Supian SUDRADJAT. editura universitaţii bucureşti

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1 Sup SUDRADJAT edu uveţ bucueş

2 Refeeţ şţfc: Pof. uv. d. Vle PREDA Pof. uv. d. Io edu uveăţ d bucueş Şo. Pdu 9-9 Bucueş-5663;Telefo/F: E-ml:edu_ubuc@yhoo.com Iee: Decee CIP Bbloec Nţoe Rome SUDRADJAT SUPIAN Mhemcl pogmmg model fo pofolo eleco / Sup Sudd Bucueş: Edu Uveăţ d Bucueş 7 ISBN :

3 Th wo would o hve bee poble whou he dvce d help of my people. Foemo I wh o epe my deep gude o: - Pofeo Vle PREDA - Pof. uv. d. Io VĂDUVA I would lo le o h ll he people who helped me dug he coue of my ude. Above ll - Reco of Buche Uvey Rom - Reco of Pdd Uvey Bdug Idoe - H.E. Nu Tu Doohe Ambdo of he Republc of Idoe Buche Rom - Puo Ww - Ilh Abdullh Dedco

4 To: - my de pe Hlmh d he le Oo SUPIAN - my wfe De SUDIARTI d - my chlde Sudd ISMAIL HASBULLAH Sudd MUHAMMAD IKHSAN d Sudd FITRIYANTI

5 Pefce Gude o he Almghy he oly God fo compleee of h boo eled Mhemcl Pogmmg Model Fo Pofolo Seleco o c be publh pled. The ubec of h boo cloe coeco o ome mhemcl echue pplco fcl modelg. Moe pecfclly mulce pofolo opmzo ed wh he Mowz me-vce model. Bclly Hy Mowz oduced he heoy of mode pofolo whch oge udc pogmmg poblem ppled fo evlug pofolo of e. The eulg model mely he me-vce model oe of he mo ued udc pogmmg model. The Mowz model w eeded vou deco. Recely ome uho mplemeed dymc veme model ode o udy log-em effec d mpove he pefomce. Coucg dymc fcl model co of hee bc compoe: ochc dffeel yem of euo fo decbg he model elev dom ue developme leve ceo e heefoe geeed; deco mulo fo fdg veo poo ech mome d 3 dymc opmzo model. I he clcl ppoch of pofolo eleco epeced uly heoy ppled bed o e of om eled o veo behvo d o ode elo bewee deemc d dom eve fom he e of poble choce. The pecfc chcec of om chcezg he uly fuco e o ccou he umpo h pobbly meue could be defed o dom eul. If ddo oe ume h he og of hee dom eul e o vey well ow he he pobbly heoy pove elf deue due o he lc of epemel fomo. I hee uo he deco poblem could be ddeed o ucey b ug dffee mhemcl ume. Fuhemoe he pefeece fuco decbg veo uly could be modfed wh epec o ucey degee.

6 The pofolo eleco poblem o ucey umpo could be fomed o deco poblem fuzzy evome. Fuzzy heoy w evely ued fom 96 fo olvg my poblem cludg fcl mgeme poblem. The cocep of fuzzy dom vble pope eeo of clcl dom vble. Ug fuzzy ppoch he epe owledge d ubecve opo of veo could be ee f pofolo eleco model. The m gol of h boo o eme he mehod fo olvg cl poblem volvg fuzzy eleme he dom epeme d m o be g po coucg pofolo eleco model of Mowz ype. Thee e peeed model whch volve ochc domce co o he eu of pofolo d ecey codo fo poble co pogmmg whch e olved by fomg hem o mul-obecve le pogmmg poblem. I he f chpe hee udeled he mpoce of he opc popoed h boo d he ome mpo eul fom he leue e peeed. Alo h chpe e lghly deled he ohe chpe of he boo d ome eul e hghlghed. I he ecod chpe Some cle of ochc poblem he elohp bewee effcecy e fo ome mul-obecve deem pogmmg poblem e peeed. Thee eul wll be ued le lyzg he cocep of effce oluo fo mul-obecve ochc pogmmg poblem. We hve o oe hee he eul obed Seco.4.6. d.7 whch eed he eul of Cbleo Ced Muoz Rez Scu-M d Whe. I hd chpe Poofolo opmzo wh ochc domce co codeed he couco of pofolo wh fe e whoe eu e decbed by dcee dbuo. A pofolo opmzo model wh ochc domce co o he eu peeed. Opmly d duly of hee model e uded d lo euvle opmzo model e couced ug uly fuco. I foh chpe The domce-coed pofolo. We em he eul fom Seco d 3.6 eedg he eul of Dechev Ruzczy Rohchld Sglz d Ogzcz.

7 I ffh chpe Pofolo opmzo ug fuzzy deco. I h chpe we oduce wh fuzzy le pogmmg model d ecve fuzzy le pogmmg. Alo epee geelzo of Chpe 4. Hee opmzo poblem wh ochc domce co ug fuzzy deco. The fuzzy le pogmmg poblem d fuzzy mul-obecve pogmmg poblem e hooughly eed. We em g he mpo eul of Seco d 5.8. d he eeo of ome eul belogg o Mowz Kl Zu Gmov L d Hwg d Seco 5.9 we uded bou mulobecve fcol pogmmg poblem ude fuzze. I h chpe A poblc poch fo pofolo eleco poblem hee codeed pogmmg poblem wh poble co whch wll be olved by fomg o mul-obecve pogmmg poblem. The eul fom Seco d 6.5 eed ome eul gve by Che Iuguch Rm Mlede Yhou d L. I eveh chpe Azbegeţ eeo of Mowz pofolo eleco epee oe bc me by whch Mowz heoy fo pofolo eleco c be eeded o ccou fo o-gu dbued eu. We he dcu how model copog fomo bou he pefomce of he e dffee me egme ove he holdg peod c be developed. Mo of he ogl eul peeed h boo wee peeed vey mpo cofeece d wohop. Alo we hve o oe he lge l of efeece codeed elbog h boo. I wh o cowledge he eche collegue d evewe who cobued o ele edo of h boo d fuhe o eed my ppeco fo he gudce d uggeo doed dug evo. Gude pcully due o Pof. DR. Vle PREDA Pof. DR. Io VĂDUVA Acd. Mu IOSIFESCU D. Io M. STANCU-MINASIAN DR. Mu BELDIMAN DR. Ro CIUMARA. I would lo le o h ll he people who helped me. Above ll Reco of Buche Uvey Rom Reco of Pdd Uvey Bdug

8 Idoe H.E. Nu Tu Doo Ambdo of he Republc of Idoe Buche Rom Ilh Abdullh Puo Ww Sm E. Mee Hy Iw Pw Ampew Ded M. Nud. Sup SUDRADJAT

9 CONTENTS Pefce Chpe Ioduco Chpe Some cle of ochc poblem Ioduco 7. Effce oluo cocep.3 Relo bewee he effce e of evel deemc mulobecve pogmmg poblem Some elo bewee epeced-vlue effce oluo mmum-vce effce oluo d epeced-vlue d-devo effce oluo Mulce poblem..6 Relo bewee cle of oluo fo P P d P3...7 Whe ppoch mulobecve weghg fco uly opmzo poblem fo P P d P Ioduco Tfomo d uly opmzo poblem oced o P P d P No-cove uly opmzo poblem 3 Chpe 3 Sochc domce Ioduco 4 3. Sochc domce The pofolo poblem... 44

10 3.4 Coecy wh ochc domce. 45 Chpe 4 The domce-coed pofolo poblem Ioducee Domce-coed Opmly d duly Splg Decompoo.. 64 Chpe 5 A fuzzy ppoch o pofolo opmzo Ioduco Fuzzy le pogmmg model Iecve fuzzy pogmmg Iecve fuzzy le pogmmg lgohm Pofolo poblem Ce of fuzzy echologcl coeffce d fuzzy gh-hd de umbe Ce of fuzzy echologcl coeffce Pofolo poblem wh fuzzy echologcl coeffce d fuzzy gh-hd-de umbe The modfed ubgde mehod Defuzzfco d oluo of defuzzfced poblem A modfed ubgde mehod o fuzzy le pogmmg Fuzzy decve e mehod Pofolo poblem wh fuzzy mulple obecve Mulobecve fcol pogmmg poblem ude fuzze.. 5

