Ind. Eng. Chem. Res. 2001, 40, 275-286 275 Sraegic Suy Chain Oimizaion for he Pharmaceuica Indusries Lazaros G. Paageorgiou,*,, Guiermo E. Rosein,, and Niay Shah Dearmen of Chemica Engineering, Universiy Coege London, Torringon Pace, London WC1E 7JE, U.K., and Cenre for Process Sysems Engineering, Imeria Coege, Prince Consor Road, London SW7 2BY, U.K. Pharmaceuica comanies are undergoing major changes o coe wih he new chaenges of he modern economy. The gobaizaion of he business, he diversiy and comexiy of new drugs, he increasing ighness of caia, and he diminishing roecion rovided by aens are some of he facors driving hese changes. A sages of he business vaue chain are affeced: from he deveomen of new drugs o he managemen of he manufacuring and mareing newors. This aer describes an oimizaion-based aroach o seecing boh a roduc deveomen and inroducion sraegy and a caaciy anning and invesmen sraegy. The overa robem is formuaed as a mixed-ineger inear rogramming (MILP) mode. This aes accoun of boh he aricuar feaures of harmaceuica acive ingredien manufacuring and he goba rading srucures. An iusraive exame is resened o demonsrae he aicabiiy of he roosed mode. 1. Inroducion Pharmaceuica comanies are undergoing major changes o coe wih he new chaenges of he modern economy. The inernaionaizaion of he business, he diversiy and comexiy of new drugs, and he diminishing roecion rovided by aens are some of he facors driving hese changes. A sages of he business vaue chain are affeced: from he deveomen of new drugs o he managemen of he manufacuring and suy newors. Mare ressures are aso forcing harmaceuica comanies o ae a more hoisic view of heir roduc orfoio. The yica ife cyces of new drugs are becoming shorer. I may ae 8 years o deveo a new roduc, and he invesmen on i mus be recovered quicy because generic equivaens can aear aer in he mare, reducing is rofiabiiy. Comanies are consany faced wih he quesion of he bes use of he imied financia resources avaiabe. For insance, consider he foowing siuaion. A comany is manufacuring and seing an esabished roduc (roduc A) and has deveoed a new roduc (roduc B) which wi be ready o be aunched in abou a year. I is aso saring o deveo anoher roduc (roduc C) which wi demand significan invesmen in research and deveomen (R&D) and require u o 6 years of deveomen ime. The comany mus somehow decide on he bes srucure for is fuure orfoio. One oion is o inves simuaneousy in manufacuring caaciy and R&D o buid u a arge orfoio wih he hree roducs. Aernaivey, i coud an o inves in R&D whie graduay hasing ou roduc A and aunching roduc B, in such a way ha no significan invesmen in caaciy is required. Severa oher oions exis, each * To whom a corresondence shoud be addressed. Te: +44-20-76792563. Fax: +44-20-73832348. E-mai:.aageorgiou@uc.ac.u. Universiy Coege London. Imeria Coege. Curren address: i2 Technoogies, The Priory, Burnham, Bershire SL1 7LL, U.K. of which differs in he invesmen required and he oenia reurn of he resuing orfoio. The chaenge is hen o oimize some erformance crierion of he orfoio whie avoiding unnecessary caia commimens. This robem has yicay been addressed in he harmaceuica indusry in a somewha simisic fashion. The vaue of he roduc orfoio is esimaed based on R&D coss and he oenia vaue esimaed for he roducs in he mare. Manufacuring coss were usuay considered o be negigibe. However, new drugs require more comex roducion ahs, and heir manufacuring coss are increasing and can nowadays ofen absorb 10-20% of he fina vaue of a drug. 1 There is herefore a need o deveo an aroach ha can simuaneousy consider (a) he R&D cos associaed wih he deveomen of oenia new roducs, (b) he commercia characerisics of each roduc (e.g., demand forecas, rice, mareing exenses, ec.), (c) he decisions associaed wih muie sies, e.g. where o insa/ exand caaciy, which roduc o roduce a which sie, ec., (d) he manufacuring coss and caaciy requiremens for he roduc orfoio [in he harmaceuica indusry, he ide ime for swiching roducion from one roduc o anoher is usuay very significan and herefore comex orfoios (i.e., wih many roducs) a a singe sie ofen resu in considerabe reducion of manufacuring efficiency], (e) he rading srucure of he comany (mos comanies nowadays manufacure and se inernaionay; differen commercia and roducion business ceners wi wor under differen ax regimes; he aocaion of he orfoio rofis among hese ceners is herefore imoran because i affecs he overa rofiabiiy execed from he orfoio). The robem is obviousy very chaenging because he aim is o bridge he ga curreny exising beween decision maers in radiionay isoaed areas, such as roduc deveomen, manufacuring, accouning, and commerciaizaion. This aer resens a mahemaica rogramming mode ha caures a of hese issues simuaneousy, 10.1021/ie990870 CCC: $20.00 2001 American Chemica Sociey Pubished on Web 12/08/2000
276 Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 hus roviding an effecive suor for a hoisic aroach o decision maing. Various asecs of his robem, each of hem in isoaion, have been considered in he as. One area is muisie anning and disribuion in he rocess indusries. Here, i is execed ha arge benefis sem from coordinaed anning across sies, in erms of coss and mare effeciveness. Mos business rocesses dicae ha a degree of auonomy is required a each manufacuring and disribuion sie, bu ressures o coordinae resonses o goba demand whie minimizing cos imy ha simuaneous anning of roducion and disribuion across ans and warehouses shoud be underaen. The need for such coordinaed anning has ong been recognized in he managemen science and oeraions research ieraure. For exame, ref 2 surveyed a series of heurisics for roducion-disribuion scheduing in muisie sysems for differen newor srucures. There are wo genera weanesses wih research in his domain: (i) seadysae demands are assumed; (ii) sime exressions or even consans are used for an caaciy. In racice, demands are usuay ime-varying and he caaciy of a fexibe manufacuring faciiy canno be nown a riori bu is raher a funcion of he roduc mix and he deais of anning. A oenia