The Definition of Optimal Solution and an Extended Kuhn-Tucker Approach for Fuzzy Linear Bilevel Programming

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1 eare rle: The Deo o Oal Solo a a Eee Kh-Tker roah The Deo o Oal Solo a a Eee Kh-Tker roah or zz ear leel Prograg Gagqa Zhag a Je sra leel eso ehqes are al eeloe or solg eeralze aagee roles wh eso akers a herarhal orgazao Orgazaoal leel eso-akg sh as lag o la-se rasorao a waer resore all a ole era aors The araeers show a lee rograg oel eher he ojee os or osras are hs oe rese whh s alle zz araeer leel rograg PP role ollowg or reos work [ ] hs s rs rooses a oel o PP I he ges he eo o oal solo or a PP role ase o he eo a relae heores hs s eelos a zz er ase Kh-Ther aroah o sole he roose PP role all a eale rher llsraes he ower o he zz er ase Kh-Ther aroah Ie Ters ear leel rograg Kh-Tker aroah zz se Ozao I INTODCTION HE eeo o a esos sesses s seqeal T ro a hgher leel leaer o a lower leel ollower; eah eeel ozes s ow ojee s aee oher s aos hrogh eerales Ths s alle leel rograg P role also alle leel eso or leel ozao roles P was rs roe Vo Sakelerg [] he oe o alae eoo arkes [ ] I a P role eah eso aker leaer or ollower res o oze hs/her ow ojee o wh arall or who oserg he ojee o he oher leel he eso o eah leel aes he ojee ozao o he oher leel [6] There hae ee earl wo oze algorhs [ 7-] roose or solg P roles se he el agh he aeo o researhers he -97s [-9] lhogh P heor a eholog hae ee ale wh rearkale sess ere oas [-] esg aroahes al sor he eso sao whh he ojee os a osras are haraerze wh rese araeers Thereore he araeers are reqre o e e a soe ales a eereal a/or sjee aer hrogh he eers ersag o he are o he araeers he role-orlao roess I has ee osere ha os hors are wh al o Iorao Teholog ers o Teholog Se PO o roawa NSW 7 srala e-al: {jel zhagg}@sea IEEE Iellge Ioras lle real-worl saos arlarl ral resore lag sh as lag o la-se rasorao a waer resore he ossle ales o hese araeers are oe ol resel or agosl kow o hese eers I resls a l o araeers he ojee os or osras o a leel rograg oel Wh hs oserao wol e eral ore arorae o erre he eers ersag o araeers as zz eral aa whh a e reresee eas o zz ses heor [] leel rograg role whh he araeers eher ojee o or osras are esre zz ales s alle a zz araeer leel rograg PP role The PP role was rs elore Sakawa e al [] Sakawa e al orlaes leel rograg roles wh zz araeers ro he ersee o eers reso a rooses a zz rograg eho or zz leel rograg roles Howeer Sakawa s work s al ase o he eo o solo or leel rograg roose ar [ ] Oe ee o ar s lear P heor s ha ol o well sole a lear leel rograg role whe he er-leel osra os are o arrar lear or Or ree researh work has eee ar s heor o leel rograg roosg a ew eo o oal solo or lear leel rograg whh a oeroe he arrar lear or role ae aoe [] We hae he roose a eee Kh-Ther aroah ase o or eo o oal solo or solg lear leel roles [] ollowg or reos researh resls show [ ] hs s as a solg a PP role raserrg o a o-zz leel rograg role Ths aer rs rooses a oel o PP role he ges a eo o he oal solo or he PP role ase o he eo a relae heores hs aer eelos a zz er ase Kh-Ther aroah o sole he roose PP role s hs aer ol eals wh lear leel role so leel rograg eas lear leel rograg hs aer ollowg he roo Seo reews relae eos heores a roeres o zz er P solo a Kh-Ther aroah or solg a P role eo o oal solo a a zz er ase Kh-Ther aroah or solg PP roles are resee Seo eral eale s show Seo or llsrag he roose zz er ase Kh-Ther aroah Colso a rher s are ssse Seo Noeer Vol6 No