11 5.9. Poblem fomulo d he oluo cocep Soluo lgohm Bc bly o fo poblem FMOFP Ulzo of Kuh-Tuce codo coepodg o poblem P Chpe 6 A poblc ppoch fo pofolo eleco poblem Ioduco Me VR pofolo eleco mulobecve model wh co co Ce of dowde Ce of popool co co model A poblc me V pofolo eleco model Poblc heoy. Some pelme Tgul d pezodl fuzzy umbe Couco of effce pofolo A weghed poblc me vlue ppoch A weghed poblc me vce d covce of fuzzy umbe... 4 Chpe 7 A eeo of Mowz pofolo eleco Ioduco Gu mue dbuo A eeo of he Mowz pofolo heoy Pofolo eleco poblem GM-PoS... 5 Bblogphy.. 54 Aped Noo. 7

12 Acoym & Abbevo 74 Ide.. 75

13 CHAPTER INTRODUCTION The poblem of opmzg pofolo of fely my e clcl poblem heoecl d compuol fce. Sce he eml wo of Mowz [] geelly geed h pofolo pefomce hould be meued wo dc dmeo: he me decbg he epeced eu d he whch meue he ucey of he eu. I he me ppoch we elec fom he uvee of ll poble pofolo hoe h e effce: fo gve vlue of he me hey mmze he o euvlely fo gve vlue of hey mmze he me. Th ppoch llow oe o fomule he poblem pmec opmzo poblem d fcle he de-off ly bewee me d. I he clcl ppoch o pofolo eleco oe ofe pple he heoy of epeced uly h deved fom e of om coceg veo behvou egd he odeg elohp fo deemc d dom eve he choce e. The pecfc ue of he om h chceze he uly fuco bed o he umpo h pobbly meue c be defed o he dom oucome. Howeve f we ume h he og of hee dom eve e o well ow he he heoy of pobbly pove deue becue of lc of epemel fomo. I uch ce oe h o ppoch he deco heoy poblem ude ucey ug dffee mhemcl ool. Fuhe he pefeece fuco h decbe he uly of he veo my elf be chgg wh he degee of ucey. Moeove oe could poule h he veo h mulple pefeece fuco ech of whch coepod o pcul vew o vou fco h fluece he fuue e of he ecoomy d he cofdece wh whch held. Ude hee codo he eg leue he feld of ecoomc heoy doe o povde he veo wh uffce ool o dde he pofolo eleco poblem. The dcuo bove hghlghed poel dffcule oe would ecoue whe ddeg he pofolo eleco poblem ude ucey. I w pouled h ude ucey he veo

14 would be cofoed wh mulple uly fuco. Ech oe of hee uly fuco my be bued o pcul me vew beg held d c be bodly decbed cpug he veo level of fco f u ou o be ue. Fo ce fud mge ucug fed-come pofolo my hve oly vgue vew egdg fuue ee e ceo d hee c bodly be decbed beg bullh beh o eul. Such vew my e ou of he ubecve d/o uve opo of he deco-me o he b of fomo vlble he gve po me. Ude hee ccumce oe mgh y o chceze he ge of ccepble oluo o he pofolo eleco poblem fuzzy e ee Bellm d Zdeh [9]. I mple em fuzzy e cl of obec whch hee o cle dco bewee hoe obec h belog o he cl d hoe h do o. Fuhe oced wh ech obec membehp fuco h defe he degee of membehp of he obec he e. I h epec fuzzy e heoy povde fmewo o del wh poblem whch he ouce of mpeco he bece of hply defed ce of cl membehp he h he peece of dom vble. Th povde he po of depue fom pobbly heoy whee he ucey e fom he dom ue of he evome he h fom y vguee of hum eog. I he coe of choog opml pofolo h ge eu bove he -fee e fo ce me ceo whle he me me gueeg mmum e of eu fuzzy deco heoy povde ecelle fmewo fo ly. Th becue he ue of he poblem eue oe o eme vou me ceo d ech uch ceo wll u gve e o obecve fuco. I he fce of ucey oe wll o be ble o g umecl vlue o he pobbly of hee ceo occug. Ude h co o cle how uble weghg veco c be deemed o olve he mul-obecve opmzo poblem. Oe wy o ovecome h dffculy o ue he membehp fuco h e fuzzy deco heoy o eve uble pefeece fuco fo fdg odeg elo fo he uce eve. I fc oe c decbe he membehp fuco he fuzzy uly of he veo whch decbe he behvou of dffeece pefeece o veo owd ucey Mheu-Nco [5]. The dvge of ug he membehp fuco h doe o ely ecely o he eece of pobbly meue bu he o he eece of elve pefeece bewee he uce eve.

15 The bove gume how how he pofolo eleco poblem ude ucey c be fomed o poblem of deco-mg fuzzy evome Bellm d Zdeh [9]. To do h oe h o model he po of he veo o he b of he egh of he vew held o vou me ceo hough uble membehp fuco of fuzzy e. Fo ce fud mge ucug fed-come pofolo my hve po level o wh he pofolo ccepble ece eu ove he -fee e hould be fo hoe ceo he/he code moe lely. The cocep of fuzzy e fuzzy gol d fuzzy deco wll be oduced d fuzzy mul-ce opmzo poblem wll be fomuled. A ed by Mowz [4 The epeced uly mm ppe eoble offhd. Bu o dd he epeced eu mm. Pehp hee ome eully og eo fo decvely eecg he epeced uly mm well. The clcl Mowz model ρ V R [ ] whee ρ he vce of he eu d R ol eu. The me pofolo opmzo poblem fomuled follow: m ρ. [ ] whee R d e defed eco 3.3. Hee oegve pmee epeeg ou deble echge e of me fo. If he h o vlue d he poblem educe o he poblem of mmzg he me. If > we loo fo compome bewee he me d he. The geel ueo of coucg me model whch e hmoy wh he ochc domce elo h bee he ubec of he ly of he ece ppe Dechev d Ruzcy [44] Rohchld d Sglz [55] Ogycz d Ruzczy [7 8]. Pofolo eleco geelly bed o de-off bewee epeced eu d d eue choce fo he meue o be mplemeed. Uully he evlued by he codol ecod-ode mome.e. codol vce o volly. Th led o he deemo of he me-vce effce pofolo oduced by Mowz [4]. I c lo be bed o fey-f ceo 3

16 pobbly of flue lly popoed by Roy [49] d he mplemeed by Levy d S []. The effce pofolo oe fo whch hee doe o e ohe pofolo h h hghe me d o hghe vce d/o h le vce d o le me he eml me T. I ohe wod effce pofolo oe h Peo opml. Nowhdg populy me vce ppoch h lo bee ubec o lo of ccm. Aleve ppoche emp o cofom he fudmel umpo o ely by dmg he omly hypohe ode o ccou fo he f-lede d he ymmey of he e eu. Coeuely ohe meue of uch Vlue R VR epeced hofll me bolue devo em-vce d o o e ued. Aohe heoecl ppoch o he pofolo eleco poblem - Sochc domce Mole d Sc [] he cocep of ochc domce eled o model of -vee pefeece Fhbu [5]. I oged fom he heoy of mozo Hdly Llewood d Poy [7] fo he dcee ce w le eeded o geel dbuo Qu d Spo[46]; Hd d Ruel [66]; Hoch d Levy [68]; Rohchld d Sel [55] d ow wdely ued ecoomc d fce Levy [99]. - The uul f ode defo of ochc domce gve pl ode he pce of el dom vble. Moe mpo fom he pofolo po of vew he oo of ecod-ode domce whch lo defed pl ode. I euvle o h eme: dom vble R dome he dom vble Y f E[ u R] E[ u Y ] fo ll o-deceg cocve fuco u fo whch hee epeced vlue e fe. Thu o -vee deco me wll pefe pofolo wh eu Y ove pofolo wh eu R. - The ochc opmzo model wh ochc domce co Dechev d Ruzcy [4 44] c be ued fo -vee pofolo opmzo. We dd o he pofolo poblem he codo h he pofolo eu ochclly dome efeece eu fo emple he eu of de. We defy cocve odeceg uly fuco whch coepod o domce co. Mmzg he 4

