Chapter 999 DUE DATE MANAGEMENT POLICIES

Chapter 999

DUE DATE MANAGEMENT POLICIES

Põnar Keskinocak∗School of Industrial and Systems Engineering

Georgia Institute of Technology, Atlanta, GA 30332

[email protected]

Sridhar TayurGraduate School of Industrial Administration

Carnegie Mellon University, Pittsburgh, PA 15213

[email protected]

∗Pınar Keskinocak is supported by NSF Career Award DMII-0093844. This research is alsosupported in part by a grant from The Logistics Institute Asia Pacific (TLI-AP).

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Introduction

To gain an edge over competitors in an increasingly global and com-petitive marketplace, companies today need to differentiate themselvesnot only in cost, but in the overall �value� of the products and the ser-vices they offer. As customers demand more and more variety of prod-ucts, better, cheaper, and faster, an essential value feature for customeracquisition and retention is the ability to quote short and reliable leadtimes. Reliability is important for customers especially in a business-to-business setting, because it allows them to plan their own operationswith more reliability and conÞdence [66].We deÞne the lead time as the difference between the promised due

date of an order (or job) and its arrival time2 Hence, quoting a lead timeis equivalent to quoting a due date. The importance of lead time quota-tion becomes even more prevalent as many companies move from massproduction to mass customization, or from a make-to-stock (MTS) to amake-to-order (MTO) model to satisfy their customers� unique needs.Hence, companies need to determine in real time if and when an ordercan be fulÞlled proÞtably. The Internet as a new sales channel furtherincreases the importance of effective lead time quotation strategies, ascustomers who place orders online expect to receive reliable lead timeas well as price quotes. For example, many customers were extremelydissatisÞed with their online purchasing experience during the 1999 hol-iday season, mainly due to unreliable delivery date quotes, lack of orderstatus updates, and signiÞcant order delays [48].Quoting unreliable lead times not only leads to potential loss of fu-

ture business, but may also result in monetary penalties. Seven e-tailers,including Toys R Us and Macy�s had to pay a total of $1.5 million to set-tle a Federal Trade Commission (FTC) action over late deliveries madeduring the 1999 holiday season [40]. According to the FTC, the e-tailerspromised delivery dates when fulÞllment was not possible and failed tonotify customers when shipments would be late. Sometimes a companymay self-impose a penalty for missed due-dates. For example, due to theincreasing insistence of many steel users on consistent reliable deliver-ies, Austin Trumanns Steel started to offer a program called TouchdownGuarantee in 1986. Under the program, if the company agrees to a re-quested delivery date at the time an order is placed, it has to deliver

2In this paper, we focus on lead times quoted to customers. Lead times can also be used forinternal purposes, e.g., planned lead times are used for determining the release times of theorders to the shop floor [69]. We do not discuss planned lead times in this paper.

Due Date Management Policies 3

on time or pays the customer 10% of the invoice value of each item notdelivered [52].A common approach to lead time quotation is to promise a constant

lead time to all customers, regardless of the characteristics of the orderand the current status of the system [65] [110]. Despite its popularity,there are serious shortcomings of Þxed lead times [62]. When the demandis high, these lead times will be understated leading to missed due datesand disappointed customers, or to higher costs due to expediting. Whenthe demand is low, they will be overstated and some customers maychoose to go elsewhere.The fundamental tradeoff in lead time quotation is between quot-

ing short lead times and attaining them. In case of multiple customerclasses with different capacity requirements or margins, this tradeoffalso includes capacity allocation decisions. In particular, one needs todecide whether to allocate future capacity to a low-margin order now, orwhether to reserve capacity for potential future arrivals of high-marginorders.Lead-time related research has developed in multiple directions, in-

cluding lead time reduction [54] [99], predicting manufacturing leadtimes, the relationship between lead times and other elements of man-ufacturing such as lot sizes and inventory [42] [61] [68], and due datemanagement (DDM) policies. Our focus in this survey is on DDM,where a DDM policy consists of a due date setting policy and a sequenc-ing policy. In contrast to most of the scheduling literature [49] [78] [88],where due dates are either ignored or assumed to be set exogenously(e.g., by the sales department, without knowing the actual productionschedule), we focus on the case where due dates are set endogenously.Most of the research reviewed here does not consider inventory decisionsand hence is applicable to MTO systems. Previous surveys in this areainclude [5] [26].Most of the research on DDM ignores the impact of the quoted due

dates on the customers� decisions to place an order. Recently, a smallbut increasing number of researches studied DDM from a proÞt maxi-mization rather than a cost minimization perspective, considering orderacceptance decisions (or the effect of quoted lead times on demand) inaddition to due date quotation and scheduling decisions. Although thisis a step forward from the earlier DDM research, it still ignores anotherimportant factor that affects the demand: price. With the goal of mov-ing towards integrated decision making, the latest advances in DDMresearch focus on simultaneous price and lead time quotation.The paper is organized as follows. In Section 1, we discuss the charac-

teristics of a DDM problem, including decisions, modeling dimensions,

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objectives and solution approaches. In Section 2, we discuss commonlyused scheduling rules in DDM policies. Offline DDM models, which as-sume that the demand and other input about the problem are availableat the beginning of the planning horizon, are discussed in Section 3.Online models, which consider dynamic arrivals of orders over time, arepresented in Section 4. Models for DDM in the presence of service levelconstraints are discussed in Section 5. We review the DDM models withorder acceptance and pricing decisions in Section 6. We conclude withfuture research directions in Section 7.We reviewed a broad range of papers in this survey; however, we do

not claim that we provide an all-inclusive coverage of all the paperspublished in the DDM literature and regret any omissions. We wouldbe delighted to receive copies of or references to any work that was notincluded in this survey.

1. Characteristics of a Due Date ManagementProblem

In this section, we discuss the characteristics of DDM problems, in-cluding the decisions, modeling dimensions, and the objectives. The no-tation that is used throughout the paper is summarized in Table 999.1.

1.1 Due Date Management Decisions

The main decisions in DDM are order acceptance (or demand man-agement), due date quotation, and scheduling. In determining a DDMpolicy, ideally one would consider due date setting and scheduling de-cisions simultaneously. There are a few papers in the literature thatfollow this approach. However, given their complexity, most papers con-sider these two decisions sequentially, where Þrst the due dates are setand then the job schedules are determined. Commonly used schedulingpolicies in DDM are discussed in Section 2. We denote a due date pol-icy by D-S, where D refers to the due-date setting rule and S refers tothe scheduling rule. While most researchers are concerned in comparingthe performance of alternative DDM policies, some focus on Þnding theoptimal parameters for a given policy [19] [87].Most of the papers in the literature assume that order acceptance de-

cisions are exogenous to DDM, and all the orders that are currently inthe system have to be quoted due dates and be processed. Equivalently,they ignore the impact of quoted due dates on demand and assume thatonce a customer arrives in the system, he will accept a due date nomatter how late it is. In reality, the quoted lead times affect whether acustomer places an order or not, and in turn, they affect the revenues. In


Indices:j: order or jobk: job classo: operation of a jobt: time period

Notationrj : arrival (or ready) time of job jpj : (total) processing time of job jRj : revenue (price) of job jwj : the weight of job j (e.g., may denote the importance of the job)gj (gjt) : number of (remaining) operations on job j (at time t)Ujt : set of remaining operations of job j at time tnt (nkt) : number of (class k) jobs in the system at time tlj : quoted lead time (or quoted ßow time) of job jCj : completion time of job jWj = Cj − rj − pj : wait time of job jFj = Cj − rj : ßow time of job jTj = (Cj − dj)+ : tardiness of job jEj = (dj − Cj)+ : earliness of job jLj = Cj − dj : lateness of job jτmax : maximum fraction of tardy jobs (or missed due dates)Tmax : maximum average tardinessρ : steady-state capacity utilizationα, β, γ : parameters

Table 999.1. Notation for DDM models. (.)jo denotes the value of parameter (.) forthe o-th operation of job j, if a job has multiple operations. E[(.)] and σ(.) denotethe expected value and the standard deviation of (.), when (.) is a random variable.

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many cases, there is a maximum limit on the lead time, either imposedby customers or by law. For example, under FTC�s mail-and-telephoneorder rule, orders must be shipped within 30 days or customers mustbe notiÞed and given the option of agreeing to a new shipping date orcanceling the order. Even if the customers accepted any kind of leadtime, due to the associated penalties for longer lead times or misseddue dates, it might be more proÞtable for a manufacturer not to ac-cept all customer orders. Hence, in contrast to most of the papers inthe literature, one needs to consider DDM from a proÞt maximizationperspective rather than a cost minimization perspective. Incorporatingorder acceptance decisions into DDM policies by considering the impactof quoted lead times and prices on demand is an important step towardsintegrated DDM policies. We discuss the literature on DDM with orderacceptance (and price) decisions in Section 6.

1.2 Dimensions of a Due Date ManagementProblem

There are several dimensions that distinguish different due date man-agement problems. Depending on the system (i.e., the manufacturingor service organization) under consideration, a combination of these di-mensions will be present and this combination will affect the appropriatemathematical model for studying DDM in that setting.Offline vs. online: In an offline setting, all the information about the

problem, such as the job arrival and processing times, are available atthe beginning of the scheduling horizon. In contrast, in an online settingfuture arrivals are not known with certainty and the information abouta job becomes available only at the time of its arrival. In general, it isassumed that the arrivals follow a known probability distribution. Anonline setting is stringent if the decisions about a job, such as its duedate, have to be made immediately at the job�s arrival time. In contrast,in an online setting with lookahead, one can wait for some time after ajob arrives before making a decision about that job, but there is usuallysome penalty for delaying the decision.Single vs. multiple servers: In a single-server setting, only one re-

source needs to be considered for satisfying customer requests. In amultiple-server setting, multiple resources are available, which may beidentical or different. In case of multiple non-identical servers, there aredifferent possible system (or shop) characteristics, such as job shop andßow shop. In a job shop, each job has a sequence of operations in dif-ferent machine groups. In a ßow shop, the sequence of operations is thesame for each job.


Preemptive vs. nonpreemptive: In a nonpreemptive setting, once theprocessing of a job starts, it must be completed without any interruption.In contrast, in a preemptive setting, interruption of the processing of ajob is allowed. In the preempt-resume mode, after an interruption, theprocessing of a job can be resumed later on without the need to repeatany work. Thus, the total time the job spends in the system is pj ,although the difference between the completion and start times can belonger than pj . In the preempt-repeat mode, once a job�s processing isinterrupted, it has to start again from the beginning. Hence, if job j isinterrupted at least once, its total processing time is strictly larger thanpj .Stochastic vs. deterministic processing times: When the processing

times are not known with certainty, it is usually assumed that theyfollow a probability distribution with known mean and variance.Setup times/costs: When changing from one order (or one customer

class) to another, there may be a transition time. Most of the currentliterature on DDM ignores setup times.Server reliability: The capacity of the resources may be known (deter-

ministic) over the planning horizon, or there may be random ßuctuations(e.g., machine breakdowns).Single vs. multiple classes of customers: Customers (or jobs) can be

divided into different classes based on the revenues (or margins) theygenerate, or based on their demand characteristics, such as (average)processing times, lead time related penalties, or maximum acceptablelead times.Service level constraints: Commonly used service level constraints in-

clude an upper bound on (1) the fraction (or percentage) of orders whichare completed after their due dates, (2) the average tardiness, and (3)the average order ßow time.Common vs. distinct due dates: In case of common due dates, all

customers who are currently in the system are quoted the same duedate. In case of distinct due dates, different due dates can be quoted todifferent customers.

1.3 Objectives of Due Date Management

When order acceptance decisions are assumed to be exogenous, thekey decisions are due date setting and scheduling. Note that these de-cisions have conßicting objectives. One would try to set the due datesas tight as possible, since this would reßect better responsiveness tocustomer demand. However, this creates a conßict with the schedulingobjectives because tight due dates are more difficult to meet than loose

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due dates. There are two approaches for resolving this conßict: (1) usean objective function that combines the objectives of both decisions,(2) consider the objective of one decision and impose a constraint onthe other decision. A relatively small number of papers follow the Þrstapproach [11] [24] [67] [87] [86] using an objective function that is theweighted sum of earliness, tardiness and lead times, where the weightsreßect the relevant importance of short lead times vs. meeeting themon time. Among the papers that follow the second approach, some tryto achieve due date tightness subject to a service level constraint, whileothers consider a service objective (e.g., minimizing tardiness) subjectto a constraint on minimum due date tightness. For example, in [6] theobjective is to minimize the average due date subject to a 100% serviceguarantee (i.e., tardiness is not allowed). In contrast, in [7] the objectiveis to minimize average tardiness subject to a constraint on the averageßow allowance (i.e., due date tightness).In general, the objective in DDM is to minimize a cost function that

measures the length and/or the reliability of quoted lead times. Commonobjectives include minimizing:

Average/total (weighted) due date (subject to service level con-straints) [1][6] [107]

Average tardiness [7] [8] [9] [10] [27] [33] [37] [41] [55] [60] [70] [102][103] [104]; number (or percentage) of tardy jobs [1] [9] [10] [14] [27][34] [60] [70] [102] [103]; maximum tardiness [60] [70]; conditionalmean tardiness [1][9] [60]

Average lateness [1] [12] [14] [27] [33] [34] [60] [102] [103] [104] [105];standard deviation of lateness [12] [27] [33] [60] [76] [82] [102] [103][105]; sum of the squares of latenesses [19] [67]; mean absolutelateness [41] [67] [76]

Average earliness [14] [55] [104]; number (or percentage) of earlyjobs [14] [33]

Total (weighted) earliness and tardiness [33] [34] [55] [77] [104]

Total (weighted) earliness, tardiness and lead times [11] [24] [67][87] [86]

Average queue length [14] [34]; waiting time [14] [34] or ßow time[14] [27] [34] [41] [60] [102] [103] [105]; standard deviation of ßowtimes [60] [105]; number of (incomplete) jobs in system (WIP) [39];total processing time of unÞnished jobs in the system (CWIP) [39];variance of queue length over all machines [39]


Note that the objectives in the last bullet focus primarily on �internal�measures of the shop ßoor, whereas the objectives in the previous bulletsfocus on �external� measures of customer service.Minimizing the mean ßow time, which is the average amount of time

a job spends in the system, helps in responding to customer requestsmore quickly and in reducing (work-in-process) inventories. Minimizingearliness helps in lowering Þnished goods inventories leading to lowerholding costs and other costs associated with completing a job beforeits due date. Earliness penalty may also capture a �lost opportunity�for better service to the customer, since a shorter lead time could bequoted to the customer if it were known that the job would completeearly. Minimizing tardiness or conditional mean tardiness, which is theaverage tardiness measured over only the tardy jobs, helps in completingthe jobs on or before their due dates, to avoid penalties due to delays,loss of goodwill, expedited shipment, etc. As noted by Baker and Kanet[9], average tardiness is equal to the product of the proportion of tardyjobs and the conditional mean tardiness, hence, looking at the two lattermeasures separately provides more information about the performancethan just looking at the aggregate measure of average tardiness. Mini-mizing due dates helps in attracting and retaining customers. Averagelateness measures whether the quoted due dates are reliable on aver-age, i.e., the �accuracy� of the due dates. Standard deviation of latenessmeasures the magnitude of digression from quoted due dates, i.e., the�precision� of the due dates. WIP and CWIP measure the congestionin the system (through work-in-process inventory) and the average in-vestment in work-in-process inventory, respectively, assuming that theinvestment in a partially Þnished job is proportional to the completedamount of processing.The performance of a due date management policy depends both on

due date setting and sequencing decisions. For certain objective func-tions, such as minimizing lateness or tardiness, due-date setting ruleshave direct and indirect effects on performance [105]. The direct ef-fect results from the type of due-date rule employed and its parameters,which determine the due date tightness. The indirect effects result fromthe due-date being a component of some of the dispatching and laborassignment rules. The due-date setting rules have only indirect effectsfor some objectives, such as minimizing the average ßow time, wheresequencing rules have more direct impact.An interesting question is which of the two decisions, due date set-

ting and sequencing, has a bigger impact on performance. The answerdepends on the measure of performance (objective function), the typeof DDM policy and its parameters, service constraints, and system con-

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ditions. For example, Weeks and Fryer [105] Þnd that due-date assign-ment rules are the most important decisions for mean lateness, varianceof lateness and proportion of tardy jobs, whereas sequencing rules arethe most important class of decision rules to impact mean ßow timeand variance of ßow time. In addition, for some objective functions (thevariance of ßow time, variance of lateness and proportion of tardy jobs)the relative inßuence of sequencing rules depends on the tightness of theassigned due dates. Wein [107] Þnds that for the objective of minimizingaverage ßow time, for the four due date setting rules he proposed, theimpact of sequencing on performance is minimal. For other due daterules, however, the impact of sequencing on performance is signiÞcant.One has to be very careful in choosing an appropriate objective func-

tion to satisfy business goals, as different objective functions may leadto remarkably different DDM policies. For example, a DDM policy thatyields a zero lateness on average would result in approximately 50% ofthe jobs early and the remaining 50% tardy. If the cost of earliness isconsiderably lower for a Þrm than the cost of tardiness, then clearly min-imizing lateness might not be an appropriate performance measure. Ingeneral, we can divide the objectives into two broad categories, depend-ing on whether they penalize only for completing the jobs after theirdue dates, or penalize both for early and late completion. There areimportant differences among the objectives within the same category aswell. For example, consider the three objective functions, minimizingaverage tardiness, the number of tardy jobs, and maximum tardiness.Consider two DDM policies, one resulting in one job that is 100 timeunits late, whereas the other one resulting in 10 jobs that are each 10units late. While these two policies are equivalent with respect to theaverage tardiness objective, the Þrst policy is better than the second onewith respect to the number of tardy jobs, and the second policy is betterthan the Þrst one with respect to maximum tardiness. Hence, dependingon the choice of the objective function, one can obtain a result that issigniÞcantly poor with respect to another objective or service constraint.For example, Baker and Bertrand [7] note that when the due date tight-ness is below a threshold for the DDM policy SLK-EDD in an M/M/1queuing system, even if the average tardiness is low, the proportion oftardy jobs is approximately equal to the utilization level in the system.While this characteristic might be quite acceptable for the objective ofminimizing average tardiness, it might be undesirable for a secondaryobjective of minimizing the number of tardy jobs. Also, see [98] for adiscussion on DDM policies which lead to unethical practice by quotinglead times where there is no hope of achieving them, while trying tominimize the number of tardy jobs.