robem wih his aroach as recognized in ref 3 is he very arge robem sizes ha wi ensue. A secondary issue is ha he deveomen of a cenra an o a very fine eve of deai is robaby unnecessary, eading o he deveomen of an aggregaion rocedure. The aim is o caure roducion and disribuion caaciies accuraey wihou considering deaied scheduing. The mehod invoves aggregaing he many discree inervas ino fewer, onger inervas nown as aggregaed ime eriods (ATPs). This echnique has been aied by ref 4 o a coninen-wide indusria case sudy. This invoved oimay anning he roducion and disribuion of a sysem wih 3 facories and 14 mare warehouses and over 100 roducs. A grea dea of fexibiiy exised in he newor which, in rincie, enabes he roducion of roducs for each mare a each manufacuring sie. I was found ha he abiiy of he echnique o caure effecs such as muiurose oeraion, inermediae sorage, and seus gave rise o counerinuiive resus, such as roducing maerias furher away from demand oins han woud be execed. A simiar robem suiabe for muie faciiies which effecivey roduce roducs on singe-sage coninuous ines for a number of geograhicay disribued cusomers is described in ref 5. Their basic mode is of muieriod inear rogramming (LP) form and aes accoun of avaiabe rocessing ime on a ines, ransoraion coss, and shorage coss. An aroximaion is used for he invenory coss, and roduc ransiions are no modeed. The mode is exended o incude minimum run enghs (which requires he incusion of binary variabes). They incude a number of addiiona suy chain reaed consrains such as singe sourcing, inerna sourcing, and ransoraion imes. This ye of aggregae mode is revaen in he ieraure. Oher anning modes of his ye do no consider each roduc in isoaion bu raher grou roducs ha ace simiar demands on resources ino famiies and base he higher eve anning funcion on hese famiies. This forms he basis of many hierarchica roducion anning sysems (see, e.g., ref 6). More sohisicaed modes exis in he rocess sysems ieraure. A mode which seecs rocesses o oerae from an inegraed newor is decribed in ref 7 whie ensuring ha he newor caaciy consrains are no exceeded. Means of imroving he souion efficiency of his cass of robems can be found in refs 8 and 9. Ahough hese aroaches have some degree of reevance o his wor, in aricuar o reae invesmens in caaciy o he overa abiiy o roduce a cerain eves, considerabe wor remains o be done o caure he saien deais reevan o he seecion and ong-erm manufacure of harmaceuica acive ingrediens a muie sies. The mode we roose is based on a famiy of sraegic anning robems ha we have wored on during he as 3 years. Mos eemens of i are quie generic, bu, of course, some (e.g., rading srucure) wi vary beween organizaions. The res of he aer is srucured as foows. In secion 2, he main characerisics of he robem are discussed. The roosed mahemaica mode ogeher wih he ey assumions are described in secion 3. An iusraive muisie exame is resened in secion 4 o demonsrae he aicabiiy of he mode. Finay, some concuding remars are given in secion 5. 2. Probem Saemen This aer considers he deveomen of modes ha can suor a hoisic aroach o roduc orfoio managemen in he harmaceuica indusry. There are hree main issues ha are o be considered during he oimizaion of he roduc orfoio of a yica harmaceuica indusry. Produc Managemen. This is concerned wih he main feaures of each roduc considered as a suiabe candidae for manufacuring and commerciaizaion. Caaciy Managemen. This is concerned wih he aocaion of he exising caaciy (a more han one sie) for he seeced roduc orfoio and decisions concerning addiiona invesmens ha may be required o saisfy fuure demands. Trading Srucure. This is concerned wih decisions reaed o he financia fows beween differen comonens of he comany, in aricuar, ransfer ricing issues among he various manufacuring and commerciaizaion business ceners. These issues are discussed in he nex hree subsecions. 2.1. Produc Managemen. Pharmaceuica roducs are made in wo main sages ( rimary or secondary manufacure). The firs sage roduces sma quaniies of he acive ingredien (AI), a high-quaiy, high-vaue chemica. The second sage convers his AI ino a roduc for fina use (e.g., as abes, vias, ec.). The rimary sage is he criica one for orfoio anning; i is his comonen ha we sha consider furher in his aer. The saring oin of he anaysis is he candidae orfoio, a se of roducs ha are being considered for deveomen, manufacuring, and commerciaizaion. The foowing daa are required for he anaysis of he roe of hese roducs wihin he orfoio: (a) Demand forecass: he quaniies of maeria ha are execed o be sod year by year wihin he ime horizon considered. These are, in he genera case, sochasic. In his wor, we rea hem as deerminisic
Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 277 and indicae how a caaciy an can be seeced for such forecass. Fuure wor wi consider sochasic demand forecass deenden on cinica ria oucomes. (b) Cinica rias suy: he quaniies of maeria ha are required for cinica rias (roduc esing) wihin he ime horizon considered. (c) Price rofie: his is he forecased seing rice for each roduc. Ofen he rice is execed o decrease wih ime as he roduc maures. (d) Manufacuring cos: his is he cos o roduce a given amoun of he roduc, usuay deermined by is bi of maerias. (e) Royaies: his is he fracion of he revenues of each roduc o be aid o hird aries. This is common in he harmaceuica comanies where roducs may be deveoed in a cooeraive fashion beween generaors of new chemica eniies, icensors, and manufacurers. (f) Deveomen cos: his is he invesmen required for he deveomen of each roduc in he candidae orfoio. 