SOLVING FUZZY LINEAR PROGRAMMING PROBLEM USING SUPPORT AND CORE OF FUZZY NUMBERS
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1 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri 7 44 SOLVING FUZZY LINER PROGRMMING PROBLEM USING SUPPORT ND CORE OF FUZZY NUMBERS Dr.S.Rmthigm K.Bmrg sst. Professor Deprtmet of mthemtics Periyr rts Coege Cddore Tmid Idi sst. Professor d Hed Deprtmet of Mthemtics chriy rts d Sciece Coege Pdcherry Emi: bmrgek@gmi.coom BSTRCT My thors sed differet types rkig fctio to sove fzzy ier progrmmig method. I this pper ew pproch hs bee proposed to sove fzzy ier progrmmed by sig spport d core of fzzy mbers withot sig membership fctio d ph ct techiqe. Differet types of probems were tke d soved by proposed method. Keywords Fzzy ier progrmmig probem Trpezoid d Trigr fzzy mber spport d core of fzzy mber.. INTRODUCTION The cocept of decisio-mkig i fzzy eviromet first proposed by R.E.Bem d L. Zdh []. Zimmer[] hs proposed the cocept of fzzy ier progrmmig. My thors proposed differet techiqe of sovig FLP probem i re time sittio. Meki [4] hs proposed ew method of sovig FLP probem sig rkig fctios. P.Rreshwri d Shy Sdh [8] proposed method to sove FFLP probem by Robst s rkig fctio d the compred their sotio with Pdi d Jykshmi [5] sotio who proposed ew method to sove FFLP probem.lter o Ide Hss ki d Frrh d [9] hs sed Meki [4] d Yker rkig fctios to sove FLP whe fzzy mbers i obective fctio coefficiets fzzy mbers i right-hd side coefficiets d fiy fzzy mbers i obective fctio coefficiets s we s i right-hd side coffeiciets..kmr d Sigh[7] proposed ew method for sovig fy fzzy ier progrmmig probems sig rkig fctio.. DEFINITIONS.. Membership fctio: Let R be the re ie. Let be the sbset of R. The fctio : R [ ] is kow s membership fctio o X. Fzzy set: Let X be set d be rbitrry eemet of X the fzzy sbset of X is mp : X [ ] (or is set of ordered pir ( where ( is membership fctio. X. Fzzy mber: fzzy set of R is sid to be trpezoid fzzy mber if the membership fctio hs the foowig chrcteristic. : R [ ] is cotios fctio.. ( ( c] [ d.. Stricty icresig o [c] d stricty decresig o [bd]. 4. ( [ b] Note: Trpezoid Fzzy mber becomes trigr fzzy mber whe =b.. ct: The ct of fzzy set is the crisp set tht cotis the eemets of X whose membership grde i re greter th or eq to the ve. i.e. (.4 Cove fzzy set: fzzy set o R is cove if ( ( y mi ( ( y d oy if y X d [ ].5 Spport of Fzzy mber: If is fzzy set the X ( d deoted by spport of Spp( or..6. Core of fzzy mber: If is fzzy set the core X ( d deoted by Cor( or of C(.
