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Fuzzy Information and Engineering

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ORIGINAL ARTICLE

Decomposition of Intuitionistic Fuzzy Matrices

T. Muthuraji • S. Sriram - P. Murugadas

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Received: 6 March 2015/ Revised: 16 May 2016/ Accepted: 20 June 2016/

Abstract In this paper, we study some properties of modal operators in intuitionistic fuzzy matrix and we introduce a new composition operator and discuss some of its algebraic properties. Finally, we obtain a decomposition of an intuitionistic fuzzy matrix by using the new composition operator and modal operators.

Keywords Intuitionistic fuzzy matrix • Intuitionistic fuzzy set © 2016 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

1. Introduction

There have been theories evolved over the years to deal with the various types of uncertainties. These evolved theories are put into practice and when found to be wanting are improved upon, paving the way for new theories to handle the tricky uncertainties. The probability theory is one such important theory concerned with the analysis of random phenomena. In 1965, Zadeh [1] came out with the concept of fuzzy set which is indeed an extension of the classical notion of set. Fuzzy set has been found

T. Muthuraji (0)

Mathematics Section, Faculty of Engineering and Technology, Annamalai University, Annamalainagar, Chidambaram, Tiunil Nadu-608002, India email: tmuthuraji@gmail.com S. Sriram

Mathematics Wing, Directorate of Distance Education, Annamalai University, Annamalainagar, Chidambaram, Tamil Nadu, India R Murugadas

Department of Mathematics, Annamalai University, Annamalainagar, Chidambaram, Tamil Nadu, India Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research Society of China.

© 2016 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/jfiae.2016.09.003

to be an effective tool to deal with fuzziness. However, it often falls short of the expected standard when describing the neutral state. As a result, a new concept namely intuitionistic fuzzy set (IFS) was worked out and the same was introduced in 1983 by Atanassov [2, 3]. Using the concept of IFS, Im et.al [4] studied intuitionistic fuzzy matrix (IFM).

IFM generalizes the fuzzy matrix introduced by Thomson [5] and has been useful in dealing with areas such as decision making, relational equations, clustering analysis etc. A number of authors [6,7] have effectively presented impressive results using fuzzy matrix. Atanassov [8], using the definition of index matrix, has paved way for intuitionistic fuzzy index matrix and has further extending it to temporal intuitionistic fuzzy index matrix. IFM is also very useful in the discussion of intuitionistic fuzzy relation [9,10]. Z.S. Xu [11,12] studied intuitionistic fuzzy value and also IFMs. He defined intuitionistic fuzzy similarity relation and also utilized it in clustering analysis.

A lot of research activities have been carried out over the years on IFMs in [13-17]. The period of powers of square IFMs is discussed at length along with some of the results for the equivalence IFM by Jeong and Park [18] while Pal et al. [19-26] made a comprehensive study and neatly developed IFM in various years. Another researcher namely Mondal [27] attempted a study on the similarity relations, invert-ibility and eigenvalues of IFM. In [28], a research was carried out on how a transitive IFM decomposed into a sum of nilpotent IFM and symmetric IFM and in [29] how an IFM gets decomposed into a product of idempotent IFM and rectangular IFM.

Atanassov introduced modal operators in [2] which are meaningless in fuzzy set theory and found a promising direction in research. The above operators for IFMs and some results are obtained in [30]. In this paper, some necessary and sufficient conditions are discussed for a transitive and c-transitive closure matrix interms of modal operators, we explore some more results using modal operators for IFM under max-min composition and discuss similarity relation, idempotents etc. Finally, using modal operators we decompose an IFM by introducing a new composition operator and some properties of that new operator are proved.

2. Preliminaries

We recollect some relevant basic definitions and results will be used later.

Definition 2.1 Let a setX = {jci, X2, ■ • ■ x„} be fixed. Then an IFS [2] can be defined as A = {(Xi,fiA(xd, va(xi)) \ Xi e X] which assigns to each element x, a membership degree HA&d and a non membership degree va(jc,), with the condition 0 < ^a(^i) + va (Xi) < 1 for every Xi e X.

