DUEDATEMANAGEMENTPOLICIES

Transcription

1 Chapter 999 DUEDATEMANAGEMENTPOLICIES Põnar Keskinocak School of Industrial and Systems Engineering Georgia Institute of Technology, Atlanta, GA Sridhar Tayur Graduate School of Industrial Administration Carnegie Mellon University, Pittsburgh, PA Pınar Keskinocak is supported by NSF Career Award DMII This research is also supported in part by a grant from The Logistics Institute Asia Pacific (TLI-AP). 1

2 2 Introduction To gain an edge over competitors in an increasingly global and competitive marketplace, companies today need to differentiate themselves not only in cost, but in the overall value of the products and the services they offer. As customers demand more and more variety of products, better, cheaper, and faster, an essential value feature for customer acquisition and retention is the ability to quote short and reliable lead times. Reliability is important for customers especially in a businessto-business setting, because it allows them to plan their own operations with 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 time 2 Hence, quoting a lead time is equivalent to quoting a due date. The importance of lead time quotation becomes even more prevalent as many companies move from mass production to mass customization, or from a make-to-stock (MTS) to a make-to-order (MTO) model to satisfy their customers unique needs. Hence, companies need to determine in real time if and when an order can be fulþlled proþtably. The Internet as a new sales channel further increases the importance of effective lead time quotation strategies, as customers who place orders online expect to receive reliable lead time as well as price quotes. For example, many customers were extremely dissatisþed with their online purchasing experience during the 1999 holiday season, mainly due to unreliable delivery date quotes, lack of order status updates, and signiþcant order delays [48]. Quoting unreliable lead times not only leads to potential loss of future 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 settle a Federal Trade Commission (FTC) action over late deliveries made during the 1999 holiday season [40]. According to the FTC, the e-tailers promised delivery dates when fulþllment was not possible and failed to notify customers when shipments would be late. Sometimes a company may self-impose a penalty for missed due-dates. For example, due to the increasing insistence of many steel users on consistent reliable deliveries, Austin Trumanns Steel started to offer a program called Touchdown Guarantee in Under the program, if the company agrees to a requested delivery date at the time an order is placed, it has to deliver 2 In this paper, we focus on lead times quoted to customers. Lead times can also be used for internal purposes, e.g., planned lead times are used for determining the release times of the orders to the shop floor [69]. We do not discuss planned lead times in this paper.

3 Due Date Management Policies 3 on time or pays the customer 10% of the invoice value of each item not delivered [52]. A common approach to lead time quotation is to promise a constant lead time to all customers, regardless of the characteristics of the order and the current status of the system [65] [110]. Despite its popularity, there are serious shortcomings of Þxed lead times [62]. When the demand is high, these lead times will be understated leading to missed due dates and disappointed customers, or to higher costs due to expediting. When the demand is low, they will be overstated and some customers may choose to go elsewhere. The fundamental tradeoff in lead time quotation is between quoting short lead times and attaining them. In case of multiple customer classes with different capacity requirements or margins, this tradeoff also includes capacity allocation decisions. In particular, one needs to decide whether to allocate future capacity to a low-margin order now, or whether to reserve capacity for potential future arrivals of high-margin orders. Lead-time related research has developed in multiple directions, including lead time reduction [54] [99], predicting manufacturing lead times, the relationship between lead times and other elements of manufacturing such as lot sizes and inventory [42] [61] [68], and due date management (DDM) policies. Our focus in this survey is on DDM, where a DDM policy consists of a due date setting policy and a sequencing 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 production schedule), we focus on the case where due dates are set endogenously. Most of the research reviewed here does not consider inventory decisions and hence is applicable to MTO systems. Previous surveys in this area include [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 small but increasing number of researches studied DDM from a proþt maximization rather than a cost minimization perspective, considering order acceptance decisions (or the effect of quoted lead times on demand) in addition to due date quotation and scheduling decisions. Although this is a step forward from the earlier DDM research, it still ignores another important factor that affects the demand: price. With the goal of moving towards integrated decision making, the latest advances in DDM researchfocusonsimultaneouspriceandleadtimequotation. The paper is organized as follows. In Section 1, we discuss the characteristics of a DDM problem, including decisions, modeling dimensions,

