mto-mts production systems in supply chains

NSF GRANT #0092854NSF PROGRAM NAME: MES/OR

MTO-MTS Production Systems in Supply Chains

Philip M. KaminskyUniversity of California, Berkeley

Onur KayaUniversity of California, Berkeley

Abstract: Increasing cost pressures have led supplychain managers to focus on running increasingly leanand efficient supply chains, with minimal inventory. In-deed, more and more firms are relying on pull or make-to-order (MTO) supply chains to minimize cost and waste.At the same time, increasing competitive pressures areleading to an increased emphasis on customer service.An important element of customer service, of course, ishaving make-to-stock (MTS) items in stock, and deliv-ering make-to-order (MTO) products quickly and by thepromised due date.

In this project, we consider a variety of models de-signed to provide insight into the operation of combinedMTS-MTO supply chains. We primarily focus on a sim-ple stylized model of a two facility supply chain featur-ing a manufacturer served by a single supplier. Initially,we model a pure MTO supply, in which customers arriveat the manufacturer and place orders. The manufacturerneeds to quote a due date to the customer when the orderis placed, and the manufacturer needs to receive a compo-nent from the supplier before completing the manufactur-ing process. We design effective algorithms for produc-tion sequencing and due date quotation in both central-ized and decentralized versions of this supply chain, char-acterize the theoretical properties of these algorithms, andcompare the performance of the centralized and decen-tralized versions of the supply chain under various condi-tions.

We also consider combined MTS-MTO versions ofthis supply chain. In these models, the manufacturer andsupplier have to decide which items to produce to order,and which items to produce to stock. Inventory levelsmust be set for the MTS items, and due dates need to bequoted for the MTO items. In addition, sequencing de-cisions must be made. We consider several versions ofthese models, design effective algorithms to find inven-tory levels, to quote due dates, and to make sequencingdecisions in both centralized and decentralized settings,and use these algorithms to assess the value of joint MTS-MTO systems, as well as the value of centralization undervarious conditions.

Finally, we consider more complex supply networks,

and employ the results from the simple two facility sup-ply chains described above to design effective heuristicsfor the MTS-MTO decision, inventory levels, sequenc-ing, and due date quotation in these more complex supplychains. We employ these heuristics to answer a varietyof questions about the value of centralization, and of us-ing a combined MTS-MTO approach in complex supplychains.

1. Introduction: Supply chain management can beviewed as the combination of the approach and informa-tion technology that integrates suppliers, manufacturers,and distributors of products or services into one cohesiveprocess in order to satisfy customer requirements. Tradi-tionally, orders have been the only mechanism for orderexchange between firms; however, information technol-ogy now allows firms to share more extensive informa-tion such as demand, inventory data, etc. quickly andinexpensively. With the help of this information sharing,firms can now coordinate their processes more easily andincrease overall system efficiency.

In this project, we analyze the inventory decisions,scheduling, and lead time quotation in a variety of supplychains structures, develop approaches to minimize the to-tal cost in these supply chains, and compare centralizedand decentralized versions of the supply chain to begin toquantify the value of centralization and information ex-change.

Minimizing inventory holding costs and quoting reli-able and short lead times to customers are clearly conflict-ing objectives in supply chains with stochastic demandand processing times. Ideally, companies would like toinitiate production every time a customer order arrivesin order to avoid inventory holding costs; however, thisstrategy is likely to lead to long waiting times for orderdelivery. Every time a customer arrives, the firm mustquote a due date for the order. Long quoted lead timeslead to customer dissatisfaction, lost sales and decreasedprofits. On the other hand, quoting short lead times in-creases the risk of missing the delivery dates, which alsohas negative implications for the firm. Also, for a com-pany that produces multiple products with different char-

Proceedings of 2006 NSF Design, Service, and Manufacturing Grantees and Research Conference, St. Louis, Missouri Grant #0092854

acteristics, the decision of when to produce each orderalso effects the completion times and thus the lead timesof many other products. In this project, we analyze thesetrade-offs, and develop approaches to effectively addressthem.

Optimal supply chain performance requires the ex-ecution of a precise set of actions, however, those ac-tions are not always in the best interest of the members inthe supply chain, i.e. the supply chain members are pri-marily concerned with optimizing their own objectives,and that self-serving focus often results in poor perfor-mance. However, optimal performance can be achievedif the firms in the supply chain coordinate by contractingon a set of transfer payments such that each firm’s objec-tive becomes aligned with the supply chain’s objective.See Cachon [6] for an extensive survey of supply chaincoordination with contracts and Chen [10] for an exten-sive survey on information sharing and supply chain co-ordination. Cachon and Lariviere [7], Li [25], Jeulandand Shugan [17], Moorthy [27], Ingene and Parry [15]and Netessine and Rudi [28] are a few of the researchersthat analyze the benefits of information sharing in supplychains and how to operate such systems for different ob-jectives. Also, Bourland et al. [5], Chen [9], Gavirneniet al. [13], Lee et al. [23] and Aviv and Federgruen [2]show that sharing demand and inventory data improvesthe supply chain performance for several different objec-tives.

As many authors have observed, supply chain man-agement and coordination has gained importance in re-cent years as businesses feel the pressure of increasedcompetition and as managers have begun to understandthat a lack of coordination can lead to decreased profitsand service levels. There is a large and growing amountof literature on this subject, but the vast majority of thisresearch focuses on make-to-stock (MTS) systems, andperformance measures built around service and inventorylevels. On the other hand, an increasing number of supplychains are better characterized as make-to-order(MTO)systems. This is particularly true as more and more sup-ply chains move to a mass customization-based approachto satisfy customers and to decrease inventory costs (seeSimchi-Levi, Kaminsky, and Simchi-Levi [30]). Masscustomization implies that at least the final details ofproject manufacturing must occur after specific ordershave been received, and must thus be completed quicklyand efficiently. MTO systems have many unique issuesand need to be operated very differently from MTS sys-tems. In an MTO system, firms need to find an effectiveapproach to scheduling their customers’ orders, and theyalso need to quote short and reliable due-dates to theircustomers.

However, not all firms employ a pure MTO or MTSsystem. Although some firms make all of their productsto order while some others make them to stock, there are

also a number of firms that maintain a middle ground,where some items are made to stock and others are madeto order. The decision on using either an MTO strategyor an MTS strategy at a facility heavily depends on thecharacteristics of the system. In supply chains using acombined system, holding inventory at some of the stagesof the chain and using an MTO strategy at other facilitiesmight decrease the costs dramatically without increasingthe lead times. Because of this, companies are starting toemploy a hybrid approach, a ”push-pull” strategy (i.e. acombined MTO-MTS system), holding inventory at someof the facilities in their supply chain and producing to or-der in others. For example, a company with a diverseproduct line and customer base can be best served with anappropriate combination of MTO-MTS systems. Also, asupplier with one primary customer and several smallercustomers might be able to operate more profitably witha combined MTO-MTS system. The importance of in-ventory management as outlined above has also increasedwith the growing prevalence of e-commerce. In today’sworld, e-commerce end customers expect high levels ofservice and speedy and on-time deliveries.

In the first part of this project, we consider a singlemanufacturer, served by a single supplier, who has toquote due dates to arriving customers in a make-to-orderproduction environment. The manufacturer is penalizedfor long lead times, and for missing due dates. In orderto meet due dates, the manufacturer has to obtain compo-nents from a supplier. We consider several variations ofthis problem, and design effective due-date quotation andscheduling algorithms for centralized and decentralizedversions of the model.

We consider stylized models of such a supply chain,with a single manufacturer and a single supplier, in orderto begin to quantify the impact of manufacturer-supplierrelations on effective scheduling and due date quotation.We analyze a make-to-order system in this simple sup-ply chain setting and develop effective algorithms forscheduling and due-date quotation in both centralized anddecentralized versions of this model. Building on thisanalysis, we explore the value of centralized control inthis supply chain, and develop schemes for managing thesupply chain in the absence of centralized control andwith only partial information exchange.

As mentioned above, researchers have introduced avariety of models in an attempt to understand effectivedue date quotation. Kaminsky and Hochbaum [19] andCheng and Gupta [31] survey due date quotation modelsin detail. The majority of earlier papers on due-date quo-tation have been simulation based. For instance, Eilonand Chowdhury [11], Weeks [32], Miyazaki [26], Bakerand Bertrand [3], and Bertrand [4] consider various duedate assignment and sequencing policies, and in generaldemonstrate that policies that use estimates of shop con-gestion and job content information lead to better shop


performance than policies based solely on job content.We extend previous work in due date quotation by ex-

ploring due date quotation in supply chains. In particular,we focus on a two member supply chain, in which a man-ufacturer works to satisfy customer orders. Customers ar-rive at the manufacturer over time, and the manufacturerproduces to order. In order to complete production, themanufacturer needs to receive a customized componentfrom a supplier. Each order takes a different amount oftime to process at the manufacturer, and at the supplier.The manufacturer’s objective is to determine a scheduleand quote due dates in order to minimize a function ofquoted lead time and lateness.

In the second part of the project, we consider a com-bined MTO-MTS supply chain composed of a manufac-turer, served by a single supplier working in a stochasticmulti-item environment. The manufacturer and the sup-plier have to decide which items to produce to stock andwhich ones to order. The manufacturer also has to quotedue dates to arriving customers for make-to-order prod-ucts. The manufacturer is penalized for long lead times,missing the quoted lead times and for high inventory lev-els. We consider several variations of this problem, anddesign effective heuristics to find the optimal inventorylevels for each item and also design effective schedulingand lead time quotation algorithms for centralized and de-centralized versions of the model.

We analyze the conditions under which an MTO orMTS strategy would be optimal to use for both the man-ufacturer and the supplier and find the optimal inventorylevels. We also design effective scheduling and lead timequotation algorithms based on the algorithms in the firstpart of this thesis. We consider several variations of thisproblem. In particular, we focus on three online mod-els. In the first model, the centralized model, both facil-ities are controlled by the same agent, who decides onthe inventory levels at both facilities, schedules the jobsand quotes a due date to the arriving customer in orderto achieve the end objective. In the second model, wedevelop a decentralized model with full information, inwhich the manufacturer and the supplier work indepen-dently from each other and make their own decisions butthe manufacturer has complete information about both hisown processes and the supplier. This model allows to ex-plore the value of information exchange, and to determineif and when the cost and difficulty of implementing a cen-tralized system are worth it. In the last model, the sim-ple decentralized model, the manufacturer has no infor-mation about the supplier and makes certain assumptionsabout in order to decide on his inventory values and quotea due date to arriving customers. In this model, each fa-cility is working to achieve its own goals, and very lit-tle information is exchanged. Indeed, in our discussionswith the managers of several small manufacturing firms,this is typical of their relationships with many suppli-

ers. We compare the centralized and decentralized mod-els through extensive computational analysis to assess thevalue of centralization and information exchange.

In literature, MTO/MTS models are generally stud-ied for single stage systems. Several researchers, such asWilliams [33], Federgruen and Katalan [12] and Carr andDuenyas [8] assume that the decision of which items toproduce-to-order and which ones to stock is made in ad-vance and they try to find the best way to operate that sys-tem. Others, like Li [24], Arreola-Risa and DeCroix [1]and Rajagopalan[29], the make-to-stock/make-to-orderdistinction is a decision variable determined within thethe model. In contrast to these models, we consider thescheduling and inventory decisions together and we an-alyze supply chain systems instead of a single facilitymodel. We also integrate lead time quotation into thesemodels in our research.

Finally, we consider more complex supply chain net-works composed of several facilities with different re-lationships to each other. In this model, we consider asupply chain composed of several facilities and managedby a single decision agent who has complete informationabout these facilities. In addition, there are external sup-pliers to this supply chain outside the control of the man-ager of the supply chain, and this manager has only lim-ited information about these external suppliers. Underthese conditions, using the results from the two-facilitysupply chain, we design effective heuristics to be used bythe manager to find optimal inventory levels, to sequencethe orders at each facility, and to quote reliable and shortlead times to customers.

In contrast to the models in literature, we integratedue-date quotation issues into combined MTO-MTS sys-tems, consider different schedules and focus on supplychain models of this system. We consider scheduling,inventory and lead time quotation decisions together inour study. We develop models that provide guidance fordeciding when to use MTS and when to use MTO ap-proaches for single facility and for supply chain models,and for how to effectively operate the system to mini-mize system wide-costs in each case. We also quantifythe value of centralization and information in this systemby building decentralized and centralized models and ob-taining good solutions to these models. To the best of ourknowledge, this is the first study that analytically exploresinventory decisions and lead time quotation together inthe context of a supply chain, and that explores the impactof the supplier-manufacturer relationship on this system.

We perform extensive computational testing for eachof the cases above to assess the effectiveness of our al-gorithms, and to compare the centralized and decentral-ized models in order to quantify the value of central-ized control and information exchange in these supplychains. Since complete information exchange and cen-tralized control is not always practical or cost-effective,


we also explore the value of partial information exchangefor these systems. In this project, we are attempting tofind answers to questions such as:

1. Which items should be produced MTO and whichones MTS at each step of the supply chain and whatare the optimal levels of inventory for MTS items?

2. Which item should be produced next when a facil-ity becomes available for production?

3. What is the optimal due-date that should be quotedto each customer at the time of arrival?

4. What is the benefit of a centralized supply chain asopposed to decentralized systems?

5. How much of the gains associated with centraliza-tion can be achieved through information exchangebetween supply chain members?

2. Scheduling and Due-Date Quotation in an MTOSupply Chain: We begin our analysis by considering apure MTO supply chain and we focus on the schedul-ing and due-date quotation issues in this system. In thissection, we consider a single manufacturer, served by asingle supplier, who has to quote due dates to arrivingcustomers in a make-to-order production environment.The manufacturer is penalized for long lead times, andfor missing due dates. We consider several variations ofthis problem, and design effective due-date quotation andscheduling algorithms for centralized and decentralizedversions of this model. We also complete an extensivecomputational experiment to evaluate the effectiveness ofour algorithms and to assess the value of centralizationand information exchange in this system.

2.1. Model and Main Results: We model two parties,a supplier and a manufacturer, working to satisfy cus-tomer orders. Customer orders, or jobs, i, i = 1, 2, .., narrive at the manufacturer at time ri, and the manufac-turer quotes a due date for each order, di, when it ar-rives. To begin processing each order, the manufacturerrequires a component specifically manufactured to orderby the supplier. The component requires processing timeps

i at the supplier, and the order requires processing timepm

i . (To clarify the remaining exposition, we will refer toan order or job with processing time ps

i at the supplierand pm

i at the manufacturer.) Recall that we are focusingon online versions of this model, in which informationabout a specific order’s characteristics (arrival time andprocessing times) is not available until the order arrivesat the manufacturer (that is, until its release time ri). Weconsider several versions of this model, which we brieflydescribe here and discuss in more detail in subsequentparagraphs. In the centralized version of the model, theentire system is operated by a single entity, who is aware

of processing times both at the supplier and at the manu-facturer. In the simple decentralized model, the manufac-turer and the supplier are assumed to work independently,and each is unaware of the job’s processing time at theother stage. Finally, in the decentralized model with ad-ditional information exchange, the supplier quotes a duedate to the manufacturer as a mechanism for limited in-formation exchange.

In this study, we propose algorithms for these models.Our goal in developing these algorithms is to provide asimple and asymptotically optimal online scheduling anddue date quotation heuristic for either the manufacturerand the supplier individually in the decentralized system,or for both, for the centralized system, that works welleven for congested systems, so that we can compare theperformance of these systems. Since firms are faced witha tradeoff between quoting short due dates, and meetingthese due dates, we consider an objective function thatcaptures both of these concerns. In particular, we are at-tempting to minimize the total cost function

n∑

i=1

(cddi + cT Ti)

where Ti = (Ci − di)+ is the tardiness of job i and cd

and cT are the unit due date and tardiness costs for themodel. Clearly, cT > cd, or otherwise it will be optimalfor all due dates to be set to 0.

As mentioned above, we use the tools of probabilisticanalysis, as well as computational testing, to characterizethe performance of these heuristics, and to compare thesemodels. In particular, we focus on asymptotic probabilis-tic analysis of this model and these heuristics. In this typeof analysis, we consider a sequence of randomly gener-ated instances of this model, with processing times drawnfrom independent identical distributions bounded aboveby some constant, and with arrival times determined bygenerating inter-arrival times drawn from identical inde-pendent distributions bounded above by some constant.Processing times are assumed to be independent of inter-arrival times. The processing time of a job at the supplierand at the manufacturer may be generated from differentdistributions.

Recall that any algorithm for this problem has to bothset due dates for arriving jobs, and determine job se-quences at the supplier and the manufacturer. In Section2.3.1, for the centralized model described above, we de-tail a series of heuristics, most notably one called (forreasons that will subsequently become clear) SPTAp −SLC, that are used to determine due dates as jobs arrive,and to sequence jobs both at the supplier and at the man-ufacturer. We define Z

SPTAp−SLCn to be the objective

function resulting from applying this heuristic to an n jobinstance, and Z∗n to be the optimal objective value for thisinstance.


Theorem 1. Consider a series of randomly generatedproblem instances of the centralized model of size n. Letinterarrival times be i.i.d. random variables boundedabove by some constant; the processing times at each fa-cility be also i.i.d random variables and bounded and theprocessing times and interarrival times be independentof each other. Also, if the processing times at the sup-plier and the manufacturer are generated from indepen-dent and exchangeable distributions, then for an n jobinstance, using SPTAp − SLC to quote due dates andsequence jobs satisfies almost surely:

limn→∞

ZSPTAp−SLCn − Z∗n

Z∗n= 0

In the simple decentralized model, recall that themanufacturer and the supplier work independently, andthey both attempt to minimize their own costs. When thecustomer arrives at the manufacturer, the manufacturerneeds to quote a due date, even though he is not awareof either the processing time of the job at the supplier,or of the supplier’s schedule. We assume that the man-ufacturer is aware of the number of jobs at the supplier(since this is equal to the number of jobs he sent there,minus the number that have returned), that he is awareof the average processing time at the supplier, and thathe is aware of the mean interarrival rate of jobs to his fa-cility. In Section 2.2, in a preliminary exploration, weconsider a single facility due date quotation model that isanalogous to our two stage model, except that jobs onlyneed to be processed at a single stage (in other words,our model, but with no supplier necessary). We developan asymptotically optimal algorithm for this single facil-ity model. For the purpose of understanding the perfor-mance of the decentralized system, we assume that thesupplier uses this asymptotically optimal single facilitymodel. Then, we explore the problem from the perspec-tive of the manufacturer, and determine an effective duedate quotation and sequencing policy for the manufac-turer, given our assumptions about the limits of the man-ufacturers knowledge about the supplier’s system. Sincethe manufacturer is unaware of ps

i values, he can’t inquireany information about the supplier’s schedule and loca-tion of a job at the supplier queue and assumes that eacharriving job is scheduled in the middle of the existing jobsin the supplier queue. Under this assumption, in section2.3.2, we propose an asymptotically optimal heuristiccalled SPTA − SLCSD. Define ZSPTA−SLCSD to bethe objective function resulting from applying this heuris-tic to an n job instance, and Z∗n to be the optimal objectivevalue for this instance given the information available tothe manufacturer.