The objective functions discussed above take a cost minimization per-spective and are appropriate when the jobs to be served are exogenouslydetermined. When order acceptance is also a decision (or equivalently,if the demand depends on quoted lead times), then the objective is tomaximize proÞt; that is, the revenues generated from the accepted ordersminus the costs associated with not completing the orders on time.

1.4 Solution Approaches for Due DateManagement Problems

To study the effectiveness of due date management policies, threeapproaches are used: simulation, analytical methods and competitiveanalysis. While they are quite interesting from a theoretical perspec-tive, the application of analytical models to DDM has been limited tosimple settings, such as special cases of offline single machine problems.To study the effect of DDM policies in more realistic multi-machine en-vironments where orders arrive over time, researchers have commonlyturned to simulation.As an alternative approach, a few researchers have used competitive

analysis to study DDM policies, where an on-line algorithm A is com-pared to an optimal off-line algorithm [93]. Given an instance I, letzA(I) and z

∗(I) denote the objective function value obtained by algo-rithm A and by an optimum off-line algorithm, respectively. We call analgorithm A c-competitive, if there exists a constant a, such that

zA(I) ≤ cAz∗(I) + afor any instance I. (Here we assume that A is a deterministic on-linealgorithm and we are solving a minimization problem.) The factor cAis also called the competitive ratio of A. The performance of off-linealgorithms can be measured in a similar way. An off-line algorithmA is called a ρ − approximation, if it delivers a solution of quality atmost ρ times optimal (for minimization problems). Although the useof competitive analysis in studying DDM problems have been limited,it has received much attention recently in the scheduling literature [43][44] [53].

2. Scheduling Policies in Due Date Management

Most DDM policies proposed in the literature propose a two-step ap-proach: assign the due dates Þrst, and then schedule the orders usinga priority dispatch policy. While it would be desirable to consider duedate and scheduling decisions simultaneously, the use of such a two-stepapproach is understandable given that scheduling problems even with

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preassigned due dates are quite difficult to solve. In a priority dispatch-ing policy for scheduling, a numerical value (priority) is computed foreach job that is currently in the system waiting to be processed. Thejob with the minimum value of priority is selected to be processed next.The priorities are usually updated at each new job arrival.Commonly considered priority dispatch sequencing rules as part of

DDM policies are summarized in Table 999.2. The simplest of the dis-patching rules, RANDOM and FCFS, consider no information about thejobs or the system, except the job arrival time. SPT, LPT, LROT/NOP,EFT, and WSPT consider the work content of the jobs. EDD, SLK,SLK", SLK/pj , SLK/gj , MDD, OPNDD and P+S/OPN(a,b) considerthe due dates as well as the work content. Some industry surveys sug-gest that due-date based rules, such as EDD and EDDo are the mostcommonly used rules, whereas other rules widely studied by researchers,e.g., SPT, are not preferred in practice [47] [50] [110].We can categorize these rules as static and dynamic. The priority

value assigned to a job by a static sequencing rule does not change whilethe job is in the system, e.g., SPT or EDD. In contrast, the priority valueassigned to a job by a dynamic sequencing rules changes over time, e.g.,SLK. It is interesting to note that sometimes a static rule and a dynamicrule can result in the same sequence, e.g., SLK and EDD3.Some of these rules are parametric, e.g., SLK and P+S/OPN have

parameters α and β. A common approach is to use simulation to Þnd theappropriate values for these parameters. The choice of the parametersdepends on the system environment factors and can have a signiÞcantimpact on the performance of the DDM policy.As in the case of SPTT, some sequencing rules Þrst divide the available

jobs into two classes, and then apply a (possibly different) sequencingrule in each class. The division of jobs into classes can be random, orbased on job characteristics. Three such rules are considered in [91]:2C-SPT, 2C-NP and 2RANDOM-SPT. In the Þrst two rules, jobs withprocessing time less than p̄ belong to class 1 and others belong to class2. In the third rule, jobs are assigned randomly into one of two classes.Higher priority is given to class 1, and within each class, jobs are se-quenced based on SPT or FCFS.The rules presented at the top portion of Table 999.2 are applied at

the job level, using aggregate information about the operations of thejob, whereas the rules presented in the lower portion of the table (witha subscript �o� denoting �operation�) are applied at the operation level.

3We are indebted to an anonymous referee for this observation.


For example, when scheduling the operations on a machine using theshortest processing time rule, one can sort the operations based on thecorresponding jobs� total processing time (

!o pjo) , or based on the

processing times of the operations to be performed on that machine(pjo). For a rule like SPT, the application either at the job or at theoperation level is straightforward, since the processing time informationexists at both levels. On the other hand, it is not obvious how a rulelike EDD can be applied at the operation level, since we usually tend tothink of due dates at the job level. One needs to keep in mind that thedue dates that are quoted to the customers do not have to be the samedue dates as the ones used for scheduling. In other words, it is possibleto have �external� and �internal� due dates, where the latter ones aredynamically updated based on shop conditions and used for schedulingpurposes only. For example, the EDDo rule is similar to EDD, butpriorities are based on internally set operation due dates.A number of researchers have focused on the performance of sequenc-

ing rules under different due-date setting rules. The performance ofscheduling rules as part of a DDM policy depend on many factors, in-cluding the due date rule, the objective function under consideration,the tightness of the due dates, the workload/congestion level in the sys-tem. The scheduling rules have especially a signiÞcant impact on theperformance of due date rules that consider shop congestion [102].No single scheduling rule is found to perform �best� for all performance

measures. Elvers [37] and Eilon and Hodgson [34] Þnd that (under theTWK rule) in general SPT performs best for minimizing job waitingtimes, ßow times, machine idle times and queue lengths, and LPT per-forms worst. Several researchers have noticed that SPT also performswell in general for the objective of minimizing average tardiness (sinceit tends to reduce average ßow time), but it usually results in highervariation. Bookbinder and Noor [14] Þnd that earliness and tardinesshave much higher standard deviation under SPT than under other se-quencing rules. Hence, SPT might not be an appropriate choice for theobjective functions that penalize both earliness and tardiness. For ex-ample, Ragatz and Mabert[82] Þnd that for the objective of minimizingthe standard deviation of lateness (σL), SPT performs worse than SLKand FCFS under eight different due date rules. Similar observationswere made in [103] for SPTT, where the average ßow time is signiÞ-cantly lower under SPTT, but the variance is higher. Weeks [104] Þndsthat for the objectives of mean earliness, tardiness and missed due dates,sequencing rules that consider due date information perform better thanthe rules that do not.

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Table 999.2. Dispatch sequencing policies

Policy Priority value References

RANDOM βj ∼Uniform[0,1] [27] [34]

First come first serve rj (or rjo) [7] [8] [14] [27] [29] [33] [34](FCFS, FCFSo) [37] [39] [67] [70] [82] [98]

[102] [103] [105] [106], [37]

Shortest processing time pj (or pjo) [7] [8] [9] [27] [33] [34](SPT, SPTo) [76] [77] [81] [82] [101] [104]

[106] [107], [37] [105]

Least remaining operation!

o∈Ujt pjo) [37]

time (LROT)

LROT/NOP!

o∈Ujt pjo/|Uj | [37]

Truncated SPT (SPTT) pjo, if Wjo < α ∀o in queue; [38] [39] [102] [103]Wjo, otherwise

Weighted SPT (WSPT) wjpjo [107]

Longest processing time 1/pj (or 1/pjo) [27] [34], [37](LPT, LPTo)

Truncated LPT (LPTT) 1/pjo, if Wjo < α ∀o in queue; [38]Wjo, otherwise

Earliest finish time (EFT) rj + pj [7] [8]

Earliest due date dj (or djo) [6] [7] [8] [27] [34] [37](EDD, EDDo) [38] [39] [41] [60] [71] [81]

[101] [107], [9] [39] [60]

Slack (SLK, SLKo) dj − t−!o∈Ujt pjo − α, djo − t− pjo [7] [8] [27] [37] [38]

[82] [106] [107], [60]

SLK# Put jobs with SLK≤ 0 into a priority [33]queue and SLK> 0 into a normal queue.

Apply SPT to each queue.

SLK/P (dj − t−!o∈Ujt pjo)/!

o∈Ujt pjo) [37] [38] [107]

SLK/OPN (dj − t−!o∈Ujt pjo)/gj [9] [37] [70] [104] [105]

SLK/RAT (dj − t−!o∈Ujt pjo)/(dj − t) [70]

COVERT cj/pjo, where cj = 0, if the job is ahead [9]of schedule; cj = 1, if the job has no slack;

and cj = c̄j , otherwise, where c̄j isthe proportion of the job’s planned

waiting time that has been consumed

Critical ratio (CR, CRo) (dj − t)/!o∈Ujt pjo, (djo − t)/pjo [39] [60], [39]

R/OPN (dj − t)/gj [10] [70]

Modified Due Date max{dj , t+ pj}, max{djo, t+ pjo} [8] [9] [10] [81] [107](MDD, MDDo) [102] [103]

Earliest operation rj + (dj − rj)(o/gj) [27]due date (OPNDD)

P+S/OPN αpjo + (1− α)dj−t−

!gjl=o

pjl(gj−o+1)β

[27]


The performance of scheduling rules also depends on the due datepolicy and how tight the due dates are. For the objective of minimiz-ing tardiness, Elvers [37] Þnds that some scheduling rules work betterunder tight due dates while others work better under loose due dates.In particular, the scheduling rules which are based on processing times,such as LROT, LROT/NOP, SPT and SPTo, perform best when thedue dates are tight. (SPT�s good performance under tight due dates isalso observed in several other papers, e.g., [105].) On the other hand,scheduling rules which consider the operation due date or slack, such asEDD, SLK, SLK/OPN and SLK/P, perform better when the due datesare loose4. Intuitively, SPT tries to minimize average ßow time withoutpaying much attention to tardiness. This might hamper SPT�s perfor-mance (with respect to minimizing tardiness) when the ßow allowanceis high where most due dates can be met using the EDD rule. But whenthe ßow allowance is low, most jobs are late and SPT is advantageousbecause shorter mean ßow times lead to less tardiness. Having noticedthe complementary strengths of different rules depending on due datetightness, Baker and Bertrand [8] propose a modiÞed due date (MDD)rule, which is a combination of EDD and SPT rules. It works like SPTwhen the due dates are very tight and like EDD when the due datesare very loose. They test the MDD rule on the same test data used in[6] and observe that MDD sequencing results in lower average tardinesscompared to SPT or EDD, under both TWK and SLK due date rules,for a large range of allowance factor values. Note that in the single ma-chine model with equal job release times, average tardiness is minimizedby MDD under either TWK or SLK due date rules [8]. Wein [107] alsoÞnds that MDD performs better than other sequencing rules in generalfor the objective of minimizing the weighted average lead time subjectto service constraints.A interesting question is whether to use job due dates or individ-

ual operation due dates for sequencing. Kanet and Hayya [60] study theeffect of considering job due dates versus operation due dates on the per-formance of scheduling rules in the context of DDM. Under the TWKdue-date setting rule (with three different parameters), they comparethree due-date based scheduling rules and their operation counterparts:EDD vs. EDDo, SLK vs. SLKo and CR vs. CRo. They considerthe objectives of minimizing: mean lateness, standard deviation of late-ness, fraction of tardy jobs, conditional mean tardiness, mean ßow time,standard deviation of ßow time, and maximum job tardiness. They Þnd

4The SLK/OPN rule can behave in undesirable ways when SLK is negative. See [59] for adiscussion on such anomalous behavior and on modifications for correcting it.

16

that operation-due-date based scheduling rules outperform their job-due-date-based counterparts for almost all performance measures. In par-ticular, operation-due-date based scheduling rules consistently result inlower ßow times, and reduce the mean and the variance of the probabil-ity density function of lateness. Among these rules, EDDo outperformsSLKo for all performance measures under all conditions. In particular,EDDo results in the minimum value of maximum tardiness in all cases.Intuitively, EDDo extends Smith�s rule [94], which states that in an of-ßine single machine problem maximum tardiness is minimized by EDDsequencing. They also look at the impact of ßow allowance on the perfor-mance. Note that when we sufficiently increase α in TWK, then the jobwith the largest processing time will always have the largest due dateregardless of when it arrives. In that case, EDD sequencing becomesequivalent to SPT sequencing. Hence, for the rules EDD, EDDo, SLKand SLKo an increase in ßow allowance results in a decrease in averageßow time.In a related study to [60], Baker and Kanet [9] compare the perfor-

mance of MDD and MDDo with a number of other well-studied schedul-ing rules under the TWK due date rule, focusing on the impact of duedate tightness and shop utilization. Using industrial data, they notethat the average manufacturing capacity utilization in the U.S. rangedfrom 80 to 90% between 1965 and 1980. Hence, they test low=80%(high=80%) utilization with α values 2.5, 5 and 7.5 (5, 10, 15, 20),where α is the parameter of the TWK rule indicating due date tight-ness. They consider three performance measures: proportion of tardyjobs, conditional mean tardiness, and average tardiness. A comparisonof MDD and MDDo indicates the superior performance of MDDo exceptin environments with high utilization and very loose due dates, in whichcase both rules perform quite well. MDDo also outperforms SPT andEDDo in all environments except under very loose due dates. Similarto [37], the observations of Baker and Kanet indicate that the tightnessof due dates has an important effect on the performance of schedulingrules. For example, the performance of S/OPN and EDDo is quite goodwith loose due dates but deteriorates as the due dates become tighter.The opposite is true for SPT. Therefore, a �hybrid� rule such as MDDtends to exhibit a more robust and superior performance over a widerange of settings.Enns [38] [39] has experimented both with job- and operation-due-

date dependent dispatch policies. He considers internal as well as ex-ternal measures of the shop ßoor. He notes that the sequencing rulehas a signiÞcant impact on internal measures such as the number of andthe investment in work-in-process jobs, smoothness of work ßow, and


the balancing of queue lengths. For example, (1) EDD and SPT leadto less amount of work completed per job at any given time comparedto FCFS, (2) jobs tend to move slower at Þrst and then faster underEDD, whereas they progress at a more steady pace under EDDo. In-tuitively, operation due dates create artiÞcial milestones along the wayrather than a single deadline, and ensure a more steady progress of thejobs. While this might be desirable from a balanced workload point ofview, it could increase work-in-process inventories and related costs.Scheduling rules based on operation due dates have received limited

attention in the literature; but the current results suggest that they arequite promising and in some cases perform better than the correspondingrules based on job due dates [8] [9] [12] [60]. In general, if the variabilityin the system is low, one can obtain better estimates of ßow times andhence quote more accurate due dates. The scheduling policy impacts thevariability of ßow times. One possible reason for the observed superiorperformance of the operation due date based scheduling rules is becausethe operation due dates result in a more steady ßow of the jobs throughthe system, reducing ßow time variability.

3. Offline Models for Due Date Management

Offline models for DDM assume that the arrival times, and possiblythe processing times, of the jobs are known at the beginning of theplanning horizon. The arrival times can be equal ([3] [11] [19] [21] [24][75] [80] [79] [86] [97], Section 3.1) or distinct ([6] [16] [45], Section 3.2).All of the papers discussed in this section consider a single machine.

3.1 Equal Order Arrival Times

Under the assumption of equal job arrival times, a number of authorsstudied the common due date problem (CON), where all the jobs areassigned a common due date d and the goal is to jointly determine thecommon due date and a sequence [3] [11] [21] [24] [75] [80] [79]. Theobjective functions considered in these papers can be characterized bythe following general function of earliness, tardiness and the due date:!j βjE

aj +γjT

bj +αjd. In most papers, except in [24], a = b = 1, i.e., the

Þrst two summations correspond to weighted earliness and tardiness. In[75], βj = β, γj = γ and αj = α; in [3], βj = β, γj = γ and αj = 0; in[21], βj = γj = wj and αj = 0; in [24], βj = γj = wj and αj = α. In thisproblem, it is easy to show that the optimal due date d is equal to thecompletion time of some job j∗, i.e., d = Cj∗ . For the general functionwith a = b = 1, the following simple condition determines the optimaldue date d = Cj∗ for a given sequence: j

∗ is the Þrst position in the

18

sequence such that!j∗k=1(βk + γk) ≥

!nk=1(γk −αk) [11] [22] [79]. Note

that all the jobs before j∗ are early, and are sequenced in non-increasingorder of the ratio Cj/βj . The jobs after j

∗ are tardy and are sequencedin non-decreasing order of the ratio Cj/γj . Such a schedule is called aV-shaped sequence [3] [11]. The optimality of V-shaped schedules forcommon due date problems have also been observed in [83]. Optimalityconditions for the case a = b = s along with an iterative procedurefor determining d (for a given sequence) are given in [24]. Enumerationalgorithms for jointly determining the due date and the optimal sequenceare presented in [3] and [21].A related problem, where customers have a common preferred due

date A, is studied in [86]. A lead time penalty is charged for lead time(or due date) delay, Aj = max{0, dj −A}, which is the amount of timethe assigned due date of a job exceeds the preferred due date, A. Theobjective is to minimize the weighted sum of earliness, tardiness andlead time penalty. The authors propose a simple policy for setting thedue dates and show that this policy is optimal when used in conjunctionwith the SPT rule.The two papers discussed next [19] [25] [45] study the TWK and SLK

due date rules presented in Table 999.3.Cheng [19] studies the TWK due date rule under the objective of

minimizing total squared lateness. Both for deterministic and randomprocessing times (with known means and the same coefficient of varia-tion), he Þnds the optimal value of the parameter α and shows that theoptimal sequencing policy is SPT. Further extensions of this work arepresented in [20] and [23].Cheng [25] studies the SLK due date rule under the objective of min-

imizing the (weighted) ßow allowance plus the maximum tardiness. Heshows that the optimal sequence is EDD and derives a simple function tocompute the optimal SLK due dates. Gordon [45] studies a generalizedversion of this problem with distinct job release times and precedenceconstraints on job completion times, allowing job preemption.Soroush [97] considers the objective of minimizing the (expected)

weighted earliness and tardiness under random processing times. First,he derives the optimal due dates for a given sequence, and shows thatthe due dates depend on the jobs� earliness and tardiness costs and themean and the standard deviation of the jobs� completion times. Next, headdresses the problem of simultaneous scheduling and due date determi-nation. Unfortunately, he is unable to provide an en efficient procedurefor obtaining the optimal sequence, and hence provides lower and upperbounds on the expected total cost as well as two heuristics. Using ex-amples, he shows that the heuristics perform well, and that treating the


processing times as deterministic could lead to signiÞcantly inaccuratedue dates and higher costs.

