091

Page 1

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in

Rank of a Type-2 Triangular Fuzzy Matrix 1

M.Manvizhi1, Dr.K.Latha2 & Dr.K.Thiripurasundari3

P.G. Student, Department of Mathematics, Poompuhar College, Melaiyur, India. 2 HOD, Department of Mathematics, Poompuhar College, Melaiyur, India, 3 Assistant Professor, Department of Mathematics, Poompuhar College, Melaiyur, India. Abstract: Type-2 fuzzy sets are fuzzy sets whose membership values are fuzzy sets on the interval [0, 1]. This concept was proposed by Zadeh, as an extension of fuzzy sets. Type-2 fuzzy sets possess a great expressive power and are conceptually quite appealing. Also fuzzy matrices play an important role in scientific developments. In this paper, rank of type-2 triangular fuzzy matrix (T2TFM) is proposed and algorithm for finding the rank of T2TFM is presented. Numerical example is also included.

1. Introduction The concept of a type-2 fuzzy set, which is an extension of the concept of an ordinary fuzzy set, was introduced by Zadeh in 1975 [13]. A type2 fuzzy set is characterized by a membership function, i.e., the membership value for each element of this set is a fuzzy set in [0, 1], unlike an ordinary fuzzy set where the membership value is a crisp number in [0, 1]. Hisdal [1] discussed the IF THEN ELSE statement and interval-valued fuzzy sets of higher type. Jhon [2] studied an appraisal of theory and applications on type-2 fuzzy sets. Stephen Dinagar and Anbalagan [7] presented new ranking function and arithmetic operations on generalized type-2 trapezoidal fuzzy numbers. The fuzzy matrices introduced first time by Thomason [11], and discussed about the convergence of powers of fuzzy matrix. Kim [3] presented some important results on determinant of square fuzzy matrices. Ragab et al [5] presented some properties of the min-max composition of fuzzy matrices. A.K.Shyamal and M.pal [6] first time introduced triangular fuzzy matrices. Recently Stephen Dinagar and Latha [8] introduced type-2 triangular fuzzy matrices. Also they presented some types and properties of type-2 triangular fuzzy matrices. The paper is organized as follows. Firstly in section-2 of this paper, we recall the definition of type-2 triangular fuzzy number and some operations on type-2 triangular fuzzy numbers. In section-3, we review the definition of type-2 triangular fuzzy matrices (T2TFM) and some operations on T2TFMs. In section-4, we define rank of T2TFM and we construct an algorithm for finding the rank of T2TFM. In section – 5, relevant

Imperial Journal of Interdisciplinary Research (IJIR)

numerical examples is presented. Finally in section - 6, conclusion is also included.

2. Type-2 Triangular Fuzzy Numbers 2.1. Definition: Fuzzy Set A fuzzy set is characterized by a membership function mapping the elements of a domain, space or universe of discourse X to the unit interval [0,1]. A fuzzy set A in a universe of discourse X is defined as the following set of pairs: A = {(x, ΟA(x)); x є X}. Here ΟA: X → [0,1] is a mapping called the degree of membership function of the fuzzy set A and ΟA(x) is called the membership value of x є X in the fuzzy set A. These membership grades are often represented by real numbers ranging from [0,1].

2.2. Definition: (Zadeh) Type-2 fuzzy set A type-2 fuzzy set is a fuzzy set whose membership values are fuzzy sets on [0,1].

2.3. Definition: The type-2 fuzzy sets are defined by functions of the form đ?œ‡ A: x → χ ([0,1]) where χ ([0,1]) denotes the set of all ordinary fuzzy sets that can be defined within the universal set [0,1]. An example [4] of a membership function of this type is given in fig.1.

Fig.1. Illustration of the concept of a fuzzy set of type-2.

Page 538

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in 2.4. Definition: Type-2 fuzzy number [7]

Let Ěƒ be a type-2 fuzzy set defined in the universe of discourse R. If the following conditions are satisfied: Ěƒ is normal, (i) Ěƒ is a convex set, (ii) (iii) The support of Ěƒ is closed and bounded, then Ěƒ is called a type-2 fuzzy number.

