the value of information sharing in a multi-product, multi-level supply chain: impact of product...

Decision Support Systems 58 (2014) 79–94

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Decision Support Systems

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The value of information sharing in a multi-product, multi-level supplychain: Impact of product substitution, demand correlation, and partialinformation sharing

Muthusamy Ganesh a, Srinivasan Raghunathan b,⁎, Chandrasekharan Rajendran c

a JDA Software India Pvt. Ltd., 9th Floor Meenakshi Tech Park, Gachibowli, Hyderabad 500 032, Indiab School of Management, The University of Texas at Dallas, Richardson, TX 75083, USAc Department of Management Studies, Indian Institute of Technology Madras, Chennai, 600036, India

⁎ Corresponding author.E-mail addresses: [email protected] (M. G

(S. Raghunathan), [email protected] (C. Rajendran).

0167-9236/$ – see front matter © 2013 Published by Elhttp://dx.doi.org/10.1016/j.dss.2013.01.012

a b s t r a c t
a r t i c l e i n f o
Available online 16 January 2013

Keywords:Information sharingProduct substitutionSupply chain collaboration

The literature on the value of information sharing within a supply chain is extensive. The bulk of the literature hasfocused on two-level supply chains that supply a single product. However, modern supply chains often havemorethan two levels and supplymany products. Becausemany of these products are variants of the same base product,they tend to be substitutes and their demands correlated. Further, achieving supply-chain-wide information shar-ing in a multi-level supply chain is challenging because different firms may have different levels of incentives toshare information.We analyze the value of information sharing using a comprehensive supply chain that hasmul-tiple levels, may have different degrees of information sharing, and supplies multiple products that may have dif-ferent levels of substitutability and whose demands could be correlated to different degrees. Our analysis showsthat substitution among the different products reduces the value of information sharing for all firms in the supplychain. The reduction is higher (i) for firms that are more upstream, (ii) when the degree of substitution is higher,(iii) when the number of substitutable products is higher, (iv) when the demands of products aremore correlated,and (v) when the degree of information sharing is higher. Our results suggest that firms, especially those that areupstream in the supply chain,may face a significant risk of over-estimating the value of information sharing if theyignore substitution, demand correlation, and partial information sharing effects.

© 2013 Published by Elsevier B.V.

1. Introduction

Information sharing is viewed as one of the key elements forsuccessful supply chain management and coordination. Informationsharing can reduce the risk brought by asymmetric and incompleteinformation, cut down lead time, mitigate bullwhip effect, and reducetotal cost and increase total supply chain profit [19,20]. Information shar-ing enables suppliers to respond to consumer demand more quickly byappropriately scheduling the replenishment of the inventory. Continu-ous Replenishment Program (CRP) and Vendor Managed Inventory(VMI) are efforts in this direction. The savings in inventory holding andshortage costs to Campbell Soup Company and its retailers because ofCRP have been documented in [8,21]. Information sharing often im-proves the accuracy of demand forecasts, which enables a better pricestructure, improved production scheduling, and better management ofconsumer demand. Schemes such as Collaborative Forecasting andReplenishment (CFAR) facilitate sharing of both long-term and short-term demand forecasts between manufacturers and retailers.

anesh), [email protected]

sevier B.V.

The value of information sharing within a supply chain has beenanalyzed extensively in prior research. Bulk of the literature hasinvestigated the case in which the supply chain manufactures anddistributes a single product to customers. However, the ability tosatisfy heterogeneous customer preferences by providing moreproduct variety is a critical success factor in retailing [18], and mod-ern supply chains often manufacture and distribute multiple varie-ties of a product [10]. Recognizing this, Ganesh et al. [11] studiedthe impact of demand substitution on the value of information shar-ing when a supply chain distributes multiple varieties of a product.That is, when a variety that a customer is looking for is unavailable,the customer may buy another variety of the same product. Theyshowed that demand substitution diminishes the value of informa-tion sharing.

Existing studies on value of supply chain information sharing includ-ing Ganesh et al. [11] have used a two-level supply chain consisting of amanufacturer and a retailer. However, depending on the complexity ofthe product and other factors such as the distance between locations ofmanufacturers and end consumers, the number of levels in the supplychain can vary. Existing studies do not offer much insight into relativeincentives of firms at various levelswithin a supply chain to share infor-mation with their trading partners.

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Information Flow

Material Flow

… 2 1NN 1

Fig. 1. N-stage supply chain model.

80 M. Ganesh et al. / Decision Support Systems 58 (2014) 79–94

We extend Ganesh et al. [11] in the following directions. First, weconsider an N-level supply chain whereas Ganesh et al. [11] considereda two level supply chain. This extension has allowed us to not onlygeneralize the findings of Ganesh et al. [11] for a more general supplychain but also examine how the impact of product substitution on thevalue of information sharing varies across different levels of a supplychain. Second, we consider different degrees of information sharing,which cannot be analyzed in a two-level supply chain. In a two-level sup-ply chain, since there are only two firms, we have only two cases: thetwo firms share information or they do not. In an N-level supply chain,various information sharing possibilities arise, depending on whichfirms have an incentive to share their information with their upstreampartners.We consider two broad types of information sharing: upstreamand downstream. In downstream (upstream) information sharing, thefirms that are closest to the customer (raw material supplier) sharetheir demand information with their suppliers, but firms that are nearthe raw material supplier (customer) of the supply chain do not sharetheir demand information. Within each type, we examine different de-grees or extents of information sharing. This analysis provides insightsinto the value of partial information sharing within supply chains.

Our analysis shows that substitution reduces the value of informa-tion sharing for all firms in the supply chain. This result generalizesthe key result of Ganesh et al. [11] to the N-level supply chain regardingthe effect of demand substitution on the value of information sharing.Additionally, we find the following new results. The reduction in thevalue of information sharing because of substitution is higher (i) if thedegree of substitution is more, (ii) if the number of products is more,(iii) if demands of products are less correlated, (iv) for firms that aremore upstream and (v) if the degree of information sharing is higher.Our results suggest that firms, especially those that are upstream inthe supply chain, may face a significant risk of over-estimating thevalue of information sharing if they ignore substitution, demandcorrelation, and partial information sharing effects.

2. Literature review

There is a significant body of literature in supply chain informationsharing. Most papers in this stream of research analyze a two-levelsupply chain with a single manufacturer and a single retailer using avariety of demand models with different sets of assumptions(e.g., [5,7,12–14,28,38]). Our work is closely related to the literaturethat uses AR(1) demand model to study value of information sharing.Lee et al. [22] showed that the benefit of demand information sharingcan be significant when either lead time or the serial correlationcoefficient is large. Raghunathan [33] showed that the results derivedby Lee et al. [22] overestimate the benefit of demand informationsharing if the manufacturer uses the entire order history to do itsforecast. A few papers have investigated models with a single manu-facturer and multiple retailers (e.g. [1,2,6,42]). Raghunathan [34] andRaghunathan and Yeh [35] examined the value of demand informa-tion sharing using an N retailer version of the model in Lee et al.[22]. Huang and Iravani [16] analyzed how information sharing byone of two retailers affects the value of information sharing. Lengand Parlar [23] studied the information sharing in a three-level serialsupply chain and analyzed sharing of the benefits from informationsharing using the Shapley value concept.

Another stream of research has investigated how information shar-ing affects pricing decisions within a supply chain. These papers do notinvestigate inventory-related issues. Li [24] analyzed a model that in-cluded a manufacturer and several competing retailers and showedthat retailers will not voluntarily share information. Zhang [41] consid-ered a model in which each retailer sells a different product developedfrom the same base product supplied by the manufacturer and allowsthese products to be either substitutes or complements. Li and Zhang[25] analyzed the impact of three information sharing scenarios be-tween retailers and the manufacturer, with varying degrees of

confidentiality. Mishra et al. [29] showed that both the manufacturerand the retailer have incentives to share distorted information.

