An efficient method for solving LR fuzzy dual matrix systems

1 Introduction

The linear system of equations plays a significant role in different areas such as social sciences, engineering, mathematics, physics and statistics [1]. Therefore it is so important to develop the system and find new methods to solve it. Since 1998, so many works have been performed on the subject. Firstly Fridman and Ming solved A x ˜ = B ˜ fuzzy system by embedding method [1]. In [2], it is discussed on dual fuzzy linear system as A x ˜ = B x ˜ + C ˜ and two necessary conditions for the solution existence are given. It is determined that the systems of A x ˜ = B ˜ and A ˜ 1 x ˜ + b 1 ˜ = A ˜ 2 x ˜ + b 2 ˜ with A ˜ = A ˜ 1 ⊖ A ˜ 2 and B ˜ = b ˜ 2 ⊖ b ˜ 1 are not the same, for the reason that there is no fuzzy symmetry. Therefore, Muzzioli and Reynarets introduced a method to solve the system in [3]. They gave the conditions under which the system has a vector solution and they showed these linear systems have the same vector solutions. Then, numerical methods applied to solve the system [4–10]. Allahvirnloo [11, 12] recently represented a work in which the system solution is achieved by solving and widening 1-cut system and in [13], the solution of fuzzy linear system (FLS) was investigated based on a 1-level expansion. LR fuzzy systems of linear algebraic equations are discussed by Abramovich et al. in [14]. In the paper, we focus on LR fuzzy dual matrix system and we suggest a simple and quick method to solve it. The paper is organized as follows: Section 2 presents the preliminary definitions of fuzzy numbers. In Section 3 the main idea for solving the LR fuzzy dual matrices system is represented. Example will be provided in Section 4 and finally, Section 5 is allocated to conclusion.

2 Basic definitions

Definition 2.1. [15] A fuzzy number is a fuzzy set like u ˜ ( x ) : [ a , d ] ⟶ [ 0 , 1 ] which satisfies

u ˜ is upper semi-continuous,

Definition 2.2. [16] A fuzzy number u ˜ is said to be an LR fuzzy number if

u ˜ ( x ) = { L ( a - x α ) , x ≤ a , α > 0 R ( x - a β ) , x ≥ a , β > 0 where a is the core of u ˜, α and β are left and right spreads, respectively, and the function L (.), which is called left shape function, satisfies:

L (x) = L (- x)

The definition of right shape function R (.) is similar to that of L (.). The core, left and right spreads, and the shape function of LR fuzzy number u ˜ are symbolically shown as: u ˜ = ( a , α , β ). Clearly, u ˜ = ( a , α , β ) is positive if and only if a - α > 0 (since L (1) =0), and u ˜ = ( a , α , β ) is a symmetric fuzzy number if and only if α = β. In [17], Dubois and Prade designed the following exact formulas for the addition of LR fuzzy numbers and scalar multiplication. They also introduced an approximate formula for multiplying the LR fuzzy numbers. For two LR fuzzy numbers u ˜ = ( a , α , β ) and v ˜ = ( b , γ , δ ), we have:

•Addition:

u ˜ + v ˜ = ( a , α , β ) + ( b , γ , δ ) = ( a + b , α + γ , β + δ )

•Scalar  multiplication:

λ ⨂ u ˜ = λ ⨂ ( a , α , β ) = { ( λ a , λ α , λ β ) , λ ≥ 0 ( λ a , - λ β , - λ α ) , λ < 0

Definition 2.3. Let A ˜ = ( a , α , β ) and B ˜ = ( b , δ , γ ) are two fuzzy numbers, then Hukuhara difference between A ˜ and B ˜ define as follows: A ˜ ⊖ H B ˜ = ( a - b ,α - δ, β - γ).

Definition 2.4. If A = [a _ij ] _n×n, 1 ≤ i, j ≤ n is a crisp matrix then we define quasi norm A as: ‖ A ‖ = max i , j { ∣ a ij ∣ }.

