The fuzzy matrix equations in which , , and are m × m, n × n, and m × n nonnegative LR fuzzy numbers matrices, respectively, are investigated. The fuzzy matrix systems is extended into three crisp systems of linear matrix equations according to arithmetic operations of LR fuzzy numbers. Based on pseudoinverse of matrix, the fuzzy approximate solution of original fuzzy systems is obtained by solving the crisp linear matrix systems. In addition, the existence condition of nonnegative fuzzy solution is discussed. Two examples are calculated to illustrate the proposed method.

1. Introduction

Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modeling, which can formulate uncertainty in actual environment. In many matrix equations, some or all of the system parameters are vague or imprecise, and fuzzy mathematics is a better tool than crisp mathematics for modeling these problems, and hence solving a fuzzy matrix equation is becoming more important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [1, 2], Dubois and Prade [3], and Nahmias [4]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [5], Goetschel, Jr. and Voxman [6], and Wu and Ma [7, 8].

In the past decades, many researchers have studied the fuzzy linear equations such as fuzzy linear systems (FLS), dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), fully fuzzy linear systems (FFLS), dual fully fuzzy linear systems (DFFLS), and general dual fuzzy linear systems (GDFLS). These works were performed mainly by Friedman et al. [9, 10], Allahviranloo et al. [11–17], Abbasbandy et al. [18–21], Wang and Zheng [22, 23], and Dehghan et al. [24, 25]. The general method they applied is the fuzzy linear equations were converted to a crisp function system of linear equations with high order according to the embedding principles and algebraic operations of fuzzy numbers. Then the fuzzy solution of the original fuzzy linear systems was derived from solving the crisp function linear systems. However, for a fuzzy matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past. In 2009, Allahviranloo et al. [26] discussed the fuzzy linear matrix equations (FLME) of the form in which the matrices A and B are known m × m and n × n real matrices, respectively; is a given m × n fuzzy matrix. By using the parametric form of fuzzy number, they derived necessary and sufficient conditions for the existence condition of fuzzy solutions and designed a numerical procedure for calculating the solutions of the fuzzy matrix equations. In 2011, Guo et al. [27–29] investigated a class of fuzzy matrix equations by means of the block Gaussian elimination method and studied the least squares solutions of the inconsistent fuzzy matrix equation by using generalized inverses of the matrix, and discussed fuzzy symmetric solutions of fuzzy matrix equations . What is more, there are two shortcomings in the above fuzzy systems. The first is that in these fuzzy linear systems and fuzzy matrix systems the fuzzy elements were denoted by triangular fuzzy numbers, so the extended model equations always contain parameter r, 0 ≤ r ≤ 1 which makes their computation especially numerical implementation inconvenient in some sense. The other one is that the weak fuzzy solution of fuzzy linear systems does not exist sometimes see; [30].

To make the multiplication of fuzzy numbers easy and handle the fully fuzzy systems, Dubois and Prade [3] introduced the LR fuzzy number in 1978. We know that triangular fuzzy numbers and trapezoidal fuzzy numbers [31] are just specious cases of LR fuzzy numbers. In 2006, Dehghan et al. [25] discussed firstly computational methods for fully fuzzy linear systems whose coefficient matrix and the right-hand side vector are LR fuzzy numbers. In 2012, Allahviranloo et al. studied LR fuzzy linear systems [32] by the linear programming with equality constraints. Otadi and Mosleh considered the nonnegative fuzzy solution [33] of fully fuzzy matrix equations by employing linear programming with equality constraints at the same year. Recently, Guo and Shang [34] investigated the fuzzy approximate solution of LR fuzzy Sylvester matrix equations . In this paper we propose a general model for solving the fuzzy linear matrix equation , where , , and are m × m, n × n, and m × n nonnegative LR fuzzy numbers matrices, respectively. The model is proposed in this way; that is, we extend the fuzzy linear matrix system into a system of linear matrix equations according to arithmetic operations of LR fuzzy numbers. The LR fuzzy solution of the original matrix equation is derived from solving crisp systems of linear matrix equations. The structure of this paper is organized as follows.