2.2. Caaciy Managemen. Once he deais of each roduc in he candidae orfoio are nown, he nex se is o idenify he caaciy needs for each individua roduc and he imac on he caaciy of a an when severa roducs are manufacured in i. The aer is of significan imorance in he harmaceuica indusry because ong seu imes beween differen roducs (e.g., 1 monh) are no unusua. The ans usuay oerae in camaigns, and each addiiona roduc manufacured in he an inroduces significan caaciy osses. The foowing daa are herefore required for an accurae anaysis of he caaciy needs: (a) Caia coss: his is he invesmen required for each new individua manufacuring faciiy and/or cos associaed wih exansion by a resecified amoun. Aso, he associaed dereciaion rae is aen ino accoun. (b) Consrucion ead imes: his is he ime required eiher for buiding of a faciiy or for fuure exansions by adding exra equimen. Noe ha, once he faciiy is in ace, fuure exansions can be reaivey fas. (c) Fixed oeraing coss: overheads required o run a aricuar faciiy in 1 year. These may be very reaivey arge in he harmaceuica indusry, where significan aboraory and anayica suor is required. (d) Producion raes er roduc: his is he rae a which a roduc can be manufacured during a camaign in a aricuar faciiy. (e) Seu ime: his is he ime required o swich a faciiy from he manufacuring of one roduc o anoher. (f) Scae-u ime: he firs ime a roduc is manufacured in a faciiy, here is some ime required o go hrough he earning curve and achieve he desired quaiy arges. This inroduces significan caaciy osses when a new roduc is firs manufacured in a an. (g) Quaificaion ime: once scae-u is comeed, a minimum amoun (nominay, five consecuive baches) of each roduc mus be manufacured o comy wih sandard reguaory requiremens. 2.2.1. Manufacuring Comonens. The manufacuring equimen a each sie is organized ino bocs. Each boc conains wo yes of eniies: suies and service ceners. A suie, in urn, comrises a roducion ine and is coued urificaion ine. These suies are Figure 1. Boc wih one service cener and four suies. avaiabe in idenica caaciies and nown fixed cos. Each boc maes use of services such as uiiies, adminisraion, and anayica/aboraory faciiies. We assume here ha a mos four suies can share one service cener, creaing a boc as shown in Figure 1. The firs suie o be aached o a boc is denoed he header suie. Then, wih a number of exising bocs, he robem is hen esseniay o seec he roducs for manufacure over he horizon as we as when o sar invesing in new bocs or in rojecs o ugrade exising bocs. Each boc need no have a of is suies consruced a he same ime, whie he invesmen sraegy mus ae accoun of he ead imes associaed wih consrucion and commissioning. Addiionay, he roduc-o-suie assignmen shoud ae accoun of he comexiy inroduced if oo many differen roducs are o be manufacured in he same faciiy. 2.2.2. Processing Consideraions. The rocessing of he roducs in he suies has a number of characerisics. Firs of a, before a camaign of a aricuar roduc is sared in a suie, he suie mus be ceaned horoughy. This aes a ong ime (e.g., 1 monh). Then, he camaign engh is usuay subjec o minimum and maximum duraions. Ahough roducs are roduced a a nomina rae during he camaign, roducion oss facors need o be incuded o accoun for acua roducion eves ha end o be ower han he nomina ones. When a roduc is made for he firs ime in a new geograhica ocaion (sie), i mus firs undergo a scaeu aciviy. This refecs ha i aes some ime for he new sie o earn o roduce he roduc in a saisfacoriy reeaabe fashion. Finay, once his has been achieved, he firs few baches of he roduc ever roduced a a sie (he quaificaion amoun) mus be sen o he reevan reguaory auhoriies for arova. 2.3. Trading Srucure. In genera, modes deaing wih he managemen of manufacuring and caaciy anning ignore he inerna rade and ransfer ricing ha aes ace wihin any cororaion. However, his inerna rading srucure ofen has a arge imac on he afer-ax rofiabiiy of he orfoio. Aso, i may argey affec decisions concerning he ocaion of a new an, for insance, when ocaions wih differenia ax raes are considered. Three main business ceners are
278 Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 Figure 2. Tyica inerna rading srucure. usuay resen in one form or anoher wihin he suy chain newor of he comany: (a) Ineecua roery (IP) owner: his is he secor of he organizaion resonsibe for he funding and deveomen of new roducs. (b) Producion sies: hese are he roducion ocaions resonsibe for roduc manufacuring. (c) Saes regions: hese are he secors of he comany resonsibe for he mareing and saes effors required o se he roducs. Significan financia fows ae ace among hese ceners, and he ransfer ricing srucure adoed (i.e., he inerna seing rices) may have a significan imac on he rofis made a each cener and herefore he axes aid a he corresonding cener ocaion. The oeraing mode of each cener can be defined as foows: Profi cener: here significan rofis are reaized. Usuay, heir roducs are sod according o a resae minus formua (i.e., he fina seing rice minus a redeermined ercenage). Cos cener: hese ceners usuay ony cover heir own coss us a sma rofi. Therefore, he inerna seing rice for he roduc is esabished according o he fuy absorbed coss (incuding caia dereciaion) us a sma ercenage. Figure 2 shows a yica rading srucure for a harmaceuica comany. In his case, a roducion sies wor as cos ceners (simiar o conrac manufacuring), whie he IP owner and he saes regions wor as business ceners and reain mos of he rofis. In he figure, he IP owner buys he roduc a cos + 20% and ses i a a rice of resae - 25% (i.e., he fina seing rice minus 25%). A Comrehensive Mode. We are now in a osiion o sae he robem formay. The foowing iems are assumed o be nown: 1. A se of oenia roducs (he candidae roduc orfoio). 2. The demand forecass (in iograms er year) for each acive ingredien roduc in he candidae orfoio. 3. The minimum demand fufimen for any roduc aunched ino he mare. 4. The forecased roduc mare rice for each year. 5. The commerciaizaion coss o be aid, such as royaies and mareing. 6. Variabe (margina) and fixed manufacuring coss. 7. Execed roducion osses. 8. The shef-ife of each roduc. 9. Oher manufacuring coss such as seus or scaeu. 10. Inerna ineres raes and execed infaion raes. 11. Producion raes for each roduc. 12. The ime required for manufacuring aciviies such as seus and scae-u. 13. Consrucion ead imes. 14. Minimum and/or maximum roducion imes. 15. The rading srucure of he comany (organizaion ino cos and rofi ceners) and he axaion rae a each cener. 