Study of Trapezoidal Fuzzy Linear System of Equations S. M. Bargir 1, *, M. S. Bapat 2, J. D. Yadav 3 1

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1 mercn Interntonl Journl of Reserch n cence Technology Engneerng & Mthemtcs vlble onlne t IN (Prnt: IN (Onlne: IN (CD-ROM: IJRTEM s refereed ndexed peer-revewed multdscplnry nd open ccess journl publshed by Interntonl ssocton of centfc Innovton nd Reserch (IIR U (n ssocton Unfyng the cences Engneerng nd ppled Reserch tudy of Trpezodl Fuzzy Lner ystem of Equtons M Brgr * M Bpt J D Ydv 3 Deprtment of Mthemtcs hvj Unversty Kolhpur [M ] Ind Chntmnro college of commerce ngl 4645 [M ] Ind 3 dguru GdgeMhrj College Krd [M ] Ind bstrct: Heren we hve reported the fuzzy lner system of equtons (FLE of nth order hvng mtrx form where s crsp mtrx B nd X re fuzzy mtrces wth trpezodl fuzzy numbers nd trpezodl fuzzy vrbles respectvely To solve these FLE we hve ntroduced two new methods Drect method nd Generl method Out of whch Drect method s tme effcent method nd provdes condtons for the exstence of the soluton However Drect method s pplcble only on ll postve or ll negtve coeffcents ths restrcton get rd off by Generl method; t permts ll types of coeffcents Keywords: Fuzzy lner system of equtons Fuzzy mtrx trpezodl fuzzy number (TrFN I Introducton precursor of fuzzy numbers s the foundton of the fuzzy lner system s they hve been found to be n dequte n the vrous felds of scences engneerng nd n recent tmes n the ndustry too s the problem crckng tool In the prctcl relzton of such problems nvolves the mprecse nd unknown prmeters therefore pproprte lner system selecton to solve the problem becomes crucl one ccordng to Moore exct numercl dt mght be unrelstc but there could be consdered uncertn dt s more spects of rel world problem [] Fuzzy dt hve been used s nturl wy to descrbe uncertn dt o we need to solve those lner systems n whch ll prmeters or some of them re fuzzy numbers Thus the crsp system of lner equtons becomes Fuzzy ystem of Lner Equtons (FLE or Fully Fuzzy ystem of Lner Equtons (FFLE The dfference between FLE nd FFLE s used coeffcent mtrx; treted s crsp n the FLE but n the FFLE ll the prmeters nd vrbles re consdered to be fuzzy numbers Fredmn et l nvestgted generl fuzzy system usng the embeddng methodology n 998[7] They derved the condtons for the exstence of unque fuzzy soluton to n n lner system of equtons e x=b where ws non-sngulr crsp mtrx Moreover they trnsferred the n x n fuzzy lner system nto n x n rel lner system s result they solved the n x n rel lner system usng the nverse mtrx Ro nd Chen proposed the soluton of smultneous lner equtons tht rse n the nlyss of engneerng systems nvolvng fuzzy nput prmeters [] The proposed methodology conssted three mjor spects computerzed selecton of fuzzness mplementton of fuzzy opertons nd development nd executon of serch bsed lgorthm In 006 Dehghn nd Hshem nvestgted generl fully fuzzy lner system of equtons nd then extended numerous well-known numercl lgorthms such s Rchrdson Jcob Jcob Over Relxton Guss-edel uccessve Over Relxton ccelerted Over Relxton ymmetrc nd Unsymmetrcl uccessve Over Relxton nd extrpolted the domn decomposton method for solvng fully fuzzy lner system of equtons wth trngulr coeffcents [4-6] fter tht Grzegorzewsk provded the Trpezodl pproxmton of fuzzy numbers perseverng the expected ntervl-lgorthms nd propertes [8] In ths pper we hve solved trpezodl FLE of n th order e X B where s crsp mtrx Here were ntroducng two new methods e Drect method nd Generl method for solvng FLE wth trpezodl fuzzy number Usng these methods we hve llustrted the numercl exmples II Prelmnres Fuzzy numbers re one wy to descrbe the dt vgueness nd mprecson They cn be regrded s n extenson of the (crsp rel numbers fuzzy number my emerge n prctcl pplcton s descrpton of the vrton n our knowledge bout the correct vlue of some mesurement when the level of our confdence n tht knowledge vres The theory of fuzzy numbers s bsed on the theory of fuzzy sets whch Zdehntroduced n 965 [3] Fuzzy number We ssume X fuzzy number s fuzzy subset of the set of rel numbers wth membershp functon : I such tht s norml convex upper sem-contnuous wth bounded support IJRTEM 9-9; 09 IJRTEM ll Rghts Reserved Pge 50