Definition 2.2 Xu and Yager called the 2-tuple a(xj) = (jia(xi), va(x,)) an intuitionistic fuzzy value (IFV) [11, 12] where ¡ia(Xi) e [0,1], va(x,)) e [0,1] and fia(xi) +

Va(Xi) < 1.

Definition 2.3 [2] Let (x, x'), <y,y') e IFS. Then we have

(i) <x,xJ) V (y,/> = <max{x,;y},min{x',/}>.

(ii) (x,x!) A(y,y') = (min {x, >>} ,max \x' ,y'}).

Fuzzy Inf. Eng. (2016) 8: 345-354_347

(iii) <x,xT = <*',*>.

Definition 2.4 [3] Let <x,x'>,<)-,/> e IFS. Then

f(l,0>, if (.x,x!)>(y,y'),

\{x,x'), <*,*-> <<y,/>. Here (x,xf) > (y, /> means x > y and x < y .

Definition 2.5 [11] Let A = {z,ij)nxn be a matrix if all its elements are IFVs. Then A is called an intuitionistic fuzzy matrix.

Definition 2.6 An IFM J = «1,0» for all entries is known as the universal matrix [14] and an IFM O = «0,1)) for all entries is known as zero matrices. Denote the set of all IFMs of order mXn by ^^ and square matrix of order n by The identity IFM I = ((Si,, S'u)) is defined by (6,j, S't]) = <1,0) ifi = j and <Si}, = <0,1) ifi * j.

Definition 2.7 [11,12] Let A = ((a^, a'J)^, B = ((bij,b'i]))mmandC = (<c, j,c^»„xp are IFMs. Then

(i) AvB = ((aij,a>j)V(bu,b'u)).

(ii) AAB = ((aij,a'ij) A (bij^'ij)).

(iii) AC(max - mincomposition) = (a'n) A (ckj> c'kj)))-

Definition 2.8 [14] LetA = (<ay, ajy»mx„ and C = (<cy, ^.»„xp are IFMs. Then we have

(i) A o C (min-max composition)= (Aa'ik) V {ckj, c^»).

(ii) A1 = «aji,a'-^)) (Transpose of A).

(iii) A<-C = (A*«a». a'ik) «- (ckj, c'kj))).

(iv) A -> C = (Ak((aik,a'ik) (ckJ,c'kj))).

(v) Ac = ((a'jj, a,;» (Complement of A).

Also we can use AC = « Ë (aikckj), + %)))• fe=i fc=i

Also A2 = AA, A4 = A*_1A for max-min composition and A'2' = A « A, A® = 0 A for min-max composition.

Definition 2.9 [14] For any IFM A e

(i) A ii reflexive if and only if A > /„.

(ii) A is symmetric if and only if A = AT.

(iii) A is transitive if and only if A > A2.

(iv) A is idempotent if and only if A = A2.

(v) A is irreflexive if {an, a'u) = <0,1> for all i = j.

(vi) A is c-transitive if A < A'2'.

Definition 2.10 [11] An IFMA is said to be an intuitionistic fuzzy equivalence matrix if it satisfy reflexivity, symmetry and transitivity.

Definition 2.11 ForlFSA, Atanassov has defined the modal operators [2] O(necessity) and 0(possibility) in the following way. OA = ((/¿aM> 1 - Ma(x)) \ x € E, the Universal set} and OA = {<1 - va(x), va(x)) | xe E}.

Proposition 2.1 [14] (A o B)c = Ac + B? for A,Be J?™.

Proposition 2.2 [14] (A + Bf = Ac o B° for A, Be

In [30], the following are discussed by the authors.

Definition 2.12 For an IFMA, we have DA = ({ay, 1 -a,y)) and OA = «1 -a'y, a'y)).

Lemma 2.1 1 - n (ait + bkj) = Z (1 - "¿0(1 - bkj)for all i, j, ay, bij e [0,1].

k=1 k=1

Lemma 2.2 1 - £ aikbkj = n ((1 - <Mk) + (1 - bkj))for all i, j, ay, by e [0,1].

k=i k=i

3. More Results of Modal Operators in IFM

Throughout this section, matrices means intuitionistic fuzzy matrices. In this section, some results about modal operators are proved and the definitions of transitive and c-transitive of an IFM A are given.

