An Approach to the Representation of Gradual Uncertainty Resolution in Stochastic Multiperiod Planning
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1 9 h Euopean mpoum on Compue Aded oce Engneeng ECAE9 J. Jeow and J. hulle (Edo 009 Eleve B.V./Ld. All gh eeved. An Appoach o he epeenaon of Gadual Uncean eoluon n ochac ulpeod lannng Vcene co-amez a gnaco E. Gomann b Boa ahan b alvado Henández-Cao c and Juan G. egova-henández c a nuo ecnologco de Celaa Av. ecnologco Gaca Cuba /N CelaaGo eco vcene@qcelaa.c.m b Canege ellon Unve bugh A 53 UA c Unvedad de Guanauao Faculad de Qumca Guanauao Go eco Abac h wo focue on he modelng of mulage ochac poblem wh endogenou (decon dependen unceane. We aume ha he pobabl dbuon of he uncean paamee ae dcee o ha a cenao ee epeenaon can be ued. A he man conbuon he pape decbe an appoach o epeen he gadual eoluon of endogenou unceane afe an nvemen n nfomaon made; paal eoluon of uncean hough me defned n em of a pecenage of vaance educon. he appoach baed on he concep of poeo and evelaon dbuon and on he paccal popoon of he heo of condonal epecaon. A mnng poducon plannng poblem wh endogenou uncean n oe qual ued a a cae-ud o how he cope of he popoed epeenaon a well a o evaluae he effec of he gadual eoluon of unceane on he opmal oluon. Kewod: ulpeod ochac plannng uncean eoluon. noducon pcal engneeng applcaon ae decon poblem ubec o he combnaon of nheen modelng and acal unceane. Uncean n plannng poblem can be dvded no wo clae: eogenou (o mae uncean and endogenou (o echncal uncean []. oblem whee ochac pocee ae ndependen of decon ae ad o have eogenou uncean wheea poblem whee ochac pocee ae affeced b decon ae ad o poe endogenou uncean. Fo eample n a mulpeod olfeld developmen poec he acual ze and he nal delveabl of a eeve geneall uncean []; howeve once a capal nvemen decon made (nepeed a an nvemen n nfomaon egadng eploaon and/o poducon he uncean wll evenuall be eolved (afe a leanng me. Hence unceane n acual ze and nal delveabl ae endogenou and he eoluon hough me nfluenced b he decon made n a gven me peod. Leaue epo ome pevou appoache o handle decon dependen unceane. Jonbaen e al. [3] aume ha decon affecng uncean eoluon occu a he f me peod. mlal Goel and Gomann [] conde mmedae eoluon of endogenou unceane afe an nvemen n nfomaon made and appl he appoach o ga feld developmen poblem. oe ecenl ahan and Gomann [4] analze he nhe of poce newo wh me-vang uncean eld n whch nvemen n plo plan can be condeed o educe uncean of he eld. 73
2 V. co-amez e al he poblem fomulaed a a mulage ochac pogam wh decon dependen elemen whee nvemen aege ae condeed o educe uncean and mevang dbuon ae ued o decbe uncean. Nevehele aumed ha full eoluon of he uncean acheved afe one me ep (peod. n h wo we decbe an appoach o epeen he gadual eoluon of endogenou unceane afe an nvemen n nfomaon made; ou eoluon aeg allow paal eoluon of uncean hough eveal me peod o ha he eoluon a each me peod defned n em of a pecenage of vaance educon. he appoach ha been ncopoaed no a mulage ochac pogam wh applcaon o mnng poducon plannng. he model nclude gadual eoluon of oe qual (uncean paamee and he concep of ndnguhabl. he followng econ decbe he gadual eoluon of uncean appoach and he mulpeod ochac model. Fuhe a cae-ud ued o how he cope of he popoed epeenaon a well a o evaluae he effec of he gadual eoluon of unceane on he opmal oluon.. Gadual eoluon of Endogenou Uncean Da [5] developed a aeg o model uncean educon a he eul of an nvemen n nfomaon. he mahemacal epeenaon baed on fou popoon fom he heo of condonal epecaon. Alo he eoluon poce nvolve