Integrating direct and indirect sales channels under ...

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Int. J. Production Economics 103 (2006) 209–229

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Integrating direct and indirect sales channels under decentralizeddecision-making

Ralf W. Seiferta,�, Ulrich W. Thonemannb, Marcel A. Siekeb

aIMD—International Institute for Management Development, Chemin de Bellerive 23, P.O. Box 915, CH-1001 Lausanne, SwitzerlandbDepartment of Supply Chain Management and Management Science, University of Cologne, 50923 Cologne, Germany

Received 20 February 2003; accepted 8 June 2005

Available online 12 October 2005

Abstract

Many manufacturers have started to use virtual stores as a direct distribution channel in addition to their existing

indirect retail channels. These companies must now decide how to integrate these channels. The alternatives are to operate

dedicated distribution channels for the virtual store and the retail stores or to extend the cooperation between the

manufacturer and the retailers to include the virtual store in the existing distribution channels. In such an integrated supply

chain, retail stores would continue to serve all in-store customers, but excess stock at retail stores could be used to fill some

online orders. We analyze this problem from a supply-chain perspective by developing and solving mathematical models

for both a dedicated and an integrated supply chain. Our mathematical models are based on a problem faced by Hewlett-

Packard Company (HP) and incorporate the major parameters of a supply chain with online and in-store customers. We

use the models to analyze how the supply chains can be coordinated, how the optimal supply-contract parameters can be

computed, and how the supply-chain profits can be allocated between a manufacturer and its retail-channel partners. Our

coordinating supply contract involves joint specification of the wholesale price, an ending-inventory subsidy, and a transfer

payment from the manufacturer’s virtual store to the retailer when using excess retail stock. We also compare the benefits

of supply-chain integration with the benefits of supply-chain cooperation. We illustrate our analytical results with

numerical examples from HP.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Supply-chain management; Internet

1. Introduction

Direct consumer sales over the Internet reached$33 billion in 2001 in the United States alone,

front matter r 2005 Elsevier B.V. All rights reserved

e.2005.06.006

ng author. Tel.: +4121 6180111;

0707.

sses: [email protected] (R.W. Seifert),

[email protected] (U.W. Thonemann),

ni-koeln.de (M.A. Sieke).

approximately 20% up from the year 2000 (USCensus Bureau, 2002). Initially, this online marketwas targeted by direct-to-the-customer pioneerssuch as Amazon and Dell. Other companies quicklyfollowed—some very successful, some having beenforced to cease operations in the meantime (e.g.,Boo.com, eToys, and Webvan; Baldwin, 2000). Inaddition, established players in all major industriesresponded to the growing significance of Internetsales and started to use virtual stores as a direct

.

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ARTICLE IN PRESSR.W. Seifert et al. / Int. J. Production Economics 103 (2006) 209–229210

distribution channel parallel to their traditionalindirect retail channels. Prominent examples ofcompanies that have added a virtual store to sellto online customers include Barnes & Noble,Compaq, Fnac, Hewlett-Packard Company (HP),IBM, Levi Strauss, Tesco, The Gap, Toys ‘‘R’’ Us,and Wal-Mart (Anders, 1998; Beck, 2000; Kauf-man, 1999; Norr, 1999).

While much uncertainty about the future pro-spects of direct sales via the Internet remains,companies continue to experiment with differentfulfillment models. Tesco, the British grocer, forexample, relies on its network of stores rather thanhigh-tech warehouses to fulfill online orders. Whena customer places an online order, the order isforwarded to the store nearest to the customer. Atthe store, an employee picks and packs the itemsordered, which are then shipped to customers’homes by vans (Beck, 2000).

Everyone said we were wrong, wrong, wrong. Wesat there mutely, thinking: ‘‘No, here’s anothergreat channel to market.’’ John Browett, CEO,Tesco.com

Likewise, Fnac, the largest retailer of books,music and software in France opened an Internetstore, which does not operate a dedicated ware-house with dedicated inventory of books, compactdiscs, and software. Instead, online orders aredirectly fulfilled from the inventory of the retailshops (Kaufman, 1999).

From a supply-chain perspective, such an inte-gration of a direct and an indirect channel isattractive because it promises significant profit gainsand inventory reductions as well as improvedcustomer service (Seifert, 2000). However, it is notwell understood if and how these benefits can beachieved if decision-making is decentralized. HP,for example, established the HP Shopping Villagebut previously did not serve the consumer marketdirectly (Wilder, 1999). Companies such as HP, thatdo not own or control their retail channels butcooperate with retail-channel partners (e.g., OfficeDepot), must decide how to integrate this directchannel into their existing supply chains. Thealternatives are to operate a dedicated (separateand independent) supply chain for the virtual storeor to extend cooperation with partners in theindirect retail channel to include the virtual storein the existing supply chain. In such an (horizon-tally) integrated supply chain, retail stores wouldcontinue to serve all in-store customers, but excess

stock at retail stores could be used to fill someonline orders. Thus, retail stores would serve aslocal distribution centers for online-order fulfill-ment. Under those conditions, however, it is notknown how to design incentive schemes and supplycontracts that would lead to optimal coordinationof the activities in the supply chain and whether theresulting terms of trade would be mutually bene-ficial for all supply-chain partners.

This paper discusses mathematical models thatwe have developed for supply chains, such as theone of HP, that operate under decentralizeddecision-making. Our mathematical models arebased on a problem faced by HP and incorporatethe major parameters of a supply chain with onlineand in-store customers. We use the models toanalyze how the supply chains can be coordinated,how the optimal supply-contract parameters can becomputed, and how the supply-chain profits can beallocated between a manufacturer and its retail-channel partners. In this context, our focus is oncoordinating supply contracts that involve jointspecification of the wholesale price, an ending-inventory subsidy, and a transfer payment from themanufacturer’s virtual store to the retailer whenusing excess retail stock. We also identify conditionsunder which it would be optimal to operate theintegrated supply chain without dedicated inventoryfor the virtual store, and we compare the benefits ofsupply-chain integration versus the benefits ofsupply-chain coordination. Decision makers canuse the results of our analysis to identify productsfor which an integrated supply chain is attractive, todesign optimal supply contracts for these products,and to determine how much inventory should beallocated to each channel.

This paper is organized as follows. In Section 2,we review the relevant literature. In Section 3, wedescribe our mathematical models in detail. Wecharacterize the optimal inventory policies anddevelop supply contracts that achieve coordinationfor supply chains under decentralized decision-making. In Section 4, we discuss the managerialimplications of our analysis and report numericalresults based on a problem faced by HP. In Section5, we summarize our key findings. Complementaryanalyses and proofs are contained in the Appendix.

2. Literature review

We analyze a dedicated and an integrated supplychain under decentralized decision-making. Two

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major lines of literature are relevant to our research:the literature on inventory pooling and the literatureon supply-chain contracting.

The dedicated supply chain represents a standardinventory problem and was studied by a number ofauthors including Bellman et al. (1955), whoestablished the optimality of a base-stock policy.Porteus (1990) and Federgruen (1993) comprehen-sively review this literature for the case whendecision-making is centralized. In the integratedsupply chain, on the other hand, excess stock at theretail stores can be used to fill some online orders,which constitutes an asymmetric transshipmentproblem. Multi-location inventory problems withgeneral transshipment among the retail stores havebeen analyzed by a number of authors. Gross(1963), Das (1975), Karmarkar and Patel (1977),Tagaras (1989), and Tagaras and Cohen (1992)analyzed two-location transshipment problems.Other researchers have analyzed transshipmentmodels for more than two locations. Karmarkar(1987) developed bounds for a general model wheretransshipments are an alternative to ordering.Robinson (1990) derived structural results for amodel with general transshipments among multipleidentical retail stores. However, Robinson’s modelcan only be solved analytically if all cost parametersare identical, thus the model is not suitable toanalyze the asymmetric structure of our integrated-supply-chain problem. Most recently, Herer et al.analyze the benefits of transshipments consideringseveral retailers, which can differ in their cost anddemand parameters in a model developed subse-quent to our research (Herer et al., 2001).

The transshipment problem has also been studiedin the literature on multi-echelon inventory modelsallowing for emergency lateral transshipments (e.g.,Axsater, 1990; Lee, 1987). Furthermore, the notionof inventory pooling as introduced in the seminalwork of Eppen (1979), Eppen and Schrage (1981),and Jackson (1988) is closely related to research onproduct substitution (e.g., Bassok et al., 1999; Smithand Agrawal, 2000) and to research on productcommonality (e.g., Dong and Lee, 1999; Yang andSchrage, 2001). However, the setups of these modelsdiffer significantly from the setup of our problemand, with few exceptions (namely Dong and Lee,1999; Parlar, 1988; Pasternack and Drezner, 1991),this literature assumes that supply chains areoperated under centralized decision-making. Thus,we do not review this literature here and refer thereader to the references.

