A New Approach to Solve Fully Fuzzy Linear Programming with Trapezoidal Numbers Using Conversion Functions

Transcription

1 valale Onlne a hp://jnsaa Vol No n 5 Jonal of Ne eseahes n Maheas Sene and eseah Banh IU Ne ppoah o Solve Flly Fzzy nea Pogang h Tapezodal Nes Usng onveson Fnons SH Nasse * Depaen of Maheaal Senes Unvesy of Mazandaan Baolsa Ian eeved Se 5 eped n 5 sa eenly fzzy lnea pogang poles have een onsdeed y any In he leae of fzzy lnea pogang seveal odels ae offeed and heefoe soe vaos ehods have een sggesed o solve hese poles One of he os poan of hese poles ha eenly has een onsdeed; ae Flly Fzzy nea Pogang FFP hh all oeffens and vaales of he pole ae he sae knd of fzzy nes One of os oon of he s he odel n hh all fzzy paaees ae dsssed y angle nes In hs pape e fs defne a flly fzzy lnea pogang h apezodal nes and hen sgges a ne ehod ased on edng he ognal pole o he pole h angle ne Speally a onveson fnon fo onveng o apezodal and angla nes o eah ohe s offeed Fnally he enoned ehod s llsaed y a neal eaple Keyods: onveson fnon Flly fzzy lnea pogang Ml-ojeve lnea pogang Tapezodal and angla fzzy nes * * oespondng aho nasse@za

2 SH Nasse/JNM Vol No n 5 Inodon Unl no os of he poles hh have een sded n he feld of fzzy lnea pogang ae he poles ha soe of he vaales and paaees ae fzzy Sh as he vale of gh sde soes age fnons o deson vaales One of he os poan of hese poles ha eenly has een onsdeed s Flly Fzzy nea Pogang FFP hh n all of pole s oeffens and vaales ae he sae knd of apezodal fzzy nes Soe knds of he odels and also ehods fo solvng hese poles s gven n [ ] Usally a onveson fnon hh s defned fo he se of all apezodal nes o he se of all angla nes sgges o solve hese poles Then sng he neaes onep of esaon of angla fzzy ne he an pole onves o o alay poles a n ha h solvng hese o poles a esponse veo onss of sye angla nes ll e aaned The esponse of azaon alay pole onsdeed as ene and he esponse of nzaon alay pole as he fnge of esponse Fnally sng a onveson fnon e defne a apezodal fzzy ne fo eah angla fzzy ne ha s deened y he poess of solvng he pole Pelnaes In hs seon soe neessay akgonds and noons of fzzy se heoy ae eveed [] Bas defnons and oneps Defnon fzzy se s alled a fzzy ne f sasfes he follong ondons: s noal ha s hg s onve Thee s ealy one h ha s oe The eeshp fnon s a leas peese onnos The vale oe hh shos he a degee of eeshp s alled he odal vale of he fzzy ne n noaonal aodane h he vale ha os os feqenly n daa saples The odal vale ay also e efeed o as peak vale ene vale o ean vale hee he las o epesson ae pefealy sed fo sye fzzy nes The paae fo of fzzy ll e

3 Ne ppoah o Solve Flly Fzzy nea Pogang h Tapezodal Nes Usng onveson Fnons shoed as = so ha and sasfy he elo ondons: Fnon fo lef onsanly s a angla fzzy ne â as a a hee a a fge s lke hs [] : a a nd s desendng onsseny Fnon fo gh onsanly s an asendng onsseny In leae of he heoy of fzzy ses angla fzzy nes and apezodal fzzy nes have any applaons see n [ 5] Hene n hs ale e also onsde he and efly defne hee Defnon We sho eah angla fzzy ne â h a a a a hee a a s onssed of he sppo of angla fzzy ne Fg â and â s s oe In hs ase e an ee he We sho he se of all angla fzzy nes n an aevaed fo of F Defnon We sho eah apezodal fzzy ne a h U a a a a a hee a U s he oe of apezodal fzzy a ne Fg a and a a s he sppo of fzzy ne a Ths e ay ee he apezodal fzzy ne a as a a a U hee U a a a a nd s fge s lke hs [] : We epesen he se of all apezodal fzzy nes n an aevaed fo F Fg Tangla fzzy ne Fg Tapezodal fzzy ne

