A Study on Triangular Type 2 Triangular Fuzzy Matrices

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1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 4, Number 2 (2014), pp Research India Publications A Study on Triangular Type 2 Triangular Fuzzy Matrices D.Stephen Dinagar 1 and K.Latha 2 1 P.G. and Research Department of Mathematics, TBML College, Porayar, India; dsdina@rediffmail.com 2 P.G.and Research Department of Mathematics, Poompuhar College, Melaiyur, India; k.latha1976@yahoo.in 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, triangular type-2 triangular fuzzy matrices (triangular T2TFM) is proposed. Some more special types and properties of triangular T2TFM are presented. Few relevant examples are also included to justify the proposed notions. Key words: Type-2 fuzzy set, Type-2 triangular fuzzy number, Type-2 triangular fuzzy matrices. 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 [12]. A type-2 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 intervalvalued 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 [10], 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]

2 146 D.Stephen Dinagar and K.Latha introduced type-2 triangular fuzzy matrices. In [9] Stephen Dinagar and Latha 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 derive some properties of triangular T2TFMs. In section-5, we define some special forms of triangular T2TFMs. 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, μ A (x)); x є X}. Here μ A : X [0, 1] is a mapping called the degree of membership function of the fuzzy set A and μ A (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] Definition: (Zadeh) Type-2 fuzzy set A type-2 fuzzy set is a fuzzy set whose membership values are fuzzy sets on [0, 1] Definition: The type-2 fuzzy sets are defined by functions of the form 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.

3 A Study on Triangular Type 2 Triangular Fuzzy Matrices Definition: Type-2 fuzzy number [7] Let be a type-2 fuzzy set defined in the universe of discourse R. If the following conditions are satisfied: is normal, is a convex set, The support of is closed and bounded, then is called a type-2 fuzzy number Definition: Type-2 triangular fuzzy number A type-2 triangular fuzzy number on R is given by = {(x, ( A 1 (x), A 2 (x), A 3 (x)); xєr} and A 1 (x) A 2 (x) A 3 (x), for all xє R. Denote = (,, ), where = (A 1 L, A 1 N, A 1 U ), = (A 2 L, A 2 N, A 2 U ) and = (A 3 L, A 3 N, A 3 U ) are same type of fuzzy numbers Arithmetic operations on type-2 triangular fuzzy numbers [9] Let = (,, ) = ((a 1 L, a 1 N, a 1 U ), (a 2 L, a 2 N, a 2 U ), (a 3 L, a 3 N, a 3 U )) and = (,, ) = ((b 1 L, b 1 N, b 1 U ), (b 2 L, b 2 N, b 2 U ), (b 3 L, b 3 N, b 3 U )) be two type-2 triangular fuzzy numbers. Then we define, (i)addition: + = ((a1 L +b 1 L, a 1 N +b 1 N, a 1 U +b 1 U ), (a 2 L +b 2 L, a 2 N +b 2 N, a 2 U +b 2 U ), (a 3 L +b 3 L, a 3 N +b 3 N, a 3 U +b 3 U )) (ii)subtraction: = ((a1 L b 3 U, a 1 N b 3 N, a 1 U b 3 L ), (a 2 L b 2 U, a 2 N b 2 N, a 2 U b 2 L ), (a 3 L b 1 U, a 3 N b 1 N, a 3 U b 1 L )) (iii)scalar multiplication: If k 0 and k є R then k = ((ka 1 L, ka 1 N, ka 1 U ), (ka 2 L, ka 2 N, ka 2 U ), (ka 3 L, ka 3 N, ka 3 U )) and if k < 0 and k є R then k = ((ka 3 U, ka 3 N, ka 3 L ), (ka 2 U, ka 2 N, ka 2 L ), (ka 1 U, ka 1 N, ka 1 L )). (iv)multiplication: Define b = b 1 L +b 1 N +b 1 U +b 2 L +b 2 N +b 2 U +b 3 L +b 3 N +b 3 U.If 0, then = ( If b<0, then = (,,,, ), ( ), (,,,, ), ( ), ( (v)division: Wheneverb 0 we define division as follows:if b>0, then =(,, ), (,, ), (,, ).,,, )., ).