4 4 objectives and solution approaches. In Section 2, we discuss commonly used scheduling rules in DDM policies. Offline DDM models, which assume that the demand and other input about the problem are available at the beginning of the planning horizon, are discussed in Section 3. Online models, which consider dynamic arrivals of orders over time, are presented in Section 4. Models for DDM in the presence of service level constraints are discussed in Section 5. We review the DDM models with order acceptance and pricing decisions in Section 6. We conclude with future 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 papers published in the DDM literature and regret any omissions. We would be delighted to receive copies of or references to any work that was not included in this survey. 1. Characteristics of a Due Date Management Problem In this section, we discuss the characteristics of DDM problems, including the decisions, modeling dimensions, and the objectives. The notation that is used throughout the paper is summarized in Table Due Date Management Decisions The main decisions in DDM are order acceptance (or demand management), due date quotation, and scheduling. In determining a DDM policy, ideally one would consider due date setting and scheduling decisions simultaneously. There are a few papers in the literature that follow this approach. However, given their complexity, most papers consider these two decisions sequentially, where Þrst the due dates are set and then the job schedules are determined. Commonly used scheduling policies in DDM are discussed in Section 2. We denote a due date policy by D-S, where D refers to the due-date setting rule and S refers to the scheduling rule. While most researchers are concerned in comparing the performance of alternative DDM policies, some focus on Þnding the optimal parameters for a given policy [19] [87]. Most of the papers in the literature assume that order acceptance decisions are exogenous to DDM, and all the orders that are currently in the system have to be quoted due dates and be processed. Equivalently, they ignore the impact of quoted due dates on demand and assume that once a customer arrives in the system, he will accept a due date no matter how late it is. In reality, the quoted lead times affect whether a customer places an order or not, and in turn, they affect the revenues. In

5 Due Date Management Policies 5 Indices: j: order or job k: job class o: operation of a job t: time period Notation r j : arrival (or ready) time of job j p j : (total) processing time of job j R j : revenue (price) of job j w j : the weight of job j (e.g., may denote the importance of the job) g j (g jt ) : number of (remaining) operations on job j (at time t) U jt : set of remaining operations of job j at time t n t (n kt ) : number of (class k) jobs in the system at time t l j : quotedleadtime(orquotedßow time) of job j C j : completion time of job j W j = C j r j p j : wait time of job j F j = C j r j : ßow time of job j T j =(C j d j ) + : tardiness of job j E j =(d j C j ) + : earliness of job j L j = C j d j : lateness of job j τ max : maximum fraction of tardy jobs (or missed due dates) T max : maximum average tardiness ρ : steady-state capacity utilization α, β, γ : parameters Table Notation for DDM models. (.) jo denotes the value of parameter (.) for the o-th operation of job j, if a job has multiple operations. E[(.)] and σ (.) denote the expected value and the standard deviation of (.), when (.) is a random variable.

6 6 many cases, there is a maximum limit on the lead time, either imposed by customers or by law. For example, under FTC s mail-and-telephone order rule, orders must be shipped within 30 days or customers must be notiþed and given the option of agreeing to a new shipping date or canceling the order. Even if the customers accepted any kind of lead time, due to the associated penalties for longer lead times or missed due dates, it might be more proþtable for a manufacturer not to accept all customer orders. Hence, in contrast to most of the papers in the literature, one needs to consider DDM from a proþt maximization perspective rather than a cost minimization perspective. Incorporating order acceptance decisions into DDM policies by considering the impact of quoted lead times and prices on demand is an important step towards integrated DDM policies. We discuss the literature on DDM with order acceptance (and price) decisions in Section Dimensions of a Due Date Management Problem There are several dimensions that distinguish different due date management problems. Depending on the system (i.e., the manufacturing or service organization) under consideration, a combination of these dimensions will be present and this combination will affect the appropriate mathematical model for studying DDM in that setting. Offline vs. online: Inanoffline setting, all the information about the problem, such as the job arrival and processing times, are available at the beginning of the scheduling horizon. In contrast, in an online setting future arrivals are not known with certainty and the information about a job becomes available only at the time of its arrival. In general, it is assumed that the arrivals follow a known probability distribution. An online setting is stringent if the decisions about a job, such as its due date,havetobemadeimmediatelyatthejob sarrivaltime. Incontrast, in an online setting with lookahead, one can wait for some time after a job arrives before making a decision about that job, but there is usually some penalty for delaying the decision. Single vs. multiple servers: In a single-server setting, only one resource needs to be considered for satisfying customer requests. In a multiple-server setting, multiple resources are available, which may be identical or different. In case of multiple non-identical servers, there are different possible system (or shop) characteristics, such as job shop and ßow shop. In a job shop, each job has a sequence of operations in different machine groups. In a ßow shop, the sequence of operations is the same for each job.

