a multi-echelon inventory system with information exchange

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A Multi-Echelon Inventory System with Information ExchangeKamran Moinzadeh,

To cite this article:Kamran Moinzadeh, (2002) A Multi-Echelon Inventory System with Information Exchange. Management Science 48(3):414-426.http://dx.doi.org/10.1287/mnsc.48.3.414.7730

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A Multi-Echelon Inventory System withInformation Exchange

Kamran MoinzadehSchool of Business, University of Washington, Seattle, Washington 98195

[email protected]

In this paper, we consider a supply-chain model consisting of a single product, one sup-plier, and multiple retailers. Demand at the retailers is random, but stationary. Each retailer

places her orders to the supplier according to the well-known �Q�R� policy. We assume thatthe supplier has online information about the demand, as well as inventory activities ofthe product at each retailer, and uses this information when making order/replenishmentdecisions. We first propose a replenishment policy for the supplier, which incorporates infor-mation about the inventory position of the retailers. Then, we provide an exact analysis ofthe operating measures of such systems. Assuming the inventory/replenishment decisionsare made centrally for the system, we compare the performance of our model with thosethat do not use information in their decision making, namely, systems that use installationstock policies via a numerical experiment. Based on our numerical results, we identify theparameter settings under which information sharing is most beneficial.(Supply-Chain Management; Inventory Management; Information Exchange)

1. IntroductionIn recent years, the field of supply-chain managementhas gone through drastic changes. Numerous com-panies in various industries have undertaken majorinitiatives such as re-engineering efforts and invest-ment in information technology (IT) to better man-age their channels and reduce inefficiencies in theirsupply system. Re-engineering efforts have resultedin more coordination and cooperation between par-ties in the supply chain in the form of alliancesand partnerships, emphasis on logistical issues whendesigning a product (e.g., delayed differentiation andpostponement), modular product and process design,agile supply networks (see Feitzinger and Lee 1997),among others. Investment in IT however, has resultedin the availability of more information on channelactivities to decision maker(s), who, in turn, mustfind ways of incorporating them in their day-to-daydecisions to achieve better material flow and on-timedeliveries.Examples of industries that have adopted and

incorporated these concepts as vehicles for improve-

ment are numerous. Kurt Salmon Associates (1993)report the adoption of Efficient Consumer Response(ECR) by the grocery industry, which requires infor-mation sharing between suppliers and the stores.Stalk et al. (1992) discuss how the distribution cen-ters at Wal-Mart, a discount retailer, obtain infor-mation on the inventory status of various productsstocked at their stores and use such information inmaking replenishment/delivery decisions. A recentarticle in Forbes (1997) reports that the distribu-tors for Heineken, a Dutch beer manufacturer, canincrease or decrease their orders via Heineken’s Webpage, and Heineken can quickly reroute shipments oradjust production. Another example that is somewhatrelated to the model studied here is Campbell Soupand its Continuous Replenishment Program. As Fisher(1997) puts it:

Campbell establishes electronic data interchange(EDI) links with retailers. Every morning, retailerselectronically inform the company of their demandfor all Campbell products and of the level of inven-tories in their distribution centers. Campbell uses

Management Science © 2002 INFORMS

Vol. 48, No. 3, March 2002 pp. 414–4260025-1909/02/4803/0414$5.00

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MOINZADEHA Multi-Echelon Inventory System with Information Exchange

that information to forecast demand and to deter-mine which products require replenishment based onupper and lower inventory limits previously estab-lished with each retailer.

In lieu of the availability of information aboutdemand activities and inventory status of the prod-ucts to parties in a supply chain, inventory andreplenishment policies that incorporate and use suchinformation need to be considered and analyzed.In this paper, we consider a standard supply chainconsisting of a single product, one supplier, and M

retailers. Demand at each retailer is random, but sta-tionary. Each retailer places her orders to the sup-plier according to the well-known �Q�R� policy. Weassume that the supplier has online information aboutthe demand, as well as inventory activities of theproduct at each retailer, which he uses in makinghis order/replenishment decisions. We first proposea supplier’s replenishment policy, which incorpo-rates the information about the inventory position ofthe retailers. Then, we provide an exact analysis ofthe operating measures of such systems. Assumingthe inventory/replenishment decisions are made cen-trally for the system, we compare the performanceof our model with those that do not use informa-tion in their decision making, namely, systems, whichuse installation stock policies (see Deuermeyer andSchwarz 1981, Lee and Moinzadeh 1987, Moinzadehand Lee 1986, Svoronos and Zipkin 1988, and Axsater1993, for examples of such systems).The literature on incorporating information in sup-