There is a vast literature on the effects of substitution among prod-ucts. For example, researchers have studied the problem of economicorder quantity and stocking level when a firm sells substitutableproducts [17,27,31,40]. Because the exact solution for the optimalinventory level is unavailable even for a simple two-product modelwith substitution, Rajaram and Tang [36] developed heuristics todetermine approximate inventory level. The literature on productsubstitution can be classified into two main categories. In the firstcategory, firms may choose to fill demand for one product using theinventory of another, perhaps, that of a higher quality product, toavoid losing the sale. Research on such “one-way substitutability”includes Bassok et al. [3], Bitran and Dasu [4], Rao et al. [37], andHonhon, Gaur, and Seshadri [15]. In the second, substitution decisionsare not directed by firms; rather, they are made by customers. Ourpaper models substitution of the second type. Mahajan and Ryzin[26] and Smith and Aggrawal [39] developed models that capturedynamic customer arrivals within a substitutable products context.Dynamic customer arrivals capture the more realistic scenario of dif-ferent products going out of stock at different time periods. Parlar[30], Pasternack and Drezner [32], and Drezner et al. [9] investigatedthe impact of substitution in a competitive setting in whichconsumers may go to another retailer when their preferred retaileris out of stock. As stated in the Introduction section, Ganesh et al.[11] considered the case in which a two-level supply chain distributesmultiple substitutable products. Our work extends this by consider-ing an N-stage supply chain with different types of informationsharing. This allows us to derive insights about the impact of substitu-tion on firms that are at different levels within a supply chain andabout the impact of substitution under partial information sharing.

3. Modeling framework

We consider an N-Stage supply chain, which distributes P products,as shown in Fig. 1.

Following earlier studies on supply chain information sharing[22,33,34], we assume that the demand faced by firm 1 for product j(or equivalently, the consumer demand for product j), D(1)j, followsan order one auto-regressive, AR(1), process. Thus, the consumerdemand for product j during period t is given by:

D 1ð Þj;t ¼ dþ ρD 1ð Þj;t−1 þ ξj;t

ð3:1Þ

where d>0,−1bρb1, and j∈{1,2, 3,…, P}. For a given t, ξj;tfollows a

normal distribution with mean zero and variance σ2, and the correla-tion coefficient between ξ

j;tand ξ

i;t, i≠ j, is ρr, −1/P−1bρrb1. Both

σ2 and ρr are independent of t and j. The condition −1/P−1≤ρr≤1guarantees that the covariance matrix of ξ

j;tis positive semi-definite.

For a given i, ξj;tare i.i.d. We assume further that σ is significantly

smaller than d, so that the probability of a negative demand for anyproduct during any period is negligible. All firms in the supplychain use an AR(1) model for forecasting their demands.

We consider a periodic review system in which, at the end ofevery period, each firm in the supply chain reviews its inventorylevel and places an order with its immediate upstream firm to

81M. Ganesh et al. / Decision Support Systems 58 (2014) 79–94

replenish the inventory for the product j. Each firm adopts the opti-mal myopic order-up-to policy in each period [22]. That is, eachfirm in the supply chain uses its own demand and any other informa-tion shared by the immediate downstream firm during the currentperiod to determine the optimal order-up-to level for the next period.The replenishment lead times are zero. So, at the end of time period t,after demand D(1)j,t has been realized, firm 1 observes its inventorylevel and places an order of sizeD(2)j,twith firm 2 tomeet the customerdemand during (t+1) for the product j. Firm 2 ships the required orderquantityD(2)j,t tofirm1 immediately after receiving the order.1 Iffirm2does not have sufficient stock to fill the order, then we assume that itwill meet the shortfall by obtaining required quantity from an alterna-tive source at an additional (shortage) cost. Then, firm 2 places itsorderD(3)j,t to firm 3 tomeet its demand from firm 1 in the next periodfor product j. The process is repeated for all firms. Thus, the inventorysystem resembles a systemwith backorders, and every firm guaranteessupply to its downstream customer.

No fixed ordering cost is incurredwhen placing the order, and inven-tory holding cost rate and shortage cost rate are stationary over time.Further, the holding cost rate and the shortage cost rate are identicalacross products and across firms. This assumption is reasonable becausethe products in our model are different varieties of the same underlyingbasic product. LetH and B, respectively, denote the holding and shortagecost per unit per time period for any firm. The profit contribution perunit of a product sold isW for all products and for all firms. We assumethatW is sufficiently large such that all firms realize profits in all periods.

We follow Ganesh et al. [11] to model substitution among products.If a product is out-of-stock, then a proportion α of excess demand forthat product will buy any of the other available products with equalprobability. That is, a customer who is willing to substitute is indiffer-ent among other products.2 The rest (1−α) portion of the excessdemand is backlogged. When α=0, products are not substitutable,and when α=1, they are perfect substitutes. The firms will not incurany shortage cost when demand substitution occurs, but each firmincurs a shortage cost when its demand is backlogged or outsourced.

We focus on the following information sharing structures.

1 D(i)j,t represents the demand for firm i and product j; further, D(i+1)j,t represents ord2 This substitution model is known in the literature as the ‘random substitution’ model [

3.1. No information sharing

In the absence of information sharing, firm i,2≤ i≤N, receives onlythe order D(i)j,t for the product j from its immediate downstream firm(i−1) at the end of period t. Firm i does not know the demandsof firms that are further downstream, i.e., it does not know D(k)j,t,k≤ i−1.

3.2. Full information sharing

Under full information sharing, every firm i,2≤ i≤N, receives forevery product j, not only the order D(i)j,t but also D(k)j,t, ∀k≤ i−1,from firm (i−1) at the end of period t.

3.3. Partial information sharing

We consider the following two types of partial information sharing.

3.3.1. Downstream information sharingIn downstream information sharing, firms 1 to n share their

demand information with their immediate upstream supplier. Thatis, for 1≤ i≤n+1, firm i receives not only the order D(i)i,t for product jbut also D(k)j,t, ∀k≤i−1, at the end of period t. However, forn+2≤ j≤N, firm i receives only the order D(i)j,t for product j at the endof period t. We denote n as the degree of downstream informationsharing. When n=N−1, downstream information sharing reduces tofull information sharing.

3.3.2. Upstream information sharingIn upstream information sharing, firms (N−n) to (N−1) share

their demand information. That is, for N−n+1≤ i≤N, firm i receivesnot only the order D(i)j,t but also D(k)j,t,N−n≤k≤ i−1, at the end ofperiod t for product j. However, for 1b i≤N−n, firm i receives onlythe order D(i)j,t for product j at the end of period t. Again, we denoten as the degree of upstream information and when n=N−1,upstream information sharing reduces to full information sharing.

4. Optimal strategies for different cases of substitution and information sharing

Let T(i)j,t denote firm i's order-up-to level for product j for period t in order to meet the demand for the product during period (t+1). Firm 1uses the following model at the end of period t to find T(1)j,t.

MaxT 1ð Þ1;t ;T 1ð Þ2;t ;…;T 1ð ÞP;t

∫∞

−∞∫∞

−∞… ∫

∞

−∞

WXPj¼1

xj; tþ1ð Þ−B 1−αð ÞXPj¼1

xj; tþ1ð Þ−T 1ð Þj;t� �þ þ α

XPj¼1

xj; tþ1ð Þ−T 1ð Þj;t� �þ

−XPj¼1

T 1ð Þj;t−xj; tþ1ð Þ� �þ0

@1Aþ0

@1A

−HXPj¼1

T 1ð Þj;t−xj; tþ1ð Þ� �þ

−αXPj¼1

xj; tþ1ð Þ−T 1ð Þj;t� �þ0

@1Aþ

0BBBBBB@

1CCCCCCA

g x1 tþ1ð Þ;…; xn0 tþ1ð Þ� �

dx1 tþ1ð Þ…dxn0 ; tþ1ð Þ

0BBBBBBBBB@

1CCCCCCCCCA

ð4:1Þ

where xj,(t+1) is the demand of product j in time period (t+1), g(x1,(t+1), x2,(t+1), …, xn',(t+1)) is the joint probability density function of

demands during time period t+1, and z+≡Max{0,z}. In the above model, WXPj¼1

xj; tþ1ð Þ is the profit contribution from all products,

1−αð ÞXPj¼1

xj; tþ1ð Þ−T 1ð Þj;t� �þ

is the fraction of demand backlogged because of those customers unwilling to substitute,

αXPi¼1

xj; tþ1ð Þ−T 1ð Þj;t� �þ−XP

j¼1

T 1ð Þj;t−xj; tþ1ð Þ� �þ0

@1Aþ

is the fraction of excess demand from customers willing to substitute but are unable to

find a substitute product, andXPj¼1

T 1ð Þi;t−xj; tþ1ð Þ� �þ−α

XPj¼1

xj; tþ1ð Þ−T 1ð Þj;t� �þ0

@1Aþ

is the inventory after satisfying demands from all customers.

er quantity by firm i for product j.26].