7 Due Date Management Policies 7 Preemptive vs. nonpreemptive: In a nonpreemptive setting, once the processing of a job starts, it must be completed without any interruption. In contrast, in a preemptive setting, interruption of the processing of a job is allowed. In the preempt-resume mode, after an interruption, the processing of a job can be resumed later on without the need to repeat any work. Thus, the total time the job spends in the system is p j, although the difference between the completion and start times can be longer than p j. In the preempt-repeat mode, once a job s processing is interrupted, it has to start again from the beginning. Hence, if job j is interrupted at least once, its total processing time is strictly larger than p j. Stochastic vs. deterministic processing times: When the processing times are not known with certainty, it is usually assumed that they follow 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 current literature on DDM ignores setup times. Server reliability: The capacity of the resources may be known (deterministic) 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) they generate, or based on their demand characteristics, such as (average) processing times, lead time related penalties, or maximum acceptable lead times. Service level constraints: Commonly used service level constraints include an upper bound on (1) the fraction (or percentage) of orders which are 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 due date. In case of distinct due dates, different due dates can be quoted to different customers. 1.3 Objectives of Due Date Management When order acceptance decisions are assumed to be exogenous, the key decisions are due date setting and scheduling. Note that these decisions have conßicting objectives. One would try to set the due dates as tight as possible, since this would reßect better responsiveness to customer demand. However, this creates a conßict with the scheduling objectives because tight due dates are more difficult to meet than loose

8 8 due dates. There are two approaches for resolving this conßict: (1) use an objective function that combines the objectives of both decisions, (2) consider the objective of one decision and impose a constraint on the other decision. A relatively small number of papers follow the Þrst approach [11] [24] [67] [87] [86] using an objective function that is the weighted sum of earliness, tardiness and lead times, where the weights reßect the relevant importance of short lead times vs. meeeting them on time. Among the papers that follow the second approach, some try to achieve due date tightness subject to a service level constraint, while others consider a service objective (e.g., minimizing tardiness) subject to a constraint on minimum due date tightness. For example, in [6] the objective is to minimize the average due date subject to a 100% service guarantee (i.e., tardiness is not allowed). In contrast, in [7] the objective is to minimize average tardiness subject to a constraint on the average ßow 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. Common objectives include minimizing: Average/total (weighted) due date (subject to service level constraints) [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]; conditional mean 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 absolute lateness [41] [67] [76] Average earliness [14] [55] [104]; number (or percentage) of early jobs [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 ßow times [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]

9 Due Date Management Policies 9 Note that the objectives in the last bullet focus primarily on internal measures of the shop ßoor, whereas the objectives in the previous bullets focus 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 requests more quickly and in reducing (work-in-process) inventories. Minimizing earliness helps in lowering Þnished goods inventories leading to lower holding costs and other costs associated with completing a job before its due date. Earliness penalty may also capture a lost opportunity for better service to the customer, since a shorter lead time could be quoted to the customer if it were known that the job would complete early. Minimizing tardiness or conditional mean tardiness, which is the average tardiness measured over only the tardy jobs, helps in completing the 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 tardy jobs and the conditional mean tardiness, hence, looking at the two latter measures separately provides more information about the performance than just looking at the aggregate measure of average tardiness. Minimizing due dates helps in attracting and retaining customers. Average lateness measures whether the quoted due dates are reliable on average, i.e., the accuracy of the due dates. Standard deviation of lateness measures the magnitude of digression from quoted due dates, i.e., the precision of the due dates. WIP and CWIP measure the congestion in the system (through work-in-process inventory) and the average investment in work-in-process inventory, respectively, assuming that the investment in a partially Þnished job is proportional to the completed amount of processing. The performance of a due date management policy depends both on due date setting and sequencing decisions. For certain objective functions, such as minimizing lateness or tardiness, due-date setting rules have direct and indirect effects on performance [105]. The direct effect results from the type of due-date rule employed and its parameters, which determine the due date tightness. The indirect effects result from the due-date being a component of some of the dispatching and labor assignment rules. The due-date setting rules have only indirect effects for some objectives, such as minimizing the average ßow time, where sequencing rules have more direct impact. An interesting question is which of the two decisions, due date setting and sequencing, has a bigger impact on performance. The answer depends on the measure of performance (objective function), the type of DDM policy and its parameters, service constraints, and system con-