ply chains is rather limited and relatively recent.Milgrom and Roberts (1990) identified information asa substitute for inventory in economic terms. Lee andWhang (1998) discuss the use of information-sharingin supply chains in practice, relate it to academicresearch, and outline the challenges facing the area.Cheung and Lee (1998) examine the impact of infor-mation availability coordination and allocation in aVendor-Managed Inventory (VMI) environment. Theyconsider a continuous review inventory/distributionsystem consisting of one supplier and multiple retail-ers. Assuming that the supplier has real-time informa-tion about the retailers’ inventory positions and retail-ers are identical, they consider coordinated shipmentand stock rebalancing initiatives. In the coordinated

shipment initiative, when the cumulative demandover all retailers reaches the truckload size, Q, then anorder is issued to the supplier to replenish the retail-ers and restore their inventory position to the tar-get level. In the rebalancing initiative, which is com-bined with shipment coordination, before its arrival,the supplier’s order will be allocated to the retailers ina manner to balance retailers’ inventory levels. Che-ung and Lee analyzed the model with the coordinatedshipment and derived bounds for the expected totalcost of the model with rebalancing when the proba-bility of the supplier being out of stock is negligible.In a numerical experiment, they show how shipmentcoordination and stock rebalancing can result in sig-nificant benefits. It should be mentioned that the poli-cies considered in Cheung and Lee will be effectiveonly when retailers are in close geographical proxim-ity to each other.Cachon and Fisher (1998) consider a setting simi-

lar to ours with one supplier and N identical retail-ers. Inventories are monitored periodically, and thesupplier has information about the inventory positionof all retailers. All locations follow an (R�nQ) order-ing policy, with the supplier’s batch size being aninteger multiple of that of the retailers. Cachon andFisher show how the supplier can use such informa-tion to better allocate stock to retailers. Specifically,they devise and analyze an optimal allocation pol-icy in which, when deciding on the allocation of thestock to the retailers, the supplier assigns a higherpriority to those retailers who are most needy (i.e.,retailers with lower inventory positions). They alsoconsider a policy that uses information to improvethe replenishment/ordering decisions at the supplier.However, unlike the one to be proposed in this paper,their policy is a complicated one because it dependson the vector of inventory positions of all retailers inthe system in a complex way. In summary, the maincontribution of Cachon and Fisher is the use of infor-mation in a more efficient allocation of stock to down-stream facilities. In this paper, we focus on the impactof using information by the supplier to improve hisreplenishment/ordering decisions. We believe thatour study complements Cachon and Fisher (2000).As discussed earlier, in our model, the retailers fol-low a standard (Q�R) policy; however, the supplier

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uses a replenishment policy that is based on retailers’inventory positions. As will be discussed later, underour proposed supplier’s policy, as a result of infor-mation exchange, the supplier attempts to predict thetime the future order is to be placed by retailer(s).This, we will show later, will result in more efficientsupply chains.Chen (1998) examines the value of information in

a continuous review serial system. He shows that theoptimal inventory policies in such systems have asimple structure similar to that of Clark and Scarf(1960). Other relevant studies that discuss the impactof the use of information in inventory/distributionsystems are Graves (1996), Moinzadeh and Aggarwal(1997), Moinzadeh and Bassok (1998), Lee et al. (1996),Hariharan and Zipkin (1995), Gavirneni et al. (1998),and Cachon and Lariviere (1999, 2001).As discussed in Cachon and Fisher (1998), the

reported benefits of information-sharing vary consid-erably. Cachon and Fisher reported an average sav-ings resulting from information-sharing of 2.2% anda maximum of 12.1%. Chen’s study indicates an aver-age savings of 1.8% and a maximum of 9%. Lee et al.found that the savings in their model can be as highas 23%. In this study, we found that the average sav-ings from the use of information is about 3.2%, witha maximum savings as high as 34.9%.The rest of the paper is organized as follows:

In the next section, we describe the model and itsassumptions. Then, we propose a supplier replen-ishment/ordering policy that incorporates informa-tion about the status of retailers’ inventory posi-tions. In §3, we derive expressions that enable us toevaluate the operating characteristics of such a sys-tem. In §4, we compare the performance of the poli-cies that use information-sharing to those withoutinformation-sharing (i.e., Svoronos and Zipkin 1988,Axsater 1993) and study the magnitude of the sav-ings, as well as behavior of their operating measuresas system parameters are varied. Finally, we close bysummarizing the results and suggestions for futureresearch.