The literature on supply-chain contracting, on theother hand, models supply chains that are operatedunder decentralized decision-making. In particular,Pasternack (1985) analyzed a manufacturer–retailermodel related to our dedicated-supply-chain model,and showed that supply-chain coordination can beachieved if the retailer is allowed to return allleftover inventory at a partial refund. Furthermore,coordinating contracts have been studied in theeconomic literature on vertical restraints (e.g., Katz,1989; Tirole, 1988) and the marketing literature onchannel coordination (e.g., Jeuland and Shugan,1983; Moorthy, 1987). Across this vast literature, avariety of mechanisms such as returns policies (e.g.,Pasternack, 1985), quantity flexibility contracts(e.g., Tsay, 1999), quantity discounts (e.g., Jeulandand Shugan, 1983), and price protection (e.g., Leeet al., 2000) have been considered to achieve supply-chain coordination. Comprehensive reviews of thisliterature can be found in Anupindi and Bassok(1999), Lariviere (1999), Tsay et al. (1998). How-ever, this literature conventionally studied the(vertical) coordination of sequential echelons in asupply chain (e.g., Chen et al., 2001; Lee andWhang, 1999), whereas we analyze (horizontal)integration within an echelon.

More recently, some researchers have started tocombine these two lines of literature and looked intotransshipment models under decentralized decision-making. Rudi et al. (2001) analyzed the transship-ment between two independent locations and deter-mined transshipment prices that induce the retailstores to choose the inventory policy that would beoptimal under centralized decision-making. Anupin-di et al. (2001) consider a more general distributionsystem with N retailers. Anupindi et al. introduce thenotion of claims to separate the ownership and thelocation of inventories in the system and analyzetheir model from a general game-theoretical perspec-tive. In addition, in a model developed subsequent toour research, Netessine and Rudi (2001a) consider amodel with one retailer and one wholesaler withemphasis on the wholesaler–retailer interaction undera drop-shipping arrangement. Netessine and Rudithen expand this framework to consider multipleidentical retailers to analyze the benefits of riskpooling through the use of drop-shipping (Netessineand Rudi, 2001b).

Finally, we would like to acknowledge theconsiderable amount of research in the marketingliterature, which explicitly considers issues ofchannel choice, channel power, and distribution

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Retail Stores

NManu- facturer

Virtual Store

1

Manu-facturer

Retail Stores

Virtual Store

N

1

(a)

(b)

Fig. 1. (a) Dedicated supply chain; (b) integrated supply chain.

R.W. Seifert et al. / Int. J. Production Economics 103 (2006) 209–229212

structures. A number of researchers specificallyconsidered the effect of channel conflict, whichmay arise between a manufacturer and its retail-channel partners when the manufacturer initiates adirect sales channel. Webb (2002) describes strate-gies for proactively managing channel conflict andargues that a supplier will experience lower levels ofchannel conflict by diverting fulfillment of ordersplaced on their website to their channel partners.Yao et al. (2005) analyze the impact of informationsharing on buyback prices, order quantities, andprofits when a direct sales channel is added to thetraditional manufacturer–retailer supply chain andthe number of direct customers is unknown. Tsayand Agrawal (1999) devise a quantitative modelwith deterministic demand functions to assess themagnitude of a potential channel conflict in moredetail. Interestingly, Tsay and Agrawal show thatadding a direct sales channel alongside an indirectretail channel is not necessarily detrimental to theretailer (Tsay and Agrawal, 1999).

We summarize the positioning and contributionof this paper as follows: we analyze a situationwherein both the retail-channel partner andthe manufacturer, while continuing to supply theretailer, have market access; this differs from theexisting literature with the exception of Tsay andAgrawal (1999). Furthermore, we acknowledge thatsuch supply chains typically operate under decen-tralized decision-making, which differs from muchof the existing literature on inventory pooling. Forthis setting, we develop a multiple-period, lost-salesmodel with stochastic demands that takes advan-tage of the asymmetric supply chain structure whenusing excess stock at the retail stores to fill onlineorders. We explicitly characterize the parties’ pre-ferences towards the contract parameters, andillustrate how profit gains can be split. Our purposeis not necessarily to advocate specific parametervalues, but to provide insight into the mechanism bywhich supply chains with direct and indirectchannels can be integrated and operated in amutually beneficial way.

3. Mathematical models

In this section, we develop and solve mathema-tical models for the dedicated and the integratedsupply chain (Fig. 1). We consider supply chainswith Nb1 identical retail stores and a single virtualstore that are operated by two independent par-ties—a retailer that operates all retail stores, and a

manufacturer that supplies the retailer and operatesthe virtual store. Individual retail stores are indexedby r and the aggregated retail channel of all N retailstores is indexed by R; the manufacturer is indexedby M and the virtual store is indexed by v. Since themarket size is restricted in most applications, thenumber of retail stores N that can be operated in theconsidered market is bounded above by Nmax.

The demand at retail store i is xri with cumulativedistribution function (c.d.f.) F rð�Þ and mean demandmr. The demand at the virtual store is xv with c.d.f.Fvð�Þ and mean demand mv. Individual retail storesserve geographically dispersed areas and thus, theirdemands are not correlated. It is important to noticethat the identical demand distribution assumption isreasonable in our setting (see Appendix A), but itmight not be appropriate in all possible situations.The virtual store serves different customer segmentsthan the retail stores (Dodd, 2001) and thus, thedemand at the retail stores and the demand at thevirtual store are not correlated. The validity of thesemodelling assumptions is supported by the correla-tion analyses of weekly sales data for nearly 200retail stores and the HP Shopping Village (seeAppendix A).


In-store customers demand instantaneous (local)product availability, and product unavailability atretail stores results in lost sales. We assume that astock-out at one retail store does not have animmediate impact on other retailers, serving othergeographical areas, or even the virtual store sincecustomers are unlikely to change their (local)purchasing behavior due to a single stock-out eventat their local store. Online customers, in contrast,expect to and are willing to wait a ‘‘regular’’shipment time for product delivery, i.e., their pointof demand realization and their point of demandfulfillment are not identical. Thus, online orders canbe filled by drawing inventory from multiple sourcesand, as a model abstraction, online orders are lostonly if they cannot be filled by the end of a giventime period (see Section 3.2 for detailed discussion).

In the dedicated supply chain, the retail storesserve all in-store customers and the virtual storeserves all online customers. In the integrated supplychain, the retail stores still serve all in-storecustomers. Online demands are filled from theinventory of the virtual store unless the virtualstore runs out of stock. Then, inventory left over atthe retail stores at the end of the period is used to fillthese orders.

We model this problem as a stationary multiple-period newsvendor problem with lost sales. At thebeginning of each period the retailer and themanufacturer must determine their order quantitiesfor the retail stores and the virtual store, respec-tively. We represent the decision variables for periodn as the base-stock levels, yrin for the retail storesi ¼ 1; . . . ;N and yvn for the virtual store. Lead timeis zero (we consider the effect of non-zero lead timein Appendix B). Thus, the order quantity for eachperiod is the difference between the base-stock leveland the on-hand inventory at the beginning of theperiod.

The unit product cost is cr for a retail store and cvfor the virtual store. The unit product cost includesa unit shipping cost from the manufacturer to thestores; thus, cr differs from cv if the shipping cost tothe retail stores is different than the shipping cost tothe dedicated warehouse that serves for the orderfulfillment of the virtual store. The retailer pays themanufacturer a wholesale price w that satisfiescrowop, where p denotes the sales price to theend-customer. The per period discount factor is a,with 0oao1. Cost discounting results in a finan-cial-inventory-holding cost of ð1� aÞc per unit ofinventory left over at the end of the period. In

addition, we charge a physical-inventory-holdingcost per unit of inventory left over at the end of theperiod of hr at the retail stores and hv at the virtualstore. The goodwill cost for lost sales is g. For easeof presentation, we restrict our analysis to pr ¼ pv

and gr ¼ gv. The model can be modified to prapv

and gragv in a static price setting, however, thedynamics of price changes and price competitionover time are not addressed.

The objectives of the retailer and the manufac-turer are to maximize their expected profits over aninfinite horizon. It is well known that, in the absenceof coordinating mechanisms, such decentralizeddecision-making results in lower profits and servicelevels as compared to centralized decision-making(Lariviere, 1999). To maximize supply-chain profitsunder decentralized decision-making, we thereforedevise a supply contract that restores supply-chaincoordination, i.e., a supply contract that induces theretailer and the manufacturer to adopt the inventorypolicy that is optimal under centralized decision-making.

In the context described above, we next analyzethe performance of dedicated supply chains (Section3.1) and integrated supply chains (Section 3.2). Inboth subsections, we first review the optimalsolution if decision-making were centralized andthen analyze the solution if decision-making weredecentralized. Our focus will be on linear supplycontracts that are widely used. To distinguishbetween centralized and decentralized decision-making, we use subscripts C and D, respectively.

3.1. Dedicated supply chain

We first consider dedicated supply chains undercentralized decision-making and then under decen-tralized decision-making. We use upper bars for allvariables and functions defined for dedicated supplychains.

3.1.1. Centralized decision-making

Summary 1 states the optimal inventory policyunder centralized decision-making (Porteus, 1990).

Summary 1. For a dedicated supply chain under

centralized decision-making, the optimal base-stock

levels are

y�ri;C ¼ y�r;C ¼ F�1r

pþ g� cr

pþ gþ hr � acr

� �8i,


y�v;C ¼ F�1v

pþ g� cv

pþ gþ hv � acv

� �.

The resulting optimal expected supply-chainprofit per period is

E½pS;Cðy�r;C; y�v;CÞ� ¼ E½pSR;Cðy�r;CÞ� þ E½pMvðy

�v;CÞ�,

where

E½pSR;Cðy�r;CÞ�

¼ Nfpmr � cry�r;C � ðhr � acrÞ

�Eðy�r;C � xrÞþ� ðpþ gÞEðxr � y�r;CÞ

þg

E½pMvðy�v;CÞ� ¼ pmv � cvy

�v;C � ðhv � acvÞ

�Eðy�v;C � xvÞþ� ðpþ gÞ

�Eðxv � y�v;CÞþ.

