Global supply chain planning for pharmaceuticals

Transcription

1 chemcal engneerng research and desgn 8 9 ( ) Conens lss avalable a ScenceDrec Chemcal Engneerng Research and Desgn journal homepage: Global supply chan plannng for pharmaceucals Ru T. Sousa a, Songsong Lu b, Lazaros G. Papageorgou b, Nlay Shah a, a Cenre for Process Sysems Engneerng, Imperal College London, London SW7 2AZ, UK b Cenre for Process Sysems Engneerng, Deparmen of Chemcal Engneerng, Unversy College London, Torrngon Place, London WC1E 7JE, UK absrac The shorenng of paen lfe perods, generc compeon and publc healh polces, among oher facors, have changed he operang conex of he pharmaceucal ndusry. In hs work we address a dynamc allocaon/plannng problem ha opmses he global supply chan plannng of a pharmaceucal company, from producon sages a prmary and secondary ses o produc dsrbuon o markes. The model explores dfferen producon and dsrbuon coss and ax raes a dfferen locaons n order o maxmse he company s ne prof value (NPV). Large nsances of he model are no solvable n realsc me scales, so wo decomposon algorhms were developed. In he frs mehod, he supply chan s decomposed no ndependen prmary and secondary subproblems, and each of hem s opmsed separaely. The second algorhm s a emporal decomposon, where he man problem s separaed no several ndependen subproblems, one per each me perod. These algorhms enable he soluon of large nsances of he problem n reasonable me wh good qualy resuls Publshed by Elsever B.V. on behalf of The Insuon of Chemcal Engneers. Keywords: Global supply chan; Pharmaceucals; Mxed neger lnear programmng; Large-scale opmsaon; Decomposon algorhm 1. Inroducon In he pas 30 years, he operang conex of he pharmaceucal ndusry has evolved and become much more challengng. The esablshmen of regulaory auhores and marke maury have led o an ncrease n he coss and me o develop new drugs, decreasng he producvy of he research and developmen (R&D) sage and shorenng he effecve paen lves of new molecules. These wo facors, n conjuncon wh he appearance of many subsue drugs n several herapeuc areas, have led o he reducon of he exclusvy perod of new producs. Anoher facor havng an mpac on he operaon of hs ndusry was he ranson of he payng responsbles from ndvduals o governmenal agences and nsurance companes, whch n assocaon wh hgh demands for pharmaceucals, due o agng populaons, pu srong pressure on prces and prescrpon polces (Shah, 2004). From he pon of vew of manufacurng, he global pharmaceucal ndusry can be dvded no fve sub-secors: large R&D based mulnaonals, generc manufacurers operang n he nernaonal marke, local companes based n only one counry, conrac manufacurers whou her own porfolo and boechnologcal companes manly concerned wh drug dscovery. The frs group, he nensve R&D based ndusres, s economcally he mos mporan and ends o have large and complex supply chans due o he global naure of s acvy. In addon, hese companes are he mos vulnerable o he global fnancal, regulaory and socal changes so hs work wll focus on her supply chans. The ndusry s preferred mechansm o overcome he producvy crses has been o ncrease nvesmen n curren busness acves, prmarly R&D and sales, he wo exreme ends of he supply chan. Ths has been mplemened by organc growh or by mergers and acqusons (M&A) o explo economes of scale. However, sascs show ha producvy connues o declne afer a decade of vgorous growh n nvesmens on hese areas (Coe, 2002). There are no sgnfcan economes of scale n sales acves. The revenues generaed by a pharmaceucal company are drecly proporonal o s sales, general and admnsrave (SG&A) expendure, suggesng ha wo merged companes wll no Correspondng auhor. Tel.: ; fax: E-mal address: n.shah@mperal.ac.uk (N. Shah). Receved 25 November 2010; Receved n revsed form 23 March 2011; Acceped 10 Aprl /$ see fron maer 2011 Publshed by Elsever B.V. on behalf of The Insuon of Chemcal Engneers. do: /j.cherd

