on optimal expediting policy for supply systems with uncertain lead-times

On Optimal Expediting Policy for Supply Systemswith Uncertain Lead-Times

Panos KouvelisOlin Business School, Washington University in St. Louis, 1 Brookings Drive, St. Louis, Missouri 63130, USA, [email protected]

Sammi Y. TangDepartment of Management, School of Business, University of Miami, 5250 University Drive, Coral Gables, Florida 33146, USA,

[email protected]

W e examine the role of expediting in dealing with lead-time uncertainties associated with global supply chains of“functional products” (high volume, low demand uncertainty goods). In our developed stylized model, a retailer

sources from a supplier with uncertain lead-time to meet his stable and known demand, and the supply lead-time is com-posed of two random duration stages. At the completion time of the first stage, the retailer has the option to expedite aportion of the replenishment order via an alternative faster supply mode. We characterize the optimal expediting policyin terms of if and how much of the order to expedite and explore comparative statics on the optimal policy to betterunderstand the effects of changes in the cost parameters and lead-time properties. We also study how the expeditingoption affects the retailer’s decisions on the replenishment order (time and size of order placement). We observe that withthe expediting option the retailer places larger orders closer to the start of the selling season, thus having this option serveas a substitute for the safety lead-time and allowing him to take increased advantages of economies of scale. Finally weextend the basic model by looking at correlated lead-time stages and more than two random lead-time stages.

Key words: supply uncertainty; random lead-time; lead-time information; expediting; safety lead-timeHistory: Received: April 2010; Accepted: February 2011, after 1 revision.

1. Introduction

With products moving over longer distances andcrossing more national borders and inspection points,the global supply chains of today are longer and morecomplex than the localized supply chains of the past.As a result, supply side risks have become a greaterconcern—particularly those that affect supply lead-times. Operating in developing countries with low-cost labor, but inadequate supply and productioninfrastructures, increases the potential for disruptionsand unexpected delays. Exposed quality and yieldproblems due to lower worker skills and higheremployee turnovers result in elongated productionflow times. Furthermore, the transportation logisticscomponents of the global supply chains are prone toserious and variable duration delays due to congestedocean ports, not only at the sourcing origins, such asthe Shanghai port, but also at the destination ports,such as Long Beach, CA and Vancouver, Canada.Additionally, suppliers may not always see a need tooffer short lead times (e.g., due to the “safety stockeffect” studied in Kraiselburd et al. 2011).The sourcing of goods from faraway locations, most

recently from China and other Southeast Asia loca-

tions, are typically for higher volume stable demandproducts (“functional products” in the supply chainliterature language, see Fisher 1997) in search of lowermaterial and production costs. However, the effi-ciency of the flows in these longer chains, and con-sequently the “leanness” of them in terms ofinventories, strongly relies on the predictability ofthese elongated lead-times, which is far from the cur-rent state of affairs. Stalk (2006) warns supply chainmanagers of the hidden costs of elongated and vari-able lead-time supply chains (e.g., the costs for stock-outs, excess inventories and write-downs, over- andunder-productions, etc.), and explicitly advises themto consider expediting options for effectively manag-ing such chains. Such options, on the transportationlogistics side, might go beyond air freight to payingpremiums for preferred treatments from ground, sea,and air shippers, port services, and other suppliers.Crone (2006) reports on dynamic rerouting practicesbased on port congestion and other traffic bottlenecks.For example, a European food manufacturer suppliesthe North American market via shipments routingthrough the Montreal port instead of the heavilycongested East Coast ports. On the production sidetypical expediting options might involve processing

309

Vol. 21, No. 2, March–April 2012, pp. 309–330 DOI 10.1111/j.1937-5956.2011.01265.xISSN 1059-1478|EISSN 1937-5956|12|2102|0309 © 2011 Production and Operations Management Society

part of the order in an expedited way, possiblythrough extra shifts and overload pay. Throughoutthe article, our use of the term “expediting” impliesshortening of lead-time, in expectation, to reach thepoint of sale, and it covers any of the above-men-tioned production and logistics expediting alterna-tives. Other alternatives we view as expeditinginclude point-to-point shipping via bypassing inter-mediate consolidation and distribution points, andpreferential treatment services by supply and logisticsintermediaries—such as “priority processing” and“unloading your goods first”—at extra payment.From an implementation perspective, we are now

more than ever capable of employing sophisticatedexpediting options. Advances in information technol-ogy have greatly facilitated information flows in sup-ply chains. The emergence of new technologies suchas radio frequency identification (RFID) furtherenables the visibility of a certain item along a supplychain. By collecting data contained in small tagsattached on items, RFID efficiently tracks in real-timewhere and when these items (product, case, or pallet)are (e.g., Gardner 2004, Wolfe et al. 2003). These tech-nologies allow recording in detail the duration of thevarious stages in the product’s journey from source todestination, thus accounting for the major sources ofunpredictable variability in the product’s productionand logistics flow path. For example, readers may beinstalled at in- and out-bound docks, loading andunloading points, custom inspection stations, ships,trucks, planes, and so on. As products pass throughintermediate production and logistics points, thereaders can identify them and broadcast this informa-tion immediately.Motivated by our discussion above, in this study,

we aim to examine the use of an expediting serviceand its implications in the cost of meeting demand,the planning of safety lead-times, and order sizes for“functional goods.” To do so, we use a stylized single-product demand model with a constant demand ratewhere the lead-time of the replenishment orders isstochastic. The lead-time consists of two stages withrandom duration. At the completion time of the firststage, the firm can decide whether to expedite someor all of the replenishment order. The expeditingservice has a deterministic lead-time but is moreexpensive.Our research clearly outlines the nature of the opti-

mal expediting policy in terms of whether and howmuch to expedite: (i) when the expediting cost is toohigh (above a defined threshold in Proposition 1), it isoptimal not to employ expediting; (ii) when the expe-diting cost is low (below a defined threshold inProposition 2), it is optimal to fully exploit the expe-diting opportunity and use it aggressively; and (iii)when the expediting cost is intermediate (i.e.,

between the two defined thresholds), it is optimal toadopt the expediting service but only leverage it in alimited way. The optimal expediting policy formal-izes some of our intuition, but it also offers insightsthat could be perceived as surprising to our “raw”intuition. Specifically, it shows that when a replenish-ment order has longer delays, it is not necessarily truethat one would expedite a larger portion. The impactfor factors such as the cost and lead-time of the expe-diting option, the inventory holding cost, backordercost, and lead-time variability on the optimal expedit-ing policy is studied.The optimal expediting policy also allows us to

explore the impact of the expediting option on boththe magnitude of the safety lead-time and size of theregular replenishment order. Numerical examplesshow that the expediting option serves as a substituteof the safety lead-time and thus with the expeditingoption, the optimal safety lead-time is smaller thanthat without the expediting option. We observe thatfrom the perspective of the expected average cost, therobustness property of the traditional EOQ model ispreserved. We extend our basic model to considercorrelated lead-time stages and also more than tworandom lead-time stages and discuss issues such asthe best time to use the one-time expediting option. Inthe next section, we position the contribution of ourstudy to the current literature.

2. Literature Review

The early research on inventory models with stochas-tic lead-times mostly assumed no informationupdating on supply conditions, and hence treated thelead-time as a whole. While some researchers (Chopraet al. 2004, Ehrhardt 1984, Kaplan 1970, Song 1994)focused on studying the effects of uncertain lead-times on inventory policies in the presence of stochas-tic demand, others focused on determining theoptimal order size and ordering times when demandis deterministic (Chang 2004, Liberatore 1979). How-ever, the literature above did not consider contingentactions, such as expediting the order upon observingthe supply conditions. Our work allows the opportu-nity for use of expediting option to respond to real-ized completion time of the first stage, by modelinglead-times as consisting of two random durationsupply stages.The second stream of related research examines

inventory models with dual sourcing or deliverymodes. In these inventory planning situations thedual sourcing flexibility is ex ante accounted.Moinzadeh and Nahmias (1988) proposed anapproximate control policy for a system where thereare two replenishment modes with different deter-ministic lead-times. Ramasesh et al. (1991) analyzed

Kouvelis and Tang: Optimal Expediting Policy Under Uncertain Lead-Times310 Production and Operations Management 21(2), pp. 309–330, © 2011 Production and Operations Management Society

dual sourcing in the context of constant demand rateand stochastic order lead-time and compared its per-formance with that of the single sourcing. Ryu andLee (2003) determined the optimal lead-time reduc-tion proportion and the optimal order splittingbetween two sources. Our work differs from theabove literature, since it studies the ex post exercisingof the contingent action (i.e., the second source) toexpedite a portion of an early placed order (i.e., thefirst source).The third stream of related research examines

inventory models with the ex post use of emergencyordering in response to delivery status information ofearly placed orders. In a recent work, Gaukler et al.(2008) studied a replenishment policy based on theclassical continuous review (Q, R) policy that allowsfor releasing emergency orders when the product pro-gress information along the production and logisticschain is observed. They showed that the optimalpolicy is given by a sequence of threshold valuesdependent on the current product progress informa-tion. Kouvelis and Li (2008) studied the use of aflexible backup supplier as an emergency response tolead-time information and its implication on the origi-nal order and on the cost of meeting demand. Ourwork differs from the above in the nature of the con-tingent action, in other words expediting a portion ofan early placed order instead of placing an emergencyorder. From a practical perspective, the obviousadvantage of expediting over emergency ordering isthat it does not introduce excess inventory units in aconstant demand rate system.The fourth stream of related research examines

the use of expediting an early placed order for fas-ter delivery. The work by Allen and D’Esopo (1968)is the first one, to our knowledge, that included thepossibility of expediting in inventory decision mak-ing. They considered a situation where on top of acontinuous review (Q, r) policy, an outstandingorder will be expedited if the inventory positiondrops below a so-called “expediting level” X.Assuming known and constant regular and expedit-ing lead-times, they characterized the optimal (Q,r, X) policy. Lawson and Porteus (2000) presented aserial multi-echelon system where orders can beexpedited or de-expedited (slowed down) betweenadjacent echelons. However, the underlying lead-times between stages were assumed to be determin-istic, unlike the stochastic lead-times in our model,and the emphasis was on the use of expediting pol-icies for managing demand uncertainty. Ex anteexpediting actions in environments of stochasticlead-times and deterministic and stationary demandwere studied in Bookbinder and Cakanyildirim(1999). They adopted the artifice of the so-called“expediting factor” � , which is a constant propor-

tion between two random variables—the expeditedlead-time and the regular lead-time, as a decisionvariable to moderate the lead-time length within acontinuous review (Q, r) model, and characterizedthe optimal (Q, r, �) policy. In Duran et al. (2004)supply lead-time is modeled as consisting of twostages, with the first one being deterministic andthe second one taking one of two values. At thecompletion time of the first stage, if the inventoryposition is below some threshold level, the firm hasthe option to expedite the regular order as a whole(no partial expediting). They proposed an algorithmto obtain the optimal policy parameters. Jain (2006)considered a periodic review system with both sto-chastic demand and stochastic lead-time. At somepredetermined point in time, it is possible to expe-dite a portion of the order. Jain (2006) characterizedthe optimal base-stock policy under different levelsof lead-time visibility. Our study differs from thework of Bookbinder and Cakanyildirim (1999) andDuran et al. (2004) by considering a more realisticlead-time model: we assume that lead-time consistsof two stages and allow each stage being stochastic,while using general lead-time distributions. Further-more, we allow the possibility of partial expeditingof the order. Relative to the Jain (2006) work, ourmodel makes the expediting decision at the comple-tion time of the first supply stage, thus capitalizingon relevant supply lead-time information, and notat a priori determined time independent of supplyconditions.The remainder of the article is organized as fol-

lows: In section 3 we set up the stylized model,develop analytic results on the optimal expeditingpolicy and on the comparative statics, and constructthe heuristic expediting policy for ease of computa-tion. In section 4 we study the impact of theexpediting option on the replenishment orderparameters (the safety lead-time and the size of theorder). We also extend our basic model to includecorrelated lead-times and more than two lead-timestages. Finally in section 5 we summarize our mainresults and conclusions.