3.2 Distinct Order Arrival Times

Unlike the previous papers discussed in this section, the two paperswe discuss next consider arbitrary (not necessarily equal) order arrivaltimes and order preemption while studying DDM in an offline setting.Baker and Bertrand [6] study DDM with job preemption under the

objective of minimizing the average due date subject to a 100% serviceguarantee, i.e., all the jobs must be completed on time. This prob-lem can be solved optimally by Þnding a schedule that minimizes theaverage completion time, and then by setting dj = Cj for each job j.Hence, using the three-Þeld notation described in [46] it is equivalentto 1|rj , pmtn|

!j Cj . When the jobs have equal release times, the ob-

jective becomes equivalent to minimizing the mean ßow time, and SPTsequencing gives the optimal solution. Unfortunately this approach re-quires that all the information about the jobs must be known in advance,which is rarely the case in practice. Hence, the authors consider threesimple due date setting rules, CON, SLK, and TWK, where the due datedecision depends only on the information about the job itself. They Þrstconsider these rules for the case of equal job arrival times and Þnd theoptimal value of the parameter α for each of these rules. Note thatwhen the arrival times are equal, EDD sequencing minimizes maximumlateness, hence, given a set of due dates, one can use the EDD rule toÞnd out if those due dates satisfy the service constraint of zero tardi-ness. Also, when the due dates are set using TWK or SLK due daterules, EDD and SPT sequences are equivalent. The authors show thatin case of equal job arrival times, CON is dominated by TWK and SLK,but no dominance relationship exists between TWK and SLK. Theyalso compare the worst case performance of these rules, by comparingthem to the optimal solution: (1) For the case of equal processing times,cTWK = cSLK = cCON = 2n/(n + 1). That is, the worst case perfor-mance of the three rules is the same, approaching 2 as the number ofjobs increases. (2) When the processing times are equal to 1 for all butone job j, for which pj > 1, cTWK = n − 1, cSLK = 1, cCON = n. Inthis case, TWK and CON have similar worst case performance, whichcan be arbitrarily large, whereas SLK produces the optimal solution. (3)When pj = j for all j, cTWK = 1.5, cSLK = 3, cCON = 3. The com-petitive ratios provide information about the worst case performanceof these rules, but they can be overly pessimistic. To gain an under-standing about the average performance of CON, TWK and SLK, the

20

authors perform simulation studies. In case of equal arrival times (withprocessing times drawn from exponential, normal or uniform distribu-tions), TWK produces the best results in terms of due date tightness.Furthermore, compared to SLK and CON, TWK produced due dateswhich are less sensitive to problem size.When the arrival times are distinct, a preemptive version of the EDD

rule minimizes the maximum lateness. Under the CON rule, since all thejobs have the same ßow allowance, EDD becomes equivalent to FCFSand preemption is not necessary. Under SLK, the waiting time allowanceis the same for all jobs, and the EDD rule will sequence the jobs in theincreasing order of rj+pj . Hence, preemption might be necessary underSLK. Note that in case of CON and SLK rules, one does not need thevalue of the parameter α to implement the EDD rule. However, incase of TWK, the value of α is needed for implementing EDD. Bakerand Bertrand [6] provide simple procedures for computing the due datesunder each of these rules. To test the performance of these rules, theyrun simulations for two different workload patterns. For the �random�workload pattern, they simulate anM/M/1 queue. For the �controlled�workload pattern, they use a scheme which releases a new job as soonas the workload in the system falls below a threshold. Experimentalresults suggest that TWK has the best average performance for therandom workload pattern. In the case of controlled workload pattern,SLK produces the best results for light workloads; however, the due datesof TWK are less sensitive to workload, suggesting that TWK might bethe preferred policy. Although TWK exhibits better performance onaverage than SLK, in unfavorable conditions it can perform considerablyworse. (This is also indicated by the arbitrarily large competitive ratioof TWK when all but one job have pj = 1.) The results suggest thatwhen the variance of processing times is low, there is little differenceamong the performances of the three rules.Charnsirisakskul et al. [16] take a proÞt-maximization rather than

a cost-minimization perspective and consider order acceptance as wellas scheduling and due-date setting decisions. In their model, an orderis speciÞed by a unit revenue, arrival time, processing time, tardinesspenalty, and preferred and latest acceptable due dates. While they al-low preemption, they assume that the entire order has to be sent to acustomer in one shipment, i.e., pieces of an order that are processed atdifferent times incur a holding cost until they are shipped. The shipmentdate of an accepted order has to be between the arrival time and thelatest acceptable due date. Orders that are completed after the prefereddue date incur a tardiness penalty. The processing and holding costs areallowed to vary from period to period. The goal is to decide how much of


each order to produce in each period to maximize proÞt (revenue minusholding, tardiness and production costs).Charnsirisakskul et al. consider both make-to-order (MTO) and make-

to-stock (MTS) environments. In the Þrst case, the processing of anorder cannot start before the order�s arrival (release) time while in thelatter case it is possible to process an order and hold in inventory forlater shipment. For both cases, they model the problem as a mixedinteger linear program and study the beneÞts of lead time and partialfulÞllment ßexibility via a numerical study. Lead time ßexibility refersto a longer lead time allowance (higher difference between the latest andpreferred due dates) and partial fulÞllment ßexibility refers to the optionof Þlling only a fraction of the ordered quantity. Their results show thatlead time ßexibility leads to higher proÞts, and the beneÞts of lead timeßexibility outweigh the beneÞts of partial fulÞllment ßexibility in bothsystems. Lead time ßexibility is more useful in MTO where early pro-duction is not an option. Numerical results also show that the beneÞtsof lead time and partial fulÞllment ßexibility depend on the attributesof the demand (demand load, seasonality, and the order sizes).

4. Online Models for Due Date Management

Online models for DDM assume that the information about a job,such as its class or processing time, becomes available only at the job�sarrival time. The arrival times are also not known in advance. Suchmodels are sometimes referred to as dynamic.Based on the �dimensions� discussed in Section 1, the papers that

study the DDM problem in an online setting can be further dividedinto two categories based on whether they study the problem in a singlemachine [7] [8] [14] [81] [107] or a multi-machine setting [12] [27] [33][37] [39] [41] [36] [38] [60] [70] [82] [101] [102] [103] [104] [105]. Althougha single vs. multi-machine categorization seems natural at Þrst, mostof the research issues are common to both settings, and the resultinginsights are often times similar. Therefore, we discuss the papers in thissection by categorizing them based on their approach to due-date settingdecisions and based on some of the other modeling dimensions and therelated research questions.For online DDM problems, very few researchers have proposed mathe-

matical models (see Section 4.3). The most common approach for settingdue dates is to use dispatch due date rules which follow the general formdj = rj + fj where fj is the ßow allowance. The tightness of the ßowallowances (and the due dates) is usually controlled by some parame-

22

ters. Alternative ßow allowances are summarized in Table 999.3 and arediscussed in Section 4.1.As in the case of dispatch rules for scheduling (Section 2), most of

the due date rules are parametric. These parameters may be constant(e.g., α in SLK) or dependent on the job or system conditions (e.g., γj inBERTRAND). The appropriate choices for these parameter values arenot straightforward and usually determined via simulation experiments.A small number of papers in the literature focus on choosing the appro-priate parameters for a given due date rule, and they are discussed inSection 4.2.Most of the DDM policies proposed in the literature do not result in

any service guarantees. A small number of researchers have proposedDDM policies with service guarantees, such as the maximum expectedfraction of tardy jobs or maximum expected tardiness. DDM policieswith service guarantees are discussed in Section 5.

Table 999.3: Due date setting rules with a flow allowance

Policy Flow allowance References

RND αβj , βj ∼Uniform[0,1] [27]

CON α [6] [8] [27] [30] [71] [72][87] [101] [107] [105]

SLK pj + α [6] [8] [101] [107]

TWK αpj [6] [7] [8] [9] [10] [19] [27][33] [34] [41] [60] [70] [71][72] [76] [77] [101] [104][105] [106] [107] [82]

TWK# α(pj)β [33]

NOP αgj [27] [76] [77] [82]

TWK+NOP αpj + βgj [76] [77] [82] [102] [103]

BN max{dj−1 − rj}+ αjpj [14]

JIQ αpj + βQj , where Qj is [14] [33] [71] [72] [76] [77]the number of jobs waiting [82] [102] [103] [107]to be processed ahead of job j

JIS αpj + βWISj , where WISj is the total [71] [82]number of jobs waiting to beprocessed in the system at time rj

WIQ αpj + βWIQj , where WIQj is [76] [77] [82]the total processing time of the jobswaiting to be processed ahead of job j

WINS αWINS, where WINS is the [41]sum of processing times of allthe jobs currently in the system

TWKCP αTWKCP , where TWKCP is [41]the sum of all operation timeson the critical path of the BOM


Table 999.3 — continued from previous pageFRY-ADD1 αTWKCP + βWINS [41]

FRY-ADD2 αpj + βWINS [41]

FRY-MULT1 αpj(WINS) [41]

FRY-MULT2 α(TWKCP )(WINS) [41]

RMR αWSPT +!k

i=1 αiWIQij + β1gj+ [76] [77][82]!ki=1 γiJIQij + β2WISj + β3WIQj+

β4JISj where WIQij and JIQij arethe work and the number of jobsin queue on the i-th machinein the routing of job j

FTDD E[F ] + ασF [10] [70]

TWK-RAGATZ pj(1 + αW #j/E[W ]), where W #

j is [81]the estimated workload inthe system when j arrives

EC3 αpj + βE[W ] [33], [104]

WEEKS6 αpj + βgjWo where [104]

W o is the expected wait timeper operation

WEEKS7 αpj + βW #, where W # [104] [82]is an estimated wait timebased on shop congestion level

OFS αF̄nj + βnj + γPj where F̄ is [72] [102] [103]the average operation time of thelast three jobs that are completed

COFS αF̄nj + β1Qj + β2nj + γPj [102] [103]

MFE (1− α)fsj + αfdj where fsj and fdj arestatic and dynamic flowtime estimates [103]

CON-BB αj , where αj = a(W #j/E[W ]) [7]

SLK-BB pj + αj , where [7]αj = E[p](a− 1)W #

j/E[W ]

TWK-BB αjpj , where αj = aW #j +E[p]/W̄ [7] [81]

BERTRAND pj + αE[pjo]gj + γj , where γj is [12]the additional flow allowance based onthe congestion/workload in the shop

WEIN-PAR I Equation (999.1) [107]

WEIN-PAR II Equation (999.2) [107]

HRS E[F (nj)] + βzασLT (nj) [55]where α is the service level andzα is the α-percentile ofthe standard normal distribution

Most of the papers that consider multiple machines in setting duedates use simulation as the method of study. Among these papers, amajority focus on job-shop environments, which are are summarized inTable 4. These simulation models assume that job preemption is notallowed and there are no service constraints. With the exception of a

24

few papers, including [12] [35] [67], they assume that the machines areperfectly reliable, i.e., there is no machine downtime.While generating routings for the jobs in the job-shop simulations,

most of these papers assume that a job is equally likely start its Þrstoperation on each machine (or machine group), and upon completion inone machine a job moves to another machine or leaves the shop withsome probability [27]. Let Pik denote the probability that when a jobcompletes operation on machine i it will move to machine k. Dependingon the processing requirements and the associated routings of the jobs,a shop has different workload characteristics, including the following:Balanced shop: In a balanced shop, the assignment of an operation to

any work center has equal probablilities (i.e., Pik = 1/m, k = 1, . . . ,m)and hence leads to approximately equal workload in each work center.Unbalanced shop: In an unbalanced shop, some work centers might

have higher loads than others.

In a bottleneck shop, one work center has a signiÞcantly higher loadthan others.

In a ßow shop, all jobs follow the same sequence of machines, thatis Pi,i+1 = 1, i = 1, . . . ,m− 1.In a central server job shop all job transfers take place thorugh acentral server, i.e., P1k > 0 and Pk1 = 1, j = 2, . . . ,m.

Most researchers base their experiments on balanced shops while a fewconsider both balanced and bottleneck shops [102] [103]. In general, twoconsecutive operations on the same machine are not permitted (excep-tions include [82]), and there are two alternatives for repeated machineson a routing: (1) a job can return to a machine after being processed onanother machine [33] [37] [39] [60] [102] [103], and (2) each job has atmost one operation in each work center [104] [105].Some papers explicitly consider multiple divisions and work centers

with multiple machines as well as dual-constrained job-shops, wherethere are labor as well as machine constraints [104] [105] [106].In the remaining part of this section, we discuss the papers with online

DDM models in more detail. We use the term �machine� to refer to anytype of resource. We say that due date management policy A dominatesB, if both policies satisfy the same service level constraints and policyA achieves a better objective function than policy B.

4.1 Due-Date Setting Rules

In quoting a lead time for a job, one can consider different types ofinformation, such as the job content, general shop congestion (workload),


Table 999.4. Simulation models for DDM in a job shop

# of Interarrival Processing # of opns. Shopmachs. times times per job utilization

[12] 5 Neg. exp. Neg. exp. Mean=4.463, Max=10 0.83, 0.9, 0.93

[27] 9 Exponential Exponential Mean=9 0.90

[33] 5 Constant Normal(20, 62) Uniform∼[1,5] 0.9

[37] 8 weekly arrivals Uniform∼ [6, 15] P(1)=.2, P(2)=.16Uniform∼ [8, 12] Triang.∼ [0.8, 1, 1.9] P(3)=.128, P(4)=.512

[41] 11 Exponential Normal depends on BOM 0.7, 0.8. 0.9

[38] 4 Exponential Neg. exp. Uniform∼ [2, 6] 0.8, 0.9

[39] 4 Neg. exp. Neg. exp. Uniform∼ [2, 6] 0.9

[9] [60] 8 Exponential Exponential Uniform∼ [1, 8] 0.8, 0.9

[67] [91]

[70] 5,7 Exponential Exponential Uniform∼ [1, 2m− 1] 0.8, 0.85, 0.9

[10] 4 Exponential Exponential 2-6 0.8, 0.9

[76] [82] 9 Neg. exp.(1) Neg. exp.(0.76) Geometric∼[1,15] 0.89

[101] 8 Exponential Exponential Uniform∼ [4, 8] 0.9

[102] 5 Exponential 2-Erlang Uniform∼ [1, 10] 0.85 - 0.95

[103] 5 Exponential Exponential Uniform∼ [1, 10] 0.85-0.95

[104] [105] 24 Neg. exp. Neg. exp. 0.9[106]

information about the routing of a job and the congestion ahead of thatjob, and the sequencing policy.The simplest due date rules are RND and CON, which do not consider

job or system characteristics in setting due dates. Given the drawbacksof these simple rules, researchers have proposed other rules (e.g., SLK,TWK, TWK", NOP, TWK+NOP and BN), which ensure that the jobcontent, such as the processing time or the number of operations, affectsthe due dates. Although these rules are one step up over the simplerules, they do not consider the state of the system, and hence, addi-tional rules are proposed to consider the workload in the system (orthe congestion) while setting due dates (e.g., EC3, JIQ, OFS, COFS).An important research question is whether the inclusion of detailed in-formation about the job content and system congestion improves theperformance of DDM policies. To answer this question, researchers havecompared the performance of various due-date setting rules in differentsettings via simulation. In this section, we discuss the due date settingrules and their relative performance.

4.1.1 Due Date Rules with Job Information. The due-date rules we discuss in this section incorporate only the information

26

about a job in quoting due dates: the processing time (SLK, TWK,TWK", BN), the number of operations (NOP), or both (TWK+NOP).One of the earliest papers that studies DDM is by Conway (1965)

[27]. He considers four due date policies, CON, NOP, TWK and RND,and tests the performance of nine priority dispatching rules for eachof these policies in a job shop. For the objective of minimizing aver-age tardiness, the author Þnds that under FCFS sequencing, the fourdue date policies exhibit similar, mediocre performance. Under EDD,SPT and P+S/OPN, the due date rules rank in the following order ofdecreasing performance: TWK, NOP, CON, RND. This suggests thatdue date rules that take the work content into account perform better.Among sequencing rules, SPT�s performance is less sensitive to the duedate policy, mainly because it does not take the due dates into account.SPT is superior to other rules when the workload in the system is high,and has the best performance in general. EDD and P+S/OPN performbetter with due date policies that take the work content into account.Baker and Bertrand [7] study combinations of three due-date rules

(CON, SLK, TWK) and Þve sequencing policies (FCFS, SPT, EDD,EFT, MST). Via simulation, they test due date management policiesin M/M/1 and M/G/1 queues under low and high utilization. Forthe objective of minimizing average tardiness (subject to a maximumaverage ßow allowance factor a, i.e., E[F ] = aE[p]), they Þnd that SPTand EDD always dominate the other sequencing rules. Furthermore,in combination with SPT or EDD, CON is always dominated by SLKwhich is dominated by TWK. Hence, they focus on the combinations ofSPT and EDD with SLK and TWK. The dominance of SLK by TWKindicates the importance of using job information in determining duedates. The authors Þnd that the performance difference between TWK,SLK and CON rules is very small when the variance of job processingtimes (σ2

p) is low. However, the difference becomes signiÞcant when σpis high, and in that case TWK dominates SLK dominates CON.The papers that are discussed in the remainder of this section con-

sider dual-constrained job shops and investigate the performance of laborassignment policies in conjunction with due date quotation and sequenc-ing.DDM policies with labor assignment decisionsIn case of dual-constrained job-shops, in addition to due date and

sequencing decisions, one also needs to consider labor assignment deci-sions. Most papers assume that laborers may be transferred betweenwork centers within a division, but not between divisions [104] [106].Under this assumption, a laborer can be considered for transfer whenhe Þnishes processing a job (CENTLAB) or when he Þnishes process-


ing a job and there are no jobs in the queue at the work center he iscurrently assigned to (DECENTLAB). The following reassignment rulesare considered in the literature in assigning a laborer to a work center:assign to the work center (1) with the job in the queue which has beenin the system for the longest period of time, (2) with the most jobs inthe queue, and (3) whose queue has the job with the least slack time peroperation remaining.Weeks and Fryer [105] investigate the effects of sequencing, due-date

setting and labor assignment rules on shop performance. They simu-late a job show with three divisions, each containing four work centersof different machine types and four laborers. There are two identicalmachines in each work center and each laborer can operate all machinetypes. For labor assignments, they consider CENTLAB and DECENT-LAB policies and the three reassignment rules discussed above. Theytest FCFS, SPTo and SLK/OPN sequencing rules with CON and TWKdue date rules and compare their performance using multiple regression.Their experiments lead to the following observations: (1) The impact ofdue-date assignment rules: Due-date assignment rules are the most im-portant decisions for mean lateness, variance of lateness and proportionof tardy jobs. Tight due dates perform better than loose due dates interms of mean ßow time, variance of ßow time, and variance of lateness,and the reverse is true for mean lateness and percent of jobs tardy. (2)The impact of sequencing rules: Sequencing rules are the most impor-tant class of decision rules to impact mean ßow time and variance of ßowtime. Sequencing rules also have pronounced effects on mean lateness,variance of lateness and proportion of tardy jobs. The relative effectsof sequencing rules are independent of the due date assignment rule forthe mean ßow time, mean lateness and total labor transfers. For thevariance of ßow time, variance of lateness and proportion of tardy jobsthe relative inßuence of sequencing rules depends on the tightness ofthe assigned due dates, but their relative performance does not change.SPTo has better performance than other rules for mean ßow time, meanlateness, proportion of jobs late and total labor transfers. FCFS per-forms best in terms of variance of ßow time and SLK/OPN performsbest in terms of variance of lateness. SPT performs best in terms of theproportion of jobs late, regardless of the due-date assignment rule. (3)The impact of labor assignment rules: By increasing the ßexibility of theshop, CENTLAB performs better than DECENTLAB in all measuresexcept total labor transfers. The rule which assigns laborers to the workcenter with the job in the queue which has been in the system for thelongest period of time performs best.