In Section 2, we recall the LR fuzzy numbers and present the concept of fully fuzzy linear matrix equation. The computing model to the positive fully fuzzy linear matrix equation is proposed in detail and the fuzzy approximate solution of the fuzzy linear matrix equation is obtained by using pseudo-inverse in Section 3. Some examples are given to illustrate our method in Section 4 and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. The LR Fuzzy Number

Definition 1 (see [1].)A fuzzy number is a fuzzy set like u : R → I = [0,1] which satisfies the following.

(1)
u is upper semicontinuous,
(2)
u is fuzzy convex, that is, u(λx + (1 − λ)y) ≥ min {u(x), u(y)} for all x, y ∈ R, λ ∈ [0,1],
(3)
u is normal, that is, there exists x₀ ∈ R such that u(x₀) = 1,
(4)
supp u = {x ∈ R∣u(x) > 0} is the support of the u, and its closure cl(supp u) is compact.

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

Definition 2 (see [3].)A fuzzy number is said to be an LR fuzzy number if

()

where m, α, and β are called the mean value, left, and right spreads of

, 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 nonincreasing on [0, ∞).

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

Clearly, is positive (negative) if and only if m − α > 0 (m + β < 0).

Also, two LR fuzzy numbers and are said to be equal, if and only if m = n, α = γ, and β = δ.

Definition 3 (see [5].)For arbitrary LR fuzzy numbers and , we have the following.

(1)
Addition:
()
(2)
Multiplication:
- (i)
  If and , then
  ()
- (ii)
  if and , then
  ()
- (iii)
  if and , then
  ()
(3)
Scalar multiplication:
()

2.2. The LR Fuzzy Matrix

Definition 4. A matrix is called an LR fuzzy matrix, if each element of is an LR fuzzy number.

will be positive (negative) and denoted by if each element of is positive (negative), where . Up to the rest of this paper, we use positive LR fuzzy numbers and formulas given in Definition 3. For example, we represent m × n LR fuzzy matrix , that with new notation , where A = (a_ij), M = (α_ij) and N = (β_ij) are three m × n crisp matrices. In particular, an n dimensions LR fuzzy numbers vector can be denoted by (x, x^l, x^r), where x = (x_i), and are three n dimensions crisp vectors.

Definition 5. Let and be two m × n and n × p fuzzy matrices; we define which is an m × p fuzzy matrix, where

()

2.3. The Fully Fuzzy Linear Matrix Equation

Definition 6. The matrix system:

()

where

, 1 ≤ i, j ≤ m,

, 1 ≤ i, j ≤ n, and

, 1 ≤ i ≤ m, 1 ≤ j ≤ n are LR fuzzy numbers, is called an LR fully fuzzy linear matrix equation (FFLME).

Using matrix notation, we have

()

A fuzzy numbers matrix

()

is called an LR fuzzy approximate solution of fully fuzzy linear matrix equation (8) if

satisfies (9).

Up to the rest of this paper, we will discuss the nonnegative solution of FFLME , where , , and .

3. Method for Solving FFLME

First, we extend the fuzzy linear matrix system (9) into three systems of linear matrix equations according to the LR fuzzy number and its arithmetic operations.

Theorem 7. The fuzzy linear matrix system (9) can be extended into the following model:

()

Proof. We denote , and , and assume , then

()

according to multiplication of nonnegative LR fuzzy numbers of Definition 2. Thus we obtain a model for solving FFLME (9) as follows:

()

Secondly, in order to solve the fuzzy linear matrix equation (9), we need to consider the crisp systems of linear matrix equation (11). Supposing A and B are nonsingular crisp matrices, we have

()

that is,

()

Definition 8. Let be an LR fuzzy matrix. If (X, Y, Z) is an exact solution of (11) such that X ≥ 0, Y ≥ 0, Z ≥ 0, and X − Y ≥ 0; we call a nonnegative LR fuzzy approximate solution of (9).

Theorem 9. Let , , and be three nonnegative fuzzy matrices, respectively, and let A and B be the product of a permutation matrix by a diagonal matrix with positive diagonal entries. Moreover, let GB ≥ CB⁻¹E + MA⁻¹C, HB ≥ CB⁻¹F + NA⁻¹C, and C + CB⁻¹E + MA⁻¹C ≥ GB. Then the systems has nonnegative fuzzy solutions.

Proof. Our hypotheses on A and B imply that A⁻¹ and B⁻¹ exist and they are nonnegative matrices. Thus X = A⁻¹CB⁻¹ ≥ 0.

