ChenGavirneni2010_ScheduledOrderingtoImprovethePerformanceofDistributionSC_MgmtSciV56n9p1615

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MANAGEMENT SCIENCEVol. 56, No. 9, September 2010, pp. 1615–1632issn 0025-1909 eissn 1526-5501 10 5609 1615

inf orms ®

doi 10.1287/mnsc.1100.1196© 2010 INFORMS

Using Scheduled Ordering to Improve the

Performance of Distribution Supply ChainsLucy Gongtao Chen

NUS Business School, National University of Singapore, Singapore 119245, [email protected]

Srinagesh Gavirneni Johnson Graduate School of Management, Cornell University, Ithaca, New York 14853,

[email protected]

W e study a supply chain with one supplier and many retailers that face exogenous end-customer demands.The supplier and the retailers all try to minimize their own inventory-related costs. In contrast to the

retailers’ newsvendor-type ordering behavior (under which retailers may place orders freely in every period),we propose two scheduled ordering policies: the scheduled balanced ordering policy (SBOP) and the scheduledsynchronized ordering policy (SSOP). Under both the SBOP and SSOP, retailers are allowed to order freely onlyin one period of an ordering cycle, and receive xed shipments in other periods. Retailers take turns to orderfreely under the SBOP, while under the SSOP all retailers order freely in the same period. With the averagesupply chain cost per period as the performance measure, we identify mathematical conditions under whichscheduled ordering policies outperform the newsvendor-type ordering. Through a large-scale numerical study,we nd that scheduled ordering policies are most effective when (i) the supplier’s holding and expediting costsare high and the retailer’s backorder cost is small, (ii) the end-customer demand variance and correlation arehigh, and (iii) the supplier’s capacity is high. In addition, we observe that the behavior of the SSOP oftencomplements that of the SBOP. Whereas the SBOP is better than SSOP when the supplier’s capacity is lowand when the end-customer demand correlation level is high, the SSOP is better when the opposite conditionsprevail.

Key words: scheduled ordering policy; distribution supply chains; inventory cost History : Received January 24, 2008; accepted April 26, 2010, by Paul H. Zipkin, operations and supply chain

management. Published online in Articles in Advance July 2, 2010.

1. IntroductionDistribution supply chains are prevalent in the globaleconomy because almost every company is faced withchallenges associated with distributing its products.In the $3 8 trillion U.S. retail sector, it is estimatedthat distribution expenses (a key portion of whichare inventory carrying costs) often exceed 10% of acompany’s gross sales (Hudson 2003). The retail sec-tor is notorious for intense competition with razor-thin margins, and rms are compelled to identifyinnovative ways to improve the efciency of their

distribution systems to gain or maintain a compet-itive edge. Distribution systems present exceptionalmanagement challenges because of the embeddedvariance and complexity, and have attracted muchattention among operations management researchers.

In a decentralized distribution supply chain withstochastic end-customer demands, newsvendor-typeordering by the retailer is often the expected outcome,and this is the premise that Lee et al. (1997), Cachonand Lariviere (1999), and Tsay (1999) start with. Leeet al. (1997) argue that the supplier will use a quan-tity exibility (QF) contract to achieve an equitable

allocation of risk. Under a QF contract, the retailerprovides the supplier a forecast of its intended pur-chase, and any adjustments to this quantity must beno larger than a prespecied percentage. Tsay (1999)describes various examples of QF in practice and per-forms a detailed analysis of its effect on supply chainperformance. In contrast to this approach and moti-vated by real-world examples, Zhu et al. (2010) pro-pose the periodic exibility (PF) strategy under whichthe retailer has full control of his inventory levels inone period of every cycle and no control in other

periods. They performed a detailed analysis and anextensive computational study to demonstrate that PFstrategies work very well. We adapt these periodicexibility strategies to the distribution supply chain,propose scheduled ordering policies, and demon-strate their effectiveness in reducing supplier uncer-tainty and improving supply chain performance.

In the supply chain management literature, thereare several articles that focus on serial supply chains(Clark and Scarf 1960, Bollapragada et al. 2004, Parkerand Kapuscinski 2004) and on assembly supply chains(Schmidt and Nahmias 1985, Rosling 1989, Decroix

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Chen and Gavirneni: Using Scheduled Ordering to Improve the Performance of Distribution Supply Chains1616 Management Science 56(9), pp. 1615–1632, ©2010 INFORMS

and Zipkin 2005, Zhang 2006). On the subject ofdistribution supply chains, although there is a largeliterature (Schwarz 1973, Eppen and Schrage 1981,Federgruen and Zipkin 1984, Roundy 1985, Jackson1988, Axsäter 1993, Graves 1996), the results are lesscomplete and robust than for those simpler systems.

Chen et al. (2001) study coordination mechanisms thatcan achieve system optimum in a decentralized sys-tem when the end-customer demand is determinis-tic. Gurnani (2001) studies quantity discount pricingstructures that encourage the buyers to coordinateor consolidate the timing of their orders, and iden-tify conditions under which this reduces the totalsystem cost. Zhang (2005) examines the impact oftransshipment on the supply chain members’ perfor-mance. Gullu et al. (2005) analyze a decentralizedsupply chain where there exists partial cooperation between the two retailers. They derive unique equi-librium order-up-to levels for the retailers under mild

conditions. Goel and Gutierrez (2008) study how theprice difference between the spot and future commod-ity markets can be utilized to enhance the efciencyof the supply chain through better procurement anddistribution policies.

We add to this growing literature by analyzingthe effectiveness of scheduled ordering policies inimproving the efciency of decentralized distributionsupply chains. Though arguably not as efcient ascentralized systems, decentralized inventory controlis still widely practiced, because managers are afraidof ceding control to a central decision maker. Decen-tralized rms are looking for ways to smooth the owof materials and information in their supply chains.Strategies that encourage retailers to follow a mutu-ally agreed upon ordering pattern have become pop-ular. Griller’s Pride (http://www.grillerspride.com), aglatt kosher meat and poultry butchery that servesAtlanta and the southeast of the United States, dividestheir customers into groups according to geograph-ical location (zip code). Products are delivered toeach group on a separate day of the week. GATX, athird-party logistics company that serves 6,500 BP andChevron service stations, divides the service stationsinto ve groups, each of which is required to orderonly on a preassigned day of the week. So 20% of theservice stations order on Monday, another 20% orderon Tuesday, etc. (Andel 1995).

Such retailer ordering patterns, known as sched-uled ordering policies, were rst studied by Leeet al. (1997) and Cachon (1999). They characterizedthem into three types, namely, randomized, balanced,and synchronized. Under the randomized policy, theretailers are free to place an order whenever theywant. When using the balanced policy, the retailersare evenly spaced out and they are only allowed toplace an order when their turn comes. On the other

hand, under the synchronized policy, all the retailersplace orders at the same time. Both Lee et al. (1997)and Cachon (1999) showed that a balanced orderingpolicy works better than the synchronized policy. Leeet al. (1997) also showed that the synchronized pol-icy is worse than the randomized policy. Whereas Lee

et al. (1997) concentrated on analyzing the scheduledordering policies in terms of the supplier’s demandvariance, Cachon (1999) placed emphasis on how theretailers’ order interval length and their batch sizeimpact the supply chain performance. However, nei-ther of these two papers mathematically compares theperformance of the three policies from the perspectiveof the total supply chain. They also do not incorpo-rate information sharing and place emphasis on sup-ply chains in which the retailers incur signicant xedordering costs.