Here, E½pSR;Cðy�r;CÞ� represents the maximal sup-ply-chain profit for the retail channel, andE½pMvðy

�v;CÞ� represents the maximal supply-chain

profit for the virtual store.

3.1.2. Decentralized decision-making

In the dedicated supply chain, the direct channelcontinues to be ‘‘coordinated’’, i.e., y�v;D ¼ y�v;C, evenwhen decision-making is decentralized. This is notsurprising because the optimal inventory policy atthe virtual store is specified by the manufacturer andindependent of the inventory policy at the retailstores. The retail channel, on the other hand, isgenerally not coordinated due to a double margin-alization effect occurring when the manufacturercharges a wholesale price above its product cost(Cachon, 1999). Consequently, supply-chain profitsand customer service suffer unless a supply contractis used to restore supply-chain coordination in theretail channel.

Researchers have analyzed a variety of coordina-tion mechanisms that can be shown to achievesupply-chain coordination (see Section 2). Ourfocus is on supply contracts that are based onexcess retail inventory such as returns or buybackpolicies. These supply contracts are widely used inmany industries (Padmanabhan and Png, 1995) andprovide a simple form of insurance to the retailer tocompensate the retailer for the risk of overstocking(Lariviere, 1999). Under such a supply contract, themanufacturer charges wholesale price w and agreesto pay an ending-inventory subsidy b to the retailerfor each unit of inventory left over at the end of aperiod (Pasternack, 1985).

If a supply contract with parameters w and b isused, the optimal base-stock level at a retail storecan be computed as

y�ri;Dðw; bÞ ¼ y�r;Dðw; bÞ

¼ F�1r

pþ g� w

pþ gþ hr � aw� b

� �8i.

The manufacturer can now induce the retailer tochoose the same inventory policy that is optimalunder centralized decision-making, i.e., y�r;Dðw; bÞ ¼y�r;C, by announcing an appropriate wholesale priceand ending-inventory subsidy combination (Sum-mary 2, adapted from Pasternack, 1985; Lariviere,1999).

Summary 2. If the manufacturer offers a supply

contract ðwðdÞ; bðdÞÞ with wðdÞ ¼ cr þ d and bðdÞ ¼ððhr þ ð1� aÞðpþ gÞÞ=ðpþ g� crÞÞd, where d 2 ½0;pþ g� crÞ, the following properties hold:

1.
The dedicated supply chain under decentralized
decision-making is coordinated and total profits

are the same as in the dedicated supply chain under

centralized decision-making.
2. The retailer’s profit is linearly decreasing in d:
E½pR;DðdÞ� ¼ E½pSR;Cðy�r;CÞ�

�N y�r;C �Eðy�r;C � xrÞ

þ

F rðy�r;CÞ

" #d.

3.
The manufacturer’s profit is linearly increasing
in d:

E½pM;DðdÞ� ¼ E½pMvðy�v;CÞ�

þN y�r;C �Eðy�r;C � xrÞ

þ

F rðy�r;CÞ

" #d.

Summary 2.3 shows that the manufacturer’sprofit function has two components: online salesand sales to the retailer. The expected profit at thevirtual store is independent of d, while the expectedprofit from supplying the retailer depends on d.Therefore, d serves as a parameter that allows us toselect a specific contract from a continuum ofcoordinating supply contracts. For d ¼ 0, theretailer captures all retail-channel profits; ford ¼ pþ g� cr, the manufacturer captures all re-tail-channel profits. For intermediate values for d,the retail-channel profit is split between the retailerand the manufacturer. Because total profits are

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s

sN

s

1v

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Allocation s1

y

ξ

ξ rNrN

yr1 r1

R.W. Seifert et al. / Int. J. Production Economics 103 (2006) 209–229 215

higher in a centralized system than in a decentra-lized system, there exists a d 2 ½0; pþ g� crÞ suchthat both the retailer and the manufacturer benefitfrom the supply contract. Additional discussion ofthe implications of Summary 2 can be found inLariviere (1999). Next we analyze integrated supplychains under centralized and decentralized decision-making.

sv

Nv facturer

Virtual Store

yv ξ v

Fig. 2. Inventory-allocation decision for an integrated supply

chain.

3.2. Integrated supply chain

In the integrated supply chain, the manufacturerand the retailer agree that excess inventory at theretail stores can be used to fill those online demandsthat the virtual store cannot meet from stock. If thetotal excess stock across all N retail stores is lessthan the unfilled online demands, all excess stock atthe retail stores is used, and the online demands thatcannot be met are lost. On the other hand, if thetotal excess stock at the retail stores exceeds thenumber of unfilled online orders, then excess stockis drawn from the retail stores in such a way that noretail store is systematically preferred (e.g., alwaysfrom the retail store closest to the delivery address).We call such an inventory-allocation decisionunbiased in that no retail store is preferred overanother one. It can then be shown that the optimalbase-stock levels are the same at all N retail stores,i.e., yri ¼ yr;8i, because retail stores have identicalcost structures and demand distributions (seeMcGavin et al., 1997).

In mathematical terms, we allocate si ¼ xri ^ yri

units to in-store customers at retail store i, sv ¼

xv ^ yv units to online customers at the virtual store,and siv units to fill online demands which the virtualstore could not meet from stock using excessinventory at retail store i. We say that retail storei ‘‘cross sells’’ siv units. The resulting inventory-allocation decision for an integrated supply chain isillustrated in Fig. 2. The decision variables for theintegrated supply chain are the base-stock levels atthe retail stores and at the virtual store.

Because the retail stores are not optimized forindividual-product shipment, the unit shipping costto an online customer is higher at the retail storesthan at the virtual store. Thus, our model includes atransshipment-penalty cost u (attributed to themanufacturer) that is charged for each unit crosssold, with max½0;�aðcr � cvÞ þ ðhr � hvÞ�puppþ

gþ hr � acr. For uo0, it would not be optimal tosatisfy demand at a retail store from on-hand

inventory instead of filling a remaining online order,while for uo� aðcr � cvÞ þ ðhr � hvÞ, it would notbe optimal to satisfy online orders using stock fromthe virtual store instead of using excess stock from aretail store. For u4pþ gþ hr � acr, an integratedsupply chain does not offer any cost savings over adedicated supply chain and the problem reduces tothe dedicated-supply-chain problem.

While all inventory accounting in our model isperformed at the end of a period to allow fortractability of the analysis, online customers wouldtypically not experience a notable delay before anorder is confirmed and a delivery date is commu-nicated. To explain in detail how online-orderfulfillment would be implemented, consider threecases:

1. A customer places an online order and thewarehouse of the virtual store has inventory onhand: the order is immediately confirmed and adelivery-time is quoted. The product is picked,packed, and shipped and the online customer willreceive the product within a few days.

2. A customer places an online order, the ware-house does not have stock on hand, but at least oneretail store has high inventory levels: then, it is veryunlikely that this retail store will run out of stock bythe end of the period and the product can beimmediately shipped. As in the previous case, theonline customer receives a confirmation and adelivery-time quote at once—the product is picked,shipped, and received within a few days.

3. A customer places an online order, the ware-house does not have stock on hand, and all retailstores are already depleted to relatively low


inventory levels: then, it is not known until the endof the period if the order can be filled or not. Theonline customer would receive an order confirma-tion and a message that the product shipment isexpected to occur shortly. At the end of the period,the inventory levels of all retail stores are reviewedand the product is shipped if at least one retail storehas stock on hand. If no retail store has stock onhand, the order cannot be filled and the customer isreleased of any order commitment.

Many virtual stores today work this way. Forinstance, Amazon.com quotes various deliverytimes from within 24 h to weeks (depending on theinventory levels of its warehouses and its distribu-tors’ warehouses) and separates the order confirma-tion process into an immediate order-receiptconfirmation and a delayed shipment confirmation.Such practise lends itself well to the order fulfillmentin an integrated supply chain as described above.

3.2.1. Centralized decision-making

If decision-making is centralized, the manufac-turer supplies the retail stores at its marginalproduct cost cr, and the total cost savings associatedwith each unit cross sold are cD ¼ pþ gþ

hr � ðacr þ uÞ. Proposition 1 summarizes the opti-mal inventory policies (Seifert, 2000).

Proposition 1. For an integrated supply chain under

centralized decision-making, the optimal base-stock

levels are

y�ri;C ¼ y�r;C ¼ F�1r

pþ g� cr

pþ gþ hr � acr

�þ

cD

Nðpþ gþ hr � acrÞq0rðN ; y

�r ; y�vÞ

�8i

y�v;C ¼ F�1v

pþ g� cv

pþ gþ hv � acv

�þ

cD

pþ gþ hv � acvq0vðN ; y

�r ; y�vÞ

�,

where

q0rðN; yr; yvÞ ¼qqyr

qðN; yr; yvÞ

q0vðN; yr; yvÞ ¼qqyv

qðN; yr; yvÞ.