Proposition 2.1. For A = [a _ij ] _n×n, where a ij ∈ R R , 1 ≤ i , j ≤ n, the quasi norm defined in Definition 2.3, satisfies the following properties:

||A||≥0, ||A||=0 ↔ A = 0

Definition 2.5. Matrix A is called a LR fuzzy matrix if each element of A is a LR fuzzy number and showing as A ˜ = [ ( a ij , α ij , β ij ) ].

Definition 2.6. The fuzzy linear system [ a 11 … a 1 n ⋮ ⋮ ⋮ a n 1 … a nn ] [ ( x 11 , α 11 , β 11 ) … ( x 1 n , α 1 n , β 1 n ) ⋮ ⋮ ⋮ ( x n 1 , α n 1 , β n 1 ) … ( x nn , α nn , β nn ) ] + [ ( b 11 , α b 11 , β b 11 ) … ( b 1 n , α b 1 n , β b 1 n ) ⋮ ⋮ ⋮ ( b n 1 , α b n 1 , β b n 1 ) … ( b nn , α b nn , β b nn ) ] = [ c 11 … c 1 n ⋮ ⋮ ⋮ c n 1 … c nn ] [ ( x 11 , α 11 , β 11 ) … ( x 1 n , α 1 n , β 1 n ) ⋮ ⋮ ⋮ ( x n 1 , α n 1 , β n 1 ) … ( x nn , α nn , β nn ) ] + [ ( d 11 , α d 11 , β d 11 ) … ( d 1 n , α d 1 n , β d 1 n ) ⋮ ⋮ ⋮ ( d n 1 , α d n 1 , β d n 1 ) … ( d nn , α d nn , β d nn ) ] is called a fuzzy dual matrix system in which A = [a _ij ] and C = [c _ij ] are n × n crisp matrices and B ˜ = [ ( b ij , α b ij , β b ij ) ], D ˜ = [ ( d ij , α d ij , β d ij ) ], X ˜ = [ ( x ij , α ij , β ij ) ] are n × n LR fuzzy matrices.

Definition 2.7. [18] The problem Minf ( x )

s . t x ∈ X ,(1)is called an optimization problem where f is a real-valued function on R ⁿ called the objective function and X is a nonempty subset of R ⁿ given by means of real-valued function g₁, …, g _n on R ⁿ , the set of all solutions of the system

g _i  (x) = b _i ,   i = 1, …, m

g _i  (x) ≤ b _i ,   i = m₁ + 1, …, m

x _j  ≥ 0,   j = 1, …, n

called constraints. The elements of X are called feasible solutions of (2), and the feasible solution x^* where f attains its global minimum over X is called the optimal solution and f (x^*) is called optimal value.

In this section, we solve the system of fuzzy Dual matrix (1). As we mentioned, the fuzzy dual matrix system in the LR form is, as follows:

[ a 11 … a 1 n ⋮ ⋮ ⋮ a n 1 … a nn ] [ ( x 11 , α 11 , β 11 ) … ( x 1 n , α 1 n , β 1 n ) ⋮ ⋮ ⋮ ( x n 1 , α n 1 , β n 1 ) … ( x nn , α nn , β nn ) ] + [ ( b 11 , α b 11 , β b 11 ) … ( b 1 n , α b 1 n , β b 1 n ) ⋮ ⋮ ⋮ ( b n 1 , α b n 1 , β b n 1 ) … ( b nn , α b nn , β b nn ) ] = [ c 11 … c 1 n ⋮ ⋮ ⋮ c n 1 … c nn ] [ ( x 11 , α 11 , β 11 ) … ( x 1 n , α 1 n , β 1 n ) ⋮ ⋮ ⋮ ( x n 1 , α n 1 , β n 1 ) … ( x nn , α nn , β nn ) ] + [ ( d 11 , α d 11 , β d 11 ) … ( d 1 n , α d 1 n , β d 1 n ) ⋮ ⋮ ⋮ ( d n 1 , α d n 1 , β d n 1 ) … ( d nn , α d nn , β d nn ) ](2)where A-C is nonsingular.