Lemma 3.1 For any two IFMs A and B,

□(<«,;■ a,j) «- (bij, b'tj)) = D«ay, a\j)) «- n((by, by)). (1)

Proof (i) If (ay, a'u) > (bu, by), then

□«%,a,,) «- (by,b\j)) = □«!,0» = <1,0>. (2)

Since <ay, a'y) > (by, by), atj > by and dy < by. Therefore, 1 - ay < 1 - by and (ay, 1 - ay) > {by, 1 - by), so □«ay, fly» > □«fcy, b'y)). Thus

□<oy, ay) «- n(by, b'^ = <1,0>. (3)

From (2) and (3), (1) holds.

(ii) If (ay, a'y) < {by, by), then

dy) (by, ¿y>) = %) = (ay, 1 - ay) (4)

n(ay, fly) <- n(by, b'u) = (ay, 1 - ay) <- (by, 1 - by) = (ay, 1 - ay) (5) Clearly, from (4) and (5), (1) holds.

Lemma 3.2 For any two IFMs A and B,

0((a,j, fly) <- (by, b'^) = 0«ay, a'y)) <- 0((by, by)). (6)

Proof (i) If (ay, a'y) > (by, by), then

<>((ay, ay) (by, by)) = 0«1,0» = <1,0). (7)

Since (ay, a'y) > (by, by), ay > by and dy < by. Therefore 1 - ay < 1 - by and (ay, 1 - ay) > (by, 1 - by), so 0((ay,a'y)) > 0(.(by,b'y)). Thus

Fuzzy Inf. Eng. (2016) 8: 345-354_349

0<ay,a;.><-0<fcy,&;.> = <l,0>. (8)

From (7) and (8), (6) holds, (ii) If (ay, fly) < <fcy,èy>, then

0«ay, ßy) «- (2>y, b'tj» = Oicnj, = <1 - a;a; ■>, (9)

o<ay, ay> «- o<&y, .> = <1 - a;.., a;.) 4- <1 - ¿;7> = <1 - ö;a;.>. ao Clearly from (9) and (10), (6) holds.

Lemma 3.3 A is reflexive matrix if and only if OA is reflexive matrix.

Proof A is reflexive (ay, a'^) > (<5y, ô'^) for all i, j.

<=> (ay, 1 - > (<îy, 1 - <5y> for all i, j DA >□/<=> OA is reflexive.

In dual way we can prove the following lemma.

Lemma 3.4 A is reflexive matrix if and only if OA is reflexive matrix.

Lemma 3.5 A is reflexive if and only ifOAc is irreflexive.

Proof It is evident that if A is reflexive if and only if Ac is irreflexive and so DAC. Similarly, OAc is irreflexive if and only if A is reflexive.

Lemma 3.6 A is symmetric matrix if and only if OA is symmetric matrix and so QAC.

Proof A is symmetric« (ay, ai}) = {aß, a'-) for all i, j <=> (ay, 1 - ay) = (aß, 1 -aß) o DA = (QA)r. Thus A is symmetric if and only if DA is symmetric.

Similarly, we can prove the following lemma.

Lemma 3.7 A is symmetric matrix if and only if OA is reflexive matrix.

Lemma 3.8 A is transitive matrix if and only if OA is transitive matrix.

, n n r /

Proof A is transitive <=> A > A2 (ay, a. ) > ( E (.a^akj), II (%+at,)) f°r allt, j <=>

1 k= 1 i=l n r n I i n n

ay > £(«ita*/>ay) < FK% + 2>t.) <=> ay > £(flikakj), 1 - ay < 1 - E^tay) <=> ¿=1 fc=l ¿=1 n n n n

{fly, 1 - ay) > (£ aikakj, 1 - 2 = (E 0((1 - "fit) + (1 - akj))) by

jfc=l k= 1 fc=l t=l

Lemma 2.2.

Similarly, we can prove the following lemma. Lemma 3.9 A is transitive matrix if and only if OA is transitive matrix. Lemma 3.10 A is idempotent matrix if and only if ÜA is idempotent matrix.

