π)) e^{i π})$ $\cup (({\tilde{R}}_{1} (0) + {\tilde{R}}_{2} (0)) e^{i 0}) .$

Proof. From proof process of Theorem 5.1, we know $[{\tilde{u}}_{1} - {\hat{u}}_{1}]^{r} = [- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)],$ $[{\tilde{u}}_{2} - {\hat{u}}_{2}]^{r} = [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$ Hence $[{\tilde{u}}_{1}]^{r} = u_{1} + [- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)],$ $[{\tilde{u}}_{2}]^{r} = u_{2} + [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$ Then $[{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r} = u_{1} + u_{2} + [- ρ_{1} (r, π), 0]$ $\cup [0, ρ_{1} (r, 0)] + [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$

It is easy to prove that $[- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)]$

$+ [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)]$ $= [- (ρ_{1} (r, π) + ρ_{2} (r, π)), 0]$ $\cup [0, (ρ_{1} (r, 0) + ρ_{2} (r, 0))] .$ In fact, for any x ∈ [-(ρ₁ (r, π) + ρ₂ (r, π)) , 0] ∪ [0, (ρ₁ (r, 0) + ρ₂ (r, 0))] , we have x ∈ [-(ρ₁ (r, π) + ρ₂ (r, π)) , 0] or x ∈ [0, (ρ₁ (r, 0) + ρ₂ (r, 0))] . If x ∈ [-(ρ₁ (r, π) + ρ₂ (r, π)) , 0] , then there exists x₁, x₂ such that -ρ₁ (r, π) ≤ x₁ ≤ 0, - ρ₂ (r, π) ≤ x₂ ≤ 0 satisfying x = x₁ + x₂ . Thus, x₁ ∈ [- ρ₁ (r, π) , 0] ∪ [0, ρ₁ (r, 0)] , x₂ ∈ [- ρ₂ (r, π) , 0] ∪ [0, ρ₂ (r, 0)] . which implies $x = x_{1} + x_{2} \in [- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)]$ $+ [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$ If x ∈ [0, (ρ₁ (r, 0) + ρ₂ (r, 0))] , it is similar to prove that $x = x_{1} + x_{2} \in [- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)]$ $+ [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$ Hence $[- (ρ_{1} (r, π) + ρ_{2} (r, π)), 0] \cup [0, (ρ_{1} (r, 0) + ρ_{2} (r, 0))]$ $\subseteq [- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)]$ $+ [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$ On the other hand, it similar to the above statement completely, we can also prove that $[- (ρ_{1} (r, π) + ρ_{2} (r, π)), 0] \cup [0, (ρ_{1} (r, 0) + ρ_{2} (r, 0))]$ $\supseteq [- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)]$ $+ [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)] .$ Therefore $[- ρ_{1} (r, π), 0] \cup [0, ρ_{1} (r, 0)]$ $+ [- ρ_{2} (r, π), 0] \cup [0, ρ_{2} (r, 0)]$ $= [- (ρ_{1} (r, π) + ρ_{2} (r, π)), 0] \cup [0, (ρ_{1} (r, 0) + ρ_{2} (r, 0))] .$

Then $[{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r} = u_{1} + u_{2} + [- (ρ_{1} (r, π)$ $+ ρ_{2} (r, π)), 0] \cup [0, (ρ_{1} (r, 0) + ρ_{2} (r, 0))] .$ Let ρ (r, φ) = ρ₁ (r, φ) + ρ₂ (r, φ) , φ ∈ {0, π} , then $[{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r} = u_{1} + u_{2} + [- ρ (r, π), 0] \cup [0, ρ (r, 0)] .$ Since $[{\tilde{u}}_{1} + {\tilde{u}}_{2}]^{r} = [{\tilde{u}}_{1}]^{r} + [{\tilde{u}}_{2}]^{r},$ we have ${\tilde{u}}_{1} + {\tilde{u}}_{2} = {\hat{u}}_{1} + {\hat{u}}_{2} + (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0}),$ where

$\begin{matrix} \tilde{R} (φ) & = & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ (r, φ)]) \\ = & ⋃_{r \in [0, 1]} (r^{*} \cap ([0, ρ_{1} (r, φ)] + [0, ρ_{2} (r, φ)])) \\ = & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ_{1} (r, φ)] + r^{*} \cap [0, ρ_{2} (r, φ)]) \\ = & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ_{1} (r, φ)]) \\ + & ⋃_{r \in [0, 1]} (r^{*} \cap [0, ρ_{2} (r, φ)]) \\ = & {\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ), φ \in {0, π} . \end{matrix}$ Therefore ${\tilde{u}}_{1} + {\tilde{u}}_{2} = {\hat{u}}_{1} + {\hat{u}}_{2} + (({\tilde{R}}_{1} (π) + {\tilde{R}}_{2} (π)) e^{i π})$ $\cup (({\tilde{R}}_{1} (0) + {\tilde{R}}_{2} (0)) e^{i 0}) .$ □