First, the 1-level system ∑ k = 1 n a ik x kj + b ij = ∑ k = 1 n c ik x kj + d ij , i , j = 1 , … , n ,(3)is solved then a minimization problem is proposed to find the spreads. Obviously, solving these problems often is difficult specially in nonlinear case. So, we have to use some norms to linearity and simplification. To this end, based on the proposed method in [10], we suppose the following minimization problem: Min { ∥ ∑ a ik ≥ 0 a ik α kj - ∑ a ik < 0 a ik β kj + α b ij - ∑ c ik ≥ 0 c ik α kj + ∑ c ik < 0 c ik β kj - α d ij ∥ + ∥ ∑ a ik ≥ 0 a ik β kj - ∑ a ik < 0 a ik α kj + β b ij - ∑ c ik ≥ 0 c ik β kj + ∑ c ik < 0 c ik α kj - β d ij ∥ s . t . α kj , β kj ≥ 0 , i , j , k = 1 , … , n ,(4)where by using the definition of quasi norm, it can be written as follows:

Min { Max i , j ∣ ∑ a ik ≥ 0 a ik α kj - ∑ a ik < 0 a ik β kj + α b ij - ∑ c ik ≥ 0 c ik α kj + ∑ c ik < 0 c ik β kj - α d ij ∣ + Max i , j ∣ ∑ a ik ≥ 0 a ik β kj - ∑ a ik < 0 a ik α kj + β b ij - ∑ c ik ≥ 0 c ik β kj + ∑ c ik < 0 c ik α kj - β d ij ∣ }

s . t . α kj , β kj ≥ 0 .(5)

By using:Max i , j ∣ ∑ a ik ≥ 0 a ik α kj - ∑ a ik < 0 a ik β kj + α b ij - ∑ c ik ≥ 0 c ik α kj + ∑ c ik < 0 c ik β kj - α d ij ∣ = z 1(6) Max i , j ∣ ∑ a ik ≥ 0 a ik β kj - ∑ a ik < 0 a ik α kj + β b ij - ∑ c ik ≥ 0 c ik β kj + ∑ c ik < 0 c ik α kj - β d ij ∣ } = z 2

we will have:Min z 1 + z 2(7) s . t . - z 1 ≤ ∑ a ik ≥ 0 a ik α kj - ∑ a ik < 0 a ik β kj + α b ij - ∑ c ik ≥ 0 c ik α kj + ∑ c ik < 0 c ik β kj - α d ij ≤ z 1 - z 2 ≤ ∑ a ik ≥ 0 a ik β kj - ∑ a ik < 0 a ik α kj + β b ij - ∑ c ik ≥ 0 c ik β kj + ∑ c ik < 0 c ik α kj - β d ij ≤ z 2 α kj , β kj , z 1 , z 2 ≥ 0 . k , i , j = 1 , . . . , n . The spreads of solution are obtained from the model (7). Finally, by allocating these spreads to solution of 1-level system, the solution is resulted.

Lemma 3.1. Every convex combination of two optimal solution the LR fuzzy dual matrix system is an optimal solution.

Proof: Assume ( α kj ′ , β kj ′ , z 1 ′ , z 2 ′ ) and ( α kj ″ , β kj ″ , z 1 ″ , z 2 ″ ) are optimal solutions, then their convex combination is as follows: ( α kj , β kj , z 1 , z 2 ) = λ ( α kj ′ , β kj ′ , z 1 ′ , z 2 ′ ) + ( 1 - λ ) ( α kj ″ , β kj ″ , z 1 ″ , z 2 ″ ) , λ ∈ [ 0 , 1 ] or α kj = λ α kj ′ + ( 1 - λ ) α kj ″ β kj = λ β kj ′ + ( 1 - λ ) β kj ″ z 1 = λ z 1 ′ + ( 1 - λ ) z 1 ″ z 2 = λ z 2 ′ + ( 1 - λ ) z 2 ″

Now, we have:

- λ z 1 ′ ≤ ∑ a ik ≥ 0 ( a ik λ α kj ′ ) - ∑ a ik < 0 ( a ik λ β kj ′ ) + λ α b ij - ∑ c ik ≥ 0 ( c ik λ α kj ′ ) + ∑ c ik < 0 ( c ik λ β kj ′ ) - ( λ α d ij ) ≤ λ z 1 ′(6) - ( 1 - λ ) z 1 ′ ≤ ∑ a ik ≥ 0 ( a ik ( 1 - λ ) α kj ′ ) - ∑ a ik < 0 ( a ik ( 1 - λ ) β kj ′ ) + ( 1 - λ ) α b ij - ∑ c ik ≥ 0 ( c ik ( 1 - λ ) α kj ′ ) + ∑ c ik < 0 ( c ik ( 1 - λ ) β kj ′ ) - ( ( 1 - λ ) α d ij ) ≤ ( 1 - λ ) z 1 ′ .(7) By adding equations (8) and (9) - z 1 ≤ ∑ a ik ≥ 0 a ik α kj - ∑ a ik < 0 a ik β kj + α b ij - ∑ c ik ≥ 0 c ik α kj + ∑ c ik < 0 c ik β kj - α d ij ≤ z 1 .

Also, similarly, we obtain - z 2 ≤ ∑ a ik ≥ 0 a ik β kj - ∑ a ik < 0 a ik α kj + β b ij - ∑ c ik ≥ 0 c ik β kj + ∑ c ik < 0 c ik α kj - β d ij ≤ z 2 .

Then, these mentioned solutions are feasible. Now, we are going to prove the optimality (α _kj , β _kj , z₁, z₂) let z^* is the optimal solution and z 1 ′ + z 2 ′ = z * and z 1 ″ + z 2 ″ = z * hence, z 1 + z 2 = ( λ z 1 ′ + ( 1 - λ ) z 1 ″ ) + ( λ z 2 ′ + ( 1 - λ ) z 2 ″ ) = λ z * + ( 1 - λ ) z * = z *.

Then the proof is completed.

Lemma 3.2. The solutions set of the system (7) is a convex set.

Proof. Assume that S is the set of solutions of (7) and X 1 ¯ , X 2 ¯ are two different solutions. according the proof of Lemma 3.1, every convex combination of two solutions is a solution, then proof is completed.

Theorem 3.1. If model (7) has more than two optimal solutions that those are fuzzy solution for system A x ˜ + B ˜ = C x ˜ + D ˜ then it has infinite number of fuzzy solutions.

Proof. Let X ˜ 1 = ( α kj ′ , β kj ′ , z 1 ′ , z 2 ′ ) and X ˜ 2 = ( α kj ″ , β kj ″ , z 1 ″ , z 2 ″ ) are the fuzzy solutions of A x ˜ + B ˜ = C x ˜ + D ˜, so for 1-cut we have AX₁ + B = CX₁ + D, AX₂ + B = CX₂ + D and z 1 ′ + z 2 ′, z 1 ″ + z 2 ″, gets the minimum value. It is sufficient that prove, Y ˜ = λ X ˜ 1 + ( 1 - λ ) X ˜ 2 is a fuzzy solution. We have { λ A X 1 + λ B = C λ X 1 + λ D ( 1 - λ ) A X 2 + ( 1 - λ ) B = C ( 1 - λ ) X 2 + ( 1 - λ ) D

⇒ A (λX₁  +   (1  -  λ) X₂) + B = C (λX₁ + (1 - λ) X₂)+D

⇒AY + B = CY + D.

Now, the proof of lemme 3.1 concludes that, if z 1 ′ + z 2 ′,z 1 ″ + z 2 ″ gets minimum value then z₁ + z₂ get minimum value too.

The proof is completed.

Theorem 3.2. If B ˜ = D ˜ = 0 ˜ then system (1) has a crisp solution.

Proof. Since B ˜ = D ˜ = 0, then system (1) is converted to A x ˜ = C x ˜, where it is a crisp system so it has a crisp solution. The proof is completed.

Lemma 3.3. If A = [a _ij ] , C = [c _ij ] and a _ij  ≥ c _ij for a _ij  ≥ 0,  c _ij  ≥ 0 and a _ij  ≤ c _ij for a _ij  ≤ 0,  c _ij  ≤ 0 and D ˜ ⊖ H B ˜ exist then the systems A x ˜ + B ˜ = C x ˜ + D ˜, ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ) have the same fuzzy solution.

