CHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD
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1 67 CHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD 7. INTRODUCTION The eso mers the setors le fl ororte lg routo lg mretg me seleto uversty lg stuet mssos helth re hostl lg et. ofte fe rolems to te esos tht otmze ertmet/euty rto roft/ost vetory/sles tul ost/str ost outut/emloyee stuet/ost urse/tet rto et. The ove rolems e solve effetly through Ler Frtol Progrmmg (LFP rolems. I LFP rolems the oeffets the rolems re ssume to e etly ow. However rte the oeffets (some or ll re ot et ue to the errors of mesuremet or vry wth mret otos et. Hee to solve these s of rolems Fuzzy Ler Frtol Progrmmg (FLFP tehues re use. 7. OBJECTIVE I ths hter roeure for solvg FLFP rolem where the ost of the oetve futo the resoures the tehologl oeffets re eresse s trgulr fuzzy umers s eveloe. Here the FLFP rolem s trsforme to euvlet etermst
2 68 Mult Oetve Ler Frtol Progrmmg (MOLFP rolem the MOLFP rolem s overte to sgle oetve ler rogrmmg rolem y usg the Tylor seres roh. Flly the soluto of FLFP rolem s ote y solvg the resultt sgle oetve ler rogrmmg rolem. 7. LITERATURE REVIEW The fferet s of FLFP rolem were vestgte y my reserhers. The FLFP rolem e lssfe to two tegores: LFP rolem wth fuzzy gols LFP rolem wth fuzzy oeffets. Most of the FLFP rolems e solve y usg fuzzy gol roh. Sw Yo (988 roose fuzzy stsfg metho to solve Mult-oetve Ler Frtol Progrmmg (MOLFP rolems. Sw et l. (99 troue geerl oet of Preto otml soluto trete two tyes of fuzzy gols (lle fuzzy eul fuzzy m. Luhul (984 solve MOLFP rolem usg lgust vrle roh. Dutt et l. (99 ote effet soluto y mofyg the lgust roh roose y L Che (996 susse FLP moel wth fuzzy oeffets. Sw Kto (998 resete tertve stsfg metho for solvg struture MOLFP rolem wth fuzzy umers. Lter Sw et l. (999 eveloe tertve fuzzy metho for two-level LFP rolems wth fuzzy rmeters y efg fuzzy gols for eets o oth levels. Stu Ms Po (00 hve resete the mrove verso of roof of Dutt's m theorem ote out tht the roh gve y Dutt et l. (99 ws effet oly whe some restrtve hyotheses re stsfe. Hshem et l. (006 roose two-hse roh for fg the otml solutos of the fully fuzzfe LFP rolem se o the
3 69 omrso of me str evto of fuzzy umers. The frst hse mmzes the osslst me vlue of fuzzy oetve futo ots set of fesle solutos. The seo hse mmzes the str evto of the orgl fuzzy oetve futo y oserg ll s fesle solutos ote t the e of the frst hse. Mehr et l. (007 roose ew metho to omute ( etle otml soluto where [0] [0] s the gre of stsfto ssote wth the fuzzy oetve futo wth the fuzzy ostrts resetvely. Po Stu Ms (008 lyze metho to solve the fully fuzzfe LFP rolem where ll the vrles rmeters re reresete y trgulr fuzzy umers. Tosr (008 roose gol rogrmmg roh to solve FMOLFP rolem se o the Tylor seres metho. 7.4 LINEAR FRACTIONAL PROGRAMMING A geerl Ler Frtol Progrmmg (LFP rolem e formulte s follows: T C X Z (7. T D X AX X 0 where C D X m A m re slrs. For some vlue of X D T X my e eul to zero. To vo suh ses
4 70 let t e ssume tht f X 0 the ether D T X 0 or D T X 0. For oveee ssume tht LFP (7. stsfes the oto tht: T X 0 D X 0 (7. To solve LFP rolem (7. trsform the LFP rolem to LP rolem usg Chres Cooer s (959 ler trsform tehue. The Chres Cooer s ler trsform tehue s summrze s follows: 7.4. Chres Cooer s Ler Trsform Tehue Now LFP rolem (7. e trsforme to LP rolem Te t T D X Y Xt wth D T X 0 (7. Multlyg oth the oetve futo the ostrts of the rolem (7. y t we ot T Z C Y t (7.4 AY t 0 D T Y t Y 0 t 0 The followg remr gves the omrso of two trgulr fuzzy umers whh s very useful for further stuy.
