A NEW ALGORITHM FOR SOLVING FULLY FUZZY BI-LEVEL QUADRATIC PROGRAMMING PROBLEMS

Transcription

1 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 A NEW AORITHM OR SOVIN UY UZZY BI-EVE QUADRATIC PRORAMMIN PROBEMS ABSTRACT A H Aer Depre of Mhes l of See Helw Uvers Cro Egp Ths pper s oered wh ew ehod o fd he f opl solo of fll f -level o-ler qdr progrg BQP proles where ll he oeffes d deso vrles of oh oeve fos d he osrs re rglr f ers TN s A ew ehod s sed o deoposed he gve prole o -level prole wh hree rsp qdr oeve fos d oded vrles osrs I order o ofe f opl solo of he BQP proles he oep of olere eershp fo s sed o develop f - deso odel for geerg ssfor f solo for BQP proles whh he pper-level deso er UDM spefes hs/her oeve fos d desos wh possle oleres whh re desred eershp fos of f se heor The he lower-level deso er DM ses hs preferee foro for UDM d solves hs/her prole se o he UDM s resros ll he deoposed ehod s llsred erl eple KEYWORDS ll Qdr No-er Progrg Proles Trglr Ners Deoposed Mehod B-evel Progrg INTRODUCTION The -level progrg BP prole s osdered sefl opo proles whh here re depede deso ers DM s d he fesle rego of he pper-level U prole s deered pll he solo se of he lower-level prole I he ps few dedes he BP prole hs ee overed he heorel d opol pos [-] d hs ee ppled dffere felds sh s fe dge rspor ewor desg [] sppl h gee [3] prpl-ge prole [] egeerg desg [5] pre orol d eler res I ree dedes he -level deso g proles ee ver hrd o fd he vles of he oeffes ese of prese foro whe fdg hese odels So f se heor hs ee ppled o hdle prese d [6] where he oeffes oh oeve fos d he osrs re desred f ers DOI: 5/or85

2 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 T e l [7] ws frs proposed he oep of f hel progrg MP o geerl level he frewor of he f deso of Bell d Zdeh [8] Zer [9] ws frs proposed he forlo of f ler progrg Also Sw e l [-3] frs forled he f -level progrg prole d developed f progrg ehod o solve M reserhers doped hs oep for solvg P proles I s oe h he deso vrles of ll he ove wors re o f d he rsp solos re elgle o desre he f dvge of he deso g proess er evroe I ree ers he fll f ler progrg P proles whh he oeffes s well s deso vrles re desred f ers hs ee rve op for he reserhers o d Wg [] proposed he oeps of rg f ers whh s plg ver por role deso g Also here re er of reserhers who hve developed d preseed ew ehods hs feld of P sh s [5-3] or he fll f o-ler progrg proles Wl Irh r [33] preses oprehesve ehodolog for solvg d lg Qdr d o-ler progrg proles fll f evroe I shold e oed h ll hese wors re osdered he se of oesgle-level P To or owledge l ow here re few reserher sdes he pe of fll f -level ler progrg BP prole whh ll oeffes d vrle of oh oeve fos d he osrs re epressed s f er sh s [3-37] The of hs pper s o develop ew ehod o del wh he fll f -level qdr progrg proles pplg he oep of olere eershp fo o show h he ssfor solo oed f - deso odel re lws f opl solo [38 39] A erl eple s gve o llsre he proposed ehod PREIMINARIES I hs seo soe s defos of he f er d f rhe operos depedg o f ers re revewed Defo [38] The hrers fo µ A of rsp se A X ssgs vle eher or o eh eer X Ths fo e geerled o fo µ A sh h he vle ssged o he elee of he versl se X fll wh spefed rge e µ : X [] The ssged vle de he eershp grde of he elee he se A The fo µ A s lled he eershp fo d he se A { µ : X} defed µ A A for eh X s lled f se A Defo [38] A f er A s sd o e rglr f er f s eershp fo s gve

