Solving Fully Fuzzy Linear Systems with Trapezoidal Fuzzy Number Matrices by Partitioning the Block Matrices

Vol. 8, No. 2, 2014, 261-267

ISSN: 2279-087X (P), 2279-0888(online) Published on 17 December 2014

www.researchmathsci.org

261

Annals of

Solving Fully Fuzzy Linear Systems with Trapezoidal

Fuzzy Number Matrices by Partitioning the Block Matrices

N.Jayanth Karthik1 and E.Chandrasekaran2

1

Department of Mathematics, D.G.Vaishnav College, Chennai 600106, India Email : [email protected]

2

Department of Mathematics, Presidency College, Chennai 600005, India Email:[email protected]

Received 28 October 2014; accepted 21 November 2014

Abstract. In this paper, an x n fully fuzzy linear system à ⊗ x = b where à is a fuzzy

matrixx and b are fuzzy vectors with trapezoidal fuzzy numbers is solved by converting theminto block matrices and there by partitioning into smaller sub-matrices. The method is illustrated with a numerical example.

Keywords: Fully fuzzy linear system, trapezoidal fuzzy number matrices, block matrices AMS Mathematics Subject Classification (2010): 08A72, 15B15

1. Introduction

Linear system of equation has applications in many areas of science, engineering, finance and economics. Fuzzy linear system whose coefficient matrix is crisp and right hand side column is an arbitrary fuzzy number was first proposed by Friedman et al [5].

A linear system is called a fully fuzzy linear system (FFLS) if all coefficients in the system are all fuzzy numbers. Several methods based on numerical algorithms were used for solving fuzzy linear systems have been introduced by many authors [1,3,4,8]. Nasseri [9] investigated linear system of equations with trapezoidal fuzzy numbers using embedding approach. Kumar [2] solved FFLS with trapezoidal fuzzy numbers using row reduced echelon form. In [7,10] the solution of FFLS was obtained by converting the FFLS into a crisp block matrix.

In this paper an x n FFLS à ⊗ x = b where à is a fuzzy matrix x and b are fuzzy vectors with trapezoidal fuzzy numbers is solved by converting them into block matrices and there by partitioning into smaller sub-matrices.

This paper is organized as follows. Some basic definitions and results on fuzzy sets and trapezoidal fuzzy numbers are given in section 2. In section 3, the method of finding inverses of partitioned matricesis presented. Section 4 introduces the method to solve FFLS using trapezoidal fuzzy number matrices by partitioning the block matrices. Illustration with a numerical example is given in section 5. Section 6 ends this paper with conclusion.

2. Preliminaries

262

assigns a real number in the interval

[ ]

shows grade membership of x in Ã.

2. A trapezoidal fuzzy number denoted by Ã=, , , has the membership function

0

1

0

3. A fuzzy number à is called positive (negative) denoted by Ã>0 (Ã<0), if its membership function (x) satisfies = 0, ∀ ≤ ∀ ≥x 0( x 0).

4. A Trapezoidal fuzzy number à = , , , is said to be zero trapezoidal fuzzy number if and only if m= 0, n =0, α =0, β=0.

5. Two fuzzy numbers , , , and N =, , , are equal if and only if .

, , , .

6. For two fuzzy numbers , , , and N =, , , the operations extended addition, extended opposite and extended multiplication are

6.1 " # , , , " , , , $ , $ , $ , $

6.2 , , , , , ,

6.3 If M>0,N>0 then

% # , , , % , , , , , $ , $

6.4 For scalar multiplication

& % & % , , , ' _{&, &, &, & & ( 0}&, &, &, & & 0

7. A matrix )* +,_-. is called a fuzzy matrix if each element of )* is a fuzzy number. A fuzzy matrix )* is positive denoted by )*>0 if each element of )* is positive. Fuzzy matrix

)* +,-. which is n x n matrix can be represented such that +,_-. /+_-., 0_-., _-., _-.1 where )* ), 2, , # where ) +_-. 2 0_-. /_-.1 # /_-.1aren x n crisp matrices.

8. Consider n x n fuzzy linear system of equations

+,33⊗_,3⊕_+,34⊗_,4⊕_{… … …}⊕_+,36⊗_{,6 03}

+,43⊗_,3⊕_+,44⊗ ₇₄⊕_{… … …}⊕_+,46⊗_{,6 04} …………..

263

The matrix of the above equation is à ⊗ x = b where coefficient matrix à = (ãij)

where 1≤i , j ≤ n is a n x n fuzzy matrix and xj, bj∈ F(R). This system is called fully

fuzzy linear system. (FFLS).