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4.1.2 Due Date Rules with Job and System Information.There are various ways of incorporating the system workload informationin setting the due dates. Ultimately, the goal is to estimate the ßow timeof a job, and quote a due date using that estimate. Some researchershave proposed using simple measures about the system conditions forestimating ßow times and quoting lead times:

the number of jobs waiting to be processed in the system (JIS),

the number of jobs waiting to be processed ahead of job j (JIQ),

the total processing time of the jobs waiting to be processed aheadof job j (WIQ),

the number and the total processing time of the jobs waiting to beprocessed on all machines on the routing of job j (RMR),

the sum of processing times of all the jobs currently in the system(WINS), and

the sum of all operation times on the critical path of the BOM(TWKCP).

The ßow time has two components: the job processing time and thewaiting (or delay) time. In general, Þnding analytical estimates of ßowtimes in job shops is very difficult, even for very special cases [57] [92];the ßow time depends on the scheduling policy, the shop structure, jobcharacteristics (e.g., the number of operations, the variability of theprocessing times) and the current state (e.g., the congestion level) ofthe system. However, for relatively simple environments, such as ex-ponential processing and interarrival times with one or two customerclasses, Þnding analytical estimates for expected job ßow (or waiting)times might be possible [7] [70] [107]. When the ßow allowance is chosento be a multiple of the average processing time, i.e., fj = aE[P ], werefer to the multiple a as the ßow allowance factor. For example, in theTWK rule, by setting α = a, where a = E[F ]/E[P ], the ßow allowancebecomes equal to the average ßow time, which would lead to an averagetardiness of zero. We denote by a∗ the minimum ßow allowance factorwhich would lead, on average, to zero tardiness for a given DDM policy.The ßow time estimates in a system can be static or dynamic, de-

pending on whether they consider time-dependent information. Thestatic estimates focus on steady-state, or average ßow time, whereas thedynamic estimates consider the current job or shop conditions. Staticestimates are usually based on queuing models or steady state simula-tion analysis. The due date rules bases on static ßow time estimates


include EC3, WEEKS6, FTDD. Several methods, ranging from simplerules to regression analysis on the information of recently completedjobs, are proposed for dynamic ßow time estimates. Due date rulesbased on dynamic ßow time estimates incorporate time-phased workloadinformation: WEIN-PAR I-II, WEIN-NONPAR I-II, JIQ, WEEKS7,OFS, COFS, BERTRAND, and EFS. For example, the ßow allowancein BERTRAND depends on the observed congestion in the shop at anytime. The ßow allowance in OFS and COFS depend on the averageoperation time of the last three jobs that are completed, as well as thenumber of operations in the queue ahead of a job.There are two types of measures that can be used in evaluating the

quality of a ßow time estimate: accuracy and precision. Accuracy refersto the closeness of the estimated and true values, i.e., the expected valueof the prediction errors. Precision refers to the variability of the predic-tion errors. Dynamic rules usually result in better precision (lower stan-dard deviation of lateness) because they can adjust to changing shopconditions; however, even if the prediction errors are small, they maybe biased in certain regions leading to poorer accuracy (deviations fromthe true mean) compared to static models. Note that if the quotedlead times are equal to the estimated ßow times, accuracy and preci-sion would measure the average and the standard deviation of lateness,respectively.A small number of due date rules are based on the probability distri-

bution of the ßow time (or the workload in the system). For example,Baker and Bertrand [7] modify the TWK rule by setting α = 2aF (W "

j)where F (x) denotes the cumulative distribution function (CDF) of theworkload. Although it is hard to determine F (x) for general systems, it isknown to be exponential with mean E[p]/(1−λE[p]), when the process-ing times are exponential. The authors Þnd that this TWK modiÞcationused in conjunction with EDD or SPT produces the best results for somea values among all the policies they tested. The idea of using a CDFfunction to estimate workload information is further explored by Udo(1993) [101] and extended to multiple machines.Given the difficulty of Þnding analytical estimates for ßow times, sev-

eral researchers have proposed approximations. For example, estimatesfor the mean and the variance of the ßow time are proposed in [89][90] and a method for approximating the ßow time distribution is pro-posed in [91]. Shantikumar and Sumita [91] develop approximations forestimating the distribution of ßow time in a job shop under a given se-quencing rule. Their approximation uses the concept of service index(SI) of a dynamic job shop, which is the squared coefficient of variationof the total service time of an arbitrary job. Depending on the value

30

of the service index, they divide dynamic job shops into three classes:G (SI << 1, e.g, ßow shop), E (SI ! 1, e.g., balanced job shop) andH (SI >> 1, e.g., central server job shop). They approximate the ßowtime distribution for the jobs in G, E and H by generalized Erlang, ex-ponential, and hyper-exponential distributions, respectively. Numericalexamples show that the approximations are good, especially when SI isclose to (σF /E[F ])

2. Recall that Seidmann and Smith [87] showed howto Þnd the optimal CON policy (for the objective of minimizing the sumof weighted earliness, tardiness and deviations from a desired due dateA) assuming that the distribution of the ßow time is known. Using anumerical example, Shantikumar and Sumita show how their estimateof the ßow time distribution can be used to Þnd the optimal CON duedate policy following the method proposed in [87].In the remainder of this section we discuss in more detail the DDM

policies which take both job and system workload information into ac-count. We Þrst present workload-dependent rules which use simple for-mulas to incorporate the information about the state of the system. Wethen continue with the rules which are based on more sophisticated butsteady-state estimates of the workload, such as the mean, standard devi-ation or distribution of ßow time. Finally, we present the due date ruleswhich are based on time- and job-dependent estimates of ßow/waitingtimes.Eilon and Chowdhury [33] are among the Þrst researchers to consider

workload-dependent due date rules. They propose EC3 and JIQ, whichextend the TWK rule by incorporating the mean waiting time in the sys-tem and the number of operations ahead of an arriving job, respectively.Via simulation, they test DDM policies which result from combinationsof three sequencing rules (FCFS, SPT and SLK") and four due datepolicies (TWK, TWK", EC3 and JIQ) for the objective of minimizing a(weighted) sum of earliness and tardiness. The simulation results suggestthat JIQ is the best due date policy in general, and SLK" performs bet-ter than FCFS and SPT. SPT seems to be the least effective, especiallywhen the penalty for late jobs is very high.Miyazaki [70] derives two formulae for estimating the mean and stan-

dard deviation of ßow times, and proposes a due date rule (FTDD)based on these estimates. He tests this policy with three sequencingrules (SLK/OPN, R/OPN and SLK/RAT), which take into account thedue dates. For comparison purposes, he also tests DDM policies TWK-FCFS and FTDD-FCFS. He considers three objectives: minimizing thenumber of tardy jobs, average tardiness and maximum tardiness. Forall these objectives, he Þnds that (1) FTDD-FCFS performs better thanTWK-FCFS, (2) sequencing rules SLK/OPN and R/OPN yield better


results than FCFS when used in conjunction with FTDD, and (3) se-quencing rule R/OPN is slightly better than SLK/OPN especially fortighter due dates. In a follow-up study, Baker and Kanet [10] compareFTDD and TWK due date rules with R/OPN and MDD scheduling rulesin a setting similar to the one in [70]. They conclude that using TWK-MDDmight be a better choice than FTDD-R/OPN or FTDD-SLK/RATfor the following reasons. (1) SLK/RAT might perform in undesirableways when the slack is negative. (2) Miyazaki claims that SLK/RAT isvery effective in reducing the number of tardy jobs when the due datesare tight. However, a relatively small proportion of jobs was tardy inMiyazaki�s experiments, and in such settings CR and SLK/RAT havesimilar performances (assuming that the slack is positive). In earlierstudies as well as this one, MDD is found to outperform various rules,including CR and R/OPN, in a variety of settings. (3) The parametersof the FTDD rule are difficult to compute, especially under any priorityrule other than FCFS.Weeks [104] proposes two new workload-dependent due date rules,

WEEKS6 and WEEKS7, and tests their performance against TWK andEC3 under SPT and SLK/gj sequencing rules for the objectives of meanearliness, tardiness, and missed due dates. He Þnds that when employedwith due-date based sequencing rules, due date rules that consider thecurrent state of the system in addition to job characteristics performbetter that the rules that only consider the job characteristics. Similarly,sequencing rules that consider due date information perform better thanthe rules that do not. He also Þnds that even though the shop size doesnot seem to affect the performance of due date management policiessigniÞcantly, increased shop complexity degrades the performance.Bertrand [12] proposes a due date policy (BERTRAND) that takes

into account the work content of a job and the workload (or congestion)in the shop at the time of the job�s arrival. He compares this policyagainst a simpler one (obtained by setting γ = 0 in BERTRAND) whichignores the workload in the shop. The mean and standard deviation ofjob lateness are used to measure the performance of DDM policies. Thescheduling (or loading) policy ensures that during any time duration(1) the load factor of a machine (the ratio of the cumulative load on amachine to the cumulative machine capacity) does not exceed a speciÞedcumulative load limit (CLL), and (2) there is a minimum waiting (or ßowtime) allowance, α, per operation. In a situation where the load factorwould exceed CLL, the ßow allowance of the job/operation is increased.Using simulation experiments, Bertrand Þnds that an increase in α or

a decrease in CLL increases the standard deviation of quoted ßow times.Similarly, an increase in α increases the standard deviation of actual

32

ßow times. The covariance of actual and quoted ßow times increase inα and decrease in CLL. He also Þnds that using time-phased workloadinformation improves the standard deviation of lateness (σL); however,the improvement decreases for larger values of α. The parameter CLLdoes not seem to have much impact on σL; however, it has a signiÞcantimpact on the mean lateness (E[L]) and its best value depends on α.For decreasing CLL values, the sensitivity of mean quoted ßow time tomean ßow time increases. These results indicate that for the objectiveof minimizing σL (1) DDM policies which use time-phased workloadand capacity information perform quite well, and (2) the performanceof BERTRAND seems to be very sensitive the choice of α and relativelyinsensitive to the type of job stream. Finally, the performance of thedue date rule is sensitive to the sequencing rule, especially if changes inthe sequencing rule shift mean ßow time.Ragatz and Mabert [82] test the performance of 24 DDM policies ob-

tained by pairwise combination of eight due date rules (TWK, NOP,TWK+NOP, JIQ, WIQ, WEEKS7, JIS, and RMR) and three sequenc-ing rules (SPT, FCFS and SLK). For the objective of minimizing thestandard deviation of lateness (σL), simulation results suggest that theparameter values of the due date rules are sensitive to the sequencingpolicy, and due date rules that consider both job characteristic and shopinformation (RMR, WIQ and JIQ) perform signiÞcantly better than therules that only consider job characteristics information (TWK, NOP,TWK+NOP). Among the rules that consider shop information, the onesthat consider information about the shop status in the routing of a job(JIQ, WIQ) perform better than the rules that only consider generalshop information (WEEKS7, JIS). It is also interesting to note that JIQand WIQ exhibit similar performance to RMR, which incorporates themost detailed information about the jobs and the shop status. This sug-gests that the use of more detailed information in setting due dates, asin RMR, only brings marginal improvement over rules, such as WIQ andJIQ, which use more aggregate information.The next three papers by Baker and Bertrand [7], Ragatz [81] and

Udo [101] modify the CON, SLK and TWK due date rules to incorpo-rate workload information. The modiÞcations in [7] and [81] are based onsimple estimates of the waiting time in the system, whereas the modiÞ-cations in [101] are based on the estimates of the cumulative distributionfunction.In a single-machine environment, Baker and Bertrand [7] modify the

CON, SLK and TWK rules, such that the parameter α depends on thewait time (estimated analytically) in the system at the arrival time ofjob j, as well as the job�s processing time and the average wait time.


In particular, they set α equal to aW "j/E[W ], E[p](a− 1)W "

j/E[W ], and

a(W "j+E[p])/W̄ , for CON, SLK, and TWK, respectively, whereW

"j is the

estimated workload at the arrival of job j and W̄ is a constant to meetthe requirement E[F ] = aE[p]. We refer to this as the proportionalworkload function (PWF). Surprisingly, simulation results show thatthe use of workload information does not always lead to better results.In particular, the inclusion of workload information frequently resultsin higher tardiness under the SPT rule when the due-dates are verytight. Similarly, in high utilization settings the DDM policy TWK-SPT without workload information produces the best results when theßow allowance factor a is between 2 and 4. On the positive side, theinclusion of workload information signiÞcantly improves the performancein some cases. For example, the DDM policy SLK-EDD, which is alwaysdominated when workload information is not included, produces the bestresults when due dates are very loose. Note that under loose due dates,i.e., when a ≥ a∗, SLK produces zero tardiness (on average). When thedue-date tightness is medium, TWK-EDD produces the best results. Insummary, for tight due dates (small a), the best policy is to use TWKor SLK with SPT, without using workload information. As the due datetightness increases, the TWK-EDD with workload information performsbetter. Finally, for really loose due dates (a ≥ a∗), the best choice isSLK-EDD.The dynamic version of the TWK rule proposed by Baker and Bertrand,

TWK-BB, adjusts the ßow allowance according to the congestion in theshop and quotes longer due dates in case of high congestion. Althoughthis rule results in lower tardiness in many settings, it usually results inhigher fraction of tardy jobs. Ragatz [81] proposes and alternative mod-iÞcation to the TWK rule, TWK-RAGATZ, which outperforms TWK-BB by reducing the number of tardy jobs, while achieving the same orlower average tardiness in most settings. Ragatz notices that when theshop is lightly loaded or if the ßow allowance parameter α is small, thenTWK-BB might result in a ßow allowance smaller than the job�s process-ing time. Hence, some jobs are a priori assigned �bad� due dates whichleads to a high number of tardy jobs. When the workload is not veryhigh (less than 90%) the percentage of tardy jobs is quite high even forhigh allowance factors (relatively loose due dates). Ragatz�s modiÞcationof the TWK rule ensures that the ßow allowance is always larger thanthe average processing time. In almost all settings, TWK-BB leads toa signiÞcantly higher fraction of tardy jobs than TWK-RAGATZ whichresults in a higher fraction of tardy jobs than TWK.Udo [101] investigates the impact of using different forms of workload

information on the performance of DDM policies. In particular, as in

34

[7], he considers two variations of the TWK and SLK rules, modiÞedbased on workload information in the form of proportional workloadfunction (PWF) and cumulative distribution function (CDF). As in [7],Udo considers the objective of minimizing average tardiness (subject toa minimum average ßow allowance a, i.e., E[F ] = aE[p]). Based onsimulations in a job-shop environment, he Þnds that the CDF modeldominates the PWF model, which dominates the basic SLK and TWKmodels (under two sequencing rules, SPT and EDD). Except in TWK-SPT, the difference between PWF and CDF models is statistically sig-niÞcant. Furthermore, the difference between these models is larger forsmall allowances and becomes smaller as the allowance factor increases(as the due dates become looser). Among all the DDM policies tested,SLK-EDD results in the best performance, followed by SLK-SPT. Theseresults suggest that (1) including workload information in DDM resultsin lower tardiness, (2) for loose due dates (low congestion), using de-tailed information about the workload in setting due dates might notbe critical, and (2) the impact of workload information depends on thechoice of the DDM policy; for certain policies (e.g., TWK-SPT) there isnot much payoff in the additional information required by more complexworkload models. Most of these observations hold for the single-machineenvironment as well, but interestingly, the SLK-EDD policy, which hasthe best performance in the job-shop setting, results in the highest tardi-ness in the single-machine environment. In addition, the mean tardinessperformance of all the DDM policies tested was much poorer in a multi-machine job-shop than in a single-machine shop. These observationsonce again indicate that the performance of a DDM policy depends onthe shop structure.Lawrence [67] uses a forecasting based approach for estimating ßow

times. In particular, he focuses on approximating the ßow time estima-tion error distribution (which is assumed to be stationary, and denotedby G) by using the method of moments [109]. He uses six methodsfor forecasting ßow times. NUL sets ßow times to zero, such that Gbecomes an estimator for the ßow time. ESF uses exponential smooth-ing, such that the ßow time estimate after the completion of k jobs is�fk = αFk + (1 − α) �fk−1. The other four rules are WIS, which is WIQwith parameters α = β = 1; EXP, which uses the (exponential) wait-ing time distribution of an M/M/1 queue; ERL, which uses Erlang-kwith mean kp, if k jobs are currently in the system; and BNH, proposedin [14], which uses a hypoexponential distribution. As due date rules,Lawrence uses (1) �f + �G−1(0.5), where �G is the current estimate of theerror distribution, (2) �f+ the sample mean of G, (3) the due date ruleproposed by Seidmann and Smith [87], and (4) �f+ �G−1(1−MFT ) where


