A review on classification methods for solving fully

A review on classification methods for solving fully fuzzy linear systems Wan Suhana Wan Daud, Nazihah Ahmad, and Khairu Azlan Abd Aziz Citation: AIP Conference Proceedings 1691, 040005 (2015); doi: 10.1063/1.4937055 View online: http://dx.doi.org/10.1063/1.4937055 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1691?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Finite solutions of fully fuzzy linear system AIP Conf. Proc. 1635, 447 (2014); 10.1063/1.4903620 Utilizing QR decomposition for solving singular fuzzy linear systems AIP Conf. Proc. 1602, 323 (2014); 10.1063/1.4882506 Numerical method for solving fuzzy wave equation AIP Conf. Proc. 1558, 2444 (2013); 10.1063/1.4826035 Software for Solving Fuzzy Linear Systems AIP Conf. Proc. 1184, 285 (2009); 10.1063/1.3271627 Solutions of The Fully Fuzzy Linear System AIP Conf. Proc. 1124, 234 (2009); 10.1063/1.3142938

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A Review on Classification Methods for Solving Fully Fuzzy Linear Systems Wan Suhana Wan Daud1,a), Nazihah Ahmad2,b) and Khairu Azlan Abd Aziz3,c) 1

2

Institute of Mathematics Engineering, Universiti Malaysia Perlis, Pauh Putra Campus, 02600, Arau, Perlis. School of Quantitative Sciences, College of Arts & Sciences, Universiti Utara Malaysia, 06010, Sintok, Kedah. 3 Department of Sciences Mathematics and Statistics, Universiti Teknologi MARA Perlis. a)

Corresponding author: [email protected] b) [email protected] c) [email protected]

Abstract. Fully Fuzzy Linear System (FFLS) exists when there are fuzzy numbers on both sides of the linear systems. This system is quite significant today since most of the linear systems play with uncertainties of parameters especially in mathematics, engineering and finance. Many researchers and practitioners used the FFLS to model their problem and they apply various methods to solve it. In this paper, we present the outcome of a comprehensive review that we have done on various methods used for solving the FFLS. We classify our findings based on parameters’ type used for the FFLS either restricted or unrestricted. We also discuss some of the methods by illustrating numerical examples and identify the differences between the methods. Ultimately, we summarize all findings in a table. We hope this study will encourage researchers to appreciate the use of this method and with that it will be easier for them to choose the right method or to propose any new method for solving the FFLS.

INTRODUCTION The use of computational algorithms to find solutions for system linear equations is an important part of numerical linear algebra, and it plays a prominent role in engineering, physics, chemistry, computer science, economics and social sciences. In a standard real system, for the sake of simplicity or for easy computation, the variables or parameters in the system are taken as crisp numbers. However, in most of the real cases, the involved parameters and variables are always uncertain or vague. It is due to the uncertainties of the model which can arise during experimentations, data collections, or measurement processes. Then, these variables can be considered as fuzzy. In other words, to overcome the uncertainty, one may use fuzzy numbers in place of crisp numbers. Thus, the system of linear equations becomes a fuzzy system of linear equations (FLS) represented as Ax
 b
where A is a crisp matrix and b
is a fuzzy number vector. Other than that, there is also a linear system, which all parameters are fuzzy and named as Fully Fuzzy Linear System (FFLS). In addition, there was a dual form of FFLS which is known as Dual Fully Fuzzy Linear System (DFFLS), commonly represented as A
1 x
 b
1  A
2 x
 b
2 . Generally, both FFLS and DFFLS are usually solved by variety of methods, which can be classified as direct and iterative methods. Direct methods which are also known as computational or classical methods in the literature are generally used for square systems, whereas the iterative methods can be used for both square and non-square linear equation systems. In addition, another classification for FFLS and DFFLS can also be made depending on whether the systems have sign restrictions on its parameters. Having sign restrictions for FFLS means that all parameters of FFLS are assumed as positive or non-negative. Therefore, in this paper, methods used for solving the FFLS are comprehensively reviewed. This review will focus on the type of parameters of the system either restricted or unrestricted. The structure of this paper is organized as follows. In Section 2, we introduced some basic concepts on fuzzy numbers and fuzzy linear system. Then, a summary of the various methods for solving FFLS are presented in

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Section 3. In Section 4, some numerical examples are illustrated by using some methods that have been discussed in the previous section and finally, conclusion remarks are presented in Section 5.