3. Model and Analysis

Consider a retailer who replenishes inventory froma supplier with uncertain lead-times in an infinitehorizon planning setting. The demand rate isassumed to be constant, reflecting “functional prod-uct” characteristics in our context, and the demandrate is normalized to be 1 without loss of general-ity. To simplify his replenishment decision, theretailer assigns an interval of demand to each orderplaced (such an assumption is an alternative to no-order-crossing commonly assumed in the stochastic

Kouvelis and Tang: Optimal Expediting Policy Under Uncertain Lead-TimesProduction and Operations Management 21(2), pp. 309–330, © 2011 Production and Operations Management Society 311

lead-time literature and has been adopted in thework of Liberatore 1979). This way the retailer hasto focus only on the placement of one specific orderfor every demand interval independently fromother orders for other intervals. We assume that thedemand interval of interest starts at time 0 andends at time T. The replenishment order for thedemand interval is placed before the demand starts,at time (�l), where l is called the safety lead-time ofthe replenishment order. The lead-time of thereplenishment order consists of two stages, eachstage taking an uncertain time to finish. The dura-tion of stage i (i = 1,2) is denoted as L1, a randomvariable with probability density function (pdf) f i(·)and cumulative density function (cdf) Fi(·). In thebasic model, we assume that L1 and L2 are indepen-dent. The case of correlated lead-time stages is dis-cussed in section 4.3.At the completion time of the first stage, denoted

by t, an expediting service can be adopted, at theretailer’s discretion, to expedite � units for deliverywith a lead-time L. By prioritizing the expeditedproduct/shipment and assigning the best resourceavailable to handle them, many expediting serviceproviders are usually able to quote a guaranteeddelivery time or ensure that product/shipment bedelivered within customers’ requested time (servicessuch as UPS Freight Urgent, UPS Freight Guaran-teed, and USPS Express Mail). Motivated by thisfact, we treat the expediting lead-time L as a con-stant in our study. In the basic model, we alsoassume that the deterministic expediting lead-time Lis shorter than the second-stage regular lead-time L2on average, that is, L � EL2. The case of guaranteedfaster expediting service (i.e., L � L2) is discussed insection 3.5. Unsatisfied demand is backlogged. Theretailer decides on how much to expedite afterobserving the first stage completion time, with theobjective of minimizing the cost of meeting demand.Relevant costs are backorder costs for unsatisfieddemand, holding costs for on-hand inventory, andexpediting costs, if any. Backorders are charged at arate of π per unit per unit of time, inventory ischarged at a rate of h per unit per unit of time, andexpediting is charged at a rate of ce per unit.

3.1. Calculation of the Expected Total CostThe calculation of the inventory holding cost andbackorder cost depends on the completion time of thefirst stage, the arrival time of the expedited order, andthe completion time of the second stage. We assumethat T > 2L. There are a total of seven cases we needto consider.(a) t � � L: This is the case when the first stage

completion time is well ahead of the demand seasonand the expedited units, if any, arrive before the

demand interval starts. Denote ETCa(t, �) as theexpected total cost, if the first stage is finished at timet and the expediting quantity is � , with the subscript“a” corresponding to case (a). Then,

ETCaðt; �Þ ¼ ce� þ h�ð�t� LÞ þ 1

2h�2

þZ ��t

0

�hðT � �Þð� � t� nÞ

þ 1

2hðT � �Þ2

�f2ðnÞdnþ

þZ T�t

��t

�1

2pðn� ð� � tÞÞ2

þ 1

2hðT � t� nÞ2

�f2ðnÞdnþ

þZ 1

T�t

�1

2pðT � �Þ2

þ pðT � �Þðn� ðT � tÞÞ�f2ðnÞdn;

where the first term is the cost for the expeditingservice; the second term is the cost to hold the expe-dited units until the demand season starts, whereasthe third term is the cost to hold the expedited unitsuntil they are depleted. The last three terms are theexpected holding and backorder costs incurred forthe rest of the regular order (T � � units, not expe-dited). For convenience, we will suppress henceforththe explicit dependence of the expected total coston t and � . It can be shown that ETCa is convex in �in the domain t ∈ (�l, �L]. Let g1(t, �) be the first-order derivative of ETCa with respect to � . Then,we have

g1ðt; �Þ : ¼ ce þ hðEL2 � LÞ� ðhþ pÞ

Z 1

��tðn� ð� � tÞÞf2ðnÞdn:

ð1Þ

The first-order condition (FOC) of ETCa for an interiorsolution �� is:

g1ðt; ��Þ ¼ 0: ðFOC1Þ(b) �L � t � 0: In this case, the expedited units willarrive at time t + L, after the demand season starts.Depending on the magnitude of the expediting quan-tity � relative to t + L (time when the expedited orderarrives, that is, the backlog if expedited order arrivesfirst) and T�(t + L), the expected total cost calculationis different in the following three cases: (b1) where� � (t + L), and thus T � � � T � (t + L), so theexpedited units cannot fulfill all the backlog if theyarrive first; (b2) where (t + L) � � � T � (t + L), andthus T � � � (t + L) and � � (t + L), so regardless of


whether the expedited or the regular units arrivefirst, they can satisfy all the backlog; and (b3)where � � T � (t + L), and thus T � � � (t + L) and� � (t + L), so if the regular order arrives first, therewill still be shortages until the expedited units arrive.For brevity of presentation, the expressions of theexpected total cost for the three cases (denoted,respectively, as ETCb1, ETCb2, and ETCb3) are includedin Appendix B.It can be shown that the expected total cost ETCb1 is

linear in � with a slope of k1 given as follows:

k1 :¼ ce þ hðEL2 � LÞ � ðhþ pÞZ 1

Lðn� LÞf2ðnÞdn: ð2Þ

The expected total cost ETCb2 is convex in � , and thefirst order condition can be characterized by (FOC1).ETCb3 can also be shown to be convex in � , and itsfirst-order condition is given as follows:

g2ðt; ��Þ :¼ ce þ hðEL2 � LÞ

þ ðhþ pÞZ L

0

ðL� nÞf2ðnÞ dn

� ðhþ pÞZ T��t

0

ðT � �� t� nÞf2ðnÞdn

� ðhþ pÞZ 1

��tðn� ð�� tÞÞf2ðnÞdn ¼ 0:

ðFOC2Þ

Similarly, we can calculate the expected total costfor other cases. We summarize the characterizingequations and slopes in Table 1. For understandingthis table we need two other first order conditions(FOC3) and (FOC4), and a constant k2 representing aslope, which are defined as follows:

g3ðt; ��Þ : ¼ ce þ hðEL2 � LÞ

þ ðhþ pÞZ L

0

ðL� nÞf2ðnÞdn

� ðhþ pÞZ 1

��tðn� ð�� tÞÞf2ðnÞdn ¼ 0:

ðFOC3Þ

g4ðt; ��Þ :¼ ce�pðEL2�LÞ

� ðhþpÞZ T��t

0

ðT� �� t� nÞf2ðnÞdn¼ 0:

ðFOC4Þk2 :¼ ce � pðEL2 � LÞ: ð3Þ

The following lemmas present the properties ofgi(t, �) (i = 1, 2, 3, and 4) and the expected total cost.

LEMMA 1.

(i) g1(t, �) is increasing in � and decreasing in t.(ii) g2(t, �) is increasing in � .(iii) g3(t, �) is increasing in � and decreasing in t.(iv) g4(t, �) is increasing in both � and t.

LEMMA 2. The expected total cost is convex in � for anygiven t.

Lemma 2 can be proved by evaluating the deriva-tives of the expected total cost function from the leftand from the right at the adjoining points of everytwo adjacent regions. Since the expected total costis piecewise convex in � and the derivative iscontinuous, the expected total cost must be convex in� in the whole domain [0, T].

Table 1 Details of the Seven Cases of ETC Calculations: First-OrderConditions and Slopes

(a) t � � L

Decision Range for � [0, T]

FOC FOC1

(b) �L � t � 0

Decision Range for � [0, (t + L)] [t + L, T � (t + L)] [T � (t + L),T]

Relations T � � � t + L T � � � t + L T � � � t + L

� � t + L � � t + L � � t + L

FOC k1 FOC1 FOC2

(c) 0 � t � T2 � L

Decision Range

for �

[0, t + L] [t + L, T � (t + L)] [T � (t + L),T � t] [T � t, T]

Relations T � � � t + L T � � � t + L t � T � � � t + L T � � � t

� � t + L � � t + L � � t + L � � t + L

FOC k1 FOC1 FOC2 FOC3

(d) T2 � L � t � T�L

2

Decision Range

for �

[0, T � (t + L)] [T � (t + L), t + L] [t + L, T � t] [T � t, T]

Relations T � � � t + L t � T � � � t + L t � T � � � t + L T � � � t

� � t + L � � t + L � � t + L � � t + L

FOC k1 FOC4 FOC2 FOC3

(e) T� L2 � t � T � L

Decision Range

for �

[0, T � (t + L)] [T � (t + L),T � t] [T � t, t + L] [t + L, T]

Relations T � � � t + L t � T � � � t + L T � � � t T � � � t

� � t � � t + L � � t + L � � t + L

FOC k1 FOC4 k2 FOC3

(f) T � L � t � T

Decision Range for � [0, T � t] [T � t, T]

Relations T � � � t T � � � t

FOC FOC4 k2

(g) t � T

Decision Range for � [0, T]

FOC k2


3.2. Optimal Expediting PolicyFirst, with a high expediting cost, the expeditingoption will not be used. The following propositioncharacterizes the threshold value of the expeditingcost above which the expediting service is consideredprohibitively expensive.