Theorem 2. Consider a series of randomly generatedproblem instances of the decentralized model of sizen. Let interarrival times be i.i.d. random variablesbounded above by some constant; the processing times

at the manufacturer be also i.i.d random variables andbounded and the processing times and interarrival timesbe independent of each other. If the manufacturer usesSPTA − SLCSD, then under manufacturer’s assump-tions about the supplier’s schedule, almost surely,

limn→∞

ZSPTA−SLCSDn − Z∗n

Z∗n= 0

When we explore our decentralized model with infor-mation exchange, we assume that when orders arrive atthe manufacturer, that the manufacturer has the same in-formation as in the simple decentralized model describedabove, and that in addition, the supplier uses our asymp-totically optimal single facility algorithm for sequencingand to quote a due date to the manufacturer. The manu-facturer in turn uses this due date in his due date quotationand sequencing heuristic, SPTA − SLCDIE . In Sec-tion 2.3.3, we explain this heuristic in detail. We defineZSPTA−SLCDIE

n to be the objective function resultingfrom applying this heuristic to an n job instance, and Z∗nto be the optimal objective value for this instance giventhe information available to the manufacturer.

Theorem 3. Consider a series of randomly generatedproblem instances of size n. Let interarrival times bei.i.d. random variables bounded above by some constant;the processing times at each facility be also i.i.d randomvariables and bounded and the processing times and in-terarrival times be independent of each other. If the man-ufacturer uses SPTA− SLCDIE , and the supplier usesa locally asymptotically optimal algorithm, then almostsurely,

limn→∞

ZSPTA−SLCDIEn − Z∗n

Z∗n= 0

In Section 2.4, we present a computational analysisof these algorithms for a variety of different problem in-stances, and compare the centralized and decentralizedversions of the model. We see that the proposed al-gorithms are effective even for small numbers of jobs,and that the objective function values approach the opti-mal values quite quickly as the number of jobs increases.Also, we characterize conditions under which the cen-tralized model performs considerably better than the de-centralized model, and calculate this “value of centraliza-tion” under various conditions. Of course, as we men-tioned previously, in many cases implementing central-ized control is impractical or prohibitively expensive, sowe also explore the value of simple information exchangein lieu of completely centralized control.

In the next section, we introduce a preliminary model,and analyze this model. In section 2.3, we present ourmodels, algorithms, and results in detail, and in section2.4, we present the computational analysis of our heuris-tics and a comparison of centralized and decentralized


supply chain due date quotation models using our heuris-tics.

2.2. Preliminary: The Single Facility Model:2.2.1. The Model: Although our ultimate goal is to

analyze multi-facility systems, we begin with a prelimi-nary analysis of a single facility system. We focus on de-veloping asymptotically optimal scheduling and due datequotation heuristics for this system, and consider caseswhen the system is congested (the arrival rate is greaterthan the processing rate), and cases when it is not.

In this model, we need to process a set of jobs, non-preemptively, on a single machine. Each job has an asso-ciated type l, l = 1, 2, ...k, and each type has an associ-ated finite processing time pl, pl < ∞. At the time thatjob i arrives at the system, ri, the operator of the systemquotes a due date di. In particular, we focus on a systemin which due dates are quoted without any knowledge offuture arrivals – an online system. However, informationabout the current state of the system and previous arrivalscan be used.

As mentioned above, we will use the tools of asymp-totic probabilistic analysis to characterize the perfor-mance of the heuristics we propose in this study undervarious conditions. In this type of analysis, we consider asequence of randomly generated deterministic instancesof the problem, and characterize the objective values re-sulting from applying a heuristic to these instances as thesize of the instances (the number of jobs) grows to infin-ity. For this probabilistic analysis, we generate probleminstances as follows. Each job has independent probabil-ity Pl, l = 1..k of being job type l, where

∑kl=1 Pl = 1

and job type l has known processing time pl. Arrivaltimes are determined by generating inter-arrival timesdrawn from identical independent distributions boundedabove by some constant, with expected value λ.

The objective of our problem is to determine a se-quence of jobs and a set of due dates such that the totalcost Zn =

∑ni=1(c

ddi + cT [Ci − di]+) is minimized,where Ci denotes the completion time of the order i.Clearly, to optimize this expression, we need to coordi-nate due date quotation and sequencing, and an optimalsolution to this model would require simultaneous se-quencing and due date quotation. However, the approachwe have elected to follow for this model (and throughoutthis thesis) is slightly different. Observe that in an opti-mal offline solution to this model, due dates would equalcompletion times since cT > cd, or otherwise it is opti-mal for all due dates to be set to 0. Thus, the problembecomes equivalent to minimizing the sum of comple-tion times of the tasks. Of course, in an online schedule,it is impossible to both minimize the sum of completiontimes of jobs, and set due dates equal to completion times,since due dates are assigned without knowledge of fu-ture arrivals, some of which may have to complete beforejobs that have already arrived in order to minimize the

sum of completion times. However, for related problems(Kaminsky and Lee, [20]), we have found that a two-phase approach is asymptotically optimal. In this typeof approach, we first determine a scheduling approachdesigned to effectively minimize the sum of completiontimes, and then we design a due date quotation approachthat presents due dates that are generally close to the com-pletion times suggested by our scheduling approach. Thisis the intuition behind the heuristic presented below.

2.2.2. The Heuristic: As mentioned above, we haveemployed a heuristic that first attempts to minimize thetotal completion times, and then sets due dates that ap-proximate these completion times in an effort to mini-mize the objective. The heuristic we propose sequencesthe jobs according to the Shortest Processing Time Avail-able (SPTA) rule. Under the SPTA heuristic, each timea job completes processing, the shortest available jobwhich has yet not been processed is selected for pro-cessing. As we observed in the introduction, althoughthe problem of minimizing completion times is NP-Hard,Kaminsky and Simchi-Levi [18] found that the SPTA ruleis asymptotically optimal for this problem. Also, note thatthis approach to sequencing does not take quoted due dateinto account, and is thus easily implemented.

Instead, the due date quotation rule takes the sequenc-ing rule into account. To quote due dates, we maintain anordered list of jobs that have been released and are wait-ing to be processed. In this list, jobs are sequenced in in-creasing order of processing time, so that the shortest jobis at the head of the list. Since we are sequencing jobsSPTA, when a job completes processing, the first job onthe list is processed, and each job moves up one positionin the list. When a job i arrives at the system at its releasetime ri with processing time pi and the system is empty, itimmediately begins processing and a due date equal to itsrelease time plus its processing time is quoted. However,if the system is not empty at time ri, job i is inserted intothe appropriate place in the waiting list. Let Ri be the re-maining time of the job in process at the time of arrival i,pos[i] be the position of job i in the waiting list and list[i]be the index of the ith job in the waiting list. Then, a duedate is quoted for this job i as follows:

di = ri + Ri +pos[i]∑

j=1

(plist[j]) + slacki (2.1)

where slacki is some additional time added to the duedate in order to account for future arrivals with process-ing times less than this job – these are the jobs that willbe processed ahead of this job, and cause a delay in itscompletion.

Throughout this thesis, we name our heuristics in twoparts, where the first part (before the hyphen) refers tothe sequencing rule, and the second part refers to the duedate quotation approach. Following this convention, wecall this approach SPTA-SL, where the SPTA refers to


the sequencing rule, and the SL refers to the due datequotation rule based on calculated completion at arrivalplus the insertion of SLack. In the next section, we an-alytically demonstrate the effectiveness of the SPTA-SLapproach.

The remainder of this section focuses on determiningan appropriate value for slacki. To do this, we need toestimate the total processing times of jobs that arrive be-fore we process job i and have processing times less thanthe processing time of job i. We complete this calculationfor one job at a time.

Define Mi to be the remaining time of the job in pro-cess plus the total processing times of all of the jobs tobe processed ahead of job i of type l, at the time of itsarrival, such that

Mi = Ri +pos[i]−1∑

j=1

(plist[j]).

Let ψl be the probability that an arriving job has process-ing time less than pl. Also, let µl = E[p|p < pl] bethe expected processing time of a job given that it is lessthan pl, and let λ be the mean interarrival time. Then,the slack for job i can be calculated using an analogousapproach to busy period analysis in queueing theory (see,e.g., Gross and Harris, [14]), where only those jobs thatare shorter than job i are considered “new arrivals” forthe analysis, since other jobs will be processed after job iand thus won’t impact job i’s completion time. Note thatthe sequence of jobs to be processed before job i doesn’timpact job i’s completion time, so for our analysis wecan assume any sequence that is convenient (even if it isnot the sequence that we will ultimately use, as long aswe consider only those jobs that will be processed beforejob i in the sequence we actually use). In particular, weassume for the purpose of our analysis that first we pro-cess all the jobs that are already there when job i arrives,which takes the amount of time Mi. During this time,suppose that K jobs with processing time shorter than iarrived. Then, at the end of Mi, we have K jobs on handthat will impact the completion time of job i, and we pickone of them arbitrarily. Now, we imagine that the job justarrived when we selected it and that there are no otherjobs in the system, and calculate that job’s busy period– the time until a queue featuring that job, and other ar-rivals shorter than it, will remain busy. We don’t considerany of the other K jobs until the busy period of this firstjob is completed. Then, we move to considering the sec-ond of the K jobs when the server becomes idle (when itfinishes the busy period of the first job) and calculate itsbusy period, and so on, until we have considered all Kjobs.

Thus, we can write the delayed busy period of job i

ALGORITHM 1: SPTA-SLScheduling: Sequence and process each job according

according to the shortest processing time available(SPTA) rule.

Due-Date Quotation:di = ri + Mi + pi + slacki

slacki =

{min{Miψlµl

λ−ψlµl, (n− i)ψlµl} if ψlµl

λ < 1(n− i)ψlµl otherwise

with Mi as:

Bi(Mi) = Mi + slacki = Mi +A(Mi)∑

j=1

Bj

where A(Mi) is the actual number of arrivals with pro-cessing time less than pi during Mi (the arrivals after ithat will be processed before job i) and Bj is the “busyperiod” of each of these jobs as defined in Gross and Har-ris [14]. Gross and Harris [14] show that for an M/G/1queue, if µi

λ/ψi< 1, then

E[Bi(Mi)] =Mi

1− µi

λ/ψi

=Miλ

λ− µiψi.

This suggests that we can approximate the slack valuewe are looking for by using this relationship:

slacki = E[Bi(Mi)]−Mi =Miµiψi

λ− µiψi

However, since we consider a problem instance of size n,it may be that all of the jobs have arrived before job i isprocessed, in which case the slack value will be equal toslacki = (n− i)ψlµl

Also, if ψlµl

λ ≥ 1, then the expected delayed waitingtime is longer than the expected time for all the remainingjobs to arrive, and thus the slack value is again equal toslacki = (n− i)ψlµl.We summarize the scheduling and due date quotation rulefor job i of type l in Algorithm 1:

2.2.3. Analysis and Results: For sets of randomlygenerated problem instances as described in precedingsections, let ZSPTA−SL

n represent the objective functionvalue obtained by applying the SPTA-SL rule for an njob instance, and let Z∗n be the optimal objective functionvalue for that instance.

Theorem 4. Consider a series of randomly generatedproblem instances of size n meeting the requirements de-scribed above. Let interarrival times be i.i.d. randomvariables bounded above by some constant; the process-ing times be also i.i.d random variables and bounded andthe processing times and interarrival times be indepen-dent of each other. Then, almost surely,

limn→∞

ZSPTA−SLn − Z∗n

Z∗n= 0


In other words, SPTA-SL is asymptotically optimalfor this problem.

2.3. Supply Chain Models, Heuristics, and Analy-sis: In this section, we analyze the scheduling and due-date quotation decisions for two-stage supply chains us-ing the results from our analysis in section 2.2, and de-velop effective algorithms for scheduling and due-datequotation for both the centralized and decentralized ver-sions of these systems. These algorithms allow us to com-pare the value of centralization and information exchangein supply chains under a variety of different conditions.

2.3.1. The Centralized Model:

The Model For the centralized case, the system can bemodeled as, in effect, a two facility flow shop. We assumethat the manufacturer and the supplier work as a singleentity and that they are both controlled by the same agentthat has complete information about both stages. The de-cisions about the scheduling at both facilities and due datesetting for the customer are made by this agent.

The Heuristic and Main Results Recall that in the sin-gle facility case, we utilized a known asymptotic optimal-ity result for the completion time problem as a basis forour sequencing rule, and then designed a due date quota-tion rule so that due dates were close to the completiontimes. For this model, we employ the same two phase ap-proach, but we first need to determine an asymptoticallyoptimal scheduling rule for the related completion timeproblem, and then design an asymptotically optimal duedate quotation heuristic for the sequence.

Xia, Shantikumar and Glynn [34] and Kaminsky andSimchi-Levi [21] independently proved that for a flowshop model with m machines, if the processing timesof a job on each of the machines are independent andexchangeable, (i.e. pj

i and pki are independent and ex-

changeable for all pairs of machines (j, k) for every jobi), processing the jobs according to the shortest total pro-cessing time pi =

∑mj=1 pj

i at the first facility (supplier)and processing the jobs on a FCFS basis at the others(manufacturer) is asymptotically optimal if all the releasetimes are 0. We extend this result in Theorem 5, focus-ing on a 2-facility flow shop model, to include the casewhere all the release times are not necessarily 0, and jobsare scheduled by shortest available total processing timeat the supplier. We denote this heuristic SPTAp (becausewe schedule the jobs based on total processing time). LetZ∗n be the minimum possible value for the total comple-tion time objective and Z

SPTApn be the total completion

time of the jobs with the heuristic explained above for ann job instance. Then, we have the following theorem forthis scheduling rule.

Theorem 5. Consider a series of randomly generatedproblem instances of size n. If the processing times ofjobs at the supplier and the manufacturer are generated

from independent and exchangeable distributions, andif the jobs are scheduled using the instance, schedulingthe jobs according to SPTAp is asymptotically optimalfor the objective of minimizing the total completion timeZn =

∑ni=1 Ci. In other words, almost surely,

limn→∞

ZSPTApn − Z∗n

Z∗n= 0

Observe that although this heuristic generates a per-mutation schedule, it is asymptotically optimal over allpossible schedules, not just permutation schedules.

We base the first phase of our Algorithm 2 on Theo-rem 5, and then generate due dates similarly to the onesfor our single facility approach. We call our due-datequotation rule SLC since it is based on the slack algo-rithm SL for the single facility case. The scheduling anddue-date quotation algorithm called SPTAp − SLC isstated as below:

The due date set of equations listed above is similarto those for the single facility case, adjusted for the cen-tralized model. Essentially, we approximate the amountof workload, both at the supplier and at the manufacturer,that will be processed before job i if the SPTAp schedul-ing rule is employed.

The due date, dmi , is equal to the sum of ds

i , the ap-proximated finish time of job i at the supplier, pm

i , theprocessing time at the manufacturer, and max{tms

i +tmmi + slackm

i − (dsi − ri), 0}, the approximate wait-

ing time of job i at the manufacturer queue. tmsi + tmm

i

denotes the sum of the processing times of the jobs thatare already in the system and scheduled before job i attime ri and slackm

i approximates the workload at man-ufacturer of future arrivals that will be scheduled beforejob i while it waits in the supplier queue. In the calcu-lation of slackm

i , min{ (dsi−ri−ps

i )λ , (n − i)} denotes the

approximate number of jobs that will arrive after ri, andmultiplying this by pr{p < pi}E{pm|p < pi} approx-imates the length of the subset of these jobs that will bescheduled before job i at supplier. These jobs will arriveat the manufacturer before job i and since we use FCFS atthe manufacturer, they will also be processed before jobi there. When setting the due date, we subtract ds

i − ri

since this is the approximate amount of work that will beprocessed at manufacturer while job i is still at the sup-plier.

Recall Theorem 1 in Section 2.1 which states thatSPTAp − SLC is asymptotically optimal.Unbalanced processing times If the processing times atthe supplier and the manufacturer are not exchangeableas assumed in the previous case, we adjust our schedul-ing and due-date quotation algorithm SPTAp − SLCto reflect the properties of the unbalanced system. Forscheduling, we approach the system to balance the work-loads at both facilities so that the total completion time isminimized. Since the processing times are unbalanced,


ALGORITHM 2: SPTAp-SLCScheduling: Process the jobs according to SPTAp at the supplier and FCFS at the

manufacturer.Due-Date Quotation:

dmi = ds

i + pmi + max{tms

i + tmmi + slackm

i − (dsi − ri), 0}

dsi = ri + ps

i + Msi + slacks

i

slacksi =

{(n− i)pr{p < pi}E{ps|p < pi} if λ− pr{p < pi}E{ps|p < pi} ≤ 0min{(n− i)pr{p < pi}E{ps|p < pi}, Ms

i pr{p<pi}E{ps|p<pi}λ−pr{p<pi}E{ps|p<pi} } otherwise

tmsi =

∑i∈A pm

i where A=set of jobs in supplier queue scheduled before job i at time ri.tmmi =

∑i∈B pm

i where B=set of jobs in manufacturer queue at time ri

slackmi = min{ (ds

i−ri−psi )

λ , (n− i)}pr{p < pi}E{pm|p < pi}

we focus on the bottleneck facility and use an SPTAbased schedule focusing on the processing times at thebottleneck facility. Then, again utilizing our two-phaseapproach, we adjust our due-date quotation algorithm forthat schedule, accordingly. Please refer to Kaya [22] formore detail on the scheduling and due-date quotation al-gorithms for unbalanced cases.