PRELIMINARIES In this section, we recall some necessary background of fuzzy numbers and also FFLS. Definition 2.1 [1] A fuzzy subset M
of X is defined by its membership function μ M
where each element x  X is mapped to the interval [0,1] , such that μ M
:

[0,1] .

Definition 2.2 [2] A fuzzy number M
 (m, α , β ) is said to be an LR fuzzy number if

 mx  L  α  , x m, α  0    μ M
( x)    R  x  m  , x  m, β  0.   β  where m is the mean value of M
, whereas α and β are right and left spreads, respectively. Definition 2.3 The arithmetic operations of two fuzzy numbers M
 (m, α , β ) and N
 (n, γ , δ ) , are as follows:

Addition:

M
 N
 (m, α , β )  (n, γ , δ ) 
m  n, α  γ , β  δ 

Multiplication: If M
 0 and N
 0, then

M
 N
 (m, α , β )  (n, γ , δ ) 
mn, mγ  nα , mδ  nβ 

(1)

(2)

Scalar multiplication:


λ m, λα , λβ  , λ N
 λ (m, α , β )   
λ m, λβ , λα  ,

λ  0, λ  0.

Definition 2.4 Consider the n n fully fuzzy linear system (FFLS) of equations (a
11  x
1 )  (a
12  x
2 )  ...  (a
1n  x
n )  b
1  (a
21  x
1 )  (a
22  x
2 )  ...  (a
2 n  x
n )  b
2    (a
 x
)  (a
 x
)  ...  (a
 x
)  b
1 n2 2 nm n n  n1


The matrix form of the above equations is Ax  B .

(3)

(4)

CLASSIFICATION METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEMS This section covers relevant methods in performing the solution of FFLS. We reviewed and classified the methods based on the type of parameters of the FFLS either restricted or unrestricted. At the end of this paper, we include the table of summary for all the papers reviewed (Refer Table 2).

Methods for Solving FFLS with Restricted Parameters The first method is LU decomposition. By using this method, the coefficient matrix, A
is factored into the product of two triangular matrices, and written as A  LU , where L is a lower triangular crisp matrix that have the