Proof: x ˜ is solution of A x ˜ + B ˜ = C x ˜ + D ˜ if and only if ( ∑ k = 1 n a ik x kj + b ij , ∑ a ik ≥ 0 a ik α kj - ∑ a ik < 0 a ik β kj + α b ij , ∑ a ik ≥ 0 a ik β kj - ∑ a ik < 0 a ik α kj + β b ij ) = ( ∑ k = 1 n c ik x kj + d ij , ∑ c ik ≥ 0 c ik α kj - ∑ c ik < 0 c ik β kj + α d ij , ∑ c ik ≥ 0 c ik β kj - ∑ c ik < 0 c ik α kj + β d ij )

if and only if

( ∑ k = 1 n ( a ik - c ik ) x kj , ∑ a ik , c ik ≥ 0 ( a ik - c ik ) α kj - ∑ a ik , c ik < 0 ( a ik - c ik ) β kj , ∑ a ik , c ik ≥ 0 ( a ik - c ik ) β kj - ∑ a ik , c ik < 0 ( a ik - c ik ) α kj ) = ( d ij - b ij , α d ij - α b ij , β d ij - β b ij )

if and only if

( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ).

Then x ˜ is solution of ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ) and proof complete.

Example 4.1. Consider the following matrix system:

[ 4 1 2 5 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 2 , 0 , 3 ) ( 1 , 0 , 0 ) ( 2 , 1 , 3 ) ( 1 , 5 , 4 ) ] = [ 3 1 2 2 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 4 , 1 , 6 ) ( 2 , 1 , 1 ) ( 5 , 3 , 7 ) ( 3 , 1 , 4 ) ]

The solution of the 1-level system is [ 2 1 1 2 3 ]. Using model (7), we have min z 1 + z 2 - z 1 ≤ α 11 - 1 ≤ z 1 - z 1 ≤ 3 α 21 - 2 ≤ z 1 - z 1 ≤ α 12 - 1 ≤ z 1 - z 1 ≤ 3 α 22 ≤ z 1 - z 2 ≤ β 11 - 3 ≤ z 2 - z 2 ≤ 3 β 21 - 4 ≤ z 2 - z 2 ≤ β 12 - 1 ≤ z 2 - z 2 ≤ 3 β 22 - 3 ≤ z 2 α i , j , β i , j , z 1 , z 2 ≥ 0 , i , j = 1 , 2 .

The solutions are [ ( α 11 , β 11 ) ( α 12 , β 12 ) ( α 21 , β 21 ) ( α 22 , β 22 ) ] = [ ( 1 , 3 ) ( 0 , 1 ) ( 0.667 , 1.33 ) ( 0 , 1 ) ], z₁ = z₂ = 0, then the fuzzy solution is X ˜ = [ ( 2 , 1 , 3 ) ( 1 , 0 , 1 ) ( 1 , 0.667 , 1.33 ) ( 2 3 , 0 , 1 ) ]. On other hand, we have a _ij  ≥ c _ij , a _ij  ≥ 0,  c _ij  ≥ 0, i, j = 1, 2, so lemma 3.3 concludes that the system ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ) has the same fuzzy solution, as X ˜ = [ ( 2 , 1 , 3 ) ( 1 , 0 , 1 ) ( 1 , 0.667 , 1.33 ) ( 2 3 , 0 , 1 ) ].

Example 4.2. Consider the following matrix system:

[ 4 1 - 1 0 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 3 , 2 , 1 ) ( 4 , 4 , 3 ) ( 1 , 1 , 1 ) ( 2 , 3 , 2 ) ] = [ 2 3 1 - 4 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 2 , 4 , 2 ) ( 1 , 3 , 2 ) ( 3 , 2 , 3 ) ( 4 , 1 , 1 ) ]

