Backlog Writeoff.pdf

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MANAGEMENT SCIENCEVol. 51, No. 10, October 2005, pp. 15051518issn 0025-1909 eissn 1526-5501 05 5110 1505

informs

doi 10.1287/mnsc.1050.0371 2005 INFORMS

The Impact of Duplicate Orders on Demand

Estimation and Capacity InvestmentMor Armony

Stern School of Business, New York University, New York, New York 10012, [email protected]

Erica L. PlambeckGraduate School of Business, Stanford University, Stanford, California 94305, [email protected]

Motivated by a $2.2 billion inventory write-off by Cisco Systems, we investigate how duplicate orders canlead a manufacturer to err in estimating the demand rate and customers sensitivity to delay, and to makefaulty decisions about capacity investment. We consider a manufacturer that sells through two distributors. Ifa customer finds that his distributor is out of stock, then he will sometimes seek to make a purchase from theother distributor; if the latter is also out of stock, the customer will order from both distributors. When hisorder is filled by one of the distributors, the customer cancels any duplicate orders. Furthermore, the customercancels all of his outstanding orders after a random period of time.

Assuming that the manufacturer is unaware of duplicate orders, we prove that she will overestimate boththe demand rate and the cancellation rate. Surprisingly, failure to account for duplicate orders can cause short-term underinvestment in capacity. However, in long-term equilibrium under stable demand conditions themanufacturer overinvests in capacity. Our results suggest that Ciscos write-off was caused by estimation errorsand cannot be blamed entirely on the economic downturn. Finally, we provide some guidance on estimation inthe presence of double orders.

Key words : maximum-likelihood estimation; duplicate ordering; distribution channels; queueing systems;reneging

History : Accepted by William S. Lovejoy, operations and supply chain management; received April 19, 2002.This paper was with the authors 3 1

2months for 2 revisions.

1. Introduction

Amid a general economic downturn, networking titanCisco Systems experienced a spectacular fall in mar-ket value from $430 billion in March of 2000 to$180 billion in March of 2001. Net income droppedfrom $0.8 billion in the first quarter of 2001 to$27 billion in the third quarter of that year as Ciscowrote off $2.2 billion worth of component inven-tory and laid off 8,500 workers (Business Week2002a).According to theWall Street Journal, Cisco executives

ignored or misread crucial warning signs that theirsales forecasts were too ambitious. Because of dupli-cate orders, Cisco executives overestimated demand

and therefore continued to expand capacity aggres-sively, even after business slowed (Thurm 2001a,p. A1). Cisco is certainly not the only technologycompany to have difficulties in forecasting because ofduplicate orders. Intel and other semiconductor man-ufacturers believe that their bookings data is irrele-vant and potentially misleading because of duplicateorders (Business Week2002b, p. 28). This paper showsthat even in a stable business environment, a man-ufacturer that fails to account for double orders will

carry excess capacity.

Ciscos policy of outsourcing all of its manufac-

turing has been lauded in the business press. Lesswidely recognized is that, since 1998 (when Ciscoachieved 65% of its revenues through direct sales),Cisco has sought to outsource sales and distribu-tion. Cisco sells networking hardware to distributors(e.g., Ingram Micro) that sell to systems integrators(including IBM and a host of smaller firms) that inturn sell to Ciscos end customers and provide ongo-ing support and maintenance. By 2001, the numberof Cisco-qualified distributors and resellers (systemsintegrators) had increased to 20,000 for the UnitedStates alone (Kothari 2001). Only 14% of Ciscossales were direct; 86% were through channels. Cisco

was using the Internet to share real-time informa-tion about inventory and production schedules withits component suppliers and contact manufacturers(Business Week2001). However, on the demand side,Ciscos information systems were relatively weak. Inparticular, Cisco had limited visibility of distributorsinventory and order backlog (Kothari 2001).

In the summer of 2000, Cisco experienced short-ages of several key components. Customers had towait for two and even three months for some ofCiscos most popular products. Some frustrated cus-

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Armony and Plambeck: Duplicate Orders on Demand Estimation and Capacity Investment1506 Management Science 51(10), pp. 15051518, 2005 INFORMS

tomers chose to cancel their orders and buy equip-ment from Ciscos competitors (Juniper Networks,Nortel Networks). Customers and resellers also beganto order from multiple distributors with the inten-tion of cancelling duplicate orders as soon as onedistributor shipped the product. Cisco failed to rec-

ognize the extent of the double orders and there-fore, although the tech economy had already begun toslow down, Cisco maintained its ambitious sales fore-casts. To avoid long lead times and lost sales, Ciscoadded workers and stockpiled components. Cisco alsoloaned $600M without interest to contract manufac-turers to buy even more parts. This expanded capac-ity did indeed serve to reduce production lead timesthroughout the fall of 2000. The order backlog disap-peared as customers cancelled duplicate orders, andnew orders anticipated by Cisco failed to materialize(Thurm 2001a). Cisco was saddled with excess capac-ity. Despite the write-off of $2.2 billion in component

inventory in April of 2001, Cisco carried $1.68 bil-lion in parts and unsold equipment on its books atthe end of fiscal year 2001 (Thurm 2001b). Accordingto Cisco Chief Strategy Officer Michelangelo Volpi,We didnt know the magnitude [of duplication inthe order backlog]. Without the misleading informa-tion we might have seen better and made better deci-sions. (Thurm 2001a).

Ciscos experiences raise several interesting ques-tions: Will order duplication cause a manufacturer tooverestimate the demand rate and the rate at whichsales will be lost if customers are forced to wait (thereneging rate)? If so, by how much? How will this

affect capacity investment? How can the manufac-turer ascertain the true demand rate, incidence ofduplicate orders, and reneging rate?

To address these questions, we analyze a stylizedmodel of a manufacturer with two independent dis-tributors (see Figure 1 for an illustration). At eachdistributor, customers arrive according to a Poissonprocess with rate . If a customer finds that his dis-tributor is out of stock, then with probabilityhe willalso place an order with the other distributor; as soonas one distributor supplies the product, the customercancels his order with the other. Furthermore, the cus-

tomer will renege,

1

cancelling all outstanding ordersafter a length of time that is exponentially distributedwith rate . The manufacturer has visibility of eachdistributors inventory level or number of outstand-ing orders, but not customer identities. (The manu-facturer can infer a distributors inventory level from

1 In most queueing models of service systems, a customer will notrenege during his service, but will renege only while waiting forservice to begin. In contrast, in our queueing model of manufactur-ing, the customer at the head of the line may renege, although hehas claim to the product in process.

Figure 1 Illustration of the Model with a Single Manufacturer and Two

Distributors

Reneging rate

Demand rate

Double order probability

Distributor 1 X1 X2

Manufacturer

Distributor 2

Production rate

1 2

her own order queue if the distributor uses a base-stock policy. Since 2001, firms in the high-tech, auto-

motive, chemical, and home appliance industries haveimplemented software to monitor distributors inven-tory levels.) A detailed model formulation is providedin 2.