4 148 D.Stephen Dinagar and K.Latha If b<0, then =(,, ), (,, ), (,, ) The proposed ranking function [9] 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 Ř:F(R) Rwhich maps each fuzzy number into R. Suppose if = (,, ), = ((A L 1, A N 1, A U 1 ), (A L 2, A N 2, A U 2 ), (A L 3, A N 3, A U 3 )) then we define Ř( ) = L (A1 +A N 1 +A U 1 +A L 2 +A N 2 +A U 2 +A L 3 +A N 3 +A U 3 ) / 9. Also we define orders on F(R) by Ř( ) Ř( ) if and only if Ř, Ř( ) Ř( ) if and only if Ř and Ř( ) = Ř( ) if and only if Ř Definition: Type-2 zero triangular fuzzy number If = ((0, 0, 0), (0, 0, 0), (0, 0, 0)) then is said to be a type-2 zero triangular fuzzy number. It is denoted by Definition: Type-2 zero-equivalent triangular fuzzy number A type-2 triangular fuzzy number is said to be a type-2 zero-equivalent triangular fuzzy number if Ř ( ) = 0. It is denoted by Definition: Type-2 unit triangular fuzzy number If = ((1, 1, 1), (1, 1, 1), (1, 1, 1)) then is said to be a type-2 unit triangular fuzzy number. It is denoted by Definition: Type-2 unit-equivalent triangular fuzzy number A type-2 triangular fuzzy number is said to be a type-2 unit-equivalent triangular fuzzy number if Ř ( ) = 1. It is denoted by TYPE-2 TRIANGULAR FUZZY MATRICES (T2TFMS) [9] 3.1. Definition: Type-2 triangular fuzzy matrix (T2TFM) A type-2 triangular fuzzy matrix (T2TFM) of order m n is defined as A = ( ) mxn where the ij th element of A is the type-2 triangular fuzzy number Operations ont2tfms: As for classical matrices we define the following operations on T2TFMs. Let A = ( ) and B = ( ) be two T2TFMs of same order. Then we have the following:

5 A Study on Triangular Type 2 Triangular Fuzzy Matrices 149 (i) A+B = ( + ) (ii) A B = ( ) (iii) For A = ( ) mxn and B = ( ) nxk then AB = ( ) mxk where =., i=1, 2,.., m and j=1, 2,.., k. (iv) A T or Aˊ= ( ) (v) ka = (k ), where k is a scalar Types of Type-2 Triangular Fuzzy Matrices Definition: Unit T2TFM or Identity T2TFM A scalar T2TFM A= ( ) is said to be an unit T2TFM or identity T2TFM if = 1 for every entry in the principal diagonal. It is denoted by I Definition: Unit-equivalent T2TFM or Identity-equivalent T2TFM A scalar - equivalent T2TFM A= ( ) is said to be an unit-equivalent T2TFM or identity-equivalent T2TFM if = 1 for every entry in the principal diagonal. It is denoted by Definition: Upper triangular T2TFM A square T2TFM A= ( ) is called an upper triangular T2TFM if all the entries below the principal diagonal are Definition: Upper triangular - equivalent T2TFM A square T2TFM A= ( ) is called an upper triangular - equivalent T2TFM if all the entries below the principal diagonal are Definition: Lower triangular T2TFM A square T2TFM A= ( ) is called a lower triangular T2TFM if all the entries above the principal diagonal are Definition: Lower triangular - equivalent T2TFM A square T2TFM A= ( ) is called a lower triangular - equivalent T2TFM if all the entries above the principal diagonal are Definition: Triangular T2TFM A square T2TFM A = ( ) is called a triangular T2TFM if it is either upper triangular T2TFM or lower triangular T2TFM Definition: Triangular - equivalent T2TFM A square T2TFM A = ( ) is called a triangular - equivalent T2TFM if it is either upper triangular equivalent T2TFM or lower triangular - equivalent T2TFM.