MFT is the mean fraction of tardy jobs. He notes that due date rules(1)-(4) minimize mean absolute lateness, mean squared lateness, the sumof earliness, tardiness and due date penalties, and due dates (subject toa service level constraint), respectively, in a single machine environment.He tests the quality of the proposed ßow time estimators as well as duedate rules via simulation both on single machine (in three environments,M/M/1, M/G/1, and G/G/1, as in [14]) and job-shop environments (asin [91]) under FCFS scheduling. For the single machine environment,simulation results indicate that ßow time estimators that use currentsystem information (WIS, ERL, BNH, and ESF) perform better thanthe estimators that use no or only long-run system information (NUL,EXP). For the multi-machine environment, in addition to using the ßowtime estimation methods discussed above by considering the system asa single aggregate work center (�aggregation� approach), he also testsvariations of these six methods by estimating ßow times (and forecasterrors) for individual work centers and then combining them (�decompo-sition� approach). In addition, he tests TWK and JIQ rules to estimateßow times. Simulation results indicate that the decomposition approachworks better for estimating ßow times and setting due dates than theaggregate approach, and WIS and ESF perform at least as good as orbetter than other ßow time estimators. These results are encouraginggiven the minimal data requirements for implementing these estimators.However, since the simulations consider only the FCFS scheduling rule,these results should be interpreted with some caution. The impact ofthe scheduling rules on ßow times is critical, and these results might notgeneralize to environments using different scheduling policies.Vig and Dooley [102] propose two due date rules, operation ßow

time sampling (OFS) and congestion and operation ßow time sampling(COFS) which use shop congestion information for estimating ßow timesand setting due dates. In particular, these rules incorporate the averageoperation time of the last three jobs that are completed. They test theperformance of due date rules OFS, COFS, JIQ and TWK+NOP in com-bination with three scheduling rules: FCFS, SPTT and MDDo. Thesethree rules are signiÞcantly different. FCFS contains no info about thejob or shop status. SPTT is a dynamic rule considering job info. MDDois also a dynamic rule but it is based on internally set operation duedates. To test the impact of work center balance on the performanceof DDM policies, they test both a balanced shop and a shop with abottleneck. To estimate the best values for the parameters of the DDMpolicies, they use multiple linear regression techniques. The coefficientsfound for each DDM policy minimize the variations of the predictedßow times from actual ßow times. In experiments, they Þnd that the

36

due-date rule, scheduling rule and shop balance levels all signiÞcantlyaffect the performance of DDM policies. For the objective of % tardyjobs, TWK+NOP performs best under all three scheduling rules andshop balance conditions. But in all other objectives, DDM policies thatcontain shop information outperform TWK+NOP. Among sequencingrules, for the objectives of average and standard deviation of lateness,MDDo performs best in general. In terms of standard deviation of late-ness (σL), i.e., due date precision, shop-information based rules OFS,COFS and JIQ perform 15-50% better than the job information ruleTWK+NOP. In addition, when MDDo rule is used, σL decreases by upto an additional 25%. In general, COFS, JIQ or OFS used in conjunctionwith MDDo result in the best performance for all objectives other than% tardy jobs. COFS and JIQ perform very good in general. OFS, whichis simpler, also performs comparably well for lateness, and provides im-provements for % tardy jobs. No statistical signiÞcance is detected inaverage lateness among the three rules, hence more detailed shop infodoes not seem to help much. The differences between the performancesof DDM policies are more pronounced in balanced shops than in un-balanced shops. Perhaps when the shop is imbalanced the performanceof all policies degrade, making it more difficult to observe signiÞcantperformance differences.In a follow-up study, Vig and Dooley [103] propose a model which

includes both static and dynamic ßow time estimates into ßow time pre-diction. Let fsj and fdj be the static and dynamic ßow time estimatesfor job j. The mixed ßow time estimate proposed by Vig and Dooley[103] is a convex combination of static and dynamic estimates, namely,fmj = (1 − α)fsj + αfdj , where the dynamic ßow time estimate fsj isbased on one of the following four due date rules, TWK+NOP, JIQ, OFSand COFS. The parameters for fsj and fdj are obtained from steady-state simulation experiments. Static and dynamic ßow time estimateshave complementary strengths: better accuracy and better precision,respectively. The mixed rule is designed with the goal of improving ßowtime accuracy (achieving a mean prediction error close to zero) withoutsacriÞcing precision (keeping the variability of ßow time errors small).As in [102], they test three scheduling rules: FCFS, SPTT and MDDo.Simulation results show that the performance of all due-date rules im-proved under the mixed strategy, for all scheduling rules, and the perfor-mance differences between the rules became less crucial. In particular,for the objective of average lateness and percent tardy jobs, the mixeddue dates perform signiÞcantly better than their dynamic counterparts,especially when used with the MDDo scheduling rule. There is no sig-niÞcant difference between dynamic and mixed rules in terms of the


standard deviation of lateness. The results also show that the improvedaccuracy was not obtained at the cost of losses in the precision of ßowtime estimates.As an alternative to the commonly studied rules (TWK, NOP, TWK+NOP,

JIQ, WIQ, and RMR) for setting due dates, Philipoom et al. [76] proposea neural networks (NN) approach and nonlinear regression. Essentially,the neural network is �trained� on sample data and then used to pre-dict ßow times. Using a simulation environment similar to [82] underSPT sequencing, for �structured� shops (where all jobs follow the samesequence of work centers) they Þnd that both NN and the nonlinear re-gression approaches outperform the conventional rules for the objectivesof minimizing mean absolute lateness and standard deviation of lateness(SDL), and the neural network outperforms nonlinear regression. Forunstructured shops, RMR slightly outperforms NN for standard devia-tion of lateness. In a follow-up study [77] they use a similar approachfor minimizing objectives that are (linear, quadratic, and cubic) func-tions of earliness and tardiness. Again, they test NN against the six duedate rules above, as well as nonlinear regression and linear programming(LP). Their results indicate that NN is fairly robust as it performs quitewell regardless of the shape of the cost function, whereas some otherapproaches show signiÞcantly poor performance under certain cost func-tions (e.g., LP�s performance with nonlinear cost functions or RMR�sperformance for linear cost functions). As in [67], one limitation of thisstudy is that it only considers a single sequencing rule, SPT. Furthertesting is needed for assessing the performance and robustness of NNunder different sequencing rules.In estimating the ßow time of a job and quoting a lead time, in addi-

tion to shop conditions and the operation processing times, the �struc-ture� of the job (or product) may also play a role. This is especiallytrue in implementing a Materials Requirements Planning (MRP) systemwhere one estimates the ßow time at different levels and then offsets thenet requirements to determine planned order releases. The total time ajob spends in the system depends on the bill of materials (BOM) struc-ture. For example, a �ßat� BOM, where there are multiple componentsor subassemblies but only a few levels, could permit concurrent process-ing of operations in multiple machines. In contrast, a �tall� BOM withmany levels would require most operations to be performed sequentially.Previous work on sequencing suggests that jobs with tall BOMs tend tohave higher tardiness than jobs with ßat BOMs [85].Few researchers have studied the impact of product structures on

ßow times, scheduling and due date management policies [41] [56]. Forassembly jobs, delays in the system occur not only due to waiting in

38

queues (due to busy resources), but also due to �staging,� i.e., waitingfor the completion of other parts of the job before assembly. Fry etal. [41] study the impact of the following job parameters on the ßowtime: total processing time (sum of all operation times, denoted by pjfor job j), sum of all operations on the critical (longest) path of theBOM (TWKPCj), total number of assembly points in the BOM (NAP),number of levels in the BOM (NL), number of components in the BOM(NB). In addition, they consider the impact of the total amount of workin the system (WINS) on ßow times. WINS is computed by adding theprocessing times of all the jobs in the system, which are currently beingprocessed or are in the queue. The authors Þnd that NAP, NL and NBare not signiÞcant. On the other hand, pj , TWKPCj and WINS aresigniÞcant. In particular, WINS becomes more important than pj orTWKCP in predicting ßow times as the congestion level increases. Theauthors propose four models (see Table 999.3), two additive [73] and twomultiplicative [4], for estimating ßow times and setting due dates basedon these parameters. They test these due date rules as well as threeother simple rules via simulation under EDD sequencing. The resultssuggest that the performance of DDM policies depend both on the duedate rule and the product structure. For the objective of minimizingmean ßow time, TWKCP and TWK, which consider only job charac-teristics, perform best for ßat or mixed product structures. For theobjective of minimizing average tardiness, FRY-ADD1 and FRY-ADD2perform best, except for high congestion levels, where WINS performsslightly better. These two rules consider both the job characteristicsand the shop congestion. For the objective of minimizing mean absolutelateness, FRY-ADD2 performs the best under all BOM types and con-gestion levels. For all objectives, FRY-MULT1 and FRY-MULT2 arethe worst performers.One of the most interesting results of [41] is the relatively poor per-

formance of TWKCP, which follows the traditional approach of settinglead times based on the critical path in the BOM. The best performerswere those rules which considered both the critical path and the systemcongestion in an additive rather than a multiplicative form.In a related study Adam et al. [1] propose dynamic coefficient esti-

mates for CON, TWK and TWKCP for multi-level assembly job shopsand test them under EDD and EDDo scheduling rules for the objectivesof minimizing the average lead time, lateness, conditional tardiness, andfraction of tardy jobs. In contrast, most of the earlier studies concern-ing assembly jobs focus in Þxed coefficients determined by a priori pi-lot simulation studies and regression. They also propose an extensionto TWKCP, called critical path ßow time (CPFT), where the ßow al-


lowance is based not only on the processing times (as in TWKCP) butalso on expected waiting times on the critical path. The dynamic es-timates of waiting times are based on the Little�s law at an aggregate(micro) level, where the estimated waiting time (at a work center i) attime t is set to W̄t = nt/λ̄t (W̄ti = nti/λ̄ti), where nt and λ̄t (nti and λ̄ti)are the number of jobs in and the estimated arrival rate to the system(work center i) at time t. Through an extensive simulation study withvarious job structures they observe that the dynamic updates improvethe overall performance of the due date rules, and reduce the perfor-mance differences among them. They also show that the performance ofthe job vs. operation-based scheduling rules (EDD vs. EDDo) heavilydepends on the job structure, the due date rule, and the objective func-tion, and the performance differences are more prevalent under complexjob structures.

4.2 Choosing the Parameters of Due Date Rules

Several researchers have investigated the impact of due date tightnessor other parameter settings on the performance of DDM policies. Forexample, some suggested α = 10 in the TWK rule claiming that a job�sßow time in practice is 10% processing and 90% waiting [73]. The �opti-mal� tightness is found to be a factor of overall shop utilization, relativeimportance of missed due dates, market and customer pressures, andrelative stability of the system [102].The choice of the parameters of a due date rule determines how tight

the due dates are. The tightness of the due dates, in turn, affect theperformance of the accompanying sequencing rule and the overall per-formance of the DDM policy. For example, in the study of Baker andBertrand [7], when the ßow allowance factor is low (i.e., due datesare tight), the choice of the sequencing rule impacts the performanceof DDM policies signiÞcantly, whereas the choice of the due date ruledoesn�t seem to matter that much. In contrast, when the due dates areextremely loose, one can obtain no tardiness only with speciÞc due daterules, but the sequencing rule does not need to be sophisticated. Finally,when the tightness of the due dates are medium, then both the due datepolicy and the sequencing policy impact the performance.Analytical studies for choosing the parameters of due date rules in-

clude [87] and [106]. Seidmann and Smith [87] consider the CON duedate policy in a multi-machine environment under the objective of min-imizing the weighted sum of earliness, tardiness and lead time penalty(the same objective function is also considered in [86]). They assumethat the shop is using a priority discipline for sequencing, such as FCFS

40

or EDD, and the probability distribution of the ßow time is known andis common to all jobs. They show that the optimal lead time is a uniqueminimum point of strictly convex functions can be found by simple nu-merical search.Weeks and Fryer [106] consider the TWK rule in a dual-constrained

job-shop along with sequencing policies FCFS, SPT and SLK. Their fo-cus is on developing a methodology (based on regression analysis) forestimating the optimal parameter α for the TWK rule. They Þrst runsimulations using different values of α and use linear and nonlinear re-gression models to estimate the resulting costs of mean job ßow time,earliness, tardiness, due date and labor transfer. They then combinethese component costs into a total cost objective function and use re-gression to estimate the total cost as a function of α. Given the estimatedtotal cost function, one can then do a numeric search to Þnd the bestvalue of α.

4.3 Mathematical Models for Setting Due Dates

All the papers discussed so far in this section propose heuristic �rules�for setting due dates. In contrast, one could possibly model the duedate setting problem as a mathematical program. Given the difficultyof developing and solving such models for DDM, the research in thisdirection has been very limited. In this section, we discuss two papers,by Elhafsi [35] and Elhafsi and Rolland [36], who propose a model fordetermining the delivery date and cost/price of an order.Elhafsi and Rolland [36] consider a manufacturing system with mul-

tiple processing centers. At the arrival of a new order, the manufacturerneeds to decide where to process the order, given the current state of thesystem, i.e., given the orders currently being processed and the queuesin the processing centers. The centers experience random breakdownsand repairs. The unit processing time and cost, the setup time and cost,and the inventory carrying cost differ across centers. The manufacturercan split an order across multiple processing centers, but the entire orderhas to be delivered at once. That is, the manufacturer incurs a holdingcost for the part of the order that is completed before the delivery date.The completion time of the order has to be between the earliest and thelatest acceptable times speciÞed by the customer. The authors proposea mathematical model to assign a newly arrived order to the process-ing centers so as to minimize the total cost associated with that order.Note that this �greedy� approach only optimizes the assignment of oneorder at a time, without considering the impact of that assignment onfuture orders. Furthermore, it does not differentiate between multiple


customer classes and assumes that all arriving orders must be acceptedand processed.Elhafsi and Rolland claim that one can estimate and quote a due

date to the customer using the results of this model. In particular,they consider two types of customers: time-sensitive and cost-sensitive.Time-sensitive customers want to have their order completed as soon aspossible, regardless of the cost, whereas cost-sensitive customers want tominimize the cost, regardless of the actual order completion time. Theauthors modify their original model according to the objectives of thesetwo customer types, and propose solution methods for solving the cor-responding models. While the models return an (expected) completiontime C for the order, the authors note that quoting C as the due datemight result in low service levels, depending on the variance of the com-pletion time. They compute an upper bound σmax on the standard de-viation of the completion time, and propose a due date quote C+ασmaxwhich contains a safety factor ασmax. Using numerical experiments,they show that for α = 0.5, the maximum average percentage tardiness(tardiness divided by the actual completion time) is 3.3% (2%) for time-sensitive (cost-sensitive) customers and the maximum safety factor isaround 11% (6.5%). These results suggest that using α = 0.5 in quotingdue dates results in fairly high service levels for both customer types. Ina follow-up paper, Elhafsi [35] extends this model by considering partialas well as full deliveries and proposes both exact algorithms and twoheuristics. A major limitation of these papers is that FCFS schedulingrule is assumed and the results do not carry over to other schedulingrules.

5. Due Date Management with ServiceConstraints

Due date setting decisions, like many other decisions in a Þrm, needto balance competing objectives. While it is desirable to quote shortlead times to attract customers, for the long-term proÞtability of theÞrm, it is also important that the quoted lead times are reliable. Thepapers discussed so far in this survey incorporate a lateness penaltyinto their models and analyses to ensure the attainability of the quotedlead times. Although this is a very reasonable modeling approach, inpractice is very difficult to estimate the actual �cost� of missing a duedate. Furthermore, unlike the common assumption in the literature,lateness penalties in practice are usually not linear in the length of thedelay. Therefore, the papers we review in this section impose service levelconstraints rather than lateness penalties to ensure lead time reliability.

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The two commonly used service constraints are upper bounds on theaverage fraction of tardy jobs (τmax) and the average tardiness (Tmax).Bookbinder and Noor [14] study DDM policies in a single machine

environment with batch arrivals (with constant interarrival times) sub-ject to τmax. For job sequencing, they test three rules: SPT, FCFS andSDD, where SDD uses SPT and FCFS for sequencing the jobs within abatch and the batches, respectively. They propose a due date rule (BN),which incorporates, through the parameter αj , the job content and shopinformation at the arrival time of a job, and ensures that a due date willbe met a given probability. For setting αj , they propose a formula whichuses information on the jobs in the queue or in process, their estimatedprocessing times, and the sequencing rule. They test the BN rule for theobjectives of mean lateness and earliness, percentage of jobs late andearly, the mean queue length, waiting time and ßow time, and compareagainst JIQ. They Þnd that RM (when used with SDD) performs worsethan JIQ in terms of earliness (i.e., quotes more conservative due dates),however, it performs better in the other objectives. The performance ofBN under sequencing rules other than SDD degrades with respect to thelateness measure, since it quotes more conservative due dates to achieveservice level constraints. This is also true for JIQ. Both due-date rulesperform worse under FCFS.Wein [107] studies DDM under the objective of minimizing the weighted

average lead time subject to τmax (Problem I) and Tmax(Problem II).He considers a multiclass M/G/1 queuing system, where order arrivalsare Poisson (with rate λk for class k), the processing times for each classare independent and identically distributed (iid) random variables withmean 1/µk, and preemption is not allowed.The due date rules proposed by are based on the estimates of the

conditional sojourn times of the jobs. The sojourn time (or ßow time) ofa job is the length of time the job spends in the system. The expectedconditional sojourn time E[Sk,t] of a class k job arriving at time t isdeÞned as the total time the job spends in the system if the WSPT(or the cµ) rule is used for sequencing. The standard deviation of theconditional sojourn time is denoted by σ[Sk,t]. Note that one needsto consider the state of the system as well as job information whileestimating the conditional sojourn time.


