A Two-Stage Modeling and Solution Framework for Multisite Midterm Planning under Demand Uncertainty

Transcription

1 Ind. Eng. Chem. Re. 2000, 39, A Two-Stage Modelng and Soluton Framework for Multte Mdterm Plannng under Demand Uncertanty Anhuman Gupta and Cota D. Marana* Department of Chemcal Engneerng, The Pennylvana State Unverty, Unverty Park, Pennylvana A two-tage, tochatc programmng approach propoed for ncorporatng demand uncertanty n multte mdterm upply-chan plannng problem. In th blevel decon-makng framework, the producton decon are made here-and-now pror to the reoluton of uncertanty, whle the upply-chan decon are potponed n a wat-and-ee mode. The challenge aocated wth the expectaton evaluaton of the nner optmzaton problem reolved by obtanng t cloed-form oluton ung lnear programmng (P) dualty. At the expene of mpong the normalty aumpton for the tochatc product demand, the evaluaton of the expected econdtage cot acheved by analytcal ntegraton yeldng an equvalent convex mxed-nteger nonlnear problem (MINP). Computatonal requrement for the propoed methodology are hown to be much maller than thoe for Monte Carlo amplng. In addton, the cot avng acheved by modelng uncertanty at the plannng tage are quantfed on the ba of a rollng horzon mulaton tudy. * Author to whom all correpondence hould be addreed. E-mal: cota@pu.edu. Phone: Fax: Introducton and Motvaton Supply-chan plannng concerned wth the coordnaton and ntegraton of key bune actvte undertaken by an enterpre, from the procurement of raw materal to the dtrbuton of the fnal product to the cutomer. In the prevalng volatle bune envronment, wth ever changng market condton and cutomer expectaton, t neceary to conder the mpact of uncertante nvolved n the upply chan. Source of uncertanty n producton plannng can be categorzed a hort-term or long-term baed on ther tme frame. 1 Short-term uncertanty may nclude dayto-day proceng varaton, canceled/ruhed order, equpment falure, etc. ong-term uncertanty refer to raw materal/fnal product unt prce fluctuaton, demand varaton, and producton rate change occurrng over longer tme frame. 2 It the latter type of uncertanty that addreed n th work. One of the key ource of uncertanty n any producton-dtrbuton ytem the product demand. Product demand fluctuaton over medum-term (1-2 year) to long-term (5-10 year) plannng horzon may be gnfcant. Determntc plannng and chedulng model may thu yeld unrealtc reult by falng to capture the effect of demand varablty on the tradeoff between lot ale and nventory holdng cot. Falure to ncorporate a tochatc decrpton of the product demand could lead to ether unatfed cutomer demand and lo of market hare or excevely hgh nventory holdng cot. 3 Recognton of th drawback of determntc model ha led to a number of publcaton devoted to tudyng proce plannng under uncertanty. Some of the key apect that have been addreed are degn and operaton of batch plant, 1,3,4-8 ue concernng flexblty and relablty n proce degn, 9-11 and long-range plannng and capacty expanon of chemcal proce network. 8,12-14 A evdent from the lterature revewed above, almot all reearch ha been lmted to () batch proceng ytem and () ngle producton te. Key feature uch a the preence of (em)- contnuou procee and multple producton te have, o far, not been condered n detal. In vew of th, the ncorporaton of demand uncertanty n (em)- contnuou, mdterm, multte plannng addreed n th work. To th end, the determntc mdterm plannng model of McDonald and Karm 15 adopted a the benchmark formulaton. Modelng and Decon Makng under Uncertanty A key component of decon makng under uncertanty the repreentaton of the tochatc parameter. Two dtnct way of repreentng uncertanty ext. The cenaro-baed approach 1,7,16 attempt to repreent a random parameter by forecatng all t poble future outcome. The man drawback of th technque that the number of cenaro ncreae exponentally wth the number of uncertan parameter, leadng to an exponental ncreae n the problem ze. To crcumvent th dffculty, contnuou probablty dtrbuton for the random parameter are frequently ued. 6,8,17 At the expene of ntroducng nonlnearte nto the problem through multvarate ntegraton over the contnuou probablty pace, a ubtantal decreae n the ze of the problem uually acheved. In th work, the latter approach ued for decrbng uncertanty. The product demand are modeled a normally dtrbuted random varable. Th approach ha been wdely nvoked n the lterature 6,17,18 a t capture the eental feature of demand uncertanty and convenent to ue. One of the mot wdely ued technque for decon makng under uncertanty two-tage tochatc programmng. 12,14,17,19-25 In th technque, the decon varable of the problem are parttoned nto two et. The frt-tage varable, alo known a degn varable, correpond to thoe decon that need to be made pror to reoluton of uncertanty ( here-and-now decon). Subequently, baed on thee decon and the realza /e CCC: $ Amercan Chemcal Socety Publhed on Web 10/02/2000