For brevity, the fuzzy number $\tilde{u} = \hat{u} + (\tilde{R} (π) e^{i π}) \cup (\tilde{R} (0) e^{i 0})$ is also written as $\tilde{u} = \hat{u} + \tilde{R} (φ) e^{i φ}, φ \in {0, π} .$ From Theorem 5.3 and 5.4, we know the addition and the scalar multiplication of fuzzy numbers are defined by the equations ${\tilde{u}}_{1} + {\tilde{u}}_{2} = {\hat{u}}_{1} + {\hat{u}}_{2} + ({\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ)) e^{i φ}, φ \in {0, π},$ $k \tilde{u} = {\begin{matrix} k \hat{u} + k \tilde{R} (φ) e^{i φ}, φ \in {0, π}, & k \geq 0, \\ k \hat{u} + (- k) \tilde{R} (φ) e^{i (φ + π)}, φ \in {0, π}, & k < 0 . \end{matrix}$ We note that $k \tilde{u} = k \hat{u} + (- k) \tilde{R} (φ) e^{i (φ + π)}$ $= k \hat{u} + (- k) \tilde{R} (φ - π) e^{i φ}, φ \in {0, π}$ if k < 0 . It follows from $\tilde{R} (φ \pm 2 k π) = \tilde{R} (φ)$ that $k \tilde{u} = k \hat{u} + (- k) \tilde{R} (φ + π) e^{i φ}, φ \in {0, π},$ where k is a integer number.

Example 5.5. Let ${\tilde{u}}_{1}, {\tilde{u}}_{2}$ be two fuzzy number and $[{\tilde{u}}_{1}]^{r} = [r, 2 - r],$ $[{\tilde{u}}_{2}]^{r} = [r + 4, 7 - 2 r] .$ Then ${\tilde{u}}_{1} = {\hat{u}}_{1} + {\tilde{R}}_{1} (φ) e^{i φ}, φ \in {0, π},$ where ${\hat{u}}_{1} = \hat{1}, [{\tilde{R}}_{1} (φ)]^{r} = [0, 1 - r], φ \in {0, π} .$ ${\tilde{u}}_{2} = {\hat{u}}_{2} + {\tilde{R}}_{2} (φ) e^{i φ}, φ \in {0, π},$ where ${\hat{u}}_{2} = \hat{5}, [{\tilde{R}}_{2} (π)]^{r} = [0, 1 - r], [{\tilde{R}}_{2} (0)]^{r} = [0, 2 (1 - r)] .$ Therefore $3 {\tilde{u}}_{1} = 3 {\hat{u}}_{1} + (3 {\tilde{R}}_{1} (φ)) e^{i φ}, φ \in {0, π},$ where $3 {\hat{u}}_{1} = \hat{3}, [3 {\tilde{R}}_{1} (φ)]^{r} = [0, 3 (1 - r)], φ \in {0, π} .$ And ${\tilde{u}}_{1} + {\tilde{u}}_{2} = {\hat{u}}_{1} + {\hat{u}}_{2} + ({\tilde{R}}_{1} (φ) + {\tilde{R}}_{2} (φ)) e^{i φ}, φ \in {0, π},$ where ${\hat{u}}_{1} + {\hat{u}}_{2} = \hat{6}, [{\tilde{R}}_{1} (π) + {\tilde{R}}_{2} (π)]^{r} = [0, 2 (1 - r)], [{\tilde{R}}_{1} (0) + {\tilde{R}}_{2} (0)]^{r} = [0, 3 (1 - r)] .$

By lemma 2.2, it is easy to establish the follow result.

Theorem 5.6. Let $\tilde{u} = \hat{u} + \tilde{R} (φ) e^{i φ}, φ \in {0, π}$ be a fuzzy number and $[\tilde{R} (φ)]^{r} = [0, ρ (r, φ)]$ . Then the following conditions are satisfied:

(1) ρ (r, φ) is a bounded left continuous non-increasing function on (0, 1] with respect to r ;

(2) ρ (r, φ) is right continuous at r = 0 ;

(3) ρ (r, φ) ≥0 .

Conversely, for any real number u, if a pair of function ρ (r, 0) and ρ (r, π) satisfy condition (1)-(3), then there exists a unique $\tilde{u} \in E^{1}$ such that $[\tilde{u}]_{r} = u + [- ρ (r, π), ρ (r, 0)]$ for each r ∈ [0, 1] .

Proof. First, we are to show the necessary condition. Since $\tilde{u} = \hat{u} + \tilde{R} (φ) e^{i φ}, φ \in {0, π}$ is a fuzzy number and $[\tilde{R} (φ)]^{r} = [0, ρ (r, φ)],$ we have ρ (r, φ) ≥0 and $[\tilde{u}]^{r} = u + [- ρ (r, π), ρ (r, 0)]$ $= [u - ρ (r, π), u + ρ (r, 0)] .$ By lemma 2.2, we know that

(1) u - ρ (r, π) is a bounded left continuous non-decreasing function on (0, 1] ;

(2) u + ρ (r, 0) is a bounded left continuous non-increasing function on (0, 1] ;

(3) u - ρ (r, π) and u + ρ (r, 0) is right at r = 0 ;

Hence ρ (r, φ) is a bounded left continuous non-increasing function on (0, 1] with respect to r and ρ (r, φ) is right continuous at r = 0 .

