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Order selection and scheduling with leadtime flexibilityKASARIN CHARNSIRISAKSKUL a , PAUL M. GRIFFIN a & PINAR KESKINOCAK aa School of Industrial and Systems Engineering, Georgia Institute of Technology , Atlanta,GA, 30332-0205, USA E-mail:Published online: 17 Aug 2010.

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IIE Transactions (2004) 36, 697–707Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408170490447366

Order selection and scheduling with leadtime flexibility

KASARIN CHARNSIRISAKSKUL, PAUL M. GRIFFIN, and PINAR KESKINOCAK

School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USAE-mail: {kasarin, pgriffin, pinar}@isye.gatech.edu

Received August 2001 and accepted August 2003

In this paper, we study integrated order selection and scheduling decisions, where the manufacturer has the flexibility to chooseleadtimes. Our main goal is to provide a mechanism for coordinating order selection, leadtime and scheduling decisions and todetermine under what conditions leadtime flexibility is most useful for increasing the manufacturer’s profits. Through numericalanalyses, we compare and contrast the benefits of leadtime flexibility and the flexibility to partially fulfill orders in different demandenvironments defined by the congestion level (or demand load), seasonality of the demand and order size. We consider both the caseswhere the manufacturer has and does not have the flexibility to produce orders early before they are committed.

1. Introduction

As business emphasis has moved from a cost focus to a valuefocus, companies can no longer solely compete on price, butmust also provide value to their customers through other“quality features.” Short and reliable leadtimes providevalue by helping customers reduce uncertainty in their busi-nesses, leading to lower inventory and more accurate pro-duction/distribution plans (Sheridan, 1999; Teresko, 2000;Rodin, 2001).

Guaranteeing leadtimes by promising to deliver productsby agreed due dates and giving discounts to the customersfor missed due dates has become a competitive leveragefor companies to attract customers. Examples are Triple Xcomposites, Tracewell Systems Inc. and Austin TrumannsSteel, who offer 15, 5 and 10% discounts, respectively, totheir customers as a percentage of the value of the prod-ucts or materials not delivered by the due dates. The dis-counts given to the customers can be thought of as tar-diness penalties. For example, in the aerospace industry, atardiness penalty as high as $1000 000 million per day is im-posed on subcontractors of aircraft components (Slotnickand Sobel, 2001). Similarly, in the Finnish forest industry,a vendor pays the buyer a delay penalty amounting to 1%of the purchase price for each week, up to a maximum of10%, plus additional damages arising from the delay.

Leadtime/due-date decisions depend on several factors,such as the manufacturer’s capacity and the customers’ de-mands and due-date preferences. In fulfilling customer or-ders, the manufacturer can benefit from various types offlexibility, including the flexibility to: (i) complete ordersafter the specified due dates (leadtime flexibility); (ii) pro-duce early before orders arrive or are committed (inventoryholding flexibility); and (iii) fulfill only part of the ordered

quantity (partial fulfillment flexibility). Leadtime decisionsmay also depend on the demand environment, such as de-mand seasonality and system congestion.

Intuitively, higher flexibility leads to higher profits. Forexample, leadtime flexibility (potentially) enables the man-ufacturer to increase profits by accepting more orders orchoosing more profitable orders that could not be com-pleted otherwise. Similarly, the flexibility to produce earlyand hold inventory provides the manufacturer with the abil-ity to better utilize her capacity and provide customers withshorter leadtimes. With leadtime or inventory holding flex-ibility, the manufacturer must tradeoff the benefits of com-pleting orders late or early and the associated costs, such astardiness and holding costs.

In this paper, we model simultaneous order selection,leadtime and scheduling decisions when each customer hasa preferred and latest acceptable due date. The manufac-turer incurs tardiness penalties for orders that cannot becompleted before the preferred due dates. In addition, weconsider the case where the manufacturer has the flexibilityto produce early and hold inventory. Through a numericalstudy, we develop useful insights regarding the benefit ofleadtime flexibility in different demand environments. Weshow that the magnitude of the benefit of leadtime flex-ibility (in terms of increasing profits) depends on variousfactors, such as demand load and seasonality, order size andthe flexibility to hold inventory. Understanding the benefitof leadtime flexibility in different demand and productionenvironments is important for a manufacturer in designingeffective due date setting/negotiation policies.

The organization of this paper is as follows. We discussthe related literature in leadtime and scheduling decisionsin Section 2. We define the problem and present a modeland a solution approach in Section 3. In Section 4, we

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present a numerical study and discuss insights drawn fromthe study. Finally, Section 5 provides conclusions andfuture research directions.

2. Literature review

The previous research that is most closely related to oursfalls into three categories: (i) scheduling; (ii) due-date man-agement; and (iii) manufacturing flexibility.

Most research in scheduling either ignores due dates orassumes that due dates are set a-priori and are an inputto the problem. For example, marketing often quotes duedates at the time of order acceptance, and manufacturingattempts to satisfy those due dates according to certain per-formance measures such as minimizing makespan or tardi-ness. Our research is related to the scheduling literature thatconsiders time windows (release times and due dates). Ex-amples include Potts and Van Wassenhove (1985, 1991),Abdul-Razaq et al. (1990), Holsenback and Russell (1992),Russell and Holsenback (1997) and Crauwels et al. (1998)who study the case where all jobs have common releasedates; and Rinnooy Kan (1976), Hariri and Potts (1983),Chu (1992) and Akturk and Ozdemir (2000), who studythe case where jobs may have different release dates. A sur-vey of the scheduling research can be found in Cheng andGupta (1989) and Koulamas (1994).