2.3.2. The Simple Decentralized Model: Whilesome supply chains are relatively easy to control in a cen-tralized fashion, most often this is not the case. Even ifthe stages in a supply chain are owned by a single firm,information systems, control systems, and local perfor-mance incentives need to be designed and implementedin order to facilitate centralized control. In many cases,of course, the supplier and manufacturer are indepen-dent firms, with relatively limited information about eachother. Implementing centralized control in these supplychains is typically even more difficult and costly, sincethe firms need to coordinate their processes, agree ona contract, implement an information technology sys-tem for their processes, etc. Thus, for either centrallyowned or independent firms, centralization might not beworth the effort if the gains from centralization are notbig enough.

Typically, if a supply chain is decentralized, the sup-plier and the manufacturer have only limited informationabout each other. The manufacturer is unaware of theprocesses at his supplier and needs to make his own de-cisions without any information from the supplier. Forexample, the manufacturer may only be aware of the av-erage time it takes for the supplier to process and deliveran order. For this type of decentralized supply chain, wedevelop an effective approach for scheduling orders andquoting due dates to customers with limited informationabout the supplier.

The Model In this section, we consider a setting inwhich the manufacturer and the supplier work indepen-dently and each tries to minimize his or her own costs.When the customer arrives at the manufacturer and placesan order, the manufacturer needs to immediately quote a

due date, although the manufacturer has limited supplier-side information. In particular, the manufacturer has toquote due dates to the customers without knowledge ofthe supplier’s schedule or knowledge of the location ofany incomplete orders in the suppliers queue, and thuswithout knowledge or control of when the materials forthat order will arrive from the supplier.

We assume that the manufacturer only knows the av-erage processing time of jobs at the supplier, as well asthe average interarrival time of orders to the system andthe processing time of jobs at his own facility. The man-ufacturer doesn’t know the processing time of jobs at thesupplier or the schedule of the supplier. Thus, the manu-facturer has to quote due dates to the customer using onlythe knowledge of his own shop, mean processing timesat the supplier, and knowledge of the number of jobs atthe supplier, since this is equal to the number of ordersthat have arrived at the manufacturer minus the numberof orders that the supplier completed and sent to the man-ufacturer.

The Heuristic and Main Results In this decentralizedcase, we focus on the manufacturer’s problem since he isthe one who quotes the due dates to the customer, and wetry to find an effective scheduling rule/due date quotationheuristic to minimize the manufacturer’s total cost givenlimited supplier information.

For this model, we employ the same two phase ap-proach that we used before, by first determining anasymptotically optimal scheduling rule to minimize thetotal completion times, and then designing a due datequotation heuristic to match the completion times withthat schedule. In this case, since the manufacturer isworking independently from the supplier and has no in-formation about the processing times or the schedulingrule used in the supplier side, to minimize the total com-pletion times, it will be asymptotically optimal for him touse the SPTA scheduling rule according to his own pro-cessing times pm.

Based on this schedule, to find an effective due datequotation heuristic for the manufacturer, we use the same


due date setting ideas as before. However, in this case,since the manufacturer is unaware of the processes at thesupplier, we use estimates of the conditions at the suppliersite – these estimates replace the information that we usedin the centralized case.

The manufacturer has no information about the sup-plier except the number of jobs at the supplier side, qs

i , attime ri. Although using an SPTA schedule according tothe processing times at the supplier, ps, is asymptoticallyoptimal to minimize the total completion times at the sup-plier, the manufacturer is unaware of ps

i values, so that hecan’t infer any information about the supplier’s scheduleor the location of a job at the supplier queue. Thus, toquote a due-date, we assume that an arriving job is lo-cated in the middle of the existing jobs in the supplierqueue, and that the future arrivals will be scheduled infront of this job with probability 1/2. This is reasonablegiven that the manufacturer doesn’t know anything aboutthe schedule used by the supplier, about the processingtime of that job, or about the other jobs at the supplierqueue. We develop an asymptotically optimal schedul-ing and due date quotation algorithm for the manufac-turer given that the manufacturer is using this assumptionabout the supplier’s status.

We call the due-date quotation heuristic SLCSD forthis simple decentralized case since it is based on theslack algorithm SL for the single facility case. Themanufacturer’s scheduling and due date setting heuristic,SPTA− SLCSD for this case follows:

In the above equations, dsi is the approximate com-

pletion time of job i in the supplier side assuming thateach arriving job is scheduled in the middle of the sup-plier queue. ωm

i denotes the approximate queue length infront of job i when it arrives to the manufacturer from thesupplier and slackm

i denotes the approximated length ofthe jobs that will arrive to the manufacturer after job i butwill be processed there before job i. The value (ds

i−ri)µs

approximates the number of jobs that will be finished be-fore job i at the supplier and will bring an extra workloadto the manufacturer and qs

i + (n− i)− (dsi−ri)µs approxi-

mates the maximum number of jobs that can arrive to themanufacturer from the supplier after job i.

We denote the objective value with this due datesetting heuristic for the simple decentralized caseZSPTA−SLCSD

n for an n job instance, and recall The-orem 2 in Section 2.1 states that SPTA − SLCSD isasymptotically optimal under the conditions describedabove.

2.3.3. The Decentralized Model with AdditionalInformation Exchange: As we will see in our compu-tational analysis in Section 2.4, there are significant gainsthat result from centralizing the control of this system.On the other hand, as we discussed above, there are fre-quently significant expenses and complexities inherent inmoving to a centralized supply chain, if it is possible at

all. Thus, firms may be motivated to consider limited orpartial information exchange to achieve some of the ben-efits of centralization. Indeed, it may be that limited in-formation exchange achieves many of the benefits of cen-tralized control, rendering complete centralization unnec-essary. In this section, we start to explore this question,by considering the case in which the supplier shares someof the information about his processes through a mecha-nism, so that presumably, the manufacturer can quote bet-ter due-dates to his customers. For this system, we findan effective scheduling and due-date quotation algorithmand analyze the gains by simple information exchange.

The Model In particular, we assume that the sup-plier shares limited information with the manufacturer byquoting intermediate due dates. In other words, when acustomer order arrives at the manufacturer, the manufac-turer immediately places an order with the supplier, andthe supplier quotes a due date for the suppliers subsystemto the manufacturer. The manufacturer can use this sup-plier due date, along with knowledge of his own shop andthe time that the order will take him, to quote a due dateto the customer. However, the model is in all other waysthe same as the simple decentralized model. The man-ufacturer still doesn’t know anything about the schedulethe supplier uses or the processing times of the jobs at thesupplier. In this case, however, the manufacturer can usethe due date given by the supplier to better estimate thecompletion times of orders at the supplier, and can thusquote more accurate due dates to the customer.

The Heuristic and Main Results Once again, we em-ploy a two-phase approach to due date quotation andscheduling, and since the manufacturer is independentfrom the supplier in this case, as in the simple decentral-ized case, it once again makes sense for the manufacturerto schedule using the SPTA rule.

Similarly, since supplier works independently fromthe manufacturer, he acts as a single facility and tries tominimize his own costs. Thus, we assume that the sup-plier uses the asymptotically optimal SPTA-SL heuristicfor scheduling and due date quotation described in Sec-tion 2.2 for the single facility case. Thus, the supplierquotes due dates to the manufacturer according to the fol-lowing rule:

dsi = ri + ps

i + Msi + slacks

i

However, the manufacturer is unaware of the sched-ule used by the supplier, and instead uses the due datesquoted by the supplier to estimate the completion timesof the orders at the supplier and sets his due dates accord-ingly.

We call the due-date quotation heuristic SLCDIE forthis decentralized case with information exchange since it


ALGORITHM 5: SPTA-SLCSD

Scheduling: Process the jobs according to shortest processing time, pmi , at the manufacturer.

Due-Date Quotation:dm

i = dsi + pm

i + ωmi + slackm

i

dsi = ri + qs

i µs

2 + slacksi

slacksi =

min{(n− i), (µs qsi2 )

λ−µs

2}µs

2 if λ− µs

2 > 0

(n− i)µs

2 otherwise

ωmi = max{tmm

i + (dsi−ri)Θi

µs − (dsi − ri), 0}

tmmi =

∑j∈B pm

j where B = set of jobs at manufacturer queue with pm < pmi at time ri

Θi = pr{pm < pmi }E{pm|pm < pm

i }λm = max{λ, µs}slackm

i =

{{(n− i + qs

i )− (dsi−ri)µs }Θi if λm −Θi ≤ 0

min{ ωmi

λm−Θi, (n− i + qs

i − (dsi−ri)µs )}Θi otherwise

ALGORITHM 6: SPTA-SLCDIE

Scheduling: Process the jobs according to shortest processing time, pm, at the manufacturer.Due-Date Quotation:

dmi = ds

i + pmi + ωm

i + slackmi

ωmi = max{tmm

i + (dsi−ri)Θi

µs − (dsi − ri), 0}

tmmi =

∑j∈B pm

j where B = set of jobs at manufacturer queue with pm < pmi at time ri

Θi = pr{pm < pmi }E{pm|pm < pm

i }λm = max{λ, µs}slackm

i =

{{(n− i + qs

i )− (dsi−ri)µs }Θi if λm −Θi ≤ 0

min{ ωmi

λm−Θi, (n− i + qs

i − (dsi−ri)µs )}Θi otherwise

is based on the slack algorithm SL for the single facilitycase. We summarize this approach below:

We denote the objective value for an n job instancewith this due date setting heuristic for this decentralizedcase with information exchange ZSPTA−SLCDIE

n . Re-call Theorem 3 in Section 2.1 which states that SPTA−SLCDIE is asymptotically optimal.

2.4. Computational Analysis: Using these schedul-ing and due date quotation heuristics, we designed com-putational experiments to explore how these asymptoti-cally optimal heuristics work even for smaller instances,and to better understand the value of centralization.

2.4.1. Efficiency of the Heuristics: To assess theperformance of the heuristics, we compared objec-tive values for various problem instance sizes to lowerbounds. Observe that in an optimal offline solution to thismodel, due dates equal completion times, and the prob-lem becomes equivalent to the problem of minimizing thesum of completion times of the orders.

Also note that for a single facility online problem,a preemptive SPTA schedule minimizes the total com-pletion times of the jobs for the preemptive version ofthe problem, and is thus a lower bound on the non-preemptive completion time problem. Therefore, if weuse that schedule and set the due dates equal to the jobcompletion times, and ignore the lateness component of

the objective, we have a lower bound on the objectivevalue of our problem.

For the supply chain models, we can use the samelower bound, focusing only on the processing times atone of the facilities, assuming that the job’s waiting timeat the queue of the other facility is zero. We use a preemp-tive SPTA schedule at our chosen facility, and then set thedue-date of a job equal to the completion time of that jobat the chosen facility plus the processing time of the jobat the other facility, once again ignoring the lateness com-ponent of our objective. For the case with exchangeableprocessing time distributions, we can select either facil-ity to focus on, and for the non-exchangeable cases, wefocus on the bottleneck facility. In all cases, we have alower bound on the completion time at one facility, andhave added only the processing time at the other facil-ity, ignoring capacity constraints at that facility, so this isclearly a lower bound on our objective.

For the single facility case, we simulate the systemusing different number of jobs arriving to the facility,and we use algorithm SPTA − SL to sequence andquote due-dates. We generate our problem sequences us-ing exponential distributions, hold the arrival rate con-stant, and vary the mean processing time, and relativedue date and tardiness costs. The ratios of ZSPTA−SL

n

to the lower bound for different combinations of n, cT


and µ values with cd = 1, λ = 1 and exponential inter-arrival and processing time distributions are presentedin Table 1. Observe that as the number of jobs, n, in-creases, ZSPTA−SL

n rapidly approaches the lower bound.The rate of convergence differs for different µ/λ (i.e.E(process time)/E(interarrival time)) ratios and for differ-ent cost values but in all cases it converges quite quicklyto the lower bound as n gets larger. The heuristic per-forms well even for small number of jobs. Note that eachentry in Table 1 reflects the average of five runs with dif-ferent random number streams.

For the centralized supply chain model, using the duedate quotation and scheduling algorithm SPTAp−SLCand its modifications for the unbalanced cases, we sim-ulate the system with different numbers of jobs for dif-ferent combinations of mean processing times µs andµm to explore the effectiveness of these heuristics, evenfor small problem instances. For these experiments, weuse exponentially distributed inter-arrival and processingtimes. The ratios of the objective function Z

SPTAp−SLCn

and the total tardiness, T , to the lower bound for differentcombinations of n, µs and µm with cd = 1, cT = 2, λ =1 and exponential inter-arrival and processing time distri-butions are in Table 2 where each entry represents the av-erage of five runs with different random number streams.As the number of jobs increases, Z

SPTAp−SLCn /LB ra-

tio approaches to 1 and, even for smaller instances, thatratio is still very close to one. Also, the T/LB ratio con-verges to 0 as the number of jobs increases, and that ratiois also very close to zero even for a small number of jobs.Thus, the asymptotically optimal due-date quotation al-gorithm and its variants seem to work well, even for smallnumbers of jobs.

2.4.2. Comparison of Centralized and Decentral-ized Models: The ultimate goal of this work is to com-pare centralized and decentralized make-to-order sup-ply chains, and to explore the value of information ex-change in this system. To that end, we prepared a com-putational study to investigate the differences betweenthe centralized and decentralized versions of this supplychain. For the centralized model we use the algorithmSPTAp − SLC for scheduling and lead time quotationand for the decentralized cases, we assume that the sup-plier uses a SPTA schedule according to his own process-ing times ps and the manufacturer uses the schedulingand due-date quotation algorithms described in section2.3. Table 3 shows the ratios of the objective values ofcentralized and decentralized models obtained by sim-ulating the system for n = 3000 jobs, where each en-try shows the average of five runs with different randomnumber streams. We used different combinations of µs

and µm with cd = 1, cT = 2, λ = 1 and exponen-tial inter-arrival and processing time distributions in thissimulation. In this table cen denotes the objective valuefor the centralized case, SD denotes the objective value

for the simple decentralized model and DIE denotes theobjective value with the decentralized model with infor-mation exchange.

Our experiments demonstrate that when the meanprocessing time at the supplier is smaller than the meaninterarrival time, i.e. when there is no congestion at thesupplier side, the centralized and decentralized modelslead to very similar performance. However, as the con-gestion at the supplier begins to increase, the value of in-formation and centralization also increases and the cen-tralized model starts to lead to much better results thanthe decentralized ones. Similarly, if the mean processingtime at the supplier is held constant, as the mean pro-cessing time at the manufacturer increases, the value ofcentralization decreases. As the supplier becomes morecongested than the manufacturer, the supplier is the bot-tleneck facility that ultimately determines system perfor-mance, but the manufacturer has limited knowledge ofthe bottleneck facility and thus gives inaccurate due datesto the customers. However, if the manufacturer is thebottleneck, the manufacturer already has the informationabout his processing times which have the majority ofimpact on system performance. The value of informa-tion about processing times at the supplier is limited be-cause they don’t have tremendous impact on system per-formance. In other words, in this case, the value of infor-mation is lower.

These experiments suggest that if there is little or nocongestion at the supplier, or if the manufacturer is sig-nificantly more congested than the supplier, centralizingcontrol of the system is likely not worth the effort (atleast, for the objective we are considering). However, ifthe congestion at the supplier increases, the value of cen-tralization increases and total costs can be dramaticallydecreased by centralizing the system.

Although centralization can significantly decreasecosts in some cases, if centralization is not possible orvery hard to implement, simple information exchangemight also help to decrease costs. As seen in Table 3,the losses due to decentralization can be cut in half bysimple information exchange. However, even with infor-mation exchange, the costs of decentralized models aremuch higher, about 80% in some cases, than centralizedones. So, if the congestion at the supplier is high, cen-tralization is worth the effort it takes to design and imple-ment information systems, design and implement supplycontracts, etc. However, if centralization is not possible,simple information exchange can also improve the levelof performance dramatically.