The function qðN; yr; yvÞ in Proposition 1 repre-sents the expected number of units cross sold. Lemma2 (Appendix C) details how to compute qðN; yr; yvÞ

and its partial derivatives. Based on Lemma 2, the

optimal base-stock levels y�r;C and y�v;C can beefficiently determined following an iterative solutionscheme (Bassok et al., 1999; Seifert, 2000). Theresulting optimal supply-chain profit per period is

E½pS;Cðy�r;C; y�v;CÞ�

¼ E½pS;Cðy�r;C; y�v;CÞ� þ cDqðN; y�r ; y

�vÞ.

3.2.2. Decentralized decision-making

If decision-making is decentralized, the manufac-turer and the retailer must agree on a wholesaleprice w and on how to divide the cost savingscD due to filling additional online orders. Weattribute these cost savings cD to the manu-facturer, and we introduce an additional contractparameter in the form of a transfer paymentt; 0ptpcD, that the manufacturer pays to theretailer as a reward for each unit cross sold fromretail stock. The manufacturer continues to pay anending-inventory subsidy b to the retailer, which isnow applied to inventory left over after cross sellinghas occurred. Proposition 2 characterizes theoptimal base-stock levels at the retail stores andthe manufacturer if a supply contract with w; b; andt is used.

Proposition 2. For an integrated supply chain under

decentralized decision-making, the optimal base-stock

levels are

y�ri;D ¼ y�r;D ¼ F�1r

pþ g� w

pþ gþ hr � aw� b

�þ

tþ hr � b� aw

Nðpþ gþ hr � aw� bÞ

� q0rðN ; y�r;D; y

�v;DÞ

�8i

y�v;D ¼ F�1v

pþ g� cv

pþ gþ hv � acv

þ½pþ g� u� tþ bþ aðw� crÞ�

pþ gþ hv � acv� q0vðN; y

�r;D; y

�v;DÞ

þNw� cr � ½bþ aðw� crÞ�F rðy

�r;DÞ

pþ gþ hv � acv�j0vðy

�v;DÞ

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA,

where

j0vðy�v;DÞ ¼

qy�r;DðyvÞ

qyv

��yv¼y�

v;C

.


Based on Propositions 1 and 2, we analyze howcoordination in an integrated supply chain can beachieved (Proposition 3).

Proposition 3. If the manufacturer offers a supply

contract ðwðdÞ; bðdÞ; tðdÞÞ, with wðdÞ ¼ cr þ d, bðdÞ ¼ðð1� aF rðy

�r;CÞÞ =F rðy

�r;CÞÞd þ ðq

0rðN; y

�r;C; y

�v;CÞ =NF r

ðy�r;CÞÞcD, and tðdÞ ¼ bðdÞ þ awðdÞ � hr, where

d 2 ½0; pþ g� crÞ, the following properties hold:

1.
The integrated supply chain under decentralized
decision-making is coordinated and total profits

are the same as in the integrated supply chain

under centralized decision-making.
2. The retailer’s profit is linearly decreasing in d:
E½pR;DðdÞ� ¼ E½pSR;Cðy�r;CÞ�

�N y�r;C �Eðy�r;C � xrÞ

þ

F rðy�r;CÞ

" #d

þEðy�r;C � xrÞ

þ

F rðy�r;CÞ

cDq0rðN; y�r;C; y

�v;CÞ.

3.
The manufacturer’s profit is linearly increasing in
d:

E½pM;DðdÞ� ¼ E½pMvðy�v;CÞ�

þN y�r;C �Eðy�r;C � xrÞ

þ

F rðy�r;CÞ

" #d

þ cDqðN; y�r;C; y�v;CÞ

� cDEðy�r;C � xrÞ

þ

F rðy�r;CÞ

q0rðN ; y�r;C; y

�v;CÞ.

We also need to analyze whether the coordinatingsupply contract is mutually beneficial, i.e., whetherboth the manufacturer and the retailer can be madebetter off than before (Proposition 4).

Proposition 4. There exists a d 2 ½0; pþ g� crÞ, such

that both the manufacturer and the retailer realize

higher profits in a coordinated integrated supply chain

than in a coordinated dedicated supply chain.

The supply contract of Proposition 3 thusachieves supply-chain coordination and allows bothparties to benefit. Next, we discuss additionalcharacteristics of this supply contract:

Proposition 5. There exist coordinating supply con-

tracts implying transfer payments that exactly offset

the retailer’s cost of filling online orders, i.e.,t ¼ bþ aw� hr. Any such supply contract thus

effectively decouples the original optimization pro-

blem, i.e., j0vðy�v;DÞ ¼ 0.

Proposition 5 implies that the retailer candetermine its optimal base-stock level directly bycalculating y�r;D ¼ F�1r ððpþ g� wðdÞÞ=ðpþ gþ hr�

awðdÞ � bðdÞÞÞ. At first glance, this result mayseem to suggest that the retailer does notrequire knowledge about the inventory policyand demand distribution at the virtual store.However, as is evident from Proposition 3.2, theretailer’s expected profit does depend on theinventory policy and on the demand distributionat the virtual store. Therefore, the retailer willrequire this information about the virtual store inorder to evaluate the attractiveness of a supplycontract.

Another characteristic is that a two-parametricsupply contract within the class of supply contracts(w, b, t) cannot achieve supply-chain coordination.Proposition 5 implies that a supply contractwithout transfer payments t cannot achieve supply-chain coordination. Furthermore, a supply contractwithout an ending-inventory subsidy b wouldrequire transfer payments t4cD to achieve supply-chain coordination and thus can be ruled outas well. Consequently, for the supply contract inquestion, non-zero supply-contract parameters w, b,and t are required to achieve supply-chain coordi-nation.

Finally, we note that the supply contract ofProposition 3 is incentive compatible (as definedby Lee and Whang, 1999), i.e., both partieshave an incentive to comply with the contract: themanufacturer will always prefer to use retailstock to fill an online order over a lost salebecause the transfer payments t are smallerthan the resulting cost savings. The retailer willalways prefer to report excess stock truthfully(thus implicitly agree to fill an online order whenexcess stock is available) because ending-inventory-subsidy payments to the retailer are contingentupon this. A successful implementation of thedeveloped supply contract, however, will ultimatelydepend on all facets of complex manufacturer–re-tailer relationships, some of which we cannotaddress within the limitations of our modellingframework.


4. Managerial insights

In this section, we discuss managerial insightsinto the benefits of supply-chain integration. InSection 4.1, we analyze the benefits of supply-chainintegration for coordinated supply chains. In Sec-tion 4.2, we compare the benefits of supply-chainintegration with the benefits of supply-chain co-ordination.

Throughout this section, we illustrate our resultswith numerical examples based on sanitized productand sales data for an HP printer (Table 1).

We use zero goodwill costs for lost sales as aconservative base-case. Product demands are ap-proximately normally distributed. To avoid apositive probability of having negative demands,we have used a truncated normal distribution forcomputing the following numerical results.

4.1. Benefits of supply-chain integration

The integration of the direct and indirect channelsimproves supply-chain performance. Specifically,supply-chain integration results in substantial chan-nel-stock reductions and reductions in lost sales thattranslate into significant profit gains for the entiresupply chain. The parameter that has the largestimpact on the magnitude of performance improve-ments is the relative channel volume—the quotientof direct and indirect sales volumes. The relativechannel volume is determined by the number ofretail stores N that are included in the integratedsupply chain because the average sales volumes ofindividual retail stores and the virtual store areexogenous to our model. Next, we will analyze howsupply-chain integration improves supply-chainperformance and how the relative channel volumeaffects the magnitude of these improvements;

Table 1

Product data for HP printer

Parameter Retail store Virtual store

Sales price p ¼ 1:18 p ¼ 1:18Product cost cr ¼ 1:03 cv ¼ 1:02Physical-inventory-holding cost hr ¼ 0:02 hv ¼ 0:01Transshipment-penalty cost u ¼ 0:03 –

Per period discount factor a ¼ 0:98 a ¼ 0:98Mean demand mr ¼ 10 mv ¼ 150

Coefficient of variation CV r ¼ 0:7 CVv ¼ 0:4

similar analysis can readily be conducted for othermodel parameters.

4.1.1. Base-stock levels

We begin our analysis of the benefits of supply-chain integration by analyzing the effect of supply-chain integration on the optimal base-stock levels(Proposition 6, Seifert, 2000).

Proposition 6. For a coordinated supply chain the

following properties hold:

1.
The optimal base-stock level at a retail store in
the integrated supply chain is higher than in the

dedicated supply chain.
2. The optimal base-stock level at a retail store in
the integrated supply chain decreases in N and

approaches the optimal base-stock level in the

dedicated supply chain as N !1.
3. The optimal base-stock level at the virtual store
in the integrated supply chain is lower than in

the dedicated supply chain.
4. The optimal base-stock level at the virtual store
in the integrated supply chain decreases in N and

approaches y�v;D;N!1¼F�1v ððuþacr�cv � hrÞ=ðuþ aðcr � cvÞ � ðhr � hvÞÞÞ as N !1.

The lower bound y�v;C;N!1 is a newsvendor-typesolution that is determined by the surcharge offilling an online order using retail stock as opposedto dedicated inventory from the virtual store, i.e.,uþ acr � cv � hr, and the overage cost at the virtualstore, i.e., hv þ ð1� aÞcv. If ðuþ acr � cv � hrÞ=ðuþ aðcr � cvÞ � ðhr � hvÞÞp0, then the optimalbase-stock level at the virtual store will be zero forsufficiently large N. Thus, Proposition 6.4 allows usto assess whether or not the manufacturer should, inthe extreme, operate its virtual store without anydedicated inventory. Fig. 3 shows how the optimalbase-stock levels for coordinated dedicated andcoordinated integrated supply chains depend on thenumber of retail stores N.

