Numerical Solution of Interval and Fuzzy System of Linear Equations

Transcription

1 Numecl Soluto of tevl d Fuzzy System of Le Equtos A THESS Submtted ptl fulfllmet of the equemets fo the wd of the degee of MASTER OF SCENCE MATHEMATCS By Sup Ds Ude the supevso of Pof S Chkvety My DEPARTMENT OF MATHEMATCS NATONAL NSTTUTE OF TECHNOLOGY ROURKELA ODSHA NDA

2 NATONAL NSTTUTE OF TECHNOLOGY ROURKELA DECLARATON heeby cetfy tht the wok whch s beg peseted the thess ettled Numecl Soluto of tevl d Fuzzy System of Le Equtos ptl fulfllmet of the equemet fo the wd of the degee of Mste of Scece submtted the Deptmet of Mthemtcs Ntol sttute of Techology Roukel s uthetc ecod of my ow wok ced out ude the supevso of D S Chkvety The mtte emboded ths thess hs ot bee submtted by me o the wd of y othe degee Dte: SUPARNA DAS Ths s to cetfy tht the bove sttemet mde by the cddte s coect to the best of my kowledge D S CHAKRAVERTY Pofesso Deptmet of Mthemtcs Ntol sttute of Techology Roukel Odsh d

3 ACKNOWLEDGEMENTS would lke to covey my deep egds to my poject supevso D SChkvety Pofesso Deptmet of Mthemtcs Ntol sttute of Techology Roukel thk hm fo hs ptece gudce egul motog of the wok d puts wthout whch ths wok eve come to futo deed the epeece of wokg ude hm ws oe of tht wll chesh foeve m vey much gteful to Pof P C Pd Decto Ntol sttute of Techology Roukel fo povdg ecellet fcltes the sttute fo cyg out esech tke the oppotuty to thk Pof G K Pd Hed Deptmet of Mthemtcs Ntol sttute of Techology Roukel fo povdg me the vous fcltes dug my poject wok would lso lke to gve my scee thks to ll of my feds fo the costt effots d ecougemets whch ws the temedous souce of spto would lke to gve hetfelt thks to Dptj Behe D Shbbt Nd d Ms Nsh fo the sptve suppot thoughout my poject wok Flly ll cedts goes to my pets my bothes d my eltves fo the cotued suppot Ad to ll mghty who mde ll thgs possble SUPARNA DAS Roll No 49MA64

4 TABLE OF CONTENT ABSTRACT 5 CHAPTER NTRODUCTON 6 CHAPTER AM OF THE PROJECT 8 CHAPTER NTERVAL AND FUZZY ARTHMATC 9 CHAPTER 4 FUZZY SOLUTON OF SYSTEM OF LNEAR EQUATON BY KNOWN METHOD CHAPTER 5 NTERVAL/ FUZZY SOLUTON OF SYSTEM OF LNEAR EQUATON BY PROPOSED METHOD 4 CHAPTER 6 NUMERCAL EXAMPLE BY KNOWN AND PROPOSED METHOD BOTH 9 CHAPTER 7 NUMERCAL EXAMPLES OF CRSP NTERVAL AND FUZZY EQUATON BY FRST SECOND AND THRD PROPOSED METHOD CHAPTER 8 CONCLUSON 54 CHAPTER 9 FUTURE DRECTON 55 REFERENCES 56 LST OF PUBLCATON 58 4

5 ABSTRACT The system of le equto hs get mpotce my el lfe poblem such s ecoomcs Optmzto d vous egeeg feld We kow tht system of le equtos geel s solved fo csp ukows Fo the ske of smplcty o fo fuzzy computto t s tke s csp vlue ctul cse the pmetes of the system of le equtos e modeled by tkg the epemetl o obsevto dt So the pmetes of the system ctully cot ucetty the th the csp oe The ucettes my be cosdeed tem of tevl o fuzzy umbe Recetly dffeet uthos hve vestgted these poblems by vous methods These methods e descbed fo the system hvg vous type of fuzzy d o-fuzzy pmetes Although solutos e obted by these methods e good but sometmes the method eques legthy pocedue d computtolly ot effcet Hee ths thess detl study of le smulteous equtos wth tevl d fuzzy pmete hve bee doe New methods hve bee poposed fo the sme The poposed methods hve bee tested fo kow poblems vz ccut lyss solved the ltetue d the esults e foud to be good geemet wth the peset Net moe emple poblems e solved usg the poposed methods to hve cofdece o these ew methods Thee ests vous type of fuzzy umbes As such hee the poblems e lso bee solved by two types of fuzzy umbes Coespodg plots of the soluto fo ll the emple poblems e cluded ths thess vew of the bove lyss of the esults t s foud tht the poposed methods e smple d computtolly effcet 5