2 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri verge of iterv: If [b] is sbset of R the verge of [ b] is defied s vg([b]= b. There re my wys of represetig the fzzy mbers. Bt most importty the fzzy mbers re represeted i two wys mey trigr fzzy mber d trpezoid fzzy mbers..8.trpezoid fzzy mber: Let s cosider the trpezoid mber s where or d its correspodig membership fctio is defied s foows ( ( ( esewhere The geometric represettio of trpezoid membership fctio for fzzy mber is show beow Fig-: Trpezoid Fzzy Nmber Here the spport of is S the core of is C (. ( d ( ( ( vg S d ( ( ( vg C ( Let s cosider the trpezoid mber s where or d its correspodig membership fctio is defied s foows ( ( ( esewhere The geometric represettio of trpezoid membership fctio for fzzy mber show beow Fig-: Trigr Fzzy Nmber Here the spport of is S core of is C(. ( vg ( is ( d the vg( C(..(.. Opertios of Trpezoid Fzzy Nmbers:- ssme tht d B b b re y two trpezoid fzzy mber the the rithmetic trpezoid fzzy mbers re defied s beow.9.trigr fzzy mber:
3 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri dditio: B b b b b... Sbtrctio: B b b b b...scr Mtipictio: if d if..opertios of Trigr Fzzy Nmbers:- ssme tht d b B re y two trpezoid fzzy mber the the rithmetic trpezoid fzzy mbers re defied s beow...dditio: B b b...sbtrctio: B b b...scr Mtipictio: if d if..rkig fctio: The rkig fctio is the most powerf techiqe to compre the fzzy mbers. My types of rkig fctios itrodced by the my thors to sove fzzy ier progrmmig probem with fzzy prmeters. Let F(R be the set of fzzy mbers. The rkig fctio is defied by : F( R R. The compriso betwee the fzzy mbers d B is show beow. B iff ( ( B. B iff ( ( B. B iff ( ( B so stisfies the ier property s foows ( c B c( ( B where c R d B F( R... Spport d core with respect to Meki Rkig Fctio for Trpezoid fzzy mbers:- If is trpezoid fzzy mber the the Meki rkig fctio[4] for trpezoid fzzy mber is ( if sp d ( ( ( ( ( ( ( ( = vg ( + vg( C(.4.Spport d core with respect to Meki Rkig Fctio for Trigr fzzy mbers:- If is trigr fzzy mber the the Meki rkig fctio[4] for trigr fzzy mber is ( if sp d 4 ( = vg ( + C(
4 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri Spport d core with respect to Robst s Rkig Fctio for Trpezoid fzzy mbers:- ( if sp d ( ( ( ( ( ( ( ( = vg( vg( C(.6.Spport d core with respect to Robst s Rkig Fctio for Trigr fzzy mbers:- If is trigr fzzy mber the the Robst s trigr fzzy mber is rkig fctio for ( if sp d 4 ( = vg( C(.6.Mthemtic formtio of Lier progrmmig Probem: The mthemtic ier progrmmig probem is stted s foows c M (or Mi z sbect to i or or b i=..m The bove mthemtic LPP is kow s crisp ier progrmmig probem. Here the prmeters re crisp ves. Some time the some or the prmeters re fzzy mbers the the crisp LPP becme fzzy ier progrmmig probem. So the mthemtic fzzy ier progrmmig probem s foows ~ i ~ c M (or Mi z sbect to ~ or or b Nmeric Empes: i=.m Where ~ c ~ i b ~ re fzzy mbers.. For symmetric trpezoid fzzy mber M z~ (48 ~ ( ~ (5 ~ sbect to ~ ~ ~ (7 ~ 4 ~ (6 ~ ~ ~ (59 ~ ~ ~ Usig Meki Rkig fctio Spport core d rkig fctio ve for the fzzy mber =(48ccted s beow S ( ( (4 8 ( ( ( ( vg S 6 48 C (