16. Dereciaion raes for caia invesmens. On he basis of he daa above, we see o deveo a mahemaica oimizaion mode which can deermine he foowing quaniies: 1. The roduc orfoio (i.e., roducs from he candidae orfoio ha are deveoed and aunched). 2. Wha manufacuring caaciy o inves in over he ime horizon. 3. Timing for scae-u and quaificaion runs for each roduc. 4. Producion ans er year for each roduc, incuding roduc aocaion o manufacuring sies and he amoun manufacured (camaign enghs). 5. Saes and invenory anning. Finay, an objecive funcion mus be saed. Here, we oimize an overa erformance cierion (e.g., maximize he ne resen vaue). 3. Mahemaica Formuaion The indices, arameers, and ses associaed wih he roduc orfoio oimizaion robem are ised. Indices ) roduc ) year i, j ) suies ) roducion ocaion ) saes region Parameers M ) maximum number of suies being served by each aboraory (boc) L ) roducion osses of roduc ζ ) ifeime of roduc
Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 279 R ) quaificaion amoun of roduc r i ) roducion rae of roduc in suie i a roducion sie H ) avaiabe roducion ime over ime eriod T,min i ) minimum roducion ime of roduc in suie i a roducion sie T,max i ) maximum roducion ime of roduc in suie i a roducion sie τj )seu ime τˆ ) scae-u ime for roduc D ) demand of roduc a eriod a saes region δ i ) consrucion ime of suie i a roducion sie Ses P ) se of roducs for saes region R ) se of saes regions where here is a mare for roduc I ) se of suies a roducion sie F ) se of header suies a roducion sie, cf. secion 2.2.1; (card(f ) ) card(i ) DIV M) N i ) se of suies ha beong o he same aboraory as header suie i F (N i ) {j I : j ) i + 1,..., i + M - 1}) Nex, he ey variabes of he formuaion are described by referring firs o he ones ha are goba (i.e., hose no associaed wih secific roducion sies or saes regions). Binary Variabes U ) 1 if roduc is seeced for deveomen and manufacuring and 0 oherwise Coninuous Variabes I ) amoun of roduc in goba invenories a he end of eriod W ) amoun of roduc wased a eriod Each roducion sie is characerized by he foowing binary variabes: V Ẑ X Z i Y i E i A i ) 1 if roduc is seeced for deveomen and manufacuring and 0 oherwise ) 1 if scae-u of roduc aes ace over eriod and 0 oherwise ) 1 if roduc is firs roduced (quaificaion run) over eriod and 0 oherwise ) 1 if scae-u of roduc aes ace in suie i over eriod and 0 oherwise ) 1 if roduc is roduced in suie i over eriod and 0 oherwise ) 1 if suie i is invesed in a eriod and 0 oherwise ) 1 if suie i is avaiabe a eriod and 0 oherwise Each roducion sie is characerized by he foowing coninuous variabes: T i T i B i ) roducion ime of suie i during eriod ) roducion ime of roduc in suie i during eriod ) amoun of roduc roduced in suie i during eriod Finay, each saes region is soey characerized by he amoun of roduc sod a eriod, S. A resource characerisics (e.g., caaciy and consrucion ime) as we as roducion requiremens are assumed o be nown. The ime horizon is discreized ino T ime inervas of equa duraion. Generay seaing, he rocess dynamics (e.g., aciviy duraions) and decision-maing cyce mean ha a 1-year discreizaion inerva is suiabe. Saru and shudown eriods are considered o be negigibe comared o he duraion of each ime inerva. For he uroses of his aer, a daa are assumed o be deerminisic. Nex, we describe he mode consrains. 3.1. Produc and Suie Exisence Consrains. If a roduc is no seeced for deveomen and manufacuring gobay (i.e., U ) 0), hen his roduc shoud be excuded from any candidae roducion sie. Mahemaicay, we have When an invesmen decision for any suie is aen, a consrucion ime is required before ha suie becomes avaiabe. This can be modeed as foows: A i ) A i,-1 V e U, (1) 3.2. Boc Consrains. Each roducion sie is organized ino many bocs. Each boc can serve u o a maximum number of suies, M (here, 4 suies/boc). Usuay, he consrucion ime of he header suie of each boc is arger han ha of he res of he suies which beong o he same boc. Therefore, a reaion beween he header suie, i F, and he oher consiuen suies of he same boc, j N i, shoud be esabished: -(δ i -δ j ) E iθ g E j θ)1 + E i,-δi, i I, (2) This ensures ha a consiuen suie j can be invesed in a ime eriod ony if he header suie i has been invesed in a eas δ i - δ j ime eriods before eriod. 3.3. Invesmen Degeneracy Consrains. Usuay, he fina oima souion wi incude fewer suies han he maximum aowed. In such cases, here wi be significan souion degeneracy because differen invesmen suie combinaions based on idenica suies woud resu in he same objecive funcion vaue. This source of degeneracy can furher be eiminaed by aowing suie i o be oeniay invesed in a ime eriod ony if suie i - 1 has aready been invesed in a any revious ime eriod. This can be wrien mahemaicay as E iθ g E i+1, θ)1, i F, j N i, (3), i ) 1,..., I - 1, (4) The above consrains imiciy assume ha a suies require he same consrucion ime. In genera, every suie may have is own consrucion ime, and herefore consrains (4) coud furher be generaized o -(δ i -δ i+1 ) E iθ g E i+1, θ)1, i ) I - 1, (5) These consrains are simiar o consrains (3), where he header of each boc is reaed o he res of he suies which beong o he same boc whie he ones described in his subsecion are wrien for successive suies which may beong o he same or differen bocs. 3.4. Producion Consrains. Each roduc can be roduced in a secific suie ony if ha suie is avaiabe:
280 Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 Of course, consrains (6) may be disaggregaed o rovide a igher aernaive a he exense of a arger robem: The amoun of each roduc roduced wihin each avaiabe suie is given by he foowing equaiies: Furhermore, when a roduc is quaified (i.e., X 1), a minimum amoun, R, mus be roduced: I shoud be added ha he minimum required amoun shoud be roduced wihin a singe eriod; oherwise, he above consrains (8) shoud be exended o incude more han one ime eriod. 3.5. Invenory Consrains. The amoun of roduc sored gobay a he end of eriod wi be equa o he amoun a he revious eriod - 1 us he ne amoun roduced during eriod (aing ino accoun roducion osses) by a roducer sies minus he amoun sod o differen cusomer regions minus he amoun wased due o he imied roduc ifeime. Thus, we obain Because we consider reaivey ong ime eriods, we ignore any ransoraion effecs. I shoud aso be added ha, desie he fac ha he ime eriod is raher ong, he above consrains are imoran by roviding anned year-end invenories, which comanies aways find usefu. 3.6. Produc Lifeime Consrains. In he revious subsecion, we inroduced he W variabes in order o mainain feasibiiy in he case of imied roduc ifeimes. However, consrains are required o guaranee ha he amoun sored in each eriod canno be used afer he nex ζ ime eriods: 3.7. Timing Consrains. Once a roduc is o be roduced, hen aroriae minimum and maximum roducion imes (camaign enghs) shoud be enforced: and Y i e card() A i Y i B i e A i ) r i T i B i g R X i I I ) I,-1 + (1 - L ) T i T i I e g T,min i Y i e T,max i Y i, i I, (6), i I,,, i I,, (7) These consrains are ony acive if he Y i variabes are equa o 1; oherwise (i.e., Y i ) 0), he T i vaues are forced o zero. ),, (8) B i - S - W i I, (9) +ζ S θ θ)+1, (10), i I,, (11), i I,, (12) The oa roducion ime for every suie over each eriod (i.e., he oa ime ha he suie is busy roducing roducs), T i, is simy he summaion of he individua roduc roducion imes, T i : Aso, he oa roducion ime, T i, canno exceed he oa avaiabe roducion ime, H, minus he ime required for any necessary seus, τj, and scae-us, τˆ: T i T i ) T i e H - τj( Y i - A i ) -, i I, (13) τˆz i 3.8. Scae-U Consrains. Every new roduc inroduced a each roducion sie mus undergo a scaeu rocedure for a cerain eriod of ime, τˆ (for exame, 3 monhs), before saring commercia roducion. This scae-u ime refecs echnoogy ransfer from a aboraory or differen roducion sie o he new one. Every roduc is aowed o be roduced during a cerain eriod ony if scae-u has aen ace u o ha eriod. This can be exressed as If scae-u occurs (i.e., Ẑ ) 1), hen i shoud be erformed wihin a singe suie: Finay, a suie can be used for scae-u uroses ony if i is avaiabe: Consrains (17) can be disaggregaed o resu in a igher form simiary o consrains (6). 3.9. Quaificaion Consrains. Afer he scae-u rocedure, each seeced roduc shoud be quaified o verify ha he an is caabe of roducing ha roduc according o he saisfacion of he reguaory auhoriies. The firs baches roduced of each roduc are caed quaificaion baches and mus coincide wih he firs ime ha roducion of ha roduc occurs: X Ẑ θ g Y i θ)1 Z i ) Ẑ i I Z i e card() A i g Y i - j -1 Y jθ θ)1 The above consrains ensure ha he firs ime a roduc is roduced (Y i ) 1 and a Y jθ ) 0 for θ ) 1,..., - 1), X mus ae a vaue of 1. A a subsequen ime inervas, he righ-hand sides of consrains (18) wi be zero or negaive. Thus, o ensure ha every roduc is quaified ony once, he foowing consains shoud be incuded: X e V I shoud be noed ha consrains (19) are no required if here are coss associaed wih he X, i I, (14), i I,, (15),, (16), i I, (17), i I,, (18), (19)
Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 281 variabes wihin he objecive funcion (see subsecion A.1.6). In his case, he oimizaion agorihm wi force as many as ossibe of hese X variabes o zero. 3.10. Saes Consrains. Differen saes sraegies coud be adoed o refec aernaive mareing oicies of he comany. Here, we simy assume ha he amoun sod a each ime eriod does no exceed he forecased demand of ha roduc: S e D, P, (20) In some cases, he forecased demand, D, may no be me by saes, S. We define heir difference as he saes ga. A conservaive mareing oicy coud be adoed by reducing he forecased demand of he nex ime eriods by he arges saes ga having been exerienced before (his refecs he fac ha i is difficu o recaure mare share). Thus, he oimizaion agorihm is forced o maximize he corresonding saes in order o minimize fuure os ooruniies. To mode he above oicy, we firs inroduce new coninuous variabes, G, aong wih he foowing consrains: G g D θ - S θ whie consrains (20) shoud reaced by S e D - G, P, θ < (21), P, I shoud be noed ha he revious oicy shoud be used wih cauion because, in cerain cases where here are shar changes in demand, consrains (21) may give rise o infeasibiiies. An aernaive mareing oicy coud require a nondecreasing ercenage of saes over demands: S D e S,+1 D,+1, P, ) ˆ + 1,..., T - 1 (22) I is imiciy assumed in he above ha, afer he firs nonzero forecased demand of roduc a ime eriod ˆ (i.e., D,ˆ * 0), hen a subsequen forecased demands are aso nonzero, i.e., D θ * 0( θ ) ˆ + 1,..., T). 3.11. Objecive Funcion. The mode can accommodae a variey of erformance crieria usuay based on economics. In his aer, we ado he maximizaion of he ne resen vaue (NPV) over a fairy ong horizon of ineres (e.g., 10-15 years) as a yica objecive funcion, before or afer axes, Φ B or Φ A, resecivey, as decribed in Aendix A. 3.12. Summary of Formuaion. In concusion, he enire formuaion described in his aer is hen ouined. max Φ A ) ɛ Σ {SR [1 - ψ F - ψ IP (1 -F )] - MC (1 - ψ )} - ɛ (RC IP + RDC IP )(1 - ψ IP ) - ɛ (OC + CSQR )[1 + ψ F - ψ IP (1 +F )] + ɛ DC (ψ IP (1 +F ) - ψ F ) - ɛ CI subjec o Produc and Suie Exisence Consrains A i ) A i,-1 V e U + E i,-δi Boc Consrains -(δ i -δ j ) θ)1 E iθ g E j, Invesmen Degeneracy Consrains -(δ i -δ i+1 ) θ)1 E iθ g E i+1, Producion Consrains Y i B i e A i ) r i T i i IB i g R X Invenory Consrains I ) I,-1 + (1 - L ) i IB i Produc Lifeime Consrains +ζ I e θ)+1s θ Timing Consrains T i T i T i T i e H - τj( Y i g T,min i Y i e T,max i Y i ) T i Scae-U Consrains θ)1ẑ θ g Y i i IZ i Z i e A i, i I,, i F, j N i,, i ) 1,..., I - 1,, i I,,, i I,,,, - S,, i I,,, i I,,, i I, - A i ) - τˆ Z i ) Ẑ Quaificaion Consrains X g Y i Saes Consrains S - -1 j θ)1y jθ X e D e V, i I,,,,, i I,, - W,, i I,, i I,,,, P, 4. An Iusraive Exame Here, we consider a comany which can roduce 7 roducs (P1-P7) over a anning horizon of 10 years (2000-2009). A discreizaion inerva of 1 year is used for our mahemaica mode, resuing in 10 ime eriods. The daa used in he exame are modified versions of acua daa based on a famiy of resiraory, dermaoogy, and arhriis roducs of a major harmaceuica comany.
