MTO-MTS Production Systems in Supply Chains

NSF GRANT #0092854 NSF PROGRAM NAME: MES/OR MTO-MTS Productio Systems i Supply Chais Philip M. Kamisky Uiversity of Califoria, Berkeley Our Kaya Uiversity of Califoria, Berkeley Abstract: Icreasig cost pressures have led supply chai maagers to focus o ruig icreasigly lea ad efficiet supply chais, with miimal ivetory. Ideed, more ad more firms are relyig o pull or make-toorder (MTO) supply chais to miimize cost ad waste. At the same time, icreasig competitive pressures are leadig to a icreased emphasis o customer service. A importat elemet of customer service, of course, is havig make-to-stock (MTS) items i stock, ad deliverig make-to-order (MTO) products quickly ad by the promised due date. I this project, we cosider a variety of models desiged to provide isight ito the operatio of combied MTS-MTO supply chais. We primarily focus o a simple stylized model of a two facility supply chai featurig a maufacturer served by a sigle supplier. Iitially, we model a pure MTO supply, i which customers arrive at the maufacturer ad place orders. The maufacturer eeds to quote a due date to the customer whe the order is placed, ad the maufacturer eeds to receive a compoet from the supplier before completig the maufacturig process. We desig effective algorithms for productio sequecig ad due date quotatio i both cetralized ad decetralized versios of this supply chai, characterize the theoretical properties of these algorithms, ad compare the performace of the cetralized ad decetralized versios of the supply chai uder various coditios. We also cosider combied MTS-MTO versios of this supply chai. I these models, the maufacturer ad supplier have to decide which items to produce to order, ad which items to produce to stock. Ivetory levels must be set for the MTS items, ad due dates eed to be quoted for the MTO items. I additio, sequecig decisios must be made. We cosider several versios of these models, desig effective algorithms to fid ivetory levels, to quote due dates, ad to make sequecig decisios i both cetralized ad decetralized settigs, ad use these algorithms to assess the value of joit MTS- MTO systems, as well as the value of cetralizatio uder various coditios. Fially, we cosider more complex supply etworks, ad employ the results from the simple two facility supply chais described above to desig effective heuristics for the MTS-MTO decisio, ivetory levels, sequecig, ad due date quotatio i these more complex supply chais. We employ these heuristics to aswer a variety of questios about the value of cetralizatio, ad of usig a combied MTS-MTO approach i complex supply chais. 1. Itroductio: Supply chai maagemet ca be viewed as the combiatio of the approach ad iformatio techology that itegrates suppliers, maufacturers, ad distributors of products or services ito oe cohesive process i order to satisfy customer requiremets. Traditioally, orders have bee the oly mechaism for order exchage betwee firms; however, iformatio techology ow allows firms to share more extesive iformatio such as demad, ivetory data, etc. quickly ad iexpesively. With the help of this iformatio sharig, firms ca ow coordiate their processes more easily ad icrease overall system efficiecy. I this project, we aalyze the ivetory decisios, schedulig, ad lead time quotatio i a variety of supply chais structures, develop approaches to miimize the total cost i these supply chais, ad compare cetralized ad decetralized versios of the supply chai to begi to quatify the value of cetralizatio ad iformatio exchage. Miimizig ivetory holdig costs ad quotig reliable ad short lead times to customers are clearly coflictig objectives i supply chais with stochastic demad ad processig times. Ideally, compaies would like to iitiate productio every time a customer order arrives i order to avoid ivetory holdig costs; however, this strategy is likely to lead to log waitig times for order delivery. Every time a customer arrives, the firm must quote a due date for the order. Log quoted lead times lead to customer dissatisfactio, lost sales ad decreased profits. O the other had, quotig short lead times icreases the risk of missig the delivery dates, which also has egative implicatios for the firm. Also, for a compay that produces multiple products with differet char-

acteristics, the decisio of whe to produce each order also effects the completio times ad thus the lead times of may other products. I this project, we aalyze these trade-offs, ad develop approaches to effectively address them. Optimal supply chai performace requires the executio of a precise set of actios, however, those actios are ot always i the best iterest of the members i the supply chai, i.e. the supply chai members are primarily cocered with optimizig their ow objectives, ad that self-servig focus ofte results i poor performace. However, optimal performace ca be achieved if the firms i the supply chai coordiate by cotractig o a set of trasfer paymets such that each firm s objective becomes aliged with the supply chai s objective. See Cacho [6] for a extesive survey of supply chai coordiatio with cotracts ad Che [10] for a extesive survey o iformatio sharig ad supply chai coordiatio. Cacho ad Lariviere [7], Li [25], Jeulad ad Shuga [17], Moorthy [27], Igee ad Parry [15] ad Netessie ad Rudi [28] are a few of the researchers that aalyze the beefits of iformatio sharig i supply chais ad how to operate such systems for differet objectives. Also, Bourlad et al. [5], Che [9], Gavirei et al. [13], Lee et al. [23] ad Aviv ad Federgrue [2] show that sharig demad ad ivetory data improves the supply chai performace for several differet objectives. As may authors have observed, supply chai maagemet ad coordiatio has gaied importace i recet years as busiesses feel the pressure of icreased competitio ad as maagers have begu to uderstad that a lack of coordiatio ca lead to decreased profits ad service levels. There is a large ad growig amout of literature o this subject, but the vast majority of this research focuses o make-to-stock (MTS) systems, ad performace measures built aroud service ad ivetory levels. O the other had, a icreasig umber of supply chais are better characterized as make-to-order(mto) systems. This is particularly true as more ad more supply chais move to a mass customizatio-based approach to satisfy customers ad to decrease ivetory costs (see Simchi-Levi, Kamisky, ad Simchi-Levi [30]). Mass customizatio implies that at least the fial details of project maufacturig must occur after specific orders have bee received, ad must thus be completed quickly ad efficietly. MTO systems have may uique issues ad eed to be operated very differetly from MTS systems. I a MTO system, firms eed to fid a effective approach to schedulig their customers orders, ad they also eed to quote short ad reliable due-dates to their customers. However, ot all firms employ a pure MTO or MTS system. Although some firms make all of their products to order while some others make them to stock, there are also a umber of firms that maitai a middle groud, where some items are made to stock ad others are made to order. The decisio o usig either a MTO strategy or a MTS strategy at a facility heavily depeds o the characteristics of the system. I supply chais usig a combied system, holdig ivetory at some of the stages of the chai ad usig a MTO strategy at other facilities might decrease the costs dramatically without icreasig the lead times. Because of this, compaies are startig to employ a hybrid approach, a push-pull strategy (i.e. a combied MTO-MTS system), holdig ivetory at some of the facilities i their supply chai ad producig to order i others. For example, a compay with a diverse product lie ad customer base ca be best served with a appropriate combiatio of MTO-MTS systems. Also, a supplier with oe primary customer ad several smaller customers might be able to operate more profitably with a combied MTO-MTS system. The importace of ivetory maagemet as outlied above has also icreased with the growig prevalece of e-commerce. I today s world, e-commerce ed customers expect high levels of service ad speedy ad o-time deliveries. I the first part of this project, we cosider a sigle maufacturer, served by a sigle supplier, who has to quote due dates to arrivig customers i a make-to-order productio eviromet. The maufacturer is pealized for log lead times, ad for missig due dates. I order to meet due dates, the maufacturer has to obtai compoets from a supplier. We cosider several variatios of this problem, ad desig effective due-date quotatio ad schedulig algorithms for cetralized ad decetralized versios of the model. We cosider stylized models of such a supply chai, with a sigle maufacturer ad a sigle supplier, i order to begi to quatify the impact of maufacturer-supplier relatios o effective schedulig ad due date quotatio. We aalyze a make-to-order system i this simple supply chai settig ad develop effective algorithms for schedulig ad due-date quotatio i both cetralized ad decetralized versios of this model. Buildig o this aalysis, we explore the value of cetralized cotrol i this supply chai, ad develop schemes for maagig the supply chai i the absece of cetralized cotrol ad with oly partial iformatio exchage. As metioed above, researchers have itroduced a variety of models i a attempt to uderstad effective due date quotatio. Kamisky ad Hochbaum [19] ad Cheg ad Gupta [31] survey due date quotatio models i detail. The majority of earlier papers o due-date quotatio have bee simulatio based. For istace, Eilo ad Chowdhury [11], Weeks [32], Miyazaki [26], Baker ad Bertrad [3], ad Bertrad [4] cosider various due date assigmet ad sequecig policies, ad i geeral demostrate that policies that use estimates of shop cogestio ad job cotet iformatio lead to better shop

performace tha policies based solely o job cotet. We exted previous work i due date quotatio by explorig due date quotatio i supply chais. I particular, we focus o a two member supply chai, i which a maufacturer works to satisfy customer orders. Customers arrive at the maufacturer over time, ad the maufacturer produces to order. I order to complete productio, the maufacturer eeds to receive a customized compoet from a supplier. Each order takes a differet amout of time to process at the maufacturer, ad at the supplier. The maufacturer s objective is to determie a schedule ad quote due dates i order to miimize a fuctio of quoted lead time ad lateess. I the secod part of the project, we cosider a combied MTO-MTS supply chai composed of a maufacturer, served by a sigle supplier workig i a stochastic multi-item eviromet. The maufacturer ad the supplier have to decide which items to produce to stock ad which oes to order. The maufacturer also has to quote due dates to arrivig customers for make-to-order products. The maufacturer is pealized for log lead times, missig the quoted lead times ad for high ivetory levels. We cosider several variatios of this problem, ad desig effective heuristics to fid the optimal ivetory levels for each item ad also desig effective schedulig ad lead time quotatio algorithms for cetralized ad decetralized versios of the model. We aalyze the coditios uder which a MTO or MTS strategy would be optimal to use for both the maufacturer ad the supplier ad fid the optimal ivetory levels. We also desig effective schedulig ad lead time quotatio algorithms based o the algorithms i the first part of this thesis. We cosider several variatios of this problem. I particular, we focus o three olie models. I the first model, the cetralized model, both facilities are cotrolled by the same aget, who decides o the ivetory levels at both facilities, schedules the jobs ad quotes a due date to the arrivig customer i order to achieve the ed objective. I the secod model, we develop a decetralized model with full iformatio, i which the maufacturer ad the supplier work idepedetly from each other ad make their ow decisios but the maufacturer has complete iformatio about both his ow processes ad the supplier. This model allows to explore the value of iformatio exchage, ad to determie if ad whe the cost ad difficulty of implemetig a cetralized system are worth it. I the last model, the simple decetralized model, the maufacturer has o iformatio about the supplier ad makes certai assumptios about i order to decide o his ivetory values ad quote a due date to arrivig customers. I this model, each facility is workig to achieve its ow goals, ad very little iformatio is exchaged. Ideed, i our discussios with the maagers of several small maufacturig firms, this is typical of their relatioships with may suppliers. We compare the cetralized ad decetralized models through extesive computatioal aalysis to assess the value of cetralizatio ad iformatio exchage. I literature, MTO/MTS models are geerally studied for sigle stage systems. Several researchers, such as Williams [33], Federgrue ad Katala [12] ad Carr ad Dueyas [8] assume that the decisio of which items to produce-to-order ad which oes to stock is made i advace ad they try to fid the best way to operate that system. Others, like Li [24], Arreola-Risa ad DeCroix [1] ad Rajagopala[29], the make-to-stock/make-to-order distictio is a decisio variable determied withi the the model. I cotrast to these models, we cosider the schedulig ad ivetory decisios together ad we aalyze supply chai systems istead of a sigle facility model. We also itegrate lead time quotatio ito these models i our research. Fially, we cosider more complex supply chai etworks composed of several facilities with differet relatioships to each other. I this model, we cosider a supply chai composed of several facilities ad maaged by a sigle decisio aget who has complete iformatio about these facilities. I additio, there are exteral suppliers to this supply chai outside the cotrol of the maager of the supply chai, ad this maager has oly limited iformatio about these exteral suppliers. Uder these coditios, usig the results from the two-facility supply chai, we desig effective heuristics to be used by the maager to fid optimal ivetory levels, to sequece the orders at each facility, ad to quote reliable ad short lead times to customers. I cotrast to the models i literature, we itegrate due-date quotatio issues ito combied MTO-MTS systems, cosider differet schedules ad focus o supply chai models of this system. We cosider schedulig, ivetory ad lead time quotatio decisios together i our study. We develop models that provide guidace for decidig whe to use MTS ad whe to use MTO approaches for sigle facility ad for supply chai models, ad for how to effectively operate the system to miimize system wide-costs i each case. We also quatify the value of cetralizatio ad iformatio i this system by buildig decetralized ad cetralized models ad obtaiig good solutios to these models. To the best of our kowledge, this is the first study that aalytically explores ivetory decisios ad lead time quotatio together i the cotext of a supply chai, ad that explores the impact of the supplier-maufacturer relatioship o this system. We perform extesive computatioal testig for each of the cases above to assess the effectiveess of our algorithms, ad to compare the cetralized ad decetralized models i order to quatify the value of cetralized cotrol ad iformatio exchage i these supply chais. Sice complete iformatio exchage ad cetralized cotrol is ot always practical or cost-effective,

we also explore the value of partial iformatio exchage for these systems. I this project, we are attemptig to fid aswers to questios such as: 1. Which items should be produced MTO ad which oes MTS at each step of the supply chai ad what are the optimal levels of ivetory for MTS items? 2. Which item should be produced ext whe a facility becomes available for productio? 3. What is the optimal due-date that should be quoted to each customer at the time of arrival? 4. What is the beefit of a cetralized supply chai as opposed to decetralized systems? 5. How much of the gais associated with cetralizatio ca be achieved through iformatio exchage betwee supply chai members? 2. Schedulig ad Due-Date Quotatio i a MTO Supply Chai: We begi our aalysis by cosiderig a pure MTO supply chai ad we focus o the schedulig ad due-date quotatio issues i this system. I this sectio, we cosider a sigle maufacturer, served by a sigle supplier, who has to quote due dates to arrivig customers i a make-to-order productio eviromet. The maufacturer is pealized for log lead times, ad for missig due dates. We cosider several variatios of this problem, ad desig effective due-date quotatio ad schedulig algorithms for cetralized ad decetralized versios of this model. We also complete a extesive computatioal experimet to evaluate the effectiveess of our algorithms ad to assess the value of cetralizatio ad iformatio exchage i this system. 2.1. Model ad Mai Results: We model two parties, a supplier ad a maufacturer, workig to satisfy customer orders. Customer orders, or jobs, i, i = 1, 2,.., arrive at the maufacturer at time r i, ad the maufacturer quotes a due date for each order, d i, whe it arrives. To begi processig each order, the maufacturer requires a compoet specifically maufactured to order by the supplier. The compoet requires processig time p s i at the supplier, ad the order requires processig time p m i. (To clarify the remaiig expositio, we will refer to a order or job with processig time p s i at the supplier ad p m i at the maufacturer.) Recall that we are focusig o olie versios of this model, i which iformatio about a specific order s characteristics (arrival time ad processig times) is ot available util the order arrives at the maufacturer (that is, util its release time r i ). We cosider several versios of this model, which we briefly describe here ad discuss i more detail i subsequet paragraphs. I the cetralized versio of the model, the etire system is operated by a sigle etity, who is aware of processig times both at the supplier ad at the maufacturer. I the simple decetralized model, the maufacturer ad the supplier are assumed to work idepedetly, ad each is uaware of the job s processig time at the other stage. Fially, i the decetralized model with additioal iformatio exchage, the supplier quotes a due date to the maufacturer as a mechaism for limited iformatio exchage. I this study, we propose algorithms for these models. Our goal i developig these algorithms is to provide a simple ad asymptotically optimal olie schedulig ad due date quotatio heuristic for either the maufacturer ad the supplier idividually i the decetralized system, or for both, for the cetralized system, that works well eve for cogested systems, so that we ca compare the performace of these systems. Sice firms are faced with a tradeoff betwee quotig short due dates, ad meetig these due dates, we cosider a objective fuctio that captures both of these cocers. I particular, we are attemptig to miimize the total cost fuctio (c d d i + c T T i ) i=1 where T i = (C i d i ) + is the tardiess of job i ad c d ad c T are the uit due date ad tardiess costs for the model. Clearly, c T > c d, or otherwise it will be optimal for all due dates to be set to 0. As metioed above, we use the tools of probabilistic aalysis, as well as computatioal testig, to characterize the performace of these heuristics, ad to compare these models. I particular, we focus o asymptotic probabilistic aalysis of this model ad these heuristics. I this type of aalysis, we cosider a sequece of radomly geerated istaces of this model, with processig times draw from idepedet idetical distributios bouded above by some costat, ad with arrival times determied by geeratig iter-arrival times draw from idetical idepedet distributios bouded above by some costat. Processig times are assumed to be idepedet of iterarrival times. The processig time of a job at the supplier ad at the maufacturer may be geerated from differet distributios. Recall that ay algorithm for this problem has to both set due dates for arrivig jobs, ad determie job sequeces at the supplier ad the maufacturer. I Sectio 2.3.1, for the cetralized model described above, we detail a series of heuristics, most otably oe called (for reasos that will subsequetly become clear) SP T A p SLC, that are used to determie due dates as jobs arrive, ad to sequece jobs both at the supplier ad at the maufacturer. We defie Z to be the objective SP T Ap SLC fuctio resultig from applyig this heuristic to a job istace, ad Z to be the optimal objective value for this istace.