3. Inverses of partitioned matrices.

In this paper, square partitioned matrices with two row blocks and two column blocks in

the form 8) 2

9 :; where ), 2, 9 and : are square sub-matrices of same order are

considered.

A square matrix partitioned such that its diagonal elements are square matrices, while all other elements are zero’s is a called a block diagonal matrix.

A block diagonal matrix with two row blocks and two column blocks is of the

form 8) 0

0 :; where ) and : are square sub-matrices.

Theorem 3.1. [6] If) and : are non singular then 8) 0_{0 :;}

?3

@)₀?3 _:0_?3A

Proof: Let 8B C _D; be the inverse of given matrix 8) 0_{0 :;} where all the sub matrices are of order .

Then 8) 0_{0 :; 8}B C _D;= @E_{: E}6 0

6A

By matrix multiplication and comparing E₆ ; C 0 ; : 0 ; :D E₆ Since ) and : are non singularB )?3; 0 ; 0 ; D :?3.

8B C _{D; @})₀?3 _:0_?3A

Theorem 3.2. [6] If9 is non singular then 89 0_{: 9;}

?3

@_9?39?3_{:9?3 9}0_?3A

Proof: Let @B 0_{C DA} be the inverse of given matrix 89 0_{: 9;} where all the sub matrices are of order .

Then 89 0_{: 9; @C BA}B 0 =@E_{: E}6 0

6A

By matrix multiplication and comparing 9B E₆ ; :B $ 9C 0 Since 9 is non singular B 9?3

9C :B :9?3_{F C 9}?3_:9?3

Hence 89 0

: 9; ?3

@_9?39?3_{:9?3 9}0_?3A .

4. Solving FFLS using trapezoidal fuzzy number matrices by partitioning the block matrices.

For solving FFLS )* % , 0where )* ), 2, , #, , = , , G, H and

0 0, I, J, Kare trapezoidal fuzzy number matrices.

), 2, , # % , , G, H 0, I, J, K

), 2, )G $ , 2H $ # 0, I, J, K (by 6.3)

264

2 I (4.2)

)G $ J (4.3) 2H $ # K. (4.4)

The above system of linear equations can be written as 4 x 4crisp block matrix.

N

) 0 0 0 0 2 0 0

0 #0 )0 20 O N

G H

O P 0 I J K

Q (4.5)

where ), 2, , and # are x crisp matrices and , , G, H, 0, I, J and karecrisp column vectors.

Let 9 8) 0_{0 2;}, : 8 0_{0 #;} , S 8_;, T 8GH;,U @_{IA +V W 8}0 J_K;. Then (4.5) may be partitioned and written as

0 0 0

0 0 0

0 0

0 0

A x b

B y g

M A z h

N B w k

                 ₌              _{   }  

F 89 0_{: 9; 8}S_{T; 8}U_{W; F 8}S_{T; 8: 9;}9 0 ?3_8UW;

8ST; @9?39?3_:9?3 ₉0?3A 8UW;(By theorem 3.2)

Now, 9?3 @)?3 0

0 2?3A(By theorem 3.1)

9?3_:9?3 _@)?3 0

0 2?3A 8 0_{0 #; @})

?3 ₀

0 2?3A= @)

?3₎?3 ₀

0 2?3_#2?3A

Hence,

1

1

1 1 1

1 1 1

0 0 0

0 0 0

0 0 0 0 A x b B y g

A MA A

z h

B NB B

w k − − − − − − − −                ₌       ₋       _{ }  −   _{ }  .(4.6) 1 1

1 1 1

1 1 1

x A b

y B g

z A MA b A h w B NB g B k

265

Another approach for solving FFLS using trapezoidal fuzzy number matrices.

From (4.1) and (4.3)

8) 0 _{); 8G; 8}0_J; (4.7)

From (4.2) and (4.4)

82 0_{# 2; 8H; 8} I_K; (4.8)

where ), 2, , and # are x crisp matrices and , , G, H, 0, I, J and kare crisp column vectors.