3. An Analysis of a Combined Make–to-Order/Make-to-Stock System: After analyzingthe pure MTO model, we consider a combined MTO-MTS supply chain composed of a manufacturer, servedby a single supplier working in a stochastic multi-item


Table 1: Ratios of ZSPTA−SLn to the lower bound

# of jobs cT =1.1 cT =1.5 cT =2 cT =5 # of jobs cT =1.1 cT =1.5 cT =2 cT =5µ=0.5 µ=1.5

10 1.00962 1.01815 1.02882 1.09278 10 1.03373 1.06192 1.09715 1.30856100 1.00258 1.00285 1.00317 1.00514 100 1.05480 1.05898 1.06421 1.095571000 1.00033 1.00036 1.00039 1.00062 1000 1.01558 1.01798 1.02098 1.038985000 1.00007 1.00008 1.00009 1.00014 5000 1.00753 1.00794 1.00846 1.01159

10000 1.00003 1.00003 1.00003 1.00006 10000 1.00541 1.00560 1.00583 1.00720µ=1 µ=210 1.10481 1.11671 1.13157 1.22079 10 1.04609 1.07794 1.11775 1.35663

100 1.01426 1.01605 1.01828 1.03165 100 1.07541 1.08011 1.08600 1.121291000 1.00677 1.00777 1.00903 1.01657 1000 1.01140 1.01636 1.02256 1.059765000 1.00270 1.00291 1.00319 1.00482 5000 1.00901 1.00925 1.00953 1.01128

10000 1.00177 1.00191 1.00219 1.00317 10000 1.00457 1.00491 1.00534 1.0079

Table 2: Ratios of the objective function ZSPTAp−SLCn and the total tardiness, T , to the lower bound

# jobs ZSP T Ap−SLCn

LB T/LB # jobs ZSP T Ap−SLCn

LB T/LB # jobs ZSP T Ap−SLCn

LB T/LBµs=1 µm=1 µs=2 µm=1 µs=5 µm=1

10 1.0202 0.0118 10 1.0344 0.0102 10 1.0462 0.0121100 1.0217 0.0059 100 1.0618 0.0096 100 1.0495 0.01221000 1.0087 0.0022 1000 1.0239 0.0052 1000 1.0127 0.00865000 1.0037 0.0010 5000 1.0102 0.0017 5000 1.0067 0.0023µs=1 µm=2 µs=2 µm=2 µs=5 µm=2

10 1.0344 0.0137 10 1.0263 0.0294 10 1.0471 0.0118100 1.0374 0.0040 100 1.0635 0.0160 100 1.0507 0.01341000 1.0133 0.0025 1000 1.0241 0.0057 1000 1.0137 0.00755000 1.0071 0.0012 5000 1.0108 0.0039 5000 1.0080 0.0024µs=1 µm=5 µs=2 µm=5 µs=5 µm=5

10 1.0552 0.0144 10 1.0553 0.0259 10 1.0467 0.0349100 1.0494 0.0053 100 1.0722 0.0159 100 1.0196 0.03381000 1.0191 0.0021 1000 1.0330 0.0046 1000 1.0117 0.01085000 1.0104 0.0016 5000 1.0143 0.0046 5000 1.0046 0.0089

Table 3: Ratios of the objective values of centralized and decentralized modelsµs µm SD/cen DIE/cen SD/DIE µs µm SD/cen DIE/cen SD/DIE0.5 0.5 1.00211627 1.00215099 0.9999653 2 0.5 2.07968368 1.50892076 1.378259030.5 1 1.0025820 1.00308953 0.99949407 2 1 2.08234862 1.5133057 1.376026380.5 2 1.01378038 1.01417906 0.99960689 2 2 1.94756315 1.44041736 1.352082530.5 5 1.00604253 1.00615847 0.99988477 2 5 1.31118576 1.05354521 1.244546261 0.5 1.02696404 1.02616592 1.00077776 5 0.5 2.4082784 1.86390096 1.292063501 1 1.02055926 1.01543491 1.00504646 5 1 2.40706065 1.86304885 1.292000821 2 1.02373349 1.01594891 1.00766237 5 2 2.35828183 1.82580914 1.291636551 5 1.00838256 1.00560849 1.00275859 5 5 2.08264443 1.63575747 1.27319878


environment. In this case, both facilities are allowed tocarry some level of inventory for each type of productinstead of operating a pure MTO system. So, in additionto designing effective scheduling and lead time quotationalgorithms, we want to find the optimal inventory levelsthat should be carried at each facility for this combinedsystem. In this system, the manufacturer and the supplierhave to decide which items to produce to stock andwhich ones to order. The manufacturer also has toquote due dates to arriving customers for make-to-orderproducts. The manufacturer is penalized for long leadtimes, missing the quoted lead times and for highinventory levels. In the following sections, we considerseveral variations of this problem, and design effectiveheuristics to find the optimal inventory levels for eachitem and also design effective scheduling and lead timequotation algorithms for centralized and decentralizedversions of this model. We also make extensive com-putational experiments to evaluate the effectiveness ofour algorithms, to analyze the benefits of the combinedMTO-MTS systems versus pure MTO or MTS systemsand to compare the centralized and decentralized supplychains.

3.1. Properties of the Model: We model two parties,a supplier and a manufacturer in a flow shop setting. Cus-tomer orders, or jobs, j, arrive at the manufacturer attime rj , and the manufacturer quotes a due date for eachorder, dj , when it arrives. We assume that there are ktypes of jobs with mean processing times µi for each typei=1,2...k. Jobs arrive to the system with rate λ and eacharriving job has a probability δi of being type i. We as-sume exponentially distributed, stationary and indepen-dent inter-arrival times, so, each job type i has an arrivalrate of λi = λδi. To begin processing each order, themanufacturer requires a component specifically manufac-tured to order by the supplier. The component type i re-quires mean processing time µs

i at the supplier, and theorder requires mean processing time µm

i at the manufac-turer.

We assume that a base-stock policy is used for inven-tory control of MTS items and starting with Ri units ofinventory, whenever a demand occurs, a production orderis sent to replenish inventory. Ri = 0 means a make-to-order production system is employed for job type i. Ifdemand is higher than the inventory level, then the extrademand is backlogged and satisfied later when it is pro-duced. It is also assumed that the system is not congestedand the interarrival times follow an exponential distribu-tion.

We aim to minimize the total expected inventory pluslead time plus tardiness costs in this system. Thus, theobjective function to minimize is

Z =k∑

i=1

{hiE[Ii] + cdi E[di] + cT

i E[Wi − di]+} (3.1)

where hi is the unit inventory holding cost, cdi is the unit

lead time cost and cTi is the unit tardiness cost for type

i. E[Ii] denotes the mean amount of inventory, E[di] de-notes the lead time mean and E[Wi − di] denotes themean tardiness.

We consider three versions of this model, which webriefly describe here and discuss in more detail in subse-quent paragraphs. In the centralized version of the model,the entire system is operated by a single entity, who isaware of the inventory levels and processing times bothat the supplier and the manufacturer. So, this single en-tity decides on the inventory levels for each class as wellas the production schedule and the lead times that shouldbe quoted to each customer. In the decentralized, full in-formation model, the manufacturer and the supplier areassumed to work independently, but the manufacturer hasfull information about both his processes and the supplier.They both try to minimize their own costs in a sequentialgame theoretic approach. The supplier acts first to deter-mine his optimal inventory levels and then the manufac-turer acts to minimize his own costs using the optimal in-ventory levels for the supplier. However, in the simple de-centralized model, the manufacturer and the supplier stillworks independently from each other but now the manu-facturer has no information about the supplier except theaverage delivery times of orders from the supplier.

The objective of our problem is to determine the op-timal inventory levels, a sequence of jobs and a set ofdue dates such that the total cost Z =

∑ki=1{hiE[Ii] +

cdi E[di] + cT

i E[Wi− di]+} is minimized. Clearly, to op-timize this expression, we need to coordinate due datequotation, sequencing and inventory management and anoptimal solution to this model would require simultane-ous consideration of these three issues. However, theapproach we have elected to follow for this model (andthroughout this thesis) is slightly different. Observe thatin an optimal off-line solution to this model, lead timeswould equal waiting times of jobs in the system. Thus,the off-line problem becomes equivalent to minimizing∑k

i=1{hiE[Ii] + cdi E[Wi]}. Of course, in an online

schedule, it is impossible to both minimize this func-tion and set due dates equal to completion times, sincedue dates are assigned without knowledge of future ar-rivals, some of which may have to complete before jobsthat have already arrived in order to minimize the sumof completion times. In this approach, we first determinea scheduling approach designed to effectively minimizethe sum of waiting times, and then based on that sched-ule, we find the optimal inventory levels to minimize∑k

i=1{hiE[Ii] + cdi E[Wi]} and design a due date quo-

tation approach that presents due dates that are generallyclose to the completion times suggested by our schedul-ing approach. This is the intuition behind the heuristicpresented below.

In section 2, we present effective scheduling and lead


time quotation algorithms for pure MTO versions of thissystem. We benefit from the properties of those algo-rithms and present modified versions of them for this sys-tem. However, note that, our results for the optimal inven-tory levels can be generalized to other schedules and leadtime quotation algorithms as long as the schedule is inde-pendent of the workload or inventory levels in the system.The schedule only effects the stationary distributions ofthe number of jobs in the system. We consider SPTA andFCFS scheduling algorithms for our calculations but theresults can be generalized to different schedules by re-calculating the stationary distributions only. We considerdifferent schedules in section 3.4 and compare them bycomputational analysis.

In the next section, we introduce a preliminary singlefacility model and analyze this model. In section 3.3, wepresent our supply chain models, algorithms, and resultsin detail, and in section 3.4, we present the computationalanalysis and a comparison of centralized and decentral-ized models.

3.2. Single Facility Model:3.2.1. The Model: Although our ultimate goal is to

analyze multi-facility systems, we begin with a prelim-inary analysis of a single facility system. We focus onfinding the optimal inventory levels for this system andthe conditions under which an MTO strategy or an MTSstrategy would be optimal for this facility. We also focuson designing effective scheduling and due date quotationheuristics for this system.

In this section, we consider a single facility that com-bines make-to-stock and make-to-order policies to mini-mize its inventory and lead time related costs. Our objec-tive in this system is to find an effective operating struc-ture to minimize the inventory and lead time related costs.In solving this problem, we need to determine the optimalvalues for base-stock levels Ri for each job type as wellas finding an effective scheduling and lead time quota-tion algorithm for these jobs to minimize the total costs.The model operates exactly as explained in the previoussection but with just a single manufacturer without anysupplier. We can think of this single facility model as aspecial case of the supply chain model where the process-ing times at the supplier are all 0, so the components areavailable to the manufacturer as soon as an order arrives.Also, the supplier doesn’t need to hold any inventory andthere is no cost related to the supplier.

Define two queues for the manufacturer in this model,the production queue and the order queue. Whenevera job arrives at the system, if that item exists in inven-tory, the demand is immediately satisfied from the inven-tory. Since that order is immediately satisfied, we don’tplace that job in the order queue. However, that job isstill placed on the production queue in order to replen-ish the inventory. When an order arrives and there isn’tany inventory of that item left, that job is placed at both

the order queue and the production queue and a lead timeneeds to be quoted for that item. Thus, the order queueincludes just the unsatisfied orders at any time while theproduction queue includes the items that are going to beproduced both to satisfy orders and to replenish inven-tory. The order queue is just a subset of the productionqueue.

The production queue operates exactly the same wayas a pure make-to-order system because even though wehave inventory at hand for an item, we place that job inthe production queue to replenish inventory. Thus, hav-ing inventory for an item does not impact the productionprocess, but does decrease the due date costs since we sat-isfy those orders immediately and don’t put them in theorder queue.

Since the production queue operates exactly the sameway as a pure MTO system, we employ the following al-gorithm, named SPTA-LTQ, for scheduling and lead timequotation, which is very similar to the algorithm SPTA-SL, designed for a pure MTO model single facility casein Section 2.2.

3.2.2. Analysis and Results: As mentioned in Sec-tion 3.1, we elect to employ a heuristic that first attemptsto find a schedule to minimize the total completion times,and then finds the optimal inventory levels to minimize afunction of the inventories and waiting times of the jobsin the system based on this schedule and then sets leadtimes that approximate the waiting times of jobs in thesystem with that schedule using the state of the system atthe time of the arrival of the order, in an effort to min-imize our objective function. The heuristic we proposesequences the jobs according to the Shortest ProcessingTime Available (SPTA) rule. Under the SPTA heuristic,each time a job completes processing, the shortest avail-able job which has yet not been processed is selected forprocessing. For a pure MTO system, although the prob-lem of minimizing completion times is NP-Hard, Kamin-sky and Simchi-Levi [18] found that the SPTA rule isasymptotically optimal for this problem, that is SPTA ruleis optimal for minimizing the sum of completion times asthe number of jobs goes to ∞. Also, note that this ap-proach to sequencing does not take quoted due date orinventory levels into account, and is thus easily imple-mented.

For the lead time quotation algorithm LTQ presentedabove, we present the following lemma.

Lemma 1. Consider a series of randomly generatedproblem instances of size n. Let interarrival times bei.i.d. random variables bounded above by some con-stant; the processing times be also i.i.d random vari-ables and bounded and the processing times and inter-arrival times be independent of each other. Also, letZSPTA

n =∑n

i=1 cdCi denote the total weighted deliverytimes of orders where Ci = ri+Wi is the delivery time oforder i and ZSPTA−LTQ

n =∑n

i=1{cdd′i+cT (Ci−d′i)+}


ALGORITHM 7: SPTA-LTQScheduling: Sequence the jobs in the production queue according to shortest processing time available

(SPTA) rule.lead time Quotation:

di =

{0 if Ii > 0 at ri

E[pi] + E[Mj ] +E[Mj ]λψiτi

1−λψiτiotherwise

where j is the job in the production queue that will be used to satisfy order i. Mj is the workload infront of job i at the time of arrival, ψi is the probability that an arriving job has processing time lessthan pi and τi = E[p|p < pi] is the expected processing time of a job given that it is less than pi

denote the total due-date plus tardiness costs with the al-gorithm SPTA-LTQ where d′i = ri + di is the quoteddue-date. Then, the lead time quotation algorithm LTQis asymptotically optimal to minimize this objective func-tion assuming SPTA scheduling rule is used, that is al-most surely,

limn→∞

ZSPTA−LTQn − ZSPTA

n

ZSPTAn

= 0

We also state the following lemma for the lead timequotation algorithm LTQ presented above:

Lemma 2. For the model explained above, for everyclass i, the expected value of the quoted lead time forclass i is equal to the expected waiting time of a class ijob, that is E[di] = E[Wi] where Wi denotes the actualwaiting time of job i in the order queue.

We use the SPTA-LTQ algorithm for scheduling andlead time quotation as above and based on this schedule,we find the optimal inventory levels for each class in thesubsequent parts of this section. However, note that dif-ferent schedules can also be considered and the resultsof our subsequent analysis can be easily modified by up-dating the stationary distributions of the number of jobsin the system. We consider the above algorithm for ourcalculations since it is an effective one to quote reliableand short due-dates. Also, for the supply chain models, inSection 3.3, we present effective algorithms for schedul-ing and lead time quotation and use them in our anal-ysis to find the optimal inventory levels, but again, ourresults regarding the optimal inventory levels can be gen-eralized to different scheduling and lead time quotationalgorithms easily.

To find the optimal inventory levels, we aim to mini-mize the following objective function:

k∑

i=1

{hiE[Ii] + cdi E[Wi]} (3.2)

where E[Ii] is the expected inventory level for job typei and E[Wi] is the expected waiting time for job type iin the order queue. There is an obvious tradeoff between

the inventory costs and the waiting time costs in this ob-jective. We can decrease the waiting times of the class ijobs by holding additional inventory of that type but thatwill increase the inventory costs. Observe that holdingadditional inventory for an item effects the order queueonly and does not effect the production queue. Thus, thewaiting times and the lead time quotation procedure forthe other classes aren’t effected by this.

Lemma 3. Objective function 3.2 is equivalent to thefunction

∑ki=1{hiE[Ii] + ciE[Ni]} where ci = cd

i /λi

and E[Ni] is the expected number of job type i in the or-der queue.

Arreola-Risa and DeCroix [1] considers the same ob-jective function

∑ki=1{hiE[Ii] + ciE[Ni] to minimize

by considering a FCFS schedule. We begin by restat-ing some of their results, expand them to include SPTAschedules and compare the two types of schedules in thissection.

Lemma 4.k∑

i=1

{hiE[Ii] + ciE[Ni]}

=k∑

i=1

hi

Ri∑x=0

(Ri − x)fi(x) + ci

∞∑

x=Ri

(x−Ri)fi(x)

(3.3)

where fi(x) denotes the probability of having x jobs oftype i in the production queue.

Lemma 5. Dividing the problem into k subproblems ac-cording to their types, and the solving for each type indi-vidually gives the optimal solution for the whole problem.

So, our problem decreases to minimizinghi

∑Ri

x=0(Ri − x)fi(x) + ci

∑∞x=Ri

(x − Ri)fi(x)for each i.

Theorem 6. The optimal level of inventory Ri is the min-imum value x ≥ 0 that satisfies

Fi(x) ≥ ci

ci + hi


Corollary 1. Produce item i MTO if and only if Fi(0) ≥ci

ci+hi

Corollary 2. An item’s production moves towards MTOif its unit lead time cost, processing time or arrival ratedecreases or unit holding cost increases.

Note that the results above regarding optimal inven-tory levels hold for a variety of queueing disciplines, notrestricted to a single server queue or an SPTA sched-ule, and they are also independent of the arrival or man-ufacturing process. However, these characteristics ef-fect Fi(x), the stationary distribution of the number inthe system. To assess the effectiveness of our schedul-ing our algorithm SPTA and to explore how the sched-ule used in the system effects Fi(x) and the objectivefunction, we analyze two different schedules SPTA andFCFS and present the following two corollaries for theseschedules. We also compare the objective function usingthese schedules through computational analysis in section3.4. Also, one can extend these results for other kinds ofschedules or queues with different characteristics by onlyconsidering the changes in Fi(x) for that queue.

Corollary 3. If FCFS scheduling rule is used in theproduction queue, instead of SPTA, assuming an M/G/1queue, to minimize 3.3, it is optimal to produce product iMTO if and only if:

k∑

j=1

δjE[e−λiµj ] ≤ (1− δi)ri

ri − (1− ρ)δi

where ri = ci

ci+hi, δi = λi

λ and ρ =∑k

i=1 λiµi

Corollary 4. If SPTA scheduling rule is used in the pro-duction queue, assuming it is an M/G/1 queue, it will beoptimal to produce product i MTO if and only if:

(1− ρ)(ξi) + λb(1− γb(ξi)λiγi(ξi)

≥ ri

where ξi = λi + λa − λaνa(λi), ri = ci

ci+hi, ρ =∑k

i=1 λiµi, λa =∑i−1

j=1 λj is the total arrival rate of

jobs shorter than class i, λb =∑k

j=i+1 λj is the totalarrival rate of jobs longer than class i, γi(z) = E[e−zpi ]is the Laplace transform associated with the processingtime of class i and νa(z) is the solution of the equationνa(z) = γa(z + λa − λaνa(z)).

3.3. Supply Chain Models: In this section, we an-alyze the inventory decisions, scheduling and due-datequotation issues for two-stage supply chains using the re-sults from our analysis in section 3.2. We develop effec-tive heuristics to find the optimal inventory levels at bothfacilities and design effective algorithms for schedulingand due-date quotation for both the centralized and de-centralized versions of these systems. These algorithms

allow us to compare the value of centralization and in-formation exchange in supply chains under a variety ofdifferent conditions.

When a manufacturer is working with a supplier, thecomponents may not be immediately available to themanufacturer at the arrival time of an order. The man-ufacturer has to wait for some time for the componentsto arrive from his supplier before he can start workingon that order. Thus, the supplier-manufacturer relation-ship effects the optimal levels of inventories that shouldbe held as well as the scheduling and lead time quotationdecisions.

We model this system as a two facility flow shopwith a manufacturer and a supplier where both partiescan choose to stock some of the items and use a make-to-order strategy for the others in a multi-item, stochasticenvironment. We assume that the supplier and the manu-facturer employs a one-to-one replenishment strategy anda base-stock policy for inventory control of their items.The manufacturer starts with an inventory of Rm

i units offinished goods and the supplier starts with an inventoryof Rs

i units of semi-finished goods that the manufacturerneeds to complete his production.

