PRO-ACTIVE SCHEDULING BY A COMBINED ROBUST OPTIMIZATION AND MULTI- PARAMETRIC PROGRAMMING APPROACH

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1 PO-ACIVE SCHEDULIG BY A COMBIED OBUS OPIMIZAIO AD MULI- PAAMEIC POGAMMIG APPOACH Mr Wm-Hohe d Ero. Poouo * Cere or Proe Sem Egeerg Derme o Chem Egeerg Imer Coege Lodo SW7 BY U.K. Ar I h wor we ddre hor-erm h roe hedug uder uer whh he hedug mode omed wh uer d he oeve uo he rgh-hd de veor d he or mr rodued re demd d roeg me or overo re uer reeve. We wo-ge rou omzo/mu-rmer rogrmmg roedure or he rome ouo o he hedug mode whh re o mu-rmer med eger er (m-milp roem. We demore h he rooed roh orue o he oruo o ro-ve hedug reg d how h rve erve o he rgorou rou omzo roh erm o rovdg gh eme o he om hedug o. Keword Proe Shedug Med Ieger Ler Progrmmg Mu-Prmer Progrmmg. Iroduo he re o hedug o hem d hrmeu roee h reeved g eo dur d dem. For e oruo h ed we reer o he eee revew o Foud d L ( L d Iererou (008 d Verderme e. (00. Mo o he wor oered wh hedug mode whh d re umed o e ve. I roee m e ue o uer rg rom vro he mre demd rodu re roeg me e. Hee he om hedug o reed o he om vue m o e om or eve ee more oe devo rom he om vue h ourred. Pro-ve hedug moved he eed o ddre uer uro order o vod reeve oe omzo reoe o dure. I he oe erure roh o ou or he reee o uer he mode rou omzo whh m dg hedug oe h re ee or oe rezo o he rmeer (L e. (004 J e. (007 L d Iererou (008. O he oher hd mu-rmer rogrmmg h o oud romg o roe hedug uder uer (u e. 007 u d Poouo (007 L d Iererou (007 whe he umer o rmeer m. I h wor we emo omed rou omzo d mu-rmer rogrmmg roh o reerred o wo-ge mehod (Wm- Hohe d Poouo (0 ro-ve hedug reg or hor-erm h roee. We how h he wo-ge mehod ( ow or he ee reme o e o uer he uderg mhem mode d ( omro wh rgorou rou omzo roh e oervve d o rem ee owrd he ororo o he rmeer oe her u vue re ow. For e he oe-o-om hedug o he derved v uo evuo whou urher eed or omzo. * o whom orreodee houd e ddreed

2 Proem Formuo We oder hor erm hedug o h roee. I rur we ue he u e eve ed mode whh ouou me ormuo eurg he oe o eve o (Iererou d Foud (998. Eve o re me reed e whh r u. Br vre h re rodued o he mode deoe he vo u o u eve o. he oeve hedug he mzo o ro hough oher rer uh he mmzo o me- m o e odered. he hedug ormuo eed o ou or mer e ewee rodu rodued d oumed d orge or eore oo d eueg or gg o ue u d u me/duro or. Mre demd mu o o e me. he deerm hedug ormuo whh orreod o med-eger er roem red oow: z I m ( ( ( ( V H w w S w ( w H ( w H ( w ( H ( w ( H V I I I S w J C J I I I J J J J ( S S I J I J I J J S S I J. he reee o uer rorm he hedug mode o he geer mu-rmer med-eger er mode (P z (θ ( H θ.. A (θ E( θ F {0} ( d L θ m (P wh mre d veor hvg rore dmeo A ( A A (ogou or E(θ d deog he veor o rmeer. For eme re uer rodue oeve uo oee (OFC uer demd uer e he ere o he rgh-hd de (HS or veor d roduo me d overo re uer e he ere o he or mre A d E reeve d rodue o ed e-hd de (LHS uer o he mode. For he e ouo o (P murmer rogrmmg ehue m o derve he om ouo whou ehuve eumerg he rmeer e (Poouo (009. he reee o LHS-uer oe hege mu-rmer med eger er rogrmmg. hereore we emo omed rou omzo/mu-rmer rogrmmg roedure or he rome ouo o (P oued he e her. wo-sge Mehod or geer m-milp Proem I he r e o he wo-ge gorhm roedure (P mmuzed g uer he or mr A whh ed r rou IMm-MILP roem eurg OFC- d HS-uer we LHS-uer reed o he or mr E where he eod e he e om ouo o he rou mode derved g ue murmer rogrmmg gorhm (Fí e. (009. he omed rou omzo/mu-rmer rogrmmg roedure omuo ee rovdg gh ower oud o z ( o (P. Here ogde he wor-e oreed ormuo we o du erve rou mode whh ow orog he degree o oervm o he ouo. he wor-e oreed r rou ouerr o (P he r ouo o (P ed r rou ee m ( : A( E( θ F or ee θ. Iororg ( o (P ed he ormuo o he r rou ouerr (C o he geer m-milp roem (P z( θ ( H θ.. u u ( (θ u {0} (d L θ e (... m... u θ τ u m...m θ θ (C m where : ( / : deoe he rge d he om vue o θ = reeve. Ever ee ouo o (C r rou ee