In contrast to the traditional scheduling literature,due-date management literature studies situations wheredue dates are set endogenously. In this case, the goal is tofind a due-date management policy, which is a combinationof due-date setting and scheduling policies. The objectiveis usually to minimize the total or average cost, whichmay be a function of earliness, tardiness, or the number ofearly/tardy orders (Eilon and Chowdhury, 1976; Weeks,1979; Conway, 1981; Seidmann and Smith, 1981; Bertrand,1983; Cheng, 1984), or to minimize the average weightedquoted due date (or leadtime) subject to a service levelconstraint on the proportion of jobs completed on-timeor on the average job tardiness (Baker and Betrand, 1981,1982; Bookbinder and Noor, 1985; Wein, 1991; Spearmanand Zhang, 1999). See Keskinocak and Tayur (2003) for areview of due date management literature.

Most of the research in scheduling and due-date man-agement ignores the impact of completion times or quoteddue dates on the customers’ decisions to place the orders.In particular, most of the scheduling literature assumesthat all orders must be processed and the due-date man-agement literature assumes that the customers accept thequoted due dates no matter how late they are. Among theliterature in scheduling and due-date management, our re-search is most closely associated with the work that consid-ers order acceptance decisions. Scheduling literature thatconsiders order acceptance decisions includes Arkin andSilverberg (1987), Liao (1992), Wester et al. (1992), Hall andMagazine (1994), Woeginger (1994), Keskinocak et al.(2001), Chuzhoy and Ostrovsky (2002) and Snoek (2002).

In these papers, the objective is to maximize the total profitfrom accepted orders subject to satisfying customer speci-fied deadlines (latest acceptable due dates). Most of thesepapers (except Keskinocak et al. (2001)) assume that cus-tomers are indifferent as to when an order is completed(i.e., due-date indifferent) as long as it is within the speci-fied deadline.

Papers in the due-date management literature that con-sider order acceptance decisions include Duenyas (1995)and Duenyas and Hopp (1995), where the probability of acustomer placing an order is a decreasing function of thequoted due date. The objective is to maximize the expectedlong-term average profit.

Most of the due-date management and scheduling re-search assumes that there is no limit on how late an ordercan be, whereas most of the research on order acceptanceassumes “tight” time windows (i.e., an order is lost if itis not processed immediately at its arrival). The reality issomewhere in between these two extreme cases of unlimitedversus no tolerance for waiting. In many manufacturing en-vironments an increase in the waiting time usually decreasessatisfaction, resulting in a decline in the revenue or an in-crease in cost (e.g., tardiness penalty). Our models capturethis practical situation by differentiating between preferredand acceptable due dates. We define the preferred due date asthe time period after which the customer’s satisfaction de-clines and the latest acceptable due of date as the date afterwhich the customer does not accept the shipment. A tardi-ness penalty is incurred if an order is completed after thepreferred due date. Additionally, all of the papers discussedabove assume that the release time and arrival time of eachorder are the same, i.e., orders cannot be produced earlyand held in inventory. In this research, we also consider thecase where early production is allowed.

The focus of most due-date related research is on findinggood due-date management policies. In contrast, our focusis on understanding when the flexibility to complete theorders after the customer specified due dates is useful. Weuse an off-line model to study the impact of this leadtimeflexibility. Clearly, using an on-line model would be morerealistic. However, since an on-line model assumes that theinformation about an order is not available until the timeof its arrival, the performance of different systems woulddepend upon the heuristic due-date management policies.In the absence of the optimal policies, we would not be ableto have a fair comparison of those systems.

Previous research on flexibility has focused on studyingthe flexibility of a manufacturer to adjust to the dynamicmarket environment, including volume, variety, process andmaterials handling flexibility (Voudouris, 1996; Sabri andBeamon, 2000; Goetschalckx et al., 2001; Duclos et al.,2002). The objective is usually to maximize system flex-ibility or profit, or minimize costs while retaining a cer-tain flexibility level. Another area of research on flexibilityfocuses on optimal investment in a flexible production re-source that is capable of adjustments to multiple systemstates (Fine and Freund, 1990; Van Mieghem, 1998; Bish

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Order selection and scheduling with leadtime flexibility 699

and Wang, 2002). While a considerable amount of researchhas been done on designing flexible manufacturing systems(i.e., supply-side flexibility), our focus is on studying flex-ibility based on demand-side factors such as customers’preferences on leadtimes.

3. The model

3.1. Problem definition

We consider a single-machine production system (e.g., aproduction line with a bottleneck and no buffers betweenstations) capable of producing multiple products, wheresetup times are negligible. The manufacturer is a price taker,who accepts the selling prices as given by the market. Cus-tomer orders differ in their arrival (or commitment) times,demanded quantities (expressed in the amount of produc-tion resources required), preferred and latest acceptable duedates, unit production, holding and tardiness costs, and thepotential revenues generated per unit resource required.

The manufacturer has the option to accept or reject or-ders. She decides which orders to accept and when to pro-duce the orders, which in turn affects the due dates. Thecustomer does not place an order if the manufacturer can-not complete the order by the latest acceptable due date.Thus, an accepted order must be completed and shipped tothe customer between the commitment time and the latestacceptable due date specified by the customer. If an order isshipped after the preferred due date, it is considered late andincurs a tardiness penalty proportional to the number of pe-riods and the quantity delayed. The tardiness penalty mayinclude a discount offered to the customer, a penalty due toloss of goodwill and the potential loss of future business.