The solution for the model depends on the type and extent of information sharing within the supply chain and the degree of substitution.Because we do not have a closed form solution for the partial substitution case, we show the optimal strategies for the no substitution andperfect substitution cases.

4.1. No substitution case (α=0)

4.1.1. No information sharingFirst order conditions for the maximization model of firm 1 yield the familiar newsvendor solution for each product because the marginal

distribution of a multi-variate normal distribution is a uni-variate normal distribution. Thus, the optimal order-up-to level for product j forperiod t under no substitution is given by the following.

T 1ð Þj;t ¼ dþ ρD 1ð Þj;t þ Kσ ð4:2Þ

where K=ϕ−1(B /B+H) for the standard normal distribution function ϕ. Note that ρr does not play any role in firm 1's ordering decision whenproducts are not substitutable. This is intuitive because even though the demands of different products during a time period may be correlated,the demand in a time period is sufficient to determine the demand distribution for the product in the following period. At the end of period t,firm 1's order D(2)j,t is given by the following.

D 2ð Þj;t ¼ D 1ð Þj;t þ T 1ð Þj;t−T 1ð Þj;t−1

� �: ð4:3Þ

Notice that firm 1's order quantity replenishes the demand during period t plus the change being made in the order-up-to level from period(t−1) to t for the product j. After substituting Eqs. (4.1) and (4.2) in Eq. (4.3) and simplifying algebraically, we obtain D(2)j,t as the following.

D 2ð Þj;t ¼ D 1ð Þj;t þ dþ ρD 1ð Þj;t þ Kσ−d−ρD 1ð Þj;t−1−Kσ� �

¼ dþ ρD 2ð Þj;t−1 þ 1þ ρð Þξ j;t−ρξ j;t−1: ð4:4Þ

Note that D(2)j,t also follows an AR(1) demand process, and therefore, using the same analysis as the one used for firm 1, we compute firm 2'sorder-up-to level for the product j to be the following:

T 2ð Þj;t ¼ dþ ρD 2ð Þj;t þ Kσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ρð Þ2 þ ρ2

q: ð4:5Þ

We show in Appendix A that the demand and the order-up-to level for firm i and product j are given by the following3:

Dno ið Þj;tþ1 ¼ dþ ρD ið Þj;t þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmξ j;tþ1−m ð4:6Þ

Tno ið Þj;t ¼ dþ ρD ið Þj;t þ KσffiffiffiffiffiffiffiffiΩi

no

qð4:7Þ

where Ωino ¼

Xi−1

m¼0i−1Cm

i−1 1þ ρð Þi−1−mρm� �2

:

4.1.2. Full information sharingWhen all firms share information with their upstream suppliers, at the end of period t, all firms know ξ j,l, 1≤ l≤ t, and the only unknown

quantity in each firm's demand expression for period (t+1) is ξ j,t+1. Consequently, we show in Appendix B the following:

Dfull ið Þj;tþ1 ¼ dþ ρD ið Þj;t−ξ j;t

Xi−1

k¼1

ρk þ ξ j;tþ1

Xi−1

k¼0

ρk ð4:8Þ

Tfull ið Þj;t ¼ dþ ρD ið Þj;t−ξ j;t

Xi−1

k¼1

ρk þ KσffiffiffiffiffiffiffiffiffiΩi

full

qð4:9Þ

where Ωifull ¼

Xi−1

k¼0

ρk

!2

:

4.1.3. Downstream information sharingAssume that firms 1 through n (n≥1) share their demand information for each product with their upstream supplier, but firms (n+1)

through N do not. So, at the end of period t, firms 1 through (n+1) know ξ j,k, 1≤k≤ t, and therefore, the demand expression and the

3nCr denotes the number of ways of choosing r out of n items.


order-up-to level expression for 1≤ i≤n+1 are identical to Eqs. (4.8) and (4.9) respectively. Appendix C shows the derivations for the following

demand and order-up-to level expressions, where X ¼Xnþ1

k¼1

ρk.

Ddownstream ið Þj;tþ1 ¼ dþ ρDdownstream ið Þj;t−ξ j;t

Xi−1

k¼1

ρk þ ξ j;tþ1

Xi−1

k¼0

ρk if 1≤i≤nþ 2

þ

dþ ρD ið Þj;t þ −1ð Þi−n−3 1þ ρð Þi−n−2ρi−n−3 1þ Xð Þξ j;tþ1

� �þ

Xi−n−2

m¼1

−1ð Þm i−n−2Cm 1þ ρð Þi−n−2−mρm 1þ Xð Þ þ i−n−2Cm−1 1þ ρð Þi−n−2− m−1ð Þρm−1X� �

ξ j;t− m−1ð Þ� �

þ −1ð Þi−n−1ρi−n−2Xξ j;t− i−n−2ð Þ

0BBBBB@

1CCCCCAif nþ 3 ≤ i ≤ N

8>>>>>>>>>>><>>>>>>>>>>>:

ð4:10Þ

Tdownstream ið Þj;t ¼dþ ρDdownstream ið Þj;t−ξ j;t

Xi−1

k¼1

ρk þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi

downstream nð Þq

; if 1≤i≤nþ 1

dþ ρDdownstream ið Þj;t þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi

downstream nð Þq

; if nþ 2ð Þ≤i≤N

8>>><>>>:

ð4:11Þ

where

Ωidownstream nð Þ ¼

Xi−1

k¼0

ρk

!2

if 1≤i≤ nþ 1ð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXi−1

k¼1

ρk

!2

þXi−1

k¼0

ρk

!2vuut i ¼ nþ 2ð Þ

dþ ρD ið Þt þ 1þ ρð Þi−n−2ρi−n−3 1þ Xð Þ� �2þ

Xi−n−2

m¼1i−n−2Cm 1þ ρð Þi−n−2−mρm 1þ Xð Þ þ i−n−2Cm−1 1þ ρð Þi−n−2− m−1ð Þρm−1X� �� 2

þ ρi−n−2X� �2

0BBBBBB@

1CCCCCCA

if nþ 3 ≤ i ≤ N:

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

ð4:12Þ

Note that when n=N−1, we have full information sharing, and Eqs. (4.10) and (4.11) reduce to Eqs. (4.8) and (4.9) respectively.

4.1.4. Upstream information sharingAssume that firms N−n through N−1 share their demand information for each product with their upstream supplier and firms 1 through

(N−n−1) do not. Therefore, the demand expression and the order-up-to level expression for 1≤ i≤N−n are identical to Eqs. (4.6) and (4.7)respectively. Appendix D shows the derivations for the following demand and order-up-to level expressions.

Dupstream ið Þj;tþ1 ¼ dþ ρDupstream ið Þj;t

þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmξ j;tþ1−m if 1≤i≤N−n

−Xi− N−nð Þ

k¼1

ρk XN−n−1

m¼0

−1ð ÞmN−n−1Cm 1þ ρð ÞN−n−1−mρmξ j;t−m

þXi− N−nð Þ

k¼0

ρk XN−n−1

m¼0

−1ð ÞmN−n−1Cm 1þ ρð ÞN−n−1−mρmξ j;tþ1−m

2666664

3777775 if N−nþ 1 ≤ i ≤ N

8>>>>>>>>>><>>>>>>>>>>:

ð4:13Þ

Tupstream ið Þj;t ¼dþ ρDupstream ið Þj;t þ Kσ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi

upstream nð Þq

if 1 ≤ i ≤ N−n

dþ ρDupstream ið Þj;t−Xi−Nþn

k¼1

ρk XN−n−1

m¼0

−1ð ÞmðN−n−1ÞCm 1þ ρð ÞN−n−1−mρmξ j;t−m þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi

upstream nð Þq

if N−nþ 1 ≤ i ≤ N

8>>><>>>:

ð4:14Þ

where

Ωiupstream nð Þ ¼

Xi−1

m¼0i−1Cm 1þ ρð Þi−1−mρm� �2

if 1 ≤ i ≤ N−n

Xi−Nþn

k¼0

ρk

!2 XN−n−1

m¼0ðN−n−1ÞCm 1þ ρð Þ N−n−1ð Þ−mρmÞ2if N−nþ 1 ≤ i ≤ N:�

8>>>><>>>>:

ð4:15Þ

Again, when n=N−1, we have full information sharing, and Eqs. (4.13) and (4.14) reduce to Eqs. (4.8) and (4.9) respectively.