2 eare rle: Gagqa Zhag a Je II PEIMINIES zz Ners I hs seo we rese soe as oes eos a heores ha are o e se he sseqe seos The work resee hs seo a also e o ro or ree aer [] e e he se o all real ers e -esoal Elea sae a T T e a wo eors a T eoes he rasose o he eor The we eoe he er ro o a or a wo eors we wre > > ; > > a ; > > Deo zz er a s ee as a zz se o whose eersh o sases he ollowg oos: a s a ag ro o he lose eral [ ]; a s oral e here ess sh ha a ; or a ] a {; a } s a lose eral eoe [ a a ] e e he se o all zz ers he eooso heore o zz se we hae a [ a a ] [] or eer a e e he se o all e zz ers o Theore e a e a zz se o he a a ol sases a [ ] a > s he rgh-oos oooe reasg o a l s he le-oos oooe ereasg o a l Corollar or eer a a [ ] he a a Deo or a a a he s o a a a he salar ro o a a are ee he eersh os s { a } a s { a } a a s a Theore or a a a α a αa [] a a [] [ a [ αa a αa [] ] ] [ a a Deo e a We ee a a a a : a a [] a T a a s alle a -esoal zz er o I a a s alle a -esoal e zz er o e a e he se o all -esoal zz ers a he se o all -esoal e zz ers o reseel Prooso or eer a a s oral Prooso or eer a he -seo o a s a -esoal lose reaglar rego or a [] Prooso or eer a a [] he a a Deo or a -esoal zz ers a we ee a a > a a > ]; a a > a a > ]; a a > a a > ] We all he ar relaos a a zz a orer a sr zz a orer a a srog zz a orer reseel The Eee Kh-Tker roah or ear leel Prograg e wre a lear rograg P as ollows sje o > s a -esoal row eor a -esoal ol eor a ar wh a e a e he al arales assoae wh osras > a reseel ar [] gae he ollowg rooso Prooso [] eessar a se oo ha soles aoe P s ha here es row eors sh ha soles: > > > > ] Noeer Vol6 No IEEE Iellge Ioras lle

3 eare rle: The Deo o Oal Solo a a Eee Kh-Tker roah or : a : a lear P role s ge ar []: a sje o sje o q q q Deo [] a Cosra rego o he lear P role: S { : } easle se or he ollower or eah e : S { : } Projeo o S oo he leaer s eso sae: S { : } ollower s raoal reao se or S : P { : arg [ ˆ : ˆ S ]} arg [ ˆ : ˆ S ] { S : ˆ ˆ S } e Ile rego: I { : S P } Deo 6 [] s sa o e a olee oal solo a ol here ess S sh ha a or all S Howeer geeral sh a olee oal solo ha slaeosl zes oh he leaer a ollower s ojee os oes o alwas es Isea o a olee oal solo a ew solo oe alle Pareo oal s roe lear P Deo 7 [] s sa o e a Pareo oal solo a ol here oes o es S sh ha a or Deo 8 oologal sae s oa eer oe oer o he ere sae has a e soer or eale [ a ] s oa he Hee-orel heore [6] To esre ha has a Pareo oal solo ar gae he ollowg asso sso a S s oe a oa or esos ake he leaer he ollower has soe roos o reso; e P φ P s a o-o-o a To esre ha s well ose we asse ha S s oe a oa a ha P s a o-o-o a The raoal reao se P ees he resose whle he le rego I rereses he se oer whh he leaer a oze hs ojee Ths ers o he aoe oaos he lear P role a e wre as { : I} 6 We also rese he ollowg heore o haraerze he oo er whh here s a Pareo oal solo or a lear P role Theore [] I S s oe a oa here ess a Pareo oal solo or a lear P role Theore [] [Eee Kh-Ther Theore] eessar a se oo ha soles he lear P role s ha here es row eors a w soles: w sh ha 7a sje o 7 7 w 7 w 7e w 7 III ZZ PMETE INE IEVE POGMMING POEM Coser he ollowg zz araeer lear leel rograg PP role: or : a : a sje o sje o q a j a j j j e e s s j q j j q j ssoae wh he PP role we ow oser he ollowg lear l-ojee l-ollower leel rograg MMP role: or : a : [ ] a [ ] sje o [ ] [ ] [ ] sje o [ ] q aj aj q q s s j j j j Theore e e he solo o he MMP IEEE Iellge Ioras lle Noeer Vol6 No