3 ouo o (P. oe h he oveo wor-eoreed rou ouerr o (P orreod o u deerm MILP roem ee L e. (004 d Be- d emrov (998 reeve whoe ouo re mmue g d vro. Cer ever ee ouo o he rou ouerr o ee or (C d oeue or (P. he r rou ouerr o (P wh due degree o oervm o he ouo I re o rmeer re e o hge rom he om vue he he wor-e oreed rou ouerr m e oo rerve. A roh h ow dg he degree o oervm o he ouo reeed Berm d Sm ( Here ded o oe roe he ouo g uer A. Aume h he ere o A re modeed deede mmer d ouded rdom vre where deoe he om vue d he rge o or reeve. We deoe J he e o uer oee o A h er he -h or o (P. he rmeer Γ ed he udge rmeer mrg he rde-o ewee roue o he ouo d oervm o he mode. I hod 0...m. J I Γ eger whh ue umo our rmewor he ouo mmue g u o Γ uer oee he -h or o A. We h he r ( r rou ee wh ree o Γ { S J S } S e ( (... m or ee θ. Iororg ( o (P ed he r rou ouerr (C Γ z( θ.. u u 0 ( H θ u u {0} θ θ u 0 (d L θ J e (...m J...m J... m...m θ θ ( (C Γ whh o IM-m-MILP roem. oe h eg Γ = J ree he mo oervve mode whh euve o he wor-e oreed roh emoed (C. A deomoo gorhm or IM-m-MILP roem We oue he e o he gorhm reeed Fí e. (009 or he ouo o (C d ogou o (C Γ. he mer roem (M derved rom (C reg he rmeer θ omzo vre. he om eger ode o o (M u o (C whh he ed m-lp u-roem (S. he r rego o (S eh ue o he ee e o rmeer whh rur rem om re uue deed he LP om odo (G (979. Bewee ever mer d u-roem ero he mer roem uded. A ew MILP roem oved o go om or eh oe o he urre r rego. Ieger u re rodued o he ormuo o (M order o eude revou ved eger ouo d rmer u eure h o eger ode h re om or (C or er rezo o he rmeer re odered. he u re gve J : { } { 0} J...K wh K deog he umer o revou deed eger ouo h rego d ( H (d L (...K where (θ he om oeve vue o (C he eger ode reed o de. he gorhm erme rego where he mer roem ee. We re eveoe o rmer roe (Du e. (00 d oe eger ode d orreodg ouou ouo h hve ee deed o e om or er o wh r rego. Fuo evuo d dre vue omro o he oeve vue or he rmer roe ored he eveoe deerme he om ouo o (C rmeer rezo. oe h he r rego o (C oed he deomoo gorhm re ohedr ove d h he ouo ored he eveoe re eewe e uo. he umer o rego ueed he umer o eger ode d or. For horough ud o he omuo ome o he gorhm he reder reerred o Fí e. (009. Comuo Sude Eme. he roduo roe whoe Srereeo gve Fgure o o hree h e e hree ere u. he rodu S3 d ured vero S4 re od o o he mre. We ume re demd d roeg me uer or he mg. he d or Eme re reeed e. Emog he wo-ge mehod he r rou mode (C eore e o he ouo or ero o he vre erm o he roeg me. he

orreodg r rou duro or red oow: ( 4. 5 θ w 5 m m m m 3 300 he r rou ouerr (C wh 5 eve o over me horzo o hour vove 85 or d 5 eger vre.

