A Simple Heuristic for Reducing the Number of Scenarios in Two-stage Stochastic Programming

Transcription

1 A Smle Heurtc for Reducng the Number of Scenaro n wo-tage Stochatc Programmng Ramumar aruah Marano Martn and gnaco E. Gromann * Deartment of Chemcal Engneerng Carnege Mellon Unverty Pttburgh PA 5 U.S.A. ABSRAC n th wor we addre the roblem of olvng multcenaro otmzaton model that are determntc equvalent of two-tage tochatc rogram. We reent a heurtc aroxmaton trategy where we reduce the number of cenaro and obtan an aroxmaton of the orgnal multcenaro otmzaton roblem. n th trategy a ubet of the gven et of cenaro elected baed on a rooed crteron and robablte are agned to the occurrence of the reduced et of cenaro. he orgnal tochatc rogrammng model converted nto a determntc equvalent ung the reduced et of cenaro. A mxed-nteger lnear rogram (MLP) rooed for the reduced cenaro electon. We aly th ractcal heurtc trategy to four numercal examle and how that reformulatng and olvng the tochatc rogram wth the reduced et of cenaro yeld an obectve value cloe to the otmum of the orgnal multcenaro roblem. eyword: wo-tage tochatc rogrammng; Multcenaro model; Scenaro reducton; Aroxmaton * Correondng author. el.: ; fax: Emal addre: gromann@cmu.edu (.E. Gromann)

2 . NRODUCON Otmzaton under uncertanty a maor ue n olvng real world roblem. Uncertanty a common feature that reent telf durng the oeraton or degn of any ytem. here an abundance of lterature n the area of otmzaton under uncertanty nvolvng everal alcaton. Some of thee nclude: roducton lannng (Clay and Gromann 997; Cheng et al. 00) chedulng (Brge and Demter 996; Balaubramanan and Gromann 00) otmal chemcal roce ynthe (Acevedo and Ptooulo 998; Lu and Sahnd 996; Rooney and Begler 00) electrcty roducton (art et al. 996; Nowa et al. 005). Uually roblem wth uncertanty are rereented a tochatc rogrammng roblem (Brge and Louveaux 997) or a determntc flexblty roblem (Gromann et al. 98). he focu of th wor on olvng two-tage tochatc rogram wth recoure where we have ome uncertan arameter that ether follow a contnuou dtrbuton or tae on a fnte et of value. he am n uch roblem to determne the t tage decon varable uch that the um of the t tage cot and the exected value of the nd tage cot mnmzed. Other aroache for olvng roblem under uncertanty nclude robut otmzaton robabltc rogrammng fuzzy otmzaton and dynamc rogrammng. Sahnd (004) reent a recent revew of roblem under uncertanty along wth the aroache ued to olve uch roblem. Algorthm for tochatc nteger rogram have been reented by Ahmed et al. (004) Carøe and nd (997) Carøe and Schultz (999) len Haneveld et al. ( ) among other author. Norn et al. (998) have develoed a branch and bound technque for global otmzaton of nonconvex nonlnear tochatc rogram where tochatc lower and uer bound are made to converge wth ome confdence level. n two-tage tochatc rogrammng wth recoure a common aroach to dcretze the uncertan arameter ace and formulate a determntc equvalent of the tochatc rogram whch lead to a multcenaro otmzaton roblem (Dantzg 96). A ngle combnaton of the value of the uncertan arameter lead to a artcular cenaro. n th dcretzaton aroach there are uually a number of uncertan arameter n a

3 ytem and thee are aumed to tae on a fnte et of value. All oble combnaton of thee value lead to an exloon n the number of cenaro. h greatly ncreae the ze of the otmzaton roblem mang t very hard to olve. o overcome th roblem aroxmaton method have been develoed to olve the tochatc rogram wth fewer cenaro and tll obtan a cloe to otmal oluton. Nova and ravana (999) have reented a reduced dmenonal tochatc otmzaton technque where they determne a ubet of the vertce of the feable olyhedral ace of the uncertan arameter and ther correondng weght to aroxmate the exected value of the obectve functon of the orgnal roblem. Duačová et al. (00) have alo rooed a cenaro reducton technque baed on a dfferent robablty metrc. Samlng method (e.g. Monte Carlo amlng) are alo qute attractve to convert the contnuou ace of uncertan arameter nto a maller dcrete ace. Samle Average Aroxmaton (SAA) ha alo been ued to olve tochatc mxed-nteger nonlnear rogram (for examle ee We and Realff 00). n th wor we addre the roblem of reducng the number of cenaro n multcenaro otmzaton roblem. We ue a mlar dea a gven n Nova and ravana (999) and Duačová et al. (00) to elect a ubet of cenaro from a gven larger et for olvng the tochatc rogram. he goal that the otmal obectve of the full cenaro roblem cloely aroxmated by the otmal obectve value of the reduced roblem. A mxed-nteger lnear rogrammng (MLP) model reented for the electon of the ubet of cenaro. he remander of the aer organzed a follow. Secton reent the roblem tatement whle the heurtc trategy to aroxmate the orgnal multcenaro roblem gven n Secton. Numercal examle on whch the aroach wa aled are gven n Secton 4 and fnally Secton 5 ummarze the concluon.. PROBLEM SAEMEN We are gven a two-tage tochatc rogram whoe determntc equvalent ha S earate cenaro wth dfferent realzaton of uncertan arameter. Each of thee