Let Dk,t denote the due date quoted to a class k job that arrives attime t. Wein (1991) proposes the following due date setting rules:

WEIN-PAR I : Dk,t = t+ αE[Sk,t] (999.1)

WEIN-PAR II : Dk,t = t+ αE[Sk,t] + βσ[Sk,t] (999.2)

WEIN-NONPAR I : Dk,t = t+ fIk,t (999.3)

WEIN-NONPAR II : Dk,t = t+ fIIk,t (999.4)

where τmax = P (Sk,t > f Ik,t), Tmax ="∞fIIk,t(x − f IIk,t)dGk,t(x) and the

random variable Sk,t has distribution Gk,t.The Þrst two of these rules are parametric and the parameters α and

β are chosen (based on simulation experiments) such that the servicelevel constraints are satisÞed. These rules apply to both problems I andII. The third and fourth rules set the shortest possible due date at anytime to satisfy the service level constraints of the maximum fraction oftardy jobs and the maximum average tardiness, and apply to problemsI and II, respectively. The proposed due date management policies usethese rules with SPT sequencing between different classes, and FCFSsequencing within each class.Note that the rules in Equations (999.1)-(999.4) do not consider the

previously set due dates. Therefore, utilizing WEIN-NONPAR I andWEIN-NONPAR II, Wein proposes two other rules, WEIN-HOT-I andWEIN-HOT-II, for problems I and II, respectively. Under these rules, ifthere is enough slack in the system, a new job can be quoted a shorterlead time and scheduled ahead of some previous jobs of the same class.The sequencing policy is still SPT for different classes, but EDD (ratherthan FCFS) within each class. Although these rules may result in shorterdue dates, they may violate the service level constraints.The rules proposed by Wein are not easy to compute for general mul-

ticlass M/G/1 queues, and hence, their applicability in practice may belimited. Wein derives these rules for a special case with only two classesof jobs, exponential processing times and equal weights for each class.To test the performance of these rules, he conducts a simulation studyon this example and compares his proposed rules against CON, SLK andTWK [6] and a modiÞcation of JIQ [33], under Þve different sequencingpolicies (i.e., a total of 40 due date management policies are tested foreach problem). The simulation results indicate that for the due datesetting policies proposed by Wein: (1) The performance is signiÞcantlybetter than the previous policies. For Problem I, the best policy pro-posed by Wein, (WEIN-HOT-I)-MDD, reduced the average lead timeby 25.2%, compared to the best of the other policies tested, which wasJIQ-EDD. JIQ-EDD�s performance was 49.7% better than the worst pol-

44

icy tested, which was CON-SLK/P. (2) There is not a large differencein performance between parametric and nonparametric rules. (3) Duedate setting has a bigger impact on performance than sequencing (how-ever, this observation does not hold for other due date setting policies).For some due date policies, however, the impact of the sequencing rulewas signiÞcant. For example, for Problem I, SLK/P worked well withSPT, and for Problem II, JIQ performed better under due-date basedsequencing policies, especially under MDD. Wein also Þnds that JIQ,which uses information about the current state of the system, performsbetter than CON, SLK and TWK, which are independent of the state.Wein�s results indicate that the performance of a DDM policy also

depends on the service constraint. For example, TWK-SPT, which per-forms well under τmax (Problem I), does not perform well under Tmax(Problem II).Spearman and Zhang [98] study the same problems as in [107], in a

multistage production system under the FCFS sequencing rule. Theyshow that the optimal policy for Problem I is to quote a zero lead timeif the congestion level is above a certain threshold, knowing that thepossibility of meeting this quote is extremely low. Intuitively, sincethe service level is on the number of tardy jobs, then knowing that anarriving job will be tardy anyway (unless a very long lead time is quoted,which would affect the objective function negatively), it does make senseto quote the shortest possible lead time to reduce the objective functionvalue. Clearly, such a policy would be quite unethical in practice andreminds us one more time that one has to be very careful in choosing theobjective function or the service measures. Using numerical examples,the authors show that a customer is more likely to be quoted a zerolead time when the service level is low or moderate, rather than high,creating service expectations completely opposite of what the systemcan deliver. For Problem II, they show that (1) the optimal policy is toquote lead times which ensure the same service level to all customers,and (2) in cases where the mean and variance of the ßow time increaseswith the number of jobs in the system, quoted lead times also increasein the number of jobs in the system. Note that the optimal policy forProblem II quotes longer lead times as the congestion level in the systemincreases. This is similar to some other rules proposed earlier in theliterature, such as JIQ, JIS and WIQ, and is more reasonable in practicethan the optimal policy found for Problem I. Also, observation (1) leadsto an easy implementation of the optimal policy for problem II, whichis to use the optimal policy for Problem I by choosing an appropriateType I service level. The information required to implement the policy


is an estimate on the conditional distribution of ßow times, which maynot be easy to determine in practice.Hopp and Roof-Sturgis [55] study the problem of quoting shortest

possible lead times subject to the service constraint τmax. In their pro-posed due date policy (HRS), which is based on control chart methods,they break the lead time quote of job j into two components: the meanßow time, which is a function of the number of jobs in the system at thearrival time of job j, plus a safety lead time, which is a function of thestandard deviation of ßow time. They use the following quadratic func-tions (applicable to a ßow shop) to estimate the mean and the standarddeviation of the ßow time

E[F (n)] =

#T0 + β1(n− 1) + β2(n− 1)2 if n ≤ 2W0

(1/rp)n if n > 2W0(999.5)

σF (n) =

#γ1 + γ2n if n ≤ 2W0

σp√n if n > 2W0

(999.6)

whereT0 = practical process time (the average time for one job to traverse theempty system)1/rp = average interdeparture time from the system (inverse of the prac-tical production rate)W0: critical WIP (the minimum number of jobs in a conveyor systemthat achieve the maximum throughput, W0 = rpT0

σp: standard deviation of the interdeparture time from the systemThe advantage of this approach is that it does not require any assump-

tions on probability distributions and it can be adjusted in response tochanging environmental conditions.Using simulation, the authors Þne tune the parameters in the above

equations as well as the parameter β in rule HRS, and show that thesefunctions indeed provide very good estimates for E[F (n)] and σF (n).They also show that zασF (n) provides a good estimate of the tail ofthe ßow time distribution. Next, they test the performance of the duedate quotes which are based on these estimates, Þrst in a simpleM/M/1system where the exact due date quotes can be computed analytically,and then in another system with nearly deterministic processing timesand random machine breakdowns. They also compare their method tothe JIS rule [104], after adjusting JIS to achieve the same service levelas their rule. For the objectives of mean earliness, mean tardiness, andthe sum of mean earliness and tardiness,they Þnd that their methodoutperforms the modiÞed JIS rule.

46

Hopp and Roof-Sturgis also propose a method inspired from statisticalprocess control for dynamically setting and adjusting the parameters ofthe lead time quote. They deÞne the model to be �out of control� if thecurrent service level (number of tardy jobs) differs from the target servicelevel by more than three standard deviations of the service values. Theservice value of a job is one, if it is early or on time, and zero, if it is tardy.Then the process is out of control it indicates that the lead time quotesare either too short or signiÞcantly too long, and the model parametersneed to be changed. They test their method in systems where two typesof changes occur over time: increasing capacity and speeding up repairs.In both cases, their method reduces the lead times in response to systemimprovements. Finally, the authors discuss possible extensions of theirmethods to more complex multi-machine systems such as multi-productsystems with intersecting or shared routing.In most of the due date setting rules in Table 999.3, the ßow allowance

is based on simple estimates of the ßow time plus a safety allowance forthe estimated waiting time. Furthermore, these allowances are usuallyparametric, and choosing the appropriate parameters is not straightfor-ward. It is especially not clear how one should set these parametersto satisfy service constraints, e.g., a maximum fraction of tardy jobs.As the congestion level increases in the shop, longer ßow allowances areneeded to achieve the same service objective, but how much longer? Toaddress these issues, Enns [38] proposes a method for estimating the ßowtimes and the variance of the forecast error, and proposes a due date rulebased on these estimates. The goal is to set the tightest possible duedates (minimize average lead time) while satisfying a service constrainton the maximum fraction of tardy jobs.The ßow time estimate proposed by Enns has a similar ßavor to

the ßow allowance in the TWK+NOP rule, where the parameter βis set equal to an estimate of the average waiting time per operation.More formally, the estimated ßow time for job j is

!o pjo + βjgj where

βj =ntE[p]mρ −E[p]. Note that here the parameter β depends on the job,

hence, the rule is dynamic and we denote it by TWK+NOPd. Via simu-lation, Enns tests the effectiveness of TWK+NOPd against TWK+NOPfor estimating ßow times. He uses these two rules for setting due dates,where the ßow allowance is equal to the estimated ßow time. Notingthat the forecast error tends to have a normal distribution with theTWK+NOPd ßow time estimates, one would expect to have approxi-mately 50% of the jobs tardy, which turns out to be the case. Com-pared to TWK+NOP with the best constant multiplier, he Þnds thatTWK+NOPd provides better estimates of the ßow times. Deviations


from estimated ßow times are higher under non-due-date dependent se-quencing rules and as utilization increases.The next step is to estimate the variance of the forecast error. Enns

proposes two estimates, based on the forecast error at the operation level(�σ2j,OLV ) and at the job level (�σ

2j,JLV ). Based on these estimates, Enns

proposes the following ßow allowances for setting due dates:

ENNS −OLV =$o

pjo + βj + γ√gj�σj,OLV (999.7)

ENNS − JLV =$o

pjo + βj + γ√gj�σj,JLV (999.8)

The Þrst two terms in equations (999.7) and (999.8) estimate the ßowtime, and the last term is a waiting time allowance based on the forecasterror. Normal probability tables can be used to choose the γ value tosatisfy the service level constraint on the maximum number of tardyjobs.Enns tests the performance of this policy via simulation under various

sequencing rules. As performance measures, he looks at the percentageof tardy jobs, and deviations from the desired service level (PTAE). HeÞnds that (1) the performance of the proposed due date setting policy isaffected more by the utilization level and the service level requirementsthan the sequencing policy, (2) due-date dependent sequencing rules ingeneral outperform non-due-date dependent rules, especially for highservice level requirements and as utilization increases.While most research in DDM focuses on �external� measures on cus-

tomer service, in a follow-up study Enns [39] focuses on internal mea-sures such as the number of and the investment in work-in-process jobs,smoothness of work ßow, and the balancing of queue lengths. To ensurea balanced workload in the shop, Enns enhances the scheduling ruleswith the queue balancing (QB) mechanism: among the top two candi-date jobs to be chosen next, pick the one that has the smaller numberof jobs in queue at the machine for the next operation. In addition tothe TWK+NOPd rule proposed in [38], he proposes TWKd, a dynamicversion of the TWK rule, by setting α = nt

mρ . Having these two ßow timeestimates, he then sets the due dates by adding to them a ßow allowanceKSD

√gj�σj,OLV or KSD�σj,JLV , where KSD is a parameter chosen to sat-

isfy the service constraint on the maximum fraction of tardy jobs (0.05in the experiments). Note that TWKd assigns tighter ßow allowances tojobs with short processing times than TWK+NOPd. Hence, when due-date dependent sequencing rules are used, short jobs get higher priorityunder TWKd (similar to the SPT rule), leading to shorter lead times.

48

Enns� computational testing of these due dates under various sequencingrules (FCFS, SPTT, EDD, CR, EDDo, and CRo and their QB versions)conÞrms this intuition and also indicates that the performance underinternal and external measures is positively correlated. Comparing jobvs. operation based due date rules, Enns Þnds that under TWKd op-eration based rules perform better in general; this observation parallelsthe Þndings of Kanet and Hayya [60] for the job- and operation-basedversions of the TWK rule. However, the superior performance of theoperation-based due dates do not hold under TWK+NOPd. About theimpact of shop ßoor balancing, Enns Þnds that combining TWK+NOPdwith the QB versions of the sequencing rules leads to reduced lead times,whereas this is not the case for TWKd. These results once again suggestthat there are strong interactions between due date and scheduling rules.We say that a DDM policy has 100% service guarantee if an order�s

processing must be completed before its quoted due date. (Note that a100% service guarantee is possible only if the processing times are de-terministic and there are no machine breakdowns). Zijm and Buitenhek[111] propose a scheduling method, which is based on the shifting bot-tleneck procedure [2], that utilizes maximum lateness as a performancemeasure and Þnds the earliest time a newcoming job can be completed,without delaying any of the existing jobs in the system (assuming thatthe existing jobs already have due dates assigned and can be completedwithout tardiness). If the due date for each arriving job is set later thanthe earliest completion time for that job, then all jobs can be completedbefore their due dates. Hence, this procedure results in 100% serviceguarantee. This approach signiÞcantly differs from most of the exist-ing approaches in the DDM literature: it �greedily� creates a tentativeschedule for a new job to minimize its completion time (and due date).Note that most due date rules (e.g., TWK, NOP, SLK, etc.) do notconsider a tentative schedule while assigning due dates.Kaminsky and Lee [58] study a single-machine model (similar to Q-

SLTQ of [64], discussed in Section 6.2) with 100% reliable due dateswhere the objective is to minimize the average due date. The authorsÞrst note that the SPT rule is asymptotically (as the number of jobs goesto inÞnity) optimal for the offline version of this problem, i.e., provides alower bound for the online version. They propose three heuristics for thisproblem. First Come First Serve Quotation (FCFSQ) heuristic simplysequences the jobs in the order of their arrival, and quotes them theirexact completion time, which can be computed easily. Sequence/SlackI (SSI) heuristic maintains a tentative sequence of the jobs in queue inan SPT-like order. A newly arriving job j is inserted at the end of thissequence and then moved forward one job at a time until the preceding


job has a smaller processing time or moving job j would make at least onejob late. After the position of the new job is established in the currentsequence, the due date is quoted by adding a slack to the completion timeof this job according to the current schedule. Note the following tradeoffin determining the slack: If the slack is too small, then in case other jobswith shorter processing times arrive later, they have to be scheduledafter job j, violating the SPT rule. On the other hand, if the slackis too large, then the quoted due date of job j might be unnecessarilylarge, degrading the value of the objective function. In SSI, the authorsdetermine the slack by multiplying the average processing time of all thejobs which have processing time smaller than pj (denoted by EPJ

", andprovides an estimate of the average processing time of a future job whichwould have to be scheduled ahead of job j in an SPT sequence) with thenumber of jobs ahead of job j in the current schedule which have smallerprocessing times than pj (which provides an estimate on the number ofjobs that might might arrive later and move ahead of job j). The SSIIheuristic is similar to SSI, but it assumes that the total number of jobsis known. In SSII, the slack is computed by multiplying EPJ " withthe expected number of yet to arrive jobs with processing time smallerthan pj . The main idea of the SSI and SSII heuristics is to build justenough slack into the schedule so as to obtain a Þnal sequence that is asclose to an SPT sequence as possible. The authors show that when theexpected processing time is shorter than the expected interarrival time,then all three heuristics are asymptotically optimal. Furthermore, SSIIis asymptotically optimal even if the expected processing time exceedsthe expected interarrival time. A nice feature of the algorithms proposedin [58] is that they �learn� the environment over time by considering theprocessing times of all the jobs that arrived until the current time.Another study that considers DDM with 100% service guarantee is

[64]. In addition to due date setting and scheduling, the authors considerorder acceptance decisions and take a proÞt maximization perspective;hence, their model and results are discussed in Section 6.1. Other papersthat consider service level constraints along with pricing decisions inDDM include [74] and [96], which are discussed in Section 6.2.

6. Due Date Management with Price and OrderSelection Decisions

Before a business transaction takes place, both the buyer and the sellerevaluate the short and long term proÞtability of that transaction andmake a decision on whether to commit or not based on that evaluation.Within the context of make-to-order environments, before placing an

50

order for a product, the buyer evaluates various attributes of the product,such as its price, quality, and lead time. Similarly, before a supplieragrees to accept an order, she evaluates the �proÞtability� of that ordergiven the resources (e.g., manufacturing capacity) required to satisfythat order, currently available resources, and other potential orders thatcould demand those resources. For an order to become a Þrm order, thebuyer and the seller have to agree on the terms of the transaction, inparticular, on the price and the lead time. If the price or the lead timequoted by the seller is too high compared to what the buyer is willingto accept, the buyer may choose not to place the order. Alternatively, ifthe price the buyer is willing to pay is low or the lead time requested bythe buyer is too short to make it a proÞtable transaction for the seller,the seller might decide not to accept the order. In either scenario, theÞnal demand the seller sees is a function of the quoted price and leadtime. In effect, by quoting prices and lead times, the seller makes anorder selection/acceptance decision by inßuencing which orders Þnallyend up the system.Most of the existing literature on DDM ignores pricing decisions and

the impact of the quoted prices and lead times on demand. Hence, itignores the order selection problem and assumes that all the customersthat arrive in the system place Þrm orders, which have to be acceptedand scheduled. In this setting, the price and order selection decisions areexogenous to the model. This would be the case, for example, if the mar-keting department quotes prices and makes order acceptance decisionswithout consulting the manufacturing department (or without consider-ing the current commitments and available resources), and taking theseÞrm orders as input, the manufacturing department makes lead timequotation and production decisions.Ideally, a manufacturer should take a global perspective and coordi-

nate its decisions on price, lead time quotation and order acceptancefor increased proÞtability. However, due to the nature of the businessenvironment or company practices, this might not always be possible.For example, prices in certain industries are largely dictated by the mar-ket, and the manufacturer may not have much ßexibility in setting theprices. Even if the manufacturer could set prices, due to the currentorganizational structure and practices, it might not be easy to integratepricing decisions with lead time quotation and production decisions. Buteven in such environments where the price is exogenous to DDM, themanufacturer would still beneÞt from incorporating order selection de-cisions into DDM. For example, in periods of high demand or tightresources, it might be more proÞtable to reject some low-margin orders.Similarly, if there are multiple classes of customers/orders with different


cost/revenue characteristics, it might be more proÞtable to reject somelow-margin orders to�reserve� capacity for potential future arrivals ofhigh-margin orders.The papers we review in this section study DDM with order selection

decisions by incorporating the impact of the quoted lead times on de-mand. In Section 6.1 we discuss papers which take price as given butconsider the impact of quoted lead times on demand. In Section 6.2,we review the new but growing area of literature on combined price andlead time quotation decisions.