In contrast to Lee et al. (1997) and Cachon (1999),who modeled a signicant xed ordering cost, we

study a supply chain without xed ordering cost.Whereas for some companies xed ordering cost isa big concern when making inventory replenishmentdecisions, for others, it is not because it is verylow compared to other inventory related costs. Forexample, Farnell UK (http://uk.farnell.com), an elec-tronic components distributor, provides free same-day dispatching (next day delivery) and chargesvery low shipping costs even if faster delivery isrequested. Another electronic components distrib-utor, Mouser Electronics (http://www.mouser.com),charges no handling fee and very low shipping costs.These companies make xed cost a very minor con-cern for retailers. As a result, retailers are ready toorder and receive shipments in every period. Wemodel such a situation. Under this setting, in contrastto the free ordering policy (FOP) where retailers orderfreely in every period, we evaluate the following twoscheduled ordering policies:

(1) Scheduled Balanced Ordering Policy(SBOP). Thispolicy operates with an ordering cycle of m periods.In every period, n retailers are allowed to order freely.(Altogether, we have N =m ·n retailers.) Each retailermay order freely only in one period of a cycle. In theremaining periods of the cycle, a predetermined xedquantity is shipped to him.

(2) Scheduled Synchronized Ordering Policy (SSOP).This policy also operates with an ordering cycle of mperiods. Different from the SBOP, here all of the N retailers are scheduled to order freely in the same andonly one period of a cycle. In the remaining periods ofthe cycle, each retailer receives a predetermined xedquantity .

In the context of a milk supply chain, Glenn et al.(2002) postulate that the supplier would like its cus-tomers to order the same amount in every period.Any changes necessary to accommodate randomness






Figure 1 Demand Standard Deviation the Supplier Faces Under FOP,SBOP, and SSOP

1FOP

(d 1,2 + D 1,1 – )+ +

(d 1,2 + D 1,1 – )+ + (d 2,2 + D 2,1 – )+

(d 2,1 + D 2,2 – )+ +

D 1,2 + D 2,2

SBOP

SSOP

Demand:

Demand:

Demand:

Std. dev.:

Std. dev.:

Std. dev.:

1

1

2

2

2

0

≈

2

2

≈

D 1,1 + D 2,1

2 2

≈

Intuitively, xing the retailers’ order amount in cer-tain periods may reduce the demand variability faced by the supplier, and different scheduled ordering maylead to different degrees of variability reduction. Letus rst take a look at how the SBOP and SSOP canpossibly be effective by considering a supply chainwith one supplier and two retailers. In this example,the end-customer demands are i.i.d. Figure 1 showsthe demand standard deviations faced by the supplierunder the FOP, SBOP, and SSOP in each period of acycle. Di j (di j ) denotes the generic random (realized)end-customer demand for retailer i in period j of acycle. Here j =1 2 because there are only two retail-ers, and thus two periods in every cycle.

Under the FOP, in each period the retailer demandis equal to the sum of the end-customer demandsincurred at the two retailers, which are assumed to beindependent. Hence, the standard deviation faced bythe supplier is √ 2 , where is the standard devia-tion of the end-customer demand. Under the SBOP, inevery period the retailer demand can be decomposedinto the following two parts. The demand from oneretailer is the xed quantity . The demand from theother retailer is the inventory reduction of that retailerover one cycle, which is the positive part of the cumu-lative two periods of end-customer demands minus . Weassume that the realized end-customer demand di j ,which is known to the retailer, is conveyed to thesupplier. Hence, the demand standard deviation thesupplier faces in every period is approximately ,which is smaller than before. Under the SSOP, thesupplier faces a demand standard deviation of 0 in thenonordering period (when the retailers receive xedshipments), and √ 2 in the free-ordering period. Sothe average demand standard deviation she faces is√ 2 / 2 per period, which is half of that seen underthe FOP.

In performing a more rigorous analysis than theabove, we make the following assumptions: (i) the

production at the supplier takes one period, whichmeans one period of lead time; (ii) the supplier hasinnite capacity; (iii) the retailers receive their ship-ments immediately; (iv) the supplier fullls all retailerorders every period. If the supplier does not haveenough inventory on hand, she will get the product

from an outside source immediately at a higher expe-diting cost, and then ship it to the retailers right away.This implies a high service standard at the supplierside. (Gavirneni et al. (1999) and Lee et al. (2000) alsomake the same assumption.)

3. Model Formulation andStructural Analysis

The end-customer demand that retailer i faces inperiod j of a cycle is assumed to be Di j . We willuse a generic random variable to refer to any Di j .The realization of the random demand Di j is di j .

The demand that the supplier faces in period t isdenoted as t . For ease of exposition, whenever thereis no confusion, we drop the time index t . Let Z zand Z z be the cumulative distribution function(c.d.f.) and probability distribution function of a ran-dom variable Z, respectively. Dene Z z =1− Z zand loss function Z R = R Z z dz . Also let yi andY i , i = f b s , be the inventory levels after order-ing/production at the beginning of a period for theretailers and the supplier, respectively. Throughoutthis paper, subscripts f , b, and s correspond to theFOP, SBOP, and SSOP, respectively, and imply that weare talking about a decentralized system.

3.1. Centralized SystemBefore we discuss the ordering strategies in a decen-tralized supply chain, we rst analyze the systemwhere a central planner makes the inventory deci-sions for all parties. Theoretically, a multiechelonsystem can achieve highest cost reduction by switch-ing to centralized decision making. We hence bench-mark the performance of scheduled ordering policieswith the centralized system in terms of cost reduction.

We dene the supplier and the retailers’ echeloninventory holding costs as H s =hs and H r =hr −hs.The supplier has the option to expedite if her inven-tory is not sufcient to raise the retailers’ inventoriesto desired levels. Note that the decision of expedit-ing will be made based on the system parameters.For example, if the expediting cost is larger than theretailer’s backorder cost, then it is not economicalto expedite in a centralized system. Let Y co denotethe supplier’s echelon inventory level at the begin-ning of every period after expediting, and let Y csand Y cr i denote the echelon inventory levels of thesupplier and retailer i, respectively, after produc-ing/ordering at the beginning of a period. Similarly,



Chen and Gavirneni: Using Scheduled Ordering to Improve the Performance of Distribution Supply ChainsManagement Science 56(9), pp. 1615–1632, ©2010 INFORMS 1619

let Xs and Xr i denote the echelon inventory levels ofthe supplier and retailer i, respectively, before expe-diting/ordering at the beginning of every period. Forease of exposition, we use boldfaced X and Yc torepresent the vector of inventory levels before andafter ordering/expediting/producing. We also dene

Lci x =H r x −Di + + r +hs x −Di −, where Di

refers to the end-customer demand at retailer i withsuppressed time index. The cost-to-go function fromperiod t to the end of the horizon is

f ct X s X r 1 X r N = min

Yc A Xvc

t Yc

vct Yc = s +hs Y co − sY cs + s −hs E

+EN

i=1Lc

i Y cr i

+f ct+1 Y cs − Y c

r 1 −D1 Y cr N −DN

f cT +1 X = sXs

where A X = Y cs ≥Y co ≥Xs Y co ≥ Y r i Y r i ≥Xr i ,and f c

T +1 is the end-of-horizon cost that initiates thedynamic formulation.