Theorem 1. Cosider a series of radomly geerated problem istaces of the cetralized model of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times at each facility be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. Also, if the processig times at the supplier ad the maufacturer are geerated from idepedet ad exchageable distributios, the for a job istace, usig SP T A p SLC to quote due dates ad sequece jobs satisfies almost surely: lim Z SP T A p SLC Z Z = 0 I the simple decetralized model, recall that the maufacturer ad the supplier work idepedetly, ad they both attempt to miimize their ow costs. Whe the customer arrives at the maufacturer, the maufacturer eeds to quote a due date, eve though he is ot aware of either the processig time of the job at the supplier, or of the supplier s schedule. We assume that the maufacturer is aware of the umber of jobs at the supplier (sice this is equal to the umber of jobs he set there, mius the umber that have retured), that he is aware of the average processig time at the supplier, ad that he is aware of the mea iterarrival rate of jobs to his facility. I Sectio 2.2, i a prelimiary exploratio, we cosider a sigle facility due date quotatio model that is aalogous to our two stage model, except that jobs oly eed to be processed at a sigle stage (i other words, our model, but with o supplier ecessary). We develop a asymptotically optimal algorithm for this sigle facility model. For the purpose of uderstadig the performace of the decetralized system, we assume that the supplier uses this asymptotically optimal sigle facility model. The, we explore the problem from the perspective of the maufacturer, ad determie a effective due date quotatio ad sequecig policy for the maufacturer, give our assumptios about the limits of the maufacturers kowledge about the supplier s system. Sice the maufacturer is uaware of p s i values, he ca t iquire ay iformatio about the supplier s schedule ad locatio of a job at the supplier queue ad assumes that each arrivig job is scheduled i the middle of the existig jobs i the supplier queue. Uder this assumptio, i sectio 2.3.2, we propose a asymptotically optimal heuristic called SP T A SLC SD. Defie Z SP T A SLC SD to be the objective fuctio resultig from applyig this heuristic to a job istace, ad Z to be the optimal objective value for this istace give the iformatio available to the maufacturer. Theorem 2. Cosider a series of radomly geerated problem istaces of the decetralized model of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times at the maufacturer be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. If the maufacturer uses SP T A SLC SD, the uder maufacturer s assumptios about the supplier s schedule, almost surely, lim Z SP T A SLC SD Z Z = 0 Whe we explore our decetralized model with iformatio exchage, we assume that whe orders arrive at the maufacturer, that the maufacturer has the same iformatio as i the simple decetralized model described above, ad that i additio, the supplier uses our asymptotically optimal sigle facility algorithm for sequecig ad to quote a due date to the maufacturer. The maufacturer i tur uses this due date i his due date quotatio ad sequecig heuristic, SP T A SLC DIE. I Sectio 2.3.3, we explai this heuristic i detail. We defie Z SP T A SLC DIE to be the objective fuctio resultig from applyig this heuristic to a job istace, ad Z to be the optimal objective value for this istace give the iformatio available to the maufacturer. Theorem 3. Cosider a series of radomly geerated problem istaces of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times at each facility be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. If the maufacturer uses SP T A SLC DIE, ad the supplier uses a locally asymptotically optimal algorithm, the almost surely, lim Z SP T A SLC DIE Z Z = 0 I Sectio 2.4, we preset a computatioal aalysis of these algorithms for a variety of differet problem istaces, ad compare the cetralized ad decetralized versios of the model. We see that the proposed algorithms are effective eve for small umbers of jobs, ad that the objective fuctio values approach the optimal values quite quickly as the umber of jobs icreases. Also, we characterize coditios uder which the cetralized model performs cosiderably better tha the decetralized model, ad calculate this value of cetralizatio uder various coditios. Of course, as we metioed previously, i may cases implemetig cetralized cotrol is impractical or prohibitively expesive, so we also explore the value of simple iformatio exchage i lieu of completely cetralized cotrol. I the ext sectio, we itroduce a prelimiary model, ad aalyze this model. I sectio 2.3, we preset our models, algorithms, ad results i detail, ad i sectio 2.4, we preset the computatioal aalysis of our heuristics ad a compariso of cetralized ad decetralized

supply chai due date quotatio models usig our heuristics. 2.2. Prelimiary: The Sigle Facility Model: 2.2.1. The Model: Although our ultimate goal is to aalyze multi-facility systems, we begi with a prelimiary aalysis of a sigle facility system. We focus o developig asymptotically optimal schedulig ad due date quotatio heuristics for this system, ad cosider cases whe the system is cogested (the arrival rate is greater tha the processig rate), ad cases whe it is ot. I this model, we eed to process a set of jobs, opreemptively, o a sigle machie. Each job has a associated type l, l = 1, 2,...k, ad each type has a associated fiite processig time p l, p l <. At the time that job i arrives at the system, r i, the operator of the system quotes a due date d i. I particular, we focus o a system i which due dates are quoted without ay kowledge of future arrivals a olie system. However, iformatio about the curret state of the system ad previous arrivals ca be used. As metioed above, we will use the tools of asymptotic probabilistic aalysis to characterize the performace of the heuristics we propose i this study uder various coditios. I this type of aalysis, we cosider a sequece of radomly geerated determiistic istaces of the problem, ad characterize the objective values resultig from applyig a heuristic to these istaces as the size of the istaces (the umber of jobs) grows to ifiity. For this probabilistic aalysis, we geerate problem istaces as follows. Each job has idepedet probability P l, l = 1..k of beig job type l, where k l=1 P l = 1 ad job type l has kow processig time p l. Arrival times are determied by geeratig iter-arrival times draw from idetical idepedet distributios bouded above by some costat, with expected value λ. The objective of our problem is to determie a sequece of jobs ad a set of due dates such that the total cost Z = i=1 (cd d i + c T [C i d i ] + ) is miimized, where C i deotes the completio time of the order i. Clearly, to optimize this expressio, we eed to coordiate due date quotatio ad sequecig, ad a optimal solutio to this model would require simultaeous sequecig ad due date quotatio. However, the approach we have elected to follow for this model (ad throughout this thesis) is slightly differet. Observe that i a optimal offlie solutio to this model, due dates would equal completio times sice c T > c d, or otherwise it is optimal for all due dates to be set to 0. Thus, the problem becomes equivalet to miimizig the sum of completio times of the tasks. Of course, i a olie schedule, it is impossible to both miimize the sum of completio times of jobs, ad set due dates equal to completio times, sice due dates are assiged without kowledge of future arrivals, some of which may have to complete before jobs that have already arrived i order to miimize the sum of completio times. However, for related problems (Kamisky ad Lee, [20]), we have foud that a twophase approach is asymptotically optimal. I this type of approach, we first determie a schedulig approach desiged to effectively miimize the sum of completio times, ad the we desig a due date quotatio approach that presets due dates that are geerally close to the completio times suggested by our schedulig approach. This is the ituitio behid the heuristic preseted below. 2.2.2. The Heuristic: As metioed above, we have employed a heuristic that first attempts to miimize the total completio times, ad the sets due dates that approximate these completio times i a effort to miimize the objective. The heuristic we propose sequeces the jobs accordig to the Shortest Processig Time Available (SPTA) rule. Uder the SPTA heuristic, each time a job completes processig, the shortest available job which has yet ot bee processed is selected for processig. As we observed i the itroductio, although the problem of miimizig completio times is NP-Hard, Kamisky ad Simchi-Levi [18] foud that the SPTA rule is asymptotically optimal for this problem. Also, ote that this approach to sequecig does ot take quoted due date ito accout, ad is thus easily implemeted. Istead, the due date quotatio rule takes the sequecig rule ito accout. To quote due dates, we maitai a ordered list of jobs that have bee released ad are waitig to be processed. I this list, jobs are sequeced i icreasig order of processig time, so that the shortest job is at the head of the list. Sice we are sequecig jobs SPTA, whe a job completes processig, the first job o the list is processed, ad each job moves up oe positio i the list. Whe a job i arrives at the system at its release time r i with processig time p i ad the system is empty, it immediately begis processig ad a due date equal to its release time plus its processig time is quoted. However, if the system is ot empty at time r i, job i is iserted ito the appropriate place i the waitig list. Let R i be the remaiig time of the job i process at the time of arrival i, pos[i] be the positio of job i i the waitig list ad list[i] be the idex of the i th job i the waitig list. The, a due date is quoted for this job i as follows: pos[i] d i = r i + R i + (p list[j] ) + slack i (2.1) j=1 where slack i is some additioal time added to the due date i order to accout for future arrivals with processig times less tha this job these are the jobs that will be processed ahead of this job, ad cause a delay i its completio. Throughout this thesis, we ame our heuristics i two parts, where the first part (before the hyphe) refers to the sequecig rule, ad the secod part refers to the due date quotatio approach. Followig this covetio, we call this approach SPTA-SL, where the SPTA refers to

the sequecig rule, ad the SL refers to the due date quotatio rule based o calculated completio at arrival plus the isertio of SLack. I the ext sectio, we aalytically demostrate the effectiveess of the SPTA-SL approach. The remaider of this sectio focuses o determiig a appropriate value for slack i. To do this, we eed to estimate the total processig times of jobs that arrive before we process job i ad have processig times less tha the processig time of job i. We complete this calculatio for oe job at a time. Defie M i to be the remaiig time of the job i process plus the total processig times of all of the jobs to be processed ahead of job i of type l, at the time of its arrival, such that pos[i] 1 M i = R i + (p list[j] ). j=1 Let ψ l be the probability that a arrivig job has processig time less tha p l. Also, let µ l = E[p p < p l ] be the expected processig time of a job give that it is less tha p l, ad let λ be the mea iterarrival time. The, the slack for job i ca be calculated usig a aalogous approach to busy period aalysis i queueig theory (see, e.g., Gross ad Harris, [14]), where oly those jobs that are shorter tha job i are cosidered ew arrivals for the aalysis, sice other jobs will be processed after job i ad thus wo t impact job i s completio time. Note that the sequece of jobs to be processed before job i does t impact job i s completio time, so for our aalysis we ca assume ay sequece that is coveiet (eve if it is ot the sequece that we will ultimately use, as log as we cosider oly those jobs that will be processed before job i i the sequece we actually use). I particular, we assume for the purpose of our aalysis that first we process all the jobs that are already there whe job i arrives, which takes the amout of time M i. Durig this time, suppose that K jobs with processig time shorter tha i arrived. The, at the ed of M i, we have K jobs o had that will impact the completio time of job i, ad we pick oe of them arbitrarily. Now, we imagie that the job just arrived whe we selected it ad that there are o other jobs i the system, ad calculate that job s busy period the time util a queue featurig that job, ad other arrivals shorter tha it, will remai busy. We do t cosider ay of the other K jobs util the busy period of this first job is completed. The, we move to cosiderig the secod of the K jobs whe the server becomes idle (whe it fiishes the busy period of the first job) ad calculate its busy period, ad so o, util we have cosidered all K jobs. Thus, we ca write the delayed busy period of job i ALGORITHM 1: SPTA-SL Schedulig: Sequece ad process each job accordig accordig to the shortest processig time available (SPTA) rule. Due-Date Quotatio: d i = r i + { M i + p i + slack i mi{ M iψ l µ l λ ψ slack i = l µ l, ( i)ψ l µ l } if ψ lµ l λ < 1 ( i)ψ l µ l otherwise with M i as: Ã(M i ) B i (M i ) = M i + slack i = M i + j=1 where Ã(M i) is the actual umber of arrivals with processig time less tha p i durig M i (the arrivals after i that will be processed before job i) ad B j is the busy period of each of these jobs as defied i Gross ad Harris [14]. Gross ad Harris [14] show that for a M/G/1 queue, if µ i λ/ψ i < 1, the E[B i (M i )] = M i 1 µi λ/ψ i = M iλ λ µ i ψ i. This suggests that we ca approximate the slack value we are lookig for by usig this relatioship: slack i = E[B i (M i )] M i = M iµ i ψ i λ µ i ψ i However, sice we cosider a problem istace of size, it may be that all of the jobs have arrived before job i is processed, i which case the slack value will be equal to slack i = ( i)ψ l µ l Also, if ψ lµ l λ 1, the the expected delayed waitig time is loger tha the expected time for all the remaiig jobs to arrive, ad thus the slack value is agai equal to slack i = ( i)ψ l µ l. We summarize the schedulig ad due date quotatio rule for job i of type l i Algorithm 1: 2.2.3. Aalysis ad Results: For sets of radomly geerated problem istaces as described i precedig SP T A SL sectios, let Z represet the objective fuctio value obtaied by applyig the SPTA-SL rule for a job istace, ad let Z be the optimal objective fuctio value for that istace. Theorem 4. Cosider a series of radomly geerated problem istaces of size meetig the requiremets described above. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. The, almost surely, lim SP T A SL Z Z Z = 0 B j

I other words, SPTA-SL is asymptotically optimal for this problem. 2.3. Supply Chai Models, Heuristics, ad Aalysis: I this sectio, we aalyze the schedulig ad duedate quotatio decisios for two-stage supply chais usig the results from our aalysis i sectio 2.2, ad develop effective algorithms for schedulig ad due-date quotatio for both the cetralized ad decetralized versios of these systems. These algorithms allow us to compare the value of cetralizatio ad iformatio exchage i supply chais uder a variety of differet coditios. 2.3.1. The Cetralized Model: The Model For the cetralized case, the system ca be modeled as, i effect, a two facility flow shop. We assume that the maufacturer ad the supplier work as a sigle etity ad that they are both cotrolled by the same aget that has complete iformatio about both stages. The decisios about the schedulig at both facilities ad due date settig for the customer are made by this aget. The Heuristic ad Mai Results Recall that i the sigle facility case, we utilized a kow asymptotic optimality result for the completio time problem as a basis for our sequecig rule, ad the desiged a due date quotatio rule so that due dates were close to the completio times. For this model, we employ the same two phase approach, but we first eed to determie a asymptotically optimal schedulig rule for the related completio time problem, ad the desig a asymptotically optimal due date quotatio heuristic for the sequece. Xia, Shatikumar ad Gly [34] ad Kamisky ad Simchi-Levi [21] idepedetly proved that for a flow shop model with m machies, if the processig times of a job o each of the machies are idepedet ad exchageable, (i.e. p j i ad pk i are idepedet ad exchageable for all pairs of machies (j, k) for every job i), processig the jobs accordig to the shortest total processig time p i = m j=1 pj i at the first facility (supplier) ad processig the jobs o a FCFS basis at the others (maufacturer) is asymptotically optimal if all the release times are 0. We exted this result i Theorem 5, focusig o a 2-facility flow shop model, to iclude the case where all the release times are ot ecessarily 0, ad jobs are scheduled by shortest available total processig time at the supplier. We deote this heuristic SP T A p (because we schedule the jobs based o total processig time). Let Z be the miimum possible value for the total completio time objective ad Z SP T A p be the total completio time of the jobs with the heuristic explaied above for a job istace. The, we have the followig theorem for this schedulig rule. Theorem 5. Cosider a series of radomly geerated problem istaces of size. If the processig times of jobs at the supplier ad the maufacturer are geerated from idepedet ad exchageable distributios, ad if the jobs are scheduled usig the istace, schedulig the jobs accordig to SP T A p is asymptotically optimal for the objective of miimizig the total completio time Z = i=1 C i. I other words, almost surely, lim Z SP T A p Z Z = 0 Observe that although this heuristic geerates a permutatio schedule, it is asymptotically optimal over all possible schedules, ot just permutatio schedules. We base the first phase of our Algorithm 2 o Theorem 5, ad the geerate due dates similarly to the oes for our sigle facility approach. We call our due-date quotatio rule SLC sice it is based o the slack algorithm SL for the sigle facility case. The schedulig ad due-date quotatio algorithm called SP T A p SLC is stated as below: The due date set of equatios listed above is similar to those for the sigle facility case, adjusted for the cetralized model. Essetially, we approximate the amout of workload, both at the supplier ad at the maufacturer, that will be processed before job i if the SP T A p schedulig rule is employed. The due date, d m i, is equal to the sum of ds i, the approximated fiish time of job i at the supplier, p m i, the processig time at the maufacturer, ad max{t ms t mm i + i + slacki m (d s i r i), 0}, the approximate waitig time of job i at the maufacturer queue. t ms i + t mm i deotes the sum of the processig times of the jobs that are already i the system ad scheduled before job i at time r i ad slacki m approximates the workload at maufacturer of future arrivals that will be scheduled before job i while it waits i the supplier queue. I the calculatio of slacki m, mi{ (ds i r i p s i ) λ, ( i)} deotes the approximate umber of jobs that will arrive after r i, ad multiplyig this by pr{p < p i }E{p m p < p i } approximates the legth of the subset of these jobs that will be scheduled before job i at supplier. These jobs will arrive at the maufacturer before job i ad sice we use FCFS at the maufacturer, they will also be processed before job i there. Whe settig the due date, we subtract d s i r i sice this is the approximate amout of work that will be processed at maufacturer while job i is still at the supplier. Recall Theorem 1 i Sectio 2.1 which states that SP T A p SLC is asymptotically optimal. Ubalaced processig times If the processig times at the supplier ad the maufacturer are ot exchageable as assumed i the previous case, we adjust our schedulig ad due-date quotatio algorithm SP T A p SLC to reflect the properties of the ubalaced system. For schedulig, we approach the system to balace the workloads at both facilities so that the total completio time is miimized. Sice the processig times are ubalaced,