PROPOSITION 1. If ce � �c, then the optimal expeditingquantity ��(t) = 0 for any t � � l, where

�c :¼ ðhþ pÞZ 1

Lðn� LÞf2ðnÞdn� hðEL2 � LÞ: ð4Þ

The proofs of all propositions are included inAppendix A. Note that �c can be written as:

h

Z L

0

ðL� nÞf2ðnÞdnþ pZ 1

Lðn� LÞf2ðnÞdn: ð5Þ

This is the standard expected loss function L (seePorteus 2002) evaluated at L. According to this wecan interpret Proposition 1 as follows: On one hand,using the expediting option we pay ce for each unitexpedited and will receive it after L time units. Onthe other hand, if the expediting service is not used,we expect to pay hEðL � L2Þþ if this unit (in theregular order) takes a time shorter than L to arrive,and pEðL2 � LÞþ if it takes longer. Proposition 1simply says that if the expediting cost ce is higherthan this expected loss, it is optimal not to expediteat all. It is easy to see that �c is increasing in both hand π.Next, we will derive the optimal expediting policy

when the expediting cost ce is below a certain thresh-old. To proceed, we define t and t2 by the followingequations:

ce þ hðEL2 � LÞ � ðhþ pÞZ 1

�tðn� ð�tÞÞf2ðnÞdn ¼ 0:

ð6Þ

ce þ hðEL2 � LÞ þ ðhþ pÞZ L

0

ðL� nÞf2ðnÞdn� ðhþ pÞ

�Z 1

T�2t2

ðn� ðT � 2t2ÞÞf2ðnÞdn ¼ 0:

ð7ÞThrough this section and the next, we assume that

t > � (T � L). This assumption implies that there isno need to use the expediting service when the firstlead-time stage finishes way ahead of the demandseason.

PROPOSITION 2. Let c :¼ pðEL2 � LÞ. If ce � c, thenthe optimal expediting quantity �� can be characterizedas follows.

��ðtÞ ¼

0; if t � t;

t� t; if t � t � T�Lþt2 ;

solution of ðFOC2Þ; ifT�Lþt

2 � t � t2;T � 2t2 þ t; if t2 � t � 2t2 ;T; if t � 2t2.

8>>>><>>>>:

Figure 1a illustrates the form of the optimal expe-diting policy when the expediting cost ce is lowerthan c. It shows that except for the interval½ðT � L þ tÞ = 2; t2� (which is referred to as the middleinterval hereafter), the optimal expediting quantity islinearly increasing in t, and eventually one wouldexpedite the whole regular order when it is gettingtoo late. The proof of Proposition 2 explains whythis is the case. When the expedited units can onlybe delivered after the demand season starts, how toallocate the order between the expedited channel withlead-time L and the regular channel with lead-time L2becomes important. When the first lead-time stagefinishes not too late, it is optimal to expedite enoughunits to cover the backlog and to leave enough unitsto the regular channel, so that if the regular orderarrives before the expedited units, there is enough tomeet the demand until the expedited units arrive. Inother words, when t is small, it is optimal to haveboth T � � (units not expedited) and � (expeditedunits) greater than t + L (the time when expeditedunits arrive). However, since the total order sizeis fixed, as time passes t + L increases and theabove allocation rule has to change. Since theexpediting cost is low, the retailer allocates relativelyless to the regular channel to take full advantage ofthe expediting opportunity. More precisely: beyondtime (T � L + t)/2 we have t � T � � � t + L and� � t+ L; and beyond time t2 it is optimal to leaveeven less to the regular channel so that T � � � tand � � t + L.We now examine the nature of the optimal

expediting policy within the so-called “middleinterval” ½ðT � L þ t; =2; t2�. It is easy to check that��(t) is convex in t in the middle interval.Furthermore, ð@��ðtÞ = @tÞjt¼t2

¼ 1 evaluated fromthe left and from the right, that is, the twosegments tangent at t2. However, ��(t) may not benecessarily monotone in the middle interval. Infact, it can be shown that:

��ðtÞ is increasing and convex, if 1 � F2ð�tÞ � F2ðLÞ;decreasing then increasing, and convex, otherwise.

�


As shown in Figure 1a, within the middle interval,the principle of “longer delay leads to more expedit-ing” (i.e., a later realization of the first stage comple-tion leads to a larger portion being expedited) mightbreak down.Next, we will derive the optimal expediting policy

when the expediting cost ce is between c and �c. Let t4be defined as follows:

ce � pðEL2 � LÞ

� ðhþ pÞZ T�2t4�L

0

ðT � 2t4 � L� nÞf2ðnÞdn ¼ 0:

ð8ÞThe following proposition characterizes the optimal

expediting policy.

PROPOSITION 3. If c\ce\�c, then the optimal expeditingquantity �� is characterized as follows:

��ðtÞ ¼

0; if t � t;

t� t; if t � t � T�Lþt2 ;

solution of ðFOC2Þ; ifT�Lþt

2 � t � t4;2t4 þ L� t; if t4 � t � 2t4 þ L;0; if t � 2t4 þ L.

8>>>><>>>>:

Figure 1b illustrates the form of the optimal expe-diting policy when c\ce\�c. Comparing to Figure 1a,we see that a higher expediting cost reduces theincentive to continue increasing the expedited quan-tity toward the end of the demand horizon. Hence inthe low expediting cost case it is optimal to “aggres-sively expedite,” while in the intermediate expeditingcost case it is optimal to “conservatively expedite”(i.e., expedite early in the demand season, but neverexpedite the whole order, and decrease the use ofexpediting toward the end of the season). The prooffor Proposition 3 explains how the allocation rulebetween the regular and expediting channel changes

as time progresses. If the first lead-time stage finishesearly, we use more aggressively the expedited serviceso that t � T � � � t + L and � � t + L. So the expe-dited units are enough to fulfill all backlog upon arri-val. This implies that, as time passes, we will expeditemore and more, since t + L is increasing in t. Beyonda certain point in time (t4), due to the high total expe-diting cost, we will reduce the expediting quantity tobe less than t + L. Eventually we will choose not toexpedite at all. Also, we note that the optimal policyhas a similar form in the middle interval as in the lowexpediting cost case, that is, ��(t) is convex in t, butmay not be monotone; and expediting the wholeorder never happens with the maximum portionexpedited being maxfðT � L � tÞ = 2; t4 þ Lg.1

3.3. Comparative StaticsFirst, we study how the cost and lead-time of theexpediting service affect the optimal expeditingpolicy.

PROPOSITION 4. In the low expediting cost case (i.e.,ce � c):

(i) Both t and t2 increase in ce: a lower expediting costinduces an earlier expediting and a higher likelihoodof expediting the whole order.

(ii) t decreases in L; t2 increases in L if F2(L) � h/(h+ π) and decreases in L if F2(L) � h / (h + π): ashorter expediting lead-time delays expediting andleads to a lower likelihood of expediting the wholeorder when L is not too large.

With everything else remaining the same, a lowerpremium makes the expediting option more profit-able and leads to its increased use. But for a smallexpediting lead-time L, a further reduction mightdampen the incentive to expedite. Speeding up thedelivery of the expedited units elongates their time ininventory and increases the corresponding holding

0

*

T

t 02

T L t2t 22t T t

2T L t0

4t 42t L T

T

*

(a) (b)

Figure 1 Illustration of Optimal Expediting Policy (L1, L2 ∼ exp(1), h = 1.5, π = 5, T = 3, L = 0.8, ce = 1 in (a) and ce = 1.2 in (b))


cost. Similarly for the intermediate expediting costcase, we have the following proposition.

PROPOSITION 5. In the intermediate expediting cost case(i.e., c\ce\�c):

(i) t increases in ce; t4 decreases in ce: a lower expeditingcost expands the expediting region.

(ii) t, t4 and 2t4 + L all decrease in L: a shorter expedit-ing lead-time leads to less expediting earlier in thehorizon and more expediting later in the horizon.

Note that although a shorter expediting lead-timedampens the incentive to expedite early in the hori-zon as in Proposition 4, as L decreases, the threshold cincreases, which makes “aggressive expediting” amore appealing option. This is exactly what weobserve later in the horizon: a shorter expeditinglead-time leads to more expediting. When L decreasesso that pðEL2 � LÞ[ce, the optimal expediting policybecomes of an “aggressive” form.Next, we study how the inventory-related costs

affect the optimal expediting behavior.

PROPOSITION 6. In the low expediting cost case (i.e.,ce � c):

(i) Both t and t2 increase in h: a higher unit holding costdelays expediting and results in a lower likelihood ofexpediting the whole order.

(ii) Both t and t2 decrease in π: a higher backorder costinduces earlier expediting and a higher likelihood ofexpediting the whole order.

We see that a higher backorder cost induces moreexpediting. However, the inventory holding cost hastwo effects: an increase in the unit holding cost willpenalize whichever of the ordered units, through reg-ular or expedited delivery, arrive first. When the expe-diting cost is low, the increase in the unit holding costimpacts the expedited units the most, thus arguing forhigher unit holding cost dampening the incentive forexpediting. This is because our optimal policy sug-gests rather “aggressive expediting” (i.e., expedite ear-lier and have a higher likelihood to expedite the wholeorder). As shown in the next result, in the case whence[c, the optimal policy, “conservatively expedites,"and the increase in the holding cost mostly affects theregular order if arriving earlier than the expeditedunits, thus leads to an increasing use of expediting.

PROPOSITION 7. In the intermediate expediting cost case(i.e., c\ce\�c), t decreases in both h and π. t4 increasesin both h and π: a higher unit holding or backorder costexpands the expediting region.

It is also interesting to study the effect of the regularlead-time variability on the expediting policy. To do

so, we introduce the notion of increasing-convexordering (see Ross 1983): Let Xa and Xb be two non-negative random variables such that EXa ¼ EXb, thenXa � icX

b (or Xa is more variable than Xb) if and onlyif E½fðXaÞ� � E½fðXbÞ� for all convex function f.

PROPOSITION 8. Let L2a and L2

b be two random variablesfor the second stage lead-time such that EL2

a ¼ EL2b. In

the intermediate expediting cost case (i.e., c\ce\�c), ifL2

a � icL2b, then ta � tb and t4

a � t4b.

The above result implies that a more variable lead-time expands the expediting region when c\ce\�c. Inthe case where ce\c, it can be analytically shown thata more variable lead-time L2 leads to a lower t as well,and through extensive numerical study, we observethat under very general conditions, t2 also decreaseswhen L2 becomes more variable. In particular, weshow an example in Figure 2, where L2 has a Gammadistribution with shape parameter k and scale parame-ter h. Keeping the mean kh fixed and decreasing k, wesee in Figure 2a that L2 becomes more variable. In Fig-ure 2b we report the optimal t2 value for various L val-ues. In all cases, t2 decreases when k decreases. So bycombining our numerical results and Proposition 8,we see that the effect of a more variable lead-time isthat it induces more use of expediting: it expands theexpediting region when expediting cost is at someintermediate level; and leads to more aggressive expe-diting when expediting cost is low.

3.4. Heuristic PolicyFrom Proposition 2 and 3, we see that within the mid-dle interval, finding the optimal expediting quantityinvolves solving an integral equation which makesthe optimal policy difficult to calculate. One reason-able heuristic that approximates the optimal expedit-ing policy is to ignore the middle interval. In the lowexpediting cost case (ce � c), let

��ðtÞ ¼0; if t � t;t� t; if t � t � T þ t;T; if t � T þ t.