2. The ModelConsider an inventory/distribution system consistingof a single item, a supplier, and M identical retail-

ers. Demand at each retailer follows a Poisson processwith a mean rate �. Each retailer carries inventoryand replenishes her stock from the supplier accord-ing to a (Q�R) policy; that is, when inventory posi-tion at the retailer reaches R, an order for Q unitsis placed. We assume that excess demand is backo-rdered at the retailers. When an order is placed by aretailer, the supplier satisfies the full order immedi-ately upon availability (i.e., no partial shipment of theorder is allowed); otherwise, the order will experiencea random delay. Delayed retailer orders are satisfiedon a first-come, first-served basis. In addition to thepossible random delay at the supplier, the transit timefrom the supplier to each of the retailers is constantand is assumed to be equal to L.We assume that the supplier has online informa-

tion on the inventory status and demand activities ofall retailers and replenishes his stock from an outsidesource in batches of Q. Furthermore, we assume thatthe replenishment lead time from the outside sourceto the supplier is constant and is equal to L0 (i.e., theoutside source has ample capacity). To take advantageof the available information, we propose the follow-ing replenishment policy for the supplier:

Supplier’s Inventory/Order PolicyStarting with m initial batches (of size Q), an order(of size Q) is placed to the outside source immedi-ately after a retailer’s inventory position reaches R+s�0≤ s ≤Q−1�.Note that m and s define the supplier’s inven-

tory/replenishment policy. Because the supplier hasonline access to the inventory position of eachretailer, as the inventory position of a retailer getscloser to R, the supplier may choose to act by order-ing from the outside source in anticipation of theplacement of a retailer’s order in the near future.This is accomplished by setting s to a positive value.By doing so, the supplier can provide more reliablereplenishments to the retailers that results in overallreduction of cost in the system. Thus, s, in a sense,guages the proactivity of the supplier from informa-tion availability. By definition, when s = 0, the sup-plier orders immediately after he receives an orderfrom a retailer. In other words, the supplier uses aclassical installation policy and the system behaves in

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a fashion similar to that considered by Axsater (1993)and Svoronos and Zipkin (1988). In the context of thisstudy, the value of information is zero in such casesas monitoring inventory position of the retailers at alltimes does not give the supplier an edge. Finally, weobserve that, when both m and s are set to zero, thesupplier does not hold any inventory and places hisorder immediately after he receives a retailer order,and his policy is in effect make to order. In such sit-uations, retailer orders face a constant delay of L0.Observe that for negative values of m, the possibledelay experienced by a retailer’s order will be largerthan L0, which results in inferior policies. Therefore,we only consider nonnegative values of m in ouranalysis.Before proceeding any further, we note that the

form of the optimal supplier policy in the contextof our model is an open question and is possiblya complex function of the different combinations ofinventory positions at all the retailers in the sys-tem. Although the policy adopted in this paper onlyconsiders the inventory position of retailers indepen-dently when making ordering decisions and basi-cally ignores the inventory positions of all otherretailers at such times, we conjecture that our pro-posed policy will capture most of the benefits frominformation-sharing. Clearly, the classical installationpolicy used in many earlier models is subsumedin our policy. We cannot make the same statementabout our policy in comparison with systems that useechelon policies where the supplier’s order is trig-gered when the system’s inventory position reachesa certain level. Previous research that compares thepure installation and echelon policies provides somehints. Axsater and Rosling (1993) show that echelonpolicies dominate installation policies in serial sys-tems. However, the dominance does not extend todistribution systems. For instance, echelon policiesshould perform poorly in systems where the sup-plier’s batch size is small, compared with that ofthe retailers (i.e., it is equal to those of retailers).Axsater and Juntti (1996) compares the performanceof echelon and installation policies in distributionsystems via a simulation study. He concludes thatechelon policies dominate installation policies in set-tings where the transportation time from the outside

source to the supplier is large, compared with thatof supplier to retailers. Cheung and Lee (1998) con-clude that echelon policies will become more attrac-tive in systems with large batch sizes and/or manyretailers.

DiscussionIn developing our model, we have made a few crit-ical assumptions that merit some further discussion.First, we assumed that all retailers are identical. Thisassumption is used in many of the previous works inthe literature (see Svoronos and Zipkin 1988, Axsater1993, Deuermeyer and Schwarz 1981, Lee and Moin-zadeh 1987, Moinzadeh and Lee 1986, Cachon andFisher 1998, Cheung and Lee 1998) and significantlysimplifies the analysis and exposition. In fact, one canapply a similar approach to that of ours to evalu-ate systems with nonidentical retailers. As seen later,both the analysis and the computational effort will betedious in such systems, as each retailer (or group ofretailers) must be considered separately in the anal-ysis. Second, we have assumed that the order lotsize, Q, is given and fixed. The assumption is reason-able in many practical situations because of packag-ing or shipping requirements and captures some ofthe economies of scale in shipping and handling. Fur-thermore, it is commonly used in the literature (seeSvoronos and Zipkin 1988, Axsater 1993, Deuermeyerand Schwarz 1981, Cachon and Fisher 2000, Cheungand Hausman 2000, Cheung and Lee 1998). Third,we have assumed that the supplier’s order quantityis equal to that of the retailers. This is probably themost restrictive assumption of our model because itonly represents cases in which there are essentiallyno economies of scale in ordering at the supplier (i.e.,the ratio of order cost to unit holding cost is negligi-ble at the supplier). This assumption is made for thesake of analytical tractability, because there are manysituations in which the assumption is violated. Thoseinclude situations in which the economies of scale inshipping and handling are significant. In cases whereshipping costs are small and electronic commerce andonline ordering are used in ordering (Forbes 1997,Fisher 1997), our assumption becomes more reason-able. Thus, a more general model in which the sup-plier’s batch size is larger than that of the retailers canconstitute an important avenue for future research. In


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the last section, we provide some closing thoughts onhow our model can be extended to such cases.Next, we define the following notation. Let

Nj�t� = number of orders placed by retailerj �j = 1� � � � �M� in �0� t� at steadystate.