Proof A idempotent <=> A = A2 <=> (ay, a. ) = ( £ (a^aty), Y\(aik + akj)) for all

k= 1 k= 1

n n n n

i,j. o (ay, 1-ay) = <£ (aiiakj), 1 - £ (aikakj)) <=> ( £ (a^a/y), FI ((1-«■*)+l-a*j)>

by Lemma 2.2 o DA = (DA)2. Thus A is idempotent <=> DA is idempotent.

The following lemma is trivial from the above. Lemma 3.11 A is idempotent matrix if and only if OA is idempotent matrix.

Remark 3.1 If A is an intuitionistic fuzzy equivalence matrix, then OA and OA are also intuitionistic fuzzy equivalence matrices.

Definition 3.1 Let A e the transitive closure and c-transitive closure of A is defined by A°° = A V A2 V A3 V • • • V A" and A„ = Ac A (Ac)[21 A • ■ • A (Ac)w respectively.

Theorem 3.1 For A e = (A°°)c.

Proof By Definition 3.1, (A°°)c = (A V A2 V A3 V • • • VA")C = (Ac A (A2)c A • • • A (A")c). First let us prove (A2)c = (Ac)[2].

We know that A2 = (Z (a,tat,), TT + an)) and so k= l k=l 1

(A2)' = ( f[ (4 + Z («ttflU». (11)

Also Ac = (fly, ay) gives by the definition of A121,

(Ac)[2] = (n(% + Z (12)

Thus by (11) and (12) (A2)c = (Ac)[2], so in general (Anf = (Ac)w. By Definition 3.1,

(A°°)c = (A V A2 V A3 V ■ ■ • V An)c = (Ac A (A2)c A ■ • • A (An'f = (Ac) A (Ac)[2] A ■ ■ ■ A (Ac)w) = Aoo.

Lemma 3.12 A is transitive if and only ifAc is c-transitive and so QAC is.

Proof It is evident from the definition of transitive and c-transitive.

Lemma 3.13 If A is reflexive IFM, then

(i) AT is reflexive.

(ii) Av Bis reflexive.

(iii) AABis reflexive if and only ifB is reflexive.

Proof (i) and (ii) are obvious from the definition of reflexive.

(iii) If B is not reflexive, then (bü, b'u) # (1,0) for at least one i, that is (b№, b'u) < <1,0). Thus {an, a'u) A (bu, b'u) < <1,0). Therefore the condition B is reflexive is necessary, the sufficient part is trivial.

Theorem 3.2 If A, B e where A is reflexive and symmetric, B is reflexive, symmetric and transitive and A < B, then A°° < B.

Proof For A = ((ay,fly)),B = ((b^b'j), AB = (<£(«,*&*;), \\(dlk + b'))) and

Jt= 1 k= 1

" , , i<l,0>, if i = j,

{Z(Akhj), n(aft + bkj)) = \\ ' , .

k= i k=l tJ \{bihby), if i*j.

Fuzzy Inf. Eng. (2016) 8: 345-354_351

Thus AB = B AA < AB = B. That is A2 < B. Continuing in this way, we have A3 < B, A4 < 5 • • ■ and also A V A2 V A3 ■ ■ ■ V A" < B and hence < B.

Lemma 3.14 If A°° is the transitive closure of A, then the transitive closure of ÜA is

Proof Now OA™ = D[A V A2 V • ■ ■ V A"] = DA V DA2 V ■ ■ • DA" = DA V (DA)2 V ■ • • (DA)" = (DA)00. Similarly, the following results are also true.

(i) DA„ = (OA)«,.

(ii) OA" = (OA)00.

(iii) OA«, = (OA)«,.

Lemma 3.15 ForanIFMA e [(nAf]00 = [(□A)«,]c.

Proof As we know (DA)C = OAc, [(nA)c]°° = [OAc]°° = OAcV(OAc)2 • • • V(OAc)".