In the HP printer example (Fig. 3), it is optimal tooperate the virtual store without any dedicatedinventory and the retail stores with base-stock levelsclose to those of a coordinated dedicated supplychain if NX50. However, for small values of N, theoptimal base-stock levels at the retail stores in thecoordinated integrated supply chain can be quitedifferent from those in the coordinated dedicatedsupply chain.

ARTICLE IN PRESS

Optimal Base-Stock Levels

0

100

200

300

10 15 20 25 30 35 40 45 50 55 60 65 70 75Number of Retailers N

at th

e V

irtua

l Sto

re

0

4

8

12

16

20

at a

Ret

ail S

tore

y* y* y* y*v v r r

Fig. 3. Effect of N on optimal base-stock levels.

Channel-Stock Reduction

0%

5%

10%

15%

20%

25%


0

50

100

150

200

250N

o. U

nits

% Reduction Absolute Reduction

as %

of T

otal

Cha

nnel

-Sto

ck

Fig. 4. Effect of N on channel-stock reduction.

Lost-Sales Reduction

0%

20%

40%

60%

80%

100%


as %

of T

otal

Los

t Sal

es

at Internet Store at Retailer Stores

Fig. 5. Effect of N on lost-sales reduction.


4.1.2. Channel stock

Next, we analyze how the differences in theoptimal base-stock levels affect total channel stock.Fig. 4 shows how the absolute and relative channel-stock reductions depend on the number of retailstores N in the HP example. Initially, the absolutechannel-stock reduction increases in N because onaverage more excess inventory at the retail storesbecomes available as N increases—allowing formore units to be cross sold. However, each retailstore adds at least y�r units to the total channel stockwhile the virtual store will stock at least y�v;N!1units. Thus, the channel-stock reduction approachesthe maximal value y�v � y�v;N!1 as N !1. Therelative channel-stock reduction, on the other hand,approaches a maximal value of 20.1% for N ¼ 45for the HP printer example. If more than 45 retailstores participate in the coordinated integratedsupply chain, the relative channel-stock reductiondecreases because the increase in the absolutechannel-stock reduction diminishes as N increases.

4.1.3. Lost sales

A coordinated integrated supply chain reduceslost sales at the virtual store and at the retail stores.

Lost sales at the virtual store can be increasinglyreduced as N increases (Fig. 5) because moreleftover retail stock can be used to fill excess onlineorders. On the other hand, lost sales at the retailstores are reduced because the optimal base-stocklevels at the retail stores in coordinated integratedsupply chains exceed those in coordinated dedicatedsupply chains. However, as N increases, the reduc-tion in lost sales at the retail stores decreases (Fig. 5)because the optimal base-stock levels at the retailstores in coordinated integrated supply chains aredecreasing in N.

4.1.4. Profit gains

The channel-stock reductions and reductions inlost sales that can be achieved by integrating thedirect and indirect channels also translate intosignificant profit gains. Next, we analyze how thenumber of retail stores N affects this ‘‘integrationprofit’’. The integration profit is computed as thedifference between the optimal supply-chain profitin a coordinated integrated supply chain and theoptimal supply-chain profit in a coordinated dedi-cated supply chain, i.e., E½DpI� ¼ E½pS;Cðy�r;C; y

�v;CÞ��

E½pS;Cðy�r;C; y�v;CÞ�. Proposition 7 shows that the

integration profit is non-negative, increasing in N,and that its maximal value can be computed basedon the lower bound for the optimal base-stock levelat the virtual store (Seifert, 2000).

Proposition 7. The integration profit is non-negative,increasing in N, and is bounded above

limN!1

E½DpI� ¼ E½pMvðy�v;N!1Þ� þ cDmX v

ðy�v;N!1Þ

� E½pMvðy�v;CÞ�.

Fig. 6 shows how the relative and absoluteintegration profit depend on the number of retailstores N. The relative profit gains are computed as

ARTICLE IN PRESS

Profit Gains

0%

2%

4%

6%

8%


0

1

2

3

4

Abs

olut

e P

rofit

Gai

ns

Relative Absolute

Rel

ativ

e P

rofit

Gai

ns

Fig. 6. Effect of N on profit gains.


percentage increase in the supply-chain profits. Inour HP printer example, the relative profit gainsapproach a maximal value of 5.2% for N ¼ 20 retailstores. If more than 20 retail stores participate in thecoordinated integrated supply chain, the absoluteprofit gains still increase, but the relative profit gainsdecrease because each additional retail store in-creases the total supply-chain profit by at leastE½prðy�r Þ�. As N !1, the absolute profit gainsapproach the limit established by Proposition 7.

4.2. Benefits of supply-chain integration versus

coordination

In this subsection, we compare the benefits ofsupply-chain integration with the benefits of supply-chain coordination. In Sections 4.2.1 and 4.2.2, weanalyze non-coordinated dedicated and non-coordi-nated integrated supply chains, respectively. Weshow how to determine the optimal wholesale priceand inventory policy if the manufacturer and theretailer maximize their respective profits in theabsence of a coordinating supply contract. Thesemodels will serve as a base-case against which tocompare the supply-chain performance of coordi-nated and non-coordinated supply chains in Section4.2.3.

4.2.1. Non-coordinated dedicated supply chain

In a dedicated supply chain under decentralizeddecision-making without coordination, the manu-facturer charges the retailer a wholesale price ofw4cr. The resulting optimal base-stock level at aretail store is

y�riðwÞ ¼ y�r ðwÞ ¼ F�1r

pþ g� w

pþ gþ hr � aw

� �8i.

If the manufacturer can assess the above responsefunction of the retailer, the manufacturer will act asa Stackelberg leader in setting the wholesale price(Lariviere, 1999). Thus, the manufacturer willchoose the wholesale price w� that maximizesE½pMRðwÞ�¼Nðw� crÞ½y

�r ðwÞ�aEðy�r ðwÞ�xrÞ

þ�. The

optimal wholesale price w� cannot be specified inclosed form because it depends on the inversec.d.f. of the retail demand distribution (Lariviereand Porteus, 2001), but w� can be determinednumerically.

The optimal inventory policy at the virtual storein a dedicated supply chain without coordination isthe same as in a coordinated dedicated supply chain,i.e., y�v;D ¼ y�v;C.

The resulting expected optimal profits for themanufacturer and the retailer are

E½pMðw�Þ� ¼ E½pMvðy�v;CÞ� þ E½pMRðw

�Þ�

and

E½pRðw�; y�r ðw�ÞÞ� ¼ Nfpmr � w�y�r ðw

�Þ

� ðhr � aw�ÞEðy�r ðw�Þ � xrÞ

þ

� ðpþ gÞEðxr � y�r ðw�ÞÞþg,

respectively.

4.2.2. Non-coordinated integrated supply chain

In an integrated supply chain under decentralizeddecision-making without coordination, the manu-facturer refunds the retailer for units drawn fromretail stock to fill online orders, but does notprovide any additional incentives. This results in anet benefit of hr þ ð1� aÞw per unit to the retailer,and the retailer’s optimal response function willnow depend on the wholesale price w and theinventory policy at the manufacturer’s virtual store,i.e.,

y�riðw; yvÞ ¼ y�r ðw; yvÞ ¼ F�1r

pþ g� w

pþ gþ hr � aw

�þ

hr þ ð1� aÞwNðpþ gþ hr � awÞ

� q0rðN; y�r ðw; yvÞ; yvÞ

�8i.

On the other hand, the manufacturer has todetermine the optimal inventory policy for thevirtual store and the optimal wholesale pricesimultaneously, taking into account the retailer’soptimal response function. The optimal base-stocklevel for the virtual store can be characterized as afunction of the wholesale price by its first-order

ARTICLE IN PRESS

Optimal Wholesale Price

1.138

1.139

1.14

1.141

1.142

1.143

1.144

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of Retailers N

Who

lesa

le P

rice

Dedicated Integrated

Supply Chain Profit

0

20

40

60

80

100

120


coor. intg. coor. ded. non-coor. intg. non-coor. ded.

Pro

fit

(a)

(b)

Fig. 7. (a,b) Effect of N on optimal non-coordinated wholesale

price and supply-chain profits.


optimality condition

y�vðwÞ ¼ F�1v

pþ g� cv

ðpþ gþ hv � acvÞ

þ½pþ g� ðacr þ uÞ � ð1� aÞw�


� q0vðN; y�r ðw; y

�vðwÞÞ; y

�vðwÞÞ

þðw� crÞ½1� aF rðy

�r ðwÞÞ�


�Nj0vðy�vðwÞÞ

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA.

Thus, the optimal expected profit to the manu-facturer can be expressed as a function of thewholesale price w, i.e.,

E½pM;DðN;wÞ� ¼ E½pMv;Dðy�vðwÞÞ�

þNðw� crÞ½y�r ðw; y

�vðwÞÞ

� aEðy�r ðw; y�vðwÞÞ � xrÞ

þ�

þ ½pþ g� aðcr þ uÞ � ð1� aÞw�

� qðN; y�r ðw; y�vðwÞÞ; y

�vðwÞÞ

and the optimal expected profit of the retailerbecomes

E½pRðwÞ� ¼ E½pRðw; y�r ðw; y�vðwÞÞÞ� þ ½hr þ ð1� aÞw�

� qðN; y�r ðw; y�vðwÞÞ; y

�vðwÞÞ.