We allow order preemption (i.e., different parts of anorder can be produced at different times), however, we re-quire that the entire order be shipped at the same time.Hence, partially finished orders (work-in-process inven-tory) incur holding costs until they are completed. Therestriction on shipping an order only after it is entirelycompleted is reasonable, for example, when the customeris responsible for the delivery cost and prefers to pay for asingle truckload shipment rather than multiple less-than-truckload shipments.

The manufacturer’s objective is to maximize the totalprofit subject to capacity and delivery time constraints. Thetotal profit is the sum of revenues received from all acceptedorders minus the production, holding and tardiness costs.

Order information represents a forecast of the future ar-rivals of orders, which may include products with knowncharacteristics or products whose characteristics are notknown in advance but whose prices and capacity require-ments are predictable. An order cannot be shipped to thecustomer before it is committed. In the case where prod-uct characteristics are predictable, the manufacturer has theflexibility to produce an order in advance and hold inven-tory until the time the customer commits to the order. In

this case, production of an order can start as early as atthe beginning of the planning horizon. The earliest possi-ble shipping time of the order is equal to the commitmenttime. In the case where product characteristics are not pre-dictable, the manufacturer does not have the flexibility toproduce early, i.e., she does not have the flexibility to holdinventory. In this case, the earliest start time for the or-der is at its commitment time. An example of this type ofproduction is in a printing business. The manufacturer canforecast the demand and production capacity requirementsfor different types of jobs, such as brochures, catalogs, orcalendars. However, the manufacturer does not know theproduct details, i.e., job specifications, until they are actu-ally committed. We consider both of these cases.

We model the problem of order acceptance, due-date set-ting and scheduling, in the presence of preferred and lat-est acceptable due dates, order preemption and aggregatedshipments as a mixed integer linear program. This problemis NP-hard; the proof of the problem’s complexity and spe-cial cases that are solvable in polynomial time are presentedin Charnsirisakskul (2003).

3.2. Notation

Sets:T = set of time periods in the planning horizon;O = set of customers/orders;O(t) = set of orders with latest acceptable due date in

or after period t .Parameters:di = order size, defined in units of capacity required

for order i;pi = unit price (or revenue per unit) of order i;ai = unit tardiness penalty per period of order i;ei, fi, li = commitment time, preferred due date, and lat-

est acceptable due date of order i;hi

t = unit holding cost of order i from period t toperiod t + 1;

hit,k = cumulative holding cost per unit of order i that

is produced in period t and delivered in periodk(=∑k−1

j=1 hij);

cit = unit production cost of order i in period t ;

Kt = units of production capacity available inperiod t .

Unit costs and quantity are measured per unit of productioncapacity.

Order i has a commitment time at the beginning ofperiod ei, requires di units of capacity, and yields a revenueof pi dollars per unit capacity. An accepted order mustbe shipped between the order commitment time andlatest acceptable due date. An order shipped within thepreferred time window [ei, fi] is considered on-time, whilean order shipped within the leadtime window [fi + 1, li]is considered late and incurs a tardiness penalty of ai perunit per period delay.

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3.3. Mixed-integer programming formulation

Decision variables:xi

tk = units of capacity in period t used to produce orderi that is delivered to the customer at the end of period k(t = 1, . . . , k, k = ei, . . . , li, i ∈ O);

Iik =

1, if order i is at the end of period k,(k = ei, . . . , li, i ∈ O).

0, otherwise.

Objective function:Maximize total profit

=∑i∈O

pi

li∑k=ei

k∑t=1

xitk −

∑i∈O

li∑k=ei

k∑t=1

cit xi

tk

−∑i∈O

ai

li∑k=fi+1

(k − fi)k∑

t=1

xitk −

∑i∈O

li∑k=ei

k−1∑t=1

hit,kxi

tk, (1)

Constraints:Demand constraints:

li∑k=ei

I ik ≤ 1 i ∈ O, (2)

k∑t=1

xitk = diIi

k i ∈ O, k ∈ [ei, li]. (3)

Capacity constraints:

∑i∈O(t)

li∑k=max{ei,t}

xitk ≤ Kt t ∈ T . (4)

Variable constraints:

xitk ≥ 0 ∀t ∈ T , k ∈ [ei, li], i ∈ O, (5)

Iik ∈ {0, 1} k ∈ [ei, li], i ∈ O. (6)

The four components of the objective function cor-respond to the total revenue, production cost, tardinesspenalty and holding cost, respectively. Constraints (2) en-sure that if an order is accepted, it is aggregated into a singleshipment within the time window specified by the customer.Constraints (3) ensure that an accepted order is fulfilledentirely. Constraints (4) ensure that the capacity in eachperiod is not exceeded. Constraints (5) are non-negativityconstraints and constraints (6) specify the binary variables.

The model can be modified to incorporate the releasetime of each order by setting the holding costs hi

t to in-finity, or equivalently the production variables xi

tk to zero,for all periods t before the release time. Note that whenearly production is not allowed, as in the case with no in-ventory flexibility, the release time of each order is equalto the commitment time. To incorporate the case with par-tial fulfillment flexibility, the model can be easily modifiedby changing the equality in the demand constraints (3) toinequality (≤).