2 chemcal engneerng research and desgn 8 9 ( ) necessarly be more profable han hey would be separaely,.e. here s no mprovemen on he reurn raes. Furher more, despe he heorecal hgher probably of successful produc developmen wh greaer scale n R&D, n realy does no ranslae no mproved ppelne value. Companes wll only mprove her prof margns f hey change he relaonshp beween volume and coss, whch can be acheved hrough producvy gans n he supply chan. Supply chan opmsaon s an excellen way o ncrease prof margns and s becomng curren pracce, no only n pharmaceucal ndusres bu also n oher areas of busness. Arnzen e al. (1995) descrbed he resrucurng of he supply chan a Dgal Equpmen Corporaon wh savngs of over US$ 100 mllon. They developed a large mxed neger lnear programmng (MILP) model ha ncorporaes a global, mul-produc bll of maerals for supply chans wh arbrary srucure and a comprehensve model of negraed global manufacurng and dsrbuon decsons. Camm e al. (1997) descrbed a projec relaed o P&G s supply chan n Norh Amerca. The man objecve of he sudy was o sreamlne he work processes o elmnae non-value added coss and duplcaon. The sudy nvolved hundreds of supplers, over 50 produc lnes, 60 plan locaons, 10 dsrbuon cenres and hundreds of cusomer zones. I allowed he company o save $200 mllon before axes. Kallrah (2000) repored on a projec n BASF where a mul-se, mul-produc, mul-perod producon/dsrbuon nework plannng model was developed, amng o deermne he producon schedule n order o mee a gven demand. Nero and Pno (2004) descrbed a peroleum supply chan plannng problem of Perobras n Brazl, whch comprses 59 peroleum exploraon ses, 11 refneres and fve ermnals, wh 20 ypes of suppled peroleum and 32 producs o local markes. Sousa e al. (2008) consdered an ndusral global supply chan of a mulnaonal agrochemcals company, comprsng wo subsysems: US nework and worldwde formulaon nework. Supply chan managemen n he process ndusres has long been used as a ool o defne producon and dsrbuon polces, as well as produc allocaon. Ths s he case of Cohen and Lee (1988) who descrbed he modellng of a supply chan composed of raw maeral vendors, prmary and secondary plans (each one wh nvenores of raw maerals and fnshed producs), dsrbuon cenres, warehouses and cusomer areas. Laer, Cohen and Moon (1991) used supply chan opmsaon o analyse he mpac of scale, complexy (he operang coss are a funcon of he ulsaon raes and number of producs beng processed n each facly) and wegh of each cos facor (e.g. producon, ransporaon and allocaon coss) on he opmal desgn and ulsaon paerns of he supply chan sysems. Tmpe and Kallrah (2000) descrbed an MILP model, whch combned producon, dsrbuon and markeng and nvolved plans and sales pons, o cover he relevan feaures requred for he complee supply chan managemen of a mul-se producon nework. Jayaraman and Prkul (2001) developed a Capacaed Plan Locaon Problem (CPLP) ype model for plannng and coordnaon of producon and dsrbuon facles for mulple commodes, comprsng raw maerals supplers, producon ses, warehouses and cusomer areas. The auhors followed a holsc approach o he supply chan, resulng n a deermnsc, seady-sae, mulechelon problem. Park (2005) consdered boh negraed and decoupled producon and dsrbuon plannng problem, conssng of mulple plans, realers, ems over mulple perods. The auhor proposed mxed neger opmsaon models and a wo-phase heursc soluon o maxmse he oal ne prof. Oh and Karm (2006) hghlghed he mporance of duy drawback regulaons n he producon-dsrbuon plannng problem, and ncorporaed hree man regulaory facors: corporae axes, mpor dues and duy drawbacks, n he proposed lnear programmng (LP) model. Tsaks and Papageorgou (2008) consdered he opmal confguraon and operaon of mul-produc, mul-echelon global producon and dsrbuon neworks, negrang producon, facly locaon and dsrbuon wh fnancal and busness ssues such as mpor dues, plan ulsaon, exchange raes and plan manenance. An MILP model was formulaed and appled o a case sudy for he coangs busness un of a global specaly chemcals manufacurer. Verderame and Floudas (2009) proposed a dscree-me mulse plannng wh producon dsaggregaon model o provde a gh upper bound on he rue capacy of daly producon and shpmen profles beween producon facles and cusomer dsrbuon cenres. Salema e al. (2010) presened a general dynamc model for he smulaneous desgn and plannng of mulproduc supply chans wh reverse flows, where me s modelled along a managemen perspecve o deal wh he sraegc desgn and he accal plannng smulaneously. The applcably of he model s proved n an example based on Poruguese glass ndusry. You e al. (2010) addressed a smulaneous capacy, producon, and dsrbuon plannng problem for a mulse supply chan nework ncludng a number of producon ses and markes and propose a mulperod MILP model and wo decomposon approaches for soluon. When performng long erm process plannng, uncerany facors (e.g. n produc demand) have o be aken no accoun n order o produce robus models whose oupu decsons wll perform well n a varey of scenaros (Verderame e al., 2010). Tsaks e al. (2001) addressed a sraegc problem of sochasc plannng for mul-echelon (alhough wh rgd srucure) supply chans. Iyer and Grossmann (1998) exended he work of Lu and Sahnds (1996), a specfc problem of long-range capacy expanson plannng n he chemcal ndusry. The npus were a se of avalable chemcal processes, an esablshed producon and dsrbuon nework and demand forecass affeced by uncerany leadng o an MILP, mul-perod plannng model wh mulple scenaros for each me perod. Ahmed and Sahnds (2003) consdered forecas uncerany parameers by specfyng a se of scenaros n a sochasc capacy expanson problem. A mulsage sochasc mxed neger programmng formulaon wh fxed-charge expanson coss was formulaed. Oh and Karm (2004) developed a deermnsc MILP model for he capacy-expanson plannng and maeral sourcng n global chemcal supply chans wh he nroducon of wo mporan regulaory facors, corporae ax and mpor duy. Gullen e al. (2005) proposed a wo-sage sochasc opmsaon approach o address a mulobjecve supply chan desgn problem. The Pareo-opmal soluon was obaned by he - consraned mehod. Pugjaner and Laínez (2008) proposed a scenaro-based MILP sochasc model consderng boh process operaons and fnance decsons wh an objecve of maxmsng he corporae value. A model predcve model sraegy was negraed wh he sochasc model for soluon. Several auhors addressed he ssue of supply chan opmsaon and long-erm process plannng n he pharmaceucal