On the other hand, because GB ≥ CB⁻¹E + MA⁻¹C and HB ≥ CB⁻¹F + NA⁻¹C, so with Y = A⁻¹(GB − CB⁻¹E − MA⁻¹C)B⁻¹ and Z = A⁻¹(HB − CB⁻¹F − NA⁻¹C)B⁻¹, we have Y ≥ 0 and Z ≥ 0. Thus is a fuzzy matrix which satisfies . Since X − Y = A⁻¹(C − GB + CB⁻¹E + MA⁻¹C)B⁻¹, the positivity property of can be obtained from the condition C + CB⁻¹E + MA⁻¹C ≥ GB.

When A or B is a singular crisp matrix, the following result is obvious.

Theorem 10 (see [35].)For linear matrix equations AXB = C, where A ∈ R^m×m, B ∈ R^n×n, and C ∈ R^m×n. Then

()

is its minimal norm least squares solution.

By the pseudoinverse of matrices, we solve model (11) and obtain its minimal norm least squares solution as follows:

()

that is,

()

Definition 11. Let be an LR fuzzy matrix. If (X, Y, Z) is a minimal norm least squares solution of (11) such that X ≥ 0, Y ≥ 0, Z ≥ 0, and X − Y ≥ 0, we call a nonnegative LR fuzzy minimal norm least squares solution of (9).

At last, we give a sufficient condition for nonnegative fuzzy minimal norm least squares solution of FFLME (9) in the same way.

Theorem 12. Let A^† and B^† be nonnegative matrices. Moreover, let GB ≥ CB^†E + MA^†C, HB ≥ CB^†F + NA^†C, and C + CB^†E + MA^†C ≥ GB. Then the systems has nonnegative fuzzy minimal norm least squares solutions.

Proof. Since A^† and B^† are nonnegative matrices, we have X = A^†CB^† ≥ 0.

Now that GB ≥ CB^†E + MA^†C and HB ≥ CB^†F + NA^†C, therefore, with Y = A^†(GB − CB^†E − MA^†C)B^† and Z = A^†(HB − CB^†F − NA^†C)B^†, we have Y ≥ 0 and Z ≥ 0. Thus is a fuzzy matrix which satisfies . Since X − Y = A^†(C − GB + CB^†E + MA^†C)B^†, the nonnegativity property of can be obtained from the condition C + CB^†E + MA^†C ≥ GB.

The following Theorems give some results for such S⁻¹ and S^† to be nonnegative. As usual, (·) ^⊤ denotes the transpose of a matrix (·).

Theorem 13 (see [36].)The inverse of a nonnegative matrix A is nonnegative if and only if A is a generalized permutation matrix.

Theorem 14 (see [37].)Let A be an m × m nonnegative matrix with rank r. Then the following assertions are equivalent:

(a)
A^† ≥ 0.
(b)
There exists a permutation matrix P, such that PA has the form
()
where each S_i has rank 1 and the rows of S_i are orthogonal to the rows of S_j, whenever i ≠ j, the zero matrix may be absent.
(c)
for some positive diagonal matrix K. In this case,
()

4. Numerical Examples

Example 15. Consider the fully fuzzy linear matrix system

()

By Theorem 7, the model of the above fuzzy linear matrix system is made of the following three crisp systems of linear matrix equations

()

Now that the matrix B is singular, according to formula (18), the solutions of the above three systems of linear matrix equations are as follows:

()

By Definition 11, we know that the original fuzzy linear matrix equations have a nonnegative LR fuzzy solution

()

since X ≥ 0, Y ≥ 0, Z ≥ 0, and X − Y ≥ 0.

Example 16. Consider the following fuzzy matrix system:

()

By Theorem 7, the model of the above fuzzy linear matrix system is made of following three crisp systems of linear matrix equations:

()

By the same way, we obtain that the solutions of the above three systems of linear matrix equations are as follows:

()

Since X ≥ 0, Y ≥ 0, Z ≥ 0, and X − Y ≥ 0, we know that the original fuzzy linear matrix equations have a nonnegative LR fuzzy solution given by

()

5. Conclusion

In this work we presented a model for solving fuzzy linear matrix equations in which and are m × m and n × n fuzzy matrices, respectively, and is an m × n arbitrary LR fuzzy numbers matrix. The model was made of three crisp systems of linear equations which determined the mean value and the left and right spreads of the solution. The LR fuzzy approximate solution of the fuzzy linear matrix equation was derived from solving the crisp systems of linear matrix equations. In addition, the existence condition of strong LR fuzzy solution was studied. Numerical examples showed that our method is feasible to solve this type of fuzzy matrix equations. Based on LR fuzzy numbers and their operations, we can investigate all kinds of fully fuzzy matrix equations in future.