(OAc)2 = « £ (1 - alk)(\ - ay), f[fe + akj))) k= 1 k=l

= «1 - fife + akj\ fife + akj))). (13)

k= 1 k= 1

tt n f f

By definition A[2] = « ü fe + akj), 2 (aitakj))) and so

k= 1 k=l

□A® = «fife + akJ), 1 - fife + akj))).

k= 1 k= 1

Which yields

(□A[2I)C = «1 - nfe + akj), fife + akj)))-t=l k=1 Therefore, (OAc)2 = (Q4[2I)C, so in general (OAcf = (□AW)C [(□A)c]°° = [OAc]°° = OAc V (OAc)2 • • • V (OAc)" = (DA)C V (GA[2I)c V • • • (GA1"1^ = (DA A DA[21 A • • • A □Aw)c = (□A00)c.

In dual fashion, one can prove the following lemma.

Lemma3.16 ForanIFMA e ((OA)«,)c = ((OA)c)°°-

Definition 3.2 For any two elements (x,x),(y,y) e IFS, we introduce the operation 'A'm as (x,x) Am <y,y) = <nnn{x,;y},min{jr',/}).

Using this definition the following lemmas are trivial.

Lemma 3.17 The operation Am is commutative on IFS.

Lemma 3.18 The operation Am is associative on IFS.

Lemma 3.19 The operation Am is distributive over addition in IFS s.

Proof For any (x, x'),(y,y), <z, z) e IFS

«x,x ) + (y,y)) Am <z,z ) = (max{x, y), min{x', y () Am <z,z'>

= (min{max{j:,y},z},min{min{j:',y},z}). (14)

Case (1) If (x,x) >(y,y) and (x,x) > (z,z), then right hand side of (14) is (z,x). Now consider

= <Z,x>.

In this case, it is distributive.

Case (2) If (x,x) <(y,y) and (x,x) < (z,z), then the left hand side of (15) reduces lo(y,y')Am(z,z),

Subcase (2.1) If (z,z) < <y,y), then (z,z) Am (y,y) = (z,/>. Now

((x, x) Am <z,z'» + «*,*') Am (z, z')) = (x,z) + (z, y ) = (z,y'>. Thus distributivity holds.

Subcase (2.2) If (z,z> > (y, /), then left hand side of (15) becomes (y,z ) and right hand side of (15) becomes

((x,x > Am (z,z'>) + ((x,x) Am (z,z)) = (x,z) + (y,z) = (y,z >. Thus it is distributive in this case also.

Case (3) If (y,y) < (x,x) < (z,z), then the left hand side becomes ((x,x) + (y,y)) Am (z,z) = (x,x) Am (z, z ) = (x,z). Also ((x,x) Am (z, z')) + ((x,x) Am (z, z )) = (x,z) + (y,z'> = (x,z). So it is distributive in this case too.

Case (4) If (z, z) > (x, x)> (y,y), then the left hand side reduces to <y,y) Am(z,z'> = (z,y>.

((x, x) Am <z,z'» + «x,x) Am (z,z'>) = (z,x) + (z, y ) = (z,/>. Thus distributivity holds for all cases.

Definition 3.3 For any two elements (x,x), (y, y) e IFS, we define the inequality ' <' as (x, x ) < (y,y ) means x < y and x < y .

Remark 3.2 The elements in the set {(y,/> e IFS \ (x,x) < (y,y>( are identity element of (x, x) with respect to Am. That is we have multiple identity element.

Remark 3.3 Any IFM A can be decomposed into two intutionistic fuzzy matrices □A and OA by means of Am. That is A = (OA) Am (OA).

Remark 3.4 For any two IFMs A and B, (A V B) Am (A A B) = (A Am B).

4. Conclusion

Any fuzzy matrix A - (aij) is an IFM in the form of A = (ay, 1 - ay). The matrices □A andOA using necessity and possibility operators denoted as □ and O are formed, in which we consider only either membership or nonmembership of any IFM A gives a fuzzy matrix. Here we present some results of the above said operators with other operators using illustration. Transitive and c-transitive closures are defined and some results are proved on IFM. Finally, using a new operator we express an IFM in terms of fuzzy matrix.

Acknowledgement

Fuzzy Inf. Eng. (2016) 8: 345-354_353

Authors would like to thank to the referees for their valuable comments to improve the presentation of the paper.

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