282 Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 Tabe 1. Tax Regions ocaion ax rae rofie reaive oeraing cos reaive caia cos A 0.28 B 0.28 1.00 1.00 C 0.00 ) 1-4 1.10 1.10 0.20 ) 5-10 1.10 1.10 D 0.30 0.27 1.30 Tabe 2. Produc Daa roduc R r i,min T i P1 6.1 2.2 1.8 4 0.10 40 0.165 P2 2.5 0.9 1.6 4 0.12 45 0.220 P3 2.5 0.9 1.7 4 0.10 25 0.220 P4 3.8 0.7 2.4 3 0.40 20 0.165 P5 5.1 1.9 1.8 2 0.20 40 0.330 P6 2.5 0.9 1.5 5 0.15 50 0.165 P7 1.4 1.5 1.1 2 0.20 60 0.550 There are four aernaive ocaions (A-D), where A and B are saes regions; A is he IP cener, whie B-D are he oenia roducion sies as shown in Figure 2. As aready menioned, in his aer, we assume ha he rading srucure is given ogeher wih he inerna ricing oicies (F ) 0.20 and F ) 0.25). Infaion and ineres raes of 3% and 15%, resecivey, are used for NPV cacuaions. The ax rae rofies as we as he reaive oeraing and caia coss for a ocaions are described in Tabe 1. Noe ha ocaion C offers a very significan incenive of no ax over he firs 4 years a he exense of increased oeraing and caia coss. Aso, ocaion D consiues an exensive invesmen aernaive. On he oher hand, he significany cheaer oeraing cos may force he oimizaion agorihm oward he seecion of ha ocaion. Every an oeraes u o 11 monhs over each year because 1 monh/year is required for genera an mainenance. Seu and scae-u duraions are 1 and 2 monhs, resecivey, which are quie common for highy reguaed and aseic rocesses. The ifeime for a roducs is given equa o 4 years, whie roducion osses of 10% are assumed for a roducion sies. Daa reaed o he quaificaion amouns, roducion raes, minimum roducion imes, and saes rices for a roducs are given in Tabe 2. Noe ha roducion raes and minimum roducion imes are assumed o be ocaion and suie invarian (i.e., heir vaues deend ony on he roduc). In addiion, we assume ha he seing rice and royaies for a roducs are he same for boh saes regions (A and B). We furher assume ha he R&D cos for a roducs is sufficieny caured by he iniia caia cos. The mareing cos fracion (µ ) for boh saes regions is ime-deenden: 0.15 from he saru o he fourh year of demand and 0.05 for he res of he anning horizon. The above mareing oicy deics a more inensive mareing effor during he firs years of saes of each roduc. ν λ σ 1 ξ i B Each consrucion boc comrises u o four roducion suies. A ead ime of 2 years is required for a suies regardess of he ocaion, whie 3 years is needed for he header suie. Two suies are assumed o be avaiabe a ocaion B a he sar of he anning horizon. Aso given are he maximum number of suies o be invesed in a each roducion ocaion: eigh, four, and four suies for ocaions B-D, resecivey. However, he oimizaion agorihm wi deermine wheher, where, and he number of addiiona suies ha are required for every roducion ocaion. A header suies a ocaion B cos 100 rmu ( reaive money unis - U.K. ounds aroriaey scaed), whie he res of he suies a he same ocaion cos 45 rmu. In addiion, he fixed oeraing coss for ocaion B, η B i, are 22 and 11 rmu/suie for he header and remaining suies, resecivey. The variabe oeraing coss, ξ B i, are given in Tabe 2. I shoud be added ha he associaed coss for oher ocaions raher han B can be found by muiying he reevan coss of B wih he reaive oeraing and caia coss as shown in Tabe 1. Scae-u and quaificaion coss are assumed o be negigibe in his aer, ahough heir aroriae imings are aen ino accoun. Finay, he demand aerns for he wo saes regions are given in Tabe 3. Noe ha roducs P1, P2, P4, and P6 are sod in saes region A, whie roducs P3, P5, and P8 are sod a B. The above exame was modeed using he GAMS modeing sysem 10 coued wih CPLEX 6.0 for he MILP oimizaion. A 5% margin of oimaiy was used during he branch-and-bound souion rocedure using a Sun Ura60 worsaion. The resuing mahemaica mode, which comrises 3008 binary and 2718 coninuous variabes, was soved in 653 s. The oima souion (wihin he 5% margin) has an NPV vaue of 837 rmu. The deaied breadown of he objecive funcion is shown in Tabe 4. The oimizaion agorihm has seeced five roducs ou of he seven oenia ones, rejecing roducs P2 and P4 as nonromising. Aso, roducion ocaions B and D have been seeced for furher caaciy invesmens wihou aocaing any caaciy a ocaion C. In oa, six addiiona suies (aar from he iniia wo a ocaion B) are going o be invesed in, hus resuing in a cumuaive roducion caaciy of eigh suies beween ocaions B and D. Two of hese suies wi be added o ocaion B and he res o ocaion D. Tabe 5 shows he scae-u and quaificaion imings as deermined by he agorihm for boh seeced roducion ocaions. Finay, he oima fows of maeria for a seeced roducs are iusraed in Figures 3-7. I can be seen from hese figures ha mos of he demand of he seeced roducs is saisfied aar from roduc P3. Tabe 3. Producs Demand roduc 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 P1 4.2 9.0 22.4 21.6 22.2 22.2 22.2 22.2 22.2 22.2 P2 0 1.0 4.4 8.6 9.0 11.0 11.0 11.0 11.0 11.0 P3 2.2 6.4 10.4 13.6 16.0 16.2 16.2 16.2 16.2 16.2 P4 0 3.8 7.2 12.2 18.6 21.4 21.4 21.4 21.4 21.4 P5 0 0 5.0 11.0 21.4 24.4 28.2 28.2 30.8 31.6 P6 4.0 7.2 9.6 15.2 20.2 23.8 23.0 22.0 20.2 20.2 P7 14.4 15.0 16.8 18.6 20.8 21.0 21.0 21.0 21.0 21.0
Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 283 Figure 3. Profie for roduc P1. Figure 5. Profie for roduc P5. Figure 4. Profie for roduc P3. Tabe 4. Objecive Funcion Breadown saes 2582 mareing cos 235 royaies and R&D coss 742 oeraing coss 325 coss of scae-u and quaificaion runs 0 dereciaion cos 66 caia invesmen 377 NPV 837 5. Concuding Remars In his aer, he main objecive was o ay mahemaica rogramming echniques so as o faciiae he sraegic suy chain decision-maing rocess for harmaceuica indusries. An oimizaion-based aroach has been resened o seec boh he oima roduc deveomen and inroducion sraegy ogeher wih ong-erm caaciy anning and invesmen sraegy a muie sies. The deveoed mahemaica Figure 6. Profie for roduc P6. mode considers many asecs eseciay reaed o harmaceuica secor (e.g., scae-u, quaificaion, roduc ifeime consrains). The overa robem has been formuaed as an MILP mode which can hen be soved o goba oimaiy rovided ha he mode size is racabe. The aicabiiy of he roosed mode has been demonsraed by one iusraive exame. In reaisic case sudies, he resuing mode size migh be rohibiivey arge and aernaive souion aroaches are worh invesigaion. Our curren research focuses on he deveomen of aggregaed mahemaica modes coued wih a hierarchica souion rocedure for he efficien souion of he resuing MILP mode. Anoher ineresing exension of our wor is o incude he considerabe demand uncerainy which exiss in his indusry. For exame, he uncerainy on he oucome of he cinica rias of a candidae roducs coud be incororaed wihin our exising Tabe 5. Scae-U and Quaificaion Timings a roducion sie B roducion sie D roduc 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 P1 X+ X+ P3 X+ X + P5 X+ X+ P6 X + X+ P7 X+ X+ a X: scae-u. +: quaificaion.