2 3800 Ind. Eng. Chem. Re., Vol. 39, No. 10, 2000 ton of the random event, the econd-tage or control decon are made ubject to the retrcton of the econd-tage recoure problem ( wat-and-ee decon). The preence of uncertanty tranlated nto the tochatc nature of the cot aocated wth the econd-tage decon. Therefore, the objectve functon cont of the um of the frt-tage decon cot and the expected econd-tage recoure cot. The man challenge aocated wth olvng two-tage tochatc problem the evaluaton of the expectaton of the nner recoure problem. For the cenaro-baed repreentaton of uncertanty, th can be acheved by explctly aocatng a econd-tage varable wth each cenaro and olvng the large-cale extenve formulaton 26 by effcent oluton technque uch a Dantzg- Wolfe decompoton 27 and Bender decompoton. 28 For contnuou probablty dtrbuton, th challenge ha been prmarly reolved through the explct/mplct dcretzaton of the probablty pace for approxmatng the multvarate probablty ntegral. The two mot commonly ued dcretzaton tratege n the chemcal engneerng lterature are Monte Carlo amplng 16,29 and Gauan quadrature. 8,9,30,31 The key advantage of thee method le n the fact that they are largely nentve to the type of probablty dtrbuton. The man dadvantage, a n the cenaro-baed approach, the harp ncreae n computatonal requrement wth ncreang number of uncertan parameter. 2 Thu far, applcaton of tochatc plannng model have been lmted compared to thoe of determntc model becaue of ther computatonally ntenve nature. One of the frt attempt at narrowng th computatonal gap the work of Petkov and Marana. 17 In th work, the problem of degnng ngle-product campagn batch plant under demand uncertanty addreed. By the explct oluton of the nner problem followed by analytcal ntegraton over all product demand realzaton, the need for dcretzaton of the probablty pace obvated. The tochatc attrbute of the problem are tranlated nto an equvalent determntc optmzaton problem at the expene of ntroducng nonlnearte nto the problem. The propoed technque hown to reult n gnfcant avng n computatonal requrement over quadrature ntegraton. A mlar treatment of uncertanty for the more complex mdterm upply-chan plannng problem purued n th work. The ret of the paper organzed a follow. In the next ecton, a two-tage tochatc formulaton for ncorporatng demand uncertanty n the mdterm plannng problem poed by McDonald and Karm 15 propoed. Frt, the pecal cae of a ngle producton te addreed to motvate the propoed oluton methodology. Subequently, utlzng the nght ganed from the analy of the ngle-te cae, the more general multte ettng dcued. A motvatng example hghlghtng the key feature of the propoed oluton methodology then preented, followed by the computatonal reult for two larger example. Fnally, the work ummarzed, and concludng remark are provded. Two-Stage Stochatc Formulaton The mdterm producton plannng model of McDonald and Karm 15 form the ba of th work. The key tradeoff captured by th model, n the prt of the clac mult-tem capactated lot-zng problem, that between fxed etup cot and nventory holdng cot. The model utlze the concept of lot wthn whch the varou upply-chan actvte are aumed to occur. The duraton of th lot range from 1 to 2 month, n accordance wth the mdterm nature of the model. Further detal about the varou model attrbute and lmtaton can be found n McDonald and Karm 15 and Gupta and Marana. 32 McDonald and Karm 15 clafy the contrant of the model nto two dtnct et. The frt et of producton contrant enure that an effcent allocaton of the producton capacty acheved at the varou producton te. Thee contrant determne the optmal operatng polce at the producton te. The econd category of contrant are referred to a upply-chan contrant. Thee contrant model the pot-producton actvte of nventory management and effectve allocaton of cutomer demand. The dea of dtnguhng between producton contrant and upply-chan contrant naturally extend to the decon varable of the model, 32 reultng n the parttonng of the varable nto producton varable and upply-chan varable. The producton varable, whch etablh the locaton and tmng of producton run, length of campagn, producton amount, and conumpton of raw materal, unquely defne the producton level and reource utlzaton n the upply chan. The upply-chan varable determne the flow of materal throughout the producton-dtrbuton ytem whle accountng for nventory management. Becaue of the conderable lead tme nvolved n the producton proce, the producton varable need to be et here-and-now, pror to demand realzaton. The upply-chan actvte, uch a nventory control and cutomer demand allocaton, on the other hand, can be performed n a wat-and-ee mode. The clafcaton of the varable and contrant of the mdterm plannng model nto two dtnct categore reult n a two-tage herarchcal decon-makng framework, whch can be effectvely utlzed for ncorporatng demand uncertanty. The two-tage mdterm plannng model under demand uncertanty formulated a (2SMP) mn P j,r j,fr fj f Y fj + f,j,fc j P j +,j,v p C +, t W,, C,W,Y fj +E θ [mn S,I t S +, ζ I +, h I +, µ I ī I,I ī ] uch that S e θ I FP I ) I 0 + P j - j W - S, I RM θ - S e I ī e θ I FP I - I e I e I, I RM S, I, I ī, I g 0 ubject to

3 Ind. Eng. Chem. Re., Vol. 39, No. 10, The frt-tage producton decon correpond to P j, R j, FR fj, C, W, and Y fj n model 2SMP. The frttage decon-makng proce repreented by the outer optmzaton problem contng of eq 1-6, whch are the producton contrant. The objectve functon of 2SMP compoed of two term. The frt nclude the cot ncurred n the producton tage. Thee are the fxed and varable cot of producton, raw materal charge, and the cot of hppng ntermedate product between producton faclte. The econd term quantfe the expected cot of the nner nventory-management recoure problem. The contrant of th embedded optmzaton problem are referred to a the upplychan contrant, and the decon varable nvolved, S, I, I, and I ī, conttute the upply-chan varable. Thee varable are akn to control varable a they can be fne-tuned to enure optmalty n the face of uncertanty. The nner problem, thu, dentfe the value of the upply-chan varable that mnmze the total upply-chan cot for a gven et of value of the producton varable and demand realzaton (θ ). The bac dea of the methodology propoed for olvng model 2SMP baed on obtanng a cloed-form oluton of th nner problem. Th proce dcued for the pecal cae of a ngle producton te n the next ecton. Sngle Producton Ste Before addreng the more relevant multte ntance of problem 2SMP, the mpler cae of a ngle producton te condered. Th analy provde the nght baed upon whch the more general multte ettng wll be addreed next. Conder the nner nventory-management problem for the ngle-te cae. Elmnaton of the nventory varable (I ) and the cutomer hortage varable (I ī ) and correpondng redundant contrant reult n the followng form for the nner problem. Note that the te ndex ha been omtted for convenence. (IP SPS ) ubject to P j ) R j R j j,, I RM (1) C ) β j P j, I FP (2) C ) W, I IP (3) FR fj ) R j j,, f (4) :λ f )1 FR fj e H j j, (5) f MR fj Y fj e FR fj e H j Y fj f, j, (6) P j, R j, FR fj, C, W g 0, Y fj {0, 1} h A + µ θ + mn I,S ζ I - (h + µ - t )S where Gven the value for S and I a obtaned from problem IP SPS, I and I ī can be calculated off-lne a The contrant of problem IP SPS can be decrbed a follow. Equaton 7 enforce no overtockng at the cutomer. Equaton 8, along wth the nonnegatvty of the nventory devaton varable, enure that the underpenalty cot ncurred only when the nventory level below the target afety tock level. Equaton 9 repreent the nonnegatvty of nventory held. Two mportant feature of IP SPS that are ueful n characterzng t optmal oluton are that () t decouple over product and () t conequently nvolve only two varable (S and I ). Th make t amenable to oluton by a graphcal approach. Note that, becaue the uncertan demand θ appear n the contrant et, the feable regon of problem IP SPS vare for dfferent demand realzaton. Th mple that, even though problem IP SPS nvolve only two varable, t not poble to a pror obtan a graphcal repreentaton of t feable regon. Th problem can, however, be reolved by conderng the dual of IP SPS. In the reultng problem, becaue all cot coeffcent are aumed to be determntc, the feable regon wll be ndependent of demand realzaton. Thu, by aocatng nonnegatve dual varable u, v, and w wth contrant 7, 8, and 9, repectvely, the dual of problem IP SPS for each product gven by (DIP ) ubject to Under the aumpton that the demand (θ ) alway nonnegatve, u can be elmnated from DIP ung eq 13 to gve the followng equvalent formulaton, after the contant term (h A + µ θ ) dropped. (DIP ) ubject to S e θ (7) S - I e A - I (8) S e A (9) S, I g 0 (10) A ) I 0 + P j (11) j I ) A - S and I ī ) θ - S (12) h A + µ θ + max u,v,w -θ u - (A - I )v - A w u + v + w g h + µ - t (13) v e ζ (14) u, v, w g 0 (15) -(h + µ - t )θ + max v,w v (θ - A + I ) + w (θ - A )