2.5. Definition: Type-2 triangular fuzzy number

A type-2 triangular fuzzy number Ěƒ on R is given by Ěƒ = {(x,(đ?œ‡ A1(x),đ?œ‡ A2(x),đ?œ‡ A3(x)); xŃ”R} and đ?œ‡ A1(x) đ?œ‡ A2(x) đ?œ‡ A3(x), for all xŃ” R. Denote Ěƒ = ( Ěƒ , Ěƒ , Ěƒ ), where Ěƒ = (A L,A N,A U), Ěƒ = 1 1 1 (A2L,A2N,A2U) and Ěƒ = (A3L,A3N,A3U) are same type of fuzzy numbers.

2.6. Arithmetic operations on type-2 triangular fuzzy numbers [9] Ěƒ =

Let

( Ěƒ , Ěƒ , Ěƒ )

=

((a1 ,a1N,a1U),(a2L,a2N,a2U),(a3L,a3N,a3U)) and Ěƒ = ( Ěƒ , Ěƒ , Ěƒ ) = ((b1L,b1N,b1U),(b2L,b2N,b2U),(b3L,b3N,b3U)) L

be two type-2 triangular fuzzy numbers. Then we define,

(i)Addition: Ěƒ + Ěƒ= ((a1L+b1L,a1N+b1N,a1U+b1U),(a2L+b2L,a2N+b2N,a2U+ b2U),(a3L+b3L,a3N+b3N, a3U+b3U))

(ii)Subtraction:

Ěƒ − Ěƒ= ((a1L−b3U,a1N−b3N,a1U−b3L),(a2L−b2U,a2N−b2N,a2U− b2L),(a3L−b1U,a3N−b1N, a3U−b1L))

(iii)Scalar multiplication:

If k 0 and k Ń” R then k Ěƒ = ((ka1L,ka1N,ka1U),(ka2L,ka2N,ka2U),(ka3L,ka3N,ka3U)) and if k < 0 and k Ń” R then k Ěƒ = ((ka3U,ka3N,ka3L),(ka2U,ka2N,ka2L),(ka1U,ka1N,ka1L)).

(iv)Multiplication: Define đ?œŽb = b1L+b1N+b1U+b2L+b2N+b2U+b3L+b3N+b3U.If đ?œŽ 0, then ĚƒĂ—Ěƒ= đ??żđ?œŽ

9

,

đ?‘ đ?œŽ

9

If đ?œŽb<0, then

,

đ?‘ˆđ?œŽ

9

,

đ??żđ?œŽ

9

,

đ?‘ đ?œŽ

9

,

đ?‘ˆđ?œŽ

9

,

đ??żđ?œŽ

9

,

đ?‘ đ?œŽ

9

ĚƒĂ—Ěƒ= đ?‘ˆđ?œŽ

9

đ?‘ đ?œŽ

,

đ??żđ?œŽ

,

9

đ?‘ˆđ?œŽ

,

9

(v)Division:

đ?‘ đ?œŽ

,

9

,

9

đ??żđ?œŽ

đ?‘ˆđ?œŽ

,

9

đ?‘ đ?œŽ

,

9

Wheneverđ?œŽb ≠0 we define division as follows:If đ?œŽb>0, then ĚƒĚƒ Ěƒ

9

= 9

đ?œŽ

đ??ż

,

9

,

9

đ?œŽ

đ?‘

,

9

,

9

đ?œŽ

If đ?œŽb<0, then ĚƒĚƒ Ěƒ

đ??żđ?œŽ

,

9

đ?‘ˆ

,

9

đ??ż

,

9

đ?‘ˆ

đ?œŽ

,

9

,

9

đ?œŽ

đ?‘

,

9

,

9

đ?œŽ

đ?‘ˆ

,

9

đ??ż

,

9

đ?‘ˆ

đ?œŽ

,

9

,

9

đ?œŽ

đ?‘

,

9

,

9

đ?œŽ

đ?‘ˆ

= 9

đ?œŽ

đ?‘ˆ

đ?œŽ

đ?‘

đ?œŽ

đ??ż

đ?œŽ

đ?œŽ

đ?‘

đ?œŽ

đ??ż

đ?œŽ

đ?œŽ

đ?‘

2.7. The proposed ranking function [8]

đ?œŽ

đ??ż

Let F(R) be the set of all type-2 normal triangular fuzzy numbers. One convenient approach for solving numerical valued problem is based on the concept of comparison of fuzzy numbers by use of ranking function. An effective approach for ordering the elements of F(R) is to define a linear ranking function Ĺ˜:F(R) → Rwhich maps each fuzzy number into R. Suppose if Ěƒ = ( Ěƒ , Ěƒ , Ěƒ ), = ((A1L,A1N,A1U),(A2L,A2N,A2U),(A3L,A3N,A3U))then we define Ĺ˜( Ěƒ )= (A1L+A1N+A1U+A2L+A2N+A2U+A3L+A3N+A3U) / 9

Also we define orders on F(R) by Ĺ˜( Ěƒ ) Ĺ˜( Ěƒ ) if and only if ĚƒĹ˜ Ěƒ , Ĺ˜( Ěƒ ) Ĺ˜( Ěƒ ) if and only if ĚƒĹ˜ Ěƒ and Ĺ˜( Ěƒ ) = Ĺ˜( Ěƒ ) if and only if Ěƒ=Ĺ˜ Ěƒ .

2.8.Definition: Type-2 zero triangular fuzzy number If Ěƒ = ((0,0,0),(0,0,0),(0,0,0)) then Ěƒ is said to be a type-2 zero triangular fuzzy number. It is denoted by 0.

2.9.Definition:Type-2 zero-equivalent triangular fuzzy number A type-2 triangular fuzzy number Ěƒ is said to be a type-2 zero-equivalent triangular fuzzy number if Ĺ˜ ( Ěƒ ) = 0. It is denoted by Ěƒ .

2.10. Definition: Type-2 unit triangular fuzzy number ,

đ?‘ˆđ?œŽ

9

Imperial Journal of Interdisciplinary Research (IJIR)

If Ěƒ = ((1,1,1),(1,1,1),(1,1,1)) then Ěƒ is said to be a type-2 unit triangular fuzzy number. It is denoted by 1.

Page 539

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in 2.11. Definition: Type-2 unit-equivalent triangular fuzzy number

Notation:

A type-2 triangular fuzzy number Ěƒ is said to be a type-2 unit-equivalent triangular fuzzy number if Ĺ˜ ( Ěƒ ) = 1. It is denoted by Ěƒ .

Let A = ( Ěƒ ) be a type-2 triangular fuzzy matrix. Suppose if we take rank for every Ěƒ in A then A is converted into a classical matrix. It is denoted by A = ( Ěƒ )c= Ĺ˜(aĚƒ )

3.1.Definition:Type-2 triangular fuzzy matrix (T2TFM)

4.1.Definition: Determinant of T2TFM

3. Type-2 Triangular Fuzzy Matrices (T2TFMs) [8]

A type-2 triangular fuzzy matrix (T2TFM) of order m Ă— n is defined as A = ( Ěƒ )mxn where the ijth element Ěƒ of A is the type-2 triangular fuzzy number.

3.2. Operations on T2TFMs: As for classical matrices we define the following operations on T2TFMs. Let A = ( Ěƒ ) and B = ( Ěƒ ) be two T2TFMs of same order. Then we

have the following: (i) A+B = ( Ěƒ + Ěƒ ) (ii) A−B = ( Ěƒ − Ěƒ ) (iii) For A = ( Ěƒ )mxn and B = ( Ěƒ )nxk then AB = ( ĚƒĚƒ )mxk where ĚƒĚƒ = ∑đ?‘?= Ěƒ đ?‘? . Ěƒđ?‘? , i=1,2,‌..,m and j=1,2,‌..,k. (iv) ATor AËŠ= ( Ěƒ ) (v) kA = (k Ěƒ ) , where k is a scalar.