6.1 Due Date Management with Order SelectionDecisions (DDM-OS)

In most businesses, the quoted lead times (or due dates) affect thecustomers� decisions on placing an order. In general, the longer the leadtime, the less likely a customer will place an order. In most cases, acustomer might have a Þrm deadline and would not consider placing anorder if the quoted due date exceeds that deadline. Within acceptablelimits, some customers might not care what the actual lead time is, whileothers might strongly prefer shorter lead times.The models proposed in the literature for capturing the impact of

quoted lead times on demand all follow these observations: demand (orthe probability that an arriving customer places an order) decreases, asthe quoted lead time increases. Let P (l) denote the probability that acustomer places an order given quoted lead time l and let lmax denote themaximum acceptable lead time to the customer. The proposed demandmodels include the following:

(D1) : P (l) = 1− l/lmax(D2) : P (l) =

#1, if l ≤ lmax0, otherwise

(D3) : P (l) = e−λl, where λ is the arrival rate of the customers(D4) : P (l) is a decreasing concave function of l

DDM models which consider order selection decisions take a proÞtmaximization perspective rather than a cost minimization perspective.In general, the revenue from an accepted order (in class j) is R (Rj)and there are earliness/tardiness penalties if the order is completed be-fore/after its quoted due date.Dellaert [29] studies DDM-OS using demand model (D1) in a single

server queuing model. Unlike most other models in the literature, asetup is needed before the production of a lot can be started. Whenall the currently available demand has been produced, the machine goesdown and a new setup is required before a new lot can be produced.

52

Assuming FCFS sequencing policy, the goal is to Þnd a combinationof a due date setting rule and a production rule (deciding on when tostart a setup) to maximize the expected proÞt (revenue minus earliness,tardiness and setup costs). Relying on earlier results in [28], the au-thor focuses on the following production rule: perform a setup only ifthe number of jobs waiting is at least n. The value of n needs to bedetermined simultaneously with the optimal lead time.Dellaert studies two lead time policies, CON and DEL, where DEL

considers the probability distribution of the ßow time in steady-statewhile quoting lead times. He models the problem as a continuous-timeMarkov chain, where the states are denoted by (n, s)=(number of jobs,state of the machine). Interarrival, service and setup times are assumedto follow the exponential distribution, although the results can also begeneralized to other distributions, such as Erlang. For both policies, hederives the pdf of the ßow time, and relying on the results in [87] (theoptimal lead time is a unique minimum of strictly convex functions), heclaims that the optimal solution can be found by binary search. For thecase of CON, computational results indicate that for small values of n,the lead time is larger than the average ßow time, while the opposite istrue for larger values of n. Intuitively, when n is large, batches and ßowtimes are also large and in order to generate revenues, one needs to quoteshorter lead times in comparison to ßow times. The results also indicatethat there is little loss in performance if the optimal CON lead time isreplaced by the average ßow time. In terms of implementation, quotinglead times equal to average ßow times (which can be determined, for ex-ample, via simulation) might be much easier than computing the pdf ofßow times, which are used to Þnd the optimal CON policy. The compar-ison of the CON policy with DEL, in which jobs can be quoted differentlead times, indicates that DEL has signiÞcantly better performance.Duenyas and Hopp [31] consider demand models (D2)-(D4) and study

DDM-OS using a queuing model. They Þrst consider a system withinÞnite server capacity. For the special case of exponential processingtimes and model (D3), they Þnd a closed form solution for the optimallead time l∗, which is the same for all customers. The structure ofl∗ implies that: (1) For higher revenues per customer, the Þrm quotesshorter lead times. This allows the Þrm to accept more orders and thehigher revenues offset the penalties in case due dates are missed. (2) Forlonger processing times, the Þrm quotes longer lead times and obtainslower proÞts.Next, they consider the capacitated case studying a single server queue

GI/GI/1 where processing times have a distribution in the form of in-creasing failure rate (IFR). That is, as the amount spent processing a job


increases, the probability that it will be completed within a certain timeduration increases as well. They use the FCFS rule for sequencing ac-cepted orders. They Þrst study the problem for model (D2), which mightbe appropriate if the lead times are Þxed for all customers, e.g., 1 hourÞlm processing. In this case, the Þrm�s main decision is to decide whichorders to accept (reject), by quoting a lead time less (greater) than lmax.They show that the optimal policy has a control-limit structure: for anyn, the number of orders currently in the system, there exists a time t(n)such that a new order is accepted if the Þrst order has been in servicefor more than t(n) time units. Next, they consider model (D4). Notethat choosing the optimal lead time l implies an effective arrival rate ofλ(l). Hence, setting lead times is equivalent to choosing an arrival ratefor the queue. Let l∗nt denote the optimal lead time quote to a new cus-tomer when there are n customers in the system and the Þrst customerhas been in service for t time units. For an M/M/1 queue, they showthat the optimal lead time to quote is increasing in n. Although theycannot Þnd general structural results for the GI/GI/1 queue, they showthat l∗nt is bounded below by b∗nt, which is the optimal lead time com-puted by disregarding the congestion effects (the customers currently inthe system). When the service and interarrival times are exponentiallydistributed, the optimal lead times increase in the number of orders inthe system.The results of Duenyas and Hopp [31] assume the FCFS sequencing

policy. When a different sequencing rule is used, the impact of acceptinga new order on the current schedule has to be considered. Dependingon the new sequence, the lead times and hence the penalties of existingorders might change. The authors show that when the penalty is pro-portional to tardiness, an optimal DDM policy processes all jobs usingthe EDD rule. This result does not hold when the penalty is Þxed foreach tardy job (e.g., as in the case of minimizing the number of tardyjobs).The models studied in [29] and [31] have a single class of customers;

the net revenue per customer is constant, customers have the same pref-erences for lead times, and the arrival and processing times follow thesame distribution. Duenyas [30] extends these results to multiple cus-tomer classes, with different net revenues and lead time preferences.Customers/jobs from class j arrive to an M/G/1 queue according to aPoisson process with rate λj . A customer from class j accepts a quotedlead time l with probability Pj(l), and generates net revenue Rj . Ifan accepted order is tardy for x time units, the Þrm pays a penaltywx. Duenyas Þrst studies the problem assuming FCFS sequencing andshows that: (1) quoted lead times increase in the number of orders in

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the system, (2) customers who are willing to pay more and/or impatientare more sensitive to quoted lead times, and (3) ignoring future arrivalsresult in lead times that are lower than optimal. Next, he considers thesimultaneous problem of due date setting and sequencing. Since the stateand action spaces of the Semi-Markov decision process (SMDP) used formodeling this problem are uncountable, he is not able to Þnd an exact so-lution, and therefore proposes a heuristic. He tests this heuristic on a setof problems with two customer classes against CON, JIQ, and two otherrules obtained from modifying WEIN-PAR I and WEIN-PAR II. Toachieve a fair comparison, the parameters of these rules are adjusted sothat they all result in the same average tardiness. Simulation results in-dicate that the proposed heuristic outperforms the other rules, especiallyat high utilization levels. The two modiÞed rules also performed verywell, and since they require less information about customers� due datepreferences, they may be a good alternative to the proposed heuristic ifsuch information is not available. Duenyas also notes that it is possibleto extend the heuristic to simultaneously decide on prices, due dates andsequencing, by assuming that a customer from class j accepts a (price,lead time) pair (Rj , lj) with probability Pj(Rj , lj). The work of Duenyas[30] suggests that collecting information on customers� preferences aboutdue dates and using this information in due date management can helpmanufacturers to achieve higher proÞtability.Keskinocak et al. [64] study DDM for orders with availability intervals

and lead time sensitive revenues in a single-server setting. In their model,revenues obtained from the customers are sensitive to the lead-time,there is a threshold of lead-time above which the customer does not placean order and the quoted lead times are 100% reliable. More formally,the revenue received from an order j with quoted lead time lj is R(d) =

wj(�lj − lj), where �lj is the maximum acceptable lead time for customer

j. Quoting a lead time longer than �lj is equivalent to rejecting order j.Note that this revenue model is equivalent to a proÞt model where thereis a Þxed revenue (or price) per order Rj = wj�lj and a penalty wjlj forquoting a lead time lj . The concept of lead-time-dependent price is alsostudied in [84], discussed in Section 6.2.Keskinocak et al. [64] study the offline (F-SLTQ) and three online

versions of this problem. In the offline models, all the orders are knownin advance whereas in the online models orders arrive over time anddecisions are made without any knowledge of future orders. F-SLTQgeneralizes the well known scheduling problem of minimizing the sumof weighted completion times subject to release times [63], denoted by1|rj |

!wjCj . They study F-SLTQ by methods from mathematical pro-

gramming and show several polynomially or pseudo-polynomially solv-


able instances. In the pure online model (O-SLTQ), decisions aboutaccepting/rejecting an order and lead time quotation can be delayed,whereas in the quotation online model (Q-SLTQ) these decisions haveto be made immediately when the order arrives. In the delayed quota-tion model (D-SLTQ), which is between the pure online and quotationmodels, decisions have to be made within a Þxed time frame after thearrival of an order. To evaluate the performance of algorithms for theon-line models, they use competitive (worst case) analysis and showlower and upper bounds on the performance of their algorithms, relativeto an optimum offline algorithm. Most of the previous research on thecompetitive analysis of on-line scheduling algorithms considers modelssimilar to O-SLTQ, where the scheduling decisions about an order canbe delayed. In many cases, this delay has no bound as the orders do nothave latest acceptable start or completions times, which is not realisticfor real-life situations. The authors show that the quotation version (Q-SLTQ), where order acceptance and due date quotation decisions haveto be made immediately when an order arrives, can be much harder thanthe traditional on-line version (O-SLTQ). Partially delaying the quota-tion decision (Q-SLTQ) can improve performance signiÞcantly. So, thedifficulty does not only lie in not knowing the demand, but in how soonone has to make a decision when an order arrives. They also observethat in order to obtain high revenues, it is important to reserve capacityfor future orders, even if there is only a single type of orders (i.e., wj = w

and �lj = �l in the revenue function).Asymptotic [58] and worst case [64] analysis of online algorithms re-

ceived very little attention so far within the DDM context. These ap-proaches are criticized for being applicable only to very simple mod-els and asymptotic analysis is further criticized for being valid onlyfor a large number of jobs. However, these criticisms hold for steady-state analysis of most queuing models within the DDM context as well.Kaminsky and Lee [58] show via computational experiments that in avariety of settings the asymptotic results are obtained in practice forless than 1000-2500 jobs; most simulation studies are based on at leastas many, and sometimes a larger number of jobs. An advantage ofworst-case analysis is that it does not assume any information about theprobability distribution of job processing or interarrival times, unlikemost queuing studies which assume M/M/1. Asymptotic analysis usu-ally assumes that the processing and interarrival times are independentdraws from known distributions, but this information is used mainly forproving results, and is not necessarily used in the algorithms. For ex-ample, the heuristics proposed in [58] do not use any information about

56

these distributions. In short, we believe that worst case and asymptoticanalysis hold promise for analytically studying DDM problems.Chatterjee et al. [18] pose DDM-OS in a decentralized marketing-

operations framework where the marketing department quotes the duedates without having full information about the shop ßoor status. Uponthe arrival of a new order, marketing learns the processing time p and se-lects a lead time l(p), and the customer places an order with probabilitye−ψl(p) (model (D3)). Let δ = 1 if the customer stays and δ = 0 oth-erwise. Manufacturing sees a Þltered stream of arrivals Λ = P (δ = 1)λwhere λ is the arrival rate observed by marketing. Marketing assumesthat the delay time W in operations has PDF FW = 1 − ρe−γx, x ≥ 0(which is the waiting time distribution in an M/M/1 queue with FCFSscheduling), where ρ = ΛE[p] and γ = (1−ρ)/E[p]. Note that γ denotesthe difference between the service rate and the arrival rate at operations.The revenue and tardiness penalty for a job with processing time p aregiven by R(p) and w(p). The authors show that the optimal lead timehas a log-linear structure and propose an iterative procedure for com-puting its parameters. For the special case of constant unit tardinesspenalty w(p) = w, they show that the optimal lead time is zero for anyjob with processing time larger than a threshold p̄, which is similar tothe �unethical� DDM practice observed by Spearman and Zhang [98].Through numerical examples, the authors illustrate that choosing leadtimes based on manufacturing objectives such as minimizing cycle timeor maximizing throughput usually reduce proÞts.

6.2 Due Date Management with Price andOrder Selection Decisions (DDM-P)

Most of the research in economics and marketing models the demandonly as a function of price, assuming that Þrms compete mainly onprice. In contrast, the operations literature usually takes the price (anddemand) as given, and tries to minimize cost and/or maximize customerservice. However, for most customers the purchasing decision involvestrading offmany factors including price and quality, where delivery guar-antees are considered among the top quality features. Hence, in mostcases demand is a function of both price and lead time and thereforea Þrm�s DDM policy is closely linked to its pricing policy. In this sec-tion, we provide an overview of the literature which considers DDM inconjunction with pricing decisions.The Þrst four papers we discuss, So and Song [96], Palaka et. al.

[74], Ray and Jewkes [84], and Boyaci and Ray [15] consider capacity se-lection/expansion decisions in addition to price and lead time decisions.


These papers study DDM-P using an M/M/1 queuing model with FCFSsequencing, where the expected demand is modeled by a linear function(Λ(R, l) = a− b1R− b2l) in [74] and [84], and a Cobb-Douglas function(Λ(R, l) = λR−al−b) in [96]. Note that the price elasticity of demand(the percentage change in demand corresponding to a 1% change inprice) is constant in the second model whereas it increases both in priceand quoted lead time in the Þrst model. These papers consider a con-stant lead time and a service level constraint (the minimum probability(s) of meeting the quoted lead time).Palaka et al. consider three types of costs: direct variable costs (pro-

portional to production volume), congestion costs (proportional to themean number of jobs waiting in the system), and tardiness costs. TheyÞrst consider the case of Þxed capacity (service rate µ), where the goal isto choose a price/lead-time pair (R, l) and a demand rate λ ≤ Λ(R, l) formaximizing the Þrm�s expected proÞts. Noting that the demand rate λwill always be equal to Λ(R, l) in the optimal solution, they focus on thetwo decision variables λ and l. They show that (i) the service constraintis binding in the optimal solution if and only if s ≥ sc = 1 − b2/b1w,where w is the tardiness penalty per unit time, (ii) when s increases,the Þrm both increases its quoted lead time and decreases its demandrate (and expected lead time). In contrast, an increase in the tardinesspenalty decreases the demand rate but the quoted lead time decreases(increases) if the service constraint is binding (non-binding). For smallvalues of the tardiness penalty, the Þrm increases the price to reduce thedemand rate, which does not decrease the probability of tardiness butreduces the tardiness penalty since there are fewer orders late overall.However, when the tardiness penalty is high, the Þrm needs to reducethe probability of tardiness and hence quotes higher lead times. Palakaet al. extend these results to the case where marginal capacity expansionis possible, that is, the Þrm can choose z, the fractional increase in theprocessing rate at a cost of c per job/unit time up to a limit of z̄. Hence,the service rate becomes µ(1+z). The authors show that in the optimalsolution, the Þrm uses both capacity expansion and a reduced arrivalrate to achieve shorter lead times. Finally, they look at the sensitivityof the proÞts to the errors in estimating the lead time and conclude thatguaranteeing a shorter than optimal lead time usually results in higherproÞt loss than guaranteeing a longer than optimal lead time.The objective function in So and Song [96] is to maximize proÞt,

which is the revenue minus direct variable costs and capacity expansioncosts. Note that this is a special case of the objective function consid-ered in Palaka et al., ignoring the congestion and tardiness costs. The

58

qualitative results of So and Song are generally consistent with Palakaet al.Ray and Jewkes [84] study a variant of the model in [74] by modeling

the market price as a function of lead time, namely, R = R̄−el where R̄is the maximum price when the lead time is zero5. Hence, the demandfunction becomes λ(R, l) = a − b1R − b2l = (a − b1R̄) − (b2 − b1e)l =a" − b"l. This model naturally leads to a distinction of lead time andprice sensitive customers: (i) when b" > 0, demand decreases in leadtime, i.e., customers are lead time sensitive (LS) and are willing to payhigher prices for shorter lead times; (ii) when b" < 0, demand increasesin lead time (as price decreases), i.e., customers are price sensitive (PS)and are willing to wait longer to get lower prices. The dependence ofprice on lead time reduces the number of variables, and the Þrm needsto choose a lead time (l) and capacity level (µ) subject to a serviceconstraint, with the goal of maximizing proÞt (revenue minus operatingand capacity costs). The authors consider both the cases of constantand decreasing convex operating costs (the latter models economies ofscale). The authors show that the optimal lead time depends stronglyon whether the customers are price or lead time sensitive, as well asoperating costs. Comparing their model with the lead-time-independentprice model, the authors show that the optimal solutions for these twomodels might look signiÞcantly different, and ignoring the dependencyof price on lead time might lead to large proÞt losses.Boyaci and Ray [15] extend the previous models to the case of two sub-

stitutable products served from two dedicated capacities. They assumethat these two products are essentially the same, and are differentiatedonly by their price and lead time. As in [84], there is a marginal capacitycost ci and a unit operating cost m, and as in [74] and [96], the priceis not a direct function of lead time. The market demand for producti = 1, 2 is given by λi = a−βRRi+ θR(Rj −Ri)−βlli+ θl(lj − li), i %= j.In this demand model, (i) θR and θl denote the sensitivity of switchoversdue to price and lead time, respectively, (ii) βR and βl denote the priceand lead time sensitivity of demand, (iii) the total market demand λ1+λ2

is independent of the switchover parameters (θ). The authors assumethat the lead time (l2) for product 2 (�regular� product) is given, andthe Þrm needs to determine a shorter lead time (l1 < l2) for product 1.Express vs. regular photo processing or package delivery are some ex-

5Recall that a lead-time-dependent price model was also studied earlier in [64], discussed inSection 6.1. Since price is not an independent decision variable in these models, we couldhave discussed [84] in Section 6.1 as well. However, due to its relevance to some of the otherpapers in this section, we decided to discuss it here.