4 4 cenaro ha a certan robablty of occurrence. he uncertan arameter that mae u thee cenaro tae on a fnte et of value. he robablte of th fnte et of value for each uncertan arameter add u to. h dcrete fnte et ether gven or ele t can be comuted from a contnuou dtrbuton (ee Luceno 999). he goal of th aer to develo an aroach where we can elect a ubet S of cenaro from the orgnal et of cenaro ( S ) wth new robablte gven to each of the S cenaro and aroxmate the otmal obectve value of orgnal multcenaro roblem a cloely a oble wth the reduced number of cenaro. h mean that on olvng the reduced dmenonal roblem (wth fewer cenaro) we get an obectve value cloe to one of the orgnal multcenaro roblem wth S cenaro. he roblem at hand to deve an MLP (or a lnear rogrammng (LP)) formulaton that allow u to elect a ubet of the cenaro and gve u ther aocated robablte that would hel n aroxmatng the orgnal otmzaton roblem. We are alo ntereted n gettng ome bound on the theoretcal error etmate.. APPROXMAON SRAEGY wo-tage tochatc rogram are often converted to determntc multcenaro otmzaton roblem by dcretzng the uncertan arameter n a fnte et of cenaro. Such roblem grow larger wth the number of cenaro. A multcenaro model wth a cenaro et S can be exreed a follow: 0 mn z f ( d) + f d x. t. where h ( d x θ ) 0 g ( d x θ ) 0 d D x X θ Θ ( x θ ) S S a ngle cenaro n the multcenaro roblem. d the et of t tage decon varable whle x the et of nd tage varable n cenaro. θ the vector of uncertan arameter n cenaro. h (.) 0 and g (.) 0 nclude the frt and econd tage contrant. Our goal to aroxmate the et wth S cenaro wth a et wth S (P)

5 5 cenaro o that we have a maller multcenaro roblem that yeld cloe to otmal exected obectve value.. Selecton of ubet of cenaro n a multcenaro roblem let θ { θ } be the vector of uncertan arameter. Let the uncertan arameter θ tae on a fnte et of value gven by { θ } robablty aocated wth the uncertan arameter θ tang on a value. he θ. Wth multle uncertan arameter thee can be combned together by conderng the correondng Cartean roduct of all the value of the uncertan arameter to yeld the et wth S cenaro. he cenaro nvolve the followng vector of uncertan arameter { θ } θ and there are a total of S cenaro. Aumng ndeendent dtrbuton the robablty aocated wth a cenaro n the orgnal et of cenaro gven by. n order to elect a mnmum ubet from the orgnal et of cenaro we rooe the followng heurtc crteron: Crteron: he um of the robablte of the new cenaro n whch the uncertan arameter value θ aear equal to. he above crteron hould hold for all of the value of each of the uncertan arameter. Furthermore the um of the robablte of the reduced et of cenaro hould be equal to. We want to re-arrange the cenaro robablte n uch a way that the overall robablty of occurrence of a artcular value of an uncertan arameter acro dfferent cenaro matche the robablty of occurrence of that value for the gven uncertan arameter. he motvaton behnd uch an aroxmaton to heurtcally reduce the aroxmaton error. f the obectve functon of a multcenaro otmzaton formulaton can be aroxmated a a um of the functon of the ndvdual uncertan arameter value multled by the reectve robablte of the cenaro n