4 3802 Ind. Eng. Chem. Re., Vol. 39, No. 10, 2000 v + w e h + µ - t (16) v e ζ (17) v, w g 0 (18) For each product, problem DIP nvolve only two varable v and w. A graphcal repreentaton of the feable regon can thu be obtaned, a hown n Fgure 1. Baed on the feable regon llutrated n Fgure 1, a cot aumpton that ha been mplctly ncorporated h + µ - t - ζ g 0 (19) Th aumpton enure economc feablty of the producton-dtrbuton ytem. For an enterpre to utan n the market, the revenue earned (µ ) for a unt of fnhed product old hould be larger than the cumulatve um of the nventory holdng (h ), underpenalty (ζ ), and tranportaton (t ) cot. The vector repreentng the gradent of the objectve functon, along wth t correpondng v and w component, alo hown n Fgure 1. The drecton of th vector repreent the drecton of maxmum acent of the objectve functon value. The gn of the component of th vector determne whch drecton, out of thoe labeled I-IV n Fgure 1, the one of maxmum acent. It not poble to determne th drecton a pror becaue the two component depend on the realzaton of the tochatc demand. Therefore, four dtnct cae hould be condered for the four poble gn combnaton of thee two component. Baed on th obervaton, the optmal oluton for problem DIP can be characterzed a follow: Fgure 1. Feable regon and drecton of teepet acent of objectve functon for problem DIP (cae I-IV). Fgure 2. Optmal upply polce for the ngle-te cae. Cae I: θ - A e 0, θ - A + I e 0 w v ) 0, w ) 0 Cae II: θ - A e 0, θ - A + I g 0 w v ) ζ, w ) 0 Cae III: θ - A g 0, θ - A + I g 0 w v ) ζ, w ) h + µ - t - ζ Cae IV: θ - A g 0, θ - A + I e 0 w Infeable combnaton The reultng optmal objectve functon value for problem DIP are Cae I: -(h + µ - t )θ Cae II: ζ (I - A + θ ) - (h + µ - t )θ Cae III: ζ I - (h + µ - t )A Baed on the dual optmal oluton obtaned for problem DIP, the correpondng prmal optmal oluton for problem IP SPS can be recontructed a Cae I: I ) 0, S ) θ Cae II: I ) θ - (A - I ), S ) θ Cae III: I ) I, S ) A Fgure 2 how the varaton of the optmal upply polce wth demand realzaton. A llutrated n the Fgure 3. Network repreentaton of problem IP SPS. fgure, the three dtnct regon correpondng to the cae I, II, and III can be vewed a regme of low-, ntermedate-, and hgh-demand realzaton, repectvely. The tranton from low- to ntermedate- and, ubequently, from ntermedate- to hgh-demand regme occur at demand realzaton ndcated by Θ fi and Θ IfH, repectvely. An ntutve nterpretaton of the optmal upply polce hown n Fgure 2 can be realzed by conderng the network repreentaton of problem IP SPS, a hown n Fgure 3. Producton varable A can be vewed a the amount of fnhed product avalable for upply at the producton node pror to demand realzaton, a llutrated n Fgure 3. Th amount equal the um of the ntal nventory of the product and the total amount produced on all proceor at the producton te. Requrement for the product ext at two dfferent demand node. There an nternal demand of I unt at the nventory node and an external demand of θ unt at the cutomer node, a hown n Fgure 3. A key dtncton between thee two demand that the nternal demand determntc, wherea the external

5 Ind. Eng. Chem. Re., Vol. 39, No. 10, demand uncertan. The flow from the producton node to the nventory node correpond to the nventory holdng varable I. Smlarly, the flow on the arc drected from the producton node to the cutomer node correpond to the cutomer upply varable S. Note that the nventory flow allowed to exceed the nternal demand (overtockng). However, overtockng at the cutomer not permtted. 15 Next, conder rewrtng eq 19 a -µ + t e - ζ + h The left-hand de of the above expreon repreent the cot of hppng a unt of product from the producton node to the cutomer node. Smlarly, the rghthand de repreent the net mnmum cot ncurred for the tranfer of one product unt to the nventory node. Th mple that, gven a choce of atfyng only one of the two demand, the external demand hould be gven prorty. Th referred to a the cutomer prorty prncple. In vew of th, conder a pecfc realzaton of the external demand θ. If th realzaton greater than the total upply A, then baed on the cutomer prorty prncple, everythng mut be hpped to the cutomer. The nventory level wll drop to zero, and thu, the maxmum poble underpenalty cot correpondng to a devaton of I unt wll be ncurred. Th upply polcy correpond exactly to the hgh-demand regme gven by cae III. In th cae, nether of the two demand completely atfed. Now conder the other extreme cae n whch the upply large enough to completely atfy both demand,.e., A g I + θ. The optmal upply polcy n th cae would nvolve hppng θ unt to the cutomer and holdng the remanng A - θ unt n nventory. Th would be equvalent to cae I. For the ntermedate-demand regme correpondng to cae II n Fgure 2, the entre external demand can be met only at the expene of ncurrng ome underpenalty cot. In th cae, after hppng θ unt to the cutomer, the remanng A - θ unt are tranferred to nventory. Th reult n a devaton of I - (A - θ ) unt below the target afety tock level. The bac dea of obtanng an explct oluton of the nner nventory-management problem mlar n prt to the parametrc programmng approach of Acevedo and Ptkopoulo. 33 The three demand regme dentfed are equvalent to the crtcal regon n the parametrc programmng framework n whch dfferent bae are optmal. Smlarly, the tranton demand level correpond to the crtcal pont at whch the change of optmal bae occur. 34 The work of Ptkopoulo and co-worker ue amplng baed numercal ntegraton technque for expectaton evaluaton. In contrat, n th work an analytcal method for calculatng the expectaton ung the explctly derved optmal oluton utlzed, a decrbed next. Expectaton Evaluaton. The calculaton of the expected value of the oluton of the nner problem requre ntegraton over all poble demand realzaton. To facltate th calculaton, defne R, R I, and R H a the probablte that meaure the lkelhood of the demand of a partcular product beng low, ntermedate, and hgh, repectvely. Therefore Applcaton of the probablty-caled addtve property of the expectaton operator to the nner problem optmal value yeld the followng recoure functon Q (A ). Q (A ) ) [ha + µθ E θ where θ m the mean demand for product. The analytcal evaluaton of the ntegral nvolved n eq 23 facltated by tandardzng the demand a and defnng R ) Pr[θ e A - I ] (20) R I ) Pr[A - I e θ e A ] (21) R H ) Pr[θ g A ] (22) + [ -(h + µ - t )θ θ e A - I ] ] + [ζ (I - A + θ ) - (h + µ - t )θ A - I θ e A ] + [ζ I - (h + µ - t )A A θ ] ) h A + µ θ m +R E θ [-(h + µ - t )θ θ e A - I ] +R I E θ [ζ (I - A + θ ) - (h + µ - t )θ A - I θ e A ] +R H E θ [ζ I - (h + µ - t )A A e θ ] (23) K 1 ) A - I m - θ σ z ) θ - θ m (24) σ and K 2 ) A - θ m (25) σ where σ the tandard devaton of the demand for product. The probablte for low-, ntermedate-, and hgh-demand realzaton are then calculated a R ) Pr[z e K 1 ] ) Φ(K 1 ) (26) R I ) Pr[K 1 e z e K 2 ] ) Φ(K 2 ) - Φ(K 1 ) (27) R H ) Pr[K 2 e z ] ) 1 - Φ(K 2 ) (28) where Φ( ) denote the tandardzed normal cumulatve dtrbuton functon. Applcaton of the defnton of expectaton for a normally dtrbuted random varable yeld the followng condtonal expectaton:

6 3804 Ind. Eng. Chem. Re., Vol. 39, No. 10, K1 z e -1/2z 2 dz E[z z e K 1-2π f(k 1 ) ] ) 1 )- K1 - e -1/2z2 dz Φ(K 1 ) 2π E[z K 1 e z e K 2 ] ) where f( ) the normal denty functon. Incorporaton of the expreon for the condtonal expectaton and the probablte n eq 23 yeld [σ ζ [K 1 Φ(K 1 ) + f(k 1 ] )] Q (A ) ) + σ (h + µ - t - ζ )[K 2 Φ(K 2 ) + f(k 2 )] - µ σ K 2 + ζ I + t A (31) Havng calculated the expectaton of the optmal value of the nner problem, the orgnal two-tage formulaton 2SMP recat a the followng ngle-tage determntc equvalent problem. (DEQ SPS ) mn P j,r j,fr fj C,A,Y fj ubject to f,j FC f Y fj + 1 K2 K1 z e -1/2z 2 dz 2π 1 ) K2 K1 e -1/2z2 dz 2π,j v j P j + eq 1-6, 11, and 25 (29) - f(k 2 ) - f(k 1 ) Φ(K 2 ) - Φ(K 1 ) (30) p C + Q (A ) Problem DEQ SPS a mxed-nteger, nonlnear problem (MINP). The nonlnear term, whch are retrcted to the objectve functon, are of the general form g(k) ) KΦ(K) + f(k) (32) The convexty of g(k) readly etablhed by recognzng that t econd dervatve alway nonnegatve. 3 Therefore, problem DEQ SPS a convex MINP and can be olved to global optmalty by technque uch a generalzed bender decompoton (GBD) 35 or outer approxmaton (OA). 36 The convexty of problem DEQ SPS a conequence of the complete recoure property of the nner nventorymanagement problem. A feable econd-tage oluton ext for any demand realzaton and any producton ettng, a the upply polcy of not hppng anythng to the cutomer and tranferrng the entre producton amount to nventory alway feable. The feable uncertanty regon contng of all poble demand realzaton, therefore, ndependent of the frt-tage decon. Th pecal problem tructure lead to the convexty of the propoed formulaton. Alternately, the convexty property can alo be nferred wthn the parametrc programmng framework by recognzng that the frt-tage varable appear only a the rght-hand de vector n the econd-tage problem. Multple Producton Ste Havng addreed the mpler ngle-te veron of the problem 2SMP, next conder the general multte Fgure 4. Network repreentaton of problem IP MPS. cae. The nner optmzaton problem for th cae gven by IP MPS. (IP MPS ), h A + µ θ + mn ubject to where I,S, ζ I - (h + µ - t )S Comparon of problem IP MPS wth the ngle-te nner optmzaton problem IP SPS ndcate that the preence of multple producton faclte reflected n eq 33, whch allocate product upply from dfferent te to meet the cutomer demand. Th contrant couple the producton te by enforcng no overtockng at the cutomer. In addton, the tranfer of ntermedate product between producton te accounted for n the defnton of A by eq 37. The eparablty over the et of product oberved for the ngle-te cae alo preerved for the multte cae. The network repreentaton of IP MPS for a gven product hown n Fgure 4. The demand at the cutomer can now be met by multple manufacturng faclte. Becaue the total number of varable nvolved n IP MPS equal to twce the number of producton te, t not amenable to drect graphcal oluton. Therefore, a dfferent oluton trategy, utlzng the nght ganed from the ngle-te cae, employed. Frt, a prmal feable oluton for problem IP MPS potulated by extendng the reult obtaned for problem IP SPS. Subequently, a dual feable oluton havng the ame objectve functon value a the potulated prmal olu-, S e θ (33) S - I e A - I (34) A ) I S e A (35) S, I g 0 (36) 0 + j P j - W (37)

7 Ind. Eng. Chem. Re., Vol. 39, No. 10, Fgure 5. Schematc llutraton of γ (OSS) and ω (USS). ton contructed. Th etablhe the optmalty of the potulated prmal oluton baed on the trong dualty theorem of lnear programmng. The contructon of the prmal feable oluton for IP MPS requre the ntroducton of addtonal notaton and aumpton. The key queton that need to be anwered for the multte cae are a follow: (1) Whch te ervce the cutomer n the three demand regme? (2) What the relatve upply rankng n the three demand regme? To anwer the frt queton, baed on the relatve magntude of A and I (the ndex dropped n all further analy for clarty of preentaton), the te are clafed a IS ) { S A - I g 0} ID { S A - I < 0} wth IS ID ) S and IS ID ) φ. Ste belongng to et IS are referred to a nternally uffcent (IS) te whle thoe conttutng et ID are termed nternally defcent (ID). Th termnology tem from the nterpretaton of A - I a the producton amount avalable n exce of the nternal demand at a partcular producton te. Note that dfferent IS and ID te clafcaton et ext for each product. To anwer the econd queton, two addtonal cot parameter need to be defned a γ ) t - h (38) ω ) t - h + ζ (39) A chematc decrpton of γ and ω, whch are defned a the over-afety tock-upply cot (OSS) and the underafety tock-upply cot (USS), repectvely, gven n Fgure 5. OSS repreent the cot ncurred after a unt of product tranferred from the nventory to the cutomer (not ncludng the revenue earned) from above the afety tock level. Alternately, f the product uppled from below the afety tock level, the cot ncurred USS. Thee two cot parameter play an mportant role n determnng the order n whch the te ervce the cutomer n a partcular demand regme. The followng cot aumpton are alo enforced. µ + h - t - ζ g 0 (40) t e h e ζ (41) Equaton 40 mply an extenon of the cutomer prorty prncple to the multte ettng. The aumpton that ζ greater than h content wth mantanng an nventory target of I. The aumpton that nventory holdng cot h exceed the tranportaton Fgure 6. ow-demand regme for the multte cae. cot t typcally hold for mot local and global upply chan. In the latter, becaue of apprecable trant lead tme, the tranportaton charge are uually vewed a urrogate for nventory holdng cot. An mportant relatonhp that reult from eq 41 γ e 0 e ω (42) Equaton 42 mple that, gven a choce for hppng a unt of product from above or below the afety tock level, the former hould be choen for mnmzng cot. Th key reult exploted next for potulatng optmal upply polce n the low-, ntermedate-, and hghdemand regme. ow-demand Regme. For the ngle-te cae, the low-demand regme cont of all poble demand realzaton for whch zero underpenalty charge are ncurred. In the multte ettng, however, becaue of the preence of the ID te, underpenalty charge may be unavodable. Therefore, for the multte cae, the low-demand regme defned a the et of all poble demand realzaton for whch mnmum total underpenalty charge are ncurred n the upply chan. Th correpond to lmtng the underpenalty charge to only the ID te and not volatng the afety tock level at any of the IS te. If the external demand zero, the avalable amount A tranferred to nventory at all te, a no overtockng permtted at the cutomer. Subequently, any nonzero demand can be met at mnmum cot by hppng the product from the IS te wth the lowet OSS cot. The maxmum demand that can be allocated to th te before wtchng to the te wth the econd lowet OSS cot the nventory n exce of the target afety tock level (.e., A - I ). Th prevent the nventory level at the frt (lowet OSS cot) te from dppng below the target afety tock level avodng underpenalty charge. Baed on thee obervaton, feable upply polce for the IS te can be obtaned by rankng them n ncreang order of OSS cot (Fgure 6). The reultng equence of te, tartng wth the te wth the lowet OSS cot, repreent the order n whch the IS te ervce the cutomer. In accordance wth the defnton of the low-demand regme, A - I unt are hpped from a gven te before wtchng to the next one to mantan the underpenalty charge at a mnmum. The entre avalable amount at the ID te tranferred to nventory to mnmze underpenalty charge. The producton te are reordered to obtan a conce mathematcal repreentaton of the above-decrbed

