Hamacher Sum and Hamacher Product of Fuzzy Matrices

Transcription

1 Intern J Fuzzy Maematial Arhive Vol 13, No, 017, ISSN: (P), (online) Published on Deember 017 wwwresearhmasiorg DOI: International Journal of amaher Sum and amaher Produt of Fuzzy Matries ISilambarasan 1 and SSriram 1 Department of Maematis, Annamalai University Annamalainagar , Tamilnadu, India sksimbuking@gmailom Maematis Wing, Diretorate of Distane Eduation Annamalai University, Annamalainagar, , Tamilnadu, India ssm_3096@yahoooin Reeived 14 November 017; aepted 30 November 017 Abstrat In is paper, we define two new operations alled amaher sum and amaher produt of fuzzy matries and investigate e algebrai properties of fuzzy matries under ese operations as well as e properties of fuzzy matries in e ase where ese new operations are ombined wi e well-known operations,, we have proved some new inequalities onneted wi fuzzy matries Keywords: Fuzzy matrix, amaher sum, amaher produt AMS Maematis Subjet Classifiation (010): 03E73, 04A7 1Introdution Among e well-known operations whih an be performed on fuzzy matries are e operations of omponent wise addition, multipliation, algebrai produt, algebrai sum and omplement Muh researh works are done onerning fuzzy matries and eir appliations to medial sienes, engineering, management environment and soial sienes In 1977, Thomason [8] initiated e study on onvergene of powers of fuzzy matrix The eory of fuzzy matrix was developed by Kim and Roush [] as an extension of Boolean matrix, wi max-min operation over e fuzzy algebra Ragab and Emam [4] presented some properties of e min-max omposition of fuzzy matries Meenakshi [3] studied e eoretial developments of fuzzy matries Shyamal and Pal [5,6] introdued two binary operators on fuzzy matries and proved several properties on em and ese operations are extended to intuitionisti fuzzy matries and studied its algebrai properties by Sriram and Boobalan [7] Zhang and Zheng [9] introdued bounded sum and bounded produt of fuzzy matries and presented several properties on ese operations The paper is organized in ree setions In setion,we give e basi definitions and operations on fuzzy matries whih will be used in is paper In setion 3, we introdue e hamaher operations on fuzzy matries and fousing on its properties The De Morgan's law for e hamaher operations are established in setion 4 Preliminaries and definitions In is setion, we give some definitions and preliminaries whih are applied in e paper 191

2 ISilambarasan and SSriram Definition 1 [3] A fuzzy matrix(fm) of order m n is defined as A ( a ), where a [0,1] Let F mn denote e set of all fuzzy matries of order m n Definition [3] For olumns and is denoted by A F mn T A, e transpose is obtained by interhanging its rows and Definition 3 [3] The m n zero matrix O is e matrix all of whose entries are zero The n n identity matrix I is e matrix ( a ) suh at a 1 if i j and a 0 if i j The m n universal matrix J is e matrix all of whose entries are 1 Definition 4 [3] Let A ( a ) and B ( b ) Fmn we write A B if a b for all i, j and we say at A is dominated by B (or) B dominates A A and B are said to be omparable, if eier A B (or) B A We define e following operators for any two matries A ( a ) and B ( b ) of order m n, ( i) A B (max( a, b )) ( ii) A B (min( a, b )) ( iii) A (1 a ) (e omplement of A ) 3 Some results on amaher sum and amaher produt of fuzzy matries amaher [1] introdued a generalized t norm and t onorm by defining as xy * x + y xy (1 γ ) xy T( x, y) and T ( x, y) γ + (1 γ )( x + y xy) 1 (1 γ ) xy From ese operations, when γ 0 ey will redue to algebrai t norm and xy * x + y xy t onorm, T ( x, y) and T ( x, y) x + y xy 1 xy In is setion, based on ese operations, amaher sum and amaher produt of fuzzy a + b ab ab matries are definied as A B and A 1 ab B a + b ab ab a + b ab Lemma 31 For a, b [0,1], a + b ab 1 ab Proof: We know at ( a + b) 4ab (11) a + b ab ( a + b ab) 4 (1+ 3( a + b ab)) ab 4ab 19 (1)

3 From (11) and (1) amaher Sum and amaher Produt of Fuzzy Matries (1+ 3( a + b ab)) ab 4 ab ( a + b) 0 ( + ) (1+ 3( + )) a b a b ab ab ( ) 3 ( ) a + b + ab ab a b ab ( ) 3 3 ( ) ab a + b + a b ab a + b ab a b a b ab a b ab ( + )( + ) ab a + b ab a + b ab 1 ab ene e result Definition 3 For any fuzzy matries A and B of e same size The amaher sum a + b ab of A and B is defined by A B The amaher produt of 1 ab ab A and B is defined by A B a + b ab Next, we have prove some results of properties of amaher sum and amaher produt Property 33 Let A and B be any two fuzzy matries of same size Then A B A Proof: By using Lemma 31, ab a + b ab j a + b ab 1 ab ene, entry of A B entry of A B Therefore, A B A Property 34 For any fuzzy matrix A, ( i) A A A ( ii) A A A a a a Proof: (i) A A is a j 1 a 1+ a Sine a a 0 a + a a 0 a + a a ene, Therefore, entry of A A A A A entry of A 193