17 epeced eu modfed by hee uly fuco guee h he opml pofolo eu wll dome he gve efeece eu. - Fuzzy e heoy ce 96 h bee wdely ued o olve my poblem cludg fcl mgeme. The cocep of fuzzy dom vble eoble eeo of he cocep of uul dom vble he my pccl pplco of dom epeme whee he mplc umpo of d peco eem o be ppope mplfco he h deue modelg of he el phycl codo. By ug fuzzy ppoche he epe owledge d he veo ubecve opo c be bee eged o pofolo eleco model. Bellm d Zdeh [9] popoed he fuzzy deco heoy. Rmwmy [4] peeed bod pofolo eleco model bed o he fuzzy deco heoy Sudd d Ped [88] peeed o pofolo opmzo ug fuzzy deco. The oo of fuzzy dom vble ee fo emple Kwe [9] Pu d Rlecu [45] Kue d Meye [89] povde vluble model h mgeble pobblc fmewo. Alo he cocep of fuzzy fomo peeed by Zdeh [6] c fomlze ehe he epemel d o he eve volvg fuzze. The cocep of fuzzy dom vble Pu d Rlecu [45] w defed ool fo eblhg elohp bewee he oucome of dom epeme d ec d Oem [8] popoed dymc pofolo mgeme model. Wd [] peeed ohe ype of pofolo eleco model bed o he fuzzy deco pcple. The model decly eled o he me-vce model whee he gol e fo epeced eu d he coepodg decbed by logc membehp fuco. - I dd pofolo model ucey eued wh dome whch cully combe boh obecvely obevble d eble dom eve wh ubecve udgme of he deco me o pobbly eme. A pu o heoy would ccep he ue of pobbly heoy o del wh obevble dom eve bu would fow upo he fomo of ubecve udgme o pobble. T e l [94] gve pecl fomulo of fuzzy deco poblem by he pobbly eve. Clo e l [6] uded he pofolo eleco model whch he e of eu of ecuy follow he pobly dbuo. Sudd Popecu d Ghc [87] uded o poblc ppoch pofolo eleco poblem. Applyg poblc dbuo my hve wo dvge: he owledge of he epe c 5

18 be ely oduced o he emo of he eu e d he educed poblem moe cble h h of he ochc pogmmg ppoch. Koe [86] poed ou h he vbly gve by wo d of ucee: dome ochc vbly d mpeco vguee. Rdome model he ochc vbly of ll poble oucome of epeme. Fuzze decbe he vguee of he gve o elzed oucome. Kwe [9] peeed ohe eplo fo he dffeece bewee dome d fuzze. He poed ou h whe we code opo poll whch umbe of people e ueoed dome occu becue o ow whch epoe my be epeced fom y gve dvdul. Oce he epoe vlble hee ll ucey bou he pece meg of he epoe. The m of h boo o eme mehod fo hdlg cl poblem volvg fuzze he eleme of he dom epeme d eve po fom whch o deve he Mowz pofolo model he peece of effce oluo cocep fo ochc mul-obecve pogmmg develop pofolo opmzo model volvg ochc domce co o he pofolo eu d ecey d uffce codo of opmly d duly we develop pofolo opmzo ug fuzzy deco cocee o fuzzy le pogmmg d flly we code mhemcl pogmmg model wh poblc co d we olve by fomg o mul-obecve le pogmmg poblem. 6

19 CHAPTER SOME CLASES OF STOCHASTIC PROBLEMS. Ioduco Sochc pogmmg del wh cl of opmzo model d lgohm whch ome of he d my be ubec o gfc ucey. Such model e ppope whe d evolve ove me d deco eed o be mde po o obevg he ee d em. Fo ce veme deco pofolo plg poblem mu be mplemeed befoe oc pefomce c be obeved. Smlly ule mu pl powe geeo befoe he demd fo eleccy elzed. Such hee ucey mplfed by echologcl ovo d me foce. A emple code he elecc powe duy. Deegulo of he elecc powe me d he pobly of peol eleccy geeo e.g. g ube e ome of he cue of ucey he duy. Ude hee ccumce py o develop model whch pl e evlued g vey of fuue ceo h epee leve oucome of d. Such model yeld pl h e bee ble o hedge g loe d cophc flue. Becue of hee popee ochc pogmmg model hve bee developed fo vey of pplco cludg elecc powe geeo Muphy [4] fcl plg Co e l [3] elecommuco ewo plg Se e l [7] d upply ch mgeme Fhe e l [5] o meo few. The wdeped pplcbly of ochc pogmmg model h ced codeble eo fom he OR/MS commuy eulg evel ece boo Kll d Wllce [77] Bge d Louveu [6] Peop [38 39] d uvey cle Bge [5] Se d Hgle [69]. Nevehele ochc pogmmg model em oe of he moe chllegg opmzo poblem. Whle ochc pogmmg gew ou of he eed o copoe ucey le d ohe opmzo model Dzg [39] Bele [8] Che d Coope [3] h cloe coeco wh ohe pdgm fo deco mg ude ucey. Fo 7

20 ce deco ly dymc pogmmg d ochc cool ll dde ml poblem d ech effecve ce dom. Deco ly uully eced o poblem whch dcee choce e evlued vew of euel obevo of dcee dom vble. Oe of he m egh of he deco lyc ppoch h llow he deco me o ue vey geel pefeece fuco compg leve coue of co. Thu boh gle d mulobecve e copoed he deco lyc fmewo. Ufouely he eed o eumee ll choce deco well oucome of dom vble lm h ppoch o deco mg poblem whch oly few egc leve e codeed. Thee lmo e ml o mehod bed o dymc pogmmg whch lo eue fe co deco d e pce. Ude Mov umpo he dymc pogmmg ppoch c lo be ued o deve opml oy polce fo fe hozo poblem of ochc cool ee lo Neuo-Dymc Pogmmg by Bee d Tl [3]. Howeve DP-bed cool em wedded o Mov Deco Poblem whee ph depedece gfc vey of emegg pplco. Sochc pogmmg povde geel fmewo o model ph depedece of he ochc poce wh opmzo model. Fuhemoe pem ucoubly my e d co ogehe wh co me-lg ec. Oe of he mpo dco h hould be hghlghed h ule dymc pogmmg ochc pogmmg epe he model fomulo cvy fom he oluo lgohm. Oe dvge of h epo h o ecey fo ochc pogmmg model o ll he me mhemcl umpo. Th led o ch cl of model fo whch vey of lgohm c be developed. O he dowde of he ledge ochc pogmmg fomulo c led o vey lge cle poblem d mehod bed o ppomo d decompoo become pmou. A whole ee of poduco pocee ecoomc yem of dffee ype d echcl obecve decbed by mhemcl model whch e mul-ce opmzo poblem Seue [77] Chog d Hme [9] d Scu-M [75]. Th uo ue uul becue feuely ecey o e o 8

21 ccou muleouly he fluece of umbe of codcoy eel fco o he yem. The mo eve developme of he heoy d he mehod of deled bblogphc decpo of whch gve Zeley [7] d Ul d Ndeu [96] e le d o le mul-ce opmzo poblem. Some clfco of he mehod of h ype oeed o he pecfc ue d mul-ce opmzo poblem wh codcoy co wee eploed e gve Sluvdze d Topchhvl [66]. Vey eeg eul geelzed o he geel domo coe fo dffee cle of oluo of mul-ce poblem e gve Sluvdze d Topchhvl [66]. Now oe of he wdely developg feld mul-ce opmzo ulve heoy; he mo mpo eul e gve Sluvdze d Topchhvl [66]. Wellow lgohm c be modfed d ew heocl eul. The obecve of h chpe o eme ome popee of dffee cle of mulce opmzo poblem oluo. Mo el-lfe egeeg opmzo poblem eue muleou opmzo of moe h oe obecve fuco. I hee ce ulely h he me vlue of deg vble wll eul he be opml vlue fo ll he obecve. Hece ome de-off bewee he obecve eeded o eue fcoy deg. A he yem effcecy dce c be dffee d muully codcoy eoble o ue he mul-obecve ppoch o opmze he ovell effcecy. Th c be doe mhemclly coecly oly whe ome opmly pcple ued. We ue Peo opmly pcple he eece of whch followg. The mul-obecve opmzo poblem oluo codeed o be Peo-opml f hee e o ohe oluo h e bee fyg ll of he obecve muleouly. Th hee c be ohe oluo h e bee fyg oe o evel obecve bu hey mu be woe h he Peo-opml oluo fyg he emg obecve. 9