In 3, we derive maximum-likelihood estimators(MLEs) forand in a system without double orders(= 0). Next, we assume that some customers willdouble order (unbeknown to the manufacturer) andwe prove that by using the MLEs for the system with = 0, the manufacturer will overestimate the demandrate and the reneging rate . The basic problemis that double orders are counted as additional cus-tomer arrivals, and cancellations of double orders are

counted as lost sales. Section 3 also contains sensi-tivity analysis of the systematic error, the difference

between the parameter values estimated by the manu-facturer (assuming that = 0) and the true parametervalues.

The MLEs are valid for an arbitrary schedule ofshipments from the manufacturer to the distributors.However, to investigate the impact of duplicate orderson capacity investment, production is modeled as aPoisson process with rate. We investigate two plau-sible allocation rules for finished goods. The basecase is that each distributor i has a fixed portion ofthe capacity i (with 1 + 2 =). This is relevant

when the distributors are located in different geo-graphical regions and transportation costs are high, sothe manufacturer serves them from different produc-tion facilities. (Cisco has regional production facilities,and some resellers, particularly those at an interme-diate location, will duplicate-order from distributorsin different regions.) The base case with 1= 2 alsoapproximates a fair division of the output betweenthe distributors. The second case is that capacities arepooled, and distributors orders are filled first infirstout (FIFO). (Cisco is concerned with fairness.)


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Armony and Plambeck: Duplicate Orders on Demand Estimation and Capacity InvestmentManagement Science 51(10), pp. 15051518, 2005 INFORMS 1507

Based on her estimation of demand, the manufac-turer chooses to minimize the cost of capacity andlost sales. She assumes that each distributor uses a

base-stock policy for inventory control with a fixedbase stock level B. In reality, capacity is a strategicdecision and inventory policy is a short-term, tacti-

cal decision that responds to capacity utilization (leadtimes). In Ciscos experience, distributors lower theirinventory levels in response to an increase in the pro-duction capacity. Moreover, distributors learn aboutdemand and adjust their inventory policies dynami-cally, and the optimal inventory policy for one distrib-utor depends on the inventory policy of the other, asthey compete for capacity and customers. Completeanalysis of the strategic interaction between the threeparties is beyond the scope of this paper. We simplycharacterize the manufacturers best response to agiven base stock level B used by both distributors.Opportunities for further research lie in integrating

strategic capacity investment with the rich litera-ture on inventory competition and Bayesian inventorymanagement.

Section 4 demonstrates how overestimating thedemand rate and the reneging rate can cause themanufacturer to purchase too much capacity (likeCisco). On the contrary, when the cost of capacity isvery high, a manufacturer that is unaware of dupli-cate orders will underinvest in capacity. For any fixedcapacity level, duplicate ordering reduces the num-

ber of lost sales and thus increases the manufacturersprofit. Unfortunately, unrecognized duplicate order-ing reduces the manufacturers profit through errors

in capacity planning. This serves as a warning to man-ufacturers: Watch out for double orders!

For the watchful manufacturer, 5 provides estima-tors for , , and , based on a general shipmentschedule and continuous observation of inventorylevels. Commonly, the manufacturer observes inven-tory levels infrequently if at all. Therefore, in 5.1 weadapt the estimators to handle discrete-time informa-tion about inventory levels.

1.1. Literature ReviewThe literature related to this research falls into fourcategories: (1) estimation of customer characteristics,

(2) dynamic inventory control under demand uncer-tainty, (3) capacity investment under uncertainty, and(4) strategic interaction between a manufacturer andcompetitive retailers with customer substitution.

For queues with impatient customers, Mandelbaumand Zeltyn (1998) and Daley and Servi (2001) deriveMLEs for the demand rate and the reneging rate.Motivated by applications in networking and call cen-ters, these authors assume that the queue length is notobservable; they use only transaction data (the pointsin time that a customer begins or completes service).

Hence, their MLEs differ from the ones derived inthis paper, where queue lengths (inventory levels) areobservable. Anupindi et al. (1998) consider a retailstore in which customers arrive according to a Pois-son process; if the desired product is not in stock,a customer may substitute it with another item, or

depart without making a purchase. Given discrete-time observations of the inventory in the store, theyderive MLEs for the demand rate and substitutionprobability. This resembles estimating the demandrate and double-order probability (the probability thata customer will substitute an alternative distribu-tor) in our model. Lee et al. (1997) observe that thevariance of orders from a distributor to a manufac-turer is larger than the variance of actual sales to endcustomers, the famous bullwhip effect. They concludethat for effective forecasting, the manufacturer needssales data. Our results are complementary: Duplicateorders distort the mean. Therefore, the manufacturer

would like to know the identity of end customers, notjust the sales quantity, to correct for duplicate orders.

Scarf (1959) introduced the problem of Bayesianinventory management: How should a retailerdynamically control his inventory level while learn-ing about the demand distribution as sales evolveover time? Many researchers have tackled this chal-lenging problem. For a variety of plausible demanddistributions, assuming linear ordering and holdingcosts and complete backordering of demand, Azoury(1985) establishes optimality of a base-stock policywith the base stock level scaled by a sufficient statistic

for observed demand. Lovejoy (1990, 1992) shows thatthe adaptive base-stock policy is optimal or near opti-mal under more general conditions, e.g., a constant(known) reneging rate, Markov-modulated demandwith cheap disposal of excess inventory. Lariviere andPorteus (1999) and Ding et al. (2002) assume unob-servable lost sales: The optimal base stock level isincreased to learn more about demand. Larson et al.(2001) incorporate a fixed cost of ordering, and derivean optimal adaptive sS policy. Toktay and Wein(2001) consider a capacity-constrained production-inventory system. The demand distribution is known,

but the forecast for actual demand in future periods

evolves dynamically. A dynamic (forecast-adjusted)base-stock policy minimizes inventory holding andbackorder costs.

These results support our assumption that the dis-tributors follow a base-stock policy, but suggest thatthe base stock level will evolve dynamically, in con-trast with our simplifying assumption that the basestock level is fixed. Indeed, capacity investment deci-sions typically occur on a quarterly or annual basisand are irreversible in the short term. Therefore, whenthe manufacturer chooses her capacity investment,