Note that this problem is similar to the one dis-cussed in Clark and Scarf (1960), except that here thesupplier has the option to expedite. As in Clark andScarf (1960), we assume that a lack of balance doesnot occur, which means that the inventory replen-ishment strategy is selected subject to A X = Y cs ≥Y co ≥Xs Y co ≥ Y r i . With this, we have the following

result for the inventory policy in a centralized system.Theorem 1. In a centralized system, the optimal inven-

tory policy is one such that the supplier expedites up toY co , produces up to Y cs , and replenishes the retailers’inventories up to Y cr i for i =1 N .

Proof. The proof follows from Clark and Scarf(1960) and Zhu et al. (2010). We omit the detailshere.

We now proceed to analyze the decentralized sys-tem under the FOP, SBOP, and SSOP.

3.2. Free Ordering PolicyUnder the FOP, it is optimal for the retailers touse a stationary order-up-to policy because they canorder freely in every period, and the end-customerdemand is stationary. Consequently, the end-customerdemands are transmitted to the supplier unaltered,and it is also optimal for the supplier to use a station-ary order-up-to policy. It is well established that theoptimal order-up-to level yf for the retailer is

yf = −1 r

r +hr

and the optimal order-up-to level Y f for the sup-plier is

Y f = −1 s

s +hs

where is the sum of N identical end-customerdemands from the distribution .

3.3. Scheduled Balanced Ordering PolicyUnder the SBOP, we assume that the ordering cyclelength is m periods, and that n retailers order freelyin each and only one period of the cycle. As such, thetotal number of retailers is N =m ·n.

Retailers’ Problem. Without loss of generality, wedene the period that a retailer can order freely asthe rst period of a cycle for him. Assume that at the beginning of any cycle, the retailer’s inventory levelis yb. In the next m −1 periods, he agrees to receive axed shipment in every period. The one-cycle costl y b for the retailer i given an inventory level of yb is

l y b =m−1

k=0hr yb +k −

k+1

j =1Di j

+

+ r

k+1

j =1Di j −k −yb

+

The cost-to-go function from cycle t to the end of thehorizon is

f t x =minyb≥x

vt y b

vt y b =E l y b +f t+1 yb + m −1 −m

j =1Di j

Supplier’s Problem.At the beginning of every period(say t − 1) when the supplier makes the produc-tion decision, she is anticipating a retailer orderat the beginning of next period ( t), which can bedecomposed into two parts. The rst part is thexed shipment to all the retailers that cannot orderfreely, n m −1 . The second part is from the retail-ers that can order freely, each of whom will makean order equal to the inventory reduction overthe periods since his last free-ordering epoch. Notethat because of information sharing, all past end-customer demands (up till the beginning of periodt −1) at every “free” retailer are known to the sup-plier. The end-customer demands unknown to thesupplier (at the beginning of period t − 1) corre-spond to the ones in the last period of an order-ing cycle at these “free” retailers, i.e., the demandin period t −1. Because the end-customer demandis stationary, the retailers will use a stationary order-up-to level. This implies that the demand from each“free” retailer is equal to the inventory reduction




at that retailer between two free-ordering epochs,which is m−1

j =1 di j +Di m − m −1 +. So the sup-plier’s demand in period t is t = i t

m−1j =1 di j +

Di m − m −1 ++n m −1 , where t is the set ofretailers scheduled to order freely in period t; that is,these are the retailers for whom t −1 is the mth period

in their cycle. The one-period cost for the suppliergiven the inventory level Y b is, therefore,

L Y b =hs Y b −i t

m−1

j =1di j +Di m − m −1

+

+n m −1+

+ si t

m−1

j =1di j +Di m − m −1

+

+n m

−1

−Y b

+

The cost-to-go function from period t to the end ofthe horizon is

F t x =minY b≥x

V t Y b

V t Y b =E L Y b +F t+1 Y b−i t

m−1

j =1di j

+Di m −m −1++n m −1

+

3.4. Scheduled Synchronized Ordering PolicyRetailers’ Problem.Because the retailer’s optimal pol-

icy and resulting cost under the SSOP are the sameas those under the SBOP, we do not reiterate themhere. However, for clarity we will use ys to denotethe order-up-to level under the SSOP to differentiateit from yb used in the SBOP scenario.

Supplier’s Problem. Under the SSOP, in the periodsthat all the retailers order freely, the supplier facesa demand of = N

i=1m−1j =1 di j +Di m − m −1 +,

whereas in other periods, =N . Thus, given inven-tory level Y s and Di m i =1 m , the supplier’s one-cycle cost is

L̃ Y s =hs Y s −N

i=1

m−1

j =1di j +Di m − m −1

+ +

+ s

N

i=1

m−1

j =1di j +Di m − m −1

+−Y s

+

+m−1

k=1hs Y s −

N

i=1

m−1

j =1di j +Di m

− m −1+−kN

+(1)

Note that the rst two terms correspond to theholding and expediting costs in the period whenthe retailers can order freely, and the last termcorresponds to the holding cost in the remaining peri-ods of a cycle when the retailers receive xed ship-ments. Only holding cost but not expediting cost can

be incurred by the supplier during those nonorderingperiods.The cost-to-go function from cycle t to the end of

the horizon is

F t x =minY s≥x

V t Y s

V t Y s =E L̃ Y s +F t+1 Y s −N

i=1

m−1

j =1di j +Di m

−m −1+−m −1 N

+

Theorem 2.vt y b and V t Y b are convex inyb and Y b,respectively; vt y s and V t Y s are convex in ys, and Y s,

respectively, and therefore order-up-to policies are optimal.Under limited capacity, modied order-up-to policies areoptimal.

Proof. The proof follows from standard induction(Hadley and Whitin 1963).

4. Analytical ResultsFor tractability reasons, we make the “regenera-tion assumption” that allows the retailers to returnunwanted product to the supplier at no cost. Lee et al.(2000) also make such an assumption. This assump-tion ensures regeneration of the system in every cycle,and therefore the average cost per cycle is equivalentto the expected one-cycle cost. Note that the retailerswill return the products only if they have inventorylevels higher than yb (or ys for the SSOP). The probability of this happening is often quite small. Considera supply chain with two retailers as an example.If the xed shipment quantity is the mean end-customer demand, then for the end-customer demandthat has different Erlang distributions with parame-ters 1 20 , 2 10 , 4 5 , and 8 2 5 , the probabili-ties of a retailer returning the product to the supplierare 0 26, 0 14, 0 05, and 0 008, respectively, and theaverage returned quantities are 10%, 3.5%, 0.8%, and0.07% of the xed shipment, respectively. We evaluatethe impact of this assumption in our computationalstudy and observe that, on average, the total supplychain cost with this assumption is different from thecost without this assumption only by 1.37% (0.8%) forthe SBOP (SSOP).