3 2398 chemcal engneerng research and desgn 8 9 ( ) ndusry. Rosen e al. (1999) sared a seres of papers dedcaed o he specfc problem of supply chan opmsaon n he pharmaceucal ndusry. Laer, Papageorgou e al. (2001) publshed a paper based on he prevous one, where he producon sage s formulaed wh hgh degree of deal and ncludng he radng srucure of he company. The proposed deermnsc model consders up o 8 possble producs n he company s porfolo. Levs and Papageorgou (2004) exended hs work o accoun for unceran demand forecass, dependen on he resuls of he clncal rals for each produc. Gaca e al. (2003) proposed an MILP approach for he problem of capacy plannng under clncal rals uncerany, where four clncal ral oucomes for each produc are consdered as s ypcal n he pharmaceucal ndusry. Amaro and Barbosa-Póvoa (2008) consdered he negraon of plannng and schedulng of generalsed supply chans wh he exsence of reverse flows. The developed approach was appled o he soluon of a real pharmaceucal supply chan case sudy. So far, mos of he problems referred n hs revew concern dealed or very dealed descrpons of supply chans of relavely small sysems (.e. 1 or 2 ses, and up o 8 producs). The long erm sraegc plannng of large pharmaceucal companes has no been addressed n any of he prevous works. In hs work we buld a model of he global supply chan of a large pharmaceucal company, wh a long ls of producs n s porfolo and an exensve nework of manufacurng ses wh locaons all over he world. The allocaon polcy of producs o ses also dffers from prevous works. In each me perod, each produc wll be produced a a sngle locaon (sngle sourcng polcy), however he produc/se assgnmen may change along he me horzon reflecng acual pracce. Ths feaure ncreases sgnfcanly he bnary varables space. In order o keep he model sze whn reasonable lms, canno be oo dealed n s descrpon of he supply chan. Neverheless, even wh hs approach, s necessary o use decomposon algorhms. The developmen of decomposon approaches s a promsng research drecon n he area (Grossmann, 2005; Maravelas and Sung, 2009). To solve a ypcal large MILP model, Iyer and Grossmann (1998) used a b-level decomposon (herarchcal) algorhm o solve he orgnal model. In he frs sep, he desgn sage, he capacy expanson varables were aggregaed n a new varable se, me ndependen, and he processes o develop are chosen. In he second level model (operaon model), only he processes chosen o be developed are subjeced o nvesmen. Bok e al. (2000) proposed a b-level decomposon. The relaxed problem makng he decsons for purchasng raw maerals generaes an upper bound o he prof, whle he subproblem yelds a lower bound by fxng he delvery from he relaxed problem. Levs and Papageorgou (2004) solved an aggregaed model, compuaonally less expensve, alhough dealed enough o make he here-and-now decsons. In he second sep, he values of he correspondng varables were fxed and he dealed model s solved. Üser e al. (2007) used a Benders decomposon approach for a mul-produc closedloop supply chan nework desgn problem. Three dfferen approaches for addng mulple Benders cus are proposed. L and Ieraperou (2009) formulaed he negraed producon plannng and schedulng as blevel opmsaon problems wh one plannng problem and mulple schedulng problems. A decomposon approach based on convex polyhedral underesmaon was proposed and successfully appled o he negraed plannng and schedulng problem of mulpurpose mulproduc bach plans. Some auhors solved he large models resulng from supply chan opmsaon problems hrough Lagrangean decomposon. In her work, Gupa and Maranas (1999) formulaed an exenson of he economc-lo-szng problem, characersed by deermnaon of he producon levels of mulple producs, n mulple ses, wh deermnsc demands and mulple me perods. Jayaraman and Prkul (2001) relaxed hree blocks of consrans concernng assgnmen of cusomers o warehouses, raw maerals avalably and maeral flows balance. Ths allowed hem o decompose he orgnal problem n hree dfferen ses of subproblems. Maravelas and Grossmann (2001) nroduced a good example of a model composed of wo (or more) ndependen submodels wh one lnkng consran. Jackson and Grossmann (2003) bul a mulperod opmsaon model for he plannng and coordnaon of producon, ransporaon and sales for a nework of geographcally dsrbued mulplan facles supplyng several markes. Two Lagrangean decomposon mehods were adoped o ackle he problem, spaal and emporal decomposons. In boh cases, he auhors followed he regular algorhm of Lagrangean decomposon o reach he opmal soluon of he orgnal problem, as descrbed n Reeves (1995). The numercal examples show he emporal decomposon o work sgnfcanly beer han he spaal decomposon. Eskgun e al. (2005) developed a Lagrangean heursc for he proposed large-scale neger lnear programmng model for supply chan nework desgn problem of an auomove company wh capacy resrcon on vehcle dsrbuon cenres. Shen and Q (2007) embedded Lagrangean relaxaon n branch and bound o solve an negraed sochasc supply chan desgn problem whch s formulaed as a nonlnear programmng (NLP) model. The Lagrangean relaxaon subproblems are hen solved by a low-order polynomal algorhm. Chen and Pno (2008) proposed several varous decomposon sraeges for he connuous flexble process nework model by Bok e al. (2000), ncludng Lagrangean decomposon, Lagrangean relaxaon, and Lagrangean/surrogae relaxaon. Among he four decomposon sraeges, was proved ha he soluons generaed from Lagrangean relaxaon are beer, alhough s CPU me s lower. Hnojosa e al. (2008) proposed an MILP formulaon for a dynamc woechelon mul-commody capacaed plan locaon problem wh nvenory and ousourcng aspecs. The auhors solved he resulng, ndependen subproblems from a Lagrangean relaxaon scheme and a dual ascen mehod o fnd a lower bound on he opmal objecve value. Nsh e al. (2008) used an augmened Lagrangean decomposon and coordnaon approach for he proposed framework for dsrbued opmsaon of supply chan plannng for mulple companes. An augmened Lagrangean approach wh a quadrac penaly funcon was used o decompose he orgnal problem no several subproblems for each company o elmnae dualy gap. Pugjaner e al. (2009) used he opmal condonal decomposon (OCD), a parcular case of Lagrangean decomposon, for he supply chan desgn-plannng model exended from he work by Pugjaner and Laínez (2008).Inhe OCD, he dfference from he classc Lagrangean decomposon s s auomac updang process of Lagrange mulplers. You and Grossmann (2010) proposed a spaal decomposon algorhm based on he negraon of Lagrangean relaxaon and pecewse lnear approxmaon o solve mxed neger

4 chemcal engneerng research and desgn 8 9 ( ) nonlnear programmng (MINLP) model for large-scale jon mul-echelon supply chan desgn and nvenory managemen problems. From he leraure dscussed above, here s a gap n he research on supply chan plannng akng boh prmary and secondary manufacurng n he pharmaceucal ndusry no accoun. Ths paper ams o fll hs gap. The res of our paper s organsed as follows. Secon 2 nroduces a bref descrpon of he ypcal supply chan and s componens n he pharmaceucal ndusry. Secon 3 s concerned wh he problem descrpon. The mahemacal formulaon of he model s presened n secon 4. In Secon 5, wo decomposon algorhms are conceved o solve he large model resulng from he problem formulaon. In Secon 6, he model as well as he performance of he developed algorhm s esed wh wo llusrave examples. Some concludng remarks are drawn n Secon Supply chans n he pharmaceucal ndusry A supply chan may be defned as an negraed process where several busness enes work ogeher o produce goods, servces, ec. (Shah, 2005; Barbosa-Póvoa, 2009; Papageorgou, 2009) Ths s a major ssue n many ndusres, as organsaons begn o apprecae he crcaly of creang an negraed relaonshp wh her supplers and cusomers. Typcally, n manufacurng ndusres he sages are: raw maerals acquson, prmary (and secondary) manufacure and dsrbuon o realers and cusomers. Each one may comprse one or more sub-sages and producs may be kep n sorage uns (e.g. warehouses) beween sages (Papageorgou, 2006). Many companes, especally mulnaonals, possess complex radng srucures where, for ax purposes, he manufacurng plans, nellecual propery and dsrbuon cenres are consdered o be dfferen enes. Ths brngs more flexbly o supply chan opmsaon as allows he praccng of several dfferen prcng polces. Supply chans n he pharmaceucal ndusry, one ypcal ndusry of producs wh a hgh added value per mass un, comprse wo manufacurng sages: prmary manufacurng for acve ngreden (AI) producon and secondary manufacurng for formulaon and packagng. As very hgh-value producs, AIs are usually produced n low amouns and n few cenralsed locaons worldwde (Sousa e al., 2007) Prmary manufacurng The cusomer-facng end s drven by orders. These ones represen dsurbances ha propagae backwards n he supply chan (n he oppose sense of maerals flow), beng amplfed as hey ge closer o he frs sages (bullwhp or Forreser effec). Prmary ses are responsble for acve AI producon and may face sgnfcan flucuaons n demand. They are characersed by long cycle mes, whch make dffcul o ensure end-o-end responsveness. The producon volumes nvolved are usually low resulng n mulpurpose plans o spread he capal cos beween producs. The changeover acves, ha ake place when he manufacured produc changes, nclude horough cleanng o avod cross-conamnaons. Ths usually akes a long me so s desrable o keep he plannng complexy (number of dfferen producs usng he same resource) a a low level Fg. 1 Supply chan srucure of an enerprse. and long campagns become he norm, oherwse equpmen ulsaon s oo low. Snce he producon volumes are low, ransporaon coss are no sgnfcan a hs end of he supply chan, so prmary ses may be locaed anywhere n he world, even dsan from secondary manufacurers. The facors rulng he choce of locaon wll be ax raes, exsence of sklled workng force, polcal and economc sably ec Secondary manufacurng Secondary manufacurers prepare and formulae he producs n a suable form for fnal consumers. Ths nvolves addng he AI o some excpen ner maerals along wh furher processng and packng. As was poned ou above, secondary ses are ofen geographcally separae from AI producers. Ths s frequenly he resul of ax and ransfer prce opmsaon whn he enerprse. A hs end of he supply chan, due o large amouns of ner producs added o he AI plus he packng, he ransporaon coss become very sgnfcan, so usually here are many more secondary han prmary locaons, servng regonal or local markes. 3. Problem descrpon As referred o before, he am of hs projec s he supply chan opmsaon of a large pharmaceucal company. The supply chan componens are prmary ses (AI manufacurers) and respecve sorage facles, secondary ses and respecve warehouses and fnal produc marke areas. The dsrbuon neworks whn each marke area are ou of he scope of hs work. Each prmary se may supply he AI o any of he secondary ses and be locaed n any place around he world. For secondary ses and markes, we consder several geographcal areas. The srucure of a supply chan example wh fve geographcal areas (ncludng Europe, Asa, Afrca and Mddle Eas, Norh Amerca and Souh Amerca) s shown n Fg. 1. Snce ransporaon coss are very sgnfcan a hs end of