Ij�t� = inventory position (net inventory +on order) of retailer j �j = 1� � � � �M�

at time t in steady state.N0�t� = total number of orders placed in the

system in �0� t� at steady state.Xk = time to have k units demanded at a

retailer.� = elapsed time between the placement

of a supplier’s order until it is re-quested to fill a retailer’s order.

L = transit time from the supplier toeach of the retailers.

L0 = replenishment lead time from theoutside source to the supplier.

� = random delay experienced by aretailer’s order.

= random time a batch (of size Q)spends in supplier’s inventory untilit is used to fill a retailer’s order.

f �·� = probability density of � .F �·� = complementary cumulative distri-

bution function of � .G�·� = complementary cumulative distri-

bution function of demand at theretailer during her order lead time.

p�x�� = probability mass function of Poissonwith mean .

h = unit holding cost/time at a retailer.h0 = unit holding cost/time at the sup-

plier.� = unit backorder cost/time at a

retailer.m = number of batches (of size Q) ini-

tially allocated to the supplier.TC�R�m� s� = expected total cost of the system/

time.

Furthermore, let g�·� denote the distribution of arandom variable. Then, we define g�k��·� as the k-fold

convolution of g�·�. Finally, denote

P�x��=�∑j=x

p�j��

and�x�+ =Max�0�x��

In the next section, we analyze the model and pro-vide expressions for computing the steady-state oper-ating characteristics of the system.

3. AnalysisWe now proceed with the exact analysis of steady-state behavior of the operating characteristics of thesystem. Our approach is to focus on a supplier’sorder and evaluate its early and tardy times. By earlytime, we mean the time an order spends in supplier’sinventory until it is used to satisfy a retailer’s order.In contrast, tardy time is the amount of time a delayedretailer’s order must wait until it is filled by an incom-ing supplier’s order.Recall that once a retailer’s inventory position

reaches R+ s (suppose this occurs at time t0), the sup-plier immediately places an order of size Q that hereceives at t0+L0. This order will be eventually usedto satisfy an order that is to be placed by one of theretailers at some random time in the future, say, t0+� .Note that � is a random variable and has been definedas the time from the supplier’s placement of an orderuntil it is used to fill a retailer’s order. As it will beseen later, once the probability distribution functionof � is obtained, the operating characteristics of thesystem can be computed.First, we derive expressions for the key operating

measures of the system. It becomes clear that to beable to compute these measures, one needs to com-pute the distribution of � . We then proceed by outlin-ing the steps to calculate the distribution of � .Based on our supplier’s inventory/replenishment

policy and considering only nonnegative values form, at steady state, we observe that (a) If � ≥ L0, thenthe supplier must carry Q units for �−L0 time units,and (b) If � ≤L0, then the retailer’s order experiences adelay equal to L0−� time units. Thus, the delay expe-rienced by a retailer’s order, �, can be expressed as

�= �L0−��+� (1)


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Furthermore E�� is given by

E��= L0−∫ L0

0F ��d�� (2)

Similarly, the total time a batch (of size Q) is heldin inventory at the supplier before it is used to fill aretailer’s order, , can be written as

= ��−L0�+� (3)

with its expected value, E��, given by

E��= E��−∫ L0

0F ��d�� (4)

Because orders are placed at a rate of M�/Q at thesupplier, the expected total inventory cost at the sup-plier will be h0�M��E��.Because the supplier fills orders on a first-come,

first-served basis and the transportation lead time, L,is constant, orders do not cross. Furthermore, the totallead times faced by the retailers are i.i.d. as demandat the retailers is stationary and a lot-for-lot orderingpolicy is used by the supplier.Since demand at the retailers is Poisson, using (1),

with some effort (using integration by parts), one canshow that the complementary cumulative distributionof demand during a retailer’s order lead time, G�·�,made up of the random order delay, �, and the fixedtransit time, L, can be written as

G�x� = P�x��L�Pr�� ≥ L0�

+∫ L0

0P�x��L+L0− ��f �� d�

= P�x��L+L0��

−�∫ L0

0p�x−1��L+L0− ��F �� d�� (5)

Finally, using the results in Hadley and Whitin(1963), we can write the average total cost rate in thesystem (Hadley and Whitin 1963, pp. 181–187, 200–204) made up of the costs at each retailer and theholding cost at the supplier as

TC�R�m� s� = M

{h

[Q+12

+R−��E��+L�

]