5 7 Remr 7.4. B ( ( ( Let A ( ( ( e y two trgulr fuzzy umers. The A B ( ( f oly f ( (. ( ( 7.5 FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM The str form of the FLFP rolem C T D X Z (7.5 T X A X X 0 where C F F m A X F m. Also ( C ( A ( A A A ( ( Core(C ( ( Core(A ( Core( ( Core( ( Core( A A ( A ( re left rght wth of C A resetvely.
6 7 The rolem (7.5 e rewrtte s Z ( ( (7.6 ( 0... m... The rolem (7.6 e reue s ( ( ( Z (7.7 ( ( ( 0... m... ( ( ( ( (
7 7 Usg Zeh s eteso rle the rolem (7.7 e wrtte s ( ( ( ( ( ( Z (7.8 m ( ( ( ( ( ( By usg the remr (7.4. the rolem (7.8 e rewrtte s ( ( ( ( ( ( Z (7.9 m ( ( ( ( (
8 74 The rolem (7.9 e reue s the followg etermst mult-oetve frtol ler rogrmmg rolem: Z Z Z ( ( ( ( ( ( (7.0 ( ( ( 0... m ( (... The rolem (7. e osere s three LFP rolems wth sgle oetve futo Z Z Z resetvely wth the otml soluto Z of eh rolem e ote usg Chres Cooer s ler trsform tehue s susse (7.4.. Usg the ote Z the Tylor seres roh the MOLFP rolem e overte to sgle oetve LP rolem whh e solve
9 75 usg lssl methos. The roeure roose y Guzel Svr (005 for overtg MOLFP rolem to sgle oetve LP rolem s susse elow Tylor Seres Aroh for MOLFP Ste Determe... whh s the vlue(s tht s use to mmze the oetve futo Z where. Ste olyoml seres. Trsform the oetve futo y usg orer Tylor Z z ( = Z... z ( Z z ( z ( Ste F stsftory... y solvg the reue rolem to sgle oetve. Tht s z ( Z
10 76 By lyg the ove roeure o the rolem (7.0 t s reue to s follows: z ( Z Z (7. ( ( ( 0... m ( ( NUMERICAL EXAMPLE Let oe oser omy tht muftures two s of routs wth roft rou 5 rou ollr er ut resetvely. However the ost for eh oe ut of the ove routs s rou 5 rou ollrs resetvely. It s ssume tht fe ost of rou ollr s to e e to the ost futo ue to eete urto through the roess of routo. Suose the rw mterl eee for mufturg rout A rout B s out uts er ou out 5 uts er ou resetvely the suly for ths rw mterl s restrte to out 5 ous. The rout A B reures M-hour of out 5 hours er ut hours er ut for mufturg ut totl M-hour vlle s out 0 hours ly. Determe how my Prouts of shoul e mufture orer to mmze the totl roft.
11 77 Let e the mout of uts of to e roue resetvely. The the ove rolem e moele s: 5 Z ( Let oe ssume tht 5 (5 5 (5 ( ( ( 5 ( (0 s trgulr fuzzy umers. The rolem (7. wrtte s (5 ( Z (7. (5 ( ( ( (5 0 5 ( (544 (0 rolem The ove FLFP rolem s euvlet to the followg MOLP
12 78 Z 6 Z Z ( Solvg eh oetve futo oe y oe we get Z = (0.7 Z = (0.65 Z = (0.7. The the oetve futo of (7. re trsforme y usg orer Tylor s seres z Z ( z Z0.7 0 (0.7.7 (0.7 Ths mles tht Z ( = z Z ( = Z (0.65 z.65 (0.65
13 79 Ths mles tht Z ( = z Z ( = Z (0.70 z.70 (0.70 Ths mles tht Z ( = 4 -. Hee the rolem (7.4 s overte to euvlet sgle oetve LP rolem s Z Solvg the ove rolem y usg lssl metho oe ot the followg soluto s =0 =.7. Hee the omy mufture.7 uts of the rout B oly.
14 CONCLUSION I ths hter metho of solvg the FLFP rolems where the ost of the oetve futo the resoures the tehologl oeffets re trgulr fuzzy umers s roose. I the roose metho FLFP rolem s trsforme to MOLP rolem the resultt rolem s overte to LP rolem usg Tylor s seres metho. A llustrtve umerl emle s gve to ustfy the roose theory.
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