6 NTRODUCTON 965 Lotf Jdeh [8] pofesso of electcl egeeg t the Uvesty of Clfo Bekley publshed the fst of hs ppes o hs ew theoy of Fuzzy sets d Systems Sce the 98s the mthemtcl theoy of ushp mouts hs bee ppled wth get successo my dffeet felds [9] The ltetue o fuzzy thmetc d ts pplctos ofte cots ctcl emks s the stdd fuzzy thmetc does ot tke to ccout of ll the fomto vlble d the obted esults e moe mpecse th ecessy o some cses eve coect emks of Zhou [][] The cocept of fuzzy umbe ses fom pheome whch c be descbed qutttvely These pheome do ot led themselves to beg chctezed Fo emple f we tke we d mesue ts legth fom dffeet vew gles we get dffeet vlues Thus fuzzy umbe s oe whch s descbed tems of umbe wod such s ppomtely ely etc System of le equtos hs vous pplctos Equtos of ths type e ecessy to solve fo gettg the volved pmetes t s smple d stght fowd whe the vbles volvg the system of equtos e csp umbe But ctul cse the system vbles cot be obted s csp Those e foud by some epemet geel So these vbles wll ethe be tevl o fuzzy umbe e the mesuemet of the legth of the we do ot gves the csp vlue ptcul As such thee wll be vgueess the esult of the epemet So to ovecome the vgueess we my use the tevl d fuzzy umbes the plce of csp umbe[7] Fuzzy le systems hve ecetly bee studed by good umbe of uthos but oly few of them e metoed hee A fuzzy le system A=b whee A s csp d b s fuzzy umbe vecto hve bee studed by Fedm et l [] M et l [] d Allhvloo [-4] 6

7 Thee e my books hvg ccut lyss such s by Bdy d Nd [5] whch gves system of le equtos whee the esstve etwoks my clude fuzzy o tevl umbe Thus we eed to solve tevl/fuzzy system of le equtos Recetly The et l [6] hs vestgted ths type of system Ds et l [7] dscussed the Fuzzy system of le equto d ts pplcto to ccut lyss ths ppe ths vestgto we hve dscussed bout tevl d ts thmetc fuzzy umbe -cut of fuzzy umbe vous type of fuzzy umbes d ts thmetc The cocepts of these hve bee used fo the umecl soluto of system of le equto As such ew methods e developed hee to hdle these fuzzy d tevl poblems specl cses the solutos e lso comped wth the kow esults tht e foud ltetue As pplcto vestgto of ccut lyss hve bee doe fo tevl/fuzzy put d tevl/fuzzy souce whch goveed by coespodg le smulteous equto 7

8 AM OF THE PROJECT Soluto of le system of equtos s well-kow whe the pmetes volved e csp As dscussed ctul pctce we my ot hve the pmetes csp fom the those e kow vgue fom tht s wth ucetty cluso of ucetty the pmete mkes the poblem comple Hee ucetty hs bee cosdeed s ethe tevl o fuzzy fom So the m of ths vestgto s to develop methodology to solve tevl d fuzzy le system of equtos 8

9 NTERVAL AND FUZZY ARTHMETC SYSTEM OF LNEAR EQUATONS: The le systems of equtos csp my be wtte s y y y Whee the coeffcet mt A j j s csp mt Ou m s hee to hve the bove system s ethe tevl o fuzzy vbles d costts d the soluto As such the followg pgphs we wll fst dscuss pelmes of tevl thmetc d the bout fuzzy set d umbes tevl thmetc: A tevl s subset of such tht A ] { t t } [ f A ] d B b b ] e two tevls the the thmetc opetos e: [ [ ADDTON: [ ] [ b b ] [ b b ] SUBTRACTON: [ ] [ b b ] [ b b ] PRODUCT: 9