On the other hand, for any real number u, if a pair of function ρ (r, 0) and ρ (r, π) satisfy condition (1)-(3), then

(1) u - ρ (r, π) is a bounded left continuous non-decreasing function on (0, 1] ;

(2) u + ρ (r, 0) is a bounded left continuous non-increasing function on (0, 1] ;

(3) u - ρ (r, π) and u + ρ (r, 0) is right at r = 0 ;

(4) u - ρ (r, π) ≤ u + ρ (r, 0) .

Therefore by lemma 2.2, we know that there exists a unique $\tilde{u} \in E^{1}$ such that $[\tilde{u}]_{r} = u + [- ρ (r, π), ρ (r, 0)]$ for each r ∈ [0, 1] . □

6 A new method for solving the fuzzy system of linear equations

The n × n fuzzy system of linear equations is written as

$\begin{matrix} \begin{matrix} a_{11} {\tilde{x}}_{1} + a_{12} {\tilde{x}}_{2} + \dots + a_{1 n} {\tilde{x}}_{n} = {\tilde{b}}_{1}, \\ a_{21} {\tilde{x}}_{1} + a_{22} {\tilde{x}}_{2} + \dots + a_{2 n} {\tilde{x}}_{n} = {\tilde{b}}_{2} \\ ⋮ \\ a_{n 1} {\tilde{x}}_{1} + a_{n 2} {\tilde{x}}_{2} + \dots + a_{nn} {\tilde{x}}_{n} = {\tilde{b}}_{n} . \end{matrix} \end{matrix}$ (6.1) In matrix notation we may write the above system as $A \tilde{X} = \tilde{B},$ (6.2) where A = (a_jk) _n×n is a crisp n × n matrix, $\tilde{B} = ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n})^{T}$ is a column vector of fuzzy numbers and $\tilde{X} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})^{T}$ is a vector of fuzzy numbers unknown.

The system (6.1) can be also represented as $\sum_{j = 1}^{n} a_{kj} {\tilde{x}}_{j} = {\tilde{b}}_{k}, k = 1, 2, \dots, n .$ (6.3) Let $a_{kj} = r_{kj}^{a} e^{i θ_{kj}^{a}}, {\tilde{b}}_{k} = {\hat{b}}_{k} + {\tilde{R}}_{k}^{b} e^{i φ},$ ${\tilde{x}}_{k} = {\hat{x}}_{k} + {\tilde{R}}_{k}^{x} e^{i φ}, φ \in {0, π},$ where $r_{kj}^{a} = | a_{kj} |, θ_{kj}^{a} = 0 or π;$ ${\hat{b}}_{k}, {\hat{x}}_{k}$ are fuzzy points on $ℝ$ which its bearing point are b_k, x_k and height is 1, ${\tilde{R}}_{k}^{b} = {\tilde{R}}_{k}^{b} (φ), {\tilde{R}}_{k}^{x} = {\tilde{R}}_{k}^{x} (φ)$ are fuzzy numbers, i is imaginary unit. Then the following system is obtained by substituting ${\tilde{b}}_{k}$ and ${\tilde{x}}_{k}$ in the system (6.3) $\sum_{j = 1}^{n} a_{kj} ({\hat{x}}_{j} + {\tilde{R}}_{j}^{x} e^{i φ}) = {\hat{b}}_{k} + {\tilde{R}}_{k}^{b} e^{i φ}, k = 1, 2, \dots, n .$ i.e. $\sum_{j = 1}^{n} (a_{kj} {\hat{x}}_{j} + r_{kj}^{a} {\tilde{R}}_{j}^{x} e^{i (φ + θ_{kj}^{a})}) = {\hat{b}}_{k} + {\tilde{R}}_{k}^{b} e^{i φ},$ $k = 1, 2, \dots, n .$

The above system can be written as $\sum_{j = 1}^{n} a_{kj} x_{j} = b_{k}, k = 1, 2, \dots, n,$ (6.4) and $\sum_{j = 1}^{n} r_{kj}^{a} {\tilde{R}}_{j}^{x} e^{i (φ + θ_{kj}^{a})} = {\tilde{R}}_{k}^{b} e^{i φ}, k = 1, 2, \dots, n,$ (6.5) where the system (6.4) is a real systems of linear equations which may be solved directly. Using matrix notation we get $AX = B,$ (6.6) where X = (x₁, x₂, ⋯ , x_n) ^T, B = (b₁, b₂, ⋯ , b_n) ^T . The system (6.5) is written as $\sum_{j = 1}^{n} r_{kj}^{a} {\tilde{R}}_{j}^{x} (φ - θ_{kj}^{a}) e^{i φ} = {\tilde{R}}_{k}^{b} e^{i φ}, k = 1, 2, \dots, n .$ i.e. $\sum_{j = 1}^{n} r_{kj}^{a} {\tilde{R}}_{j}^{x} (φ - θ_{kj}^{a}) = {\tilde{R}}_{k}^{b} (φ), k = 1, 2, \dots, n,$ (6.7) where $r_{kj}^{a}$ are positive real numbers, for every φ, ${\tilde{R}}_{k}^{b} (φ)$ are fuzzy numbers known and ${\tilde{R}}_{k}^{x} (φ)$ fuzzy numbers unknown.