Acknowledgments

The work is supported by the Natural Scientific Funds of China (no. 71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).

References

1 Zadeh L. A., Fuzzy sets, Information and Computation. (1965) 8, 338–353, MR0219427, ZBL0139.24606.
Google Scholar
2 Zadeh L. A., The concept of a linguistic variable and its application to approximate reasoning. I, Information Science. (1975) 8, 199–249, MR0386369, ZBL0397.68071.
10.1016/0020-0255(75)90036-5
Web of Science® Google Scholar
3 Dubois D. and Prade H., Operations on fuzzy numbers, International Journal of Systems Science. (1978) 9, no. 6, 613–626, MR0491199, ZBL0383.94045.
10.1080/00207727808941724
Web of Science® Google Scholar
4 Nahmias S., Fuzzy variables, Fuzzy Sets and Systems. An International Journal in Information Science and Engineering. (1978) 1, no. 2, 97–110, MR0498158, https://doi.org/10.1016/0165-0114(78)90011-8, ZBL0383.03038.
10.1016/0165-0114(78)90011-8
Google Scholar
5 Puri M. L. and Ralescu D. A., Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications. (1983) 91, no. 2, 552–558, https://doi.org/10.1016/0022-247X(83)90169-5, MR690888, ZBL0528.54009.
10.1016/0022-247X(83)90169-5
Web of Science® Google Scholar
6 Goetschel,R.Jr. and Voxman W., Elementary fuzzy calculus, Fuzzy Sets and Systems. (1986) 18, no. 1, 31–43, https://doi.org/10.1016/0165-0114(86)90026-6, MR825618, ZBL0626.26014.
10.1016/0165-0114(86)90026-6
Web of Science® Google Scholar
7 Wu C. X. and Ma M., Embedding problem of fuzzy number space. I, Fuzzy Sets and Systems. (1991) 44, no. 1, 33–38, https://doi.org/10.1016/0165-0114(91)90030-T, MR1133980, ZBL0757.46066.
10.1016/0165-0114(91)90030-T
Web of Science® Google Scholar
8 Wu C. X. and Ma M., Embedding problem of fuzzy number space. III, Fuzzy Sets and Systems. (1992) 46, no. 2, 281–286, https://doi.org/10.1016/0165-0114(92)90142-Q, MR1167324, ZBL0774.54003.
10.1016/0165-0114(92)90142-Q
Web of Science® Google Scholar
9 Friedman M., Ming M., and Kandel A., Fuzzy linear systems, Fuzzy Sets and Systems. (1998) 96, no. 2, 201–209, https://doi.org/10.1016/S0165-0114(96)00270-9, MR1614810, ZBL0929.15004.
10.1016/S0165-0114(96)00270-9
Web of Science® Google Scholar
10 Ma M., Friedman M., and Kandel A., Duality in fuzzy linear systems, Fuzzy Sets and Systems. (2000) 109, no. 1, 55–58, https://doi.org/10.1016/S0165-0114(98)00102-X, MR1718465, ZBL0945.15002.
10.1016/S0165-0114(98)00102-X
Web of Science® Google Scholar
11 Allahviranloo T. and Ghanbari M., On the algebraic solution of fuzzy linear systems based on interval theory, Applied Mathematical Modelling. (2012) 36, no. 11, 5360–5379, https://doi.org/10.1016/j.apm.2012.01.002, MR2956750.
10.1016/j.apm.2012.01.002
Web of Science® Google Scholar
12 Allahviranloo T. and Salahshour S., Fuzzy symmetric solutions of fuzzy linear systems, Journal of Computational and Applied Mathematics. (2011) 235, no. 16, 4545–4553, https://doi.org/10.1016/j.cam.2010.02.042, MR2805696, ZBL1220.65033.
10.1016/j.cam.2010.02.042
Web of Science® Google Scholar
13 Allahviranloo T. and Ghanbari M., A new approach to obtain algebraic solution of interval linear systems, Soft Computing. (2012) 16, no. 1, 121–133, https://doi.org/10.1007/s00500-011-0739-7.
10.1007/s00500-011-0739-7
Web of Science® Google Scholar