10 10 ditions. For example, Weeks and Fryer [105] Þnd that due-date assignment rules are the most important decisions for mean lateness, variance of lateness and proportion of tardy jobs, whereas sequencing rules are the most important class of decision rules to impact mean ßow time and variance of ßow time. In addition, for some objective functions (the variance of ßow time, variance of lateness and proportion of tardy jobs) the relative inßuence of sequencing rules depends on the tightness of the assigned due dates. Wein [107] Þnds that for the objective of minimizing average ßow time, for the four due date setting rules he proposed, the impact of sequencing on performance is minimal. For other due date rules, however, the impact of sequencing on performance is signiþcant. One has to be very careful in choosing an appropriate objective function to satisfy business goals, as different objective functions may lead to remarkably different DDM policies. For example, a DDM policy that yields a zero lateness on average would result in approximately 50% of the jobs early and the remaining 50% tardy. If the cost of earliness is considerably lower for a Þrm than the cost of tardiness, then clearly minimizing lateness might not be an appropriate performance measure. In general, we can divide the objectives into two broad categories, depending on whether they penalize only for completing the jobs after their due dates, or penalize both for early and late completion. There are important differences among the objectives within the same category as well. For example, consider the three objective functions, minimizing average tardiness, the number of tardy jobs, and maximum tardiness. Consider two DDM policies, one resulting in one job that is 100 time units late, whereas the other one resulting in 10 jobs that are each 10 units late. While these two policies are equivalent with respect to the average tardiness objective, the Þrst policy is better than the second one with respect to the number of tardy jobs, and the second policy is better than the Þrst one with respect to maximum tardiness. Hence, depending on the choice of the objective function, one can obtain a result that is signiþcantly poor with respect to another objective or service constraint. For example, Baker and Bertrand [7] note that when the due date tightness is below a threshold for the DDM policy SLK-EDD in an M/M/1 queuing system, even if the average tardiness is low, the proportion of tardy jobs is approximately equal to the utilization level in the system. While this characteristic might be quite acceptable for the objective of minimizing average tardiness, it might be undesirable for a secondary objective of minimizing the number of tardy jobs. Also, see [98] for a discussion on DDM policies which lead to unethical practice by quoting lead times where there is no hope of achieving them, while trying to minimize the number of tardy jobs.

11 Due Date Management Policies 11 The objective functions discussed above take a cost minimization perspective and are appropriate when the jobs to be served are exogenously determined. When order acceptance is also a decision (or equivalently, if the demand depends on quoted lead times), then the objective is to maximize proþt; that is, the revenues generated from the accepted orders minus the costs associated with not completing the orders on time. 1.4 Solution Approaches for Due Date Management Problems To study the effectiveness of due date management policies, three approaches are used: simulation, analytical methods and competitive analysis. While they are quite interesting from a theoretical perspective, the application of analytical models to DDM has been limited to simple settings, such as special cases of offline single machine problems. To study the effect of DDM policies in more realistic multi-machine environments where orders arrive over time, researchers have commonly turned to simulation. As an alternative approach, a few researchers have used competitive analysis to study DDM policies, where an on-line algorithm A is comparedtoanoptimaloff-line algorithm [93]. Given an instance I, let z A (I) andz (I) denote the objective function value obtained by algorithm A andbyanoptimumoff-line algorithm, respectively. We call an algorithm A c-competitive, if there exists a constant a, such that z A (I) c A z (I)+a for any instance I. (Here we assume that A is a deterministic on-line algorithm and we are solving a minimization problem.) The factor c A is also called the competitive ratio of A. The performance of off-line algorithms can be measured in a similar way. An off-line algorithm A is called a ρ approximation, if it delivers a solution of quality at most ρ times optimal (for minimization problems). Although the use of 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 approach: assign the due dates Þrst, and then schedule the orders using a priority dispatch policy. While it would be desirable to consider due date and scheduling decisions simultaneously, the use of such a two-step approach is understandable given that scheduling problems even with

12 12 preassigned due dates are quite difficult to solve. In a priority dispatching policy for scheduling, a numerical value (priority) is computed for each job that is currently in the system waiting to be processed. The job 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 The simplest of the dispatching rules, RANDOM and FCFS, consider no information about the jobs 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/p j,slk/g j, MDD, OPNDD and P+S/OPN(a,b) consider the due dates as well as the work content. Some industry surveys suggest that due-date based rules, such as EDD and EDD o are the most commonly 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 while the job is in the system, e.g., SPT or EDD. In contrast, the priority value assigned 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 dynamic rule can result in the same sequence, e.g., SLK and EDD 3. 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 the appropriate values for these parameters. The choice of the parameters depends on the system environment factors and can have a signiþcant impact on the performance of the DDM policy. AsinthecaseofSPTT,somesequencingrulesÞrst divide the available jobs into two classes, and then apply a (possibly different) sequencing rule in each class. The division of jobs into classes can be random, or based on job characteristics. Three such rules are considered in [91]: 2C-SPT, 2C-NP and 2RANDOM-SPT. In the Þrst two rules, jobs with processing time less than p belong to class 1 and others belong to class 2. 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 sequenced based on SPT or FCFS. The rules presented at the top portion of Table are applied at the job level, using aggregate information about the operations of the job, whereas the rules presented in the lower portion of the table (with a subscript o denoting operation ) are applied at the operation level. 3 We are indebted to an anonymous referee for this observation.