+ �h+��B�R�m� s�

}

+Mh0�E�� (6)

where

B�R�m� s� = 1Q

{ �∑u=R+1

�u−R−1�G�u�

−�∑

u=R+Q+1

�u−R−Q−1�G�u�

}� (7)

Replacing (4) in (6) and using (2), we get

TC�R�m� s� = M

{h

[Q+12

+R−��L+L0�

]

+ �h+��B�R�m� s�+h0�E��

+ �h−h0��∫ L0

0F ��d�

}� (8)

As previously described, we assume that Q is givenand fixed; therefore, we have ignored the retailers’order cost term in (8). We have also excluded termslike the variable retailers’ replenishment costs and thesupplier’s replenishment costs, because they are allconstants and play no role in finding the parametersof the optimal policy. However, to avoid nonnegativecost figures, the term hM�Q+1�/2 is included in theaverage total cost expression in (8).To be able to evaluate (7) and then (8), we need

to compute the complementary cumulative distribu-tion function of ��F �·�. Recall that a supplier’s orderis placed immediately after the inventory position atone of the retailers reaches R+ s. At that instant, theinventory positions at all (M − 1) other retailers areeither (i) less than or equal to, or (ii) greater than R+s.Let

A0�t�= j ∈ �1�2� � � � �M�� Ij�t�≤ R+ s��

A1�t�= j ∈ �1�2� � � � �M�� Ij�t� > R+ s��(9)

Therefore, when a retailer’s inventory positionreaches R+ s, all other retailers are either in A0 or A1.Note that, before this time, a supplier’s order wasplaced for each of those retailers in A0. However,these retailers are yet to place their orders to the sup-plier. This means that the inventory position at thesupplier will be equal to m plus the number of retail-ers in A0. Because the number of retailers is M , thenthe inventory position at the supplier can take on thevalues m�m+1� � � � �m+M�.Consider an instance when the retailer’s inven-

tory position reaches R+ s (suppose this occurs attime zero). This will be immediately followed by the


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supplier placing an order of size Q, which he receivesat L0. The order will be then used to satisfy an orderto be placed by one of the retailers at some ran-dom time in the future, say, � . Suppose there arek �k = 0�1� � � � �M − 1� retailers in A0 other than theone whose inventory position just reached R+s (thus,M −1−k retailers in A1). Then, the order just placedby the supplier will be used to satisfy the (m+k+1�−St future order to be placed by the retailers. (Recallthat the inventory position at the supplier takes on thevalues m�m+ 1� � � � �m+M�.) Therefore, � , definedas the time from the supplier’s placement of an orderuntil it is used to fill a retailer’s order, will be the timeuntil �m+k+1� orders are placed at the supplier. Ascan be seen, to derive the distribution of � , namelyF �·�, we need to characterize the order process of theretailers in A0 and A1 from the time an order is placedby the supplier (instances when inventory position ata retailer reaches R+ s) until the �m+k+1�−St sub-sequent retailers’ order is placed at the supplier.Suppose at time 0 the inventory position at retailer

j reached R+ s; this immediately triggers a supplier’sorder. Denote ��i� t� as the probability of having i

orders placed by a retailer, say retailer n, in t timeunits and that n∈A0�0�, given that the inventory posi-tion at retailer j �j �= n� hit R+ s at time 0. Similarly,let ��i� t�, be the probability of having i orders placedby a retailer, say retailer n, in t time units and thatn ∈A1�0�, given that the inventory position at retailerj �j �= n� hit R+ s at time 0. This means that

��i� t� = Pr Nn�t�= i�n ∈A0�0� � Ij�0�= R+ s��

��i� t� = Pr Nn�t�= i�n ∈A1�0� � Ij�0�= R+ s��

Furthermore, let ��i� t� be the probability of hav-ing i orders placed by retailer j in t time units andthat her inventory position at time 0 hit R+s. Becauseall retailers are identical, �� , and �� areindependent of the choice of the retailer, j.It can be easily verified that

��i�t� = 1Q

s∑l=1

Pr Xl+�i−1�Q≤ t<Xl+iQ��

= 1Q

s∑l=1

P�l+�i−1�Q��t�−P�l+iQ��t�� (10)

��i�t� = 1Q

Q∑l=s+1

Pr Xl+�i−1�Q≤ t<Xl+iQ�

= 1Q

Q∑l=s+1

P�l+�i−1�Q��t�−P�l+iQ��t�� (11)

and

��i� t� = P Xs+�i−1�Q ≤ t < Xs+iQ�

= P�s+ �i−1�Q��t�−P�s+ iQ��t�� (12)