10 [ ] [ b b ] [m b b b b m b b b b ] DVSON: [ ]/[ b b ] [m / b / b / b / b m / b / b / b / b ] b b DEFNTON OF A FUZZY SET: A fuzzy set c be defed s the set of odeed ps such tht A { / X [ ]} whee A s clled the membeshp fucto o gde of membeshp of [] A A 4 DEFNTON OF A FUZZY NUMBER: A fuzzy umbe s cove omlzed fuzzy set of the csp set such tht fo oly oe X A d A s pecewse cotuous [] 5 TYPES OF A FUZZY NUMBER: Hee we hve dscussed bout types of fuzzy umbe mely Tgul fuzzy umbe: Fg Tgul fuzzy umbe A tgul fuzzy umbe TFN s show Fg s specl type of fuzzy umbe d ts membeshp

11 fucto s gve by A such tht ] [ ] [ A Tpezodl fuzzy umbe Fg Tpezodl fuzzy umbe A tpezodl fuzzy umbe T F N s show Fg s specl type of fuzzy umbe d ts membeshp fucto s gve by A such tht ] [ ] [ ] [ A 6 DEFNTON OF ALPHA CUT:

12 The csp set of elemets tht belog to the fuzzy set t lest to the degee s clled the - level set []: 7 CONVERSON FROM FUZZY NUMBER TO NTERVAL USNG ALPHA CUT: Tgul fuzzy umbe to tevl Let tgul fuzzy umbe defed s A We c wte the fuzzy tevl tems of -cut tevl s: A [ ] Tpezodl fuzzy umbe to tevl Let tpezodl fuzzy umbe defed s A 4 We c wte the fuzzy tevl tems of -cut tevl s: A [ 4 4] 8 Fuzzy Athmetc: As A s ow tevl so fuzzy ddto subtcto multplcto d dvso e sme s tevl thmetc

13 4 FUZZY SOLUTON OF SYSTEM OF LNEAR EQUATON BY KNOWN METHOD Now f the system s fuzzy The et l[6] the th equto of the system my be wtte s We hve y y y y whee otto α s cosdeed fo -cut of fuzzy umbe Fom we hve two csp le system fo ll tht c be eteded to csp le system s follows The et l[6] SX Y S S S S X X Y Y S S X X S X S X Y Y S X S X Y o 4 S X S X Y Now let X vecto U u u s u thefuzzy soluto of m{ m{ { u { u wek fuzzy soluto [4] u } } SX } Yf }deote theuque soluto of SX defed by d Us clled e ll tgul YThe fuzzy umbe fuzzy umbes the stog fuzzy soluto OthewseUs

14 4 5 NTERVAL/FUZZY SOLUTON OF SYSTEM OF LNEAR EQUATONS 5 FRST METHOD Let the system of equtos be: Whee ll e The bove equtos my be wtte equvletly s 6 Equtos 6 c ow be wtte mt fom s:

15 5 = 7 Fo cle udestdg we ow gve the pocedue wth equtos d ukows So fo equtos d ukows we hve: 8-9 Sml to equto 7 oe my wte 8 d 9 s

16 Ths s csp system of equto d the bove mt equto my esly be solved ow s below 5 SECOND METHOD ths method equto 8 d 9 e fst wtte tkg LEFT: RGHT: Solvg d ; d we get 6

17 Although methods d e sme but ths s show the bove methods tht we my solve fo the left d ght dvdully too 5 THRD METHOD Hee we fst cosde the equtos d Wte the bove s AX b tht s Fom 4 we my hve f the vese of the coeffcet mt ests the gve soluto 7

18 Followg the sme wy equtos d my be used to get the soluto 6 8

19 6 NUMERCAL EXAMPLE BY KNOWN AND PROPOSED METHOD 6 NTERVAL AND FUZZY EQUATONS: Let us fst cosde the followg two tevl equtos two ukows Fst of ll we wll stt wth emple gve The et l[6] Hee thee dffeet cses tkg souce d esstce s csp o tevl o fuzzy e cosdeed the ccut lyss These cses e med s cse to cse the followg pgph whee cuet s tke s fuzzy cse d cse d csp cse 6CASE-: Method of The et l[6]: Fst ccut cosdeg fuzzy system of le equtos ppled to ccut lyss The et l[6] hvg souce s well s cuet s fuzzy d esstce s csp hs bee cosdeed Relted fgue s show Fg Fg A ccut wth fuzzy cuet fuzzy souce d csp esstce Fom the fst d secod loop Kchhoff s secod lw gves 9