13 Due Date Management Policies 13 For example, when scheduling the operations on a machine using the shortest processing time rule, one can sort the operations based on the corresponding jobs total processing time (! o p jo), or based on the processing times of the operations to be performed on that machine (p jo ). For a rule like SPT, the application either at the job or at the operation level is straightforward, since the processing time information exists at both levels. On the other hand, it is not obvious how a rule like EDD can be applied at the operation level, since we usually tend to think of due dates at the job level. One needs to keep in mind that the due dates that are quoted to the customers do not have to be the same due dates as the ones used for scheduling. In other words, it is possible to have external and internal due dates, where the latter ones are dynamically updated based on shop conditions and used for scheduling purposes only. For example, the EDD o rule is similar to EDD, but priorities are based on internally set operation due dates. A number of researchers have focused on the performance of sequencing rules under different due-date setting rules. The performance of scheduling rules as part of a DDM policy depend on many factors, including the due date rule, the objective function under consideration, the tightness of the due dates, the workload/congestion level in the system. The scheduling rules have especially a signiþcant impact on the performance 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 the TWK rule) in general SPT performs best for minimizing job waiting times, ßow times, machine idle times and queue lengths, and LPT performs worst. Several researchers have noticed that SPT also performs well in general for the objective of minimizing average tardiness (since it tends to reduce average ßow time), but it usually results in higher variation. Bookbinder and Noor [14] Þnd that earliness and tardiness have much higher standard deviation under SPT than under other sequencing rules. Hence, SPT might not be an appropriate choice for the objective functions that penalize both earliness and tardiness. For example, Ragatz and Mabert[82] Þnd that for the objective of minimizing the standard deviation of lateness (σ L ), SPT performs worse than SLK and FCFS under eight different due date rules. Similar observations were made in [103] for SPTT, where the average ßow time is signiþcantly lower under SPTT, but the variance is higher. Weeks [104] Þnds that for the objectives of mean earliness, tardiness and missed due dates, sequencing rules that consider due date information perform better than the rules that do not.

14 14 Table Dispatch sequencing policies Policy Priority value References RANDOM β j Uniform[0,1] [27] [34] First come first serve r j (or r jo) [7] [8] [14] [27] [29] [33] [34] (FCFS, FCFS o ) [37] [39] [67] [70] [82] [98] [102] [103] [105] [106], [37] Shortest processing time p j (or p jo ) [7][8][9][27][33][34] (SPT, SPT o ) [76] [77] [81] [82] [101] [104] [106] [107], [37] [105] Least remaining operation time (LROT) LROT/NOP! o U jt p jo) [37]! o U jt p jo/ U j [37] Truncated SPT (SPTT) p jo,ifw jo < α o in queue; [38] [39] [102] [103] W jo, otherwise Weighted SPT (WSPT) w j p jo [107] Longest processing time 1/p j (or 1/p jo ) [27] [34], [37] (LPT, LPT o ) Truncated LPT (LPTT) 1/p jo,ifw jo < α o in queue; [38] W jo,otherwise Earliest finish time (EFT) r j + p j [7] [8] Earliest due date d j (or d jo) [6][7][8][27][34][37] (EDD, EDD o ) [38] [39] [41] [60] [71] [81] Slack (SLK, SLK o ) d j t! o U jt p jo α, d jo t p jo [101] [107], [9] [39] [60] [7] [8] [27] [37] [38] [82] [106] [107], [60] SLK # Put jobs with SLK 0intoapriority [33] queue and SLK> 0intoanormalqueue. Apply SPT to each queue. SLK/P (d j t! o U jt p! jo)/ o U jt p jo) [37] [38] [107] SLK/OPN SLK/RAT (d j t o U jt p jo )/g j (d j t! o U jt p jo )/(d j t) [9][37][70][104][105] [70] COVERT c j/p jo, wherec j = 0, if the job is ahead [9] Critical ratio (CR, CR o) of schedule; c j = 1, if the job has no slack; and c j = c j,otherwise,where c j is the proportion of the job s planned waiting time that has been consumed (d j t)/! o U jt p jo, (d jo t)/p jo [39] [60], [39] R/OPN (d j t)/g j [10] [70] Modified Due Date max{d j,t+ p j },max{d jo,t+ p jo } [8] [9] [10] [81] [107] (MDD, MDD o) [102] [103] Earliest operation r j +(d j r j )(o/g j ) [27] duedate(opndd) P+S/OPN αp jo +(1 α) d j t!g j l=o p jl (g j o+1) β [27]