14 Allahviranloo T., Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation. (2004) 155, no. 2, 493–502, https://doi.org/10.1016/S0096-3003(03)00793-8, MR2077064, ZBL1067.65040.
10.1016/S0096-3003(03)00793-8
Web of Science® Google Scholar
15 Allahviranloo T., Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation. (2005) 162, no. 1, 189–196, https://doi.org/10.1016/j.amc.2003.12.085, MR2107267, ZBL1062.65037.
10.1016/j.amc.2003.12.085
Web of Science® Google Scholar
16 Allahviranloo T., The Adomian decomposition method for fuzzy system of linear equations, Applied Mathematics and Computation. (2005) 163, no. 2, 553–563, https://doi.org/10.1016/j.amc.2004.02.020, MR2121810, ZBL1069.65025.
10.1016/j.amc.2004.02.020
Web of Science® Google Scholar
17 Nuraei R., Allahviranloo T., and Ghanbari M., Finding an inner estimation of the so- lution set of a fuzzy linear system, Applied Mathematical Modelling. (2013) 37, no. 7, 5148–5161, https://doi.org/10.1016/j.apm.2012.10.020.
10.1016/j.apm.2012.10.020
Web of Science® Google Scholar
18 Abbasbandy S., Ezzati R., and Jafarian A., LU decomposition method for solving fuzzy system of linear equations, Applied Mathematics and Computation. (2006) 172, no. 1, 633–643, https://doi.org/10.1016/j.amc.2005.02.018, MR2197928, ZBL1088.65023.
10.1016/j.amc.2005.02.018
Web of Science® Google Scholar
19 Abbasbandy S., Jafarian A., and Ezzati R., Conjugate gradient method for fuzzy symmetric positive definite system of linear equations, Applied Mathematics and Computation. (2005) 171, no. 2, 1184–1191, https://doi.org/10.1016/j.amc.2005.01.110, MR2199697, ZBL1121.65311.
10.1016/j.amc.2005.01.110
Web of Science® Google Scholar
20 Abbasbandy S., Otadi M., and Mosleh M., Minimal solution of general dual fuzzy linear systems, Chaos, Solitons & Fractals. (2008) 37, no. 4, 1113–1124, https://doi.org/10.1016/j.chaos.2006.10.045, MR2411537, ZBL1146.15002.
10.1016/j.chaos.2006.10.045
Web of Science® Google Scholar
21 Asady B., Abbasbandy S., and Alavi M., Fuzzy general linear systems, Applied Mathematics and Computation. (2005) 169, no. 1, 34–40, https://doi.org/10.1016/j.amc.2004.10.042, MR2169131, ZBL1119.65325.
10.1016/j.amc.2004.10.042
Web of Science® Google Scholar
22 Wang K. and Zheng B., Inconsistent fuzzy linear systems, Applied Mathematics and Computation. (2006) 181, no. 2, 973–981, https://doi.org/10.1016/j.amc.2006.02.019, MR2269976, ZBL1122.15004.
10.1016/j.amc.2006.02.019
Web of Science® Google Scholar
23 Zheng B. and Wang K., General fuzzy linear systems, Applied Mathematics and Computation. (2006) 181, no. 2, 1276–1286, https://doi.org/10.1016/j.amc.2006.02.027, MR2270761, ZBL1122.15005.
10.1016/j.amc.2006.02.027
Web of Science® Google Scholar
24 Dehghan M. and Hashemi B., Iterative solution of fuzzy linear systems, Applied Mathematics and Computation. (2006) 175, no. 1, 645–674, https://doi.org/10.1016/j.amc.2005.07.033, MR2216371, ZBL1137.65336.
10.1016/j.amc.2005.07.033
Web of Science® Google Scholar
25 Dehghan M., Hashemi B., and Ghatee M., Solution of the fully fuzzy linear systems using iterative techniques, Chaos, Solitons & Fractals. (2007) 34, no. 2, 316–336, https://doi.org/10.1016/j.chaos.2006.03.085, MR2327410, ZBL1144.65021.
10.1016/j.chaos.2006.03.085
Web of Science® Google Scholar
26 Allahviranloo T., Mikaeilvand N., and Barkhordary M., Fuzzy linear matrix equation, Fuzzy Optimization and Decision Making. (2009) 8, no. 2, 165–177, https://doi.org/10.1007/s10700-009-9058-1, MR2511473, ZBL1177.15016.