The manufacturer can determine the optimalwholesale price w� by maximizing E½pM;DðN;wÞ�numerically. The corresponding optimal base-stocklevels y�ri and y�v can be efficiently determinedfollowing an iterative solution scheme (Bassok etal., 1999; Seifert, 2000). The optimal wholesale pricew� then also determines the retailer’s profit.

4.2.3. Coordination versus integration

In this subsection, we compare the benefits ofsupply-chain coordination with the benefits ofsupply-chain integration. The optimal wholesaleprices for the non-coordinated dedicated and thenon-coordinated integrated supply chain weredetermined as outlined in Sections 4.2.1 and 4.2.2.Fig. 7a shows how the optimal wholesale prices areaffected by the relative channel volume. Initially,the optimal wholesale price for the non-coordinatedintegrated supply chain increases in N because themanufacturer can induce the retailer to stock moreby stocking less at the virtual store. For large N,however, the optimal wholesale price decreases andapproaches the wholesale price that is optimal for

the non-coordinated dedicated supply chain becausethe retailer’s response functions converge, i.e.,limN!1 q0rðN ; y

�r ; y�vÞ=N ¼ 0.

Fig. 7b shows the resulting supply-chain perfor-mance of coordinated and non-coordinated dedi-cated and integrated supply chains for the HPprinter example. Both coordinated supply chainsyield higher supply-chain profits than the non-coordinated supply chains, regardless of the relativechannel volume. Similarly, we can see from Fig. 8(using N ¼ 45) that in this example, supply-chaincoordination should precede supply-chain integra-tion because coordination achieves 74% of theprofit gains that can be achieved by coordinationplus integration.

5. Conclusion

Many manufacturers have started to use virtualstores as a direct channel in addition to theirindirect retail channel. In this paper, we haveanalyzed this development from a supply-chainperspective. We discussed our development andsolutions of mathematical models for a dedicated

ARTICLE IN PRESS

Dedicated Integrated

Coordinated

Non-Coordinated

72.28

64.23

75.10

66.45

26%

74% 80%

20%

Fig. 8. Supply-chain coordination versus supply-chain integra-

tion.


and an integrated supply chain under decentralizeddecision-making. We elaborated on linear supplycontracts that achieve supply-chain coordinationand that allow the allocation of the expected profitgains between the manufacturer and its retail-channel partner in a mutually beneficial way. Thecoordinating supply contract for an integratedsupply chain was shown to require a combinationof an appropriate wholesale price, an ending-inventory subsidy, and a transfer payment. Ourmodel can be used by manufacturers to determine towhat extent, and under what conditions, to rely onretail stock for online-order fulfillment.

Based on our model, we quantified the expectedbenefits that can be achieved by an integratedsupply chain over a dedicated supply chain andprovided numerical illustration of our results usingspecific HP product data. The profit gains that weobserved in our example were due to a significantchannel-stock reduction and a reduction in lostsales. In our HP example supply-chain coordinationshould also precede supply-chain integration.

Acknowledgements

The authors thank Professors W.H. Hausmanand H.L. Lee, Stanford University, for severalsuggestions that improved this paper as well asJ.P. Bowman, Hewlett-Packard Company, forhis support in conducting correlation analyses.The authors would also like to thank the anon-ymous referees for their constructive and detailedfeedback.

Appendix A. Correlation analysis

In the supply chain we looked at the fact that, oneretail chain was operating more than 600 stores inthe US, with average sales for a specific productvarying between 1

2of a unit up to 14 units per week.

The manufacturer categorized these retail storesinto four groups based on the average sales. The‘‘A-group’’ consisted of more than 100 retail storesthat were selling at least 8 units per week and, as agroup, sold an average of 10 units per week andstore. The A-group retail stores accounted for morethan 50% of the total sales volume and would belikely candidates to serve for ‘‘local’’ online fulfill-ment. In consequence, we motivate the assumptionof identical retail stores in the context of these A-retailers.

We analyzed the correlation of weekly sales dataat 178 retail stores of this retail chain and the HPShopping Village. The reference provided herein isbased on sales data of an HP printer for severalmonths in 1999. We first analyzed the between-storedemand correlation to determine whether there wassignificant demand correlation between retail stores.We calculated the correlation coefficients for all1795 pairs of retail stores. Although the correlationcoefficients themselves are not normally distributed,the distribution of their sample mean will be verywell approximated by a normal distribution giventhe large number of observations. Thus, we cancalculate a z-statistic to test the null hypothesis thatthe mean correlation is equal to zero. For thereferenced data set, the mean correlation was0.0158, which is representative of other estimates,the detail of which are not included here. Thestandard deviation was 0.307, which correspondswell with the asymptotic standard deviation of acorrelation coefficient from independent normalvariates based upon 10 observations (0.316). Thez-score is 0:0158=ð0:307=

ffiffiffiffiffiffiffiffiffiffi1795p

Þ ¼ 2:18, which, gi-ven the large number of observations, is weaklyindicative of a small amount of positive correlation.The close match of the sample standard deviation tothe asymptotic standard deviation furthermoresuggests that we do not see a pattern of most retailstores having no inter-store correlation with a fewhaving a large inter-store correlation; such a patternwould increase the standard deviation of thecorrelation coefficients significantly faster than themean. Thus, we conclude that correlations betweenretail stores are very small, i.e., on the order of0–3%.

ARTICLE IN PRESS

Profit Gains (N = 45)

0%

2%

4%

6%

0 1 2

Lead Time in Number of Periods

0

3

6

9

Abs

olut

e P

rofit

Gai

ns

Relative Absolute

Rel

ativ

e P

rofit

Gai

ns

Fig. 9. Effect of lead time on profit gains.


The analysis of correlation between retail storessales and Shopping Village sales followed the samelogic as outlined above. The sample mean was0.0494, and the sample standard deviation was0.304. The same testing procedure as above resultsin a z-score of 2.16, which is weakly indicative of asmall amount of positive correlation. The samplestandard deviation again closely matched theasymptotic standard deviation of a correlationcoefficient from independent normal variates. Thus,we have no indication that a few retail stores wouldbe highly correlated with Shopping Village saleswith the bulk of stores having no correlation. Insummary, we conclude that correlations betweenretail stores and the HP Shopping Village are verysmall, i.e., on the order of 5%.

Appendix B. Replenishment lead time

The impact of lead time on the operationalperformance of a supply chain can be substantial(e.g., Morton, 1971; Nahmias, 1979). The originalformulation of the periodic-review lost-sales modelwith lead time l was given by Karlin and Scarf,1958, who analyzed the structure of the optimalpolicy for a system with lead time of one period.Karlin and Scarf showed that the optimal inventorypolicy is a complicated function of the entire vectorof on-hand and on-order inventories. A number ofauthors then developed myopic approximations forthe optimal inventory policy (e.g., Morton, 1969,1971; Moses and Seshadri, 2000; Nahmias, 1979).

In this paper, we adopt Morton’s (1971) ap-proach, which has proven to be an accurateapproximation for problems with lead times ofone and two periods. Morton (1971) suggests thatthe base-stock level optimal for a positive-lead-timemodel with back-ordering be used, but to then limitthe actual order quantity in any period to be lessthan or equal to the optimal base-stock level of thezero-lead-time model with lost sales. In addition, webuild on the results of McGavin et al. (1997).McGavin et al. (1997) analyzed a system with N

identical retail stores and showed that it is optimalto balance the available on-hand inventory acrossall N retail stores to the best extent possible.

To analyze the profit of the dedicated supplychain, we use Morton’s (1971) approach directly. Toanalyze the profit of the integrated supply chain, weuse a modification of Morton’s (1971) approach.We scale the base-stock levels for the positive-lead-time case by the ratio of the optimal base-stock

levels in the dedicated and in the integrated supplychain for the zero-lead-time case. In addition, wefollow the approach of McGavin et al. (1997) andbalance the on-hand inventory levels at the retailstores to the best extent possible when units arecross sold. Thus, in filling excess orders at thevirtual store we use excess inventory from thoseretail stores with the highest level of on-handinventory.

To evaluate the profit gains for lead times of oneand two periods, we used simulation. We validatedour simulation model by comparing the simulationresults for the zero-lead-time case with the results ofour mathematical model. In our experiments, theerror in expected cost was typically less than 0.1%.

Fig. 9 shows how the profit gains depend on thelead time. Both the absolute profit gains and therelative profit gains increase as the lead timeincreases. The integrated supply chain thus offershigh profit gains when the lead time is long.

Appendix C. Expected number of units cross sold

The expected number of units cross sold can beexpressed as

qðN; yr; yvÞ

¼ E½YR ^ X v�

¼

Z 1�R¼0

�R½1� F X vð�RjyvÞ�dFYR

ð�RjN; yrÞ

þ

Z 1dv¼0

dv½1� FYRðdvjN; yrÞ�dF X v

ðdvjyvÞ,

where YR denotes the total excess stock across all N

retail stores and X v denotes the number of excessorders at the virtual store. YR is the sum of N i.i.d.truncated random variables (YR ¼

PNi¼1ðyr � xriÞ

þ)


and its c.d.f. F YRð�Þ depends on the truncation

points yr and on the number of retail stores N.Therefore, closed-form solutions for qðN; yr; yvÞ

become analytically intractable. To evaluateFYRð�jN; yrÞ by a closed-form expression we use

the Central Limit Theorem (Lemma 1).