20 The bove my be wtte s ˆ Y S S S S S Now wtg Y S ˆ s we hve s wtte the soluto tevl fom my be So

21 Coespodg plot of the solutos e gve Fg4 Fg4The soluto of the system fo cse 6 METHOD : Now the bove poblem s solved usg the poposed methods vz fst to thd The poblem s gve s Ths c be wtte s

22 Smplfyg the bove equtos we c get Ad Fst method We c get the mt dectly s So thesoluto tevl fom my be wtte s

23 Secod method Fom 7 ; 9 d 8 ; we get the esult s: So As metoed ctully both method tus out to be sme d so the esults e lso ectly sme Thd method Fom equto 7 d 9 we c wte AX b 4 X 4 6 A b Smlly fom equto 8 d we get X A b CASE-: Net emple gves the soluto fo ccut wth souce cuet d esstce ll tevl Coespodg fgue s show Fg5

24 Fg5A ccut wth cuet souce d esstce tevl Fom the bove ccut we c wte Fom the fst d secod loops d smplfyg the tevl equtos we hve the followg equtos Soluto of the equtos to 4 e: 4

25 Hee lowe odes e lge th the uppe odes d t s clled s wek soluto f the ght hd sde s cosdeed s tevl whch cludes zeo fo emple s [-] the the equtos fte some clculto c be wtte s Soluto of 5 to 8 gves ths cse fst we get wek soluto The stog soluto s obted whe the ght hd sde btly chged to tevl whch cludes Ths cse s the smulted by educg the tevl wdth s show Fg6 The coespodg solutos e gve s 5

26 Fg6A ccut wth tevl cuet souce d esstce CASE-: The bove emple tkg esstce d voltge both fuzzy hs bee dscussed ow d the coespodg ccut s show Fg 7 The soluto my esly be wtte by followg legthy clcultos s Cse d whch e gve Fg 8 6

27 Fg7 A ccut wth fuzzy souce fuzzy cuet d fuzzy esstce So we c wte the equto s Tsfeg tevl fom d smplfyg we get the equto s Fst method Fom the equto 9 d 4 oe my wte s: 7

28 Secod method Fom 4-44 we get The soluto c be obted s

29 Thd method Fom equto 4 d 44 we c wte AX b X A b Smlly fom equto 4 d 4 we get X A b Coespodg gph s show fgue 8 9

30 Fg8 the soluto of the system fom the bove emple

31 7 NUMERCAL EXAMPLES OF CRSP NTERVAL AND FUZZY EQUATON BY FRST SECOND AND THRD PROPOSED METHOD 7 EXAMPLE : ths poblem tkg thee csp equto wth thee ukows: These equtos we c wte s Fst method Fom we get we get the soluto s

32 6 6 Secod method Solvg d the solutos comes s 6 6 Thd method

33 AX X A b b X 6 Smlly X 6 EXAMPLE : ths emple tevl equtos ukows e cosdeed Fst method

34 Equto we c wte mt fom s: Soluto s obted s Secod method Fom we get by dspesg left d ght

35 Solvg the bove s equto 57-6 we hve Thd method Fom we c wte AX b X A b X 5 6 X Smlly 58 6 d 6 gves X

36 6 EXAMPLE : Cosde the equtos s Tsfeg the fuzzy equtos to tevl fom we get Fst Method Equto 6-65 we c wte mt fom s / / / / Secod method

37 We get fom d espectvely / / / / Thd method We c wte the equtos s AX B X A B Fo we hve X /5 66 /

38 Also fo we hve X / 54 4 / Coespodg dvdul gph s show fgue 9 d espectvely Fo Fg 9 8

39 Fo Fg Fo 9

40 Fg Plots fo d e gve ll togethe fgue Fg 4

41 Specl cse: Oe my ote tht esult fo csp cse c be obted smply by puttg fuzzy cse Moeove by substtutg fuzzy cse we c obt the esults fo the tevl cse EXAMPLE 4: Let the equtos be Tsfomg the tpezodl fuzzy umbes to tevl Fst method We c wte the bove equtos mt fom s: 4

42 / / / / Secod method Fom d 77 we c wte

43 4 Thd method Equto 78-8 we c wte s: / / / /

44 AX X X X A B B / 5 66 / / / Coespodg dvdul gph s show fgue 4 d 5 espectvely Gph fo 44 Fg

45 Gph fo Fg 4 Gph fo Fg 5 45

46 Plots fo gve ll togethe fgue 6 Fg 6 Specl Cses: Oe my ote tht fo we get dffeet vlue wth membeshp vlue fuzzy cse Moeove by substtutg fuzzy cse we c obt the esults fo the tevl cse But t the pot we get the sme fo tgul fuzzy umbe 46