Note that in Equation (1) the third term is an obsta-cle for mathematical analysis, because the associatedinventory is not in the same form as those in the rst




two terms, and hence we are not able to write out theoptimal decision and the cost in a simple form. Care-ful thinking suggests that the optimal decision of Y swould be mainly governed by the rst two terms, because the third term 1 simply captures the inventorycarried over from one nonordering period to the next

nonordering period after using the leftover inventoryin the free-ordering period as part or all of the xedshipments. In a simulation where hr =hs =1, s =19,N =3, the end-customer demands are perfectly corre-lated and follow an Erlang distribution with param-eters 2 10 , we nd that the carryover inventory ispositive only 4% of the time. Thus, from now on,when we look at the supplier’s one-cycle cost underthe SSOP, we will deal with L̃ instead of L̃.

L̃ Y s =hs Y s −N

i=1

m−1

j =1di j +Di m −m −1

+ +

+ sN

i=1

m−1

j =1di j +Di m −m −1 +−Y s +

4.1. Average CostBecause all the retailers are identical, the average costper period of every retailer under the SBOP (same forthe SSOP, but with cs and ys as the notation) is asfollows:

cb =1m

minyb

m−1

k=0hr E yb +k −

k+1

j =1Di j

+

+ r Ek

+1

j =1Di j −k −yb

+(2)

With the regeneration assumption, the supplier’saverage cost per period under the SBOP is

C b =minY b

hsE Y b −n

i=1

m−1

j =1di j +Di m

+

+ sEn

i=1

m−1

j =1di j +Di m −Y b

+(3)

1 It is positive only if N i=1 Di m < Y s +N m −1−k − N

i=1m−1j =1 di j

(with regeneration assumption) for some k 1 m −1 , theprobability of which decreases as it is further away from the free-ordering period (in other words, as k increases), and is the highestfor k =1. For the case where is equal to the mean end-customerdemand , and Y s takes a value such that Y s + N m − 1 −N

i=1m−1j =1 di j is equal to N +z√ N , the probability of the above-

mentioned inequality holding for k =1 becomes z −√ N , astandard Normal probability because N

i=1 Di m can be approxi-mated as a Normally distributed random variable especially forlarge N , which decreases as N and / increases. As such, wehave, for example, if N =3, z =2, and / =2, then this probabilityis 0.07.

and the supplier’s average cost per period under theSSOP is

C s =1m

minY s

hsE Y s −N

i=1

m−1

j =1di j +Di m −m −1

+

+ sE

N

i=1

m−1

j =1di j +Di m −m −1 −Y s

+(4)

Theorem 3. Under both the SBOP and SSOP, the opti-mal xed shipment quantity chosen by solely minimizingthe retailers’ costs also minimizes the total supply chaincost.

Proof. See the appendix.

4.2. Special Case: Two RetailersIn this section, we focus on a supply chain with onesupplier and two retailers. For consistency of compar-ison between the different ordering policies, we con-sider the case where the cycle length is equal to thenumber of retailers, i.e., m =N =2. The end-customerdemand is assumed to be exponentially distributedwith mean 1 / and c.d.f. z =1−e−z . We will con-sider two different scenarios, i.e., independent andperfectly correlated end-customer demand.

Note that for all the three ordering strategies (FOP,SBOP, and SSOP), the retailer’s cost is the same forindependent and perfectly correlated demand.

4.2.1. Retailer’s Average Cost Under FOP. For aretailer, the average cost per period is

cf =minyf h r E y f − + + r E −yf + (5)

Let yf be the optimal order quantity for anyretailer. It is well known that yf = −1 r / r +hr .Because is exponentially distributed, yf = −1/ lnh r / r +hr , and the loss function is

y f = yf

z dz = yf

e−z dz =1

e−y f

=hr

r +hr

Plugging yf back into Equation (5), we havecf =hr y f −E + y f + r y f

=hr −1

lnhr

r +hr −1

+ r +hr y f

= −hr ln

hr

r +hr

4.2.2. Supplier’s Average Cost Under FOP.Independent Demand. When the retailers use station-

ary order-up-to policies, the end-customer demands






equal to the cycle length, the average cost per periodof all retailers is equal to one retailer’s average costper cycle (2 cb).

Let =1 −ln h r / r +hr + . By (9) we have

=e , and

e −

=hr

e r +hr (10)

Also let = Y f +1, then by (7) we have

e − =hs

e s +hs(11)

Theorem 6 below lays some groundwork for theproofs of theorems in the upcoming sections.

Theorem 6. For and dened in Equations (10)and (11), we have

(i) −lnhr

r +hr ≤ +1

−2 ≤ −2 lnhr

r +hr ,

and

(ii) −lnhs

s +hs ≤ +1

−2 ≤−2 lnhs

s +hs .

Proof. See the appendix.4.2.4. Supplier’s Average Cost Under SBOP.

Under the SBOP, the supplier’s average cost is indif-ferent between the two cases when the end-customerdemands are independent and when they are corre-lated across the retailers. This is because the demandvariability faced by the supplier originates from theend-customer demands at one retailer over multipleperiods, which are assumed to be independent overtime.

We say the supplier is in state k if the retailerwho is ordering freely next has an inventory level ofyb −k before the arrival of the xed shipment, wherek is the realized end-customer demand at this retailerand, hence, nonnegative. By the discussion in §3.3,we know that the supplier faces a demand of =k +Di 2 − ++ if retailer i is the next one to orderfreely. With the regeneration assumption, = k +Di 2 − + =k +Di 2.

The supplier’s average cost is given by

C b =minY b

hsE Y b − k +Di 2 +

+ sE k +Di 2 −Y b + (12)

The optimal order quantity Y b satises Di 2 Y b −k= s/ s +hs , and

Y b =k −1

lnhs

s +hs

Plugging Y b back into Equation (12), we can verifythat the supplier’s average cost is

C b =hs Y b −k −E D i 2 + s +hs Di 2 Y b −k

= −hs ln

hs

s +hs

4.2.5. Supplier’s Average Cost Under SSOP.Under the SSOP, it is necessary to differentiate thecases when the end-customer demands are inde-pendent and perfectly correlated, because here thedemand randomness comes from both the retailers.

Independent Demand. We say the supplier is in state

u v if the two retailers to order freely have inven-tory levels of ys −u and ys −v, respectively, beforethe arrival of the xed shipment. Note that u and vare the realized end-customer demands and, hence,nonnegative. By the discussion in §3.4, we know thatthe supplier faces an uncertain demand of = u +D1 2 − ++ v +D2 2 − + if the retailers are to orderfreely. With the regeneration assumption, =D1 2 +D2 2 +u +v −2 . So the supplier’s cost per period is

C s =minY s

12 hsE Y s − D 1 2 +D2 2 +u +v −2 +

+ sE D 1 2 +D2 2 +u +v −2 −Y s +

Given solved by (9), we have Y s satisfyinghs − s +hs D1 2+D2 2 Y s +2 −u −v =0 (13)

Hence,

hs − s +hs Y s +2 −u −v +1 e− Y s +2 −u−v

=0 (14)

Note that Y s +2 −u −v +1 = . With and Y ssatisfying (9) and (14), we can compute the supplier’saverage cost as

C s

=1

2hs Y s

+2

−u

−v

−hs

+ s +hs e− Y s +2 −u−v

=hs

2 +1−2

Perfectly Correlated Demand. Now in (13), D1 2 andD2 2 are perfectly correlated. So we have Y s satisfying

hs − s +hs e−2 Y s +2 −u−v =0 (15)

With Y s as the solution to (15), we compute thesupplier’s average cost per period as

C s =−hs ln

hs

s

+hs

We summarize the supplier and the retailers’respective average costs under different scenarios inTable 1, and compare the performance of differentpolicies in the following theorems.