4 orreodg r rou duro or red oow: ( 4. 5 θ w 5 m m m m he r rou ouerr (C wh 5 eve o over me horzo o hour vove 85 or d 5 eger vre. he deomoo gorhm reure he ouo o 9 MILP mer roem oved o go om d 3 m-lp uroem. hree eger ode hve ee deed o e om or er rego o he ee rmeer e. he r rego re deed Fgure d he oeve uo vue oed o he ouo ored he eveoe o rmer roe re gve e. For he o e he ouo ored he eveoe re o gve. I u oe r rego he om r rou hedug o h ee deed. I rego C he eveoe o wo dde ouo mog whh or ever rmeer rezo he om r rou hedue o Eme w uded. For eme oder θ * =( C. A h o uo evuo ed z =8.6 oed wh he r roe d z =6 wh he eod roe reeve Dre omro dee he r roe o e he om r rou hedug o θ *. he orreodg G-Chr o he om r rou hedug o θ * reeed Fgure 3. I omro he e om vue o Eme θ * z=44. he deerm wor-e oreed rou ouerr o Eme however ee. C Fgure. Cr ego o Eme wh wo-sge Mehod e. Eveoe o Pr ou Pro o Eme Cr ego {0 θ θ 0.5} Pro -8.3θ θ +6.θ -0.6 θ θ -6θ C {-0.9 θ θ 0.5} -8.3θ θ +6.θ -0.6 θ C 3 C 4 C 5 {-θ +0.4θ 0.6 θ θ 0.5} {-θ +0.9θ θ θ -0.4 θ θ } {θ -0.9θ -0.7-θ + 0.8θ 0.9- θ θ -0.4 θ -0.6θ 0.5} -0θ +46θ -9θ θ +50θ -6θ θ +50θ -6θ Fgure. S-ereeo o Eme e. D or Eme U C Proeg me (τ U 00 Mg 4.5+θ U 75 eo 3.0 U 3 50 Sero.5 Se Sorge I Amou Pre S S S Demd S4-0 +θ 50-0θ θ θ ge - θ θ 0.5 Fgure 3. G-Chr o he Om Pr ou Shedug Po o Eme θ * =( Eme. he roe vove he roduo o wo rodu d ever ermede rodu deed he S-rereeo Fgure 4.he reo d 3 reeve e e oe o wo u U d U d here re ere u or heg d ero. Beue o he o e or he d o Eme we reer o Iererou (998. he re d he demd o he rodu S8 d S9 vr d here uer he roduo re o S8 d he ermede rodu S7:

C C ρ S 8 S 9 S 8 0 0 θ 5 5θ 0. 4 0. θ 3 S 8 S 9 ρ S 7 50 40 θ 60 0 θ 0. θ 4 0 θ 0 θ 0 θ 3 4 U U. Furhermore we ow h mo oe o he roduo re or S7 d S8 reeve U d U e o hge rom he om vue.

For ever re { S 7 S 8} we rodue he udge rmeer. he orreodg orge ( or mmuzed g overo re uer red oow: I ( J I I ( J J S I J ( ( { S 7 S 8} Fgure 4.

5 C C ρ S 8 S 9 S θ 5 5θ θ 3 S 8 S 9 ρ S θ 60 0 θ 0. θ 4 0 θ 0 θ 0 θ 3 4 U U. Furhermore we ow h mo oe o he roduo re or S7 d S8 reeve U d U e o hge rom he om vue. Hee wh he wo-ge mehod we emed he r rou mode (C Γ wh due degree o oervm o he ouo. Covero re uer e he or oug or he mer e. he mou o e (rw ermede or rodu eve o h o e e h he mum orge d ur e w o-egve. For ever re { S 7 S 8} we rodue he udge rmeer. he orreodg orge ( or mmuzed g overo re uer red oow: I ( J I I ( J J S I J ( ( { S 7 S 8} Fgure 4. S-ereeo o Eme wh om Covero e ( ( ρ ( I J { S 7 S 8} where he uerr d deoe he om vue d he rge reeve o he roduo re. I mr w we re e o derve he roued or reed o he o-egv o he mou o S7 d S8 ever eve o. oe h he uer oee o he or mr re o deede. hereore ee hoe o he udge rmeer eer o eure oe ewee dere or d eve o order o derve megu rou mode.e.. ( ( Eorg h he ouo roeed g he dervo o mo oe roduo re rom he om vue we e. he wo-ge mehod ed o ( Eme wh 5 eve o over me horzo o 8 hour reure he ouo o 7 MILP roem d oe m-lp roem. he ouo o he r rou mode re deede o θ 3 d θ 4. I o 6 r rego hve ee deed whh re deed Fgure 5. I eh o he rego oe ouo ored. hereore he om r rou hedug o o Eme h ee deermed d we de he over r rou ro Fgure 6. oe h eg reeme he wor-e ( oreed roh.e. roduo re re e o hge rom he om vue where eg 0 o devo uored. I omro he ( rgorou wor-e oreed rou MILP mode o Eme ed ro o z=65.3 whh ower oud o he om r rou ro wh ree o deed Fgure 6 u o o he ( om r rou ro or 0. ( Fgure 5. Cr ego o Eme wh wo- Sge Mehod Fgure 6. Om Pr ou Pro (Sure Po d Om Wor-Ce Oreed ou Pro (Grd Po o Eme Couo I h wor we hve ddreed hor-erm h roe hedug omed wh uer he d ug wo-ge mehod whh ome e-ohe-r rou omzo d mu-rmer rogrmmg ehue. We o om r rou hedug o h ed gh ower oud o