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diagonal of 1 and U is an upper triangular crisp matrix with the general diagonal. It was a well-known method since a long time ago and frequently used to solve linear system of equations. Due to its efficiency and practicability, it has been modified and extended for solving fuzzy linear system and also for FFLS. Beginning 2006, Dehghan et al. [3], used this method to solve FFLS. They showed that, LU decomposition method has provided a better computation technique since it was faster than Gaussian elimination method. Later in 2008, Nasseri et al. [4] did some modifications on the algorithm proposed by Dehghan et al., which was simpler but still produced the same solution. Mashadi [5] showed that LU decomposition method can also be applied in solving DFFLS, A
 x
 b
 A
 x
 d
. Besides that, QR decomposition method was also widely used in the literature. For QR decomposition method, A
 ( A, M , N ) in the FFLS is considered a full rank crisp matrix and it is a decomposition of A  QR , where Q is an orthonormal crisp matrix and R is an upper triangular crisp matrix. Nasseri and Sohrabi [6] applied QR decomposition method with LR fuzzy numbers, and they used Gram-Schmidt approach to orthogonalize the basis matrix A. Nasseri et al. [7] solved the positive FFLS by considering that whereas in another publication [8], they ( A, M , N )  (Q1 , 0, Q3 )  ( R1 , R2 , 0)  (Q1 R1 , Q1 R2 , Q3 R1 ) , used ( A, M , N )  (Q1 , Q2 , Q3 )  ( R1 , 0, 0)  (Q1 R1 , Q2 R1 , Q3 R1 ) . Both algorithms were just slightly different, but the solutions obtained were the same. Similar to LU decomposition, QR decomposition was also used to solve the trapezoidal FFLS by considering trapezoidal fuzzy numbers, ( A, B, M , N )  (Q1 , Q2 , 0, 0)  ( R1 , R2 , R3 , R4 ) [9]. Another decomposition method that had been applied in solving FFLS with restricted coefficient is Signed decomposition method by Allahviranloo et al. [10]. In this work, the authors employed the parametric form of linear system. Firstly, they solved 0-cut system of A0 X 0  b0 and replaced the coefficient matrix of n n by 2n 4n parametric coefficient matrix and then finally transformed the matrix to 2n 2n system. However, this algorithm was only limited to the non-zero fuzzy solution. Besides that, ST decomposition method had also been applied in solving FFLS [11] and DFFLS [12]. By this decomposition, every non-singular fuzzy matrix can be represented as a product of a fuzzy symmetric matrix S and a fuzzy triangular matrix T. Besides that, Vijayalakshmi & Sattanathan [13], had proposed ST decomposition method for solving non-singular fuzzy matrix and they used Gauss Jordan method in finding the corresponding inverses. This method was quite special compared to the others, because it was able to obtain both positive and negative solutions. Other than that, Jaikumar & Sunantha [14] had modified ST decomposition to be Secondary Symmetric Triangular (SsT) decomposition method, which decomposed the coefficient matrix A to secondary symmetric matrix and triangular matrix. By this method, the equation was reduced to diagonal matrix by Gauss elimination method and applied the back substitution to get the corresponding unknown values in the form of triangular fuzzy numbers. Then in the same year, there is another method introduced in solving FFLS, which only allowed positive definite matrix to be solved, known as Cholesky decomposition [15]. By using this method, the n n positive definite A was converted to be lower triangular matrix L that multiplied by its matrix transpose LT , such that A  LLT . Apart from that, some iterative methods had also been applied in solving positive and non-negative FFLS. Dehghan & Hashemi [16], applied the method of Adomian decomposition which considered the non-negative of FFLS. This method was usually applied for a big class of linear and non-linear problems which could involve square and non-square matrix. Instead of just using the Adomian decomposition technique, the authors also had applied Jacobi iterative method in solving the iteration matrix. Besides that, there were some other well-known iterative methods such as Jacobi, Gauss–Seidel, SOR, AOR, SSOR, USSOR and EMA [17], also gradient iterative algorithm and least-squares iterative algorithm by [18] which were effectively applied in solving the FFLS. Later in 2010, Nasseri and Zahmatkesh [19] proposed a new method based on the Abaffy-Broyden-Spedicato (ABS) class of algorithm (note that ABS class of algorithm can be used to solve m linear equations in n unknowns with m n ), which was derived by analogy with the quasi-Newton methods and it’s named Huang method. In addition, a method of Singular Value decomposition was studied by Karthik & Chanrasekaran [20] in solving positive trapezoidal fuzzy numbers. On top of that, two methods known as fast iterative method (FIM) [21] and Chebyshev semi-iterative method [22] had been applied by Abdolmaleki & Edalapanah, and they had shown that their methods were more reliable and efficient compared with other iterative method such as Jacobi, Gauss-Seidel and SOR. Later, a method named median interval defuzzification was introduced in obtaining the approximate nonnegative symmetric solution of FFLS where the coefficient matrix and the right hand side vector were non-negative fuzzy numbers. By this method, Huang method was applied but with some modifications to solve the m n real crisp system of equations [23].

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In 2014 also, an alternative method called block matrix method had been applied as the alternative way of using classical inversion method which sometimes are complicated to solve if it involve large size matrices. This method is better compared to the conventional one, because it can deal with any size of system and also able to solve the FFLS with unknown coefficient [24]–[26].

Methods for Solving FFLS with Unrestricted Parameters Although there are varieties of methods in solving FFLS introduced year by year, there are very little studies that consider the coefficient matrices are unrestricted. Based on our review, Mosleh et al. [27], didn’t put any restriction for the coefficient as well as the fuzzy solution obtained, so the method proposed were applicable for any real applications. Kumar et al. [28], [29] in their consecutive papers were much accentuated on the type of coefficient used in applying their proposed methods. They introduced a new algorithm to derive the FFLS with arbitrary coefficient and solved by using any direct method such as Cramer’s rule, LU decomposition, Linear Programming and etc, in order to obtain non-negative fuzzy solution. In the same year, Allahviranloo and Mikaeilvand [30], eliminated the restriction on the parametric form of fuzzy numbers which in some situations the system also able to produce non-zero solutions. A year later, Kumar et al. again came out with another two computational methods and presented it in two different publications. In [31], the algorithm developed was to solve DFFLS with arbitrary triangular fuzzy numbers, meanwhile in [32], they had another two algorithms, which the first one was to obtain non-negative solution from arbitrary parameters, and the second one was conversely. Later, Moloudzadeh et al. [33], constructed a new method for solving an arbitrary FFLS which was based on the extending 0-cut and 1-cut solution of the original FFLS. Then in 2014, Radhakrishnan & Gajivaradhan [34] applied QR decomposition but this time they eliminated the positive constraint on the coefficient in order to obtain the nonnegative solution of FFLS. At the same time, Sedaghatfar et al. [35] presented a solution of DFFLS which considered arbitrary type of fuzzy numbers.