2 Brgr et l mercn Interntonl Journl of Reserch n cence Technology Engneerng & Mthemtcs5( December 08- Februry 09 pp Trpezodl fuzzy numbers (TrFN If left hnd nd rght hnd curves re strght lnes then the fuzzy numbers re clled trpezodl fuzzy numbers mong the vrous shpes of fuzzy numberstrfn re most populr trpezodl fuzzy number s denoted by nd defned s 3 4 where the membershp functon s x ( 0 x 0 x x x 3 4 x 3 x x 4 We denote the bove trpezodl fuzzy number by new nottons s [ ] of the fuzzy number [] The membershp functon of ε-δ trpezodl fuzzy number s gven by [ b]( x The α-cut of [ ] If the left nd rght spreds of x ( ε when ε< x ε when x b ( b x when b < x b 0 otherwse b s gven by( b b [ ] ( ( [0] b where ε nd re left nd rght spreds fuzzy number re equls then t s clled symmetrc trpezodl fuzzy numbers: denoted by [ b ] or [ b ] Note tht for ny fuzzy number [ x y ] [ x y ] 0 nd [ x y ] / [ x y ] 3 rthmetc operton of TrFN [] We defned the rthmetc operton of epslon-delt TrFN [] ddton [ x y] [ b] =[ x y b] Negton [ x y ] [ y x ] 3 ubtrcton [ x y] [ b] =[ x b y ] [ kx ky] k 0 k k 4 clr multplcton k[ x y] [ ky kx] k 0 k k 5 Trpezodl multplcton If [ x y] 0 nd [ b] > 0 then [ x y] [ b] [ x by] 6Equlty of trpezodl fuzzy numbers let[ x y] nd [ b] [ x y ] = [ b ] x = y = b x x re equl TrFNf 4 Fuzzy Mtrx The mtrx s clled fuzzy mtrx f ech element of s fuzzy number fuzzy mtrx s clled nonnegtve f ech element of s non-negtve fuzzy number ] Defnton If [ ] j n s crsp mtrx then we defne mtrx j n j j Tht s mtrx whose ll elements re non-negtve Let ( nd ( ( ( Note tht every mtrx cn be wrtten s ] Condtonl dstrbutve property 0 0 Mtrces x nd B sd to be condtonl dstrbutve property hold f 0 y b 0 x 0 0 p q x 0 p q 0 p q 0 y b 0 r s 0 y r s b 0 r s III Trpezodl Fuzzy lner system of equton Here we hve solved the FLE wth TrFNs of order n Moreover they trnsformed the n x n FLE to n x n system of lner equton Ths cn be solved by nverse mtrx or ny lgebrc methods IJRTEM 9-9; 09 IJRTEM ll Rghts Reserved Pge 5

3 Brgr et l mercn Interntonl Journl of Reserch n cence Technology Engneerng & Mthemtcs5( December 08- Februry 09 pp Defnton:- fuzzy lner system of equtons of order n s gven by [ X Y] P Q [ C D] [ ] j n nd j T T T n n n n [ X Y ] [ x y ][ x y ][ x y ] P [ ] Q [ ] nd R where T T T n n n n [ C D ] [ c d ][ c d ][ c d ] R [ ] [ ] tht s [ x y ] [ x y ] [ c d ] n n n n n [ x y ] [ x y ] [ c d ] nn n n n n n n n n n 3 Drect Method for FLE In ths method we hve solved the FLE wth TrFN of order In whch ll coeffcents re ether postve or negtve Ths theorem gves the condtons for the exstence of the solutons of FLE Theorem:- The system of fuzzy lner equtons [ X Y] P Q [ C D] R of order hs solutonexsts f ether ll sclrs re postve or negtve nd 0 f f Proof :- Let tht s 0 we hve 0 we hve [ x y ] [ c d ] x y c d [ ] [ ] FLE cn be wrtten s X Y C D coeffcents s follows Cse I: - ll sclrs re postve P Q R X Y C D we hve solved n two cses on the bss of the sgns of the P Q R X Y C D ( by postve sclr multplcton ( by equlty of fuzzy numbers P Q R X C Y D nd P R Q X C X C x c c c ( c c ( c c x x x c c c ( ( nd Y D Y D y d d d ( d d ( d d y y y d d d ( ( Next P R P R ( ( ( ( Fnlly Q Q ( ( ( ( nce = re non-negtve we must hve If 0 0 nd 0 then smlrly 0 nd 0 If 0 0 nd 0 then IJRTEM 9-9; 09 IJRTEM ll Rghts Reserved Pge 5