5 2400 chemcal engneerng research and desgn 8 9 ( ) he supply chan, maeral flows beween wo dfferen geographcal areas are no allowed. We assume he followng nformaon s known: mul-perod demand forecas profle of he company s porfolo; producon, ransporaon and allocaon specfc coss; ax raes; he company s supply chan srucure, wh prmary and secondary locaons avalable. The model ams o make he followng supply chan decsons: where o allocae he manufacure of prmary and secondary producs and how o manage he avalable resources durng he whole me horzon; wha producon amouns and nvenory levels shall be se for each manufacurng se; how o esablsh he produc flows beween echelons,.e. beween prmary and secondary ses and beween secondary ses and markes. Each geographc area produces and consumes some, bu no all, produc famles from he company s secondary (fnal) producs porfolo. Whn each geographc area, was decded ha a sngle sourcng polcy would be followed,.e. each produc (boh prmary and secondary) wll be produced n one and only one se on each me perod. The produc/se assgnmen may change beween dfferen me perods ( allocaon ransfer ), alhough he number of exchanges s lmed. In lne wh he sngle sourcng polcy, hs ransfer can only ake place beween ses n he same geographcal area. The model ncludes he occurrence of changeover acves as hese sgnfcanly affec he equpmen avalably. However, due o he global scope of he projec and he dmenson of he problem, some producon deals as scale-up mes and qualfcaon runs are no consdered. The locaon of he several members n he SC s an exremely mporan ssue o large mulnaonal enerprses, as he prof afer axes may change sgnfcanly due o he dfferen ax raes n dfferen counres. So, he objecve s se as he maxmsaon of he enerprse s ne prof value (NPV). The erms of he objecve funcon are lsed below. A cos for unfulflled demand has also been consdered. Revenues come from he secondary producs sales. The cos erms nclude: producon coss; ransporaon coss; nvenory handlng coss; producs allocaon coss; unme demand coss; ax coss. 4. Mahemacal formulaon 4.1. Noaon Indces c j l prmary ses prmary producs geographcal areas prmary ses resources (manufacurng equpmens) m marke locaons p secondary producs r secondary ses resources (manufacurng equpmens) s secondary ses, me perods Ses M j P j S j markes n geographcal area j secondary producs n geographcal area j secondary ses n geographcal area j Parameers A sr Avalably of resource r n me perod n secondary se s (hour) AP cl Avalably of resource l n me perod n prmary se c (hour) CIV Invenory handlng coss ($/kg) COT Changeover me n secondary ses resources (hour) COTP Changeover me n prmary ses resources (hour) CPP c Producon cos of prmary produc n se c ($/kg) CPS sp Producon cos of secondary produc p n se s ($/kg) CTA sp Cos of ransferrng he allocaon of secondary produc p o secondary se s ($) CTAP c Cos of ransferrng he allocaon of prmary produc o prmary se c ($) CTP c Average ransporaon coss from prmary se c o secondary ses ($/kg) CTS sm Transporaon cos of secondary producs from secondary se s o marke m ($/kg) CU p Cos of no sasfyng he whole demand of secondary produc p ($/kg) D pm Demand forecas of produc p n marke m n me perod (kg) K pr Equals 1 f produc p uses resource r and 0 oherwse K l Equals 1 f produc uses resource l and 0 oherwse Max Upper lm of he producon amoun n one me perod (kg) MT pr Manufacurng me demand of secondary produc p n resource r (hour/kg) MTP l Manufacurng me demand of prmary produc n resource l (hour/kg) PF p Produc p formulaon as a funcon of prmary produc T Number of me perods TF p Transformaon facor for fnal produc p (yeld) TRP c Tax rae on prmary se locaon TRS Tax rae on secondary se locaon V1 Inernal prce of prmary produc ($) V2 pm Sellng prce of produc p n marke m ($) XPTN Allocaon ransfer lm of prmary producs n each me perod XTN j Allocaon ransfer lm of secondary producs n geographcal area j n each me perod Bnary varables X sp Equals 1 f secondary produc p s produced a se s n me perod, 0 oherwse XP c Equals 1 f prmary produc s produced a se c, 0 oherwse XPT c Allocaon ransfer decson varable (equals 1 f XP c, +1 = 1 and XP c =0) XT sp Allocaon ransfer decson varable (equals 1 f X sp, +1 = 1 and X sp =0)