NUMERICAL EXAMPLES In this section, we choose the most preferable methods namely as LU decomposition and Cholesky decomposition to be employed on some numerical examples of FFLS. We have observed the advantages and limitation of these two methods and compare each other (refer Table 1).



 B
with triangular fuzzy numbers written as follows: Example 3.1 Let us have a FFLS AX (12,1.5,1.5) (6, 0.5, 0.2)   x
1   (1897, 427.7,536.2)   (19,1,1)      (1.5, 0.2, 0.2)   x
2    (434.5, 76.2,109.3)  (5)  (2, 0.1, 0.1) (4, 0.1, 0.4)  (2, 0.1, 0.2) (2, 0.1, 0.3)      (4.5, 0.1, 0.1)   x
3   (535.5,88.3,131.9)   B
 (b, g , h) , then In this case, we assume that A
, X
, B
 0 , where A
 ( A, M , N ) , X
 ( x, y, z ) and ( A, M , N )  ( x, y, z )  (b, g , h) . By arithmetic operations on fuzzy numbers we have,  Ax  b  x  A1b  1  Ay  Mx  h  y  A (h  Mx)  1  Az  Nx  g  z  A ( g  Nx)

 B
is decomposed to be Now, we are going to solve Eq. (5) by using LU decomposition method, thus AX


 B
. In [3], they take L
 ( L , L , L ) and U
 (U ,U ,U ) , but for the sake of simplicity, we apply the LUX 1 2 3 1 2 3

algorithm improved by [4], which take L  ( L , 0, 0) and U  (U ,U ,U ) . Then, we have 1

 19 12 6   1    A   2 4 1.5   L1U1   0.1052  2 2 4.5   0.1052    and U 2 , U 3 can be obtained as follows

1

2

0 1 0.2693

3

0 19  0  0 1   0

12 2.7368 0

6   0.8684  3.6346 

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1 0 0  1 1.5 0.5   1 1.5 0.5        U 2  L M    0.1052 1 0  0.1 0.1 0.2     0.0052 0.1474   0.0578       0.0768696  0.2693 1 0.1 0.1 0.1   0.00379964  0.0422345 0.00770518  1 0 0  1 1.5 0.2   1 1.5 0.2        1 U 3  L1 N    0.1052 1 0  0.1 0.4 0.2     0.0052 0.2422 0.17896   0.0768696  0.2693 1 0.2 0.3 0.1   0.0962004 0.0769755 0.0307661      Therefore x  (U1 ) 1 L11b 1 1

y  (U1 ) 1 L11 ( g  L1U 2 x) z  (U1 ) 1 L11 (h  L1U 3 x) The solutions obtained are x
1  ( x1 , y1 , z1 )  (36.9691, 6.99395,13.2966) , x
2  ( x2 , y2 , z2 )  (62.039,5.50482, 4.588237) and x
3  ( x3 , y3 , z3 )  (75.0199,10.2033,13.924) . Example 3.2 For this example, we are going to show how Cholesky decomposition method has been applied in solving FFLS. We use a new FFLS which is the coefficient A is a symmetric positive definite matrix, since Cholesky only can solve this type of matrices. Consider the following FFLS:

 (4, 2, 6)   (12,10,14)  (18,16,18) 

(12,12,14) (45, 45,50) (78, 75,80)

(18,16, 20)   x
1   (124,178,320)      (78, 74,80)   x
2    (495, 741,1222)  (146,146,150)   x
3   (890,1349, 2164) 

where x
1  ( x1 , y1 , z1 )  0 , x
2  ( x2 , y2 , z2 )  0 and x
3  ( x3 , y3 , z3 )  0 . Now, we solve the above system by using algorithm as in [15]. Firstly, we obtain decomposition for symmetric matrix A as follows:  4 12 18   2 0 0  2 6 9       T A  12 45 78   LL   6 3 0  0 3 8  18 78 146   9 8 1  0 0 1       Then, we compute matrices P and Q as follows: 1     2 0 0  1 6 8     2 12 16    1 4 26    0  10 45 74    3 P  L1 M   1    3 3 3    16 75 146    11 14  7 8 1   3       3 3   3 2  1     2 0 0  3 7 10     6 14 20    1 4 8 20      1 Q  L N  1 0  14 50 80        3 3 3 3     18 80 150    7 8 5 13 20     1       3 3 3  3 2  The solutions will be obtained by x  ( LT ) 1 L1b y  ( LT ) 1 L1 ( g  LPx ) z  ( LT ) 1 L1 (h  LQx )