22 . Effce oluo cocep Code model whch he deg/deco oced wh yem pecfed v veco. Ude ucey he yem opee evome whch hee e ucoollble pmee whch e modeled ug dom vble. Coeuely he pefomce of uch yem c lo be vewed dom vble. Accodgly ochc pogmmg model povde fmewo whch deg c be choe o opmze ome meue of he pefomce dom vble. I heefoe ul o code meue uch he wo ce pefomce epeco d ohe mome of pefomce o eve he pobbly of g pedeemed pefomce gol. Le u code he ochc mul-obecve pogmmg poblem Cblleo e l [] m D z c ~... z c ~ whee he followg oo d umpo hee compc e D. R of feble co; R heveco of deco vble of he poblem d c ~ dom veco whoe compoe e dom coou vble defed o he e E R. We ume gve he fmly F of eve h ube of E d he dbuo of pobbly P defed o F o h fo y ube of E A E A F he pobbly PA ow. Alo we ume h he dbuo of pobbly P depede of he deco vble... ; hee e obecve fuco { f } wh f R fo ll D d c ~ dom veco whoe compoe e dom couou vble; eued o fd membe of he effce veco mml e E of D wh epec o he ode elo o R whee by defo E { D : y D f y f f y f }. Le z he epeced vlue of he h obecve fuco d le σ be dd devo {... }. Le u ume h fo evey {... } d fo evey feble veco of he ochc mul-obecve pogmmg poblem he dd devo σ fe. I h eco we wll how he defo d

23 elo bewee epeced vlue dd devo effce oluo d effce oluo. Ne he followg defo by Cblleo e l [] Defo. [] Epeced-Vlue Effce Soluo. The po D epecedvlue effce oluo of he ochc mul-obecve poblem f Peo effce o he followg poblem : PE : m z... z. D Le E PE be he e of epeced-vlue effce oluo of he ochc mul-obecve poblem. Defo. [] Mmum-Vce Effce Soluo. The po D mmum-vce effce oluo fo he ochc mul-obecve poblem f Peo effce oluo fo he poblem : Pσ : m D σ... σ. Le E be he e of effce oluo of he poblem P σ. Pσ Defo.3 [] Epeced-Vlue Sdd-Devo Effce Soluo o E σ Effce Soluo. The po D epeced-vlue dd-devo effce oluo fo he ochc mul-obecve pogmmg poblem f Peo effce oluo o he poblem PEσ : m z... z σ... σ. Le D E PEσ be he e of epeced-vlue dd-devo effce oluo of he ochc mul-obecve pogmmg poblem.. Now we gve he cocep of effcecy fo wo ce of mmum pobbly. A we wll ee e ode o defe hee wo cocep he mmum- ceo cocep of mmum- effcecy d Ko ceo effcecy pobbly e ppled epecvely o ech ochc obecve. Defo.4 [] Mmum-R Effce Soluo fo he Level u...u. See Scu-M d Tg 8. The po fo level D mmum vecol oluo u...u f Peo effce oluo o he poblem:

24 PRM u : m D P ~ z c~ u... P ~ z c~ u Le E PRM u be he e of effce oluo fo he poblem PMRu. Defo.5 [] Effce Soluo wh Pobble β... β o β -Effce Soluo. The po D effce oluo wh pobble β... β f hee e u R uch h u Peo effce oluo o poblem: PP β Le E PPβ u D u u m... P{ ~ z c ~ u } β R be he e of effce oluo wh pobble β... β ochc mul-obecve pogmmg poblem.. fo he I my be oed h hee defo of effce oluo e obed by pplyg he me fomo ceo o ech oe of he obecve epely epeced vlue mmum vce ec. d by buldg fe wod he eulg deemc mulobecve poblem. I h ee ecey o he followg eul. Rem. The cocep of epeced vlue mmum vce ec. we d popely effce oluo c be defed ul wy. Rem. The cocep of mmum- effcecy d β -effcecy eue eg po veco of po level u o pobbly veco β. Th mple h boh ce he effce e obed deped o he pedeemed veco uch wy h geel dffee level d pob bly veco gve e o dffee effce e ' u u ' β β E E PRM PP u E β E PP ' u ' β. PRM Rem.3 The cocep of epeced dd-devo effce oluo eeo o mulobecve ce of he cocep of he me-vce effce oluo h Mowz [4] defe fo he ochc gle obecve poblem of pofolo eleco. I h wy we hve he wo cl mome coepodg o ech ochc obecve he me meug u. Sce he ue oo fuco

25 cly ceg he e of effce oluo doe o vy poblem f we ubue dd devo fo vce Whe [9]. Rem.46 The effcecy pobbly ceo geelzo of he oe peeed by Gocoeche He d Duce [63] who defe he me cocep g he me pobbly β fo ll he ochc obecve d wh he pobblc euly co g he fom P ~ z c ~ u } β. { Th oo w oduced by Scu-M [79] codeg he Ko poblem he ce of mulple ce..3 Relo bewee he effce e of evel of deemc mulobecve pogmmg poblem We pee ome elo bewee he effce e of evel poblem of deemc mul-obecve pogmmg poblem. Thee eul wll be ued le fo ly of cocep of effce oluo fo mul-obecve ochc poblem. Codeed f d g be vecol fuco defed o he me e H R wh f : H R R d g : H R R d le γ α be oull veco wh el compoe h α γ R d α γ mulobecve poblem: m f... f γ g... γ g PD D PD m f... f D PD 3 m γ g... γ g D. Le u code he followg wh γ R γ. Le E E E3 be he e of wely effce effce d pope effce po of poblem PD epecvely. The followg heoem ele hee poblem PD PD d PD 3 poblem o ech ohe. 3

26 Theoem. We ume h g > fo evey D. The: E E3 E w E E3 E 3 Poof: w w w E E3 E E E3 Le u how h E by educo d budum. We ume h E. The hee e D uch h f f d γ g γ g fo evey {... } hee beg {... } fo whch he euly c f < f o γ g < γ g. Theefoe E o E3 ce γ g γ g mple γ γ coy o E E3. g g w E E3 E Le E E3. Le u ee h w E by educo de budum. We ume h w E. The hee e veco D h wely dome d o vefe f < f d γ g < γ g fo evey {... }. Thu E d ce γ g < γ g mple γ < γ E3 coy o E E3. g g 3 Le w w w E E3 E w w E E3. Le u ee h w E by educo de budum. We ume h w E. The hee e veco D h wely dome he veco d heefoe vefe h f < f d γ g < γ g fo evey {... }. Thu w E d ce γ g < γ g mple γ < g g w E3 coy o w w E E3. 4

27 Thu c be deduced fom 3 Now we code he followg poblem f α g... f α g m D.6 whee α α... R R. α : G Le E d E deoe he effce oluo e d he popely effce 4 α 4 α oluo e epecvely fo poblem.6. We wll ow pee ome elo bewee hee e d he e of effce oluo d popely effce oluo fo poblem PD. Theoem. [] Fo γ γ... γ : R R α 5 α... α : R R α γ d g α g γ he followg elo hold : E α E. 4 Poof: Le E 4 α. We ume h E. I h ce hee oluo dome he oluo h wh h f f d γ g γ g fo evey {... } d hee e le oe {... } fo whch he euly c h f < f o γ g < γ g Fom h po owd ce f f γ g γ g mple α g α g he followg eule e vefed: f f α g f g fo evey {... }.7 α g f α g fo evey {... }..8 Fom.7 d.8 we ob f I pcul fo f f < f f α g f α g fo evey {... }. we hve he eul bellow: α g < f α g

28 d he followg euly obed fom.8: f α g < f α g ; b f α g < α g f α g < f α g d ce f f we ob f α g < f α g. Theefoe fo evey {... } f α g f α g d hee le ubcp {... } fo whch f α g < f α g whch mple h he oluo dome he oluo ; heefoe we ech codco wh he hypohe of beg he effce oluo o poblem.6. Ne we pove h ome codo h elohp hold fo he e of popely effce oluo. Fo h pupoe we defe poblem P d P ξ f γg obed by pplyg he weghg mehod o poblem.3-.6 epecvely follow: f γ g : m f g D P γ Pα ξ : m ξ f α g D. We ue he eul vlble he leue bou he elohp bewee he opml oluo o he weghg poblem d he effce oluo o he mulobecve poblem. Some eul ee Chog d Hme [9] ppled o poblem.3 d oced weghed poblem P e follow : f γg α 6

