How To Design A Supply Chain

Design of Responsive rocess upply Chains under Demand Uncerainy Fengqi You and Ignacio E. Grossmann * Cener for Advanced rocess Decision-making, Deparmen of Chemical Engineering, Carnegie Mellon Universiy, isburgh, A 15213 ABTRACT This paper addresses he opimizaion of supply chain design and planning under he crieria of responsiveness and economics wih he presence of demand uncerainy. The supply chain consiss of muli-sie processing faciliies and corresponds o a muli-echelon producion nework wih boh dedicaed and muliproduc plans. The economic crierion is measured in erms of ne presen value, while he crierion for responsiveness accouns for ransporaion imes, residence imes, cyclic schedules in muliproduc plans, and invenory managemen. By using a probabilisic model for sockou, he expeced lead ime is proposed as he quaniaive measure of supply chain responsiveness. The probabilisic model can also predic he safey sock levels by inegraing sockou probabiliy wih demand uncerainy. These are all incorporaed ino a muli-period mixedineger nonlinear programming (MINL) model, which akes ino accoun he selecion of manufacuring sies and disribuion ceners, process echnology, producion levels, scheduling and invenory levels. The problem is formulaed as a bi-crierion opimizaion model ha maximizes he ne presen value and minimizes he expeced lead ime. The model is solved wih he ε-consrain mehod and produces a areo-opimal curve ha reveals how he opimal ne presen value, supply chain nework srucure and safey sock levels, change wih differen values of he expeced lead ime. A hierarchical algorihm is also proposed based on he decoupling of differen decision-making levels (sraegic and operaional) in he problem. The applicaion of his model and he proposed algorihm are illusraed wih wo examples of polysyrene supply chains. Keywords: upply Chain Managemen, Responsiveness, Lead Time, Demand Uncerainy, afey ock, MINL * To whom all correspondence should be addressed. Email: grossmann@cmu.edu

1. INTRODUCTION There is a growing recogniion ha individual businesses no longer compee as sandalone eniies, bu raher as supply chains whose success or failure is ulimaely deermined in he markeplaces by he end consumers (Chrisopher and Towill, 2001). For beer cusomer saisfacion and marke undersanding, companies are sriving o achieve he bes performance from heir supply chains by differen measures, of which accurae demand forecasing, invenory and responsive supply chain are hree key componens (Fisher, 1997). Quick response enables supply chains o mee he cusomer demands for ever-shorer lead imes, and o synchronize he supply o mee he peaks and roughs of demand (abah, 1998). Nowadays responsive supply chains have become keys o compeiive success and survival (Fisher, 1997; Chrisopher, 2000, 2005) due o he increasing pressure o reduce coss and invenories for compeiions in he global markeplace (Grossmann, 2005). Alhough sophisicaed mehods such as ime series have been developed o improve he forecasing accuracy, uncerainies in demand are unavoidable due o ever changing marke condiions. In supply chains, invenory improves he service by helping deal wih demand uncerainy and providing flexibiliy, alhough i can be cosly (Chase and Aquilano, 1995). In his work, we consider he design of a responsive supply chain wih inegraion of invenory and safey sock under demand uncerainy. The supply chain consiss of mulisie processing faciliies and corresponds o a muli-echelon producion nework wih boh dedicaed and muliproduc faciliies. The major goal is o deermine he processes ha are o be inegraed in he supply chain nework wih heir corresponding suppliers, disribuion ceners and he associaed ranspor links beween hem. The major consideraions in he design are he supply chain responsiveness and profiabiliy. rofiabiliy is expressed in erms of ne presen value, while responsiveness accouns for ransporaion imes, residence imes, cyclic schedules in muliproduc plans, and invenory managemen. By using a probabilisic model for sockou, a quaniaive characerizaion of responsiveness for supply chain neworks is presened, which measures he expeced response ime or expeced lead ime o changes under uncerain demands wih inegraion of invenory and safey sock level. The probabilisic model can also predic he safey sock levels by inegraing sockou probabiliy wih demand uncerainy. These are incorporaed ino a muli-period mixed-ineger non-linear - 2 -

programming (MINL) model, which akes ino accoun he selecions of manufacuring sies and disribuion ceners, process echnology, producion levels, scheduling and invenory levels. The problem is formulaed as a bi-crierion opimizaion model in which he objecives are o maximize he ne presen value and o minimize he expeced lead ime. Aside from relying on he ε-consrain mehod o generae he areo-opimal curve, a hierarchical algorihm is also proposed for he soluion of he resuling large-scale nonconvex MINL model based on he decoupling of he differen decision-making levels (sraegic and operaional) idenified in our problem. The applicaion of his model is illusraed hrough wo examples of polysyrene supply chains. The res of he paper is organized as follows. We briefly review relaed lieraure in he nex secion, and he main quaniaive characerisics of responsiveness of process supply chain neworks are discussed in ecion 3. A formal problem saemen along wih he key assumpions is given in ecion 4, while he proposed mahemaical model is described in ecion 5. ecion 6 presens a hierarchical soluion approach, and is applicabiliy is demonsraed by wo illusraive examples in ecion 7. Finally, concluding remarks are presened in ecion 8. 2. LITERATURE REVIEW Mos of he responsiveness lieraure for supply chains ends o be qualiaive and concepual, and has no been subjeced o he kind of quaniaive analysis ha his paper inends o address. There are, however, several relaed works ha offer relevan insighs. Forreser (1961) firs illusraed in a series of case sudies he effec of dynamics in indusrial sysems, which gives rise o he bullwhip effec. Lee e al (1997) furher demonsraed ha he bullwhip effec is a consequence of he informaion delay due o he srucure of supply chains, and he severiy of his effec is posiively relaed o lead imes. Responsiveness in he wider supply chain conex has been discussed by Fisher (1997), who argues ha he produc characerisics (innovaive or funcional) and life cycles need o be linked o he layou and funcions (conversion and marke mediaion) of he supply chain. He also poined ou he need of reducing he lead ime, which enables quick response o unpredicable demand o minimize sockous, markdowns and obsolee invenory. Mason e al (1999) discussed conceps and issues associaed wih responsiveness in producion and illusrae he audi ools hey proposed from a case sudy in he seel indusry. Recenly, several concepual models on supply chain responsiveness have been proposed. Chrisopher and Towill (2001) inegrae lead ime and agiliy o - 3 -

highligh he differences in heir approach, and combined hem o propose an inegraed hybrid sraegy for designing cos-effecive responsive supply chains wih seamless connecion beween manufacuring and logisics. In a laer work, Yusuf e al (2004) have reviewed emerging paerns for creaing responsive supply chains based on survey research driven by a concepual model. Holweg (2005) proposed in his paper ha produc, process and volume are hree key facors ha deermine he responsiveness of a supply chain sysem, and provided guidelines on how o align he supply chain sraegy o hese hree facors in order o balance responsiveness o cusomer demand and supply chain efficiency. An examinaion of supply chain sysems in process indusries from a responsiveness view poin was carried ou by haw e al (2005). These auhors also proposed a concepual managemen sraegy o improve responsiveness. Anoher group of relevan papers o be considered are on supply chain design and operaion. A general review of his area is given in Kok and Graves (2003), and a specific review for supply chains in process indusries is presened by hah (2005). ome recen works include he following. Tsiakis e al (2001) presened a supply chain design model for he seady-sae coninuous processes. Their supply chain model was developed based on deermining he connecion beween muliple markes and muliple plans wih fixed locaions. Jackson and Grossmann (2003) presened a emporal decomposiion scheme based on Lagrangean decomposiion for a nonlinear programming problem model for muli-sie producion planning and disribuion, where nonlinear erms arise from he relaionship beween producion and physical properies or blending raios. chulz e al (2005) described wo muli-period MINL models for shor erm planning of perochemical complexes. Linearizaion echniques were applied o reformulae he nonconvex bilinear consrains as MIL models. Recenly, ousa e al (2006) presened a wo sage procedure for supply chain design wih responsiveness esing. In he firs sage, hey design he supply chain nework and opimize he producion and disribuion planning over a long ime horizon. In he second sage, responsiveness of he firs sage decisions are assessed using he service level o he cusomers (i.e. delay in he order fulfillmen). However, all hese models consider supply chain neworks wih only dedicaed processes. Muli-produc bach plans or flexible processes were no aken ino accoun, and hence no scheduling models were included. There are works on supply chain opimizaion wih consideraion of flexible processes in he producion nework, bu mos of hem are resriced o planning and scheduling for a given faciliy in a fixed locaion wihou exension o he mulisie supply chain nework - 4 -

design problems. Bok e al (2000) proposed a muliperiod supply chain opimizaion model for operaional planning of coninuous flexible process neworks, where sales, inermien deliveries, producion shorfalls, delivery delays, invenory profiles and job changeovers are aken ino accoun. A bilevel decomposiion algorihm was proposed, which reduced he compuaional ime significanly. Kallrah (2002) described a ool for simulaneous sraegic and operaional planning in a muli-sie producion nework, where key decisions include operaing modes of equipmen in each ime period, producion and supply of producs, minor changes o he infrasrucure and raw maerial purchases and conracs. A muli-period model is formulaed where equipmen may undergo one change of operaion mode per period. The sandard maerial balance equaions are adjused o accoun for he fac ha ransporaion imes are much shorer han he period duraions. Chen e al (2003) presened a muli-produc, mulisage and muliperiod producion and disribuion planning model. They also proposed a wo-phase fuzzy decision making mehod o obain a compromise soluion among all paricipans of he muli-enerprise supply chain. To accoun for produc demand flucuaion and o obain a beer undersanding of how uncerainy affecs he supply chain performance, a number of approaches have been proposed in he chemical engineering lieraure for he quaniaive reamen of uncerainy in he design, planning and scheduling problems. A classificaion of differen areas of uncerainy for bach chemical plan design is suggesed by ubrahmanyam e al (1994), where uncerainy in prices and demand, equipmen reliabiliy and manufacuring are aken ino accoun. The auhors used he popular scenario-based approach, which aemps o capure uncerainy by represening i in erms of a number of discree realizaions of he sochasic quaniies, consiuing disinc scenarios. Each complee realizaion of all uncerain parameers gives rise o a scenario and all he possible fuure oucomes are forecased and aken ino accoun hrough he use of scenarios. The objecive is o find a robus soluion which performs well under all scenarios. The scenario-based approach provides a sraighforward way o implicily accoun for uncerainy (Liu and ahinidis, 1996). Is main drawback is ha i ypically relies on eiher a priori forecasing of all possible oucomes, or he explici/implici discreizaions of a coninuous mulivariae probabiliy disribuion by mehods of Gaussian quadraure inegraion or Mone Carlo sampling, which can resul in an exponenial number of scenarios. Anoher popular mehod o address he uncerainy is using probabilisic approaches, which consider he uncerainy aspec of he supply chain by reaing one or more - 5 -