Although in the numerical study we require an entireorder to be shipped at the same time, the above model canbe easily modified to accommodate the case where disag-gregation is allowed as follows. The variables xi

tk and Iik are

replaced by xit (non-negative) and Ii (binary), where xi

t isthe amount of capacity used to produce order i in period tand Ii is the binary order acceptance variable, equal to oneif order i is accepted and zero otherwise. Assuming that thequantity produced (in units of capacity consumed) beforethe commitment time ei is delivered in ei and the quantityproduced in [ei,li] is delivered immediately after its comple-tion in each period, the objective function (1) is replaced by∑

i∈O pi∑li

t=1 xit − ∑

i∈O∑li

t=1 cit xi

t − ∑i∈O ai

∑lit=fi+1 (t −

fi)xit − ∑

i∈O∑ei−1

t=1 hit,ei−1xi

t . Constraints (2) are removed.Constraints (3) are replaced by

∑lit=1 xi

t = diIi, ∀i ∈ O.

The term∑li

k=max{ei,t} xitk in constraints (4) is replaced by

xit .

4. A numerical study

A manufacturer’s order acceptance, scheduling and due-date decisions and profitability depend on the flexibilityshe has in fulfilling orders, which comes partly from ne-gotiations with the customers. In a typical manufacturingenvironment, the negotiation process usually takes placebefore orders are accepted and the production plan is exe-cuted. In such cases, analyzing the benefit of different lead-time flexibility levels based on currently available demandinformation or forecasts can provide insights to the man-ufacturer regarding the level of flexibility to negotiate forfuture orders. In reality, offering longer leadtimes (throughhigher leadtime flexibility) may result in additional costs tothe manufacturer, such as the loss of goodwill or a decreasein the probability of the customer placing an order. Suchcosts can be incorporated into the analysis by consideringa fixed cost of leadtime flexibility.

In order to develop an efficient negotiation policy, themanufacturer must have a thorough understanding of thebenefits of different types of flexibility that she could nego-tiate. The objective of this numerical study is to develop in-sights that serve as guidelines for manufacturers in due-datenegotiations. We compare the benefits of leadtime flexibilityand partial fulfillment flexibility in different environmentsto answer questions such as: In which environments is theflexibility important? How much or what type of flexibilityis required to significantly benefit the manufacturer?

4.1. Experimental design

The manufacturing environment can be described byseveral factors such as demand load, order size andseasonality of the demand. In order to test how the benefitof flexibility depends on these conditions, we simulatedifferent demand environments defined by the abovefactors. We use a 90-period forecast horizon. Note thatthe production period can assume any time unit. In our

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Order selection and scheduling with leadtime flexibility 701

numerical study, a production period represents 4 days andthe 90-period forecast horizon is approximately 1 year. Forsimplicity, we assume that the manufacturer has one unitof capacity per period (since the planning period can bediscretized and other parameters scaled accordingly, thereis no loss of generality).

Demand load dictates how congested the system is and isdefined as the expected ratio of the total requested amountof production capacity over the total available capacity dur-ing the forecast horizon. Four levels of demand load areincluded in the study: low (0.75), moderate (1.0), high (1.5)and extremely high (2.0). The total number of order arrivalsgenerated during the forecast horizon is equal to:

(demand load) × (the length of the forecast horizon)(the average order size)

.

To test the impact of order size on the benefit of flexibility,we used two distributions to model small (di ∼ uniform[2,6]) and large order sizes (di ∼ uniform[8, 12]). Four differentdistributions of commitment times are tested: uniform (U:ei ∼ uniform[1, 90]), high number of order commitmentsnear the beginning (LT: ei ∼ triangular[1, 20, 90]), the mid-dle (MT: ei ∼ triangular[1, 45, 90]) and the end (RT: ei ∼triangular[1, 70, 90]) of the forecast horizon. The processingtime requirement for each order (pti) is equal to the ordersize (in units of capacity required) divided by the produc-tion capacity per period. Since we assume unit capacity perperiod, the processing time requirement for each order isequal to the order size (i.e., pti = di).

The length of the planning horizon is set to the max-imum value of the latest acceptable due date (i.e., |T | =maxi{li}). To illustrate the benefit of leadtime flexibility ex-cluding the effect of leadtime preferences among differentorders/customers, we assume that the preferred due date ofeach order is equal to the commitment time plus the pro-cessing time (fi = ei + pti). The latest acceptable due date(li = fi + R × pti) is equal to the preferred due date plusthe leadtime window, which is proportional to the leadtimeflexibility factor (R) and the processing time (pti). In eachexperiment, we assume that R is the same for all orders.

Unit price (in dollars) of each order is generated from auniform[1000, 2000] distribution. Unit holding and tardi-ness costs are 0.5 and 2% of the selling price per period,respectively. The unit production cost is zero and prices,unit costs and order sizes have integer values.

We simulated a complete combination of all factor levels(32 environments), each with nine levels of leadtime flexibil-ity: R ∈ {0, 0.3, 0.6, 1, 2, 3, 4, 5, 6}. For each scenario (envi-ronment and R), we generated five replications of probleminstances and solved the corresponding mixed integer pro-grams using CPLEX7.0. The time spent to solve an instance(Pentium III, 500 MHz) of the mixed-integer program in theexperiments varied from less than 1 second to more than15,000 seconds, with an average of 2,500 seconds. The solu-tion time tends to be shorter when order commitment timesare uniformly distributed than when the commitment times

Fig. 1. Average percentage profit increase over the base case(NOINV-NOPF with R = 0) for the combinations of inventoryand partial fulfillment flexibility at different levels of leadtimeflexibility.

exhibit seasonality, and tends to increase as the order size,the demand load and leadtime flexibility increase.