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Thus, x
1  ( x1 , y1 , z1 )  (4,1,3) , x
2  ( x2 , y2 , z2 )  (3, 4, 6) and x
3  ( x3 , y3 , z3 )  (4,1,5) . Now, based on these two methods discussed, we summarize and compare the characteristics of each other. TABLE 1. Summarization and comparison on methods discussed in Section 4

LU Decomposition Represented as A  LU , which are products of a lower triangular matrix L and upper triangular matrix U. Able to solve all type of positive definite matrix. Provides smaller operations count that is n 3 3  O ( n 2 ) and the storage required is only n 2  O ( n) , both for solution with any number of right-hand side, and also for matrix inversion.

Cholesky Decomposition Represented as A  LLT , which are products of a lower triangular matrix L and its conjugate. Only able to solve symmetric positive definite matrix. Operation times required is only n3 6  O ( n 2 ) which is faster than LU decomposition, and the storage locations needed is only n(n  1) 2 . This is because, this method only needs one factor to be calculated as the other factor is simply the transpose of the first.

DISCUSSION AND CONCLUSION The use of fuzzy undeniably brought many advantages in handling various linear algebra systems, especially in an uncertain world. Because of that fuzzy method has been widely used in solving FFLS. In this paper, we have shown that FFLS was completed by a variety of direct and iterative methods such as LU decomposition, Cholesky decomposition, QR decomposition, Jacobian, Adomian decomposition and Conjugate gradient method. The numerical examples are illustrated, which have shown their ability to solve FFLS. Based on this survey, we noticed that:

Direct methods such as matrix decomposition methods (LU, ST, SsT, Cholesky) and also block matrix method can only solve square linear systems which provide a simpler computation compared to iterative methods.

Iterative methods such as Adomian decomposition, Jacobian, Gauss–Seidel, SOR, AOR, SSOR, USSOR and EMA are mostly used for solving large size of linear systems and also able to solve non-square linear systems.

The number of studies which considered an unrestricted type of fuzzy numbers for both coefficients and solutions are also smaller than the restricted one. On top of that, in the future, it is more interesting if further research could be undertaken to explore the possibility of other methods (direct or iterative) in resolving FFLS. Moreover, studies should also consider other types of fuzzy numbers. Finally, it is hoped that we can overcome most of the weaknesses found in the existing methods especially that we have to expand the type of parameters used for FFLS, which is not only positive numbers, but also negative, zero or complex fuzzy numbers. Therefore, the proposed method would be more practical and useful for many fields.

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TABLE 2. Summary of the various methods reviewed Papers (author’s name)

Dehghan et al. [3] Nasseri et al. [4] Mashadi [5] Nasseri & Sohrabi [6] Nasseri et al. [7],[8] Radhakrishnan & Gajivaradhan [9] Allahviranloo et al. [10] Mosleh et al. [11] Mosleh et al. [12] Vijayalakshmi & Sattanathan [13] Jaikumar & Sunantha [14] Senthilkumar & Rajendran [15] Dehghan & Hashemi [16] Dehghan et al. [17] Jing & Qiang [18] Nasseri & Zahmatkesh [19] Karthik & Chanrasekaran [20] Abdolmaleki & Edalapanah [21],[22] Ezzati et al. [23] Ghassan et al. [24], [25] Radhakrishnan & Gajivaradhan [26] Mosleh et al. [27] Kumar et al. [28] Kumar et al. [29] Allahviranloo & Mikaeilvand [30] Kumar et al. [31], [32] Moloudzadeh et al. [33] Radhakrishnan & Gajivaradhan [34] Sedaghatfar et al. [35]

Type of system FFLS DFFLS √ √ √ √ √ √

Type of Fuzzy numbers Triangular Trapezoi Parametric dal form √ √ √ √ √ √

√

√

Solution methods Direct Iterative method method √ √ √ √ √ √ √

Type of parameters Restricted Unrestri cted √ √ √ √ √ √

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