parameers as random variables wih known probabiliy disribuion. By inroducing a cerain number of nonlinear erms from coninuous disribuion, his approach can lead o a reasonable size of he deerminisic equivalen represenaion of he probabilisic model, circumvening he need for explici/implici discreizaion or sampling. As argued by Zimmermann (2000), he choice of he appropriae mehod is conex-dependen, wih no single heory being sufficien o model all kinds of uncerainy. Recenly, hen e al (2003) proposed a novel approach o deal wih he demand uncerainy for faciliy locaion problems. In heir work, demand uncerainy is hedged by holding cerain amoun of safey socks, and he safey sock level is decided by he demand variance and a specific service level. By adding he safey sock erm in he model, he recourse problem for uncerain parameers is avoided. Thus, hese papers eiher focus only on he long-erm sraegic design models, or else are resriced o shor-erm planning and scheduling models. Hence, no quaniaive analyses are available for responsive supply chains under demand uncerainy. 3. ULY CHAIN REONIVENE A major goal of his paper is o develop a quaniaive definiion of supply chain responsiveness wih inegraion of invenory under demand uncerainy. Responsiveness is defined as he abiliy of a supply chain o respond rapidly o he changes in demand, boh in erms of volume and mix of producs (Chrisopher, 2000; Holweg, 2005). ince he definiion is qualiaive, we need o find a quaniaive measure for supply chain responsiveness. Lead ime is he ime of a supply chain nework o respond o cusomer demands. We will consider have he lead ime o he one corresponding o he longes ime for all pahs. Furhermore, in he wors case lead ime corresponds o he response ime when here are zero invenories. This was used as a measure of responsiveness in our previous work (You and Grossmann, 2007). As shown in Figure 1, a supply chain nework wih long lead ime implies ha is responsiveness is low, and vice versa. In his work, since we consider uncerain demands and safey socks, expeced lead ime will be used as he measure of supply chain responsiveness. Thus, he challenge is o quaniaively define he expeced lead ime wih inegraion of he supply chain nework srucure, invenory and operaion deails under demand uncerainy. In he following secions we will firs review some key definiions, and hen presen our new proposal for expeced lead ime. - 6 -

3.1. Time Delays in imple Linear upply Chains Consider firs he case of a simple linear supply chain as given in Figure 2 ha consiss of one supplier, several manufacuring plans, one disribuion cener and one cusomer. The maerial flow sars from he supplier by way of manufacuring plans and disribuion cener(s) 1 and ends a he cusomer. The informaion ransfers in he reverse direcion. In his work, we assume ha informaion ransfers insananeously, hus he ime delay for he enire supply chain comes from he ime delay incurred by he ransfer of maerials. From Figure 2, we can see ha he ransfer of maerial flow is delayed by boh ransporaion and producion. The ransporaion delays beween supplier, plans, disribuion cener and cusomer are equal o he associaed ransporaion imes beween hem (Figure 3). The producion delay by each single-produc plan is equal o he residence ime of he producs. The producion delay in a muliproduc plan is more complicaed as i needs o accoun for scheduling deails. Therefore, we will leave his o he discussion in ecion 5.3 cyclic scheduling. The ime delay of he enire linear supply chain can be pariioned ino wo pars, delivery lead ime and producion lead ime. The delivery lead ime is defined as he ime o ransfer he produc from disribuion cener o he cusomer, and he producion lead ime is he ime ha he maerial flow akes o ransfer from supplier o he disribuion cener (Figure 4). Thus, he delivery lead ime is equal o he ransporaion ime from he disribuion cener o he cusomer; he producion lead ime is equal o he summaion of all he ime delays incurred by ransporaion and producion from he supplier o he disribuion cener. Noe ha his characerizaion of he ime delay of produc aciviies is similar o he lean ool value sream mapping (Voekel and Chapman, 2003). 3.2. Expeced Lead Time of imple Linear upply Chain If here is sufficien invenory in he disribuion cener o handle he demand changes, he lead ime should be equal o he ime o ransfer producs from disribuion cener o cusomers, which is he delivery lead ime. If here is no sufficien invenory in he disribuion cener o handle he demand changes (i.e. he produc is ou of sock), he wors case is when here is no exra sock for raw maerials or inermediae producs, and he only way is o go back o reorder he raw maerials from he supplier. In his way, afer a series of ransporaion and producion seps, he produc will be finally shipped o he 1 A linear supply chain can have more han one disribuion ceners. For a supply chain nework, he disribuion sysem can be muli-echelon. For simpliciy, we only consider one disribuion cener for he linear supply chain in his work. - 7 -

cusomer. Therefore, in his case he lead ime would be equal o he producion lead ime plus he delivery lead ime. Because he demand is uncerain, here is a probabiliy ha he produc will be ou of sock. We denoe rob as he probabiliy of sock ou, L D as he delivery lead ime, and L as he producion lead ime. Thus, if he produc is ou of sock, he lead ime is he producion lead ime plus delivery lead ime ( L + LD) wih he sock ou probabiliy rob. If he produc is no socked ou, he lead ime is delivery lead ime L wih he probabiliy of (1 rob ). Therefore, we have ha he expeced lead ime D EL ( ) of his simple linear supply chain is given by: E( L) = rob ( L + L ) + (1 rob ) L which can be arranged as, EL ( ) = robl + LD D D This implies ha he expeced lead ime is equal o he delivery lead ime plus he expeced producion lead ime (he sock ou probabiliy imes producion lead ime). 3.3. Expeced Lead Time of rocess upply Chain Nework Alhough a general process supply chain nework is more complex han a simple linear supply chain, we can sill decompose he supply chain nework ino pahs of maerial flows ha sar from a supplier, and end a a cusomer, by way of several plans and disribuion ceners (as shown in Figure 5). For simple supply chain neworks we can deermine all he pahs by inspecion. For complex supply chain neworks, various pahfinding algorihms (such as he one by Lengauer and Tarjan, 1979) ha can be used o figure ou all he possible pahs in he supply chain nework. I is worh menioning ha value sream mapping (Voekel and Chapman, 2003), a lean manufacuring ool widely used in indusries, can also analyze he process and find all he possible pahs in a supply chain nework auomaically. Thus, each pah is equivalen o a simple linear supply chain for which he expeced lead ime can be easily deermined. We define he maximum expeced lead ime of all possible pahs as he oal expeced lead ime of he enire supply chain. One could also consider a weighed expeced lead ime according o he imporance of cusomers. Bu for simpliciy, we consider he former definiion in his work. - 8 -

4. ROBLEM TATEMENT An inegraed approach is needed in order o consider simulaneously supply chain nework design, producion planning and scheduling, demand uncerainy and invenory managemen o resolve he rade-offs beween economics and responsiveness in an opimal manner. The problem of responsive supply chain design under demand uncerainy can be formally saed as follows. Given is a poenial process supply chain nework (CN) ha includes possible suppliers, manufacuring sies, disribuion ceners and cusomers as shown in Figure 6. Also, a se of processes and a ime horizon consising of a number of ime periods are given. The processes may be dedicaed or flexible. Flexible processes are muli-produc processes ha operae under differen producion schemes using differen raw maerials and/or producing differen producs. Furhermore, changeovers are incurred beween producs (Figure 7). For all he producion schemes, mass balances are expressed linearly in erms of he main produc s producion. The invesmen coss for insalling he plans and disribuion ceners are expressed by a cos funcion wih fixed charges. There could be differen ransporaion modes, coninuous (e.g. pipelines) or discree (e.g. barges, rail cars, vessel), for each pah ha connecs he suppliers, plan sies, disribuion ceners and cusomers. For simpliciy, we will assume ha he ransporaion of maerials in his problem is coninuous. Thus, fixed charge cos funcions provide good esimaions of ransporaion coss. The ransporaion ime of each roue and he residence ime of each produc are assumed o be known. The CN involves a se of chemicals, which includes raw maerials, inermediae producs and final producs. rices for raw maerials and final producs are assumed o be known over he enire ime horizon. Raw maerials are subjec o availabiliy consrains (i.e., wihin lower and upper bounds). Demands in each ime period are uncerain and are described by a specified coninuous probabiliy disribuion funcion. Mos of he invenories and all he safey socks are hold in disribuion ceners, while plan sies also mainain a cerain amoun of invenory. Uni invenory cos for raw maerials, inermediae producs and final producs are also given. In order o design a responsive supply chain, one objecive is o minimize he expeced lead ime of he enire supply chain nework. From he economic aspec, he oher objecive funcion is o maximize he ne presen value (NV) over he specified ime horizon. The income from sales, along wih he invesmen, operaing, ransporaion and purchase coss are aken ino accoun in he NV objecive funcion. - 9 -

ince he wo conflicing objecives need o be opimized simulaneously, he corresponding problem yields an infinie se of rade-off soluions. These soluions are areo-opimal in he sense ha i is impossible o improve boh objecive funcions simulaneously (Halsall and Thibaul, 2006). This implies ha any designs for which he expeced ne presen value and he expeced lead ime can be improved simulaneously are inferior soluions ha do no belong o he areo-opimal curve. The aim of his problem is o deermine he supply chain nework configuraions and operaional decisions ha define he areo opimal curve by maximizing NV and minimizing he expeced lead ime. 5. MODEL The model will be formulaed as a muli-period MINL problem, which will predic he deailed design decisions, producion and invenory profiles and schedules of he CN wih differen specificaions of he expeced lead ime. A lis of indices, ses, parameers and variables are given in he Appendix. Four ypes of consrains are included in his model. They are nework srucure consrains, operaional planning consrains, cyclic scheduling consrains and probabilisic consrains. Consrains (1) o (12) deermine he nework srucure, consrains (13) o (23) refer o he operaional planning consrains, consrains (24) o (36) are used for he cyclic scheduling of muli-produc plans, consrains (37) o (40) are probabilisic consrains. Finally, inequaliies (41) o (44) define he expeced lead ime and equaion (46) defines he ne presen value, boh of which are objecive funcions o be opimized. 5.1. Nework rucure Consrains To deermine he opology of he nework srucure and model he selecion of suppliers, plan sies, disribuion ceners, ogeher wih he ransporaion links beween m I N O hem, he binary variables ( Y ki,, Y, Y kls,, Y k', k, Y km,, Y, mld ) for plans, disribuion ceners and ransporaion links are inroduced for he design decisions. Four ypes of nework srucure consrains are applied o represen he relaionships beween each node in he supply chain nework. 5.1.1. upplier lan (ie) The firs ype of relaionship is beween suppliers and plan sies (Figure 8). A I ransporaion link for raw maerial j from supplier ls o plan sie k exiss ( Y kls, ), only if - 10 -