8<: ð9Þ

Figure 3a illustrates the form of the heuristic policyand Figure 3b illustrates its performance. The solidline represents the optimal expediting policy and thecorresponding expected total cost. The dashed lineshows the heuristic policy and the correspondingexpected total cost. The dotted line on Figure 3bshows the expected cost if the expediting option is notavailable. Here, we assume that, under both the opti-mal and heuristic policy, the regular replenishmentorder is placed at time �l, where l is the optimalsafety lead-time without the expediting opportunity(optimally choosing a safety lead-time will be


discussed in section 4). As shown, the heuristic policycan achieve 92–94% of the overall cost savingsbrought by the optimal policy. The proofs of Proposi-tion 4 and 6 imply that the heuristic works the bestwhen ce and h are big or when L and π are small (sincethe length of the middle interval will be small). This iswhat we observe in Figure 3b: the gap between theheuristic policy and the optimal policy is smaller thehigher ce is.Similarly, a simple heuristic policy for the interme-

diate expediting cost case (c\ce\�c) ignores themiddle interval and sets:

��ðtÞ ¼0; if t � t or t � 2t4 þ L ;

t� t; if t � t � 2t4þLþt2 ;

2t4 þ L� t; if2t4þLþt

2 � t � 2t4 þ L.

8<:

The form of the heuristic policy is shown in Figure4a. The solid line represents the optimal expeditingpolicy and the dashed line the heuristic policy. Sincethe middle interval is narrower in the intermediateexpediting cost case, the heuristic policy behaves evenbetter than its counterpart in the low expediting costcase. It can achieve 95–98% of the overall cost savings.The proofs of Propositions 5 and 7 imply that the heu-ristic performs the best when ce is big, or h and π aresmall, since the length of the middle intervaldecreases in ce and increases in h and π.

3.5. Guaranteed Faster Expediting ServiceIn the above analysis, we assumed that the deter-ministic expediting lead-time is shorter than theregular lead-time L2 on average. This implies thatthe expediting service may take longer than theregular lead-time. Now, we consider the case thatthe expediting service guarantees a shorter lead-time.Let L2 = L + e, where e is a non-negative random

variable with pdf g2(·) and cdf G2(·). Define t similarlyas in Equation (6) using the new notation:

ce þ hEe� ðhþ pÞZ 1

�ðtþLÞðtþ Lþ eÞg2ðeÞde ¼ 0: ð10Þ

Then the optimal expediting policy can be character-ized as follows.

PROPOSITION 9. If L2 = L + e, where e is a non-negativerandom variable, then the optimal expediting policy canbe given as follows:

1. If ce[c, then �� = 0: expediting is never used.2. If ce � c, then,

��ðtÞ ¼0; if t � t;t� t; if t � t � T þ t;T; if t � T þ t.

8<: ð11Þ

2t0 1 2 3 4 5 6 7 8

0

0.1

0.2

0.3

0.4

0.51, 22, 13, 2 / 34, 1 / 25, 2 / 56, 1 / 3

kkkkkk

(a) (b)

Figure 2 Impact of Variability of L2 on t2 for Gamma Distributions (h = 1.5, π = 5, T = 4, L = 0.8, kh = 2)

tec

Exp.TotalCost

(a) Illustration of the heuristic policy (b) Cost comparison of the optimal and the heuristic policy

0 T0

T

optimalheuristic

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 18

8.5

9

9.5

10

10.5

11

optimalheuristicoriginal

Figure 3 Illustration of the Heuristic Policy for Low Expediting Cost Case (L1, L2 ∼ exp(1), h = 1.5, π = 5, T = 3, L = 0.8)


Note that the optimal policy (11) is equivalent tothe heuristic policy (9) and is illustrated in Figure 3a.The fact that the expediting service guarantees ashorter lead-time changes the optimal expeditingpolicy in two ways. First, the “conservatively expe-dite” region disappears. We have seen in Proposition1 that, when the expediting cost ce is higher than theexpected loss of not expediting, the expediting servicewill not be used at all. Here, since the expeditingservice guarantees a shorter lead-time, the expectedloss of not expediting only includes the expectedbackorder cost p

R10 eg2ðeÞde ¼ pEe which equals c.

So the region c\ce\�c collapses. Second, we see thatwhen the expediting service guarantees faster deliv-ery, we restore the intuition of “longer delay leads tomore expediting.”

4. Discussion

4.1. Effect of Expediting on the Safety Lead-TimeWithout the presence of the expediting service, inchoosing the safety lead-time, the retailer minimizesthe expected total cost:

ECoðlÞ ¼Z l

0

½hTðl� nÞ þ 1

2hT2� fðnÞdn

þZ lþT

l

1

2pðn� lÞ2 þ 1

2hðlþ T � nÞ2

� �fðnÞdn

þZ 1

lþT

1

2pT2 þ pTðn� l� TÞ

� �fðnÞdn;

ð12Þwhere f(·) is the probability density function for thetotal lead-time L1 + L2 and can be calculated by theconvolution of f1 and f2. The subscript “o” stands for“original” (in the absence of expediting). For a fixeddemand horizon T, the expected total cost can beshown to be convex in l and the optimal safetylead-time lo can be characterized by the followingfirst-order condition:

Fðlo þ TÞ �Z loþT

lo

ðn� loÞT

fðnÞdn ¼ phþ p

: ð13Þ

In the presence of the expediting service, if theretailer uses the service according to the optimalpolicy derived in section 3, then the expected totalcost viewed at the time when the replenishmentorder is placed, that is, at time �l, can be calcu-lated as follows: If the expediting cost is too high,that is, ce � �c, then the expediting service willnever be used and the expected total cost is thesame as Equation (12). In the low expediting costregion, that is, ce � c, the expected total cost canbe calculated as follows.

EClowðlÞ ¼Z lþt

0

ETCað�lþ l1; 0Þf1ðl1Þdl1

þZ l�L

lþtETCað�lþ l1;�lþ l1 � tÞf1ðl1Þdl1

þZ l

l�LETCb2ð�lþ l1;�lþ l1 � tÞf1ðl1Þdl1

þZ lþT�Lþt

2

lETCc2ð�lþ l1;�lþ l1 � tÞf1ðl1Þdl1

þZ lþt2

lþT�Lþt

2

ETCc3ð�lþ l1; ��ð�lþ l1ÞÞf1ðl1Þdl1

þZ lþ2t2

lþt2

ETCd4ð�lþ l1;T � 2t2 � lþ l1Þf1ðl1Þdl1

þZ lþT�L

lþ2t2

ETCe4ð�lþ l1;TÞf1ðl1Þdl1

þZ lþT

lþT�LETCf2ð�lþ l1;TÞf1ðl1Þdl1

þZ 1

lþTETCgð�lþ l1;TÞf1ðl1Þdl1;

where the subscript “low” represents low expeditingcost case, and ��(·) is determined by (FOC2) witht = �l + l1. The expected total cost for the intermedi-ate expediting cost case c\ce\�c can be calculatedin a similar way:

tT-Lec

Exp.TotalCost

1.2 1.4 1.6 1.8 2 2.2 2.4 2.69.7

9.8

9.9

10

10.1

10.2

10.3

10.4

10.5

10.6

optimalheuristicoriginal

optimalheuristic

T00

T

(a) (b)

Figure 4 Illustration of the Heuristic Policy for Intermediate Expediting Cost Case (L1, L2 ∼ exp(1), h = 1.5, π = 5, T = 3, L = 0.8)


ECmedðlÞ¼Z lþt

0

ETCað�lþ l1;0Þf1ðl1Þdl1

þZ l�L

lþtETCað�lþ l1;�lþ l1�tÞf1ðl1Þdl1

þZ l

l�LETCb2ð�lþ l1;�lþ l1�tÞf1ðl1Þdl1

þZ lþT�Lþt

2

lETCc2ð�lþ l1;�lþ l1�tÞf1ðl1Þdl1

þZ lþt4

lþT�Lþt

2

ETCc3ð�lþ l1;��ð�lþ l1ÞÞf1ðl1Þdl1

þZ lþT�L

lþt4

ETCd2ð�lþ l1;2t4þLþ l� l1Þf1ðl1Þdl1

þZ lþ2t4þL

lþT�LETCf1ð�lþ l1;2t4þLþ l� l1Þf1ðl1Þdl1

þZ lþT

lþ2t4þLETCf1ð�lþ l1;0Þf1ðl1Þdl1

þZ 1

lþTETCgð�lþ l1;0Þf1ðl1Þdl1;

where the subscript “med” represents medium(intermediate) expediting cost case, and ��(·) isdetermined in the same way as above. One wouldexpect that the introduction of the expediting servicewill result in a shorter safety lead-time because onecan respond to unusual delays through expeditingan order. Since it is analytically hard to show theunimodality of EClow,med(l) and determine the opti-mal safety lead-time through first-order conditions,below we conduct numerical studies and draw man-agerial insights based upon the numerical results.The following parameters are used in Figure 5. The

expediting lead-time is fixed at L = 2; the first lead-time component has a Gamma distribution withshape parameter k1 = 4 and scale parameter h1 = 1,and the second lead-time component has a Gammadistribution with shape parameter k2 = 2.5 and scaleparameter h2 = 1; the order size is T = 6; and theinventory-related costs are h = 1.5, π = 5. The twothreshold values c and �c which separate the three

expediting policy regions are: c = 2.5 and �c ¼ 4:78.Figure 5a shows the optimal safety lead-time l* as afunction of the expediting cost ce. As we can see,expediting reduces the safety lead-time from itsoriginal level lo = 5.64. This relationship holds withineach policy region and across the two regions. In thissense, expediting and safety lead-time can be thoughtof as strategic substitutes because the expeditingoption hedge against the extreme lead-time realiza-tions, thus reducing the need for very conservativeplanning of the safety lead-time. It is also observedthat as ce increases, the benefit of expediting dimin-ishes and the usage decreases, thus a longer safetylead-time is needed.Figure 5b compares the expected total cost. As a

benchmark, the expected total cost without expeditingand with the original safety lead-time determined byEquation (13) is shown by the dotted horizontal line.The solid line is the expected total cost with expedit-ing and the optimal safety lead-time obtained inFigure 5a. The gap between these two curves indicatesthe value of expediting. Though choosing the optimalsafety lead-time results in the least expected total cost,it requires higher computational effort. Figure 5b alsoshows the comparison of the expected total cost underthe optimal safety lead-time l* (the solid line) andunder the original safety lead-time lo (the dashed line).As we can see most of the cost savings result fromusing the expediting option in an optimal way. Theadditional benefit from optimizing the safety lead-time is significantly smaller, in most cases < 10%, as inour example.

