A new model for solving fuzzy linear fractional programming problem with ranking function

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J. ppl. Res. Ind. Eng. Vol. 4 No. 07 89 96 Journl of pplied Reserch on Indusril Engineering www.journl-prie.com new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion Spn Kumr Ds * Trni Mndl Deprmen of Mhemics Nionl Insiue of Technology Jmshedpur Indi P P E R I N F O S T R C T Chronicle: Received: 0 My 07 Revised: 0 July 07 cceped: 0 ugus 07 ville : 0 ugus 07 Keywords : Tringulr fuzzy numer. Fuzzy liner frcionl progrmming. Rning funcion. Muli Ojecive Progrmming. In his pper we sudied fuzzy liner frcionl progrmming FFP prolems wih rpezoidl fuzzy numers where he ojecive funcions re fuzzy numers nd he consrins re rel numers. In his sudy in order o oin he fuzzy opiml soluion wih unresriced vriles nd prmeers new efficien mehod for FFP prolem hs een proposed. These proposed mehods re sed on crisp liner frcionl progrmming nd newly rnsformion echnique is lso used. compuionl procedure hs een presened o oin n opiml soluion. To show he efficiency of our proposed mehod rel life emple hs een illusred.. Inroducion iner frcionl progrmming FP prolem is one of he mos imporn echniques in operion reserch. Mny rel world prolems cn e rnsformed o liner frcionl progrmming model; hence his model is n indispensle ool for ody s pplicions such s finncil secor hospiliy indusril secor ec. I is mhemicl echnique for opiml llocion o severl civiies on he sis of given decision of opimliy. This ype of prolem is evidenly n uncerin opimizion prolem due o is decision-sed sysem. So i leds o proposiion of new concep in fuzzy opimizion y ellmn nd Zdeh [5]. ofi e l. [3] inroduced mehod o oin he pproime soluion of fully fuzzy liner progrmming prolems. mi Kumr e l. [] proposed mehod for solving fully fuzzy liner progrmming prolems using ide of crisp liner progrmming nd rning funcion. Recenly Veermni nd Sumhi [4] eslished new mehod for solving fuzzy liner frcionl Progrmming prolem nd hey hve rnsformed he prolem ino muli ojecive liner progrmming prolem. Gnesn nd Veermni [6] inroduced he fuzzy liner progrmming prolem * Corresponding uhor E-mil ddress: cool.spnumr@gmil.com DOI: 0.05/jrie.07.48543

Ds nd Mndl / J. ppl. Res. Ind. Eng. 4 07 89-96 90 wih rpezoidl fuzzy numers wihou convering hem o crisp liner progrmming prolem. Erhimnejd nd Tvn [] proposed new concep in which he coefficiens of ojecive funcion nd he vlues of he righ hnd side re represened y rpezoidl fuzzy numers nd oher prs re represened y rel numers. They convered he fuzzy liner progrmming prolem ino n equivlen crisp liner progrmming prolem nd solved y simple mehod. Ds e l. [7 8] hve proposed so mny mehods for solving fuzzy liner frcionl progrmming prolem y using vrious mehods. Ds e l. [7] proposed generl form of fuzzy liner frcionl progrmming prolem wih rpezoidl fuzzy numers. Hmi nd Kzemipoor [9] hve solved fuzzy liner frcionl progrmming prolem y using ig-m mehod. Seri Njfi nd Edlpnh [0] hve proposed mehod for solving liner progrmming prolems y using Homoopy perurion mehod. Seri Njfi e l. [] hve proposed mehod nonliner model for fully fuzzy liner progrmming wih fully unresriced vriles nd prmeers. Hosseinzdeh nd Edlpnh [] hve proposed mehod for solving fully fuzzy liner progrmming y using he leicogrphy mehod; see lso [3-]. In his pper new mehod is proposed for finding he fuzzy opiml soluion of FFP prolems wih inequliy consrins. The coefficiens of he ojecive funcion re represened y rpezoidl fuzzy numers nd he consrins re represened y rel numers. We inroduce new ype of fuzzy rihmeic for symmeric rpezoidl fuzzy numers nd propose mehod for solving fuzzy liner frcionl progrmming prolem wih convering hem o crisp liner frcionl progrmming prolems. fer h we used new rnsformion echnique o solve hese crisp liner frcionl progrmming prolems. To illusre he proposed mehod numericl emples re solved. This sudy is orgnized s follows: In Secion some sic definiions of fuzzy symmeric rpezoidl fuzzy numer nd some rihmeic resuls re presened. In Secion 3 formulion of FFP prolems nd pplicion of rning funcion for solving FFP prolems re eslished. new mehod is proposed for solving FFP prolems in secion 4. In Secion 5 we give numericl emple including symmericl rpezoidl fuzzy numers o illusre he heory developed in his pper. Finlly in Secion 6 we presen he conclusion pr.. Preliminries In his secion some noions nd resuls of fuzzy se heory re presened nd discussed: Definiion. []. conve fuzzy se on R is fuzzy numer if he following condiions hold: Is memership funcion is piecewise coninuous. There eis hree inervls [ ] [ c] nd [c d] such h is incresing on [ ] equl o on [ c] decresing on [c d] nd equl o 0 elsewhere. Definiion. []. The rihmeic operions on wo symmeric rpezoidl fuzzy numers = nd = re given y:

9 new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion u Where }. m{ } min{ 0. 0. Definiion.3 []. e nd = e wo symmeric rpezoidl fuzzy numers. The relions nd re defined s follows: i h is in his cse we my wrie or ii in his cse we sy or iii in his cse we sy. Definiion.4 []. fuzzy se on R is clled symmeric rpezoidl fuzzy numer if is memership funcion is defined s follows: else u 0

3. Proposed mehod o conver FP ino P Ds nd Mndl / J. ppl. Res. Ind. Eng. 4 07 89-96 9 In his secion we refer o [8] rnsformion mehod o conver liner frcionl o equivlen liner progrmming s follows: Trnsformion of Ojecive: M Z= M Z= = c c d c d d * * d M Z = p y g Where p c d * y g d Trnsformion of consrins: 0 d * * d = Gy h Where G d * h We ge he liner progrmming prolem s M Z= p y g Sujec o Gy h y 0. = 0 4. Proposed mehod o find he fuzzy opiml soluion of FFP prolems In his secion new mehod is proposed o find he fuzzy opiml soluion of he following ype of fuzzy liner frcionl progrmming FFP prolems: c Mimize or Minimize = d Sujec o 0. Where [ c c ] is y n mri; d [ d ] is y n mri; ] j is m y n mri; ] is m y mri; [ ] [ ij j [ j is n y mri; ] nd [ ] re he sclrs. j j [ ij

93 new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion sed on definiion.3 we define rn for ech symmeric rpezoidl fuzzy numer for comprison purposes. ssuming h is symmeric rpezoidl fuzzy numer hen R. This equion llows us o conver he fuzzy liner frcionl progrmming FFP prolem in o crisp liner frcionl progrmming CFP prolem. We susiue he rn order of ech fuzzy numer for he corresponding fuzzy numer in he fuzzy prolem under considerion. This leds o n equivlen crisp liner frcionl progrmming prolem which cn e solved y sndrd mehod. In he following pr we re going o inroduce n lgorihm o find n ec opiml soluion of FFP prolem. The seps of he proposed lgorihm re given s follows: Sep. Wih respec o he generl form of our prolem i cn e wrien s follows: c Mimize or Minimize = d Sujec o 0. Sep. Regrding definiions.3 he prolem is convered o he crisp liner frcionl progrmming prolem my e wrien s follows: c Mimize or Minimize = d Sujec o 0. 3 Sep 3. In erms of he ojecive funcion he rnsformion mehod will e used ino n equivlen crisp liner progrmming prolem so we hve: M or Min = p y g 4 Sujec o Gy h y 0. Sep 4. Solve he prolem 4 wih he help of ingo sofwre: we ge he opiml soluion. 5. Numericl Emple: In his secion we illusre he proposed lgorihm using rel life prolem. iner frcionl progrmming prolem is evidenly n uncerin opimizion prolem due o is vriions in he mimum dily requiremens. So he moun of ech produc of ingredien will e uncerin. Hence we will model he prolem s Fuzzy liner frcionl progrmming FFP prolem. We use rpezoidl fuzzy numers for ech uncerin vlue. lso he mhemicl progrmming prolem will e solved y ingo.

Ds nd Mndl / J. ppl. Res. Ind. Eng. 4 07 89-96 94 5.. Emple Producion Plnning compny mnufcures wo inds of producs nd wih profi round 4633 nd round 5 dollr per uni respecively. However he cos for ech one uni of he ove producs is round 37 nd round 3 dollrs respecively. I is ssumed h fied cos of round dollrs is dded o he cos funcion due o he epeced durion hrough he process of producion. Supposing he rw meril needed for mnufcuring produc nd ou 3 unis per pound nd ou 5 unis per pound respecively he supply for his rw meril is resriced o ou 5 pounds. Mn-hours per uni for he produc is ou 5 hours nd produc is ou hours per uni for mnufcuring u ol he Mn-hour ville is ou 0 hour dily. Deermine how mny producs nd should e mnufcured in order o mimize he ol profi. This rel life prolem cn e formuled o he following FFP prolem: M 4 633 5 37 3 3 5 5 s.. 5 5 0 0. So wih respec o Sep we conver he prolem ino n equivlen crisp liner progrmming prolem s follows: 5 3 M 5 6 3 5 5 s.. 5 0 0. Now he crisp liner frcionl progrmming prolem 6 is convering ino n equivlen crisp liner progrmming prolem y using Sep 3. M 5y 3y 7 s.. 78y 35y 5 55y y 0 y y 0. The prolem 7 is he crisp liner progrmming prolem. Now solved y simple mehod we ge he resul is: y =0 y =0.4 nd he ojecive funcion vlue is Z=.8. y compring he resul of proposed mehod wih Erhimnejd mehod e l. we conclude h our resul is more effecive ecuse:.5=z Erhimnejd mehod el.<z proposed mehod=.8.