6 chemcal engneerng research and desgn 8 9 ( ) Connuous varables IV sp Invenory of secondary produc p n se s a me perod (kg) IVP Invenory of prmary produc a me perod (kg) PR sp Producon of produc p n se s n me perod (kg) PRP c Producon of prmary produc n se c a me perod (kg) TP s Amoun of prmary produc ranspored o se s n me perod (kg) TS spm Amoun of produc p ranspored from se s o marke m n me perod (kg) SL pm Sales of produc p n marke m n me perod (kg) U pm Unsasfed demand of produc p n marke m n me perod (kg) Z NPV, objecve funcon ($) 4.2. Secondary produc allocaon consrans s S j X sp = 1 j,, (1) XT sp, 1 X sp X sp, 1 s, p, > 1 (2) X sp 1 XT sp s, p, (3) XT sp XTN j j, (4) s S j Eq. (1) guaranees ha, whn each geographcal area, each produc s allocaed o one and only one secondary se n each me perod. Eqs. (2) and (3) guaranee ha each allocaon ransfer wll only ake place afer he acual decson has been aken. Eq. (4) lms he number of allocaon ransfers occurrng n each me perod Prmary produc allocaon consrans XP c = 1, (5) c XPT c, 1 XP c XP c, 1 c,, > 1 (6) XP c 1 XTP c c,, (7) XPT c XPTN (8) c Eqs. (5) (8) play he same role n prmary producs allocaon as Eqs. (1) (4) respecvely for secondary producs Capacy consrans ( ) MT pr PR sp A sr K pr X sp 1 COT s, r, (9) p p ( ) MTP l PRP c AP cl KP l XP c 1 COTP c, l, (10) Eqs. (9) and (10) lm he producon of secondary and prmary producs n accordance wh resource avalably, respecvely. Noe ha alhough more general resources can be consdered n he proposed model, he resources n he examples dscussed laer refer o he manufacurng equpmens n he producon ses Mass/flow balance consrans PR sp Max X sp s, p, (11) SL pm = s S j TS spm j,,m M j, (12) SL pm D pm j,,m M j, (13) U pm = D pm SL pm j,,m M j, (14) Each secondary produc wll only be produced n he secondary se where was allocaed, as shown n Eq. (11). Eq. (12) s he flow balance beween sorage facles (n he secondary ses) and he markes; Eq. (13) ensures ha he sales of each produc n each me perod s less han or equal o he respecve demand and Eq. (14) esmaes he unme demand n order o nroduce a penalsaon erm n he objecve funcon. The lnk beween producon and ougong flows from secondary ses (Eqs. (11) and (12)) s provded by he nvenory consran, Eq. (17). PRP c Max XP c c,, (15) PR sp PF p TP s = TF p s,, (16) p Eq. (15) gves he producon relaons n he prmary ses. Eq. (16) esablshes he prmary producs flow o secondary ses based on producon amouns, produc formulaons and manufacurng losses of secondary producs Invenory consrans IV sp = IV sp, 1 + PR sp TS spm j, s S j,p P j, (17) m M j IVP = IVP, 1 + PRP c TP s, (18) c s Eqs. (17) and (18) are he nvenory balances for secondary and prmary ses, respecvely Non-negavy consrans IV sp, IVP,PR sp, PRP c,sl pm,tp s,ts spm,u pm 0 (19) 4.8. Objecve funcon The oal NPV mus nclude: Sales revenue; = V2 pm SL pm (20) p m Prmary and secondary producs producon coss (per se); KP c = CPP c PRP c c (21)

7 2402 chemcal engneerng research and desgn 8 9 ( ) KP s = CPS sp PR sp j, s S j (22) Prmary and secondary producs ransporaon coss (per se); KT c = CTP c PRP c c (23) KT s = CTS sm TS spm j, s S j (24) m M j Prmary and secondary ses nvenory handlng coss; KI prm = CIV IVP (25) KI s = CIV IV sp j, s S j (26) Prmary and secondary producs allocaon coss (per se); KA c = CTAP c XPT c c (27) KA s = CTA sp XT sp j, s S j (28) For each se, he oal coss before axes wll be as follows. Here, he average nvenory coss are consdered for prmary ses (see below). K c = KP c + KT c + KI prm + KA c c #c (29) K s = KP s + KT s + KI s + KA s s (30) Tax coss on prmary and secondary se locaons; [( ) ] c TRP c V1 PRP c K c c (31) s TRS s V2 pm SL pm Ks j, s Sj p Pj m M j (32) The NPV s gven by Eq. (33): [ ] Z = (K c + c ) + (K s + s ) U pm CU c s p m (33) As menoned above, ransporaon coss a he prmary end of he supply chan are no sgnfcan, so average coss for ransporaon of prmary producs beween secondary and prmary ses are used, dependng on he prmary se locaon and s dsances o he secondary ses. Ths reduces he number of varables snce s possble o express he prmary producs flow varables, TP s, wh one dmenson less. One consequence of hs procedure s ha he prmary produc nvenory coss canno be assgned o a specfc prmary XT X SL IV TS U PR TP PRP IVP XP XPT (1) x (2) x x (3) x x (4) x (11) x x (9) x x (12) x x (13) x (17) x x x (14) x x (16) x x (10) x x (15) x x (18) x x x (5) x (6) x x (7) x x (8) x Fg. 2 Srucural marx. Lnes correspond o he model consrans and he columns correspond o he varables. se (because he model formulaon does no allow he ncluson of he se ndex n he varable IVP ), whch prevens he accurae calculaon of ax coss on hese locaons. Anoher consequence s ha he produc flow from one specfc prmary se o one specfc secondary se s no unknown from he model. Thus, he prmary produc producon amoun s used o calculae he ransporaon cos n any specfc prmary se. 5. Soluon mehods Wh he purpose of esng he model, wo ses of daa, wh dfferen szes, were generaed o smulae hypohecal problems. The smaller problem s solvable n an hour; however, he larger one does no ermnae n a reasonable me (CPU < 50,000 s). Ths movaed us o develop heursc procedures o solve he larger nsances of hs model, correspondng o real-world problems. Accordng o prevous works by oher auhors, (see Secon 1) he mos suable approaches for hs knd of models are decomposon and/or aggregaon mehods and Lagrangean decomposon procedures. In hs paper, we propose wo decomposon mehods. The frs one consss of separang he SC n s wo echelons, prmary and secondary (spaal decomposon algorhm). In he second mehod he model s decomposed no several ndependen subproblems, one per each me perod (emporal decomposon algorhm) Spaal decomposon algorhm Durng he las a few years, compuaonal hardware has experenced enormous mprovemen. Faser compuer processors allow he soluon of ever-larger problems whn reasonable CPU mes. Neverheless, he sae-of-he-ar of opmsaon ools always ends o lag he requremens of realsc problems. The prncple underpnnng decomposon mehods s problem sze reducon, hrough decomposon no smaller subproblems, deleon of hard consrans, or, as ofen happens, a combnaon of boh. Our model s srucural marx (Fg. 2) shows ha can be composed of wo ready decomposable subproblems, correspondng o prmary and secondary echelons conneced by a lnkng consran (Eq. (16)), he flow balance beween he wo areas of he supply chan.