a leas one plan ha consumes raw maerial j exiss in sie k ( Y ki, ). The relaionships discussed above can be expressed by he following logic proposiion: Y Y, k K, ls L, j J (1.a) I kls, i Ij ki, These logic proposiions can be furher ransformed ino inequaliies as described in Raman and Grossmann (1993). Y Y, k K, ls L, j J (1) I kls, ki, i I j On he plan side, if a plan ha consumes raw maerial j is se up ( Y, ), a leas one I ransporaion link from he supplier ls o sie k mus be seleced ( Y, proposiions are: Y I ki, ls Lj kls, j j ki kls ). The logic Y, k K, i I, j J (2.a) which can be ransformed ino inequaliies: Y Y, k K, i I, j J (2) I ki, kls, ls L j 5.1.2. Inpu and oupu relaionship of a plan The second ype of nework srucure relaionship is he inpu and oupu relaionship of a plan (Figure 9). This ype of relaionship is somewha more complicaed han he previous one because he iner-sie ransporaion mus be aken ino accoun. If an inersie ransporaion link from sie k ' o sie k is insalled for chemical j ( Y ha a leas one plan i ' in sie i i j j k ' is insalled ha produces chemical j ( Y leas one plan i in sie k is insalled ha consumes chemical j ( Y, ), ki N k', k k', i' ), i implies ), and also a Y N k', k Yk', i', k k K j Jk', i' JM k, i i' Oj, ', ( ) (3) Y Y, k, k ' K, j ( Jk', i' JM k, i) (4) N k', k k, i i I j If a plan i in sie k is insalled ( Y, ), ha consumes chemical j, hen sie k is I conneced o one of he suppliers of chemical j denoed as ls ( Y, anoher sie k ' ha produces chemical j ( Y produces chemical j ( Y ki,' I N ki, ls Lj kls, k' Ki' k', k i' Oj ki, ' ki N k', k kls ), or conneced o ), or here is anoher plan i ' in sie k ha ). The logic proposiions can be wrien as follows: Y Y Y Y, k Ki, i I j, j JMk, i (5.a) - 11 -

which can be ransformed ino linear inequaliies as,, k Ki, i I j, j JMk, i Y Y + Y + Y I N ki, kls, k', k ki, ' ls L j k ' Ki' i ' O j (5) imilarly, if he chemical j is produced by plan i in sie k, hen a leas one oher plan i ' in he same sie is insalled ha consumes chemical j ( Y ki,' ), or here is a leas O N one ransporaion link o a disribuion cener ( Y km, ), or o anoher sie ( Y kk, ') ha consumes chemical j :, k Ki, i Oj, j Jk, i Y Y + Y + Y O N ki, km, kk, ' ki, ' m M k' Ki' i' I j (6) Consrains (5) are defined for raw maerials and inermediae producs, and consrains (6) are defined for inermediae producs and final producs. 5.1.3. lan (sie) Disribuion Cener The hird ype of relaionship is beween plan sies and disribuion ceners as shown in Figure 10. A ransporaion link for produc j from plan sie k o disribuion cener O m exis ( Y km, ), only if a leas one plan ha consumes raw maerial j exiss in sie k ( Y ki, ). On he plan side, if a plan ha consumes raw maerial j is se up, here should be a leas one link from he disribuion cener m o sie k exiss. imilarly, ransforming he corresponding logic proposiions, leads o he following inequaliies: Y Y Y, k K, m M, j Jki, (7) O km, ki, i Oj Y, k K, i Oj, j Jk, i (8) O ki, km, m M 5.1.4. Inpu and oupu relaionship of disribuion cener The las ype of nework srucure relaionship is he inpu and oupu relaionship of a disribuion cener as in Figure 11. A ransporaion link from plan sie k o disribuion cener m exiss, only if he disribuion cener m exiss ( Y disribuion cener m o cusomer ld exiss ( Y, exiss. The relaionships can be expressed as, Y O km, m Y,, mld m ). A ransporaion link from ), only if he disribuion cener m k K m M (9) Y mld, m Y,, m M ld LD (10) - 12 -

On he oher hand, if a disribuion cener m is se up, a leas one ransporaion link O from he plan sie k o disribuion cener m ( Y km, ) and a leas one ransporaion link from disribuion cener m o cusomer ld ( Y, Y Y m m k K O km, mld ) mus exis. Y, m M (11) Y, m M (12) ld LD mld, 5.2. Operaional lanning Consrains In he operaional planning model invesmen in plan capaciy, and purchases, sales, producion, ransporaion and mass balance relaionships are considered ogeher wih he consrains of hese aciviies due o he supply chain srucure. 5.2.1. roducion Consrains All he chemical flows W ki,, js,, of chemical j associaed wih producion scheme s in plan i of sie k, oher han he main produc j ', are given by he mass balance coefficien μ i, j, s. The following equaion relae inpu or inle flow of chemical j ( W ki,, js,, ) wih he oupu of a main produc j ' ( W ki,, j', s, W ki,, js,, i, js, ki,, j', s, ) of each process, = μ W, k Ki, i I j, j Ji', s, j' Ji, s, s i, T (13) The producion amoun should no exceed he design capaciy Q ki, defined by he main produc j for each process: W ρ Q Lenp, k Ki, i Oj, j Jk, i, s i, T (14) 1 ki,, js,, is, ki, where Lenp is he lengh of ime period and ρ is he relaive producion amoun of main produc j of producion scheme s in plan i in urns of capaciy The formulaion is based on he assumpion ha here are no capaciy expansions over he enire ime horizon. However, muli-period capaciy planning evens and decisions can readily be considered by suiably modifying he formulaion consrains along wih he deailed capaciy invesmen consrains (ahinidis e al, 1989) which are no deailed here. For flexible processes, he maximum producion rae r kis,, of he each produc s in plan i of sie k is proporional o he capaciy of he plan (see Noron and Grossmann, 1994), r = ρ Q, k Ki, i Oj, j Jk, i, s i (15) 2 kis,, is, ki, 1 is, - 13 -

where ρ is he relaive maximum producion rae of main produc of producion scheme 2 is, s in urns of he capaciy of plan i 5.2.2. Mass Balance Consrains The mass balance for chemical j in manufacuring sie k a ime period is given as follows: + TR + W = F + TR + W, k, j, ls, k, k', j, k, i, j, s, k, j, m, k, k', j, k, i', j, s, ls L k ' K i O j s m M k K i' I j s where k, j, ls, k K, j J, T (16) is purchase amoun, TR kk, ', j, is he iner-sie shipping amoun and W ki,, j, is he producion amoun. The mass balance for chemical j in disribuion cener m a ime period is given as follows: F =, j Jm, M, T (17) k K k, j, m, j, m, ld, ld LD where F k, j, m, is he shipping amoun from producion sie o disribuion cener and jmld,,, is he sale amoun. 5.2.3. Invenory Consrains The oal available amoun of chemical j for cusomer ld ( Q jld,, ) should be equal o he safey socks ( I jmld,,,) commied o his cusomer in all disribuion ceners plus he sale amoun. Q I + =,,, jld,, jmld,,, jmld,,, m M m M i j Jld LD T (18) The sale amoun of chemical j o each cusomer ld a ime period should be equal m o he associaed arge demand d jld,, (usually he arge is equal o he mean value of uncerain demand), m jmld,,, = d, jld,, j Jld, LD, T (19) m M The oal available amoun of chemical j will be less han he upper bound of he demand, Q d, j Jld, LD, T (20) U jld,, jld,, - 14 -

The working invenories of he plan sies ( WI k, j, ) and he disribuion ceners DC ( WI m, j, ) are relaed linearly o he inle flows of maerials ha hey handle (Tsiakis e al, 2001). This is expressed via he consrains, WI WI α, k K, j JM, T (21) k, j, k, j, k, j, ls, ls L β F, m M, j J, T (22) DC m, j, m, j, k, j, m, k K where α k, j, and β m, j, are given coefficiens coming from empirical sudies (such as he probabiliy of machines broken down or supply limiaion). 5.2.4. Upper Bound Consrains urchases k, j, ls, from supplier ls o plan sie k ake place only if he ransporaion link beween hem is se up, Y, k, j, ls, (23.1) U I k, j, ls, k, j, ls, k, ls Iner-sie ransporaion TR kk, ', j, from sie k o sie k ' ake place only if he ransporaion link beween hem is se up, TR TR Y, kk, ', j, (23.2) U N kk, ', j, kk, ', j, kk, ' ales jmld,,, from disribuion cener m o cusomer ld ake place only if he ransporaion link beween hem is seleced, Y, j, mld,, (23.3) U j, mld,, jmld,,, mld, Nonzero producion flows W ki,, js,, are allowed in plan i of sie k only if he plan is insalled, W W Y, ki,, js,, (23.4) Q U ki,, js,, ki,, js,, ki, Q Y, ki, (23.5) U ki, ki, ki, The ransporaion amoun F k, j, m, from plan sie k o disribuion cener m akes place only if he ransporaion link beween hem is se up, F F Y, k, j, m, (23.6) U O k, j, m, k, j, m, k, m k k 5.3. Cyclic cheduling Consrains To address deailed operaions of he muli-produc plans, we have considered a cyclic scheduling policy (ino and Grossmann, 1994). Under his policy, he sequences o produce each produc are decided ogeher wih he cycle ime, and hen he idenical - 15 -

schedule is repeaed over each ime period (Figure 12). The rade-offs beween invenories and ransiions are esablished by opimizing he cycle imes. Imporan decisions in cyclic scheduling including he sequence of producion ( Y kissl,,,, ) and precedence relaionship for changeovers beween pairs of producs ( Z kiss,,, ', sl, ), are deermined hrough he following assignmen and sequencing consrains. 5.3.1. Assignmen Consrains The assignmen consrains sae ha exacly one ime slo mus be assigned o one produc and vice versa. The oal number of ime slos will be exacly equal o he oal number of producs, Ykissl,,,, = 1, k Ki, i I j, s i, T (24) sl Li Ykissl,,,, = 1, i, j, i, s i 5.3.2. equence Consrains k K i I sl L T (25) The sequence consrains sae ha exacly one ransiion from produc s occurs in he beginning of any ime slo if and only if s was being processing during he previous ime slo. On he oher hand, exacly one ransiion o produc s occurs in he ime slo if and only if produc s is being processed during ha ime slo. As suggesed in Wolsey (1997), he ransiion variables Z kiss,,, ', sl, can be replaced by coninuous variables beween 0 and 1, insead of binary variables. This significanly reduces he number of discree variables and improves he compuaional efficiency. Zk,, i s, s', sl, = Yk,, i s', sl 1,, k Ki, i I j, s ' i, sl Li, T (26) s i Z = Y, k Ki, i I j, s i, sl Li, T (27) s' ' i k, i, s, s', sl, k, i, s, sl, 0 1, Z kiss,,, ', sl, k,, iss, ', sl, (28) 5.3.3. roducion Consrains The producion amoun of produc s in a cycle ( W kis,,, ) is equal o he processing rae r kis,, imes he processing ime δ kissl,,,, : W = r δ, k K, i I, s, T (29) kis,,, kis,, kissl,,,, sl Li i j i - 16 -