6 150 D.Stephen Dinagar and K.Latha 4. PROPERTIES OF TRIANGULAR T2TFMs Property 4.1 The sum of two lower triangular T2TFMs of order n is also a lower triangular T2TFM of order n. Let A = ( ) and B = ( ) be two lower triangular T2TFMs. Since A and B are lower triangular T2TFMs, = 0 and = 0 for all i < j ; i, j = 1, 2,, n. Let A+B = C. Then ( + ) = ( ). For all i < j ; i, j = 1, 2,, n, = + = = 0. Hence C is also a lower triangular T2TFM of order n. Property 4.2 The sum of two upper triangular T2TFMs of order n is also an upper triangular T2TFM of order n. Let A = ( ) and B = ( ) be two upper triangular T2TFMs. Since A and B are upper triangular T2TFMs, = 0 and = 0 for all i > j; i, j = 1, 2,, n. Let A+B = C. Then ( + ) = ( ). For all i > j; i, j = 1, 2,, n, = + = = 0. Hence C is also an upper triangular T2TFM of order n. Property 4.3 The product of a lower triangular T2TFM by a constant is also a lower triangular T2TFM. Let A = ( ) be a lower triangular T2TFM. Since A is a lower triangular T2TFM, = 0 for all i < j ; i, j = 1, 2,, n. Let k be a scalar and ka = B. Then (k ) = ( ). For all i < j; i, j = 1, 2,, n, = k = k0 = 0. Hence B is also a lower triangular T2TFM. Property 4.4 The product of an upper triangular T2TFM by a constant is also an upper triangular T2TFM. Let A = ( ) be an upper triangular T2TFM. Since A is an upper triangular T2TFM, = 0 for all i > j; i, j = 1, 2,, n.

7 A Study on Triangular Type 2 Triangular Fuzzy Matrices 151 Let k be a scalar and ka = B. Then (k ) = ( ). For all i > j; i, j = 1, 2,, n, = k = k0 = 0. Hence B is also an upper triangular T2TFM. Property 4.5 The product of two lower triangular T2TFMs of order n is also a lower triangular T2TFM of order n. Let A = ( ) and B = ( ) be two lower triangular T2TFMs. Since A and B are lower triangular T2TFMs, = 0 and = 0 for all i < j ; i, j = 1, 2,, n. Let AB = C = ( ) where =.. Now we will show that = 0 for all i < j ; i, j = 1, 2,, n. For i < j we have = 0 for k = i+1, i+2,, n, and similarly = 0 for k = 1, 2,, i. Therefore =. =. +. = 0 Now =. = =. Hence the result follows. Property 4.6 The product of two upper triangular T2TFMs of order n is also an upper triangular T2TFM of order n. Let A = ( ) and B = ( ) be two upper triangular T2TFMs. Since A and B are upper triangular T2TFMs, = 0 and = 0 for all i > j ; i, j = 1, 2,, n. Let AB = C = ( ) where =.. Now we will show that = 0 for all i > j ; i, j = 1, 2,, n. For i > j we have = 0 for k = 1, 2,, i 1, and similarly = 0 for k = i, i+1,, n. Therefore =. =. +. = 0 Now =. =