Let $[{\tilde{R}}_{k}^{b}]^{r} = [{\tilde{R}}_{k}^{b} (φ)]^{r} = [0, b_{k} (r, φ)],$ $[{\tilde{R}}_{k}^{x}]^{r} = [{\tilde{R}}_{k}^{x} (φ)]^{r} = [0, ρ_{k} (r, φ)],$ $b_{k} (φ) = b_{k} (r, φ), ρ_{k} (φ) = ρ_{k} (r, φ) .$ Then the system (6.7) is equivalent to following system $\sum_{j = 1}^{n} r_{kj}^{a} ρ_{j} (φ - θ_{kj}^{a}) = b_{k} (φ), k = 1, 2, \dots, n,$ (6.8) where ρ_i (φ) are functions unknown period 2π, b_i (φ) are functions known period 2π.

The system (6.8) is also written as a real system of linear equation $\begin{matrix} {\begin{matrix} r_{11} ρ_{1} (0) + r_{12} ρ_{2} (0) + \dots + r_{1 n} ρ_{n} (0) \\ + r_{1, n + 1} ρ_{1} (π) + r_{1, n + 2} ρ_{2} (π) + \dots + r_{1, 2 n} ρ_{n} (π) = b_{1} (0), \\ ⋮ \\ r_{n 1} ρ_{1} (0) + r_{n 2} ρ_{2} (0) + \dots + r_{nn} ρ_{n} (0) \\ + r_{n, n + 1} ρ_{1} (π) + r_{n, n + 2} ρ_{2} (π) + \dots + r_{n, 2 n} ρ_{n} (π) = b_{n} (0), \\ r_{n + 1, 1} ρ_{1} (0) + r_{n + 1, 2} ρ_{2} (0) + \dots + r_{n + 1, n} ρ_{n} (0) \\ + r_{n + 1,, n + 1} ρ_{1} (π) + r_{n + 1, n + 2} ρ_{2} (π) + \dots + r_{n + 1, 2 n} ρ_{n} (π) = b_{1} (π), \\ ⋮ \\ r_{2 n, 1} ρ_{1} (0) + r_{2 n, 2} ρ_{2} (0) + \dots + r_{2 n, n} ρ_{n} (0) \\ + r_{2 n, n + 1} ρ_{1} (π) + r_{2 n, n + 2} ρ_{2} (π) + \dots + r_{2 n, 2 n} ρ_{n} (π) = b_{n} (π), \end{matrix} \end{matrix}$ where r_ij are determined as follows: $\begin{matrix} \begin{matrix} r_{ij} = a_{ij}, r_{i + n, j + n} = a_{ij} if a_{ij} \geq 0, \\ r_{i, j + n} = - a_{ij}, r_{i + n, j} = - a_{ij} if a_{ij} < 0 \end{matrix} \end{matrix}$ (6.9) and any r_ij which is not determined by Equations 6.9 is zero. Using matrix notation we get ${RX}_{ρ} = B_{ρ},$ (6.10) where R = (r_ij) _2n×2n and $X_{ρ} = (\begin{matrix} ρ_{1} (0) \\ ⋮ \\ ρ_{n} (0) \\ ρ_{1} (π) \\ ⋮ \\ ρ_{n} (π) \end{matrix}), B_{ρ} = (\begin{matrix} b_{1} (0) \\ ⋮ \\ b_{n} (0) \\ b_{1} (π) \\ ⋮ \\ b_{n} (π) \end{matrix}) .$

The above discussion leads to the following theorem.

Theorem 6.1. A vector $\tilde{X} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})^{T}$ is a solution (a unique solution) of Equation (6.2) if only if the vectors X = (x₁, x₂, ⋯ , x_n) ^T, X_ρ = (ρ₁ (0) , ⋯ , ρ_n (0) , ρ₁ (π) , ⋯ , ρ_n (π)) ^T are solutions (two unique solution) of Equation (6.6) and (6.10) respectively, and $[{\tilde{x}}_{k}]^{r} = x_{k} + [- ρ_{k} (π), ρ_{k} (0)],$ where ρ_i (φ) , φ ∈ {0, π} satisfy conditions (1)-(3) of Theorem 5.6 .