4. Rank of a Type-2 Triangular Fuzzy Matrix

The determinant of a nxn T2TFM A = ( Ěƒ ) is denoted by | A | or det(A) and is defined as follows: | A | = ∑đ?‘?∈đ?‘†đ?‘› đ?‘ đ?‘”đ?‘› đ?‘? âˆ? = Ěƒ đ?‘? =∑đ?‘?∈đ?‘†đ?‘› đ?‘ đ?‘”đ?‘› đ?‘? Ěƒ đ?‘? Ěƒ đ?‘? ‌ Ěƒ đ?‘? Where Ěƒ đ?‘? are type-2 triangular fuzzy numbers and denotes the symmetric group of all permutations of the indices {1, 2, 3, . . . , n} and sign p = 1 or −1 according as the permutation p = ‌‌ đ?‘› ( ) is even or odd đ?‘? đ?‘? ‌‌ đ?‘? đ?‘› respectively.

4.2. Definition: Minor Let A = ( Ěƒ ) be a square T2TFM of order n. The minor of an element Ěƒ in A is a determinant of order (n – 1) x (n – 1) which is obtained by deleting the ith row and the jth column Ěƒ . from A and is denoted by đ?‘€

3.3. Definition: Diagonal T2TFM

4.3. Definition: Cofactor

3.4. Definition: Diagonal - equivalent T2TFM

Let A = ( Ěƒ ) be a square T2TFM of order n. The cofactor of an element Ěƒ in A is denoted Ěƒ . by Ěƒ and is defined as Ěƒ = (−1)i+jđ?‘€

A square T2TFM A= ( Ěƒ ) is said to be a diagonal T2TFM if all the elements outside the principal diagonal are 0.

A square T2TFM A= ( Ěƒ ) is said to be a diagonal - equivalent T2TFM if all the elements outside the principal diagonal are Ěƒ

3.5. Definition: Equal type-2 triangular fuzzy matrices

Two type-2 triangular fuzzy matrices A = Ěƒ ( ) and B = ( Ěƒ ) of the same order are said to be equal if the rank of their elements in the corresponding positions are equal. Also it is denoted by A = B.

Imperial Journal of Interdisciplinary Research (IJIR)

4.4. Definition: Singular T2TFM

A square T2TFM A = ( Ěƒ ) is said to be singular T2TFM if | A | = Ěƒ .

4.5. Definition: Non-singular T2TFM

A square T2TFM A = ( Ěƒ ) is said to be non-singular T2TFM if | A | â‰ Ěƒ .

4.6. Definition: Rank

The rank of T2TFM A = ( Ěƒ ) is the highest order of a non-zero minor of A.

Page 540

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in Note: If A = ( Ěƒ ) is a square T2TFM of order n, then the rank k satisfies the relation k ≤ n. If k = n, then the T2TFM is a non-singular T2TFM. If k < n, then the T2TFM is a singular T2TFM.

4.7. Elementary transformations We use the term elementary transformations of a T2TFM A = ( Ěƒ ) for the following transformations: a. Interchange of two rows or columns. b. Multiplication of a row (or a column) by an arbitrary type-2 triangular fuzzy number â‰ Ěƒ . c. Addition of a multiple of one row (or column) by type-2 triangular fuzzy Ěƒ number ≠to another row (column).

Step 5: Performing the same manipulations (step 1 to step 4) with the sub-matrix that remains in the lower right corner, and so on, we finally-after a finite number of manipulations-arrive at a diagonalequivalent T2TFM with the same rank as the original T2TFM A. Thus, to find the rank of T2TFM it is necessary to convert the T2TFM, by means of elementary transformations, to diagonal-equivalent T2TFM. Finally, we convert this diagonalequivalent T2TFM into classical matrix and count the number of units in the principal diagonal. The number of units gives the rank of the given T2TFM.

Clearly these elementary transformations do not change the rank of T2TFM.

4.8. Algorithm for finding the rank of T2TFM Ěƒ Ěƒ = â‹Ž (Ěƒ

Suppose Ěƒ ‌ Ěƒ ‌ â‹Ž ‌ Ěƒ ‌

Ěƒ Ěƒ Ěƒ

we

â‹Ž

have

T2TFM

A

)

Step1: If Ĺ˜( Ěƒ ) = 0 then an interchange of rows and columns will change element in the position Ěƒ . So that Ĺ˜( Ěƒ ) ≠0. Step 2: Convert the element Ěƒ multiplying the first rowby â „ Ěƒ .

to Ěƒ by

Step 3:Subtract from the đ?‘Ąâ„Ž row, i > 1, the first row multiplied by Ěƒ , whenever Ĺ˜( Ěƒ ) ≠0, then element Ěƒ (i > 1) will be replaced by Ěƒ .