4 eare rle: Gagqa Zhag a Je role The s also a solo o he PP role ee Proo The roo s oos ro Deo ea I here s sh ha > > a > or a a soseles ragle zz ers a he > > or a a are he ere o a reseel Proo s -seo o soseles ragle zz ers a are a a Thereore we hae > ro > a > we a roe > ro slar reaso Theore or I all he zz oees a e s j j j j a hae ragle eersh os o he PP role z z z z z z z z z z z z z z eoes a e s j j j j a a z are he ere o z reseel The s he solo o he role ha sasg a sje o sje o Proo ro ea sases a a he sases a a The we ee ol roe sases a he sases a I a or a we hae a j j a a j j j j a j j a a a j j j j a a j j j j a a j j j j ro Slarl we a roe or a ro a The roo s olee Theore [Eee Kh-Ther Theore] eessar a se oo ha soles he PP role wh ragle zz ers s ha here es row eors a w sh ha w soles: a sje o w w e w Proo: ro Theore we kow ha we ee ol o sole he role I a o sole he role we a se he eho o weghg [7] o hs role sh ha s he ollowg role: 6a sje o 6 6 Noeer Vol6 No IEEE Iellge Ioras lle

5 eare rle: The Deo o Oal Solo a a Eee Kh-Tker roah sje o 6 Thereore he lear P role a e wre as { : I} 7 e s ge a el eresso o 7 a rewre 7 as ollows: sje o I We hae sje o S P Deo e The we hae sje o S arg [ ˆ : ˆ S ] Deo We rewre as: sje o S sje o S We hae sje o S sje o S Deo Coseqel we a hae 8a sje o 8 sje o 8 8 Deo a Ths sle rasorao has show ha solg he zz lear P s eqale o solg 8 Neess s oos ro 8 Se I s he oal solo o 6 we ee o show ha here es row eors a w sh ha w o sole Gog oe se arher we ol ee o roe ha here es row eors a w sh ha w sas he ollows w 9a e 9 9g w 9h q w a he are o egae arales ease s he oal solo o 6 we hae I 7 Ths we hae P Deo e s he oal solo o he ollowg role : S Deo ewre as ollows sje o S ro Deo we hae a sje o e g h > To sl we a hae g a sje o > > > > e > > g > h IEEE Iellge Ioras lle Noeer Vol6 No

6 6 eare rle: Gagqa Zhag a Je Noeer Vol6 No IEEE Iellge Ioras lle e we oe a We rewre sg a we ge g a sje o > > Now we see ha s he oal solo o whh s a P role Prooso here ess eor sh ha sas a sse elow a > q a e q a w a ee w Ths we hae w ha sas 9 Or roo s olee Theore eas ha he os re aroah o solg s o sole he eqale aheaal rogra ge Oe aaage ha oers s ha allows or a ore ros oel o e sole who rog a ew oaoal les IV N ISTTIVE EMPE Eale Coser he ollowg PP role wh a } { } { a sje o sje o e Se The role s raserre o he ollowg MMP role sg Theore sje o sje o Se The role s raserre o he ollowg lear P role sg eho o weghg [7] 6 sje o sje o Se Sole hs lear P role 6 sje o