8 152 D.Stephen Dinagar and K.Latha =. Hence the result follows. Property 4.7 The transpose of an upper triangular T2TFM is a lower triangular T2TFM and vice versa. Let A = ( ) be an upper triangular T2TFM. Since A is an upper triangular T2TFM, = 0 for all i > j; i, j = 1, 2,, n. Let B be the transpose of A. Then Aˊ = B. i.e, ( ) = ( ). For all i > j; i, j = 1, 2,, n, = 0 =. That is for all i < j ; i, j = 1, 2,, n, = 0. Hence B is a lower triangular T2TFM. Remark: However, operations mixing upper and lower triangular T2TFMs do not in general produce triangular T2TFMs. For instance, the sum of an upper and lower triangular T2TFM can be any T2TFM. The product of a lower triangular T2TFM with an upper triangular T2TFM is not necessarily triangular T2TFM either. 5. SPECIAL FORMS OF TRIANGULAR T2TFMs Definition: Unitriangular T2TFM If the entries on the main diagonal of a triangular T2TFM are all 1, then the matrix is called unitriangular T2TFM. The Identity T2TFM is the only matrix which is both upper and lower unitriangular T2TFM. 1 For instance, A = Definition: Unitriangular - equivalent T2TFM If the entries on the main diagonal of a triangular - equivalent T2TFM are all 1, then the matrix is called unitriangular - equivalent T2TFM. The Identity equivalent T2TFM is the only matrix which is both upper and lower unitriangular - equivalent T2TFM Definition: Strictly triangular T2TFM If the entries on the main diagonal of a triangular T2TFM are all 0, then the matrix is called strictly triangular T2TFM.

9 A Study on Triangular Type 2 Triangular Fuzzy Matrices For instance, A = Definition: Strictly triangular - equivalent T2TFM If the entries on the main diagonal of a triangular - equivalent T2TFM are all 0, then the matrix is called strictly triangular - equivalent T2TFM Definition: Atomic triangular T2TFM An atomic triangular T2TFM is a special form of unitriangular T2TFM, where all the off-diagonal entries are 0, except for the entries in a single column. 1 For instance, A = Definition: Atomic triangular - equivalent T2TFM An atomic triangular - equivalent T2TFM is a special form of unitriangular - equivalent T2TFM, where all the off-diagonal entries are 0, except for the entries in a single column. 6. CONCLUSION In this article triangular type-2 triangular fuzzy matrices are defined and also some special properties of triangular T2TFMs are proved. Also it is noted that the discussed notions are very useful to solve the fuzzy solutions of the type-2 fuzzy matrix equations and the theories of the discussed T2TFMs may be utilized in further works. 7. REFERENCES [1] Hisdal, E., The IF THEN ELSE Statement and Interval-valued Fuzzy sets of Higher type, Int.J.Man-Machine studies 15(1981) [2] Jhon, R.I., Type-2 Fuzzy sets; an Appraisal of Theory and Applications, Int.J.Fuzziness Knowledge-Based Systems 6(6) (1998) [3] Kim, J.B., Determinant theory for Fuzzy and Boolean Matrices, Congressus Numerantium, (1988) [4] Klir, G.J., Yuan, B., Fuzzy sets and Fuzzy Logic: Theory and Applications, Prentice- Hall, Englewood cliffs, NJ, [5] Ragab, M.Z., and Emam, E.G., On the Min-Max Composition of Fuzzy Matrices, Fuzzy sets and Systems, 75(1995), [6] Shyamal, A.K., and Pal, M., Triangular Fuzzy Matrices, Iranian Journal of Fuzzy Systems, Vol.4, No.1, (2007), pp

10 154 D.Stephen Dinagar and K.Latha [7] Stephen Dinagar, D., Anbalagan, A., Fuzzy Programming based on Type-2 Generalized Fuzzy Numbers, International J.of Math. Sci. &Engg. Appls. Vol.5, No.IV (July 2011), pp [8] \Stephen Dinagar, D., and Latha, K., A Note on Type-2 Triangular Fuzzy Matrices, International J.of Math. Sci. &Engg. Appls. Vol.6, No.I (Jan 2012), pp [9] Stephen Dinagar, D., and Latha, K., Some types of Type-2 Triangular Fuzzy Matrices, International Journal of Pure and Applied Mathematics, Volume 82, No. 1(Jan 2013). [10] Thomason, M.G., Convergence of Powers of a Fuzzy Matrix, J.Math Anal. Appl, 57 (1977), [11] Zadeh, L.A., Fuzzy sets, Information and Control, 8 (1965), [12] Zadeh, L.A., The Fuzzy concept of a Linguistic Variable and its Application to Approximate Reasoning 1, Inform. Sci. 8 (1975),