hee dffeen pobabl dbuon; namel po dbuon poeo dbuon and evelaon dbuon. he auho appled h appoach b aumng connuou pobabl dbuon and epeed he educon of uncean a a pecenage of vaance educon. Ou appoach o eolve endogenou uncean Hee we have eended he appoach decbed b Da [5] o cae whee he pobabl dbuon ae dcee o ha we can appl a cenao ee epeenaon o mulpeod ochac plannng poblem. he uncean eoluon poce a follow. We aume ha a po (ognal dcee dbuon of he endogenou paamee nown o ha he pobable π of n pobable value (z ae gven ( n. hen afe an nvemen n nfomaon made a an me peod he eoluon poce fo he paamee a and wll connue fo a leanng me nvolvng eoluon ep wh a educon of vaance a each eoluon ep (he vaance educon epeed n em of a facon V. We alo aume ha a each eoluon ep we ma eceve m meage (each wh pobabl θ m o ha he ognal dbuon change eulng n m poeo dbuon. he me vang pofle fo he endogenou paamee fnall gven b ung he evelaon dbuon a each eoluon ep. he evelaon dbuon defned b he mean value (each wh pobabl θ of he poeo dbuon. Fgue how a epeenaon of he hee dffeen pobabl dbuon fo he cae of hee meage. n he paccal mplemenaon of he appoach ou man aumpon ae: he numbe of meage equal o he numbe of pobable value (mn all of he meage ae equall pobable (θ /m and each of he poeo dbuon how a hgh pobabl (p z fo one of he pobable value. We deved a geneal epeon o calculae he pobabl p z of each poeo dbuon whch eul n a educon of vaance V. he devaon no peened hee bu baed on he heoecal popoon whch ae ha he educon of vaance of he po dbuon equal o he (epeced mean vaance of he poeo dbuon. 74
3 An Appoach o he epeenaon of Gadual Uncean eoluon n ochac ulpeod lannng 3 eage 3 oeo Dbuon o Dbuon Good New θ Value p z θ z π Neual Value p z θ oeo Dbuon θ nvemen on nfomaon Bad New θ 3 Value p z 3 θ 3 Fgue. o poeo and evelaon dbuon Afe applng he aumpon enled above he geneal epeon educe o Eq.. n ( z z ( V ( p z ( n n n ( p z ( z z n ( n whee z he mean of he po dbuon. Alo n Fgue poeo dbuon. Eq. can be olve o oban p z. z he mean of each 3. ulpeod odel fo nng oducon lannng h econ decbe he cae-ud and he mulpeod model whch ncopoae he gadual eoluon of endogenou uncean. eeng mplc due o pace lmaon onl one eample and he man elemen of he model ae decbed. Cae-ud and gadual eoluon pofle he cae-ud con of a mnng poducon plannng poblem nvolvng hee mne (adaped fom Wllam [6]. he mneal poduced b each of he mne med n ode o af he demand and oe qual a each me peod. oale ae pad f a mne fo poducon. Opmal decon nclude he mne poducon pofle and pevou o ha whehe a mne hould be (and oale pad o no. he oe qual of one of he mne nown (equal o bu uncean fo he ohe wo. egadng he mne wh uncean oe qual alo aumed ha afe a mne a aon (an nvemen n nfomaon made hough he pamen of oale he oe qual wll eolve gaduall fo one mne bu wll mmedael be eolved fo he ohe one (ehe 0.8 o 0.9. he po dbuon fo he endogenou oe qual con of wo equall pobable value (nm; π π 0.5; z 0.6; z 0.7 and he leanng me nclude hee eoluon ep (he vaance educon of 33% a each ep. able how he mean value of he poeo dbuon (evelaon dbuon a each eoluon ep calculaed b ung Eq. ; he nvemen n nfomaon a me. can be obeved ha hee a ucceve educon a he epeced value of vaance unl full eoluon acheved. 75