The amoun produced for each produc in ime period ( N ki,, cycles in he ime period) should be no less han he oal producion prediced from operaional planning in his ime period: Wkis,,, Nki,, = Wki,, js,,, k Ki, i Oj, j Jk, i, s i, T (30) 5.3.4. Timing Consrains Consrains (31) o (34) are used o resric he iming issues in he cyclic scheduling. The processing ime δ kissl,,,, in a cerain ime slo is equal o he summaion of he processing imes assigned o all he producs in his ime slo, δ = δ, k K, i I, sl L, T (31) k,, i sl, k,, i s, sl, s i i j i The cycle ime TC ki,, is equal o he summaion of all he processing imes in each ime slo plus he summaion of ransiion imes in his cycle, TC = δ + Z τ, k Ki, i I j, T (32) ki,, kisl,,, kiss,,, ', sl+ 1, iss,, ' sl Li s i s' i sl Li The oal producion ime should no exceed he duraion of each ime period H ki,,, TC N H, k K, i I, T (33) ki,, ki,, ki,, i j The producion for scheme s in ime slo sl can ake place only if he ime slo is assigned o he producion scheme: δ δ Y, k K, i I, s, sl L, T (34) U kissl,,,, kissl,,,, kissl,,,, 5.3.5. Cos Consrains i j i i To inegrae he cyclic scheduling wih he sraegic planning, he invenory and ransiion coss from cyclic scheduling are considered as par of he operaing cos. Consrain (35) represens ha cos from scheduling in a ime period for a cerain plan. The firs erm on he righ hand side of he equaion sands for he oal ransiion cos in a ime period. The second erm is he invenory cos for all he chemicals involved in he producion. The change of invenory level in a ime period is given in Figure 13. In he work by ino and Grossmann (1994), hese auhors consider invenory only for final producs, as heir model is for single plan. In our case, each manufacuring sie may have more han one producion plan, and invenory for maerials of muli-produc plans mus be also aken ino accoun. ince we assume ha maerial balances are expressed linearly in erms of he main produc s producion, he cumulaive invenory levels for raw maerials are also relaed linearly o he cumulaive invenory level of main produc in - 17 -

each producion scheme and he coefficiens of he linear relaionships are equal o he mass balance coefficiens. This leads o he second erm on he righ hand side of he following consrain. Thus, he invenory and ransiion coss of muliproduc processes are given by, COT = CTR Z N + ( μ ε )( r H W N ) δ /2 ki,, iss,, ' kiss,,, ', sl, ki,, i, js, k, j, kis,, ki,, kis,,, ki,, kisl,,, s i s' i sl Li s i j Ji sl Li k K, i I, T (35) This consrain is nonlinear and nonconvex, wih bilinear and rilinear erms. If all he processes in he producion nework are dedicaed, cyclic scheduling need no be aken ino accoun, and his consrain can be discarded. 5.3.6. Upper Bound Consrains As a muli-sie problem, we need o make sure ha if a plan i in sie k is no insalled, here are no producion cycles. To model his, we inroduce he upper bound consrain (36) for he number of cycles N ki,, in each ime period for each muliproduc plan in each manufacuring sie: N N Y, k K, i I, T (36) U ki,, ki,, ki, i j Also he assignmen consrains are reformulaed o accoun for he fac ha all he scheduling aciviies can ake place only if he plan is insalled: Ykissl,,,, = Yki,, k Ki, i I j, s i, T (24) sl Li Ykissl,,,, = Yki,, i, j, i, s i k K i I sl L T (25) i j 5.4. robabilisic Consrains for Demand Uncerainy A key componen of decision making under uncerainy is he represenaion of he sochasic parameers. There are wo disinc ways of represening uncerainies. The scenario-based approach (ubrahmanyam e al, 1994; Liu and ahinidis, 1996) aemps o capure he uncerainies by represening hem in erms of a number of discree realizaions of he sochasic parameers where each complee realizaion of all uncerain parameers gives rise o a scenario. In his way all he possible fuure oucomes are aken ino accoun hrough he use of scenarios. This approach provides a sraighforward way o formulae he problem, bu is major drawback is ha he problem size increases exponenially as he number of scenarios increases. This is paricularly rue when using coninuous mulivariae probabiliy disribuion wih Gaussian quadraure inegraion schemes. - 18 -

Alernaively, Mone Carlo sampling could be used, alhough i also requires a raher large number of poins o achieve a desired level of accuracy. These difficulies can be somewha circumvened by analyically inegraing coninuous probabiliy disribuion funcions for he random parameers (Wellons e al, 1989; ekov e al, 1998). This approach can lead o a reasonable size deerminisic equivalen represenaion of he probabilisic model, a he expense of inroducing cerain amoun of nonlineariies ino he model hrough mulivariae inegraion over he coninuous probabiliy space. In his work, his approach is used for describing he demand uncerainy. The probabilisic descripion of demand uncerainy renders some operaional planning variables (amoun of sale, shorfall and salvage) o be sochasic. A radiional way o deal wih his is o allow correcive acions by adding recourse acions in he model. ochasic programming problems wih recourse are usually complicaed in is naure and difficul o solve. Based on he special characerisics of demand uncerainy, hen e al (2003) recenly proposed a novel approach o hedge unexpeced demand by holding a cerain amoun of safey sock before demand realizaion. The amoun of safey sock is esimaed by using a chance consrain linking service level o a demand probabiliy disribuion. The need for recourse is obviaed by aking he proacive acion wih safey sock, and he sochasic aribues of he problem are ranslaed ino a deerminisic opimizaion problem a he expense of inroducing nonlinear erms ino he model. In his work, we use a similar approach. Insead of specifying he service level (or over-socked probabiliy), we rea he sock-ou probabiliy as a variable, and inegrae i wih supply chain responsiveness. The deailed model formulaions are given below. 5.4.1. ock-ou robabiliy If a paricular produc demand realizaion d jld,, is higher han is oal available amoun Q jld,,, we are under-socked, i.e. sock-ou will happen. If a paricular produc demand realizaion d jld,, is less han is oal available amoun Q jld,,, we are oversocked. The probabiliy of over-sock is defined as β -service level in manufacuring lieraures. The probabiliy of sock-ou plus he service level should be always equal o 1 (as Figure 14). Thus he sock-ou probabiliy can be expressed as, rob = r( Q d ) = 1 r( Q d ), j Jld, LD, T (37) jld,, jld,, jld,, jld,, jld,, The form of his consrain is very similar o a chance consrain (Charnes and Cooper, 1963), which suggess ha he above equaion can be ransformed ino a deerminisic - 19 -

form. I is worh menioning ha his general definiion for esimaing safey sock level can be applied for all ypes of demand disribuions. In real world cases, demand is ofen assumed o be normally disribued when here are sufficien samples, or riangularly disribued when limied sample applied. In he following secions we consider he cases when he demand follows a normal disribuion and a riangular disribuion. 5.4.2. Normal Disribuion Due o he cenral limi heorem, he produc demands are ofen modeled as random parameers ha are normally disribued (Wellons e al, 1989; ekov e al, 1998). For a normal disribued demand, we are given he mean ( μ jld,, ) and sandard deviaion ( σ jld,, ) of he demand. To faciliae he calculaion of he sock-ou probabiliy rob jld,,, sandardizaion of he variables is firs performed. Normal random variables can be recas ino he sandardized normal form, wih a mean of zero and a variance of 1, by subracing heir mean and dividing by heir sandard deviaion. For he deerminisic variables Q jld,,, he sandardized normal variables K jld,, are given as: K jld,, Q μ jld,, jld,, = (38) σ jld,, ubsiuing (38) ino he general definiion of sock-ou probabiliy defined in (37), we have he sock-ou probabiliy of normally disribued demand for produc j cusomer ld a ime period is, rob where = 1 Φ ( K ), j Jld, LD, T (39.a) jld,, jld,, Φ (x) denoes he cumulaive probabiliy funcion of sandard normal random variable in he form of, 2 1 x x Φ (x) = exp( )d x (39.b) 2π 2 The sock-ou probabiliy for he enire planning horizon is considered as he wors case for all he individual sock-ou probabiliies of all he ime periods. Hence we have ha he sock-ou probabiliy ( rob jld, ) for produc j and cusomer ld is: rob rob = 1 Φ ( K ), j Jld, LD, T (39) jld, jld,, jld,, - 20 -

5.4.3. Triangular Disribuion U d For riangular disribuion, we are given he demand s lower bound L d, upper bound, and he mos likely value d M (as in Figure 15). Due o he non-differeniable characerisics of riangular disribuion, he sock-ou probabiliy has differen represenaions when he oal available amoun Q is less or more han he mos likely demand M d. Their relaionship can be represened by he following disjuncion (for simpliciy, he subscrips j, ld, are omied) y y L M U M d Q d d Q d L 2 U 2 ( Q d ) ( d Q) rob = 1 ( U L )( M L rob = ) U L U M d d d d ( d d )( d d ) Using he convex hull reformulaion, he disjuncion can be ransformed ino MINL consrains as discussed by Lee and Grossmann (2000): Q = Q + Q (40.1) 1 2 jld,, jld,, jld,, d Y Q d Y (40.2) L d 1 M d jld,, jld,, jld,, jld,, jld,, d (1 Y ) Q d (1 Y ) (40.3) M d 2 U d j, ld, j, ld, j, ld, j, ld, j, ld, rob = rob + rob (40.4) rob rob 1 2 jld,, jld,, jld,, 1 d jld,, jld,, (40) Y (40.5) 1 Y (40.6) 2 d j, ld, j, ld, rob rob ( Q d ) 1 L 2 1 d jld,, jld,, jld,, Yjld,, U L M L d jld,, d jld,, d jld,, d jld,, ( )( ) ( ydu Q ) 2 2 2 jld,, jld,, jld,, U L U M d jld,, d jld,, d jld,, d jld,, ( )( ) (40.7) (40.8) d M where Y j, ld, is a binary variable ha equal o 1 if Q jld,, is less han d j, ld,. 5.5 Lead Time Definiion As discussed in ecion 3, for each pah he expeced lead ime is equal o he delivery lead ime, plus he producion lead ime, imes he sock ou probabiliy. The delivery lead ime and he producion lead ime are in urn equal o he summaion of all he producion delays and ransporaion delays incurred in he corresponding pah. The expeced lead ime of he whole supply chain nework is equal o he maximum expeced lead ime of - 21 -