4.2. Effect of Expediting on the Order SizeIn our discussion so far, the order size T was exo-genously given. When there are economies of scale inordering (e.g., a fixed ordering cost), choosing theright order size becomes important. Denote the fixedordering cost to be K and the variable cost v. Then theexpected average cost per period is:

EACðl;TÞ ¼ K

Tþ vþ ETC

T:

(a) (b)

Figure 5 Effect of Expediting on Safety Lead-Time and Expected Cost (h = 1.5, π = 5, L1 ∼ Gamma(4, 1), L2 ∼ Gamma(2.5, 1), T = 6, L = 2)


Figure 6 shows the per period expected cost whenexpediting is used and the safety lead-time is chosenoptimally for different values of T in the followingsetting: L1 ∼ Gamma(4, 1) and L2 ∼ Gamma(2.5, 1),expediting lead-time L = 2, expediting cost ce = 3,inventory-related cost h = 1.5, π = 5, fixed orderingcost K = 10, and variable ordering cost v = 1.5. Weobserve that the expected average cost preserves therobustness property of the traditional EOQ model.That is, a small deviation from the optimal size willnot increase the expected cost by much. Hence somesimple near-optimal heuristic may perform well. Onecandidate is the EOQ quantity. In our numericalexample, the EOQ quantity is TEOQ = 7.6012, which is25% higher than the optimal order size T* = 6.0941.But the expected cost under TEOQ is only 1.3% higherthan the minimum expected cost per period.The small table next to Figure 6 reports the optimal

safety lead-time and order size with and withoutexpediting. We observe that the presence of the expe-diting service leads to an increase in the order sizeand at the same time a decrease in the safety lead-time. In other words, the optimal use of expeditingservice will support the placement of larger replenish-ment orders, thus taking increased advantage of econ-omies of scales and using smaller safety lead-times,which adds to the system responsiveness.

4.3. Correlated Lead-Time StagesIn section 3, we analyzed the model in which the twolead-time random variables L1 and L2 are indepen-dently distributed. This is reasonable when the twolead-time stages have very different nature and thecorresponding processes are managed by differentparties. For example, the first stage represents pro-duction and the second stage represents transporta-tion part of the lead-time. However, when themajority of the lead-time consists of transportationtime and L1 and L2 correspond to two phases of thetransportation activity, these two lead-time stages areusually correlated. They may be negatively corre-lated, for instance when the logistics manager sees adelay in the first phase, he may assign the best

resource to handle the second phase so that the reali-zation of L2 is likely to be small. In other cases, L1 andL2 may be positively correlated, when the uncertain-ties in the two lead-time stages are due to the same ora similar factor (e.g., union strike, weather condition,safety inspection, etc).To analyze the effect of correlated lead-times, we

adopt the simple discrete joint distribution frame-work considered in Babich et al. (2007) in modelingsupply risk. We assume that L1 can take one of twovalues l1L and l1S, where l1L > l1S and the subscript Land S represent long and short, respectively. For sim-plicity, we use the same assumption as section 3.5 forL2. That is, L2 = L + e, where L is the deterministicexpediting lead-time and e is a positive random vari-able. Similar to L1, we assume that e can take one oftwo values eL and eS, where eL > eS. Use pij, (i, j = 0, 1)to represent the joint probability of (L1, e). If the lead-time takes the longer time, the corresponding value ofi or j equals 1. For example, p01 denotes the joint prob-ability of L1 = l1S and e = eL. This joint distribution iscompletely characterized by the two marginal proba-bilities /1 = p10 + p11, /2 = p01 + p11, and the jointprobability p11. By setting /1 ¼ /2 ¼ 1

2 (this impliesp01 = p10) and allowing p11 to vary between 0 and 1

2,we can capture the entire range of lead-time co-dependence from perfectly negative to perfectly posi-tive correlation. Higher value of p11 corresponds tohigher degree of lead-time correlation (in fact, the cor-relation coefficient ρ = 4p11 � 1).In Proposition 9, we see that the expediting decision

depends on the realized value of L1, or equivalently t(t = �l + L1). So let us first assume that the first lead-time stage takes the longer time, that is, L1 = l1L.Given this, the conditional probability for L2 can beexpressed as:

PrðL2 ¼ Lþ eL j L1 ¼ l1LÞ ¼ p11p11 þ p10

;

PrðL2 ¼ Lþ eS j L1 ¼ l1LÞ ¼ p10p11 þ p10

:

By Proposition 9, the optimal expediting quan-tity also depends on the threshold value t, which is

Figure 6 Expected per Period Cost for Different Order Size T and Comparison of Optimal and Original Safety Lead-Time and Order Size (h = 1.5,π = 5, K = 10, m = 1.5, L1 ∼ Gamma(4, 1), L2 ∼ Gamma(2.5, 1), L = 2, ce = 3)


characterized by Equation (10). Under the discretejoint distributions specified above, Equation (10) canbe written as:

ce þ hEe� ðhþ pÞ p11p11 þ p10

ðtþ Lþ eLÞ�

þ p10p11 þ p10

ðtþ Lþ eSÞ�

¼ 0;

and t can be calculated as2:

t ¼ ce þ hEehþ p

� p11eL þ p10eSp11 þ p10

� L: ð14Þ

It can be easily seen that @t = @p11 ¼�2pðeL � eSÞ = ðh þ pÞ\0. This says when L1 takesa long time, an increase of lead-time correlation leadsto a decreased value for t. By the optimal policystated in Proposition 9, this means when it is likelythat L2 also takes a long time (due to high correlationof L1 and L2 and the fact that L1 = l1L), one ismore likely to use the expediting service andexpedite a larger portion of the order (��increases inEquation (11)).Similarly, if L1 = l1S, the conditional probability for

L2 can be expressed as PrðL2 ¼ L þ eL j L1 ¼l1SÞ ¼ p01 = ðp01 þ p00Þ and PrðL2 ¼ L þ eS j L1 ¼l1SÞ ¼ p00 = ðp01 þ p00Þ. Now Equation (10) can bewritten as:

ce þ hEe� ðhþ pÞ p01

p01 þ p00ðtþ Lþ eLÞ þ p00

p01 þ p00ðtþ Lþ eSÞ

� �

¼ 0;

and t can be calculated as:

t ¼ ce þ hEehþ p

� p01eL þ p00eSp01 þ p00

� L: ð15Þ

Since @t = @p11 ¼ 2pðeL � eSÞ = ðh þ pÞ[0, anincrease of lead-time correlation leads to an increasedvalue for t. Note that when L1 takes a short time, ahigher correlation means a higher likelihood that L2also takes a short time. This reduces the need forexpediting. An increased value for t exactly leads tothis expediting decision by Equation (11). Therefore,we see that the expediting option will be used moreheavily to handle shipment delays when the lead-time stages are positively correlated, and less fornegative correlation.

4.4. Three Lead-Time StagesFinally, we extend our basic model to include morethan two lead-time stages reflecting the case wherethe regular shipment stops more than once during thesupply process. The question we address is: If there isa one-time opportunity to expedite part or all of the

shipment, should it be used earlier or later? Due tocomputational complexity, we restrict ourselves tothe case where the expediting service guarantees ashorter lead-time. Particularly, let Li = mi + ei be theith stage of the lead-time (i = 1, 2, 3), where mi is aconstant and ei is a non-negative random variable. Ifone decides to expedite at the first breakpoint whenL1 is realized, then the expediting lead-time isassumed to be m2 + m3. If one waits until the ship-ment goes through the second breakpoint and makesthe expediting decision at that time, then the expedit-ing lead-time is denoted by m3.In our numerical examples below, the following

parameters are used: ei is normally distributed withmean mi and standard deviation mi/3 (so the proba-bility of ei < 0 is small). The cost parameters are:ce = 4, h = 4, π = 3, T = 10, and l = 12. To study thetiming issue of expediting, we first fix the location ofthe first breakpoint and show the optimal expeditingdecision for different locations of the second break-point. For that we fix m1 = 2 and consider the follow-ing three scenarios: case (1) m2 = 1, m3 = 5; case (2)m2 = 3, m3 = 3; and case (3) m2 = 5, m3 = 1, corre-sponding to the second breakpoint closer and closerto the destination. Figure 7a compares the optimalexpediting decisions for the three scenarios, where“0” denotes “shipment has no delay so no need forexpediting,” “1” denotes “expedite right now,” and“2” denotes “shipment has delay but it is more profit-able to wait until the next stop and make a decisionthen.” We observe that if the shipment has only asmall delay when it goes through the first breakpoint,it is worthwhile to hold the expediting opportunityuntil more information about the lead-time isrevealed at the second stop. However, the wait optionbecomes less and less attractive when the secondbreakpoint is located closer and closer to the destina-tion. The information simply becomes less valuable asthe shipment gets close to the destination. Any expe-diting may not help much.We also perform the same analysis when the loca-

tion of the second breakpoint is fixed while the loca-tion of the first breakpoint varies. For that we fixm3 = 2 and consider the following three scenarios:case (1) m1 = 1, m2 = 5; case (2) m1 = 3, m2 = 3; andcase (3) m1 = 5, m2 = 1, corresponding to the firstbreakpoint further and further away from the origin.Figure 7b shows the optimal expediting decisions forthe three scenarios. For each scenario, the graphshows the period when L1 2 ½lL1 � 6rL1 ; lL1 þ 6rL1 �.We find that the observations are consistent withthose obtained in Figure 7a. In particular, we see thatthe closer the first breakpoint is to the origin, the moreincentive to wait and expedite later. Our studyimplies that the one-time expediting option should beexercised to respond to shipment delays when


enough information is revealed and when it is not toolate.

5. Conclusions

In contemporary global supply chains of “functionalgoods,” low-cost-driven outsourcing from far awaysuppliers results in challenges to managing long andhighly variable lead-times. In our research we studiedthe use of order expediting in response to partiallyresolved lead-time information in a situation wherethe lead-time of a retailer’s replenishment orders tomeet his constant demand rate consists of two stages.Each stage has a random duration. At the completiontime of the first supply stage, the retailer can expediteat a cost premium a portion of his earlier placedorder, with the expedited portion arriving in a deter-ministic lead-time that is shorter than the averageduration of the second stage. To facilitate our study,we assumed that the retailer assigns an interval ofdemand to each order and thus we focused our studyonly in a demand interval that is planned to be satis-fied from an order placed earlier before the demandstarts.For the situation considered, we formulated the

expected total cost of meeting demand for a givensafety lead-time, whose calculation depends on thecompletion time of the first stage, viewed at the timeof the expediting decision. We showed that theexpected total cost is convex in the expediting orderquantity and that the optimal expediting policy canbe characterized explicitly. Two thresholds on theunit expediting cost define three regions of the expe-diting policy. When expediting is too expensive, itwill never be used. When expediting is very cheap, itis optimal to fully exploit the expediting opportunityand use it aggressively (expediting the whole orderfor extremely long delays). When the expediting costis in between, it is optimal to adopt the expeditingservice but only leverage it in a limited way (beyond acertain delay level, decreasing the expedited portionas delay increases). It was observed that the optimalexpediting policy does not necessarily have simplemonotonicity in the first stage completion time,

particularly when the completion of the first stagefalls in the so-called “middle-interval.” Heuristic poli-cies were presented to simplify the calculation of theexpediting policy and numerical examples suggestedthat the performance of the heuristic policies is satis-factory. It was noted that if the expediting optionguarantees a shorter lead-time than the second stage,then the intuition of “longer delay leads to moreexpediting” restores.The impact of some basic model parameters on the

optimal expediting policy was studied. In particular,a lower cost of the expediting service, a higher back-order cost, or a more variable supply lead-time leadsto a higher propensity to use the expediting service(i.e., expedite earlier and higher portions). The impactof unit holding cost on the optimal expediting policydepends on the cost of the expediting service. For alow expediting cost, an increase in inventory holdingcost reduces the use of expediting (i.e., expedite laterand smaller portions). In the intermediate cost case,the opposite occurs with expediting becoming moreof an attractive option. Another seemingly counter-intuitive result is that a decreasing expediting lead-time might reduce the incentive to expedite and leadto delayed expediting and lower likelihood to expe-dite the whole order, especially so for small delays inthe first supply stage.We also formulated the expected cost of meeting

demand for a given safety lead-time viewed at thetime of placing the order. Numerical examplesshowed that with the expediting option, the optimalsafety lead-time is smaller than that without the expe-diting option, and the difference between the twosafety lead-times could be significant. Our study alsosuggested that the presence of the expediting optionsupports the placement of larger replenishmentorders. However, the marginal benefit of simulta-neous optimization of the safety lead-time and ordersize is insignificant. So from a planning perspective,the timing and quantity decision for the regularreplenishment order can be made without taking intoaccount the expediting option. The focus should beon the careful exercise of the expediting option, asoutlined in the optimal expediting policy (or the