Let i i = f b s be the average total supply chaincost per period. Theorems 7 and 8 below compare theperformance of the SBOP and FOP.

Theorem 7. When the end-customer demands are independent, if r / s ≤hr /h s ≤1, then b ≤ f .

Proof. See the appendix.






Theorem 12. When end-customer demands are perfectly correlated, then C s =C b, i.e., the SSOP and theSBOP have the same performance.

Proof. When end-customer demands are perfectlycorrelated, C s =C b =−hs/ ln h s/ s +hs .

Theorems 11 and 12 show that the SSOP is a better alternative for improving the supply chainperformance when the end-customer demands areindependent. This result is consistent with our numer-ical observation in §5.3 where we expand the demandcorrelation to include values that lie within these twoextreme cases.

Theorems 7 to 12 provide the following interest-ing and possibly useful insights into the behavior ofscheduled ordering policies: (i) The supplier shouldhave cost parameters that are larger than those of theretailers in order to consider implementation of sched-uled ordering policies; (ii) when the end-customer

demands are independent across the retailers, thenscheduled synchronized ordering policies are moreattractive than scheduled balanced ordering policies;and (iii) when the end-customer demands are corre-lated, then both the types of scheduled ordering poli-cies have the same effectiveness.

5. Numerical StudyWe conduct an extensive numerical study to examine(1) how well the mathematical conditions developedpredict the effectiveness of the SBOP and SSOP (byeffective we mean the SBOP (SSOP) results in lower

total supply chain cost than the FOP); (2) to whatdegree the SBOP (SSOP) is effective; and (3) how dif-ferent supply chain parameters (such as end-customerdemand variance) affect the performance of sched-uled ordering policies.

In the numerical study we drop the regenerationassumption. For the SSOP, the supplier carryoverinventory, if any, between xed shipment periods isalso tabulated. Furthermore, the number of retailersis not restricted to two. However, because of the longcomputation time needed to nd an optimal cyclelength, we examine the performance of the SBOP andSSOP for the case where the cycle length is equal tothe number of retailers. In §5.4, we conduct a simu-lation to examine how the performance of the SBOPand SSOP is impacted by the cycle length.

It is very hard to nd closed-form solutions forthe optimal order-up-to levels and their associatedcosts. So we use innitesimal perturbation analysis(IPA) to compute the optimal order-up-to levels. Thevalidation and implementation of IPA is similar tothat in Kapuscinski and Tayur (1998). More details onthe IPA procedures can be found in Glasserman andTayur (1995).

To cover the cases when the SBOP (SSOP) is effec-tive and when it is not, we test a wide range of sup-ply chain parameter combinations. We x hr =1. Theretailer’s backorder cost, the supplier’s holding cost,and the supplier’s expediting cost can be any valuein the following sets, respectively: r = 6 9 12 15 ,

hs = 0 5 0 75 1 1 25 , and s = 9 19 39 99 . Thenumber of retailers (also the number of periods inan ordering cycle) can be N = 2 3 4 5 . The end-customer demand follows an Erlang distribution withpossible parameters 1 20 , 2 10 , 4 5 , and 8 2 5 .End-customer demands can have different correla-tion levels represented by correlation coefcient =0 0 25 0 5 0 75 1 .

We compute the percentage cost difference betweenthe different ordering policies as

fi =f − i

f ×100%

where i b s c , and c refers to the supply chaincost in a centralized system.

For example, the bigger fs is, the better the SSOPperforms. Given different values of the supply chainparameters, we test 5 120 different combinations. Inreporting the results, we take the average over all theexperiments related to a specic level of the parame-ter on which we are focused. For example, if we wantto see the performance of the SSOP when the numberof retailers is ve, we take an average of the percent-age cost differences over all the 1 280 experiments inwhich the number of retailers is ve.

5.1. Impact of the Regeneration AssumptionWe have developed the mathematical results basedon the regeneration assumption in §4. To check theimpact of the regeneration assumption, we exam-ine how much the cost difference is between thecases with and without it. We test all the combina-tions of the parameters and nd that over the 5 120instances, on average, the difference of the total sup-ply chain costs per period with and without thisassumption is 1 37% for the SBOP and 0.8% for theSSOP. Furthermore, the results in §5.2 also indicatethat this regeneration assumption has little impact

on our mathematical analysis. We thus conclude thatthe mathematical conditions we develop based on theregeneration assumption are quite robust.

5.2. Predictive Ability of the MathematicalConditions

We examine in the numerical study how often thefollowing cases happen: (1) mathematical conditionsare satised and the SBOP (SSOP) is effective (YY);(2) mathematical conditions are not satised butthe SBOP (SSOP) is effective (NY); (3) mathematicalconditions are satised but the SBOP (SSOP) is not




effective (YN); and (4) mathematical conditions arenot satised and the SBOP (SSOP) is not effective(NN). It turns out that among all of the 128 combi-nations (two retailers, exponential demand, indepen-dent, or fully correlated demand), under the SBOP thenumbers of instances for the four cases are 70 , 38, 0,

and 20 (YY, NY, YN, and NN, respectively). The numbers under the SSOP are 49, 76, 0, and 3, respectively.Among the four scenarios (YY, NY, YN, NN), YN

is considered bad because it represents the situationin which the mathematical conditions recommendimplementation of scheduled ordering, and that even-tually leads to an increase in cost. We are pleased tosee that the number of such instances is 0 for boththe SBOP and SSOP. Note that a large portion of theinstances fall into the category NY. This is not verysurprising because the conditions we provide are suf-cient but not necessary ones. However, compared tothe average cost reduction (which is 17 19% for the

SBOP and 18 54% for the SSOP) of the instances in cat-egory YY, the average cost reduction of those in cate-gory NY is only 8 66% for the SBOP and 9 33% for theSSOP. So we can see that the mathematical conditionswe develop are quite useful because they recommendimplementing scheduled ordering policies when theyare in fact very benecial.

5.3. Effectiveness of Scheduled OrderingWe observe that among the 5 120 instances we tested,the SBOP is effective 59 10% of the time, and the SSOPis effective 72 66% of the time. Figure 3 shows the his-togram of the percentage cost reduction over all the

experiments of interest. When better than the FOP,the cost reduction using the SBOP can be as highas 43 03%, with an average of 13 11%. And the costreduction using the SSOP can be as high as 47 44%,with an average of 14 74%.

We also observe that among all of the 3 720 in-stances that the SSOP is effective, the SSOP outper-forms the SBOP 89 52% of the time. By switching fromthe SBOP to the SSOP, the cost can be further reduced by as much as 20 04%, with an average of 3 51%.