each pah. As a supply chain design problem, we need o consider he case ha if a plan or a ransporaion link is no seleced, he associaed delay is 0. Binary variables are used o model he lead ime T wih he following inequaliies: n n 1 I I N N O O j, ld k, lsλk, ls j, ld kx, i θ x kx, ix j, ld kx, k λ x 1 kx, kx 1 j, ld kn, mλkn, m m, ldλ + + m, ld x= 1 x= 1 T rob Y + rob Y + rob Y + rob Y + Y ( ls, k, k... k, m, ld) ah (41) 1 2 n ls, k, m, ld where rob jld, is he sock-ou probabiliy, all he Y are binary variables for design decisions, λ denoes ransporaion delays and θ represens producion delay. The superscrip (*) I denoes he ranspor link from supplier o plan sie, he superscrip (*) denoes he plan, (*) N is for iner-sie ranspor link, (*) O represens he ranspor link from plan sie o disribuion cener, and (*) is for he ranspor link from disribuion cener o cusomer. The se ah ls, k, m, ld includes all he possible pahs in a given poenial CN nework. All he elemens in he se ah ls, k, m, ld are in he form of ( ls, k1, k2... kn, m, ld ), where supplier ls is he sar of he pah, k 1, k 2... k n are he manufacuring sies and m is he disribuion cener ha he associaed sream goes hrough, and cusomer ld is a he end of he pah. In equaion (41) he ransporaion delay of each roue and he producion delay of each single produc plan are given parameers. The producion ime delay for a muliproduc plan is no so obvious. Before inroducing our definiions, consider he moivaing example shown in Figure 7, 12 and 13. A muli-produc plan produces hree chemicals A, B and C. Assume here is a demand change of chemical A. The wors case is when we jus finished producing A, and here is no exra invenory of A besides he one commied o he former demand. There are wo operaing policies ha can be implemened o deal wih his siuaion. If he demand of chemical A has a large change, one would usually sop he curren producion for chemical C as soon as possible and skip all he oher producs (Chemical B) o produce chemical A direcly. In his case, he producion delay is equal o he residence ime of chemical A (Figure 16). o we have he producion delay θ ki, for muliproduc R plan i in sie k is equal o he maximum residence ime ( θ is, ) of all he producs produced by his plan, - 22 -

θ θ, k K, i I (42.a) R ki, is, If he demand change of chemical A is no very significan, one will wai unil he plan produces A again, so ha we can adjus he producion o mee he demand change. This akes some ime which is given by he processing ime of chemical B and C, plus residence ime of A. In his way we define for muliproduc plan, he ime delay for each produc as cycle ime plus residence ime minus is processing ime (Figure 17). imilarly, he producion delay for a muliproduc plan is equal o he maximum ime delay for each produc: θ TC + θ δ, k K, i I (42.b) R ki, ki,, is, kissl,,,, sl Li In his definiion cycle imes of each plan are aken ino accoun as par of he delay due o producion, so ha we have inegraed he producion deails ino he quaniaive definiion of responsiveness. The erms robjld, Y (sock-ou probabiliy imes binary variable omiing he subscrips for simpliciy) in he lead ime definiion can be linearized. We use a coninuous variable Y o replace he robjld, Y erm in he lead ime consrain: robjld, Y = Y, j, ld (43.1) n n 1 I I N N O O k, j, ls, ldλk, ls kx, ix, j, ldθkx, ix kx, kx 1j, ldλkx, kx 1 kn, j, m, ldλkn, m m, ldλ + + m, ld x= 1 x= 1 T Y + Y + Y + Y + Y i i j j ( ls, k, k... k, m, ld) ah (43) 1 2 n ls, k, m, ld The equaion (43.1) is equivalen o he following disjuncion, Y Y Y = rob jld, Y = 0, j, ld (44.1) Applying he convex hull reformulaion (Balas, 1985) o he above disjuncive consrain leads o: Y + Y = rob j, ld Y Y Y2 1 Y 2 jld, j, ld j, ld (44.2) (44.3) (44.4) where Y 2 is a new coninuous variable inroduced as a slack variable. The consrains (33) are applied for all he erms wih superscrip (*) I, (*), (*) N, (*) O, (*) in he expeced lead ime definiion. - 23 -

5.6. Nonnegaive Consrains All coninuous variables mus be nonnegaive and he binary variables should be ineger: Q, W,, TR, F,, I, Q, T, θ 0 (45.1) k, i k, i, j, k, j, ls, k, k ', j, k, j, m, j, m, ld, j, m, ld, j, ld, Z, W, r, δ, TC, Te, Ts, N, θ, COT 0 (45.2) k,, i s, s', sl, k,, i s, k,, i s k,, i sl, k,, i k,, i sl, k,, i sl, k,, i k, i k,, i Y, Y, Y, Y, Y, Y {0,1} (45.3) I O m kls, ki, km, mld, kissl,,,, 5.7. Ne resen Value The NV of he supply chain nework is given by he following equaions, NV = Income C purch Coper Cranp Cinves Cinvenory (46) Income C = j ld ϕ = a jld,, jld,, ϕ purch j, ls, k, j, ls, k j ls C = σ W + COT operae i, s, k, i, j, s, k, i, k i s j Jis, k i C = ω + ω TR + ω F I N O ranp k, j, ls, k, j, ls, k, k ', j, k, k ', j, k, j, m, k, j, m, k j ls k k' j k j m + ω jmld,,, jmld,,, j m ld C = ω Q + γ Y + γ Y + γ Y + γ Y I I O O N N inves ki, ki, ki, ki, kls, kls, km, km, kk, ' kk, ' k i k i k ls k m k k' m m + γ Y + γkld, Ykld, m m ld C = ε I + ε ( WI + WI ) DC invenory j, m, j, m, ld, k, j, k, j, m, j, j m ld k j All he parameers in he above formulaion are discouned a a specified ineres rae and include he effec of axes and ineres rae on he ne presen value. 6. OLUTION ROCEDURE 6.1. oluion rocedure for Muli-objecive Opimizaion In order o obain he areo-opimal curve for he bi-crierion opimizaion problem2, one of he objecives is specified as an inequaliy wih a fixed value for he bound which is reaed as a parameer. There are wo major approaches o solve he problem in erms of 2 Two objecives are given by (43) and (46), consrains are given by (1)-(42), (44)-(45) - 24 -

his parameer. One is o simply solve i for a specified number of poins o obain an approximaion of he areo opimal curve. The oher is o solve i as a parameric programming problem (Dua and isikopoulos, 2004), which yields he exac soluion for he areo opimal curve. While he laer provides a rigorous soluion approach, he former is simpler o implemen for nonlinear models. For his reason we have seleced his approach. The procedure includes he following hree seps. The firs one is o minimize he expeced lead ime T o obain he shores expeced lead ime T, which in urn yields he lowes areo opimal NV. The second sep is o maximize NV ha in urn yields he longes areo opimal expeced lead ime TL. In his case he objecive funcion is se as, NV ε T (47) where ε is a very small value (e.g., i is on he order of 0.001). The las sep is o fix he expeced lead ime T o discree values beween T and T L, and opimize he model by maximizing NV a each seleced poin. In his way we can obain an approximaion o he areo-opimal curve, ogeher wih he opimal configuraions of CN for differen values of lead ime. 6.2. hores Opimal Expeced Lead Time In he aforemenioned soluion procedure for his bi-crierion opimizaion problem, one of he imporan seps is o find he shores opimal expeced lead ime. Insead of minimizing he expeced lead ime by solving he enire problem direcly, we use he following soluion sraegy o improve he compuaional efficiency. The expeced lead ime of each pah of chemical flow as defined in secion 3.2 is given by EL ( ) = robl + LD, which equals o he delivery lead ime ( L D ) plus sock-ou probabiliy ( rob ) imes producion lead ime ( L ). From he above equaion, we can see ha as he sock-ou probabiliy decreases, he expeced lead ime will decrease. In he sep for deermining he shores opimal expeced lead ime, we do no accoun for he economic objecive. Therefore, if here are sufficienly high safey sock levels in all he disribuion ceners o hedge he uncerain demands, he sock-ou probabiliy will be 0. Then he expeced lead ime of he supply chain nework will be reduced o he maximum delivery lead ime of each pah of chemical flow. The delivery lead ime is equal o he ransporaion delay from a disribuion cener o a cusomer. Because we can selec wha disribuion ceners o insall and wha ransporaion links o se up, we are able o choose - 25 -

a subse of all he possible ranspor links beween he cusomers and disribuion ceners, such ha each cusomer has a leas one ranspor link conneced o, and he maximum ransporaion delay of all hese ranspor links is minimized. In summary, he minimum expeced lead ime can be calculaed by: T = max{min{ λ }} (48) ld LD m M m, ld where λ mld, is he shipping ime from disribuion cener m o cusomer ld. Equaion (48) defines he shores opimal expeced lead ime. I can be inerpreed as follows. For each cusomer ld, we only choose one disribuion cener ha have minimum shipping ime o his cusomer, and se up he ranspor link beween hem. In his way, we se up as many ranspor links as he number of he cusomers (number of elemens in se ld ). ince he expeced lead ime of he supply chain nework is equal o he longes expeced lead ime for each pah, he minimum expeced lead ime of he supply chain nework will be equal o he maximum shipping ime for hose ranspor links ha have been se up. Insead of solving he large scale nonconvex MINL problem, we can obain he opimal soluion wih much less compuaional effor by using he proposed mehod. However, his mehod works only for he shores expeced lead ime case. To obain oher poins in he areo curve, we need o solve a series of large scale nonconvex MINL problems. To reduce he compuaional ime, a heurisic hierarchical soluion approached is proposed. 6.3. Heurisic Hierarchical Approach The soluion of each problem for he poins in he areo curve can be very compuaionally demanding asks due o he large number of discree decisions and he highly nonlinear nonconvex erms3. In his secion we propose a heurisic hierarchical soluion approach ha is able o handle he combinaorial and nonconvex naure of he responsive supply chain design problem and o reduce he compuaional effor needed. The basic idea is o exploi he fac ha he operaional cos arising form scheduling (invenory coss and ransiion coss) makes up only a small par of he oal NV. Therefore, we can use a wo-sage decision approach as follows. We firs deermine he supply chain nework srucure and sraegic planning decisions (producion and shipping amouns), neglecing changeovers and ransiions for muliproduc plans. In he second 3 The nonlinear nonconvex erms arise from he cyclic scheduling consrains (29), (30), (35), sock-ou probabiliy definiion (37)-(40) and expeced lead ime definiion (41), (43). - 26 -