284 Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 The reevan cos of header suies (i.e., i F ) is higher han hose of he ohers in order o ae accoun of ab consrucion cos required for each new boc. A.1.4. R&D Cos. Aso, he cos sen a he IP associaed wih he R&D over eriod, RDC IP, shoud be incuded: RDC IP ) σ U (27) Figure 7. Profie for roduc P7. framewor (see, for exame, ref 12, which deas wih he singe-sie case). Finay, we have assumed a given rading srucure wih fixed arameers. In some cases, eiher or boh of hese may be oimized as we. Aendix: Derivaion of he Objecive Funcion A.1. Objecive Funcion (before Taxes). The obecive funcion incudes revenues from saes, caia invesmen, R&D coss, oeraing coss, and coss of scae-u and quaificaion runs. Nex, we consider each of hese erms in sequence. A.1.1. Saes Revenue. The revenue due o roduc saes in region over eriod, SR, can simy be saed as: where ν is he saes rice for roduc a region. A.1.2. Saes-Based Coss. Furhermore, he above amoun, SR, shoud be reduced accordingy in order o caure royaies for IP, RC IP, usuay o hird ary comanies, as we as mareing conribuions for saes cener, MC. Boh of hese erms are assumed o be rooriona o he amouns sod. Therefore, we obain and SR ) ν S P RC IP ) λ ν S P MC ) µ ν S P where λ and µ are he royaies and mareing coss (as fracions of he seing rice) for each roduc a each saes region, resecivey. The aer cos coefficien is ime-deenden o caure he case where a more inensive mareing effor is required during he aunch of each roduc. A.1.3. Caia Invesmen. The caia invesmen reaed o roducion sie in eriod, CI, deends on he cos of each suie i of each roducion sie, ω i : CI ) ω i E i i I, (23) (24), (25), (26) where σ is he corresonding ime-varying R&D cos of roduc. Ahough he above summaion indicaes a imevarying invesmen on R&D, i is usuay considered as an iniia caia cos (σ 1 * 0 and σ ) 0 for > 1). A.1.5. Oeraing Cos. The oeraing cos of roducion sie in eriod, OC, associaed wih each suie i a each ocaion comrises wo erms. The firs one is a fixed amoun, η i, aid each ime eriod if suie i exiss (i.e., A i ) 1): The second erm deends on he acua amoun of roduc roduced a suie i a ocaion during eriod, B i : where ξ i is he corresonding cos of roducing roduc in suie i of ocaion. Therefore, he oa oeraing cos er ime eriod wi be given by A.1.6. Cos of Scae-U and Quaificaion Runs. The cos associaed wih scae-u and quaificaion runs a roducion sie over eriod, CSQR, can be described mahemaicay as where β and γ η i A i i I ξ i i I OC ) (η i A i i I B i + ξ i,, B i ), (28) CSQR ) (β Ẑ + γ X ), (29) reresen scae-u and quaificaion coss, resecivey. Finay, for NPV cacuaions, we mus inroduce a discoun facor, ɛ. This facor is associaed wih he infaion rae, f, and ineres rae, g, according o he foowing formua: ɛ ( 1 + f 1 + g) -1 NPV is based on a common currency (e.g., US$). Hence, f and g are assumed o be indeenden of ocaion. Overa, he objecive funcion, Φ B, being maximized is he foowing: Φ B max ɛ [ (SR - MC ) - (CI + OC + CSQR ) - RC IP - RDC IP ] (30)
Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 285 So far, we have no considered axes, and genera issues associaed wih rading srucures have no been aen ino consideraion. In he nex secion, we describe how axes and he rading srucure affec he erformance of he suy chain. A.2. Objecive Funcion (afer Taxes). In his aer, we sha ado he rading srucure iusraed in Figure 2. I can ceary be seen from he figure ha here are hree disincive cener yes: roducion sie (), ineecua roery owner (IP), and saes region (). The rading srucure and eseciay he ransfer ricing srucure adoed may have a significan imac on he axes aid by each cener, hus affecing he afer-ax rofis made a each cener. I shoud be recaed ha he roducion sies are considered as cos ceners and he saes regions as rofi ceners. In genera, he objecive funcion afer axes, Φ A, wi be equa o Φ B minus he axes aid by each cener, Th, Th, and Th IP. Overa, we have Φ A ) Φ B - ɛ ( Th + Th + Th IP ) (31) Nex, we examine each cener in urn o deermine is individua ax comonen. A.2.1. Saes Region. Each saes cener has associaed saes revenue (SR ) and mareing cos (MC ) comonens as described in he revious secion. In addiion, an inerna cos is aid from each saes cener o IP based on he roduc seing rice minus a redeermined margin, F, i.e., (1 -F )SR. Therefore, he rofi of each saes cener over eriod, Π, is given by Π ) SR - MC - (1 -F )SR, (32) Then, he ax aid by each cener a eriod, Th, can be cacuaed by Th ) ψ Π, (33) or by using consrains (32) o eiminae he Π variabes: Th ) ψ (F SR - MC ), (34) where ψ is he ax rae for saes region