amples motivating this model. As in the previous papers, the goal is tomaximize proÞts subject to service level constraints 1− e−(µi−λi)ti ≥ si,i = 1, 2. The authors Þrst study the uncapacitated model (Model 0,c1 = c2 = 0) with a given l1 (it turns out that in the uncapacitatedcase the optimal l1 is zero), and differentiate between price and timesensitive customers as in [84]: (i) Price and time difference sensitive(PTD): When βRθl > θRβl, the market is more sensitive to price butswitchovers are mainly due to the difference in lead times. Note thatthis condition is equivalent to [θR/(βR + θR)] < [θl/(βl + θl)], i.e., thefraction of customers that switch due to price is smaller than the fractionof customers that switch due to lead time. (ii) Time and price differ-ence sensitive (TPD): When βRθl < θRβl, the market is more sensitiveto lead time but switchovers are mainly due to the difference in prices.When θR ≈ θl, we are back to the standard price and lead time sensitivemarkets. Next, they study the case where only product 1 is capacitated(Model 1, c1 > 0, c2 = 0). Comparing Model 1 to Model 0, they showthat the change in the optimal prices depend on the market type (TPDvs. PTD) as well as c1. Interestingly, while price and time differentia-tion go hand in hand in Model 0, in Model 1 the Þrm price differentiatesonly if the marginal cost of capacity is sufficiently high. Finally, theauthors consider the case where both products are capacitated (Model2, c1 ≥ c2 > 0). They Þnd that for the same c1, the optimal l1 is shorterin Model 2, i.e., the Þrm offers a better delivery guarantee even thoughits cost is higher for the regular product. This is because a shorter l1lures customers away from the regular product and attracts them to themore proÞtable express product. The authors conclude their study bycomparing their results to the case where substitution is ignored or notpresent.So [95] extends the work in [96] to a competitive multi-Þrm setting,

where the Þrms have given capacities (µi) and unit operating costs (γi),i.e., they are differentiated by their size and efficiency, and choose prices(Ri) and delivery time guarantees (li). As in [74] and [96], each Þrmchooses a uniform delivery time guarantee. The market size (λ) is as-sumed to be Þxed and the market share of each Þrm is given by

λi = λ

%αiR

−ai t

−bi!n

j=1 αjR−aj t

−bj

&In this model, the parameter αi denotes the overall �attractiveness� ofÞrm i for reasons other than price and lead time, such as reputationand convenience of the service location. Parameters a and b denote theprice and time sensitivity of the market, respectively. To ensure thereliability of the quoted lead times, the Þrms seek to satisfy a service

60

constraint, which states that on average an s fraction of the orders willbe on time. Using an M/M/1 queue approximation, this constraint ismodeled as 1−e−(µi−λi)ti ≥ s, or equivalently, (µi−λi)ti ≥ − log(1−s).Focusing only on the case where s is given and the same for each Þrm, SoÞnds the �best response� of Þrm i in closed form given the other Þrms�price and lead time decisions. He shows that the price is decreasingin lead time. He also shows that (1) in the combined solution of priceand lead time, the Þrm always charges the highest price R̄ if a ≤ 1,and (2) the optimal price and lead time are increasing in αi. Theseresults suggests that Þrms compete only based on delivery time when themarket is not price sensitive, and Þrms with lower attractiveness needto compete by offering better prices and lead times. So characterizesthe Nash equilibrium in closed form for N identical Þrms, and proposesan iterative solution procedure for identifying the equilibrium in caseof non-identical Þrms. Comparing the results to the single-Þrm casestudied in [96], a capacity-increase in the multi-Þrm case leads to lowerprices, whereas the reverse is true in the single-Þrm case. This is quiteintuitive since an increased capacity in the multi-Þrm case leads to moreintense competition in a Þxed-size market. Numerical results with twoÞrms, s = 0.95 and equal αi�s indicate that (1) the advantage of highercapacity increases as the market becomes more time sensitive, (2) lowcost Þrm offers a lower price, longer lead time, and captures more of themarket share and proÞts, (3) an increase in price sensitivity leads to lowerprices overall, and shorter and longer lead times for the high capacityand low cost Þrms, respectively, (4) an increase in time sensitivity leadsto an increase in prices, and shorter and longer lead times by the low costand high capacity Þrms, respectively, reducing the difference between thelead times offered by the two Þrms, (5) as the time (price) sensitivityincreases, proÞts and market share of the high capacity (low cost) Þrmincrease. So also conducts experiments in a three-Þrm setting, whereone of the Þrms dominates the others both in terms of capacity andoperating cost. Interestingly, in such a setting the dominant Þrm offersneither the lowest price, nor the shortest lead time, while the other twoÞrms strive to differentiate themselves along those two dimensions byoffering either the lowest price (low cost Þrm) or the shortest lead time(high capacity Þrm).The papers discussed so far assume that once the Þrm quotes a cus-

tomer a lead time (and price), the customer immediately makes a de-cision as to whether to accept or reject the offer. In reality, customersmight �shop around�, i.e., request quotes from multiple Þrms, and/orneed some time to evaluate an offer. Hence, there might be a delay be-fore a customer reaches a decision on whether or not to accept an offer.


Easton and Moodie [32] study DDM-P in the face of such contingent or-ders, which add variability to the shop congestion and hence to the leadtime of a new order. In their model, the probability that the customerwill accept a quoted price/due-date pair (R, l) follows an S-shaped logitmodel

P (R, l) =

'1 + β0 exp

(β1l − pp

+ β2

(R

cp− 1))*−1

where p is the estimated work content of the order, c is the cost per unitwork content, and β0, β1 and β2 are parameters to be estimated fromprevious bidding results. The two terms in this expression refer to thelead time as a multiple of total work content, and the markup embed-ded in the quoted price. For choosing the price R and the lead timel for a new order (assuming FCFS sequencing), they propose a modelthat myopically maximizes the expected proÞt (revenue minus tardinesspenalties) from that order considering both Þrm and contingent ordersin the system but ignoring any future potential orders. The solutionmethod they propose involves evaluating all possible accept/reject out-comes for contingent orders, i.e., 2N scenarios if there are N contingentorders, which is clearly not efficient for a large number of contingentorders. Via a numerical example, the authors show that their modeloutperforms simple due-date setting rules based on estimates on theminimum, maximum or expected shop load.All the papers we discussed in this survey so far assume that the

due date (and price) decisions are made internally by the Þrm. Thisis in contrast to the scheduling literature where it is assumed that thedue dates are set externally, e.g., by the customer. In practice, mostbusiness-to-business transactions involve a negotiation process betweenthe customer and the Þrm on price and due date, i.e., neither the cus-tomer nor the Þrm is the sole decision maker on these two importantissues. With this in mind, Moodie [71] and Moodie and Bobrowski [72]incorporate the negotiation process into price and due date decisions. Intheir model, both the customer and the Þrm have a reservation tradeoffcurve between price and due date, which is private information. Cus-tomer arrivals depend on the delivery service reputation (SLR) of theÞrm, which depends on the Þrm�s past delivery performance as follows:SLRnew = (1−α)SLRold+αs, where s is the fraction of jobs completedon time in the last period. If the Þrm�s service level is below the cus-tomer�s requirement, the customer does not place an order. Hence, bychoosing its due dates, the Þrm indirectly impacts the demand throughits service level. Given a new customer, the Þrm Þrst establishes an ear-liest due date. The Þrm then chooses one of the four negotiation frame-

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works for price and lead time: negotiate on both, none, or only one of thetwo. Third, the Þrm chooses a price (a premium price for early deliveryand a base price for later delivery) and due date to quote. And Þnally,if the order is accepted, it needs to be scheduled. Moodie [71] proposesand tests four Þnite-scheduling based due date methods, as well as fourof the well-known rules from the literature: CON, TWK, JIQ, and JIS.Simulation results (under EDD scheduling) suggest that (1) due datemethods based on the jobs� processing times (especially TWK and JIS)perform better than the proposed Þnite-scheduling based methods, (2)negotiating on both price and lead time provides higher revenues, and(3) it may be proÞtable to refuse some orders even if the capacity is nottoo tight. A more extensive study of this model is performed by Moodieand Bobrowski [72]. They Þnd that full bargaining (both on price andlead time) is useful if there is a large gap between the quoted and thecustomers� preferred due dates. If this gap is small, then price-onlybargaining seems more beneÞcial.Charnsirisakskul et al. [17] study the beneÞts of lead time ßexibil-

ity (the willingness of the customers to accept longer lead times) to themanufacturer in an offline setting with 100% service guarantees. Theyconsider a discrete set of prices {R1

j , . . . , Rnjj } the manufacturer can

charge for order j. The demand quantity (expressed in units of capac-ity required, or processing time) corresponding to price Rkj is p

kj . Each

customer has a preferred and acceptable lead time, denoted by fkj and

lkj , respectively, if the manufacturer quotes price Rkj . There is also an

earliest start time for starting the production of and an earliest deliverytime for each order. If an order�s production (partially) completes beforethe earliest delivery time, the manufacturer incurs a holding cost. If thequoted lead time is between fkj and l

kj , i.e., after the customer�s preferred

due date, the manufacturer incurs a lead time penalty. The authorsmodel this problem as a linear mixed integer program where the deci-sions are: which orders to accept, which prices to quote to the acceptedorders, and when to produce each order. Note that this model simulta-neously incorporates order acceptance, pricing and scheduling decisions.The authors also consider a simpler model where the manufacturer mustquote a single price (again, chosen from a discrete set of prices) to allcustomers. They propose heuristics for both models, since the solutiontime can be quite large for certain instances. To establish the link be-tween prices and order quantities in their computational experiments,they consider the case where each customer is a retailer who adopts anewsvendor policy for ordering. They model the retailer�s demand bya discrete distribution function and compute the retailer�s optimal or-der quantity as a function of the manufacturer�s quoted price. In an


extensive computational study, they test the impact of price (single vs.multiple prices), lead time, and inventory holding (whether or not themanufacturer can produce early and hold inventory) ßexibility on themanufacturer�s proÞts. They Þnd that lead time ßexibility is useful ingeneral both with and without price ßexibility. The beneÞt of lead timeßexibility is higher if there is no inventory ßexibility, suggesting thatlead time and inventory ßexibilities are complementary. However, theyalso observe that in certain environments price, leadtime, and inventoryßexibilities can be synergistic.

7. Conclusions and Future Research Directions

Lead-time/due-date guarantees have undoubtedly been among themost prominent factors determining the service quality of a Þrm. Theimportance of due date decisions, and their coordination with schedul-ing, pricing and other related order acceptance/fulÞllment decisions in-creased further in recent years with the increasing popularity of make-to-order (MTO) over make-to-stock (MTS) environments.In this paper, we provided a survey of due date management (DDM)

literature. The majority of the research in this area focuses solely ondue-date setting and scheduling decisions, ignoring the impact of thequoted lead times on demand. Assuming that the orders arriving inthe system are exogenously determined and must all be processed, thesepapers study various DDM policies with the goal of optimizing one ora combination of service objectives such as minimizing the quoted duedates, average tardiness/earliness, or the fraction of tardy jobs. Clearly,no single DDM policy performs well under all environments. Severalfactors inßuence the performance of DDM policies, such as the due daterule, the sequencing rule, job characteristics (e.g., product structures,variability of processing times), system utilization (the mean and vari-ance of load levels), shop size and complexity, and service constraints.Due date policies that consider job and shop characteristics (e.g., JIQ)in general perform better than the rules that only consider job charac-teristics (e.g., TWK) which in turn perform better than the rules thatignore both job and shop characteristics (e.g., RND, CON) [7] [33] [82][104]. However, the performance of a due-date rule also depends on theaccompanying sequencing policy. For example, due-date based sequenc-ing rules perform better than other rules, such as SPT, in conjunctionwith due date policies that consider work content and shop congestion[33] [104]. Scheduling rules that use operation due-dates may result inimprovements over similar rules that use job-due-dates. The best valuesof the parameters for the due date rules also depend on the sequencing

64

rule [33] [82]. While shop size does not seem to affect the performanceof due date management policies signiÞcantly, increased shop complex-ity degrades the performance [101] [104]. Due date performance seemsto be fairly robust to utilization levels of balanced shops but changessigniÞcantly in shops with unbalanced machine utilization [102].Very few studies in the DDM literature use real-world data for testing

purposes, or report implementations of DDM policies in practice. Mostpapers use hypothetical job-shop environments in simulations or useanalytical models with very limiting assumptions. Exceptions include[67], which tests the proposed methods in a real-world example takenfrom [13]; [108], which discusses order acceptance and DDM practicesat Lithonia Lighting, a lighting Þxture manufacturer; and [100], whichreports results from a survey of 24 make-to-order Þrms (subcontractingcomponent manufacturers and manufacturers of capital equipment) inthe U.K on these Þrms� pricing and lead time quotation practices.There is a limited but growing body of literature on DDM that con-

siders the impact of quoted due-dates on demand. By modeling thedemand as a function of quoted lead times, these papers endogenize or-der selection decisions in addition to due-date setting and scheduling.Taking this approach one step further, a small number of papers modeldemand as a function of both price and lead time, and consider simul-taneous pricing, order acceptance, lead time quotation and schedulingpolicies.While there is a large body of literature on DDM, there are several

directions for research that are still open for exploration. One area thatreceived a lot of attention but is still open for research is the developmentand testing of new due date rules, which are based on readily availabledata, easy to compute and implement. A related research topic is theselection of the parameters for the due-date rules. The due-date rulescurrently available in the literature use static parameters, that is, theparameters are estimated once at the beginning and remain constantthroughout the implementation. One could possibly change the para-meters of a due date or scheduling rule in response to the changes in thesystem, which is an area that received very little attention so far [55]. Al-ternatively, one could use different predictive models or due-date settingor scheduling rules depending on the state of the system, e.g., measuredby the number of operations of an arriving job or shop congestion. Suchmixed DDM strategies might result in better performance.To ensure that the quoted lead times are short and reliable, some

of the DDM models impose service constraints, such as the maximumfraction of tardy jobs or the maximum average tardiness. The servicelevel is assumed to be Þxed, regardless of the changes in the system. In


practice, Þrms do not necessarily offer the same service guarantees allthe time. For example, most e-tailers guarantee delivery times only fororders that are placed sufficiently in advance of Christmas. It wouldbe interesting to study models with variable service guarantees, e.g.,making the service level a function of the number of jobs in the system.In this case, one needs to simultaneously optimize the service levels andthe due date quotes dynamically.Another interesting question facing a Þrm is what kind of lead time

guarantee to offer. Many companies offer constant lead times, and in caseof lateness, they offer a discount or free service. For example, Bennigan�s,a restaurant chain promised service within 15 minutes at lunch time, orthe meal was free [51]. Since the choice of the service guarantee and theassociated lateness penalties directly impact the delivery performance ofthe Þrm, it is important to understand what kind of service guaranteesare appropriate in different business environment. In particular, a Þrmshould not commit to a hard to meet service guarantee unless such aguarantee would positively impact its demand. For example, a restau-rant�s guarantee of less than 10 minutes wait might be valuable duringlunch, but not necessarily at dinner time. In general, a Þrm needs to un-derstand the impact of lead times or lead time guarantees on demand,especially on future demand. For example, the Touchdown guaranteeprogram of Austin Trumanns Steel provided promotional value in ap-proaching new customers, and it helped the company to win, and keepmany new customers [52].The papers reviewed in this survey assume that all (accepted) orders

must be completed even when the system is very congested and thereare signiÞcant lateness penalties. Alternatively, one could consider thepossibility of �outsourcing� an order if the system is highly congestedand completing all the orders in-house would lead to very high tardinesspenalties. In some industries, outsourcing might be necessary if the de-mand is time-sensitive. For example, in the air cargo industry if a carrierdoes not have enough capacity to ship all its contracted cargo within thepromised lead time (within 3 days for most time-sensitive cargo), theypurchase capacity from competing carriers in the spot market, usuallyat a high cost. While the option of outsourcing might be quite costly,it might actually help lower the overall costs by reducing the congestionin the system and lowering additional future delays.Note that outsourcing is just one form of (short-term) capacity expan-

sion. Very few papers in the DDM literature consider ßexible capacity[74] [96], where a Þrm can increase its (short-term) capacity by incurringsome additional cost. More research is needed to better understand un-

66

der what conditions capacity ßexibility is most useful and how it impactsthe performance of DDM policies.Most of the research in DDM considers MTO environments and hence,

does not consider the option of producing early and holding inventory.However, it may be possible to keep inventory of partially Þnished prod-ucts (vanilla boxes) or fast moving items in a MTO or hybrid MTO/MTSenvironment. The development of DDM policies for such hybrid systemsis for the most part an untapped research area.As we pointed out earlier, the incorporation of order selection and

pricing decisions into DDM is an important area that needs further ex-ploration. In addition to just reacting to demand, a Þrm can inßuencethe demand via its choice of prices and lead times. Hence, rather thanassuming that the demand and the associated revenues are input to theDDM policy and taking a cost minimization perspective, the Þrm cantake a proÞt maximization perspective and coordinate its decisions onpricing, order selection, lead time quotation and scheduling. So far, verylittle res reach has been done on DDM policies with pricing decisions,and the available models have signiÞcant limitations, such as focusing onquoting a single price/lead time pair to all customers and/or ignoringmultiple customer classes. Given that customers have different sensi-tivity to prices and lead time, it is important to incorporate multiplecustomer classes into the models. We believe that the development ofDDM policies with pricing and order selection decisions is an importantresearch direction with a signiÞcant potential impact in practice.In their quest towards proÞtably balancing supply and demand, com-

panies traditionally focused on the supply side. Among the decisionsthat inßuence supply availability over time are inventory policies, ca-pacity selection and production scheduling. In contrast, price and leadtime decisions help companies manage the demand side, that is, to shiftor adjust the demand over time to better match supply. In essence, acompany achieves different types of ßexibility by being able to adjust�levers� such as price, lead time or inventory. An interesting researchdirection is to investigate which of these ßexibility levers have more im-pact on proÞts in different types of business environments. Furthermore,are the beneÞts from having multiple types of ßexibility subadditive orsuperadditive?Most of the research in scheduling either ignores due dates or assumes

that they are set exogenously (e.g., by the customer). Most of the re-search in DDM assumes that the due dates are set internally by theÞrm. The reality is in between these two extremes. In most business-to-business transactions, due dates are set through a negotiation processbetween the customer and the Þrm. DDM research that incorporates


buyer-seller negotiation is still at its infancy [71] [72] and we believe thisto be an interesting and fertile area for future research. Other interest-ing extensions to the existing literature include substitutable productsand competition.

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Chapter 999 DUE DATE MANAGEMENT POLICIES