8 3806 Ind. Eng. Chem. Re., Vol. 39, No. 10, 2000 upply polce uch that the condton hold. Alo, defne te l / IS a * l -1 γ -1 e γ (43) (A - I ) e θ e (A - I ) (44) Note that the ummaton n eq 44 conder only the IS te. The upply polce for the dfferent te are then gven by For a low-demand realzaton gven by eq 44, eq ummarze the upply polce obtaned by equentally allocatng demand to the IS te. Rankng of the IS te on the ba of the OSS cot acheved through eq 43, and no hortage at the cutomer (I - ) 0) ext n the low-demand regme. The tranton demand level Θ fi thu gven by a hown n Fgure 6. Intermedate-Demand Regme. Demand realzaton exceedng Θ fi compre the ntermedate-demand regme. For a demand realzaton of Θ fi, baed on the analy preented for the low-demand regme, the upply polce cont of hppng A - I unt from each of the IS te. A a reult, the nventory level at all of the IS te are drven to I. Thu, upply of an addtonal unt of product to the cutomer caue the nventory level of one of the te to fall below the target afety tock level. Th mple that, rrepectve of the type of te (.e., IS or ID) choen to hp th addtonal l * S ) 0 I ) A I ) I - A } ID (45) S ) A - I I ) I I ) 0 * l -1 S ) θ - (A - I ) 1 )1 l * -1 I ) A - θ + (A - I ) I ) 0 } e / l - 1, IS (46) } ) l /, IS (47) S ) 0 I ) A } g / l + 1, IS (48) I ) 0 Θ fi ) (A - I ) unt, a USS cot ncurred. Therefore, to mnmze cot, the canddate te for upplyng th addtonal unt (over and above Θ fi ) to the cutomer the one wth the lowet USS cot. The above obervaton ugget that a procedure baed on rankng the te wth repect to the USS cot can be ued to allocate the porton of demand n exce of Θ fi. Thu, for an ntermedate demand θ g Θ fi, the allocaton of demand can be acheved n two phae. In the frt phae, Θ fi unt are allocated to the IS te baed on ther OSS cot rank. In the econd phae, the remanng θ - Θ fi unt are allocated by re-rankng all of the te n term of ther USS cot (Fgure 7). In the prt of the low-demand regme, rankng of the te on the ba of ther USS cot acheved by reorderng the producton te uch that the condton ω -1 e ω (50) hold. et / IS ID be the te for whch / -1 Θ fi + ID * -1 A + I e θ e Θ fi + A + ID For th ntermedate-demand realzaton, the upply polce for the ID and IS te are gven by Equaton repreent the upply polce for an ntermedate-demand realzaton gven by eq 51. Conequently, the tranton demand level Θ IfH gven by a hown n Fgure 7. A n the low-demand regme, there no hortage at the cutomer (I - ) 0). * * I (51) S ) A I ) 0 } e / - 1 (52) I ) I * -1 S ) θ - A - (A - I ) g * +1 * -1 I ) A - θ + A + g * +1 * -1 I ) I - A + θ - A - S ) 0 I ) A I ) I - A S ) A - I I ) I I ) 0 (A - I ) (A - I ) g * +1 } * ) (53) } g / + 1, ID (54) } g / + 1, IS (55) Θ IfH ) A (56)

9 Ind. Eng. Chem. Re., Vol. 39, No. 10, Becaue IS and ID are djont et, the condton δ + + δ - ) 1 (59) mut hold. The djuncton extended to the producton varable A by defnng A ) A + + A - (60) n conjuncton wth the retrcton Fgure 7. Intermedate-demand regme for the multte cae. I δ + e A + e A UP δ + 0 e A - e I δ - (61) (62) Hgh-Demand Regme. In the low- and ntermedate-demand regme, the entre cutomer demand met, and no ale are lot (.e., I - ) 0). For hgh-demand realzaton (.e., θ g Θ IfH ), however, th no longer poble. Therefore, baed on the cutomer prorty paradgm, the entre producton amount from all of the te tranferred to the cutomer locaton reultng n the followng optmal upply polce. S ) A I ) 0 I ) I I - ) θ - } ɛ IS ID (57) A ID Complete nventory depleton at all of the manufacturng faclte reult n maxmum underpenalty charge n the upply chan. Next, the feablty and optmalty of the potulated upply polce are acertaned. The methodology adopted for allocatng demand to the varou producton te enure that the amount uppled from a te doe not exceed the total amount avalable for upply (A ). Th guarantee nonnegatve nventory level at the te. The no-overtockng retrcton alo enforced by not hppng n exce of the demand n the low- and ntermedate-demand regme, thereby enurng the feablty of the contructed prmal oluton. The optmalty of th oluton etablhed baed on P dualty (ee Appendx A for the proof). Subequently, a for the ngle-te cae, the expectaton evaluaton carred out analytcally. The derved expreon are gven n Appendx B. Determntc Equvalent Formulaton The dervaton of the determntc equvalent formulaton of the multte cae requre the clafcaton of te nto type ID and IS. To th end, the djunctve programmng approach of Bala 37 utlzed. Th nvolve aocatng bnary varable δ + and δ - defned a δ + 1f ) { 0 otherwe and δ - 1f ID ) { 0 otherwe (58) wth each of the producton te n the upply chan. where A UP denote an upper bound on A and gven by baed on the frt-tage producton ettng contrant. Under th tranformaton, the tandardzed varable K 1 l * and K 2 * are gven by Conequently, after the expected econd-tage cot are ncluded (eq 74 n Appendx B), the determntc equvalent formulaton for the multte cae gven by (DEQ MPS ) mn P j,r j,fr fj f,j, C,Y fj,a,w ubject to σk l * A UP ) I 0 + R j H j (63) j * l 1 ) [ (A + - I δ + ) - θ m ] (64) * σk ) [ΘfI + * (A - + I δ + ) - θ m ] (65) FC f Y fj + v j P j + p C +,j,,, Note that the product ndex ha been rentroduced n the formulaton. Illutratve Example The propoed methodology frt hghlghted wth a mall three-te upply-chan example. A ngle product produced at each one of thee te on a ngle proceor. The parameter characterzng th upply chan are lted n Table 1. The ntal nventory at all three te zero, and the revenue earned per unt product µ equal to 5.0. The uncertan product demand θ gven by N(110,30). The determntc ngle-tage equvalent problem DEQ MPS olved for the example upply chan ung OA. The optmal objectve value obtaned 291 wth, eq 1-6, 37, 58-65, and 74 t W + Q (A )