We again define the production and order queues sim-ilar to the way we did in the single facility model, butin this case, for both the supplier and the manufacturer.When an order arrives, if that item is in the manufac-turer’s inventory, the order is immediately satisfied anda lead time of 0 is quoted. However, a production orderof that class is sent to both the supplier and the manufac-turer to replenish the inventory of the manufacturer. Thatorder is not placed in the manufacturer’s order queue untilthe semi-finished goods are delivered to the manufacturerby the supplier. If the supplier also has inventory of thatclass, he sends it directly to the manufacturer and that or-der appears in the manufacturer’s production queue im-mediately. The supplier still places that order in his pro-duction queue to replenish his own semi-finished goodsinventory. If that item is neither in the manufacturer’s northe supplier’s inventory, then a lead time is quoted to thecustomer and a production order is sent to both facilitiesto satisfy this order. This order appears immediately inthe supplier’s production queue and after it is deliveredfrom the supplier, it appears in the manufacturer’s pro-duction queue. The manufacturer has to wait for sometime for the semi-finished goods to be delivered to himby the supplier to put that order in his production queue.However, if the supplier has this item in its inventory, thenthat order immediately appears in the manufacturer’s pro-duction queue as well as the supplier’s production queue.A shorter lead time is quoted in this case since the cus-tomer only needs to wait for the production at the manu-facturer and waiting time at the supplier is 0.

Let xsi and xm

i denote the amount of jobs of type i inthe supplier’s and manufacturer’s production queue, re-


spectively. Then, let Nsi denote the amount of jobs of

class i waiting in supplier’s order queue that should bedelivered to the manufacturer, Is

i denote the amount ofsemi-finished goods inventory of class i, Nm

i denote thetotal number of customers of class i waiting in the systemfor their orders to be delivered and Im

i denote the amountof finished goods inventory at the manufacturer of classi. Then,

Nsi = max{xs

i −Rsi , 0}

Isi = max{Rs

i − xsi , 0}

Nmi = max{xm

i + max(xsi −Rs

i , 0)−Rmi , 0}

Imi = max{Rm

i − xmi −max(xs

i −Rsi , 0), 0}

(3.4)

3.3.1. The Centralized Supply Chain Model:

The Model In some systems, the manufacturer has aclose relationship and perhaps even complete control overhis supplier. For example,the manufacturer and the sup-plier may belong to the same firm. In those cases, a cen-tral agent that has complete information about both par-ties and makes all the decisions about both firms will ob-viously be much more effective in minimizing the totalcosts in the system than the individual parties would be.

In this section, we assume that the manufacturer andthe supplier work as a single entity and they are bothcontrolled by the same agent that has all the informationabout both sides. The decisions about the scheduling atboth facilities and lead time quotation for the customer aswell as the inventory levels for each party are made bythis agent.

In the centralized model, our objective function tominimize is:

k∑

i=1

hsi E[Is

i ] + hmi E[Im

i ] + cdi E[di] + cT

i E[Wi − di]+

(3.5)

where hsi is the unit holding cost of semi-finished goods

at the supplier and hmi is the unit holding cost of finished

goods at the manufacturer.

Analysis and Results In this system, we use an ap-proach that is similar to the one we employed for the sin-gle facility case. We first find the optimal inventory levelsfor a schedule that is independent of the workload or in-ventory levels in the system (e.g. using a FCFS schedulein both facilities is such a schedule), to minimize the ob-jective function

Z(Rs, Rm) =k∑

i=1

{hsi E[Is

i ] + hmi E[Im

i ] + cdi E[Wi]}

(3.6)

Then, we present an effective scheduling algorithm con-sistent with this model to minimize the total waiting timesof the jobs in the system and a lead time quotation algo-rithm that matches these waiting times.

Using the definitions 3.4 and the equations cdi = λci

and E[Wi] = E[Nmi ]/λi due to Little’s law, and writing

them explicitly, we get the objective function 3.7 to min-imize in terms of the inventory amounts and the numberof jobs in the production queues of the supplier and themanufacturer both of which operate as pure MTO sys-tems.

Z(Rs, Rm) =k∑

i=1

Z(Rsi , R

mi )

=k∑

i=1

{hsi E[Is

i ] + hmi E[Im

i ] + ciE[Nmi ]}

=K∑

i=1

{hsi

Rsi∑

ysi =0

(Rsi − ys

i )P (xsi = ys

i )

+ hmi [

Rsi−1∑

ysi =0

Rmi∑

ymi =0

(Rmi − ym

i )P (xsi = ys

i , xmi = ym

i )

+Rs

i +Rmi∑

ysi =Rs

i

Rsi +Rm

i −ysi∑

ymi =0

(Rsi + Rm

i − ysi − ym

i )

∗P (xsi = ys

i , xmi = ym

i )]

+ ci[Rs

i−1∑ys

i =0

∞∑

ymi =Rm

i

(ymi −Rm

i )P (xsi = ys

i , xmi = ym

i )

+∞∑

ysi =Rs

i

∞∑

ymi =Rs

i +Rmi −ys

i

(ysi + ym

i −Rsi −Rm

i )

∗P (xsi = ys

i , xmi = ym

i )]} (3.7)

where Rs and Rm are the array of inventory levels atthe supplier and the manufacturer and xs

i and xmi are

the number of class i jobs at the supplier’s and manu-facturer’s production queue, respectively.

Observe that due to our assumption about the sched-ule used in the facilities, both the supplier and the manu-facturer’s production queue operates independent of Rm.In addition, the supplier’s production queue is indepen-dent of Rs. However, the manufacturer’s productionqueue depends on Rs, thus fm(x), the stationary distribu-tion of number of jobs at the manufacturer, is a functionof Rs.

Theorem 7. For fixed inventory levels Rsi for each class

i at the supplier, the optimal levels of inventory for themanufacturer are the minimum Rm

i values that satisfy:

P (xsi > Rs

i , xsi + xm

i ≤ Rsi + Rm

i )

+ P (xsi ≤ Rs

i , xmi ≤ Rm

i ) ≥ ci

ci + hmi

(3.8)


Corollary 5. If the manufacturer is working with a pureMTO supplier, then the manufacturer’s optimal inventorylevels for each class are the minimum Rm

i values thatsatisfy:

P (xsi + xm

i ≤ Rmi ) ≥ ci

ci + hmi

(3.9)

When we look at the supplier inventory levels, ob-serve that fm(x), the probabilities of the number of jobsat the manufacturer’s production queue, depends on theinventory levels Rs at the supplier which makes the prob-lem very hard to solve analytically.

However, we can find the optimal solutions undersome special conditions. We state the following theoremfor one of these conditions.

Theorem 8. For the centralized model, if hsi ≥ hm

i for aproduct type i, then an MTO strategy for type i at the sup-plier, that is holding no inventory of type i at the supplier,is optimal.

For other cases, finding the optimal solution is veryhard since the manufacturer’s production queue dependson the inventory levels Rs at the supplier which makesthe problem very hard to trace analytically. To find anapproximation on the optimal Rs values, we assume thatthe change in the stationary distributions of the numberof jobs at the manufacturer is negligible w.r.t. a change inthe amount of the inventory levels at the supplier and tryto find the optimal Rs values using this approximation. Inthat case, we can divide the problem into K subproblemsand analyze each class separately. Still, we can’t state aresult similar to Theorem 7 for the inventory values at thesupplier, since the objective function 3.7 doesn’t possessthe convexity structure in Rs

i for fixed Rmi . (i.e. Z(Rs

i +1, Rm

i )−Z(Rsi , R

mi ) is not nondecreasing in Rs

i for everyRm

i .)So, to determine the optimal levels of Rs, we employ

a one-dimensional search on Rsi . For each Rs

i , we calcu-late the optimal values of Rm

i , calculate the total cost us-ing the objective function 3.7 and pick the pair with min-imum cost for each class i. However, we can decreasethe search space using some properties of the objectivefunction.

Lemma 6. For this model, the function Z(Rsi , R

mi ) as

given in 3.7 is supermodular for every class i.

Theorem 9. Let Rsi ≥ 0 be the minimum value that sat-

isfies

P (xsi ≤ Rs

i ) ≥ci

ci + hsi

Then, the optimal level of inventory Rsi∗ ≤ Rs

i

So, there is no need to search for Rsi∗ beyond Rs

i .

Corollary 6. It is optimal for the supplier to use an MTOstrategy to produce product type i if F s

i (0) ≥ ci

ci+hsi

In general, for the fixed inventory value Rmi at the

manufacturer, if for every Rsi , ∆2Z(Rs

i ,Rmi )

∆2Rsi

≥ 0, then thefollowing theorem holds. An example of this case occurs,when hs

i ≥ hmi .

Theorem 10. For fixed inventory levels Rmi at the man-

ufacturer, if ∆2Z(Rsi ,Rm

i )∆2Rs

i≥ 0, the optimal levels of in-

ventory at the supplier are the minimum value Rsi that

satisfies:

(hmi + ci)P (xs

i > Rsi , x

si + xm

i ≤ Rsi + Rm

i )+ (hs

i + ci)P (xsi ≤ Rs

i ) ≥ ci (3.10)

We present effective scheduling and lead time quota-tion algorithms in Section 2 for a pure MTO supply chainsystem. Using the same ideas as in those algorithms, wedesign modified versions of them for our system with in-ventories. For the centralized supply chain model, assum-ing independent and exchangeable processing times, thealgorithm SPTAp − LTQC is outlined below.

Note that this algorithm is consistent with our as-sumptions for this model since the schedule is indepen-dent of the workload or inventory in the system and thelead times quoted with this algorithm satisfies the relationE[di] = E[Wi].

For the lead time quotation algorithm LTQC pre-sented above, we present the following lemma.

Lemma 7. Consider a series of randomly generatedproblem instances of size n. Let interarrival times bei.i.d. random variables bounded above by some constant;the processing times at each facility be also i.i.d randomvariables and bounded and the processing times and in-terarrival times be independent of each other. Also, letZ

SPTApn =

∑ni=1 cdCi denote the total weighted de-

livery times of orders with the SPTAp schedule whereCi = ri + Wi is the delivery time of order i andZ

SPTAp−LTQCn =

∑ni=1{cdd′i + cT (Ci − d′i)

+} denotethe total due-date plus tardiness costs with the algorithmSPTAp − LTQC where d′i = ri + di is the quoted due-date. Then, the lead time quotation algorithm LTQC isasymptotically optimal to minimize this objective func-tion for this centralized system assuming that SPTAp

scheduling rule is used to sequence jobs, that is almostsurely,

limn→∞

ZSPTAp−LTQCn − Z

SPTApn

ZSPTApn

= 0

The schedule used in the system only effects the sta-tionary distributions of the number of jobs in the system,(i.e.f(x)). For the schedule we presented above, the sup-plier is using an SPTA rule w.r.t. total processing time


ALGORITHM 8: SPTAp-LTQC

Scheduling: Process the jobs according to SPTAp (SPTA based on total processing time pi = psi + pm

i ) at thesupplier and FCFS at the manufacturer.

lead time Quotation:

dmi =

0 if Imi > 0 at ri

E[pmi ] + tmm

i if Imi = 0, Is

i > 0 at ri

dsi + E[pm

i ] + max{tmsi + tmm

i + slackmi − ds

i , 0} otherwise

whereds

i = E[psi ] + E[Ms

i ] + E[Msi ]λpr{p<pi}E{ps|p<pi}

1−λpr{p<pi}E{ps|p<pi}tmsi =

∑j∈A E[pm

j ] where A=set of jobs in supplier queue scheduled before job i and will be sent tomanufacturer.

tmmi =

∑j∈B E[pm

j ] where B=set of jobs in manufacturer queue at time ri

slackmi = (ds

i − psi )λpr{p < pi}E{pm|p < pi}+

∑j∈L min{(ds

i − psi )λj , I

sj }E[pm

j ] where L is the setof jobs that will be scheduled after i at the supplier

of the jobs and the manufacturer is scheduling his jobsaccording to FCFS. Because the production queue at thesupplier is independent of Rs and Rm and it operates justlike the single facility described in Section 3.2, the sup-plier can find the stationary distribution of the number inhis system using the probability generating function for asingle facility queue.

However, when we look at the manufacturer side, theinventory levels at the supplier effects the processes at themanufacturer, since the whole supply chain is describedas an inventory queue. This makes the problem very dif-ficult analytically. We use the common decompositionapproach to approximate the stationary distributions ofthe number of jobs in the manufacturer side. Note that,when the interarrival times are exponentially distributedin a queueing model like the well known Jackson networkmodel presented first in Jackson[16], the departure pro-cess is poisson distributed. Since the inter-arrival time toour system is exponentially distributed, we approximatethe departure process from the supplier with a poissondistribution and thus we assume that the arrivals to themanufacturer are poisson. So, we treat the processes atthe manufacturer as a single facility with multiple classeswith poisson arrivals for FCFS schedule.

3.3.2. The Decentralized Supply Chain, Full In-formation Model: While some supply chains are rela-tively easy to control in a centralized fashion, most oftenthis is not the case. Even if the stages in a supply chainare owned by a single firm, information systems, controlsystems, and local performance incentives need to be de-signed and implemented in order to facilitate centralizedcontrol. In many cases, of course, the supplier and man-ufacturer are independent firms, with relatively limitedinformation about each other. Implementing centralizedcontrol in these supply chains is typically even more dif-ficult and costly, since the firms need to coordinate theirprocesses, agree on a contract, implement an information

technology system for their processes, etc. Thus, for ei-ther centrally owned or independent firms, centralizationmight not be worth the effort if the gains from centraliza-tion are not big enough.

Although centralization in supply chains is generallya difficult and costly thing to do, at the same time, with adecentralized system, companies might lose a lot of theirprofits. In some cases, instead of completely centralizingthe system, the companies might just choose to share alltheir information with each other to increase their prof-its. Thus, we are motivated by the fact that informationexchange in some supply chains might increase the prof-its high enough so that complete centralization will beunnecessary. In this case, the manufacturer has all theinformation about the whole system but has no controlover the supplier’s decisions. For this system, we findthe optimal inventory levels for the manufacturer and thesupplier as well as an effective scheduling and due-datequotation algorithm. We also analyze the differences be-tween this decentralized model and the centralized modelthrough computational analysis in the next section.

The Model In this decentralized supply chain model,we assume that the two parties work independently fromeach other and aim to minimize their own costs. How-ever, the manufacturer has full information about the pro-cesses at the supplier as well as his own processes. Sincethe supplier works independently from the manufacturerand tries to minimize his own costs, the results fromthe single facility case applies for the supplier to deter-mine the optimal inventory levels at his facility and to se-quence the orders to minimize his own completion times.Then, for that sequence and inventories at the supplier,we find the optimal inventory levels and design an effec-tive scheduling and lead time quotation algorithm for themanufacturer.


Analysis and Results Since the supplier is indepen-dent from the manufacturer, he operates exactly thesame way as the single facility explained in the previ-ous section. Thus, the supplier’s objective is to minimize∑k

i=1{hsi E[Is

i ] + cdi E[ds

i ] + cTi E[W s

i − dsi ]

+}. Sincethe supplier works independently from the manufacturer,he uses the scheduling and lead time quotation algorithmas explained in the single facility case and the results insection 3.2 holds for the supplier. Thus, the optimal in-ventory levels for class i jobs at the supplier, Rs

i , are theminimum x ≥ 0 that satisfy

F si (x) ≥ ci

ci + hsi

Using the relations given in section 3.2, the optimal in-ventory levels for the supplier can be obtained.

Corollary 7. The optimal inventory levels for the sup-plier in the decentralized, full information case will begreater than or equal to the optimal level in the central-ized case if the same schedules are used for both cases.

Assuming that the supplier uses the optimal inventorylevels for his facility, the manufacturer tries to minimizehis own costs for those fixed Rs values. So, we try tominimize the objective function:

k∑

i=1

{hmi E[Im

i ] + ciE[Nmi ]}

Using the definitions 3.4 and writing them explicitly, forfixed Rs, we will get the objective function for the man-ufacturer to minimize as

Z(Rs, Rm) =k∑

i=1

{hmi E[Im

i ] + ciE[Nmi ]}

=K∑

i=1

{hmi

Rsi−1∑

ysi =0

Rmi∑

ymi =0

(Rmi − ym

i )

∗P (xsi = ys

i , xmi = ym

i )

+Rs

i +Rmi∑

ysi =Rs

i

Rsi +Rm

i −ysi∑

ymi =0

(Rsi + Rm

i − ysi − ym

i )

∗P (xsi = ys

i , xmi = ym

i )

+ ci[Rs

i−1∑ys

i =0

∞∑

ymi =Rm

i

(ymi −Rm

i )P (xsi = ys

i , xmi = ym

i )

+∞∑

ysi =Rs

i

∞∑

ymi =Rs

i +Rmi −ys

i

(ysi + ym

i −Rsi −Rm

i )

∗P (xsi = ys

i , xmi = ym

i )]} (3.11)

Observe that both the supplier and the manufacturer’sproduction queue operates independent of Rm for fixedRs.

Then, for fixed inventory levels Rsi for each class i

at the supplier, the optimal levels of inventory Rmi for

the manufacturer can be found by Theorem 7. Also, theoptimal inventory levels for a manufacturer working witha pure MTO supplier satisfy corollary 5.

Corollary 8. The optimal inventory level for the manu-facturer in the decentralized, full information case will beless than or equal to the optimal level in the centralizedcase if the same schedules are used both cases.

Corollary 9. Using an MTO strategy for class i jobs atthe manufacturer is optimal iff

P (xsi ≤ Rs

i , xmi ≤ 0) ≥ ci

ci + hmi

(3.12)

where Rsi is the optimal inventory level for class i at the

supplier.

Since the manufacturer is working independentlyfrom the supplier, he also uses an SPTA schedule accord-ing to his own processing times to sequence his jobs andemploys a lead time quotation algorithm similar to thecentralized model since he has full information about thesupplier. So, we use the following scheduling and leadtime quotation algorithm SPTA−LTQDFI for the man-ufacturer which satisfies E[di] = E[Wi].