15 Due Date Management Policies 15 The performance of scheduling rules also depends on the due date policy and how tight the due dates are. For the objective of minimizing tardiness, Elvers [37] Þnds that some scheduling rules work better under 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 SPT o,performbestwhenthe due dates are tight. (SPT s good performance under tight due dates is also observed in several other papers, e.g., [105].) On the other hand, scheduling rules which consider the operation due date or slack, such as EDD, SLK, SLK/OPN and SLK/P, perform better when the due dates are loose 4. Intuitively, SPT tries to minimize average ßow time without paying much attention to tardiness. This might hamper SPT s performance (with respect to minimizing tardiness) when the ßow allowance is high where most due dates can be met using the EDD rule. But when the ßow allowance is low, most jobs are late and SPT is advantageous because shorter mean ßow times lead to less tardiness. Having noticed the complementary strengths of different rules depending on due date tightness, Baker and Bertrand [8] propose a modiþed due date (MDD) rule, which is a combination of EDD and SPT rules. It works like SPT when the due dates are very tight and like EDD when the due dates are very loose. They test the MDD rule on the same test data used in [6] and observe that MDD sequencing results in lower average tardiness compared 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 machine model with equal job release times, average tardiness is minimized by MDD under either TWK or SLK due date rules [8]. Wein [107] also Þnds that MDD performs better than other sequencing rules in general for the objective of minimizing the weighted average lead time subject to service constraints. A interesting question is whether to use job due dates or individual operation due dates for sequencing. Kanet and Hayya [60] study the effect of considering job due dates versus operation due dates on the performance of scheduling rules in the context of DDM. Under the TWK due-date setting rule (with three different parameters), they compare three due-date based scheduling rules and their operation counterparts: EDD vs. EDD o, SLK vs. SLK o and CR vs. CR o. They consider the objectives of minimizing: mean lateness, standard deviation of lateness, fraction of tardy jobs, conditional mean tardiness, mean ßow time, standard deviation of ßow time, and maximum job tardiness. They Þnd 4 The SLK/OPN rule can behave in undesirable ways when SLK is negative. See [59] for a discussion on such anomalous behavior and on modifications for correcting it.

16 16 that operation-due-date based scheduling rules outperform their job-duedate-based counterparts for almost all performance measures. In particular, operation-due-date based scheduling rules consistently result in lower ßow times, and reduce the mean and the variance of the probability density function of lateness. Among these rules, EDD o outperforms SLK o for all performance measures under all conditions. In particular, EDD o results in the minimum value of maximum tardiness in all cases. Intuitively, EDD o extends Smith s rule [94], which states that in an ofßine single machine problem maximum tardiness is minimized by EDD sequencing. They also look at the impact of ßow allowance on the performance. Note that when we sufficiently increase α in TWK, then the job with the largest processing time will always have the largest due date regardless of when it arrives. In that case, EDD sequencing becomes equivalent to SPT sequencing. Hence, for the rules EDD, EDD o,slk and SLK o an increase in ßow allowance results in a decrease in average ßow time. In a related study to [60], Baker and Kanet [9] compare the performance of MDD and MDD o with a number of other well-studied scheduling rules under the TWK due date rule, focusing on the impact of due date tightness and shop utilization. Using industrial data, they note that the average manufacturing capacity utilization in the U.S. ranged from 80 to 90% between 1965 and 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 tightness. They consider three performance measures: proportion of tardy jobs, conditional mean tardiness, and average tardiness. A comparison of MDD and MDD o indicates the superior performance of MDD o except in environments with high utilization and very loose due dates, in which case both rules perform quite well. MDD o also outperforms SPT and EDD o in all environments except under very loose due dates. Similar to [37], the observations of Baker and Kanet indicate that the tightness of due dates has an important effect on the performance of scheduling rules. For example, the performance of S/OPN and EDD o is quite good with loose due dates but deteriorates as the due dates become tighter. The opposite is true for SPT. Therefore, a hybrid rule such as MDD tends to exhibit a more robust and superior performance over a wide range of settings. Enns [38] [39] has experimented both with job- and operation-duedate dependent dispatch policies. He considers internal as well as external measures of the shop ßoor. He notes that the sequencing rule has a signiþcant impact on internal measures such as the number of and the investment in work-in-process jobs, smoothness of work ßow, and