47 7 EXAMPLE : Fst method We hve used equto 7 fo ths emple s whose soluto my esly be obted s Secod method Fom 84 d 85 we get

48 86-89 Solvg 86 d 88; 87 d 89 we hve Thd method Equto 86 d 88 e obted fo the peset emple s: AX b 4 8 X A b Smlly equtos 87 d 89 gve the dect soluto s X EXAMPLE : Let the equtos be 48

49 Afte tsfeg the fuzzy tevl to -cut tevl we hve Fst method Equto 9 d 9 c be wtte mt fom s Secod method Fom 9 d 9 we c wte 49

50 Solvg the bove we hve Thd method Fom 94 d 96 we c wte AX X X A b 4 b Smlly fom 95 d 97 we hve 5

51 X Coespodg dvdul gph s show fgue 7 Fo Fg 7 Fo 5

52 Fg 8 Plots of d e show fgue 9 ll togethe 5

53 Fg 9 5

54 8 CONCLUSON Peset wok demosttes ew method fo tevl d fuzzy soluto of fuzzy system of le equtos Ths s ppled fst kow poblem of ccut lyss As dscussed ele the cocepts of fuzzy umbe tht s tgul fuzzy umbe tpezodl fuzzy umbe hve bee used hee to solve the umecl poblems of system of le equtos Few othe emple poblems e lso solved to hve the effcecy of the poposed method Thee dffeet cses e cosdeed the bove ccut lyss poblem tkg souce d esstce s: Cse : Csp Cse : tevl Cse : Fuzzy whee cuet s tke s fuzzy cse d cse d csp cse As metoed bove the othe emple poblems e solved wth tevl tgul fuzzy umbe d tpezodl fuzzy umbe to hve the elblty d powefuless of the poposed methods Ths vestgto gves ew de of solvg the tevl/fuzzy system of le equtos wth smple computtos 54

55 9 FUTURE DRECTON ths poject methodologes hve bee developed to solve fuzzy system of le equtos Smulto wth dffeet emple poblems shows tht the poposed methods e ese d smple to hdle compso wth the estg methods The poposed method my vey well be ppled to othe poblems whee we get le smulteous equtos tevl o fuzzy fom Some subject e my be metoed s bo-mthemtcs electcl egeeg compute scece egeeg mechcl egeeg etc whee we my get the tevl o fuzzy le system of equtos d those my be solved by the poposed methods 55

56 REFERENCES [] M Fedm M M A Kdel Fuzzy le systems Fuzzy Sets d Systems [] M M M Fedm A Kdel Dulty fuzzy le systems Fuzzy Sets d Systems [] T Allhvloo Numecl methods fo fuzzy system of le equtos Appl Mth Comput [4] T Allhvloo The Adom decomposto method fo fuzzy system of le equtos Appl Mth Comput [5] S Bdy AUsh Nd Electc Ccut Theoy [6] The Rhgooy Hd Sdogh Yzd Rez Mosef Fuzzy Comple System of Le Equtos Appled to Ccut Alyss tetol Joul of Compute d Electcl Egeeg Vol No Decembe 9 [7] S Ds d S Chkvety Fuzzy system of le equto d ts pplcto to ccut lyss 8 th Aul Cofeece of Oss Mthemtcl Socety TER Bhubesw 8-9 th J [8] LAZdeh fuzzy sets fomto d cotol 8: [9] Rudolf Sesg A hstoy of the theoy of fuzzy sets d systems d ts pplctos to medcl phlosophy d dgoss [] Mchel Hss Appled fuzzy thmtc pt

57 [] Aold Kufm d Md M Gupt Fuzzy mthemtcl models egeeg d mgemet scece pt [] H J Zmmem Fuzzy set theoy d ts pplcto secod edto996 57

58 LST OF PUBLCATONS/To be commucted: Ds S d Chkvety S Fuzzy system of le equto d ts pplcto to ccut lyss 8 th Aul Cofeece of Oss Mthemtcl Socety TER Bhubesw 8-9 th J ; Ds S d Chkvety S Numecl soluto of tevl d fuzzy system of le equto tetol Joul of Computtol d Mthemtcl Sceces To be commucted 58