6 6 whch they occur t value wll be cloe to. f ( θ ). f ( θ ). he reagnment of robablte to elected cenaro baed on the above-mentoned crteron and olvng the otmzaton roblem wll then gve an obectve functon that can be aroxmated by f ( θ ).. f ( θ ). Snce the obectve functon of the orgnal and the reduced cenaro roblem can be aroxmated by the ame exreon the dfference between ther value alo exected to be mall. o llutrate the cenaro reducton aroach wth a mall examle conder two uncertan arameter θ and θ where each arameter can tae on two value. Let θ tae the two value {5} each occurrng wth a robablty of 0.5 and let arameter θ tae a value of 0 wth a robablty of 0.5 and a value of 70 wth a robablty of 0.5. We obtan a et of four cenaro {(0) (70) (50) (570)} that reult from the Cartean roduct of {5} and {070}. hee are hown n Fg. and denoted by () () () and (4). he robablty of occurrence of each of thee cenaro 0.5 whch obtaned by multlyng the robablte of the uncertan arameter value n each cenaro. θ 70 (4) 0.5 () () 0.5 () θ Fg. Scenaro n llutratve examle Loong at the θ ax we fnd that θ tae a value of n cenaro () and (4) where each of thee cenaro occur wth a robablty of 0.5 thu mang the overall

7 7 robablty of occurrence of the value for θ to be 0.5. he creaton of cenaro ha earated the value of taen by θ nto dfferent cenaro. However the creaton of cenaro ha enured that the um of the robablte of the cenaro n whch θ tae a value of the ame a the occurrence robablty of θ whch 0.5. he ame analy true for the value of 5 taen on by θ. When loong at the θ ax we can fnd an dentcal analy for the value taen on by θ. o reduce the number of cenaro we ue the dea of reverng th daggregaton of uncertan arameter value and ther robablte and combnng bac the cenaro o that the robablte of occurrence of the ndvdual uncertan arameter value reman ntact. One oble re-combnaton hown n Fg. a where cenaro () combned wth cenaro () whle cenaro (4) combned wth cenaro () leadng to the new cenaro ( ) and ( ) n Fg. b. θ 70 (4) 0.5 () () 0.5 () θ Fg. a Re-combnaton of cenaro n llutratve examle

8 8 θ 70 ( ) ( ) θ Fg. b Scenaro wth modfed robablte n llutratve examle n Fg. b the ndvdual uncertan arameter value have the ame robablty of occurrence through the cenaro ( ) and ( ) a when the cenaro had not even been created. For ntance loong at the θ ax we ee that θ tae a value of only n cenaro ( ) whch ha a robablty of 0.5 enurng the robablty of occurrence of the value for θ to be 0.5. Smlarly for θ 5 th value now occur only n cenaro ( ) whoe robablty 0.5 meanng that θ 5 occur wth a robablty of 0.5 n the overall ytem whch the robablty of occurrence of the value 5 for θ. A mlar analy hold for the value on the θ ax. he mnmum number of reduced cenaro that we can obtan deend alo on the ndvdual robablte of the value taen on by the uncertan arameter. o llutrate th let θ now tae two value {5} each occurrng wth correondng robablte of 0. and 0.7 reectvely. he arameter θ tae a value of 0 wth a robablty of 0.6 and a value of 70 wth a robablty of 0.4. Combnng thee two uncertan arameter we obtan four cenaro a hown n able.

9 9 able. Scenaro for two uncertan arameter n llutratve examle Scenaro θ θ Probablty of cenaro Now on electng a ubet of cenaro from the gven et n able we obtan a reduced et of cenaro S (ee able ) that can be ued for aroxmatng the orgnal multcenaro roblem (P). able. Reduced et of cenaro for two uncertan arameter n llutratve examle Scenaro ' θ θ Probablty of cenaro ' ' 0 0. ' ' he aforementoned crteron atfed by the cenaro S. For ntance n the orgnal roblem θ tae a value of 5 wth a robablty 0.7. n the reduced et of cenaro S θ tae a value of 5 n ' ' and n ' ' and the um of the robablte of occurrence of ' ' and ' ' 0.7 ( ) enurng that the rooed crteron hold. he ame logc hold for each of the value of both the uncertan arameter. o model the electon of cenaro let the new robablty agned to a cenaro be ˆ a contnuou varable. he bnary varable w correond to the extence of the cenaro n the new et of cenaro. he MLP formulaton to determne the mnmum number of cenaro atfyng the rooed crteron a follow:

10 0 { } w w t w f 0 ˆ 0 ˆ ˆ ˆ ˆ ˆ.. mn M (SG) On olvng the MLP model (SG) we obtan the mnmum et of cenaro and ther aocated robablte. he numercal value of the robablty correondng to a cenaro wth the uncertan arameter { } θ θ θ n the reduced et of cenaro * ˆ. Remar. Snce the roblem (SG) yeld a very large MLP roblem we can conder ntead a lnear rogrammng relaxaton (SG-L) to obtan a et of cenaro atfyng the rooed crteron. h can be done by elmnatng the bnary varable and modfyng the obectve functon a gven n the formulaton below:

11 t f ˆ 0 ˆ ˆ ˆ ˆ.. ˆ ). ( mn M (SG-L) he weght n the new obectve functon nvolve the nown robablte of the extng et of cenaro and are reent to drve the otmzaton to reduce the number of cenaro whle tryng to ee the orgnal et of cenaro that had relatvely larger robablte. he oluton wll be a ubet of the ntal et of S cenaro although t not guaranteed to be the mnmum number of cenaro nce the roblem (SG) and (SG-L) are not equvalent.. t alo oble to agn weght to the ndvdual term n (SG) to hel elect cenaro wth new robablte cloe to ther orgnal robablte.. n th method for determnng a maller number of cenaro t may be oble to dentfy the wort-cae cenaro from among the gven dcrete et of cenaro (that guarantee feablty of degn for all the gven dcrete cenaro). f uch cenaro ext and are ealy dentfed they can be ncluded n the reduced et of cenaro. n the MLP formulaton (SG) th would mean agnng a lower bound on the robablty of the wort-cae cenaro (f nown) a

12 ε where ε a mall otve number le than or equal to. he wortcae cenaro bnary varable w are alo fxed to a value of. wort cae cenaro 4. he theoretcal mnmum number of cenaro that can be obtaned ung th method the maxmum of the number of ndeendent value that each uncertan arameter can tae. he other lmtng cae that f no value of an uncertan arameter occur n more than a ngle cenaro n the et S the number of cenaro cannot be reduced wth th method.. Reduced cenaro otmzaton he tochatc otmzaton roblem (P') that ue a reduced et of cenaro S a follow: mn z f d x. t. n (P). 0 d D x ( d) + h ( d x θ ) 0 g ( d x θ ) 0 X θ Θ f ( x θ ) S All the functon n (P') have the exact ame form a the correondng functon the robablty of a elected cenaro obtaned by olvng (SG) or (SG-L). he otmal value of the degn varable vector obtaned by olvng (P') denoted by dˆ and the otmal exected obectve value by * z. n cae the wort-cae cenaro from the et S are ncluded n the et S then dˆ wll be feable for every cenaro n the orgnal cenaro et S (Gromann et al. 98). We can alo olve the orgnal roblem (P) by fxng the degn varable n (P) to the value dˆ. Note that th mae the model (P) decomoable nto S earate otmzaton ubroblem wth each ubroblem correondng to a ngle cenaro. Solvng (P) by fxng the degn varable to the value dˆ gve u a locally otmal oluton to the orgnal roblem where the otmal obectve * value obtaned ung th method ~ z. (P')

13 For ractcal uroe we can obtan a bound on the error n uch an aroxmaton a follow: (a) he exected obectve value comuted by olvng each cenaro (SP ) earately (e.g. wat-and-ee aroach) or may be aroxmated by tang a ubet of cenaro wth larger robablte. he otmal obectve value of all condered cenaro B S are ummed to obtan z z * *. * (b) he obectve value of the bet found feable oluton o far ~ z (th value ~ B * * ether the global mnmum or hgher than t). he value of EB z z calculated whch an uer bound on the error ung the aroxmaton technque. Note that th bound may be looe. 4. NUMERCAL EXAMPLES he rooed cenaro reducton and aroxmaton aroach aled to four examle. he otmzaton roblem are formulated ung GAMS (Brooe et al. 998) and olved on an ntel Pentum V Wndow machne wth 5 MB memory. he LP and MLP roblem are olved ung GAMS/CPLEX 9.0 whle GAMS/ CONOP.0 ued for the nonlnear rogrammng (NLP) roblem. Examle h mall examle taen from Clay and Gromann (997) and correond to roblem (EX) n that aer. t a tochatc rogram wth uncertan arameter { θ θ } and one t tage decon varable (d). n order to convert th roblem nto t determntc equvalent the uncertan arameter are agned three value each whch lead to the creaton of the LP determntc equvalent wth 9 cenaro. he two uncertan arameter are gven by θ and θ each wth value and aocated robablte a follow: θ {.5} { } θ {.5} { } he LP determntc equvalent a follow:

14 4 mn z d +. t. x x x d R x x R x θ θ θ ( x + θ R + d θ + x + θ ) (E) n the above formulaton x and x are the contnuou nd tage varable and.. he LP formulaton for the determntc equvalent (E) wth 9 cenaro ha 9 contnuou varable and 7 contrant. Solvng th model yeld an otmal obectve value of 0. where the otmal value of the t tage varable 4.0. We aly the cenaro reducton technque to th roblem and obtan the 4 cenaro n able. able. Reduced number of cenaro for examle Scenaro ' θ θ Probablty of cenaro ' ' 0. ' ' 0. 4'.5 0. he correondng reduced LP roblem ha 4 cenaro 9 contnuou varable and contrant. t to be noted that an necton of the value of the uncertan arameter and ther correondng robablte ued n determnng the reduced et of cenaro uch that the crteron gven n Secton. atfed. he MLP formulaton (SG) not ued to elect the cenaro n th examle. On olvng the reduced cenaro roblem we obtan the otmum value of 0. and the value of the t tage varable agan 4.0. h mean that we have a zero aroxmaton error n th cae. Note however that the reduced et of cenaro not unque. Solvng the model (P') wth dfferent cenaro wth dfferent robablte could otentally lead to a value of the degn varable that nfeable for the orgnal roblem (P). A degn varable obtaned by

15 5 olvng the aroxmate model (P') wll be feable for the orgnal roblem only f the wort-cae cenaro from the orgnal et of 9 cenaro are ncluded n the reduced et of cenaro ued n formulatng (P'). Examle We olve the model (EXP) taen from Clay and Gromann (997) a a next examle. h an LP wth 0 contnuou varable and 8 contrant. t ha uncertan arameter that are aumed to tae on value each leadng to a total of 9 cenaro. he two uncertan arameter and ther dtrbuton are gven below: θ { } { } θ { } { } he formulaton correondng to th examle a follow mn z d +. t. d + x d R x d + x + R + θ θ. x (E) where d and x are the contnuou t and nd tage varable reectvely and the robablty.. Solvng th model we obtan an otmum exected value of.6 and the otmal value of the t tage varable Ung the rooed cenaro electon aroach we can obtan a mnmum of 4 cenaro atfyng the rooed robablty crteron n Secton.. he formulaton (SG) correondng to th examle gven below

16 6 mn f. t. ˆ ˆ ˆ ˆ ˆ ˆ ˆ + ˆ + ˆ + ˆ + ˆ + ˆ + ˆ ˆ w + ˆ + ˆ + ˆ + ˆ + ˆ + ˆ w (SG-E) 0 w ˆ { 0} able 4 how the cenaro obtaned by olvng the above formulaton (SG-E). able 4. Reduced number of cenaro for examle obtaned from olvng model (SG-E) Scenaro ' θ θ Probablty of cenaro ' ' 0. ' 0.5 ' 0. 4' 0. On olvng the reduced dmenonal model (P') wth the four cenaro hown n * able 4 obtaned by olvng model (SG-E) we obtan an otmal value of z.68 and the otmal value of the degn varable found to be Fxng the value of the degn varable d to n model (E) and re-olvng t we obtan the otmal obectve * value of ~ z.6 whch the ame a the otmum of model (P). We alo ee f we can refne the oluton by generatng the cenaro ung model (SG-L). We fnd that we obtan the ame et of 4 cenaro a hown n able 4 by olvng (SG-L) correondng to th examle.

17 7 Examle he thrd examle a larger cae tudy and taen from Nova and ravana (999) wth ome modfcaton. h roblem correond to the degn of a heat exchanger networ wth 5 heat exchanger hot tream cold tream and utlte. he networ tructure gven Fg.. Fg. Heat exchanger networ for examle he three temerature 5 and 9 are uncertan arameter that change durng networ oeraton. Each of thee uncertan arameter aumed to tae on 5 value wth robablte ( ) gven n able 5. able 5 Value and robablte for uncertan arameter 5 9 ( C) ) ( 5 ( C) ) ( 5 9 ( C) ) ( 9 he otmzaton roblem formulated a a two-tage tochatc rogram whch converted to t multcenaro equvalent. here are a total of 5 cenaro n th roblem. he goal of the degn roblem to mnmze the exected total cot that nclude the catal cot of the heat exchanger and the exected utlty cot. he heat

18 8 exchanger area are the t tage degn varable whle the heat load and the temerature that are not fxed are the nd tage varable. he multcenaro model (E) a follow:

19 9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r A m W U m W U U U U r r U A t A A z r r r r r r r r + + Δ Δ Δ Δ Δ Δ ) /( 000 ) /( ) max( ln ln 9 9.ln.ln ln 5 ) 000( / 50) 500( ) 000(9 ) 000( ) 000( ) 000( ) 000( ) 500( mn ln ln ln ln 4 4 ln ln φ φ φ φ φ φ φ φ φ φ (E)

20 0 n the above model the ubcrt correond to a artcular cenaro. φ r ertan to the heat load n heat exchanger r n cenaro. he t tage degn varable A r ertan to the area of heat exchanger r. 5 9 are the value of the reectve uncertan arameter 5 and 9 n cenaro. he coeffcent denote the robablty of occurrence of cenaro and calculated by multlyng the ndvdual robablte of the value of the uncertan arameter whch occur n that cenaro. he model (E) wth 5 cenaro a nonconvex nonlnear rogram wth 005 contnuou varable and 75 contrant. On olvng th model we obtan the otmal oluton of $45.07 wth the followng otmal value of the degn varable: A 5.4 m A.7 m A 6. m A 4.99 m A 5. m. Alyng our cenaro reducton aroach to th examle where we frt olve the MLP model (SG) for th roblem we obtan 5 cenaro (ee able 6). Scenaro able 6. Reduced number of cenaro for examle ( C) 5 ( C) 9 ( C) ' ' ' ' ' On ung the above 5 cenaro n roblem (E) reformulatng t and olvng t we obtan a nonconvex NLP model wth 86 varable and 96 contrant. he otmal obectve of th reduced roblem $ whch 0.% hgher that the otmum of the orgnal multcenaro roblem wth 5 cenaro. he otmal value of the degn varable o obtaned are  5.8 m Â.4 m  6.8 m  4.98 m  5. m. On olvng the 5 cenaro model (E) by fxng the degn varable to the otmal value obtaned by olvng the reduced model we obtan an exected cot * of ~ z $455.6 whch almot the exact oluton of the orgnal tochatc rogram. n term of the comutatonal tme for olvng the otmzaton roblem t tae 0. CPU to olve (E) whle the reduced dmenonal roblem wth 5 cenaro olved n

21 ut 0.4 CPU. On fxng the degn varable (area of heat exchanger) n the model * (E) we are able to olve t n 9.8 CPU to obtan ~ z $ Examle 4 he lat examle a modfed veron of the one ued by Acevedo and Ptooulo (998) and We and Realff (004). he orgnal roblem nvolve the roducton of 5 roduct from 5 raw materal ung dfferent rocee (Fg. 4). n th roblem the uncertan arameter are the maxmum avalablte of raw materal and the demand for roduct. he contnuou decon varable are the caacte for the rocee wherea the bnary varable denote the electon of the requred rocee. Fg. 4 Proce networ for examle 4 he determntc model for th examle baed on bac ma balance and decrbed a follow.

22 n Fg. 4 the node are ether ltter or mxer. F( unt unt) the ma flow rate from a ource unt to a detnaton unt. For a ltter lt connected to a ource unt and detnaton unt q the ma balance gven by F ( unt lt) F( lt unt q ) q For a mxer mx wth nut connecton from unt q and an outut to unt the ma balance are q F( unt mx) F( mx unt) q A raw materal wth flowrate RM aumed to come from an nlet ource F( ource unt) RM A roduct wth ma flow rate P aumed to go out to a detnaton out F( unt out ) P S he um of the ma flow to a roce from nlet ource unt q equal to F( untq roce ) S q he ma flow from a roce to a detnaton unt equal to OS F( roce unt) S Other balance nclude: Yeld relaton OS PC S Dered roducton P D Avalablty of raw materal RM Max RM Logc contrant S Q M Q 0 MaxQ y 0 he obectve functon gven by

23 5 5 mn z β P α RM OC S [ DCQ + FC y ] he ymbol n the revou equaton are ummarzed a follow: D the uncertan demand for roduct (arameter) DC cot for roce F unt unt ) the ma flow rate n the tream between unt and unt ( FC the fxed cot of roce (arameter) S the ma flow n nut tream to roce (varable) Max Q the maxmum volume caacty of roce (arameter) Max RM the maxmum avalablty of raw materal whch uncertan (arameter) M the ma flow to volume relatonh contant for roce (arameter) OC the oeratng cot of roce (arameter) OS the ma flow n outut tream to roce (varable) P the ma flow of roduct (varable) PC the yeld contant for roce (arameter) Q the caacty of roce (varable) RM the ma flow of raw materal (varable) y the bnary varable for electon of roce (bnary varable) α the cot of raw materal (arameter) β the rce of roduct (arameter) Snce D and Max RM are uncertan arameter the above model converted nto a two-tage tochatc rogram whch then re-formulated a a determntc multcenaro model by dcretzng the uncertan arameter. Q and y are the frt tage decon varable whle the flow n the ytem raw materal conumton and