Figure 3 Histogram of the Percentage Cost Difference Between Using SBOP (SSOP) and FOP

1,400

1,200

1,000

800

600

400

200

0–40 – 30 –20 –10 0 10 20 30 40 50 More

Percentage cost difference (%)

SBOP SSOP

–40 –30 –20 –10 0 10 20 30 40 50 More

Percentage cost difference (%)

F r e q u e n c y

1,600

1,400

1,200

1,000

800

600

400

200

0

F r e q u e n c y

Note that in both the mathematical analysis andthe numerical study, we show that the SSOP can be better than the FOP, and that the SSOP can also be better than the SBOP, which is different from theobservations in Cachon (1999) and Lee et al. (1997).Clearly, information sharing plays a role (especially

in determining the magnitudes of the benets) inthese results. We believe that information sharingenables the supplier to see much lower variance in thedemands she faces, which leads to increased savingsat her location resulting in the fact that these sched-uled ordering policies are effective more often than isobserved by Cachon (1999) and Lee et al. (1997).

5.3.1. Scheduled Ordering vs. Centralized Con-trol. Note that scheduled ordering is a strategy thatcan potentially reduce the total supply chain costand yet maintain decentralized decision making.However, it is necessary to benchmark it againstthe centralized system to accurately appreciate itseffectiveness.

In Figures 4 and 5, we illustrate the cost reduc-tions of using the two scheduled ordering policies aswell as using centralized control. We can see that, asexpected, the performance of a centralized system isalways better than the scheduled ordering. To bet-ter assess the performance of the SSOP and SBOP,we dene i − f / c − f (i b s ) as the cap-tured cost reduction due to using the SSOP or SBOP.We nd that when the SBOP is effective, it captures31 38% of the maximum possible cost reduction onaverage, and that when the SSOP is effective, the aver-

age captured cost reduction is 36 07%. The capturedcost reduction is as large as 67.29% for the SBOP andas high as 75.93% for the SSOP. Although the pro-posed policies do not coordinate the supply chain (asChen et al. (2001) are able to do for the deterministicsystem), their effectiveness is similar to that reportedin Zhu et al. (2010).

We next discuss how the performance of the SBOPand SSOP changes with different supply chain pa-rameters. It is easy to see that the SBOP and SSOP




Figure 4 Percentage Cost Difference Between SBOP (SSOP)and FOP vs. the End-Customer Demand Variance

50 100

40

30

20

10

0

–10

150

SBOP

SSOPCentralized

200 P e r c e n

t a g e c o s t d

i f f e r e n c e

( % )

2

perform better as hs and s increase, and as r decreases. We therefore focus our attention only on

other parameters.5.3.2. Effect of End-Customer Demand Variance.

The performance of the SBOP (SSOP) as a functionof the end-customer demand variance is shown inFigure 4. The average cost reduction by using theSBOP (SSOP) with respect to (w.r.t.) the end-customerdemand variance is as high as 4 52% (9 06%). Itappears that the cost reduction is larger as thedemand variance increases. The reason is probablyas follows. When the demand variance increases,the cost reduction at the supplier from using sched-uled ordering policies increases faster than the cost

increase at the retailers. Therefore, the benet of usingscheduled ordering on the collective system perfor-mance is more signicant when the demand varianceis higher. We hence conclude that the SBOP and SSOPare more effective when the end-customer demand varianceis high.

5.3.3. Effect of End-Customer Demand Correla-tion. Figure 5 shows that when the end-customerdemand correlation increases, the performance of the

Figure 5 Percentage Cost Difference Between SBOP (SSOP)and FOP vs. the End-Customer Demand Correlation

40

30

20

10

0

–10

–20

SBOPSSOP

0 0.50 0.75 1.00

Centralized

P e r c e n

t a g e c o s t d

i f f e r e n c e

( % )

0.25

SBOP and SSOP becomes better. The cost reductioncan be as high as 17 80% (16 34%) for the SBOP(SSOP). This is because the main benet of the SBOPand SSOP is in reducing the average demand vari-ance faced by the supplier. When the end-customerdemand correlation level is higher, this benet is more

signicant. For example, for the case N =3, hs =1, r =9, s =39, and Erlang (2, 10), using the SBOPincreases the supplier’s cost by 4 69% when =0 andreduces the supplier’s cost by 27 44% when =1. Wealso observe that the SSOP is better than the SBOPwhen the demand correlation level is relatively low,and the opposite holds when the demand is highlycorrelated. This is probably because when the end-customer demand is highly correlated, the risk pool-ing of the SSOP results in a more signicant demandvariability surge in the free-ordering periods. Con-sequently, on average, the demand variability perperiod faced by the supplier will be higher than that

if the SBOP is used, where the demand variability ismore evenly distributed and is not affected by thedemand correlation level. Thus, we conclude that theSBOP and SSOP are more effective than the FOP whenthe end-customer demands have higher correlation. Fur-thermore, the SSOP is better than the SBOP when thedemand correlation level is low, and the SBOP is preferred for highly correlated demand.

5.4. Optimal Cycle LengthThe mathematical conditions of the SBOP and SSOP being effective and the numerical results we ana-lyze so far are based on the assumption that thecycle length is equal to the number of retailers. How-ever, this is not necessarily the best choice for thecycle length. In this section, we examine how theperformance of the SBOP and SSOP changes withdifferent cycle lengths. Here we have hr = hs = 1, r =9, s =39, and =0 5, and the end-customerdemand follows an Erlang distribution with parame-ters 2 10 . For the SBOP, the number of retailers is 40,and the cycle lengths are 2 4 5 8 10 . For the SSOP,the number of retailers is N = 2 4 5 8 10 . For eachpossible value of N , the possible cycle length can beany value from m = 2 4 5 8 10 . Figure 6 showsthat the performance of both the SBOP and SSOPhas an approximately concave pattern w.r.t. the cyclelength. It rst gets better and then worse as the cyclelength increases. The reason is probably as follows.When the cycle length is small, the average demandvariance the supplier faces is still quite high. So the benet of scheduled ordering policies is not that sig-nicant. However, when the cycle length is too big,although the supplier’s average demand variance isgreatly reduced, the retailers incur a very large cost,which cannot be compensated for by the cost reduc-tion at the supplier. Therefore, a medium cycle lengthappears to be the best choice.




Figure 6 Performance of SBOP and SSOP vs. Cycle Length

SBOP

2 4 5

75 25

20

15

10

5

0

70

65

60

55

50

458

Cycle length

P e r c e n

t a g e c o s

t d i f f e r e n c e

( % )

P e r c e n

t a g e c o s


( % )

10 2 4 5 8

Cycle length

N = 2N = 4N = 5N = 8N = 10

10

SSOP

6. Extensions6.1. Capacity LimitWe conduct a small-scale simulation to study theeffect of the capacity limit. Here we have hr =hs =1, r = 9, s = 39, and =0 5, and the end-customer demand follows an Erlang distribution withparameters 2 10 . The number of retailers is equalto 2. Because of the capacity limit, the supplier mayneed to produce extra in those nonordering periods just to ensure that the optimal order quantity can be achieved in the free-ordering period. We use aheuristic (similar to the one in Gavirneni 2001) tocalculate how much should be produced in thosenonordering periods. As mentioned in §3.3, we denethe rst period of an ordering cycle as the free-ordering period. So the periods for back produc-tion are in the previous cycle, and we dene them

as period −m −m +1 −2. On top of the xedshipments, the extra quantity the supplier needs toproduce in those periods is dened as z−i i = m m −1 2 . Let be the capacity limit. Our heuris-tic is to let z−m =min −m Y − + , and fori =m −1 2, we let z−i =min −m Y − −m−i

j =1 z−i−j + .Figure 7 shows that both the SBOP and SSOP per-

form better as the supplier’s capacity becomes larger.