Consider one of such instances described previ-ously in which at time 0, the inventory position atretailer M reaches R+ s. Suppose at this time, inven-tory positions of retailers 1 through k are smaller orequal to R+ s, and inventory positions at retailers�k+1� through �M−1� are greater than R+ s �A0�0�= j = 1�2� � � � � k��A1�0� = j = k+1� k+2� � � � �M −1��.Then, as discussed previously, � will be the time from0 until the �m+k+1�−St subsequent system order isplaced. Because all retailers are identical, the probabil-ity distribution of � for the above case is the same forall cases with k retailers in A0�0� and �M−k−1� retail-ers in A1�0�, irrespective of which retailers belong toeither A0�0� or A1�0�. Thus, we can write the comple-mentary cumulative distribution of ��F �t�, by consid-ering all the possible combinations of k ranging from0 to M −1 as

F �t� =M−1∑k=0

Pr{k retailers belong to A0�0��

�M −1−k� retailers belong to

A1�0��N0�t�≤ k+m � IM�0�= R+ s}

=M−1∑k=0

{(M −1

k

)Qk�k+m�t�

}� (13)

In (13), Qk�k+m�t� is the probability of having lessthan or equal to k+m system orders in t time unitswhen there are k retailers in A0�0�, given that retailerM ’s inventory position reached R+ s at time 0, andcan be expressed by summing the probabilities of allthe possible combinations as

Qk�k+m�t�=k+m∑l=0

qk�l� t�� (14)


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where

qk�l�t� = Pr N0�t�= l��j∈A0�0��j=1�� k�

k+i∈A1�0��i=1�� M−k−1 � IM�0�=R+s��

=l∑

i=0

l−i∑j=0

��k��i�t��M−k−1��j�t��l−i−j�t��

(15)

Replacing (12) in (15), (14) reduces to

Qk�k+m�t� =k+m∑i=0

k+m−i∑j=0

��k��i� t��M−1−k��j� t�

× 1−P�s+ �k+m− i− j�Q��t�� (16)

Using (16), we are now able to compute F �·� from (13)and eventually evaluate (6), (7), and (8). The analysisof the model is now complete.

4. Computational ResultsTo examine the impact of information-sharingachieved by our model when compared with systemsthat are based on pure installation stock ordering,we resorted to a numerical experiment, the resultsof which will be discussed later in this section. Welike to clarify that, although we provide an exactframework for the operating characteristics of ourmodel, to evaluate them, we resorted to numeri-cal approximation. This mainly involved evaluatingthose terms, which required the evaluation of inte-gral values. Although it is clear that our system willoutperform the traditional systems that are based oninstallation stock policies (i.e., Axsater 1993, Svoronosand Zipkin 1988), we are most interested in identi-fying the settings where the savings resulting fromour model would be most significant. In doing so, weused the best average system cost rate found in ourmodel as the benchmark and compared it with thatwhere no information-sharing is employed. Unfortu-nately, because of the complex nature of the problem,it is not clear that the average total cost rate is convexin the policy parameters. Therefore, to find the bestpolicy parameters and its associated average total costrate in our model, we resorted to a standard searchprocedure, which we will describe next. Note that itcan be shown that, for fixed values of m and s, theaverage total cost rate in the system is convex in R,

and the optimal reorder point at the retailer can befound easily. The optimal values of m and s werefound by examining values of m starting with m = 0and s = 0�1� � � � �Q−1 and obtaining the optimal val-ues of R for each combination of m and s. The searchcontinued by incrementing m in units of one until alocal minimum was achieved. To ensure that the solu-tion found is the best solution to the problem underconsideration, we continued examining our search upto twice the value of m, which resulted in the bestsolution. The best solution to the systems with noinformation-sharing was found in the same fashionby limiting the search to values of s = 0. In all theproblems examined, the local optimum found was thebest solution.Before summarizing our findings, we would like to

point out two important factors that impact the choiceof supplier’s stocking policy: (a) Risk-Pooling, and (b)Information-Sharing. In our model, both information-sharing and risk-pooling impact the choice of the sup-plier’s policy; in contrast, in systems with no infor-mation exchange, only risk-pooling is present. In all,we considered 600 problems made up of all the com-binations of the following parameters:

� 1�5L0 1L 0�0�5�1h 1h0/h 0�1�0�2�0�4�0�6�0�8�/h 10Q 2�5�10�20M 1�2�4�8�16�

For each problem studied, we expressed our find-ings as % deviation between average total cost ratesbetween the two systems, in which:

% deviation

= 100∗ E�TC∗w/o information�−E�TC∗

w/information�

E�TC∗w/information�

�

First, we examined the benefits resulting from ourmodel for different values of the supplier’s lead-time as the number of the retailers in the systemwas varied. Figure 1 depicts the typical behavior fordifferent demand levels (� = 1 and 5) at the retail-ers. As can be seen, in cases with a small number


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Figure 1 Different Average Demand Rates

01

23

45

67

8

M

Ave. %dev(L/(L+L0)=0)Ave. %dev(L/(L+L0)=0.33)Ave. %dev(L/(L+L0)=0.5)

λ = 1

λ = 5

0

2

4

6

8

10

M

Ave. %dev(L/(L+L0)=0)Ave. %dev(L/(L+L0)=0.33)Ave. %dev(L/(L+L0)=0.5)