6 he over ro d e oervve h he rgorou rou omzo roh. he wo-ge mehod e o de wh e o uer he hedug mode rur wh dure he ere o he or mre whh re he mo hegg e o uer mu-rmer med-eger er rogrmmg. he ee o mu-rmer rogrmmg oo or ro-ve hedug h he mode oved oe wh he rmer roe eg ored oo-u e. Hee oe he rue vue o he rmeer re ow he om r rou hedug o red oed v uo evuo rom he roe ored he oo-u e. Wh he wo-ge mehod erve o he wor-e oreed r rou mode we reeed he ormuo o r rou mode h ow orog he roue o he mode d oervm o he ouo whh he h ee ed o h roe hedug uded Eme. Aowedgme F uor rom EPSC (EP/G05907/ EP/I04640/ d rom he Euroe eerh Cou (MOBILE EC Adved Gr o 646 greu owedged. eeree Be- A. emrov A. (000. ou ouo o er rogrmmg roem omed wh uer d. Mh. Progrm. 88( Berm D. Sm M. (003. ou dree omzo d ewor ow. Mh. Progrm. 98( Berm D Sm M. (004. he re o roue. Oer. e. 5( Du V. Boz.A. Poouo E.. (00. A murmer rogrmmg roh or medeger udr egeerg roem. Comu. Chem. Eg. 6( Fí.P. Komd V.D. uem B. Poouo E. (009. Go omzo o mu-rmer MILP roem. J. Go Om 45( 3-5. Foud C.A. L X. (004. Couou-me veru dreeme rohe or hedug o hem roee: revew. Comu. Chem. Eg. 8( Foud C.A. L X. (005. Med eger er rogrmmg roe hedug: modeg gorhm d o. A. Oer. e. 39( 3-6. G. (979. Poom e rmer rogrmmg d reed o MGrw-H I.US. Iererou M.G. Foud C.A. (998. Eeve ououme ormuo or hor-erm hedug.. Muuroe h roee. Id. Eg. Chem. e. 37( J S.L. L X. Foud C.A. (007. A ew rou omzo roh or hedug uder uer: II. Uer wh ow ro druo. Comu. Chem. Eg. 3( L Z. Iererou M.G. (007. Proe hedug uder uer ug murmer rogrmmg. AIChE J. 53( L Z. Iererou M.G. (008. Proe hedug uder uer: evew d hege. Comu. Chem. Eg. 3( L Z. Iererou M.G. (008. ou omzo or roe hedug uder uer. Id. Eg. Chem. e. 47( L X. J S.L. Foud C.A. (004. A ew rou omzo roh or hedug uder uer:: I. Bouded uer. Comu. Chem. Eg. 8( Poouo E.. (009. Pereve murmer rogrmmg d e mode redve oro. AIChE J. 55( u J.-H. Du V. Poouo E.. (007. Prove hedug uder uer: A rmer omzo roh. Id. Eg. Chem. e. 46( u J.-H. Poouo E.. (007. A ove roh o hedug o zero-w h roee uder roeg me vro. Comu. Chem. Eg. 3( Verderme P.M. E J.A. L J. Foud C.A. (00. Pg d hedug uder uer: A revew ro mue eor. Id. Eg Chem. e. 49( Wm-Hohe M. Poouo E.. (0. A rou omzo ed roh o he geer ouo o m-milp roem I Proeedg o he Euroe Smoum o Comuer Aded Proe Egeerg ESCAPE Aed oo H I I I me horzo erormed u /roeg e J U u ue or erormg J S Se Eve o Amou o e od o mre eve o Amou o e eve o mou Amou o mer uderg u eve o Srg/hg me o u reed o eve o w Avo u o u eve o C Pre o e Demd o e Prooro o e oumed/rodued durg S Mmum orge o e V m V Mmum/mum or u Me roeg me o u Co/vre erm o roeg me or u wh : : 3 3 V V m

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