4.2. Perfect substitution case (α=1)

When products are perfect substitutes, though a customer may have a preference for a specific product, he is willing to buy an alternate prod-uct when his preferred product is out of stock. This is likely to occur when either the customers' preferences are weak or the customers' cost tosearch for the preferred product in another retailer is high. If products are perfect substitutes of each other, then the retailer is indifferentbetween stocking just one or any number of product varieties.

Substituting α=1 in the maximization model for firm 1, we find that its cost is a function of the total demand and the total stocking level forall products. Consequently, it is sufficient for firm 1 to determine the optimal total stocking level based on the total demand. That is, firm 1 cantreat the set of all products as a single entity and determine the order-up-to level for this entity as a whole. Once the total order (for the entireset of products) for a period is computed, the total order can be divided into orders for individual products based on any allocation scheme. Asimilar reasoning applies for other firms also. We assume that firm 1 will allocate the total order equally among all products in our analysis.

The total demand for all products for firm 1 is given by the following.

XPj¼1

D 1ð Þj;t ¼ Pdþ ρXPj¼1

D 1ð Þj;t−1 þXPj¼1

ξj;t : ð4:16Þ

Notice that VarXPj¼1

ξj;t

0@

1A ¼ σ2 P 1þ P−1ð Þρrð Þð Þ. Thus, the sum of order-up-to levels for all products for firm 1 can be derived using

newsvendor analysis as the following.

XPj¼1

T 1ð Þj;t ¼ Pdþ ρXPj¼1

D 1ð Þj;t þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þ

p: ð4:17Þ

Note that, unlike the no substitution case, firm 1's order-up-to level in the substitution case depends on the correlation between product de-mands. This is because while demand correlation does not affect the variance of demand for a single product, it affects the variance of the totaldemand; the variance of total demand is increasing in the demand correlation. We derive the sum of order-up-to levels for all products for otherfirms in a manner similar to that in the no substitution case.

4.2.1. No information sharingThe total order for firm i is given by

XPj¼1

Dno ið Þj;tþ1 ¼ Pdþ ρXPj¼1

Dno ið Þj;t þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmXPj¼1

ξj;tþ1−m: ð4:18Þ

Thus, we have

XPj¼1

Tno ið Þj;t ¼ Pdþ ρXPj¼1

Dno ið Þj;t þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þ

p ffiffiffiffiffiffiffiffiΩi

no

qð4:19Þ

where Ωino ¼

Xi−1

m¼0i−1Cm 1þ ρð Þi−1−mρmÞ2:�

4.2.2. Full information sharingUsing an analysis similar to that in the no substitution case, we get the following for the total order-up-to level under full information sharing

for perfect substitution case.

XPj¼1

Dfull ið Þj;tþ1 ¼ Pdþ ρXPj¼1

Dfull ið Þj;t−XPj¼1

ξj;tXi−1

k¼1

ρk

0@

1Aþ

XPj¼1

ξj;tþ1

Xi−1

k¼0

ρk

0@

1A ð4:20Þ

XPj¼1

Tfull ið Þj;t ¼ Pdþ ρXPj¼1

Dfull ið Þj;t−XPj¼1

ξj;tXi−1

k¼1

ρk

0@

1Aþ Kσ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þ

p Xi−1

k¼0

ρk ð4:21Þ


4.2.3. Downstream information sharingWe get the following for the total order-up-to level under full information sharing for perfect substitution case in the case of downstream

sharing.

XPj¼1

Ddownstream ið Þj;tþ1 ¼ Pdþ ρXPj¼1

Ddownstream ið Þj;t

−XPj¼1

ξj;tXi−1

k¼1

ρk þXPj¼1

ξj;tþ1

Xi−1

k¼0

ρk if 1 ≤ i ≤ nþ 2

þ

Pdþ ρXPj¼1

Ddownstream ið Þj;t þ −1ð Þi−n−3 1þ ρð Þi−n−2ρi−n−3 1þ Xð ÞXPj¼1

ξj;tþ1

0@

1Aþ

Xi−n−2

m¼1

−1ð Þm i−n−2Cm 1þ ρð Þi−n−2−mρm 1þ Xð Þ þ i−n−2Cm−1 1þ ρð Þi−n−2− m−1ð Þρm−1X� �XP

j¼1

ξj;t− m−1ð Þ

0@

1A

þ −1ð Þi−n−1ρi−n−2XXPj¼1

ξj;t− i−n−2ð Þ

0BBBBBBBBBBB@

1CCCCCCCCCCCAif nþ 3≤i≤N

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð4:22Þ

XPj¼1

Tdownstream ið Þj;t ¼Pdþ ρ

XPj¼1

Ddownstream ið Þj;t−XPj¼1

ξj;tXi−1

k¼1

ρk þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ωidownstream nð Þ

q; if 1 ≤ i ≤ nþ 1

Pdþ ρXPj¼1

Ddownstream ið Þj;t þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ωidownstream nð Þ

q; if nþ 2ð Þ ≤ i ≤ N:

8>>>>><>>>>>:

ð4:23Þ

4.2.4. Upstream information sharingUsing an analysis similar to that in the no substitution case, we get the following for the total order-up-to level under upstream partial in-

formation sharing for perfect substitution case.

XPj¼1

Dupstream ið Þj;tþ1 ¼

Pdþ ρXPj¼1

Dupstream ið Þj;t

þXi−1

m¼0

−1ð Þmi−1Cmi−1 1þ ρð Þi−1−mρm

XPj¼1

ξj;tþ1−m if 1≤i≤N−n

−Xi− N−nð Þ

k¼1

ρk XN−n−1

m¼0

−1ð ÞmN−n−1Cm 1þ ρð ÞN−n−1−mρmXPj¼1

ξj;t−m

þXi− N−nð Þ

k¼0

ρk XN−n−1

m¼0

−1ð ÞmN−n−1Cm 1þ ρð ÞN−n−1−mρmXPj¼1

ξj;tþ1−m

2666664

3777775 if N−nþ 1≤i≤N

8>>>>>>>>>>><>>>>>>>>>>>:

ð4:24Þ

XPj¼1

Tupstream ið Þj;t ¼

Pdþ ρXPj¼1

Dupstream ið Þj;t þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ωiupstream nð Þ

qif 1≤i≤N−n

Pdþ ρXPj¼1

Dupstream ið Þj;t−Xi−Nþn

k¼1

ρk XN−n−1

m¼0

−1ð ÞmðN−n−1ÞCm 1þ ρð ÞN−n−1−mρmXPj¼1

ξj;t−m

0@

1Aþ

KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ωiupstream nð Þ

q� �0BBB@

1CCCA if N−nþ 1 ≤ i ≤ N:

8>>>>>>>>><>>>>>>>>>:

ð4:25Þ


5. Theoretical analysis of the impact of substitution on the value ofinformation sharing

This section theoretically analyzes the value of informationsharing. We define the value of information sharing to firm i whenthe degree of information sharing is n and information sharing is ofType∈{Upstream,Downstream}, VType(n),α

i , as

ViType nð Þ;α ¼

Profit of firmiunder nth

degree ofinformation sharingof given type anddegree of substitution α

26666664

37777775−

Profitof firm iunderno information sharinganddegree of substitution α

26666664

37777775

, Profit of firmiunder no information sharinganddegree of substitition α

2664

3775:

ð5:1Þ

For all q and t, and for all degrees of substitution, E[D(q)t] under anyinformation sharing scenario is identical to that under the no informa-tion scenario. Therefore, the expected revenue during any period for afirm remains the same regardless of information sharing. Consequent-ly, the benefit from information sharing arises solely from a reductionin inventory holding and shortage costs. Following Lee et al. [22], weimpose the restriction that the service level of all supply chain partnersfor all products is sufficiently high for analytical tractability. When theservice level is high, the expected cost to firm i under information shar-

ing scenario of degree n is given by HKPσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi

Type nð Þq

when products are

not substitutable andHKσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ΩiType nð Þ

qwhen products

are perfectly substitutable, and the expected revenue is given by WPd1−ρ

regardless of the level of product substitution. Therefore, the value ofinformation sharing reduces to the following.