8 chemcal engneerng research and desgn 8 9 ( ) START Relax secondary bnary varables Solve MILP ALLOCATIONS1 W = 0 XT = 0; X = 0; TS psm = SL ps /#M j SL pm = Σ sj SL ps /#M j U pm = U pj /#M j Fg. 4 Evoluon of Z along he several eraons of he spaal decomposon algorhm. W > n JC = 1 no JC <= 5 yes Fx secondary varables j <> JC; (TP, PR, TS, SL, IV, U, X, XT) Fx prmary bnary varables; Solve MILP ALLOCATIONS2 Updae TP, PR, SL, IV, U, X, XT JC = JC + 1 yes no STOP The frs sep uses an aggregaed verson of he model,.e. ALLOCATIONS1, (see Appendx) whose soluon process s faser han he orgnal model. Beween he frs and he second seps a se of operaons s performed o conver he resuls obaned wh he aggregaed model no values ha can be used n he dealed model. In he second sep, each geographcal area, j, s opmsed separaely n he orgnal model wh prmary bnary varables and secondary varables n oher areas fxed (model ALLOCATIONS2), unl a complee se of secondary bnary varables s generaed. These varables, n urn, are fxed n he orgnal model (MILP model ALLO- CATIONS3) n he hrd sep of he procedure, where prmary producs are reallocaed opmally. The second and hrd seps may be repeaed eravely unl he objecve funcon s no longer mproved, or a specfed number of eraons are reached. On he one hand, here s no guaranee ha he opmum wll be found, snce boh soluons, from he second and hrd seps, are lower bounds o he problem. On he oher hand, n each sep, s possble o calculae an upper bound o he dfference o he opmum neger soluon, gven by he dfference beween he resul of sep 1 and he acual value of Z (Fg. 4). Fx secondary bnary varables Accoun for changeover Solve MILP ALLOCATIONS3 W = W+1 Fg. 3 Flowchar of he spaal decomposon algorhm. In he followng descrpon, prmary bnary varables, refers o hose varables concerned wh prmary producs allocaon whle secondary bnary varables are relaed o secondary producs allocaon. Ths s a hree-sep mehod, based on he srucure of he supply chan model, as llusraed n Fg. 3. Sep 1: he secondary bnary varables are relaxed and an neger se of prmary varables s generaed. Smulaneously, hs provdes an esmae of he producon amouns, sales and nvenory for boh prmary and secondary producs, as well as an upper bound o he problem. Sep 2: he prmary bnary varables are fxed and he program opmses each of he secondary geographcal areas, j, separaely, unl a complee se of neger secondary bnary varables s generaed. Sep 3: he secondary bnary varables are fxed and he model reallocaes he prmary producs and adjuss he producon amouns Temporal decomposon algorhm Ths work s orened owards long-erm plannng, where each me perod has he duraon of several monhs. Under hese crcumsances, he nvenory relaons (consrans) do no sgnfcanly nfluence he fnal allocaon decsons. A possble approach s o decompose he problem no several ndependen subproblems, one per me perod ha can be opmsed separaely. Two knds of consrans lnk he dfferen perods: allocaon ransfer and nvenory consrans (Eqs. (2), (3), (6), (7), (17) and (18)). Seng he nvenory varables o 0 a he end of each me perod leads o Eq. (34) and (35). Each one of he new equaons may now be wren ndependenly for each me perod,.e. whou dependng on varables referrng o adjacen me perods. 0 = PR ps m M j TS psm j,,s S j, (34) 0 = PRP c TP s, (35) c s Eqs. (2), (3), (6) and (7) need o be reformulaed more carefully; he allocaon decsons nformaon has o be passed beween me perods. Ths s acheved hrough modfcaon of he allocaon consrans, as dealed below. In fac, each sub-model (correspondng o each me perod) s no

9 2404 chemcal engneerng research and desgn 8 9 ( ) fully ndependen; uses some nformaon (allocaon decsons) from prevous (or subsequen) me perods, as here s a lm on he produc/se allocaon ransfers ha can occur n each me perod. Accordng o hs, he opmsaon process has o be performed followng eher he drec or nverse chronologcal sequence of me perods. In order o reduce he relaons beween me perods, each allocaon ransfer decson s assumed o ake place n he same me perod as he ransfer self. Usng he varables subsuon expressed n Eq. (36), Eqs. (2), (3), (6) and (7) are reformulaed for each me perod = (usng nverse chronologcal sequence). Varable subsuon START = T STEP 1 ALLOCATIONT1 Opmsaon of me perod XXS ps = X ps, XXP c = XP c, X sp,+1 = XXS sp and XP c,+1 = XXP c s, p, c, l, = and <T (36) = - 1 Secondary ses no = 0 X sp 1 XT sp s, p, = and <T (37) XT sp XXS sp X sp s, p, = and <T (38) Prmary ses XP c 1 XPT c c,, = and <T (39) yes X2 ps = X1 ps XT2 ps = XT1 ps XP2 c = XP1 c XPT2 c = XPT1 c XPT c XXP c XP c c,, = and <T (40) The ncluson of bnary allocaon varables n he capacy balance consrans (Eqs. (9) and (10)) o accoun for he changeover operaons may resul n he generaon of nfeasble capacy balance consrans n he followng me perod beng opmsed n he soluon process (see relaonshp below). ( ) A sr, 1 < K pr X sp 1 COT s, r, (41) p ( ) AP cl, 1 < KP l XP cl 1 COTP c, l, (42) If he soluon of hese nfeasbles nvolves he reallocaon of more producs han he allocaon ransfer upper lm (Eqs. (4) and (8)), hen he model wll be nfeasble. In order o preven hs, exra consrans are ncluded: ( ) A sr, K pr X sp 1 COT s, r,, < (43) p ( ) AP cl, KP l XP cl 1 COTP s, r,, < (44) In he frs sep of he algorhm, each me perod s opmsed separaely and sequenally. The nformaon concernng produc/se assgnmen s passed o he followng me perod and so on, unl all me perods have been opmsed and a complee se of bnary varables s generaed. In hs sep, he objecve funcon s a funcon of. Inhe STEP 2 ALLOCATIONT2 LP recalculaon of he connuous varables STOP Fg. 5 Flowchar of he emporal decomposon algorhm. Indces 1 and 2 of he bnary varables n he 7h box refer o ALLOCATIONT1 and ALLOCATIONT2, respecvely. second sep, he bnary varables are fxed and he connuous varables are recalculaed n order o mprove he fnal NPV. ALLOCATIONT1 s he MILP model used o calculae he bnary varables n each me perod. I comprses he producon and sales consrans and modfed allocaon consrans,.e. Eqs. (36) (40). ALLOCATIONT2 s he orgnal model, where all he me perods are opmsed smulaneously, wh he bnary varables fxed a he values calculaed on sep 1. So ALLOCA- TIONT2 s a LP model. Fg. 5 shows he flowchar of he emporal decomposon algorhm, n whch he opmsaon process n he frs sep s performed followng he nverse chronologcal sequence of me perods. 6. Illusrave examples Two examples movaed by ndusral processes were generaed n order o es he model as well as he performance of he developed decomposon algorhms. The dmensons of each example are presened n Table 1.