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she should ideally account for future dynamics in dis-tributors base stock levels. Van Mieghem (2003) pro-vides an extensive review of the literature on capacityinvestment under uncertainty, and observes that fewof these papers consider capacity investment anddemand estimation. A notable example, Ryan (2003)

analyzes capacity expansion with an autocorrelateddemand process, and discrete capacity incrementswith long lead times. Our contribution is to show howan erroneous belief about system structure (disregard-ing duplicate orders) produces an error in estimation,and how the error in estimation both influences andis influenced by the capacity investment decision. Weshow how this results in excess capacity in equilib-rium. This is strikingly similar to the heuristic equilib-rium with excess quantity in the newsvendor model

by Cachon and Kok (2002), where the procurementquantity depends on the estimated salvage value, andthe estimated salvage value depends on the quantity

remaining at the end of the season.In making a capacity investment, the manufacturer

should ideally anticipate how each distributor willmodify his base stock level in response to the pro-duction lead time and the base stock level of theother distributor. Researchers have used game the-ory to analyze strategic inventory management in set-tings with consumer substitution, assuming that thesubstitution probability and distribution of demandis known. Parlar (1988), Lippman and McCardle(1997), and Mahajan and van Ryzin (2001) character-ize Nash equilibria in a single-period (newsvendor)game. Anupindi and Bassok (1999) and Netessine

et al. (2001) consider the stocking decisions of tworetailers in a multiperiod problem with stationarydemand. In each period, if a customer finds that hisretailer is out of stock, he may purchase from theother retailer. Assuming that the manufacturer hasunlimited production capacity, they prove existenceof myopic Nash equilibria, i.e., the multiperiod gamereduces to a static (newsvendor) problem. Li (1992)analyzes a queueing model very similar to ours, in thespecial case = 1 and = 0; customers arrive accord-ing to a Poisson process and attempt to buy fromone ofn competing firms. If the firm has a queue ofcustomer orders, the customer places an order withevery firm, buys from the one that delivers first, thencancels all other orders. Li characterizes the condi-tions under which all the firms will choose to maketo order rather than carry inventory. The firms, act-ing selfishly, may choose to carry inventory even ifexpected profit would be greater if all firms choseto make to order. Indeed, a common conclusion inthese papers is that, in competition for customers,firms will stock more than is optimal. Netessine andRudi (2003) give a counterexample with asymmetricfirms, and one stocking less than is globally optimal.

In contrast, Cachon (2001) shows that when retail-ers compete for supply from a common manufacturer

but do not compete for customers, they may carryless inventory than would be optimal for the supplychain as a whole. Cachon and Lariviere (1999) showhow retailers order quantities depend on the alloca-

tion scheme chosen by the manufacturer, not just hiscapacity investment.

2. Model FormulationConsider a manufacturer that sells a single productthrough two independent distributors. For brevity, wewill at times use the pronoun he to refer to a dis-tributor and she to refer to the manufacturer. Ateach distribution center, customers arrive according toa Poisson process with rate (which is independentof customer arrivals at the other distribution center),and each customer demands one unit of the prod-

uct. Let Xit denote the inventory level for distrib-utor i i = 1 2 at time t; Xit

= minXit 0indicates the number of outstanding orders from cus-tomers. If a customer arrives when his distributor isout of stock Xit 0, then with probability thecustomer buys immediately from the other distribu-tor (if the other distributor has inventory) or ordersthe product from both distributors. As soon as one ofthe distributors delivers the product to him, the cus-tomer will cancel the duplicate order. With probability1 , the customer orders from his original distribu-tor only. The customer is impatient; after waiting for

a time that is exponentially distributed with rate ,he will cancel all outstanding orders and leave thesystem without making a purchase. (One may inter-pret this waiting time before reneging as the time foran alternative manufacturer to deliver. Note that thecustomer at the head of the line may renege althoughhe has claim to the product in process. This contrastswith queueing models of service systems, in whicha customer will not renege during his service, onlywhile waiting for service to begin.) We will denote

by Dt the number of duplicate orders that are out-standing at time t. Clearly, the number of customerswaiting for the product at time t is given byX1t

+

X2t Dt and if Xit 0 for either i= 1 or 2,then Dt = 0.

Each distributor follows a base-stock policy. In par-ticular, each distributor orders one unit from themanufacturer every time a customer orders a unitfrom him, and cancels an order with the manufac-turer every time a customer cancels an order withhim. Let Yit denote the number of outstandingorders from distributor i to the manufacturer. Then,Yit = B Xit, where B is the base stock level. Themanufacturer does not hold inventory and has a total


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production capacity of rate. When she has outstand-ing orders from distributor i Yit >0, the manu-facturer delivers the product according to a Poissonprocess with rate i, which is independent of theproduction process for the other distributor. The man-ufacturer knows the base-stock policy used by the

distributors, and can therefore infer the inventorylevel and the number of customer orders outstand-ing for each distributor from her own order processY1tY2t. Furthermore, the manufacturer knowswhen a downward transition in Yit corresponds toan order cancellation and when it corresponds to anorder fulfillment, and therefore effectively observesthe orders and cancellations made by customers.

To completely describe the system dynamics, itremains to specify the sequence in which customerorders are filled. We will assume that each distributorknows which of his customers have placed a dupli-cate order, and gives priority to serving these cus-

tomers (to avoid losing a sale to the other distributor).Hence,X1 X2 Dis a continuous-time Markov chain.The assumption that distributors can identify doubleorders is plausible because any customer that doubleorders has an incentive to reveal this to the distribu-tors to shorten his lead time. Furthermore, softwarefor channel management enables distributors to shareinformation in real time about customer identity andpurchasing behavior. The most plausible alternativeassumption is that distributors serve customers on aFIFO basis. For most of the propositions in this paper,we have an analogous result for the system with FIFO

sequencing. Under FIFO sequencingX1 X2 Dis nota continuous-time Markov chain; one must keep trackof the precise position of double orders in the cus-tomer order queue to obtain a Markov process. Wecomment on how to extend each proof from the sim-ple case with priority sequencing to the complicatedcase with FIFO sequencing.

For brevity, we focus on the above system in whichthe manufacturer dedicates a fraction of her capacityto each distributor. To demonstrate that our resultsare robust, we have also analyzed a system in whichthe manufacturer uses a FIFO sequencing policy (withsimultaneous orders placed in front of each other in

the queue with equal probabilities) and in which thedistributors prioritize duplicate orders. Let DC=dc1c2cn be the state descriptor, with dthe number of outstanding duplicate orders, n thetotal number of outstanding orders (counting dou-

ble orders twice), andck 1 2 the distributor thatmade the order which is currently in position k inthe manufacturers queue (k = 1 n). Then, underthese two sequencing and prioritizing assumptionsDC is a continuous-time Markov chain. All theresults in this paper also hold for this FIFO system,

with one minor exception: We have proven Proposi-tion 3 only for = 1. We comment briefly within thepaper on adapting our proofs to this FIFO system;details are in the online appendix (available at http://mansci.pubs.informs.org/ecompanion.html).

Finally, to guarantee that the Markov process

X1 X2 D is ergodic, we assume that the number ofdedicated orders at each distributor and the numberof duplicate orders are bounded by a very large num-

ber M; that is, Xi D M for i = 1 2 and D M.Throughout, we omit the time index t whenever werefer to the whole process, and write t = to refer tothe process in steady state.