ALGORITHM 2: SPTA p -SLC Schedulig: Process the jobs accordig to SP T A p at the supplier ad FCFS at the maufacturer. Due-Date Quotatio: d m i = d s i + pm i + max{t ms i d s i = r i + p s i + M i s + slacks i + t mm i + slack m i (d s i r i), 0} { ( i)pr{p < slacki s = pi }E{p s p < p i } if λ pr{p < p i }E{p s p < p i } 0 mi{( i)pr{p < p i }E{p s p < p i }, M s i pr{p<pi}e{ps p<p i} λ pr{p<p i }E{p s p<p i } } otherwise t ms i = i A pm i where A=set of jobs i supplier queue scheduled before job i at time r i. t mm i = i B pm i where B=set of jobs i maufacturer queue at time r i slacki m = mi{ (ds i ri ps i ) λ, ( i)}pr{p < p i }E{p m p < p i } we focus o the bottleeck facility ad use a SPTA based schedule focusig o the processig times at the bottleeck facility. The, agai utilizig our two-phase approach, we adjust our due-date quotatio algorithm for that schedule, accordigly. Please refer to Kaya [22] for more detail o the schedulig ad due-date quotatio algorithms for ubalaced cases. 2.3.2. The Simple Decetralized Model: While some supply chais are relatively easy to cotrol i a cetralized fashio, most ofte this is ot the case. Eve if the stages i a supply chai are owed by a sigle firm, iformatio systems, cotrol systems, ad local performace icetives eed to be desiged ad implemeted i order to facilitate cetralized cotrol. I may cases, of course, the supplier ad maufacturer are idepedet firms, with relatively limited iformatio about each other. Implemetig cetralized cotrol i these supply chais is typically eve more difficult ad costly, sice the firms eed to coordiate their processes, agree o a cotract, implemet a iformatio techology system for their processes, etc. Thus, for either cetrally owed or idepedet firms, cetralizatio might ot be worth the effort if the gais from cetralizatio are ot big eough. Typically, if a supply chai is decetralized, the supplier ad the maufacturer have oly limited iformatio about each other. The maufacturer is uaware of the processes at his supplier ad eeds to make his ow decisios without ay iformatio from the supplier. For example, the maufacturer may oly be aware of the average time it takes for the supplier to process ad deliver a order. For this type of decetralized supply chai, we develop a effective approach for schedulig orders ad quotig due dates to customers with limited iformatio about the supplier. The Model I this sectio, we cosider a settig i which the maufacturer ad the supplier work idepedetly ad each tries to miimize his or her ow costs. Whe the customer arrives at the maufacturer ad places a order, the maufacturer eeds to immediately quote a due date, although the maufacturer has limited supplierside iformatio. I particular, the maufacturer has to quote due dates to the customers without kowledge of the supplier s schedule or kowledge of the locatio of ay icomplete orders i the suppliers queue, ad thus without kowledge or cotrol of whe the materials for that order will arrive from the supplier. We assume that the maufacturer oly kows the average processig time of jobs at the supplier, as well as the average iterarrival time of orders to the system ad the processig time of jobs at his ow facility. The maufacturer does t kow the processig time of jobs at the supplier or the schedule of the supplier. Thus, the maufacturer has to quote due dates to the customer usig oly the kowledge of his ow shop, mea processig times at the supplier, ad kowledge of the umber of jobs at the supplier, sice this is equal to the umber of orders that have arrived at the maufacturer mius the umber of orders that the supplier completed ad set to the maufacturer. The Heuristic ad Mai Results I this decetralized case, we focus o the maufacturer s problem sice he is the oe who quotes the due dates to the customer, ad we try to fid a effective schedulig rule/due date quotatio heuristic to miimize the maufacturer s total cost give limited supplier iformatio. For this model, we employ the same two phase approach that we used before, by first determiig a asymptotically optimal schedulig rule to miimize the total completio times, ad the desigig a due date quotatio heuristic to match the completio times with that schedule. I this case, sice the maufacturer is workig idepedetly from the supplier ad has o iformatio about the processig times or the schedulig rule used i the supplier side, to miimize the total completio times, it will be asymptotically optimal for him to use the SPTA schedulig rule accordig to his ow processig times p m. Based o this schedule, to fid a effective due date quotatio heuristic for the maufacturer, we use the same

due date settig ideas as before. However, i this case, sice the maufacturer is uaware of the processes at the supplier, we use estimates of the coditios at the supplier site these estimates replace the iformatio that we used i the cetralized case. The maufacturer has o iformatio about the supplier except the umber of jobs at the supplier side, qi s, at time r i. Although usig a SPTA schedule accordig to the processig times at the supplier, p s, is asymptotically optimal to miimize the total completio times at the supplier, the maufacturer is uaware of p s i values, so that he ca t ifer ay iformatio about the supplier s schedule or the locatio of a job at the supplier queue. Thus, to quote a due-date, we assume that a arrivig job is located i the middle of the existig jobs i the supplier queue, ad that the future arrivals will be scheduled i frot of this job with probability 1/2. This is reasoable give that the maufacturer does t kow aythig about the schedule used by the supplier, about the processig time of that job, or about the other jobs at the supplier queue. We develop a asymptotically optimal schedulig ad due date quotatio algorithm for the maufacturer give that the maufacturer is usig this assumptio about the supplier s status. We call the due-date quotatio heuristic SLC SD for this simple decetralized case sice it is based o the slack algorithm SL for the sigle facility case. The maufacturer s schedulig ad due date settig heuristic, SP T A SLC SD for this case follows: I the above equatios, d s i is the approximate completio time of job i i the supplier side assumig that each arrivig job is scheduled i the middle of the supplier queue. ωi m deotes the approximate queue legth i frot of job i whe it arrives to the maufacturer from the supplier ad slacki m deotes the approximated legth of the jobs that will arrive to the maufacturer after job i but will be processed there before job i. The value (ds i ri) µ s approximates the umber of jobs that will be fiished before job i at the supplier ad will brig a extra workload to the maufacturer ad qi s + ( i) (ds i r i) µ approximates the maximum umber of jobs that ca arrive to the s maufacturer from the supplier after job i. We deote the objective value with this due date settig heuristic for the simple decetralized case Z SP T A SLC SD for a job istace, ad recall Theorem 2 i Sectio 2.1 states that SP T A SLC SD is asymptotically optimal uder the coditios described above. 2.3.3. The Decetralized Model with Additioal Iformatio Exchage: As we will see i our computatioal aalysis i Sectio 2.4, there are sigificat gais that result from cetralizig the cotrol of this system. O the other had, as we discussed above, there are frequetly sigificat expeses ad complexities iheret i movig to a cetralized supply chai, if it is possible at all. Thus, firms may be motivated to cosider limited or partial iformatio exchage to achieve some of the beefits of cetralizatio. Ideed, it may be that limited iformatio exchage achieves may of the beefits of cetralized cotrol, rederig complete cetralizatio uecessary. I this sectio, we start to explore this questio, by cosiderig the case i which the supplier shares some of the iformatio about his processes through a mechaism, so that presumably, the maufacturer ca quote better due-dates to his customers. For this system, we fid a effective schedulig ad due-date quotatio algorithm ad aalyze the gais by simple iformatio exchage. The Model I particular, we assume that the supplier shares limited iformatio with the maufacturer by quotig itermediate due dates. I other words, whe a customer order arrives at the maufacturer, the maufacturer immediately places a order with the supplier, ad the supplier quotes a due date for the suppliers subsystem to the maufacturer. The maufacturer ca use this supplier due date, alog with kowledge of his ow shop ad the time that the order will take him, to quote a due date to the customer. However, the model is i all other ways the same as the simple decetralized model. The maufacturer still does t kow aythig about the schedule the supplier uses or the processig times of the jobs at the supplier. I this case, however, the maufacturer ca use the due date give by the supplier to better estimate the completio times of orders at the supplier, ad ca thus quote more accurate due dates to the customer. The Heuristic ad Mai Results Oce agai, we employ a two-phase approach to due date quotatio ad schedulig, ad sice the maufacturer is idepedet from the supplier i this case, as i the simple decetralized case, it oce agai makes sese for the maufacturer to schedule usig the SPTA rule. Similarly, sice supplier works idepedetly from the maufacturer, he acts as a sigle facility ad tries to miimize his ow costs. Thus, we assume that the supplier uses the asymptotically optimal SPTA-SL heuristic for schedulig ad due date quotatio described i Sectio 2.2 for the sigle facility case. Thus, the supplier quotes due dates to the maufacturer accordig to the followig rule: d s i = r i + p s i + M s i + slack s i However, the maufacturer is uaware of the schedule used by the supplier, ad istead uses the due dates quoted by the supplier to estimate the completio times of the orders at the supplier ad sets his due dates accordigly. We call the due-date quotatio heuristic SLC DIE for this decetralized case with iformatio exchage sice it

ALGORITHM 5: SPTA-SLC SD Schedulig: Process the jobs accordig to shortest processig time, p m i, at the maufacturer. Due-Date Quotatio: d m i = d s i + pm i + ωi m + slacki m d s i = r i + qs i µs 2 + slacki s slacki s = mi{( i), (µs q s i 2 ) } µs λ µs 2 if λ µs 2 > 0 2 ( i) µs 2 otherwise ωi m = max{t mm i + (ds i ri)θi µ (d s s i r i), 0} t mm i = j B pm j where B = set of jobs at maufacturer queue with p m < p m i at time r i Θ i = pr{p m < p m i }E{pm p m < p m i } λ m = max{λ, { µ s } {( i + q s slacki m = i ) (ds i r i) µ }Θ s i if λ m Θ i 0 mi{ ωm i λ m Θ i, ( i + qi s (ds i ri) µ )}Θ s i otherwise ALGORITHM 6: SPTA-SLC DIE Schedulig: Process the jobs accordig to shortest processig time, p m, at the maufacturer. Due-Date Quotatio: d m i = d s i + pm i + ωi m + slacki m ω m i = max{t mm i + (ds i ri)θi µ s (d s i r i), 0} t mm i = j B pm j where B = set of jobs at maufacturer queue with p m < p m i at time r i Θ i = pr{p m < p m i }E{pm p m < p m i } λ m = max{λ, µ s } { {( i + q s slacki m = i ) (ds i r i) µ }Θ s i if λ m Θ i 0 mi{ ωm i λ m Θ i, ( i + qi s (ds i r i) µ )}Θ s i otherwise is based o the slack algorithm SL for the sigle facility case. We summarize this approach below: We deote the objective value for a job istace with this due date settig heuristic for this decetralized case with iformatio exchage Z SP T A SLC DIE. Recall Theorem 3 i Sectio 2.1 which states that SP T A SLC DIE is asymptotically optimal. 2.4. Computatioal Aalysis: Usig these schedulig ad due date quotatio heuristics, we desiged computatioal experimets to explore how these asymptotically optimal heuristics work eve for smaller istaces, ad to better uderstad the value of cetralizatio. 2.4.1. Efficiecy of the Heuristics: To assess the performace of the heuristics, we compared objective values for various problem istace sizes to lower bouds. Observe that i a optimal offlie solutio to this model, due dates equal completio times, ad the problem becomes equivalet to the problem of miimizig the sum of completio times of the orders. Also ote that for a sigle facility olie problem, a preemptive SPTA schedule miimizes the total completio times of the jobs for the preemptive versio of the problem, ad is thus a lower boud o the opreemptive completio time problem. Therefore, if we use that schedule ad set the due dates equal to the job completio times, ad igore the lateess compoet of the objective, we have a lower boud o the objective value of our problem. For the supply chai models, we ca use the same lower boud, focusig oly o the processig times at oe of the facilities, assumig that the job s waitig time at the queue of the other facility is zero. We use a preemptive SPTA schedule at our chose facility, ad the set the due-date of a job equal to the completio time of that job at the chose facility plus the processig time of the job at the other facility, oce agai igorig the lateess compoet of our objective. For the case with exchageable processig time distributios, we ca select either facility to focus o, ad for the o-exchageable cases, we focus o the bottleeck facility. I all cases, we have a lower boud o the completio time at oe facility, ad have added oly the processig time at the other facility, igorig capacity costraits at that facility, so this is clearly a lower boud o our objective. For the sigle facility case, we simulate the system usig differet umber of jobs arrivig to the facility, ad we use algorithm SP T A SL to sequece ad quote due-dates. We geerate our problem sequeces usig expoetial distributios, hold the arrival rate costat, ad vary the mea processig time, ad relative SP T A SL due date ad tardiess costs. The ratios of Z to the lower boud for differet combiatios of, c T

SP T A SL ad µ values with c d = 1, λ = 1 ad expoetial iterarrival ad processig time distributios are preseted i Table 1. Observe that as the umber of jobs,, icreases, Z rapidly approaches the lower boud. The rate of covergece differs for differet µ/λ (i.e. E(process time)/e(iterarrival time)) ratios ad for differet cost values but i all cases it coverges quite quickly to the lower boud as gets larger. The heuristic performs well eve for small umber of jobs. Note that each etry i Table 1 reflects the average of five rus with differet radom umber streams. For the cetralized supply chai model, usig the due date quotatio ad schedulig algorithm SP T A p SLC ad its modificatios for the ubalaced cases, we simulate the system with differet umbers of jobs for differet combiatios of mea processig times µ s ad µ m to explore the effectiveess of these heuristics, eve for small problem istaces. For these experimets, we use expoetially distributed iter-arrival ad processig times. The ratios of the objective fuctio Z SP T A p SLC ad the total tardiess, T, to the lower boud for differet combiatios of, µ s ad µ m with c d = 1, c T = 2, λ = 1 ad expoetial iter-arrival ad processig time distributios are i Table 2 where each etry represets the average of five rus with differet radom umber streams. As the umber of jobs icreases, Z SP T A p SLC /LB ratio approaches to 1 ad, eve for smaller istaces, that ratio is still very close to oe. Also, the T/LB ratio coverges to 0 as the umber of jobs icreases, ad that ratio is also very close to zero eve for a small umber of jobs. Thus, the asymptotically optimal due-date quotatio algorithm ad its variats seem to work well, eve for small umbers of jobs. 2.4.2. Compariso of Cetralized ad Decetralized Models: The ultimate goal of this work is to compare cetralized ad decetralized make-to-order supply chais, ad to explore the value of iformatio exchage i this system. To that ed, we prepared a computatioal study to ivestigate the differeces betwee the cetralized ad decetralized versios of this supply chai. For the cetralized model we use the algorithm SP T A p SLC for schedulig ad lead time quotatio ad for the decetralized cases, we assume that the supplier uses a SPTA schedule accordig to his ow processig times p s ad the maufacturer uses the schedulig ad due-date quotatio algorithms described i sectio 2.3. Table 3 shows the ratios of the objective values of cetralized ad decetralized models obtaied by simulatig the system for = 3000 jobs, where each etry shows the average of five rus with differet radom umber streams. We used differet combiatios of µ s ad µ m with c d = 1, c T = 2, λ = 1 ad expoetial iter-arrival ad processig time distributios i this simulatio. I this table ce deotes the objective value for the cetralized case, SD deotes the objective value for the simple decetralized model ad DIE deotes the objective value with the decetralized model with iformatio exchage. Our experimets demostrate that whe the mea processig time at the supplier is smaller tha the mea iterarrival time, i.e. whe there is o cogestio at the supplier side, the cetralized ad decetralized models lead to very similar performace. However, as the cogestio at the supplier begis to icrease, the value of iformatio ad cetralizatio also icreases ad the cetralized model starts to lead to much better results tha the decetralized oes. Similarly, if the mea processig time at the supplier is held costat, as the mea processig time at the maufacturer icreases, the value of cetralizatio decreases. As the supplier becomes more cogested tha the maufacturer, the supplier is the bottleeck facility that ultimately determies system performace, but the maufacturer has limited kowledge of the bottleeck facility ad thus gives iaccurate due dates to the customers. However, if the maufacturer is the bottleeck, the maufacturer already has the iformatio about his processig times which have the majority of impact o system performace. The value of iformatio about processig times at the supplier is limited because they do t have tremedous impact o system performace. I other words, i this case, the value of iformatio is lower. These experimets suggest that if there is little or o cogestio at the supplier, or if the maufacturer is sigificatly more cogested tha the supplier, cetralizig cotrol of the system is likely ot worth the effort (at least, for the objective we are cosiderig). However, if the cogestio at the supplier icreases, the value of cetralizatio icreases ad total costs ca be dramatically decreased by cetralizig the system. Although cetralizatio ca sigificatly decrease costs i some cases, if cetralizatio is ot possible or very hard to implemet, simple iformatio exchage might also help to decrease costs. As see i Table 3, the losses due to decetralizatio ca be cut i half by simple iformatio exchage. However, eve with iformatio exchage, the costs of decetralized models are much higher, about 80% i some cases, tha cetralized oes. So, if the cogestio at the supplier is high, cetralizatio is worth the effort it takes to desig ad implemet iformatio systems, desig ad implemet supply cotracts, etc. However, if cetralizatio is ot possible, simple iformatio exchage ca also improve the level of performace dramatically. 3. A Aalysis of a Combied Make to-order/make-to-stock System: After aalyzig the pure MTO model, we cosider a combied MTO- MTS supply chai composed of a maufacturer, served by a sigle supplier workig i a stochastic multi-item

SP T A SL Table 1: Ratios of Z to the lower boud # of jobs c T =1.1 c T =1.5 c T =2 c T =5 # of jobs c T =1.1 c T =1.5 c T =2 c T =5 µ=0.5 µ=1.5 10 1.00962 1.01815 1.02882 1.09278 10 1.03373 1.06192 1.09715 1.30856 100 1.00258 1.00285 1.00317 1.00514 100 1.05480 1.05898 1.06421 1.09557 1000 1.00033 1.00036 1.00039 1.00062 1000 1.01558 1.01798 1.02098 1.03898 5000 1.00007 1.00008 1.00009 1.00014 5000 1.00753 1.00794 1.00846 1.01159 10000 1.00003 1.00003 1.00003 1.00006 10000 1.00541 1.00560 1.00583 1.00720 µ=1 µ=2 10 1.10481 1.11671 1.13157 1.22079 10 1.04609 1.07794 1.11775 1.35663 100 1.01426 1.01605 1.01828 1.03165 100 1.07541 1.08011 1.08600 1.12129 1000 1.00677 1.00777 1.00903 1.01657 1000 1.01140 1.01636 1.02256 1.05976 5000 1.00270 1.00291 1.00319 1.00482 5000 1.00901 1.00925 1.00953 1.01128 10000 1.00177 1.00191 1.00219 1.00317 10000 1.00457 1.00491 1.00534 1.0079 SP T Ap SLC Table 2: Ratios of the objective fuctio Z Z SP T A p SLC Z SP T A p SLC ad the total tardiess, T, to the lower boud Z SP T A p SLC LB # jobs LB T/LB # jobs LB T/LB # jobs T/LB µ s =1 µ m =1 µ s =2 µ m =1 µ s =5 µ m =1 10 1.0202 0.0118 10 1.0344 0.0102 10 1.0462 0.0121 100 1.0217 0.0059 100 1.0618 0.0096 100 1.0495 0.0122 1000 1.0087 0.0022 1000 1.0239 0.0052 1000 1.0127 0.0086 5000 1.0037 0.0010 5000 1.0102 0.0017 5000 1.0067 0.0023 µ s =1 µ m =2 µ s =2 µ m =2 µ s =5 µ m =2 10 1.0344 0.0137 10 1.0263 0.0294 10 1.0471 0.0118 100 1.0374 0.0040 100 1.0635 0.0160 100 1.0507 0.0134 1000 1.0133 0.0025 1000 1.0241 0.0057 1000 1.0137 0.0075 5000 1.0071 0.0012 5000 1.0108 0.0039 5000 1.0080 0.0024 µ s =1 µ m =5 µ s =2 µ m =5 µ s =5 µ m =5 10 1.0552 0.0144 10 1.0553 0.0259 10 1.0467 0.0349 100 1.0494 0.0053 100 1.0722 0.0159 100 1.0196 0.0338 1000 1.0191 0.0021 1000 1.0330 0.0046 1000 1.0117 0.0108 5000 1.0104 0.0016 5000 1.0143 0.0046 5000 1.0046 0.0089 Table 3: Ratios of the objective values of cetralized ad decetralized models µ s µ m SD/ce DIE/ce SD/DIE µ s µ m SD/ce DIE/ce SD/DIE 0.5 0.5 1.00211627 1.00215099 0.9999653 2 0.5 2.07968368 1.50892076 1.37825903 0.5 1 1.0025820 1.00308953 0.99949407 2 1 2.08234862 1.5133057 1.37602638 0.5 2 1.01378038 1.01417906 0.99960689 2 2 1.94756315 1.44041736 1.35208253 0.5 5 1.00604253 1.00615847 0.99988477 2 5 1.31118576 1.05354521 1.24454626 1 0.5 1.02696404 1.02616592 1.00077776 5 0.5 2.4082784 1.86390096 1.29206350 1 1 1.02055926 1.01543491 1.00504646 5 1 2.40706065 1.86304885 1.29200082 1 2 1.02373349 1.01594891 1.00766237 5 2 2.35828183 1.82580914 1.29163655 1 5 1.00838256 1.00560849 1.00275859 5 5 2.08264443 1.63575747 1.27319878