4.2. Analysis of numerical results

In this section we discuss the numerical results obtainedfrom the simulations.1

4.2.1. Preliminary analysisWe consider four different combinations of inventoryand partial fulfillment flexibility: (i) no inventory and nopartial fulfillment (NOINV-NOPF); (ii) inventory andno partial fulfillment (INV-NOPF); (iii) no inventorybut with partial fulfillment (NOINV-PF); and (iv) bothinventory and partial fulfillment (INV-PF). For each ofthese combinations, Fig. 1 shows the total average per-centage profit increase due to different levels of leadtimeflexibility, compared to the base case of NOINV-NOPFwith R = 0. From Fig. 1, for all combinations we observethat leadtime flexibility is useful, however, it exhibitsdiminishing marginal returns. The benefit of partialfulfillment flexibility is relatively high under NOINV forlow values of R, however, the benefits are smaller underINV or higher values of R.

An interesting question is which of these three types offlexibility, namely leadtime, partial fulfillment and inven-tory flexibility, is most useful. For any level of leadtime

1For detailed numerical and statistical test results, please contactany of the authors.

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Table 1. Profit and costs as percentages of potential revenue and rejected orders as a percentage of total orders for different levels ofleadtime flexibility, under NOPF

Leadtime flexibility (R)

0 0.3 0.6 1 2 3 4 5 6

INV%Profit 74.34 76.64 77.28 77.88 78.27 78.30 78.32 78.33 78.34%Holding cost 3.05 2.91 2.81 2.79 2.69 2.60 2.58 2.56 2.56%Tardiness cost 0.00 0.42 0.90 1.48 2.92 4.18 5.12 5.59 5.86%Lost revenue 22.61 20.03 19.01 17.85 16.12 14.92 13.98 13.52 13.21%Rejected orders 25.81 24.27 22.98 21.75 19.04 17.25 16.08 15.44 15.09

NOINV%Profit 52.04 58.66 61.70 64.04 66.43 67.31 67.78 67.90 67.90%Holding cost 1.54 1.53 1.52 1.50 1.51 1.51 1.51 1.51 1.51%Tardiness cost 0.00 1.31 2.72 4.24 7.88 10.05 11.13 11.79 12.22%Lost revenue 46.42 38.55 34.06 30.22 24.18 21.13 19.58 18.80 18.37%Rejected orders 49.47 41.87 37.52 33.93 27.54 23.96 22.61 21.73 21.02

flexibility, the profits under INV-NOPF are higher than theprofits under NOINV-PF. This suggests that if the manu-facturer has to choose between inventory and partial fulfill-ment flexibilities, she would prefer the former. In comparingthe benefits of partial fulfillment and leadtime flexibilities,the profit increase under NOINV-NOPF, R ≥ 0.6 is higherthan the profit increase under NOINV-PF, R = 0, suggest-ing that a leadtime flexibility of R ≥ 0.6 is more usefulthan partial fulfillment flexibility under NOINV (Fig. 1).Similarly, the profit increase under INV-NOPF, R ≥ 0.3 ishigher than the profit increase under INV-PF, R = 0, sug-gesting that a leadtime flexibility of R ≥ 0.3 is more usefulthan partial fulfillment flexibility under INV. These obser-vations suggest that even a small amount of leadtime flex-ibility is more useful than partial fulfillment flexibility. Incomparing the benefits of inventory and leadtime flexibil-ity, we see that the percentage profit increase due to inven-tory flexibility only (INV-NOPF, R = 0) is higher than theprofit increase due to the maximum tested leadtime flex-ibility (NOINV-NOPF, R = 6), suggesting that inventoryflexibility is more useful than leadtime flexibility. In sum-mary, the three types of flexibility are ranked as inventory,leadtime and partial fulfillment in decreasing order of theirusefulness.

Table 2. Average percentage profit increase due to leadtime flexibility over the base case. For each row, the base case is when there isno leadtime flexibility (R = 0) for that combination

Leadtime flexibility (R)

0.3 0.6 1 2 3 4 5 6

NOINV-NOPF 12.76 19.87 24.24 30.33 33.11 33.88 34.24 34.50INV-NOPF 1.80 3.15 4.04 6.09 6.95 7.34 7.54 7.66NOINV-PF 5.94 9.07 11.33 15.25 16.81 17.36 17.68 17.87INV-PF 1.59 2.73 3.56 5.24 5.99 6.31 6.52 6.61

To better understand the impact of leadtime flexibilityon profit, costs and accepted orders, in Table 1 we summa-rize the (average) decomposition of the potential revenue(the total revenue that would be achieved if all orders wereaccepted and completed by the preferred due dates) intoprofit, holding cost, tardiness cost and lost revenue at dif-ferent flexibility levels under NOPF. Lost revenue is thepotential revenue minus the actual revenue received. FromTable 1 we see that as leadtime flexibility increases, the per-centages of holding cost, lost revenue and rejected ordersdecrease, while the percentages of the profit and the tardi-ness cost increase. Intuitively, the increase in profits due tohigher leadtime flexibility is primarily due to the manufac-turer’s ability to accept more profitable orders. The grad-ual decrease in lost revenue, which parallels the decreasein the percentage of rejected orders, as leadtime flexibilityincreases supports this hypothesis. Although the decreasein lost revenue (equivalently the increase in the revenue re-ceived) is accompanied by an increase in tardiness cost, theincrease in tardiness cost is smaller than the correspondingincrease in revenue.

