MTOMTS Production Systems in Supply Chains


 Arlene Strickland
 3 years ago
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1 NSF GRANT # NSF PROGRAM NAME: MES/OR MTOMTS 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 maketoorder (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 maketostock (MTS) items i stock, ad deliverig maketoorder (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 MTSMTO 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 MTSMTO 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 MTSMTO 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 MTSMTO 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
2 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 tradeoffs, 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 selfservig 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 maketostock (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 maketoorder(mto) systems. This is particularly true as more ad more supply chais move to a mass customizatiobased approach to satisfy customers ad to decrease ivetory costs (see SimchiLevi, Kamisky, ad SimchiLevi [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 duedates 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 pushpull strategy (i.e. a combied MTOMTS 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 MTOMTS systems. Also, a supplier with oe primary customer ad several smaller customers might be able to operate more profitably with a combied MTOMTS system. The importace of ivetory maagemet as outlied above has also icreased with the growig prevalece of ecommerce. I today s world, ecommerce ed customers expect high levels of service ad speedy ad otime 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 maketoorder 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 duedate 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 maufacturersupplier relatios o effective schedulig ad due date quotatio. We aalyze a maketoorder system i this simple supply chai settig ad develop effective algorithms for schedulig ad duedate 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 duedate 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
3 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 MTOMTS supply chai composed of a maufacturer, served by a sigle supplier workig i a stochastic multiitem 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 maketoorder 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 producetoorder 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], ArreolaRisa ad DeCroix [1] ad Rajagopala[29], the maketostock/maketoorder 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 twofacility 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 duedate quotatio issues ito combied MTOMTS 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 widecosts 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 suppliermaufacturer 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 costeffective,
4 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 duedate 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 DueDate Quotatio i a MTO Supply Chai: We begi our aalysis by cosiderig a pure MTO supply chai ad we focus o the schedulig ad duedate 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 maketoorder 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 duedate 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 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 iterarrival 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.
5 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
6 supply chai due date quotatio models usig our heuristics Prelimiary: The Sigle Facility Model: The Model: Although our ultimate goal is to aalyze multifacility 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 iterarrival 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 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 NPHard, Kamisky ad SimchiLevi [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 SPTASL, where the SPTA refers to
7 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 SPTASL 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: SPTASL Schedulig: Sequece ad process each job accordig accordig to the shortest processig time available (SPTA) rule. DueDate 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: 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 SPTASL 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
8 I other words, SPTASL is asymptotically optimal for this problem Supply Chai Models, Heuristics, ad Aalysis: I this sectio, we aalyze the schedulig ad duedate quotatio decisios for twostage supply chais usig the results from our aalysis i sectio 2.2, ad develop effective algorithms for schedulig ad duedate 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 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 SimchiLevi [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 2facility 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 duedate quotatio rule SLC sice it is based o the slack algorithm SL for the sigle facility case. The schedulig ad duedate 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 duedate 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,
9 ALGORITHM 2: SPTA p SLC Schedulig: Process the jobs accordig to SP T A p at the supplier ad FCFS at the maufacturer. DueDate 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 twophase approach, we adjust our duedate quotatio algorithm for that schedule, accordigly. Please refer to Kaya [22] for more detail o the schedulig ad duedate quotatio algorithms for ubalaced cases 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
10 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 duedate, 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 duedate 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 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 duedates to his customers. For this system, we fid a effective schedulig ad duedate 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 twophase 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 SPTASL 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 duedate quotatio heuristic SLC DIE for this decetralized case with iformatio exchage sice it
11 ALGORITHM 5: SPTASLC SD Schedulig: Process the jobs accordig to shortest processig time, p m i, at the maufacturer. DueDate 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: SPTASLC DIE Schedulig: Process the jobs accordig to shortest processig time, p m, at the maufacturer. DueDate 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 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 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 duedate 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 oexchageable 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 duedates. 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
12 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 iterarrival 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 iterarrival 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 duedate quotatio algorithm ad its variats seem to work well, eve for small umbers of jobs Compariso of Cetralized ad Decetralized Models: The ultimate goal of this work is to compare cetralized ad decetralized maketoorder 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 duedate 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 iterarrival 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 toorder/maketostock 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 multiitem
13 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= µ=1 µ= 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 = µ s =1 µ m =2 µ s =2 µ m =2 µ s =5 µ m = µ s =1 µ m =5 µ s =2 µ m =5 µ s =5 µ m = 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
14 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 maketoorder 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 MTOMTS systems versus pure MTO or MTS systems ad to compare the cetralized ad decetralized supply chais 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 iterarrival 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 basestock 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 maketoorder 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 offlie solutio to this model, lead times would equal waitig times of jobs i the system. Thus, the offlie 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
15 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 Sigle Facility Model: The Model: Although our ultimate goal is to aalyze multifacility 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 maketostock ad maketoorder 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 basestock 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 maketoorder 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 SPTALTQ, 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 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 NPHard, Kamisky ad SimchiLevi [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 )+ }
16 ALGORITHM 7: SPTALTQ 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 duedate plus tardiess costs with the algorithm SPTALTQ where d i = r i + d i is the quoted duedate. 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 SPTALTQ 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 duedates. 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. ArreolaRisa 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
17 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)) Supply Chai Models: I this sectio, we aalyze the ivetory decisios, schedulig ad duedate quotatio issues for twostage 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 duedate 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 suppliermaufacturer 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 maketoorder strategy for the others i a multiitem, stochastic eviromet. We assume that the supplier ad the maufacturer employs a oetooe repleishmet strategy ad a basestock 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 semifiished 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 semifiished 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 semifiished 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 semifiished 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
18 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 semifiished 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} 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 semifiished 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)
19 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 oedimesioal 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 duedate 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
20 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 iterarrival 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 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 duedate 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.
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