eviromet. I this case, both facilities are allowed to carry some level of ivetory for each type of product istead of operatig a pure MTO system. So, i additio to desigig effective schedulig ad lead time quotatio algorithms, we wat to fid the optimal ivetory levels that should be carried at each facility for this combied system. I this system, the maufacturer ad the supplier have to decide which items to produce to stock ad which oes to order. The maufacturer also has to quote due dates to arrivig customers for make-to-order products. The maufacturer is pealized for log lead times, missig the quoted lead times ad for high ivetory levels. I the followig sectios, we cosider several variatios of this problem, ad desig effective heuristics to fid the optimal ivetory levels for each item ad also desig effective schedulig ad lead time quotatio algorithms for cetralized ad decetralized versios of this model. We also make extesive computatioal experimets to evaluate the effectiveess of our algorithms, to aalyze the beefits of the combied MTO-MTS systems versus pure MTO or MTS systems ad to compare the cetralized ad decetralized supply chais. 3.1. Properties of the Model: We model two parties, a supplier ad a maufacturer i a flow shop settig. Customer orders, or jobs, j, arrive at the maufacturer at time r j, ad the maufacturer quotes a due date for each order, d j, whe it arrives. We assume that there are k types of jobs with mea processig times µ i for each type i=1,2...k. Jobs arrive to the system with rate λ ad each arrivig job has a probability δ i of beig type i. We assume expoetially distributed, statioary ad idepedet iter-arrival times, so, each job type i has a arrival rate of λ i = λδ i. To begi processig each order, the maufacturer requires a compoet specifically maufactured to order by the supplier. The compoet type i requires mea processig time µ s i at the supplier, ad the order requires mea processig time µ m i at the maufacturer. We assume that a base-stock policy is used for ivetory cotrol of MTS items ad startig with R i uits of ivetory, wheever a demad occurs, a productio order is set to repleish ivetory. R i = 0 meas a make-toorder productio system is employed for job type i. If demad is higher tha the ivetory level, the the extra demad is backlogged ad satisfied later whe it is produced. It is also assumed that the system is ot cogested ad the iterarrival times follow a expoetial distributio. We aim to miimize the total expected ivetory plus lead time plus tardiess costs i this system. Thus, the objective fuctio to miimize is Z = k {h i E[I i ] + c d i E[d i ] + c T i E[W i d i ] + } (3.1) i=1 where h i is the uit ivetory holdig cost, c d i is the uit lead time cost ad c T i is the uit tardiess cost for type i. E[I i ] deotes the mea amout of ivetory, E[d i ] deotes the lead time mea ad E[W i d i ] deotes the mea tardiess. We cosider three versios of this model, which we briefly describe here ad discuss i more detail i subsequet paragraphs. I the cetralized versio of the model, the etire system is operated by a sigle etity, who is aware of the ivetory levels ad processig times both at the supplier ad the maufacturer. So, this sigle etity decides o the ivetory levels for each class as well as the productio schedule ad the lead times that should be quoted to each customer. I the decetralized, full iformatio model, the maufacturer ad the supplier are assumed to work idepedetly, but the maufacturer has full iformatio about both his processes ad the supplier. They both try to miimize their ow costs i a sequetial game theoretic approach. The supplier acts first to determie his optimal ivetory levels ad the the maufacturer acts to miimize his ow costs usig the optimal ivetory levels for the supplier. However, i the simple decetralized model, the maufacturer ad the supplier still works idepedetly from each other but ow the maufacturer has o iformatio about the supplier except the average delivery times of orders from the supplier. The objective of our problem is to determie the optimal ivetory levels, a sequece of jobs ad a set of due dates such that the total cost Z = k i=1 {h ie[i i ] + c d i E[d i] + c T i E[W i d i ] + } is miimized. Clearly, to optimize this expressio, we eed to coordiate due date quotatio, sequecig ad ivetory maagemet ad a optimal solutio to this model would require simultaeous cosideratio of these three issues. However, the approach we have elected to follow for this model (ad throughout this thesis) is slightly differet. Observe that i a optimal off-lie solutio to this model, lead times would equal waitig times of jobs i the system. Thus, the off-lie problem becomes equivalet to miimizig k i=1 {h ie[i i ] + c d i E[W i]}. Of course, i a olie schedule, it is impossible to both miimize this fuctio ad set due dates equal to completio times, sice due dates are assiged without kowledge of future arrivals, some of which may have to complete before jobs that have already arrived i order to miimize the sum of completio times. I this approach, we first determie a schedulig approach desiged to effectively miimize the sum of waitig times, ad the based o that schedule, we fid the optimal ivetory levels to miimize k i=1 {h ie[i i ] + c d i E[W i]} ad desig a due date quotatio approach that presets due dates that are geerally close to the completio times suggested by our schedulig approach. This is the ituitio behid the heuristic preseted below. I sectio 2, we preset effective schedulig ad lead

time quotatio algorithms for pure MTO versios of this system. We beefit from the properties of those algorithms ad preset modified versios of them for this system. However, ote that, our results for the optimal ivetory levels ca be geeralized to other schedules ad lead time quotatio algorithms as log as the schedule is idepedet of the workload or ivetory levels i the system. The schedule oly effects the statioary distributios of the umber of jobs i the system. We cosider SPTA ad FCFS schedulig algorithms for our calculatios but the results ca be geeralized to differet schedules by recalculatig the statioary distributios oly. We cosider differet schedules i sectio 3.4 ad compare them by computatioal aalysis. I the ext sectio, we itroduce a prelimiary sigle facility model ad aalyze this model. I sectio 3.3, we preset our supply chai models, algorithms, ad results i detail, ad i sectio 3.4, we preset the computatioal aalysis ad a compariso of cetralized ad decetralized models. 3.2. Sigle Facility Model: 3.2.1. The Model: Although our ultimate goal is to aalyze multi-facility systems, we begi with a prelimiary aalysis of a sigle facility system. We focus o fidig the optimal ivetory levels for this system ad the coditios uder which a MTO strategy or a MTS strategy would be optimal for this facility. We also focus o desigig effective schedulig ad due date quotatio heuristics for this system. I this sectio, we cosider a sigle facility that combies make-to-stock ad make-to-order policies to miimize its ivetory ad lead time related costs. Our objective i this system is to fid a effective operatig structure to miimize the ivetory ad lead time related costs. I solvig this problem, we eed to determie the optimal values for base-stock levels R i for each job type as well as fidig a effective schedulig ad lead time quotatio algorithm for these jobs to miimize the total costs. The model operates exactly as explaied i the previous sectio but with just a sigle maufacturer without ay supplier. We ca thik of this sigle facility model as a special case of the supply chai model where the processig times at the supplier are all 0, so the compoets are available to the maufacturer as soo as a order arrives. Also, the supplier does t eed to hold ay ivetory ad there is o cost related to the supplier. Defie two queues for the maufacturer i this model, the productio queue ad the order queue. Wheever a job arrives at the system, if that item exists i ivetory, the demad is immediately satisfied from the ivetory. Sice that order is immediately satisfied, we do t place that job i the order queue. However, that job is still placed o the productio queue i order to repleish the ivetory. Whe a order arrives ad there is t ay ivetory of that item left, that job is placed at both the order queue ad the productio queue ad a lead time eeds to be quoted for that item. Thus, the order queue icludes just the usatisfied orders at ay time while the productio queue icludes the items that are goig to be produced both to satisfy orders ad to repleish ivetory. The order queue is just a subset of the productio queue. The productio queue operates exactly the same way as a pure make-to-order system because eve though we have ivetory at had for a item, we place that job i the productio queue to repleish ivetory. Thus, havig ivetory for a item does ot impact the productio process, but does decrease the due date costs sice we satisfy those orders immediately ad do t put them i the order queue. Sice the productio queue operates exactly the same way as a pure MTO system, we employ the followig algorithm, amed SPTA-LTQ, for schedulig ad lead time quotatio, which is very similar to the algorithm SPTA- SL, desiged for a pure MTO model sigle facility case i Sectio 2.2. 3.2.2. Aalysis ad Results: As metioed i Sectio 3.1, we elect to employ a heuristic that first attempts to fid a schedule to miimize the total completio times, ad the fids the optimal ivetory levels to miimize a fuctio of the ivetories ad waitig times of the jobs i the system based o this schedule ad the sets lead times that approximate the waitig times of jobs i the system with that schedule usig the state of the system at the time of the arrival of the order, i a effort to miimize our objective fuctio. The heuristic we propose sequeces the jobs accordig to the Shortest Processig Time Available (SPTA) rule. Uder the SPTA heuristic, each time a job completes processig, the shortest available job which has yet ot bee processed is selected for processig. For a pure MTO system, although the problem of miimizig completio times is NP-Hard, Kamisky ad Simchi-Levi [18] foud that the SPTA rule is asymptotically optimal for this problem, that is SPTA rule is optimal for miimizig the sum of completio times as the umber of jobs goes to. Also, ote that this approach to sequecig does ot take quoted due date or ivetory levels ito accout, ad is thus easily implemeted. For the lead time quotatio algorithm LTQ preseted above, we preset the followig lemma. Lemma 1. Cosider a series of radomly geerated problem istaces of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. Also, let Z SP T A = i=1 cd C i deote the total weighted delivery times of orders where C i = r i +W i is the delivery time of SP T A LT Q order i ad Z = i=1 {cd d i +ct (C i d i )+ }

ALGORITHM 7: SPTA-LTQ Schedulig: Sequece the jobs i the productio queue accordig to shortest processig time available (SPTA) rule. lead time Quotatio: { 0 if Ii > 0 at r i d i = E[p i ] + E[M j ] + E[M j]λψ i τ i 1 λψ iτ i otherwise where j is the job i the productio queue that will be used to satisfy order i. M j is the workload i frot of job i at the time of arrival, ψ i is the probability that a arrivig job has processig time less tha p i ad τ i = E[p p < p i ] is the expected processig time of a job give that it is less tha p i deote the total due-date plus tardiess costs with the algorithm SPTA-LTQ where d i = r i + d i is the quoted due-date. The, the lead time quotatio algorithm LTQ is asymptotically optimal to miimize this objective fuctio assumig SPTA schedulig rule is used, that is almost surely, lim SP T A LT Q Z Z SP T A Z SP T A = 0 We also state the followig lemma for the lead time quotatio algorithm LTQ preseted above: Lemma 2. For the model explaied above, for every class i, the expected value of the quoted lead time for class i is equal to the expected waitig time of a class i job, that is E[d i ] = E[W i ] where W i deotes the actual waitig time of job i i the order queue. We use the SPTA-LTQ algorithm for schedulig ad lead time quotatio as above ad based o this schedule, we fid the optimal ivetory levels for each class i the subsequet parts of this sectio. However, ote that differet schedules ca also be cosidered ad the results of our subsequet aalysis ca be easily modified by updatig the statioary distributios of the umber of jobs i the system. We cosider the above algorithm for our calculatios sice it is a effective oe to quote reliable ad short due-dates. Also, for the supply chai models, i Sectio 3.3, we preset effective algorithms for schedulig ad lead time quotatio ad use them i our aalysis to fid the optimal ivetory levels, but agai, our results regardig the optimal ivetory levels ca be geeralized to differet schedulig ad lead time quotatio algorithms easily. To fid the optimal ivetory levels, we aim to miimize the followig objective fuctio: k {h i E[I i ] + c d i E[W i ]} (3.2) i=1 where E[I i ] is the expected ivetory level for job type i ad E[W i ] is the expected waitig time for job type i i the order queue. There is a obvious tradeoff betwee the ivetory costs ad the waitig time costs i this objective. We ca decrease the waitig times of the class i jobs by holdig additioal ivetory of that type but that will icrease the ivetory costs. Observe that holdig additioal ivetory for a item effects the order queue oly ad does ot effect the productio queue. Thus, the waitig times ad the lead time quotatio procedure for the other classes are t effected by this. Lemma 3. Objective fuctio 3.2 is equivalet to the fuctio k i=1 {h ie[i i ] + c i E[N i ]} where c i = c d i /λ i ad E[N i ] is the expected umber of job type i i the order queue. Arreola-Risa ad DeCroix [1] cosiders the same objective fuctio k i=1 {h ie[i i ] + c i E[N i ] to miimize by cosiderig a FCFS schedule. We begi by restatig some of their results, expad them to iclude SPTA schedules ad compare the two types of schedules i this sectio. Lemma 4. = k {h i E[I i ] + c i E[N i ]} i=1 k i=1 (R i x)f i (x) + c i h i R i x=0 x=r i (x R i )f i (x) (3.3) where f i (x) deotes the probability of havig x jobs of type i i the productio queue. Lemma 5. Dividig the problem ito k subproblems accordig to their types, ad the solvig for each type idividually gives the optimal solutio for the whole problem. So, our problem decreases to miimizig Ri h i x=0 (R i x)f i (x) + c i x=r i (x R i )f i (x) for each i. Theorem 6. The optimal level of ivetory R i is the miimum value x 0 that satisfies c i F i (x) c i + h i

Corollary 1. Produce item i MTO if ad oly if F i (0) c i c i+h i Corollary 2. A item s productio moves towards MTO if its uit lead time cost, processig time or arrival rate decreases or uit holdig cost icreases. Note that the results above regardig optimal ivetory levels hold for a variety of queueig disciplies, ot restricted to a sigle server queue or a SPTA schedule, ad they are also idepedet of the arrival or maufacturig process. However, these characteristics effect F i (x), the statioary distributio of the umber i the system. To assess the effectiveess of our schedulig our algorithm SPTA ad to explore how the schedule used i the system effects F i (x) ad the objective fuctio, we aalyze two differet schedules SPTA ad FCFS ad preset the followig two corollaries for these schedules. We also compare the objective fuctio usig these schedules through computatioal aalysis i sectio 3.4. Also, oe ca exted these results for other kids of schedules or queues with differet characteristics by oly cosiderig the chages i F i (x) for that queue. Corollary 3. If FCFS schedulig rule is used i the productio queue, istead of SPTA, assumig a M/G/1 queue, to miimize 3.3, it is optimal to produce product i MTO if ad oly if: k δ j E[e λ iµ j ] (1 δ i)r i r i (1 ρ)δ i j=1 where r i = c i c i+h i, δ i = λ i λ ad ρ = k i=1 λ iµ i Corollary 4. If SPTA schedulig rule is used i the productio queue, assumig it is a M/G/1 queue, it will be optimal to produce product i MTO if ad oly if: (1 ρ)(ξ i ) + λ b (1 γ b (ξ i ) λ i γ i (ξ i ) r i where ξ i = λ i + λ a λ a ν a (λ i ), r i = c i c i+h i, ρ = k i=1 λ iµ i, λ a = i 1 j=1 λ j is the total arrival rate of jobs shorter tha class i, λ b = k j=i+1 λ j is the total arrival rate of jobs loger tha class i, γ i (z) = E[e zp i ] is the Laplace trasform associated with the processig time of class i ad ν a (z) is the solutio of the equatio ν a (z) = γ a (z + λ a λ a ν a (z)). 3.3. Supply Chai Models: I this sectio, we aalyze the ivetory decisios, schedulig ad due-date quotatio issues for two-stage supply chais usig the results from our aalysis i sectio 3.2. We develop effective heuristics to fid the optimal ivetory levels at both facilities ad desig effective algorithms for schedulig ad due-date quotatio for both the cetralized ad decetralized versios of these systems. These algorithms allow us to compare the value of cetralizatio ad iformatio exchage i supply chais uder a variety of differet coditios. Whe a maufacturer is workig with a supplier, the compoets may ot be immediately available to the maufacturer at the arrival time of a order. The maufacturer has to wait for some time for the compoets to arrive from his supplier before he ca start workig o that order. Thus, the supplier-maufacturer relatioship effects the optimal levels of ivetories that should be held as well as the schedulig ad lead time quotatio decisios. We model this system as a two facility flow shop with a maufacturer ad a supplier where both parties ca choose to stock some of the items ad use a maketo-order strategy for the others i a multi-item, stochastic eviromet. We assume that the supplier ad the maufacturer employs a oe-to-oe repleishmet strategy ad a base-stock policy for ivetory cotrol of their items. The maufacturer starts with a ivetory of Ri m uits of fiished goods ad the supplier starts with a ivetory of Ri s uits of semi-fiished goods that the maufacturer eeds to complete his productio. We agai defie the productio ad order queues similar to the way we did i the sigle facility model, but i this case, for both the supplier ad the maufacturer. Whe a order arrives, if that item is i the maufacturer s ivetory, the order is immediately satisfied ad a lead time of 0 is quoted. However, a productio order of that class is set to both the supplier ad the maufacturer to repleish the ivetory of the maufacturer. That order is ot placed i the maufacturer s order queue util the semi-fiished goods are delivered to the maufacturer by the supplier. If the supplier also has ivetory of that class, he seds it directly to the maufacturer ad that order appears i the maufacturer s productio queue immediately. The supplier still places that order i his productio queue to repleish his ow semi-fiished goods ivetory. If that item is either i the maufacturer s or the supplier s ivetory, the a lead time is quoted to the customer ad a productio order is set to both facilities to satisfy this order. This order appears immediately i the supplier s productio queue ad after it is delivered from the supplier, it appears i the maufacturer s productio queue. The maufacturer has to wait for some time for the semi-fiished goods to be delivered to him by the supplier to put that order i his productio queue. However, if the supplier has this item i its ivetory, the that order immediately appears i the maufacturer s productio queue as well as the supplier s productio queue. A shorter lead time is quoted i this case sice the customer oly eeds to wait for the productio at the maufacturer ad waitig time at the supplier is 0. Let x s i ad xm i deote the amout of jobs of type i i the supplier s ad maufacturer s productio queue, re-