10 chemcal engneerng research and desgn 8 9 ( ) Table 1 Example problems. Problem 1 Problem 2 6 acve ngredens 10 acve ngredens 6 prmary ses 10 prmary ses 5 secondary geographcal areas 5 secondary geographcal areas 30 secondary produc famles 100 secondary produc famles 33 secondary ses 70 secondary ses 10 marke areas 10 marke areas 12 me perods (4 monhs each) 12 me perods (4 monhs each) Table 3 Decomposon algorhms performance for problem 1 (COT = 100). Z op Gap a (%) CPU (s) LP 437,995 Full space 426, Spaal dec. alg. 420, Temporal dec. alg. 416, a Wh respec o he LP model soluon. CPU (s) Table 2 Comparson beween he number of bnary varables n he full varable space model and decomposon algorhms. Problem 1 Problem 2 Full space 12,480 84,096 Spaal dec. alg. 1s Sep nd Sep (mn/max) 1080/ /24,626 3rd Sep 864 2,400 Temporal dec. alg (1) 4 5 Problem Sze Fg. 6 Influence of he problem s sze n he solvng me. CPU me ncreases exponenally wh he sze of he problem (1) and (2) faser processors allow movng he curve o he rgh. (2) Table 4 Decomposon algorhm s performance for problem 2 (COT = 100). Z op Gap a (%) CPU (s) LP 1,008,746 Full space 954,454 b ,022 Spaal dec. alg 959, ,165 Temporal dec. alg. 974, a Wh respec o he LP model soluon. b Termnaed by he CPU me lm of 50,000 s. Table 5 Sascs for he spaal decomposon algorhm. The changeover me (COT = 300) represens up o 10% of he avalably of each resource. CPU (s) Opmum Gap (%) Sep ,002, Sep 2 a j = , j = 2 1, , j = , j = 4 4, , j = , Sep , Fnal 7, , a j refers o secondary geographcal areas. CPU (s) j=4 All he ess were performed on a Wndows XP based machne wh 1 GB RAM and 3.4 GHz Penum 4 processor, runnng he GAMS 22.8 (Brooke e al., 2008) wh CPLEX 11.1 solver (ILOG, 2007). Sascs concernng he number of neger (bnary) varables n he full space models and n he dfferen seps of he decomposon algorhm are shown n Table 2. Takng no accoun he relaonshp beween sze and CPU expressed n Fg. 6 and he values n Table 2, s o be expeced ha he sum of he CPU mes o solve he hree seps of he decom- 500 j=1 0 0 j= j=5 Excess Capacy (%) Fg. 7 CPU dependence on excess capacy of sep 2, of he spaal decomposon algorhm, for each geographcal regon, j. CPU and EC values are an average over many runs wh dfferen values of COT j=3 800 poson algorhm wll be lower han he me o solve he full model. Tables 3 and 4 show he resuls of he decomposon algorhms. The performance of hese mehods s hghly dependen on he se of daa beng opmsed. The spaal decomposon mehod s parcularly sensve o he parameer excess capacy as defned n Eq. (45) ha relaes demand and avalable manufacurng capacy. Ths parameer s affeced by he me aken on each changeover operaon, as hs affecs he manufacurng equpmen avalables. Table 6 COT nfluence on he soluon me of decomposon algorhms for problem 2. Spaal dec. alg. Temporal dec. alg. COT CPU (s) Opmum CPU (s) Opmum , , , , , , , , ,330

11 2406 chemcal engneerng research and desgn 8 9 ( ) Fg. 8 Prmary produc allocaons from emporal decomposon algorhm for problem 2 (COT = 100) T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Prmary Produc on (kg) C1 C2 C3 C4 C5 C6 Prmary Se C7 C8 C9 C10 Fg. 9 The prmary producs producons from he emporal decomposon algorhm for problem 2 (COT = 100). A rs s S j EC rj = 1 MT pr D pm m M j 100 r,, j = 1, 2, 3, 4, 5 (45) Table 5 presens he CPU mes for he hree seps of hs mehod, where he frs and hrd seps represen abou 15% of he oal soluon me when COT = 300. Table 6 shows how he changeover me (COT) affecs he CPU me o solve problem 2 wh he spaal decomposon algorhm, by changng he avalable capacy. Fg. 7 relaes he average compuaonal Fg. 10 Breakdown of he oal cos from he emporal decomposon algorhm for problem 2 (COT = 100).