0

1

2

12 10 8 6 4The me when L1 is realized

2,1,5

2,3,3

2,5,1

(a) Op mal expedi ng me whenthe 1st breakpoint is fixed

(“0” denotes: shipment has no delay so no need for expediting)

(“1” denotes: expedite right now)

(“2” denotes: shipment has delay, but wait until the next stop and make decision then)

m1, m2, m3

0

1

2

12 8 4 0 4 8Timewhen L1 is realized

1,5,2

3,3,2

5,1,2

(b) Op mal expedi ng me whenthe 2nd breakpoint is fixed

m1, m2, m3

Figure 7 Optimal Expediting Decision when there Are Three Lead-Time Stages


heuristic policy for practical purposes). We extendour basic model to consider correlated lead-timestages and in a simple discrete joint distributionmodel, we see that for positively (negatively) corre-lated lead-times, one would count more (less) on theexpediting service when there is a delay. Finally, weshow that when the replenishment lead-time consistsof more than two stages, the one-time expeditingoption should be exercised to respond to shipmentdelays when enough information is revealed andwhen it is not too late.Our research prescribes for logistics managers the

optimal way of utilizing an expediting service andoffers important insights on factors that may affecttheir expediting decisions. More importantly, ourwork provides a useful model to help quantify thevalue of detailed supply information, hence can beused to evaluate whether and where to install someadvanced information technology equipment, such asRFID, in an effort to obtain such real-time supplyinformation. Further research should try to betterunderstand how differently an expediting serviceshould be used to manage “innovative” products(Fisher 1997) rather than “functional” products stud-ied in this research. Innovative products are charac-terized by less predictable demand, hence it isnecessary to factor demand uncertainty in the analy-sis. Also, Wang et al. (2010) suggests that demandand lead-time may be correlated. This may also affectthe use of expediting service. Finally, we assume thatthe expediting service has a deterministic lead-time. Itcould be interesting to see how to make use of a ser-vice with random expediting lead-time.

Acknowledgments

We are grateful to the anonymous reviewers, theSenior Editor, and Departmental Editor J. GeorgeShanthikumar for their valuable comments. We alsowould like to thank Jian Li for his help in improvingthe paper, and the support from the Boeing Center forTechnology, Information and Manufacturing(BCTIM).

Appendix A: Proofs

PROOF OF PROPOSITION 1. We show that in each ofthe seven regions when ce � �c, ��(t) = 0.

First, in region (a) t � �L, we have g1(t,0) � g1(�L, 0)by Lemma 1, and also g1(�L, 0) � 0 by the assump-tion that ce � �c. So we have that (FOC1) evaluated at� = 0 is non-negative since the expected total costETCa is convex in � . Hence, we obtain a corner solu-tion with ��(t) = 0 for ∀t � �L.

In region (b) through (e), the expected total cost isconvex in � and the first piece is linear with slope k1.Since k1 � 0 by the assumption that ce � �c, again��(t) = 0 for ∀t ∈ [�L, T�L].In region (f), the expected total cost is convex in �

andg4ðt; 0Þ � g4ðT � L; 0Þ ¼ ce � pðEL2 � LÞ � ðh þ pÞR L0 ðL � nÞf2ðnÞdn ¼ k1 � 0. So ��(t) = 0 for ∀t ∈[T�L, T].In region (g), the expected total cost is linear in �

with slope k2. Since

k2 ¼ ce � p EL2 � Lð Þ¼ ce þ h EL2 � Lð Þ � ðhþ pÞ

Z 1

Lðn� LÞf2ðnÞdn

þ ðhþ pÞZ L

0

ðL� nÞf2ðnÞdn

[ce þ h EL2 � Lð Þ � ðhþ pÞZ 1

Lðn� LÞf2ðnÞdn;

that is, k2 > k1. Hence k2 > 0. The optimal � thatachieves the minimum is 0. □

PROOF OF PROPOSITION 2. First we show that t and t2exist when ce < c. In the proof of Proposition 1 wehave seen that k1 < k2 hence k1 < 0 when k2 < 0.Since function g1 is decreasing in t, with g1(�L, 0) =k1 < 0 and g1(�∞, 0) > 0, there must exist a t suchthat t < �L and it satisfies g1ðt; 0Þ ¼ 0. Denote theleft hand side of Equation (7) as a function g5(t) sothat g5(t2) = 0. It is easy to show that g5(t) is decreas-ing in t. Since g5((T � L)/2) = k2 � 0, there mustexist a t2 such that t2 � ((T � L)/2) and it satisfiesg5(t2) = 0. Also, notice that g5(t) = g2(t, T � t) >g2(t, T � (t + L)) = g1(2t + L � T, 0). The inequalityfollows by the fact that g2 is increasing in � . Sinceg5(t2) = 0, g1(2t2 + L � T, 0) < 0. Recall thatg1ðt; 0Þ ¼ 0 and g1(t,�) is decreasing in t. Thereforet\2t2 þ L � T, that is, t2 is in fact greater than(T � L + t)/2).

Next, we prove the optimal policy in each of theseven regions. By Lemma 2 we see that the expectedtotal cost is convex in � . So the essence of theproof is to find in which interval the minimumexpected cost is achieved. We will explain thedetails for the first three regions and sketch theproof for the rest.

(a) t � �L: For any t � t we have g1ðt; 0Þ �g1ðt; 0Þ ¼ 0 by Equation (6). Since ETCa is convex in� for � ∈ [0, T], we get ��(t) = 0 for ∀t � t. Forany t ∈ [t, �L], since g1ðt; 0Þ � g1ðt; 0Þ ¼ 0 andg1ðt;TÞ[g1ð�L;TÞ ¼ g1ð�ðT þ LÞ;0Þ[g1ðt;0Þ ¼ 0, the


minimum expected cost is achieved by the solution to(FOC1).(b)–(c) �L � t � T

2 � L: First when t � ((T � L +t)/2), in the domain � ∈ [t + L, T � (t + L)]the derivative of the expected cost w.r.t. � is capturedby function g1(t,�). Since g1(t, t + L) = k1 < 0 andg1ðt;T � ðt þ LÞÞ ¼ g1ð2t þ L � T; 0Þ � g1ðt; 0Þ ¼ 0,the minimum expected cost is achieved in this inter-val and by the solution to (FOC1). When t is in region(c) and ððT � L þ tÞ=2Þ � t � minðt2; ðT=2Þ � LÞ, inthe domain � ∈ [T�(t + L), T � t] the derivative ofthe expected cost w.r.t. � is captured by functiong2(t, �). Since g2ðt;T � ðt þ LÞÞ ¼ g1ð2t þ L � T; 0Þ� g1ðt; 0Þ ¼ 0 and g2(t, T � t) = g5(t) > g5(t2) = 0, theminimum expected cost is achieved in this intervaland by the solution to (FOC2). If t2 < (T/2) � L, thenwhen t2 � t � (T/2) � L the minimum will beachieved in interval � ∈ [T � t,T]. The derivativeg3(t, �) satisfies g3(t, T � t) = g5(t) � g5(t2) = 0, thatis, g3(t, T � t) � 0, and g3(t, T) � 0 for t � 2t2while g3(t, T) � 0 for t � 2t2. So the minimumexpected cost is achieved by the solution to (FOC3)for t2 � t � min(2t2, (T/2) � L) and by ��ðtÞ ¼ T formin(2t2, (T/2) �L) � t � (T/2) � L.(d) (T/2) � L � t � ((T � L)/2): If t2 < (T/2) � L,

then from the proof for (c) we immediatelyhave (FOC3) is the governing equation. If t2 � T

2 � L,then for t � t2 the result follows from the proof for(c) and the optimal �� is given by (FOC3). Fort � t2 and in the interval � ∈ [t + L, T � t], the deriv-ative is captured by g2(t, �) and g2(t, T � t) > 0. Since

g2ðt; tþ LÞ ¼ ce � p EL2 � Lð Þ

� ðhþ pÞZ T�2t�L

0

ðT� 2t� L� nÞf2ðnÞdn

\ ce � p EL2 � Lð Þ � hþ pÞZ L

0

ðL� nÞf2ðnÞdn¼ k1;

that is, g2(t, t + L) < 0. The minimum is achieved bythe solution to (FOC2).(e) The minimum is achieved in the domain

� ∈ [t + L, T]. The derivative is captured by g3(t, �)and g3(t, t + L) < 0. The minimum is achieved by thesolution to (FOC3). For t � 2t2, we get a corner solu-tion and ��(t) = T.(f)–(g) The expected cost is convex in � and the

derivative at � = T is k2. Since k2 < 0 we get a cornersolution and ��(t) = T.Finally, comparing Equations (6) and (FOC1), we

have ��ðtÞ ¼ t � t. Comparing Equations (7) and(FOC3), we get ��(t) = T � 2t2 + t. □

PROOF OF PROPOSITION 3. First we show that t4 existswhen c\ce\�c. Define the left hand side of Equation

(8) as g6(t) so that g6(t4) = 0. It is easy to show thatg6(t) is increasing in t. Since

g6T

2� L

� �¼ ce þ hðEL2 � LÞ

� ðhþ pÞZ 1

Lðn� LÞf2ðnÞdn\0;

and g6T � L

2

� �¼ ce � pðEL2 � LÞ[0:

There must exist t4 ∈ ((T/2) � L, ((T � L)/2) suchthat g6(t4) = 0. So T � L < 2t4 + L < T.

The proof for region (a)–(b) is the same as that forProposition 2. In region (c), the minimum is achievedby the solution to (FOC1) for 0 � t � ((T � L + t)/2)and by the solution to (FOC2) for ((T � L + t)/2) �t � (T/2) � L (since t4 > (T/2) � L).