spectively. The, let Ni s deote the amout of jobs of class i waitig i supplier s order queue that should be delivered to the maufacturer, Ii s deote the amout of semi-fiished goods ivetory of class i, Ni m deote the total umber of customers of class i waitig i the system for their orders to be delivered ad Ii m deote the amout of fiished goods ivetory at the maufacturer of class i. The, N s i = max{x s i R s i, 0} I s i = max{r s i x s i, 0} N m i = max{x m i + max(x s i R s i, 0) R m i, 0} I m i = max{r m i x m i max(x s i R s i, 0), 0} 3.3.1. The Cetralized Supply Chai Model: (3.4) R The Model I some systems, the maufacturer has a K s i close relatioship ad perhaps eve complete cotrol over = {h s i (Ri s yi s )P (x s i = yi s ) i=1 yi his supplier. For example,the maufacturer ad the supplier may belog to the same firm. I those cases, a ce- i 1 R s=0 R i m tral aget that has complete iformatio about both parties ad makes all the decisios about both firms will ob- yi s=0 yi m=0 + h m i [ (Ri m yi m )P (x s i = yi s, x m i = yi m ) viously be much more effective i miimizig the total R s i +Rm i R i +Rm i ys i costs i the system tha the idividual parties would be. + (Ri s + Ri m yi s yi m ) I this sectio, we assume that the maufacturer ad yi s=rs i yi m=0 the supplier work as a sigle etity ad they are both P (x s i = yi s, x m i = yi m )] cotrolled by the same aget that has all the iformatio R about both sides. The decisios about the schedulig at s i 1 + c both facilities ad lead time quotatio for the customer as i [ (yi m Ri m )P (x s i = yi s, x m i = yi m ) yi well as the ivetory levels for each party are made by yi m=rm i this aget. + (yi s + yi m Ri s Ri m ) I the cetralized model, our objective fuctio to yi miimize is: i yi m=rs i +Rm i ys i P (x s i = yi s, x m i = yi m )]} k (3.7) h s i E[Ii s ] + h m i E[Ii m ] + c d i E[d i ] + c T i E[W i d i ] + where R s ad R m are the array of ivetory levels at i=1 the supplier ad the maufacturer ad x s i ad x m i are (3.5) the umber of class i jobs at the supplier s ad maufacturer s productio queue, respectively. where h s i is the uit holdig cost of semi-fiished goods Observe that due to our assumptio about the schedule used i the facilities, both the supplier ad the mau- at the supplier ad h m i is the uit holdig cost of fiished goods at the maufacturer. facturer s productio queue operates idepedet of R m. I additio, the supplier s productio queue is idepedet Aalysis ad Results I this system, we use a approach of R s. However, the maufacturer s productio that is similar to the oe we employed for the sigle facility case. We first fid the optimal ivetory levels for a schedule that is idepedet of the workload or ivetory queue depeds o R s, thus f m (x), the statioary distributio of umber of jobs at the maufacturer, is a fuctio of R s. levels i the system (e.g. usig a FCFS schedule i both facilities is such a schedule), to miimize the objective fuctio Theorem 7. For fixed ivetory levels Ri s for each class i at the supplier, the optimal levels of ivetory for the maufacturer are the miimum Ri m values that satisfy: k Z(R s, R m ) = {h s i E[Ii s ] + h m i E[Ii m ] + c d i E[W i ]} P (x s i > Ri s, x s i + x m i Ri s + Ri m ) i=1 (3.6) The, we preset a effective schedulig algorithm cosistet with this model to miimize the total waitig times of the jobs i the system ad a lead time quotatio algorithm that matches these waitig times. Usig the defiitios 3.4 ad the equatios c d i = λc i ad E[W i ] = E[Ni m ]/λ i due to Little s law, ad writig them explicitly, we get the objective fuctio 3.7 to miimize i terms of the ivetory amouts ad the umber of jobs i the productio queues of the supplier ad the maufacturer both of which operate as pure MTO systems. = Z(R s, R m ) = k Z(Ri s, Ri m ) i=1 k {h s i E[Ii s ] + h m i E[Ii m ] + c i E[Ni m ]} i=1 + P (x s i Ri s, x m i Ri m ) c i + h m i c i (3.8)

Corollary 5. If the maufacturer is workig with a pure MTO supplier, the the maufacturer s optimal ivetory levels for each class are the miimum Ri m values that satisfy: P (x s i + x m i c i Ri m ) c i + h m i (3.9) Whe we look at the supplier ivetory levels, observe that f m (x), the probabilities of the umber of jobs at the maufacturer s productio queue, depeds o the ivetory levels R s at the supplier which makes the problem very hard to solve aalytically. However, we ca fid the optimal solutios uder some special coditios. We state the followig theorem for oe of these coditios. Theorem 8. For the cetralized model, if h s i hm i for a product type i, the a MTO strategy for type i at the supplier, that is holdig o ivetory of type i at the supplier, is optimal. For other cases, fidig the optimal solutio is very hard sice the maufacturer s productio queue depeds o the ivetory levels R s at the supplier which makes the problem very hard to trace aalytically. To fid a approximatio o the optimal R s values, we assume that the chage i the statioary distributios of the umber of jobs at the maufacturer is egligible w.r.t. a chage i the amout of the ivetory levels at the supplier ad try to fid the optimal R s values usig this approximatio. I that case, we ca divide the problem ito K subproblems ad aalyze each class separately. Still, we ca t state a result similar to Theorem 7 for the ivetory values at the supplier, sice the objective fuctio 3.7 does t possess the covexity structure i Ri s for fixed Rm i. (i.e. Z(Rs i + 1, Ri m) Z(Rs i, Rm i ) is ot odecreasig i Rs i for every Ri m.) So, to determie the optimal levels of R s, we employ a oe-dimesioal search o Ri s. For each Rs i, we calculate the optimal values of Ri m, calculate the total cost usig the objective fuctio 3.7 ad pick the pair with miimum cost for each class i. However, we ca decrease the search space usig some properties of the objective fuctio. Lemma 6. For this model, the fuctio Z(R s i, Rm i ) as give i 3.7 is supermodular for every class i. Theorem 9. Let R i s 0 be the miimum value that satisfies c i P (x s i R i s ) c i + h s i The, the optimal level of ivetory R s i R s i So, there is o eed to search for R s i beyod R s i. Corollary 6. It is optimal for the supplier to use a MTO strategy to produce product type i if Fi s(0) c i c i+h s i I geeral, for the fixed ivetory value Ri m at the maufacturer, if for every Ri s, 2 Z(R s i,rm i ) 2 R 0, the the i s followig theorem holds. A example of this case occurs, whe h s i hm i. Theorem 10. For fixed ivetory levels Ri m at the maufacturer, if 2 Z(R s i,rm i ) 2 R 0, the optimal levels of ivetory at the supplier are the miimum value Ri s that i s satisfies: (h m i + c i )P (x s i > R s i, x s i + x m i R s i + R m i ) + (h s i + c i )P (x s i R s i ) c i (3.10) We preset effective schedulig ad lead time quotatio algorithms i Sectio 2 for a pure MTO supply chai system. Usig the same ideas as i those algorithms, we desig modified versios of them for our system with ivetories. For the cetralized supply chai model, assumig idepedet ad exchageable processig times, the algorithm SP T A p LT Q C is outlied below. Note that this algorithm is cosistet with our assumptios for this model sice the schedule is idepedet of the workload or ivetory i the system ad the lead times quoted with this algorithm satisfies the relatio E[d i ] = E[W i ]. For the lead time quotatio algorithm LT Q C preseted above, we preset the followig lemma. Lemma 7. Cosider a series of radomly geerated problem istaces of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times at each facility be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. Also, let SP T Ap Z = i=1 cd C i deote the total weighted delivery times of orders with the SP T A p schedule where C i = r i + W i is the delivery time of order i ad Z SP T A p LT Q C = i=1 {cd d i + ct (C i d i )+ } deote the total due-date plus tardiess costs with the algorithm SP T A p LT Q C where d i = r i + d i is the quoted duedate. The, the lead time quotatio algorithm LT Q C is asymptotically optimal to miimize this objective fuctio for this cetralized system assumig that SP T A p schedulig rule is used to sequece jobs, that is almost surely, Z SP T Ap LT Q C lim Z SP T A p SP T Ap Z = 0 The schedule used i the system oly effects the statioary distributios of the umber of jobs i the system, (i.e.f(x)). For the schedule we preseted above, the supplier is usig a SPTA rule w.r.t. total processig time

ALGORITHM 8: SPTA p -LTQ C Schedulig: Process the jobs accordig to SP T A p (SPTA based o total processig time p i = p s i + pm i ) at the supplier ad FCFS at the maufacturer. lead time Quotatio: 0 if Ii m > 0 at r i d m i = E[p m i ] + tmm i if Ii m = 0, Ii s > 0 at r i d s i + E[pm i ] + max{tms i + t mm i + slacki m d s i, 0} otherwise where d s i = E[ps i ] + E[M i s] + E[M s i ]λpr{p<p i}e{p s p<p i } 1 λpr{p<p i}e{p s p<p i} t ms i = j A E[pm j ] where A=set of jobs i supplier queue scheduled before job i ad will be set to maufacturer. t mm i = j B E[pm j ] where B=set of jobs i maufacturer queue at time r i slacki m = (d s i ps i )λpr{p < p i}e{p m p < p i } + j L mi{(ds i ps i )λ j, Ij s}e[pm j ] where L is the set of jobs that will be scheduled after i at the supplier of the jobs ad the maufacturer is schedulig his jobs accordig to FCFS. Because the productio queue at the supplier is idepedet of R s ad R m ad it operates just like the sigle facility described i Sectio 3.2, the supplier ca fid the statioary distributio of the umber i his system usig the probability geeratig fuctio for a sigle facility queue. However, whe we look at the maufacturer side, the ivetory levels at the supplier effects the processes at the maufacturer, sice the whole supply chai is described as a ivetory queue. This makes the problem very difficult aalytically. We use the commo decompositio approach to approximate the statioary distributios of the umber of jobs i the maufacturer side. Note that, whe the iterarrival times are expoetially distributed i a queueig model like the well kow Jackso etwork model preseted first i Jackso[16], the departure process is poisso distributed. Sice the iter-arrival time to our system is expoetially distributed, we approximate the departure process from the supplier with a poisso distributio ad thus we assume that the arrivals to the maufacturer are poisso. So, we treat the processes at the maufacturer as a sigle facility with multiple classes with poisso arrivals for FCFS schedule. 3.3.2. The Decetralized Supply Chai, Full Iformatio Model: While some supply chais are relatively easy to cotrol i a cetralized fashio, most ofte this is ot the case. Eve if the stages i a supply chai are owed by a sigle firm, iformatio systems, cotrol systems, ad local performace icetives eed to be desiged ad implemeted i order to facilitate cetralized cotrol. I may cases, of course, the supplier ad maufacturer are idepedet firms, with relatively limited iformatio about each other. Implemetig cetralized cotrol i these supply chais is typically eve more difficult ad costly, sice the firms eed to coordiate their processes, agree o a cotract, implemet a iformatio techology system for their processes, etc. Thus, for either cetrally owed or idepedet firms, cetralizatio might ot be worth the effort if the gais from cetralizatio are ot big eough. Although cetralizatio i supply chais is geerally a difficult ad costly thig to do, at the same time, with a decetralized system, compaies might lose a lot of their profits. I some cases, istead of completely cetralizig the system, the compaies might just choose to share all their iformatio with each other to icrease their profits. Thus, we are motivated by the fact that iformatio exchage i some supply chais might icrease the profits high eough so that complete cetralizatio will be uecessary. I this case, the maufacturer has all the iformatio about the whole system but has o cotrol over the supplier s decisios. For this system, we fid the optimal ivetory levels for the maufacturer ad the supplier as well as a effective schedulig ad due-date quotatio algorithm. We also aalyze the differeces betwee this decetralized model ad the cetralized model through computatioal aalysis i the ext sectio. The Model I this decetralized supply chai model, we assume that the two parties work idepedetly from each other ad aim to miimize their ow costs. However, the maufacturer has full iformatio about the processes at the supplier as well as his ow processes. Sice the supplier works idepedetly from the maufacturer ad tries to miimize his ow costs, the results from the sigle facility case applies for the supplier to determie the optimal ivetory levels at his facility ad to sequece the orders to miimize his ow completio times. The, for that sequece ad ivetories at the supplier, we fid the optimal ivetory levels ad desig a effective schedulig ad lead time quotatio algorithm for the maufacturer.

Aalysis ad Results Sice the supplier is idepedet from the maufacturer, he operates exactly the same way as the sigle facility explaied i the previous sectio. Thus, the supplier s objective is to miimize k i=1 {hs i E[Is i ] + cd i E[ds i ] + ct i E[W i s d s i ]+ }. Sice the supplier works idepedetly from the maufacturer, he uses the schedulig ad lead time quotatio algorithm as explaied i the sigle facility case ad the results i sectio 3.2 holds for the supplier. Thus, the optimal ivetory levels for class i jobs at the supplier, Ri s, are the miimum x 0 that satisfy c i Fi s (x) c i + h s i Usig the relatios give i sectio 3.2, the optimal ivetory levels for the supplier ca be obtaied. Corollary 7. The optimal ivetory levels for the supplier i the decetralized, full iformatio case will be greater tha or equal to the optimal level i the cetralized case if the same schedules are used for both cases. Assumig that the supplier uses the optimal ivetory levels for his facility, the maufacturer tries to miimize his ow costs for those fixed R s values. So, we try to miimize the objective fuctio: k {h m i E[Ii m ] + c i E[Ni m ]} i=1 Usig the defiitios 3.4 ad writig them explicitly, for fixed R s, we will get the objective fuctio for the maufacturer to miimize as = + Z(R s, R m ) = K i=1 {h m i R s i 1 y s i =0 k {h m i E[Ii m ] + c i E[Ni m ]} i=1 R m i y m i =0 (R m i y m i ) P (x s i = yi s, x m i = yi m ) R s i +Rm i R s i +Rm i ys i y s i =Rs i y m i =0 (R s i + R m i y s i y m i ) P (x s i = yi s, x m i = yi m ) R s i 1 + c i [ (yi m Ri m )P (x s i = yi s, x m i = yi m ) + yi s=0 yi m=rm i yi s=rs i yi m=rs i +Rm i ys i (y s i + y m i R s i R m i ) P (x s i = y s i, x m i = y m i )]} (3.11) Observe that both the supplier ad the maufacturer s productio queue operates idepedet of R m for fixed R s. The, for fixed ivetory levels Ri s for each class i at the supplier, the optimal levels of ivetory Ri m for the maufacturer ca be foud by Theorem 7. Also, the optimal ivetory levels for a maufacturer workig with a pure MTO supplier satisfy corollary 5. Corollary 8. The optimal ivetory level for the maufacturer i the decetralized, full iformatio case will be less tha or equal to the optimal level i the cetralized case if the same schedules are used both cases. Corollary 9. Usig a MTO strategy for class i jobs at the maufacturer is optimal iff P (x s i R s i, x m i c i 0) c i + h m i (3.12) where Ri s is the optimal ivetory level for class i at the supplier. Sice the maufacturer is workig idepedetly from the supplier, he also uses a SPTA schedule accordig to his ow processig times to sequece his jobs ad employs a lead time quotatio algorithm similar to the cetralized model sice he has full iformatio about the supplier. So, we use the followig schedulig ad lead time quotatio algorithm SP T A LT Q DF I for the maufacturer which satisfies E[d i ] = E[W i ]. For the lead time quotatio algorithm LT Q DF I preseted above, we preset the followig lemma. Lemma 8. Cosider a series of radomly geerated problem istaces of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times at each facility be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. Also, let Z SP T A = i=1 cd C i deote the total weighted delivery times of orders with the SP T A schedule where C i = r i + W i is the delivery time of order i ad Z SP T A LT Q DF I = i=1 {cd d i +ct (C i d i )+ } deote the total due-date plus tardiess costs with the algorithm SP T A LT Q DF I where d i = r i +d i is the quoted duedate. The, the lead time quotatio algorithm LT Q DF I is asymptotically optimal to miimize the objective fuctio for this decetralized system with full iformatio assumig that SPTA schedule is used for sequecig jobs, that is almost surely, lim Z SP T A LT Q DF I Z SP T A Z SP T A = 0 I this case, the supplier is workig as a sigle facility ad usig SPTA rule w.r.t. his ow processig times. So, the statioary distributios of the umber of jobs at the supplier ca be foud usig the pgf for a sigle facility queue. The maufacturer is also schedulig his jobs accordig to SPTA. We agai use the decompositio approach

ALGORITHM 9: SPTA-LTQ DF I Schedulig: Schedule the jobs usig SPTA rule accordig to p m. lead time Quotatio: 0 if Ii m > 0 at r i d m i = E[p m i ] + tmm i + slacki m if Ii m = 0, Ii s > 0 at r i where slacki m = tmm d s i + E[pm i ] + M i m + slacki m otherwise i λpr{p m <p m i }E{pm p m <p m i } 1 λpr{p m <p m i }E{pm p m <p m i } where d s i = E[ps i ] + E[M i s] + slacks i slacki s = E[M s i ]λpr{ps <p s i }E{ps p s <p s i } 1 λpr{p s <p s i }E{ps p s <p s i } Mi m = max{t ms i + t mm i + sli m d s i, 0} t ms i = j A E[pm j j A] where A=set of jobs of class k at supplier queue that will be set to maufacturer s.t. E[p m k ] < E[pm i ] ad E[ps k ] < E[ps i ]. t mm i = j B E[pm j j B] where B=set of jobs at maufacturer queue at time r i with E[p m ] < E[p m i ]. sli m = (slacki s + E[M i s])λpr{ps < p s i, pm < p m i }E{pm p m < p m i } + j L mi{(slacks i + E[M i s])λ j, Ij s}e[pm j where L is the set of jobs of class k s.t. E[p m k ] < E[pm i ] ad E[ps k ] > E[ps i ]. slacki m = M m i λpr{pm <p m i }E{pm p m <p m i } 1 λpr{p m <p m i }E{pm p m <p m i } j L] to approximate the statioary distributio of the umber of jobs at the maufacturer s site, similar to the cetralized model ad assume that the arrivals to the maufacturer are expoetially distributed. Thus, the statioary distributio of the umber of jobs at the maufacturer ca also be foud by usig the pgf for a sigle facility queue. However, sice the maufacturer ad supplier have differet processig times ad thus differet schedules, their statioary distributios will also be differet although both are foud through the same pgf. 3.3.3. The Simple Decetralized Model: I may real-world systems, the supplier ad the maufacturer are idepedet firms ad they have very little iformatio or o iformatio at all about each other. The maufacturer is uaware of the processes at his supplier ad eeds to make his ow decisios without ay iformatio from the supplier. Most of the time, the maufacturer is oly aware of the average time it takes for the supplier to process a order ad sed it to him. I such a decetralized supply chai system, we fid the optimal ivetory levels for the maufacturer ad a effective way to schedule the orders ad to quote due-dates to the customers without ay iformatio from the supplier. The Model I this model, we agai assume that the maufacturer ad the supplier work idepedetly from each other, but distict from the full iformatio model, the maufacturer has o iformatio about the supplier, ad thus he ca t deduce the productio schedule or ivetory levels at the supplier. The supplier still behaves the same way as before ad the results obtaied i the previous sectio for the supplier still hold. However, i this case, the maufacturer oly kows the average time it takes for the supplier to deliver a job type i, deoted by E[d s i ]. Thus, for each job, the maufacturer acts as if each job of type i is goig to be delivered to him from the supplier after E[d s i ] time uits. Based o this assumptio, we determie the optimal ivetory levels for the maufacturer ad desig a effective schedulig ad lead time quotatio algorithm. Aalysis ad Results Sice the supplier acts as a sigle facility as i the previous sectio, the optimal ivetory levels for class i jobs, Ri s are the miimum values that satisfy Fi s (Ri s ) c i c i + h s i However, i this case, sice the maufacturer oly kows the average time it takes for a order of type i to be delivered by the supplier, whe a order arrives, he assumes that order will be delivered by the supplier to him after E[d s i ] time uits. Observe that if we model the supplier as a M/D/ queue with determiistic processig times E[d s i ] for type i ad without ay ivetories, each job of type i will take exactly E[d s i ] time uits to be delivered from the supplier to the maufacturer as assumed by the maufacturer i our system for this case ad we ca use the same aalysis ad the results i previous sectios for this model. Thus, the objective fuctio for the