Lemma 1. For N which are sufficiently large holds

F YRð�jN; yrÞ ffi FYR

ð�jN; yrÞ

¼ Normal ½NmY rðyrÞ;Ns2Y r

ðyrÞ�,

where

mY rðyrÞ ¼

Z yr

xr¼0ðyr � xrÞdF rðxrÞ

s2Y rðyrÞ ¼

Z yr

xr¼0½yr � xr � mY r

ðyrÞ�2 dF rðxrÞ

þ ½mY rðyrÞ�

2½1� F rðyrÞ�.

In Lemma 1, we require N to be large. To analyzewhich values of N are ‘‘sufficiently’’ large weperformed a Kolmogorov–Smirnov test for differ-ent product demand functions and different coeffi-cients of variation. For a 5% level of significance,the results show that the normal approximation canbe used for NX10. To indicate when the approx-imation is used, we use the symbol fð�Þ for the p.d.f.(probability density function) and the symbol Fð�Þfor the c.d.f. of the total excess stock across all N

retail stores.

Lemma 2. The expected number of units cross sold

can be computed as

qðN; yr; yvÞ ¼ NmY rðyrÞ �

Z 1�R¼0

�RFvðyv þ �RÞ

�dFYRð�RjN; yrÞ

þ mX vðyvÞ �

Z 1dv¼0þ

dvFYRðdvjN; yrÞ

� f vðyv þ dvÞddv,

where

FYRð�jN; yrÞ ¼ Normal ½NmY r

ðyrÞ;Ns2Y rðyrÞ�

mX vðyvÞ ¼

Z 1xv¼yv

ðxv � yvÞdF vðxvÞ

Lemma 2 allows us to evaluate q0rðN; yr; yvÞ andq0vðN; yr; yvÞ numerically. To gain structural insights

into the relationship between the expected numberof units cross sold and the base-stock levels at theretail stores and at the virtual store, Seifert (2000)also derived closed-form solutions for q0rðN; yr; yvÞ

and q0vðN; yr; yvÞ (Lemma 3).

Lemma 3. The partial derivatives of the expected

number of units cross sold can be computed as

q0rðN; yr; yvÞ ¼1

2p

XN

k¼1

kN

k

� �½1� F rðyrÞ�

N�k

"

�

Z dv

y¼0dyZ 1o¼�1

doeioðyþkyrÞ

�

Z yr

t¼0

dteiotf rðyr � tÞ

� �k#X0

q0vðN; yr; yvÞ ¼ FvðyvÞ �

Z 1�R¼0

F vðyv þ �RÞ

�dFYRð�RjN; yrÞp0.

Proof of Lemma 3 follows from direct algebraicmanipulation, application of Fourier Transforms,and use of the Convolution Theorem (Seifert, 2000).

Appendix D. Proofs

Note on Proposition 2. Note that for all admissablevalues for d, wðdÞ4bðdÞX0 given hroaðpþ gÞ.

Proof of Proposition 2. The dynamic inventorymodel for the retailer’s decision problem can bewritten as

f RnðxrnjyvnÞ

¼ Npmr �min wðNyrn � xRnÞ þN½ðhr � bÞ

(�Eðyrn � xrnÞ

þþ ðpþ gÞEðxrn � yrnÞ

þ�

� ðtþ hr � bÞEXN

i¼1

sivn

" #

� aE f n�1

XN

i¼1

½ðyrn � xrinÞþ� sivn�

!" #)

s:t: yrnXxRn=N

0psivnpðyrn � xrinÞþ8iPN

sivnpðxvn � yvnÞþ

i¼1


and f R0ðxR0Þ ¼ wxR0. The expected profit functionfor the retailer becomes

E½pR;DðN; yr; yvÞ�

¼ N½pmr � wyr � ðhr � bÞEðyrn � xrnÞþ

� ðpþ gÞEðxrn � yrnÞþ�

þ ðtþ hr � b� wÞqðN; y�r;D; yvÞ.

Similarly, the dynamic inventory model for themanufacturer’s decision problem can be written as

f MnðxRn;xvnjyrnÞ

¼ pmv þ ðw� crÞ½Nyrn � xRn�

� bXN

i¼1

Eðyrn � xrniÞþ�min cvðyvn � xvnÞ

(þ hvEðyvn � xvnÞ

þþ ðpþ gÞEðxvn � yvnÞ

þ

� ðpþ g� u� tþ bÞEXN

i¼1

sivn

" #

� aE f n�1

XN

i¼1

fðyrn � xrinÞþ� sivng

" #;

("

ðyvn � xvnÞþ

)#)

s:t: yvnXxvn

0psivnpðyrn � xrinÞþ8iPN

i¼1

sivnpðxvn � yvnÞþ

and f M0ðxR0;xv0Þ ¼ �ðw� crÞxR0 þ cvxv0. The ex-pected profit function for the manufacturer becomes

E½pM;DðN; yr; yvÞ ¼ E½pMv;DðyvÞ�

þNfðw� crÞy�r;Dðyv;w; b; tÞ

� ½bþ aðw� crÞ�Eðy�r;D � xrÞ

þg

þ ½pþ gþ aðw� crÞ � u� tþ b�

� qðN; y�r;D; yvÞ.

E½pR;DðN; yr; yvÞ and E½pM;DðN ; yr; yvÞ can be shownto be concave in yr and yv, respectively. The optimalbase-stock levels follow from differentiation in yr

and yv. &

Proof of Proposition 3. Substitution of the specifiedsupply contract parameters w; b; and t into Proposi-tion 2 yields equality in optimal base-stock levels ofProposition 1. Similarly, we can substitute w; b; and

t into E½pR;DðN; yr; yvÞ and E½pM;DðN; yr; yvÞ (de-rived in the proof of Proposition 2). &

Proof of Proposition 4. We want to show that boththe retailer and the manufacturer can be better off ina coordinated integrated supply chain than in acoordinated dedicated supply chain for somed� 2 ½0; pþ g� crÞ. Proposition 7 states that thetotal supply-chain profit in a coordinated integratedsupply chain is higher than in a coordinateddedicated supply chain, i.e., DpI40. Furthermore,it is easy to show that the optimal base-stock level ata retail store in a coordinated integrated supply chainwill be higher than in a coordinated dedicated supplychain (Proposition 6). From Proposition 2, we knowthat both the retailer and the manufacturer canbenefit from supply-chain coordination in a dedi-cated supply chain for some d 2 ½0; pþ g� crÞ. FromProposition 5, it follows that bðdÞXbðdÞmust hold. Ifd is kept constant, then a retailer in a coordinatedintegrated supply chain will always be better offthan a retailer in a coordinated dedicated supplychain, i.e.,

E½pR;Dðd; bðdÞ; y�r;CÞ�

pE½pR;Dðd; bðdÞ; y�r;CÞ�

pE½pR;Dðd; bðdÞ; ey�r;CÞ�pE½pR;Dðd; bðdÞ; ey�r;CÞ�þ

Eðey�r;C � xrÞþ

F rðey�r;CÞ cDq0rðN; ey�r;C; y�v;CÞpE½pR;Dðd; bðdÞ; y�r;CÞ�

þEðy�r;C � xrÞ

þ

F rðy�r;CÞ

cDq0rðN; y�r;C; y

�v;CÞ,

where ey�r;C maximizes E½pR;Dðd; bðdÞ; yrÞ�. Let usdenote the retailer’s profit increase for d byDpRðdÞ40.

Moreover, the manufacturer can also benefitfrom the supply-chain integration, which will betrue whenever 0oDpRðd

�ÞoDpI. First, we note that

this condition is satisfied for N large given thatDpRðdÞ is decreasing in N with limN!1 DpRðdÞ ¼ 0.On the other hand, if DpRðdÞXDpI40 forsmaller values of N, then we can always find dod�

such that 0oDpRðd�ÞoDpI because E½pR;DðdÞ�

(and thus DpRðdÞ) decreases linearly in d. Tocomplete the proof, we have to verify that d�

is always admissible, i.e., d�opþ g� cr. This


condition is met because

E½pR;Dðd ¼ pþ g� crÞ�

¼ NE½pSr;Cðy�r;CÞ� �N y�r;C �Eðy�r;C � xrÞ

þ

F rðy�r;CÞ

" #

�ðpþ g� crÞ þEðy�r;C � xrÞ

þ

F rðy�r;CÞ

cD

� q0rðN; y�r;C; y

�v;CÞ

¼ Nfðpþ gþ hr � acrÞEðy�r;C � xrÞ

þ

þ E½pSr;Cðy�r;CÞ� � ðpþ g� crÞy�r;Cg

¼ Nfðpþ gÞEðy�r;C � xrÞþ� ðpþ gÞ

�Eðxr � y�r;CÞþþ pmr � ðpþ gÞy�r;Cg

¼ �Ngmr

j0vðy�v;DÞ ¼ lim

D!0

F�1r

pþ g� cr

pþ gþ hr � acrþ

cD

N

yy� 1

q0rðN; y�r;D; y

�v;D þ DÞ

� �� y�r;D

D.

i.e., E½pR;Dðd¼ pþg� crÞ� ¼E½pR;Dðd¼ pþ g� crÞ�

and thus, DpRðd ¼ pþ g� crÞ ¼ 0. &

Proof of Proposition 5. The supply contract given inProposition 3 can be viewed as a special case (wherej0vðy

�v;DÞ ¼ 0) of a class of supply contracts sum-

marized by Lemma 4.