95 new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion 6. Conclusion: In he ps few yers growing ineres hs een shown in Fuzzy liner frcionl progrmming nd severl mehods for solving FFP prolem hve een suggesed. In his pper new efficien mehod hs een proposed in order o oin he fuzzy opiml soluion of fuzzy liner frcionl progrmming prolems wih inequliy consrins occurring in dily rel life prolem. We showed h he mehod proposed in his pper is highly relile nd pplicle. To illusre he proposed mehod numericl emples re solved. References [] Erhimnejd. & Tvn M. 04. novel mehod for solving liner progrmming prolems wih symmeric rpezoidl fuzzy numers. pplied Mhemicl Modelling 387 4388-4395. [] Kumr. Kur J. & Singh P. 0. new mehod for solving fully fuzzy liner progrmming prolems. pplied Mhemicl Modelling 35 87-83. [3] ofi F. H. llhvirnloo T. Jondeh M.. & lizdeh. 009. Solving full fuzzy liner progrmming using leicogrphy mehod nd fuzzy pproime soluion. pplied Mhemicl Modelling 337 35-356. [4] Veermni C. & Sumhi M. 04. Fuzzy mhemicl progrmming pproch for solving fuzzy liner frcionl progrmming prolem. RIRO-Operions Reserch 48 09-. [5] ellmn R. E. & Zdeh.. 970. Decision-ming in fuzzy environmen. Mngemen science 74-4. [6] Gnesn K. & P. Veermni V. 006. Fuzzy liner progrms wih rpezoidl fuzzy numers. nn. Oper. Res. 43 305-35. [7] Ds S. K. Mndl T. & Edlpnh S.. 07. new pproch for solving fully fuzzy liner frcionl progrmming prolems using he muli-ojecive liner progrmming. RIRO-Operions Reserch 5 85-97. [8] Ds S. K. & S.. Edlpnh 06. Generl Form of Fuzzy iner Frcionl Progrms wih Trpezoidl Fuzzy Numers. Inernionl Journl of D Envelopmen nlysis nd *Operions Reserch* 6-9. [9] Hmi. & Kzemipoor H. 03. Fuzzy ig-m mehod for solving fuzzy liner progrms wih rpezoidl fuzzy numers. Inernionl Journl of Reserch in Indusril Engineering 3-9. [0] Njfi H. S. & Edlpnh S.. 04. Homoopy perurion mehod for liner progrmming prolems. pplied Mhemicl Modelling 385 607-6. [] Seri Njfi H. Edlpnh S.. nd Du H. 06. nonliner model for fully fuzzy liner progrmming wih fully unresriced vriles nd prmeers. lendri Engineering Journl 553 589-595. [] Hosseinzdeh. & Edlpnh S.. 06. New pproch for Solving Fully Fuzzy iner Progrmming y sing he eicogrphy Mehod. dv. Fuzzy Sys. hp://d.doi.org/0.55/06/538496 6 pges 538496. [3]Mlei H. R T M. Mshinchi M. 996. Fuzzy numer liner progrmming ProcInern Conf. on inelligen nd Cogniive Sysems FSS Sponsored y IEE nd ISRF Tehrn Irn 45-48. [4] Tn H. Oud T. & si K. 974. On fuzzy mhemicl progrmming J. Cyern 34 37-46. [5] Zimmermnn H. J. 978. Fuzzy progrmming nd liner progrmming wih severl ojecive funcions. Fuzzy ses nd sysems 45-55. [6] Zimmermnn H. J. 98. Trends nd new pproches in Europen operionl reserch. Journl of he Operionl Reserch Sociey 597-603. [7] Ds S. & Mndl T. 05. single sge single consrins liner frcionl progrmming prolem. Oper Res nd ppl: n Inern J -5. [8] Spn K. D. Mndl T. & Edlpnh S.. 05. noe on new mehod for solving fully fuzzy liner frcionl progrmming wih ringulr fuzzy numers. [9] Ds S. K. Mndl T. & Edlpnh S.. 07. mhemicl model for solving fully fuzzy liner progrmming prolem wih rpezoidl fuzzy numers. pplied Inelligence 463 509-59.

Ds nd Mndl / J. ppl. Res. Ind. Eng. 4 07 89-96 96 [0] Tori N.& Njfi S. E. 05. New model for rning sed on Sum Weighs Dispriy Inde in d envelopmen nlysis in fuzzy condiion. Journl of pplied Reserch on Indusril Engineering -9. [] Sheihi H. 0. Novel pproch for Solving Fuzzy Muli-Ojecive Zero-One iner Progrmming Prolems. Inernionl Journl of Reserch in Indusril Engineering 4-63.

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