Figure 7 Performance of SBOP vs. SSOP with Capacity Limit

P e r c e n

t a g e c o s t

d i f f e r e n c e

( % )

1.5 2.0

25

20

15

10

5

0

–5

–10

2.5

Capacity parameter

3.0 3.5 4.0 4.5

SBOPSSOP

1.0

This is because when the supplier has higher capac-ity, both the SBOP and SSOP are able to more ef-ciently reduce the demand variability faced by thesupplier. Also note that when the capacity limit isloose, the SSOP performs better than the SBOP. How-

ever when the capacity limit is tight, then the SSOPperforms worse than the SBOP. The reason is prob-ably as follows. When the capacity limit is loose,the SSOP greatly reduces the average demand vari-ance the supplier faces by pooling the demand riskinto certain periods and leaving other periods free ofuncertainty. However, when the capacity limit is tight,pooling all the demand risk into certain periods isa bad idea because the supplier may not react wellto the high demand surge in those specic periods because of the capacity limit. It thus results in hugeexpediting cost, which cannot be compensated for by being risk-free in other periods. We conclude that boththe SSOP and SBOP will perform better when the supplierhas a higher capacity. Whereas the SSOP may be preferredwhen the supplier has a higher capacity, the SBOP is thebetter choice when the capacity limit is tight.

6.2. Positive Lead TimeIn our analysis, we are looking at a special case wherethe retailers receive the product right away and thesupplier gets the product one period after her inven-tory replenishment decision is made. Although we believe the modeling framework is the same if wehave positive lead times, we do recognize that thelead time can have an impact on the performance ofscheduled ordering policies, and we therefore conducta simulation to study the effect. The study is for thecase where N =2, hr =hs =1, r =9, s =39, =0 5,and the end-customer demand follows an Erlang dis-tribution with parameters 2 10 . The supplier andthe retailer lead times vary from 0 to 5. Figure 8shows how the performance of the SBOP and SSOPchanges with the lead time. In the legend, ls refersto the supplier lead time. We can see that both theSBOP and SSOP perform worse when the supplierlead time increases. This is not surprising, because the




Figure 8 Performance of SBOP and SSOP with Lead Time

P e r c e n t a g e c o s


( % )

P e r c e n

t a g e c o s t d

i f f e r e n c e

( % )

0

18 20

–20

–40

–60

–80

–100

–120

–140

–160

016141210

86420

1 2 3

Retailer’s lead time

SBOP SSOP

Retailer’s lead time4 5

0 1 2 3 4 5I S = 0I S = 1I S = 2I S = 3I S = 4I S = 5

slow responsiveness at the supplier would dampenthe effectiveness of the scheduled ordering. We alsoobserve that if the supplier lead time is zero (mean-ing only one period of natural production lead time),then the performance of both the SBOP and SSOP getsworse when the retailer lead time increases. How-ever, if the supplier lead time is positive, then theSBOP and SSOP perform better as the retailer leadtime increases. The reason is probably as follows. Theretailer lead time impacts the performance of sched-uled ordering in two ways. A higher retailer lead timeimplies more variable retailer orders under the FOP,leaving a poorer system for scheduled ordering toimprove on, and hence makes the scheduled order-ing more effective. On the other hand, a larger retailerlead time also means that the retailer incurs a highercost under the scheduled ordering compared to the

FOP, and consequently, the scheduled ordering is lesseffective. Our results suggest that the former effectdominates when the supplier lead time is positive,and that the latter effect dominates when the supplierlead time is zero. Finally, we nd that the SSOP isnot effective when the supplier lead time is positive.This is because the nonresponsiveness of the suppliermagnies the drawback of the SSOP, that is, the surgeof demand uncertainty in the free-ordering periods.

6.3. Backlogging at the SupplierHere we assume that when the supplier does nothave sufcient inventory to satisfy all of the retailerdemands, she allocates her inventory among theretailers and backlogs the unsatised demand. Thespecic inventory allocation scheme that the sup-plier uses plays a critical role in dening the retailer behavior. To avoid any order ination by the retail-ers, we assume that she uses the lexicographic alloca-tion rule, which, as Cachon and Lariviere (1999) haveshown, induces truth telling by the retailers. Underthis scheme, retailers are ranked independent of theirorder sizes (say, alphabetically) and allocated inven-tory according to that ranking.

Because of the added complexity that backlogging brings into an already intricate distribution supplychain, we are not able to mathematically compare theperformance of scheduled ordering policies with theFOP. If one were to follow the framework developed by Aviv (2003) and Aviv (2007), mathematical char-acterization of the system behavior may be possible.We leave that as an avenue for future research, andhere we conduct an extensive simulation to obtainsome insights. The simulation is done for all the 5 120instances studied under the expediting case, and theresults show that all the observations we make (per-formance change with the system parameters) underexpediting still hold here.

For the SBOP, we nd that it is effective 62 87% ofthe time under backlogging, similar to that (59 10%of the time) under expediting. The average cost reduc-tion under backlogging is 13 99%, also very close tothe one under expediting, 13 10%. Finally, we noticethat 66 91% of the time the benet of the SBOP ismore signicant under backlogging than it is underexpediting, which often happens when the supplier’spenalty cost 2 ( s) is small. This is probably becausea smaller penalty cost makes the supplier less incen-tivized to carry high inventory to satisfy all retailerdemand, which results in a less efcient supply chainto begin with under backlogging than expediting.

For the SSOP, our simulation shows that it is effec-tive only 2 11% of the time under backlogging, asopposed to 76 66% of the time under expediting. Fur-thermore, the average cost reduction is only 3 19%compared to 14 74% under expediting. The reasonfor the poor performance of the SSOP under back-logging is as follows. The risk pooling of the SSOPresults in supplier backlogging large retailer demandin the free-ordering periods, which causes the retail-ers to incur large backorder costs because of shortageof inventory.

2 The penalty cost s refers to the dollars per expedited unit underexpediting and the dollars per backlogged unit per period under backlogging.




7. ConclusionIn this paper, we examine the effectiveness of twoscheduled ordering policies—the scheduled balancedordering policy and the scheduled synchronizedordering policy—in improving the performance of adecentralized distribution supply chain. We consider

them when there is information sharing in the supplychain because the absence of information makes thesepolicies considerably less effective. We provide easy-to-evaluate mathematical conditions under which theSBOP (SSOP) reduces the total supply chain cost com-pared to the traditional free ordering policy.