4 V. co-amez e al able. evelaon dbuon fo he endogenou oe qual oeo Dbuon me ep Low qual Hgh qual E(vaance Vaance educon % % % A heoecall ancpaed he evelaon dbuon a full eoluon equal o he po dbuon. Boh of he poeo dbuon ae aumed a equall pobable. Noce ha due o he numbe of eoluon ep 6 combnaon of value fo he uncean paamee eul (86; 3 8 fo he endogenou paamee and fo he mmedael eolved uncean paamee. Hence f boh of he mne wh ochac oe qual ae aed 6 dffeen cenao hould be condeed. ochac ulpeod odel he L ochac mulpeod model defned b Eq. hough 6. he me hozon dvded n N me peod. he e of me peod ( N he e of mne ( epeen he e of cenao J he e of mne wh endogenou uncean paamee K he e of mne whou eogenou unceane and he numbe of eoluon ep. Bna vaable epeen decon abou ng and aon of a mne a me and cenao. N N N q p f η ρ α ( q J w p J w K Q C q U 0 (3 76
5 An Appoach o he epeenaon of Gadual Uncean eoluon n ochac ulpeod lannng ( ( ( ( Z < (... (4 [ ] ( Z Z < (5 ( ( [ ] J p Q Q p ( (6 epeen he oe poducon of a mne and q he oal oe poducon. Q he endogenou paamee (oe qual he mne poducon of he valued mneal U he mamum achevable oe poducon of a mne and C he mnmum equed qual of he poduced maeal. Ve mpoan Z a bna vaable epeenng ndnguhabl of cenao and. Fnall p and w ae bna vaable ued u o epeen he logcal elaonhp among decon. he obecve funcon (Eq. nclude hee em; he pof he oal co and he aon co. eul and dcuon Fo me hozon nvolvng 8 me peod he model conan 88connuou vaable and 984 bna vaable whch how he combnaoal comple of he appoach. A pe he eul he dffeence beween he deemnc and ochac obecve (value of he ochac oluon V almo 48% fo he eample whch how he gnfcance of ncopoang unceane n he model paamee. neengl he opmal decon fo all he cenao and he deemnc cae nclude he ng and aon of he mne wh endogenou uncean nce he f me peod. Howeve becaue of poo oe qual he deemnc cae uggeed oppng aon afe me peod. A an eample of he numecal eul able how he oe poducon (0 6 on/ea n 4 me peod of he mne wh mmedae eoluon of oe qual n cenao hough Concluon and Fuue Wo h pape decbe an appoach o model he gadual eoluon of endogenou unceane epeened b dcee pobabl dbuon on he cone of L mulpeod plannng poblem. 77
6 V. co-amez e al able. Numecal eul ( fo mne wh mmedae eoluon of uncean me peod he devaon baed on he heoecal popoon whch ae ha he educon of vaance of he po dbuon equal o he (epeced mean vaance of he poeo dbuon. n fac paal eoluon of uncean hough me defned n em of a pecenage of vaance educon. Due o combnaoal comple he numbe of cenao apdl nceae wh boh he numbe of eoluon ep and he numbe of pobable value of he uncean paamee. Hence he eulng L model can onl be olved hough an L-baed banch and bound fo malle nance. he dffeence beween he deemnc and ochac obecve (value of he ochac oluon V almo 48% fo he cae ud whch how he gnfcance of ncopoang unceane n he model paamee. Dual-baed banch and bound algohm fo olvng lage (lnea and nonlnea poblem ae cuenl beng developed and eed. Fuhemoe a pape decbng a genealzed appoach o model he me vang pofle of gaduall eolved endogenou paamee alo n pepaaon. 5. Acnowledgemen V. co-amez han he fnancal uppo povded b he Fulbgh cholahp pogam and b CONACY eco. efeence [] Goel V. and. E. Gomann. A cla of ochac pogam wh decon dependen uncean. ahemacal ogammng - ee B [] Goel V. and. E. Gomann. A ochac pogammng appoach o plannng of offhoe ga feld developmen unde uncean n eeve. Compue & Chemcal Engneeng [3] Jonbaen.W.. J. B. We and D. L Wooduff A cla of ochac pogam wh decon dependen andom elemen. Annal of Opeaon eeach [4] ahan B. and. E. Goman. A mulage ochac ogammng Appoach wh aege fo Uncean educon n he nhe of oce Newo wh Uncean Yeld Compue & Chemcal Engneeng [5] Da. A. G. nvemen n nfomaon n peoleum: eal opon and evelaon n he poceedng of he 6h Annual nenaonal Confeence on eal Opon Cpu [6] Wllam H.. odel buldng n mahemacal pogammng 4h Edon John Wle and on London
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