For the lead time quotation algorithm LTQDFI pre-sented above, we present the following lemma.

Lemma 8. Consider a series of randomly generatedproblem instances of size n. Let interarrival times bei.i.d. random variables bounded above by some con-stant; the processing times at each facility be also i.i.drandom variables and bounded and the processing timesand interarrival times be independent of each other. Also,let ZSPTA

n =∑n

i=1 cdCi denote the total weighted de-livery times of orders with the SPTA schedule whereCi = ri + Wi is the delivery time of order i andZSPTA−LTQDF I

n =∑n

i=1{cdd′i +cT (Ci−d′i)+} denote

the total due-date plus tardiness costs with the algorithmSPTA−LTQDFI where d′i = ri +di is the quoted due-date. Then, the lead time quotation algorithm LTQDFI

is asymptotically optimal to minimize the objective func-tion for this decentralized system with full information as-suming that SPTA schedule is used for sequencing jobs,that is almost surely,

limn→∞

ZSPTA−LTQDF In − ZSPTA

n

ZSPTAn

= 0

In this case, the supplier is working as a single facilityand using SPTA rule w.r.t. his own processing times. So,the stationary distributions of the number of jobs at thesupplier can be found using the pgf for a single facilityqueue.

The manufacturer is also scheduling his jobs accord-ing to SPTA. We again use the decomposition approach


ALGORITHM 9: SPTA-LTQDFI

Scheduling: Schedule the jobs using SPTA rule according to pm.lead time Quotation:

dmi =

0 if Imi > 0 at ri

E[pmi ] + tmm

i + slackmi if Im

i = 0, Isi > 0 at ri where slackm

i = tmmi λpr{pm<pm

i }E{pm|pm<pmi }

1−λpr{pm<pmi }E{pm|pm<pm

i }ds

i + E[pmi ] + Mm

i + slackmi otherwise

whereds

i = E[psi ] + E[Ms

i ] + slacksi

slacksi = E[Ms

i ]λpr{ps<psi}E{ps|ps<ps

i}1−λpr{ps<ps

i}E{ps|ps<psi}

Mmi = max{tms

i + tmmi + slmi − ds

i , 0}tmsi =

∑j∈A E[pm

j |j ∈ A] where A=set of jobs of class k at supplier queue that will be sent to manufacturer s.t.E[pm

k ] < E[pmi ] and E[ps

k] < E[psi ].

tmmi =

∑j∈B E[pm

j |j ∈ B] where B=set of jobs at manufacturer queue at time ri with E[pm] < E[pmi ].

slmi = (slacksi + E[Ms

i ])λpr{ps < psi , p

m < pmi }E{pm|pm < pm

i }+∑

j∈L min{(slacksi + E[Ms

i ])λj , Isj }E[pm

j |j ∈ L]where L is the set of jobs of class k s.t. E[pm

k ] < E[pmi ] and E[ps

k] > E[psi ].

slackmi = Mm

i λpr{pm<pmi }E{pm|pm<pm

i }1−λpr{pm<pm

i }E{pm|pm<pmi }

to approximate the stationary distribution of the numberof jobs at the manufacturer’s site, similar to the central-ized model and assume that the arrivals to the manufac-turer are exponentially distributed. Thus, the stationarydistribution of the number of jobs at the manufacturercan also be found by using the pgf for a single facil-ity queue. However, since the manufacturer and supplierhave different processing times and thus different sched-ules, their stationary distributions will also be differentalthough both are found through the same pgf.

3.3.3. The Simple Decentralized Model: In manyreal-world systems, the supplier and the manufacturer areindependent firms and they have very little information orno information at all about each other. The manufactureris unaware of the processes at his supplier and needs tomake his own decisions without any information from thesupplier. Most of the time, the manufacturer is only awareof the average time it takes for the supplier to process anorder and send it to him. In such a decentralized supplychain system, we find the optimal inventory levels for themanufacturer and an effective way to schedule the ordersand to quote due-dates to the customers without any in-formation from the supplier.

The Model In this model, we again assume that themanufacturer and the supplier work independently fromeach other, but distinct from the full information model,the manufacturer has no information about the supplier,and thus he can’t deduce the production schedule or in-ventory levels at the supplier. The supplier still behavesthe same way as before and the results obtained in theprevious section for the supplier still hold.

However, in this case, the manufacturer only knowsthe average time it takes for the supplier to deliver a job

type i, denoted by E[dsi ]. Thus, for each job, the man-

ufacturer acts as if each job of type i is going to be de-livered to him from the supplier after E[ds

i ] time units.Based on this assumption, we determine the optimal in-ventory levels for the manufacturer and design an effec-tive scheduling and lead time quotation algorithm.

Analysis and Results Since the supplier acts as a singlefacility as in the previous section, the optimal inventorylevels for class i jobs, Rs

i are the minimum values thatsatisfy

F si (Rs

i ) ≥ci

ci + hsi

However, in this case, since the manufacturer onlyknows the average time it takes for an order of type i tobe delivered by the supplier, when an order arrives, he as-sumes that order will be delivered by the supplier to himafter E[ds

i ] time units. Observe that if we model the sup-plier as a M/D/∞ queue with deterministic processingtimes E[ds

i ] for type i and without any inventories, eachjob of type i will take exactly E[ds

i ] time units to be de-livered from the supplier to the manufacturer as assumedby the manufacturer in our system for this case and wecan use the same analysis and the results in previous sec-tions for this model. Thus, the objective function for the


manufacturer to minimize is

Z(Rm) =k∑

i=1

{hmi E[Im

i ] + ciE[Nmi ]}

=K∑

i=1

{hmi [

Rmi∑

ysi +ym

i =0

(Rmi − ys

i − ymi )P (xs

i + xmi = ys

i + ymi )]

+ci[∞∑

ymi +ys

i =Rmi

(ysi + ym

i −Rmi )P (xs

i + xmi = ys

i + ymi )]}

For this case, since we model the supplier as a M/D/∞queue with no inventories, using Corollary 5, the optimallevels of inventory are the minimum Rm

i values that sat-isfy:

P (xsi + xm

i ≤ Rmi ) ≥ ci

ci + hmi

We find the stationary distributions of the jobs at the sup-plier using the M/D/∞ queue model with processingtimes E[ds

i ]. Since we assume that each order is deliveredto the manufacturer at time ri +E[ds

i ] by the supplier, theprocesses at the supplier doesn’t effect the arrival processto the manufacturer and the distribution of the interarrivaltime of jobs to the manufacturer follows the same expo-nential distribution of the interarrival time of jobs to thesystem. Thus, the arrival process of the jobs to the man-ufacturer is also poisson. Both the supplier and manufac-turer are working as a single facility in this case, and thususing the relations for a single facility queue, we can findthe stationary distributions of the number of jobs and theinventory levels for the manufacturer and the supplier.

For this case, since the manufacturer focuses only onhis individual objective, he uses an SPTA schedule ac-cording to his own processing times. We assume that themanufacturer has no information about the supplier andis unaware of the schedule or inventory levels there, butthat from his previous experience, he can deduce an av-erage delivery time for each class of products. So, heuses this piece of information to quote due dates to hiscustomers. Thus, we use the following scheduling andlead time quotation algorithm SPTA− LTQSD for thissimple decentralized model:

For the lead time quotation algorithm LTQSD pre-sented above, we present the following lemma.

Lemma 9. Consider a series of randomly generatedproblem instances of size n. Let interarrival times bei.i.d. random variables bounded above by some con-stant; the processing times at the manufacturer be alsoi.i.d random variables and bounded and the process-ing times and interarrival times be independent of eachother. Also, let ZSPTA

n =∑n

i=1 cdCi denote the totalweighted delivery times of orders with the SPTA sched-ule where Ci = ri+Wi is the delivery time of order i andZSPTA−LTQSD

n =∑n

i=1{cdd′i + cT (Ci − d′i)+} denote

the total due-date plus tardiness costs with the algorithm

SPTA−LTQSD where d′i = ri + di is the quoted due-date. Then, the lead time quotation algorithm LTQSD

is asymptotically optimal to minimize this objective func-tion for this simple decentralized system assuming thatSPTA schedule is used for sequencing jobs, that is almostsurely,

limn→∞

ZSPTA−LTQSDn − ZSPTA

n

ZSPTAn

= 0

3.4. Computational Analysis: Using the inventoryvalues determined by the results in the previous sectionsand the scheduling and lead time quotation algorithms,we designed several computational experiments to assessthe effectiveness of our algorithms, how the combinedMTS/MTO systems differ from pure MTO or MTS sys-tems and the differences between centralized and decen-tralized supply chains. For the centralized system, weassume that both parties cooperate and use the heuristicsexplained in section 3.3.1 for finding the inventory levels,scheduling and lead time quotation. For the decentralizedsystems, we assume that each facility acts independentlyfrom each other and use the heuristics that will minimizetheir own costs as explained in sections 3.3.2 and 3.3.3for finding the inventory levels, scheduling and lead timequotation. We implement our heuristics using C++ andpresent the results in the following sections.

3.4.1. Effectiveness of a Combined System and Ef-ficiency of Heuristics for a Single Facility: First, weconsider a single facility system using n = 1000 jobsfor each simulation. Then, for each of these n job in-stances, the ratios of the total costs Z =

∑ki=1{hiE[Ii]+

cdi E[di] + cT

i E[Wi − di]+}, on average, are displayed inTable 4 where E[Ii] denotes the average inventory, E[di]denotes the average quoted lead time and E[Wi−di]+ de-notes the average tardiness in a simulation run for producttype i.

We consider different scenarios regarding differentnumber of job types k and different multipliers h, cd andcT for cost functions to evaluate the effect of these pa-rameters on our system. For each of these cases, we de-signed 10 scenarios with different arrival and processingrates, each having exponential distributions. Each valuein Table 4 denotes the average of these 10 scenarios.

We present the comparison of combined MTO-MTSsystem with pure systems and with different schedulesfor a single facility using cd = 2 and different combi-nations of h, cT and k in Table 4. The first and sec-ond columns contain the comparison of the combinedMTS/MTO system using SPTA-LTQ algorithm with pureMTS and MTO systems, respectively. Observe that thecombined system, on average, provides a 20% decreasein costs as opposed to the pure MTS systems and a 15%decrease as opposed to the pure MTO systems. Also, inthe third column, we compare the costs with the SPTA-LTQ algorithm for this combined system with a relevant


ALGORITHM 10: SPTA-LTQSD

Scheduling: Process the jobs according to shortest processing time at the manufacturer.lead time Quotation:

di =

0 if Imi > 0 at ri

E[pmi ] + tmm

i + slackmi if Im

i = 0, Isi > 0 at ri where slackm

i = tmmi λpr{pm<pm

i }E{pm|pm<pmi }


i }E[ds

i ] + E[pmi ] + Mm

i + slackmi otherwise

whereE[ds

i ] is the average delivery time of class i products from the supplier.Mm

i = max{tmmi + E[ds

i ]pr{pm<pmi }E{pm|pm<pm

i }E[ps] − E[ds

i ], 0}slackm

i = Mmi λpr{pm<pm

i }E{pm|pm<pmi }


i } }

lead time quotation algorithm for a FCFS schedule. Wesee that SPTA-LTQ algorithm decreases the costs about15% on average as opposed to a FCFS-LTQ algorithm.

We also see the effect of the parameters on the sys-tem performance in Table 4. Unsurprisingly, as the unitinventory holding cost increases while all other param-eters remain constant, the combined system moves to-ward a pure MTO system and gives much better resultsthan pure MTS systems since holding inventory becomesmuch more costly as h increases. The system is effectedthe same way as the unit due-date cost, cd, decreases be-cause in that case lead times become less important andan MTO system becomes much more attractive as cd de-creases. We also see that SPTA-LTQ algorithm givesmuch better results than FCFS-LTQ algorithm, as h in-creases (alternatively cd decreases) because the schedulehas an important effect on completion times, thus due-dates, in MTO systems, while for MTS systems, sincethe orders are satisfied from the inventory, minimizing thecompletion times of the jobs is not so important. On theother hand, as the unit tardiness cost, cT , increases whileall other parameters remain constant, MTO systems giveworse results and pure MTS systems become much moreattractive. Also, we see that the performance differencebetween SPTA-LTQ algorithm and FCFS-LTQ algorithmbecomes less and less as cT increases and FCFS-LTQ al-gorithm start to give better results as cT becomes veryhigh. This happens because, with FCFS-LTQ algorithm,the future arrivals don’t effect the completion times ofthe jobs that have already arrived, thus we can quote thelead times exactly as the completion times of the jobs andthere will be no tardiness. However, with SPTA-LTQ al-gorithm, there will be some tardiness cost and as cT in-creases, the cost of tardiness overcomes the gains in totalcompletion times with SPTA-LTQ algorithm and the totalcosts with FCFS-LTQ becomes lower than the total costswith SPTA-LTQ algorithm.

3.4.2. Effect of Inventory Decisions on SupplyChains: We next explore the effect of inventory deci-sions on our system without considering the lead time

quotation. So, for a fixed schedule, we compare the ob-jective functions Z ′ =

∑ki=1{hs

i E[Isi ] + hm

i E[Imi ] +

cdi E[Wi]} to explore only the effect of inventory deci-

sions in this system unaffected by lead time quotation.Recall that we made some assumptions about the interac-tion between the supplier and the manufacturer regardingthe stationary distributions of the number of jobs at themanufacturer and we also assumed that inventory valuesof different types at the supplier don’t effect the stationarydistributions of other types at the manufacturer. We ex-plore how these assumptions effect the optimal inventorylevels and the objective function with this computationalstudy. We also compare the centralized and decentralizedversions of the combined MTS/MTO supply chain withthis objective function. Table 5 shows the ratios of ob-jective functions Z ′ for different cases using cd = 2 anddifferent combinations of hs, hm and k.

To explore the effectiveness of our algorithms, wecompare the objective function using our inventory lev-els for the centralized model with the minimum objectivefunction for that case. In our simulations, we find theminimum objective functions by trying all the possiblecombinations of inventories of each type for the supplierand the manufacturer and selecting the one that gives thebest objective value. We use these minimum objectivevalues as lower bounds in our simulations. However, thisprocess takes a very long time, especially when the num-ber of types are big. Although we can find the optimalinventories by trying all possible solutions for the caseswe analyzed in this experiment since they are relativelyof small size, note that this method becomes almost im-possible to apply as the problem size gets bigger and big-ger. For example consider the case when there are 10different jobs and the inventory of each job type at thesupplier and the manufacturer can take a value from 0to 5. In that case we need to evaluate (5 ∗ 5)10 differ-ent possible solutions for the trial and error method andif each of them takes a nanosecond(10−9 seconds), thewhole process takes more than a day and becomes almostimpossible to apply for even bigger problems. However,


Table 4: Comparison of combined MTO-MTS system with pure systems and with different schedules for a singlefacility

h=0.5cT =2.5

ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

h=1cT =2.1

ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

k=3 0.963 0.526 0.942 k=3 0.829 0.880 0.894k=5 0.972 0.613 0.973 k=5 0.798 0.844 0.850

k=10 0.947 0.734 0.926 k=10 0.778 0.831 0.724h=1

cT =2.5ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

h=1cT =3

ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

k=3 0.832 0.906 0.913 k=3 0.847 0.873 0.922k=5 0.814 0.827 0.897 k=5 0.825 0.829 0.908

k=10 0.795 0.872 0.762 k=10 0.808 0.820 0.795h=2

cT =2.5ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

h=1cT =5

ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

k=3 0.724 0.936 0.846 k=3 0.880 0.845 0.931k=5 0.719 0.934 0.823 k=5 0.896 0.796 0.911

k=10 0.683 0.967 0.695 k=10 0.913 0.753 0.924h=5

cT =2.5ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

h=1cT =10

ZMT O−MT S

ZMT S

ZMT O−MT S

ZMT O

ZSP T A−LT Q

ZF CF S−LT Q

k=3 0.573 1 0.837 k=3 0.925 0.743 1.032k=5 0.551 1 0.776 k=5 0.946 0.719 1.131

k=10 0.472 0.987 0.672 k=10 0.953 0.651 1.096average 0.754 0.858 0.838 0.866 0.799 0.926

with our heuristic, there is no interaction between the dif-ferent types and we consider them separately. Also, weonly do a one-dimensional search over the supplier’s in-ventory levels up to an upper bound. Thus, for the samecase, we only need to evaluate 5 ∗ 10 possible solutionsat the worst case which can be done immediately and canbe easily applied to problems of much bigger size.

We construct our lower bound by using the inventoryvalues found by the trial and error method that gives theminimum objective value and compare it with the objec-tive value obtained by using the inventory values foundby our heuristic. The first column in Table 5 comparesthese two objective functions for the centralized supplychain model. We see that the inventory values found byour heuristic are very close to the optimal inventory val-ues and there is only a 5% difference, on average, be-tween the minimum costs and the costs obtained by usingour inventory values. The difference is mainly due to ourassumption that having inventory of type i at the supplierdon’t effect the stationary distributions of the number ofjobs at the manufacturer which isn’t the case in reality.

When we compare the centralized and decentralizedmodels, we see that the costs with the centralized modelis, on average, 10% less than the decentralized modelwith full information. The cost savings due to inventorydecisions increase to more than 15% when we comparethe centralized model with the simple decentralized case.We also see that, without centralization, if the manufac-turer has full information about the supplier, he can de-crease the costs by about 7% by only adjusting his owninventory levels without changing anything else. We seethat if the manufacturer had control over the supplier, hecould cut his costs significantly. However, even if the

manufacturer didn’t have control over the supplier buthad full information about the whole system, he still cancut his costs and increase his profits.

We can also see the effects of the parameters hs, hm,cd and k on the system performance. We see that asthe inventory holding cost at the supplier, hs, increaseswhile everything else remains the same, our heuristicgives closer results to the lower bound. A intuitive expla-nation for this is; as hs increases, the optimal inventorylevels at the supplier and their effect on the manufacturerdecreases as we assumed in our approximation. Besides,if the supplier uses a pure MTO strategy, then our heuris-tic finds the optimal solution since our approximationsbecome exact for a pure MTO supplier. The same effectoccurs as hm decreases because in that case it would bebetter to carry inventories at the manufacturer instead ofthe supplier and the inventory levels at the supplier andtheir effect on the manufacturer decreases as above. Sim-ilarly, as cd decreases, the inventory levels at the supplierdecreases, too and our heuristic gives closer results to theoptimal solution.