29 If f d... γ g g γ e cove fuco D cove d popely effce oluo fo he mul-obecve poblem.3 hee e ome wegh veco wh cly pove compoe uch h fo weghed poblem P. f γg he opml oluo b Fo ech veco of wegh wh cly pove compoe he opml oluo o he weghed poblem P popely effce fo he mul-obecve f γg poblem P. Popoo. If f d γ g... γ g e cove fuco D cove e d hee e g α g fo evey α α... α : R R γ {... } he G G E4 E α. Poof: If f d γ g... γ g e cove fuco d f D cove e he he e of popely effce oluo o poblem PD d.6 e obed fom he oced weghed poblem fo cly pove wegh veco. We wll pove h y oluo o he opmzo poblem P ξ wh ξ > oluo o poblem P fo ome veco >. f γg α G Le E. The gve he eblhed hypohee hee e veco ξ > fo 4 α whch he oluo fo poblem P ξ. Le u ume h fo evey {... } α γ. The we e ξ ξ α / γ > Sce ξ > we ob h opml oluo o poblem P α f g. Fo ome {... } f α γ he he poof would be he me ce poblem.3 he fuco g o volved d ce poblem.6 he fuco h obecve would be f. I geel he vee cluo doe o hold how by he followg emple. 7

30 Emple.. Le u code he followg poblem: /. m y y 9y y wh f y g y y u. The e of effce po fo h poblem { y R / 4y y > } d epeeed Fg... We oule he oluo of he poblem /. m y 9y y αy wh α >. Fo ech fed α > he opml oluo of he eulg poblem oe of popey effce oluo o he ogl beceo poblem. y ε D Fgu. 3 Popoo. If f d γ g... γ g e cove fuco he U G G E E α α Ω 4 wh Ω { α α... α : R R g α g γ }. Poof: A he pevou ce he poof of he popoo ced ou by demog h y oluo o he poblem P f γg oluo o he poblem P α ξ fo ome veco α R wh g α g γ {... } d fo ome ξ >. 8

31 Code G E. Sce f d... γ g g γ e cove fuco hee e veco > uch h oluo o poblem P. Becue ξ > we pu ξ γ α ξ ce ξ > heefoe we ob h lo oluo o he poblem P α ξ. Fom Popoo. d Popoo. f f d fug g... γ g γ e cove fuco d f g α g γ α. γ > fo evey {... } he e of popely effce oluo o poblem.3 d.6 vefy he followg popee:. Evey popely effce oluo o poblem.6 popely effce fo poblem.3; b. Seg γ R wh oull compoe he e of popely effce oluo o poblem.3 ube of he uo α of he e of popely effce oluo fo poblem.6..4 Some elo bewee epeced-vlue effce oluo mmumvce effce oluo d epeced-vlue dd devo effce oluo Code poblem. d e effce oluo epeced vlue E PE mmum vce E d epeced vlue dd devo E PEσ PEσ oced wh he poblem. Le E w PE E E be he e of wely effce oluo oced wh w PEσ w PEσ he poblem Defo.-.3 epecvely. If we code f z g σ Ad f we chooe γ gve h fo {... } vefed h σ : R R he he elo bewee hee effce e e deduced decly fom Theoem 9

32 .5 Mul-ce poblem Code he followg model of mul-ce opmzo poblem: m F... F.8 D.9 whee D oempy e of ll feble oluo D R m ;... R F F : D. Sed befly mul-ce opmzo poblem co he choce of pcul oluo D fo whch ll of he uly fuco F muleouly ppoch bgge vlue o le do o decee. Le u ecll ome cocep of mul-ce opmzo poblem oluo; Zeley [7] d Ul d Ndeu [96] Sluvdze d Topchhvl [66]. Defo.6 The oluo P D clled Peo-opml o effce fo he poblem.8-.9 f d oly f fo evey D he yem of eule P F < F whee le oe euly c coe. Defo.7 The oluo w D clled wely effce o Sle-opml fo he poblem.8-.9 f d oly f fo evey D he yem of c eule w F < F coe. Defo.8 The oluo G D clled pope effce o Geoffo-opml fo he poblem.8-.9 f d oly f Peo-opml oluo fo he poblem.8-.9 d hee e pove umbe θ > uch h fo ech p we hve G G [ F F ]/[ F F ] θ G G fo ome uch h F > F whee D d F < F coe. w Le E E G E deoed he e of wely-effce effce d pope effce oluo epecvely fo he poblem I obvou h G E E E w.

33 Ne we wll uded ome elo bewee he effce e of evel poblem of deemc mul-obecve pogmmg. Le f d g be vecol fuco defed o he me e H R wh f H R : d g H R :. Le u code he followg mul-obecve poblem: P f... f u g... u g m D P m f... f D P 3 m u g... u g D... wh D H u u u u d el umbe. : R R... >.6 Relo bewee cle of oluo fo P P d P 3 We pee ome elo bewee he effce e of bove codeed deemc mul-obecve pogmmg poblem. Thee eul wll be ued le fo ly of cocep of effce oluo fo mul-obecve ochc poblem. Thee eul eed Seco.4. Fo 3 le E E E be he e of wely effce effce d pope w G effce po of poblem P epecvely. The followg heoem ele hee poblem P P d P 3 poblem o ech ohe. Theoem.3 We ume h g > fo evey D d fo d we hve u u mple h u < u. The: < E E3 E > w E E3 E w w w E E3 E Poof: E E3

34 Le u how h E by educo d budum. We ume h E. The hee e D uch h f f d u g u g fo evey {... } hee beg {... } fo whch he euly c f < f o u g < u g. Theefoe E o E3 ce u g u g mple u g u g coy o E E3. w E E3 E Le E E3. Le u ee h w E by educo de budum. We ume h w E. The hee e veco D h wely dome d o vefe f < f d u g < u g fo evey {... }. Thu E d ce u g < u g mple u g < u g coy o E E3. E 3 Le w w w E E3 E w w E E3. Le u ee h w E by educo de budum. We ume h. The hee e veco D h wely dome he veco d w E heefoe vefe h f < f d u g < u g fo evey {... }. Thu w E d ce u g < u g mple u g < u g w E3 Thu c be deduced fom coy o w w E E3. Rem.5 Alo we ee h c be deduced fom. I obvou h w w w E E3 E lo vefed. Fuhemoe w E E d w E3 E3

35 he w w E E3 E E3. Rem.6 We oe h > fy codo fom Theoem.3. p u : R R wh u o u p > d Now we code he followg poblem m f u~... ~ g f u g D whee u ~ u ~... ~ R R. u : Le E ~ d ~ 4 u E G 4 u.3 deoe he effce oluo e d he popely effce oluo e epecvely fo poblem.3. We wll ow pee ome elo bewee hee e d he e of effce oluo d popely effce oluo fo poblem P. Theoem.4 Fo u u... : R R d u ~ u ~... ~ R R uch h u u : u < u mple u~ ~ < u he followg elo hold : E4 u ~ E. Poof: Le E u~ 4. We ume h E. I h ce hee oluo h dome he oluo h f f d u g u g fo evey {... } d hee e le oe {... } fo whch he euly c h f < f o u g < u g Fom h po owd ce f f u g u g mple u ~ g u ~ g he followg eule e vefed: ~ ~ f u g f u g fo evey {... }.4 ~ f u g f u~ g fo evey {... }..5 Fom.4 d.4 we ob 3

36 f ~ u g f u~ g fo evey {... }. I pcul fo we hve he eul bellow: c f f < f f ~ ~ u g < f u g d he followg euly obed fom.5: ~ f u g < f u~ g ; d f u ~ ~ g < u g f ~ u g < f u~ g d ce f f we ob f ~ u g < f u~ g. Theefoe fo evey {... } f ~ u g f u~ g d hee le ubcp {... } fo whch f ~ u g < f u~ g whch mple h he oluo dome he oluo ; heefoe we ech codco wh he hypohe of beg he effce oluo o poblem.3. Ne we pove h ome codo h elohp hold fo he e of popely effce oluo. Fo h pupoe we defe poblem P d P ~ ξ obed by pplyg he weghg mehod o poblem. d.3 epecvely follow: fug D f u g P : m u D P ~ ~ ξ : m ξ f u g. We ue he eul vlble he leue bou he elohp bewee he opml oluo o he weghg poblem d he effce oluo o he mul- 4 fug u