7 eare rle: The Deo o Oal Solo a a Eee Kh-Tker roah K Se The resl s a 6 Coseqel we hae he solo o he role a V CONCSION Ma orgazaoal eso roles a e orlae leel rograg oels ollowg or reos researh [ ] hs aer rooses he eo o oal solo a relae heores or zz araeer ase lear leel rograg sg he roose eo a heores hs s eelos a zz er ase Kh-Ther aroah o sole roose PP role eral eale llsraes he ower a eals o he roose aroah rher s les he eeloe o he oel a relae solg aroahes or zz araeer ase l-ollower leel rograg roles CKNOWEDGMENTS The work resee hs aer was sore srala esearh Col C er soer gras DP7 a DP9 [] C Sh J a G Zhag " eee Kh-Tker aroah or lear leel rograg" le Maheas a Coao ol I ress [] H Sakelerg The Theor o he Marke Eoo New ork Oor: Oor ers Press 9 [] G aalga a T resz "Herarhal ozao: roo" als o Oeraos esearh ol - 99 [] J ar Praal leel Ozao: lgorhs a laos: Klwer ae Plshers 998 [6] J a "Herarhal ozao: sasaor solo" zz Ses a Sses ol [7] J rake a J MGll "Maheaal rogras wh ozao roles he osras" Oeraos esearh ol 7-97 [8] E osh a K Shz "Herarhal eeralze sses a s ew solo a arrer eho" IEEE Trasaos o Sses Ma a Ceres ol [9] W alas a M Karwa "Mlleel lear rograg Tehal eor 78-" Sae ers o New ork alo Oeraos esearh Progra 978 [] W Caler a Towsle " lear woleel rograg role" Coers a Oeraos esearh ol [] W alas a M Karwa "Two leel lear rograg" Maagee See ol - 98 [] Che M lora a S W " ese al aroah or lear leel rogras Tehal eor CT866" Cere e eherhe sr les Trasors 99 [] S Dee " sle algorh or he lear leel rograg role" Ozao ol [] G Paaassloolos "lgorhs or sa Sakelerg gaes wh lear oss a olheral osras" resee a he s IEEE Coeree o Desos a Corol 98 [] J ar a J alk " el solo o he rograg role" Coers a Oeraos esearh ol [6] J a a MCarl " rereseao a eoo erreao o a woleel rograg role" Joral o he Oeraoal esearh Soe ol [7] P Hase Jaar a G Saar "New rahao rles or lear leel rograg" SIM Joral o Se a Sasal Cog ol [8] W alas M Karwa a J Shaw " araer oleear o aroah or woleel lear rograg Tehal eor 8" Sae ers o New ork a alo Oeraos esearh Progra 98 [9] D Whe a G aalga " eal o aroah or solg leel lear rogras" Joral o Gloal Ozao ol [] ela a D oe " leel rograg algorh or ea solo o he ework esg role wh seroal lows" Trasorao esearh ol [] P Maroe "Nework ozao wh oos orol araeers" Trasorao See ol [] T Mller T resz a To "Hers algorhs or elere re saall oee ework al loao roles" als o Oeraos esearh ol [] Zaeh "zz ses" Ior & Corol ol [] M Sakawa I Nshzak a era "Ierae zz rograg or lleel lear rograg roles wh zz araeers" zz Ses a Sses ol 9-9 [] G Zhag W M ea a J "orlao o zz lear rograg roles as or-ojee osrae roles" le Maheas a Coao ol [6] ers o Carge h://hesarsahsorg/oar/a/wor/7 [7] M Sakawa "Ierae lojee lear rograg wh zz araeers" zz ses a erae lojee ozao New ork: Ple Press 99 EEENCES [] C Sh G Zhag a J "O he eo o lear leel rograg solo" le Maheas a Coao ol IEEE Iellge Ioras lle Noeer Vol6 No