sage, we hen consider he deailed producion scheduling afer he sraegic design and planning decisions are made. The proposed mehodology employs a simplified model (firs sage) o deermine he sraegic design and planning decisions, which are hen fed ino he deailed model (second sage) in order o derive he operaional and scheduling decision variables. I is imporan o noe ha his soluion approach can be applied when one defines he producion delay wih consrain (42.a), which assumes ha he demand is undergoing large changes and he producion delay is irrelevan o he cycle ime (please refer o secion 5.5 for deails). Clearly his heurisic approach may no yield opimal or nearopimal soluions if he model includes consrain (42.b) for he producion delays. The reason is given in secion 6.3.1. 6.3.1. implified Model The simplified model formulaion is an approximaion of he deailed model formulaion. The main advanage of he simplified model is ha i does no include he large amoun of he nonconvex erms and binary variables for cyclic scheduling. In his way, he problem can be solved more efficienly and sill capure he main rends in he supply chain design and planning. Furhermore, he model provides a valid upper bound by overesimaing he objecive funcion of he original problem. As ransiions and changeovers for muliproduc plans are no aken ino accoun in he simplified model, all he consrains for cyclic scheduling are no included. The objecive funcions are he same for each sep as in he aforemenioned soluion procedure for muli-objecive opimizaion problem. In summary, he simplified model includes he following consrains: (1)-(23), (37)-(40) (depends on he associaed demand probabiliy disribuion), (42.a), (44)-(45). The objecive funcions are (43) and (46). In he simplified model we define he producion delay wih consrain (42.a), which considers producion delay equal o he maximum residence ime of all he producs. Therefore, he expeced lead ime of a supply chain depends on he design and planning decisions only, and unrelaed o producion scheduling. Thus, he expeced lead ime obained from he simplified model is exacly he same as wha we can ge by solving he original model. Based on his, in he second sep of ε-consrain mehod, which is o calculae he longes opimal expeced lead ime, we only need o solve he simplified model for he soluion. - 27 -

On he oher hand, he simplified model migh no provide a good approximaion o he original model, if he model includes consrain (42.b) o define he producion delays insead of (42.a). Because consrain (42.b) considers producion delay equal o he cycle ime minus processing ime plus residence ime. However, cycle imes and processing imes are decisions for cyclic scheduling, which have been negleced in he simplified model. Missing informaion of scheduling may lead o inaccuracies when opimizing he expeced lead ime. Thus, his heurisic approach may no yield near-opimal soluions if he model includes consrain (42.b) for he producion delays 6.3.2. olving Deailed Model In he deailed model, he design variables (binary variables for design decisions) and planning variables (producion and ransporaion amouns, invenory levels) are fixed o heir values as deermined from he simplified model. The original deailed model is hen solved in he reduced variable space in order o deermine he opimal levels for he scheduling decision variables. However, he deailed model is sill a large scale nonconvex MINL model. To obain a good saring poin for he near-opimal soluions we formulae a heurisic subproblem and a convexified subproblem o selec he iniial values of he scheduling variables. The heurisic subproblem is used for he producion sequence of muliproduc plans. I is based on he fac ha ransiion coss are usually proporional o he ransiion imes. Thus, a producion sequence ha can minimize he oal ransiion ime in a producion cycle is ofen he opimal producion sequence (ahinidis and Grossmann, 1991). Therefore, he heurisic subproblem is o minimize he oal ransiion imes in a producion cycle (consrain (32)), subjec o he consrains (24) (28). In he convexified subproblem, all he bilinear erms in he deailed model are replaced by convex envelopes (McCormick, 1976; Quesada and Grossmann, 1995). For example, he equaion W kis,,, Nki,, = Wki,, js,, (30) was replaced by he following consrains, W N + W N W N W (49.1) LO LO LO LO kis,,, ki,, kis,,, ki,, kis,,, ki,, ki,, js,, W N + W N W N W (49.2) U U U U kis,,, ki,, kis,,, ki,, kis,,, ki,, ki,, js,, W N + W N W N W (49.3) LO U LO U kis,,, ki,, kis,,, ki,, kis,,, ki,, ki,, js,, W N + W N W N W (49.4) U LO U LO kis,,, ki,, kis,,, ki,, kis,,, ki,, ki,, js,, where LO and U represen lower and upper bounds on he variables, respecively. - 28 -

The soluions of he wo subproblems are hen used as he saring poin o solve he deailed model. By doing his, we increase he chance o obain a near-opimal soluion. 6.3.3. Algorihm In summary, he hierarchical algorihm comprises of he following seps: ep 1: olve he simplified nonconvex MINL model (objecive funcions are (43) and (46), subjec o consrains (1)-(23), (37)-(40), (42), (44)-(45)) by neglecing ransiions and changeovers of muliproduc plans and hen fix he design and planning decisions. ep 2: For each muliproduc plan, solve he MIL heurisic subproblem (objecive funcion is (32), consrains are (24)-(28)) o minimize he oal ransiion ime in a producion cycle o obain he iniial values of he scheduling variables. ep 3: olve he convexified model ha uses convex envelopes o replace he bilinear consrains (30), (33) and (35) ep 4: Use he soluions from he subproblems in ep 2 and 3 as he saring poins, solve he deailed MINL model in he reduced variables space o obain he scheduling and operaional decisions. 7. NUMERICAL EXAMLE In order o illusrae he applicaion of he proposed model and is corresponding soluion sraegy, we consider wo examples for he design of polysyrene supply chains. The firs example is a medium size problem ha is solved wih differen producion delay definiions under differen demand uncerainies (riangular disribuion and normal disribuion), and using differen soluion approaches (direc approach and hierarchical approach). The second example is a large scale problem moivaed by a real world applicaion of which wo insances wih and wihou considering safey socks are solved. The summary of all he insances is given in Table 1. In boh examples he ime horizon is 10 years, and hree ime periods are considered wih lenghs of 2 years, 3 years and 5 years, respecively. An annual ineres rae of 10% and a ax rae of 45% have been considered for he calculaion of he ne presen value. All he oher inpu daa are available upon reques. All he insances are modeled wih GAM (Brooke e al, 1998) and solved on an IBM T60 lapop wih an Inel Core Duo 1.83 GHz CU and 1GB RAM. Due o he nonconvexiy of he MINL problems, he global opimizaion solver BARON (ahinidis, 1996) was used for he insances where he demands are riangular disribued, and he - 29 -

BB solver was used for he insances where he demands are normal disribued (because BARON does no suppor error funcion). All he insances are firs solved wih he ouerapproximaion algorihm (Duran and Grossmann, 1986) solver in DICOT for obaining a lower bound before solving wih BARON or BB solvers. 7.1. Example 1 This example has a producion nework wih hree ypes of candidae plans (Figure 18). lan I is used o produce syrene monomers from ehylene and benzene; lan II is a muliproduc plan for he producion of hree differen ypes of solid polysyrene () resins; lan III is also a muliproduc plan for he producion of wo differen ypes of expandable polysyrene (E) resins. The enire supply chain nework includes hree poenial suppliers, hree poenial producion sies, wo poenial disribuion ceners, hree cusomers and he associaed poenial ranspor links beween hem. The supersrucure of he poenial process supply chain nework for Example 1 is given in Figure 19. Two raw maerials (benzene and ehylene), one inermediae (syrene monomer) and seven producs (hree ypes of resins and wo ypes of E resins) are included in he supply chain nework. 7.1.1. oluions for Differen roducion Delay Definiions We firs consider wo insances for Example 1 where he demand follows riangular disribuions, bu differen producion delay definiions are needed for muli-produc plans. Insance 1 considers ha he demand has large changes, and hus he producion delay is irrelevan o he cycle ime (Consrain 42.a); Insance 2 considers ha he demand changes are small and ha he producion delay is closely relaed o cycle ime (Consrain 42.b). Boh insances consis of 126 binary variables, 2,970 coninuous variables and 3,438 consrains and hey are solved direcly by using GAM/BARON wih 0% opimaliy olerance. ix poins in he areo opimal curve require 4,063 CU seconds for Insance 1 and 16,893 seconds for Insance2. The areo curve is shown in Figure 20. We can see ha for Insance 1 he areo curve ranges from 1.5 o 4.8 days in he expeced lead ime, and from $532 million o $667 million for he NV. For Insance 2, he areo curve ranges from 1.5 o 9.32 days in he expeced lead ime, and from $491 million o $667 million for he NV. For boh curves he NV increases as he expeced lead ime increases. This means ha he price o reduce he expeced lead ime and o improve he supply chain responsiveness is o decrease he NV. From he rends of boh curves, we can see ha he - 30 -

rae of increase of NV decreases as he expeced lead ime increases. This means ha he cos o reduce he expeced lead ime increases when he expeced lead ime becomes smaller. Alhough hese wo insances use differen producion delay definiions, hey sill have he same maximum NV, because differen definiions of producion delay have only an impac on he expeced lead imes and he NV for inermediae poins. Because he lead ime definiion (42.b) no only akes ino accoun he residence ime, bu also considers he cycle imes and processing imes, he producion delay we can obain from consrain (42.b) will always be greaer han he one we can ge from (42.a). Thus, he longes opimal expeced lead ime for Insance 2 is greaer han he one for Insance 1. I is reasonable ha he wo insances have he same minimum expeced lead ime, because as long as here are sufficien safey socks in he disribuion ceners, he expeced lead ime will be equal o he expeced delivery lead ime regardless of he producion delay. However, for he minimum expeced lead ime he opimal NV from Insance 2 is less han he NV form Insance 1. The reason is ha we define he expeced lead ime of he supply chain as he longes expeced lead ime for all he possible pahs. Alhough hese wo cases have he same expeced lead ime for he enire supply chain, he lead ime for each individual pah is no he same. As Insance 2 always has a longer producion lead ime han Insance 1 under he same circumsances, more invenories need o be held o reduce he sock ou probabiliy so ha he longer producion lead ime of he pahs excep he longes one could be raded off. Figure 21 shows he change of cumulaive invenories for all he E resins in disribuion cener DC1 for differen expeced lead ime specificaions for he wo producion delay definiions. The safey sock levels are boh zero a he longes expeced lead ime case. As he expeced lead ime decreases, he invenory level increases. More invenories are required for Insance 2 o obain he same expeced lead ime as Insance 1. This is also because he producion delay definiion allows Insance 1 o have shorer producion lead ime. The opimal nework srucure (Figure 22) is he same for all he poins in he areo curves in boh insances. All he hree manufacuring sies are seleced, bu only sie 2 insalls plan I o produce syrene monomers, and hen he monomers are shipped o plan sies II and III as he raw maerials for differen polysyrene resins. lan sie 1 connecs o - 31 -