When we compare the supply chain models, we seethat as hs decreases, decentralized models give closer re-sults to the centralized one. This is because as hs de-creases, the optimal inventory levels at the supplier forthe centralized model become close to the upper boundwe presented in Theorem 9 which is also the optimal in-ventory level for the supplier in the decentralized modelsassuming that the same schedule is used in both cases.Thus, as hs decreases, the inventory levels at the sup-plier, thus the inventory levels at the manufacturer forthe centralized and decentralized models become closeto each other and the objective values with the decentral-


ized models become closer to the centralized model ob-jective value. We can see the same effect as hm increases,because in the centralized model as hm increases, the in-ventory levels at the manufacturer decreases and the in-ventory levels at the supplier increases becoming close tothe upper bound. Thus, as explained above, the decen-tralized models give closer results to the centralized one.

3.4.3. Effectiveness of a Combined System andEfficiency of Heuristics for Supply Chains: We thencomplete a similar study which includes lead timequotation, and compare the objective functions Z =∑k

i=1{hsi E[Is

i ]+hmi E[Im

i ]+cdi E[di]+cT

i E[Wi−di]+}.We use the multipliers hs = 0.5, hm = 1, cd = 2 andcT = 2.5 for this case. Through this analysis, we com-pare the costs for the whole system for the centralizedand decentralized models. In the first column in Table 6,we compare the total costs for the centralized model us-ing the algorithm SPTAp − LTQC with the algorithmthat schedules the jobs according to FCFS in both sys-tems and quotes lead times similar to our heuristic. Weconclude that the schedule used to produce the jobs hasan important effect on the total costs and we see that, onaverage, our heuristic performs about 20% better than thecommonly used schedule FCFS in the industry. Observefrom Table 4 that this was also the case for the single fa-cility case.

Also, when we compare the second columns of bothtables, we see that having full information about the sup-plier helps the manufacturer a lot to quote better leadtimes and there is very little difference between costs dueto lead time quotation for centralized and decentralizedwith full information cases. The difference between costsis mainly because of inventory decisions in this compari-son.

However, when we compare the simple decentralizedmodel with the centralized model, we see that the costdifference increases significantly. Since the manufac-turer has very little information about the supplier, he canno longer quote reliable due-dates and the costs increasedramatically and the costs with the simple decentralizedmodel are about 40% worse than the centralized model.Observe that, although there weren’t much difference be-tween the decentralized model with full information andthe simple decentralized model in Table 5, in Table 6,this difference increases significantly to 30% mainly dueto lead time quotation. We conclude that having full in-formation about the supplier is critical to the manufac-turer for lead time quotation. In addition, the inventorycosts can be decreased dramatically if the manufacturerhas complete control over the supplier in addition to thefull information case.

4. Complex Supply Chain Networks: In this section,we consider more complex supply chain networks com-posed of several facilities with different classes of rela-

tionships. Utilizing from the results from the previoussections, we design effective heuristics for determininginventory levels, sequencing the orders at each facilityand quoting reliable and short lead times to customersfor these complex networks. We also complete extensivecomputational experiments to evaluate the effectivenessof our algorithms and to analyze the benefits of the com-bined MTO-MTS systems versus pure MTO or MTS sys-tems.

4.1. Model: We model a multi-facility supply chainthat is composed of a main manufacturer and its internaland external suppliers. The manufacturer receives ordersfrom the customers and delivers it immediately if it hasthat product in its inventory or quotes a due date to thecustomer if it doesn’t. We assume that there are a total ofN facilities in the supply chain and there are K types ofdifferent products this firm offers to its customers, eachwith a different demand rate. The manufacturer wantsto quote short and reliable due-dates and doesn’t want tokeep too much inventory either. Thus, we want to min-imize the objective function

∑Ki=1{(

∑Nj=1 hijE[Iij ]) +

cdi E[di] + cT

i E[Wi − di]+} composed of the inventorycosts, lead time costs and tardiness costs.

If the processing times are short and the total time re-quired to produce a product is less than the desired leadtime, then the company doesn’t need to keep any inven-tory and uses an MTO approach. However, if the totalprocessing time is more than the desired lead time, due tothe stochastic nature of demand and lead times, the com-pany has to start processing before the orders actually ar-rive and keep some work-in-process or finished goods in-ventory to achieve the desired lead time. Instead of keep-ing only finished goods inventory at the manufacturer, wecan choose to store inventory at other facilities and use anMTO approach at others. We aim to find the optimal lo-cations to store inventory in this chain to minimize thesystem-wide inventory costs and design effective algo-rithms to quote short and reliable lead times.

We consider a supply chain like in Figure 1 wherethe same firm owns some of the facilities at different lo-cations drawn with rectangles and also has independentoutside suppliers drawn with circles. The company hastotal control over its own facilities; however, it has nocontrol over outside suppliers. We can also think of theinternal suppliers as different machines in a facility wherea single decision agent makes all of the decisions and theexternal suppliers as the suppliers of this company whichthis decision agent can not control. The decision agentcan choose to stock some WIP inventory in this systeminstead of choosing to stock only the finished goods in-ventory.

Since the demand is stochastic, the firm needs to carrysome inventory to respond to customer orders in a shorttime. We assume that an initial inventory amount (i.e.safety stocks) of xij for raw materials and yi for finished


Table 5: Effect of inventory decisions on the centralized and decentralized systemshs=0.5hm=2

Z′LB

Z′Cen

Z′Cen

Z′DF I

Z′Cen

Z′SD

Z′DF I

Z′SD

hs=0.5hm=0.6

Z′LB

Z′Cen

Z′Cen

Z′DF I

Z′Cen

Z′SD

Z′DF I

Z′SD

k=3 0.916 0.963 0.896 0.930 k=3 0.964 0.834 0.794 0.952k=5 0.933 0.925 0.849 0.917 k=5 0.959 0.877 0.808 0.921

k=10 0.929 0.923 0.832 0.901 k=10 0.967 0.892 0.836 0.937hs=1hm=2

Z′LB

Z′Cen

Z′Cen

Z′DF I

Z′Cen

Z′SD

Z′DF I

Z′SD

hs=0.5hm=1

Z′LB

Z′Cen

Z′Cen

Z′DF I

Z′Cen

Z′SD

Z′DF I

Z′SD

k=3 0.921 0.935 0.868 0.928 k=3 0.943 0.945 0.890 0.942k=5 0.949 0.914 0.855 0.935 k=5 0.909 0.856 0.788 0.920

k=10 0.952 0.922 0.862 0.934 k=10 0.918 0.875 0.808 0.923hs=1.9hm=2

Z′LB

Z′Cen

Z′Cen

Z′DF I

Z′Cen

Z′SD

Z′DF I

Z′SD

hs=0.5hm=5

Z′LB

Z′Cen

Z′Cen

Z′DF I

Z′Cen

Z′SD

Z′DF I

Z′SD

k=3 0.969 0.856 0.802 0.937 k=3 0.922 0.961 0.881 0.917k=5 1 0.839 0.784 0.934 k=5 0.911 0.975 0.874 0.896

k=10 1 0.803 0.771 0.960 k=10 0.936 0.953 0.841 0.882average 0.952 0.898 0.835 0.931 0.939 0.908 0.836 0.921

Table 6: Comparison of centralized and decentralized supply chains for combined MTO-MTS systemZSPTAp−LTQC /ZFCFS−LTQ ZCen/ZDFI ZCen/ZSD ZDFI/ZSD

k=3 0.859 0.954 0.679 0.712k=5 0.775 0.857 0.575 0.671k=10 0.682 0.893 0.584 0.654

Average 0.772 0.902 0.613 0.680

Customer

Figure 1: Supply Chain Structure


goods are held at facility i. A customer order triggersa production order in the system and the orders have towait a lead time depending on the amount of initial in-ventories. We desire short lead times without carryingtoo much inventory, so we aim to minimize a cost func-tion:

Z =K∑

i=1

{(N∑

j=1

hijE[Iij ]) + cdi E[di] + cT

i E[Wi − di]+}

that is composed of the lead times and the safety stocksin the whole system.

Each order k requires a processing time of pik at facil-ity i and a transshipment time tijk to move from facilityi to j. Also for each individual order l, wil denotes thewaiting time of order l at facility i, i.e. wil is the timestarting with the arrival of the required parts for order lto facility i and ending with the processing of that orderat facility i. Note that if there is no queue at facility iwhen order l receives to facility i, wil = pik. Other-wise, wil = pik + {waiting time in queue of facility i}.Also, we assume that each facility quotes a due-date fil

for each order to its downstream member and eventuallythe manufacturer to its customers.

For this system, we need to decide on a schedul-ing rule and a due-date quotation algorithm as well ashow much inventory to store at each of the facilities.The objective of our problem is to determine the opti-mal inventory levels, a sequence of jobs and a set of duedates such that the total cost

∑Ki=1{

∑Nj=1 hijE[Iij ] +

cdi E[di] + cT

i E[Wi− di]+} is minimized. Clearly, to op-timize this expression, we need to coordinate due datequotation, sequencing and inventory management and anoptimal solution to this model would require simultane-ous consideration of these three issues. However, theapproach we have elected to follow for this model (andthroughout this project) is slightly different. Observethat in an optimal offline solution to this model, leadtimes would equal actual waiting times of jobs in thesystem. Thus, the problem becomes equivalent to mini-mizing

∑Ki=1{(

∑Nj=1 hijE[Iij ])+cd

i E[Wi]}. Of course,in an online schedule, it is impossible to both minimizethis function and set due dates equal to completion times,since due dates are assigned without knowledge of fu-ture arrivals, some of which may have to complete be-fore jobs that have already arrived in order to minimizethe sum of completion times. In this approach, we firstdetermine a scheduling approach designed to effectivelyminimize the sum of completion times, and then basedon that schedule, we find the optimal inventory levels tominimize

∑Ki=1{(


i E[Wi]} and thenwith these inventories, we design a due date quotationapproach that presents due dates that are generally closeto the completion times suggested by our scheduling ap-proach.

4.2. Analysis and Results: For this model, we use thesame ideas for scheduling and lead time quotation as weuse in previous sections and adjust them for this morecomplex situation. However, when we first attempted touse the same analysis that we did in section 3 to find theoptimal inventory levels at each of the facilities, we seethat the model becomes very complex and difficult to ana-lyze. Thus, we develop another approach to approximatethe optimal inventory levels that should be stored at eachfacility for each product type for this system. We explainour approach to find the inventory levels as well as thealgorithms we use for scheduling and lead time quotationin detail in the following sections.

4.2.1. Single Product Type Model: In this section,we assume that the firm produces only one type of prod-uct and that they use FCFS scheduling rule at all facilitiessince there is no difference between orders.

To find the optimal inventory levels at each of the fa-cilities, assuming x ≤ p, we use the relation l = p− x ateach facility where p is the time units required to processan order, l is the lead time and x is the length of time thatthe safety stock that we carry is worth for a specified ser-vice level. Every time a customer order arrives, we startto produce one item to replenish inventory or to satisfyfuture orders. Then, the demand for the first x time unitsis satisfied from the inventory and since the first order fin-ishes processing at time p, after the inventory is depletedat time x, the first order that arrives, can only be satisfiedat time p causing a lead time l = p − x. Note that thisis a conservative approach and overestimates the amountof inventory that we need to carry. In reality, the samelead time might be achieved by carrying much less in-ventory but holding this amount of inventory ensures thatthe specified service level is achieved for this lead timeand every order is satisfied with a lead time less than l forsure at this service level.

We define the parameters and variables and write theLP formulation of the model below:

Parameters:i = subscript used to describe the facilities in the sup-

ply chain starting with 1 denoting the manufacturer.vi =unit value of parts produced at facility ipi = processing time of parts at facility itij = transshipment time to move parts from facility

i to jh1

ij = unit raw material inventory cost at facility i forthe parts produced at facility j

h2i = unit finished part inventory cost at facility i.

cd = cost of a unit increase in response time to orders.S = set of facilities that belong to the firmE = set of external suppliersPi = set of facilities that are immediate predecessors

of facility ifi for ∀i ∈ E = committed response time of orders

from external supplier i ∈ E


Variables:xij = units of time that the safety stock of raw ma-

terials received from facility j and stored at facility i toachieve the desired service level.

yi = units of time that the finished parts stored atfacility i that achieves the desired service level.

wi = waiting time of orders at facility i, i.e. wi is theaverage time starting with the arrival of the required partsfor the production of an order at facility i and ending withthe processing of that order at facility i. Note that if thereis no queue at facility i when an order receives to facilityi, wi = pi. Otherwise, wi = pi + {waiting time in queueof facility i}.

fi for ∀i ∈ S = total lead time of orders up to facilityi ∈ S, i.e lead time between the arrival of a customer tothe system and the completion of the parts required forthat order at facility i.

Min

N∑

i=1

[N∑

j=1

{h1ijxij}+ h2

i yi] + cdf1

s.t. max{maxi∈Pj{max{fi + tij − xij , 0}}+wi − yi, 0} ≤ fj for ∀j ∈ S

All variables ≥ 0 (4.1)

In this system, if we assume that there is no conges-tion or queues in the system (e.g. there are assembly linesor infinite servers at all the facilities and each order is puton the assembly line and starts processing as soon as therequired materials for that order receives to that facility),there will be no queues and we use only the processingtimes as the waiting times at the facilities. However, if weassume that there is a single server (or M < ∞ servers)at a facility, then there will be congestion and queues inthe system. In that case, we approximate the waiting timeof orders at facility i by using the expected waiting timeof a job in that queue with arrival rate Di and mean pro-cessing time pi. We use E[Wi] instead of pi and use thesevalues in the LP formulation.

Also, if we assume that the manufacturer has a com-mitted response time to its customers, which is generallythe case in competitive markets, we fix the committed re-sponse time and try to minimize the total inventory costswith the above LP model with the fixed lead time as aparameter instead of a variable.

For due-date quotation, observe that if there is inven-tory at a facility, the components are immediately satis-fied from that facility and the order doesn’t have to waitany time for the processing at or before that facility. If thewaiting time of an order at a facility is constant and is noteffected by the other orders (e.g. an assembly line systemor an infinite server queue system), then the due-dates arequoted by the following algorithm.

However, if there are queues in the system, then thelead time of an order is effected by the other orders and

the state of the system at time rk, the arrival time of orderk to the system. For this model, since there is only oneproduct and a FCFS scheduling rule is used, we don’tneed to consider future arrivals and we can quote the duedates by only considering the current state of the systemat the time of arrival. An order has to wait only for theprocessing of orders that are already at the queue of a fa-cility. Let wk

i denote the waiting time of an order k at fa-cility i.(Note that wk

i = pi for the assembly line system)and Uk

i denote the set of orders that arrived to the sys-tem before rk but not yet finished processing at facility i.Then, the due-date for an order k is quoted according tothe following algorithm.

4.2.2. Multiple Product Types Model: In this sec-tion, we assume that the firm produces multiple prod-ucts with different characteristics, (i.e different process-ing times, arrival rates, and supply chain architecture).In this case, in addition to deciding on inventory lev-els and due-date quotation, we also design an effectivescheduling algorithm to process the jobs at each facilitysince products have different characteristics and the se-quence of jobs will effect the completion times and re-sponse times to customers.

For the multiple product model, again if we assumethat there is no congestion or queues in the system (e.g.there are assembly lines at all the facilities and each or-der is put on the assembly line and starts processing assoon as the required materials for that order receives tothat facility), then the model becomes exactly the sameas the single product case because the different orders andproduct types don’t effect each other and don’t cause anycongestion in the system. In this case, we use the sameLP model 4.1 to decide on the inventory amounts andthe due-date quotation algorithm 11 to quote due-dates.There is no need for a scheduling algorithm since eacharriving job is immediately placed under process withoutwaiting in this system.

However, if we assume that there is a single server(or M < ∞ servers at a facility), then there will be con-gestion and queues in the system. In that case, we ap-proximate the waiting time of orders at facility i by usinga similar approach as in the single product model. Forthe schedule used at facility i, we find the expected wait-ing time of an order type l in that queue with arrival rateDl

i and mean processing time pli. In this case, the ex-

pected waiting time for type l, E[W li ], depends on the

schedule used in that facility and other products. We usewl

i = E[W li ] as the waiting time of jobs of type l at facil-

ity i and use these values in the LP formulation 4.1.For this case, we also use the same idea for due-date

quotation as in Algorithm 12. However, in this case, thescheduling algorithm doesn’t have to be FCFS and thefacilities might choose other sequences to minimize totalcompletion times. Although the problem of minimizingcompletion times at a single facility is NP-Hard, Kamin-


ALGORITHM 11:Step 1: At the time of arrival of an order k to the system, denoted by rk, form a new subgraph,

G′, of the supply chain considering the inventories at time rk. Starting from themanufacturer, do a depth-first search on the graph and add a facility to the subgraphif there is no inventory at hand at that facility. If there is inventory at a facility do notadd that to the subgraph and do not go further from that node.

Step 2: Let the length between two facilities i and j be lij = pi + tij .Step 3: Add a node 0 at the end of the subgraph and connect it with all

the facilities that don’t have a predecessor and let l0j = 0. Using these lij values, findthe longest path from node 0 to the manufacturer.

Step 4: Set the length of the longest path as the lead time for that order.

ALGORITHM 12:Step 1: Set dk

i = 0 if there is inventory of type l at facility i and di = fi if i ∈ E.Step 2: Form the subgraph G′ as in Algorithm 11 and put a facility i into set F if facility i ∈ G′

and i ∈ S. Set F denotes the set of facilities that belong to the firm and haven’t beenquoted a due-date yet.

Step 3: If facility i ∈ F and j /∈ F for ∀j ∈ Pi, then set due-date for order k at facility i asbelow and delete facility i from set F .

dki = maxj∈Pi{dk

j + tkji}+ wki

wki = max{sumj∈Uk

ipj

i −maxj∈Pi{dkj + tkji}, 0}+ pk

i

Step 4: Stop if set F is empty and set dk1 as the lead time for order k. Return to step 3 otherwise.

sky and Simchi-Levi [18] shows that the SPTA rule isasymptotically optimal for this problem. Under the SPTAheuristic, each time a job completes processing, the short-est available job which has yet not been processed is se-lected for processing. Also, note that this approach tosequencing does not take quoted due date into account,and is thus easily implemented.