24 4 roduct flow are the econd tage varable. he obectve to mnmze the negatve of the roft functon n the multcenaro formulaton the uncertan arameter are 4 and Max RM 4 and each of thee aumed to taen to two value. D 5 and Max RM 5 are aumed to be nown and contant. he value for all the arameter ued n th examle 4 can be een n able 7 where the two level of the eght uncertan arameter and ther robablte can alo be found. n th examle we obtan an exact oluton conderng all the cenaro and comare that otmal value of the obectve functon wth thoe obtaned ung the rooed aroach and by ung Samle Average Aroxmaton (SAA). D able 7. Parameter ued n the model for examle 4 Proce PC M OC DC FC Max Q Product 4 5 D (D ) Raw 4 5 materal Max RM (Max RM ) he multcenaro roblem wth 56 cenaro cont of bnary varable 765 contnuou varable and 9 contrant. By olvng the full multcenaro roblem we fnd that only rocee and are n oeraton y ( 4780 ) and the otmal obectve functon value h

25 5 MLP model olve n only 0. CPU. able 8 how the value for the degn caacty varable Q. able 8. Degn varable value obtaned from oluton of full multcenaro roblem Q Ung the rooed method n the aer a reduced et of cenaro along wth ther robablte obtaned by olvng model (SG) correondng to th examle and the reult are hown n able 9. he reduced cenaro roblem ha only 5 cenaro and bnary varable 48 contnuou varable and 87 contrant. able 9. Reduced et of cenaro for examle 4 Scenaro Max Max Max Max RM RM RM RM 4 D D D D On ung the reduced et of cenaro reformulatng and olvng the roblem the otmal obectve value found to be he oluton tme for the reduced cenaro model 0.0 CPU. f we fx the value of Q and y n the full multcenaro model to thoe obtaned by olvng the reduced cenaro roblem we obtan an obectve functon value of whch wthn 0.05% of actual otmal obectve functon value. We hould note that though the oluton tme for the orgnal and the reduced cenaro roblem are very mall they how the otental of the rooed aroach to reduce comutatonal tme for much larger roblem. Fnally we fnd the oluton rovded by the SAA method and an nterval for the oluton wth a confdence lmt of 95% (ee We and Realff 004). A tattcal lower lmt on th nterval (-675.4) found by olvng the tochatc rogram 0 tme each

26 6 wth 0 randomly elected amle (cenaro). he tattcal uer lmt ( ) found by formulatng a multcenaro roblem wth 50 cenaro randomly elected from the gven et of 56 cenaro and olvng t wth fxed value of the frt tage decon varable obtaned durng calculaton of the lower tattcal lmt on the confdence nterval for the oluton. Snce we amled cenaro from a fnte oulaton adutment were made n the calculaton of the tattcal lmt. he otmal obectve value ung the rooed aroach alo le between the tattc lmt comuted by the SAA method. 6. CONCLUSONS h wor ha reented a new ractcal heurtc trategy for olvng two-tage tochatc rogrammng roblem formulated a determntc multcenaro otmzaton roblem. he dea cont of relacng a gven et of cenaro obtaned by dcretzaton of the uncertan arameter ace by a maller et of cenaro and thu aroxmatng the otmzaton roblem n a reduced ace. he rooed crteron for electng a ubet of gven et of cenaro that the overall robablty of occurrence of a artcular realzaton of any uncertan arameter n the fnal et of cenaro hould be equal to the robablty of the uncertan arameter tang on that artcular value. h crteron ha to be atfed for each uncertan arameter n the model. We reented an MLP formulaton a well a a relaxed LP model for determnng a mnmum ubet of cenaro from a gven cenaro et uch that th crteron atfed. he tochatc rogram were reformulated wth the maller et of cenaro n order to obtan aroxmate model. he alcaton of th heurtc technque on numercal examle ha hown that we obtan cloe to otmal oluton ung the aroxmate model wth the maller number of cenaro. h method would alo comlement other amlng baed otmzaton method a th heurtc can be aled to the amle collected from an nfnte ace to further mlfy the roblem.