boh disribuion ceners, bu plan sie 2 only connecs o he firs disribuion cener. And each cusomer is served by only one disribuion cener. Insance 2 requires much longer compuaional ime han Insance 1 due o he large number of bilinear erms in he lead ime definiion (because he producion delays are variables for insance 2). For compuaional simpliciy, we will use Consrain 42.a as he producion delay definiion in he following examples, i.e. producion delay irrelevan o cycle ime. 7.1.2. Normal Disribued Demand For Insance 3, we solve Example 1 wih demand following a normal disribuion. This insance consiss of 105 binary variables, 2,466 coninuous variables and 2,963 consrains, and is solved wih GAM/BB and 5% margin of opimaliy. The reason we used he BB solver insead of BARON is ha he laer one canno handle he nonlinear erm (error funcion) arising from he cumulaive probabiliy disribuion of normal disribuion (Consrain 39). ix poins in he areo opimal curve require 753.57 CU seconds. The areo curve is shown in Figure 23. I has he same range as Insance 1 in he expeced lead ime, bu a wider range of NV from $190 million o $591 million. This curve shows a similar rend as he areo curve for Insance 1, bu he firs poin (he opimal NV for he shores lead ime case) is much lower han expeced. A possible reason is ha he normal disribuion has a bell shape curve. To ensure ha he sock ou probabiliy is zero for he longes pah, a sufficienly large amoun of safey sock should be held due o he long ail of he normal disribuion, which leads o significan invenory cos. Figure 24 represens he safey sock level for hree ypes of and wo ypes of E in he wo disribuion ceners. imilar o he case of he riangular disribuion, he invenory level goes down as he expeced lead ime increases. The opimal nework srucure (Figure 22) is he same as we obained from Insance 1. 7.1.3. Hierarchical oluion Approach To es he performance of our proposed soluion algorihm, we solve Insance 1 wih he proposed hierarchical soluion approach. The solver we used is GAM/BARON, and he opimaliy margin is se o be 0% for he simplified model and 5% for he deailed model. - 32 -

The compuaional resuls are showed in Table 2. We can see ha for all he poins, he compuaional imes are smaller han he imes required for he direc approach. ince Insance 1 is solved wih a 0% opimaliy olerance, he resuls are globally opimized. Afer comparing wih he resuls from hierarchical approach, he differences for he soluions of hese wo insances are small (around 3%). As we can see from his insance, he proposed hierarchical algorihm can obain a good near-opimal soluion in shor compuaional imes. Anoher imporan advanage of using he hierarchical algorihm is ha we can obain near opimal soluions in reasonable ime for very large scale problems ha are unable o be solved direcly wih GAM/BARON. In he nex example, we are going o solve an indusrial size problem ha GAM/BARON failed o converge for more han wo weeks if solving i direcly. However, wih our hierarchical approach, soluions for six poins of he areo curves are found wihin 5 hours. The deails will be discussed nex. 7.2. Example 2 Example 2 is a large scale problem which is moivaed by a real world applicaion. I has he same producion nework as in Figure 18. The poenial supply chain nework (Figure 25) includes hree possible ehylene suppliers locaed in Illinois, Texas and Mississippi, and hree poenial benzene suppliers locaed in Texas, Louisiana, Alabama. Four poenial manufacuring sies can be locaed in Michigan, Texas, California and Louisiana. The Michigan manufacuring sie can se up all he hree ypes of plans, he Texas manufacuring sie can only insall lan I, he California manufacuring sie can only se up lans II and III, and he Louisiana manufacuring sie can only se up lans I and II. The supply chain can have five disribuion ceners, locaed in Nevada, Texas, Georgia, ennsylvania and Iowa. Cusomers are pooled ino nine sale regions across he counry based on heir geographical proximiy. The corresponding supersrucure of he supply chain nework is given in Figure 26. We assume ha he demands follow riangular disribuions, and solve his problem wih consideraion of safey sock (Example2, Insance 1) and wihou holding any safey sock (Example2, Insance 2). Boh insances consis of 215 binary variables, 8,216 coninuous variables and 14,617 consrains and hey are solved wih he proposed hierarchical algorihm by using GAM/BARON wih 0% opimaliy olerance for he simplified model and 5% for he deailed model. ix poins in he areo opimal curve require 15,396 CU seconds for Insance 1 and 16,927 seconds for Insance 2. The reason - 33 -

i akes longer o solve Insance 2 is because for he sep for calculaing he shores expeced lead ime we need o solve he problem by minimizing he expeced lead ime insead of using Equaion 48 direcly, because safey socks are no considered in Insance 2, and hus Equaion 48 is no applicable for his case. The areo opimal curves are given in Figure 27. The areo curve for Example2, Insance 1 (wih safey sock) has a similar rend as he Insance 1 of Example 1, and ranges from 1.6 o 5 days in he expeced lead ime and from $409 million o $683 million for NV. The areo curve of Example 2, Insance 2 (wihou safey sock) ranges from 4.3 o 5 days and is very similar o he curve obained for he deerminisic supply chain design repored in You and Grossmann (2007). Alhough wih differen ranges of NV, hese wo curves has he same opimal longes expeced lead ime and he associaed NV (due o he opimaliy margin, here is a small difference beween heir opimal NV). This is because in he longes expeced lead ime case, he supply chain needs o reduce cos by seing he safey sock levels o zero for Insance 1, which is equivalen o he case of no safey sock in Insance 2. ince here is no invenory held in Insance 2, he only facor ha can change he expeced lead ime is he nework srucure. No maer how he nework srucure changes, he expeced lead ime in Insance 2 always includes he producion lead ime. Due o his reason, he range of expeced lead ime for Insance 2 is much smaller han ha of Insance 1. On he oher hand, changing he supply chain nework srucure is always much more expensive han holding a cerain amoun of safey sock. Thus, he areo curve for Insance 2 is below he curve for Insance 1. The opimal nework srucures under differen expeced lead imes for Insance 1 are shown in Figure 28-30. I is ineresing o see ha all he four sies are seleced, and ha differen ypes of plans are insalled in he nework srucures. Wih he shores expeced lead ime, 1.6 days (NV = $489.39 MM), (Figure 28) eigh plans in he four sies are insalled, and all he four suppliers are seleced and conneced o he associaed neares plan sies. The CA sie is only supplied by he TX sie for syrene monomer. As he expeced lead ime increases o 2.96 days (NV = $644.46 MM), he supplier in LA is seleced o provide benzene o he TX sie, which leads o cheaper raw maerial, in urn increasing he NV (Figure 29). As shown in Figure 30 and 31, a new iner-sie ransporaion link from LA sie o CA sie for he shipping of syrene monomer is added. The change of nework srucure increases he expeced lead ime, and leads o he highes NV up o $690 MM. These examples shows he imporance of esablishing rade-offs - 34 -

beween responsiveness and economics in he design and planning of a CN for he improvemen of overall earning and performance of a company. 8. CONCLUION In his paper we have presened a quaniaive approach for designing responsive supply chains under demand uncerainy. The expeced lead ime was proposed as a measure of process supply chain responsiveness, and defined quaniaively wih inegraion of supply chain nework srucure and invenory level. A muliperiod mixed ineger nonlinear programming (MINL) mahemaical model was developed for he bicrierion opimizaion of economics and responsiveness, while considering cusomer demand uncerainy. The model inegraes he long-erm sraegic decisions (e.g. insallaion of plans, selecion of suppliers, manufacuring sies, disribuion ceners and ranspor links) wih he shor-erm operaional decisions (e.g. produc ransiions and changeovers) for he muli-sie muli-echelon process supply chain nework. The model also includes a novel approach o predic he safey sock levels wih consideraion of responsiveness, demand uncerainy and economic objecives. A bi-crierion opimizaion model was implemened o obain he rade-offs beween responsiveness and economics using he ε-consrain mehod. A hierarchical algorihm was furher presened for he soluion of he resuling large-scale MINL problem based on decoupling of he decision-making levels (sraegic and operaional). Wihou compromising he soluion qualiy, significan savings in compuaional effor was achieved by employing he proposed algorihm in he illusraive examples. Two examples relaed o syrene producion were solved o illusrae he indusrial applicaion of his model. The resuls show ha small changes in expeced lead ime can lead o significan changes in he ne presen value and he nework srucure, which in urn suggess he imporance of inegraing responsiveness ino he design and operaions of process supply chain nework. Acknowledgemens The auhors graefully acknowledge financial suppor from Naional cience Foundaion under Gran No. DMI-0556090. - 35 -

Nomenclaure Indices/es k Manufacuring ies i lans j Chemicals m Disribuion ceners ls uppliers ld Cusomers Time periods s, s ' roducion schemes K e of sies ha can se up plan i i J is, e of chemicals involved in scheme s of plan i J is, J ki, e of main producs for producion scheme s of plan i e of main producs for plan i in sie k JM ki, e of maerials of plan i in sie k i L i L j LD j I j O j e of producion schemes for plan i e of ime slos for plan i in he producion scheduling e of suppliers ha supply chemical j e of cusomers ha need chemical j e of plans ha consume chemical j e of plans ha produce chemical j ah e of possible pah of chemical flow from a supplier o some sies and disribuion ls, k, m, ld cener, finally ends a a cusomer. Elemens are in he form of ( ls, k1, k2... kn, m, ld ) arameers Lenp Lengh of each ime period L d Lower bound of demand of chemical j in marke ld during ime period jld,, U d Upper bound of demand of chemical j in marke ld during ime period jld,, M d Mos likely demand of chemical j in marke ld during ime period jld,, m d Targe demand of chemical j in marke ld during ime period jld,, α k, j, Coefficien for hroughpu working invenory amoun of chemical j for sie k β m, j, Coefficien for hroughpu working invenory of chemical j for disribuion cener m ϕ jld,, elling price of chemical j in marke ld during ime period ϕ jls,, urchase price of chemical j in marke ls during ime period ε jm,, Invenory cos of chemical j in disribuion cener m in ime period ε k, j, Invenory cos of chemical j in plan sies k in ime period θ Residence ime of he main produc for producion scheme s of plan i R is, - 36 -