4 SH Nasse/JNM Vol No n 5 eak Fo eah apezodal fzzy ne e say f hee ae h a as a a h h and a a h h and a s ll e shon y I s oh enonng ha s eqal h [ ] We an easly see ha f y hen y If hen e say ha s a apezodal fzzy ne eqal h Slaly defne as angla zeo Flly fzzy lnea pogang pole Flly Fzzy Tapezodal nea Pogang FFTP pole s defned as follos: a s hee all eleens of elong o F and ae Noe ha he pole of flly fzzy angla lnea pogang defnes sh as elo []: hee all eleens of elong o F and ĉ ae onveson fnon fo apezodal fzzy ne o angla In hs pa y nodng a fnon T e nend o ansfo eah apezodal fzzy ne o a angla fzzy ne; eans ha: T: F F In hs ase f a F hen T a a F U T a a a a a hee a a a a a U a U a and The follong fge eploes he enoned onep Fg Fg ansfoaon of he apezodal fzzy ne o he angla fzzy ne

5 Ne ppoah o Solve Flly Fzzy nea Pogang h Tapezodal Nes Usng onveson Fnons Neaes sye angla defzzfaon Ns Hee e sae he onep of he neaes sye angla defzzfaon hh s gven n [] Defnon If e sppose ha s a sye angla ne s paae fo ll e as ha enaly n and odes sh as elo: hee Noe ha e sho he se of all he sye angla fzzy nes h T S Defnon If e sppose ha s an asye angla fzzy ne s paae fo s sh as elo: oe Those ae espevely he ene and he lef and gh fnge of asye angle fzzy ne as: The sye angla fzzy ne s a speal fo of asye angla fzzy ne hee s s We sho he se of all asye angla fzzy nes h T S Defnon The donan elaons on he se of asye angla fzzy nes ll e defned sh as elo: K K K K K K k k K K ST l Defnon If and hen h h h hee } n{ } a h h Fo eaple; he lplaon of o asye angla fzzy nes s sh as elo:

6 SH Nasse/JNM Vol No n 5 Defnon 5 Sppose ha û s a angla fzzy ne n he paae fo as In hs ase fo ganng a sye angla fzzy ne hh s lose o he follong fnon s e nzed: D s [ ] s [ s [ ] ] d d If s[ ] e he nze of D s[ ] hen s[ ] s a non-fzzy sp fo û o he ene of and fnge of Theefoe e have fo he nze of D s[ ] : D s [ ] D s [ ] 5 So fo solvng he pevos eqaon e have: d d 7 The neaes sp ne s asye angla o û o he ene of and ned o o poles of lnea pogang Then e sho ha sng fnon e an n he pole o he pole No y sng he ehod of neaes angla esaon e solve he oaned alay pole and hen gan o an esaon fo of he solon of he an pole sng hs esponse Theefoe e onsde he follong pole hh s onssed of sye angla fzzy nes sla o he pole hh s dsssed n [] a s N s We an see ha he pevos pole s a pole of Ml Ojeve nea Pogang MOP ha fo azng of he ojeve fnons n pole e offe he azaon of pole fo ene and as nze pole fo ls oleane of s yo kno n seon a fnon as noded ha he onveson possly of a apezodal fzzy ne o a angla fzzy ne has een pepaed Hee fs e sho ha ho eah pole of angla flly fzzy lnea pogang as has vefed n a F X s n F s a hee s N s 9