ViType nð Þ;0 ¼

ffiffiffiffiffiffiffiffiΩi

no

q−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi

Type nð Þq� �,

WHKσ

d1−ρ

� �−


no

q� �ð5:3Þ

ViType nð Þ;1 ¼


no

q−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩi


WffiffiffiP

p

HKσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ P−1ð Þρrð Þp d

1−ρ

� �−


no

q !:

ð5:4Þ

Now, we show the following result.

Proposition 1. Under both upstream and downstream informationsharing, and for all degrees of information sharing,

(i) for every firm, the value of information sharing is lower whenproducts are perfect substitutes compared to when they are notsubstitutes,

(ii) the reduction in the value of information sharing for any firm dueto substitution is increasing in the degree of information sharing,

(iii) the reduction in the value of information sharing for any firm dueto substitution is decreasing in correlation, and

(iv) the reduction in the value of information sharing for any firm dueto substitution is increasing in number of products.

(Proofs for the proposition in Appendix E).Proposition 1(i) generalizes the finding by Ganesh et al. [11] that

substitution reduces the value of information sharing in a two-levelsupply chain. Proposition 1(i) shows that the value of informationsharing is smaller when the products are perfect substitutes thanwhen they are not substitutes for every level in an N-level supplychain. The reasons for the decrease in the value of information shar-ing under substitution are two-fold. The first is the demand poolingeffect of substitution. It is well known that the pooled demand has alower standard deviation than the sum of standard deviations of indi-vidual demands. Thus, the safety stock as well as inventory holdingand shortage costs are lower when products are more substitutableand the value of information sharing is lower when the degree of sub-stitution is higher. Second, because demand pooling caused by substi-tution reduces inventory holding and shortage costs even wheninformation is not shared, the base level profit is higher when thedegree of substitution is higher. A higher base level also reduces thevalue of information sharing, which is expressed as a percentage in-crease in profit.

Proposition 1(ii) shows that the reduction in the value of in-formation sharing due to substitution is higher for more upstreamfirms. It is the direct result of bullwhip effect, which results in ahigher demand variance for more upstream firms in the supplychain. Consequently, demand pooling effect of substitution reducesthe demand variance more for more upstream firms, leading to alarger reduction in the value of information sharing because ofsubstitution.

Proposition 1(iii) shows that the reduction in the value of infor-mation sharing due to substitution is more when product demandsare less correlated. This result occurs because the reduction indemand variance because of pooling is higher when demands areless correlated.

Proposition 1(iv) shows that, the reduction in the value of informa-tion sharing due to substitution is more when the number of productsis higher. Again, as the number of products increases, the pooling effectof demands will also increase.

6. Simulation analysis

For the theoretical analysis presented in Sections 4 and 5, we considered only the reduction in the inventory holding cost to compute thevalue of information sharing. For the simulation analysis, we analyzed the “true” value of information sharing that includes both inventory hold-ing and shortage costs. Further, we used results from the simulation analysis to obtain additional insights that were not possible from the the-oretical analysis, such as the impact of the degree of substitution on the value of information sharing and reduction in the value of informationdue to various degree of substitution. For our simulation analysis, we considered a six stage supply chain model. We used the following parameters:σ=2, ρ=0.7, H=1, B=25, d=40, and W=3.We used the following ranges for other parameters.

P∈ 5;10;25;50f g; ρr∈ 0;0:2;0:4;0:6;0:8;1:0f g; α∈ 0;0:2;0:4;0:6;0:8;1:0f g; n∈ 0;1;2;3;4;5f g:

We had a total of 1440 scenarios. For each scenario, we conducted the simulation for 10,000 periods.

Table 1Regression results.a

Independent variables Standardized beta coefficient t-value p-value F-value Adjusted R-squared

α 0.214 06.659 0.000 1143.77 0.66n 0.282 20.873 0.000∂ −0.090 −11.814 0.000N 0.926 68.485 0.000ρr −0.016 −01.164 0.244P 0.001 00.088 0.930(α×N) −0.401 −15.733 0.000(α×ρr) 0.173 09.867 0.000(α×P) −0.016 −00.929 0.353(α×n) −0.099 −04.570 0.000

a Number of observations in the regression model=5760.


Since the number of scenarios is very large, we used a regression model to summarize the qualitative impact of various parameters on thevalue of information sharing. We used the following regression model.

Value of Information Sharing ¼β0 þ β1 degree of substitutionð Þ þ β2 degree of information sharingð Þ þ β3 type of information sharingð Þþβ4 level of firm in the supply chainð Þ þ β5 demand correlation coefficientð Þ þ β6 number of productsð Þþβ7 degree of substitution � level of firm in the supply chainð Þþβ8 degree of substitution � demand correlation coefficientð Þþβ9 degree of substitution � number of productsð Þþβ10 degree of substitution � degree of information sharingð Þ þ ε

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;:

ð6:1Þ

Table 1 shows the regression statistics.All independent variables, except those that relate to number of products and demand correlation, were significant. The regression analysis

shows that (i) for all firms, the value of information sharing decreases when the degree of substitution is higher (note that the sum of coefficientof α and the coefficient of α×N is negative for all α and N), and, for the same degree of substitution, (ii) the reduction in the value of informationsharing because of substitution is higher for more upstream firms (i.e., the coefficient of α×N is negative), the products are less correlated(i.e., the coefficient of α×ρr is positive), and the degree of information sharing is higher (i.e., the coefficient of α×n is positive). The regressionanalysis confirms our theoretical result. We discuss the impact of model parameters on the value of information sharing in more detail in thefollowing paragraphs.

6.1. Effect of degree of substitution on the impact of substitution

We discuss the impact of degree of substitution using Fig. 2 for the following parameter values: N=25, ρr=0.6 and degree of sharing isfive. The reduction in the value of information sharing for firm 1 because of substitution ranges from 0.01 (for α=0.0) to 0.04 (for α=1.0),whereas the reduction ranges from 4.44 (for α=0.0) to 18.42 (for α=1.0) for firm 6. We find that the reduction in the value of informationsharing due to substitution increases with the degree of substitution for all firms, which is consistent with our theoretical proposition.

0

2

4

6

8

10

12

14

16

18

20

Firm 2 Firm 3 Firm 4 Firm 5 Firm 6

Firms

Reduction in the

Value ofInformation

Sharing

α = 0.2α = 0.4α = 0.6α = 0.8α = 1.0

Fig. 2. Impact of degree of substitution.


Moreover, the rate of increase in the reduction is higher when the firm is more upstream. The most important implication of our observationabout the impact of substitution on the value of information sharing relates to the extent by which the value of information sharing can beover-stated if substitution effects are ignored. The over-estimation risk is more significant for more upstream firms.

6.2. Impact of number of products

Fig. 3 shows the impact of number of products. For this figure, we use ρr=0.2 and a degree of sharing of five. Our simulation results are con-sistent with our theoretical result. We find that the reduction in the value of information sharing for firm 2 ranged from 0.00 (when α=0.2) to0.06 (when α=1.0) when P=5, but the same range was from 0.01 to 0.1 when P=50. The range for firm 6 was 5.71 (when α=0.2) to 31.69(when α=1.0) when P=5 and 5.70 to 38.46 when P=50. This observation suggests that the reduction in the value of information sharingbecause of product substitution is smaller when the degree of substitution is low and the firm is more downstream. An interesting observationis that the marginal reduction is higher when the firms are more upstream and the degree of substitution is higher.

6.3. Impact of correlation coefficient

Fig. 4 shows the impact of correlation for N=50 and the degree of sharing is five. The figure is generally consistent with our theoreticalresult that the reduction in the value of information sharing because of product substitution decreases when the demand correlation in-creases. We find that the reduction in the value of information sharing for firm 2 ranged from 0.01 (when α=0.2) to 0.14 (when α=1.0) when ρr=0.0, but the same range was from 0.00 to 0.00 when ρr=1.0. The range for firm 6 was 5.4 (when α=0.2) to 52.56 (whenα=1.0) when ρr=0.0. The rate of reduction in the value of information sharing is higher when the degree of substitution is higher andthe firm is more upstream.