3. Maximum-Likelihood EstimationWhen = 0 (in the ManufacturersOpinion)

We begin by analyzing the basic system in which

each distributor has a dedicated stream of customers = 0, and derive MLEs for the demand rate andreneging rate from the manufacturers point of view.Then, we evaluate the systematic error (difference

between the limiting estimator and the true parame-ter value) when some customers double order > 0,

but the manufacturer is unaware of this and usesthe estimator for the system with = 0. We provethat the manufacturer overestimates the demand rateand the reneging rate. Finally, we investigate how thesystematic error varies with the underlying systemparameters. Business-press pundits (Business Week2001, Thurm 2001a) attribute Ciscos forecast error

to shortages in the summer of 2000. Therefore, wepay particular attention to how the systematic errorvaries with capacity . We prove that the system-atic error in estimating the demand rate is decreasingin and converges to zero in the limit as (as the production capacity becomes much larger thanthe demand rate). However, the systematic error inestimating the reneging rate is initially increasingin , and may be strictly positive in the limit .Even if capacity is much greater than demand, sothat backordering rarely occurs, a manufacturer thatis unaware of duplicate ordering will make a signifi-cant error in estimating the reneging rate.

Our first proposition introduces estimators of thedemand and the reneging rates. The MLEs are valid

for an arbitrary production and shipment schedule. Theydepend only on the number of customers that haveordered from distributor i , NiT , and the number ofthese orders that have been cancelled, ZiT .

Proposition 1. For the system with = 0, the MLEsfor and are given by

T =N1T + N2T

2T


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and

T= Z1T + Z2T T

0 Y1t B+ + Y2t B+ dt

= Z1T + Z2T

T0 X1t + X2tdt

respectively.

Proof. This problem can be viewed as estimatingthe transition rate parameters in a continuous-timeMarkov chain. In particular, Y1 and Y2 are twoindependent continuous-time Markov chains withgenerators (transition rate matrices)Q1,Q2, which sat-isfy fory 0 Qiyy + 1 = , Qiyy 1 = i1y>0 +y B+. For a given distributor i, we count thenumber of transitions out of state y during the timeinterval 0 T , including Niy T arrivals, Ziy Torder cancellations, and Eiy T service completions.In addition, let iy T denote the total amount of

time during this interval that the queue length of out-standing orders from distributor i is equal to y . Thus,the likelihood function given the observation ofYican be written as follows:

i=y0

exp + i1y>0 + y B+iyT

NiyTy B+ZiyTEiyTi

=exp

T+

y>B

y Biy T

NiT ZiT Ci

where Ci stands for a constant that involves onlyterms that are not a function of or . Let =1 2. Then, the values of and thatmaximize are T and T as given in thestatement of the proposition.

3.1. Systematic Error in Maximum-LikelihoodEstimation

Now suppose that some customers double order > 0 unbeknown to the manufacturer, who usesthe MLE for the system with = 0. To compute theresulting systematic error in estimation, some addi-tional notation is needed. The superscript 0 willindicate that the distributor is out of stock, and thesuperscript 1 will indicate that the correspondingdistributor has items in stock. The first superscriptwill refer to Distributor 1, and the second to Dis-tributor 2. For example, N01i T denotes the numberof orders placed with distributor i up to time T,when Distributor 1 is out of stock, and Distributor 2has some items in inventory immediately prior tothe arrival. Similarly, N00i T denotes the number oforders placed with distributor i while both distribu-tors are out of stock (immediately before the customer

arrives). N10i T and N11

i T are defined in an analo-gous fashion. Also, let00T be the total time the sys-tem spends in states where both distributors are outof stock during the corresponding time interval, andlet P00 = P X1 0 X2 0 be the steady-stateprobability that both distributors are out of stock.

Finally, recall that D is the steady-state (random)number of duplicate orders in the system and EXi isthe expected backlog level at distributor i in steadystate.

The next proposition characterizes the systematicerror, establishing that the manufacturer will overes-timate the demand rate and the reneging rate.

Proposition 2. Suppose that customers double orderwith positive probability >0, but the manufactureruses the MLE for the system with = 0given in Proposi-tion 1. Then, the systematic error in estimating the demandrate is given by

= P

00

> 0and the systematic error in estimating the reneging rate is

given by

=PD > 0

EX1 + EX2

> 0

where = limTT and = limT T.

Proof. Given the notation introduced above, thesystematic error in estimating the demand rate is

= limT

T

= limT

N1T + N2T 2T

= limT

N001 T + N00

2 T

200T

00T

T

+N011 T + N

012 T + N

101 T + N

102 T + N

111 T + N

112 T

2T 00T

T 00T

T

= + P00 + 1 P00 = P00

where the last equality follows from the strong lawof large numbers (SLLN) for renewal processes. Theabove equalities indicate that the systematic error isstrictly positive because double orders are counted astrue customer arrivals. In addition, they imply that asthe production capacity increases to , the system-atic error goes to 0.

Calculation of the systematic error in the estimatorfor is more involved. Because distributors prioritizedouble orders, whenever Dt >0 each service com-pletion is coupled with an order cancellation. Hence,one cancellation is seen whenever a nonduplicateorder is cancelled or a service completion occurs for


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a duplicate order. The resulting rate of one cancella-tion at time t is X1t + X

2t 2Dt + 1Dt>0.

Twosimultaneous cancellations will be observed withrateDt. LetZix1 x2 d T be the number of ordercancellations during the time interval 0 T , whenthe state immediately prior to the cancellation is

X1tX2tDt = x1 x2 d. We deal first with thenumerator of the expression for T:

limT

Z1T + Z2T

T

= limT

x1B

x2B

d0

Z1x1 x2 d T + Z2x1 x2 d T

x1 x2 d T

x1 x2 d T

T

=

x1B

x2B

d0

x1 + x2 2d + 1d>0 + 2d

P Xi = xi i = 1 2 D = d

= EX1 + EX2 + PD > 0

where the second equality follows from the SLLN forrenewal processes. The denominator of the expressionfor T is simpler to analyze. Specifically,

limT

T0

X1t + X2tdt

T = EX1 + EX

2

from ergodicity. Hence,

= limT

T

= limT

Z1T + Z2T /TT

0 X1t + X

2tdt

/T

=PD > 0

EX1 + EX2

The systematic error expressions in Proposition 2are exactly the same for the system in which the man-ufacturer uses a FIFO policy and distributors priori-tize double orders. When distributors serve customerson a FIFO basis, Proposition 2 is true except thatPD > 0 in the numerator of the error expressionfor is replaced by the probability that a job at thehead of the line corresponds to a double order.

3.2. Sensitivity Analysis of Systematic ErrorsOverestimation of and occurs because the manu-facturer fails to recognize the potential for duplicateorders. One might therefore expect that as the pro-duction capacity increases, the systematic error willdecrease because there is less opportunity for doubleordering to occur. In this section, we prove that, asexpected, the systematic error in the estimator for decreases with . However, the error in the estima-tor for is not so well behaved. In fact, we observesituations in which first increases with , and

only then starts to decrease. We explain this behav-ior by teasing apart the various drivers of system-atic error in the reneging rate estimator. Finally, wepresent numerical results to illustrate that the system-atic error increases with .