Example 6.2. Consider the 2 × 2 fuzzy system of linear equations

$\begin{matrix} \begin{matrix} {\tilde{x}}_{1} - {\tilde{x}}_{2} = {\tilde{b}}_{1}, \\ {\tilde{x}}_{1} + 3 {\tilde{x}}_{2} = {\tilde{b}}_{2}, \end{matrix} \end{matrix}$ (6.11) where ${\tilde{b}}_{1} = \hat{1} + {\tilde{R}}_{1}^{b} (φ) e^{i φ}, [{\tilde{R}}_{1}^{b} (φ)]^{r} = [0, 1 - r], φ \in {0, π},$

${\tilde{b}}_{2} = \hat{5} + {\tilde{R}}_{2}^{b} (φ) e^{i φ}, [{\tilde{R}}_{2}^{b} (0)]^{r} = [0, 2 (1 - r)],$ $[{\tilde{R}}_{2}^{b} (π)]^{r} = [0, 1 - r] .$

Let ${\tilde{x}}_{k} = {\hat{x}}_{k} + {\tilde{R}}_{k}^{x} e^{i φ}, [{\tilde{R}}_{k}^{x}]^{r} = [{\tilde{R}}_{k}^{x} (φ)]^{r} = [0, ρ_{k} (r, φ)], ρ_{k} (φ) = ρ_{k} (r, φ), i = 1, 2 .$ The system (6.11) is written as $\begin{matrix} \begin{matrix} ({\hat{x}}_{1} + {\tilde{R}}_{1}^{x} e^{i φ}) - ({\hat{x}}_{2} + {\tilde{R}}_{2}^{x} e^{i φ}) = \hat{1} + {\tilde{R}}_{1}^{b} (φ) e^{i φ}, \\ ({\hat{x}}_{1} + {\tilde{R}}_{1}^{x} e^{i φ}) + 3 ({\hat{x}}_{2} + {\tilde{R}}_{2}^{x} e^{i φ}) = \hat{5} + {\tilde{R}}_{2}^{b} (φ) e^{i φ} . \end{matrix} \end{matrix}$ The above system is equivalent to systems $\begin{matrix} \begin{matrix} x_{1} - x_{2} = 1, \\ x_{1} + 3 x_{2} = 5 \end{matrix} \end{matrix}$ (6.12) and $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{x} e^{i φ} - {\tilde{R}}_{2}^{x} e^{i φ} = {\tilde{R}}_{1}^{b} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{x} e^{i φ} + 3 {\tilde{R}}_{2}^{x} e^{i φ} = {\tilde{R}}_{2}^{b} (φ) e^{i φ} . \end{matrix} \end{matrix}$ (6.13) The solution of system (6.12) are $x_{1} = 2, x_{2} = 1 .$ (6.14) The system (6.13) is written as $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{x} e^{i φ} + {\tilde{R}}_{2}^{x} e^{i (φ + π)} = {\tilde{R}}_{1}^{b} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{x} e^{i φ} + 3 {\tilde{R}}_{2}^{x} e^{i φ} = {\tilde{R}}_{2}^{b} (φ) e^{i φ}, \end{matrix} \end{matrix}$ i.e. $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{x} (φ) e^{i φ} + {\tilde{R}}_{2}^{x} (φ - π) e^{i φ} = {\tilde{R}}_{1}^{b} (φ) e^{i φ}, \\ {\tilde{R}}_{1}^{x} (φ) e^{i φ} + 3 {\tilde{R}}_{2}^{x} (φ) e^{i φ} = {\tilde{R}}_{2}^{b} (φ) e^{i φ} . \end{matrix} \end{matrix}$ That is $\begin{matrix} \begin{matrix} {\tilde{R}}_{1}^{x} (φ) + {\tilde{R}}_{2}^{x} (φ - π) = {\tilde{R}}_{1}^{b} (φ), \\ {\tilde{R}}_{1}^{x} (φ) + 3 {\tilde{R}}_{2}^{x} (φ) = {\tilde{R}}_{2}^{b} (φ) . \end{matrix} \end{matrix}$ It follows that $\begin{matrix} \begin{matrix} ρ_{1} (φ) + ρ_{2} (φ - π) = b_{1} (φ), \\ ρ_{1} (φ) + 3 ρ_{2} (φ) = b_{2} (φ), \end{matrix} \end{matrix}$ (6.15) where ρ_i (φ) are functions unknown with period 2π, b_i (φ) are functions known with period 2π, and $b_{1} (φ) = 1 - r, φ \in {0, π},$ $b_{2} (0) = 2 (1 - r), b_{2} (π) = 1 - r .$

The system (6.15) is also written as a real system of linear equation ${\begin{matrix} ρ_{1} (0) + ρ_{2} (π) = b_{1} (0), \\ ρ_{1} (0) + 3 ρ_{2} (0) = b_{2} (0), \\ ρ_{2} (0) + ρ_{1} (π) = b_{1} (π), \\ ρ_{1} (π) + 3 ρ_{2} (π) = b_{2} (π) . \end{matrix}$ It is easy to obtain that