Next, we analyze the marginal benefit of: (i) leadtimeflexibility under each of the four combinations of inventoryand partial fulfillment flexibility (Table 2); and (ii) partial

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Table 3. Average percentage profit increase due to partial fulfillment flexibility for different levels of leadtime flexibility under NOINVand INV (NOINV-PF compared to NOINV-NOPF and under INV-PF compared to INV-NOPF)

Leadtime flexibility (R)

0 0.3 0.6 1 2 3 4 5 6

NOINV-PF over NOINV-NOPF 15.78 8.78 5.35 3.75 2.38 1.60 1.49 1.50 1.40INV-PF over INV-NOPF 1.24 1.03 0.83 0.77 0.43 0.33 0.27 0.28 0.25

fulfillment flexibility under INV and NOINV for variouslevels of leadtime flexibility (Table 3).

4.2.1.1. Leadtime flexibility. Table 2 shows the (average)percentage profit increase for different R values comparedto R = 0 for each of the four combinations of inventoryand partial fulfillment flexibilities. For instance, the valueof 4.04 under INV-NOPF means that there was a 4.04%increase in profits by having a leadtime flexibility of R = 1compared to R = 0 in INV-NOPF. Since this value is amarginal benefit, it is over and above the percentage profitincrease due to inventory flexibility alone.

The analysis of the results in Table 2 show that the ad-ditional benefit due to leadtime flexibility is statisticallysignificant2 even when both inventory and partial fulfill-ment flexibilities are available. The additional benefit ofleadtime flexibility is even more pronounced when eitherinventory or partial fulfillment flexibility or both are notavailable. Pairwise t-tests also show that the (percentage)profit difference between INV and NOINV decreases sig-nificantly as leadtime flexibility (R) increases. These resultssuggest that leadtime flexibility and inventory holding flex-ibility are partial substitutes.

4.2.1.2. Partial fulfillment flexibility. Table 3 shows themarginal benefit of partial fulfillment flexibility. In this ta-ble, the average percentage profit increase of NOINV-PFover NOINV-NOPF and INV-PF over INV-NOPF is givenfor all values of R. The additional benefit due to partialfulfillment flexibility is statistically significant for all casesalthough there are diminishing marginal returns as R in-creases. Partial fulfillment flexibility allows a manufacturerto satisfy a large fraction of an otherwise rejected order.Since inventory flexibility allows the manufacturer to pro-duce orders before they are committed and, thus, acceptmore orders (entirely), we see a higher benefit of partial ful-fillment flexibility under NOINV-PF. As can be seen fromTable 3 even the smallest increase of 1.40% for NOINV-PFover NOINV-NOPF (R = 6) is greater than the largest in-crease of 1.24% for INV-PF over INV-NOPF(R = 0).

In the following sections, we discuss how the benefits ofleadtime and partial fulfillment flexibility depend on differ-ent environmental factors. Note that in order to distinguish

2For detailed t-test results, please contact any of the authors.

between the effects of the factors on the benefits of leadtimeflexibility and partial fulfillment flexibility, we do not con-sider the cases where both types of flexibility are availablesimultaneously.

4.2.2. The impact of environmental factors on the benefitof leadtime flexibility

In this section we consider leadtime and partial fulfillmentflexibilities separately to analyze the impact of the environ-mental factors. To test statistically the significance of theeffect of each factor, we performed an Analysis Of Vari-ance (ANOVA) using a 10% significance level. We refer tothe effects of order commitment time distribution, demandload and order size as the main factor effects and refer to thecombined effects between two or more factors as the interac-tion effects. The percentage profit increase due to leadtimeflexibility under NOPF is summarized in Table 4 for variousdemand environments. ANOVA results show that all mainfactors have a significant impact on the benefit of leadtimeflexibility at all flexibility levels (R) in INV. For NOINV,the significance of different factors depends on the level ofleadtime flexibility. At a low flexibility level (R = 0.3), nomain effects are significant, but the combined (interaction)effect between order size and demand load is significant.At R = 0.6 and 1, order size is the only significant factor.At R = 2, order size and demand load are significant. AtR ≥ 3, order size and commitment time distribution aresignificant.

4.2.2.1. The impact of demand load. When the demand loadis high, a manufacturer with insufficient production capac-ity to accept and complete all orders on-time must rejectseveral orders. As leadtime flexibility increases, the man-ufacturer has more flexibility to adjust the due dates andaccept more orders. Thus, we expect leadtime flexibility tobe more useful when demand load is high. Our observationsin the case of INV confirm this intuition. From Table 4, weobserve that the benefit of leadtime flexibility (at any flexi-bility level) increases as demand load increases. The highestprofit increase is achieved at extremely high load (2.0) exceptwhen R = 0.6, where the benefit is highest at high load (1.5).We also observe that under INV the diminishing marginalreturns of leadtime flexibility are more pronounced at lowerdemand loads.

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704 Charnsirisakskul et al.