The solution of the 1-level system is [ 0 - 2 1 2 - 1 2 ]. By using model (7), we have

min z 1 + z 2 - z 1 ≤ 2 α 11 - 2 α 21 - 2 ≤ z 1 - z 1 ≤ 2 α 12 - 2 α 22 + 1 ≤ z 1 - z 1 ≤ β 11 - α 11 - 4 β 21 - 1 ≤ z 1 - z 1 ≤ β 12 - α 12 - 4 β 22 + 2 ≤ z 1 - z 2 ≤ 2 β 11 - 2 β 21 - 1 ≤ z 2 - z 2 ≤ 2 β 12 - 2 β 22 + 1 ≤ z 2 - z 2 ≤ α 11 - β 11 - 4 α 21 - 2 ≤ z 2 - z 2 ≤ α 12 - 4 α 22 - β 12 + 1 ≤ z 2 α i , j , β i , j , z 1 , z 2 ≥ 0 , i , j = 1 , 2 .

The solutions are [ ( α 11 , β 11 ) ( α 12 , β 12 ) ( α 21 , β 21 ) ( α 22 , β 22 ) ] = [ ( 1 , 2 ) ( 0 , 0 ) ( 0 , 0 ) ( 1 2 , 1 2 ) ], z₁ = 0, z₂ = 3, then the fuzzy solution is X ˜ = [ ( 0 , 1 , 2 ) ( - 2 , 0 , 0 ) ( 1 2 , 0 , 0 ) ( - 1 2 , 1 2 , 1 2 ) ]. We observe that D ˜ ⊖ H B ˜ does not exist, so we can not construct system as ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ).

Example 4.3. Consider the following matrix system:

[ - 2 - 3 - 1 - 4 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 1 , 2 , 1 ) ( 2 , 3 , 5 ) ( - 4 , 2 , 1 ) ( - 2 , 2 , 4 ) ] = [ - 1 - 2 - 1 - 3 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 2 , 4 , 3 ) ( - 2 , 5 , 6 ) ( 7 , 7 , 2 ) ( 4 , 3 , 6 ) ]

The solution of the 1-level system is [ 10 10 - 11 - 6 ]. By using model (7), we have min z 1 + z 2 - z 1 ≤ β 1 + β 21 - 2 ≤ z 1 - z 1 ≤ β 12 + β 22 - 2 ≤ z 1 - z 1 ≤ β 21 - 5 ≤ z 1 - z 1 ≤ β 22 - 1 ≤ z 1 - z 2 ≤ α 11 + α 21 - 2 ≤ z 2 - z 2 ≤ α 12 + α 22 - 1 ≤ z 2 - z 2 ≤ α 21 - 1 ≤ z 2 - z 2 ≤ α 22 - 2 ≤ z 2 α i , j , β i , j , z 1 , z 2 ≥ 0 , i , j = 1 , 2 .

The solutions are [ ( α 11 , β 11 ) ( α 12 , β 12 ) ( α 21 , β 21 ) ( α 22 , β 22 ) ] = [ ( 2 , 0 ) ( 0 , 7 2 ) ( 1 2 , 7 2 ) ( 3 2 , 0 ) ], z 1 = 3 2 , z 2 = 1 2, then the fuzzy solution is X ˜ = [ ( 10 , 2 , 0 ) ( 10 , 0 , 7 2 ) ( - 11 , 1 2 , 7 2 ) ( - 6 , 3 2 , 0 ) ].

On other hand, we have a _ij  ≤ c _ij , a _ij  ≤ 0,  c _ij  ≤ 0, so lemma 3.3 concludes that system ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ) has the same fuzzy solution, as X ˜ = [ ( 10 , 2 , 0 ) ( 10 , 0 , 7 2 ) ( - 11 , 1 2 , 7 2 ) ( - 6 , 3 2 , 0 ) ].