γ Fixed cos of ranspor link from suppliers ls o plan sies k I kls, O km, mld, N kk, ' ki, I k, j, ls, O k, j, m, jmld,,, N kk, ', j, γ Fixed cos of ranspor link from plan sies k o disribuion cener m γ Fixed cos of ranspor link from disribuion cener m o cusomer ld γ Fixed cos of iner-plan sie ransporaion γ Fixed cos of insallaion of plan i in sie k ω Uni shipping cos for chemical j from suppliers ls o plan sies k in ime period ω Uni shipping cos for chemical j from plan sies k o DC m in period ω Uni shipping cos for chemical j from disribuion cener m o cusomer ld ω Uni shipping cos for chemical j for iner-plan sie ransporaion ω Variable cos of insallaion of plan i in sie k ki, λ Transporaion ime from supplier ls o plan sie k I kls, λ Transporaion ime from plan sie k o k ' N kk, ' λ Transporaion ime from plan sie k o disribuion cener m O km, λ Transporaion ime from disribuion cener m o cusomer ld mld, ρ relaive producion amoun of main produc j of producion scheme s in plan i 1 is, ρ relaive maximum producion rae of main produc of producion scheme s of plan i 2 is, μ i, j, s Mass balance coefficiens of chemical j in scheme s of plan i σ is,, Uni operaing cos of scheme s of plan i during period τ iss,, ' Transiion ime from produc s ' o s in plan i CTR Transiion cos from produc s o s ' in plan i iss,, ' H ki,, Toal available producion ime in plan i of sie k in period U Upper bound of purchase of chemical j from supplier ls o sie k during period k, j, ls, U TR Upper bound of shipmen of produc j from sie k o ' kk, ', j, F U jmld,,, U k, j, m, U ki,, js,, k in period Upper bound of sales of produc j o marke ld from disribuion cener m in ime period Upper bound of shipmen of produc j from sie k o disribuion cener m in ime period W Upper bound of producion of chemical j in plan i of sie k in period Q U ki, U kissl,,,, Upper bound of capaciy of each plan δ Upper bound of processing ime for produc s in slo sl of plan i in sie k Coninuous Variables: Q Capaciy of plan i in sie k ki, r kis,, roducion rae of produc s in plan i of sie k W ki,, js,, Amoun of chemical j produced in plan i of sie k in period - 37 -

k, j, ls, urchase of chemical j from supplier ls o sie k during period jmld,,, ales of produc j o marke ld from disribuion cener m during ime period F k, j, m, hipping amoun of chemical j from sie k o disribuion cener m in ime period TR hipping amoun of chemical j from sie k o k ' in period kk, ', j, Q jld,, Toal available amoun of chemical j for cusomer ld in ime period I jmld,,, afey sock of chemical j for marke ld in disribuion cener m during period WI Working invenory of chemical j in sie k during ime period k, j, DC m, j, WI Working invenory of chemical j in disribuion cener m during ime period T Expeced lead ime of he whole supply chain nework NV Ne presen value of he supply chain nework θ Time delay by producion of plan i in sie k ki, Ts kisl,,, aring ime of slo sl in plan i of sie k in period Te kisl,,, End ime of slo sl in plan i of sie k in period δ kissl,,,, rocessing ime of scheme s in slo sl of plan i in sie k δ kisl,,, rocessing ime of he ime slo sl of plan i in sie k TC ki,, Cycle ime of plan i in sie k in period W Amoun produced of main produc in scheme s of plan i of sie k in period kis,,, N ki,, Number of cycle in plan i of sie k in period COT Toal cos for invenories and ransiions of plan i in sie k in period ki,, K jld,, andardized normal variables of produc j for cusomer ld in ime period rob jld,, sock-ou probabiliy for produc j cusomer ld a ime period Binary Variables I Y 1 if a ransporaion link from supplier ls o plan sie k is se up kls, Y 1 if plan i in sie k is insalled ki, Y 1 if an iner-sie ransporaion link from sie k ' o sie k is se up N k', k Y 1 if ransporaion link from sie k o disribuion cener m is se up O km, m Y 1 if disribuion cener m is insalled mld, Y 1 if a ransporaion link from disribuion cener m o cusomer ld is se up Y kissl,,,, 1 if he slo sl is assigned o he produc s in plan i sie k in period Z kiss,,, ', sl, 1 if produc s is preceded by s ' in ime slo sl of plan i sie k ime period - 38 -

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Responsiveness Lead Time Figure 1. Concepual relaionship beween lead ime and responsiveness Figure 2. A simple linear supply chain Figure 3. Time delays of a simple linear supply chain Figure 4. roducion lead ime and delivery lead ime - 41 -

Figure 5. A pah of chemical flow in a CN Figure 6. rocess supply chain nework roduc A Transiion roduc B roduc C Figure 7. Changeovers of flexible processes - 42 -

ls1 ls2... lsn lan ie k Figure 8. Relaionship beween suppliers and manufacuring sies Figure 9. Inpu and oupu relaionship of a plan k1 k2... DC m kn Figure 10. Relaionship beween manufacuring sies and disribuion ceners ld1 DC m ld2... Figure 11. Inpu and oupu relaionship of a disribuion cener ldn - 43 -

Transiion rocessing jobs One Time eriod Cycle Time Cycle Time Cycle Time Figure 12. Cyclic scheduling of each ime period Invenory Level a ime period Time cycle ime processing ime Figure 13. Invenory level change in cyclic scheduling afey ock Figure 14. ervice level and sock-ou probabiliy - 44 -

Targe Demand d L d M afey ock d U Figure 15. afey sock for riangular disribuion Time delayed by roducion Inpu Oupu Residence Time of C Residence Time of A Figure 16. Time delay by producion (large change of demand) Time delayed by roducion rocessing Time Residence Time Cycle Time Figure 17. Time delay by producion (small change of demand) - 45 -

- 1 II - 2 Ehylene yrene Muli roduc - 3 I Benzene ingle roduc E - 1 III E - 2 Muli roduc Figure 18. roducion nework for polysyrene supply chains A1 Ehylene I Benzene lan ie 1 yrene II III E DC 1 D A2 B Ehylene I II III yrene yrene Benzene E lan ie 2 lan ie 3 uppliers lan ies Disribuion Ceners Cusomers Figure 19. oenial process supply chain nework supersrucure for Example 1 DC 2 E1 E2 700 650 NV (M$) 600 550 500 Insance 1 Insance 2 450 1 2 3 4 5 6 7 8 9 10 Expeced Lead Time (day) Figure 20. areo opimal curve for differen producion delay definiions in Insance 1 and 2 of Example 1 (demand uncerainy boh follow riangular disribuion) - 46 -

180 160 afey ock (10^4 Ton) 140 120 100 80 60 40 Insance 1 Insance 2 20 0 0 1 2 3 4 5 6 7 8 9 10 Expeced Lead Time (days) Figure 21. afey sock levels for E resins in DC1 for Insance 1 and 2 of Example 1 (riangular disribued demand for boh insances) lan ie 1 A1 II III E DC 1 D A2 B Ehylene I II III yrene yrene Benzene E lan ie 2 lan ie 3 DC 2 E1 E2 uppliers lan ies Disribuion Ceners Cusomers Figure 22. Opimal nework srucure for Insance 1 and 2 of Example 1-47 -

700 600 500 NV (M$) 400 300 200 100 1 1.5 2 2.5 3 3.5 4 4.5 5 Expeced Lead Time (day) Figure 23. areo opimal curve for Insance 3 of Example 1 (normal disribued demand) 250 afey ock (10^4 T) 200 150 100 50 E in DC2 in DC2 E in DC1 in DC1 0 1.51 2.17 2.83 3.48 4.14 4.8 Expceed Lead Time (day) Figure 24. afey sock level for normal disribued demand in Insance 3 of Example 1-48 -

ossible lan ie upplier Locaion Disribuion Cener Cusomer Locaion Figure 25. Locaion map for Example 2 IL TX lan ie MI Ehylene yrene II I Benzene III E NV TX WA CA AZ TX M LA II Ehylene I III yrene yrene Benzene E lan ie TX lan ie CA Ehylene yrene I Benzene III E lan ie LA GA A OK NC FL OH AL IA MA MN uppliers lan ies Disribuion Ceners Cusomers Figure 26. oenial process supply chain nework supersrucure for Example 2-49 -

750 700 650 600 NV (M$) 550 500 450 400 350 wih safey sock wihou safey sock 300 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Expeced Lead Time (day) Figure 27. areo curve for Example 2 wih safey sock and wihou safey sock (Insance 1 and 2 of Example 2) IL lan ie MI yrene II NV WA CA TX III E TX AZ TX M LA II Ehylene I III yrene yrene Benzene E lan ie TX lan ie CA Ehylene yrene I Benzene III E lan ie LA GA A OK OH FL NC AL IA MA MN uppliers lan ies Disribuion Ceners Cusomers Figure 28. Opimal nework srucure of Example 2 a expeced lead ime = 1.5 days, NV = $489.39 MM - 50 -

IL lan ie MI yrene II NV WA CA TX III E TX AZ TX M LA II Ehylene I III yrene yrene Benzene E lan ie TX lan ie CA Ehylene yrene I Benzene III E lan ie LA GA A OK OH FL NC AL IA MA MN uppliers lan ies Disribuion Ceners Cusomers Figure 29. Opimal nework srucure of Example 2 a expeced lead ime = 2.96 days, NV = $644.46 MM IL lan ie MI yrene II NV WA CA TX III E TX AZ TX M LA II Ehylene I III yrene yrene Benzene E lan ie TX lan ie CA Ehylene yrene I Benzene III E lan ie LA GA A OK OH FL NC AL IA MA MN uppliers lan ies Disribuion Ceners Cusomers Figure 30. Opimal nework srucure of Example 2 a expeced lead ime = 5.0 days, NV = $690 MM - 51 -

ossible lan ie upplier Locaion Disribuion Cener Cusomer Locaion Figure 31. Maerial flows in he locaion map for he longes expeced lead ime (5.0 days) case of Example 2 Insances Demand Uncerainy roducion Delay oluion Approach afey ock Example 1, Insance 1 Triangular Consrain 42.a Direc Yes Example 1, Insance 2 Triangular Consrain 42.b Direc Yes Example 1, Insance 3 Triangular Consrain 42.a Hierarchical Yes Example 1, Insance 4 Normal Consrain 42.a Hierarchical Yes Example 2, Insance 1 Triangular Consrain 42.a Hierarchical Yes Example 2, Insance 2 Triangular Consrain 42.a Hierarchical No Table 1. ummary of all he numerical examples and insances for case sudies oins on areo Curve Expeced Lead Time Direc Approach ( Example 1, Insance 1) Hierarchical Approach ( Example 1, Insance 3) (days) NV (M$) CU(s) NV (M$) CU(s) Difference from Opimum I 1.5 532.82 316.48 514.68 104.25 3.40% II 2.16 586.78 484.08 568.65 70.52 3.09% III 2.82 622.07 649.85 603.94 106.49 2.91% IV 3.48 651.73 1292.35 633.6 168.33 2.78% V 4.14 662.69 960.83 644.56 138.99 2.74% VI 4.8 667.79 359.65 649.65 101.34 2.72% Table 2. Comparison for six poins in he areo curve wih differen soluion approaches (Insance 1 and 3 of Example 1). In he direc approach (Insance 1), he opimaliy margin is se o be 0%, which is he global opimum; in he hierarchical approach (Insance 3), he opimaliy margin is se o 0% for simplified model and 5% for he deail model. - 52 -