37 obecve poblem. Some eul ee Chog d Hme [9] ppled o poblem. d oced weghed poblem P e follow : c If f d u g... u g e cove fuco D cove d fug 5 popely effce oluo fo he mul-obecve poblem. hee e ome wegh veco wh cly pove compoe uch h fo weghed poblem P. fug he opml oluo d Fo ech veco of wegh wh cly pove compoe he opml oluo o he weghed poblem P popely effce fo he mul-obecve fug poblem P. Popoo.3 If f d u g... u g e cove fuco D cove e d hee e u~ {... } he ~ u ~... u : R R wh ~. G G E4 u E u. ~ u > fo evey d Poof: If f d u g... u g e cove fuco d f D cove e he he e of popely effce oluo o poblem P d.3 e obed fom he oced weghed poblem fo cly pove wegh veco. We wll pove h y oluo o he opmzo poblem P ~ ξ wh ξ > oluo o poblem P fo ome veco >. fug u G Le E ~ 4 u. The gve he eblhed hypohee hee e veco ξ > fo whch he oluo fo poblem P ~ ξ. Le u ume h fo evey {... } u he we e ξ u ~ d uch h ξ u ~ u fo {... }. Sce ξ > we ob h opml oluo o poblem P u fug. Fo ome {... } f u he he poof would be he me ce poblem. he fuco g o volved d ce poblem.3 he h obecve would be f. Popoo.4 If f d u g... u g e cove fuco he

38 E U E u~ G G 4 u~ Ω wh ~ ~ ~ Ω { u u... : ~ u R R u. u > }. Poof: Ug he Popoo.3 we ee h he poof of h popoo ced ou by demog h y oluo o he poblem P oluo o he poblem P ~ ξ wh u~ ξ >. u ~ u ~... u : R R. fug u ~ u > d fo ome Code G E. Sce f d u g... u g e cove fuco hee e veco > uch h oluo o poblem P. Becue ξ > we fug pu ξ. u u ~ ξ fo... } R {. Theefoe we ob h lo oluo o he poblem P ~ ξ. Fom Popoo.3 d Popoo.4 f f d u g... u g e cove u fuco d f u~ u ~ u~ u~... : R R. u > he he e of popely effce oluo o poblem. d.3 vefy he followg popee:. Evey popely effce oluo o poblem.3 popely effce fo poblem.; b. Seg u : R R \ {} he e of popely effce oluo o poblem.3 ube of he uo u ~ of he e of popely effce oluo fo poblem.6. By combg boh eul he followg coolly: Coolly.4 If f d u g... u g e cove fuco he E U ~. G G E4 u u~ Ω Coolly.5 If f d u g... u g e cove fuco d u. ~ u > fo evey d {... } he 6 ~. G G E4 u E

39 Coolly.6 Cblleo e l [] If α R \ {} α α... α d u α he poblem.3 o educed fom poblem 4 Cblleo e l [].. Coolly.7 If γ R \ {} γ γ... γ u γ he. P -P 3 o educed fom -3 Cblleo e l []. I h ce we ob ue coeuece fom Theoem 3. d Theoem 3. Cblleo e l []. Fuhe f we hve vew Coolly.6 we ee h we ge coeuely Popoo 3. Popoo 3. d Coolly 3. Cblleo e l []. Fo emple ug Chog d Hme [9] we c ppled o poblem P d oced weghed wh poblem P. Thu we hve: fug Popoo.5 If f d u g... u g e cove fuco D cove e d E he hee e oegve el umbe uch h opml oluo fo poblem P. fug Popoo.6 Le oegve el umbe d of opml oluo fo poblem P fug he E..7 Whe ppoch mulobecve weghg fco uly opmzo poblem fo P P d P 3 We code he geeo of effce ochc mul-obecve oluo ug weghg fco h-powe ppoch fo ome o-cove uly fuco opmzo poblem. We wll oduce ou cl of uly fuco d gve ome ow dd eul d we wll how ee cle of o-cove uly opmzo poblem gvg cocvy-peevg fomo fo he h powe of cocve fuco fo ge oluo fo uly opmzo poblem..7. Ioduco Code he followg cl of mul-obecve poblem: 7

40 hee compc e D R of feble co; hee e obecve fuco { F } wh F R fo ll D ; eued o fd membe of he effce veco mml e E of D whee by defo E { D : y D F y F F y F }.6 whch F F... F. The mo commo of hee uly fom he povely weghed fom gve by he cl of uly fuco wh ν ; δ δ F.e. ; δ F F ν δ F.7 δ Δ δ R : δ o { δ Δ : > } δ Δ δ. We defe M δ F g mν ; δ F. D The ug well ow eul mul-obecve pogmmg ee fo emple Kl [8] we hve he followg popoo. Popoo.7 U M δ F E ; δ Δ b If F cove veco fuco d D cove he Kl[8] U E M δ F. δ Δ Accodg o Kl [8] hee e wo cel ue g fom h cl of uly fuco mely. h cl of uly fuco my o be cpble of geeg eough po E 8

41 b. f F o cove veco fuco o D o cove hee my be o cue mehod fo fdg M δ F..7. Tfomo d uly opmzo poblem oced o P P d P 3 We code he fomo gve by θ θ... R θ : R Relve o P P d P 3 we code he followg fomed poblem.8.9 d. epecvely. PS PS m D υ υ θ f θ u g.8 υ m θ f.9 D PS 3 m υ θ u g >. D wh υ > u u... u : R R. We ow defe fo poblem.8 uly opmzo poblem AP θ follow fo Z \ {} Δ whee Δ R R /. Fd M θ g mψ he e of opml oluo fo.8 θ D whee ψ ; θ : D R gve by Fo ψ ; θ θ f θ u g. elo. eplced by ψ ; θ m m θ f m θ u g. 9

42 By ug he le of Whe [7 9] Bowm [9] Kl [8] elve o he poblem.8 we ob he followg eul: Theoem.5 If he M θ E f Δ ; If ce ufom domce codo hold Bowm [9] he M E fo θ Δ ; 3 If { θ f } e ll cove o D d D cove he U M θ Δ E ; E ; 4 U M θ Δ 5 If D fe he hee e Z \{} uch h E U M θ Δ. Fo poblem.9 uly opmzo poblem AP follow fo Z whee \ {} Δ Δ R /. θ Fd M θ g mψ whee ψ ; θ : D R gve by θ D Fo ψ ; θ θ f..3 euo.3 eplced by ψ ; θ m θ f.5 By ug he le of Whe [7 9] Bowm [9] Kl [8] elve o he poblem.9 we ob he followg eul: Theoem.6 If he M θ E f Δ ; 3

43 If ce ufom domce codo hold Bowm [9] he M θ E f Δ ; 3 If { θ f } e ll cove o D d D cove he E M θ ; U Δ 4 U Δ E M θ ; 5 If D fe he hee e Z \{} uch h E U Δ M θ. Fo poblem. uly opmzo poblem AP follow fo Z \ {} Δ 3 θ Fd M 3 θ g mψ 3 θ whee ψ 3 ; θ : D R gve by D ψ ; θ θ u g.5 3 Fo euo.5 eplced by ψ θ m θ u g.9 3 Flly fo poblem. we ob he followg eul: Theoem.7 If he M 3 θ E3 f Δ If ce ufom domce codo hold Bowm [9] he M θ E f 3 3 Δ 3 If { θ f } e ll cove o D d D cove he E 3 M 3 θ 4 E 3 U Δ U Δ M 3 θ 3

44 5 If D fe he hee e Z \{} uch h E 3 U Δ M 3 θ. Rem.7 Some poof of hee eul e gve Sudd d Ped [89]..7.3 No-cove uly opmzo poblem The poblem o fdg po D whch e E o cloe o po E. We lo wh o ue covey d cocvy popee. Fo hee o be megful we eed ppope cove e wh whch o embed ou ly. R oo lge becue we hve pule meely h F R fo ll D D cove ll h we eed e h F defed o D. I he followg we ue le gve by Whe [49]. d o fo ll R. Whe D wh F R fo ll Ce of cocve { f }{ u g } We ume h he { f } e ll cocve o { θ d D d loo he choce of } d oced lgohm fo uly opmzo poblem AP AP d AP. θ 3 θ θ Geelly eve f θ cocve o D o ecely ue h f θ f cocve o D. We eed o chooe f } { θ o h le fo ome { θ f } e ll cocve o ce whch wll do wh eued mely θ ϕ log ϕ ove he ge ϕ whee } R D. The followg fom of } { { θ wll povde Lemm. If { f } e cocve o { θ e ll cocve o D he f } D fo ll Z \ {} d m uch h 3