maufacturer to miimize is = Z(R m ) = K {h m i [ i=1 +c i [ k {h m i E[Ii m ] + c i E[Ni m ]} i=1 R m i (Ri m yi s yi m )P (x s i + x m i = yi s + yi m )] yi s+ym i =0 (yi s + yi m Ri m )P (x s i + x m i = yi s + yi m )]} yi m+ys i =Rm i For this case, sice we model the supplier as a M/D/ queue with o ivetories, usig Corollary 5, the optimal levels of ivetory are the miimum Ri m values that satisfy: P (x s i + x m i Ri m c i ) c i + h m i We fid the statioary distributios of the jobs at the supplier usig the M/D/ queue model with processig times E[d s i ]. Sice we assume that each order is delivered to the maufacturer at time r i +E[d s i ] by the supplier, the processes at the supplier does t effect the arrival process to the maufacturer ad the distributio of the iterarrival time of jobs to the maufacturer follows the same expoetial distributio of the iterarrival time of jobs to the system. Thus, the arrival process of the jobs to the maufacturer is also poisso. Both the supplier ad maufacturer are workig as a sigle facility i this case, ad thus usig the relatios for a sigle facility queue, we ca fid the statioary distributios of the umber of jobs ad the ivetory levels for the maufacturer ad the supplier. For this case, sice the maufacturer focuses oly o his idividual objective, he uses a SPTA schedule accordig to his ow processig times. We assume that the maufacturer has o iformatio about the supplier ad is uaware of the schedule or ivetory levels there, but that from his previous experiece, he ca deduce a average delivery time for each class of products. So, he uses this piece of iformatio to quote due dates to his customers. Thus, we use the followig schedulig ad lead time quotatio algorithm SP T A LT Q SD for this simple decetralized model: For the lead time quotatio algorithm LT Q SD preseted above, we preset the followig lemma. Lemma 9. Cosider a series of radomly geerated problem istaces of size. Let iterarrival times be i.i.d. radom variables bouded above by some costat; the processig times at the maufacturer be also i.i.d radom variables ad bouded ad the processig times ad iterarrival times be idepedet of each other. Also, let Z SP T A = i=1 cd C i deote the total weighted delivery times of orders with the SP T A schedule where C i = r i +W i is the delivery time of order i ad Z SP T A LT Q SD = i=1 {cd d i + ct (C i d i )+ } deote the total due-date plus tardiess costs with the algorithm SP T A LT Q SD where d i = r i + d i is the quoted duedate. The, the lead time quotatio algorithm LT Q SD is asymptotically optimal to miimize this objective fuctio for this simple decetralized system assumig that SPTA schedule is used for sequecig jobs, that is almost surely, lim Z SP T A LT Q SD Z SP T A Z SP T A = 0 3.4. Computatioal Aalysis: Usig the ivetory values determied by the results i the previous sectios ad the schedulig ad lead time quotatio algorithms, we desiged several computatioal experimets to assess the effectiveess of our algorithms, how the combied MTS/MTO systems differ from pure MTO or MTS systems ad the differeces betwee cetralized ad decetralized supply chais. For the cetralized system, we assume that both parties cooperate ad use the heuristics explaied i sectio 3.3.1 for fidig the ivetory levels, schedulig ad lead time quotatio. For the decetralized systems, we assume that each facility acts idepedetly from each other ad use the heuristics that will miimize their ow costs as explaied i sectios 3.3.2 ad 3.3.3 for fidig the ivetory levels, schedulig ad lead time quotatio. We implemet our heuristics usig C++ ad preset the results i the followig sectios. 3.4.1. Effectiveess of a Combied System ad Efficiecy of Heuristics for a Sigle Facility: First, we cosider a sigle facility system usig = 1000 jobs for each simulatio. The, for each of these job istaces, the ratios of the total costs Z = k i=1 {h ie[i i ]+ c d i E[d i] + c T i E[W i d i ] + }, o average, are displayed i Table 4 where E[I i ] deotes the average ivetory, E[d i ] deotes the average quoted lead time ad E[W i d i ] + deotes the average tardiess i a simulatio ru for product type i. We cosider differet scearios regardig differet umber of job types k ad differet multipliers h, c d ad c T for cost fuctios to evaluate the effect of these parameters o our system. For each of these cases, we desiged 10 scearios with differet arrival ad processig rates, each havig expoetial distributios. Each value i Table 4 deotes the average of these 10 scearios. We preset the compariso of combied MTO-MTS system with pure systems ad with differet schedules for a sigle facility usig c d = 2 ad differet combiatios of h, c T ad k i Table 4. The first ad secod colums cotai the compariso of the combied MTS/MTO system usig SPTA-LTQ algorithm with pure MTS ad MTO systems, respectively. Observe that the combied system, o average, provides a 20% decrease i costs as opposed to the pure MTS systems ad a 15% decrease as opposed to the pure MTO systems. Also, i the third colum, we compare the costs with the SPTA- LTQ algorithm for this combied system with a relevat

ALGORITHM 10: SPTA-LTQ SD Schedulig: Process the jobs accordig to shortest processig time at the maufacturer. lead time Quotatio: 0 if Ii m > 0 at r i d i = E[p m i ] + tmm i + slacki m if Ii m = 0, Ii s > 0 at r i where slacki m = tmm E[d s i ] + E[pm i ] + M i m + slacki m otherwise where E[d s i ] is the average delivery time of class i products from the supplier. Mi m = max{t mm i + E[ds i ]pr{pm <p m i }E{pm p m <p m i } E[p s ] E[d s i ], 0} slacki m = M m i λpr{pm <p m i }E{pm p m <p m i } 1 λpr{p m <p m i }E{pm p m <p m i } } i λpr{p m <p m i }E{pm p m <p m i } 1 λpr{p m <p m i }E{pm p m <p m i } lead time quotatio algorithm for a FCFS schedule. We see that SPTA-LTQ algorithm decreases the costs about 15% o average as opposed to a FCFS-LTQ algorithm. We also see the effect of the parameters o the system performace i Table 4. Usurprisigly, as the uit ivetory holdig cost icreases while all other parameters remai costat, the combied system moves toward a pure MTO system ad gives much better results tha pure MTS systems sice holdig ivetory becomes much more costly as h icreases. The system is effected the same way as the uit due-date cost, c d, decreases because i that case lead times become less importat ad a MTO system becomes much more attractive as c d decreases. We also see that SPTA-LTQ algorithm gives much better results tha FCFS-LTQ algorithm, as h icreases (alteratively c d decreases) because the schedule has a importat effect o completio times, thus duedates, i MTO systems, while for MTS systems, sice the orders are satisfied from the ivetory, miimizig the completio times of the jobs is ot so importat. O the other had, as the uit tardiess cost, c T, icreases while all other parameters remai costat, MTO systems give worse results ad pure MTS systems become much more attractive. Also, we see that the performace differece betwee SPTA-LTQ algorithm ad FCFS-LTQ algorithm becomes less ad less as c T icreases ad FCFS-LTQ algorithm start to give better results as c T becomes very high. This happes because, with FCFS-LTQ algorithm, the future arrivals do t effect the completio times of the jobs that have already arrived, thus we ca quote the lead times exactly as the completio times of the jobs ad there will be o tardiess. However, with SPTA-LTQ algorithm, there will be some tardiess cost ad as c T icreases, the cost of tardiess overcomes the gais i total completio times with SPTA-LTQ algorithm ad the total costs with FCFS-LTQ becomes lower tha the total costs with SPTA-LTQ algorithm. 3.4.2. Effect of Ivetory Decisios o Supply Chais: We ext explore the effect of ivetory decisios o our system without cosiderig the lead time quotatio. So, for a fixed schedule, we compare the objective fuctios Z = k i=1 {hs i E[Is i ] + hm i E[Im i ] + c d i E[W i]} to explore oly the effect of ivetory decisios i this system uaffected by lead time quotatio. Recall that we made some assumptios about the iteractio betwee the supplier ad the maufacturer regardig the statioary distributios of the umber of jobs at the maufacturer ad we also assumed that ivetory values of differet types at the supplier do t effect the statioary distributios of other types at the maufacturer. We explore how these assumptios effect the optimal ivetory levels ad the objective fuctio with this computatioal study. We also compare the cetralized ad decetralized versios of the combied MTS/MTO supply chai with this objective fuctio. Table 5 shows the ratios of objective fuctios Z for differet cases usig c d = 2 ad differet combiatios of h s, h m ad k. To explore the effectiveess of our algorithms, we compare the objective fuctio usig our ivetory levels for the cetralized model with the miimum objective fuctio for that case. I our simulatios, we fid the miimum objective fuctios by tryig all the possible combiatios of ivetories of each type for the supplier ad the maufacturer ad selectig the oe that gives the best objective value. We use these miimum objective values as lower bouds i our simulatios. However, this process takes a very log time, especially whe the umber of types are big. Although we ca fid the optimal ivetories by tryig all possible solutios for the cases we aalyzed i this experimet sice they are relatively of small size, ote that this method becomes almost impossible to apply as the problem size gets bigger ad bigger. For example cosider the case whe there are 10 differet jobs ad the ivetory of each job type at the supplier ad the maufacturer ca take a value from 0 to 5. I that case we eed to evaluate (5 5) 10 differet possible solutios for the trial ad error method ad if each of them takes a aosecod(10 9 secods), the whole process takes more tha a day ad becomes almost impossible to apply for eve bigger problems. However,

Table 4: Compariso of combied MTO-MTS system with pure systems ad with differet schedules for a sigle facility h=0.5 c T =2.5 Z SP T A LT Q Z F CF S LT Q h=1 c T =2.1 Z MT O MT S Z MT S Z MT O MT S Z MT O Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q k=3 0.963 0.526 0.942 k=3 0.829 0.880 0.894 k=5 0.972 0.613 0.973 k=5 0.798 0.844 0.850 k=10 0.947 0.734 0.926 k=10 0.778 0.831 0.724 h=1 c T =2.5 Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q h=1 c T =3 Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q k=3 0.832 0.906 0.913 k=3 0.847 0.873 0.922 k=5 0.814 0.827 0.897 k=5 0.825 0.829 0.908 k=10 0.795 0.872 0.762 k=10 0.808 0.820 0.795 h=2 c T =2.5 Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q h=1 c T =5 Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q k=3 0.724 0.936 0.846 k=3 0.880 0.845 0.931 k=5 0.719 0.934 0.823 k=5 0.896 0.796 0.911 k=10 0.683 0.967 0.695 k=10 0.913 0.753 0.924 h=5 c T =2.5 Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q h=1 c T =10 Z MT O MT S Z MT S Z MT O MT S Z MT O Z SP T A LT Q Z F CF S LT Q k=3 0.573 1 0.837 k=3 0.925 0.743 1.032 k=5 0.551 1 0.776 k=5 0.946 0.719 1.131 k=10 0.472 0.987 0.672 k=10 0.953 0.651 1.096 average 0.754 0.858 0.838 0.866 0.799 0.926 with our heuristic, there is o iteractio betwee the differet types ad we cosider them separately. Also, we oly do a oe-dimesioal search over the supplier s ivetory levels up to a upper boud. Thus, for the same case, we oly eed to evaluate 5 10 possible solutios at the worst case which ca be doe immediately ad ca be easily applied to problems of much bigger size. We costruct our lower boud by usig the ivetory values foud by the trial ad error method that gives the miimum objective value ad compare it with the objective value obtaied by usig the ivetory values foud by our heuristic. The first colum i Table 5 compares these two objective fuctios for the cetralized supply chai model. We see that the ivetory values foud by our heuristic are very close to the optimal ivetory values ad there is oly a 5% differece, o average, betwee the miimum costs ad the costs obtaied by usig our ivetory values. The differece is maily due to our assumptio that havig ivetory of type i at the supplier do t effect the statioary distributios of the umber of jobs at the maufacturer which is t the case i reality. Whe we compare the cetralized ad decetralized models, we see that the costs with the cetralized model is, o average, 10% less tha the decetralized model with full iformatio. The cost savigs due to ivetory decisios icrease to more tha 15% whe we compare the cetralized model with the simple decetralized case. We also see that, without cetralizatio, if the maufacturer has full iformatio about the supplier, he ca decrease the costs by about 7% by oly adjustig his ow ivetory levels without chagig aythig else. We see that if the maufacturer had cotrol over the supplier, he could cut his costs sigificatly. However, eve if the maufacturer did t have cotrol over the supplier but had full iformatio about the whole system, he still ca cut his costs ad icrease his profits. We ca also see the effects of the parameters h s, h m, c d ad k o the system performace. We see that as the ivetory holdig cost at the supplier, h s, icreases while everythig else remais the same, our heuristic gives closer results to the lower boud. A ituitive explaatio for this is; as h s icreases, the optimal ivetory levels at the supplier ad their effect o the maufacturer decreases as we assumed i our approximatio. Besides, if the supplier uses a pure MTO strategy, the our heuristic fids the optimal solutio sice our approximatios become exact for a pure MTO supplier. The same effect occurs as h m decreases because i that case it would be better to carry ivetories at the maufacturer istead of the supplier ad the ivetory levels at the supplier ad their effect o the maufacturer decreases as above. Similarly, as c d decreases, the ivetory levels at the supplier decreases, too ad our heuristic gives closer results to the optimal solutio. Whe we compare the supply chai models, we see that as h s decreases, decetralized models give closer results to the cetralized oe. This is because as h s decreases, the optimal ivetory levels at the supplier for the cetralized model become close to the upper boud we preseted i Theorem 9 which is also the optimal ivetory level for the supplier i the decetralized models assumig that the same schedule is used i both cases. Thus, as h s decreases, the ivetory levels at the supplier, thus the ivetory levels at the maufacturer for the cetralized ad decetralized models become close to each other ad the objective values with the decetral-

ized models become closer to the cetralized model objective value. We ca see the same effect as h m icreases, because i the cetralized model as h m icreases, the ivetory levels at the maufacturer decreases ad the ivetory levels at the supplier icreases becomig close to the upper boud. Thus, as explaied above, the decetralized models give closer results to the cetralized oe. 3.4.3. Effectiveess of a Combied System ad Efficiecy of Heuristics for Supply Chais: We the complete a similar study which icludes lead time quotatio, ad compare the objective fuctios Z = k i=1 {hs i E[Is i ]+hm i E[Im i ]+cd i E[d i]+c T i E[W i d i ] + }. We use the multipliers h s = 0.5, h m = 1, c d = 2 ad c T = 2.5 for this case. Through this aalysis, we compare the costs for the whole system for the cetralized ad decetralized models. I the first colum i Table 6, we compare the total costs for the cetralized model usig the algorithm SP T A p LT Q C with the algorithm that schedules the jobs accordig to FCFS i both systems ad quotes lead times similar to our heuristic. We coclude that the schedule used to produce the jobs has a importat effect o the total costs ad we see that, o average, our heuristic performs about 20% better tha the commoly used schedule FCFS i the idustry. Observe from Table 4 that this was also the case for the sigle facility case. Also, whe we compare the secod colums of both tables, we see that havig full iformatio about the supplier helps the maufacturer a lot to quote better lead times ad there is very little differece betwee costs due to lead time quotatio for cetralized ad decetralized with full iformatio cases. The differece betwee costs is maily because of ivetory decisios i this compariso. However, whe we compare the simple decetralized model with the cetralized model, we see that the cost differece icreases sigificatly. Sice the maufacturer has very little iformatio about the supplier, he ca o loger quote reliable due-dates ad the costs icrease dramatically ad the costs with the simple decetralized model are about 40% worse tha the cetralized model. Observe that, although there were t much differece betwee the decetralized model with full iformatio ad the simple decetralized model i Table 5, i Table 6, this differece icreases sigificatly to 30% maily due to lead time quotatio. We coclude that havig full iformatio about the supplier is critical to the maufacturer for lead time quotatio. I additio, the ivetory costs ca be decreased dramatically if the maufacturer has complete cotrol over the supplier i additio to the full iformatio case. 4. Complex Supply Chai Networks: I this sectio, we cosider more complex supply chai etworks composed of several facilities with differet classes of relatioships. Utilizig from the results from the previous sectios, we desig effective heuristics for determiig ivetory levels, sequecig the orders at each facility ad quotig reliable ad short lead times to customers for these complex etworks. We also complete extesive computatioal experimets to evaluate the effectiveess of our algorithms ad to aalyze the beefits of the combied MTO-MTS systems versus pure MTO or MTS systems. 4.1. Model: We model a multi-facility supply chai that is composed of a mai maufacturer ad its iteral ad exteral suppliers. The maufacturer receives orders from the customers ad delivers it immediately if it has that product i its ivetory or quotes a due date to the customer if it does t. We assume that there are a total of N facilities i the supply chai ad there are K types of differet products this firm offers to its customers, each with a differet demad rate. The maufacturer wats to quote short ad reliable due-dates ad does t wat to keep too much ivetory either. Thus, we wat to miimize the objective fuctio K i=1 {( N j=1 h ije[i ij ]) + c d i E[d i] + c T i E[W i d i ] + } composed of the ivetory costs, lead time costs ad tardiess costs. If the processig times are short ad the total time required to produce a product is less tha the desired lead time, the the compay does t eed to keep ay ivetory ad uses a MTO approach. However, if the total processig time is more tha the desired lead time, due to the stochastic ature of demad ad lead times, the compay has to start processig before the orders actually arrive ad keep some work-i-process or fiished goods ivetory to achieve the desired lead time. Istead of keepig oly fiished goods ivetory at the maufacturer, we ca choose to store ivetory at other facilities ad use a MTO approach at others. We aim to fid the optimal locatios to store ivetory i this chai to miimize the system-wide ivetory costs ad desig effective algorithms to quote short ad reliable lead times. We cosider a supply chai like i Figure 1 where the same firm ows some of the facilities at differet locatios draw with rectagles ad also has idepedet outside suppliers draw with circles. The compay has total cotrol over its ow facilities; however, it has o cotrol over outside suppliers. We ca also thik of the iteral suppliers as differet machies i a facility where a sigle decisio aget makes all of the decisios ad the exteral suppliers as the suppliers of this compay which this decisio aget ca ot cotrol. The decisio aget ca choose to stock some WIP ivetory i this system istead of choosig to stock oly the fiished goods ivetory. Sice the demad is stochastic, the firm eeds to carry some ivetory to respod to customer orders i a short time. We assume that a iitial ivetory amout (i.e. safety stocks) of x ij for raw materials ad y i for fiished