17 Due Date Management Policies 17 the balancing of queue lengths. For example, (1) EDD and SPT lead to less amount of work completed per job at any given time compared to FCFS, (2) jobs tend to move slower at Þrst and then faster under EDD, whereas they progress at a more steady pace under EDD o. Intuitively, operation due dates create artiþcial milestones along the way rather than a single deadline, and ensure a more steady progress of the jobs. While this might be desirable from a balanced workload point of view, 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 are quite promising and in some cases perform better than the corresponding rules based on job due dates [8] [9] [12] [60]. In general, if the variability in the system is low, one can obtain better estimates of ßow times and hence quote more accurate due dates. The scheduling policy impacts the variability of ßow times. One possible reason for the observed superior performance of the operation due date based scheduling rules is because the operation due dates result in a more steady ßow of the jobs through the system, reducing ßow time variability. 3. Offline Models for Due Date Management Offline models for DDM assume that the arrival times, and possibly the processing times, of the jobs are known at the beginning of the planning 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 authors studied the common due date problem (CON), where all the jobs are assigned a common due date d and the goal is to jointly determine the common due date and a sequence [3] [11] [21] [24] [75] [80] [79]. The objective functions considered in these papers can be characterized by the! following general function of earliness, tardiness and the due date: j β jej a +γ jtj b +α jd. In most papers, except in [24], a = b =1,i.e.,the Þrst two summations correspond to weighted earliness and tardiness. In [75], β j = β, γ j = γ and α j = α; in[3],β j = β, γ j = γ and α j =0;in [21], β j = γ j = w j and α j = 0; in [24], β j = γ j = w j and α j = α. Inthis problem, it is easy to show that the optimal due date d is equal to the completion time of some job j,i.e.,d = C j. For the general function with a = b = 1, the following simple condition determines the optimal due date d = C j for a given sequence: j is the Þrstpositioninthe

18 18 sequence such that! j k=1 (β k + γ k )! n k=1 (γ k α k ) [11] [22] [79]. Note that all the jobs before j are early, and are sequenced in non-increasing order of the ratio C j /β j. The jobs after j are tardy and are sequenced in non-decreasing order of the ratio C j /γ j. Such a schedule is called a V-shaped sequence [3] [11]. The optimality of V-shaped schedules for common due date problems have also been observed in [83]. Optimality conditions for the case a = b = s along with an iterative procedure for determining d (for a given sequence) are given in [24]. Enumeration algorithms for jointly determining the due date and the optimal sequence are 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, A j =max{0,d j A}, which is the amount of time the assigned due date of a job exceeds the preferred due date, A. The objective is to minimize the weighted sum of earliness, tardiness and lead time penalty. The authors propose a simple policy for setting the due dates and show that this policy is optimal when used in conjunction with the SPT rule. The two papers discussed next [19] [25] [45] study the TWK and SLK due date rules presented in Table Cheng [19] studies the TWK due date rule under the objective of minimizing total squared lateness. Both for deterministic and random processing times (with known means and the same coefficient of variation), he Þnds the optimal value of the parameter α and shows that the optimal sequencing policy is SPT. Further extensions of this work are presented in [20] and [23]. Cheng [25] studies the SLK due date rule under the objective of minimizing the (weighted) ßow allowance plus the maximum tardiness. He shows that the optimal sequence is EDD and derives a simple function to compute the optimal SLK due dates. Gordon [45] studies a generalized version of this problem with distinct job release times and precedence constraints 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 that the due dates depend on the jobs earliness and tardiness costs and the mean and the standard deviation of the jobs completion times. Next, he addresses the problem of simultaneous scheduling and due date determination. Unfortunately, he is unable to provide an en efficient procedure for obtaining the optimal sequence, and hence provides lower and upper bounds on the expected total cost as well as two heuristics. Using examples, he shows that the heuristics perform well, and that treating the

19 Due Date Management Policies 19 processing times as deterministic could lead to signiþcantly inaccurate due dates and higher costs. 3.2 Distinct Order Arrival Times Unlike the previous papers discussed inthissection,thetwopapers we discuss next consider arbitrary (not necessarily equal) order arrival times 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% service guarantee, i.e., all the jobs must be completed on time. This problem can be solved optimally by Þnding a schedule that minimizes the average completion time, and then by setting d j = C j for each job j. Hence, using the three-þeld notation described in [46] it is equivalent to 1 r j,pmtn! j C j. When the jobs have equal release times, the objective becomes equivalent to minimizing the mean ßow time, and SPT sequencing gives the optimal solution. Unfortunately this approach requires that all the information about the jobs must be known in advance, which is rarely the case in practice. Hence, the authors consider three simple due date setting rules, CON, SLK, and TWK, where the due date decision depends only on the information about the job itself. They Þrst consider these rules for the case of equal job arrival times and Þnd the optimal value of the parameter α for each of these rules. Note that when the arrival times are equal, EDD sequencing minimizes maximum lateness, hence, given a set of due dates, one can use the EDD rule to Þnd out if those due dates satisfy the service constraint of zero tardiness. Also, when the due dates are set using TWK or SLK due date rules, EDD and SPT sequences are equivalent. The authors show that in case of equal job arrival times, CON is dominated by TWK and SLK, but no dominance relationship exists between TWK and SLK. They also compare the worst case performance of these rules, by comparing them to the optimal solution: (1) For the case of equal processing times, c TWK = c SLK = c CON =2n/(n + 1). That is, the worst case performance of the three rules is the same, approaching 2 as the number of jobs increases. (2) When the processing times are equal to 1 for all but one job j, forwhichp j > 1, c TWK = n 1, c SLK =1,c CON = n. In this case, TWK and CON have similar worst case performance, which can be arbitrarily large, whereas SLK produces the optimal solution. (3) When p j = j for all j, c TWK =1.5, c SLK =3,c CON =3. Thecompetitive ratios provide information about the worst case performance of these rules, but they can be overly pessimistic. To gain an understanding about the average performance of CON, TWK and SLK, the