$ρ_{1} (0) = \frac{7}{8} (1 - r), ρ_{1} (π) = \frac{5}{8} (1 - r),$ $ρ_{2} (0) = \frac{3}{8} (1 - r), ρ_{2} (π) = \frac{1}{8} (1 - r) .$ (6.16)

From Theorem 6.1 and Equation (6.14) , (6.16) , we obtain that

$[{\tilde{x}}_{1}]^{r} = [2 - \frac{5}{8} (1 - r), 2 + \frac{7}{8} (1 - r)]$ $= [\frac{11}{8} + \frac{5}{8} r, \frac{23}{8} - \frac{7}{8} r],$ $[{\tilde{x}}_{2}]^{r} = [1 - \frac{1}{8} (1 - r), 1 + \frac{3}{8} (1 - r)]$ $= [\frac{7}{8} + \frac{1}{8} r, \frac{11}{8} - \frac{3}{8} r] .$

The results obtained are equal to the results of the example problem taken from [22].

From Example 6.2, we know that the structure of R in Equation 6.10 implies that r_i,j ≥ 0, 1 ≤ i, j ≤ 2n and that

$R = (\begin{matrix} C & D \\ D & C \end{matrix}),$ where C contains the positive entries of A, D the absolute values of the negative entries of A and A = C - D is the coefficient matrix of Equation 6.2.

Example 6.3. Consider the 2 × 2 fuzzy system of linear equations $\begin{matrix} \begin{matrix} {\tilde{x}}_{1} - {\tilde{x}}_{2} = {\tilde{b}}_{1}, \\ {\tilde{x}}_{1} + 3 {\tilde{x}}_{2} = {\tilde{b}}_{2}, \end{matrix} \end{matrix}$ From Example 6.2, we have $R = (\begin{matrix} C & D \\ D & C \end{matrix}),$ where $C = (\begin{matrix} 1 & 0 \\ 1 & 3 \end{matrix}), D = (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) .$

7 The conditions of existing solutions for the fuzzy system of linear equations

In this section, we will give some conditions of existing solution for the fuzzy system of linear equations. Let r (A) denotes the rank of matrix A, R^-1 = (s_i,j) _2n×2n denotes inverse matrix of R . We write X_ρ = R^-1B_ρ = (ρ₁ (0) , ⋯ , ρ_n (0) , ρ₁ (π) , ⋯ , ρ_n (π)) ^T, T = (A, B) , S = (R, B_ρ) .

Theorem 7.1. A vector $\tilde{X}$ is a unique solution of Equation (6.2) for arbitrary $\tilde{B}$ if only if A, R nonsingular and R^-1 = (s_i,j) _2n×2n is nonnegative, i.e. $s_{ij} \geq 0, 1 \leq i, j \leq 2 n .$

Proof. By Theorem 6.1, we know that the vector $\tilde{X} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})^{T}$ is a unique solution of Equation (6.2) if only if the vectors $X = (x_{1}, x_{2}, \dots, x_{n})^{T},$ $X_{ρ} = (ρ_{1} (0), \dots, ρ_{n} (0), ρ_{1} (π), \dots, ρ_{n} (π))^{T}$ are two unique solution of Equation (6.6) and (6.10) respectively, and ρ_i (φ) , φ ∈ {0, π} satisfy the conditions (1)-(3) of Theorem 5.6 .

Since the vectors X, X_ρ are two unique solution of Equation (6.6) and (6.10) if only if the matrix A, R nonsingular, we only to prove ρ_i (φ) , φ ∈ {0, π} satisfy the conditions (1)-(3) of Theorem 5.6 .

In fact, since for arbitrary $\tilde{B},$ $ρ_{i} (φ) = s_{i 1} b_{1} (0) + s_{i 2} b_{2} (0) + \dots + s_{in} b_{n} (0)$ $+ s_{i, n + 1} b_{1} (π) + s_{i, n + 2} b_{2} (π) + \dots + s_{i, 2 n} b_{n} (π) \geq 0$ if only if s_i,j ≥ 0 . In addition, it is obvious that ρ_i (φ) are bounded left continuous non-increasing functions on (0, 1] with respect to r and right continuous at r = 0 if only if b_i (φ) , φ {0, π} are bounded left continuous non-increasing functions on (0, 1] with respect to r and right continuous at r = 0 . Therefore ρ_i (φ) , φ ∈ {0, π} satisfy the conditions (1)-(3) of Theorem 5.6 if only if the matrix R^-1 = (s_i,j) _2n×2n is nonnegative.

Therefore, the vector $\tilde{X} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})^{T}$ is a unique solution of Equation (6.2) for arbitrary $\tilde{B}$ if only if A, R nonsingular and R^-1 = (s_i,j) _2n×2n is nonnegative. □

From the proof process of Theorem 7.1, we may prove the follow theorem.

Theorem 7.2. A vector $\tilde{X}$ is a unique solution of Equation (6.2) if only if A, R nonsingular, ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where φ ∈ {0, π} , 1 ≤ i ≤ 2n .