We consider FCFS and SPTA scheduling algorithmsto use at the facilities. If FCFS schedule were used at allfacilities, then the due-date quotation algorithm would beexactly the same as Algorithm 12. However, if SPTAschedule is used at a facility, then we consider future ar-rivals, too, in addition to the current state of the systembecause future arrivals might be scheduled before previ-ous orders and increase their delivery times.

Xia, Shantikumar and Glynn [34] and Kaminsky andSimchi-Levi [21] independently proved that for a flowshop model with n machines, if the processing times ofa job on each of the machines are independent and ex-changeable, processing the jobs according to the shortesttotal processing time pi =

∑nj=1 pj

i at the first facilityand processing the jobs on a FCFS basis at the others isasymptotically optimal if all the release times are 0.

We design a scheduling algorithm for our systembased on this result. For each product type, we find thelongest path in the production network of that productand find the total processing time of each product typeby summing the processing types on this path. Then, weschedule the jobs according to shortest total processingtimes at the facilities that are at the end of the supplychain network and use a FCFS schedule at the other facil-

ities. We explain our scheduling algorithm with due-datequotation below in algorithm 13.

In this system let Prl denote the arrival probabilitiesfor each product type l = 1..K with sum equal to 1. Letset Ml denote the set of product types that are going to bescheduled before product type l and ψl =

∑k∈Ml

Prk isthe probability that an arriving job is going to be sched-uled before job type l. µl

i =∑

k∈Ml{PrkE[pk

i ]} is theexpected processing time of such a job at facility i andλ is the mean inter-arrival time of orders. Also, Nk

i de-notes the number of orders that arrived after order k butscheduled before it at facility i and wk

i denotes the wait-ing time of order k at facility i. Note that wk

i = pi for theassembly line system . Then, the scheduling and due-datequotation algorithm for an order k is as below:

4.3. Computational Analysis: We performed severalcomputational experiments in order to evaluate the per-formance of the algorithms explained above for differ-ent cases, and we also compared the objective of solu-tions based on our heuristics with the objective functionsachieved using traditional methods, and with a lowerbound. We design a complex supply chain network usingdifferent processing times, transshipment times betweenfacilities, unit holding costs and unit waiting costs andimplement our heuristics in C++. Whenever needed, wesolve the LP model 4.1 using ILOG AMPL/CPLEX 7.0.

Consider a supply chain network for a single producttype as shown in figure 2. The meanings of the numbersin figure 2 are explained in figure 3. In this network, Si

denotes the facilities that belong to the same firm and Ei

denotes the external suppliers.


ALGORITHM 13:Scheduling:

Step 1: Find the total time required to process each product type l(i.e. the longest path fromthe suppliers at the end of the chain to the manufacturer in the supply chain for producttype l.) and denote it by Tl.

Step 2: Define set L to be set of facilities that have no internal supplier. Schedule the jobsaccording to shortest Tl at a facility i if i ∈ L and use FCFS at the other facilities.

lead time Quotation:Step 3: Set Nk

i = 0 for all i, dki = 0 if there is finished goods inventory of type l at facility i

and di = fi if i ∈ E.Step 4: Form the subgraph G′ as in Algorithm 11 and put a facility i into set F if facility i ∈ G′

and i ∈ S. Set F denotes the set of facilities that belong to the firm and haven’t beenquoted a due-date yet.

Step 5: If facility i ∈ F and j /∈ F ∀j ∈i, then set due-date for order k at facility ias below and delete facility i from set F . Let Uk

i denote the set of orders that arealready in the system at time rk and is scheduled before job k but not yet finishedprocessing at facility i

If facility i ∈ L, thendk

i = maxj∈Pi{dkj + tkji}+ pk

i + wki + slackk

i

wki = max{sumj∈Uk

ipj

i −maxj∈Pi{dk

j + tkji}, 0}

slackki =

{min{wk

i ψlµli

λ−ψlµli

, (n− i)ψlµli} if λ− ψlµ

li > 0

(n− i)ψlµli otherwise

Nki = slackk

i /µli

If facility i /∈ Ldk

i = maxj∈Pi{dkj + tkji}+ pk

i + wki

wki = max{sumj∈Uk

ipj

i + maxj∈Pi{Nj}µli −maxj∈Pi{dk

j + tkji}, 0}Ni = maxj∈PiNj

Step 6: Stop if set F is empty and set dk1 as the lead time to the customer. Return to step 5

otherwise.

E1 20

5

0.5

3

1.2

1.5

10

1 E2 30

S2 15 1

8

1.5 S1 10 1

4

2

S3 6

1.8

12

3.5 S4 30 3

120

151.5

S5 15

1.2

S6 10 8

Figure 2: Supply Chain Network Example

tij

hij1

Fi pi hi

2 Fj pj

Figure 3: Explanation of the values


Table 7: Comparison of combined strategy with purestrategies for an assembly line system

h=8,cd=10 h=8,cd=5 h=4, cd=5ZMTO−MTS/ZMTS 0.526 0.451 0.814ZMTO−MTS/ZMTO 0.424 0.710 0.655

4.3.1. Effect of Inventory Positioning withoutCongestion Effects in the System: We start by exam-ining the case where the waiting time of a job at facil-ity i is deterministic and equal to pi as in an assemblyline model or an infinite server model with determinis-tic processing times. Since, the demand is stochastic, thefirm needs to keep some safety stock to achieve the de-sired service level. In Table 7, we compare the optimalobjective values of the formulation 4.1 allowing to holdinventory at every facility as opposed to holding no in-ventory at all or holding only finished good inventories.The ratios of the objective function values of formula-tion 4.1 with the combined strategy over the costs withpure MTS and MTO strategies for different combinationsof inventory holding cost at the last facility, h, and unitlead time cost, cd are shown in Table 7. Traditionally,firms either use an MTO strategy without keeping any in-ventory or an MTS strategy keeping only finished goodsinventory. However, as we show in Table 7, using a com-bined strategy is much more beneficial for minimizingthe cost function

∑Nj=1 hjE[Ij ] + cdE[W ] where E[Ij ]

denotes the average inventory at facility j and E[W ] de-notes the average waiting time of customers in the wholesystem. For example, for the case where the holdingcosts are as shown in figure 2 except that the holding costfor finished goods at the last facility is 4 (h = 4) andcd = 5, with a pure MTS strategy, we need to keep afinished goods inventory of 95 units with a cost of 380.With a pure MTO strategy, the lead time will be 95 andthe cost is 475. However, with a combined strategy withyS1 = 10, yS2 = 15, yS4 = 12, yS6 = 40, xE1,S2 = 25,xE2,S5 = 40 xS1,S4 = 20 xS2,S4 = 3, the total cost willbe 307.1. The cost with the combined strategy is signifi-cantly lower than the costs with pure strategies.

Consider another example in which the industry leadtime is 30 and the firm tries to achieve this lead time bykeeping some inventory. If the firm only keeps finishedgoods inventory, then yS6 = 65 and the total inventorycost is 260. However, by keeping inventory at other fa-cilities, with the same lead time, the total inventory costcan be decreased to 187.1 with yS1 = 10, yS2 = 15,yS6 = 22, xE1,S2 = 25, xE2,S5 = 28 xS1,S4 = 20xS2,S4 = 3. In addition, the firm will be able to cut thelead time by half to 15 with a cost of 247.1 which is evenless than the cost with the initial strategy.

4.3.2. Effect of Inventory Positioning with Con-gestion Effects for Single Product Type Model: If weconsider a single server model for each facility with

Table 8: Comparison of combined strategy with purestrategies for a single server model


stochastic processing times, we find the mean waitingtime of a job at each facility and use these values as thewaiting times. In that case, the ratios of the objective val-ues in formulation 4.1 with the combined strategy overthe pure strategies are shown in Table 8.

Note that these costs are found through the objectivefunctions in our LP formulation which are calculated us-ing the maximum lead time for the specified service levelfor all jobs and using the corresponding safety stock lev-els. However, in reality each job has a different lead timedepending on the congestion in the system and the inven-tory levels fluctuate due to the stochastic nature of de-mand and production rates. We simulate this stochasticsystem assuming that there is a single server at each facil-ity with mean processing times and other values as givenin figure 2 and mean inter-arrival time 50. We calculatethe costs considering the fluctuations in the inventory val-ues throughout the simulation and using the waiting timeof each job in the system as the lead time which is differ-ent for each job. We assume exponential inter-arrival andprocessing times at each facility independent from eachother. Using our heuristics, we make 10 runs for eachof the simulations with different random number streamsand using n = 5000 jobs and present the comparison ofobjective values

∑Nj=1 hjE[Ij ]+cdE[d]+cT E[W−d]+

on average in the following tables.For the single product type model, we start our sim-

ulations with the initial inventory levels found by our LPformulation and every time a customer order arrives tothe system, a new production order is given at that time.We compare the objective functions

∑Nj=1 hjE[Ij ] +

cdE[d]+cT E[W−d]+ with this combined model to pureMTO and MTS models where both systems are operatedas the combined system but with the initial inventories all0 for the MTO model and there is only finished goods in-ventory for the MTS model. The ratios of the costs are in9 for different h and cd combinations with cT = 12.

In this simulation analysis, to assess the effectivenessof our lead time quotation algorithm, we also compare thelead times quoted for this single type system to the actualwaiting times of the jobs in the system using cd = 5 andcT = 7. Let Zn

LT =∑n

i=1{cddi+cT (Wi−di)+} denotethe total lead time plus tardiness costs, Zn

DD = ZLT +∑ni=1 ri denote the total due-dates plus tardiness costs,

ZnW =

∑ni=1{cdWi} denote the total waiting times of

the jobs in the system and ZnC = ZW +

∑ni=1 ri denote

the total completion times of the jobs. We present ratiosfor these values for different number of jobs, n, in Table


Table 9: Simulation analysis of combined strategy com-pared to pure strategies


Table 10: Comparison of lead times and due-dates to ac-tual waiting times and completion times

n=10 n=100 n=1000 n=5000Zn

W /ZnLT 0.891 0.950 0.964 0.962

ZnC/Zn

DD 0.925 0.967 0.992 0.996

10. As we see in Table 10, the lead times quoted with ouralgorithm are very close to the actual waiting times andZn

DD approach to ZnC as n gets bigger since the effect of

the release times increase with n.4.3.3. Effectiveness of the Algorithms for Multiple

Product Types: We then considered multiple producttypes and designed a simulation study to assess how ouralgorithms work for the multiple product type case. Inaddition to the single product type we considered above,now assume that there are 4 more product types with ar-rival probabilities and mean processing times as shownin Table 11, everything else remaining the same for allproduct types. We first consider a 3-product type modelusing the first two product types in Table 11 and then a5-product type model considering all the product types.

In this case, we use an SPTA schedule and a LTQ al-gorithm as explained in algorithm 13. To explore the ef-fectiveness of the SPTA schedule, we compared the totalwaiting times of jobs with the SPTA schedule to that of alower bound. If we consider only the bottleneck facility(the facility with slowest processing rate) and use a SPTAschedule with preemption in that facility and assume thatthe waiting time of any job in a queue of any other facilityis zero, then the total weighted waiting time of jobs at thissystem will be a lower bound for our model. Let ZLB de-note the lower bound for the total weighted waiting timesof jobs in the system and ZSPTA =

∑ni=1{cdWi} denote

the total weighted waiting times with the SPTA-basedschedule. The comparison of the total waiting times withour heuristic to that of the lower bound are presentedin Table 12 for different number of jobs. Also, to ex-plore the effectiveness of the LTQ algorithm, we presentthe ratios of the total quoted lead times plus tardinesscosts, ZSPTA−LTQ =

∑ni=1{cddi + cT (Wi − di)+}

for this case over ZSPTA and ZLB in Table 12 usingthe same weights for different product types, cd = 5and cT = 7. Note that the optimal off-line LTQ al-gorithm quotes lead times that are exactly equal to thewaiting times of the jobs in the system which is equal toZSPTA, thus ZSPTA is a lower bound for the LTQ algo-rithm with the SPTA-based schedule and ZLB is a lowerbound among all schedules. As seen in this table, the

difference between ZSPTA and the lower bound are lessthan 20% and the lead time quotation algorithm gives re-sults that are less than 7% worse than ZSPTA. Thus, weconclude that our scheduling and lead time quotation al-gorithms are effective in minimizing the objective func-tion

∑ni=1{cddi + cT (Wi − di)+}

In Table 13, we present the ratios of the total costs∑Ki=1{(


i E[di]+cTi E[Wi−di]+} by

using the inventory values obtained from our LP formu-lation and using the scheduling and lead time quotationalgorithm as explained in algorithm 13 to that of the totalcosts with pure strategies and with a FCFS schedule, forn = 5000 jobs using cT = 12 and different weights forh and cd. As we see in Table 13, the costs with the com-bined strategy is about 20% less on average than the purestrategies. Also, the SPTA based schedule used for thismodel decreases the total costs by about 10% comparedto a FCFS schedule. Also, we see that as cd decreases, thecombined system moves toward an MTO system while ash decreases, an MTS system gives better results.

5. Conclusion: In this project, we considered a supplychain in a stochastic, multi-item environment and de-signed effective algorithms for the optimal inventory lev-els, scheduling of the jobs and lead time quotation to cus-tomers. We also analyzed the value of centralization andinformation exchange by considering centralized and de-centralized versions of this supply chain.

In the first part of the project, we consider stylizedmodels of a supply chain, with a single manufacturerand a single supplier, in order to quantify the impactof manufacturer-supplier relations on effective due datequotation. In our models, we consider several variationsof scheduling and due-date quotation problems in MTOsupply chains in order to minimize a function of the to-tal quoted due-dates plus tardiness. We present due-datequotation and scheduling algorithms for centralized anddecentralized versions of this model that are asymptoti-cally optimal, and that are computationally found to beeffective for relatively small problem instances. We alsoinvestigate the value of coordination schemes involvinginformation sharing between supply chain members forthis system. Through computational analysis, we see thatif the processing rate at the supplier is larger than the ar-rival rate, i.e. when there is no congestion at the sup-plier side, the centralized and decentralized models per-form similarly. However, as the congestion at the supplierstarts to increase, the centralized model performs signif-icantly better, and the value of information and central-ization increases dramatically. Also, if centralization isnot possible, a simple information exchange in the decen-tralized model can also improve the level of performancedramatically, although not as significantly as centralizedcontrol.

In the second part of the project, we consider stylized


Table 11: Arrival probabilities and mean processing times of product typesP(l) pE1 pE2 pS1 pS2 pS3 pS4 pS5 pS6

l=2 0.3 15 10 25 5 45 15 20 10l=3 0.15 5 15 10 10 15 20 25 15l=4 0.25 10 20 30 15 10 5 10 20l=5 0.1 20 5 5 10 5 10 30 15

Table 12: Comparison of SPTA schedule and the LTQ with the lower bound for K=3 and K=5 product typesK=3 n=10 n=100 n=1000 n=5000 K=5 n=10 n=100 n=1000 n=5000

ZLB/ZSPTA 0.962 0.813 0.847 0.833 0.859 0.790 0.822 0.816ZSPTA/ZLT 0.874 0.933 0.952 0.947 0.941 0.922 0.925 0.931ZLB/ZLT 0.848 0.753 0.802 0.786 0.807 0.735 0.766 0.751

models of a combined MTO-MTS supply chain, with asingle manufacturer and a single supplier, in order to findthe optimal inventory values that should be carried at eachfacility and to assess the impact of manufacturer-supplierrelations on inventory decisions and effective lead timequotation. In our models, we consider several variationsof inventory, scheduling and lead time quotation prob-lems in order to minimize a function of the total inven-tory, lead times and tardiness. We derive the optimalityconditions for both the centralized and decentralized ver-sions, under which an MTO or MTS system should beused for each product at each facility and we present al-gorithms to find the optimal inventory levels. We alsopresent effective lead time quotation and scheduling algo-rithms for centralized and decentralized versions of thismodel that are computationally found to be effective. Wealso investigate the value of coordination schemes involv-ing information sharing between supply chain membersfor this system. Through computational analysis, we seethat costs can be cut dramatically by using a combinedsystem instead of pure MTO or MTS systems and infor-mation exchange between the supplier and the manufac-turer is critical for effective lead time quotation. Also, wesee that if centralization is not possible, an informationexchange in the decentralized model can also improve thelevel of performance substantially.

In the final part of the project, we consider stylizedmodels of more complex MTO-MTS supply chains withmultiple facilities in order to find the optimal inventoryvalues that should be carried at each facility and to designeffective scheduling and lead time quotation algorithms.Through computational analysis, we see that combinedMTO-MTS systems give much better performance thanpure MTO or MTS systems and an SPTA based algo-rithm for scheduling the jobs performs much better thanthe generally used FCFS approach. We also assess thatinformation exchange is critical for short and reliable leadtime quotation.

Of course, these are stylized models, and real worldsystems have many more complex characteristics that arenot captured by these models. Nevertheless, this is to

the best of our knowledge the first study that analyticallyexplores inventory decisions, scheduling and lead timequotation together in the context of an MTO/MTS sup-ply chain, and that explores the impact of the supplier-manufacturer relationship on this system.

In the future, we intend to expand this research toconsider different functions of lead time in the objec-tive function. In some systems, the manufacturer doesn’thave to accept all orders and has the option to reject cer-tain orders. Pricing and capacity decisions can also beincorporated into these model. In all of these models andvariants, the manufacturer needs to develop strategies forsystem design, and for scheduling and and due-date quo-tation.

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Table 13: Comparison of combined strategy with pure strategies for multiple product type modelsK=3 h=8,cd=10 h=8,cd=5 h=4, cd=5 K=5 h=8,cd=10 h=8,cd=5 h=4, cd=5

ZMTO−MTS/ZMTS 0.732 0.695 0.887 0.762 0.717 0.869ZMTO−MTS/ZMTO 0.666 0.851 0.789 0.724 0.871 0.812

ZSPTA/ZFCFS 0.943 0.918 0.930 0.885 0.874 0.891

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