The steady-state probability distribution (and hence

the systematic error) can be expressed in closed formonly in the special cases = 0 and = 1. There-fore, our method of proof involves sample-path argu-ments and coupling. By this method, one can provestatements that are stronger than what we need.Specifically, our sensitivity analysis is concerned withcomparisons of certain quantities in the limit as timegoes to infinity. Instead, the sample-path argumentsestablish stochastic ordering of the relevant quantities

for every time t. This approach works when varyingthe capacity , but not the double-order probabil-ity (the relevant quantities are not ordered in theprelimit).

Proposition 3. The systematic error in estimating thedemand rate is decreasing in for any fixed allocation1 = p and 2 = 1 p, where0 < p < 1.

Proof. From Proposition 2, the systematic error inthe demand rate is = P00 =PX1 0X2 0. In three steps, we will prove thatP X1 0 X2 0, the steady-state probabil-ity that both distributors are out of stock, decreaseswith . First, consider the uniformized discrete-time Markov chain with one-step transition proba-

bilities equal to the corresponding transition rates ofthe continuous-time Markov chain, divided by v

2 + + + 3M. Transitions from a state to itselfare allowed to ensure that the transition probabilitiessum up to 1. The steady-state distribution of the uni-formized discrete-time chain is identical to that of thecontinuous-time Markov chain.

Second, letL< H, and denote by XL1 X

L2 D

Lthe state of the system when = L. Similarly,denote by XH1 X

H2 D

H the corresponding statedescriptor when = H. We use sample-path cou-

pling arguments to show that XL1 XL2 D

L st

XH1 XH2 D

Hwherest denotes stochastic order-ing. This will imply, in particular, that P XL1 0XL2 0 P X

H1 0 X

H2 0. The sample-

path coupling argument works as follows: Let ZL =XL1 + D

L XL2 + DL DL and ZH = XH1 +

DH XH2 + DH DH. Note that ZL M and

ZH M (where the inequalities hold component-wise). We can construct versions of ZL and ZH(which for notational simplicity are denoted the sameas the original processes), such thatZL ZHwithprobability 1. Specifically, we letv = 2 + + H +3M and assume that ZL0 ZH0. Then, by cou-pling the transitions of both chains and using induc-tion onn, it follows thatZ Ln ZHn.


8/14


Figure 2 Maximum-Likelihood Estimator for the Demand Rate as a

Function of the Production Capacity for the System with = 095, = 01,B= 5, and1 = 2 = /2

0.95

1.05

1.15

1.25

1.6 2.4 3.2 4.0 4.8

= 0.5

= 0.2

Third, the relationship ZL st ZH impliesthat XL1 X

L2 D

L st XH1 XH2 D

H, becauseX1 X2 Dcan be expressed as an increasing func-tion ofZ = X1 + D X2 + DD.

The second step in this proof breaks down ifthe manufacturer fills distributors orders FIFO. Theonline appendix gives an alternative proof for FIFOassuming = 1.

Figure 2 illustrates how the systematic error inestimating the demand rate decreases with thecapacity.

In contrast, the systematic error in estimating thereneging rate initially increases with .

Proposition 4. Let = limT T be the limit ofthe reneging rate MLE as the time horizon grows to infin-ity. Then, is increasing in at = 0

Proof. It is easy to see that when = 0 = 0.However, > 0 for all > 0.

However, the systematic error in estimating thereneging rate may subsequently decrease with asshown in Figure 3.

To understand this nonmonotonicity, one mustclosely examine the source of the error. Orders arecancelled for one of the following reasons: (1) a cus-tomer reneges and consequently cancels all outstand-ing orders (two orders are cancelled if he has placed

Figure 3 Maximum-Likelihood Estimator for the Reneging Rate as

a Function of the Production Capacity for the System with = 095, = 01,B= 5, and1 = 2 = /2

0.1

0.2

0.3

1.6 2.4 3.2 4.0 4.8

= 0.5

= 0.2

an order with both distributors), and (2) a customerreceives the product from one distributor and can-cels a duplicate order from the other distributor (oneorder is cancelled). Order cancellations of Type 1 donot contribute to the systematic error because cus-tomers that double order are counted twice as part

of the backlog and twice as an order cancellation.Order cancellations of Type 2 are the ones that causethe systematic error. That is, the systematic erroroccurs because the cancellation of a duplicate order iscounted as a reneging customer, but no customer isactually reneging.

The effect of on order cancellations of Type 2 istwofold. Having greater production capacity reducesthe proportion of time that the system spends in the

backordered states, so fewer duplicate orders occur.This tends to reduce order cancellations of Type 2. Onthe other hand, as the capacity increases, a largerfraction of all duplicate orders are cancelled due toservice completion (Type 2) rather than because ofreneging (Type 1). This tends to increase the numberof order cancellations of Type 2. This second effect isthe one that tends to increase the error. Note that can be written as/EX1 + EX

2 PD > 0, where

the first term in the product is increasing in , whilethe second term is decreasing. This explains why isnot monotone in .

We have observed numerically that the systematicerror is increasing with the double-order probabil-ity as shown in Figure 4. Furthermore, the system-atic error in the demand rate is only slightly higherif the distributors use FIFO sequencing, rather thangiving priority to the customers that double order.However, giving priority to the customers that doubleorder increases the error in estimating the renegingrate. That is because giving priority to customers thatdouble order increases the frequency of order cancel-lations of Type 2 (in which a customer receives theproduct and cancels a duplicate order).

4. The Manufacturers OptimalCapacity Investment

The conventional wisdom following Ciscos notori-ous inventory write-off is that duplicate ordering bycustomers will lead a manufacturer to overinvest in

capacity. This is not necessarily true. In this section,we show that, in fact, a manufacturer that mistakesdouble orders for true customers orders may buy toolittle capacity.

Suppose that the manufacturer chooses capacityaccording to2

min

cEX1 + X2 D + k (1)

2 More generally, one may consider choosing capacities 1, 2to minimize the cost function cEX1 +X

2 D + k1 + 2.

However, we have observed in all numerical experiments that the


9/14


Figure 4 Maximum-Likelihood Estimator for the Demand Rate and Reneging Rate as a Function of the Double-Order Probabilityfor the System with = 095 = 01andB= 5

1.0

1.1

1.2

1.3

0 0.2 0.4 0.6 0.8 1.0

FIFO

Priority

0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0

FIFO

Priority

where the decision variable is the total capacitydevoted to both distributors; c is the manufacturerscontribution per unit sold, so the first term in theobjective function is the expected cost of lost sales;the second term is the cost of capacity. (Without lossof generality, we will assume that c= 1.) In practice,a distributors inventory policy depends on the deliv-ery lead time and hence upon the manufacturers

capacity. However, in solving for the optimal capacityin (1) we disregard strategic interaction, and assumethat the base stock level B per distributor is fixed.The solution to (1) can be interpreted as the manu-facturers best response to the distributors inventorypolicies.