3 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 3 A µ Defo 3[38] A rglr f er s sd o e o-egve f er ff Defo [38] Arhe Operos e d g f e B A e wo rglr f ers The lger operos ewee wo rglr f ers A d B e defed : g f e B A A 3 Θ A e f g B < A 5 e A e rrr rglr f er d le g f e B e oegve rglr f er he < < e f g g f g g f e B A 3 ORMUATION O THE UY UZZY BI-EVE QUADRATIC PRORAMMIN PROBEM Cosder he followg BQP prole whh ll he oeffes d he deso vrles re f ers: UDM: where solves DM: s

4 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 { 3 } R R where s -desol f deso veor of he pper-level d s -desol f deso veor of he lower-level N N e : R R R e he pper-level oeve fos d : R R R e he lower-level oeve fos N N elees of deso veors re o-egve f rglr f ers; d re -desol f veors; elees s d re f ers; s he osr rego of prole -3 s 3 Deoposo Mehod of BQ P prole I hs seo ew ehod s proposed o fd he f opl solo of BQP prole I hs w oe frs ges he f ssfor solo s eple o UDM he UDM gves f deso vrles d f gols wh soe foro o he DM for h/her o see he f ssfor solo d o rrve he f solo whh s loses o he f ssfor solo of he UDM Ths de o he DM shold o ol ope hs/her oeve fos lso r o ssf he UDM s gols d preferee s h s possle [] e he f preers d e he rglr f ers whh s represeed he prole 3 e wre s: UDM: where solves DM: 5

5 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 5 s 6 Now sg he rhe operos see def he prole -6 e deoposed o hree rsp -level qdr progrg BQP prole s: UDM: 7 vle of lower vle of ddle vle of pper where solves DM: 8 of vle lower vle of ddle

6 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 6 of vle pper s of vle lower 9 ddle vle of pper vle of Defo 5 [] or { } gve UDM f he f deso g vrle { } he lower- level s he f opl solo of DM he s f fesle solo of prole 3 Defo 6 [] * * s f fesle solo of BQP prole -3; o oher f fesle solo ess sh h * * f s s f ; les oe s s N s sr eql he * * s he f opl solo of prole -3 3 Deso Models for BQP Prole I hs seo he solo ehod splfes BQP prole rsforg o sepre qdr deso g QDM proles pper d lower- levels s:

7 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M UDM Prole The UDM solves he followg f l-oeve deso g MODM prole s: s where R We shold frs fd dvdl es f solos d dvdl wors f solos for eh oeve of where []: d ols d oleres he e resol se for dvdl es f solos d he dfferees of he es d wors f solos respevel Ths d he e forled s he followg eershp fos of f se heor [8 9] s: [ ] > f f f µ Also we fd he f solo of he UDM prole solvg he followg Theheff prole [89] s: [ ] ] [ µ s 3 where s ssfor level d he f solo s ssed o e d for he pper-level 3 DM Prole I he se w he DM depedel solves: s where R The dvdl es f solos d dvdl wors f solos for eh oeve of re:

8 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 8 d 5 ro hs foro he eershp fos e forled sg f heore s: [ ] > f f f µ 6 Now we o he f solo of he DM prole solvg he followg Theheff prole s [8 9]: [ ] ] [ µ s 7 Whose f solo s ssed o e d for he lower-level d s ssfor level 33 BQDM Prole ll order o geere he f ssfor solo whh s lso f opl solo wh overll ssfo for oh DM s we solve he followg Theheff prole [ 8 9] s: s 8 [ ] [ ] [ ] [ ] [ ] [] d / / ] [ ]/ [ / ] [ µ µ I I where s he overll ssfo I s he ol veor wh ll elees eql o s d he se deso s ; s he f opl deso vrle of he UDM s oleres of he rge of he deso vrle rod d he eershp fo of e sed s follows:

9 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 9 f f f oherwse µ 9 The followg eershp fos of he UDM e sed s: [ ] > f f f µ where Also he DM hs he followg eershp fos for hs/her gols s: [ ] > f f f µ where B solvg prole 8 for hree deoposo proles f he UDM s ssfed wh hs solo he f ssfor solo s rehed Oherwse he/she shold provde ew eershp fos for he f orol vrle d oeves o DM l f ssfor solo s rehed A NEW AORITHM OR SOVIN BQP PROBEM The seps for he opo of BQP prole e sred s: Sep : Cosder he vrles d preers of BQP prole s rglr f er Sep : orle he BQP prole -3 Sep 3: Trsfor prole -3 o prole -6 Sep : Use he deoposo ehod o over prole -6 o hree rsp BQP proles 7-9 sg he rhe operos o fll f Sep 5: Clle he dvdl f es d wors f solos for eh oeve of prole d prole s d 5 Sep 6: orle he eershp fo of he f UDM d f DM proles s d 6 Sep 7: d he f solo of he f BQP prole solvg he Theheff prole s 3 d 7

10 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 Sep 8: orle he eershp fo of he f BQP prole s 9 d fer defed he vle of f orol deso vrles d he olere Sep 9: orle Theheff prole for BQDM prole s 8 o reh he f ssfor solo Sep : If >5 sop d he f oprose solo s oed s he overll ssfo for ll DM s oherwse go o sep8 fer hgg of he vles 5 NUMERICA EXAMPE To deosre he solo ehod for BQP prole le s osder he followg eple: 35 solves where s where re rglr f ers Asse h d Aordg o rhe operos of rglr f ers he BQP prole 7-9 e rewre s: UDM: 3 5 where solves DM: 3 s 5 7 8

11 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 { } where d re rglr f ers rs: The UDM solves prole s follows: s - d dvdl es f solo d dvdl wors f solo solvg we ge Upper level Bes f solo Wors f solo - Use o ld he eershp fos ] [ µ d solve 3 s follows: 3 3 s s s whose solo s: d Seod: I he se w he DM solves prole s follows: - d dvdl es f solos d dvdl wors f solos solvg 5 we ge

12 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 ower level Bes solo Wors solo f f B sg 6 ld he eershp fos µ [ ] d solve 7 we ge: Thrd: I order o geere he f ssfor solo - Asse he UDM' s f orol deso s rod 5667 wh he olere - B 9 d d fro eershp fos µ µ [ ] [ µ ] he DM solves he followg Theheff prole 8 s: s s s Therefore he oprose f solos re d overll ssfo for oh DM s 6 CONCUSION I hs pper fll f -level qdr progrg prole whh ll he oeffes d deso vrles re f ers s roded I order o o f opl solo of he BQP prole he oeps of olere eershp fos eh level o develop f - deso odel for geerg f ssfor solo for BQP prole The he fll f -level qdr progrg e overed o deers -level progrg prole sg he od d deoposo ehod Also ew lgorh s sed o he f deso pproh od d deoposo ehod

13 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 re roded o o he f ssfor solo for BQP prole ll llsrve erl eple hs ee gve o lrf he proposed solo ehod REERENCES [] HS Shh YJ d E Sle Approh for Ml-evel Progrg Proles Copers d Operos Reserh [] X Sh d H X Ierve B-evel Ml Oeve Deso Mg Jorl of Operol Reserh Soe [3] MA Ao-S d AH Aer Eeso of TOPSIS for Ml-Oeve rge-sle No- er Progrg Proles Appl Mh Cop [] M A Ao-S A B-level No-er Ml-Oeve Deso Mg Uder ess J Oper Res So Id OPSEARCH [5] MS Os MAAo-S AHAer d OEE A Ml-evel No-er Ml- Oeve Uder ess Appled Mhes d Copos [6] AH Aer d MA Ao-S A Ierve D ol Progrg for B-evel Ml-Oeve er rol Progrg Proles Jorl of Adves Mhes [7] MA Ao S d AH Aer TOPSIS Approh for Solvg B-evel rol MODM Proles Jorl of Adves Mhe [8] A H Aer Ipleeo of he -Cosr Mehod Spel Clss of Ml-Oeve B-evel No-er Proles PJ S Oper Res vol XIII o [9] AH Aer O he Krsh-Kh-Ter Reforlo of B-evel eoer Progrg Prole wh Iervl Coeffes s Mlple Preers Jorl of Progressve Reserh Mhes JPRM [] J Brd Prl B-evel Opo: Algorhs d Applos Klwer Ade Plshers Dordreh he Neher-ds 998 [] B Colso P Mroe d Svrd A Overvew of B-evel Opo Als of Operos Reserh vol 53 pp [] r A Cg Ple Approh for B-evel Hrdos Merl Trspor Newor Desg Operos Reserh eers vol o pp-6 3 [3] TKs d A Kovs E solo Approhes for B-evel o-sg Er J Oper Res [] M W X d JJ e A soohg Ageed grg Mehod for Solvg Sple B-evel Progrs Cop Op Appl [5] V Klshov Tl Ms JC Vlleo d SV Kv B-evel Progrg Eqlr d Coorl Proles wh Applos o Egeerg Mh Prol Eg vol 5 5 [6] A Zdeh Ses Iforo d Copo vol 8 o 3pp [7] H T T Od d K As O Mhel Progrg Jorl of Ceres [8] RE Bell d A Zdeh Deso Mg Evroe Mgee See [9] H J Zer Progrg d er Progrg wh Severl Oeve os Ses d Sses