10.1007/s10700-009-9058-1
Web of Science® Google Scholar
27 Guo X. and Gong Z., Block Gaussian elimination methods for fuzzy matrix equations, International Journal of Pure and Applied Mathematics. (2010) 58, no. 2, 157–168, MR2590168, ZBL1192.65034.
Google Scholar
28 Gong Z. and Guo X., Inconsistent fuzzy matrix equations and its fuzzy least squares solutions, Applied Mathematical Modelling. (2011) 35, no. 3, 1456–1469, https://doi.org/10.1016/j.apm.2010.09.022, MR2740923, ZBL1211.15038.
10.1016/j.apm.2010.09.022
Web of Science® Google Scholar
29 Guo X. and Shang D., Fuzzy symmetric solutions of fuzzy matrix equations, Advances in Fuzzy Systems. (2012) 2012, 9, https://doi.org/10.1155/2012/318069, 318069, MR2950305, ZBL1250.15023.
10.1155/2012/318069
Google Scholar
30 Allahviranloo T., Ghanbari M., Hosseinzadeh A. A., Haghi E., and Nuraei R., A note on ‘Fuzzy linear systems’, Fuzzy Sets and Systems. (2011) 177, 87–92, https://doi.org/10.1016/j.fss.2011.02.010, MR2812834.
10.1016/j.fss.2011.02.010
Web of Science® Google Scholar
31 Yeh C.-T., Weighted semi-trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems. (2011) 165, 61–80, https://doi.org/10.1016/j.fss.2010.11.001, MR2754586, ZBL1226.03058.
10.1016/j.fss.2010.11.001
Web of Science® Google Scholar
32 Allahviranloo T., Lotfi F. H., Kiasari M. K., and Khezerloo M., On the fuzzy solution of LR fuzzy linear systems, Applied Mathematical Modelling. (2013) 37, no. 3, 1170–1176, https://doi.org/10.1016/j.apm.2012.03.037.
10.1016/j.apm.2012.03.037
Web of Science® Google Scholar
33 Otadi M. and Mosleh M., Solving fully fuzzy matrix equations, Applied Mathematical Modelling. (2012) 36, no. 12, 6114–6121, https://doi.org/10.1016/j.apm.2012.02.005.
10.1016/j.apm.2012.02.005
Web of Science® Google Scholar
34 Guo X. B. and Shang D. Q., Approximate solutions of LR fuzzy Sylvester matrix equations, Journal of Applied Mathematics. In press.
Google Scholar
35 Ben-Israel A. and Greville T. N. E., Generalized Inverses: Theory and Applications, 2003, 2nd edition, Springer-Verlag, New York, NY, USA, MR1987382.
Web of Science® Google Scholar
36 Berman A. and Plemmons R. J., Nonnegative Matrices in the Mathematical Sciences, 1979, Academic Press, New York, NY, USA, MR544666.
Google Scholar
37 Plemmons R. J., Regular nonnegative matrices, Proceedings of the American Mathematical Society. (1973) 39, 26–32, MR0313274, https://doi.org/10.1090/S0002-9939-1973-0313274-6, ZBL0273.20051.
10.1090/S0002-9939-1973-0313274-6
Web of Science® Google Scholar

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Fuzzy Approximate Solution of Positive Fully Fuzzy Linear Matrix Equations

Abstract

1. Introduction

2. Preliminaries

2.1. The LR Fuzzy Number

2.2. The LR Fuzzy Matrix

2.3. The Fully Fuzzy Linear Matrix Equation

3. Method for Solving FFLME

4. Numerical Examples

5. Conclusion

Acknowledgments

References

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Fuzzy Approximate Solution of Positive Fully Fuzzy Linear Matrix Equations

Abstract

1. Introduction

2. Preliminaries

2.1. The LR Fuzzy Number

2.2. The LR Fuzzy Matrix

2.3. The Fully Fuzzy Linear Matrix Equation

3. Method for Solving FFLME

4. Numerical Examples

5. Conclusion

Acknowledgments

References

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