20 20 authors perform simulation studies. In case of equal arrival times (with processing times drawn from exponential, normal or uniform distributions), TWK produces the best results in terms of due date tightness. Furthermore, compared to SLK and CON, TWK produced due dates which 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 the jobs have the same ßow allowance, EDD becomes equivalent to FCFS and preemption is not necessary. Under SLK, the waiting time allowance is the same for all jobs, and the EDD rule will sequence the jobs in the increasing order of r j +p j. Hence, preemption might be necessary under SLK. Note that in case of CON and SLK rules, one does not need the value of the parameter α to implement the EDD rule. However, in case of TWK, the value of α is needed for implementing EDD. Baker and Bertrand [6] provide simple procedures for computing the due dates under each of these rules. To test the performance of these rules, they run simulations for two different workload patterns. For the random workload pattern, they simulate an M/M/1 queue. For the controlled workload pattern, they use a scheme which releases a new job as soon as the workload in the system falls below a threshold. Experimental results suggest that TWK has the best average performance for the random workload pattern. In the case of controlled workload pattern, SLK produces the best results for light workloads; however, the due dates of TWK are less sensitive to workload, suggesting that TWK might be the preferred policy. Although TWK exhibits better performance on average than SLK, in unfavorable conditions it can perform considerably worse. (This is also indicated by the arbitrarily large competitive ratio of TWK when all but one job have p j = 1.) The results suggest that when the variance of processing times is low, there is little difference among 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 well as scheduling and due-date setting decisions. In their model, an order is speciþed by a unit revenue, arrival time, processing time, tardiness penalty, and preferred and latest acceptable due dates. While they allow preemption, they assume that the entire order has to be sent to a customer in one shipment, i.e., pieces of an order that are processed at different times incur a holding cost until they are shipped. The shipment date of an accepted order has to be between the arrival time and the latest acceptable due date. Orders that are completed after the prefered due date incur a tardiness penalty. The processing and holding costs are allowed to vary from period to period. The goal is to decide how much of

21 Due Date Management Policies 21 each order to produce in each period to maximize proþt (revenueminus holding, tardiness and production costs). Charnsirisakskul et al. consider both make-to-order (MTO) and maketo-stock (MTS) environments. In the Þrst case, the processing of an order cannot start before the order s arrival (release) time while in the latter case it is possible to process an order and hold in inventory for later shipment. For both cases, they model the problem as a mixed integer linear program and study the beneþts of lead time and partial fulþllment ßexibility via a numerical study. Lead time ßexibility refers to a longer lead time allowance (higher difference between the latest and preferred due dates) and partial fulþllment ßexibility refers to the option of Þlling only a fraction of the ordered quantity. Their results show that lead time ßexibility leads to higher proþts, and the beneþts of lead time ßexibility outweigh the beneþts of partial fulþllment ßexibility in both systems. Lead time ßexibility is more useful in MTO where early production is not an option. Numerical results also show that the beneþts of lead time and partial fulþllment ßexibility depend on the attributes of 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 s arrival time. The arrival times are also not known in advance. Such models 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 divided into two categories based on whether they study the problem in a single machine [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]. Although a single vs. multi-machine categorization seems natural at Þrst, most of the research issues are common to both settings, and the resulting insights are often times similar. Therefore, we discuss the papers in this section by categorizing them based on their approach to due-date setting decisions and based on some of the other modeling dimensions and the related research questions. For online DDM problems, very few researchers have proposed mathematical models (see Section 4.3). The most common approach for setting due dates is to use dispatch due date rules which follow the general form d j = r j + f j where f j is the ßow allowance. The tightness of the ßow allowances (and the due dates) is usually controlled by some parame-