14 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 [] M Sw d H Yo Ierve Deso Mg for Ml-Oeve No-er Progrg Proles wh Preers Ses d Ss [] M Sw I Nsh d Y Uer Ierve Progrg for l-evel er Progrg Proles wh Preers Se d Ss [] M Sw I Nsh d Y Uer Ierve Progrg for Two-evel er rol Progrg Proles wh Preers Ses d Ss [3] M Sw I Nsh d Y Uer Ierve Progrg for Two-evel er d er rol Prodo d Assge Proles: A Cse Sd Erope Jorl of Oper Res [] TS o d MJ Wg Rg ers wh Iegrl Vle Ses d Sses vol 5 o3pp [5] T Allhvrloo H of MK Ksr NA K d Aldeh Solvg ll er Progrg Prole he Rg o Appled Mhel Sees [6] H of T Allhvrloo MA Jodeh d Aldeh Solvg ll er Progrg Usg eogrph Mehod d Approe Solo Appled Mhel Modelg [7] A Kr J Kr d P Sgh Solvg ll er Progrg Proles wh Ieql Cosrs Ierol Jorl of Phsl d Mhel Sees 6-8 [8] A Kr J Kr d P Sgh Opl Solo of ll er Progrg Proles wh Ieql Cosrs Ierol Jorl of Appled Mhes d Coper Sees 6 37 [9] A Kr J Kr Ad P Sgh A New Mehod for Solvg ll er Progrg Proles Appled Mhel Modelg [3] M De Ad P K De Opl Solo of ll er rol Progrg Prole Usg rded Me Iegro Represeo Mehod A Ierol Jorl of Applos d Appled Mhes Je 5 [3] R E E Khorr d R E A New Algorh o Solve ll er Progrg Proles Usg he MOP Proles Appled Mhel Modelg [3] H S Nf S A Edlph d H D A No er Model for ll er Progrg wh ll Uresred Vrles d Preers Aledr Egeerg Jorl [33] W I r Qdr d No er Progrg Proles Solvg d Alss ll Evroe Aledr Egeerg Jorl [3] A Re Reserh Arle Solvg he ll B evel er Progrg Prole hrogh Devo Degree Mesres d Rg o Mehod Mhel Proles Egeerg Vole 6 Arle ID 7698 Pges 6 [35] A Re Reserh Arle A Novel Mehod for Solvg he ll B evel er Progrg Prole Mhel Proles Egeerg Vole 5 Arle ID 838 Pges 5 [36] A Re Y Wg d X Xe Ierve Progrg Approh for Solvg he ll B evel er Progrg Prole Kowledge Bsed Sses

15 Operos Reserh d Applos : A Ierol Jorl ORAJ Vol5 No M 8 [37] N Sfe d M Sr A New Mehod for Solvg ll er B evel Progrg Proles Ierol Jorl of Appled Operol Reserh vol o pp 39 6 [38] H J Zer Ses Deso Mg d Eper Sses Klwer Ade Boso 987 [39] D Dos d H Prde Ses d Sses: Theor d Applos Ade Press New Yor 98 5