How To Design A Supply Chain

Transcription

1 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 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. grossmann@cmu.edu

2 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 -

3 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 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 -

4 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 -

5 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 -

6 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

7 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) 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

8 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) 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

9 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

10 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 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 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

11 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 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)

12 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 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 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)

13 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 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,

14 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 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 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,,

15 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) 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

16 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 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 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) 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

17 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) 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,,,, 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

18 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 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

19 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 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

20 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 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,,

21 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 ) 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

22 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,

23 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

24 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)

25 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 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