(d) (T/2) � L � t � ((T � L)/2): When t � t4,g2(t, t + L) = g6(t) < 0 by Equation (8). Also, g2(t,T � t) = g5(t) > g5((T � L)/2) = k2 > 0. The first in-equality follows by the fact that g5 being decreasingin t. Hence the minimum expected cost is achievedin this interval and the solution is characterized by(FOC2). When t � t4, since g4(t, T � (t + L)) =k1 < 0 and g4(t, t + L) = g6(t) > g6(t4) = 0, the mini-mum point is the solution to (FOC4).

(e) T� L2 � t � T � L: g4(t,T � (t + L)) > 0 and

g4(t, T � t) = k2 > 0, the minimum expected cost isachieved in the interval � ∈ [T � (t + L), T � t] atthe solution to (FOC4).

(f) T � L � t � T: when t � 2t4 + L,

g4ðt; 0Þ ¼ ce � pðEL2 � LÞ

� ðhþ pÞZ T�t

0

ðT � t� nÞf2ðnÞdn\ce � pðEL2 � LÞ

� ðhþ pÞZ T�ð2t4þLÞ

0

ðT � ð2t4 þ LÞ � nÞf2ðnÞdn

hence g4(t, 0) < 0. Also, g4(t, T � t) = k2 > 0. So theminimum point is the solution to (FOC4). Whent � 2t4 + L, g4(t, 0) > g6(t4) = 0, Since g4(t, 0) > 0 weget a corner solution with ��(t) = 0.

(g) t � T: k2 > 0 the expected cost is increasinghence is minimized at �� = 0.

Finally, comparing Equations (8) and (FOC4), we get��(t) = 2t4 + L � t whenever (FOC4) is the character-izing equation. □

PROOF OF PROPOSITION 4. Take total derivative ofEquations (6) and (7) w.r.t. ce, we have:


@t

@ce¼ 1

ðhþ pÞ 1� F2ð�tÞð Þ[0

@t2@ce

¼ 1

2ðhþ pÞ 1� F2ðT � 2t2Þð Þ[0

Part (i) follows.

Take total derivative of Equation (6) and (7) w.r.t. L,we have:

@t

@L¼ � h

ðhþ pÞ 1� F2ð�tÞð Þ\0

@t2@L

¼ �hþ ðhþ pÞF2ðLÞ2ðhþ pÞ 1� F2ðT � 2t2Þð Þ

So (ot2/oL) > 0 if F2(L) > (h/(h + π)) and (ot2/oL)< 0 otherwise. Part (ii) follows. □

PROOF OF PROPOSITION 5. The partial derivative of tw.r.t. ce and L have been derived in the proof ofProposition 4. Take total derivative of t4 w.r.t. ce andL, respectively, we have:

@t4@ce

¼ � 1

2ðhþ pÞF2ðT � 2t4 � LÞ\0

@t4@L

¼ � pþ ðhþ pÞF2ðT � 2t4 � LÞ2ðhþ pÞF2ðT � 2t4 � LÞ \0

@ð2t4 þ LÞ@L

¼ � pðhþ pÞF2ðT � 2t4 � LÞ\0

□

PROOF OF PROPOSITION 6. Take total derivative ofEquation (6) w.r.t. h, we have:

@t

@h¼

EL2 � L� R1�t n� ð�tÞð Þf2ðnÞdn

ðhþ pÞ 1� F2ð�tÞð Þ

¼ EL2 � L� ceþhðEL2�LÞhþp

ðhþ pÞ 1� F2ð�tÞð Þ ¼pðEL2 � LÞ � ce

ðhþ pÞ2 1� F2ð�tÞð Þ

where the second equality follows from Equation(6). Since ce � c we have (ot/oh) > 0.

Take total derivative of Equation (7) w.r.t. h, wehave:

where the second equality follows from Equation(7). Again since ce � c we have (ot2/oh) > 0.

Similarly, take total derivative of Equations (6) and(7) w.r.t. π, we have:

@t

@p¼

� R1�t n� ð�tÞð Þf2ðnÞdn

ðhþ pÞ 1� F2ð�tÞð Þ \0

@t2@p

¼R L0 ðL� nÞf2ðnÞdn�

R1T�2t2

n� ðT � 2t2Þð Þf2ðnÞdn2ðhþ pÞ 1� F2ðT � 2t2Þð Þ

¼ � ce þ hðEL2 � LÞ2ðhþ pÞ2 1� F2ðT � 2t2Þð Þ\0:

PROOF OF PROPOSITION 7. The partial derivative of tw.r.t. h and π have been derived in the proof ofProposition 6. Notice that

@t

@h¼ pðEL2 � LÞ � ce

ðhþ pÞ2 1� F2ð�tÞð Þ\0

in the intermediate expediting cost case ðc\ce\�c).

Take total derivative of t4 w.r.t. h and π, respec-tively, we have:

@t4@h

¼R T�2t4�L0 ðT � 2t4 � L� nÞf2ðnÞdn

2ðhþ pÞF2ðT � 2t4 � LÞ [0

@t4@p

¼ EL2 � Lþ R T�2t4�L0 ðT � 2t4 � L� nÞf2ðnÞdn

2ðhþ pÞF2ðT � 2t4 � LÞ [0:

□

@t2@h

¼ EL2 � Lþ R L

0 ðL� �Þf2ð�Þd� �R1T�2t2

� � ðT � 2t2Þð Þf2ð�Þd�2ðhþ �Þ 1� F2ðT � 2t2Þð Þ

¼ EL2 � L� ceþhðEL2�LÞhþ�

2ðhþ �Þ 1� F2ðT � 2t2Þð Þ ¼�ðEL2 � LÞ � ce

2ðhþ �Þ2 1� F2ðT � 2t2Þð Þ


PROOF OF PROPOSITION 8. Define function J(L2, t):=(L2 � t)+, which is convex in L2. Since L2

a � icL2b, we

have for each t:

E½JðLa2; tÞ� � E½JðLb2; tÞ�that is

Z 1

tðn� tÞf2aðnÞdn �

Z 1

tðn� tÞf2bðnÞdn:

Denote ta and tb to be the corresponding t under L2a

and L2b respectively. Combining with Equation (6)

and the assumption that EL2a ¼ EL2

b, we have:

0 ¼ ce þ hðEL2a � LÞ � ðhþ pÞZ 1

�tan� ð�taÞð Þf2aðnÞdn

� ce þ hðEL2b � LÞ � ðhþ pÞZ 1

�tan� ð�taÞð Þf2bðnÞdn:

Since the left hand side of Equation (6) is decreasingin t, we have ta � tb.

Define function K(L2, t) := (t � L2)+, which is convex

in L2. Denote t4a and t4

b to be the corresponding t4under L2

a and L2b respectively. Since L2

a�icL2b, we get

E½KðL2a; tÞ� � E½KðL2b; tÞ�. Combine with Equation (8)and the assumption that EL2

a ¼ EL2b, we have:

0 ¼ ce � pðEL2a � LÞ

� ðhþ pÞZ T�2t4

a�L

0

ðT � 2t4a � L� nÞf2aðnÞdn

� ce � pðEL2b � LÞ

� ðhþ pÞZ T�2t4

a�L

0

ðT � 2t4a � L� nÞf2bðnÞdn:

Since g6(t) is increasing in t, we get t4a � t4

b. □

PROOF OF PROPOSITION 9. When t < �L, the expectedtotal cost is:

ETCa ¼ ce� � h�ðtþ LÞ þ 1

2h�2þ

þZ ��ðtþLÞ

0

�hðT � �Þ

�� ðtþ Lþ eÞ

�

þ 1

2hðT � �Þ2

�g2ðeÞde

þZ T�ðtþLÞ

��ðtþLÞ

�1

2p

�tþ Lþ e� �

�2

þ 1

2h

�T � ðtþ Lþ eÞ

�2�g2ðeÞde

þZ 1

T�ðtþLÞ

�1

2pðT � �Þ2

þ pðT � �Þðtþ Lþ e� TÞ�g2ðeÞde

with first-order condition

@ETCa

@�¼ ce þ hðEL2 � LÞ

þ ðhþ pÞZ 1

��ðtþLÞð� � ðtþ LÞ � eÞg2ðeÞde ¼ 0:

ð1Þ

The optimal expediting quantity when t < �L is:�� = 0 if ce[pðEL2 � LÞ; otherwise,

�� ¼ t� t; if t[t0; if t\t:

�

When �L < t < T � L, the expedited units willarrive within the demand season. When � < t + L,the expedited units cannot satisfy the backlog whenthey arrive. The expected total cost is:

ETCb1 ¼ ce� þ 1

2pðtþ LÞ2þ

þZ T�ðtþLÞ

0

�1

2pðtþ L� � þ tþ L� � þ eÞe

þ 1

2h

�T� ðtþ Lþ eÞ

�2�g2ðeÞde

þZ 1

T�ðtþLÞ

�1

2pðtþ L� � þT� �Þ

�T� ðtþ LÞ

�

þ pðT� �Þðtþ Lþ e�TÞ�g2ðeÞde

with the first-order derivative @ETCb1=@� ¼ ce

�pðEL2 � LÞ. When � > t + L, the expedited unitscan satisfy the backlog when they arrive. Theexpected cost is:

ETCb2 ¼ ce� þ 1

2pðtþ LÞ2 þ 1

2h � � ðtþ LÞð Þ2

þZ ��ðtþLÞ

0

�hðT � �Þ

�� ðtþ Lþ eÞ

�

þ 1

2hðT � �Þ2

�g2ðeÞde

þZ T�ðtþLÞ

��ðtþLÞ

�1

2pðtþ Lþ e� �Þ2

þ 1

2h

�T � ðtþ Lþ eÞ

�2�g2ðeÞde

þZ 1

T�ðtþLÞ

�1

2pðT � �Þ2

þ pðT � �Þðtþ Lþ e� TÞ�g2ðeÞde

with the first-order condition (1). Since the expectedcost function is continuous, if ce[pðEL2 � LÞ, then


�� = 0. Otherwise, �� ¼ maxft � t;Tg. Whent > T � L, the expedited units will arrive after thedemand season ends. The regular units will arriveeven later. The expected cost is:

ETCc ¼ ce� þ 1

2pT2 þ pTðtþ L� TÞ

þZ 1

0

pðT � �Þeg2ðeÞde

with the first order derivative @ETCc=@� ¼ce � pðEL2 � LÞ. The optimal expediting quantity is�� = 0 if ce[pðEL2 � LÞ; and �� = T otherwise. □

Appendix B: Expressions for Expected Total Costs

ETCaðt; �Þ ¼ce� þ h�ð�t� LÞ þ 1

2h�2 þ

Z ��t

0

hðT � �Þð� � t� nÞ þ 1

2hðT � �Þ2

� �f2ðnÞdn

þZ T�t

��t

1

2p n� ð� � tÞð Þ2þ 1

2hðT � t� nÞ2

� �f2ðnÞdnþ

Z 1

T�t

1

2pðT � �Þ2 þ pðT � �Þ n� ðT � tÞð Þ

� �f2ðnÞdn:

ETCb1ðt; �Þ ¼ ce� þZ �t

0

hðT � �Þð�t� nÞ þ 1

2hðT � �Þ2 þ h�ðT � � � t� LÞ þ 1

2h�2

� �f2ðnÞdn

þZ L

�t

1

2pðnþ tÞ2 þ 1

2hðT � � � n� tÞ2 þ h�ðT � � � t� LÞ þ 1

2h�2

� �f2ðnÞdn

þZ T�t

L

1

2pðtþ LÞ2 þ 1

2p 2ðtþ L� �Þ þ n� Lð Þðn� LÞ þ 1

2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2pðtþ L� � þ T � �ÞðT � t� LÞ þ pðT � �Þðn� T þ tÞ

� �f2ðnÞdn:


0


2hðT � �Þ2 þ h�ðT � � � t� LÞ þ 1

2h�2

� �f2ðnÞdn

þZ L

�t

1

2pðnþ tÞ2 þ 1

2hðT � � � n� tÞ2 þ h�ðT � � � t� LÞ þ 1

2h�2

� �f2ðnÞdn

þZ ��t

L

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ hðT � �Þð� � t� nÞ þ 1

2hðT � �Þ2

� �f2ðnÞdn

þZ T�t

��t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1

2pðn� � þ tÞ2 þ 1

2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1

2pðT � �Þ2 þ pðT � �Þðn� T þ tÞ

� �f2ðnÞdn:


0


2hðT � �Þ2 þ 1

2pðtþ L� T þ �Þ2 þ 1

2hðT � t� LÞ2

� �f2ðnÞdn

þZ T��t

�t

1

2pðtþ nÞ2 þ 1

2hðT � � � t� nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ L

T��t

1

2pðtþ nÞ2 þ 1

2p 2ðtþ n� T þ �Þ þ L� nð ÞðL� nÞ þ 1

2hðT � t� LÞ2

� �f2ðnÞdn

þZ ��t

L

1

2pðtþ LÞ2 þ 1


2hðT � �Þ2

� �f2ðnÞdn

þZ T�t

��t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1

2pðnþ t� �Þ2 þ 1

2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1

2pðT � �Þ2 þ pðT � �Þðnþ t� TÞ

� �f2ðnÞdn:


ETCc1ðt; �Þ ¼ ce� þZ L

0

1

2pðtþ nÞ2 þ 1

2hðT � � � t� nÞ2 þ h�ðT � � � t� LÞ þ 1

2h�2

� �f2ðnÞdn

þZ T�t

L

1

2pðtþ LÞ2 þ 1


2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2pðtþ L� � þ T � �ÞðT � t� LÞ þ pðT � �Þðnþ t� TÞ

� �f2ðnÞdn:


0

1

2pðtþ nÞ2 þ 1

2hðT � � � t� nÞ2 þ h�ðT � � � t� LÞ þ 1

2h�2

� �f2ðnÞdn

þZ ��t

L

1

2pðtþ LÞ2 þ 1


2hðT � �Þ2

� �f2ðnÞdn

þZ T�t

��t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1

2pðnþ t� �Þ2 þ 1

2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1


� �f2ðnÞdn:

ETCc3ðt; �Þ ¼Z T��t

0

1

2pðtþ nÞ2 þ 1

2hðT � � � t� nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ L

T��t

1

2pðtþ nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ ��t

L

1

2pðtþ LÞ2 þ 1


2hðT � �Þ2

� �f2ðnÞdn

þZ T�t

��t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1


2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1


� �f2ðnÞdnþ ce�:


0

1

2pðtþ nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ ��t

L

1

2pðtþ LÞ2 þ 1


2hðT � �Þ2

� �f2ðnÞdn

þZ T�t

��t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1


2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1

2hð� � t� LÞ2 þ 1


� �f2ðnÞdn:

ETCd1(t, �) same as ETCc1(t, �).

ETCd2ðt; �Þ ¼Z T��t

0

1

2pðtþ nÞ2 þ 1

2hðT � � � t� nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ L

T��t

1

2pðtþ nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ T�t

L

1

2pðtþ LÞ2 þ 1


2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1


� �f2ðnÞdnþ ce�:

ETCd3(t, �) same as ETCc3(t, �).ETCd4(t, �) same as ETCc4(t, �).


ETCe1(t, �) same as ETCc1(t, �).ETCe2(t,�) same as ETCd2(t, �).

ETCe3ðt; �Þ ¼ ce� þZ L

0

1

2pðtþ nÞ2 þ 1


2hðT � t� LÞ2

� �f2ðnÞdn

þZ T�t

L

1

2pðtþ LÞ2 þ 1

22ðtþ L� �Þ þ n� Lð Þðn� LÞ þ 1

2hðT � t� nÞ2

� �f2ðnÞdn

þZ 1

T�t

1

2pðtþ LÞ2 þ 1


� �f2ðnÞdn:

ETCe4(t, �) same as ETCc4(t, �).

ETCf1ðt; �Þ ¼ ce� þZ T�t��

0

1

2pðtþ nÞ2 þ 1

2hðT � � � t� nÞ2 þ 1

2p�2 þ p�ðtþ L� TÞ

� �f2ðnÞdn

þZ T�t

T�t��

1

2pðtþ nÞ2 þ 1

2pðtþ n� T þ � þ �ÞðT � t� nÞ þ p�ðtþ L� TÞ

� �f2ðnÞdn

þZ L

T�t

1

2pT2 þ pTðnþ t� TÞ þ p�ðL� nÞ

� �f2ðnÞdn

þZ 1

L

1

2pT2 þ pTðtþ L� TÞ þ pðT � �Þðn� LÞ

� �f2ðnÞdn:

ETCf2ðt; �Þ ¼ ce� þZ T�t

0

1

2pðtþ nÞ2 þ 1

2pðtþ n� T þ 2�ÞðT � t� nÞ þ p�ðtþ L� TÞ

� �f2ðnÞdn

þZ L

T�t

1

2pT2 þ pTðnþ t� TÞ þ p�ðL� nÞ

� �f2ðnÞdn

þZ 1

L

1

2pT2 þ pTðLþ t� TÞ þ pðT � �Þðn� LÞ

� �f2ðnÞdn:

ETCgðt; �Þ ¼ ce� þZ L

0

1

2pT2 þ pTðt� T þ nÞ þ p�ðL� nÞ

� �f2ðnÞdn

þZ 1

L

1

2pT2 þ pTðt� T þ LÞ þ pðT � �Þðn� LÞ

� �f2ðnÞdn:

Notes1It can be shown that ðT � L� tÞ=2 < T, since t > �ðT � LÞby assumption. Also, in the proof for Proposition 3, it isshown that t4 < ðT � LÞ=2. So ðt4 þ LÞðT þ LÞ=2 < T, wherethe last inequality follows from the model assumptionT > 2L.

2Expression (14) holds when eS[ðpEe � ceÞ = ðh þ pÞ. How-ever, one can show that the second term in Equation (14)changes to p11eL/(p11 + p10) when the condition does nothold. Therefore the insights derived continue to hold. Asimilar thing is true for the case when L1 = l1S.

ReferencesAllen, S. G., D. A. D’Esopo. 1968. An ordering policy for stock

items when delivery can be expedited. Oper. Res. 16(4): 880–883.

Babich, V., A. N. Burnetas, P. H. Ritchken. 2007. Competition anddiversification effects in supply chains with supplier defaultrisk. Manuf. Ser. Oper. Manage. 9(1): 123–146.

Bookbinder, J. H., M. Cakanyildirim. 1999. Random lead timesand expedited orders in (Q, r) inventory systems. Eur. J. Oper.Res. 115(2): 300–313.

Chang, H.-C. 2004. A note on the EPQ model with shortages andvariable lead time. Inf. Manage. Sci. 15(1): 61–67.

Chopra, S., G. Reinhardt, M. Dada. 2004. The effect of lead timeuncertainty on safety stocks. Decision Sci. 35(1): 1–24.

Crone, M. 2006. Are global supply chains too risky? A practitioner’sperspective. Supply Chain Manage. Rev. May–June: 28–35.

Duran, A., G. Gutierrez, R. I. Zequeira. 2004. A continuous reviewinventory model with order expediting. Int. J. Prod. Econ. 87(2): 157–169.

Ehrhardt, R. 1984. (s, S) policies for a dynamic inventory modelwith stochastic lead times. Oper. Res. 32(1): 121–132.

Fisher, M. L. 1997. What is the right supply chain for your prod-uct? Harvard Bus. Rev. March–April: 105–116.

Gardner, C. 2004. Automotive aftermarket RFID. White paper,MEMA Information Services Council, Research Triangle Park,NC.

Gaukler, G. M., O. Ozer, W. H. Hausman. 2008. Order progressinformation: Improved dynamic emergency ordering policies.Prod. Oper. Manag. 17(6): 599–613.


Jain, A. 2006. Responding to shipment delays: The roles of opera-tional flexibility and lead-time visibility. Working Paper,Indian School of Business, Hyderabad, India.

Kaplan, R. S. 1970. A dynamic inventory model with stochasticlead times. Manage. Sci. 16(7): 491–507.

Kouvelis, P., J. Li. 2008. Flexible backup supply and themanagementof lead-time uncertainty. Prod. Oper. Manag. 17(2): 184–199.

Kraiselburd, S., R. Pibernik, A. Raman. 2011. The manufacturer’sincentive to reduce lead times. Prod. Oper. Manage, doi:10.1111/j.1937-5956.2010.01192.x.

Lawson, D. G., E. L. Porteus. 2000. Multistage inventory manage-ment with expediting. Oper. Res. 48(6): 878–893.

Liberatore, M. J. 1979. The EOQ model under stochastic lead time.Oper. Res. 27(2): 391–396.

Moinzadeh, K., S. Nahmias. 1988. A continuous review model foran inventory system with two supply modes. Manage. Sci. 34(6): 761–773.

Porteus, E. L. 2002. Foundations of Stochastic Inventory Theory. Stan-ford University Press, Palo Alto, CA.

Ramasesh, R. V., J. K. Ord, J. C. Hayya, A. Pan. 1991. Sole versusdual sourcing in stochastic lead-time (s, Q) inventory models.Manage. Sci. 37(4): 428–443.

Ross, S. 1983. Stochastic Processes. John Wiley, New York.

Ryu, S. W., K. K. Lee. 2003. A stochastic inventory model of dualsourced supply chain with lead-time reduction. Int. J. Prod.Econ. 81–82: 513–524.

Song, J-S. 1994. The effect of leadtime uncertainty in a simplestochastic inventory model. Manage. Sci. 40(5): 603–613.

Stalk, G. Jr. 2006. Surviving the China riptide. Supply ChainManage. Rev. May–June: 18–26.

Wang, P., W. Zinn, K. L. Croxton. 2010. Sizing inventory when leadtime and demand are correlated. Prod. Oper. Manage. 19(4):480–484.

Wolfe, E., P. Alling, H. Schwefel, S. Brown. 2003. Supply chaintechnology—tracking to the future. Bear Stearns equityresearch report, Bear Stearns.


on optimal expediting policy for supply systems with uncertain lead-times

Documents