10 3808 Ind. Eng. Chem. Re., Vol. 39, No. 10, 2000 Table 3. Varaton of EEV, RP, and VSS wth Demand Standard Devaton for the Illutratve Example σ EEV RP VSS VSS RP Fgure 8. Supply polce for motvatng example. Table 1. Parameter for the Motvatng Example FC v R H MR t h ζ I Table 2. Optmal Frt-tage Producton Polce for the Motvatng Example Y R P the optmal frt-tage plannng decon gven n Table 2. The optmal producton run length R and amount P are lted n Table 2. The detaled upply polce for the three producton te, hown n Fgure 8, llutrate the flow on the upply and nventory arc for all demand regme. The afety tock level I mply that te 1 of type ID, whle te 2 and 3 are of type IS. Therefore ID ) {1} and IS ) {2,3}. The rankng of the IS te on the ba of ther OSS cot determne the upply polce n the low-demand regme. The frt 57 product demand unt are allocated to te 2, whch the IS te wth the lowet OSS cot. Product demand between 57 and 76 unt agned to te 3, a hown n Fgure 8. Th demand level of 76 unt correpond to the tranton demand level Θ fi. By not allowng the nventory level to fall below the afety tock level at te 2 and 3, the underpenalty charge are retrcted to only te 1 (where they are unavodable). The ntermedate-demand regme compoed of product demand order between 76 and 166 unt. Rankng of the three te wth repect to ther USS cot determne the optmal upply polcy n the ntermedate regme. Ste 2 the te wth the lowet USS cot. Demand ntally allocated to te 2 untl the nventory at te 2 completely depleted. At th pont, upply redrected γ ω to the te wth the econd lowet USS cot (te 3). Upon depleton of nventory n te 3, product demand allocated to te 1, a hown n Fgure 8. Th determne the tranton demand level Θ IfH. The upply polce n the hgh-demand regme cont of hppng the entre producton amount at the three te to the cutomer. By parametrcally olvng the nner optmzaton problem, the optmal upply polce for the entre range of demand realzaton are dentfed. Gven any demand realzaton, the upply polcy that mnmze the expected cot can thu be acertaned. In lght of thee reult, t mportant to quantfy the mpact of uncertanty on the plannng decon. Th can be accomplhed on the ba of the value of the tochatc oluton (VSS), 26 whch evaluate the cot of gnorng uncertanty. By replacng all random parameter by ther expected value and olvng the reultng determntc expected-value (EV) problem, the EV oluton obtaned. Subequently, the recoure problem (RP) olved wth the frt-tage decon fxed at the EV oluton. The reultng optmal value known a the expected reult of ung the EV oluton (EEV). 26 EEV meaure how the EV oluton perform n the face of uncertanty. The VSS then defned by Table 3 lt the EEV and the VSS value obtaned for the llutratve example a the tandard devaton of the demand vared. The monotoncally ncreang value of the VSS ndcate that the cot of neglectng uncertanty ncreae wth the degree of uncertanty. Under hgh-rk condton, avng of approxmately 9% are acheved, jutfyng the ncluon of uncertanty n the plannng decon. In the followng example, a computatonal comparon between the dcretzaton method and the propoed oluton technque provded for a larger upply chan. Example 1 VSS ) EEV - RP The frt example, ntally propoed by McDonald and Karm, 15 cont of 34 product that are manufactured at two conecutve producton te. The frt te produce 23 product grouped nto 11 product famle. Some of thee 23 product are hpped a ntermedate to the econd te, whch produce the remanng 11 product. Demand for all product are preent at the begnnng of each one of the 12 tme perod of 1 month duraton. A detaled decrpton of the problem can be found n McDonald and Karm. 15 Frt, the ngle-perod veron of th problem ntance condered. The product demand are aumed to be normally dtrbuted wth a tandard devaton of 20% of the expected demand. The reultng problem DEQ MPS olved by a cutomzed mplementaton of

11 Ind. Eng. Chem. Re., Vol. 39, No. 10, Fgure 11. ettng. Smulaton procedure adopted for the multperod Fgure 9. Comparon of a Monte Carlo amplng mplementaton wth the propoed oluton procedure. Fgure 10. Varaton of computatonal tme wth number of cenaro ampled. the OA algorthm. 36 The lower and upper bound obtaned are and , repectvely. Thee are obtaned n only 5 teraton of the OA mplementaton utlzng a total of 2 CPU. For comparon, the ngleperod problem alo olved ung Monte Carlo amplng. 26 Th nvolve the generaton of a large number of demand cenaro and the ncorporaton of upplychan varable for each one of thee cenaro n the nner optmzaton problem, yeldng an MIP formulaton. Th MIP olved for an ncreang number of cenaro, and the reult are hown n Fgure 9. A expected, the Monte Carlo optmal value approache the exact optmal objectve functon a the number of condered cenaro ncreae. However, over 1000 of them are needed for good agreement. Fgure 10 how the computatonal reource expended n obtanng thee reult, whch cale exponentally wth the number of cenaro, n accordance wth the NP-hard nature of the MIP problem. Computatonal avng of almot 2 order of magntude over Monte Carlo amplng (2 CPU a compared to 1067 CPU for 700 cenaro) clearly hghlght the beneft of the propoed methodology. Next examned what quanttatve beneft, f any, acheved by ncorporatng a decrpton of uncertanty for a multperod plannng framework. To anwer th queton, the followng multperod mulaton tudy conducted. Conder two planner: a tochatc planner (S), who ha nformaton about both the mean and the tandard devaton of the demand; and a determntc planner (D), who ha nformaton only about the mean demand. Both of thee planner plan on a rollng horzon ba, a hown n Fgure 11. In the frt perod, planner S olve the tochatc formulaton, whle planner D Fgure 12. Multperod mulaton reult for the tochatc and the determntc planner. olve the determntc formulaton. Th reult n two alternatve optmal producton polce for the frt perod. Baed on the optmal value taken by the producton varable, the two planner dentfy the three demand regme for each product. Subequently, randomly generated demand realzaton are revealed to both planner (Fgure 11). Baed on whether the demand realzed low, ntermedate, or hgh, the upply polce for each product are determned by the two planner, along wth the actual econd-tage cot. The optmal upply polce defne the ntal condton for the econd perod. For example, the nventory level a determned by the upply polce defne the ntal nventory for the econd perod. Smlarly, the hortage at the cutomer at the end of the frt perod ncorporated nto the mean demand for the econd perod. Th procedure carred out n a rollng horzon manner for the 12-month plannng perod. It repeated a number of tme to average over the randomly generated demand revealed to the two planner. The performance of the two planner hown n Fgure 12, where the runnng average optmal expected cot for the two planner are plotted agant an ncreang number of demand randomzaton. Clearly, the tochatc planner contently outperform the determntc planner by dentfyng better plannng polce. In the lmt, expected value of the multperod cot obtaned reach and for planner S and D, repectvely. Thee repreent cot avng of approxmately 2% by the tochatc planner. Th dfference n the expected cot can be nterpreted a the avng acheved olely by ncludng a decrpton of demand varablty, quantfed n term of the tandard devaton of the uncertan demand, nto the plannng proce. Example 2 The econd example problem cont of a larger upply chan nvolvng x producton te manufactur-