Results from an extensive numerical study indicatethat the mathematical conditions we develop are quiterobust and predict signicant cost reduction due tousing the SBOP and SSOP. We also nd that there isa large number of experiments for which the SBOPand SSOP are not effective. However, there is a cleardistinction between when the SBOP and SSOP work

and when they do not. We observe that the SBOPand SSOP are most effective when (i) the supplier hashigh holding and expediting costs and the retailer hassmall backorder cost; (ii) the end-customer demandsare highly correlated and highly variable; and (iii) thesupplier’s capacity limit is high. Although the perfor-mance of the SBOP and that of the SSOP w.r.t. thesupply chain parameters have a similar trend, their behaviors also complement each other; that is, theSBOP is more effective than the SSOP for low capacityand high demand correlation, and the SSOP is pre-ferred under the opposite conditions.

When benchmarking the performance of the SBOPand SSOP with a centralized system, we nd thatwhen scheduled ordering is effective, on average, both the SBOP and SSOP can capture about 31%–36%of the cost reduction that can be achieved by a cen-tralized system.

We also show that the benet of the SBOP andSSOP rst increases and then decreases as the cyclelength increases. This observation suggests that theperformance of the SBOP and SSOP (equivalently thesupply chain cost incurred under the SBOP and SSOP)might be concave in the cycle length. Finding themathematical evidence of this behavior would be aninteresting avenue to explore for future research.

Our results indicate that the effectiveness of theSSOP is very much dependent on the supplier’s oper-ational environment. The SSOP is most effective whenthe supplier’s lead time is zero, capacity is unlimited,and she has the ability to meet, via expediting, all ofretailers’ demands. These conditions capture the set-ting when the supplier is most exible, and this ex-ibility enables the supplier to react effectively to therandomness in the system as well as the informationavailable to her. Because of the accumulation of allthe randomness into one period, the SSOP especially

requires the supplier to be exible, and when the sup-plier exibility is restricted, its effectiveness drops aswell. Thus, when the supplier lead time increases orcapacity decreases, the SSOP is less attractive. Fur-thermore, if the supplier has to backlog demand shecannot satisfy from on-hand inventory, then the effec-

tiveness of the SSOP is further reduced.We can identify a few directions to extend researchon this topic. One generalization is to allow a differentxed shipment quantity in every nonordering period.That will make these policies more attractive to theretailers while not impacting supplier performance.It is, however, not clear if the additional benet ofthis generalization will justify the mathematical com-plexity. The other extension would be to incorporatecapacity limits more rigorously.

Our current analysis makes extensive use of heuris-tics, and it is not clear how that has impacted theresults. It is possible that optimal quantities alter the

relative performance of the SBOP and SSOP. This isprobably worth clarifying in future research.

AcknowledgmentsThe authors thank the department editor, the associate edi-tor, and two anonymous reviewers for their comments andsuggestions, which have greatly improved the exposition ofthis paper. Financial support was provided by NUS AcRFResearch Grant, project number R-314-000-074-133.

AppendixProof of Theorem 3. By (3) we can see that under the

SBOP, the supplier’s cost is not impacted by the choice of .Therefore, the optimizing for the retailers is also optimalfor the whole supply chain.

Under the SSOP, the retailers’ costs are interrelated to thesupplier’s only through the xed shipment quantity . ByEquation (4), we can see that the supplier’s cost is deter-mined by a z =Y s +N m −1 , but not affected by thespecic individual value of Y s or . Hence, in minimizingthe system cost, the optimization problem can actually bedecomposed into two problems, one that determines theand ys that are best for the retailers, and the other that deter-mines z =Y s +N m −1 that optimizes the supplier’scost. In other words, the best thing for the system to do isto nd an optimal that minimizes retailer’s cost (whichcan be done solely by the retailer), and then adjust Y s basedon .

Proof of Theorem 4. If we look at the optimizationproblem in (8), we can see that it can be decomposed intotwo separate problems, (i) deciding an optimal yb with ran-dom demand Di 1 and (ii) deciding an optimal z =yb +when facing a demand of Di 1 +Di 2. This is because there isno other cost attached to . So the retailers can choose a ybto ensure the same minimum cost as under the FOP at leastfor one period, and then do the best in the second period by picking an optimal z . Therefore, can be determined by the choices of yb and z .

Proof of Theorem 6. Taking the logarithm of (10),we have ln − = −1 + ln h r / r +hr . Therefore,




+ ln 1/ =1 − ln h r / r +hr . By Taylor’s expansion,we know ln 1/ ≤1/ −1. We then have +1/ −1 ≥

+ ln 1/ = 1 − ln h r / r +hr . Hence, +1/ −2 ≥−ln h r / r +hr .

Because ln h r / r +hr = ln − +1, we have +1/ −2+2 ln h r / r +hr = +1/ −2+2 ln − +1 =1/

− +2 ln . Letting g1

=1/

− +2 ln , we

have g1 1 =0. Furthermore, g1 = −1/ 2 − 1 + 2/ =− 1/ −1 2 ≤0. Hence, g1 ≤0 for all ≥1. This means

+1/ −2 +2 ln h r / r +hr ≤0. The second part of thetheorem holds with a similar argument.

Proof of Theorem 7. Let g2 z = ln z +1/z −1. Notethat g 2/ z =1/z 1−1/z ≥0 when z ≥1. So g2 z is increas-ing if z ≥1. Now if r / s ≤hr /h s, it’s clear that ≤ . Also,we can see that ≥1 and ≥1. Hence, g2 ≤g2 .

The difference of the total supply chain’s average cost between the SBOP and the FOP is

b − f = 2cb +C b − 2cf +C f

=hr

+1

−2−ln

hr

r +hr

+ −hs ln

hs

s +hs

− −2hr ln

h r

r +hr +hs

+1

−2

=hr

+1

−2+lnhr

r +hr

−hs

+1

−2+lnhs

s +hs

=hr

+1

−2+ln − +1

−hs+ 1 −2+ln − +1

=hr ln +

1−1 −

hs ln +1

−1

=hr g2 −

hs g2

≤hr −hs g2 ≤0

because hr ≤hs and g2 ≥0.Proof of Theorem 8.

b

−f

=hr

+1

−2

−ln

h r

r +h r

+ −hs ln

hs

s +hs

− −2hr ln

hr

r +hr + −2hs ln

hs

s+hs

=hs ln

hs

s +hs +h r

+1−2+ln

h r

r +hr

≤hs ln

hs

s +hs −h r ln

h r

r +h r

≤ 0

by Theorem 6, and

hs

s+hs

hs

≤hr

r +hr

hr

Proof of Theorem 9.

s

−f

=2cs

+C s

−2cf

+C f

=h r

+1

−2−lnhr

r +hr +hs

2 +1

−2

− −2h r ln

hr

r +h r +hs

+1

−2

=h r ln

hr

r +h r +hr

+1

−2

−hs

2 +1

−2

≤hs

2ln

hs

s +hs −h r ln

hr

r +h r

≤0

becausehs

s +hs

hs / 2

≤hr

r +h r

hr

Proof for Theorem 11. When end-customer demandsare independent, by §4.2.4, we know the supplier’s costunder the SBOP is

C b =−hs ln

hs

s +hs

We also know from §4.2.5 that the supplier’s cost under theSSOP is

C s

=

hs

2 +

1

−2

≤−

hs lnhs

s +hs

=C b

by Theorem 6.

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