(4.7) and (4.8) are 2n x 2n crisp block matrices and can be solved by partitioning

.8G; 8 _);) 0

?3 80J;

8G; 8 ))?3?3₎?3 ₎0?3; 80J; (using theorem 3.2)

@_)?3))_?3?3_{0 $ )}0 _?3JA (4.9)

8H; 8# 2;2 0 ?3

8IK;

8H; 82?32?3_#2?3 ₂0?3; 8IK; (using theorem 3.2)

@_2?3#22_?3?3_{I $ 2}I _?3KA (4.10)

Combining (4.9) and (4.10) the solution of (4.5) is

1

1

1 1 1

1 1 1

x A b

y B g

z A MA b A h w B NB g B k

−

−

− − −

− − −

 

 

 

 

 

 ₌

 

  _{− +}

 

  _{− +} 

   

5. Numerical example

Example 5.1. Consider the FFLS

266

Solution: The given fully fuzzy linear system can be written as

) 82 3_{5 7;},2 83 4_{4 5;}, 81 9_{4 2;},# 85 1_{2 6;} ;

0 825_60;,I 842_54;,J 862_59;, K 856_74;

)?3 _{87 3}

5 2;; 2?3 85 4 4 3;

)?3₎?3 _{87 3}

5 2; 81 94 2; 87 3 5 2; 8 226 91 ;320 129

2?3_#2?3 _{85 4}

4 3; 85 12 6; 85 4 4 3; 8161 125 126 98;

1

1

1 1 1

1 1 1

0 0 0

0 0 0

0 0 0 0 A x b B y g

A MA A

z h

B NB B

w k − − − − − − − −                ₌       ₋       _{ }  −   _{ }  _ ` ` ` ` ` a3₄

3 4 G3 G4 H3 H4b

c c c c c d _ ` ` ` ` ` `

a 7_{5 2}3 0₀ 0₀ 0₀ 0_{0 0}0 0₀

0 0 5 4 0 0 0 0

0 0 4 3 0 0 0 0

320 129 0 0 7 3 0 0

226 91 0 0 5 2 0 0

0 0 161 125 0 0 5 4

0 0 126 98 0 0 4 3bc

c c c c c d _ ` ` ` ` ` ` a25₆₀

42 54 62 59 56 74bc c c c c c d _ ` ` ` ` ` ` a5₅

6 6 3 2 4 2bc c c c c c d

Hence , @5,6,3,4_5,6,2,2A.

Solving by another approach

8) 0 _{); 8G; 8}0_{J; F 8G; 8 );}) 0 ?3_{80J; F 8G; 8)?3})?3_{)?3 )}0_{?3; 80J;}

N 3 4 G3 G4

O N

7 3

5 2 0 00 0 320 129

226 91 7 3 5 2

O N 25 60 62 59

O N 5 5 3 2 O

267

N 3 4 H3 H4

O N

5 4

4 3 0 00 0 161 125

126 98 5 4 4 3 O N

42 54 56 74

O N 6 6 4 2 O

Hence 7 @5,6,3,4_5,6,2,2A.

6. Conclusion

In this paper an x FFLS Ã ⊗ x = b is transformed to a 4 x 4 crisp block matrix or two 2 x 2 crisp block matrices and solution is obtained bypartitioning the block matrices. When is large, computing inverse for such cases become complicated and impractical because they consume much computer time and space. By partitioning, the present method is easier to implement as the order of the system reduces to sub-matrices of order x .

REFERENCES

1. T.Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation, 155(2) (2004) 493-502.

2. A.Kumar, Neetu and A.Bansal, A new method to solve fully fuzzy linear systems with trapezoidal fuzzy numbers, Canadian Journal on Science and Engineering Mathematics, 1(3) (2010) 45-56.

3. T.Asady and P.Mansouri, Numerical solution of fuzzy linear system, International Journal of Computer Mathematics, 86(1) (2009) 151-162.

4. M.Dehghan , B.Hashemi and M.Ghatee, computational methods for solving fully fuzzy linear system, Applied Mathematics and Computation, 179 (2006) 328-343. 5. M.Friedman, M.Ming and A.Kandel, Fuzzy linear systems, Fuzzy Sets and Systems,

96 (1998) 201-209.

6. K.M.Abadir and J.R.Magnus, Matrix Algebra, Cambridge University Press, Cambridge, 2005.

7. G.Malkawi, Nazihah Ahmad and Haslinda Ibrahim, Solving fully fuzzy linear system with the necessary and sufficient condition to have a positive solution, Appl. Math. Inf. Sci., 8(3) (2014) 1003-1019.

8. S.H.Nasseri, M.Sohrabi and E.Ardil, Solving fully fuzzy linear systems by use of a certain decomposition of the coefficient matrix, International Journal of Computational and Mathematical Sciences, 2 (2008) 140−142.

9. S.H.Nasseri and M.Gholami, Linear system of equations with trapezoidal fuzzy numbers, The Journal of Mathematics and Computer Science, 3 (2011) 71-79. 10. S.Radhakrishnan and P.Gajivaradhan, A new approach to solve fully fuzzy linear