12 3810 Ind. Eng. Chem. Re., Vol. 39, No. 10, 2000 Fgure 13. Supply chan for econd example. Table 4. Monte Carlo (MC) Reult for Example 2 # cenaro MC optmal CPU a g b g10000 a 2% optmalty gap. b 4% optmalty gap. ng a total of 30 product, a llutrated n Fgure 13. Ste 1 and 2 (3 and 4) produce the ame product, 1-10 (11-20). However, thee te are characterzed by dfferent producton charactertc and cot parameter. Thee product are ether hpped a fnhed product to the cutomer or a ntermedate product to te 5 and 6, a hown n Fgure 13. An aembly-type product tructure ext at te 5 and 6, where each fnhed product produced from two ntermedate product. All te cont of a ngle proceor that capacty contraned, and fxed etup charge are ncurred at each te. To ae the computatonal complexty of the problem, the determntc veron of the problem olved frt. The optmal determntc plan ncur a total cot of obtaned after 145 CPU. Subequently, the tochatc problem olved wth the cutomzed OA algorthm. Th dentfe lower and upper bound of and repectvely n 9 teraton of the algorthm and 2372 of CPU tme. The ncreaed objectve value over the determntc optmal objectve value reflect the cot of uncertanty at the plannng tage. The ame problem ntance alo olved ung Monte Carlo amplng. The reult obtaned (ee Table 4) llutrate the wdenng gap between analytcal ntegraton and tochatc amplng for larger problem ntance. Summary and Concluon In th paper, a two-tage modelng and oluton framework wa propoed for ncorporatng demand uncertanty n mdterm plannng problem. The mdterm plannng model of McDonald and Karm wa adopted a the reference model. Specfcally, the upply chan condered were characterzed by (em)contnuou procee and multple producton te. The parttonng of the varable and contrant of the model nto producton and upply chan provded the approprate tructure for a two-tage tochatc programmng formulaton. The producton decon, becaue of ther apprecable lead tme, were made n a here-and-now fahon before the uncertanty n demand wa reolved. Subequently, the wat-and-ee upply-chan decon were made on the ba of the producton decon and the realzaton of the demand. The expectaton evaluaton of the nner recoure problem wa reolved n two tep. The frt tep nvolved obtanng a cloed-form oluton of the nner problem ung P dualty. Th analy led to three dfferent optmal upply polce dependng on whether the product demand wa wthn the low-, ntermedate-, or hgh-demand regme. Baed on the nght obtaned from the ngle-te cae, the multte cae wa ubequently reolved. Two key ue dentfed n the analy for the multte cae were () the clafcaton of te nto type IS and ID and () the rankng of te on the ba of the OSS and USS cot n the low- and ntermedate-demand regme. The econd tep wa the computaton of the expectaton of the econd-tage cot by analytcal ntegraton. The reultng ngletage determntc equvalent MINP wa hown to have a convex contnuou part. A cutomzed veron of OA 36 wa mplemented. Computatonal reult for multte problem ndcated that the propoed analy and oluton framework wa at leat an order of magntude more effcent than amplng method uch a Monte Carlo ntegraton at the expene of retrctng the modelng of uncertanty to normal. In addton, a compartve tudy between the plannng uggeton of a determntc model and the propoed two-tage tochatc model howed that plannng avng can be realzed by recognzng and ncorporatng demand uncertanty n the decon-makng framework. It mportant to note that the normalty aumpton for the uncertan demand play a key role n the expectaton evaluaton tep of the propoed methodology. Extenon of th work to account for a general probablty dtrbuton under conderaton. Incorporaton of uncertanty n the econd-tage cot parameter uch a revenue, tranportaton cot, and underpenalty cot wthn the propoed analytcal framework alo beng explored. Furthermore, applcaton of the propoed methodology to the more general multperod and multcutomer problem under nvetgaton. Acknowledgment We acknowledge ueful dcuon wth Dr. Conor McDonald and fnancal upport by NSF-GOAI Grant CTS and DuPont Educatonal Ad Grant 1999/ Notaton Set I ) {} ) et of product I RM I ) {} ) et of raw materal I IP I ) {} ) et of ntermedate product I FP I ) {} ) et of fnhed product F ) {f} ) et of product famle J ) {j} ) et of proceor S ) {} ) et of producton te Parameter p ) prce of raw materal I RM at te FC f ) etup cot for famly f v j ) varable producton cot for product I RM on proceor j at te

13 Ind. Eng. Chem. Re., Vol. 39, No. 10, h ) nventory holdng cot for product I FP at te ζ ) penalty for dppng below target afety tock level of product I FP at te t /t ) tranportaton cot to move a unt of product from te to te /cutomer locaton µ ) revenue per unt of product I FP R j ) rate of producton of product I RM on proceor j at te β ) the yeld adjuted amount of raw materal or ntermedate product I FP that mut be conumed to produce a unt of I RM at te H j ) amount of tme avalable for producton on proceor j at te λ f ) bnary parameter ndcatng whether product belong to famly f MR fj ) mnmum run length for famly f on proceor j at te θ ) uncertan demand for product I FP I 0 ) nventory of product at te at the tart of the plannng horzon I ) afety tock target for product at te Varable Y fj ) { 1 f famly f proceed on proceor j at te 0 otherwe P j ) producton amount of I RM on proceor j at te R j ) run length of product I RM on proceor j at te FR fj ) run length for famly f on proceor j at te C ) conumpton of raw materal or ntermedate product I FP at te W ) flow of ntermedate product I IP from te to S ) upply of fnhed product I FP from faclty to the cutomer locaton I ) nventory level for I RM at te I ī ) amount of hortage of fnhed product I FP at the cutomer locaton I ) devaton below target afety tock level for product I RM at te Appendx A Conder the P dual of problem IP MPS. By aocatng nonnegatve dual varable u, v, and w wth eq 33, 34, and 35, repectvely, th can be formulated a ubject to max -θu + u,v,w (I - A )v - A w S The followng dual oluton potulated for the three S u + v + w g µ - t + h v e ζ u, v, w g 0 demand regme: ow-demand Regme u ) µ - γ l * {ζ ID v ) γ l * - γ e / l - 1, IS / 0 g l, IS w ) 0 Intermedate-Demand Regme u ) µ - ω l * {ζ / e l v ) ζ g / l + 1, ID ω * - γ g / l + 1, IS w ) { ω * l - ω e / l g -γ Hgh-Demand Regme u ) 0 v ) ζ w ) µ - ω The feablty of th oluton etablhed by ubttutng nto the contrant of the dual problem and ung the cot aumpton (eq 40-42) and the rankng cheme n the varou demand regme. In addton, th dual feable oluton reult n the ame objectve functon value for the three demand regme a the potulated prmal feable oluton. Dual oluton feablty and equalty of prmal and dual objectve mple the optmalty of the potulated upply polce baed on the prncple of trong P dualty. Appendx B et R l *, R I l *, and R H be the probablte that the product demand le n the low-, ntermedate-, and hgh-demand regme, repectvely. I R l * R l * ) Pr[ * l -1 * -1 ) Pr[Θ fi + ID (A - I ) e θ e (A - I )] (66) * -1 Subequently, the recoure functon Q(A ), whch quantfe the expected econd-tage cot for the multte cae, gven by l * A + I e θ e * Θ fi + ID * A + I ] (67) R H ) Pr[θ g Θ IfH ] (68) Q(A ) ) µθ m + Sh A + E (A ) + E I (A ) + E H (A ) (69)