6.4. Impact of degree of sharing

This section analyzes how the reduction in the value of information sharing due to substitution changes as we move up the supply chain.Fig. 5 shows the reduction in the value of information sharing for N=50 and ρr=0.0. The reduction in the value of information due to substi-tution is higher for more upstream firms under both upstream and downstream information sharing. Further, the reduction in the value in-creases in a convex fashion as we move up the supply chain for the same degree of substitution, implying that the risk of ignoringsubstitution effects can be significantly higher for upstream firms. This observation is the direct result of the amplification of bullwhip effectfrom downstream to upstream, which results in a higher demand variance for more upstream firms in the supply chain.

In summary, our numerical simulations, in addition to confirming our theoretical results, show that upstream firms face a significantly higherrisk of overestimation of value if they ignore product substitution, compared to downstream firms. This risk is enhanced when more down-stream firms share information among themselves, the demand correlation is higher, the number of products is higher, and the degree ofsubstitution among the products is higher.

0

5

10

15

20

25


Firms

Reduction inthe Value ofInformation

Sharing

P = 5

P = 10

P = 25

P = 50

0

5

10

15

20

25

30

35

Firm 2 Firm 3 Firm 4 Firm 5 Firm 6Firms

Reduction inthe Value ofInformation

Sharing

P = 5

P = 10

P = 25

P = 50

a)

b)

Fig. 3. a. Impact of number of products (α=0.6). b. Impact of number of products (α=0.8).

0

5

10

15

20

25

1 2 3 4 5

Firms

ρr = 1.0

ρr = 0.8

ρr = 0.6

ρr = 0.4

ρr = 0.2

ρr = 0.0

0

5

10

15

20

25

30

35

40


ρr = 0.0

ρr = 0.2

ρr = 0.4

ρr = 0.6

ρr = 0.8

ρr = 1.0

Firms

Reductionin the

Value ofInformation

Sharing

Reductionin the

Value ofInformation

Sharing

a)

b)

Fig. 4. a. Impact of correlation coefficient (α=0.6). b. Impact of correlation coefficient(α=0.8).


7. Conclusion

This paper analyzed the impact of substitution in a multi-levelsupply chain that supplies multiple substitutable products. The re-sults reveal that ignoring product substitution can lead to a signifi-cant overestimation of value of information sharing, especially forfirms that are more upstream in the supply chain. We made severalsimplifying assumptions in our analysis. For instance, we assumedthat the lead time for all firms is zero. We made this assumption be-cause the impact of lead time has been considered by prior literaturethat does not consider substitution. We do not anticipate that leadtime will alter the impact of substitution qualitatively. However,an increase in lead time is likely to increase the value of informationsharing in general, and increase the reduction in value of because ofproduct substitution. We assumed an AR(1) model for the demandforecasting process. However, the qualitative nature of our resultsis not dictated by the specific demand model. While the magnitudeof the value of information sharing under substitution and no sub-stitution will depend critically on the demand model and the levelof demand uncertainty, the impact of substitution is not likely tochange. Future research should address other demand forecastingmodels with substitution. Our model assumes that all products aresymmetric; that is, demand models and costs are identical acrossproducts. This was done to isolate the effects of substitution sothat factors such as demand sizes and costs do not contaminateour findings. However, products could differ in their inventory hold-ing and shortage costs, production costs, mean demands, and otherparameters. We believe that firms that have higher inventory-holding and shortage costs will benefit more from information shar-ing, and correspondingly realize a smaller value when products are

substitutable. Future research can look into relaxing the above lim-itations and identify the exact impact of these limitations.

Appendix A. No information sharing: No substitution

We prove Eqs. (4.6) and (4.7) by induction.

D 2ð Þj;tþ1 ¼ dþ ρD 2ð Þj;t þ 1þ ρð Þξj;tþ1−ρξj;t

T 2ð Þj;t ¼ dþ ρD 2ð Þj;t þ Kσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ρð Þ2 þ ρ2

q:

ðA:1Þ

Substituting D(2)t+1 and T(2)t in D(3)j,t+1=D(2)j,t+1+(T(2)j,t+1−T(2)j,t), we obtain

D 3ð Þj;tþ1 ¼ dþ ρD 3ð Þj;t þ 1þ ρð Þ2ξj;tþ1−2ρ 1þ ρð Þξj;t þ ρ2ξj;t−1

T 3ð Þj;t ¼ dþ ρD 3ð Þj;t þ Kσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ρð Þ4 þ 4ρ2 1þ ρð Þ2 þ ρ4

q:

ðA:2Þ

So; Dno ið Þj;tþ1 ¼ dþ ρDno ið Þj;tþXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmξj;t− m−1ð Þ ðA:3Þ

is satisfied when i≤3. So, if Eq. (A.3) is true for a general i, then

Tno ið Þj;t ¼ dþ ρDno ið Þj;t þ KσffiffiffiffiffiffiffiffiΩi

no

qwhere Ωi

no

¼Xi−1

m¼0i−1Cm 1þ ρð Þi−1−mρmÞ2�

ðA:4Þ

Now, consider firm (i+1). Dno(i+1)j,t+1=Dno(i)j,t+1+(Tno(i)j,t+1−Tno(i)j,t). Using Eq. (A.4), we get Dno(i+1)j,t+1=(1+ρ)Dno(i)j,t+1−ρDno(i)j,t. Therefore,

Dno iþ 1ð Þj;tþ1 ¼ 1þ ρð Þ dþ ρDno ið Þj;t þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmξj;t− m−1ð Þ

!−

ρ dþ ρDno ið Þj;t−1 þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmξj;t−1− m−1ð Þ

!

¼ dþ ρ 1þ ρð ÞDno ið Þj;t−ρDno ið Þj;t−1

� �þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−mρmξj;t− m−1ð Þ−

Xi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmþ1ξj;t−1− m−1ð Þ

¼ dþ ρDno iþ 1ð Þj;t þXi−1

m¼0

−1ð Þm 1þ ρð Þi−mρmi−1Cmþi−1Cm−1ð Þξj;t− m−1ð Þ

− −1ð Þi−1ρiξj;tþ1−i ¼ dþ ρDno iþ 1ð Þj;t þXim¼0

−1ð Þm 1þ ρð Þi−mρmiCmξj;t− m−1ð Þ:

Appendix B. Full information sharing: No substitution

We have, from Eqs. (A.1), D(2)t+1=d+ρD(2)t+(1+ρ)ξt+1−ρξt, and hence T(2)t=d+ρD(2)t−ρξt+Kσ(1+ρ), since ξt isknown under full information sharing.

D(3)t+1=D(2)t+1+(T(2)t+1−T(2)t). Substituting D(2)t andT(2)t in the above equation and simplifying, we obtain

D 3ð Þtþ1 ¼ dþ ρD 3ð Þt− ρþ ρ2� �

ξt þ 1þ ρþ ρ2� �

ξtþ1: ðB:1Þ

So; T 3ð Þt ¼ dþ ρD 3ð Þt− ρþ ρ2� �

ξt þ Kσ 1þ ρþ ρ2� �

:

So; Dfull ið Þtþ1¼dþρDfull ið Þt−ξtXi−1

k¼1

ρkþξtþ1

Xi−1

k¼0

ρk is satisfied when i≤2:

ðB:2Þ

So, if Eq. (B.2) is true for a general i, then

Tfull ið Þt ¼ dþ ρDfull ið Þt−ξtXi−1

k¼1

ρk þ KσXi−1

k¼0

ρk: ðB:3Þ

0

5

10

15

20

25


Firms

Degree of Sharing = 1





0

5

10

15

20

25

30

35







0

5

10

15

20

25


Firms

Firms Firms

Reduction in theValue of Information

Sharing


Sharing


Sharing


Sharing






0

5

10

15

20

25

30

35







Downstream Sharing Upstream Sharingα = 0.6

α = 0.8

Fig. 5. Impact of degree of sharing.


Now, consider firm (i+1).