4 Brgr et l mercn Interntonl Journl of Reserch n cence Technology Engneerng & Mthemtcs5( December 08- Februry 09 pp smlrly 0 nd 0 Cse II:- ll sclrs re negtve thenfrom negtve sclr multplcton then we get ( d d ( d d X D X D x x ( ( ( c c ( c c Y C Y C y y ( ( Next ( ( Q Q ( R R ( ( Fnlly ( ( P P ( ( ( nce = re non-negtve we must hve If 0 then nd If 0 then nd Note:bove propostons evdently s so restrctve whch s pplcble only when ll sclrs re ether postve or negtve n FLE of order o we ntroduced generl method for solvng FLE of order n wth no such restrcton 3 Generl Method for FLE fuzzy lner system of equtons X Y C D X Y x y nd C D c d where [ ] s sclr mtrx P Q R P Q R [ j ] j n crsp mtrx nd [ j ] j n nd j re mtrces whose elements re TrFNnce ll postve crsp coeffcents Let clerly contns ll postve sclrs Clerly we get Therefore FLE X Y C D becomes P Q R ( X Y C D P Q R X Y P Q X Y P Q C D R X Y X Y C D P Q P Q R X Y Y X C D X Y Y X C D (by defnton of (by condtonl dstrbutve property P Q Q P R P Q P Q R X Y C Y X D P Q R nd P Q Now X Y C Y X D gves system of n equtons wth n vrbles nd lso P Q R P Q gves n equton n n vrbles These systems cn be solved by ny known lgebrc method If we get non-negtve soluton for system P Q R P Q then gven FLE hs soluton Otherwse the soluton for FLE does not exst Exmple 5 consder the FLE x y x y IV Numercl Exmples x y 4 x y 4 7 Gven FLE contns ll postve coeffcents so we hve used cse I of the drect method IJRTEM 9-9; 09 IJRTEM ll Rghts Reserved Pge 53

5 Brgr et l mercn Interntonl Journl of Reserch n cence Technology Engneerng & Mthemtcs5( December 08- Februry 09 pp thus 4 0 nd Clerly ll condtons holds Thus soluton of the FLE s exsts whch s s follows ( c c ( c c ( d d ( d d x 0 5 x y 4 y 4 ( ( ( ( nd 7 7 ( ( ( ( The requred soluton 5 sx y 0 nd x y Exmple 5 consder the FLE s x y 9 x y 5 8 x y 7 x y Gven FLE contns mxed sgn coeffcents Here we hve used generl method s follows nd from bove the method we hve nd 0 y 0 9 x 5 y 9x 0 7 y 8 0 x 7 7 y 8x 7 0 x 0 9 y 5 x 9y 5 X Y C 0 7 x 8 0 y 7 7x 8y 7 Y X D Then we get [ x y ] [ ][ x y ] [ ] mlrly P Q R nd P Q From bove system we get [ ] [ ][ ] [ ] [ ] [ 9 9 ] [ 4 ] [ 9 x y 9 ] x y Thus the requred soluton s V Concluson In ths rtcle fuzzy lner systems of equtons (FLE re nvestgted drect method nd generl method re ppled for solvng them Here we found the soluton of FLE by the drect method long wth the condton for the exstence of the solutons The generl method for fndng the soluton of FLE offers no such restrctons on the selecton of coeffcents nd vrbles Fnlly we llustrted some numercl exmples References [] bbsbndy Ezzt R LU Decomposton method for solvng fuzzy system of lner equtons ppled mthemtcs nd computton 7( [] Bpt M Ydv N Kmble P N Trngulr pproxmtons of fuzzy numbers Interntonl Journl of ttstk nd Mthemtk 7 (3 ( [3] M Brgr J D Ydv M Bpt olutons Of Fully Fuzzy Lner Equtons x+b=c Interntonl Journl of Ltest Reserch n cence nd Technology Vol 4 No 6 pp6-305 [4] Dehghn M Hshem BGhtee M oluton of the fully fuzzy lner systems usng the decomposton procedure ppled Mthemtcs nd computton 8 ( [5] Dehghn M Hshem B Itertve soluton of fuzzy lner systems ppled Mthemtcs nd computton 75 ( [6] Dehghn M Hshem B Ghtee M oluton of the fully fuzzy lner systems usng tertve technques ppled Mthemtcs nd computton 34 ( [7] Fredmn M Mng M Kndel Fuzzy lner systems Fuzzy ets nd ystems 96 ( [8] Grzegorzewsk P Trpezodl pproxmton of fuzzy numbers perseverng the expected ntervl-lgorthms nd propertes Fuzzy ets nd ystems59 ( [9] Horck R oluton of system of lner equton for wth fuzzy number Fuzzy et ystem 59 ( [0] Kumr bhnv Bnslb Neetu method for solvng fully fuzzy lner system wth trpezodl fuzzy numbers Irnn Journl of Optmzton ( [] Moore RE Methods nd pplctons of Intervl nlyss IM Phldelph979 [] Ro nd Chen Numercl soluton of fuzzy lner equtons n engneerng nlyss Interntonl Journl for Numercl Methods n Engneerng 43 ( [3] L Zdeh Fuzzy sets Inform Control 8 ( IJRTEM 9-9; 09 IJRTEM ll Rghts Reserved Pge 54