Example 7.3. Consider the 2 × 2 fuzzy system of linear equations $\begin{matrix} \begin{matrix} {\tilde{x}}_{1} - {\tilde{x}}_{2} = {\tilde{b}}_{1}, \\ {\tilde{x}}_{1} + 3 {\tilde{x}}_{2} = {\tilde{b}}_{2}, \end{matrix} \end{matrix}$ (7.1) where ${\tilde{b}}_{1} = \hat{1} + {\tilde{R}}_{1}^{b} (φ) e^{i φ}, [{\tilde{R}}_{1}^{b} (φ)]^{r} = [0, 1 - r], φ \in {0, π},$ ${\tilde{b}}_{2} = \hat{5} + {\tilde{R}}_{2}^{b} (φ) e^{i φ}, [{\tilde{R}}_{2}^{b} (0)]^{r} = [0, 2 (1 - r)],$ $[{\tilde{R}}_{2}^{b} (π)]^{r} = [0, 1 - r] .$

Obviously, $A = (\begin{matrix} 1 & - 1 \\ 1 & 3 \end{matrix}), R = (\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 3 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 3 \end{matrix}),$ $B_{ρ} = (\begin{matrix} 1 - r \\ 2 (1 - r) \\ 1 - r \\ 1 - r \end{matrix}) .$ Since |A|=4 ≠ 0, |R|=8 ≠ 0, we know A and R nonsingular and $R^{- 1} = (\begin{matrix} \frac{9}{8} & - \frac{1}{8} & \frac{3}{8} & - \frac{3}{8} \\ - \frac{3}{8} & \frac{3}{8} & - \frac{1}{8} & \frac{1}{8} \\ \frac{3}{8} & - \frac{3}{8} & \frac{9}{8} & - \frac{1}{8} \\ - \frac{1}{8} & \frac{1}{8} & - \frac{3}{8} & \frac{3}{8} \end{matrix}) .$ Hence by X_ρ = R^-1B_ρ, we have $(\begin{matrix} ρ_{1} (0) \\ ρ_{2} (0) \\ ρ_{1} (π) \\ ρ_{2} (π) \end{matrix}) = (\begin{matrix} \frac{7}{8} (1 - r) \\ \frac{3}{8} (1 - r) \\ \frac{5}{8} (1 - r) \\ \frac{1}{8} (1 - r) \end{matrix}) .$ Apparently, ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where i = 1, 2, φ ∈ {0, π} . Therefore the vector $\tilde{X}$ is the unique solution of Equation (7.1) .

Theorem 7.4. If r (T) = r (A) , R nonsingular and R^-1 = (s_i,j) _2n×2n is nonnegative, i.e. $s_{ij} \geq 0, 1 \leq i, j \leq 2 n,$ then the vector $\tilde{X}$ is a solution of Equation (6.2) .

Proof. By Theorem 6.1, we know that the vector $\tilde{X} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})^{T}$ is a solution of Equation (6.2) if only if the vectors $X = (x_{1}, x_{2}, \dots, x_{n})^{T},$ $X_{ρ} = (ρ_{1} (0), \dots, ρ_{n} (0), ρ_{1} (π), \dots, ρ_{n} (π))^{T}$ are two solutions of Equation (6.6) and (6.10) respectively, and ρ_i (φ) , φ ∈ {0, π} satisfy conditions (1)-(3) of Theorem 5.6 .

Since the vector X_ρ is a solution of Equation (6.10) if only if matrix R nonsingular, similar to proof of Theorem 7.1, we also may prove ρ_i (φ) , φ ∈ {0, π} satisfy conditions (1)-(3) of Theorem 5.6 if only if matrix R^-1 = (s_i,j) _2n×2n is nonnegative.

Additional, since r (T) = r (A) , the vector X is a solution of Equation (6.6) . Therefore, the vector $\tilde{X}$ is a solution of Equation (6.2) . □

From the proof process of Theorem 7.4, we may prove the follow theorem.

Theorem 7.5. If r (T) = r (A) , R nonsingular, ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where φ ∈ {0, π} , 1 ≤ i ≤ 2n, then the vector $\tilde{X}$ is a solution of Equation (6.2) .

The Theorem 1 in the reference [22] tell us that if R is nonsingular, then A is also nonsingular, that is to say, r (T) = r (A) . Therefore, Theorem 7.5 is same with Theorem 7.2 .

Theorem 7.6. If r (S) = r (R) , A nonsingular, ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where φ ∈ {0, π} , 1 ≤ i ≤ 2n, then the vector $\tilde{X}$ is a solution of Equation (6.2) .

Since matrix A nonsingular, the vector X is a unique solution of Equation (6.6) .