Step 4: Subtract from the đ?‘Ąâ„Ž column, j > 1, the first column multiplied by Ěƒ , whenever Ĺ˜( Ěƒ ) ≠0, then element Ěƒ (j> 1) will be replaced by Ěƒ . At this stage we get a T2TFM of the form ĚƒĚƒ Ěƒ ‌ ËŠ Ěƒ ĚƒËŠ ‌ Ěƒ AĚ = ‌ â‹Ž â‹Ž â‹Ž ĚƒËŠ ‌ ĚƒËŠ Ěƒ ( )

Imperial Journal of Interdisciplinary Research (IJIR)

Page 541

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in

5. Numerical example [ , ,, , − , , , − , , ] ] Let A = [ − , , , − , , , − , − , [ , , , , , , , , ] R1 →

R

[ − , , , , ,, , − , , ] [ , , , , , , , , ] [ , , , , , , , , ] [ , , , , , , , , ] [ , , , , , , , , ] [ , , , , , , , , ]

⁄[ , , , , − , , , − , , ], where Ř([ , , , , − , , , − , , ] ≠ 0.

[ , ,, , − , , , − , , ] [ − , , , , ,, , − , , ] ] [ , , , , , , , , ,] ~ [ − , , , − , , , − ,− , [ , , , , , , , , ] [ , , , , , , , , ] R2 →R2 − [ − , , , − , , , − , − , R3 →R3 − [ , , ,

, , ,

R2 →

~

⁄[ −

~

,− ,

, , ,

, , ]) ≠ 0.

[ − , , , , ,, , − , , ] [ − ,− , , − ,− , , − , , [ − ,− , , − ,− , , − , ,

,− , , − ,− , , − , ,

[[ , , , − , , , − , , ]] [ − , , , − , , , − , , ] [ − ,− , , − ,− , , − , , ] R3 → R3 − [ −

] R1, where Ř([ − , , , − , , , − , − ,

, , ] R1, where Ř([ , , ,

[[ , , , − , , , − , , ]] ~ [ − ,− , , − , , ] , − ,− , [ − ,− , , − ,− , , − , , ]

[ , , , , , , , , ] [ , , , , , , , , ] [ , , , , , , , , ]

], where Ř([ −

[ , , , , , , , , ] [ − ,− , , − ,− , , − ,− , ] [ − ,− ,− , − ,− , , − ,− , ]

] ]

,− , , − ,− , , − , ,

[ − , , , , ,, , − , , ] [ − ,− , , − , , , − , , ] [ − ,− , ] , − ,− , , − , ,

, − ,− , , − , ,

] R2, where Ř([ −

, , , , − , , ] C1, where Ř([ − , , ,

Imperial Journal of Interdisciplinary Research (IJIR)

]) ≠ 0.

[ , , , , , , , , ] [ − , , ] , − , , , , , [ − ,− ,− , − ,− , , − ,− , ]

,− ,

[ − , , , , ,, , − , , ] [[ , , , − , , , − , , ]] [ − , , , − , , [ − ,− , , − , , , − , , ] , − , , ] [ − , , ] [ , − − , , − , , − ,− , , − ,− , , − , , C2 → C2 − [ − , , ,

]) ≠ 0.

, − ,− , , − , ,

]

]) ≠ 0.

[ , , , , , , , , ] [ − , , , − , , , , , [ − ,− , , − ,− , , −

, , , , − , , ]) ≠ 0.

]

, ,

]

Page 542

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in C3 → C3 − [ , , ,

, , ,

, , ] C1, where Ř([ , , ,

, , ,

, , ]) ≠ 0.

[ − ,− , , − , , , − , , ] [[ , , , − , , , − , , ]] ~ [ − , , , − , , [ − ,− , , − , , ] , − , , , − ,− , [ − , , ] [ − ,− , , − − , , − , , , − , , , − , , C3 → C3 − [ −

,− ,

, −

, ,

[ , , , − , , , − , , ] [ − ,− , , − , , , − , , ] [ − ,− , , − , , , − , , [ − , , , − , , [ − ,− , , − , , [ − , , , − , ,

~

, − , , ] , − ,− , , − , ,

[ − , , , − ,− , [ − ,− , , − , , ([ − ,− , , − ,− ,

R3→

⁄[ −

,−

,

, −

, −

, −

] C2, where Ř([ −

,− ,

, −

, ,

, −

,− ,

] ≠ 0.