Lemma 4. The following parameterized supply con-

tract restores supply-chain coordination for an

integrated supply chain under decentralized-decision-

making

wðdÞ ¼ cr þ d,

bðdÞ ¼1

1� y

q0rðN; y�r;C; y

�v;CÞ

NF rðy�r;CÞ

cD þ1� aF rðy

�r;CÞ

F rðy�r;CÞ

d

tðdÞ ¼ bðdÞ þ awðdÞ � hr þy

y� 1cD,

where

y ¼q0rðN; y

�r;C; y

�v;CÞ

q0vðN; y�r;C; y

�v;CÞ

j0vðy�v;DÞ.

Based on Lemma 4 we can rewrite Proposition 2as follows:

y�r;D ¼ F�1r

pþ g� w

pþ gþ hr � aw� bþ

yy� 1

��

cD

Nðpþ gþ hr � aw� bÞq0rðN; y

�r;D; y

�v;DÞ

�,

y�v;D ¼ F�1v

pþ g� cv

pþ gþ hv � acv�

1

y� 1

�cD

pþ gþ hv � acv� q0vðN; y

�r;D; y

�v;DÞ

þNw� cr � ½bþ aðw� crÞ�F rðy

�r;DÞ

pþ gþ hv � acv�j0vðy

�v;DÞ

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA,

where

Integral to the computation of the supply-contract parameters in Lemma 4 is the calculationof y. Consider three cases:

Case 1. y=ðy� 1Þo0; this implies j0vðy�v;DÞ40

because qq0rðN; y�r;D; y

�v;DÞ=qyv;Do0. Furthermore,

we have j0vðy�v;DÞ403yo0 because q0rðN; y

�r;C;

y�v;CÞ40 and q0vðN; y�r;C; y

�v;CÞo0. However, y=

ðy� 1Þo030oyo1. Thus, yo0 constitutes adirect contradiction.

Case 2. y=ðy� 1Þ40; this implies j0vðy�v;DÞo0 and

we have j0vðy�v;DÞo03y40 (as in Case 1). On the

other hand, y=ðy� 1Þ403yo0 or y41. Thus,yo0 constitutes a direct contradiction. Moreover,y41 results in y=ðy� 1Þ41 and thus, would resultin tðdÞ4bðdÞ þ awðdÞ � hr þ cD (Lemma 4), i.e., arequired transfer payment t in excess of the costsavings cD.

Case 3. y=ðy� 1Þ ¼ 0; this implies y ¼ 03j0vðy

�v;DÞ ¼ 0 and thus exhibits the properties

stated in Proposition 5. Case 3 is the only feasiblearrangement. &

Proof of Proposition 6. Propositions 6.1 and 6.3follow directly from Lemma 3. To proof Proposi-tions 6.2 and 6.4, we have to show that ðq=qNÞy�rp0and ðq=qNÞy�vp0. To make the dependency of


the profit function on N explicit we writeE½pS;CðN; yr; yvÞ�. Because E½pS;CðN ; yr; yvÞ� is jointlyconcave in yr and yv, it suffices to show that (a)ðq=qyrÞE½pS;CððN þ 1Þ; y�r ðNÞ; y

�vðNÞÞ�p0, and (b)

ðq=qyvÞE½pS;CððN þ 1Þ; y�r ðNÞ; y�vðNÞÞ�p0.

(a) When there are N retail stores and the virtualstore has a base-stock level y�vðNÞ, the total expectedprofit can alternatively be expressed as

E½pS;CðN; yr; y�vðNÞÞ� ¼ NE½epRðyrÞ� þ E½pMvðy

�vðNÞÞ�,

where

E½epRðyrÞ� ¼ E½pRðyrÞ� þ cDE½ðyr � xrÞþ^ edrv�.

Here, edrv represents the random number of excessonline orders that are passed on to a single retailstore. We use ðyr � xrÞ

þ^ edrv ¼ ðyr � xrÞ

þ� ðyr �

xr � edrvÞþ and define egr ¼ xr þ edrv, and thus,

e1=2 ¼Nð1þNÞfmY r

ðy�r ðNÞÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifmY r½y�r ðNÞ�g

2 þ 2ð1þ 2NÞfsY r½y�r ðNÞ�g

2 Log½1þ 1=N�q

g

1þ 2N.

E½epRðyrÞ� ¼ pmr � cryr þ ðpþ g� uÞEðyr � xrÞþ

þ ðpþ gÞEðxr � yrÞþ� cDEðyr �egrÞþ.

The c.d.f of egr is Fegrð�jN; yvÞ. Because E½epRðyrÞ� isconcave in yr, differentiation yields necessary andsufficient conditions for the optimal base-stock levelat a retail store, i.e.,qqyr

E½pS;CðN ; yr; y�vðNÞÞ�

¼ Nqqyr

E½epRðyrjN; y�vðNÞÞ�

� �¼ Nf�cr � ðpþ gÞ � uF rðyrÞ

� cDFegrðyrjN; y�vðNÞÞg ¼ 0.

For given N and y�vðNÞ, y�r ðNÞ must satisfy this

optimality criteria. Because edrvX0, egr is stochasti-cally larger than xr. Furthermore, Fegrð�jN; yvÞpFegr ð�jðN þ 1Þ; yvÞpF rð�Þ. For y�vðNÞ ¼ const:, includ-

ing one additional retail store thus results inqqyr

E½pS;CððN þ 1Þ; y�r ðNÞ; y�vðNÞÞ�

¼qqyr

E½epRðy�r ðNÞjðN þ 1Þ; y�vðNÞÞ�p0

given cD40.

(b) For N retail stores, y�r ðNÞ and y�vðNÞ areoptimal base-stock levels. We include one additionalretail store and consider q=qyvE½pS;CððN þ 1Þ; y�r ðNÞ;y�vðNÞÞ� for y�r ðNÞ ¼ const:, i.e.,

qqyv

E½pS;CððN þ 1Þ; y�r ðNÞ; y�vðNÞÞ�

¼ �cv þ ðpþ gÞ � ½ðhv � acrÞ þ ðpþ gÞ�

�F vðy�vðNÞÞ þ cDq0v½ðN þ 1Þ; y�r ðNÞ; y

�vðNÞ�

¼ cDfq0v½ðN þ 1Þ; y�r ðNÞ; y

�vðNÞ�

� q0v½N ; y�r ðNÞ; y

�vðNÞ�g

using Proposition 1. Since cD40, we need to showthat q0v½N; y

�r ðNÞ; y

�vðNÞ� � q0v½ðN þ 1Þ; y�r ðNÞ; y

�vðNÞ�

X0 or equivalently that,R1�R¼�1

Fv½y�vðNÞ þ �R�

ffYR½�RjðN þ1Þ; y�r ðNÞ��fYR

½�R jN; y�r ðNÞ�gd�RX0.The roots of fYR

½ejðN þ 1Þ; y�r ðNÞ� ¼ fYR½ejN;

y�r ðNÞ� are given by

It is easy to verify that fYR½ejðN þ 1Þ; y�r ðNÞ�

ofYR½ejN; y�r ðNÞ� for e1o0pepmY r

½y�r ðNÞ�oe2.Furthermore, we have FYR

½0jðN þ 1Þ; y�r ðNÞ� �FYR½0jN; y�r ðNÞ� ¼ 0 for N large and Fvð�Þ is an

increasing function with Fvð0Þ ¼ 0. By the virtue ofTheorem 2 of Milne and Neave (1994) it follows that

E½F vfy�vðNÞ þ YR½ðN þ 1Þ; y�r ðNÞ�g�

XE½F vfy�vðNÞ þ YR½ðNÞ; y

�r ðNÞ�g�

and thus

qqyv

E½pS;CððN þ 1Þ; y�r ðNÞ; y�vðNÞÞ�p0.

Lemma 3 can be used to verify thatlimN!1 q0rðN ; y

�r ; y�vÞ ¼ 0 and thus, it follows from

Proposition 1 that y�r;N!1 ¼ y�r .Lemma 3 can be used to verify that

q0vðN; y�r ; y�vÞXFvðy

�vÞ�1 and limN!1 q0vðN; y

�r ; y�vÞ¼

Fvðy�vÞ � 1. Thus, it follows from Proposition 1 that

F vðy�v;N!1Þ

uþ aðcr � cvÞ � ðhr � hvÞ

pþ gþ hv � acv

� �¼

u� hr þ acr � cv

pþ gþ hv � acv

¼uþ acr � cv � hr

uþ aðcr � cvÞ � ðhr � hvÞ

because cD ¼ pþ gþ hv � acv40. &


Proof of Proposition 7.

E½DpIðN; y�r ðNÞ; y�vðNÞ; y

�r ; y�vÞ�

pE½DpIðN; y�r ðNÞ; y�vðNÞ; y

�r ; y�vÞ�

þ cD EXN

i¼1

ðy�r ðNÞ � xriÞþ

("(

þðy�r � xr;Nþ1Þþ

)^ ðxvn � y�vðNÞÞ

þ

#

�qðN; y�r ðNÞ; y�vðNÞÞ

)pE½DpIððN þ 1Þ; y�r ðN þ 1Þ; y�vðN þ 1Þ,

y�r ; y�vÞ� &

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