Conventional wisdom is that consumers areincreasingly impatient, and this increases capacityrequirements. However, the optimal total capacity is not monotone increasing in the reneging rate .Intuitively, when customers become extremely impa-tient (as ), the optimal capacity may drop down

to zero. This is because the capacity required to cap-ture a certain amount of sales from customers with anincreasing reneging rate may become too high, andhence prohibitively expensive.3

Therefore, whereas overestimating always leadsthe manufacturer to buy more capacity, overestimat-ing may lead the manufacturer to buy less capac-

solution to this optimization problem is symmetric:1= 2= /2.Intuitively, if the manufacturer dedicates greater capacity to onedistributor, that distributor tends to have greater inventory. Whenhis inventory reaches the base stock level the greater capacity isidled, while the other distributor typically has outstanding orders.

Hence, asymmetric capacity is inefficient. Only in the degeneratecase that all customers duplicate order = 1 and distributors donot carry inventory B= 0 does asymmetric capacity perform aswell as symmetric capacity. In this case, the expected rate of lostsales is constant for all 1 and 2 such that 1+ 2= . Asym-metric capacity might, however, yield greater system profit if thedistributors choose their inventory levels in Nash equilibrium afterobserving capacity.3 This result is proven in Armony et al. (2005) for the special casethat B = 0 and = 0, a make-to-order system without doubleorders. We have observed nonmonotonicity in numerical exampleswith B > 0 and 0 1, and conjecture that the result is true ingeneral.

ity.4 Furthermore, for fixed and , the optimal levelof capacity investment may be increasing in . Forthese two reasons, a manufacturer that is unaware ofduplicate ordering may purchasetoo littlecapacity. Wedemonstrate this result through numerical examples.

Consider a system in which = 1 (every customerthat must wait for the product will place a dupli-cate order), but the manufacturer believes that = 0.

For this system, we have the steady-state probabil-ity distribution in closed form (see Appendix A), andcan therefore compute the limiting MLEs and the costfunction exactly. The expected rate of lost sales isstrictly lower in the system with = 1 than in thesystem with =0 because, in choosing to doubleorder, each customer increases the likelihood that hewill obtain the product before reneging. Effectively,inventory and capacity are pooled in the system with = 1. Furthermore, as illustrated by Figure 5, whenthe capacity is very small, the marginal value ofcapacity is greater in the system with = 1 than inthe system with = 0 (i.e., increasing does more

to reduce lost sales when = 1 than when = 0.However, if the capacity is sufficiently large, addi-tional capacity is more beneficial when = 0 thanwhen = 1. Therefore, if the manufacturer knows thetrue demand rate and reneging rate, but incorrectlyassumes that = 0, she will underinvest when thecost of capacity k is large and overinvest when thecost of capacity k is small. When the manufacturerscapacity is pooled and distributors orders are filledFIFO, duplicate orders are still beneficial in pool-ing the distributors inventories. The pooling effectsof duplicate orders are less pronounced with pooledcapacity than with dedicated capacities, but remain

significant and qualitatively the same as the effectsillustrated in Figure 5.Suppose that the manufacturer has been operating

the system at some fixed initial level of capacity ,and uses the MLEs and (which depend on thelevel of capacity ) to compute his optimal capac-ity investment. Figure 6 shows that when the ini-tial capacity level is larger than the demand rate and

4 In fact, it is plausible that under any setting in which one overes-timates the reneging rate, underinvestment in capacity may occur(even in the absence of duplicate orders).


10/14


Figure 5 The Difference in the Expected Rate of Lost Sales in the Case = 0 and the Case = 1 for a System with = 1, = 02, B= 10, and1 = 2 = /2

Capacity Capacity 1.6 2.0 2.4 2.8 2.0 2.4 2.8

0.2

0.4

0

= 0

= 1 0.01

0.02

0.03

1.6

Differenceinrateoflostsales

Rateoflostsales(EXiD)

the cost of capacity is relatively large, this optimalcapacity investment will be strictly smaller than thetrue optimal capacity. That is, the manufacturer willunderinvest in capacity.

One might suspect that this underinvestment phe-

nomenon occurs because the manufacturer devotesa fraction of her production capacity exclusivelyto each distributor. However, underinvestment alsooccurs with resource pooling. Figures 5 and 6 can

be essentially reproduced for a system in which themanufacturer uses a FIFO sequencing policy andthe distributors prioritize their double orders. Here,the steady-state probabilities are easily calculated for = 1, but when = 0 things are more intricate; how-ever, one can obtain fairly simple expressions forthose steady-state probabilities in the pure loss model(i.e., = ; see Appendix B).

Now, let us suppose that the manufacturer repeat-edlyruns the system for long enough to compute theestimators and , and then adjusts capacity to theoptimal level. In all of our numerical experiments,the capacity converges to an equilibrium that appearsto be optimal if the manufacturer assumes that and (evaluated at the current capacity level) are thetrue demand rate and reneging rate. Figure 7 shows

Figure 6 True Optimal Capacity and the Optimal Capacity Invest-

ment for a Manufacturer Who Assumes that = 0, = , and =for the System with = 1, = 02, = 1 B= 10, and

Initial Capacities1 = 2 = 18

0

0.4

0.8

1.2

1.6

2.0

1.0 1.2 1.4 1.6 1.8

Cost of capacity k

Capacityinvestment() True optimal

Optimal assuming = 0

that in equilibrium the manufacturer overinvests incapacity.

5. Maximum-Likelihood Estimation

When > 0If the manufacturer is aware of the potential for dou-ble orders and observes the system continuously, shecan recognize a double order whenever both distribu-tors order simultaneously, or cancel an order simulta-neously. One may argue that in reality no two eventswill occur at exactly the same time. As a practicalalternative, if the manufacturer has visibility of end-customers identities, she can pair two orders made

by the same customer at approximately the sametime, and recognize a double order. In this section, wespell out the MLEs of , and in the case of fullinformation (continuous observation or visibility of

customers identities). These estimators are valid for ageneral shipment schedule from the manufacturer tothe distributors.

To write down the MLEs for these three parame-ters, we need to introduce some additional notation.Let N T denote the total number of orders made by

both distributors in the period0 T (accounting onlyonce for those orders that are immediately switched

Figure 7 True Optimal Capacity and Equilibrium Optimal Capacity

Investment for a Manufacturer Who Assumes that = 0, = , and =, for the System with = 1, = 02, = 1,

andB= 10

0

1

2

3

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

Cost of capacity k

True optimal

Equilibrium optimal

assuming = 0

Capacityinvestment()


11/14


from an out-of-stock distributor to a distributor withthe item in inventory, but twice for double orders).Also, letZT correspond to the total number of ordercancellations from both distributors in the same timeinterval. Let DinT be the total number of duplicateorders made between time 0 and time T (counting

only those duplicate orders that occur while both dis-tributors are out of stock), and let Dout T be the totalnumber of double orders (counted in DinT thathave both been cancelled by timeT. Finally, letSwT

be the number of customers who switch from an out-of-stock distributor to one with positive inventory inthe time interval 0 T , and denote byNT the totalnumber of arriving customers who find the first dis-tributor they turn to being out of stock. As in the casewith = 0 analyzed in 3, maximizing the likelihoodfunction given continuous-time transition informationyields the MLEs described in the following proposi-tion. The proof of Proposition 5 is very similar to that

of Proposition 1, and hence is omitted.Proposition 5. The MLEs of , and are given by

T =N T DinT

2T

T = ZT DoutT T

0 X1t + X

2t Dtdt

T =DinT + SwT

NT

These estimators are consistent: T T T as T .