Table 5: Effect of ivetory decisios o the cetralized ad decetralized systems h s =0.5 h m =2 Z LB Z Ce Z Ce Z DF I Z Ce Z SD Z DF I Z SD h s =0.5 h m =0.6 Z LB Z Ce Z Ce Z DF I Z Ce Z SD Z DF I Z SD k=3 0.916 0.963 0.896 0.930 k=3 0.964 0.834 0.794 0.952 k=5 0.933 0.925 0.849 0.917 k=5 0.959 0.877 0.808 0.921 k=10 0.929 0.923 0.832 0.901 k=10 0.967 0.892 0.836 0.937 h s =1 h m =2 Z LB Z Ce Z Ce Z DF I Z Ce Z SD Z DF I Z SD h s =0.5 h m =1 Z LB Z Ce Z Ce Z DF I Z Ce Z SD Z DF I Z SD k=3 0.921 0.935 0.868 0.928 k=3 0.943 0.945 0.890 0.942 k=5 0.949 0.914 0.855 0.935 k=5 0.909 0.856 0.788 0.920 k=10 0.952 0.922 0.862 0.934 k=10 0.918 0.875 0.808 0.923 h s =1.9 h m =2 Z LB Z Ce Z Ce Z DF I Z Ce Z SD Z DF I Z SD h s =0.5 h m =5 Z LB Z Ce Z Ce Z DF I Z Ce Z SD Z DF I Z SD k=3 0.969 0.856 0.802 0.937 k=3 0.922 0.961 0.881 0.917 k=5 1 0.839 0.784 0.934 k=5 0.911 0.975 0.874 0.896 k=10 1 0.803 0.771 0.960 k=10 0.936 0.953 0.841 0.882 average 0.952 0.898 0.835 0.931 0.939 0.908 0.836 0.921 Table 6: Compariso of cetralized ad decetralized supply chais for combied MTO-MTS system Z SP T Ap LT Q C /Z F CF S LT Q Z Ce /Z DF I Z Ce /Z SD Z DF I /Z SD k=3 0.859 0.954 0.679 0.712 k=5 0.775 0.857 0.575 0.671 k=10 0.682 0.893 0.584 0.654 Average 0.772 0.902 0.613 0.680 Customer Figure 1: Supply Chai Structure

goods are held at facility i. A customer order triggers a productio order i the system ad the orders have to wait a lead time depedig o the amout of iitial ivetories. We desire short lead times without carryig too much ivetory, so we aim to miimize a cost fuctio: Z = K N {( h ij E[I ij ]) + c d i E[d i ] + c T i E[W i d i ] + } i=1 j=1 that is composed of the lead times ad the safety stocks i the whole system. Each order k requires a processig time of p ik at facility i ad a trasshipmet time t ijk to move from facility i to j. Also for each idividual order l, w il deotes the waitig time of order l at facility i, i.e. w il is the time startig with the arrival of the required parts for order l to facility i ad edig with the processig of that order at facility i. Note that if there is o queue at facility i whe order l receives to facility i, w il = p ik. Otherwise, w il = p ik + {waitig time i queue of facility i}. Also, we assume that each facility quotes a due-date f il for each order to its dowstream member ad evetually the maufacturer to its customers. For this system, we eed to decide o a schedulig rule ad a due-date quotatio algorithm as well as how much ivetory to store at each of the facilities. The objective of our problem is to determie the optimal ivetory levels, a sequece of jobs ad a set of due dates such that the total cost K i=1 { N j=1 h ije[i ij ] + c d i E[d i] + ci T E[W i d i ] + } is miimized. Clearly, to optimize this expressio, we eed to coordiate due date quotatio, sequecig ad ivetory maagemet ad a optimal solutio to this model would require simultaeous cosideratio of these three issues. However, the approach we have elected to follow for this model (ad throughout this project) is slightly differet. Observe that i a optimal offlie solutio to this model, lead times would equal actual waitig times of jobs i the system. Thus, the problem becomes equivalet to miimizig K j=1 h ije[i ij ])+c d i E[W i]}. Of course, i=1 {( N i a olie schedule, it is impossible to both miimize this fuctio ad set due dates equal to completio times, sice due dates are assiged without kowledge of future arrivals, some of which may have to complete before jobs that have already arrived i order to miimize the sum of completio times. I this approach, we first determie a schedulig approach desiged to effectively miimize the sum of completio times, ad the based o that schedule, we fid the optimal ivetory levels to miimize K i=1 {( N j=1 h ije[i ij ])+c d i E[W i]} ad the with these ivetories, we desig a due date quotatio approach that presets due dates that are geerally close to the completio times suggested by our schedulig approach. 4.2. Aalysis ad Results: For this model, we use the same ideas for schedulig ad lead time quotatio as we use i previous sectios ad adjust them for this more complex situatio. However, whe we first attempted to use the same aalysis that we did i sectio 3 to fid the optimal ivetory levels at each of the facilities, we see that the model becomes very complex ad difficult to aalyze. Thus, we develop aother approach to approximate the optimal ivetory levels that should be stored at each facility for each product type for this system. We explai our approach to fid the ivetory levels as well as the algorithms we use for schedulig ad lead time quotatio i detail i the followig sectios. 4.2.1. Sigle Product Type Model: I this sectio, we assume that the firm produces oly oe type of product ad that they use FCFS schedulig rule at all facilities sice there is o differece betwee orders. To fid the optimal ivetory levels at each of the facilities, assumig x p, we use the relatio l = p x at each facility where p is the time uits required to process a order, l is the lead time ad x is the legth of time that the safety stock that we carry is worth for a specified service level. Every time a customer order arrives, we start to produce oe item to repleish ivetory or to satisfy future orders. The, the demad for the first x time uits is satisfied from the ivetory ad sice the first order fiishes processig at time p, after the ivetory is depleted at time x, the first order that arrives, ca oly be satisfied at time p causig a lead time l = p x. Note that this is a coservative approach ad overestimates the amout of ivetory that we eed to carry. I reality, the same lead time might be achieved by carryig much less ivetory but holdig this amout of ivetory esures that the specified service level is achieved for this lead time ad every order is satisfied with a lead time less tha l for sure at this service level. We defie the parameters ad variables ad write the LP formulatio of the model below: Parameters: i = subscript used to describe the facilities i the supply chai startig with 1 deotig the maufacturer. v i =uit value of parts produced at facility i p i = processig time of parts at facility i t ij = trasshipmet time to move parts from facility i to j h 1 ij = uit raw material ivetory cost at facility i for the parts produced at facility j h 2 i = uit fiished part ivetory cost at facility i. c d = cost of a uit icrease i respose time to orders. S = set of facilities that belog to the firm E = set of exteral suppliers P i = set of facilities that are immediate predecessors of facility i f i for i E = committed respose time of orders from exteral supplier i E

Variables: x ij = uits of time that the safety stock of raw materials received from facility j ad stored at facility i to achieve the desired service level. y i = uits of time that the fiished parts stored at facility i that achieves the desired service level. w i = waitig time of orders at facility i, i.e. w i is the average time startig with the arrival of the required parts for the productio of a order at facility i ad edig with the processig of that order at facility i. Note that if there is o queue at facility i whe a order receives to facility i, w i = p i. Otherwise, w i = p i + {waitig time i queue of facility i}. f i for i S = total lead time of orders up to facility i S, i.e lead time betwee the arrival of a customer to the system ad the completio of the parts required for that order at facility i. Mi N N [ {h 1 ijx ij } + h 2 i y i ] + c d f 1 i=1 j=1 s.t. max{max i Pj {max{f i + t ij x ij, 0}} +w i y i, 0} f j for j S All variables 0 (4.1) I this system, if we assume that there is o cogestio or queues i the system (e.g. there are assembly lies or ifiite servers at all the facilities ad each order is put o the assembly lie ad starts processig as soo as the required materials for that order receives to that facility), there will be o queues ad we use oly the processig times as the waitig times at the facilities. However, if we assume that there is a sigle server (or M < servers) at a facility, the there will be cogestio ad queues i the system. I that case, we approximate the waitig time of orders at facility i by usig the expected waitig time of a job i that queue with arrival rate D i ad mea processig time p i. We use E[W i ] istead of p i ad use these values i the LP formulatio. Also, if we assume that the maufacturer has a committed respose time to its customers, which is geerally the case i competitive markets, we fix the committed respose time ad try to miimize the total ivetory costs with the above LP model with the fixed lead time as a parameter istead of a variable. For due-date quotatio, observe that if there is ivetory at a facility, the compoets are immediately satisfied from that facility ad the order does t have to wait ay time for the processig at or before that facility. If the waitig time of a order at a facility is costat ad is ot effected by the other orders (e.g. a assembly lie system or a ifiite server queue system), the the due-dates are quoted by the followig algorithm. However, if there are queues i the system, the the lead time of a order is effected by the other orders ad the state of the system at time r k, the arrival time of order k to the system. For this model, sice there is oly oe product ad a FCFS schedulig rule is used, we do t eed to cosider future arrivals ad we ca quote the due dates by oly cosiderig the curret state of the system at the time of arrival. A order has to wait oly for the processig of orders that are already at the queue of a facility. Let wi k deote the waitig time of a order k at facility i.(note that wi k = p i for the assembly lie system) ad Ui k deote the set of orders that arrived to the system before r k but ot yet fiished processig at facility i. The, the due-date for a order k is quoted accordig to the followig algorithm. 4.2.2. Multiple Product Types Model: I this sectio, we assume that the firm produces multiple products with differet characteristics, (i.e differet processig times, arrival rates, ad supply chai architecture). I this case, i additio to decidig o ivetory levels ad due-date quotatio, we also desig a effective schedulig algorithm to process the jobs at each facility sice products have differet characteristics ad the sequece of jobs will effect the completio times ad respose times to customers. For the multiple product model, agai if we assume that there is o cogestio or queues i the system (e.g. there are assembly lies at all the facilities ad each order is put o the assembly lie ad starts processig as soo as the required materials for that order receives to that facility), the the model becomes exactly the same as the sigle product case because the differet orders ad product types do t effect each other ad do t cause ay cogestio i the system. I this case, we use the same LP model 4.1 to decide o the ivetory amouts ad the due-date quotatio algorithm 11 to quote due-dates. There is o eed for a schedulig algorithm sice each arrivig job is immediately placed uder process without waitig i this system. However, if we assume that there is a sigle server (or M < servers at a facility), the there will be cogestio ad queues i the system. I that case, we approximate the waitig time of orders at facility i by usig a similar approach as i the sigle product model. For the schedule used at facility i, we fid the expected waitig time of a order type l i that queue with arrival rate Di l ad mea processig time pl i. I this case, the expected waitig time for type l, E[Wi l ], depeds o the schedule used i that facility ad other products. We use wi l = E[W i l ] as the waitig time of jobs of type l at facility i ad use these values i the LP formulatio 4.1. For this case, we also use the same idea for due-date quotatio as i Algorithm 12. However, i this case, the schedulig algorithm does t have to be FCFS ad the facilities might choose other sequeces to miimize total completio times. Although the problem of miimizig completio times at a sigle facility is NP-Hard, Kami-

ALGORITHM 11: Step 1: At the time of arrival of a order k to the system, deoted by r k, form a ew subgraph, G, of the supply chai cosiderig the ivetories at time r k. Startig from the maufacturer, do a depth-first search o the graph ad add a facility to the subgraph if there is o ivetory at had at that facility. If there is ivetory at a facility do ot add that to the subgraph ad do ot go further from that ode. Step 2: Let the legth betwee two facilities i ad j be l ij = p i + t ij. Step 3: Add a ode 0 at the ed of the subgraph ad coect it with all the facilities that do t have a predecessor ad let l 0j = 0. Usig these l ij values, fid the logest path from ode 0 to the maufacturer. Step 4: Set the legth of the logest path as the lead time for that order. ALGORITHM 12: Step 1: Set d k i = 0 if there is ivetory of type l at facility i ad d i = f i if i E. Step 2: Form the subgraph G as i Algorithm 11 ad put a facility i ito set F if facility i G ad i S. Set F deotes the set of facilities that belog to the firm ad have t bee quoted a due-date yet. Step 3: If facility i F ad j / F for j P i, the set due-date for order k at facility i as below ad delete facility i from set F. d k i = max j P i {d k j + tk ji } + wk i w k i = max{sum j U k i pj i max j P i {d k j + tk ji }, 0} + pk i Step 4: Stop if set F is empty ad set d k 1 as the lead time for order k. Retur to step 3 otherwise. sky ad Simchi-Levi [18] shows that the SPTA rule is asymptotically optimal for this problem. Uder the SPTA heuristic, each time a job completes processig, the shortest available job which has yet ot bee processed is selected for processig. Also, ote that this approach to sequecig does ot take quoted due date ito accout, ad is thus easily implemeted. We cosider FCFS ad SPTA schedulig algorithms to use at the facilities. If FCFS schedule were used at all facilities, the the due-date quotatio algorithm would be exactly the same as Algorithm 12. However, if SPTA schedule is used at a facility, the we cosider future arrivals, too, i additio to the curret state of the system because future arrivals might be scheduled before previous orders ad icrease their delivery times. Xia, Shatikumar ad Gly [34] ad Kamisky ad Simchi-Levi [21] idepedetly proved that for a flow shop model with machies, if the processig times of a job o each of the machies are idepedet ad exchageable, processig the jobs accordig to the shortest total processig time p i = j=1 pj i at the first facility ad processig the jobs o a FCFS basis at the others is asymptotically optimal if all the release times are 0. We desig a schedulig algorithm for our system based o this result. For each product type, we fid the logest path i the productio etwork of that product ad fid the total processig time of each product type by summig the processig types o this path. The, we schedule the jobs accordig to shortest total processig times at the facilities that are at the ed of the supply chai etwork ad use a FCFS schedule at the other facilities. We explai our schedulig algorithm with due-date quotatio below i algorithm 13. I this system let P r l deote the arrival probabilities for each product type l = 1..K with sum equal to 1. Let set M l deote the set of product types that are goig to be scheduled before product type l ad ψ l = k M l P r k is the probability that a arrivig job is goig to be scheduled before job type l. µ l i = k M l {P r k E[p k i ]} is the expected processig time of such a job at facility i ad λ is the mea iter-arrival time of orders. Also, Ni k deotes the umber of orders that arrived after order k but scheduled before it at facility i ad wi k deotes the waitig time of order k at facility i. Note that wi k = p i for the assembly lie system. The, the schedulig ad due-date quotatio algorithm for a order k is as below: 4.3. Computatioal Aalysis: We performed several computatioal experimets i order to evaluate the performace of the algorithms explaied above for differet cases, ad we also compared the objective of solutios based o our heuristics with the objective fuctios achieved usig traditioal methods, ad with a lower boud. We desig a complex supply chai etwork usig differet processig times, trasshipmet times betwee facilities, uit holdig costs ad uit waitig costs ad implemet our heuristics i C++. Wheever eeded, we solve the LP model 4.1 usig ILOG AMPL/CPLEX 7.0. Cosider a supply chai etwork for a sigle product type as show i figure 2. The meaigs of the umbers i figure 2 are explaied i figure 3. I this etwork, S i deotes the facilities that belog to the same firm ad E i deotes the exteral suppliers.

ALGORITHM 13: Schedulig: Step 1: Fid the total time required to process each product type l(i.e. the logest path from the suppliers at the ed of the chai to the maufacturer i the supply chai for product type l.) ad deote it by T l. Step 2: Defie set L to be set of facilities that have o iteral supplier. Schedule the jobs accordig to shortest T l at a facility i if i L ad use FCFS at the other facilities. lead time Quotatio: Step 3: Set Ni k = 0 for all i, d k i = 0 if there is fiished goods ivetory of type l at facility i ad d i = f i if i E. Step 4: Form the subgraph G as i Algorithm 11 ad put a facility i ito set F if facility i G ad i S. Set F deotes the set of facilities that belog to the firm ad have t bee quoted a due-date yet. Step 5: If facility i F ad j / F j i, the set due-date for order k at facility i as below ad delete facility i from set F. Let Ui k deote the set of orders that are already i the system at time r k ad is scheduled before job k but ot yet fiished processig at facility i If facility i L, the d k i = max j P i {d k j + tk ji } + pk i + wk i + slackk i wi k = max{sum j Ui kpj i max j P i {d k j + tk ji }, 0} { slacki k = mi{ wk i ψl µ l i, ( i)ψ λ ψ l µ l l µ l i } if λ ψ lµ l i > 0 i ( i)ψ l µ l i otherwise Ni k = slacki k/µl i If facility i / L d k i = max j P i {d k j + tk ji } + pk i + wk i wi k = max{sum j Ui kpj i + max j P i {N j }µ l i max j P i {d k j + tk ji }, 0} N i = max j Pi N j Step 6: Stop if set F is empty ad set d k 1 as the lead time to the customer. Retur to step 5 otherwise. S1 10 1 1 8 20 1.5 S3 6 1.8 4 E1 20 5 0.5 S2 15 1 3 1.5 1.2 S4 30 3 12 15 2 3.5 1.5 S6 10 8 E2 30 10 1 S5 15 1.2 Figure 2: Supply Chai Network Example F i t ij p i 2 h i h 1 ij F j p j Figure 3: Explaatio of the values