14 3812 Ind. Eng. Chem. Re., Vol. 39, No. 10, 2000 where [- [µ - γl *]θ + ζ [I - A ] ID l )] * -1 + [γ E (A ) ) R l * - γ ][I - A ] l * E θ * l IS l -1 * l (A - I ) e θ e (A - I (70) E I (A ) ) I R l * * IS ID [- [µ - ωl *]θ + ζ l * [I l * - A l *] - -1 [ω * - ω ]A + -1 ζ [I - A ] ] ID IS ID IS E θ + ζ [I - A ] + [ω * - γ ][I - A ] g * +1 g * +1 ID Θ fi + -1 A + -1 I e θ e Θ fi + -1 A + -1 I ID ID (71) and E H (A ) ) R H E θ [-(µ - ω )A + ζ (I - A ) S θ g Θ IfH ] (72) repreent the expected econd-tage cot n the low-, ntermedate-, and hgh-demand regme, repectvely. Analytcal ntegraton of eq 70-72, facltated by the tandardzaton of the demand parameter and the defnton of K 1 l * and K 2 l * a σk l * 1 ) [ * l -1 reult n (A - I ) - θ m ] σk l * * 2 ) [Θ fi + ID A + I - θ m ] (73) IS -1 Q(A ) ) σ[γ l *+1 -γ l *][K 1 l * Φ(K 1 l *) + f(k 1 l *)] * l )1 + σ[ω l *)1 - γ l *) IS ][K l *) IS Φ(K l *) IS ) + f(k l *) IS )] S -1 + σ[ω l *+1 - ω ][K 2 * * * l l Φ(K 2 l *) + f(k 2 l *)] )1 + σ[µ - γ l *) S ][K 2 l *) S 2 - σµk l *) S 2 Φ(K l *) S + ζ I + S S where IS denote the IS te wth the hghet OSS cot and S repreent the te (IS or ID) wth the hghet * 2 ) + f(k l *) S )] t A (74) USS cot. The convexty of the nonlnear term n eq 74 preerved becaue they retan the ame form a the ngle-te cae. terature Cted (1) Subrahmanyam, S.; Pekny, J. F.; Reklat, G. V. Degn of batch chemcal plant under market uncertanty. Ind. Eng. Chem. Re. 1994, 33, (2) Shah, N. Sngle and Multte Plannng and Schedulng: Current Statu and Future Challenge. Foundaton of Computer Aded Proce Operaton Conference (FOCAPO), Snowbrd, Utah, July 5-10, (3) Petkov, S. B.; Marana, C. D. Multperod Plannng and Schedulng of Multproduct Batch Plant under Demand Uncertanty. Ind. Eng. Chem. Re. 1997, 36, (4) John, W. R.; Marketo, G.; Rppn, D. W. T. The optmal degn of chemcal plant to meet tme-varyng demand n the preence of techncal and commercal uncertanty. Tran. Int. Chem. Eng. 1978, 56, 249. (5) Rehhart, H. J.; Rppn, D. W. T. Degn of flexble multproduct plant - A new procedure for optmal equpment zng under uncertanty. Amercan Inttute of Chemcal Engneer Meetng, New York, (6) Wellon, H. S.; Reklat, G. V. The degn of multproduct batch plant under uncertanty wth taged expanon. Comp. Chem. Eng. 1989, 13, 115. (7) Shah, N.; Pantelde, C. C. Degn of multpurpoe batch plant wth uncertan producton requrement. Ind. Eng. Chem. Re. 1992, 31, (8) Ierapetrtou, M. G.; Ptkopoulo, E. N. Batch plant degn and operaton under uncertanty. Ind. Eng. Chem. Re. 1996, 35, 772. (9) Straub, D. A.; Gromann, I. E. Integrated tochatc metrc of flexblty for ytem wth dcrete tate and contnuou parameter uncertante. Comp. Chem. Eng. 1990, 14, 967. (10) Straub, D. A.; Gromann, I. E. Degn optmzaton of tochatc flexblty. Comp. Chem. Eng. 1993, 17, 339. (11) Georgad, M. C.; Ptkopoulo, E. N. An ntegrated framework for robut and flexble proce ytem. Comp. Chem. Eng. 1999, 38, 133. (12) u, M..; Sahnd, N. V. Optmzaton n Proce Plannng under Uncertanty. Ind. Eng. Chem. Re. 1996a, 35, (13) Clay, R..; Gromann, I. E. A daggregaton algorthm for the optmzaton of tochatc plannng model. Comp. Chem. Eng. 1997, 21, 751. (14) u, M..; Sahnd, N. V. Robut proce plannng under uncertanty. Ind. Eng. Chem. Re. 1998, 37, (15) McDonald, C.; Karm, I. A. Plannng and Schedulng of Parallel Semcontnuou Procee. 1. Producton Plannng. Ind. Eng. Chem. Re. 1997, 36, (16) u, M..; Sahnd, N. V. ong Range Plannng n the Proce Indutre: A Projecton Approach. Comp. Oper. Re. 1996, 23, 237. (17) Petkov, S. B.; Marana, C. D. Degn of Sngle-Product Campagn Batch Plant under Demand Uncertanty. AIChE J. 1998, 44, 896. (18) Nahma, S. Producton and Operaton Analy; Rchard D. Irwn Inc.: Homewood, I, (19) Dantzg, G. B. near Programmng Under Uncertanty. Manage. Sc. 1955, 1, 197. (20) Ierapetrtou, M. G.; Ptkopoulo, E. N.; Flouda, C. A. Operatonal Plannng under Uncertanty. Comp. Chem. Eng. 1994, S18, S553. (21) Ierapetrtou, M. G.; Ptkopoulo, E. N. Smultaneou ncorporaton of flexblty and economc rk n operatonal plannng under uncertanty. Comp. Chem. Eng. 1994a, 18, 163. (22) Ierapetrtou, M. G.; Ptkopoulo, E. N. Novel Optmzaton Approach of Stochatc Plannng Model. Ind. Eng. Chem. Re. 1994, 33, (23) Ptkopoulo, E. N. Uncertanty n proce degn and operaton. Comp. Chem. Eng. 1995, S19, S553. (24) Ptkopoulo, E. N.; Ierapetrtou, M. G. Novel approach for optmal proce degn under uncertanty. Comp. Chem. Eng. 1995, 19, (25) Clay, R..; Gromann, I. E. Optmzaton of Stochatc Plannng Model. Tran. Int. Chem. Eng. 1994, 72, 415.