Table 4. Percentage profit increase from the case with no leadtime flexibility (R = 0), due to different levels of leadtime flexibilityunder NOPF

Leadtime flexibility (R)

Factor Factor level 0.3 0.6 1 2 3 4 5 6

INVDemand load Low 0.40 1.00 1.03 1.11 1.11 1.11 1.11 1.11

Moderate 1.83 3.40 4.79 6.71 6.98 7.04 7.07 7.10High 2.44 4.15 5.16 8.23 9.84 10.48 10.82 11.01Ext. high 2.53 4.06 5.17 8.30 9.86 10.74 11.16 11.41

Commitment time U 1.24 2.22 3.24 4.61 5.16 5.38 5.51 5.58LT 2.54 4.05 4.99 7.57 8.51 8.89 9.05 9.15MT 1.82 3.38 4.00 6.50 7.43 7.87 8.11 8.24RT 1.60 2.97 3.91 5.67 6.69 7.24 7.49 7.65

Order size Small 1.29 2.34 3.02 4.78 5.76 6.37 6.74 6.96Large 2.31 3.96 5.05 7.39 8.13 8.32 8.34 8.35

NOINVDemand load Low 13.52 20.33 26.10 32.83 35.25 35.43 35.52 35.61

Moderate 11.78 19.80 25.25 31.91 34.90 35.48 35.76 35.92High 13.57 20.51 23.98 29.88 32.98 34.12 34.64 34.98Ext. high 12.17 18.84 21.62 26.71 29.30 30.48 31.04 31.50

Commitment time U 11.53 19.97 23.13 28.08 29.61 30.16 30.32 30.51LT 14.09 19.82 24.48 29.98 32.63 33.27 33.57 33.75MT 12.48 19.93 23.95 30.75 33.93 34.64 35.05 35.33RT 12.94 19.76 25.38 32.52 36.27 37.45 38.02 38.42

Order size Small 12.78 18.40 21.14 26.06 28.47 29.72 30.41 30.93Large 12.74 21.35 27.34 34.60 37.75 38.04 38.07 38.07

Contrary to our intuition, for suffciently high leadtimeflexibility (R ≥ 1), the highest benefit under NOINV isachieved when the demand load is either low or moder-ate. (However, the effect of the demand load on the benefitof leadtime flexibility is not statistically significant exceptat R = 2.) One possible explanation for this observation isthat under INV, when the demand load is low to moder-ate, the manufacturer can complete most of the orders byholding inventory, i.e., the additional benefit due to lead-time flexibility is relatively small. However, under NOINV,the manufacturer can significantly benefit from leadtimeflexibility even for a low to moderate demand load. For ahigh demand load, since the manufacturer already uses asignificant portion of its capacity and captures most of thedemand, the benefit of leadtime flexibility may not be ashigh as in the case of low or moderate demand loads.

4.2.2.2. The impact of commitment time distribution(demand seasonality). We observe from Table 4 that thebenefit of leadtime flexibility is smallest when order com-mitment times are uniformly distributed throughout theforecast horizon both in the cases of INV and NOINV (atany flexibility level, except for R = 0.6 in NOINV). Intu-itively, when order commitment times are evenly distributedover the horizon, the manufacturer can complete an orderthat is being processed before a new attractive order is com-mitted. Thus, having a higher leadtime flexibility does notadd a significant benefit in terms of accepting more prof-

itable orders. On the other hand, in a highly seasonal de-mand situation most of the profitable orders are committedduring the same periods, making it harder for the manufac-turer to accept and complete many orders without leadtimeflexibility. In such cases, having a higher leadtime flexibilityallows the manufacturer to satisfy more orders and achievehigher profits.

Under INV, the benefit of leadtime flexibility (at any flex-ibility level) is highest when commitment times of most or-ders are near the beginning of the forecast horizon (LT),followed by high number of commitments near the middle(MT) and the end of the horizon (RT). This observationcan be explained as follows. Under RT, the manufacturercan produce several orders during earlier periods where thecongestion is low and hold inventory until the orders arecommitted. Therefore, the additional benefit of leadtimeflexibility is relatively small. Under LT, however, since thecongestion is towards the beginning of the planning hori-zon, the flexibility to produce early is not very beneficialand therefore the additional benefit of leadtime flexibilityis relatively high. Under NOINV, for R ≥ 1, the pattern isthe reverse of what we observe under INV, namely, the ben-efit of leadtime flexibility is highest under RT, followed byMT and LT.

To understand the impact of the commitment time dis-tribution on the benefit of leadtime flexibility under INVand NOINV, we compute for each R the percentage differ-ence between the largest and the smallest profit increase due

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Order selection and scheduling with leadtime flexibility 705

to leadtime flexibility across all commitment time patterns.For example, under INV, R = 0, the percentage differenceis:

2.54 − 1.241.24

× 100 = 104.8%.

These percentage differences are 104.8, 82.4, 54.0, 64.2,64.9, 65.2, 64.2 and 65.9% under INV and 22.2, 1.1, 9.7,15.8, 22.4, 24.2, 25.4 and 25.9% under NOINV, for R = 0.3,0.6, 1, 2, 3, 4, 5 and 6 respectively. A pairwise comparisonof these percentages for each R value shows that the dif-ference between the maximum and the minimum benefit ishigher under INV than NOINV. Hence, our results suggestthat the impact of the commitment time distribution onthe benefit of leadtime flexibility is higher under INV thanNOINV.

4.2.2.3. The impact of order size. Intuitively, we would ex-pect the number of rejected orders to be higher when or-ders are large, since large orders require more time periodsto complete (provided that the demand load is adequatelyhigh such that some orders are rejected). Accordingly, wewould expect leadtime flexibility to be more useful in theenvironments with large orders. From the numerical re-sults (Table 4), we observe that this intuition holds underboth INV and NOINV, except for the low flexibility level(R = 0.3) in NOINV. In the R = 0.3 case, however, the re-sult is not statistically significant.

4.2.3. The impact of environmental factors on the benefitof partial fulfillment flexibility

In this section, we discuss how the benefit of partial fulfill-ment depends on the environmental factors for the cases ofINV and NOINV. Table 5 summarizes the percentage profitincrease due to partial fulfillment flexibility under INV andNOINV when R = 0.