during ime eriod. Noe ha differen axaion oicies over ocaion and ime may ay a very imoran roe during oimizaion. A.2.2. Producion Sie. Each roducion suie incurs he oeraing coss (OC ), cos for scae-u and quaificaion runs (CSQR ), and caia invesmen (CI ) coss as described in he revious secion. To cacuae he ax reaed o each roducion sie a each ime eriod, Th, caia invesmen is no incuded bu dereciaion cos is aen ino accoun as exained nex. In afer-ax cacuaions, we need o incude dereciaion comonens, DC. Dereciaion can simy be considered as a governmena ax incenive which reresens a ax-free annua exense due o equimen decay. Each governmen aows comanies o deduc ar of he caia invesmen for a given ax ife eriod, φ, from heir rofis. Here, we ado he sraigh ine dereciaion mode for φ ime eriods afer an equimen iem becomes avaiabe. Therefore, he corresonding dereciaion, DC, comonen wi be The inerna seing rice from he roducion sie cener o IP is based on incurred cos us a redeermined margin, F, i.e., (1 +F )(DC + OC + CSQR ). Therefore, he rofi of each roducion sie cener a eriod, Π, is given by The ax aid by each roducion cener a eriod, Th, is as foows: or by using consrains (36) o eiminae Π variabes: where ψ is he reevan ax rae. A.2.3. Ineecua Proery. The rofi generaed a he ineecua roery, IP, cener is due o he inerna roduc seing, firs, from IP o saes regions (i.e., Σ (1 -F )SR and, second, from roducion sies o IP [i.e., Σ (1 +F )(DC + OC + CSQR )]. Of course, he coss associaed wih royaies and R&D shoud aso be incuded. Therefore, he rofi for he IP cener over eriod, Π IP, is given by whie he corresonding ax, Th IP, aid by IP wi be or -δ i ω i DC ) i I θ)-δ i -φ +1 φ E iθ, (35) Π ) (1 +F )(DC + OC + CSQR ) - DC - OC - CSQR, (36) Π IP ) Th IP ) ψ IP [ Th ) ψ Π, (37) Th ) ψ F (DC + OC + CSQR ), (38) (1 -F )SR - (1 +F )(DC + OC + CSQR ) - RC IP - RDC IP Th IP ) ψ IP Π IP (39) (40) (1 -F )SR - (1 +F )(DC + OC + CSQR )] - ψ IP (RC IP + RDC IP ) (41) where ψ IP is he IP ax rae. In concusion, he objecive funcion, Φ A, o be maximized, by aroriaey eiminaing he axaion erms from consrains (31) and aying consrains (34), (38), and (41), is he foowing: Φ A ) Φ B - ɛ ψ (F SR - MC ) - ɛ ψ F (DC + OC + CSQR ) - ɛ {ψ IP [ (1 - F )SR - (1 +F )(DC + OC + CSQR )]} + ɛ [ψ IP (RC IP + RDC IP )] (42) Finay, if we subsiue Φ B by reaions described in
286 Ind. Eng. Chem. Res., Vo. 40, No. 1, 2001 secion A.1, hen eq 42 can be rewrien as Φ A ) Saes ɛ SR [1 - ψ F - ψ IP (1 -F )] Mareing Cos - ɛ MC (1 - ψ ) Royaies and R&D Coss - ɛ (RC IP + RDC IP )(1- ψ IP ) Oeraing Cos - ɛ OC [1 + ψ F - ψ IP (1 +F )] Coss of Scae-U and Quaificaion Runs - ɛ CSQR [1 + ψ F - ψ IP (1 +F )] Dereciaion Cos + ɛ DC [ψ IP (1 +F ) - ψ F ] Caia Invesmen - ɛ CI I shoud be noed ha if a ceners are ocaed a he same ace (i.e., ψ ) ψ IP ) ψ ψ, hen consrain (43) becomes max Φ A ) ɛ OC - CSQR )(1 - ψ ) + (SR - MC - RC IP - RDC IP - ɛ DC ψ - ɛ (43) CI (44) Lieraure Cied (1) Pisano, G. P. The Deveomen FacorysUnocing he Poenia of Process Innovaion, 3rd ed.; Harvard Business Schoo Press: Boson, 1997. (2) Wiiams, J. F. Heurisic echniques for simuaneous scheduing of roducion and disribuion in muiecheon srucuresheory and emirica comarisons. Man. Sci. 1981, 27, 336-352. (3) Wiinson, S. J.; Shah, N.; Paneides, C. C. Aggregae modeing of muiurose an oeraion. Comu. Chem. Eng. 1995, S19, S583-S588. (4) Wiinson, S. J.; Corier, A.; Shah, N.; Paneides, C. C. Inegraed roducion and disribuion scheduing on a euroewide basis. Comu. Chem. Eng. 1996, S20, S1275-S1280. (5) McDonad, C. M.; Karimi, I. A. Panning and scheduing of arae semiconinuous rocesses. 1. Producing anning. Ind. Eng. Chem. Res. 1997, 36, 2691-2700. (6) Biran, G. R.; Haas, E. A.; Hax, A. C. Hierarchica roducion anning: A wo-sage sysem. Oer. Res. 1982, 30, 232-251. (7) Sahindis, N. V.; Grossmann, I. E.; Fornari, R. E.; Chahranhi, M. Oimizaion mode for ong-range anning in he chemica indusry. Comu. Chem. Eng. 1989, 13, 1049-1063. (8) Sahindis, N. V.; Grossmann, I. E. Reformuaion of muieriod oimizaion mode for ong range anning in he chemica indusry. Comu. Chem. Eng. 1991, 15, 255-272. (9) Miu, M. L.; Sahinidis, N. V. Comuaiona rends and effecs of aroximaions in an mi mode for rocess anning. Ind. Eng. Chem. Res. 1995, 34, 1662-1673. (10) Brooe, A.; Kendric, D.; Meeraus, A.; Raman, R. GAMS: A User s Guide; GAMS Deveomen Cororaion, Washingon, 1998. (11) Rosein, G. E.; Paageorgiou, L. G.; Shah, N.; Murhy, D. C.; Musafa, R. A roduc orfoio aroach in he harmaceuica indusry. Comu. Chem. Eng. 1999, S23, S883-S886. Received for review December 1, 1999 Acceed Ocober 10, 2000 IE990870T