5 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri 7 48 ( ( ( (4 8 vg C 6 Meki Rkig fctio ve for =(48 is( vg( vgc( 6 6 Simir mer crisp ves for the remiig fzzy mber fod d ist i the tbe beow ( Fzzy C( vg vg mber( ( (C( ( (4 [] 4 (5 (-6 [5] 6 (7 (- [7] (6 (8 [6] (59 ( [59] So the crisp LPP for the bove FLPP s give beow M z 4 sbect to Where z re crisp vribes correspodig to fzzy vribes ~ z ~ ~ ~ Sovig the bove LPP sig simpe method we get =5.48 =.857 = d z=76.57 Usig Robst s Rkig fctio Spport core d rkig fctio ve for the fzzy mber =(48ccted s beow S ( ( (4 8 ( ( ( ( vg S 6 48 C ( ( ( ( (4 8 vg C 6 Robst s Rkig fctio ve for =(48 is( vg( vgc( Simir mer crisp ves for the remiig fzzy mber fod d ist i the tbe beow ( Fzzy C( vg vg mber( ( (C( ( (4 [] (5 (-6 [5] (7 (- [7] (6 (8 [6] (59 ( [59] So the crisp LPP for the bove FLPP s give beow M z 6 sbect to Where z re crisp vribes correspodig to ~ ~ ~ fzzy vribes ~ z Sovig the bove LPP sig simpe method we get =.74 =.486 = d z= For o-symmetric trpezoid fzzy mber M z~ (48 ~ ( ~ (5 ~ sbect to ~ ~ ~ (74 ~ 4 ~ (6 ~ ~ ~ (59 ~ ~ ~ Usig Meki Rkig fctio Spport core d rkig fctio ve for the fzzy mber =(48ccted s beow S ( ( (4 8 (9 ( 9 ( ( vg S 5 48 C (
6 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri 7 49 ( ( ( (4 8 vg C 6 ( vg( vgc( 5 6 Simir mer crisp ves for the remiig fzzy mber fod d ist i the tbe beow ( Fzzy mber( C( vg ( vg (C( ( (4 [] 4 (5 (-6 [5] 5 (74 (-9 [7] 4 7 (6 (9 [6] (59 (4 [59] So the crisp LPP for the bove FLPP s give beow M z 4 sbect to re crisp vribes correspodig to ~ ~ Where fzzy vribes ~ Sovig the bove LPP sig simpe method we get =5.4 =.56 = d z=69.74 Usig Robst s Rkig fctio Spport core d rkig fctio ve for the fzzy mber =(48ccted s beow S ( ( (4 8 (9 ( 9 ( ( vg S 5 48 C ( ( ( ( (4 8 vg C 6 ( vg( vgc( Simir mer crisp ves for the remiig fzzy mber fod d ist i the tbe beow Fzzy mber( C( vg ( vg (C( ( (4 [] ( (5 (-6 [5].5 (74 (-9 [7] 4.5 (6 (9 [6] (59 (4 [59] So the crisp LPP for the bove FLPP s give beow M z 5.5. sbect to re crisp vribes correspodig to Where fzzy vribes ~ ~ ~ Sovig the bove LPP sig simpe method we get =.574 =.649 = d z= Now cosider trigr Fy Fzzy ier progrmmig probem the probem discssed i Rreshwri[8] M z~ ( ~ ~ (4 sbect to ( ~ ( ~ (7 ( ~ ( ~ (8 ~ ~ Sotio:- By Meki rkig fctiowe get the crisp LPP s foows M z 4 6 sbect to The sotio is = ; =.6667; Z= By Robst s rkig fctio we get the crisp LPP s foows M z sbect to The sotio is = ; =.6667; Z=.. CONCLUSION I this pper we proposed ew method of sovig y kid Fzzy ier progrmmig probem withot sig ph ct method. Here we itrodced the ew techiqe
7 Itertio Jor of Scietific Reserch Egieerig & Techoogy (IJSRET ISSN Vome 6 Isse 4 pri 7 5 spport d core of trpezoid d trigr fzzy mbers with differet types of probem. Whe compre with other proposed methods by vrios thors this wi be the simpest method of sovig y FLPP. We hve tke Meki rkig fctio d Robst s Rkig fctio for sovig FLPP. REFERENCES [] BemR.E d ZdhL. 97.Decisio mkig i fzzy eviromet.mgemet Scieces7(97 pp [] ZimmermH.J 978.Fzzy progrmmig d ier progrmmig with sever obective fctios Fzzy sets d systems.(978pp:45-55 [] Yger R.R 98. procedre for orderig fzzy sbsets of the it iterv.iformtio secieces vo.4o:pp:4-6. [4] MekiH.R..Rkig fctios d their ppictios to fzzy ier progrmmig.fr Est Jor Mthemtics Scieces4(pp: 8- [5]Jykshmi. M Pdi P. New Method for fidig optim Fzzy Sotio for Fy Fzzy Lier Progrmmig Probems. Itertio Jor of Egieerig Reserch d ppictios (IJER ISSN: vo.isse 4 Jy- gst pp [6]ZdehL.(965Fzzy sets iformtio d cotro 8(8-5. [7] Kmr d SighP. ew method for sovig fy fzzy ier progrmmig probems.s of fzzy Mthemtics d Iformtio vo. o. Jry pp:-8. [8]P.Rreswri d.shy Sdh-4 Sovig fy fzzy ier progrmmig probem by rkig. Itertio or of mthemtics treds d techoogy-vome 9 mber-my 4. [9]Ide Hss ki d Frrh d- 4.Rkig fctio methods for sovig fzzy ier progrmmig probems.mthemtictheory d modeig.issn 4-584(pper Vo 4No.44.
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