4.2.3.1. The impact of demand load. We observe that thebenefit of partial fulfillment flexibility depends on the de-

Table 5. Percentage profit increase for INV-PF over INV-NOPFand NOINV-PF over NOINV-NOPF when there is no leadtimeflexibility (R = 0)

INV-PF over NOINV-PF overFactor Factor level INV-NOPF NOINV-NOPF

Demand load Low 0.37 14.93Moderate 1.57 14.43High 1.49 17.24Ext. high 1.52 16.50

Commitment U 0.95 14.14time LT 1.91 16.26

MT 1.12 15.74RT 0.96 16.96

Order size Small 0.34 11.64Large 2.13 19.91

mand load as follows. For INV, the benefit of partial ful-fillment is significant in all environments with moderate toextremely high load. When the demand load is low, partialfulfillment is only useful when orders are large and com-mitment time distribution is LT. For NOINV, the benefit ofpartial fulfillment is significant at all load levels, althoughthe benefit under low and moderate demand loads is signifi-cantly smaller than under high and extremely high demandloads. As expected, for both INV and NOINV, partial ful-fillment flexibility is more useful for higher levels of demandload, although clearly as demand load gets high enough, thebenefit of partial fulfillment will not increase further sinceall capacity will be utilized.

4.2.3.2. The impact of commitment time distribution. Un-der INV, the benefit of partial fulfillment is highest on av-erage when the order commitment time distribution is LT,followed by MT, RT and uniform. Note that this is verysimilar to what we observed in Section 4.2.2 for the benefitof leadtime flexibility. Intuitively, when most orders haveearly commitment times, inventory flexibility is not verybeneficial and, hence, the benefit of leadtime or partial ful-fillment flexibility is more pronounced. Under NOINV, thebenefit of partial fulfillment flexibility is highest when thecommitment time distribution is RT, followed by LT, MTand uniform.

4.2.3.3. The impact of order size. On average, the benefit ofpartial fulfillment is highest when orders are large underboth INV and NONV. In congested systems, it is harderto accept and fully complete large orders than to completesmall orders fully. Thus, partial fulfillment flexibility is es-pecially beneficial when orders are large.

5. Conclusions

In this paper, we consider simultaneous order selection,scheduling, and due date decisions and provide a frame-work for analyzing the benefits of leadtime and partial ful-fillment flexibility. Through numerical analyses, we developinsights on how leadtime flexibility benefits the manufac-turer in various demand and production environments.

Our results suggest that leadtime flexibility is benefi-cial and more so than partial fulfillment flexibility. This istrue both when the manufacturer has (INV) and does nothave (NOINV) inventory holding flexibility. The benefit ofhigher leadtime flexibility is higher in NOINV comparedto INV although it exhibits diminishing marginal returnsunder both cases. While the benefit in NOINV is purely at-tributed to the capability to accept more profitable orders,the benefit in INV may also be attributed to the tradeoffsbetween the options to produce orders late or early.

The magnitude of the benefit of leadtime flexibility de-pends on several factors, including demand load, seasonal-ity of demand and order size. The impact of these factors

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becomes consistent when leadtime flexibility reaches a mod-erate level and can be summarized as follows. In INV, lead-time flexibility is more useful when the demand load is high.On the contrary, in NOINV the benefit is higher when theload is low or moderate. In both cases, leadtime flexibilityis more useful when order commitment times exhibit someseasonality. In INV, the benefit increases as the periods ofhigh order commitments are closer to the beginning of theforecast horizon. The converse is true for NOINV. For bothcases, leadtime flexibility is more useful when orders arelarge.

In our model, demand is deterministic, setup costs andsetup times are negligible and order preemption is allowed.Future research directions include alternate models that re-lax these assumptions, such as considering probabilistic de-mand, order setup times and no order preemptions. Whilethis paper, like most of the research in the literature, assumesthat prices are exogenous (i.e., the manufacturer takes pricesas given by the market), the case where the manufacturerhas some power to set prices and incorporate price deci-sions into production decisions is an interesting area toexplore. We have taken a first attempt in this direction inCharnsirisakskul et al. (2003).

Acknowledgements

Pinar Keskinocak is supported by NSF CAREER awardDMI-0093844. The authors are grateful to two anonymousreviewers for their helpful comments and would like to givespecial thanks to Professor S. Rajagopalan for his effortsduring the revision of this paper.

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Biographies

Kasarin Charnsirisakskul received her undergraduate degree in Indus-trial Engineering from Sirindhorn Institute of Technology, ThammasatUniversity, Thailand, in 1997. She received her M.S. and Ph.D. degrees inIndustrial Engineering from the School of Industrial and Systems Engi-neering at Georgia Institute of Technology in 1998 and 2003, respectively.

Paul M. Griffin is an Associate Professor in the School of Industrial andSystems Engineering at the Georgia Institute of Technology. He receivedhis Ph.D. in Industrial Engineering from Texas A&M University. Histeaching and research interests are in manufacturing systems, logisticssystems and economic decisions analysis.

Pınar Keskinocak is the Coca Cola Assistant Professor in the School ofIndustrial and Systems Engineering at Georgia Institute of Technology.Before joining Georgia Tech, she worked at IBM T.J. Watson ResearchCenter, Yorktown Heights, New York. She holds a Ph.D. degree in Oper-ations Research from Carnegie Mellon University. Her research focuseson supply chain management, with an emphasis on lead time and pric-ing decisions. She is also interested in the applications of optimizationtechniques in real world environments.

Contributed by the Scheduling/Production Planning/Capacity PlanningDepartment

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