45 m m[log f m m[log u D D g ] ].7 povded h m[ f ].8 D Poof: Fo z R z d θ z dz log Thu fo y gve z θ d z dz f log z. z [ log z ]. z Replcg z by f d u g we ee h θ cocve o h.6 d.7 hold. D povded b Ce of cove { f } { u } d fe D g If { θ } e ll cove o R d { f } d { u } e ll cove o g D he { θ f } e ll cove o θ ϕ ϕ ϕ R. D. Th pple fo emple whe I h ce he uly opmzo poblem AP become oe of θ mmzg cove fuco ove fe e D lo fo AP d AP. 3 θ Th lo hold he ce of. θ Le u ow ume h ψ ψ ; ψ ; o ψ ; epecvely d h ψ cove o θ θ 3 θ D. The code he followg lgohm gve he ulfe { θ} { θ} o { θ} epecvely fo ee of 33

46 epoo : ψ he ubdffeel of ψ d ψ φ Rocfell [5]; S y fe o-empy ube of ψ obed by ome pecfed mehod; he f compoe of. Algohm. Selec ε R \ {}. Se D D. Se. v v v Aume h we hve deved Fd S d g mψ Se D D { D D. : y ε y S v If D φ e A g m ψ [... ] d op. v If D φ go o ep v. We hve he followg heoem. Theoem.8 Algohm. eme fe umbe of eo. If ψ he mml vlue of ψ o D he ψ ψ A ψ ε whee ψ ψ ψ }. { ψ 3 Poof: Le u uppoe h he lgohm o fe. Becue D fe hee e e } Z \ {} uch h { whee ome membe of D d 34

47 D he Thu y y ε y ψ. Th poble. Le D φ d ε y ψ. Y D \ D he Le D U Y. Y. The D d y ε fo ome y ψ. Becue ψ ψ ψ } e cove Hece { ψ 3 ψ ψ y ε. ψ ψ A ψ ε. Rem.8 Sep v elly oly eue fdg feble oluo D. The ue of obecve fuco meely o fcle h. Rem.9 Sep v ubpoblem of mmzg le fuco ove hoe oluo defed by polyope y e lo D. Whe ll fuco e dffeeble ψ. I he geel ce ψ Z geeed by he ubgde co whch d hece ubdffeel of he fuco { θ } Rocfell [5]. f Rem. If D he vee e of polyope vee of D gleo gde veco fo D my be foud em of he D he y oluo fom D lo D Z. Thu vee ech ublgohm uch h of Muy [4] c 35

48 be developed. Ohe pocedue fo eumeg he vece of polyope my be dpble fo h poblem Mhe d Rub [6]. Rem. If D egl e of polyope he ubpoblem ep v ege le pogmmg poblem fo whch ge of lgohm e. Wheve mehod ue o olve he uly poblem he covey of ψ helpful povdg lowe boud becue f { } y e of oluo geeed he ψ m m D y ψ [ ψ y Rem. If D ube of polyope ]. D we oe h ψ m ψ.. D The deemo of gh-hd de of. cove pogmmg poblem. Lowe boud my be ueful deemg how cloe he be oluo o de o opml oluo o h compuo my be emed ely f whed. c Ce of mue of cocve d cove { f }{ u g } Fom globl opmzo po of vew h he hde poblem of ll. Code M M ' M d M e o-empy ube of {... } uch h ' ' ' ' M M M M {... } M φ d M φ ; { θ f } be M 36 ' M cocve o D fo M;{ θ f } be cove o D fo M lo h { θ u g } be cocve o fo M ; ψ ; θ M θ f D fo d { θ u g } be cove o ' M ' M θ u g ψ ; θ θ f θ u g M ' M The doppg he ulfe { θ} fo ee of epoo we hve ψ ψ ψ whee ψ d ψ e epecvely cocve d cove o D. D

49 The followg lgohm eeo of lgohm of Whe [6]. S y fe ube of he ub-dffeel ψ of ψ obed by ome pecfed mehod. Algohm. Selec ε R \ {}. Se D D. Se. v Aume h we hve deved v v Fd S d g mψ Se D D D. { D : y ε y S v If D φ e A g d op m ψ [... ] v If D φ go o ep v. We hve he followg heoem whee ψ he ubdffeel of ψ po. Theoem.9 37 If U D ψ compc he Algohm. eme fe umbe of eo. If ψ he mml vlue of ψ o D he ψ ψ B ψ ε Poof: Le u uppoe h he lgohm o fe. Becue of he umpo bou he ub-dffeel d he compce of D hee e y R d e } Z \ {} uch h {

50 Thu y y ε / ε / Ieule whch o poble. Le D φ d Y D \ D he U D Y. Le Y. The D d y ε fo ome ψ. Becue ψ cove ψ ψ y ε. Alo ψ ψ. Thu ψ ψ B ψ ε. Rem.3 Sep v ubpoblem volvg he mmzo of cocve fuco ove D. Whe D polyope o D d lgohm e fo olvg uch poblem e.g. Glove d Klgm [6]; Fl d Hoffm [49]; Co [3]. We oe h h ce ep v dd cug co whch eclude he cue oluo d fo whch he dul mple mehod ueful Hdley [65]. Rem.4 If D he egl e of polyope D he he ubpoblem e ege pogmmg fom fo whch lgohm e. Rem.5 If D he vece of polyope D he ecep pecl ce e.g. whe vece e egl he gme poblem ome ew lgohm eued. 38

51 CHAPTER 3 STOCHASTIC DOMINANCE 3. Ioduco. The elo of ochc domce fudmel cocep of deco heoy d ecoomc Dechev d Ruzczy [4 4] Hoch d Levy [68] Qu d Spo [46] d Rohchld [55]. A dom vble dome ohe dom vble Y he ecod ode whch we we f Y f E[ u ] E[ u Y ] fo evey cocve odeceg fuco u fo whch hee epeced vlue e fe. A bc model of ochc opmzo c be fomuled follow: m E[ ϕ z ω]. 3. z Z I h fomulo ω deoe elemey eve pobbly pce Ω F P z deco veco ppope pce Z d ϕ : Z Ω R. The e Z Z defed ehe eplcly o v ome co h my volve he elemey eve ω d mu hold wh ome pecbed pobbly. The f ochc opmzo model wh epeced vlue wee oduced by Lehm [96] d Hoch d Levy [68]. Mhemcl heoy of epeco model volvg wo-ge d mulge deco h bee developed by We [4 5] d Bge [6]. Model volvg co o pobbly wee oduced by Che d Coope [3] Peop [39] Dechev d Ruzczy [4] dcue del he heoy d umecl mehod fo le model wh oe pobblc co o fely my eule. Aohe wy o loo poblem 3. o code he e C of dom vble uch h fo ome z Z oe h ω ϕ z ω.. The we c we he model m E[ ]. C 39

52 I pcce howeve lmo mpoble o elc he uly fuco of deco me eplcly. Addol dffcule e whe hee goup of deco me wh dffee uly fuco who hve o come o coeu. I ome pplco efeece oucome Y L Ω F P vlble. I my hve he fom Y ω ϕ z ω ω Ω fo ome polcy z. Ou eo o hve he ew oucome pefeble ove Y. Theefoe we code he followg opmzo poblem: m f 3. ubec o f Y 3.3 C. 3.4 Hee Y dom vble L Ω F P he e C L Ω F P cove d cloed d f : C R cocve couou fucol. Co 3.3 guee h fo y deco me whoe uly fuco u cocve d odeceg he oluo of he poblem wll fy he elo E [ u ] > E[ u Y ]. Aohe cl of model h ecely ced much eo e me- model. I ou oo hey e he fom m{e [ ] ρ }. C I h poblem > d ρ fucol whch deped o he ee dbuo of d g o cl meue of vbly. Fo emple he epeced hofll below he me ρ E E[ ] [ ] my be ued he fucol. Hee m. Me- model e lo cloely eled o ochc domce elo. If we ue ppope meue ρ d he pmee wh ce ge he he opml oucome ˆ o ochclly domed by y ohe feble oucome Dechev d Ruzczy [4] Ogycz d Ruzczy [7 8 9]. 4

53 Ohe ochc opmzo model volvg geel fucol wee codeed by Dechev d Ruzczy [4] Rocfell d Uyev [5]. Model coepod o ew ppoch ochc opmzo poblem. 3. Sochc domce I he ochc domce ppoch dom eu e comped by po-we compo of ome pefomce fuco couced fom he dbuo fuco. Fo el dom vble V f pefomce fuco defed he ghcouou cumulve dbuo fuco of V : F V η P { V η} fo η R. A dom eu V d ochclly dome ohe dom eu S o he f ode Dechev d Ruzczy [4] Lehm [96] d Qu d Spo [46] deoed V f FSD S f F V; η F S; η fo ll η R. Defe he fuco F V ; η F V; η F V; α dα fo η R 3.5 egl of odeceg fuco cove fuco of η d defe he we elo of he ecod-ode ochc domce SSD. Th he dom eu V ochclly dome S o he ecod ode deoed V f F V; η F S; fo ll η R. η SSD S f The coepodg c domce elo fo uul wy: V f S f d oly f V f S S f/ V. f FSD d f SSD e defed he Fuhemoe fo V L m Ω F P we c defe ecuvely he fuco V; η F V α F η dα fo η R 3 m. 3.6 Fuhemoe fo V L m Ω F P we c defe ecuvely he fuco 4