Ave

. % d

evia

tion

Ave

. % d

evia

tion

1 2 4 8

1 2 4 8

16

Note. Performance of systems without information exchanged, compared with that of ours for different average demand rates because the supplier’s leadtimes and the number of retailers in the system are varied.

of retailers in which the effect of risk-pooling is rel-atively small, information-sharing has more impactwhen transit time from the supplier to the retailersis small, compared with the maximum possible leadtime for an order, L+L0. (Note that the supplier’s leadtime was set to unity in all the problems considered.)In other words, the savings is decreasing in L/�L+L0�.The behavior is intuitive as in systems with few retail-ers and small transit times (relative to the maximumpossible lead time). Most of the safety stock in thesystem will be allocated to the supplier who has thesmaller unit holding cost/time, and only a small por-tion of safety stock will be kept at the retailers tohedge against the demand uncertainty during therelatively small transit times. Therefore, the savingsresulting from reduction of the supplier’s safety stock(which is accomplished through information-sharing)will have a larger impact. Similarly, the savings result-ing from our policy will decrease as the maximum

possible lead time for an order, L+ L0, increases.We also note from Figure 1 that, the savings result-ing from information-sharing diminishes for systemswith many retailers. (In Figure 1, the % deviation is 0for all cases when �= 1 and M = 16.) This observationhas interesting managerial implications with respectto supply-chain design, because it provides one withthe settings (in terms of the number of retailers ina system) in which the savings from using informa-tion while utilizing risk-pooling is maximized. Forinstance, in systems with only one retailer, in whichthere is no risk-pooling, all the resulting savings fromour model are solely from information exchange. Con-versely, in systems with many retailers, risk-poolingat the supplier will dictate the form of the supplier’spolicy, and information exchange will have minimal(or no) impact at all. (In fact, as the number of retail-ers approaches infinity, the savings from our modelgoes to zero.)


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Next, we examined the benefits of information-sharing with respect to the order quantity �Q�. FromFigure 2, we observe that the value of information-sharing will be minimal when Q is either small orvery large, compared with the mean demand rate. In

Figure 2 Order Quantities �Q�

λ = 5

0

5

10

15

20

25

%dev (Q=2,L=0)%dev (Q=5,L=0)%dev (Q=10,L=0)%dev (Q=20,L=0)

λ = 5

0

2

4

6

8

10

12

14

M

%dev (Q=2,L=0.5)%dev (Q=5,L=0.5)%dev (Q=10,L=0.5)%dev (Q=20,L=0.5)

λ = 5

0

1

23

4

5

67

8

9

M

%dev (Q=2,L=1)%dev (Q=5,L=1)%dev (Q=10,L=1)%dev (Q=20,L=1)

M

1 2 4 8

1 2 4 8

1 2 4 8

Ave

. % d

evia

tion

Ave

. % d

evia

tion

Ave

. % d

evia

tion

Note. Performance of systems without information exchange, compared with that of ours for order quantities (Q) because the supplier’s lead times and thenumber of retailers in the system are varied.

fact, it can be shown that the % deviation betweencosts goes to 0 either when Q = 1 (when the order-for-order policy is in effect) or as Q approachesinfinity. This observation can be explained as fol-lows: Recall that through information-sharing, the


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supplier obtains real-time information on the inven-tory position of all retailers. Now, in systems withno information exchange, the supplier’s informationabout inventory position at a retailer becomes avail-able only at times when a retailer places an order(inventory position at the retailer is Q+R). Thus, insuch systems, the supplier lacks information aboutthe status of a retailer’s inventory position during thetime between placement of orders. Thus, in systemswith small order quantity, the time between retailer’sorders is relatively small and the supplier, in effect,gets frequent updates on the status of the retailers’inventory position and the role of having informa-tion is reduced. In systems with very a large orderquantity, the times between retailer’s orders are longand thus, there is opportunity to take advantage ofinformation exchange. However, inventory levels insuch systems tend to be very large (as Q is large), andthe cost associated with safety stocks in the systemplay a very small role in the total cost figure. Clearly,depending on system parameters, there are intermedi-ate values of Q in which information-sharing is mostadvantageous.Another important issue that needs some investi-

gation is the benefits of information-sharing in rela-tionship to the unit holding cost at the supplier tothat of the retailer �h0/h�. Figure 3 depicts a typ-ical behavior of the % deviation between costs. Weobserve in Figure 3 that, as the unit holding cost at thesupplier is approaching the unit cost at the retailers(h0/h approaches 1), the value of information-sharingbecomes marginal as the % deviation between costsgoes to zero. This can be explained as follows: Whenthe unit holding cost at the supplier is large, com-pared with that at the retailers (i.e., the ratio h0/h