Furthermore, by r (S) = r (R) , we know the vector X_ρ is a solution of Equation (6.10) , and similar to proof of Theorem 7.1, we get ρ_i (φ) , φ ∈ {0, π} satisfy conditions (1)-(3) of Theorem 5.6 if ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where φ ∈ {0, π} , 1 ≤ i ≤ 2n . □

Example 7.7. Consider the 2 × 2 fuzzy system of linear equations $\begin{matrix} \begin{matrix} {\tilde{x}}_{1} - {\tilde{x}}_{2} = {\tilde{b}}_{1}, \\ {\tilde{x}}_{1} + {\tilde{x}}_{2} = {\tilde{b}}_{2}, \end{matrix} \end{matrix}$ (7.2) where ${\tilde{b}}_{1} = \hat{1} + {\tilde{R}}_{1}^{b} (φ) e^{i φ}, [{\tilde{R}}_{1}^{b} (0)]^{r} = [0, 1 - r],$ $[{\tilde{R}}_{1}^{b} (π)]^{r} = [0, 2 (1 - r)] .$ ${\tilde{b}}_{2} = \hat{5} + {\tilde{R}}_{2}^{b} (φ) e^{i φ}, [{\tilde{R}}_{2}^{b} (0)]^{r} = [0, 2 (1 - r)],$ $[{\tilde{R}}_{2}^{b} (π)]^{r} = [0, 1 - r] .$

Obviously, $A = (\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}), R = (\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix}),$ $B_{ρ} = (\begin{matrix} 1 - r \\ 2 (1 - r) \\ 2 (1 - r) \\ 1 - r \end{matrix}) .$ Since |A|=2 ≠ 0, |R|=0, we know A is nonsingular, R is singular and r (S) = r (R) . Furthermore,

$(\begin{matrix} ρ_{1} (0) \\ ρ_{2} (0) \\ ρ_{1} (π) \\ ρ_{2} (π) \end{matrix}) = (\begin{matrix} 1 - r \\ 1 - r \\ 1 - r \\ 0 \end{matrix}) .$ Apparently, ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where i = 1, 2, φ ∈ {0, π} . Therefore the vector $\tilde{X}$ is a solution of Equation (7.2) .

By the proof process of Theorem 7.5 and 7.6, we also get follow result.

Theorem 7.8. If r (T) = r (A) , r (S) = r (R), ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where φ ∈ {0, π} , 1 ≤ i ≤ 2n, then the vector $\tilde{X}$ is a solution of Equation (6.2) .

Example 7.9. Consider the 2 × 2 fuzzy system of linear equations $\begin{matrix} \begin{matrix} {\tilde{x}}_{1} - {\tilde{x}}_{2} = {\tilde{b}}_{1}, \\ 2 {\tilde{x}}_{1} - 2 {\tilde{x}}_{2} = {\tilde{b}}_{2}, \end{matrix} \end{matrix}$ (7.3) where ${\tilde{b}}_{1} = \hat{1} + {\tilde{R}}_{1}^{b} (φ) e^{i φ}, [{\tilde{R}}_{1}^{b} (φ)]^{r} = [0, 1 - r], φ \in {0, π} .$ ${\tilde{b}}_{2} = \hat{2} + {\tilde{R}}_{2}^{b} (φ) e^{i φ}, [{\tilde{R}}_{2}^{b} (φ)]^{r}$ $= [0, 2 (1 - r)], φ \in {0, π} .$

Obviously, $A = (\begin{matrix} 1 & - 1 \\ 2 & - 2 \end{matrix}), R = (\begin{matrix} 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 2 \\ 0 & 1 & 1 & 0 \\ 0 & 2 & 2 & 0 \end{matrix}),$ $B_{ρ} = (\begin{matrix} 1 - r \\ 2 (1 - r) \\ 1 - r \\ 2 (1 - r) \end{matrix}) .$ Since |A|=0, |R|=0, we know A, R are singular, and $r (T) = r (A), r (S) = r (R) .$ Furthermore, $(\begin{matrix} ρ_{1} (0) \\ ρ_{2} (0) \\ ρ_{1} (π) \\ ρ_{2} (π) \end{matrix}) = (\begin{matrix} 1 - r \\ 1 - r \\ 0 \\ 0 \end{matrix}) .$ Apparently, ρ_i (φ) = ρ_i (r, φ) are non-increasing functions on [0, 1] with respect to r and $ρ_{i} (φ) \geq 0,$ where i = 1, 2, φ ∈ {0, π} . Therefore the vector $\tilde{X}$ is a solution of Equation (7.3) .

8 Inclusion

We have discussed a new representation of the two-dimensional fuzzy numbers and its operational properties. The background of this paper focuses on solving the two-dimensional fuzzy systems of linear equations. Applying the theory proposed in this paper, the problems mentioned above are solved. It includes the new representation of the two-dimensional fuzzy numbers, operations of the two-dimensional fuzzy numbers and the new method to solving the two-dimensional fuzzy systems of linear equations. Finally, the solutions of fuzzy linear system with the one-dimensional fuzzy number are discussed with same method and compared with known solutions. In the future, the research in this field will focus on analysis of two-dimensional fuzzy numbers and application of two-dimensional fuzzy linear system.

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