27 7 Acnowledgment he author gratefully acnowledge fnancal uort from the Natonal Scence Foundaton under Grant CS We are alo grateful to Dr. evn Furman at ExxonMobl Reearch and Engneerng for h helful comment and uggeton. Marano Martn alo whe to acnowledge a MCNN / Fulbrght fellowh. REFERENCES. Acevedo.; Ptooulo E. N. (998). Stochatc Otmzaton baed Algorthm for Proce Synthe under Uncertanty. Comuter and Chemcal Engneerng Ahmed S.; awarmalan M.; Sahnd N. (004). A Fnte Branch-and-Bound Algorthm for wo-tage Stochatc nteger Program. Mathematcal Programmng Balaubramanan.; Gromann. E. (00). A Novel Branch and Bound Algorthm for Schedulng Flowho Plant wth Uncertan Proceng me. Comuter and Chemcal Engneerng Brge. R.; Demter M. A. H. (996). Stochatc Programmng Aroache to Stochatc Schedulng. ournal of Global Otmzaton Brge. R.; Louveaux F. V. (997). ntroducton to Stochatc Programmng. Srnger New Yor. 6. Brooe A.; endrc D.; Meerau A; Raman R. (998). GAMS: A Uer Gude Releae.50. GAMS Develoment Cororaton. 7. Carøe C. C.; nd. (997). A Cuttng-lane Aroach to Mxed 0- Stochatc nteger Program. Euroean ournal of Oeratonal Reearch Carøe C. C.; Schultz R. (999). Dual Decomoton n Stochatc nteger Programmng. Oeraton Reearch Letter Cheng L.; Subrahmanan E.; Weterberg A. W. (00). Degn and Plannng under Uncertanty: ue on Problem Formulaton and Soluton. Comuter and Chemcal Engneerng Clay R.; Gromann.. E. (997). A Daggregaton Algorthm for the Otmzaton of Stochatc Plannng Model. Comuter and Chemcal Engneerng Dantzg G. B. (96). Lnear rogrammng and extenon. Prnceton N: Prnceton unverty re.. Duačová.; Gröwe-ua N.; Römch W. (00). Scenaro Reducton n Stochatc Programmng: An Aroach ung Probablty Metrc. Mathematcal Programmng Gromann. E.; Halemane. P.; Swaney R. E. (98). Otmzaton Stratege for Flexble Chemcal Procee. Comuter and Chemcal Engneerng

28 8 4. Hort R.; uy H. (996). Global Otmzaton Determntc Aroache ( rd ed.). Berln: Srnger-Verlag. 5. len Haneveld W..; Stouge L.; van der Vler M. H. (995). On the Convex Hull of the Smle nteger Recoure Obectve Functon. Annal of Oeraton Reearch len Haneveld W..; Stouge L.; van der Vler M. H. (996). An Algorthm for the Contructon of Convex Hull n Smle nteger Recoure Programmng. Annal of Oeraton Reearch Lu M. L.; Sahnd N. V. (996). Otmzaton n Proce Plannng under Uncertanty. ndutral and Engneerng Chemtry Reearch Luceno A. (999). Dcrete Aroxmaton to Contnuou Unvarate Dtrbuton An Alternatve Smulaton. ournal of the Royal Stattcal Socety (Ser B) Norn V..; Pflug G. Ch.; Ruzczyn A. (998). A Branch and Bound method for Stochatc Global Otmzaton. Mathematcal Programmng Nowa M.P.; Schultz R.; Wethalen M.(005). A tochatc nteger rogrammng model for ncororatng day-ahead tradng of electrcty nto hydro-thermal unt commtment. Otmzaton and Engneerng Nova Z.; ravana Z. (999). Mxed-nteger Nonlnear Programmng Problem Proce Synthe under Uncertanty by Reduced Dmenonal Stochatc Otmzaton. ndutral and Engneerng Chemtry Reearch Rooney W. C.; Begler L.. (00). Otmal Proce Degn wth Model Parameter Uncertanty and Proce Varablty. AChE ournal Sahnd N. (996). BARON: A General Puroe Global Otmzaton Software Pacage. ournal of Global Otmzaton 8 () Sahnd N. V. (004). Otmzaton under Uncertanty: State-of-the-art and Oortunte. Comuter and Chemcal Engneerng art S.; Brge. R.; Long E. (996). A Stochatc Model for the Unt Commtment Problem. EEE ranacton on Power Sytem We.; Realff M.. (004). Samle Average Aroxmaton Method for Stochatc MNLP. Comuter and Chemcal Engneerng 8-46.