approaches 1), then inventory will be mostly heldat the retailers. Thus, the use of information, whichtends to shift inventory from the retailers to the sup-plier, will not result in significant cost savings.We also observe a similar behavior as h0/h

approaches 0 that can be explained in the follow-ing way: Consider the system with no information-sharing when the holding cost at the supplier is low.Then, it is optimal to keep most of the inventory atthe supplier and very little inventory at the retailers.Although the use of information will result in less

inventory costs in the system, this reduction mainlyoccurs at the supplier, which has low holding cost.Therefore, the reduction in the expected total cost willbe small. In fact, when h0/h is 0, then one can elim-inate the possible order delay at the warehouse byallocating an infinite amount of inventory at no cost.In that case, the system will be equivalent to M inde-pendent �Q�R� systems with constant lead time, L,and clearly, information will have no value. The sav-ings from information-sharing is most significant forintermediate values of h0/h, as proper allocation ofinventory between supplier and the retailers becomescritical. The reader may note that, as discussed ear-lier, the % deviation values in Figures 1, 2, and 3 willbe far greater if the constant term hMQ/2 is excludedin the computations. We would like to point outthat, whereas Figures 2 and 3 summarize the resultsfor � = 5, we observed similar behavior for � = 1as well.

5. ConclusionsIn this paper, we studied the benefits of information-sharing in a supply chain characterized as theavailability of online information of retailers’ inven-tory positions to the supplier. Specifically, we con-sidered a standard supply-chain model consistingof a single product, one supplier, and M identicalretailers. Demand at each retailer follows a Poissonprocess. Each retailer places her orders to the sup-plier according to the �Q�R� policy. We assumedthat the supplier has online information about thedemand, as well as inventory activities of the prod-uct at each retailer, which he uses in making hisorder/replenishment decisions. We first proposed asupplier’s replenishment policy, which incorporatesthe information about the inventory position of theretailers. Then, we provided an exact analysis forthe operating measures of such systems. Assum-ing that inventory/replenishment decisions are madecentrally for the system, we compared the perfor-mance of our model to those that do not use infor-mation in their decision making, namely, systems thatuse installation stock policies via a numerical experi-ment. Based on our numerical results, we found that


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Figure 3 Different Number of Retailers

λ = 5

λ = 5

λ = 5

0

2

4

6

8

10

12

M=1(L=0)

M=2(L=0)

M=4(L=0)

M=8 (L=0 )

0

1

2

3

4

5

6

7

M=1(L=0.5)

M=2(L=0.5)M=4(L=0.5)M=8(L=0.5)

0

1

2

3

4

5

0.1 0.2 0.4 0.6 0.8

h0/h

0.1 0.2 0.4 0.6 0.8h0/h

0.1 0.2 0.4 0.6 0.8

h0/h

M=1(L=1)

M=2(L=1)

M=4(L=1)

M=8 (L=1 )

Ave

. % d

evia

tion

Ave

. % d

evia

tion

Ave

. % d

evia

tion

Note. Performance of systems without information exchange, compared with that of ours for systems with a different number of retailers because the supplier’slead times and the ratio of holding cost of the supplier to that of the retailers are varied.

information-sharing is most beneficial in systems,which exhibit the following characteristics:(a) Systems where the supplier’s lead time is long

compared with other lead times (i.e., transit timesfrom the supplier to the retailers) in the system.(b) Systems where the number of retailers is not

large (this is clearly a function of system parameters,such as costs, demand rates, and lead times).(c) Systems where the order quantities are either

not too small or too large.

(d) Systems where the ratio of the unit holding costof the retailers to that of the supplier is either not toosmall or too large.There are a number of important future extensions

to this work. One includes the analysis to the casewhen the order quantity at the supplier is a multipleinteger to that of the retailers. In such cases, our pol-icy can be extended in the following fashion. Definethe batch size as Q. Let Q0 = qQ be the order quantityat the supplier. Then, the supplier who starts with m


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initial batches of size Q will place orders of size Q0

to the outside supplier immediately after the inven-tory position of the qth retailer since the supplier’slast order reaches R+ s. The analysis of this policyis much more complicated as (to compute the distri-bution of �) one needs to derive the distribution ofthe times between the ith retailer’s inventory position�i = 1�2� � � � � q� in the supplier’s order reaching R+ s

until the supplier’s order is placed.Another interesting extension is the study of

a decentralized system and examining whetherchannels can be coordinated followed by devisingincentive-compatible mechanisms under which chan-nel coordination can be achieved. Finally, consideringthe benefits of information-sharing in systems whendemand among retailers are correlated is an impor-tant extension to this work.

AcknowledgmentsThe author thanks Professors Hau Lee and Apurva Jain and theanonymous referees for their helpful suggestions. Funding for thisresearch was provided by the Burlington Northern/BurlingtonResources Foundation.

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Accepted by Hau Lee; received May 30, 2000. This paper was with the authors 7 months for 2 revisions.


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a multi-echelon inventory system with information exchange

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