54 V ; η F V α F η dα fo η R 3 m. 3.6 Fgue 3.3 F ode domce η F ; η F ; α dα E η fo η R Fgue 3. Scod-ode domce They e lo cove d odeceg fuco of he ecod gume. Defo 3.. A dom vble L - Ω F P dome he h ode ohe dom vble Y L - Ω F P f F ; η F Y; η fo ll η R. 3.7 We hll deoe elo 3.7 f Y 3.8 d he e of fyg h elo A Y { L Ω F P : f Y}

55 By chgg he ode of ego we c epe he fuco F V ; he epeced hofll Rocfell d Uyev [5]: fo ech ge vlue η we hve [ η ] F V; η E 3. V whee η V m η V. The fuco F V ; couou cove oegve d odeceg. I well defed fo ll dom vble V wh fe epeced vlue. 3.3 The pofolo poblem Le R... R be dom eu of e. We ume h [ R ] < E fo ll. Ou m o ve ou cpl hee e ode o ob ome deble chcec of he ol eu o he veme. Deog by he l cpl veed e... he fco of... epecvely we c ely deve he fomul fo he ol eu: R R... R. 3. Clely he e of poble e lloco c be defed follow: { :... } R whee... }. { I ome pplco oe my oduce he pobly of ho poo.e. llow ome o become egve. Ohe eco my lm he epoue o pcul e o he goup by mpog uppe boud o he o o he pl um. Oe c lo lm he bolue dffeece bewee he d ome efeece veme whch my epee he eg pofolo ec. Ou ly wll o deped o he deled wy h e defed; we hll oly ue he fc h cove polyhedo. All modfco dcued bove defe ome cove polyhedl feble e d e heefoe coveed by ou ppoch. 43

56 The m dffculy fomulg megful pofolo opmzo poblem he defo of he pefeece ucue mog feble pofolo. If we ue oly he me eu E R [ ] he he eulg opmzo poblem h vl d megle oluo: ve eveyhg e h hve he mmum epeced eu. Fo hee eo he pcce of pofolo opmzo eo uully o wo ppoche. I he f ppoch we oce wh pofolo ome meue ρ epeeg he vbly of he eu R. I he clcl Mowz model ρ he vce of he eu [ R ] ρ V bu my ohe meue e poble hee well. The me pofolo opmzo poblem fomuled follow: m[ ρ ] 3. Hee oegve pmee epeeg ou deble echge e of me fo. If he h o vlue d he poblem educe o he poblem of mmzg he me. If > we loo fo compome bewee he me d he. The geel ueo of coucg me model whch e hmoy wh he ochc domce elo h bee he ubec of he ly of he ece ppe Dechev d Ruzczy [4 4] Rohchld d Sglz [55] Ogycz d Ruzczy [7 8]. We hve defed hee evel pml meue mo obly cel emdevo d dul meue bed o he Loez cuve whch e coe wh he ochc domce elo. The ecod ppoch o elec ce uly fuco u : R R d o fomule he followg opmzo poblem m E u R 3.3 [ ] I uully eued h he fuco u cocve d odeceg hu epeeg pefeece of -vee deco me. The chllege hee o elec 44

57 he ppope uly fuco h epee well ou pefeece d whoe pplco led o o-vl d megful oluo of Coecy wh ochc domce The cocep of ochc domce eled o omc model of -vee pefeece Fhbu [5]. I oged fom he heoy of mozo Hdy Llewood d Poly [7] Mhll d Ol [9] fo he dcee ce d w le eeded o geel dbuo Qu d Spo [46] Hd d Ruell [66] Hoch d Levy [68] Rohchld d Sglz [55]. I owdy wdely ued ecoomc d fce Bw[7] Levy [99]. I he ochc domce ppoch dom eu e comped by po-we compo of ome pefomce fuco couced fom he dbuo fuco. Fgue 3.3. Me ly. Pofolo bee h pofolo y he me ee bu oe of hem effce. Fo el dom vble V f pefomce fuco defed he ghcouou cumulve dbuo fuco of V : F V η P{ V η} fo η R. A dom eu V d Lehm[94] Qu d Spo [44] o ochclly dome ohe dom eu S o he f ode deoed V f FV η FS η fo ll η R. FSD S f The ecod pefomce fuco F gve by e below he dbuo fuco F F η V; η F V ; α dα fo η R

58 d defe he we elo of he ecod-ode ochc domce SSD. Th dom eu V ochclly dome S o he ecod ode deoed V f SSD S f F V; η F S; fo η R. 3.5 η ee Hd d Ruell [66] Hoch d Levy [99]. The coepodg c domce elo f FSD d p SSD e defed he uul wy V f S V p S. d V f S 3.6 FSD SSD SSD Fo pofolo he dom vble ueo e he eu defed by 3.. To vod plcg he deco veco ubcp epeo we hll mply we I wll o led o y cofuo we beleve. Thu we y h pofolo dome pofolo y ude he FSD ule f F η ; F η; y fo ll η R whee le oe c euly hold. Smlly we y h dome y ude he SSD ule R f R y f F η ; F η; y fo llη R wh le oe euly c. SSD R R F η; F η d F η; F. η Fgue 3.4. The epeced hofll fuco. Sochc domce elo e of cucl mpoce fo deco heoy. I ow h R f R y f d oly f E[ U R ] E[ U R y] fo y odeceg FSD fuco U fo whch hee epeced vlue e fe. Alo R f R y f d oly f E[ U R ] E[ U R y] fo evey odeceg d cocve U fo whch hee epeced vlue e fe ee e.g. Levy [97]. 46 SSD

59 Fo e P of pofolo pofolo P f hee o P clled SSD-effce o FSD-effce y P uch h R y f R o R y f R. SSD We hll focu ou eo o he SSD elo becue of coecy wh vee pefeece: f R f R y he pofolo pefeed o y by ll - SSD vee deco me. By chgg he ode of ego we c epe he fuco F ; he epeced hofll Ogycz g Ruczy [9]: fo ech ge vlue η we hve F η ; E[m η R ]. 3.7 The fuco F ; couou cove oegve d odeceg. I gph llued Fgue 3.4. Followg [4 43] we oduce he followg defo. Defo. Ruzczy d Vdebe [56] The me- model ρ coe wh SSD wh coeffce α > f he followg elo ue fo ll R f SSDR y ρ y ρ y fo ll α. α I fc we hll ee he poof below uffce o hve he bove euly fed fo α ; vldy fo ll α follow fom h. FSD The cocep of coecy u ou o be fuful. I [9] we hve poved he followg eul. Theoem 3.. The me model whch he defed he bolue emdevo δ E {m R } 3.8 coe wh he ecod-ode ochc domce elo wh coeffce. We povde ey leve poof hee. Poof. F cle fom 3.7 h he le η he ympoe of F ; fo η. Theefoe R f R y mple h SSD y

60 Secodly eg η 4 we ob δ E {m R y}. Sce y we hve m R y m y y R y y m R y Tg he epeced vlue of boh de d combg wh he pecedg euly we ge δ y δ y whch c be ewe δ y δ y. 3. Combg eule 3.9 d 3. wh coeffce [] we ob he eued eul. 48 d whee A decl eul ude he codo of fe ecod mome h bee obed Ogycz d Ruzczy [8] fo he dd emdevo d fuhe eeded Ogycz d Ruzczy [9] o cel emdevo of hghe ode d ochc domce elo of hghe ode ee lo Gooh d Koo [64]. Elemey clculo how h fo y dbuo δ δ whee δ he me bolue devo fom he me: δ E R. 3. Thu δ coe meue wh he coeffce α. The me bolue devo model h bee oduced covee le pogmmg me model by Koo d Ymz [86]. Aohe ueful cl of meue c be obed by ug ule of he dbuo of he eu R. Le p deoe he p-h ule I he fcl leue he uy p W whee W he l veme omeme clled he Vlue R of he dbuo of he eu R.e.