12 chemcal engneerng research and desgn 8 9 ( ) me o solve each of he secondary areas and her excess capacy, defned n Eq. (45). The emporal decomposon algorhm has, by far, he bes performance of boh soluon algorhms developed n hs work; s sensvy o he excess capacy parameer s much lower han ha for he spaal decomposon algorhm. One downsde of hs algorhm s ha he qualy of s resuls s hghly dependen on he demand profle,.e. large changes n demand levels beween consecuve me perods wll renforce he mporance of he nvenory relaons and lead o poorer qualy resuls, whle Eqs. (43) and (44) wll nroduce many lmaons on he possble produc allocaon. Fg. 8 shows he allocaon of he prmary producs (I1 I10) o prmary ses (C1 C10) obaned by he emporal decomposon algorhm for problem 2, from whch we can see no only he produc allocaons, bu also he allocaon ransfer decsons. Prmary producs I1, I3, I8, I9 and I10 are all allocaed o one prmary se, whle oher prmary producs make allocaon ransfers, n whch boh I4 and I7 have he maxmum number of allocaon ransfers (wo ransfers). Fg. 9 shows he producon a each prmary se n each me perod. C5 and C8 are he mos producve prmary ses, whle C6, C7 and C10 are he hree ones wh he lowes producon amouns. Whn each se, we can also see ha he producons vary hroughou all he me perods. Fg. 10 gves he percenage of each cos erm n he oal cos gven by he emporal decomposon algorhm. The ax cos on he secondary ses, he ransporaon cos (of prmary and secondary producs), he producon coss (of prmary and secondary producs) and he unme demand cos accoun for over 99% of he oal cos, whle he oher coss, ncludng he nvenory handlng coss (on prmary and secondary ses) and allocaon coss (of prmary and secondary producs), occupy a ny poron of he oal cos. Parcularly, he ax cos on he prmary ses are zero, as he cos on prmary ses s hgher han he revenue, bu he ax cos on he secondary ses accouns for around 40%. 7. Concludng remarks In hs work, we have defned a problem mporan o he pharmaceucal ndusry, revewed relevan relaed publshed works, developed a model o cover he global nework allocaons and allocaon ransfers and nvesgaed wo soluon algorhms o ackle hs parcular large MILP problem. In he spaal decomposon mehod, he sensvy analyss o he changeover me shows ha, for hs parcular se of daa, he opmum value s no affeced sgnfcanly, snce he boleneck of he supply chan s he capacy of prmary ses. On he oher hand, he CPU me ncreases sgnfcanly. Ths s parcularly he case for he crcal secondary geographcal areas, where j = 2 and j = 4. These correspond o areas where he capacy used s close o he lm, manly due o a lower excess capacy (Eq. (45)), whch becomes more crcal as he changeover me ncreases. The emporal decomposon mehod performs well wh hese ses of daa alhough he qualy of he resuls may be poor n oher cases. Neverheless, opens new possbles o explore even larger nsances of he problem, such as s sochasc verson, where demand and oher parameers may be unceran. Acknowledgemens The fundng for R.T.S. from he Poruguese Scence and Technology Foundaon (FCT), and for S.L. from he Overseas Research Suden Award Scheme (ORSAS) and he Cenre for Process Sysems Engneerng (CPSE) s graefully acknowledged. Appendx A. Appendx: Aggregae Model The varables space of he dealed model may become very large for realsc problems, manly because of he eradmensonal ransporaon varables of secondary producs from ses o markes, TS spm. We develop an aggregae verson of he model, whou ransporaon varables ha, n spe of beng less dealed, s more racable and suable for he developmen of algorhms (he number of varables s reduced by 30%) A.1. Parameers The aggregaed model uses all he parameers of he dealed model excep demands, secondary producs ransporaon coss and secondary producs sellng prce ha are subsued by aggregaed or average parameers. DJ pj CTSJ spj V2J pj DJ pj = Toal demand of secondary produc p n area j n me perod, Eq. (A1). Average secondary produc p ransporaon cos from se s o markes n area j, Eq. (A2). Average prce of secondary produc p n markes n area j, Eq. (A3). m M j D pm, j, (A1) ( ) CTS sm D pm m M j CTSJ spj = m M j V2J pj = A.2. Varables m M j D pm ( ) V2 pm D pm m M j D pm j,,s S j j, (A2) (A3) Almos all he varables from he dealed model are kep n he aggregaed model, bu some of hem have o be redefned n order o f he dmensonal changes n he parameers. Varable TS psm s smply removed. SL sp Sales of produc p from se s n me perod ; U pj Unsasfed demand of produc p n area j n me perod ;

13 2408 chemcal engneerng research and desgn 8 9 ( ) A.3. Consrans Eq. (12) has o be subsued and Eqs. (13), (14) and (17) have o be modfed o be accordng wh he redefned se of varables. PR sp DJ pj j, s S j SL sp DJ pj j, s S j,p P j, U pj = DJ pj SL sp j,, s S j (A4) (A5) (A6) Tax coss on prmary and secondary se locaons; [( ) ] c TRP c V1 PRP c K c c s TRS s V2J pj SL sp Ks j, s Sj p Pj The NPV s gven by Eq. (A21): [ ] Z = (K c + c) + (K s + s) U pj CU (A19) (A20) (A21) IV sp = IV sp 1 + PR sp SL sp s, p, A.4. Objecve funcon Sales revenues; = V2J pj SL sp j s S j (A7) (A8) Prmary and secondary producs producon coss (per se); KP c = CPP c PRP c c KP s = CPS sp PR sp j, s S j (A9) (A10) Prmary and secondary producs producon coss (per se); KT c = CTP c PRP c c KT s = CTSJ psj SL sp j, s S j Prmary and secondary ses nvenory handlng coss; KI prm = CIV IVP KI s = CIV IV sp j, s S j (A11) (A12) (A13) (A14) Prmary and secondary producs allocaon coss (per se); KA c = CTAP c XPT c c KA s = CTA sp XT sp j, s S j (A15) (A16) For each se, he oal coss before axes wll be as follows: K c = KP c + KT c + KI prm #c K s = KP s + KT s + KI s + KA s s + KA c c (A17) (A18) c s Beween he frs and second seps of he spaal decomposon algorhm, he converson of he resuls obaned wh he aggregaed model o values ha can be used n he dealed model s performed as follows: TS spm = SL sp #M j j, s S j,m M j,p, (A22) SL sp s Sj SL pm = j, m M #M j,p, (A23) j U pm = U pj #M j j, m M j,p, (A24) References Ahmed, S., Sahnds, N.V., Scheme for sochasc neger programs arsng n capacy expanson. Oper. Res. 51, Amaro, A.C.S., Barbosa-Póvoa, A.P.F.D., Plannng and schedulng of ndusral supply chans wh reverse flows: a real pharmaceucal case sudy. Compu. Chem. Eng. 32, Arnzen, B.C., Brown, G.G., Harrson, T.P., Trafon, L.L., Global supply chan managemen a dgal equpmen corporaon. Inerfaces 25 (1), Barbosa-Póvoa, A.P., Susanable supply chans: key challenges. In: Bro Alves, R.M., Oller Nascmeno, C.A., Bscaa, E.C. (Eds.), 10h Inernaonal Symposum on Process Sysems Engneerng: Par A, Compuer Aded Chemcal Engneerng, vol. 27. Elsever, Amserdam, pp Bok, J.K., Grossmann, I.E., Park, S., Supply chan opmzaon n connuous flexble process neworks. Ind. Eng. Chem. Res. 39, Brooke, A., Kendrck, D., Meeraus, A., Raman, R., GAMS A User s Gude. GAMS Developmen Corporaon, Washngon, D.C. Camm, J.D., Chorman, T.E., Dll, F.A., Evans, J.R., Sweeney, D.J., Wegryn, G.W., Blendng OR/MS, judgmen, and GIS: resrucurng PG s supply chan. Inerfaces 27, Chen, P., Pno, J.M., Lagrangean-based echnques for he supply chan managemen of flexble process neworks. Compu. Chem. Eng. 32, Coe, J., Neworked Pharma: Innovave sraeges o overcome margn deeroraon. Conrac Pharma, June. Cohen, M.A., Lee, H.L., Sraegc analyss of negraed producon-dsrbuon sysems models and mehods. Oper. Res. 36, Cohen, M.A., Moon, S., An negraed plan loadng model wh economes of scale and scope. Eur. J. Oper. Res. 50, Eskgun, E., Uzsoy, R., Preckel, P.V., Beaujon, G., Krshnan, S., Tew, J.D., Oubound supply chan nework desgn wh mode j

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