Remark. To ensure consistency of the estimatorsfor an arbitrary shipment schedule, all that needs

to be verified is that as T , the system spendsenough time in both the backordered and the positiveinventory states. This condition is guaranteed to holdunder our Poisson production assumption with ded-icated capacities or with pooled capacity and FIFOfulfillment of distributors orders.

Before the manufacturer can compute the estima-tors for T and T, some customers must expe-rience a stockout. By reducing the production rate and risking some lost sales, the manufacturer canobtain an improved estimate of her customers toler-

ance for delay and willingness to switch to an alter-native distributor of her product.

5.1. Periodic Observation of Inventory LevelsIn practice, production occurs in batches, distribu-tors order by the truckload, and the manufacturerobserves distributors inventory levels infrequently,if at all. Specifically, prior to the spring of 2001,Cisco did not have information systems in placeto track distributors inventory levels. Managementmaintained a record of all shipments, and could, byplacing a phone call to a distributor, check on the

aggregate dollar value of Cisco products in inventory.Such check-ups occurred on an infrequent, ad hoc

basis (Kothari 2001). Recently, Cisco and many otherhigh-tech manufacturers, including Sony, HP, Toshiba,and Sun, have installed software that enables them toreview distributors inventory levels at the SKU level

on a weekly, and in some cases daily, basis (Chua2003).

Let us assume that at discrete times tk with 0 < t1 0 units of the product to distributorik orthe manufacturer observes the inventory level Xik tk(in the latter caseSk 0. Recall thatXi may take neg-ative values: Xi is the number of backorders for dis-tributor i, including double orders. We assume thatthe manufacturer cannot identify double orders (D ishidden from the manufacturer). On the contrary, eachdistributor knows which of his customers have placed

a duplicate order, and gives priority to serving thesecustomers (to avoid losing a sale to the other distrib-utor). Therefore,

Xik tk = Xik tk + Sk and

Xjtk = Xjtk + Sk Dt

k forj= ik

Dtk = Dtk S

k+

We will derive the manufacturers MLE for , ,and , given the discrete-time observations of inven-tory levels and the schedule of deliveries. For each

k= 1 K and t tk1 tk, the stochastic processX1tX2tDtis a continuous-time Markov chainwith generator matrixA given by

AX1 X2DX11 X2 D = AX1 X2DX1 X21 D =

ifX1> 0 and X2> 0

AX1 X2DX1 1 X2 D = 1 + ifX1> 0 and X2 0

AX1 X2DX1 X21 D = 1 + ifX1 0 and X2> 0

AX1 X2DX11 X2 D = AX1 X2DX1 X21 D = 1

ifX1 0 and X2 0

AX1 X2DX11 X21 D+1 = 2 ifX1 0 andX2 0

AX1 X2DX1+1 X2 D = X1 D+ ifX1< 0

AX1 X2DX1 X2+1 D = X2 D+ ifX2< 0

AX1 X2DX1+1 X2+1 D1 = D ifD > 0

AX1 X2DX1 X2 D = 0 otherwise

with the obvious modification to reflect our state spacetruncation:D M and Xi D M. The continuous-time Markov chain is completely characterized by the


12/14


initial distribution and the generatorA. In particular,the transition matrixPtis given by

Pt = expAt =

n=0

Atn

n!

that is, PX1 X2DX1 X2 Dt is the conditional proba-bility that X

1s + tX

2s + tDs + t = X

1 X

2 D

given thatX1sX2sDs = X1 X2 D and tk1 s < s + t < tk. Let q

0 denote the initial distribution ofX10 X20D0. For example, if the distributionchannel is empty at time zero, then

q0X1 X2 D =1 ifX1 = X2 = D = 0

q0X1 X2 D =0 otherwise.

Because the deliveries Sk1 Sk2 k=1K are known,

the likelihood function can be computed recur-sively, as follows. For k = 1 2 we will com-pute k, the unnormalized conditional distributionfor X1t

k X2t

kDt

k given the manufacturers

observations and deliveries up to time tk1, andthen compute qk, the unnormalized conditional dis-tribution for X1tk X2tkDtk given the manu-facturers observations and deliveries up to time tk.Definet0 = 0 and

k qk1P tk tk1

In the case that a delivery is made at time tk Sk > 0,

then

qkX1 X2 0

Skd=0

kX1Sk X2dd ifik = 1

Skd=0

k

X1d X2Sk d ifik = 2

qkX1 X2 D kX1Sk X2Sk Sk

forD > 0

In the case that an observation is made at time tkSk = 0,

qkX1 X2 D

kX1 X2 D ifXik = Xik tk

0 otherwise

Then, the likelihood function is given by

= qK

is the probability that under the delivery

schedule {Sk ik tkk=1K, the distributorik has inven-tory levelXik tkat timetkfor whichSk = 0. This estab-lishes our main result:

Proposition 6. Suppose that the manufacturer can-not identify double orders, and observes each distributorsinventory level at discrete points in time. In this case, the

MLEs of , and are given by

tK tKtK =argmaxR3+

In extensive simulation experiments, we have foundthat for systems with a small base stock level B 10

andM= 10, the estimator is consistent. Unfortunately,the time required to compute the matrix exponentialexpAt grows exponentially with B and M, and wehave been unable to directly compute the MLE forsystems with B 20 or M20. Clearly, a more effi-cient estimator will be needed in practice. The expecta-

tion maximization (EM) algorithm (Elliott et al. 1995)can be used to compute a series of parameter val-ues that converges to tKtKtKin a mannerthat avoids direct computation of the matrix exponen-tial. An alternative, efficient method of moments esti-mators based on periodic sampling of a continuous-time Markov process has recently been developed inthe finance literature (Hansen and Scheinkman 1995,Duffie and Glynn 2004).

6. Concluding RemarksOur results suggest that Ciscos write-off was caused

by estimation errors and cannot be blamed entirely

on the economic downturn. Any manufacturer thatfails to account for duplicate orders will overestimatethe demand rate and the reneging rate, and there-fore err in capacity planning. Business-press punditshave attributed Ciscos multibillion-dollar overinvest-ment in capacity to a severe component shortage fol-lowed by a drop in demand, and failure by Ciscoto recognize this downturn because of duplication inthe order backlog. However, our analysis shows thatexcess capacity can be an insidious, chronic problemeven under stable demand conditions. An acute dropin demand just exacerbates the problem. When thecost of capacity is relatively low, so that the produc-

tion rate is greater than the demand rate, the error inestimating the demand rate is small, but the overesti-mate in the reneging rate is large. Because customersappear to be very sensitive to delay, the manufacturerdoes not realize that she has too much capacity.