] ]

] , − , , ] , − , , , − , ,

,− ,

,− ,

[ − ,− , , − , , , − , , ] ] [ − ,− , ] , − , , , − ,− , ] [ − ] ,− , , − , , , − , ,

,

Imperial Journal of Interdisciplinary Research (IJIR)

]

,

])

], where Ř([ −

,−

,

, −

,− ,

, −

,

,

]) ≠ 0.

Page 543

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in [ , , , − , , , − , , ] [ − ,− , , − , , , − , , ] [ − ,− , , − , , , − , , [ − , , , − , , [ − ,− , , − , , [ − , , , − , ,

~

[

([

c= (

[

− 9

−

−

,−

,

,

−

−

,

,

,

, − ,

,

−

−

, ,

,

−

, ,

]

, − , , ] , − ,− , , − , , ,

, − , 9

,

−

−

, −

]

,

, , ,

) which is a unit matrix of order 3.

]

] ,

]

])

Hence the rank of the given T2TFM A is 3.

Imperial Journal of Interdisciplinary Research (IJIR)

Page 544

Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in

6. Conclusion In this paper rank of type-2 triangular fuzzy matrices is defined. Using this rank of T2TFM, some important results like the nature of system of simultaneous fuzzy linear equations can be studied in future. Also the theories of the discussed T2TFMs may be utilized in further works.

[11] Thomason, M.G., “Convergence of Powers of a Fuzzy Matrix”, J.Math Anal. Appl, 57 (1977), 476-480. [12] Zadeh, L.A., “Fuzzy sets, Information and Control”, 8 (1965), 338-353. [13] Zadeh, L.A., “The Fuzzy concept of a Linguistic Variable and its Application to Approximate Reasoning1”, Information Sciences, 8 (1975), 199-249.

7. References [1] Hisdal, E., “The IF THEN ELSE Statement and Interval-valued Fuzzy sets of Higher type”, International Journal of Man-Machine Studies,15(1981) 385-455. [2] Jhon, R.I., “Type-2 Fuzzy sets; an Appraisal of Theory and Applications”, International Journal of FuzzinessKnowledge-Based System, 6(6) (1998) 563576. [3] Kim, J.B., “Determinant theory for Fuzzy and Boolean Matrices”, Congressus Numerantium, (1988) 273-276. [4] Klir, G.J., Yuan, B., “Fuzzy sets and Fuzzy Logic: Theory and Applications”, Prentice-Hall, Englewood cliffs, NJ, 1995. [5] Ragab, M.Z., and Emam, E.G., “On the Min-Max Composition of Fuzzy Matrices”, Fuzzy sets and Systems, 75(1995), 83-92. [6] Shyamal, A.K., and Pal, M.,“Triangular Fuzzy Matrices”, Iranian Journal of Fuzzy Systems, Vol.4, No.1, (2007), 75-87. [7] Stephen Dinagar, D., Anbalagan, A., “Fuzzy Programming based on Type-2 Generalized Fuzzy Numbers”, International Journal of Mathematical Science & Engineering Applications, Vol.5, No.IV (2011), 317-329. [8] Stephen Dinagar, D., and Latha, K., “A Note on Type-2 Triangular Fuzzy Matrices”, International Journal of Mathematical Science& Engineering Applications,Vol.6, No.I (2012), 207-216. [9] Stephen Dinagar, D., and Latha, K., “Some types of Type-2 Triangular Fuzzy Matrices”, International Journal of Pure and Applied Mathematics, Vol 82, No.1(2013), 21-32. [10] Stephen Dinagar, D., and Latha, K., “A study on Determinant of Type-2 Triangular Fuzzy Matrices”, The Journal of Fuzzy Mathematics, Vol 23, No.4, (2015), 895-906.

Imperial Journal of Interdisciplinary Research (IJIR)

Page 545

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.