7 Ne ppoah o Solve Flly Fzzy nea Pogang h Tapezodal Nes Usng onveson Fnons 5 Tha he oe ene esponse has odnal poy ahe han ls o oleane esponse ha a s an op vale of age fnon 9 The fnal ondon n fo hs eason s onsdeed ha he op esponse e se p n 9 a l s In hs ase he pole of 9 and ll e ee sh as elo: n s * a s a s he op vale of ojeve fnon If pole had a nqe esponse hen one pa of op anse has aaned If pole had an alenave anse s he op anse of also f he op anse e pole The defnon of onveson fnon of angla o apezodal Hee y nodng of fnont e nend o onve evey angla fzzy ne o apezodal fzzy ne eans ha: : F F T U U n } a {

8 SH Nasse/JNM Vol No n 5 The nonal analyzng of pole Eaple seel ll anfae on eas n shape of I n fo szes; sall ed lage vey lage The ogh lengh of anfaed eas y ahne appoaely n evey 5 o 7 nes s lke hs: Tale nd also he sale s pof of eah one s oghly 5 and dollas espevely Fheoe ahnes oghly n ho pode appoaely 5 and foo of vaos szes of eas espevely The ojeve of shedlng he ahnes s azaon he pofs odng o he vepon of he anage of he faoy he fzzy nes h he follong fo: = 5 =9 5 = 7 = =7 9 7 = = = =7 9 = = 5 9 The aove pole an e folaed as follos: a s Tha nde he fnon ned o elo daa:

9 Ne ppoah o Solve Flly Fzzy nea Pogang h Tapezodal Nes Usng onveson Fnons 7 Tale Vey lage age Med Sall pod ahne B ne Ths e oan he solon as Then he elaed flly fzzy lnea pogang poles h angla nes s: a 5 s No y solvng he poles 9 and e have he follong esls: The op anse of azaon pole s eqal o a a = 7 The op anse of nzaon pole s eqal o a n = The anse of alay pole s eqal h â = and 79 Tha s a sye angla fzzy ne No sng he fnon e onve he solon o apezodal fzzy follos: onlson In he oon saegy fo solvng flly fzzy lnea pogang sally he ahos had een sed he lassfaon fnons o solve he pole of fzzy lnea pogang B n hs pape e se a ne ehod ho sng any lassfaon fnon and js y applyng he os paly fzzy oneps o fnd he solon of he fzzy pogang pole s ell as he noded fnons n hs pape e an hoose an appopae fzzy daa fo solvng a flly fzzy lnea pogang pole We ay se he enoned appoah n hs pape fo solvng he odels hh s appeaed n he eal old hen onss of he apezodal fzzy nes n all paaees knoledgen

10 SH Nasse/JNM Vol No n 5 The aho hank o he anonyos efeees of Indsal Engneeng 7- fo vaos sggesons hh have led o an poveen n oh he qaly and lay of he pape efeenes [] Ezza E Khoa and Enaya ne algoh o solve flly fzzy lnea pogang poles sng he MOP 9 [] SH Nasse E Behanesh F Taleshan M dolalpoo N Tagh- Nezhad Flly fzzy lnea pogang h neqaly onsans Inenaonal Jonal of Indsal Maheas 5 9- pole ppled Maheaal Modellng [] F Hossenzadeh of T llahvanloo M ladan Jondaeh and lzadeh Solvng a fll fzzy lnea pogang sng leogaphy ehod and fzzy appoae solon ppled Maheaal Modelng [] J Ka and Ka Meha s ehod fo solvng flly fzzy lnea pogang poles h - fzzy paaees ppled Maheaal Modellng [] N Mahdav- and SH Nasse Daly esls and a dal sple ehod fo lnea pogang poles h fzzy vaales Fzzy Ses and Syses [5] SH Nasse and Eahnejad fzzy pal sple algoh and s applaon fo solvng he flele lnea pogang poles Eopean Jonal