Surprisingly, we have also observed that when thecost of capacity is very high and the manufactureris unaware of duplicate ordering, she may investtoo little in capacity. At a low level of capacity, cus-tomers tendency to switch to an alternative distribu-tor reduces the number of lost sales and increases themarginal value of capacity.

For the manufacturer that is monitoring doubleorders, we give MLEs for the demand rate, the reneg-ing rate, and the probability that a customer will dou-

ble order when forced to wait. These are valid forany production, transportation, and inventory policy.The basic MLE assumes that the manufacturer hasreal-time visibility of distributors inventory levels,either through sophisticated software or because thedistributor follows a base-stock policy (so the manu-facturer can infer the inventory level from her ownorder queue). More commonly, distributors order in

batches and the manufacturer observes distributors


13/14


inventory levels infrequently if at all; we also providethe MLE for this setting.

An important insight is that the manufacturermust experience backorders and lose some customers

before she can estimate her customers tolerance fordelay. In models with lost sales, other researchers

have shown that carrying more inventory results ina better estimate of the demand rate. Our model isdistinctive in that customers will wait, although onlyfor a limited time, before the sale is lost. Carrying lessinventory and/or reducing the production capacityresults in a better estimate of the reneging rate and thedouble-order probability, without affecting the esti-mate of the demand rate.

We have assumed a fixed probability that a cus-tomer duplicates orders, given that his distribu-tor is out of stock. In reality, duplicate orderingis history dependent. Confronted with a long leadtime, some customers will make the effort to seek

out an alternative distributor. Having learned abouttheir alternatives, these customers are more likely toduplicate-order in the future (Kothari 2001). This can

be modeled as a Markov process with the double-order probability as a hidden state variable. Then,maximum-likelihood estimation requires repeatednumerical evaluation of an exponential function ofthe generator matrix (a computationally intensiveprocedure). This is impractical for industrial-sizedproblems. Ongoing research will develop efficientmethod-of-moments estimators based on periodicsampling of a continuous-time (partially observed)Markov process, drawing on methods from the

finance literature (Duffie and Glynn 2004).Since 2001, Cisco has increased visibility andtightened control of its distribution channels. Newinformation systems provide visibility of distribu-tors inventory levels and the ability to control thepurchase price for resellers (www.comergent.com).Resellers that share demand information and providea high service level to customers are rewarded with areduced price for Cisco hardware. With survey infor-mation from resellers and corporate customers, Ciscohas improved its demand forecasting (Business Week2002a). In another new program, Cisco owns the hard-ware in the distribution channel, sets the price for theend customer, and pays channel partners. Resellerscharge the end customer for the service of configura-tion, rather than selling hardware. Adoption of thisnew program has been slow, perhaps because chan-nel partners are unwilling to cede information andcontrol.

High-tech manufacturers including Sony, HP,Toshiba, and Sun have recently implemented iGINEsoftware for monitoring channel inventory. Accordingto Chua I-Pin, Vice-President of iGINE, manufacturersmay need to provide significant financial incentivesfor distributors to share inventory data. Validation

is challenging, as the data sets are complex. Forexample, in the Asia-Pacific region, Hewlett-Packardmonitors more than 100,000 SKUs at 100 Tier-1 whole-salers, and 300 Tier-2 resellers. Distributors typicallywill not share the identity of backordered customersor price information. In shortage conditions, distrib-

utors typically increase their prices, so demand esti-mation is complicated by the hidden variable of priceas well as duplicate orders. With visibility of inven-tory levels, manufacturers are moving beyond simpleFIFO order fulfillment or proportional allocation, and

beginning to replenish inventory based on the over-stock or understock conditions at various distributors.Will distributors reduce their inventory to the detri-ment of the manufacturer? Further research is neededto address dynamic strategic interaction with infor-mation asymmetry and estimation.

An online appendix to this paper is available athttp://mansci.pubs.informs.org/ecompanion.html.

AcknowledgmentsThe authors thank Tushar Kothari, Vice-President of Distri-

bution at Cisco; and Chua I-Pin, Vice-President of Opera-tions at iGINE for discussions on duplicate ordering andinventory visibility in channel management. They thank LiChen for numerical analysis, Halina Frydman for advice onstatistical estimation, and Sridhar Seshadri for guidance inthe sensitivity analysis. Finally, they thank Bill Lovejoy andthe anonymous associate editor and referee for suggestionson model formulation, presentation, and literature.

Appendix A. The Steady-State ProbabilitiesWhen = 1

We describe how to derive the steady-state probabilitiesfor the system with = 1, and the exclusive productioncapacity devoted to each distributor. Note that the statesX1 X2 where X1 < 0 and X2 > 0 (and vice versa) can-not be accessed. Therefore, one can partition the completestate space into the following three mutually exclusive sets:(1) S =n n n >0, (2) S0+ =0 0 1 B 0 1 B 0, and (3) S+ = 1 B 1 B . Itis straightforward to calculate the steady-state probabilitiesfor the Markov chains that are restricted to each of thesethree sets, and hence to calculate the steady-state probabili-ties for the whole chain, using the transition rates betweenthese sets. The resulting expressions are cumbersome. Forexample, the steady-state probability that both distributors

have outstanding orders is

S =+0 02

+0 02 + 1 1 +

where

1 1 =

n=1

2n1nj=2 + j

1and +0 0 =

q1q1 + q2

with

q1 =

1 11 B1

B11 +1 21 B2

B12


14/14


q2 = c

1

21B1 1/22

B

221 1/22 + 2

1 21B

1 21

c =

1 21

B+1

1 21+

21B1 1/22

B

221 1/22

1 i=

i

i= 1/2 1,i = 1 2. Similar expressions apply when i= 1/2or 1 for i = 1 or 2.

Appendix B. The Steady-State Probabilitieswith FIFOIn this appendix, we describe how to calculate the steady-state probabilities when the manufacturer uses a FIFO pol-icy, the distributors prioritize their double orders, ordersmade when distributors are out of stock are lost (that is, = ), and there are no double orders = 0. Under theseassumptions, the state descriptor C=c1c2cn(with n the total number of outstanding orders, and ck 1 2 is the distributor who made the order that is cur-rently in position k in line k= 1 n ) is a continuous-time Markov chain. Let YiC =

nk=1 1ck=i, i = 1 2, be the

number of orders from distributori currently in queue; then

YiC is constrained to the set 0 B . Without this con-straint, the steady state of the system has a product form,and hence it has this form on the constrained state space aswell. In summary, the steady-state distribution is as follows:

C = b

A

2

n

where = 2/, b = 1/12B+1 if =1, and b =1/2B + 1when = 1. Also,

A = 1 b2B

n=B+1

Mn

2

nwithMn = 2

nk=B+1

n

k

The loss rate in this system is

PY1C = B + PY2C = B = 2PY1C = B

= 22B

n=B

n

B

b

A

2

n

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