Dfull iþ 1ð Þtþ1 ¼ Dfull ið Þtþ1 þ Tfull ið Þtþ1−Tfull ið Þt� �

: ðB:4Þ

Using Eq. (B.3), we get

Tfull ið Þtþ1−Tfull ið Þt ¼ ρ Dfull ið Þtþ1−Dfull ið Þt� �

−ρ 1þ ρþ ::::þ ρi−1� �

ξtþ1−ξt� �

ðB:5Þ

Substituting Eq. (B.5) in Eq. (B.4), we obtain

Dfull iþ 1ð Þtþ1 ¼�dþ ρDfull iþ 1ð Þt− ρþ ρ2 þ ρ3 þ :::::þ ρi

� �ξt

þ 1þ ρþ ρ2 þ :::::þ ρi þ ρi� �

ξtþ1

�

¼ dþ ρDfull iþ 1ð Þt−Xik¼1

ρkξt þXik¼0

ρkξtþ1:

Appendix C. Downstream partial information sharing: No substitution

Using results for full information sharing up to firm (n+1), we have

Dfull ið Þj;tþ1 ¼ dþ ρDfull ið Þj;t−ξj;tXi−1

k¼1

ρk þ ξj;tþ1

Xi−1

k¼0

ρk; 1 ≤ i ≤ nþ 1 From Eq: 4:8ð Þð Þ ðC:1Þ

Tfull ið Þj;t ¼ dþ ρDfull ið Þj;t−ξj;tXi−1

k¼1

ρk þ KσXi−1

k¼0

ρk; 1 ≤ i ≤ nþ 1 From Eq: 4:9ð Þð Þ: ðC:2Þ

Define X ¼Xnþ1

k¼1

ρk and D(n+2)j,t+1=D(n+1)j,t+1+(T(n+1)j,t+1−T(n+1)j,t).

Substituting Eqs. (C.1) and (C.2) in the above equation and simplifying, we get

D nþ 2ð Þj;tþ1 ¼ dþ ρD nþ 2ð Þj;t þ 1þ Xð Þξj;tþ1−Xξj;t :

Since (n+2) does not know ξj,t, T nþ 2ð Þj;t ¼ dþ ρD nþ 2ð Þj;tþ Kσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Xð Þ2 þ X2

q:


Substituting the above in D(n+3)j,t+1=D(n+2)j,t+1+(T(n+2)j,t+1−T(n+2)j,t) and simplifying, we obtain

D nþ 3ð Þj;tþ1 ¼ dþ ρD nþ 3ð Þj;t þ 1þ ρð Þ 1þ Xð Þð Þξj;tþ1− ρ 1þ Xð Þ þ X 1þ ρð Þð Þξj;t þ ρXξj;t−1

¼ dþ ρD nþ 3ð Þj;t þ−1ð Þ0 1C0 1þ ρð Þ1ρ0 1þ Xð Þ

h iξj;tþ1þ

−1ð Þ1 1C1 1þ ρð Þ0ρ1 1þ Xð Þ� �

þ 1C0 1þ ρð Þ1ρ0X� �h i

ξj;t

þ −1ð Þ2 1C1 1þ ρð Þ0ρ1Xh i ξj;t−1

8>>><>>>:

9>>>=>>>;

So; Ddownstream nð Þ ið Þj;tþ1 ¼

dþ ρDdownstream nð Þ ið Þj;t þ −1ð Þi−n−3 1þ ρð Þi−n−2ρi−n−3 1þ Xð Þξj;tþ1

� �þ

Xi−n−2

m¼1

−1ð Þm i−n−2Cm 1þ ρð Þi−n−2−mρm 1þ Xð Þ þ i−n−2Cm−1 1þ ρð Þi−n−2− m−1ð Þρm−1X� �

ξj;t− m−1ð Þ� �

þ −1ð Þi−n−1ρi−n−2Xξj;t− i−n−2ð Þ

0BBBBB@

1CCCCCA

ðC:3Þ

is true when i=n+3.Further, because none of the ξs are known to firm (n+3), we have

Tdownstream nð Þ nþ 3ð Þj;t ¼ dþ ρDdownstream nð Þ ið Þj;t þ KσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩnþ3

downstream nð Þq

where

Ωidownstream nð Þ ¼ 1þ ρð Þi−n−2ρi−n−3 1þ Xð Þ

� �2þXi−n−2

m¼1i−n−2Cm 1þ ρð Þi−n−2−mρm 1þ Xð Þ þ i−n−2Cm−1 1þ ρð Þi−n−2− m−1ð Þρm−1X� �� 2 þ ρi−n−2X

� �2:

Assume that Eq. (C.3) holds for an arbitrary i≥(n+3).Now, consider firm (i+1)

Ddownstream nð Þ iþ 1ð Þj;tþ1 ¼ Ddownstream nð Þ ið Þj;tþ1 þ Tdownstream nð Þ ið Þj;tþ1−Tdownstream nð Þ ið Þj;t� �

¼ 1þ ρð ÞDdownstream nð Þ ið Þj;tþ1−ρDdownstream nð Þ ið Þj;t :

Substituting Eq. (C.3) in the above equation and simplifying, we obtain

Ddownstream nð Þ iþ 1ð Þj;tþ1 ¼

dþ ρDdownstream nð Þ iþ 1ð Þj;t þ −1ð Þiþ1−n−3 1þ ρð Þiþ1−n−2ρiþ1−n−3 1þ Xð Þξj;tþ1

� �þ

Xiþ1−n−2

m¼1

−1ð Þm iþ1−n−2Cm 1þ ρð Þiþ1−n−2−mρm 1þ Xð Þ þ iþ1−n−2Cm−1 1þ ρð Þiþ1−n−2− m−1ð Þρm−1X� �

ξj;t− m−1ð Þ� �

þ −1ð Þiþ1−n−1ρiþ1−n−2Xξj;t− iþ1−n−2ð Þ

0BBBBB@

1CCCCCA:

So, Eq. (C.3) holds for firm (i+1).

Appendix D. Upstream information sharing: No substitution

Assume that firms N−n through N−1 share their demand infor-mation with their upstream supplier and firms 1 through (N−n−1)do not.

For 1≤ i≤N−n, using results from the no information sharingcase, we have

Dno ið Þj;tþ1 ¼ dþ ρDno ið Þj;t þXi−1

m¼0

−1ð Þmi−1Cm 1þ ρð Þi−1−mρmξj;t− m−1ð Þ

Tno ið Þj;t ¼ dþ ρDno ið Þj;t þ KσffiffiffiffiffiffiffiffiΩi

no

q:

Defineξ0j;tþ1 ¼

XN−n−1

m¼0

−1ð ÞmN−n−1Cm 1þ ρð ÞN−n−1−mρmξj;tþ�

1−mÞ:

Now, firms (N−n) through N share ξj,t' with each other, but do notknow the individual ξ components within it.

Thus, we can apply the full information sharing results to the sup-ply chain that has only firms N−n through N in which the mostdownstream firm (N−n) faces an AR(1) demand model given by

D N−nð Þj;tþ1 ¼ dþ ρD N−nð Þj;t þ ξ0

j;tþ1:

Thus, we have, for N−n+1≤ i≤N

Dupstream nð Þ ið Þj;tþ1 ¼ dþ ρDupstream nð Þ ið Þj;t−Xi− N−nð Þ

k¼1

ρkξ0

j;t þXi− N−nð Þ

k¼0

ρkξ0

j;tþ1

Tupstream nð Þ ið Þj;t ¼ dþ ρDupstream nð Þ ið Þj;t−Xi− N−nð Þ

k¼1

ρkξ0

j;t

þKσXi− N−nð Þ

k¼0

ρk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN−n−1

m¼0N−n−1Cm 1þ ρð Þ N−n−1ð Þ−mρmÞ2:�vuut


Appendix E. Proof for the proposition for the partial information sharing

Proposition 1 (i). For this proposition, it is sufficient to prove VType(n),α=0j >VType(n),α=1

j

ffiffiffiffiffiffiffiffiΩj

no

q−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩj


W

HKjnoσ

d1−ρ

� �−


no

q !>


no

q−



WP

HKjnoσ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp d

1−ρ

� �−


no

q !

WP=HKjnoσ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp

d= 1−ρð Þð Þ−ffiffiffiffiffiffiffiffiΩj

no

q� �>

W

HKjnoσ

d= 1−ρð Þð Þ−ffiffiffiffiffiffiffiffiΩj

no

q !

WP= HKjnoσ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp� �

d= 1−ρð Þð Þ > W=HKjnoσ

� �d= 1−ρð Þð Þ

P >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þð Þp

and P > 1þ Pρr−ρr

P−1>ρr (P−1) which is true.