Example 4.4. Consider the following matrix system: [ 1 2 3 1 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 3 , 0 , 2 ) ( 4 , 1 , 2 ) ( 1 , 1 , 0 ) ( 2 , 5 , 4 ) ] = [ 4 3 1 - 1 ] [ x ˜ 11 x ˜ 12 x ˜ 21 x ˜ 22 ] + [ ( 1 , 1 , 2 ) ( - 4 , 4 , 3 ) ( 0 , 2 , 1 ) ( 12 , 5 , 5 ) ]

The solution of the 1-level system is [ 5 4 3 2 - 7 4 7 2 ]. By using model (7), we have min z 1 + z 2 - z 1 ≤ - 3 α 11 - α 21 - 1 ≤ z 1 - z 1 ≤ 2 α 11 + α 21 - β 21 - 1 ≤ z 1 - z 1 ≤ - 3 α 12 - α 22 - 3 ≤ z 1 - z 1 ≤ 2 α 12 + 2 α 22 - β 22 ≤ z 1 - z 2 ≤ - 3 β 11 - β 21 ≤ z 2 - z 2 ≤ 2 β 11 + β 21 - α 21 - 1 ≤ z 2 - z 2 ≤ - 3 β 12 - β 22 - 1 ≤ z 2 - z 2 ≤ 2 β 12 + 2 β 22 - α 22 - 1 ≤ z 2 α i , j , β i , j , z 1 , z 2 ≥ 0 , i , j = 1 , 2 .

The solutions are [ ( α 11 , β 11 ) ( α 12 , β 12 ) ( α 21 , β 21 ) ( α 22 , β 22 ) ] = [ ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ], z₁ = z₂ = 0, then the fuzzy solution is X ˜ = [ ( 5 4 , 0 , 0 ) ( 3 2 , 0 , 0 ) ( - 7 4 , 0 , 0 ) ( 7 2 , 0 , 0 ) ].

Now we consider ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ) form of above system as follows: [ - 3 - 1 2 3 ] [ x 11 ~ x 12 ~ x 21 ~ x 21 ~ ] = [ ( - 2 , 1 , 0 ) ( - 8 , 3 , 1 ) ( - 1 , 1 , 1 ) ( 10 , 0 , 1 ) ]

By using model (7) we have min z 1 + z 2 - z 1 ≤ 3 β 11 + β 21 - 1 ≤ z 1 - z 1 ≤ 2 α 11 + 2 α 21 - 1 ≤ z 1 - z 1 ≤ 3 β 12 + β 22 - 3 ≤ z 1 - z 1 ≤ 2 α 12 + 2 α 22 ≤ z 1 - z 2 ≤ 3 α 11 + α 21 ≤ z 2 - z 2 ≤ 2 β 11 + 2 β 21 - 1 ≤ z 2 - z 2 ≤ 3 α 12 + α 22 - 1 ≤ z 2 - z 2 ≤ 2 β 12 + 2 β 22 - 1 ≤ z 2 α i , j , β i , j , z 1 , z 2 ≥ 0 , i , j = 1 , 2 . By solving, we obtain z₁ = 0.6, z₂ = 0.2 and [ ( α 11 , β 11 ) ( α 12 , β 12 ) ( α 21 , β 21 ) ( α 22 , β 22 ) ] = [ ( 0 , 0.4 ) ( 0.267 , 0.8 ) ( 0.2 , 0 ) ( 0 , 0 ) ] .

Then the fuzzy solution is X ˜ = [ ( 5 4 , 0 , 0.4 ) ( 3 2 , 0.267 , 0.8 ) ( - 7 4 , 0.2 , 0 ) ( 7 2 , 0 , 0 ) ].

We observe that since the condition of lemma 3.3 is not hold so the systems A x ˜ + B ˜ = C x ˜ + D ˜, ( A - C ) x ˜ = ( D ˜ ⊖ H B ˜ ) do not have the same fuzzy solution.

5 Conclusion

In this paper, we proposed a model for solving the LR fuzzy dual matrix system A x ˜ + B ˜ = C x ˜ + D ˜ in which A and B are n × n crisp matrices and C ˜, D ˜ are LR fuzzy matrices. Firstly, we solved 1-level for calculating the core and obtained spreads by solving an optimization problem. In addition, we explained about the relationship between the systems of A x ˜ + B ˜ = C x ˜ + D ˜ and ( A - C ) x ˜ = ( D ˜ ⊖ h B ˜ ). Numerical examples showed that our method is feasible to solve this type of fuzzy matrix system.