Table 7: Compariso of combied strategy with pure strategies for a assembly lie system Table 8: Compariso of combied strategy with pure strategies for a sigle server model h=8,c d =10 h=8,c d =5 h=4, c d =5 h=8,c d =10 h=8,c d =5 h=4, c d =5 Z MT O MT S /Z MT S 0.526 0.451 0.814 Z MT O MT S /Z MT S 0.581 0.515 0.842 Z MT O MT S /Z MT O 0.424 0.710 0.655 Z MT O MT S /Z MT O 0.492 0.769 0.677 4.3.1. Effect of Ivetory Positioig without Cogestio Effects i the System: We start by examiig the case where the waitig time of a job at facility i is determiistic ad equal to p i as i a assembly lie model or a ifiite server model with determiistic processig times. Sice, the demad is stochastic, the firm eeds to keep some safety stock to achieve the desired service level. I Table 7, we compare the optimal objective values of the formulatio 4.1 allowig to hold ivetory at every facility as opposed to holdig o ivetory at all or holdig oly fiished good ivetories. The ratios of the objective fuctio values of formulatio 4.1 with the combied strategy over the costs with pure MTS ad MTO strategies for differet combiatios of ivetory holdig cost at the last facility, h, ad uit lead time cost, c d are show i Table 7. Traditioally, firms either use a MTO strategy without keepig ay ivetory or a MTS strategy keepig oly fiished goods ivetory. However, as we show i Table 7, usig a combied strategy is much more beeficial for miimizig the cost fuctio N j=1 h je[i j ] + c d E[W ] where E[I j ] deotes the average ivetory at facility j ad E[W ] deotes the average waitig time of customers i the whole system. For example, for the case where the holdig costs are as show i figure 2 except that the holdig cost for fiished goods at the last facility is 4 (h = 4) ad c d = 5, with a pure MTS strategy, we eed to keep a fiished goods ivetory of 95 uits with a cost of 380. With a pure MTO strategy, the lead time will be 95 ad the cost is 475. However, with a combied strategy with y S1 = 10, y S2 = 15, y S4 = 12, y S6 = 40, x E1,S 2 = 25, x E2,S 5 = 40 x S1,S 4 = 20 x S2,S 4 = 3, the total cost will be 307.1. The cost with the combied strategy is sigificatly lower tha the costs with pure strategies. Cosider aother example i which the idustry lead time is 30 ad the firm tries to achieve this lead time by keepig some ivetory. If the firm oly keeps fiished goods ivetory, the y S6 = 65 ad the total ivetory cost is 260. However, by keepig ivetory at other facilities, with the same lead time, the total ivetory cost ca be decreased to 187.1 with y S1 = 10, y S2 = 15, y S6 = 22, x E1,S 2 = 25, x E2,S 5 = 28 x S1,S 4 = 20 x S2,S 4 = 3. I additio, the firm will be able to cut the lead time by half to 15 with a cost of 247.1 which is eve less tha the cost with the iitial strategy. 4.3.2. Effect of Ivetory Positioig with Cogestio Effects for Sigle Product Type Model: If we cosider a sigle server model for each facility with stochastic processig times, we fid the mea waitig time of a job at each facility ad use these values as the waitig times. I that case, the ratios of the objective values i formulatio 4.1 with the combied strategy over the pure strategies are show i Table 8. Note that these costs are foud through the objective fuctios i our LP formulatio which are calculated usig the maximum lead time for the specified service level for all jobs ad usig the correspodig safety stock levels. However, i reality each job has a differet lead time depedig o the cogestio i the system ad the ivetory levels fluctuate due to the stochastic ature of demad ad productio rates. We simulate this stochastic system assumig that there is a sigle server at each facility with mea processig times ad other values as give i figure 2 ad mea iter-arrival time 50. We calculate the costs cosiderig the fluctuatios i the ivetory values throughout the simulatio ad usig the waitig time of each job i the system as the lead time which is differet for each job. We assume expoetial iter-arrival ad processig times at each facility idepedet from each other. Usig our heuristics, we make 10 rus for each of the simulatios with differet radom umber streams ad usig = 5000 jobs ad preset the compariso of objective values N j=1 h je[i j ]+c d E[d]+c T E[W d] + o average i the followig tables. For the sigle product type model, we start our simulatios with the iitial ivetory levels foud by our LP formulatio ad every time a customer order arrives to the system, a ew productio order is give at that time. We compare the objective fuctios N j=1 h je[i j ] + c d E[d]+c T E[W d] + with this combied model to pure MTO ad MTS models where both systems are operated as the combied system but with the iitial ivetories all 0 for the MTO model ad there is oly fiished goods ivetory for the MTS model. The ratios of the costs are i 9 for differet h ad c d combiatios with c T = 12. I this simulatio aalysis, to assess the effectiveess of our lead time quotatio algorithm, we also compare the lead times quoted for this sigle type system to the actual waitig times of the jobs i the system usig c d = 5 ad c T = 7. Let ZLT = i=1 {cd d i +c T (W i d i ) + } deote the total lead time plus tardiess costs, ZDD = Z LT + i=1 r i deote the total due-dates plus tardiess costs, ZW = i=1 {cd W i } deote the total waitig times of the jobs i the system ad ZC = Z W + i=1 r i deote the total completio times of the jobs. We preset ratios for these values for differet umber of jobs,, i Table

Table 9: Simulatio aalysis of combied strategy compared to pure strategies tha 20% ad the lead time quotatio algorithm gives re- differece betwee Z SP T A ad the lower boud are less h=8,c d =10 h=8,c d =5 h=4, c d sults that are less tha 7% worse tha Z SP T A. Thus, we =5 coclude that our schedulig ad lead time quotatio algorithms are effective i miimizig the objective fuc- Z MT O MT S /Z MT S 0.833 0.810 0.892 Z MT O MT S /Z MT O 0.785 0.871 0.827 Table 10: Compariso of lead times ad due-dates to actual waitig times ad completio times =10 =100 =1000 =5000 ZW /Z LT 0.891 0.950 0.964 0.962 ZC /Z DD 0.925 0.967 0.992 0.996 10. As we see i Table 10, the lead times quoted with our algorithm are very close to the actual waitig times ad ZDD approach to Z C as gets bigger sice the effect of the release times icrease with. 4.3.3. Effectiveess of the Algorithms for Multiple Product Types: We the cosidered multiple product types ad desiged a simulatio study to assess how our algorithms work for the multiple product type case. I additio to the sigle product type we cosidered above, ow assume that there are 4 more product types with arrival probabilities ad mea processig times as show i Table 11, everythig else remaiig the same for all product types. We first cosider a 3-product type model usig the first two product types i Table 11 ad the a 5-product type model cosiderig all the product types. I this case, we use a SPTA schedule ad a LTQ algorithm as explaied i algorithm 13. To explore the effectiveess of the SPTA schedule, we compared the total waitig times of jobs with the SPTA schedule to that of a lower boud. If we cosider oly the bottleeck facility (the facility with slowest processig rate) ad use a SPTA schedule with preemptio i that facility ad assume that the waitig time of ay job i a queue of ay other facility is zero, the the total weighted waitig time of jobs at this system will be a lower boud for our model. Let Z LB deote the lower boud for the total weighted waitig times of jobs i the system ad Z SP T A = i=1 {cd W i } deote the total weighted waitig times with the SPTA-based schedule. The compariso of the total waitig times with our heuristic to that of the lower boud are preseted i Table 12 for differet umber of jobs. Also, to explore the effectiveess of the LTQ algorithm, we preset the ratios of the total quoted lead times plus tardiess costs, Z SP T A LT Q = i=1 {cd d i + c T (W i d i ) + } for this case over Z SP T A ad Z LB i Table 12 usig the same weights for differet product types, c d = 5 ad c T = 7. Note that the optimal off-lie LTQ algorithm quotes lead times that are exactly equal to the waitig times of the jobs i the system which is equal to Z SP T A, thus Z SP T A is a lower boud for the LTQ algorithm with the SPTA-based schedule ad Z LB is a lower boud amog all schedules. As see i this table, the tio i=1 {cd d i + c T (W i d i ) + } I Table 13, we preset the ratios of the total costs K i=1 {( N j=1 h ije[i ij ])+c d i E[d i]+c T i E[W i d i ] + } by usig the ivetory values obtaied from our LP formulatio ad usig the schedulig ad lead time quotatio algorithm as explaied i algorithm 13 to that of the total costs with pure strategies ad with a FCFS schedule, for = 5000 jobs usig c T = 12 ad differet weights for h ad c d. As we see i Table 13, the costs with the combied strategy is about 20% less o average tha the pure strategies. Also, the SPTA based schedule used for this model decreases the total costs by about 10% compared to a FCFS schedule. Also, we see that as c d decreases, the combied system moves toward a MTO system while as h decreases, a MTS system gives better results. 5. Coclusio: I this project, we cosidered a supply chai i a stochastic, multi-item eviromet ad desiged effective algorithms for the optimal ivetory levels, schedulig of the jobs ad lead time quotatio to customers. We also aalyzed the value of cetralizatio ad iformatio exchage by cosiderig cetralized ad decetralized versios of this supply chai. I the first part of the project, we cosider stylized models of a supply chai, with a sigle maufacturer ad a sigle supplier, i order to quatify the impact of maufacturer-supplier relatios o effective due date quotatio. I our models, we cosider several variatios of schedulig ad due-date quotatio problems i MTO supply chais i order to miimize a fuctio of the total quoted due-dates plus tardiess. We preset due-date quotatio ad schedulig algorithms for cetralized ad decetralized versios of this model that are asymptotically optimal, ad that are computatioally foud to be effective for relatively small problem istaces. We also ivestigate the value of coordiatio schemes ivolvig iformatio sharig betwee supply chai members for this system. Through computatioal aalysis, we see that if the processig rate at the supplier is larger tha the arrival rate, i.e. whe there is o cogestio at the supplier side, the cetralized ad decetralized models perform similarly. However, as the cogestio at the supplier starts to icrease, the cetralized model performs sigificatly better, ad the value of iformatio ad cetralizatio icreases dramatically. Also, if cetralizatio is ot possible, a simple iformatio exchage i the decetralized model ca also improve the level of performace dramatically, although ot as sigificatly as cetralized cotrol. I the secod part of the project, we cosider stylized

Table 11: Arrival probabilities ad mea processig times of product types P(l) p E1 p E2 p S1 p S2 p S3 p S4 p S5 p S6 l=2 0.3 15 10 25 5 45 15 20 10 l=3 0.15 5 15 10 10 15 20 25 15 l=4 0.25 10 20 30 15 10 5 10 20 l=5 0.1 20 5 5 10 5 10 30 15 Table 12: Compariso of SPTA schedule ad the LTQ with the lower boud for K=3 ad K=5 product types K=3 =10 =100 =1000 =5000 K=5 =10 =100 =1000 =5000 Z LB /Z SP T A 0.962 0.813 0.847 0.833 0.859 0.790 0.822 0.816 Z SP T A /Z LT 0.874 0.933 0.952 0.947 0.941 0.922 0.925 0.931 Z LB /Z LT 0.848 0.753 0.802 0.786 0.807 0.735 0.766 0.751 models of a combied MTO-MTS supply chai, with a sigle maufacturer ad a sigle supplier, i order to fid the optimal ivetory values that should be carried at each facility ad to assess the impact of maufacturer-supplier relatios o ivetory decisios ad effective lead time quotatio. I our models, we cosider several variatios of ivetory, schedulig ad lead time quotatio problems i order to miimize a fuctio of the total ivetory, lead times ad tardiess. We derive the optimality coditios for both the cetralized ad decetralized versios, uder which a MTO or MTS system should be used for each product at each facility ad we preset algorithms to fid the optimal ivetory levels. We also preset effective lead time quotatio ad schedulig algorithms for cetralized ad decetralized versios of this model that are computatioally foud to be effective. We also ivestigate the value of coordiatio schemes ivolvig iformatio sharig betwee supply chai members for this system. Through computatioal aalysis, we see that costs ca be cut dramatically by usig a combied system istead of pure MTO or MTS systems ad iformatio exchage betwee the supplier ad the maufacturer is critical for effective lead time quotatio. Also, we see that if cetralizatio is ot possible, a iformatio exchage i the decetralized model ca also improve the level of performace substatially. I the fial part of the project, we cosider stylized models of more complex MTO-MTS supply chais with multiple facilities i order to fid the optimal ivetory values that should be carried at each facility ad to desig effective schedulig ad lead time quotatio algorithms. Through computatioal aalysis, we see that combied MTO-MTS systems give much better performace tha pure MTO or MTS systems ad a SPTA based algorithm for schedulig the jobs performs much better tha the geerally used FCFS approach. We also assess that iformatio exchage is critical for short ad reliable lead time quotatio. Of course, these are stylized models, ad real world systems have may more complex characteristics that are ot captured by these models. Nevertheless, this is to the best of our kowledge the first study that aalytically explores ivetory decisios, schedulig ad lead time quotatio together i the cotext of a MTO/MTS supply chai, ad that explores the impact of the suppliermaufacturer relatioship o this system. I the future, we ited to expad this research to cosider differet fuctios of lead time i the objective fuctio. I some systems, the maufacturer does t have to accept all orders ad has the optio to reject certai orders. Pricig ad capacity decisios ca also be icorporated ito these model. I all of these models ad variats, the maufacturer eeds to develop strategies for system desig, ad for schedulig ad ad due-date quotatio. 6. Refereces: [1] Arreola-Risa, A. ad DeCroix, G.A. (1998), Make-to-Order versus Make-to-Stock i a Productio-Ivetory System with Geeral Productio Times. IIE Trasactios 30, o.8, pp.705-713. [2] Aviv, Y. ad A. Federgrue (1998) The Operatioal Beefits of Iformatio Sharig ad Vedor Maaged Ivetory (VMI) Programs Workig paper,washigto Uiversity ad Columbia Uiversity. [3] Baker, K.R. ad J.W.M. Bertrad (1981), A Compariso of Due-Date Selectio Rules. AIIE Trasactios 13, pp. 123-131. [4] Bertrad, J.W.M. (1983), The Effect of Workload Depedet Due-Dates o Job Shop Performace. Maagemet Sciece 29, pp. 799-816. [5] Bourlad, K.E., S. Powell ad D. Pyke (1996) Exploitig Timely Demad Iformatio to Reduce Ivetories Europea Joural of Oper. Research 92, pp. 239-253. [6] Cacho, G.P. (2002) Supply Chai Coordiatio with Cotracts Hadbooks i Operatios

Table 13: Compariso of combied strategy with pure strategies for multiple product type models K=3 h=8,c d =10 h=8,c d =5 h=4, c d =5 K=5 h=8,c d =10 h=8,c d =5 h=4, c d =5 Z MT O MT S /Z MT S 0.732 0.695 0.887 0.762 0.717 0.869 Z MT O MT S /Z MT O 0.666 0.851 0.789 0.724 0.871 0.812 Z SP T A /Z F CF S 0.943 0.918 0.930 0.885 0.874 0.891 Research ad Maagemet Sciece: Chai Maagemet, North-Hollad. Supply [7] Cacho, G.P. ad M.A. Lariviere (1997) Cotractig to Assure Supply: How to Share Demad Forecasts i a Supply Chai Maagemet Sciece 47(5), pp. 629-646. [8] Carr, S. ad Dueyas, I.(2000), Optimal Admissio Cotrol ad Sequecig i a Make-to Stock, Make-to-Order Productio System. Operatios Research 48, o.5, pp.709-720. [9] Che, F. (1998) Echelo Reorder Poits, Istallatio Reorder Poits, ad the Value of Cetralized Demad Iformatio Maagemet Sciece 44, pp. S221-S234. [10] Che, F. (2002) Iformatio Sharig ad Supply Chai Coordiatio Hadbooks i Operatios Research ad Maagemet Sciece. [11] Eilo, S. ad I.G. Chowdhury (1976), Due Dates i Job Shop Schedulig. Iteratioal Joural of Productio Research 14, pp. 223-237. [12] Federgrue, A. ad Katala, Z. (1999), Impact of Addig a Make-to-Order Item to a Make-to- Stock Productio System. Maagemet Sciece 45, o.7, pp.980-994. [13] Gavirei, S., R. Kapusciski ad S. Tayur (1999) Value of Iformatio i Capacitated Supply Chais Maagemet Sciece 45, pp. 16-24. [14] Gross, D. ad Harris C. (1985) Fudametals of Queueig Theory Joh Wiley ad Sos: New York. [15] Igee, C. ad M. Parry (1995) Coordiatio ad Maufacturer Profit Maximizatio: The Multiple Retailer Chael Joural of Retailig 71, pp. 129-151. [16] Jackso, J.R. (1963) Job Shop-like Queuig Systems. Maagemet Sciece 10 pp.131-142. [17] Jeulad, A.P. ad S.M. Shuga (1983) Maagig Chael Profits Marketig Sciece 2, pp. 239-272. [18] Kamisky, P. ad D. Simchi-Levi (2001), Probabilistic Aalysis of a O-lie Algorithm for the Sigle Machie Completio Time Problem with Release Dates. Operatios Research Letters 29 pp. 141-148. [19] Kamisky, P ad D. Hochbaum (2004) Due Date Quotatio Models ad Algorithms. To be published as a chapter i the forthcomig book Hadbook o Schedulig Algorithms, Methods ad Models, Joseph Y. Leug (ed.), Chapma Hall/CRC. [20] Kamisky, P. ad Z-H. Lee (2005) Asymptotically Optimal Algorithms for Reliable Due Date Schedulig. Submitted for publicatio. [21] Kamisky, P. ad D. Simchi-Levi (2001) The Asymptotic Optimality of the Shortest Processig Time Rule for the Flow Shop Completio Time Problem. Operatios Research 49 pp. 293-304. [22] Kaya, O. (2006). Schedulig, Due-Date Quotatio ad Ivetory Issues i MTO-MTS Supply Chai Systems. PhD Thesis, Uiversity of Califoria, Berkeley, USA. [23] Lee, H.L.,K.C. So ad C.S. Tag (2000) The Value of Iformatio Sharig i a Two-Level Supply Chai Maagemet Sciece 46(5), pp. 626-643. [24] Li, L. (1992) The Role of Ivetory i Delivery Time Competitio Maagemet Sciece 38, o.2, pp.182-197. [25] Li, L. (2001) Iformatio Sharig i a Supply Chai with Horizotal Competitio Maagemet Sciece 48(9), pp. 1196-1212. [26] Miyazaki, S. (1981), Combied Schedulig System for Reducig Job Tardiess i a Job Shop. Iteratioal Joural of Productio Research 19, pp. 201-211. [27] Moorthy, K. (1987) Maagig Chael Profits: Commet Marketig Sciece 6, pp. 375-379. [28] Netessie, S., N. Rudi (2004)Supply Chai Structures o the Iteret ad the Role of Marketig-Operatios Iteractio Hadbook of

Quatitative Supply Chai Aalysis: Modelig i the E-Busiess Era, Kluwer Academic Publishers, pp.607-642 [29] Rajagopala, S. (2002) Make-to-Order or Make-to-Stock: Model ad Applicatio. Maagemet Sciece 48, o.2, pp.241-256. [30] Simchi-Levi, D., P. Kamisky ad E. Simchi- Levi(2004)Maagig The Supply Chai: The Defiitive Guide for the Busiess Professioal, McGraw-Hill Trade: New York. [31] Cheg, T.C.E. ad M.C. Gupta (1989), Survey of Schedulig Research Ivolvig Due Date Determiatio Decisios. Europea Joural of Operatioal Research 38, pp. 156-166. [32] Weeks, J.K. (1979), A Simulatio Study of Predictable Due-Dates. Maagemet Sciece 25, pp. 363-373. [33] Williams, T.M. (1984), Special Products ad Ucertaity i Productio/Ivetory Systems. Europea Joural of Operatios Research 15, pp.46-54. [34] Xia, C., G. Shathikumar, ad P. Gly (2000), O The Asymptotic Optimality of The SPT Rule for The Flow Shop Average Completio Time Problem. Operatios Research 48, pp. 615-622.