Proposition 1 (ii). Let RjType nð Þ;1 ¼


no

q−


Type nð Þq� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P1þ P−1ð Þρrð Þ

q�−1Þ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

1þ P−1ð Þρrð Þq

−ffiffiffiffiffiffiΩj

no

pWH

�Kjnoσ

d1−ρ

� �Þ 1−

ffiffiffiffiffiffiΩj

no

pW

HKjnoσ

d1−ρð Þ

!: It is sufficient to prove


Type nð Þq

≥ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩj

Type nþ1ð Þq

.

Downstream sharingIt is sufficient to show ΩDownstream(n)

i ≥ΩDownstream(n+1)i , 1≤n≤N−2, ∀ i.

Case 1. i≤n+1From Eq. (4.12), we know that ΩDownstream(n)

i =ΩDownstream(n+1)i when i≤n+1.

Case 2. i=n+2

Ωnþ2Downstream nð Þ ¼

Xnþ1

k¼1

ρk

!2

þXnþ1

k¼0

ρk

!2

> Ωnþ2Downstream nþ1ð Þ ¼

Xnþ1

k¼0

ρk

!2

:

Case 3. i=n+3

Ωnþ3Downstream nð Þ ¼ 1þ ρð Þ 1þ

Xnþ1

k¼1

ρk

! !2

þ ρ 1þXnþ1

k¼1

ρk

!þ 1þ ρð Þ

Xnþ1

k¼1

ρk

!2

þ ρXnþ1

k¼1

ρk

!2

Ωnþ3Downstream nþ1ð Þ

Xnþ2

k¼1

ρk

!2

þXnþ2

k¼0

ρk

!2

:

Case 4. i>n+3

ΩiDownstream nð Þ ¼

1þ ρð Þi−n−2ρi−n−3 1þXnþ1

k¼1

ρk

! !2

þ

Xi−n−2

m¼1i−n−2Cm 1þ ρð Þi−n−2−mρm 1þ

Xnþ1

k¼1

ρk

!þ i−n−2Cm−1 1þ ρð Þi−n−2− m−1ð Þρm−1Xnþ1

k¼1

ρk

! !2

þ ρi−n−2Xnþ1

k¼1

ρk

!2

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

>

ΩiDownstream nþ1ð Þ ¼

1þ ρð Þi−n−3ρi−n−4 1þXnþ2

k¼1

ρk

! !2

þ

Xi−n−3

m¼1i−n−3Cm 1þ ρð Þi−n−3−mρm 1þ

Xnþ2

k¼1

ρk

!þ i−n−3Cm−1 1þ ρð Þi−n−3− m−1ð Þρm−1Xnþ2

k¼1

ρk

! !2

þ ρi−n−3Xnþ2

k¼1

ρk

!2

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;:


Upstream sharing

Ωiupstream nð Þ ¼

Xi−1

m¼0i−1Cm 1þ ρð Þi−1−mρm� �2

if 1≤i≤N−n

Xi−Nþn

k¼0

ρk

!2 XN−n−1

m¼0ðN−n−1ÞCm 1þ ρð Þ N−n−1ð Þ−mρmÞ2 if N−nþ 1≤i≤N:�

8>>>><>>>>:

It is sufficient to show that ΩUpstream(n)i >ΩUpstream(n+1)

i , 1≤n≤N−1, ∀i.

Case 1. i≤N−n−1.From Eq. (4.15), we know that ΩUpstream(n)

i =ΩUpstream(n+1)i .

Case 2. i=N−n

ΩN−nUpstream nð Þ ¼

XN−n−1

m¼0N−n−1Cm 1þ ρð ÞN−n−1−mρm� �2

> ΩN−nUpstream nþ1ð Þ ¼

X1k¼0

ρk

!2 XN−n−2

m¼0N−n−2Cm 1þ ρð Þ N−n−2ð Þ−mρm� �2

:

Case 3. N−n+1≤ i≤N

ΩiUpstream nð Þ ¼

Xi−Nþn

k¼0

ρk

!2 XN−n−1

m¼0ðN−n−1ÞCm 1þ ρð Þ N−n−1ð Þ−mρm� �2

>

ΩiUpstream nþ1ð Þ ¼

Xi−Nþnþ1

k¼0

ρk

!2 XN−n−2

m¼0ðN−n−2ÞCm 1þ ρð Þ N−n−2ð Þ−mρm� �2

:

Proposition 1 (iii).

∂RjType nð Þ;1∂ρr

¼


no

q−


Type nð Þq� �

1−ffiffiffiffiffiffiΩj

no

pW

HKjnoσ

d1−ρð Þ

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

1þ P−1ð Þρrð Þ

s−


no

qW

HKjnoσ

d1−ρ

� �0BBB@

1CCCA − P−1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P 1þ P−1ð Þρrð Þp2

!26664

37775

− − P−1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1þ P−1ð Þρrð Þp2

! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP


s−1

!




no

pW

HKjnoσ

d1−ρð Þ

!2


¼− P−1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P 1þ P−1ð Þρrð Þp ffiffiffiffiffiffiffiffiΩj

no

q−


Type nð Þq� �

2 1−ffiffiffiffiffiffiΩj

no

pW

HKjnoσ

d1−ρð Þ

!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


q−


no

pW

HKjnoσ

d1−ρð Þ

!−



−1� �" #




no

pW

HKjnoσ

d1−ρð Þ

!2


¼− P−1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P 1þ P−1ð Þρrð Þp ffiffiffiffiffiffiffiffiΩj

no

q−


Type nð Þq� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


q−


no

pW

HKjnoσ

d1−ρð Þ

!−



−1� �" #

2 1−ffiffiffiffiffiffiΩj

no

pW

HKjnoσ

d1−ρð Þ

! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP



no

pW

HKjnoσ

d1−ρð Þ

!2 b0:

Proposition 1 (iv).

∂RjType nð Þ;1∂P ¼


no

q−


Type nð Þq� �


no

pW

HKjnoσ

d1−ρð Þ

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP


s−


no

qW

HKjnoσ

d1−ρ

� �0BBB@

1CCCA

1−ρr

2ffiffiffiP

p1þ P−1ð Þρrð Þ3=2

!

2666666664

3777777775−

1−ρr

2ffiffiffiP


!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


s−1

!266664

377775




no

pW

HKjnoσ

d1−ρð Þ

!2


∂RjType nð Þ;1∂P ¼


no

q−


Type nð Þq� �

1−ρr

2ffiffiP


� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


s−


no

qW

HKjnoσ

d1−ρ

� �0BBB@

1CCCA

−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


s−1

!" #

2666666664

3777777775


no

pW

HKjnoσ

d1−ρð Þ

! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP



no

pW

HKjnoσ

d1−ρð Þ

!2 > 0:

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Muthusamy Ganesh is working as a Functional Architect at JDA Software India Pvt. Ltd,Hyderabad, India. He obtained a Ph.D. in the area of Supply Chain from the IndianInstitute of Technology Madras, India. His areas of research include Supply ChainManagement, Inventory Management and Simulation. He has publication in IIETransactions and International Journal of Logistics Systems and Management.

Srinivasan Raghunathan is a Professor of Information Systems in the School ofManagement, The University of Texas at Dallas. He obtained B. Tech degree in ElectricalEngineering from IIT, Madras, Post Graduate Diploma in Management from IIM, Calcut-ta, and Ph.D. in Business Administration from the University of Pittsburgh. His currentresearch interests are in the economics of information security and the value of collab-oration in supply chains. His papers have been published in journals such as Manage-ment Science, Information Systems Research, Journal of MIS, various IEEE transactions,IIE transactions, and Production and Operations Management.

Chandrasekharan Rajendran is a Professor of Operations Management in the IndianInstitute of Technology Madras, India. His areas of interest include scheduling, simula-tion, meta-heuristics, total quality management and supply chain management. He haspublished several articles in international journals. He has publications in NavalResearch Logistics, IIE Transactions, European Journal of Operational Research, Interna-tional Journal of Production Research, Journal of the Operational Research Society andInternational Journal of Production Economics. He serves as referee for many journals.He is a recipient of the Alexander von Humboldt Fellowship of Germany.

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the value of information sharing in a multi-product, multi-level supply chain: impact of product...

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