4 a a a a a (ii) A A is a Sine 1 a ( ie ), a a ene, Therefore, entry of A A A A A ISilambarasan and SSriram a entry of A 194 j The following properties are obvious The operations and well as assoiative Existene of e identity elements wi respet to determined in e following Theorems Property 35 Let A, B and C be any ree fuzzy matries Then ( i) A B B A, ( ii)( A B) C A ( B C), ( iii) A B B A, ( iv)( A B) C A ( B C) Property 36 For a fuzzy matrix A, ( i) A O O A A, ( ii) A J J A A, ( iii) A J J, ( iv) A O O are ommutative as and are Thus ( Fmn, ) and ( Fmn, ) form ommutative monoids The operators and do not obey e De Morgan's laws over transpose Property 37 For any two fuzzy matries A and B of same size, T T T ( i)( A B) A B, T T T ( ii)( A B) A B where T A is e transpose of A Property 38 For any two fuzzy matries A and B of same size, if A B, en A C B C Proof: Let a b j a b a + a b (1 ) b + a b (1 )

5 amaher Sum and amaher Produt of Fuzzy Matries a + a b a b b + a b a b a ( + b b ) b ( + a a ) a b a + a b + b Therefore, e entry of A ene e result C entry of B C Property 39 For any two fuzzy matries A and B of same size, if A B, en A C B C Proof: Let a b j a (1 ) b (1 ) a b (1 + ) (1 + ) a a + a b b + b a a a a b a b b b b a b a b + + ( + ) + ( + ) a + a a b b + a b b + b a b a + a b a + a b ( a + a ) b + b a ( b + b ) ( a + a )(1 b ) ( b + b )(1 a ) a + a b + b 1 a 1 b Therefore, e entry of A C ene e result 195 entry of B C Property 310 For any two fuzzy matries A and B of same size, Then ( i)( A B) ( A B) A B, ( ii)( A B) ( A B) A Proof: ( i)( A B) ( A B) min( a, b ) max( a, b ) ( ) ( ) ( min( a, b ) + max( a, b ) min( a, b ) max( a, b )) (1 min( a, b ) max( a, b )) a + b ab 1 ab A

6 ISilambarasan and SSriram ( ) ( ) ( ii)( A B) ( A B) min( a, b ) max( a, b ) ene e result min( a, b ) max( a, b ) ( min ( a, b ) + max ( a, b ) min ( a, b ) max ( a, b )) ab a b ab + A 4 Results on omplement of fuzzy matrix In is setion, e omplement of a fuzzy matrix is used to analyse e omplementary nature of any system For example, if A represents e rowdness of e roads at a partiular time period en its omplement A represents e learness of e roads Using e following results we an study e omplement nature of a system wi e help of original fuzzy matrix The operator omplement obey e De Morgan's law for e operator and This is established in e following property Property 41 For any two fuzzy matries A and B of same size, ( i)( A B) A B, ( ii) ( A B) A B, ( iii) ( A B) A B, ( iv) ( A B) A B Proof: ( i) A B (1 a ) (1 b ) ( ) (1 a )(1 b ) (1 a ) + (1 b ) (1 a )(1 b ) (1 b a + ab ) (1 a ) + (1 b ) (1 + b + a ab ) 1 b a + ab a b 1 + b + a a b 196

7 amaher Sum and amaher Produt of Fuzzy Matries 1+ a b a b 1 ab (1 a )(1 b ) 1 ab a + b a b 1 1 ab ( A B) ( ) ( ii) A B (1 a ) (1 b ) (1 a ) + (1 b ) (1 a )(1 b ) 1 (1 a )(1 b ) a b (1 b a + ab ) 1 (1 b a + ab ) a b + b + a ab 1 1 b a ab + + a + b ab a b ab + ab 1 a + b a b ( A B) (iii) From Property 1 A B A Then ( A B) ( A B) A B ( iv)( A B) ( A B) ene e result A B 5 Conlusions In is artile, amaher sum and amaher produt of fuzzy matries are defined and some properties are proved The set of all fuzzy matries form a ommutative monoids wi respet to ese operations Thus e hamaher sum and hamaher produt are very useful to furer works Aknowledgements The auor is very grateful and would like to express his anks to e 197

8 ISilambarasan and SSriram referees for eir valuable omments and suggestions provided in revising and improve e presentation of e paper REFERENCES 1 amaher, Uber logishe verknunpfungenn unssharfer Aussagen undderen Zugenhorige Bewertungsfunktione In:Trappl, Klir, Riardi (eds) Progress in Cybernatis and systems researh, vol 3 emisphere, Washington, pp (1978) KKim and FWRoush, Generalized Fuzzy matries, Fuzzy Sets and Systems, 4 (1980) ARMeenakshi, Fuzzy matrix Theory and Appliations, MJP Publishers, Chennai, India, (008) 4 MGRagab and EGEmam, On e min-max omposition of fuzzy matries, Fuzzy Sets and Systems, 75 (1995) AKShyamal and MPal, Distanes between intuitionisti fuzzy matries, JPhysial Sienes, 8 (00) AKShyamal and MPal, Two new operators on fuzzy matries, J Appl Ma Computing, 15(1-) (004) SSriram and JBoobalan, Arimeti operations on Intuitionisti fuzzy matries, Proeedings of International Conferene on Maematial Sienes-014 (Published by Elsevier) organized by Sayabama university, Chennai, pp , (014) 8 MGThomason, Convergene of powers of fuzzy matrix, J Maematial Analysis and Appliations, 57 (1977) YZhang and MZheng, New operators on fuzzy matries, Proeedings of Four International Workshop on Advaned Computational Intelligene IEEE (011) Wuhan,